text
stringlengths 14
1.76M
|
|---|
# Joint Secure Transmit Beamforming Designs for Integrated Sensing and
Communication Systems ††thanks: Part of this paper has been presented in the
IEEE Wireless Communications and Networking Conference (WCNC), 2022 [1].
††thanks: J. Chu, R. Liu, M. Li, and Y. Liu are with the School of Information
and Communication Engineering, Dalian University of Technology, Dalian 116024,
China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]). ††thanks: Q. Liu is with the School
of Computer Science and Technology, Dalian University of Technology, Dalian
116024, China (e-mail: [email protected]).
Jinjin Chu, Rang Liu, Ming Li,
Yang Liu, and Qian Liu
###### Abstract
Integrated sensing and communication (ISAC), which allows individual radar and
communication systems to share the same spectrum bands, is an emerging and
promising technique for alleviating spectrum congestion problems. In this
paper, we investigate how to exploit the inherent interference from strong
radar signals to ensure the physical layer security (PLS) for the considered
multi-user multi-input single-output (MU-MISO) communication and colocated
multi-input multi-output (MIMO) radar coexistence system. In particular, with
known eavesdroppers’ channel state information (CSI), we propose to jointly
design the transmit beamformers of communication and radar systems to minimize
the maximum eavesdropping signal-to-interference-plus-noise ratio (SINR) on
multiple legitimate users, while guaranteeing the communication quality-of-
service (QoS) of legitimate transmissions, the requirement of radar detection
performance, and the transmit power constraints of both radar and
communication systems. When eavesdroppers’ CSI is unavailable, we develop a
joint artificial noise (AN)-aided transmit beamforming design scheme, which
utilizes residual available power to generate AN for disrupting malicious
receptions as well as satisfying the requirements of both legitimate
transmissions and radar target detection. Extensive simulations verify the
advantages of the proposed joint beamforming designs for ISAC systems on
secure transmissions and the effectiveness of the developed algorithms.
###### Index Terms:
Integrated sensing and communication (ISAC), physical layer security (PLS),
multi-user multi-input single-output (MU-MISO), artificial noise (AN),
interference exploitation.
## I Introduction
With the explosive growth of wireless devices, exponentially increased
bandwidth is required to support a variety of high data-rate services.
Consequently, spectrum resources have been increasingly scarce, which
motivates the development of advanced spectrum sharing technologies [2]. Since
large portions of the spectrum are available at radar frequency bands,
spectrum sharing between radar and communication systems has led to
substantial research interest [3]-[6]. This line of research is referred to as
integrated sensing and communication (ISAC), which is also known as joint
radar-communication (JRC), joint communication and radar (JCR), joint
communication and radar sensing (JCAS), etc. ISAC allows radar and
communication systems to share the same spectrum bands, which can
significantly improve the spectrum efficiency and thus alleviates the spectrum
congestion problem. It is foreseeable that ISAC will become a critical
enabling technology for future wireless networks, supporting various vital
applications, including vehicular networks [7], Internet of Things (IoT) [8],
etc.
Research on ISAC can be generally categorized into two main directions: Dual-
functional radar-communication (DFRC) and radar and communication coexistence
(RCC). The former focuses on using the same signals transmitted from one
fully-shared hardware platform to simultaneously perform communication and
radar sensing functionalities [9]-[12]. Although it has the benefits of low
power consumption and smaller-size platform, the trade-off between the radar
and communication functionalities requires sophisticated optimizations on the
unified dual-functional waveform. Moreover, the resulting hardware complexity
significantly hinders practical applications. On the contrary, RCC enables
separately deployed communication and radar platforms to cooperatively perform
their respective functions using independent transmitted signals. Therefore,
RCC is more suitable for many existing scenarios such as sharing spectrum
between air-borne early warning radars and 3.5 GHz time-division duplex long-
term evolution (TDD-LTE) [13], [14], between battlefield/ground surveillance
and vessel traffic service (VTS) radars and WLAN networks [15], [16], etc.
In RCC systems, the interference management between the non-colocated base
station (BS) and radar transmitter is vital for achieving good communication
and radar sensing performance. Therefore, the cooperative design of these two
systems is necessary for practical applications [17]-[28]. Meanwhile, the
multi-input multi-output (MIMO) architecture has been widely deployed in both
radar and communication systems to provide additional spatial degrees of
freedom (DoFs) for pursuing more considerable beamforming gains.
Various signal processing techniques have been proposed to design transmit
beamformers for multi-antenna BS and MIMO radar to realize efficient
interference management. A null space projection (NSP) method was proposed in
[29], [30], where the radar transmit beamforming is designed to project the
radar signals onto the null space of the effective interference channels to
eliminate the interference at the communication receivers. However, the target
echo may fall into the row space of the interference channels, resulting in
degraded radar performance. As a step further, the authors of [31] expanded
the projection space to include the subspace corresponding to the small non-
zero singular values under a specified threshold, expecting to control the
interference to communication systems and achieve different trade-offs between
radar and communication performance. However, these NSP-based beamforming
designs significantly reduce the DoFs for optimization and cannot guarantee to
meet specific radar sensing requirements. In order to overcome these
disadvantages, investigations on the joint design of the transmit beamformers
for both radar and communication systems have been proposed in [32], [33],
where different performance metrics of these two systems are satisfied with
controllable constraints. In addition to the above scenarios with perfectly
known channel state information (CSI), a robust beamforming design was
proposed in [34] under the assumption of imperfect CSI.
As mentioned above, the existing literature on beamforming designs for RCC
systems essentially aims at suppressing the interference between radar and
communication systems. While radar interference is usually deemed as the most
significant harmful component to the communication system, from another
perspective, it can be utilized to disrupt illegal receivers to safeguard
confidential information against eavesdropping. Utilizing interfering/jamming
signals to disrupt potential eavesdroppers has been widely considered in the
literature on physical layer security (PLS) [35]-[44]. In [35], constructive
interference was leveraged to implement secure beamforming using a symbol-
level precoding approach. In [36], a multi-antenna cooperative jammer was
employed to assist the secure communications. In order to confuse the
eavesdropper, the authors in [37] utilized the idea of destructive
interference to push the received symbols at the eavesdropper towards the
destructive region where the wrong symbol will be detected. The authors in
[38] proposed a cooperative secure transmission scheme, in which the
legitimate information and interference signals lie in different subspaces at
the destination of the confidential transmission, but are aligned along the
same subspace at the eavesdropper.
Noting that the above secure beamformer designs require the knowledge of the
eavesdropper’s CSI, which is not always available in practical applications.
In such cases, artificial noise (AN) technology was introduced to realize PLS
[39], which uses a large amount of additional energy to generate interfering
signals for disrupting potential malicious receptions. The existing
communication literature generally forces the AN to be uniformly distributed
onto the null space of the confidential transmission channels to disturb the
eavesdropper’s reception but not harm legitimate users [40]. Inspired by this
concept, in addition to realizing the respective performance requirements of
communication and radar systems, the residual power can be used to generate
AN. However, this null space projection design can only exploit limited
spatial DoFs to generate AN, especially for a system with many users. In
recent works [41], [42], the AN and transmit beamforming were jointly
optimized to implement secure DFRC transmissions, making full use of available
DoFs for generating AN. In addition, the authors in [43] proposed to enhance
secure performance by deploying an RIS in DFRC systems. However, how to ensure
the physical layer security in RCC systems remains an open problem. Instead of
consuming additional power to generate AN, the authors in [44] proposed to
exploit the inherent multi-user interference as a helpful resource by
converting it to act like AN or distributed friendly jammers to improve the
security performance. Inspired by the concept of interference exploitation,
the radar signal, which usually has very strong signal power, is an up-and-
coming candidate as the jamming signal to enhance the physical layer security
performance for the considered RCC system.
Motivated by the above findings, we investigate the PLS problem for multi-user
multi-input single-output (MU-MISO) communication and colocated MIMO radar
coexistence systems in this paper. In particular, the considered RCC system
includes a multi-antenna BS serving multiple single-antenna users in the
presence of multiple eavesdroppers and a colocated MIMO radar attempting to
detect a point-like target. We aim to exploit the coexisted strong radar
signals as inherent jamming signals to disrupt the eavesdroppers’
reception111Since the radar signal that is only used to detect the target does
not contain any confidential information, it will not cause security/privacy
problem to the considered RCC system. . The transmit beamformers of
communication and radar systems and radar receive filter are jointly designed
to ensure security performance and satisfy the requirements of legitimate
transmissions and radar target detection. The main contributions of this paper
can be summarized as follows:
* •
We consider the physical layer security in an ISAC system and innovatively
propose to exploit the coexisted strong radar signals as inherent
jamming/interfering signals to weaken the reception of potential eavesdroppers
and enhance the transmission security. Joint secure transmit beamforming and
radar receive filter designs are investigated to achieve this goal.
* •
With the knowledge of eavesdroppers’ CSI, we jointly design the transmit
beamforming and radar receive filter to minimize the maximum eavesdropping
signal-to-interference-plus-noise ratio (SINR) on multiple legitimate users,
while satisfying the quality-of-service (QoS) requirements of the legitimate
users, the radar output SINR constraint, and the transmit power budgets of
radar and communication systems. An efficient algorithm based on the block
coordinate descent (BCD), fractional programming (FP), and semi-definite
relaxation (SDR) methods is developed to solve the resulting non-convex
optimization problem.
* •
When eavesdroppers’ CSI is unavailable, we propose to jointly design the AN-
aided transmit beamforming and radar receive filter to maximize the power of
AN under the same constraints. In this case, all available transmit power of
the BS and radar should be utilized to generate AN as much as possible to
destruct eavesdroppers’ reception. A double-loop BCD and SDR based algorithm
is employed to convert the resulting complicated non-convex optimization
problem into two more tractable sub-problems that can be alternatively solved.
* •
Extensive simulation results show that the eavesdropping SINR is generally
several orders of magnitude smaller than the SINR of legitimate communication
users, which verify the significant advancement of utilizing radar signals as
inherent jamming/interference signals to enhance the secure transmissions for
ISAC systems and the effectiveness of the proposed joint secure transmit
beamforming design algorithms.
The rest of this paper is organized as follows. Section ii@ introduces the
system model of the ISAC system in the presence of eavesdroppers and develops
a joint transmit beamforming and radar receive filter design with known
eavesdroppers’ CSI. Section iii@ investigates the joint AN-aided transmit
beamforming and radar receive filter design without eavesdroppers’ CSI.
Simulation results are demonstrated in Section iv@, and finally, conclusions
are provided in Section<EMAIL_ADDRESS>
Notations: Boldface lower-case and upper-case letters indicate column vectors
and matrices, respectively. $(\cdot)^{H}$ and $(\cdot)^{-1}$ denote the
transpose-conjugate and inverse operations, respectively. $\mathbb{C}$ denotes
the set of all complex numbers. $|a|$, $\|\mathbf{a}\|$, and
$\|\mathbf{A}\|_{F}$ are the magnitude of a scalar $a$, the norm of a vector
$\mathbf{a}$, and the Frobenius norm of a matrix $\mathbf{A}$, respectively.
$\mathbb{E}\\{\cdot\\}$ represents statistical expectation.
$\text{Tr}\\{\mathbf{A}\\}$ and $\text{Rank}\\{\mathbf{A}\\}$ are the trace
and rank of matrix $\mathbf{A}$, respectively.
Figure 1: An ISAC system at the presence of eavesdroppers.
## II Joint Transmit Beamforming Design with Known Eavesdroppers’ CSI
### II-A System Model and Problem Formulation
We consider an ISAC system in which an MU-MISO communication system coexists
with a colocated MIMO radar system operating on the same frequency band, as
shown in Fig. 1. In particular, a BS equipped with $N$ antennas in a uniform
linear array (ULA) serves $K$ single-antenna users in the presence of $I$
eavesdroppers who attempt to intercept the confidential information
transmissions between the BS and the legitimate users. Meanwhile, a colocated
MIMO radar with $M$ transmit/receive antennas in a ULA attempts to detect a
point-like target. For convenience, the BS, the legitimate communication
users, and the eavesdroppers are referred to as Alice, Bobs, and Eves,
respectively. In this paper, we aim to exploit the coexisted strong radar
signals as inherent jamming signals to disrupt eavesdroppers’ reception under
different assumptions about the availability of Eves’ CSI.
We first assume that the CSI of Eves is perfectly known. This assumption is
valid, for example, Eves may be legitimate users who intend to overhear other
users’ confidential information. With known Eves’ CSI, we jointly design the
transmit beamformers of the BS and the radar to ensure secure transmissions
and satisfactory radar target detection performance for the ISAC system.
The received signal of the $k$-th Bob can be written as
$\displaystyle
y_{k}=\mathbf{h}_{k}^{H}\mathbf{W}\mathbf{s}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{c}+n_{k},$
(1)
where $\mathbf{h}_{k}\in\mathbb{C}^{N}$ is the channel vector between the BS
and the $k$-th Bob,
$\mathbf{W}\triangleq\left[\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{K}\right]\in\mathbb{C}^{N\times
K}$ is the beamforming matrix with $\mathbf{w}_{k}$ representing the
beamforming vector of the $k$-th Bob, $\mathbf{s}\in\mathbb{C}^{K}$ is the
symbol vector with $\mathbb{E}\\{\mathbf{s}\mathbf{s}^{H}\\}=\mathbf{I}_{K}$,
$\mathbf{g}_{k}\in\mathbb{C}^{M}$ is the channel vector between the radar and
the $k$-th Bob, $\mathbf{F}\in\mathbb{C}^{M\times M}$ is the radar beamforming
matrix, $\mathbf{c}\in\mathbb{C}^{M}$ is the radar transmit waveform with
$\mathbb{E}\\{\mathbf{c}\mathbf{c}^{H}\\}=\mathbf{I}_{M}$, and $n_{k}$ is the
additive white Gaussian noise (AWGN) with
$n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$. We assume that the information
symbol vector $\mathbf{s}$ is statistically independent with the radar
waveform $\mathbf{c}$. Thus, the SINR of the $k$-th Bob can be calculated as
$\displaystyle\mathrm{SINR}_{k}$
$\displaystyle=\frac{|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}}{\sum\limits_{j\neq
k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}+\|\mathbf{g}_{k}^{H}\mathbf{F}\|^{2}+\sigma_{k}^{2}}$
(2)
$\displaystyle=\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{k}+\sigma_{k}^{2}}.$
Similarly, the received signal of the $i$-th Eve can be expressed as
$\displaystyle
y_{\mathrm{e},i}=\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{W}\mathbf{s}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{c}+n_{\mathrm{e},i},$
(3)
where $\mathbf{h}_{\mathrm{e},i}\in\mathbb{C}^{N}$ is the channel vector
between the BS and the $i$-th Eve,
$\mathbf{g}_{\mathrm{e},i}\in\mathbb{C}^{M}$ is the channel vector between the
radar and the $i$-th Eve, and $n_{\mathrm{e},i}$ is the AWGN with
$n_{\mathrm{e},i}\sim\mathcal{CN}(0,\sigma_{\mathrm{e},i}^{2})$. The
eavesdropping SINR of the $i$-th Eve on the $k$-th Bob is thus given by
$\displaystyle\mathrm{SINR}_{i,k}^{\mathrm{e}}$
$\displaystyle=\frac{|\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}|^{2}}{\sum\limits_{j\neq
k}^{K}|\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}|^{2}+\|\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\|^{2}+\sigma_{\mathrm{e},i}^{2}}$
(4)
$\displaystyle=\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}}.$
To ensure secure communications, we aim to minimize the maximum eavesdropping
SINR on the $K$ communication users while guaranteeing the communication QoS
of legitimate transmissions.
In the radar system, the received signal, which includes the echo signal from
the target, interference from the BS, and noise, can be written as
$\displaystyle\mathbf{y}_{\mathrm{r}}=\alpha\mathbf{a}_{\mathrm{r}}(\theta_{0})\mathbf{a}_{\mathrm{t}}^{H}(\theta_{0})\mathbf{F}\mathbf{c}+\mathbf{Q}^{H}\mathbf{W}\mathbf{s}+\mathbf{n}_{\mathrm{r}},$
(5)
where $\alpha$ is the complex target amplitude with
$\mathbb{E}\\{|\alpha|^{2}\\}=\sigma_{0}^{2}$. The vectors
$\mathbf{a}_{\mathrm{t}}(\theta_{0})\in\mathbb{C}^{M}$ and
$\mathbf{a}_{\mathrm{r}}(\theta_{0})\in\mathbb{C}^{M}$ denote the transmit and
receive steering vectors of the radar antenna array,
$\displaystyle\mathbf{a}_{\mathrm{t}}(\theta_{0})=\mathbf{a}_{\mathrm{r}}(\theta_{0})\triangleq[1,e^{j\frac{2\pi}{\lambda}\Delta\mathrm{sin}(\theta_{0})},\ldots,e^{j\frac{2\pi}{\lambda}(M-1)\Delta\mathrm{sin}(\theta_{0})}]^{T},$
(6)
where $\theta_{0}$ represents the azimuth angle of the target222In radar
related literature, the direction of the target is typically known to the
transmitter since it can be readily estimated [45]-[47] at previous
observations, or given by the center of angular sector-of-interest. , $\Delta$
denotes the antenna spacing and $\lambda$ the wavelength. The matrix
$\mathbf{Q}\in\mathbb{C}^{N\times M}$ denotes the interfering channel between
the BS and the radar receiver, and $\mathbf{n}_{\mathrm{r}}\in\mathbb{C}^{M}$
is the AWGN with
$\mathbf{n}_{\mathrm{r}}\sim\mathcal{CN}(0,\sigma_{\mathrm{r}}^{2}\mathbf{I})$.
By defining
$\mathbf{A}\triangleq\mathbf{a}_{\mathrm{r}}(\theta_{0})\mathbf{a}_{\mathrm{t}}^{H}(\theta_{0})$,
the received signal at the radar can be concisely re-written as
$\mathbf{y}_{\mathrm{r}}=\alpha\mathbf{A}\mathbf{F}\mathbf{c}+\mathbf{Q}^{H}\mathbf{W}\mathbf{s}+\mathbf{n}_{\mathrm{r}}.$
(7)
To achieve better radar sensing performance, the radar utilizes a receive
filter $\mathbf{u}\in\mathbb{C}^{M}$ to suppress the interference from the BS
and the noise. The signal after filtering is
$\displaystyle
y_{\mathrm{r}}=\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{c}+\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{W}\mathbf{s}+\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}},$
(8)
and radar output SINR can thus be written as
$\displaystyle\mathrm{SINR}_{\mathrm{r}}$
$\displaystyle=\frac{\|\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{F}\|^{2}}{\sum\limits_{k=1}^{K}|\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{w}_{k}|^{2}+\mathbb{E}\left\\{|\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}}|^{2}\right\\}}$
(9)
$\displaystyle=\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}.$
For radar systems, the radar SINR is widely utilized as the metric to evaluate
radar sensing performance. Thus, we attempt to design the transmit beamforming
and the radar receive filter to guarantee that the radar SINR is no less than
a pre-defined threshold for achieving satisfactory radar target detection
performance.
In the considered MU-MISO communication and MIMO radar coexisted system, the
CSI $\mathbf{h}_{k}$, $\mathbf{g}_{k}$, $\forall k$ and $\mathbf{Q}$ are
assumed to be perfectly known by appropriate channel estimation approaches333
The CSI $\mathbf{h}_{k}$, $\forall k$ can be obtained by conventional uplink
training, i.e., users send orthogonal pilot sequences and the BS estimates CSI
by classic channel estimation algorithms. Utilizing the same pilot signals
sent by users, the CSI $\mathbf{g}_{k}$, $\forall k$ can also be acquired by
the radar without additional signaling overhead. The channel $\mathbf{Q}$
between the BS and the radar requires specific pilot signaling for channel
estimation. Fortunately, since the geometric locations of the BS and the radar
are fixed, channel $\mathbf{Q}$ is generally quasi-static and requires less
estimation, which allows acceptable signaling overhead. . Our objective, in
this case, is to jointly design the BS transmit beamformer $\mathbf{W}$, the
radar transmit beamformer $\mathbf{F}$ and the radar receive filter
$\mathbf{u}$ to minimize the maximum eavesdropping SINR of the $I$
eavesdroppers on the $K$ communication users. Meanwhile, the communication QoS
requirements of the legitimate users, the radar target detection performance
constraint and the power constraints of both communication and radar systems
are satisfied. Therefore, the optimization problem can be formulated as
$\displaystyle\underset{\mathbf{W},\mathbf{F},\mathbf{u}}{\min}\quad$
$\displaystyle\underset{i,k}{\max}\quad\mathrm{SINR}_{i,k}^{\mathrm{e}}$ (10a)
s.t. $\displaystyle\mathrm{SINR}_{k}\geq\Gamma_{k},~{}\forall k,$ (10b)
$\displaystyle\mathrm{SINR}_{\mathrm{r}}\geq\Gamma_{\mathrm{r}},$ (10c)
$\displaystyle\left\|\mathbf{W}\right\|_{F}^{2}\leq P_{\mathrm{c}},$ (10d)
$\displaystyle\left\|\mathbf{F}\right\|_{F}^{2}\leq P_{\mathrm{r}},$ (10e)
where $\Gamma_{k}$ is the SINR requirement of the $k$-th Bob,
$\Gamma_{\mathrm{r}}$ is the pre-defined threshold for achieving required
target detection performance, $P_{\mathrm{c}}$ and $P_{\mathrm{r}}$ denote the
total power budgets of the BS and the radar, respectively. Due to the
quadratic fractional objective (10a), the quadratic fractional constraints
(10b) and (10c), and the coupled variables, problem (10) is a complicated non-
convex problem that cannot be directly solved. In order to tackle these
difficulties, in the next subsection, we first utilize the BCD method to
convert the original problem into two sub-problems, and then employ efficient
algorithms based on FP and SDR methods to iteratively solve them.
### II-B Proposed Joint Transmit Beamforming Design
In order to decouple the transmit beamformers $\mathbf{W}$ and $\mathbf{F}$
and the radar receive filter $\mathbf{u}$ in the non-convex constraint (10c),
we adopt the BCD method to iteratively solve them, which are described in
detail as follows.
Update $\mathbf{W}$ and $\mathbf{F}$: With fixed $\mathbf{u}$, the sub-problem
for updating $\mathbf{W}$ and $\mathbf{F}$ is re-arranged as
$\displaystyle\underset{\mathbf{w}_{k},\forall
k,\mathbf{F}}{\min}~{}~{}\underset{i,k}{\max}~{}~{}\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}}$
(11a)
$\displaystyle\text{s.t.}~{}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (11b)
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(11c)
$\displaystyle~{}~{}~{}~{}~{}~{}\sum\limits_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P_{\mathrm{c}},$ (11d)
$\displaystyle~{}~{}~{}~{}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}\leq
P_{\mathrm{r}},$ (11e)
which is a complicated min-max problem. To tackle the min-max problem, we
introduce an auxiliary variable $z$ to re-formulate it into a more favorable
form as
$\displaystyle\underset{\mathbf{w}_{k},\forall k,\mathbf{F},z}{\min}\quad z$
(12a)
$\displaystyle\text{s.t.}~{}~{}\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}}\leq
z,~{}\forall i,k,$ (12b) $\displaystyle\qquad(\ref{eq:known update W and
F}\text{b})-(\ref{eq:known update W and F}\text{e}),$ (12c)
which is a minimization problem but still difficult to solve due to the
fractional constraint (12b) and the non-convex constraints (11b) and (11c).
Noting that problem (12) has a similar form as the max-min-ratio fractional
programming problems [48], Dinkelbach’s transform can be applied to convert it
into a more tractable form [49]. Specifically, the fractional constraint (12b)
can be converted into a polynomial expression (13) as shown on the top of this
page
$\displaystyle\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}-c_{i,k}\big{(}\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}\big{)}\leq
z,~{}\forall i,k,$ (13)
by introducing an auxiliary variable $c_{i,k}$, which essentially represents
the eavesdropping SINR of the $i$-th Eve on the $k$-th Bob and is
alternatively updated with the transmit beamformers $\mathbf{w}_{k},\forall
k$, and $\mathbf{F}$.
With given $\mathbf{w}_{k},\forall k$, and $\mathbf{F}$, the optimal
$c_{i,k}^{\star}$ can be easily obtained by
$\displaystyle
c_{i,k}^{\star}=\frac{\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{\mathrm{e},i}}{\sum\nolimits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}},~{}\forall
i,k.$ (14)
Then, the optimization problem for updating $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{F}$ can be formulated as
$\displaystyle\underset{\mathbf{w}_{k},\forall k,\mathbf{F},z}{\min}\quad z$
(15a) $\displaystyle\text{s.t.}~{}~{}\eqref{eq:c}$
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{F}\mathbf{F}^{H}\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (15b)
$\displaystyle~{}~{}~{}~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(15c)
$\displaystyle~{}~{}~{}~{}~{}~{}\sum\limits_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2}\leq
P_{\mathrm{c}},$ (15d)
$\displaystyle~{}~{}~{}~{}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}\leq
P_{\mathrm{r}}.$ (15e)
It is easy to see that the constraints (13), (15b), and (15c) are non-convex
with respect to variables $\mathbf{w}_{k},\forall k$, and $\mathbf{F}$ and are
hard to tackle. Therefore, we apply the SDR method to convert them into
primary variables for an easier solution. Specifically, by defining
$\displaystyle\mathbf{W}_{k}\triangleq\mathbf{w}_{k}\mathbf{w}_{k}^{H},~{}\forall
k,$ (16)
$\displaystyle\mathbf{R}_{\mathrm{F}}\triangleq\mathbf{F}\mathbf{F}^{H},$
the quadratic terms $\mathbf{w}_{k}\mathbf{w}_{k}^{H}$ and
$\mathbf{F}\mathbf{F}^{H}$ are converted into the primary variables
$\mathbf{W}_{k}$ and $\mathbf{R}_{\mathrm{F}}$, respectively. In the meantime,
the rank-one Hermitian positive semidefinite matrices $\mathbf{W}_{k}$,
$\forall k$, and the Hermitian positive semidefinite matrix
$\mathbf{R}_{\mathrm{F}}$ should satisfy
$\displaystyle\mathbf{W}_{k}=\mathbf{W}_{k}^{H},~{}\mathbf{W}_{k}\succeq
0,~{}\forall k,$ (17a) $\displaystyle\mathrm{Rank}(\mathbf{W}_{k})=1,$ (17b)
$\displaystyle\mathbf{R}_{\mathrm{F}}=\mathbf{R}_{\mathrm{F}}^{H},~{}\mathbf{R}_{\mathrm{F}}\succeq
0.$ (17c)
For simplicity, we define the set of all $N\times N$-dimensional Hermitian
positive semidefinite matrices as
$\mathbb{S}_{N}\triangleq\\{\mathbf{S}|\mathbf{S}=\mathbf{S}^{H},~{}\mathbf{S}\succeq
0\\}$. Afterwards, problem (15) is transformed into
$\displaystyle\underset{\mathbf{W}_{k},\forall
k,\mathbf{R}_{\mathrm{F}},z}{\min}\quad z$ (18a)
$\displaystyle\text{s.t.}~{}~{}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{W}_{k}\mathbf{h}_{\mathrm{e},i}$
(18b) $\displaystyle\quad~{}~{}~{}-c_{i,k}\big{(}\sum\limits_{j\neq
k}^{K}\mathbf{h}^{H}_{\mathrm{e},i}\mathbf{W}_{j}\mathbf{h}_{\mathrm{e},i}+\mathbf{g}^{H}_{\mathrm{e},i}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{\mathrm{e},i}+\sigma_{\mathrm{e},i}^{2}\big{)}\leq
z,~{}\forall i,k,$
$\displaystyle\quad~{}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{W}_{k}\mathbf{h}_{k}}{\sum\limits_{j\neq
k}^{K}\mathbf{h}_{k}^{H}\mathbf{W}_{j}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},~{}\forall
k,$ (18c)
$\displaystyle\quad~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{R}_{\mathrm{F}}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{W}_{k}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(18d)
$\displaystyle\quad~{}~{}~{}\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{W}_{k})\leq
P_{\mathrm{c}},$ (18e)
$\displaystyle\quad~{}~{}~{}\mathrm{Tr}(\mathbf{R}_{\mathrm{F}})\leq
P_{\mathrm{r}},$ (18f)
$\displaystyle\quad~{}~{}~{}\mathbf{W}_{k}\in\mathbb{S}_{N},~{}\forall
k,~{}\mathbf{R}_{\mathrm{F}}\in\mathbb{S}_{M},$ (18g)
$\displaystyle\quad~{}~{}~{}\mathrm{Rank}(\mathbf{W}_{k})=1,~{}\forall k.$
(18h)
It is evident that the rank-one constraint (18h) tremendously hinders finding
a straightforward solution. Thus, we apply the SDR algorithm by dropping the
rank-one constraint (18h) and relaxing the problem (18) as
$\displaystyle\underset{\mathbf{W}_{k},\forall
k,\mathbf{R}_{\mathrm{F}},z}{\min}~{}~{}$ $\displaystyle z$ (19a) s.t.
$\displaystyle(\ref{eq:SDR}\text{b})-(\ref{eq:SDR}\text{g}),$ (19b)
which is a semi-definite programming (SDP) problem that can be efficiently
solved by various existing algorithms and toolboxes such as CVX. Since the
rank-one constraint is temporarily neglected, the optimal objective value of
problem (19) only serves as a lower bound. After obtaining
$\mathbf{W}_{k},\forall k$, the eigenvalue decomposition (EVD) is usually used
to obtain the optimal solution $\mathbf{w}_{k}$ if the resulting
$\mathbf{W}_{k}$ satisfies the rank-one constraint. Otherwise, Gaussian
randomization is required to convert the high-rank solution to a feasible
rank-one solution to the problem (11). In our considered case, the rank-1
solution can be guaranteed, whose proof is given in Appendix A. On the other
hand, with resulting optimal $\mathbf{R}_{\text{F}}$, the radar beamforming
matrix $\mathbf{F}$ can be obtained by utilizing Cholesky decomposition.
Algorithm 1 Joint transmit beamforming design algorithm for solving problem
(10)
0: $\mathbf{h}_{\mathrm{e},i}$, $\mathbf{g}_{\mathrm{e},i}$,
$\sigma_{\mathrm{e},i}^{2}$, $\forall i$, $\mathbf{h}_{k}$, $\mathbf{g}_{k}$,
$\sigma_{k}^{2}$, $\Gamma_{k}$, $\forall k$, $\mathbf{A}$, $\mathbf{Q}$,
$\sigma_{\text{0}}^{2}$, $\sigma_{\text{r}}^{2}$,
$~{}~{}~{}~{}~{}\Gamma_{\mathrm{r}}$, $P_{\mathrm{c}}$, $P_{\mathrm{r}}$.
0: $\mathbf{w}_{k}^{\star}$, $\forall k$, $\mathbf{F}^{\star}$, and
$\mathbf{u}^{\star}$.
1: Initialize $\mathbf{F}$, $\mathbf{u}$, and $c_{i,k}$, $\forall i,k$.
2: while the objective value (10a) does not converge do
3: while no convergence do
4: Calculate $\mathbf{W}_{k},\forall k$, and $\mathbf{R}_{\mathrm{F}}$ by
solving (19).
5: Update $\mathbf{w}_{k}$ from $\mathbf{W}_{k},\forall k$, by EVD.
6: Update $\mathbf{F}$ from $\mathbf{R}_{\mathrm{F}}$ by Cholesky
decomposition.
7: Update $c_{i,k}$, $\forall i,k$, by (14).
8: end while
9: Update $\mathbf{u}$ by solving (20).
10: end while
11: Return $\mathbf{w}_{k}^{\star},\forall k$, $\mathbf{F}^{\star}$, and
$\mathbf{u}^{\star}$.
Update $\mathbf{u}$: It can be seen that the variable $\mathbf{u}$ only exists
in the constraint (10c) of the problem (10). Therefore, with given
$\mathbf{W}$ and $\mathbf{F}$, problem (10) is transformed into a feasibility-
check problem. For the sake of leaving enough freedoms for solving
$\mathbf{W}$ and $\mathbf{F}$ in the subsequent optimization process and
accelerating the convergence, we propose to optimize $\mathbf{u}$ to maximize
the radar output SINR. The optimization problem is formulated as
$\mathbf{u}^{\mathrm{opt}}=\arg\mathop{\max}\limits_{\mathbf{u}}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sum\limits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}.$
(20)
We observe that problem (20) is a typical generalized Rayleigh quotient, whose
optimal solution can be easily obtained as the generalized eigenvector
corresponding to the largest eigenvalue of the matrix
$\sigma_{\mathrm{0}}^{2}\big{(}\sum\nolimits_{k=1}^{K}\mathbf{Q}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\big{)}^{-1}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}$.
### II-C Summary, Initialization, Convergence, and Complexity Analysis
#### II-C.1 Summary
Based on the above derivations, the proposed joint transmit beamforming and
radar receive filter design algorithm is straightforward and summarized in
Algorithm 1. With appropriate initialization, problems (11) and (20) are
iteratively solved to respectively update $\mathbf{w}_{k}$, $\forall k$,
$\mathbf{F}$, and $\mathbf{u}$ until the objective value (10a) converges. For
updating $\mathbf{w}_{k}$, $\forall k$, and $\mathbf{F}$, we iteratively
update the communication beamforming $\mathbf{w}_{k}$, $\forall k$, and the
radar beamforming $\mathbf{F}$ by solving the SDP problem (19) and then
recovering feasible solutions using EVD and Cholesky decomposition, and the
auxiliary variables $c_{i,k},\forall i,k$, until the convergence of problem
(11) is found.
#### II-C.2 Initialization
In order to solve sub-problem (15) for updating $\mathbf{w}_{k}$, $\forall k$,
and $\mathbf{F}$, we need to initialize $\mathbf{u}$ and the auxiliary
variables $c_{i,k},~{}\forall i,k$. The initial value of the radar receive
filter is selected as $\mathbf{u}=\mathbf{a}_{\mathrm{r}}(\theta_{0})$ via a
phase alignment operation for better radar detection performance. Since
$c_{i,k}$ represents the eavesdropping SINR, which is generally several orders
of magnitude smaller than the communication SINR, initializing $c_{i,k}$ with
the pre-defined threshold of communication SINR $\Gamma_{k}$ can guarantee the
feasibility. The obtained $\mathbf{w}_{k}$, $\forall k$, and $\mathbf{F}$ by
solving (15) can be set as the initial value for solving sub-problem (20).
#### II-C.3 Convergence
We will briefly prove the convergence of the proposed algorithm as follows.
Denote $\eta\big{(}\mathbf{W},\mathbf{F},\mathbf{u}\big{)}$ as the objective
value of the original problem (10). First, in the transmit beamformer design,
we apply Dinkelbach’s transform to convert it into a more tractable form and
transform it into the problem (15). According to [50], it is easy to prove the
convergence of the algorithm given the non-increasing property of the
auxiliary variable $c_{i,k}$. Since the optimal solution of problem (11) is
obtained with given $\mathbf{u}^{t}$, we have
$\eta\big{(}\mathbf{W}^{t},\mathbf{F}^{t},\mathbf{u}^{t}\big{)}\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t}\big{)},$
(21)
where the superscript $t$ denotes the index of iterations. Second, with fixed
$\left\\{\mathbf{W},\mathbf{F}\right\\}$, the sub-problem for updating
$\mathbf{u}$ is a feasibility-check problem. After solving the problem (20), a
better radar output SINR than the original requirement is achieved with the
obtained radar receive filter $\mathbf{u}$ in the current iteration, i.e., the
feasible domain of the original problem (10) is expanded while the objective
value is fixed. In other words, with given
$\left\\{\mathbf{W}^{t+1},\mathbf{F}^{t+1}\right\\}$, we have
$\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t}\big{)}=\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t+1}\big{)}.$
(22)
Based on the above analysis, we have the relationship of the objective values
between iterations as
$\eta\big{(}\mathbf{W}^{t},\mathbf{F}^{t},\mathbf{u}^{t}\big{)}\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{F}^{t+1},\mathbf{u}^{t+1}\big{)},$
(23)
which indicates that the objective value of problem (10) is non-increasing
during the iterations of Algorithm 1. Since the objective value of problem
(10) is greater than zero, the proposed Algorithm 1 can converge to a local
optimum point.
#### II-C.4 Complexity Analysis
In this subsection, the computational complexity of Algorithm 1 is analyzed as
follows. We first analyze the computational complexity of solving for
$\mathbf{w}_{k}$, $\forall k$, and $\mathbf{F}$. Problem (19) is a convex
problem with $K$ $N\times N$-dimensional and an $M\times M$-dimensional
variable to be optimized, $(I+1)K+1$ second-order cone (SOC) constraints and
$K+1$ linear matrix inequality (LMI) constraints. Using the CVX solver, the
computational complexity is of order
$\mathcal{O}\\{\text{ln}(1/\xi)2\sqrt{5}IK^{1.5}M^{6}\\}$, where $\xi$ is the
convergence threshold. The computational complexity of updating $c_{i,k}$,
$\forall i,k$, is of order $\mathcal{O}\\{M^{3}\\}$. Other calculations have
much lower complexities. For example, updating $\mathbf{w}_{k}$, $\forall k$,
has negligible computational of order $\mathcal{O}\\{N^{3}\\}$. Thus, the
total complexity to obtain $\mathbf{w}_{k},\forall k$, and $\mathbf{F}$ is of
order $\mathcal{O}\\{N_{\text{FP}}\text{ln}(1/\xi)2\sqrt{5}IK^{1.5}M^{6}\\}$,
where $N_{\text{FP}}$ is the number of iterations of the inner loop. The
computational complexity of updating $\mathbf{u}$ is of order
$\mathcal{O}\\{M^{3}\\}$. Therefore, the total computational complexity of the
proposed BCD-FP-SDR algorithm is of order
$\mathcal{O}\\{N_{\text{tot}}N_{\text{FP}}\text{ln}(1/\xi)2\sqrt{5}IK^{1.5}M^{6}\\}$,
where $N_{\text{tot}}$ is the number of iterations of the outer loop.
## III Joint AN-Aided Transmit Beamforming Design without Eavesdroppers’ CSI
### III-A System Model and Problem Formulation
When Eves are pure passive eavesdroppers, Alice is unaware of Eves’ CSI or
even their existence. In the sequel, the proposed design in the previous
section cannot be adopted to ensure security performance. In such cases, AN is
a very effective method to improve the physical layer security by disrupting
Eves’ reception. Specifically, in addition to transmitting confidential
information or probing signals, the BS and the radar also use the available
transmit power to generate AN for disturbing potential eavesdroppers as much
as possible. Therefore, the transmit beamforming and AN of both the BS and the
radar are jointly designed to guarantee good communication and radar sensing
performance while safeguarding the communication system against potential
malicious eavesdroppers.
Based on the above descriptions, the received signal of the $k$-th Bob can be
written as
$\displaystyle
y_{k}=\mathbf{h}_{k}^{H}(\mathbf{W}\mathbf{s}+\mathbf{z})+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{c}+\mathbf{v})+n_{k},$
(24)
where $\mathbf{z}\sim\mathcal{CN}(0,\mathbf{R}_{\mathrm{z}})$ and
$\mathbf{v}\sim\mathcal{CN}(0,\mathbf{R}_{\mathrm{v}})$ are AN vectors
generated by the BS and the radar, respectively. We assume that the
information symbol vector $\mathbf{s}$, the AN vector $\mathbf{z}$ generated
by the BS, the radar waveform $\mathbf{c}$, and the AN vector $\mathbf{v}$
generated by the radar are statistically independent of each other. Thus, the
SINR of the $k$-th Bob can be calculated as
$\displaystyle\mathrm{SINR}_{k}$ (25)
$\displaystyle=\frac{|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}}{\sum\limits_{j\neq
k}^{K}|\mathbf{h}_{k}^{H}\mathbf{w}_{j}|^{2}+\mathbb{E}\left\\{|\mathbf{h}_{k}^{H}\mathbf{z}|^{2}\right\\}+\|\mathbf{g}_{k}^{H}\mathbf{F}\|^{2}+\mathbb{E}\left\\{|\mathbf{g}_{k}^{H}\mathbf{v}|^{2}\right\\}+\sigma_{k}^{2}}$
$\displaystyle=\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{F}^{H}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}.$
From the communication perspective, the proposed joint AN-aided transmit
beamforming design aims to guarantee the communication QoS requirements of
legitimate transmission while interfering with Eves as much as possible.
On the radar side, the echo wave received by the radar is expressed as
$\mathbf{y}_{\mathrm{r}}=\alpha\mathbf{A}(\mathbf{F}\mathbf{c}+\mathbf{v})+\mathbf{Q}^{H}(\mathbf{W}\mathbf{s}+\mathbf{z})+\mathbf{n}_{\mathrm{r}}.$
(26)
After passing through the receive filter $\mathbf{u}$, the radar output is
$y_{\mathrm{r}}=\alpha\mathbf{u}^{H}\mathbf{A}(\mathbf{F}\mathbf{c}+\mathbf{v})+\mathbf{u}^{H}\mathbf{Q}^{H}(\mathbf{W}\mathbf{s}+\mathbf{z})+\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}}.$
(27)
The radar output SINR is thus given by
$\displaystyle\mathrm{SINR}_{\mathrm{r}}$ (28)
$\displaystyle=\frac{\|\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{F}\|^{2}}{\mathbb{E}\left\\{|\alpha\mathbf{u}^{H}\mathbf{A}\mathbf{v}|^{2}\right\\}+\sum\limits_{k=1}^{K}|\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{w}_{k}|^{2}+\mathbb{E}\left\\{|\mathbf{u}^{H}\mathbf{Q}^{H}\mathbf{z}|^{2}+|\mathbf{u}^{H}\mathbf{n}_{\mathrm{r}}|^{2}\right\\}}$
$\displaystyle=\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}.$
From the radar perspective, the radar output SINR is guaranteed to be no less
than a pre-defined threshold for achieving satisfactory target detection
performance. Meanwhile, as much power as possible is used to generate AN for
interfering with Eves.
It is intuitive that higher transmission power of the confidential information
poses a higher risk of being intercepted, since Eves’ eavesdropping SINR is
directly proportional to the transmission power. Thus, the BS should minimize
the transmit power to satisfy Bobs’ QoS and use the residual power to generate
AN signals. Similarly, for the radar system, the minimum required power is
allocated to generate directional signals, whose main lobe points to the
direction of the target for achieving satisfactory detection performance. The
huge residual power is used to generate omni-directional AN signals, which
will bring excellent security performance in the presence of potential
eavesdroppers. Therefore, our objective is to jointly design the BS transmit
beamformer $\mathbf{W}$, the covariance $\mathbf{R}_{\mathrm{z}}$ of the AN
vector $\mathbf{z}$ generated by the BS, the radar transmit beamformer
$\mathbf{F}$, the covariance $\mathbf{R}_{\mathrm{v}}$ of the AN vector
$\mathbf{v}$ generated by the radar, and the radar receive filter $\mathbf{u}$
to minimize the total transmit power used by the BS and radar beamformers.
Meanwhile, the communication QoS requirements of the legitimate users, the
radar output SINR constraint, and the power constraints of both communication
and radar systems are satisfied. Therefore, the optimization problem is
formulated as
$\displaystyle\underset{\mathbf{W},\mathbf{R}_{\mathrm{z}},\mathbf{F},\mathbf{R}_{\mathrm{v}},\mathbf{u}}{\min}\quad$
$\displaystyle\left\|\mathbf{W}\right\|_{F}^{2}+\left\|\mathbf{F}\right\|_{F}^{2}$
(29a) s.t. $\displaystyle\mathrm{SINR}_{k}\geq\Gamma_{k},~{}\forall k,$ (29b)
$\displaystyle\mathrm{SINR}_{\mathrm{r}}\geq\Gamma_{\mathrm{r}},$ (29c)
$\displaystyle\left\|\mathbf{W}\right\|_{F}^{2}+\mathrm{Tr}(\mathbf{R}_{\mathrm{z}})=P_{\mathrm{c}},$
(29d)
$\displaystyle\left\|\mathbf{F}\right\|_{F}^{2}+\mathrm{Tr}(\mathbf{R}_{\mathrm{v}})=P_{\mathrm{r}},$
(29e)
$\displaystyle\mathbf{R}_{\mathrm{z}}\in\mathbb{S}_{N},~{}\mathbf{R}_{\mathrm{v}}\in\mathbb{S}_{M}.$
(29f)
We observe that problem (29) is a non-convex problem that is difficult to
solve for the following two reasons. First, the variables are intricately
coupled in the constraints (29b) and (29c). Second, these two constraints are
quadratic and fractional. In order to efficiently solve this problem, we
employ the BCD and SDR algorithms to convert it into two more tractable sub-
problems and then alternately solve each of them until convergence is
achieved.
### III-B Proposed AN-aided Joint Transmit Beamforming Design
In this subsection, we propose an efficient double-loop BCD-SDR algorithm to
tackle the non-convex problem (29). It can be seen that problem (29) is very
complicated due to the coupling variables in the SINR constraints (29b) and
(29c). To this end, we divide this problem into two sub-problems with respect
to the radar and communication systems, and utilize a two-block BCD algorithm
to iteratively solve them.
#### III-B.1 The Sub-problem for Radar System
Given the variables $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{R}_{\mathrm{z}}$ of the communication system, the transmit beamformer
$\mathbf{F}$, the covariance $\mathbf{R}_{\mathrm{v}}$ of the AN vector
$\mathbf{v}$, and the receive filter $\mathbf{u}$ of the radar system are
jointly optimized. The problem for updating $\mathbf{F}$,
$\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$ can be formulated as
$\displaystyle\underset{\mathbf{F},\mathbf{R}_{\mathrm{v}},\mathbf{u}}{\min}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}$
(30a)
$\displaystyle\text{s.t.}~{}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{F}^{H}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (30b)
$\displaystyle~{}~{}~{}~{}~{}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(30c)
$\displaystyle~{}~{}~{}~{}~{}\left\|\mathbf{F}\right\|_{F}^{2}+\mathrm{Tr}(\mathbf{R}_{\mathrm{v}})=P_{\mathrm{r}},$
(30d) $\displaystyle~{}~{}~{}~{}~{}\mathbf{R}_{\mathrm{v}}\in\mathbb{S}_{M}.$
(30e)
Since the variables $\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$
are highly coupled in constraint (30c), which makes problem (30) very
difficult to solve, we adopt a two-block BCD scheme to iteratively solve them
as follows.
Update $\mathbf{F}$ and $\mathbf{R}_{\mathrm{v}}$: With fixed $\mathbf{u}$,
the sub-problem for updating $\mathbf{F}$ and $\mathbf{R}_{\mathrm{v}}$ is re-
arranged as
$\displaystyle\underset{\mathbf{F},\mathbf{R}_{\mathrm{v}}}{\min}~{}~{}$
$\displaystyle\left\|\mathbf{F}\right\|_{F}^{2}$ (31a) s.t.
$\displaystyle(\ref{eq:bcd radar}\text{b})-(\ref{eq:bcd radar}\text{e}).$
(31b)
Considering that the constraints (30b) and (30c) are still non-convex due to
the quadratic terms with respect to $\mathbf{F}$, we transform problem (31)
into
$\displaystyle\underset{\mathbf{R}_{\mathrm{F}},\mathbf{R}_{\mathrm{v}}}{\min}~{}~{}\mathrm{Tr}(\mathbf{R}_{\mathrm{F}})$
(32a)
$\displaystyle\text{s.t.}~{}\frac{\mathbf{h}_{k}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{R}_{\mathrm{F}}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (32b)
$\displaystyle\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{R}_{\mathrm{F}}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(32c)
$\displaystyle\mathrm{Tr}(\mathbf{R}_{\mathrm{F}})+\mathrm{Tr}(\mathbf{R}_{\mathrm{v}})=P_{\mathrm{r}},$
(32d)
$\displaystyle\mathbf{R}_{\mathrm{F}}\in\mathbb{S}_{M},~{}\mathbf{R}_{\mathrm{v}}\in\mathbb{S}_{M}.$
(32e)
Obviously, problem (32) is an SDP problem and can be solved by convex tools,
e.g., CVX. After solving $\mathbf{R}_{\text{F}}$, the radar beamforming matrix
$\mathbf{F}$ can be obtained by utilizing Cholesky decomposition.
Update $\mathbf{u}$: Notice that the variable $\mathbf{u}$ only exists in the
constraint (30c) of the problem (30). Therefore, with given $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$, problem (30) is transformed into a feasibility-
check problem. In order to leave more freedoms for minimizing
$\left\|\mathbf{F}\right\|_{F}^{2}$ in the next iteration, we propose to
optimize $\mathbf{u}$ to maximize the radar output SINR. The optimization
problem is formulated as
$\displaystyle\mathbf{u}^{\mathrm{opt}}=$ (33)
$\displaystyle\arg\mathop{\max}\limits_{\mathbf{u}}\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}},$
whose optimal solution can be easily obtained as the generalized eigenvector
corresponding to the largest eigenvalue of matrix
$\sigma_{\mathrm{0}}^{2}\big{[}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}\\!+\\!\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\\!+\\!\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\big{]}^{-1}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}$.
Algorithm 2 Joint AN-aided transmit beamforming design algorithm for solving
problem (29)
0: $\mathbf{h}_{k}$, $\mathbf{g}_{k}$, $\Gamma_{k}$, $\sigma_{k}^{2}$,
$\forall k$, $\mathbf{A}$, $\mathbf{Q}$, $\sigma_{\mathrm{0}}^{2}$,
$\sigma_{\text{r}}^{2}$, $\Gamma_{\mathrm{r}}$, $P_{\mathrm{c}}$,
$P_{\mathrm{r}}$.
0: $\mathbf{w}_{k}^{\star}$, $\forall k$, $\mathbf{R}_{\mathrm{z}}^{\star}$,
$\mathbf{F}^{\star}$, $\mathbf{R}_{\mathrm{v}}^{\star}$, and
$\mathbf{u}^{\star}$.
1: Initialize $\mathbf{w}_{k}$, $\forall k$, $\mathbf{R}_{\mathrm{z}}$, and
$\mathbf{u}$.
2: while the objective value (29a) does not converge do
3: while the objective value (30a) does not converge do
4: Calculate $\mathbf{R}_{\mathrm{F}}$ and update $\mathbf{R}_{\mathrm{v}}$ by
solving (32).
5: Update $\mathbf{F}$ from $\mathbf{R}_{\mathrm{F}}$ by Cholesky
decomposition.
6: Update $\mathbf{u}$ by (LABEL:eq:update_u).
7: end while
8: Calculate $\mathbf{W}_{k}$, $\forall k$, and update
$\mathbf{R}_{\mathrm{z}}$ by solving (35).
9: Update $\mathbf{w}_{k}$ from $\mathbf{W}_{k}$, $\forall k$, by EVD.
10: end while
11: Return $\mathbf{w}_{k}^{\star}$, $\forall k$,
$\mathbf{R}_{\mathrm{z}}^{\star}$, $\mathbf{F}^{\star}$,
$\mathbf{R}_{\mathrm{v}}^{\star}$, and $\mathbf{u}^{\star}$.
#### III-B.2 The Sub-problem for Communication System
With fixed $\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$, the sub-
problem for updating the transmit beamformer $\mathbf{W}$ and the covariance
$\mathbf{R}_{\mathrm{z}}$ of the AN vector $\mathbf{z}$ is re-arranged as
$\displaystyle\underset{\mathbf{w}_{k},\forall
k,\mathbf{R}_{\mathrm{z}}}{\min}~{}~{}$
$\displaystyle\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{w}_{k}\mathbf{w}_{k}^{H})$
(34a) s.t. $\displaystyle(\ref{eq:bcd radar}\text{b}),~{}(\ref{eq:bcd
radar}\text{c})$ (34b)
$\displaystyle\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{w}_{k}\mathbf{w}_{k}^{H})+\mathrm{Tr}(\mathbf{R}_{\mathrm{z}})=P_{\mathrm{c}},$
(34c) $\displaystyle\mathbf{R}_{\mathrm{z}}\in\mathbb{S}_{N}.$ (34d)
Similarly, we utilize the SDR algorithm to convert this problem into an SDP
problem by using the definitions in (16) and dropping the rank-one constraint
(17b) as
$\displaystyle\underset{\mathbf{W}_{k},\forall
k,\mathbf{R}_{\mathrm{z}}}{\min}~{}~{}\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{W}_{k})$
(35a) s.t.
$\displaystyle\frac{\mathbf{h}_{k}^{H}\mathbf{W}_{k}\mathbf{h}_{k}}{\mathbf{h}_{k}^{H}(\sum\limits_{j\neq
k}^{K}\mathbf{W}_{j}+\mathbf{R}_{\mathrm{z}})\mathbf{h}_{k}+\mathbf{g}_{k}^{H}(\mathbf{F}\mathbf{F}^{H}+\mathbf{R}_{\mathrm{v}})\mathbf{g}_{k}+\sigma_{k}^{2}}\geq\Gamma_{k},\forall
k,$ (35b)
$\displaystyle\frac{\sigma_{\mathrm{0}}^{2}\mathbf{u}^{H}\mathbf{A}\mathbf{F}\mathbf{F}^{H}\mathbf{A}^{H}\mathbf{u}}{\mathbf{u}^{H}\Big{(}\sigma_{\mathrm{0}}^{2}\mathbf{A}\mathbf{R}_{\mathrm{v}}\mathbf{A}^{H}+\mathbf{Q}^{H}(\sum\limits_{k=1}^{K}\mathbf{W}_{k}+\mathbf{R}_{\mathrm{z}})\mathbf{Q}+\sigma_{\mathrm{r}}^{2}\mathbf{I}\Big{)}\mathbf{u}}\geq\Gamma_{\mathrm{r}},$
(35c)
$\displaystyle\sum_{k=1}^{K}\mathrm{Tr}(\mathbf{W}_{k})+\mathrm{Tr}(\mathbf{R}_{\mathrm{z}})=P_{\mathrm{c}},$
(35d) $\displaystyle(\ref{eq:Wk constraint}\text{a}),~{}(\ref{eq:bcd
communication}\text{d}),$ (35e)
and then solve the resulting SDP problem by various existing algorithms or
toolboxes, e.g., CVX. The obtained $\mathbf{W}_{k}$, $\forall k$ can also be
guaranteed to satisfy the rank-1 constraint in this case. The proof is given
in Appendix A. We use the same method as in the previous section to obtain
$\mathbf{w}_{k}$, $\forall k$.
### III-C Summary, Initialization, Convergence, and Complexity Analysis
#### III-C.1 Summary
Based on the above derivations, the proposed joint AN-aided transmit
beamforming design is straightforward and summarized in Algorithm 2. In the
inner loop, we iteratively solve problems (31) and (LABEL:eq:update_u) to
respectively update $\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$
until the objective value (30a) converges. In the outer loop, we iteratively
solve problems (30) and (34) for updating $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{R}_{\text{z}}$ until the objective value (29a) converges.
#### III-C.2 Initialization
In order to solve sub-problem (31) for updating $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$, we investigate to properly initialize $\mathbf{W}$,
$\mathbf{R}_{\mathrm{z}}$, and $\mathbf{u}$. The concept of NSP is utilized to
initialize $\mathbf{W}$ and $\mathbf{R}_{\mathrm{z}}$. Specifically, we
jointly design the beamforming without AN aiming to guarantee the performance
requirements of communication and radar systems, and then project the AN of
the communication system onto the null space of the effective interference
channels between the BS and Bobs. Similarly, the initial value of the radar
receive filter is selected as $\mathbf{u}=\mathbf{a}_{\mathrm{r}}(\theta_{0})$
via a phase alignment operation. The obtained $\mathbf{F}$,
$\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$ by solving (30) can be set as the
initial value for solving sub-problem (34).
#### III-C.3 Convergence
The convergence of the proposed algorithm will be briefly proven as follows.
Denote
$\eta\big{(}\mathbf{W},\mathbf{R}_{\mathrm{z}},\mathbf{F},\mathbf{R}_{\mathrm{v}},\mathbf{u}\big{)}$
as the objective value of the original problem (29). First, for the radar
system, the sub-problem for updating $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$ is transformed into an SDP problem. It is obvious
that the transmit power is non-increasing between iterations, we have
$\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t},\mathbf{R}_{\mathrm{v}}^{t},\mathbf{u}^{t}\big{)}\geq\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t}\big{)},$
(36)
with fixed
$\left\\{\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{u}^{t}\right\\}$.
Given
$\left\\{\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1}\right\\}$,
the sub-problem for updating $\mathbf{u}$ is a feasibility-check problem,
which follows that
$\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t}\big{)}=\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)}.$
(37)
Second, the sub-problem for the communication system is also transformed into
an SDP problem. With given
$\left\\{\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\right\\}$,
it follows that
$\displaystyle\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)}$
(38)
$\displaystyle\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{R}_{\mathrm{z}}^{t+1},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)}.$
Based on the above analysis, we have the relationship of the objective values
between iterations as
$\displaystyle\eta\big{(}\mathbf{W}^{t},\mathbf{R}_{\mathrm{z}}^{t},\mathbf{F}^{t},\mathbf{R}_{\mathrm{v}}^{t},\mathbf{u}^{t}\big{)}$
(39)
$\displaystyle\geq\eta\big{(}\mathbf{W}^{t+1},\mathbf{R}_{\mathrm{z}}^{t+1},\mathbf{F}^{t+1},\mathbf{R}_{\mathrm{v}}^{t+1},\mathbf{u}^{t+1}\big{)},$
which indicates that the objective value of problem (29) is non-increasing
during iterations of Algorithm 2. Since the objective value of problem (29) is
greater than zero, the proposed Algorithm 2 can converge to a local optimum
point.
#### III-C.4 Complexity Analysis
In the inner loop, problem (32) is a convex problem with two $M\times
M$-dimensional variables to be optimized, $K+1$ SOC constraints and an LMI
constraint. Using the CVX solver, the complexity for updating $\mathbf{F}$ and
$\mathbf{R}_{\mathrm{v}}$ is of order
$\mathcal{O}\\{\text{ln}(1/\xi)8\sqrt{2}K^{1.5}M^{6}\\}$, and solving problem
(LABEL:eq:update_u) for updating $\mathbf{u}$ has the complexity of order
$\mathcal{O}\\{M^{3}\\}$. Thus, the total complexity for obtaining
$\mathbf{F}$, $\mathbf{R}_{\mathrm{v}}$, and $\mathbf{u}$ is of order
$\mathcal{O}\\{N_{\text{inn}}\text{ln}(1/\xi)8\sqrt{2}K^{1.5}M^{6}\\}$, where
$N_{\text{inn}}$ is the number of iterations of the inner loop. Similarly,
problem (35) is a convex problem with $K+1$ $N\times N$-dimensional variables
to be optimized, $K+1$ SOC constraints, and an LMI constraint. The
computational complexity for updating $\mathbf{w}_{k}$, $\forall k$, and
$\mathbf{R}_{\text{z}}$ is of order
$\mathcal{O}\\{\text{ln}(1/\xi)\sqrt{2}K^{4.5}N^{6}\\}$. Other lower
complexity calculations are omitted. Thus, the total computational complexity
for solving the problem (29) is of order
$\mathcal{O}\\{N_{\text{out}}N_{\text{inn}}\text{ln}(1/\xi)8\sqrt{2}K^{1.5}M^{6}\\}$,
where $N_{\text{out}}$ is the number of iterations of the outer loop.
## IV Simulation Results
In this section, extensive simulation results are provided to demonstrate the
performance of our proposed joint secure beamforming designs for the
considered ISAC systems under the assumption that Eves’ CSI is available or
not. Except for the radar-target link adopting the AoA model, the Rayleigh
fading channel model is adopted for all links, i.e., each entry of the channel
matrices is assumed to obey the standard complex Gaussian distribution. The
number of eavesdroppers is $I$ = 2. The antenna spacing of the radar is
$\Delta=\lambda/2$. The noise power at Eves and Bobs are set as
$\sigma_{\mathrm{e},i}^{2}=\sigma_{k}^{2}=10\mathrm{dBm},~{}\forall i,k$. The
total power budget of the BS is $P_{\mathrm{c}}$ = 10W. For simplicity, we
assume the communication QoS requirement of each Bob is the same and denoted
by $\Gamma_{\mathrm{c}}$. The target is located at the azimuth angle
$\theta_{0}=0^{\circ}$ and the radar cross section (RCS) is
$\sigma_{\mathrm{0}}^{2}=1$. The noise power to radar echoes is set as
$\sigma_{\mathrm{r}}^{2}=0\mathrm{dB}$444The radar echo passes through the
round-trip distance between the radar and the target, while the signal the
user receives only passes through the one-way distance from the BS to the
user. Thus, the noise power to radar echoes is usually a little larger than
that of the users.. Besides, the SINR requirement thresholds should be set
appropriately to prevent an infeasible case.
Figure 2: Convergence of the proposed algorithm. Figure 3: SINRs of Eve and
Bob versus the communication QoS requirement $\Gamma_{\mathrm{c}}$ ($M$ = 16,
$\Gamma_{\mathrm{r}}$ = 10dB, $P_{\mathrm{r}}$ = 500W). Figure 4: SINRs of Eve
and Bob versus the number of radar transmit/receive antennas $M$
($\Gamma_{\mathrm{c}}$ = 5dB, $\Gamma_{\mathrm{r}}$ = 10dB, $P_{\mathrm{r}}$ =
500W). Figure 5: SINRs of Eve and Bob versus the radar system transmit power
$P_{\mathrm{r}}$ ($M$ = 16, $\Gamma_{\mathrm{c}}$ = 5dB, $\Gamma_{\mathrm{r}}$
= 10dB).
### IV-A Known Eavesdroppers’ CSI
In this subsection, we illustrate the performance of our proposed joint secure
beamforming design assuming perfect knowledge of Eves’ CSI. We assume that the
BS is equipped with $N=4$ antennas to serve $K=4$ Bobs. In all simulations,
the SINR of Eve denotes the maximum eavesdropping SINR and the SINR of Bob
denotes the minimum legitimate SINR, respectively.
We first illustrate the convergence performance of the proposed algorithm in
Fig. 2. It can be observed that our proposed algorithm converges very quickly
under different settings, which reveals the effectiveness of the proposed
algorithm and its great potential in practical applications.
The SINRs of Eve and Bob versus the communication QoS requirement
$\Gamma_{\mathrm{c}}$ are plotted in Fig. 3 to illustrate the performance of
secure communication. The “proposed” scheme denotes our proposed joint
transmit beamforming design for secure ISAC systems. For comparisons, we also
include the “no radar” scheme, which denotes that there is one single
communication system without radar interference, and the “separate” scheme,
which represents that the transmit beamformers of the communication system and
radar system are separately designed without considering the interference
between them. From Fig. 3, we can observe that in the scenario of “no radar”,
the required communication QoS of Bobs can be guaranteed, but the security of
the communication system is not very satisfactory since Eves’ SINR is not low
enough. When the radar joins the system without cooperative joint beamforming
design, Eves’ SINR is significantly reduced owing to the strong radar
interfering signals, which provides good confidentiality. However, at the same
time, legitimate transmissions are also severely damaged by radar
interference. Compared with these two scenarios, our proposed joint transmit
beamforming design significantly decreases Eves’ SINR to a minimum value,
while always maintaining Bobs’ SINR at a required level to satisfy the QoS
requirements of the legitimate transmissions.
In Fig. 4, we show the SINRs of Eve and Bob versus the number of radar
antennas $M$. It is natural that the number of radar antennas does not affect
the security and legitimate transmission in the “no radar” and “separate”
scenarios. With the proposed joint beamforming design algorithm, Eves’ SINR
decreases with increasing $M$ since additional spatial DoFs can be exploited
to provide stronger interference to Eve. This phenomenon verifies that a
higher DoF of MIMO radar is of great help for secure communication.
Next, we present the SINRs of Eve and Bob versus the radar system power budget
$P_{\mathrm{r}}$ in Fig. 5, where the same relationship between different
scenarios can be observed as that in Figs. 3 and 4. In addition, we observe
that the eavesdropping SINRs of both the “proposed” and “separate” schemes
decrease with a larger $P_{\mathrm{r}}$ due to stronger jamming signals.
Moreover, the proposed scheme achieves the least eavesdropping SINR thanks to
the cooperative joint beamforming design, which guarantees the most favorable
security performance.
### IV-B Unknown Eavesdroppers’ CSI
Figure 6: Convergence of the proposed algorithm.
In this subsection, we demonstrate the performance of our proposed joint AN-
aided transmit beamforming design for the case that Eves’ CSI is unknown. The
BS is equipped with $N=8$ antennas. We first illustrate the convergence
performance of the proposed algorithm in Fig. 6. It can be observed that the
objective value (29a) monotonically converges within a limited number of
iterations under different settings, which verifies the effectiveness of the
proposed algorithm.
Next, we present the security performance of the proposed algorithm in Figs.
7-9. We also include a scheme that does not consider the physical layer
security. In other words, it only uses the least power to satisfy the
communication QoS requirements of the legitimate users and the radar output
SINR constraint (denoted as “No PLS”), and the scheme that utilizes the
tremendous residual power to generate AN in the null space of Bobs’ channels
(denoted as “Null space”). The security of the communication system is
evaluated in terms of the maximum eavesdropping SINR of the potential
eavesdroppers on the $K$ communication users.
Figure 7: SINR of Eves versus the communication QoS requirement
$\Gamma_{\mathrm{c}}$ ($M$ = 16, $\Gamma_{\mathrm{r}}$ = 10dB,
$P_{\mathrm{r}}$ = 1000W). Figure 8: SINR of Eves versus the number of radar
antennas $M$ ($\Gamma_{\mathrm{c}}$ = 5dB, $\Gamma_{\mathrm{r}}$ = 10dB,
$P_{\mathrm{r}}$ = 500W). Figure 9: SINR of Eves versus the radar system
transmit power $P_{\mathrm{r}}$ ($M$ = 16, $\Gamma_{\mathrm{c}}$ = 5dB,
$\Gamma_{\mathrm{r}}$ = 10dB).
In Fig. 7, we plot the maximum SINR of Eves versus the communication QoS
requirement $\Gamma_{\mathrm{c}}$. Not surprisingly, the maximum SINR of Eves
increases with increasing communication QoS requirements, because the stricter
communication QoS requirements require more power for the beamforming and
leave less power to generate AN, which brings higher risks of being
eavesdropped. In addition, we observe a considerable performance gap between
the “No PLS” scheme and the schemes with PLS design (“Null space” and
“Proposed”), since the generated AN greatly confuses the eavesdropper.
Moreover, it is seen that the proposed algorithm achieves notably better
security performance than the “Null space” scheme since more DoFs are
exploited in designing AN. These findings verify the advantages of the
proposed AN-aided transmit beamforming design in preventing potential
eavesdropping without the knowledge of Eves’ CSI.
Fig. 8 shows the maximum SINR of Eves versus the number of radar antennas $M$.
Since more antennas can exploit additional spatial DoFs, the transmit power
required to achieve the pre-set radar SINR becomes less. Therefore, for the
“No PLS” scheme, the jamming/interference from the radar to the eavesdropper
becomes weak and the security performance is degraded. On the other hand, for
the “Null space” scheme and the proposed scheme, more power is available for
generating AN, which can significantly improve the security performance. It is
noticed that, compared with Fig. 4, Eves’ SINR will not visibly decrease when
the number of radar antennas is greater than 16. This is because the spatial
DoFs of a radar system with 16 antennas are sufficient for providing enough
signal processing ability. More radar antennas would not significantly reduce
the transmit power for achieving the required radar SINR. Consequently, the
available power for generating AN will be relatively small with more radar
antenna than 16, and the Eves’ SINR will be maintained at almost the same
extremely low level, e.g. close to -44dB. These results can still verify that
the proposed AN-aided secure beamforming design can provide superior security
performance even when the MIMO radar has a moderate number of antennas.
Fig. 9 presents the maximum SINR of Eve versus the radar system power budget
$P_{\mathrm{r}}$. The same relationship between different design schemes can
be observed in Figs. 7 and 8. In addition, since more additional power can be
utilized for generating AN, the eavesdropping SINR decreases with the increase
of $P_{\mathrm{r}}$ for the “Null space” and the proposed schemes.
## V Conclusions
This paper investigated joint secure transmit beamforming designs for ISAC
systems. When the eavesdroppers’ channels are available, the maximum
eavesdropping SINR was minimized under the communication QoS constraints, the
radar detection performance constraint, and the communication and radar power
constraints. An efficient BCD-FP-SDR-based algorithm was proposed to solve the
non-convex optimization problem. When the eavesdroppers’ channels are
unavailable, a joint AN-aided transmit beamforming design was developed to
disrupt the eavesdroppers’ reception while guaranteeing the legitimate
communication SINR and the radar output SINR requirements by utilizing the
available power of radar and communication systems to generate as much AN as
possible. We proposed a double-loop BCD-SDR-based algorithm to solve the
resulting non-convex optimization problem. Simulation results illustrated the
advantages of ISAC systems on secure transmissions and the effectiveness of
the proposed algorithms.
## Appendix A
Let $\widetilde{\mathbf{W}}_{k}$, $\forall k$, be an arbitrary global optimal
solution to problem (18). Then, we construct a new solution
$\widehat{\mathbf{W}}_{k}$, $\forall k$, based on
$\widetilde{\mathbf{W}}_{k}$, $\forall k$, as
$\widehat{\mathbf{W}}_{k}=\widetilde{\mathbf{W}}_{k},~{}\widehat{\mathbf{w}}_{k}=(\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k})^{-1/2}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k},~{}\widehat{\mathbf{W}}_{k}=\widehat{\mathbf{w}}_{k}\widehat{\mathbf{w}}_{k}^{H}.$
(40)
Clearly, $\widehat{\mathbf{W}}_{k}$, $\forall k$, is rank-one and positive
semidefinite which means that constraints (18g) and (18h) hold. Next, we show
that $\widehat{\mathbf{W}}_{k}$ is also a global optimal solution to problem
(18). Based on (40), we have the following equality
$\sum_{k=1}^{K}\widetilde{\mathbf{W}}_{k}=\sum_{k=1}^{K}\widehat{\mathbf{W}}_{k},$
(41)
which implies that the constraints (18d) and (18e) hold. Moreover, the
constraint (18f) and the objective function hold since they do not contain the
variables $\widetilde{\mathbf{W}}_{k}$, $\forall k$.
In addition, since
$\mathbf{h}_{k}^{H}\widehat{\mathbf{W}}_{k}\mathbf{h}_{k}=\mathbf{h}_{k}^{H}\widehat{\mathbf{w}}_{k}\widehat{\mathbf{w}}_{k}^{H}\mathbf{h}_{k}=\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k},$
(42)
the SINR constraint in (18c) can be re-written as
$\displaystyle\frac{(\Gamma_{k}+1)}{\Gamma_{k}}\mathbf{h}_{k}^{H}\widehat{\mathbf{W}}_{k}\mathbf{h}_{k}=\frac{(\Gamma_{k}+1)}{\Gamma_{k}}\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{k}\mathbf{h}_{k}$
(43)
$\displaystyle\geq\sum_{j=1}^{K}\mathbf{h}_{k}^{H}\widetilde{\mathbf{W}}_{j}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{k}+\sigma_{k}^{2}$
$\displaystyle=\sum_{j=1}^{K}\mathbf{h}_{k}^{H}\widehat{\mathbf{W}}_{j}\mathbf{h}_{k}+\mathbf{g}_{k}^{H}\mathbf{R}_{\mathrm{F}}\mathbf{g}_{k}+\sigma_{k}^{2},$
namely (18c) holds for $\widehat{\mathbf{W}}_{k}$, $\forall k$. Similarly, the
SINR constraint (18b) also holds for $\widehat{\mathbf{W}}_{k}$, $\forall k$.
Therefore, it is proven that the constructed $\widehat{\mathbf{W}}_{k}$,
$\forall k$ in (40) is a feasible and global optimal rank-1 solution to the
problem (18).
For problem (35), we also let $\widetilde{\mathbf{W}}_{k}$, $\forall k$, be an
arbitrary global optimal solution to problem (35). In the same way as (40), we
construct a new solution $\widehat{\mathbf{W}}_{k}$, $\forall k$, which
implies that (35e) holds. Similar to the previous proof, constraints
(35a)-(35d) hold. Thus, the constructed $\widehat{\mathbf{W}}_{k}$, $\forall
k$, in (40) being a feasible and global optimal solution to problem (35) is
verified.
## References
* [1] J. Chu, R. Liu, Y. Liu, M. Li, and Q. Liu, “Joint transmit beamforming design for secure communication and radar coexistence systems,” in Proc. IEEE Wireless Commun. Network Conf. (WCNC), Austin, USA, Apr. 2022.
* [2] A. R. Chiriyath, B. Paul, and D. W. Bliss, “Radar-communication convergence: Coexistence, cooperation, and co-design,” IEEE Trans. Cogn. Commun. Netw., vol. 3, no. 1, pp. 1-12, Mar. 2017.
* [3] F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” IEEE Trans. Commun., vol. 68, no. 6, pp. 3834-3862, Jun. 2020.
* [4] N. C. Luong, X. Lu, D. T. Hoang, D. Niyato, and D. I. Kim, “Radio resource management in joint radar and communication: A comprehensive survey,” IEEE Commun. Surv. Tut., vol. 23, no. 2, pp. 780-814, 2nd Quart. 2021.
* [5] J. A. Zhang, M. L. Rahman, K. Wu, X. Huang, Y. J. Guo, S. Chen, and J. Yuan, “Enabling joint communication and radar sensing in mobile networks - A survey,” IEEE Commun. Surv. Tut., vol. 24, no. 1, pp. 306-345, 1st Quart. 2022.
* [6] A. Liu et al., “A survey on fundamental limits of integrated sensing and communication,” IEEE Commun. Surv. Tut., vol. 24, no. 2, pp. 994-1034, 2nd Quart. 2022.
* [7] F. Liu and C. Masouros, “A tutorial on joint radar and communication transmission for vehicular networks—Part I: Background and fundamentals,” IEEE Commun. Lett., vol. 25, no. 2, pp. 322-326, Feb. 2021.
* [8] Y. Cui, F. Liu, X. Jing, and J. Mu, “Integrating sensing and communications for ubiquitous IoT: Applications, trends, and challenges,” IEEE Netw., vol. 35, no. 5, pp. 158-167, Sep./Oct. 2021.
* [9] F. Liu, L. Zhou, C. Masouros, A. Li, W. Luo, and A. Petropulu, “Toward dual-functional radar-communication systems: Optimal waveform design,” IEEE Trans. Signal Process., vol. 66, no. 16, pp. 4264-4279, Aug. 2018.
* [10] F. Liu, C. Masouros, A. Li, H. Sun, and L. Hanzo, “MU-MIMO communications with MIMO radar: From co-existence to joint transmission,” IEEE Trans. Wireless Commun., vol. 17, no. 4, pp. 2755-2770, Apr. 2018.
* [11] R. Liu, M. Li, Q. Liu, and A. L. Swindlehurst, “Joint waveform and filter designs for STAP-SLP-based MIMO-DFRC systems,” IEEE J. Sel. Areas Commun., vol. 40, no. 6, pp. 1918-1931, Jun. 2022.
* [12] R. Liu, M. Li, Y. Liu, Q. Wu, and Q. Liu, “Joint transmit waveform and passive beamforming design for RIS-aided DFRC Systems,” IEEE J. Sel. Topics Signal Process., vol. 16, no. 5, pp. 995-1010, Aug. 2022.
* [13] J. H. Reed et al., “On the co-existence of TD-LTE and radar over 3.5 GHz band: An experimental study,” IEEE Wireless Commun. Lett., vol. 5, no. 4, pp. 368-371, Aug. 2016.
* [14] F. Hessar and S. Roy, “Spectrum sharing between a surveillance radar and secondary Wi-Fi networks,” IEEE Trans. Aerosp. Electron. Syst., vol. 52, no. 3, pp. 1434-1448, Jun. 2016.
* [15] H. Griffiths et al., “Radar spectrum engineering and management: Technical and regulatory issues,” IEEE Proc., vol. 103, no. 1, pp. 85-102, Jan. 2015.
* [16] E. Grossi, M. Lops, and L. Venturino, “Joint design of surveillance radar and MIMO communication in cluttered environments,” IEEE Trans. Signal Process., vol. 68, no. 1, pp. 1544-1557, Dec. 2020.
* [17] L. Zheng, M. Lops, X. Wang, and E. Grossi, “Joint design of overlaid communication systems and pulsed radars,” IEEE Trans. Signal Process., vol. 66, no. 1, pp. 139-154, Jan. 2018.
* [18] H. Deng and B. Himed, “Interference mitigation processing for spectrum-sharing between radar and wireless communications systems,” IEEE Trans. Aerosp. Electron. Syst., vol. 49, no. 3, pp. 1911-1919, Jul. 2013.
* [19] Z. Geng, H. Deng, and B. Himed, “Adaptive radar beamforming for interference mitigation in radar-wireless spectrum sharing,” IEEE Signal Process. Lett., vol. 22, no. 4, pp. 484-488, Apr. 2015.
* [20] B. Li and A. Petropulu, “MIMO radar and communication spectrum sharing with clutter mitigation,” in Proc. IEEE Radar Conf. (RadarConf), Philadelphia, USA, May 2016.
* [21] F. Liu, C. Masouros, A. Li, T. Ratnarajah, and J. Zhou, “MIMO radar and cellular coexistence: A power-efficient approach enabled by interference exploitation,” IEEE Trans. Signal Process., vol. 66, no. 14, pp. 3681-3695, Jul. 2018.
* [22] B. Li, A. P. Petropulu, and W. Trappe, “Optimum co-design for spectrum sharing between matrix completion based MIMO radars and a MIMO communication system,” IEEE Trans. Signal Process., vol. 64, no. 17, pp. 4562-4575, Sep. 2016.
* [23] N. Nartasilpa, A. Salim, D. Tuninetti, and N. Devroye, “Communications system performance and design in the presence of radar interference,” IEEE Trans. Commun., vol. 66, no. 9, pp. 4170-4185, Sep. 2018.
* [24] L. Zheng, M. Lops, Y. C. Eldar, and X. Wang, “Radar and communication coexistence: An overview: A review of recent methods,” IEEE Signal Process. Mag., vol. 36, no. 5, pp. 85-99, Sep. 2019.
* [25] Z. Cheng, B. Liao, S. Shi, Z. He, and J. Li, “Co-design for overlaid MIMO radar and downlink MISO communication systems via Cramér-Rao bound minimization,” IEEE Trans. Signal Process., vol. 67, no. 24, pp. 6227-6240, Dec. 2019.
* [26] M. Bic and V. Koivunen, “Radar waveform optimization for target parameter estimation in cooperative radar-communications systems,” IEEE Trans. Aerosp. Electron. Syst., vol. 55, no. 5, pp. 2314-2326, Oct. 2019.
* [27] J. Qian, M. Lops, L. Zheng, X. Wang, and Z. He, “Joint system design for coexistence of MIMO radar and MIMO communication,” IEEE Trans. Signal Process., vol. 66, no. 13, pp. 3504-3519, Jul. 2018.
* [28] J. Qian, L. Venturino, M. Lops, and X. Wang, “Radar and communication spectral coexistence in range-dependent interference,” IEEE Trans. Signal Process., vol. 69, pp. 5891-5906, Oct. 2021.
* [29] S. Sodagari, A. Khawar, T. C. Clancy, and R. McGwier, “A projection based approach for radar and telecommunication systems coexistence,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Anaheim, USA, Dec. 2012, pp. 5010-5014.
* [30] A. Babaei, W. H. Tranter, and T. Bose, “A nullspace-based precoder with subspace expansion for radar/communications coexistence,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Atlanta, USA, Dec. 2013.
* [31] J. A. Mahal, A. Khawar, A. Abdelhadi, and T. C. Clancy, “Spectral coexistence of MIMO radar and MIMO cellular system,” IEEE Trans. Aerosp. Electron. Syst., vol. 53, no. 2, pp. 655-668, Apr. 2017.
* [32] M. Rihan and L. Huang, “Optimum co-design of spectrum sharing between MIMO radar and MIMO communication systems: An interference alignment approach,” IEEE Trans. Veh. Technol., vol. 67, no. 12, pp. 11667-11680, Dec. 2018.
* [33] X. Liu, T. Huang, N. Shlezinger, Y. Liu, J. Zhou, and Y. C. Eldar, “Joint transmit beamforming for multiuser MIMO communications and MIMO radar,” IEEE Trans. Signal Process., vol. 68, pp. 3929-3944, Jun. 2020.
* [34] F. Liu, C. Masouros, A. Li, and T. Ratnarajah, “Robust MIMO beamforming for cellular and radar coexistence,” IEEE Wireless Commun. Lett., vol. 6, no. 3, pp. 374-377, Jun. 2017.
* [35] M. Khandaker, C. Masouros, and K.-K. Wong, “Constructive interference based secure precoding: A new dimension in physical layer security,” IEEE Trans. Inf. Forensics Security, vol. 13, no. 9, pp. 2256-2268, Sep. 2018.
* [36] Z. Chu, K. Cumanan, Z. Ding, M. Johnston, and S. Y . Le Goff, “Secrecy rate optimizations for a MIMO secrecy channel with a cooperative jammer,” IEEE Trans. Veh. Technol., vol. 64, no. 5, pp. 1833-1847, May 2015.
* [37] N. Su, F. Liu, Z. Wei, Y.-F. Liu, and C. Masouros, “Secure dual-functional radar-communication transmission: Exploiting interference for resilience against target eavesdropping,” IEEE Trans. Wireless Commun., vol. 21, no. 9, pp. 7238-7252, Sep. 2022.
* [38] L. Li, A. P. Petropulu, Z. Chen, and J. Fang, “Improving wireless physical layer security via exploiting co-channel interference,” IEEE J. Sel. Topics Signal Process., vol. 10, no. 8, pp. 1433-1448, Dec. 2016.
* [39] W.-C. Liao, T.-H. Chang, W.-K. Ma, and C.-Y. Chi, “QoS-based transmit beamforming in the presence of eavesdroppers: An optimized artificial-noise-aided approach,” IEEE Trans. Signal Process., vol. 59, no. 3, pp. 1202-1216, Mar. 2011.
* [40] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 2180-2189, Jun. 2008.
* [41] N. Su, F. Liu, and C. Masouros, “Secure radar-communication systems with malicious targets: Integrating radar, communications and jamming functionalities,” IEEE Trans. Wireless Commun., vol. 20, no. 1, pp. 83-95, Jan. 2021.
* [42] Y. Zhou, H. Zhou, F. Zhou, S. Ma, and R. Q. Hu, “Artificial-noise aided precoding design for MIMO integrated radar and communication systems,” in Proc. Int. Conf. Wireless Commun. Signal Process. (WCSP), Xi’an, China, Dec. 2019.
* [43] K. V. Mishra, A. Chattopadhyay, S. S. Acharjee, and A. P. Petropulu, “OptM3Sec: Optimizing multicast IRS-aided multiantenna DFRC secrecy channel with multiple eavesdroppers,” in Proc. IEEE Int. Conf. Acoustics Speech Signal Process. (ICASSP), Singapore, Singapore, May 2022.
* [44] M. Li, G. Ti, and Q. Liu, “Secure beamformer designs in MU-MIMO systems with multiuser interference exploitation,” IEEE Trans. Veh. Technol., vol. 67, no. 9, pp. 8288-8301, Sep. 2018.
* [45] I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 3873-3883, Oct. 2006.
* [46] E. Aboutanios and A. Hassanien, “Low-cost beamforming-based DOA estimation with model order determination,” in Proc. IEEE Sensor Array Multichannel Signal Process. Workshop (SAM), Hangzhou, China, Jun. 2020.
* [47] P. Stoica and K. C. Sharman, “Maximum likelihood methods for direction-of-arrival estimation,” IEEE Trans. Acousti. Speech Signal Process., vol. 38, no. 7, pp. 1132-1143, Jul. 1990.
* [48] K. Shen and W. Yu, “Fractional programming for communication systems—Part I: Power control and beamforming,” IEEE Trans. Signal Process., vol. 66, no. 10, pp. 2616-2630, May 2018.
* [49] W. Dinkelbach, “On nonlinear fractional programming,” Manage. Sci., vol. 133, no. 7, pp. 492-498, Mar. 1967.
* [50] J.-P. Crouzeix, “Algorithms for generalized fractional programming,” Math. Program., vol. 52, no. 1, pp. 191-207, May 1991.
|
# Controlled-source electromagnetic modelling using high order finite-
difference time-domain method on a nonuniform grid
Pengliang Yang1 and Rune Mittet2
1School of Mathematics, Harbin Institute of Technology, Harbin, China, 150001
E-mail<EMAIL_ADDRESS>
2Norwegian University of Science and Technology (NTNU), Norway
E-mail<EMAIL_ADDRESS>
###### Abstract
Simulation of 3D low-frequency electromagnetic fields propagating in the Earth
is computationally expensive. We present a fictitious wave domain high-order
finite-difference time-domain (FDTD) modelling method on nonuniform grids to
compute frequency-domain 3D controlled-source electromagnetic (CSEM) data. The
method overcomes the inconsistency issue widely present in the conventional
2nd order staggered grid finite difference scheme over nonuniform grid,
achieving high accuracy with arbitrarily high order scheme. The finite-
difference coefficients adaptive to the node spacings, can be accurately
computed by inverting a Vandermonde matrix system using efficient algorithm. A
generic stability condition applicable to nonuniform grids is established,
revealing the dependence of the time step and these finite-difference
coefficients. A recursion scheme using fixed point iterations is designed to
determine the stretching factor to generate the optimal nonuniform grid. The
grid stretching in our method reduces the number of grid points required in
the discretization, making it more efficient than the standard high-order FDTD
with a densely sampled uniform grid. Instead of stretching in both vertical
and horizontal directions, better accuracy of our method is observed when the
grid is stretched along the depth without horizontal stretching. The
efficiency and accuracy of our method are demonstrated by numerical examples.
## 1 Introduction
Marine controlled source electromagnetics (CSEM) provides valuable information
about subsurface resistivities and therefore potentially about pore fluids or
rocks. It is very useful to decipher subsurface properties to assist energy
exploration, in particular when combined with seismic data. The CSEM
technology relies on low-frequency electromagnetic field propagation to probe
the subsurface. The low-frequency electromagnetic (EM) field propagation does
not lend itself to an intuitive understanding in the same manner as seismic
field propagation does due to the diffusive nature of the EM field in
conductive media. Thus, three-dimensional modeling becomes an important tool
for the interpretation of CSEM data. Imaging of marine CSEM data is today
mainly done by inversion of the observed electric and/or magnetic fields.
The kernel of CSEM inversion is the numerical simulation of 3D electromagnetic
field propagation, which is computationally expensive. Reducing the simulation
time without compromising the accuracy is important. It can shorten the
turnaround time for an imaging project while reducing the investments in
computer hardware. The implementation of nonuniform grid schemes is a well
known strategy to reduce simulation time. To retain good accuracy we propose a
high-order finite-difference approach.
There are many studies on diffusive electromagnetic modelling using different
methods. Examples are the frequency-domain finite-difference method (Newman
and Alumbaugh,, 1995; Smith, 1996a, ; Mulder,, 2006; Streich,, 2009), the
frequency-domain finite-element method (Li and Key,, 2007; da Silva et al.,,
2012; Key,, 2016; Rochlitz et al.,, 2019), and the time-domain finite-
difference method (Oristaglio and Hohmann,, 1984; Wang and Hohmann,, 1993;
Taflove and Hagness,, 2005). A key fact in all numerical modelling methods is
that the computational cost and the memory requirement are connected and
cannot be splitted. A method can be very efficient if more computer memory is
available. The efficiency and accuracy of the modelling can be dramatically
hampered when the available computer resources are restricted.
Due to the diffusive nature of low-frequency CSEM fields, most of the 3D CSEM
modelling schemes resort to the frequency-domain solution of the Maxwell
equation to avoid the high computational cost dictated by the restrictive
stability condition for the direct solution in the time domain. Time-domain
methods are attractive options because they require less amount of memory than
frequency-domain modelling within a model of the same size. Another advantage
with time-domain solutions is that multiple frequencies can be extracted from
the same simulation. Both the frequency-domain finite-difference method and
the frequency-domain finite-element method formulate Maxwell equation as a
linear equation system, which may be solved using direct (Streich,, 2009) or
iterative (Smith, 1996b, ; Mulder,, 2006; Puzyrev et al.,, 2013) solvers. A
nice feature with a direct solver is that multiple right-hand sides are fast
to calculate after the system matrix has been factorized or inverted. However,
there are significant implementation challenges with this approach when
realistic size marine CSEM surveys are simulated. The memory requirements are
large even if the equation system is sparse.
The finite-difference time-domain (FDTD) modelling based on the staggered grid
proposed by Yee, (1966) has for several decades been a main workhorse for many
EM applications. The implementation of the numerical core is straight forward
and the computational efficiency is good for wave phenomena. The computational
efficiency for diffusive phenomena is rather poor in the time domain. The
system of partial differential equations can be considered stiff in this case
(Mittet,, 2010) and a very small time step is required to retain stability.
The computational efficiency can be improved significantly due to a
correspondence principle for wave and diffusion fields (Lee et al.,, 1989; de
Hoop,, 1996; Mittet,, 2010). Maaø, (2007) proposed a mixed wave and diffusion-
domain FDTD method to perform numerically efficient CSEM modelling. This
method allowed for large time steps compared to a purely diffusion-domain
solution. Mittet, (2010) proposed a high-order FDTD scheme by utilizing the
fictitious wave to diffusion-domain transformation. The simulation is
performed in the wave domain where the propagation velocity is proportional to
the square root of resistivity.
The Yee grid (staggered grid) FDTD scheme is often the method of choice due to
a good agreement with physics. It gives divergence free magnetic fields and
electric currents (Smith, 1996a, ). This standard scheme proposed by Yee,
(1966) is based on the second-order approximation of the first derivatives
assuming an equispaced mesh. On the uniform grid, moving from second-order
FDTD to high-order FDTD is straight forward (Mittet,, 2010), and gives
improved modelling accuracy due to the reduction of spatial dispersion errors.
To improve the modelling efficiency, the use of nonuniform grid is widespread
(Newman and Alumbaugh,, 1995; Mulder,, 2006), however it results in
inconsistencies for the grid staggering. As illustrated in Figure 1, this
inconsistency leads to only first-order local truncation error, even though
the global accuracy may be up to second order (Monk and Süli,, 1994). The
problem is persistent and has remained unresolved for high-order schemes.
Figure 1: The 2nd order staggered-grid finite-difference scheme on (a) uniform
grid and (b) nonuniform grid. The grid points on staggered grid are obtained
by taking the midpoints from nonuniform grid to ensure the 2nd order accuracy.
This accuracy is not guaranteed on nonuniform grid since the midpoints is
inconsistent. For example, $x_{3/2}=(x_{1}+x_{2})/2$,
$x_{5/2}=(x_{2}+x_{3})/2$, but $\tilde{x}_{2}=(x_{3/2}+x_{5/2})/2\neq x_{2}$
due to uniform grid spacing. This inconsistency leads to first-order local
truncation error using staggered-grid finite-difference scheme on the
nonuniform grid.
We propose an efficient 3D CSEM simulation method with high accuracy using
_high-order FDTD on a staggered, nonuniform grid_ , following the fictitious
wave domain approach (Mittet,, 2010). To resolve the inconsistency issue in
conventional 2nd order staggered grid approach, our key recognition is that
the order of local truncation error can be arbitrarily high also on a
nonuniform grid if the finite-difference operator coefficients are adapted
properly to the variable grid spacing. The derivative operator coefficients
are calculated by inverting a Vandermonde matrix system.
To gain good modeling efficiency, we transforms the diffusive Maxwell equation
into the fictitious wave domain. The efficiency of the method is restricted by
stability condition: the stepsize in time is proportional to the inverse of
the propagation velocity of the field, affected by the node spacing. The
gridding of the same physical domain leads to different number of gridpoints,
affecting the size of the linear system to be solved. The use of nonuniform
grid helps to reduce the number of gridpoint, thus reducing the computational
cost. Unfortunately, the stability condition for high order FDTD over non-
uniform grid is non-trivial. An important contribution of this paper is to
establish a new stability condition valid for arbitrarily high order FDTD
scheme on nonuniform grid. The stability condition shows the strong dependence
between time step and the finite difference coefficients computed by inverting
the Vandermonde matrix. This is not known in EM geophysics community, as far
as we know.
To generate the optimal grid using a power law, we design a recursion scheme
using fixed point iterations to find the optimal stretching factor. We prove
the recursion scheme is guaranteed to converge. The optimal factor found by
the recursive scheme allows accurate matching of the computational domain
using given number of mesh points. The high accuracy and efficiency of this
high-order FDTD method on a nonuniform grid will be exemplified by a number of
numerical tests using reference solutions.
## 2 Theory
We utilize the correspondence principle for electromagnetic wave and diffusion
fields (Lee et al.,, 1989; de Hoop,, 1996; Mittet,, 2010) to calculate the
CSEM response efficiently. The key to a high-order local truncation error on a
nonuniform grid relies on the solutions to a Vandermonde system giving
derivative operator coefficients that adapts to local grid properties. The
stability condition is then established for FDTD modelling on the nonuniform
grid.
The Maxwell equations in a quasi-static regime (i.e., with negligible effect
of displacement currents) are written in the time domain as
$\displaystyle\nabla\times\mathbf{E}+\mu\partial_{t}\mathbf{H}$
$\displaystyle=$ $\displaystyle-\mathbf{M},$ (1)
$\displaystyle-\nabla\times\mathbf{H}+\mathbf{\sigma}\mathbf{E}$
$\displaystyle=$ $\displaystyle-\mathbf{J},$
or in the frequency domain as
$\displaystyle\nabla\times\mathbf{E}-\mathrm{i}\omega\mu\mathbf{H}$
$\displaystyle=$ $\displaystyle-\mathbf{M},$ (2)
$\displaystyle-\nabla\times\mathbf{H}+\mathbf{\sigma}\mathbf{E}$
$\displaystyle=$ $\displaystyle-\mathbf{J},$
where $\mathbf{E}=(E_{x},E_{y},E_{z})^{T}$ and
$\mathbf{H}=(H_{x},H_{y},H_{z})^{T}$ are electric and magnetic fields. The
magnetic permeability is $\mu$. The conductivity is a symmetric $3\times 3$
tensor: $\sigma_{ij}=\sigma_{ji},\;i,j\in\\{x,y,z\\}$. An isotropic medium
means that only the diagonal elements of the conductivity tensor are non-zeros
and the same in all directions: $\sigma_{xx}=\sigma_{yy}=\sigma_{zz}$;
$\sigma_{ij}=0,i\neq j$. The vertical transverse isotropic (VTI) medium
implemented here still has only diagonal elements, but the vertical and the
horizontal conductivities may differ, i.e.,
$\sigma_{h}:=\sigma_{xx}=\sigma_{yy}$, $\sigma_{v}=\sigma_{zz}$. We use the
following Fourier transform convention,
$\partial_{t}\leftrightarrow-\mathrm{i}\omega$.
To speed up the FDTD modelling, we transform the above system from the
diffusion to the wave domain, following Mittet, (2010). The idea is to define
a fictitious dielectric permittivity in equation 2 as
$\sigma=2\omega_{0}\varepsilon$, yielding
$\displaystyle\nabla\times\mathbf{E}-\mathrm{i}\omega\mu\mathbf{H}$
$\displaystyle=$ $\displaystyle-\mathbf{M},$ (3)
$\displaystyle-\nabla\times\mathbf{H}+2\omega_{0}\varepsilon\mathbf{E}$
$\displaystyle=$ $\displaystyle-\mathbf{J},$
which gives the following relation after multiplying the second equation with
$\sqrt{-\mathrm{i}\omega/2\omega_{0}}$
$\displaystyle\nabla\times\mathbf{E}+\underbrace{\sqrt{-\mathrm{i}2\omega\omega_{0}}}_{-\mathrm{i}\omega^{\prime}}\mu\underbrace{\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\mathbf{H}}_{\mathbf{H}^{\prime}}$
$\displaystyle=$ $\displaystyle-\mathbf{M},$ (4)
$\displaystyle-\nabla\times\underbrace{\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\mathbf{H}}_{\mathbf{H}^{\prime}}+\underbrace{\sqrt{-\mathrm{i}2\omega\omega_{0}}}_{-\mathrm{i}\omega^{\prime}}\varepsilon\mathbf{E}$
$\displaystyle=$
$\displaystyle-\underbrace{\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\mathbf{J},}_{\mathbf{J}^{\prime}}$
which translates into the wave and simulation domain as the time dependent
system
$\displaystyle\nabla\times\mathbf{E}^{\prime}+\mu\partial_{t}\mathbf{H}^{\prime}$
$\displaystyle=$ $\displaystyle-\mathbf{M}^{\prime},$ (5)
$\displaystyle-\nabla\times\mathbf{H}^{\prime}+\varepsilon\partial_{t}\mathbf{E}^{\prime}$
$\displaystyle=$ $\displaystyle-\mathbf{J}^{\prime}.$
We have introduced a prime to identify the fields in the wave domain.
From the electromagnetic fields in the wave domain, the frequency-domain
fields can be computed on the fly during modelling using the fictitious wave
transformation, exemplified by the electric field here,
$\displaystyle\mathbf{E}^{\prime}(\mathbf{x},\omega^{\prime})=\int_{0}^{T_{\max}}\mathbf{E}^{\prime}(\mathbf{x},t)e^{\mathrm{i}\omega^{\prime}t}\mathrm{d}t,$
(6)
where $T_{\max}$ is the final time until the field
$E^{\prime}(\mathbf{x},\omega^{\prime})$ reaches its steady state and where
$\displaystyle\omega^{\prime}=(1+\mathrm{i})\sqrt{\omega\omega_{0}}.$ (7)
In order to have results valid for the frequency domain we need to calculate
the Green’s functions. We need the following relation,
$\displaystyle\mathbf{E}^{\prime}$ $\displaystyle=$ $\displaystyle\mathbf{E},$
(8) $\displaystyle\mathbf{H}^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\mathbf{H},$
$\displaystyle\mathbf{M}^{\prime}$ $\displaystyle=$ $\displaystyle\mathbf{M},$
$\displaystyle\mathbf{J}^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\mathbf{J}.$
The Green’s functions are then obtained by normalizing the transformed
electric and magnetic fields with the source current,
$\displaystyle
G_{kj}^{E|J}(\mathbf{x},\omega|\mathbf{x}_{s})=\frac{E_{k}(\mathbf{x},\omega|\mathbf{x}_{s})}{J_{j}(\omega)}=\sqrt{\frac{-\mathrm{i}\omega}{2\omega_{0}}}\frac{E_{k}^{\prime}(\mathbf{x},\omega|\mathbf{x}_{s})}{J_{j}^{\prime}(\omega)},$
(9) $\displaystyle
G_{kj}^{H|J}(\mathbf{x},\omega|\mathbf{x}_{s})=\frac{H_{k}(\mathbf{x},\omega|\mathbf{x}_{s})}{J_{j}(\omega)}=\frac{H_{k}^{\prime}(\mathbf{x},\omega|\mathbf{x}_{s})}{J_{j}^{\prime}(\omega)},$
where $G_{kj}^{E|J}(\mathbf{x},\omega|\mathbf{x}_{s})$ and
$G_{kj}^{H|J}(\mathbf{x},\omega|\mathbf{x}_{s})$ stand for the electrical and
magnetic Green’s function for angular frequency $\omega$ at spatial location
$\mathbf{x}$ with the source located at $\mathbf{x}_{s}$.
Equation 4 is a pure wave-domain equation and the time integration can easily
be discretized using the leap-frog method. We let the time be $t_{n}=n\Delta
t$ with $n$ the integer time variable and $\Delta t$ the time step. We also
introduce $N=n+\frac{1}{2}$ such that,
$\displaystyle{\mathbf{H}^{\prime}}^{N}$ $\displaystyle=$
$\displaystyle{\mathbf{H}^{\prime}}^{N-1}+\Delta
t\mu^{-1}(-\nabla\times{\mathbf{E}^{\prime}}^{n}-{\mathbf{M}^{\prime}}^{n}),$
(10) $\displaystyle{\mathbf{E}^{\prime}}^{n+1}$ $\displaystyle=$
$\displaystyle{\mathbf{E}^{\prime}}^{n}+\Delta
t\epsilon^{-1}(\nabla\times{\mathbf{H}^{\prime}}^{N}-{\mathbf{J}^{\prime}}^{N}).$
The time integration of these equations is second-order accurate. It is shown
in Mittet, (2010) that the calculation of the desired fields in the “real
world” diffusive domain is independent of the frequency content of the source
term used for calculating the fictitious fields. We exploit this fact and
achieve good accuracy for the time integration by transmitting a low-frequency
signal in the fictitious wave domain. Here we are concerned with the spatial
part of the simulation scheme so we turn to this topic next.
We use a similar notation for the space variables as for the time variables
where we write $x_{i}=x_{i-1}+\Delta x_{i}$ where $\Delta x_{i}$ is the node
separation between node $x_{i-1}$ and node $x_{i}$. Likewise, we assume a
forward staggered grid such that $x_{I}=x_{I-1}+\Delta x_{I}$ where $\Delta
x_{I}$ is the node separation between node $x_{I-1}$ and node $x_{I}$. For a
uniform staggered grid we have a constant node separation such that $\Delta
x_{I}=\Delta x_{i}=\Delta x$ and $I=i+\frac{1}{2}$. The $y$ and $z$ directions
can be described in the same way with lower case and upper case integer
arguments.
Calculation of the partial derivative of the field $f(x)$ can, in the
continuous case, be formulated as an integral operator by
$\displaystyle\partial_{x}f(x)=\int_{-\infty}^{\infty}\mathrm{d}x^{\prime}f(x+x^{\prime})\\{-\partial_{x^{\prime}}\delta(x^{\prime})\\}=\int_{-\infty}^{\infty}\mathrm{d}x^{\prime}f(x+x^{\prime})\alpha(x^{\prime}).$
(11)
The discrete formulation, with $f(x_{i}):=f(i)$, is,
$\displaystyle\partial_{x}f(x_{i})\approx
D_{x}f(i)=\sum_{l=-L}^{L}f(i+l)\alpha_{l}(i),$ (12)
where $\alpha_{l}(i)$ is a band-limited approximation to the operator
$\alpha(x^{\prime})$ in equation 11. The half length of the operator is $L$.
The argument $i$ is used to explicitly show that this operator will vary with
location for a nonuniform grid. For the staggered grid, we can then define
discretized forward, $D_{x}^{+}$, and backward, $D_{x}^{-}$, derivative
operators as,
$\displaystyle
D_{x}^{+}f(i)=\partial_{x}f(I)=\sum_{l=1}^{L}f(i+l)\alpha_{l}(i)-f(i-l+1)\alpha_{-l}(i),$
(13) $\displaystyle
D_{x}^{-}f(I)=\partial_{x}f(i)=\sum_{l=1}^{L}f(I+l-1)\alpha_{l}(I)-f(I-l)\alpha_{-l}(I),$
which is the form we need for nonuniform grids and which is investigated here.
The $f^{{}^{\prime}}$ implies spatial derivative in equation 13.
The operator simplifies for uniform grids where the operator becomes
independent of spatial location such that
$\alpha_{l}(i)=\alpha_{-l}(i)=\alpha_{l}$
$\displaystyle
D_{x}^{+}f(i)=\partial_{x}f(I)=\sum_{l=1}^{L}(f(i+l)-f(i-l+1))\alpha_{l},$
(14) $\displaystyle
D_{x}^{-}f(I)=\partial_{x}f(i)=\sum_{l=1}^{L}(f(I+l-1)-f(I-l))\alpha_{l}.$
If we use $L=1$ we have that $\alpha_{1}=1/\Delta x$ and equation 14
formulates the well known second-order accurate partial derivative operations,
$\displaystyle D_{x}^{+}f(i)=\partial_{x}f(I)=(f(i+1)-f(i))/\Delta x,$ (15)
$\displaystyle D_{x}^{-}f(I)=\partial_{x}f(i)=(f(I)-f(I-1))/\Delta x.$
Let us assume a non-magnetic subsurface so that $\mu$ has the same value as in
the vacuum, $\mathbf{M}=0$, while $\epsilon$ is a diagonal tensor
$\epsilon=\mbox{diag}(\epsilon_{ii}),i=x,y,z$. The staggering is as in Mittet,
(2010),
$\displaystyle{{H}^{\prime}}_{x}^{N}(i,J,K),\quad{{H}^{\prime}}_{y}^{N}(I,j,K),\quad{{H}^{\prime}}_{z}^{N}(I,J,k),$
(16)
$\displaystyle{{E}^{\prime}}_{x}^{n}(I,j,k),\quad{{E}^{\prime}}_{y}^{n}(i,J,k),\quad{{E}^{\prime}}_{z}^{n}(i,j,K),$
$\displaystyle{{J}^{\prime}}_{x}^{n}(I,j,k),\quad{{J}^{\prime}}_{y}^{n}(i,J,k),\quad{{J}^{\prime}}_{z}^{n}(i,j,K),$
$\displaystyle{\varepsilon}_{xx}(I,j,k),\quad{\varepsilon}_{yy}(i,J,k),\quad{\varepsilon}_{zz}(i,j,K),$
and the scheme implemented is,
$\displaystyle{{H}^{\prime}}_{x}^{N}$ $\displaystyle=$
$\displaystyle{{H}^{\prime}}_{x}^{N-1}-\frac{\Delta
t}{\mu}(D_{y}^{+}{{E}^{\prime}}_{z}^{n}-D_{z}^{+}{{E}^{\prime}}_{y}^{n}),$
(17) $\displaystyle{{H}^{\prime}}_{y}^{N}$ $\displaystyle=$
$\displaystyle{{H}^{\prime}}_{y}^{N-1}-\frac{\Delta
t}{\mu}(D_{z}^{+}{{E}^{\prime}}_{x}^{n}-D_{x}^{+}{{E}^{\prime}}_{z}^{n}),$
$\displaystyle{{H}^{\prime}}_{z}^{N}$ $\displaystyle=$
$\displaystyle{{H}^{\prime}}_{z}^{N-1}-\frac{\Delta
t}{\mu}(D_{x}^{+}{{E}^{\prime}}_{y}^{n}-D_{y}^{+}{{E}^{\prime}}_{x}^{n}),$
$\displaystyle{{E}^{\prime}}_{x}^{n+1}$ $\displaystyle=$
$\displaystyle{{E}^{\prime}}_{x}^{n}+\frac{\Delta
t}{\epsilon_{xx}}(D_{y}^{-}{{H}^{\prime}}_{z}^{N}-D_{z}^{-}{{H}^{\prime}}_{y}^{N}-{{J}^{\prime}}_{x}^{N}),$
$\displaystyle{{E}^{\prime}}_{y}^{n+1}$ $\displaystyle=$
$\displaystyle{{E}^{\prime}}_{y}^{n}+\frac{\Delta
t}{\epsilon_{yy}}(D_{z}^{-}{{H}^{\prime}}_{x}^{N}-D_{x}^{-}{{H}^{\prime}}_{z}^{N}-{{J}^{\prime}}_{y}^{N}),$
$\displaystyle{{E}^{\prime}}_{z}^{n+1}$ $\displaystyle=$
$\displaystyle{{E}^{\prime}}_{z}^{n}+\frac{\Delta
t}{\epsilon_{zz}}(D_{x}^{-}{{H}^{\prime}}_{y}^{N}-D_{y}^{-}{{H}^{\prime}}_{x}^{N}-{{J}^{\prime}}_{z}^{N}).$
The conventional second-order staggered grid FDTD scheme discretizes the
spatial derivatives as follows (Newman and Alumbaugh,, 1995; Mulder,, 2006):
$\displaystyle
D_{y}^{-}H^{\prime}_{z}=\frac{H^{\prime}_{z}(I,J,k)-H^{\prime}_{z}(I,J-1,k)}{\Delta
y_{J}},\quad
D_{z}^{-}H^{\prime}_{y}=\frac{H^{\prime}_{y}(I,j,K)-H^{\prime}_{y}(I,j,K-1)}{\Delta
z_{K}},$ $\displaystyle
D_{z}^{-}H^{\prime}_{x}=\frac{H^{\prime}_{x}(i,J,K)-H^{\prime}_{x}(i,J,K-1)}{\Delta
z_{K}},\quad
D_{x}^{-}H^{\prime}_{z}=\frac{H^{\prime}_{z}(I,J,k)-H^{\prime}_{z}(I-1,J,k)}{\Delta
x_{I}},$ $\displaystyle
D_{x}^{-}H^{\prime}_{y}=\frac{H^{\prime}_{y}(I,j,K)-H^{\prime}_{y}(I-1,j,K)}{\Delta
x_{I}},\quad
D_{y}^{-}H^{\prime}_{x}=\frac{H^{\prime}_{x}(i,J,K)-H^{\prime}_{x}(i,J-1,K)}{\Delta
y_{J}},$ $\displaystyle
D_{y}^{+}E^{\prime}_{z}=\frac{E^{\prime}_{z}(i,j+1,K)-E^{\prime}_{z}(i,j,K)}{\Delta
y_{j}},\quad
D_{z}^{+}E^{\prime}_{y}=\frac{E^{\prime}_{y}(i,J,k+1)-E^{\prime}_{y}(i,J,k)}{\Delta
z_{k}},$ $\displaystyle
D_{z}^{+}E^{\prime}_{x}=\frac{E^{\prime}_{x}(I,j,k+1)-E^{\prime}_{x}(I,j,k)}{\Delta
z_{k}},\quad
D_{x}^{+}E^{\prime}_{z}=\frac{E^{\prime}_{z}(i+1,j,K)-E^{\prime}_{z}(i,j,K)}{\Delta
x_{i}},$ $\displaystyle
D_{x}^{+}E^{\prime}_{y}=\frac{E^{\prime}_{y}(i+1,J,k)-E^{\prime}_{y}(i,J,k)}{\Delta
x_{i}},\quad
D_{y}^{+}E^{\prime}_{x}=\frac{E^{\prime}_{x}(I,j+1,k)-E^{\prime}_{x}(I,j,k)}{\Delta
y_{j}},$ (18)
where $\Delta x_{i}$, $\Delta y_{j}$ and $\Delta z_{k}$ are distances between
nodes on the reference grid, while $\Delta x_{I}$, $\Delta y_{J}$ and $\Delta
z_{K}$ are the distance between the grid points $(I,j,k)$ and $(I-1,j,k)$, the
distance between the grid points $(i,J,k)$ and $(i,J-1,k)$, and the distance
between the grid points $(i,j,K)$ and $(i,j,K-1)$. This scheme is second-order
accurate on a uniform grid with $\Delta x_{i}=\Delta x_{I}$ and likewise for
the other spatial directions. The scheme is consistent with using equation 14
with $L=1$ to approximate the derivatives.
A standard discretization method for nonuniform grids is to use the same
formulation as above, but where the node distance may vary along the same
spatial direction. It is well known that there are accuracy issues with this
implementation. To achieve second-order accuracy using equation 14 is not
possible. The midpoints between nodes do not align after going from the
reference grid to the staggered grid and back again. Since $\Delta
x_{i}\neq\Delta x_{I}$, we have a situation where the cell center does not
match on a nonuniform grid, as is illustrated in Figure 1b. Consequently, the
local truncation error of the resulting scheme can only reach first order.
Reduced accuracy will also be a problem if we implement a high-order FDTD
scheme ($L>1$) on a nonuniform grid, using derivative-operator coefficients
designed for a regular grid. This is unfortunate since the nonuniform grid is
potentially attractive for efficient modelling due to significant reduction of
the number of grid point. However, good accuracy can be restored if equation
13 is used instead of equation 14. The problem that remains is to calculate
the operator coefficients for equation 13.
### 2.1 Vandermonde matrix
The major difference between FDTD implementations on a uniform grid and on a
nonuniform grid lies in the design of the spatial-derivative operator
coefficients. The position of each field component on the nonuniform staggered
grid has been illustrated in Figure 2, which is similar to the staggered FDTD
on a uniform grid.
Figure 2: Electrical and magnetic fields on staggered grid (Yee,, 1966).
In order to compute the electromagnetic field as well as its derivatives with
arbitrary grid spacing, we have to do a polynomial interpolation using a
number of knots $x_{0},x_{1},\cdots,x_{n}$. According to the Taylor expansion,
we have
$\displaystyle f(x_{i})$ $\displaystyle=$ $\displaystyle
f(x)+f^{\prime}(x)(x_{i}-x)+\frac{1}{2}f^{\prime\prime}(x)(x_{i}-x)^{2}+\cdots+\frac{1}{n!}f^{(n)}(x)(x_{i}-x)^{n}+\cdots$
(19) $\displaystyle i$ $\displaystyle=$ $\displaystyle 0,1,\cdots,n.$
By defining $a_{i}(x):=f^{(i)}(x)/i!$, we end up with a polynomial of the
Newton form,
$\displaystyle f(x_{i})$ $\displaystyle=$ $\displaystyle
a_{0}(x)+a_{1}(x)(x_{i}-x)+a_{2}(x)(x_{i}-x)^{2}+\cdots+a_{n}(x)(x_{i}-x)^{n}+\cdots$
(20) $\displaystyle i$ $\displaystyle=$ $\displaystyle 0,1,\cdots,n.$
Let us consider $n+1$ distinct nodes $x_{0},x_{1},\cdots,x_{n}$ and drop the
terms $O((x_{i}-x)^{n+1})$. This builds a matrix system
$\displaystyle\underbrace{\begin{pmatrix}f(x_{0})\\\ f(x_{1})\\\ \vdots\\\
f(x_{n})\end{pmatrix}}_{\mathbf{f}}=\underbrace{\begin{pmatrix}1&x_{0}-x&(x_{0}-x)^{2}&\cdots&(x_{0}-x)^{n}\\\
1&x_{1}-x&(x_{1}-x)^{2}&\cdots&(x_{1}-x)^{n}\\\ \cdots\\\
1&x_{n}-x&(x_{n}-x)^{2}&\cdots&(x_{n}-x)^{n}\\\
\end{pmatrix}}_{\mathbf{V}^{T}(x_{0}-x,\cdots,x_{n}-x)}\underbrace{\begin{pmatrix}a_{0}(x)\\\
a_{1}(x)\\\ \vdots\\\ a_{n}(x)\end{pmatrix}}_{\mathbf{a}},$ (21)
where $\mathbf{V}^{T}(x_{0}-x,\cdots,x_{n}-x)$ is the transpose of a
Vandermonde matrix determined by $x_{0}-x,\cdots,x_{n}-x$. The above
expression implies that the function $f(x)$ and its derivatives up to the
$n$-th order at arbitrary location $x$ can be found by inverting the
Vandermonde matrix:
$(f(x),f^{\prime}(x),\cdots,f^{(n)}(x)/n!)^{T}=\mathbf{a}=[\mathbf{V}^{T}]^{-1}\mathbf{f}$.
It is well known that the Vandermonde matrix is highly ill-conditioned and
direct matrix inversion by Gaussian elimination should be avoided due to
numerical instabilities when the matrix size becomes large. Fortunately, there
exists an efficient algorithm based on the method of Björck and Pereyra,
(1970) to invert the Vandermonde matrix. In fact, the algorithm circumvents
the curse of severe ill-conditioning of the Vandermonde matrix to arrive at
arbitrarily high accuracy for the inversion (Demmel and Koev,, 2005). Compared
with Gauss-elimination of complexity $O(n^{3})$ , the Vandermonde matrix
inversion algorithm reduces the computational complexity to $O(n^{2})$.
Because the elements of the Vandermonde matrix are fully determined by the
interpolation nodes, there is no need to explicitly construct the matrix and
store it before inversion. The detailed implementation of this fast algorithm
is available in Golub, (1996, Algorithm 4.6.1).
Let the $i$-th row, $j$-th column of the inverse matrix
$[\mathbf{V}^{T}]^{-1}$ be $w_{ij}$, i.e.,
$([\mathbf{V}^{T}]^{-1})_{ij}=w_{ij},i,j=0,\cdots,n$. It also follows that
$\displaystyle\underbrace{\begin{pmatrix}a_{0}(x)\\\ a_{1}(x)\\\ \vdots\\\
a_{n}(x)\end{pmatrix}}_{\mathbf{a}}:=\underbrace{\begin{pmatrix}w_{00}&w_{01}&\cdots
w_{0n}\\\ w_{10}&w_{11}&\cdots w_{1n}\\\ \vdots\\\ w_{n0}&w_{n1}&\cdots
w_{nn}\\\
\end{pmatrix}}_{[\mathbf{V}^{T}]^{-1}}\underbrace{\begin{pmatrix}f(x_{0})\\\
f(x_{1})\\\ \vdots\\\ f(x_{n})\end{pmatrix}}_{\mathbf{f}}.$ (22)
The $i$-th row gives the explicit expression to find the $i$-th derivative
$\displaystyle\frac{1}{i!}f^{(i)}(x)=a_{i}(x)=\sum_{j=0}^{n}w_{ij}f(x_{j})=\langle
w_{i\cdot}|\mathbf{f}\rangle,\quad i=0,\cdots,n.$ (23)
The first row of the matrix $[\mathbf{V}^{T}]^{-1}$ (i.e., $w_{0j}$,
$j=0,\cdots,n$) is $f(x)$ which in analogy with equation 11 can be written in
discrete form as
$\displaystyle
f(x)=\int_{-\infty}^{\infty}\mathrm{d}x^{\prime}f(x+x^{\prime})\\{\delta(x^{\prime})\\}=\int_{-\infty}^{\infty}\mathrm{d}x^{\prime}f(x+x^{\prime})\beta(x^{\prime}).$
(24)
The discrete formulation is,
$\displaystyle f(x_{i})=\sum_{l=-L}^{L}f(i+l)\beta_{l}(i),$ (25)
where $\beta_{l}(i)$ are the coefficients for an interpolation operator
adapted to a nonuniform grid and identical to the $w_{0j}$ coefficients. The
second row of the matrix $[\mathbf{V}^{T}]^{-1}$ (i.e., $w_{1j}$,
$j=0,\cdots,n$) is $\partial_{x}f(x)$ and the continuous and discrete
representations are given in equations 11 and 12. The $\alpha_{l}(i)$
coefficients are for a derivative operator adapted to a nonuniform grid and
identical to the $w_{1j}$ coefficients. To be explicit, consider the staggered
finite-difference approximation of the first derivatives in $x$ direction
using $2L$ non-equidistant nodes. The finite-difference coefficients
$\alpha_{l}(x_{i})$ and $\alpha_{l}(x_{I})$, $l=-L+1,\cdots,L$ are the 2nd row
of the inverse of the matrices
$\mathbf{V}^{T}(x_{i+L}-x_{I},\cdots,x_{i-L+1}-x_{I})$ and
$\mathbf{V}^{T}(x_{I+L}-x_{i},\cdots,x_{I-L+1}-x_{i})$. Using the $2L$ nodes,
we achieve accuracy up to $2L$-th order in space.
In general we find that the operator coefficients (interpolation weights) for
$f^{(i)}(x)$ are $i!w_{ij}$. Given the points $x_{0},\cdots,x_{n}$ and $x$,
the Vandermonde matrix is determined and the operator coefficients can be
calculated. For the simulation we only need the derivative-operator
coefficients. It is noteworthy to mention that the Vandermonde matrix must be
non-singular to be inverted. For the derivative-operator coefficients we have
that the Vandermonde matrix is non-singular by construction of the staggered
grids. A finite-difference scheme on a staggered grid implies that the node
$x_{I}$, whose derivative is computed, will stay in the middle between the
selected nodes $x_{i}$, which ensures that the resulting Vandermonde matrix is
invertible.
The interpolation operators are useful for recording fields at arbitrary
locations and for the distribution of source contributions (Mittet,, 2017).
Just as for the derivative-operator coefficients, the interpolation
coefficients can be pre-calculated and then reused every time step. For the
interpolation operator we may find that some of the coordinates $x_{i}$
matches the interpolation points $x$, the inversion of Vandermonde matrix is
then not necessary. This might happen when the source or receiver positions
coincide with a finite-difference node. In this case, the interpolation
weights $w_{0i}$ to evaluate $f(x)$ should be exactly 1 at $x_{i}$ and $0$
elsewhere. In order to do 2D/3D simulation on nonuniform grid,
multidimensional interpolant is simply constructed by tensor products of many
1D interpolants.
The above procedure is significant as it allows us to use arbitrarily high-
order finite-difference scheme to accurately compute the electromagnetic
fields and their derivatives, typically with arbitrary grid spacing in the
rectilinear grid. This opens the door for CSEM modelling using high-order FDTD
on a nonuniform grid in a consistent framework. The computed finite-difference
coefficients may also be used to do high-order frequency-domain modelling on a
nonuniform grid, while the resulting sparse banded matrix has to be solved
accurately if sufficient computational resources are available.
### 2.2 Stability condition
Let us write down the FDTD scheme in equation 10 without source terms as
follows:
$\displaystyle\begin{cases}\mathbf{E}^{{}^{\prime}n+1}=\mathbf{E}^{{}^{\prime}n}+\Delta
t\epsilon^{-1}\nabla\times\mathbf{H}^{{}^{\prime}N}\\\
\mathbf{H}^{{}^{\prime}N+1}=\mathbf{H}^{{}^{\prime}N}-\Delta
t\mu^{-1}\nabla\times\mathbf{E}^{{}^{\prime}n+1}\end{cases},$ (26)
leading to
$\displaystyle\begin{split}\begin{bmatrix}\mathbf{E}^{{}^{\prime}n+1}\\\
\mathbf{H}^{{}^{\prime}N+1}\end{bmatrix}=\underbrace{\begin{bmatrix}\mathbf{I}&\Delta
t\epsilon^{-1}\nabla\times\\\ -\Delta t\mu^{-1}\nabla\times&\mathbf{I}-\Delta
t^{2}\epsilon^{-1}\mu^{-1}\nabla\times\nabla\times\end{bmatrix}}_{\mathbf{A}}\begin{bmatrix}\mathbf{E}^{{}^{\prime}n}\\\
\mathbf{H}^{{}^{\prime}N}\end{bmatrix}.\end{split}$ (27)
The numerical stability requires the eigenvalues of the amplification matrix
$\mathbf{A}$ to be less than or equal to 1. Assume the eigenvalue
decomposition for the amplification matrix is
$\mathbf{A}=\mathbf{\bar{V}}\mathbf{\Lambda}\mathbf{\bar{V}}^{T}$, where
$\mathbf{\bar{V}}=(\mathbf{V}_{E},\mathbf{V}_{H})^{T}$ is an unitary matrix
such that $\mathbf{\bar{V}}^{T}\mathbf{\bar{V}}=\mathbf{I}$. Then we have
$\mathbf{A}\mathbf{V}=\mathbf{V}\mathbf{\Lambda}$, yielding
$\displaystyle\begin{bmatrix}\mathbf{I}&\Delta t\epsilon^{-1}\nabla\times\\\
-\Delta t\mu^{-1}\nabla\times&\mathbf{I}-\Delta
t^{2}\epsilon^{-1}\mu^{-1}\nabla\times\nabla\times\end{bmatrix}\begin{bmatrix}\mathbf{V}_{E}\\\
\mathbf{V}_{H}\end{bmatrix}=\begin{bmatrix}\mathbf{V}_{E}\\\
\mathbf{V}_{H}\end{bmatrix}\Lambda.$ (28)
That is,
$\displaystyle\begin{cases}\Delta
t\epsilon^{-1}\mu^{-1}\nabla\times\mathbf{V}_{H}=\mathbf{V}_{E}(\mathbf{\Lambda}-\mathbf{I})\\\
-\Delta t\mu^{-1}\nabla\times\mathbf{V}_{E}-\Delta
t^{2}\epsilon^{-1}\mu^{-1}\nabla\times\nabla\times\mathbf{V}_{H}=\mathbf{V}_{H}(\mathbf{\Lambda}-\mathbf{I}).\end{cases}$
(29)
Multiplying the second sub equation $\mathbf{\Lambda}-\mathbf{I}$ from the
right and inserting the first sub equation gives
$-\Delta
t^{2}\mu^{-1}\epsilon^{-1}\nabla\times\nabla\times\mathbf{V}_{H}\mathbf{\Lambda}=\mathbf{V}_{H}(\mathbf{\Lambda}-\mathbf{I})^{2}.$
(30)
Denote $\mathbf{V}_{H,j}$ the $j$th column of $\mathbf{V}_{H}$ and
$\lambda_{j}$ the $j$th eigenvalue in $\mathbf{\Lambda}$. The above equation
reads
$\mu^{-1}\epsilon^{-1}\nabla\times\nabla\times\mathbf{V}_{H,j}=-\frac{(\lambda_{j}-1)^{2}}{\Delta
t^{2}\lambda_{j}}\mathbf{V}_{H,j},$ (31)
which shows that $-\frac{(\lambda_{i}-1)^{2}}{\Delta t^{2}\lambda_{i}}$ is the
eigenvalue of the matrix $(\mu\epsilon)^{-1}\nabla\times\nabla\times$
associated with the eigenvector $V_{H,j}$. This leads to
$(\lambda_{j}^{2}+(-2+\Delta
t^{2}c^{2}\nabla\times\nabla\times)\lambda_{j}+1)\mathbf{V}_{H,j}=(\lambda_{j}^{2}-(2+\Delta
t^{2}c^{2}\Delta)\lambda_{j}+1)\mathbf{V}_{H,j}=0,$ (32)
where we denote $c:=1/\sqrt{\mu\epsilon}$ and have applied
$\nabla\times\nabla\times\mathbf{F}=\nabla\nabla\cdot\mathbf{F}-\nabla\cdot\nabla\mathbf{F}=-\Delta\mathbf{F}$
due to Gauss theorem $\nabla\cdot\mathbf{F}=0$,
$\mathbf{F}=\mathbf{E},\mathbf{H}$ in the homogeneous, source free medium. The
roots of the above equation are
$\lambda_{j;1,2}=1+\frac{\Delta
t^{2}c^{2}\Delta}{2}\pm\frac{\mathrm{i}}{2}\sqrt{-\Delta
t^{2}c^{2}\Delta(4+\Delta t^{2}c^{2}\Delta)},$ (33)
which requires the following condition to be satisfied
$0\leq-\Delta t^{2}c^{2}\Delta\leq 4,$ (34)
in order to ensure $|\lambda_{j;1,2}|\leq 1$. Finally, we arrive at the same
stability condition as equation 41 of Mittet, (2010),
$\Delta
tc_{\max}\sqrt{(D_{x}^{\max})^{2}+(D_{y}^{\max})^{2}+(D_{z}^{\max})^{2}}\leq
2,$ (35)
where $D_{x}^{\max}$, $D_{y}^{\max}$ and $D_{z}^{\max}$ are the maximum value
of the the discretized first derivatives along $x$, $y$ and $z$ directions.
Let us emphasize this condition applies to both uniform and nonuniform grid.
The difference lies in the spatial derivative operator.
To proceed with the stability analysis, we represent the fields on the grid
via time harmonic plane waves
$\mathbf{E}^{\prime},\mathbf{H}^{\prime}\propto e^{-\mathrm{i}(\omega
t-k_{x}x-k_{y}y-k_{z}z)},$ (36)
where the amplitude has been omitted. Equation 13 becomes
$\begin{cases}D_{x}^{+}u(x_{I})=(\alpha_{L}(x_{I})e^{\mathrm{i}k_{x}(x_{i+L}-x_{I})}+\alpha_{L-1}(x_{I})e^{\mathrm{i}k_{x}(x_{i+L-1}-x_{I})}+\cdots+\alpha_{-L+1}(x_{I})e^{\mathrm{i}k_{x}(x_{i-L+1}-x_{I})})u(x_{I})\\\
D_{x}^{-}u(x_{i})=(\alpha_{L}(x_{i})e^{\mathrm{i}k_{x}(x_{I+L-1}-x_{i})}+\alpha_{L-1}(x_{i})e^{\mathrm{i}k_{x}(x_{I+L-1}-x_{i})}+\cdots+\alpha_{-L+1}(x_{i})e^{\mathrm{i}k_{x}(x_{I-L}-x_{i})})u(x_{i})\end{cases}.$
(37)
Hence, we end up with the maximum possible values for discrete first
derivative operators
$D_{x}^{\max}=\max\left(\sum_{l=-L+1}^{L}|\alpha_{l}(x_{I})|,\sum_{l=-L+1}^{L}|\alpha_{i}(x_{l})|\right)$
(38)
and similar estimations for $D_{y}^{\max}$ and $D_{z}^{\max}$. In case of a
uniform grid, the above expressions becomes much simpler
$\begin{cases}D_{x}^{+}u(x_{I})=(\alpha_{L}e^{\mathrm{i}k_{x}(L-1/2)\Delta
x}+\alpha_{L-1}e^{\mathrm{i}k_{x}(L-3/2)\Delta
x}+\cdots+\alpha_{-L+1}e^{\mathrm{i}k_{x}(-L+1/2)\Delta x}u(x_{I})\\\
D_{x}^{-}u(x_{i})=(\alpha_{L}e^{\mathrm{i}k_{x}(L-1/2)\Delta
x}+\alpha_{L-1}e^{\mathrm{i}k_{x}(L-3/2)\Delta
x}+\cdots+\alpha_{-L+1}e^{\mathrm{i}k_{x}(-L+1/2)\Delta x}u(x_{i})\\\
\end{cases},$
where $\Delta x$ stands for the uniform grid spacing in $x$ direction, while
the coefficients $\alpha_{l}$ (which can be computed by inverting a
Vandermonde matrix system according to Appendix AComputing uniform staggered-
grid finite difference coefficients via Vandermonde matrix inversion) are
independent of the location $x_{i}$.
### 2.3 Grid stretching
Our finite-difference modelling is carried out on a rectilinear mesh, which
can be generated from the tensor (outer) product of 1D non-equispaced meshes.
We use the geometrical progression to generate the 1D nonuniform grid,
following the work of Mulder, (2006, Appendix C). This is also often referred
to as power law grid stretching since the cell sizes stretch exponentially to
guarantee a smooth extension of the grid.
Assume we have the total grid length $L_{x}$ divided into $n$ intervals ($n+1$
nodes) with a common ratio $r>1$. Denote the smallest interval $\Delta
x=x_{1}-x_{0}$. Thus, the relation between $L_{x}$ and $\Delta x$ is
$L_{x}=(x_{1}-x_{0})+(x_{2}-x_{1})+\cdots+(x_{n}-x_{n-1})=\Delta
x(1+r+\cdots+\cdot r^{n-1})=\Delta x\frac{r^{n}-1}{r-1}.$ (39)
Given the total distance $L_{x}$, the smallest grid spacing $\Delta x$ and the
stretching factor $r$, we can compute an approximate value for the number of
nodes $n=\left\lceil\frac{\ln(1+\frac{L_{x}}{\Delta
x}(r-1))}{\ln(r)}\right\rceil$ following Mulder, (2006), where
$\left\lceil\cdot\right\rceil$ takes the ceiling integer value. This strategy
yields approximate solution as the value of $L_{x}$ is not exactly preserved.
Due to the stability requirement and the resulting computational cost in the
modelling, we are restricted to the smallest interval $\Delta x$ and a given
number of intervals $n$ to discretize over a certain distance $L_{x}$. The
question boils down to finding an optimal growth factor $r$. This problem is
more complicated since equation 39 does not yield an explicit expression for
the stretching factor $r$.
The relation in equation 39 is equivalent to
$r=\underbrace{\left(\frac{L_{x}}{\Delta
x}(r-1)+1\right)^{\frac{1}{n}}}_{g(r)},$ (40)
which inspires us to carry out a number of fixed point iterations until
convergence:
$r^{k+1}=g(r^{k}),\quad k=0,1,\cdots.$ (41)
Assume $r^{*}$ is the analytic solution such that $r^{*}=g(r^{*})$. Thanks to
Lagrange mean value theorem, the error estimation at ($k+1$)-th iteration is
linked with the error at $k$-th iteration via
$|e^{k+1}|=|r^{k+1}-r^{*}|=|g(r^{k})-g(r^{*})|=|g^{\prime}(\xi)(r^{k}-r^{*})|=|g^{\prime}(\xi)||e^{k}|,\quad\xi\mbox{
between }r^{k}\mbox{ and }r^{*}.$ (42)
It becomes evident that $e^{k}\rightarrow 0\;(k\rightarrow\infty)$ provided
that $|g^{\prime}(r)|<1$. Starting from any initial guess $r^{0}>1$, the fixed
point iteration scheme in equation 41 is guaranteed to converge since
$|g^{\prime}(r)|=\frac{L_{x}}{n\Delta x}\left(\frac{L_{x}}{\Delta
x}(r-1)+1\right)^{1/n-1}=\frac{1}{n}(r^{-(n-1)}+r^{-(n-2)}+\cdots+1)<1,$
thanks to the relations in equations 39 and 40.
We note that within areas of constant $r$, the derivative-operator
coefficients in equation 13 can be calculated from the $2L$ coefficients
$a_{-l}$ and $a_{l}$, such that
$\displaystyle\alpha_{-l}(i)$ $\displaystyle=$
$\displaystyle\frac{a_{-l}}{\Delta x\>r^{i}},$ (43)
$\displaystyle\alpha_{l}(i)$ $\displaystyle=$
$\displaystyle\frac{a_{l}}{\Delta x\>r^{i}}.$
Two solutions of the Vandermonde system are required in this case, one for
operators valid on the reference grid and one for operators valid on the
staggered grid.
### 2.4 Implementation
A number of techniques have been applied to achieve efficient and accurate 3D
CSEM simulation. According to equation 6, the frequency-domain CSEM response
is obtained by a time to frequency transform. The transform is using the
complex frequency given in equation 7. This gives exponential damping of late
arrivals as is discussed in more detail in Mittet, (2015). The lowest
frequency experiences the least damping in this transform and by that requires
the longest simulation time. This allows us to bound the number of times steps
to terminate the simulation when the lowest frequency component has converged.
The convergence means that the frequency-domain field obtained by time
integration has reached its steady state and later arrivals are damped to the
degree that they do not contribute to the time integral.
The source and receiver locations may be arbitrarily distributed over the
whole computational domain, not necessarily located at the nodes of the
finite-difference grid. In case they do, interpolation is not required and we
directly take the field from the grid; otherwise, we need interpolation
operators extracted from the first row of the relevant $[\mathbf{V}^{T}]^{-1}$
matrices.
The CFL condition to achieve stable FDTD modelling dictates the timestep for a
given spatial sampling. The air-wave travels at extremely high speed which
does not allow us to use the local finite-difference stencil to simulate it.
The air-water boundary condition is implemented in the Fourier-wavenumber
domain following the method proposed by Oristaglio and Hohmann, (1984) and
Wang and Hohmann, (1993) and using the extension to high-order schemes given
in Mittet, (2010).
To mimic wave propagation in unbounded domain, we use a truncated computation
mesh surrounded by an artificial absorbing boundary using convolutional
perfectly matched layer (PML) technique (Roden and Gedney,, 2000), except the
top air-water interface. To ease the implementation, we extend the domain with
equal spacing based on the last finite-difference cell size in the interior
domain. Our CPML implementation is very standard using all the parameter
setups given in Komatitsch and Martin, (2007).
## 3 Numerical examples
We now present three examples to demonstrate the merits of high-order finite
differences using nonuniform staggered grids. The first two examples are using
1D resistivity models under shallow water and deep water scenarios. In the 1D
case, the semi-analytic solution can be computed in the frequency-wavenumber
domain as a reference to benchmark our results. The third example takes into
account varying seafloor topography, in which a reference solution can be
computed using `emg3d` software (Werthmüller et al.,, 2019).
In all of our modelling, we use x-directed electrical dipole source. The
$E_{x}$ component at three commonly used frequencies - 0.25 Hz, 0.75 Hz and
1.25 Hz, are modelled. In total 12 layers of CPML are sufficient to achieve
nearly perfect absorbing effect. The computed electromagnetic fields are
normalized by the source current. The convergence check has been conducted
every 100 time steps to avoid modelling after the frequency-domain EM fields
stop evolving. We examine the amplitude error by inspecting the ratio of the
modelled field to the reference solution, $|E_{x}^{FD}|/|E_{x}^{ref}|$, which
should be close to unity if the modelling is precise. The phase difference is
computed by $\angle E_{x}^{FD}-\angle E_{x}^{ref}$ in degrees.
### 3.1 Moving to high-order schemes
The model shown in Figure 3 includes 5 layers: the top is the air, then a 325
m layer of sea water with a resistivity of 0.3 $\Omega$m, followed by three
layers of formation with increasing resistivity in the depth direction. The
whole model extends down to 5 km depth below the sea surface. A horizontal
electrical dipole source is deployed at 275 m water depth, its lateral
position being in the middle of the model. The EM fields are recorded using
201 receiver positions at the seafloor. The offsets range from -10 km to +10
km (receiver separation equals 100 m).
Figure 3: The resistivity model with air ($\rho=10^{12}$ $\Omega$m), shallow
column (325 m) of 0.3 $\Omega$m and 3 sediment layers ($\rho=1$ $\Omega$m in
[325, 1025] m; $\rho=2$ $\Omega$m in [1025, 1525] m; $\rho=4$ $\Omega$m
downwards).
We first compare the 3D CSEM response simulated by FDTD with the reference
solution calculated using `empymod` program (Werthmüller,, 2017). To limit the
factors affecting the modelling accuracy, this experiment has been done using
isotropic modeling and uniform grid spacing ($\Delta x=\Delta y=150$ m,
$\Delta z=50$ m). Panels (a) and (b) in Figure 4 shows the amplitude and the
phase of the modelled EM fields by FDTD of 2nd order, which is highly
consistent to the reference solution. At the offset beyond 1 km, the agreement
between finite-difference and the analytical solution is good. The kink in the
transition of the near-to-far offset in the amplitude response for all
frequencies is a manifestation of the strong air-wave effect. In the very near
field, where the receivers are located just below the transmitter, the phase
exhibits a $180^{o}$ jump due to the change in the direction of the electric
field immediately below the electric dipole source. Due to the extension of
finite-difference stencil, this phase rollover becomes smeared compared to the
reference solution. For the same reason, finite difference method introduces
significant errors in the near field in amplitude (which becomes more evident
for high-order schemes). Fortunately it is not an issue for practical 3D CSEM
applications since the CSEM inversion to deduce the subsurface resistivity is
mainly driven by far offset refractions. We thus focus on the analysis of the
amplitude and phase error beyond 1 km offset in the following.
Figure 4: Comparison between 2nd order FDTD (solid line) and reference
solution (dash line) for 3D CSEM simulation in the shallow water scenario. The
horizontal coordinates are offsets, while the vertical coordinates are (a)
Amplitude; (b) Phase.
The modelling results by FDTD of higher orders are not displayed in Figure 4,
since they are visually very similar to the reference solution. Instead, the
amplitude and phase error are computed using 1D reference solution to examine
the accuracy of our methods. In Figure 5a, c and e, we clearly see that the
2nd order FDTD gives the largest amplitude error for all frequencies; moving
from 2nd order to 4th order significantly reduces the amplitude error, while
moving to 6th order behaves even better, although the accuracy improvement
becomes less. The phase errors exhibit a similar behavior.
Figure 5: The amplitude and phase error of FDTD compared with 1D reference
solution for 0.25 Hz (a,b), 0.75 Hz (c,d) and 1.25 Hz (e,f).
In principle, the computational cost of the 4th order and the 6th order finite
differences will be double and triple of that for the 2nd order scheme. Table
1 lists the CPU time for these modelling exercises, running on a laptop
possessing Intel(R) Core(TM) i7-4710HQ CPU @ 2.50GHz. It shows the increase of
the computing cost by increasing the order of FDTD scheme. The computing time
of the 4th order scheme is less than two times the computing time of the 2nd
order scheme. Further increase of the FDTD order results in significant
increase of CPU time. Since moving to the 6th order scheme demands more
computation while the accuracy improvement is marginal, we stay with the 4th
order scheme from now on, as it gives the best compromise between increased
accuracy and computational load.
Table 1: Comparison of computing time using FDTD of orders 2, 4 and 6. FDTD | Order-2 | Order-4 | Order-6
---|---|---|---
Time (s) | 422.3 | 715.5 | 1090.9
### 3.2 The impact of nonuniform grid
The above example demonstrates the importance of higher order scheme to
achieve accurate CSEM modeling in the presence of strong air-wave. We now
examine the impact of grid non uniformity in achieving computational
efficiency for high-order staggered grid FDTD. To get rid of the impact of
air-wave, we consider a resistivity model in deep water scenario. As shown in
Figure 6a, the model has 1020 m water column of 0.3 $\Omega$m, followed by
formation of $1\Omega\mbox{m}$ down to 1900 m, and 120 m thickness of resistor
of 50 $\Omega$m. The background resistivity below the resistor is 2.5
$\Omega$m. To mimic a vertical transverse isotropic (VTI) Earth, all layers
below the seabed are assigned with an anisotropy ratio (defined as
$\lambda=\rho_{v}/\rho_{h}$ in this paper) $\lambda=1.5$. The source is placed
in the middle of the model, 40 m above the receivers sitting on the seabed.
The resistivity in the water column and the formation above 1200 m was
discretized with constant grid spacing: $\Delta x=\Delta y=150$ m, $\Delta
z=40$ m. From 1200 m down to the bottom of the resistivity model, the grid has
been stretched with different growing factors.
Figure 6: (a) The resistivity model in deep water: the 1020 m water column of
0.3 $\Omega$m followed by formation of $1\Omega\mbox{m}$ down to 1900 m, and
120 m thickness of resistor of 50 $\Omega$m, while the background resistivity
below the resistor is 2.5 $\Omega$m; (b) VTI anisotropy below seabed is 1.5.
Figure 7a displays the stretching factor $r$ of the nonuniform grid for
different number of grid points $n_{z}$. Note that the cell size grows very
fast with the factor $r$. For example, with $r=1.05$ after 40 cells we obtain
$r^{40}>7$ times the size of the first cell. Let us now increase $r$ (hence
decrease $n_{z}$ correspondingly) and analyze how it changes the total
modelling time for the resistivity model of the same physical length in $z$
direction. Figure 7b displays a significant decrease of the modelling time
associated with the reduction of the number of grid points $n_{z}$. We indeed
see that modelling using $n_{z}=66$ takes almost half of the simulation time
compared to modelling using $n_{z}=126$.
Figure 7: (a) The exponentially growing factor with the decreasing number of
grid points $n_{z}$; (b) With the decreasing number of grid points $n_{z}$,
the modelling time decreases dramatically.
Figure 8 overlays the FDTD modelled EM fields with the reference solution. We
see a very good agreement between the two in terms of both the amplitude and
phase. To gain an idea of the modelling accuracy reduction when using less
computing effort with nonuniform grid, we plotted the amplitude and phase
errors in Figure 9 corresponding to three different frequencies. These figures
clearly show that both the amplitude error and the phase error increase with
the increase of the frequencies. It is also interesting to note that
stretching the grid does not necessarily increase the amplitude error, but
does increase the phase discrepancy. This highlights the importance of
combination in examing both amplitude and phase.
Figure 8: Comparison of (a) amplitude and (b) phase between 4th order FDTD
(solid line) and reference solution (dash line) for 3D CSEM simulation in the
deep water scenario. Figure 9: The amplitude and phase error of FDTD compared
with 1D reference solution for 0.25 Hz (a,b), 0.75 Hz (c,d) and 1.25 Hz (e,f)
in deep water.
### 3.3 3D modelling with seafloor bathymetry
The above 1D example highlights the importance of nonuniform grid in
combination with high-order schemes to achieve efficient numerical modelling
with sufficient accuracy. Here we consider a more realistic 3D resistivity
model with bathymetry variations in horizontal directions, as shown in Figure
10a.
The model has seawater of 0.3 $\Omega$m, followed by 2 formation layers of
resistivity - 1 $\Omega$m and 2 $\Omega$m. A resistor of 50 $\Omega$m was
buried in the last formation layer to mimic a hydrocarbon bearing formation
located between 1800 m and 2000 m in depth, with the offset expanding from
-3000 m to 3000 m in both x and y directions.
Figure 10: The 3D resistivity model with seafloor bathymetry, followed by 2
formation layers of resistivity values 1 $\Omega$m and 2 $\Omega$m. The last
sediment layer includes a strong resistor of 100 $\Omega$m mimicking a
hydrocarbon bearing formation located within the depth range [1800, 2000] m
and the offset range [-3000, 3000] m in both x and y directions. The source
was placed at the center of the model while 101 receivers with 200 m spacing
are deployed along x direction.
In order to validate the modelling accuracy of the proposed method, we
simulated a reference solution by finite integration method in frequency
domain using `emg3d` (Werthmüller et al.,, 2019) The finite-integration
modelling extends the model tens of kilometers in each direction, to mimic
that the EM fields propagate to very far distance while avoiding possible
edge/boundary effects. In our finite-difference modelling, the PML boundary
condition attenuates the artificial reflections in the computational domain
within ten grid nodes to achieve the same behavior. A dipole source was placed
at 650 m depth in the middle of the model, while 101 receivers are sitting on
the curved seabed.
Figure 11 displays the comparison of the 3D CSEM modelling between our method
and the result from `emg3d`. From the Figures 11a and 11b, both the amplitude
and the phase from our method match very well with the reference solution. The
maximum amplitude discrepancy for all frequencies in Figure 11c is bounded
within 5% at most of the relevant offset. The maximum mismatch in phase is
less than 1 degree for 0.25 Hz and 2 degrees for 0.75 Hz and 1.25 Hz for the
EM field above the noise level (1e-15 V/m2), as can be seen in Figure 11d.
These demonstrate the good accuracy achieved by our method.
Figure 11: Comparison of the modelling results between the proposed method and
the reference solution.
## 4 Discussion
A natural idea to achieve higher modelling accuracy is to use denser sampling.
However, increasing the number of grid points in each dimension will lead to
exponentially growing computational overhead, i.e., double the sampling in x,
y and z coordinates results in eight times more nodes in the simulation. Over
the same domain the node separations are reduced with a factor of two. By
that, the stability criterion dictates a reduction in the time step by a
factor of two. The net result is an increase of computational cost with a
factor of sixteen. This is significantly more costly than considering high-
order FDTD. For a fixed error requirement, high-order FDTD has been
demonstrated to be much more efficient than a lower-order scheme with dense
sampling (Yefet and Petropoulos,, 2001). Assuming a linear scaling dependency
between computing time and the number of grid points, doubling the length of
the finite-difference operator in x, y and z directions will simply double the
computational cost.
Stretching along $z$-direction seems always to be beneficial in terms of
computational efficiency. It reduces the computational cost in two ways:
first, it leads to increased grid spacing, hence less grid points and larger
node spacing to discretize the resistivity model for simulation on the same
physical size. Meanwhile, larger grid spacing permits to use a larger temporal
step in terms of stability condition. This decreases the number of time steps
needed to reach the steady state of the frequency-domain EM fields.
It is natural to stretch the nonuniform grid in all three spatial directions,
with the motivation to decrease the computational cost further. In case there
is no a priori knowledge about the subsurface, a possible practice is to start
stretching from a given offset from the source location in the horizontal
directions. This approach is not followed up here due to the fact that grid
stretching complicates the calculation of the air-wave for time-domain codes.
Grid stretching is applicable also for frequency-domain finite-difference
codes. The airwave implementation is very different for frequency-domain
codes, where the air layer is part of the simulation domain. Frequency-domain
codes lend themselves easily to both vertical and horizontal grid stretching.
More research must be invested with respect to increasing the simulation
efficiency of time-domain finite-difference schemes focusing on airwave
implementation combined with horizontally nonuniform grids. The root of the
problem is that the most common airwave implementations require fast Fourier
transform (FFT) on regular grids. The fields are transformed to the wavenumber
domain, propagated into the air layer in this domain and then transformed back
to the space domain. A straight forward approach is to use interpolation
between the uniform grid and the nonuniform one, as illustrated in Figure 12.
The grid stretching in $x$\- and $y$\- directions certainly complicates the
implementation while introducing additional computational cost, which is
opposite to what we want to achieve. We have tested horizontal stretching for
the previous 1D model using the stretching factor 1.05 in both $x$\- and $y$\-
directions. The resulting amplitude and phase error in Figure 13 shows that
the numerical accuracy is highly degraded (maximum amplitude error for 0.25 Hz
is around 4%) compared with the result for a horizontally uniform grid
(maximum amplitude error is less than 1.5% for all calculated frequencies).
The running time is in fact longer than for the horizontally uniform grid
which has a higher number of nodes. Due to the nonuniform grid staggering,
several nodes on the uniform grid may reside in the same interval between two
neighboring nodes on the nonuniform grid. The error panel displays an
unsymmetrical pattern in Figure 13. Interpolating the fields from the uniform
grid back to nonuniform grid can thus produce less accurate solutions. No
stretching is therefore recommended along $x$\- and $y$\- for efficiency and
accuracy considerations until a more accurate solution to this problem is
developed.
Figure 12: Horizontal grid stretching requires interpolating between the
coarse nonuniform grid and a dense uniform grid, due to the equidistance
requirement of FFT in airwave manipulation. Figure 13: The amplitude and phase
error after horizontal stretching of the 1D model with a stretching factor
1.05 along both $x$\- and $y$\- directions.
It is noteworthy that fictitious wave domain is simply a mathematical tool to
compute correct frequency domain EM response efficiently by time stepping. The
fictitious time is different from real time. To compute the true EM time
series correctly, one needs to simulate a large number frequencies of the CSEM
fields and then perform inverse Fourier transform, as has been done in Mittet,
(2010) and Rochlitz et al., (2021).
## 5 Conclusion
We have presented a 3D CSEM modelling method using high-order FDTD on a
nonuniform grid. The key problem addressed in this work is the low accuracy
and inconsistency issue in standard 2nd order staggered FDTD scheme on
nonuniform grid. The strategy we propose is to adapt the interpolation weights
depending on the nodal distance. These finite difference coefficients can be
computed by inverting a Vandermonde matrix in an accurate and efficient
manner. This makes our approach different from the commonly used EM modelling
approaches. A new yet more generic stability condition has been established in
order to achieve stable FDTD modelling. In designing the nonuniform grid based
on geometrical progression, we develop a fixed point iteration to compute the
optimal growing factor which allows good match of the modelling domain in case
of grid stretching. The numerical examples demonstrate that there is a
significant improvement in accuracy by using a high-order FDTD scheme, while
combining it with a nonuniform grid reduces the computational cost without a
significant sacrifice of accuracy. We conclude that high-order finite
differences on nonuniform grid is a viable tool for full scale 3D CSEM
modelling applications. Since the key idea is to use high order finite-
difference coefficients adaptive to the node spacing, the method is expected
to be applicable also for finite-difference frequency-domain schemes.
## Acknowledgments
Pengliang Yang was supported by Chinese Fundamental Research Funds for the
Central Universities (AUGA5710010121) and National Natural Science Fundation
of China (42274156). Pengliang Yang thanks Dieter Werthermullter for the
assistance to produce the reference solution using `empymod` and `emg3d` to
validate the accuracy of the proposed method. The source code of this work can
be found in the github repository: https://github.com/yangpl/libEMM.
## Computing uniform staggered-grid finite difference coefficients via
Vandermonde matrix inversion
The method to invert Vandermonde matrices gives us a generic approach to
compute finite-difference coefficients with arbitrary grid spacing. A special
case is the regular grid spacing. In what follows, we show how the standard
staggered grid finite difference coefficients can be accurately computed also
within the same framework.
The Taylor series expansion of a function $f(x)$ can be written as
$\begin{cases}f(x+h)=f(x)+\frac{\partial f(x)}{\partial
x}h+\frac{1}{2!}\frac{\partial^{2}f(x)}{\partial
x^{2}}h^{2}+\frac{1}{3!}\frac{\partial^{3}f(x)}{\partial x^{3}}h^{3}+\ldots\\\
f(x-h)=f(x)-\frac{\partial f(x)}{\partial
x}h+\frac{1}{2!}\frac{\partial^{2}f(x)}{\partial
x^{2}}h^{2}-\frac{1}{3!}\frac{\partial^{3}f(x)}{\partial
x^{3}}h^{3}+\ldots\end{cases}.$ (44)
It leads to
$\begin{cases}\frac{f(x+h)+f(x-h)}{2}&=f(x)+\frac{1}{2!}\frac{\partial^{2}f(x)}{\partial
x^{2}}h^{2}+\frac{1}{4!}\frac{\partial^{4}f(x)}{\partial x^{4}}h^{4}+\ldots\\\
\frac{f(x+h)-f(x-h)}{2}&=\frac{\partial f(x)}{\partial
x}h+\frac{1}{3!}\frac{\partial^{3}f(x)}{\partial
x^{3}}h^{3}+\frac{1}{5!}\frac{\partial^{5}f(x)}{\partial
x^{5}}h^{5}+\ldots\end{cases}.$ (45)
Let $h=\Delta x/2$. This implies the 2nd order accuracy of centered finite
difference scheme using only two staggered nodes
$\begin{cases}\frac{\partial f(x)}{\partial x}=\frac{f(x+\Delta
x/2)-f(x-\Delta x/2)}{\Delta x}+O(\Delta x^{2})\\\ f(x)=\frac{f(x+\Delta
x/2)+f(x-\Delta x/2)}{2}+O(\Delta x^{2})\end{cases}.$ (46)
To approximate the 1st order derivatives as accurate as possible, we express
it using more consecutive nodes. Due to regular grid staggering, the
coefficients lying on symmetric positions should have the same coefficients.
This means the first-order derivative reads in the following form
$\begin{split}\frac{\partial f}{\partial x}=&b_{1}\frac{f(x+\Delta
x/2)-f(x-\Delta x/2)}{\Delta x}+\\\ &b_{2}\frac{f(x+3\Delta x/2)-f(x-3\Delta
x/2)}{3\Delta x}+\\\ &b_{3}\frac{f(x+5\Delta x/2)-f(x-5\Delta x/2)}{5\Delta
x}+\cdots.\end{split}$ (47)
Substituting the $f(x+h)$ and $f(x-h)$ with equation 44 for $h=\Delta
x/2,3\Delta x/2,\ldots$ results in
$\begin{split}\frac{\partial f}{\partial x}=&b_{1}\cdot 2\left(\frac{\Delta
x}{2}\frac{\partial f}{\partial x}+\frac{1}{3!}(\frac{\Delta
x}{2})^{3}\frac{\partial^{3}f}{\partial x^{3}}+\cdots\right)/{\Delta x}\\\
&+b_{2}\cdot 2\left(\frac{3\Delta x}{2}\frac{\partial f}{\partial
x}+\frac{1}{3!}(\frac{3\Delta x}{2})^{3}\frac{\partial^{3}f}{\partial
x^{3}}+\cdots\right)/{3\Delta x}\\\ &+b_{3}\cdot 2\left(\frac{5\Delta
x}{2}\frac{\partial f}{\partial x}+\frac{1}{3!}(\frac{5\Delta
x}{2})^{3}\frac{\partial^{3}f}{\partial x^{3}}+\cdots\right)/{5\Delta
x}+\ldots\\\ =&(b_{1}+b_{2}+b_{3}+b_{4}+\cdots)\frac{\partial f}{\partial
x}\\\ &+\frac{\Delta x^{2}}{3!\cdot
2^{2}}(b_{1}+3^{2}b_{2}+5^{2}b_{3}+7^{2}b_{4}+\cdots)\frac{\partial^{3}f}{\partial
x^{3}}\\\ &+\frac{\Delta x^{4}}{5!\cdot
2^{4}}(b_{1}+3^{4}b_{2}+5^{4}b_{3}+7^{4}b_{4}+\cdots)\frac{\partial^{5}f}{\partial
x^{5}}+\cdots.\end{split}$ (48)
Thus, taking first $L$ terms (corresponding to using $2L$ nodes) requires
$\begin{cases}b_{1}+b_{2}+b_{3}+\cdots+b_{L}&=1\\\
b_{1}+3^{2}b_{2}+5^{2}b_{3}+\cdots+(2L-1)^{2}b_{L}&=0\\\
b_{1}+3^{4}b_{2}+5^{4}b_{3}+\cdots+(2L-1)^{4}b_{L}&=0\\\ \cdots&\\\
b_{1}+3^{2L-2}b_{2}+5^{2L-2}b_{3}+\cdots+(2L-1)^{2L-2}b_{L}&=0\\\
\end{cases},$ (49)
which again builds up a Vandermonde-like system
$\displaystyle\underbrace{\begin{bmatrix}1&1&\ldots&1\\\
x_{1}&x_{2}&\ldots&x_{L}\\\ \vdots&&\ddots&\vdots\\\
x_{1}^{L-1}&x_{2}^{L-1}&\ldots&x_{L}^{L-1}\\\
\end{bmatrix}}_{\textbf{V}}\underbrace{\begin{bmatrix}b_{1}\\\ b_{2}\\\
\vdots\\\ b_{L}\\\ \end{bmatrix}}_{\textbf{b}}=\underbrace{\begin{bmatrix}1\\\
0\\\ \vdots\\\ 0\\\ \end{bmatrix}}_{\textbf{z}},$ (50)
in which $x_{i}=(2i-1)^{2}$, $i=1,\cdots,L$. The finite difference weights can
then be easily computed using Golub, (1996, Algorithm 4.6.2), see an
Octave/Matlab script for computing them in Yang, (2014, section 2.5). These
numerically computed weights may be cross-validated with the generic method by
Fornberg, (1988, Table 2). With regular grid spacing $\Delta x$, the weights
$b_{i}$ are dimensionless and can be connected to the coefficients
$\alpha_{i}$ in equation 13 involving a scaling factor $\Delta x$.
## References
* Björck and Pereyra, (1970) Björck, A., and V. Pereyra, 1970, Solution of Vandermonde systems of equations: Mathematics of computation, 24, 893–903.
* da Silva et al., (2012) da Silva, N. V., J. V. Morgan, L. MacGregor, and M. Warner, 2012, A finite element multifrontal method for 3D CSEM modeling in the frequency domain: Geophysics, 77, E101–E115.
* de Hoop, (1996) de Hoop, A. T., 1996, A general correspondence principle for time-domain electromagnetic wave and diffusion fields: Geophysical Journal International, 127, 757–761.
* Demmel and Koev, (2005) Demmel, J., and P. Koev, 2005, The accurate and efficient solution of a totally positive generalized vandermonde linear system: SIAM Journal on Matrix Analysis and Applications, 27, 142–152.
* Fornberg, (1988) Fornberg, B., 1988, Generation of finite difference formulas on arbitrarily spaced grids: Mathematics of Computation, 51, 699–706.
* Golub, (1996) Golub, G. H., 1996, Matrix computation, third edition: Johns Hopkins Studies in Mathematical Sciences.
* Key, (2016) Key, K., 2016, MARE2DEM: a 2-D inversion code for controlled-source electromagnetic and magnetotelluric data: Geophysical Journal International, 207, 571–588.
* Komatitsch and Martin, (2007) Komatitsch, D., and R. Martin, 2007, An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation: Geophysics, 72, SM155–SM167.
* Lee et al., (1989) Lee, K. H., G. Liu, and H. Morrison, 1989, A new approach to modeling the electromagnetic response of conductive media: Geophysics, 54, 1180–1192.
* Li and Key, (2007) Li, Y., and K. Key, 2007, 2D marine controlled-source electromagnetic modeling: Part 1—an adaptive finite-element algorithm: Geophysics, 72, WA51–WA62.
* Maaø, (2007) Maaø, F., 2007, Fast finite-difference time-domain modeling for marine subsurface electromagnetic problems: Geophysics, 72, A19–A23.
* Mittet, (2010) Mittet, R., 2010, High-order finite-difference simulations of marine CSEM surveys using a correspondence principle for wave and diffusion fields: Geophysics, 75, F33–F50.
* Mittet, (2015) ——–, 2015, Seismic wave propagation concepts applied to the interpretation of marine controlled-source electromagnetics: Geophysics, 80, E63–E81.
* Mittet, (2017) ——–, 2017, On the internal interfaces in finite-difference schemes: Geophysics, 82, T159–T182.
* Monk and Süli, (1994) Monk, P., and E. Süli, 1994, A convergence analysis of Yee’s scheme on nonuniform grids: SIAM Journal on Numerical Analysis, 31, 393–412.
* Mulder, (2006) Mulder, W., 2006, A multigrid solver for 3D electromagnetic diffusion: Geophysical prospecting, 54, 633–649.
* Newman and Alumbaugh, (1995) Newman, G. A., and D. L. Alumbaugh, 1995, Frequency-domain modelling of airborne electromagnetic responses using staggered finite differences: Geophysical Prospecting, 43, 1021–1042.
* Oristaglio and Hohmann, (1984) Oristaglio, M. L., and G. W. Hohmann, 1984, Diffusion of electromagnetic fields into a two-dimensional earth: A finite-difference approach: Geophysics, 49, 870–894.
* Puzyrev et al., (2013) Puzyrev, V., J. Koldan, J. de la Puente, G. Houzeaux, M. Vázquez, and J. M. Cela, 2013, A parallel finite-element method for three-dimensional controlled-source electromagnetic forward modelling: Geophysical Journal International, 193, 678–693.
* Rochlitz et al., (2021) Rochlitz, R., M. Seidel, and R.-U. Börner, 2021, Evaluation of three approaches for simulating 3-D time-domain electromagnetic data: Geophysical Journal International, 227, 1980–1995.
* Rochlitz et al., (2019) Rochlitz, R., N. Skibbe, and T. Günther, 2019, custem: Customizable finite-element simulation of complex controlled-source electromagnetic data: Geophysics, 84, F17–F33.
* Roden and Gedney, (2000) Roden, J. A., and S. D. Gedney, 2000, Convolution PML (CPML): An efficient FDTD implementation of the CFS–PML for arbitrary media: Microwave and optical technology letters, 27, 334–339.
* (23) Smith, J. T., 1996a, Conservative modeling of 3-D electromagnetic fields, Part i: Properties and error analysis: Geophysics, 61, 1308–1318.
* (24) ——–, 1996b, Conservative modeling of 3-D electromagnetic fields, Part II: Biconjugate gradient solution and an accelerator: Geophysics, 61, 1319–1324.
* Streich, (2009) Streich, R., 2009, 3D finite-difference frequency-domain modeling of controlled-source electromagnetic data: Direct solution and optimization for high accuracy: Geophysics, 74, F95–F105.
* Taflove and Hagness, (2005) Taflove, A., and S. C. Hagness, 2005, Computational electrodynamics: The finite-difference time-domain method, 3rd ed.: Artech House.
* Wang and Hohmann, (1993) Wang, T., and G. W. Hohmann, 1993, A finite-difference, time-domain solution for three-dimensional electromagnetic modeling: Geophysics, 58, 797–809.
* Werthmüller, (2017) Werthmüller, D., 2017, An open-source full 3D electromagnetic modeler for 1D VTI media in Python: empymod: Geophysics, 82, WB9–WB19.
* Werthmüller et al., (2019) Werthmüller, D., W. Mulder, and E. Slob, 2019, emg3d: A multigrid solver for 3d electromagnetic diffusion: Journal of Open Source Software, 4, 1463\.
* Yang, (2014) Yang, P., 2014, A numerical tour of wave propagation: Technical report, Xi’an Jiaotong University.
* Yee, (1966) Yee, K. S., 1966, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media: IEEE Transactions on Antennas and Propagation, 14, 302–307.
* Yefet and Petropoulos, (2001) Yefet, A., and P. G. Petropoulos, 2001, A staggered fourth-order accurate explicit finite difference scheme for the time-domain maxwell’s equations: Journal of Computational Physics, 168, 286–315.
|
# Tight Conditional Lower Bounds for Vertex Connectivity Problems
Zhiyi Huang Tsinghua University Yaowei Long University of Michigan
Thatchaphol Saranurak Supported by NSF CAREER grant 2238138. University of
Michigan Benyu Wang Tsinghua University
(April, 2023)
###### Abstract
We study the fine-grained complexity of graph connectivity problems in
unweighted undirected graphs. Recent development shows that all variants of
_edge connectivity_ problems, including single-source-single-sink, global,
Steiner, single-source, and all-pairs connectivity, are solvable in
$m^{1+o(1)}$ time, collapsing the complexity of these problems into the
almost-linear-time regime. While, historically, _vertex connectivity_ has been
much harder, the recent results showed that both single-source-single-sink and
global vertex connectivity can be solved in $m^{1+o(1)}$ time, raising the
hope of putting all variants of vertex connectivity problems into the almost-
linear-time regime too.
We show that this hope is impossible, assuming conjectures on finding
4-cliques. Moreover, we essentially settle the complexity landscape by giving
tight bounds for _combinatorial_ algorithms in dense graphs. There are three
separate regimes:
1. 1.
all-pairs and Steiner vertex connectivity have complexity
$\hat{\Theta}(n^{4})$,
2. 2.
single-source vertex connectivity has complexity $\hat{\Theta}(n^{3})$, and
3. 3.
single-source-single-sink and global vertex connectivity have complexity
$\hat{\Theta}(n^{2})$.
For graphs with general density, we obtain tight bounds of
$\hat{\Theta}(m^{2})$, $\hat{\Theta}(m^{1.5})$, $\hat{\Theta}(m)$,
respectively, assuming Gomory-Hu trees for element connectivity can be
computed in almost-linear time.
## 1 Introduction
Vertex connectivity and edge connectivity are central concepts in graph
theory. In an unweighted undirected graph $G$ with $n$ vertices and $m$ edges,
the _vertex connectivity_(_edge connectivity_) between two vertices $u,v$ is
the maximum number of internally vertex-disjoint (edge-disjoint) paths from
$u$ to $v$. Efficient algorithms for variants of both problems have been
extensively studied for at least half a century [FF56, Kle69, GH61, GR98,
CKM+11, She13, KLOS14, Mad16, vdBLN+20, CKL+22].
Until very recently, the graph algorithm community has reached a very
satisfying conclusion on the complexity of edge connectivity problems: _all
variants of edge connectivity problems can be solved in almost-linear time_.
There are five variants of the connectivity problems studied in the
literature, including (1) global, (2) single-source single-sink, (3) Steiner,
(4) single-source, and (5) all-pairs connectivity. See the detailed
definitions in Section 2. Among these problems, all-pairs connectivity is the
hardest via straightforward reductions. For the global edge connectivity
problem, Karger showed in 2000 that the problem admits a near-linear time
algorithm [Kar00]. In the recent line of work [LP20, AKT21b, LP21, AKT21a,
LPS21, AKL+22], Abboud et al. [AKL+22] finally showed that even the all-pairs
edge connectivity case could be reduced to the single-source single-sink case,
which solvable in almost-linear time via the recent max flow result by Chen et
al. [CKL+22]. This finally puts all five problems on edge connectivity into
the almost-linear time regime and raises the following hope:
_Can we solve all variants of vertex connectivity problems in almost-linear
time too?_
Historically, the vertex connectivity problems have been much more difficult
than their edge connectivity counterpart. Furthermore, Abboud et al. [AKT20]
showed that the all-pairs vertex connectivity in _weighted_ graphs with
$\tilde{O}(n)$ edges requires $\hat{\Omega}(n^{3})$ time assuming SETH.111We
use $\tilde{O}(\cdot)$ to hide a $\mathrm{polylog}(n)$ factor and
$\hat{O}(\cdot)$ to hide a $n^{o(1)}$ factor. In _directed_ unweighted graphs,
Abboud et al. [AGI+19] also showed that the problem even requires
$\hat{\Omega}(n^{4})$ for combinatorial algorithms and
$\hat{\Omega}(n^{\omega+1})$ time for general algorithms. Thus, at least in
weighted or directed graphs, the problem does not admit almost-linear
algorithms.
But, again, the recent development on vertex connectivity in the standard
_unweighted undirected_ graphs raises the hope for almost-linear time
algorithms. Li et al. [LNP+21] showed how to compute global vertex
connectivity using polylogarithmic calls to max flows, which implied an
$\hat{O}(m)$-time algorithm via the max flow algorithm of [CKL+22] and
improved upon the known $\tilde{O}(mn)$ bound by [HRG00]. Indeed, the recent
max flow algorithm [CKL+22] also implies a $\hat{O}(m)$-time algorithm for the
$(s,t)$-vertex connectivity problem. Until now, there are still no nontrivial
lower bounds for Steiner, single-source, and all-pairs vertex connectivity in
unweighted graphs, and the technique of [AKT20] is quite specific for weighted
graphs. The complexity landscape for vertex connectivity problems is still
very unclear.
### 1.1 Our Results
We give a firm negative answer to the above open problem. Based on well-known
conjectures, we settle the complexity landscape for all five vertex
connectivity problems in dense graphs by giving tight bounds for
_combinatorial_ algorithms, which generally refer to algorithms that do not
use fast matrix multiplication. We can obtain tight bounds even in general
graphs by assuming a possible hypothesis about computing Gomory-Hu trees for
element connectivity. Below, we discuss our results in more detail.
#### All-pairs vertex connectivity (Section 3).
Our first result is a conditional lower bound of the all-pair vertex
connectivity (APVC) problem based on the _4-clique conjecture_ , which
postulates that the running time for deciding the existence of a 4-clique in a
graph must be at least $\hat{\Omega}(n^{4})$ time for combinatorial algorithms
[BGL17] and $\hat{\Omega}(n^{\omega(1,2,1)})=\Omega(n^{3.25})$ time for
general algorithms [DW22].222$\omega(1,2,1)$ is the exponent of multiplying an
$n\times n^{2}$ matrix by an $n^{2}\times n$ matrix.
###### Theorem 1.1.
Assuming the 4-clique conjecture, the all-pairs vertex connectivity problem on
an undirected unweighted graph with $n$ vertices requires
$\hat{\Omega}(n^{4})$ time for combinatorial algorithms, and
$\Omega(n^{3.25})$ time for general algorithms.
Theorem 1.1 gives the first super cubic lower bounds for all-pairs vertex
connectivity problems in the standard undirected case. Moreover, the bound is
tight for _combinatorial_ algorithms. Indeed, a naive algorithm is to call max
flow $O(n^{2})$ times, one for each pair of vertices, which takes
$\hat{O}(mn^{2})=\hat{O}(n^{4})$ time.
It is interesting to compare APVC with the _all-pairs shortest paths_ (APSP)
problem, a central problem in fine-grained complexity. It is conjectured that
the right complexity of APSP in weighted graphs is $\hat{\Theta}(n^{3})$ (see
e.g. [WW18]). Theorem 1.1 shows that, for general algorithms, APVC in
unweighted graphs is strictly harder than APSP in weighted graphs, assuming
$\omega>2$,
#### Steiner vertex connectivity (Section 4).
Next, we study the _Steiner vertex connectivity_ problem. In this problem,
given a set of vertices $T$, we need to compute the minimum vertex
connectivity among all pairs of vertices in $T$. Even though Steiner vertex
connectivity looks much easier than APVC, we can extend the same lower bound
of Theorem 1.1 for APVC to work for the Steiner case too.
Towards this hardness, we propose a variant of the 4-clique conjecture, the
edge-universal 4-clique conjecture, which postulates that, given any subset of
demand edges $E_{\mathrm{dem}}\subseteq E(G)$, checking if every edge in
$E_{\mathrm{dem}}$ is contained in a 4-clique requires $\hat{\Omega}(n^{4})$
time for combinatorial algorithms (see 2.3 for details).333Note that this
problem is at least as hard as the problem when $E_{\mathrm{dem}}=E(G)$, i.e.
when we need to check if every edge is contained in a 4-clique.
###### Theorem 1.2.
Assuming the edge-universal 4-clique conjecture, the Steiner vertex
connectivity problem on an undirected unweighted graph with $n$ vertices
requires $\hat{\Omega}(n^{4})$ time for combinatorial algorithms.
We note that our reduction would imply hardness for general algorithms too if
we conjectured the hardness of the edge-universal 4-clique problem for general
algorithms.
#### Upper bounds for general density (Section 5).
On sparse graphs, we observe that there is still some discrepancy between our
lower bounds and the naive algorithm. More precisely, we observe that our
lower bounds for combinatorial algorithms for all-pairs and Steiner vertex
connectivity can be easily extended to $\hat{\Omega}(m^{2})$ in a graph with
$m$ edges. However, the naive algorithm requires $\hat{O}(mn^{2})$ time. For
example, in sparse graphs, the naive algorithm takes $\hat{O}(n^{3})$ time,
while our lower bound is $\hat{\Omega}(n^{2})$.
We fix the above discrepancy by showing improved algorithms in sparse graphs
via the following result.
###### Theorem 1.3.
Given an $n$-vertex $m$-edge unweighted graph, there is an algorithm to solve
the APVC problem in $\hat{O}(m^{11/5})$ time with high probability. Assuming
that the _element connectivity Gomory-Hu tree_ can be constructed in
$\hat{O}(m)$ time, the running time can be improved to $\hat{O}(m^{2})$.
Theorem 1.3 gives the first subcubic algorithm for computing all-pairs vertex
connectivity in sparse graphs. Moreover, the bound $\hat{\Theta}(m^{2})$ is
tight with our lower bounds for all density regimes, assuming the almost-
linear-time construction of the element connectivity Gomory-Hu tree.
We note that the _element connectivity Gomory-Hu tree_ is a generalization of
the well-known _edge connectivity Gomory-Hu tree_ [GH61], whose almost-linear-
time construction was very recently shown by Abboud et al. [AKL+22]. It is
quite believable that an element connectivity Gomory-Hu tree can also be
constructed in almost-linear time.
#### The complexity landscape of vertex connectivity.
Based on these main results, we obtain some other results and corollaries on
all variants of vertex connectivity problems, as summarized in Table 1. In
contrast to the edge connectivity problems which can all be solved in almost-
linear time, there are three separate regimes for vertex connectivity.
1. 1.
all-pairs and Steiner vertex connectivity have complexity
$\hat{\Theta}(n^{4})$,
2. 2.
single-source vertex connectivity has complexity $\hat{\Theta}(n^{3})$, and
3. 3.
single-source-single-sink and global vertex connectivity have complexity
$\hat{\Theta}(n^{2})$.
For graphs with general density, we obtain tight bounds of
$\hat{\Theta}(m^{2})$, $\hat{\Theta}(m^{1.5})$, $\hat{\Theta}(m)$,
respectively, assuming Gomory-Hu trees for element connectivity can be
computed in almost-linear time.
Table 1: Upper bounds and lower bounds for connectivity problems | Global | Single-Source | all-pairs | Steiner
---|---|---|---|---
edge connectivity, unweighted graphs | $\tilde{O}(m)$ [Kar00] | $\hat{O}(m)$ [AKL+22] | $\hat{O}(m)$ [AKL+22] | $\hat{O}(m)$ [LP20]
vertex connectivity, unweighted graphs with general density | | $\hat{\Omega}(m^{1.5})$ for comb. algo., Corollary 3.11 | $\hat{\Omega}(m^{2})$ for comb. algo., Corollary 3.10 | $\hat{\Omega}(m^{2})$ for comb. algo., Corollary of Theorem 4.1
$\hat{O}(m)$ [LNP+21] | $\hat{O}(m^{1.5})$, assuming 2.5, Theorem 5.2 | $\hat{O}(m^{2})$, assuming 2.5, Theorem 5.1 | $\hat{O}(m^{2})$, assuming 2.5, Theorem 5.1
| | $\hat{O}(m^{5/3})$, Theorem 5.2 | $\hat{O}(m^{11/5})$, Theorem 5.1 | $\hat{O}(m^{11/5})$, Theorem 5.1
vertex connectivity, dense unweighted graphs $m=\Theta(n^{2})$ | | $\hat{\Omega}(n^{3})$ for comb. algo., Corollary 3.9 | $\hat{\Omega}(n^{4})$ for comb. algo., Theorem 3.1 | $\hat{\Omega}(n^{4})$ for comb. algo. Theorem 4.1
| | $\hat{\Omega}(n^{\omega(1,2,1)})$ for all algo., Remark 3.8 |
$\hat{O}(n^{2})$ [LNP+21] | $\hat{O}(n^{3})$ trivially | $\hat{O}(n^{4})$ trivially | $\hat{O}(n^{4})$ trivially
vertex connectivity, sparse weighted graphs, $m=\tilde{O}(n)$ | | | $\hat{\Omega}(n^{3})$ [AKT20] |
#### Discussions and Open Problems.
Our lower bounds for combinatorial algorithms are particularly relevant to the
context of vertex connectivity since basically all algorithms for the problems
are indeed combinatorial. There are a few exceptions [LLW88, AGI+19], but
these algorithms are still far from optimal (even cannot break our
combinatorial lower bounds). It is a very interesting open problem whether one
can bypass our combinatorial lower bounds using fast matrix multiplications,
or show conditional lower bounds for general algorithms that match our
combinatorial lower bounds.
### 1.2 Technical Overview
To prove Theorem 1.1, we are will reduce the 4-clique problem to the APVC
problem. Previously, Abboud et al. [AGI+19] showed the hardness of all-pairs
vertex connectivity in _directed_ graphs using the 4-clique problem, which
inspired our reduction. However, the techniques are not strong enough to work
on undirected graphs. In more details, the hard instance of [AGI+19] is a
directed acyclic graph with four layers, and so they only need to consider
directed paths of length at most 3. In contrast, when we consider undirected
graphs, paths connecting sources and sinks can be much more complex, which
requires more advanced techniques and more careful arguments. Let us sketch
our construction below.
Starting from a 4-clique instance $G$, it is helpful to consider its 4-partite
version $G_{4p}$, which is simply constructed by duplicating $V(G)$ into 4
groups $A,B,C,D$ and copying $E(G)$ for each pair of different groups (see
Definition 2.2 for a formal definition). A natural way to answer the 4-clique
problem on $G$ is then checking for each pair of adjacent $a\in A$ and $d\in
D$, whether there exists an adjacent pair of vertices $b\in B$ and $c\in C$
that is adjacent to both $a$ and $d$ (call such $(b,c)$ a _4-clique witness_
of $(a,d)$).
To correspond this to a vertex connectivity problem, for each pair of $a$ and
$d$, consider a 4-layer graph $\hat{H}_{ad}$ defined as follows. Let $B_{a}$
denote the set of vertices in $B$ adjacent to $a$ (also define $B_{d}$,
$C_{a}$ and $C_{d}$ similarly). Then $B_{a}\cap B_{d}$ (resp. $C_{a}\cap
C_{d}$) are vertices in $B$ (resp. $C$) adjacent to both $a$ and $d$. The
vertices of $\hat{H}_{ad}$ are $V(\hat{H}_{ad})=\\{a\\}\cup(B_{a}\cap
B_{d})\cup(C_{a}\cap C_{d})\cup\\{d\\}$, where the first layer (resp. the last
layer) has only a single vertex $a$ (resp. $d$), and the second layer (resp.
the third layer) has vertices $B_{a}\cap B_{d}$ (resp. $C_{a}\cap C_{d}$). The
edge set of $\hat{H}_{ad}$ is $E(\hat{H}_{ad})=\\{(a,b)\mid b\in B_{a}\cap
B_{d}\\}\cup E_{G_{4p}}(B_{a}\cap B_{d},C_{a}\cap C_{d})\cup\\{(c,d)\mid c\in
C_{a}\cap C_{d}\\}$444Here $E_{G_{4p}}(B_{a}\cap B_{d},C_{a}\cap
C_{d})\subseteq E(G_{4p})$ denote the set of edges connecting $B_{a}\cap
B_{d}$ and $C_{a}\cap C_{d}$ in $G_{4p}$., which connects vertex $a$ (resp.
$d$) to each second-layer (resp. third layer) vertex, and connects the second
layer $B_{a}\cap B_{d}$ and the third layer $C_{a}\cap C_{d}$ using the same
edges in $G_{4p}$. One can simply observe that a 4-clique witness $(b,c)$ of
$(a,d)$ exists if and only if $\kappa_{\hat{H}_{ad}}(a,d)\geq 1$555In the
overview, we use $\kappa_{H}(a,d)$ to denote the vertex connectivity between
$a$ and $d$ in a graph $H$.. To check the existence of 4-clique witnesses for
all pairs $(a,d)$ simultaneously, our final APVC instance $H$ will be a
combination of all $\hat{H}_{ad}$, and the main challenge is to make the
combination compact.
We overcome this challenge by introducing two modules called the _source-sink
isolating gadgets_ and the _set-intersection filter_. Interestingly, our
technique for proving time lower bounds is inspired by the space-lower bound
techniques. More specifically, the source-sink isolating gadget is inspired by
the construction in [PSY22].
The intuition behind these two modules is as follows. When considering
$\kappa_{\hat{H}_{ad}}(a,d)$ of a specific pair of $a\in A$ and $d\in D$, we
will somehow “remove” redundant vertices in $H\setminus\hat{H}_{ad}$ so that
$\kappa_{\hat{H}_{ad}}(a,d)$ can be derived from $\kappa_{H}(a,d)$. The high-
level idea to remove a redundant vertex $v$ in $H$ is to create a “flow” path
from $a$ to $d$ through $v$ (e.g. a simple path made up of two edges $(a,v)$
and $(v,d)$). By a simple flow-cut argument, this path will enforce that $v$
appears in any $(a,d)$-vertex cut, while bringing some additive deviations to
estimate $\kappa_{\hat{H}_{ad}}(a,d)$ as $\kappa_{H}(a,d)$. These modules
apply this simple rule in a more general way. The source-sink isolating gadget
will remove all vertices in $A$ and $D$ except $a$ and $d$, and the
intersection patterns will generate $B_{a}\cap B_{d}$ and $C_{a}\cap C_{d}$ by
removing other vertices in $B$ and $C$. The remaining graph will then be
exactly $\hat{H}_{ad}$, and the additive deviations between
$\kappa_{\hat{H}_{ad}}(a,d)$ and $\kappa_{H}(a,d)$ can be computed and
subtracted easily.
The proof of Theorem 1.2 extends the above idea to reduce a Steiner vertex
connectivity problem to an edge-universal 4-clique problem. Consider an edge-
universal 4-clique instance $G$. Based on the above construction of $H$, by
creating more “flow” paths, we can guarantee that the additive deviations
between $\kappa_{\hat{H}_{ad}}(a,d)$ and $\kappa_{H}(a,d)$ for all pair of
$(a,d)$ will be the same, say a value $K$. Therefore, for each pair $(a,d)$,
$\kappa_{H}(a,d)$ is either at least $K+1$ or equal to $K$, and there is no
4-clique containing $a$ and $d$ in the original graph $G$ if and only if the
latter case happens. Finally, checking the Steiner vertex connectivity of
terminal set $A\cup D$ suffices to answer the edge-universal 4-clique problem
on $G$.
Towards the upper bound of the APVC problem on sparse graphs in Theorem 1.3,
our algorithm uses the following scheme. Let the input graph be $G$ and let
$k$ be a degree threshold to separate vertices into two parts, the high-degree
part $V_{h}=\\{v\in V(G)\mid\deg_{G}(v)>k\\}$ and the low-degree part
$V_{l}=\\{v\in V(G)\mid\deg_{G}(v)\leq k\\}$. For pairs $(u,v)$ such that
$u,v\in V_{h}$, we can compute $\kappa_{G}(u,v)$ by simply calling max flows,
which takes totally $O(m^{2}/k^{2})$ calls since $|V_{h}|=O(m/k)$. For other
pairs $(u,v)$ with $u\in V_{l}$ or $v\in V_{l}$, there will be
$\kappa_{G}(u,v)\leq k$, because the vertex connectivity of $u$ and $v$ is
upper bounded by their degrees. The vertex connectivity oracle in [PSY22] can
exactly capture all-pairs vertex connectivity bounded by $k$, which takes
$\tilde{O}(k^{2})$ black-box calls to Gomory-Hu trees for element connectivity
whose construction time is currently $\hat{O}(mk)$. The whole algorithm takes
$\hat{O}(m^{11/5})$ time by choosing a proper $k$, and the running time will
be immediately improved to $\hat{O}(m^{2})$ if Gomory-Hu trees for element
connectivity can be constructed in almost linear time.
### 1.3 Organization
We will start with some basic notations and introduce conjectures in Section
2. In Section 3, we will show the conditional lower bound of the APVC problem.
In Section 4, we will extend the approach in Section 3 to show the conditional
lower bound of the Steiner vertex connectivity problem. In Section 5, we will
show a simple APVC algorithm for graphs with general density.
## 2 Preliminaries
In this paper, we use standard graph theoretic notations. For a graph $G$, we
use $V(G)$ and $E(G)$ to denote its vertex set and edge set. For any vertex
set $V^{\prime}\subseteq V(G)$, we let $G[V^{\prime}]$ denote the subgraph
induced by $V^{\prime}$ and let $G\setminus V^{\prime}$ be a short form of
$G[V(G)\setminus V^{\prime}]$. For two graphs $G_{1}$ and $G_{2}$ which vertex
sets $V(G_{1})$ and $V(G_{2})$ may intersect, we let $G_{1}\cup G_{2}$ denote
the graph with vertex set $V(G_{1})\cup V(G_{2})$ and edge set $E(G_{1})\cup
E(G_{2})$. For two subset of vertices $V_{1},V_{2}\subseteq V(G)$, we let
$E_{G}(V_{1},V_{2})$ denote the set of edges directly connecting $V_{1}$ and
$V_{2}$. For a vertex $v\in G$, we let $N_{G}(v)=\\{u\mid(u,v)\in E(G)\\}$
denote its neighbor set, and let $\bar{N}_{G}(v)=V(G)\setminus N_{G}(v)$
denote its non-neighbors.
#### Vertex connectivity.
In a graph $G$, the vertex connectivity for two vertices $u,v\in V(G)$,
denoted by $\kappa_{G}(u,v)$, is the maximum number of internally vertex-
disjoint paths from $u$ to $v$. By Menger’s theorem, $\kappa_{G}(u,v)$ is
equal to the size of minimized subsets $C\subseteq(V(G)\setminus\\{u,v\\})\cup
E(G)$ deleting which from $G$ will disconnect $u$ and $v$.
#### Vertex Connectivity problems.
In this paper, we will consider four vertex connectivity problems, i.e. the
global, single-source, all-pairs, and Steiner vertex connectivity problems.
The edge connectivity versions of these problems are analogous.
* •
The global vertex connectivity problem. Given an undirected unweighted graph
$G$, the global vertex connectivity problem (the global-VC problem) asks the
global vertex connectivity, denoted by $\kappa_{G}$, where
$\kappa_{G}=\min_{u,v\in G}\kappa_{G}(u,v)$.
* •
The single-source vertex connectivity problem. Given an undirected unweighted
graph $G$ with a source vertex $s$, a single source vertex connectivity
problem (the SSVC problem) asks $\kappa_{G}(s,v)$ for all other vertices $v\in
G$.
* •
The all-pairs vertex connectivity problem. Given an undirected unweighted
graph $G$, the all-pairs vertex connectivity problem (the APVC problem) asks
$\kappa_{G}(u,v)$ for all pairs of $u,v\in G$.
* •
The Steiner vertex connectivity problem. Given an undirected unweighted graph
$G$ with a terminal set $T\subseteq V(G)$, the Steiner vertex connectivity
problem (the Steiner-VC problem) asks the Steiner vertex connectivity of $T$,
denoted by $\kappa_{G}(T)$, where $\kappa_{G}(T)=\min_{u,v\in
T}\kappa_{G}(u,v)$.
#### The 4-clique conjecture.
Given an $n$-vertex undirected graph $G$, the $k$-clique problem is to decide
whether there is a clique with $k$ vertices in $G$. The $k$-clique problem can
be solved in $O(n^{k})$ time trivially, and a more efficient combinatorial
algorithm takes running time $O(n^{k}/\log^{k}n)$ [Vas09]. The popular
$k$-clique conjecture (see e.g. [BGL17]) suggests that there is no
combinatorial algorithm for the $k$-clique problem with $O(n^{k-\epsilon})$
running time for any constant $\epsilon>0$. In Section 3, we will use this
conjecture in the case $k=4$.
###### Conjecture 2.1 (4-clique conjecture).
There is no combinatorial algorithm that solves the 4-clique problem for
$n$-vertex graphs in $O(n^{4-\epsilon})$ time for any constant $\epsilon>0$.
For each 4-clique instance $G$, it is equivalent to consider its 4-partite
form $G_{4p}$ as defined below. Note that a 4-clique in a 4-partite graph
should contain exactly one vertex from each group, and the original graph has
a 4-clique if and only if the 4-partite graph $G_{4p}$ has a 4-clique.
###### Definition 2.2 (4-partite graph $G_{4p}$).
Given an undirected graph $G$, the 4-partite graph $G_{4p}$ of $G$ has vertex
set $V(G_{4p})=\\{v_{A},v_{B},v_{C},v_{D}\mid v\in V(G)\\}$, and we let
$A=\\{v_{A}\mid v\in V(G)\\},B=\\{v_{B}\mid v\in V(G)\\},C=\\{v_{C}\mid v\in
V(G)\\},D=\\{v_{D}\mid v\in V(G)\\}$ be four groups partitioning $V(G_{4p})$.
The edge set $E(G_{4p})=\\{(u_{X},v_{Y})\mid(u,v)\in
E(G),X,Y\in\\{A,B,C,D\\},X\neq Y\\}$.
#### The edge-universal 4-clique conjecture.
We consider a variant of the 4-clique problem, called the _edge-universal
4-clique problem_. Given an undirected graph $G$ and a subset of _demand_
edges $E_{\mathrm{dem}}\subseteq E(G)$, this problem asks if every edge in
$E_{\mathrm{dem}}$ is contained by a 4-clique. We suggest the following
conjecture on this problem.
###### Conjecture 2.3 (Edge-universal 4-clique conjecture).
There is no combinatorial algorithm that answers the all edge 4-clique problem
for $n$-vertex graphs in $O(n^{4-\epsilon})$ time for any constant
$\epsilon>0$.
We note our formulation of the edge-universal 4-clique problem is at least as
hard as the problem of checking if every edge is contained in some 4-clique by
fixing $E_{\mathrm{dem}}=E(G)$. We choose to present this formulation that
allows any $E_{\mathrm{dem}}\subseteq E(G)$ because it only strengthens our
hardness result and shows the flexibility of our reduction.
The difference between the edge-universal 4-clique conjecture vs. the 4-clique
conjecture is that we switch the quantifier in the definition of the problems.
This allows us to obtain new hardness results. This strategy for proving
conditional lower bounds has been studied and done several times in the
literature of fine-grained complexity [GIKW19, AWW16, ABHS22]. For example,
the relationship between the edge-universal 4-clique problem vs. the 4-clique
problem is the same as the relationship between the well-known _orthogonal
vector_ problem vs. the _hitting set_ problem introduced in [AWW16], and
_SETH_ vs. _quantified SETH_ introduced in [ABHS22].
#### Gomory-Hu trees for element connectivity
Gomory-Hu trees are cut-equivalent trees introduced by Gomory and Hu [GH61] to
_capture_ all-pairs edge connectivity in weighted graphs. More precisely,
given a Gomory-Hu tree, the edge connectivity of any given pair of vertices
can be queried in nearly constant time. Very recently, a breakthrough by
[AKL+22] showed that a Gomory-Hu tree can be constructed in $\tilde{O}(n^{2})$
time for a weighted graph and $\hat{O}(m)$ time for an unweighted graph. For
vertex connectivity, it has been shown by [Ben95] that there are no such cut-
equivalent trees. See also [PSY22] for a more general space lower bound
forbidding the existence of such trees for vertex connectivity.
Element connectivity is the notion of connectivity that is related to vertex
connectivity, and yet Gomory-Hu trees have been shown to exist for element
connectivity (see e.g. [CRX15]). Given an undirected graph $G$ and a terminal
set $U\subseteq V(G)$, the element connectivity for two vertices $u,v\in U$,
denoted by $\kappa^{\prime}_{G,U}(u,v)$, is the size of minimized set
$C\subseteq E(G)\cup(V(G)\setminus U)$ whose removal will disconnect $u$ and
$v$. An element connectivity Gomory-Hu tree for the graph $G$ and terminal set
$U$ will capture $\kappa^{\prime}_{G,U}(u,v)$ for all pairs of $u,v\in U$.
In [PSY22], they consider a variant called $k$-Gomory-Hu tree for element
connectivity, which given an additional parameter $k$, will capture the value
$\min\\{\kappa^{\prime}_{G,U}(u,v),k\\}$ for all $u,v\in U$ (namely we can
query $\min\\{\kappa^{\prime}_{G,U}(u,v),k\\}$ for each given $u,v\in U$ in
nearly constant time). By generalizing the $(1+\epsilon)$-approximate Gomory-
Hu tree algorithm by [LP21] to the element connectivity setting, the following
result was obtained by [PSY22].
###### Theorem 2.4.
Given an $n$-vertex $m$-edge undirected unweighted graph $G$, a terminal set
$U\subseteq V(G)$ and a parameter $k$, there is a randomized algorithm to
construct a $k$-Gomory-Hu tree for element connectivity in $\hat{O}(mk)$ time
with high probability.
Given the similarity of Gomory-Hu trees for edge connectivity and element
connectivity, and the recent breakthrough of $\hat{O}(m)$-time construction
algorithm for edge connectivity Gomroy-Hu tree, it seems reasonable to
conjecture that element connectivity Gomory-Hu tree can also be constructed in
$\hat{O}(m)$ time.
###### Conjecture 2.5.
Given an $n$-vertex $m$-edge undirected unweighted graph $G$ and a terminal
set $U\subseteq V(G)$, an element connectivity Gomory-Hu tree can be
constructed in $\hat{O}(m)$ time.
We leave this conjecture as a very interesting open problem.
## 3 The Lower Bound for the APVC Problem
In this section, we will prove Theorem 3.1, a conditional lower bound of the
APVC problem in undirected unweighted graphs conditioning on the 4-clique
conjecture. Concretely, we will show a reduction from the 4-clique problem to
the APVC problem.
###### Theorem 3.1.
Assuming 2.1, for $n$-vertex undirected unweighted graphs, there is no
combinatorial algorithm that solves the APVC problem in $O(n^{4-\epsilon})$
time for any constant $\epsilon>0$.
Given an $n$-vertex 4-clique instance $G$, let $G_{4p}$ be the corresponding
4-partite graph defined in Definition 2.2 (where $V(G_{4p})$ is partitioned
into 4 groups $A,B,C,D$). We start with some notations. For each vertex $a\in
A$, we use $B_{a}=\\{b\in B|(a,b)\in E(G_{4p})\\}$ to denote the neighbors of
$a$ in $B$ and let $\bar{B}_{a}=B\setminus B_{a}$. Analogously, $C_{a}$ is the
set of vertices in $C$ adjacent to $a$ and $\bar{C}_{a}=C\setminus C_{a}$. For
each $d\in D$, we define $B_{d},\bar{B}_{d},C_{d},\bar{C}_{d}$ in a similar
way.
As discussed in Section 1.2, our 4-clique instance $H$ will be constructed
using two kinds of modules, the source-sink isolating gadgets and the set-
intersection filters, which will be introduced in Section 3.1 and Section 3.2
respectively. After that, the final construction of $H$ and the proof of
Theorem 3.1 will be completed in Section 3.3.
### 3.1 The Source-Sink Isolating Gadget
We first introduce the _source-sink isolating gadget_. Basically, for an
undirected graph $R$ and two disjoint groups of vertices $X,Y\subseteq V(R)$,
a source-sink isolating gadget $Q(X,Y)$ (or just $Q$ for short) is a graph on
vertices $X\cup Y$ with additional vertices outside $R$. Its formal guarantee
is as follows.
###### Lemma 3.2.
Given an undirected graph $R$ and two disjoint groups of vertices
$X,Y\subseteq V(R)$, there is a graph $Q$ with $V(Q)\cap V(R)=X\cup Y$ and
$|V(Q)|=O(|X|+|Y|)$ such that for any $x\in X,y\in Y$ with $(x,y)\notin E(R)$,
$\kappa_{R\cup Q}(x,y)=\kappa_{R_{xy}}(x,y)+|X|+|Y|,$
where $R_{xy}=R\setminus((X\cup Y)\setminus\\{x,y\\})$. Such a graph $Q$ is
called a source-sink isolating gadget, and moreover, the construction of $Q$
is independent from $R$.
The reason we call the graph $Q$ a source-sink isolating gadget is that by
adding $Q$ into the input graph $R$ the vertex connectivity between any pair
of source $x\in X$ and sink $y\in Y$ in $R\cup Q$, i.e., $\kappa_{R\cup
Q}(x,y)$, can be derived from their connectivity in $R_{xy}$, i.e.,
$\kappa_{R_{xy}}(x,y)$. But the graph $R_{xy}$, as defined in Lemma 3.2, is
just the graph $R$ after removing all source and sink vertices in $X$ and $Y$
except $x$ and $y$. That is, the gadget “isolates” the pair $x$ and $y$ from
the rest. We will use this gadget in Section 3.3.
###### Proof.
We construct $Q$ in the following way. We create duplicated sets $X_{1},X_{2}$
of $X$, and also $Y_{1},Y_{2}$ of $Y$. For each vertex $x\in X$, we let
$\hat{x}_{1}\in X_{1}$ and $\hat{x}_{2}\in X_{2}$ denote copies of $x$ in
$X_{1}$ and $X_{2}$ respectively if there is no other specification (for each
$y\in Y$, $\hat{y}_{1},\hat{y}_{2}$ are defined similarly). The vertex set of
$Q$ is $V(Q)=X\cup X_{1}\cup X_{2}\cup Y\cup Y_{1}\cup Y_{2}$, and the edge
set is
$\displaystyle E(Q)=$ $\displaystyle\\{(x,\hat{x}_{1})\mid x\in
X\\}\cup\\{(x_{1},y)\mid x_{1}\in X_{1},y\in Y\\}\cup$
$\displaystyle\\{(y,\hat{y}_{1})\mid y\in Y\\}\cup\\{(y_{1},x)\mid y_{1}\in
Y_{1},x\in X\\}\cup$ $\displaystyle\\{(x,x_{2})\mid x\in X,x_{2}\in
X_{2}\\}\cup$ $\displaystyle\\{(y,y_{2})\mid y\in Y,y_{2}\in Y_{2}\\}.$
The construction of $Q$ is illustrated in Figure 1.
Figure 1: The source-sink isolating gadget $Q$ and the whole graph $R\cup Q$.
There are bipartite cliques between $X$ and $Y_{1},X_{2}$, as well as $Y$ and
$X_{1},Y_{2}$, and there are perfect matchings between $X$ and $X_{1}$, $Y$
and $Y_{1}$.
Fixing some $x\in X$ and $y\in Y$, we let $\kappa=\kappa_{R_{xy}}(x,y)$ for
short. We first show $\kappa_{R\cup Q}(x,y)\geq\kappa+|X|+|Y|$. From the flow
view of vertex connectivity, there are $\kappa$ internal vertex-disjoint paths
from $x$ to $y$ in $R_{xy}$. Combining 3.3 and $V(Q)\cap V(R_{xy})=\\{x,y\\}$,
there are $\kappa+|X|+|Y|$ internal vertex-disjoint paths in $R\cup Q$.
Therefore, $\kappa_{R\cup Q}(x,y)\geq\kappa+|X|+|Y|$.
###### Claim 3.3.
There are $|X|+|Y|$ internal vertex-disjoint paths from $x$ to $y$ in $Q$.
###### Proof.
The first path is $x\to\hat{x}_{1}\to y$. Each of the next $|X|-1$ paths
corresponds to each $x^{\prime}\in X$ s.t. $x^{\prime}\neq x$, the path
$x\to\hat{x}^{\prime}_{2}\to x^{\prime}\to\hat{x}^{\prime}_{1}\to y$
concretely, where $\hat{x}^{\prime}_{1}$ and $\hat{x}^{\prime}_{2}$ are copies
of $x^{\prime}$ in $X_{1}$ and $X_{2}$. Symmetrically, there is a path $x\to
y_{1}\to y$ and $|Y|-1$ paths, each of which corresponds to each
$y^{\prime}\in Y$ s.t. $y^{\prime}\neq y$ (namely the path
$x\to\hat{y}^{\prime}_{1}\to y^{\prime}\to\hat{y}^{\prime}_{2}\to y$, where
$\hat{y}^{\prime}_{1}$ and $\hat{y}^{\prime}_{2}$ are copies of $y^{\prime}$
in $Y_{1}$ and $Y_{2}$). Observe that these $|X|+|Y|$ paths are internal
vertex-disjoint. ∎
We then argue from the cut view that $\kappa_{R\cup
Q}(x,y)\leq\kappa+|X|+|Y|$. Consider the vertex set $S_{Q}=\\{x^{\prime}\mid
x^{\prime}\in X,x^{\prime}\neq x\\}\cup\\{y^{\prime}\mid y^{\prime}\in
Y,y^{\prime}\neq y\\}\cup\\{\hat{x}_{1},\hat{y}_{1}\\}$. After removing
$S_{Q}$ from $R\cup Q$, observe that vertices in both $R$ and $Q$ are only $x$
and $y$, so a simple path from $x$ to $y$ in graph $(R\cup Q)\setminus S_{Q}$
will be totally inside subgraphs $Q\setminus S_{Q}$ or $R\setminus S_{Q}$.
Note that $x$ and $y$ are disconnected in $Q\setminus S_{Q}$. Moreover,
subgraph $R\setminus S_{Q}$ is exactly $R_{xy}$, so removing $\kappa$ vertices
can disconnect $x$ and $y$ in $R\setminus S_{Q}$. In conclusion, in graph
$R\cup Q$, $x$ and $y$ can be disconnected by removing $|S_{Q}|+\kappa$
vertices, so $\kappa_{R\cup Q}(x,y)\leq\kappa+|X|+|Y|$.
Finally, the size of $Q$ follows directly from the construction.
∎
### 3.2 The Set-Intersection Filter
We now introduce the set-intersection filter. For each $a\in A$, $d\in D$, the
set-intersection filter $P_{ad}^{B}$ is a subgraph of the final $H$, which
will “filter” the intersection $B_{a}\cap B_{d}$ from the whole set $B$ as
Lemma 3.4 shows. It is constructed as follows. Let $V(P_{ad}^{B})=\\{a\\}\cup
B\cup B^{\prime}\cup\\{d\\}$, where $B^{\prime}$ duplicates vertices in $B$.
For each vertex $b\in B$, we use $\hat{b}^{\prime}$ to denote its copy in
$B^{\prime}$, and for each (non-)neighbor sets
$B_{a},\bar{B}_{a},B_{d},\bar{B}_{d}\subseteq B$, we use
$B^{\prime}_{a},\bar{B}^{\prime}_{a},B^{\prime}_{d},\bar{B}^{\prime}_{d}\subseteq
B^{\prime}$ to denote their counterparts respectively. The edge set of
$P_{ad}^{B}$ is constructed by
$\displaystyle E(P_{ad}^{B})=$ $\displaystyle\\{(a,b)\mid b\in
B\\}\cup\\{(b,d)\mid b\in\bar{B}_{d}\\}\cup$
$\displaystyle\\{(a,\hat{b}^{\prime})\mid b\in
B_{a}\\}\cup\\{(b^{\prime},d)\mid b^{\prime}\in B^{\prime}\\}\cup$
$\displaystyle\\{(b,\hat{b}^{\prime})\mid b\in B\\}.$
See Figure 2 for an illustration of $P^{B}_{ad}$.
Figure 2: The set-intersection filter. The sets $P_{ad}^{B}$ and $R$ are the
areas surrounded by dotted lines.
The construction of $E(P^{B}_{ad})$ can be interpreted in the following
intuitive way.
* •
First, the edges $\\{(a,b)\mid b\in B\\}$ and $\\{(b,d)\mid
b\in\bar{B}_{d}\\}$ create vertex-disjoint paths of the format $a\to b\to d$
for all $b\in\bar{B}_{d}$, which implies $\bar{B}_{d}$ will be cut from $B$ in
every vertex cut of $(a,d)$.
* •
Second, the edges $\\{(a,\hat{b}^{\prime})\mid b\in B_{a}\\}$ and
$\\{(b^{\prime},d)\mid b^{\prime}\in B^{\prime}\\}$ create vertex-disjoint
paths of the format $a\to\hat{b}^{\prime}\to d$ for all $b\in B_{a}$, which
analogously implies that $B^{\prime}_{a}$ will be cut from $B^{\prime}$ in
every $(a,d)$-vertex cut.
* •
Third, for every $(a,d)$-vertex cut, after $\bar{B}_{d}$ and $B^{\prime}_{a}$
are cut from $B$ and $B^{\prime}$ respectively from the above discussion,
either $b$ or $\hat{b}^{\prime}$ should be cut for all $b\in
B_{d}\cap\bar{B}_{a}$. The reason is that the edges
$\\{(b,\hat{b}^{\prime})\mid b\in B\\}$ form a matching between $B$ and
$B^{\prime}$, which will create vertex-disjoint paths of the format $a\to
b\to\hat{b}^{\prime}\to d$ for all $b\in B_{d}\cap\bar{B}_{a}$.
Therefore, suppose that in the third step we choose to cut vertex $b$ of all
$b\in B_{d}\cap\bar{B}_{a}$. (In the formal proof of Lemma 3.4, we will see
that, cutting $b$ rather than $\hat{b}^{\prime}$ for all $b\in
B_{d}\cap\bar{B}_{a}$ is always a better choice when considering vertex min
cut between $a$ and $d$.) Now $\bar{B}_{d}$ is cut in the first step and
$B_{d}\cap\bar{B}_{a}$ is cut in the third step, so vertices in $B$ that
survive are $B\setminus\bar{B}_{d}\setminus(B_{d}\cap\bar{B}_{a})=B_{a}\cap
B_{d}$. Therefore, the set-intersection filter indeed obtain $B_{a}\cap B_{d}$
as desired.
###### Lemma 3.4.
For each $a\in A,d\in D$, the set-intersection filter $P^{B}_{ad}$ has the
following property. Let $R$ be an undirected graph such that $V(R)\cap
V(P^{B}_{ad})=\\{a\\}\cup B\cup\\{d\\}$, then
$\kappa_{R\cup
P^{B}_{ad}}(a,d)=\kappa_{R^{B}_{ad}}(a,d)+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|,$
where $R^{B}_{ad}=(R\cup P^{B}_{ad})\setminus((B\setminus(B_{a}\cap
B_{d}))\cup B^{\prime})$ equivalent to $(R\setminus(B\setminus(B_{a}\cap
B_{d})))\cup\\{(a,b)\mid b\in B_{a}\cap B_{d}\\}$, that is, the graph starting
from $R$, removing non $B_{a}\cap B_{d}$ vertices and then adding edges
connecting $a$ and each $b\in B_{a}\cap B_{d}$.
###### Proof.
Let $\kappa=\kappa_{R^{B}_{ad}}(a,d)$ for short. We first show $\kappa_{R\cup
P^{B}_{ad}}(a,d)\geq\kappa+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|$.
There are $\kappa$ internal vertex-disjoint paths from $a$ to $d$ in
$R^{B}_{ad}$. Because $V(P^{B}_{ad})\cap V(R^{B}_{ad})=(B_{a}\cap
B_{d})\cup\\{a,d\\}$, the paths from $a$ to $d$ in
$P^{B}_{ad}\setminus(B_{a}\cap B_{d})$ are internal disjoint with those in
$R^{B}_{ad}$, and there are $|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|$ of
them by 3.5. Therefore, we have
$\kappa+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|$ internal vertex-disjoint
paths from $a$ to $d$ in $R\cup P^{B}_{ad}$.
###### Claim 3.5.
There are $|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|$ internal vertex-
disjoint paths from $a$ to $d$ in $P^{B}_{ad}\setminus(B_{a}\cap B_{d})$.
###### Proof.
The first $|\bar{B}_{d}|$ paths correspond to vertices $b\in\bar{B}_{d}$, each
of which has a path $a\to b\to d$. The next $|B_{a}|$ paths correspond to
vertices $b\in B_{a}$, each of which has a path $a\to\hat{b}^{\prime}\to d$.
The last $|B_{d}\cap\bar{B}_{a}|$ paths correspond to vertices $b\in
B_{d}\cap\bar{B}_{a}$, each of which has a path $a\to b\to\hat{b}^{\prime}\to
d$. ∎
We then complete the proof by showing $\kappa_{R\cup
P^{B}_{ad}}(a,d)\leq\kappa+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|$. Let
$S_{P}=\\{b\in B\mid b\in\bar{B}_{d}\\}\cup\\{\hat{b}^{\prime}\in
B^{\prime}\mid b\in B_{a}\\}\cup\\{b\in B\mid b\in B_{d}\cap\bar{B}_{a}\\}$ be
a vertex cut of $(a,d)$ in $P^{B}_{ad}$. Observe that in $(R\cup
P^{B}_{ad})\setminus S_{P}$, a path from vertex $a$ to a vertex $b^{\prime}\in
B^{\prime}\setminus S_{P}$ must go through vertex $d$, because each
$b^{\prime}\in B^{\prime}\setminus S_{P}$ only connects to $d$ after removing
$S_{P}$. Therefore, it is safe to ignore $B^{\prime}\setminus S_{P}$ when
considering the vertex connectivity between $a$ and $d$ in graph $(R\cup
P^{B}_{ad})\setminus S_{P}$, i.e.
$\kappa_{(R\cup P^{B}_{ad})\setminus S_{P}}(a,d)=\kappa_{(R\cup
P^{B}_{ad})\setminus(S_{P}\cup
B^{\prime})}(a,d)=\kappa_{R^{B}_{ad}}(a,d)=\kappa,$
which means by further removing $\kappa$ vertices, we can disconnect $a$ and
$d$ in $(R\cup P_{ad}^{B})\setminus S_{P}$.
In conclusion, we can remove $|S_{P}|+\kappa$ vertices to disconnect $a$ and
$d$ in $R\cup P^{B}_{ad}$, which implies $\kappa_{R\cup
P^{B}_{ad}}(a,d)\leq\kappa+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|$.
∎
For each $a\in A,d\in D$, we also define the set-intersection filter
$P^{C}_{ad}$ similarly but symmetrically. Let $V(P^{C}_{ad})=\\{a\\}\cup
C^{\prime}\cup C\cup\\{d\\}$, where $C^{\prime}$ duplicates vertices in $C$.
For each $c\in C$, let $\hat{c}^{\prime}$ denote its copy in $C^{\prime}$. For
each (non-)neighbor sets $C_{a},\bar{C}_{a},C_{d},\bar{C}_{d}\subseteq C$, let
$C^{\prime}_{a},\bar{C}^{\prime}_{a},C^{\prime}_{d},\bar{C}^{\prime}_{d}\subseteq
C^{\prime}$ denote their counterparts respectively. The edge set is
$\displaystyle E(P^{C}_{ad})=$ $\displaystyle\\{(a,c)\mid
c\in\bar{C}_{a}\\}\cup\\{(c,d)\mid c\in C\\}\cup$
$\displaystyle\\{(a,c^{\prime})\mid c^{\prime}\in
C^{\prime}\\}\cup\\{(\hat{c}^{\prime},d)\mid c\in C_{d}\\}\cup$
$\displaystyle\\{(\hat{c}^{\prime},c)\mid c\in C\\}.$
The pattern $P^{C}_{ad}$ will also have similar properties as shown below.
###### Lemma 3.6.
For each $a\in A,d\in D$, the set-intersection filter $P^{C}_{ad}$ has the
following property. Let $R$ be an undirected graph such that $V(R)\cap
V(P^{C}_{ad})=\\{a\\}\cup C\cup\\{d\\}$, then
$\kappa_{R\cup
P^{C}_{ad}}(a,d)=\kappa_{R^{C}_{ad}}(a,d)+|C_{d}|+|\bar{C}_{a}|+|C_{a}\cap\bar{C}_{d}|,$
where $R^{C}_{ad}=(R\cup P^{C}_{ad})\setminus((C\setminus(C_{a}\cap
C_{d}))\cup C^{\prime})$ equivalent to $(R\setminus(C\setminus(C_{a}\cap
C_{d})))\cup\\{(c,d)\mid c\in C_{a}\cap C_{d}\\}$, that is, the graph starting
from $R$, removing non $C_{a}\cap C_{d}$ vertices and then adding edges
connecting $d$ and each $c\in C_{a}\cap C_{d}$.
### 3.3 The Final Construction of the APVC Instance
We are now ready to construct the final APVC instance $H$. For each $a\in
A,d\in D$, we first construct a graph $H_{ad}$ as follows. Let $P^{B}_{ad}$
and $P^{C}_{ad}$ be the set-intersection filters defined in Section 3.2. Then
the graph $H_{ad}$ will be defined by
$H_{ad}=P^{B}_{ad}\cup P^{C}_{ad}\cup G_{4p}[B\cup C],$
which is the union of two set-intersection filters with edges in the graph
$G_{4p}$ connecting $B$ and $C$. The final graph then will be constructed by
$H=\bigcup_{a\in A,d\in D}H_{ad}\cup Q(A,D),$
where $Q(A,D)$ is the source-sink isolating gadget from Lemma 3.2 given
$R=\bigcup_{a\in A,d\in D}H_{ad}$ and the sets $A,D$. See Figure 3 below for
an illustration.
Figure 3: The graph $H$. The edges connecting $B$ and $C$ (i.e. $G_{4p}[B\cup
C]$) are omitted in the highlighted part.
###### Lemma 3.7.
For each $a\in A$ and $d\in D$ of $G_{4p}$, we have
$\kappa_{H}(a,d)\geq 4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap
N_{G}(d)|.$ (1)
Furthermore, without ambiguity let $a$ and $d$ also denote their original
vertices in $G$. Then there is a 4-clique in $G$ containing $a$ and $d$ if and
only if $a$ and $d$ are adjacent in $G_{4p}$ and
$\kappa_{H}(a,d)\geq 4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap
N_{G}(d)|+1.$ (2)
###### Proof.
Fixing some $a\in A,d\in D$, we first apply Lemma 3.2 on $H$ and vertex sets
$A,D$, which gives $\kappa_{H}(a,d)=\kappa_{R_{ad}}(a,d)+|A|+|D|$, where
$R_{ad}=(\bigcup_{a\in A,d\in D}H_{ad})\setminus((A\cup
D)\setminus\\{a,d\\})$. In fact, the graph $R_{ad}$ is exactly $H_{ad}$,
because the induced subgraphs $H_{ad}[B\cup B^{\prime}\cup C\cup C^{\prime}]$
are the same for all $a\in A,d\in D$ by construction. Therefore, we have
$\kappa_{H}(a,d)=\kappa_{H_{ad}}(a,d)+|A|+|D|.$ (3)
Recall that $H_{ad}=P^{B}_{ad}\cup P^{C}_{ad}\cup G_{4p}[B\cup C]$. Let
$R_{1}=P^{C}_{ad}\cup G_{4p}[B\cup C]$. We apply Lemma 3.4 on $P^{B}_{ad}$ and
$R_{1}$, which gives
$\kappa_{H_{ad}}(a,d)=\kappa_{R_{1}\cup
P^{B}_{ad}}(a,d)=\kappa_{R^{\prime}_{1}}(a,d)+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|,$
(4)
where
$\displaystyle R^{\prime}_{1}$
$\displaystyle=(R_{1}\setminus(B\setminus(B_{a}\cap B_{d})))\cup\\{(a,b)\mid
b\in B_{a}\cap B_{d}\\}$ $\displaystyle=P^{C}_{ad}\cup G_{4p}[(B_{a}\cap
B_{d})\cup C]\cup\\{(a,b)\mid b\in B_{a}\cap B_{d}\\}.$
Let $R_{2}=G_{4p}[(B_{a}\cap B_{d})\cup C]\cup\\{(a,b)\mid b\in B_{a}\cap
B_{d}\\}$. We apply Lemma 3.6 on $P^{C}_{ad}$ and $R_{2}$, which gives
$\kappa_{R^{\prime}_{1}}(a,d)=\kappa_{R_{2}\cup
P^{C}_{ad}}(a,d)=\kappa_{R^{\prime}_{2}}(a,d)+|C_{d}|+|\bar{C}_{a}|+|C_{a}\cap\bar{C}_{d}|,$
(5)
where
$\displaystyle R^{\prime}_{2}=$
$\displaystyle(R_{2}\setminus(C\setminus(C_{a}\cap C_{d})))\cup\\{(c,d)\mid
c\in C_{a}\cap C_{d}\\}$ $\displaystyle=$ $\displaystyle G_{4p}[(B_{a}\cap
B_{d})\cup(C_{a}\cap C_{d})]\cup\\{(a,b)\mid b\in B_{a}\cap B_{d}\\}\cup$
$\displaystyle\\{(c,d)\mid c\in C_{a}\cap C_{d}\\}.$
We use $\hat{H}_{ad}$ to denote $R^{\prime}_{2}$ in the remaining proof and
note that it is equivalent to the definition of $\hat{H}_{ad}$ in Section 1.2.
Because $\kappa_{\hat{H}_{ad}}(a,d)\geq 0$, combining Equations 3, 4 and 5, we
get
$\kappa_{H}(a,d)\geq|A|+|D|+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|+|C_{d}|+|\bar{C}_{a}|+|C_{a}\cap\bar{C}_{d}|.$
(6)
We now prove the second part of the lemma. If $a$ and $d$ are not adjacent in
$G_{4p}$, they are not adjacent in $G$ either, so there is no 4-clique in $G$
containing them. Otherwise, $a$ and $d$ are adjacent, we claim that there is a
4-clique containing $a$ and $d$ in $G_{4p}$ (which is equivalent to the
existence of a 4-clique containing $a$ and $d$ in $G$ by Definition 2.2) if
and only if $\kappa_{\hat{H}_{ad}}(a,d)\geq 1$. If the 4-clique exists, let
$b\in B$ and $c\in C$ be the other vertices in the 4-clique. Then there is a
path $a\to b\to c\to d$ in $\hat{H}_{ad}$ from the construction, which implies
$\kappa_{\hat{H}_{ad}}(a,d)\geq 1$. On the other hand, if
$\kappa_{\hat{H}_{ad}}(a,d)\geq 1$, there is a path $a\to b\to c\to d$, and
$(a,b,c,d)$ forms a 4-clique in $G_{4p}$. Combining this claim with Equations
3, 4 and 5 gives that there is a 4-clique containing $a$ and $d$ in $G$ if and
only if
$\kappa_{H}(a,d)\geq|A|+|D|+|\bar{B}_{d}|+|B_{a}|+|B_{d}\cap\bar{B}_{a}|+|C_{d}|+|\bar{C}_{a}|+|C_{a}\cap\bar{C}_{d}|+1.$
(7)
Finally, by the construction of $G_{4p}$, we have $|A|=|B|=|C|=|D|=n$,
$|B_{a}|=|C_{a}|$, $|B_{d}|=|C_{d}|$,
$|B_{d}\cap\bar{B}_{a}|=|N_{G}(d)\cap\bar{N}_{G}(a)|$ and
$|C_{a}\cap\bar{C}_{d}|=|N_{G}(a)\cap\bar{N}_{G}(d)|$. Combining them with
Inequalities (6) and (7) completes the proof.
∎
###### Proof of Theorem 3.1.
Assume for contradiction that there exists a combinatorial algorithm
$\mathcal{A}$ for the APVC problem with running time $O(n^{4-\epsilon})$ for
some constant $\epsilon>0$. Let $G$ be an arbitrary 4-clique instance. We
first construct the 4-partite graph $G_{4p}$ and the graph $H$ following the
construction in this section. Note that $V(H)=V(Q)\cup A\cup B\cup
B^{\prime}\cup C\cup C^{\prime}\cup D$, so $|V(H)|=O(n)$ by the construction
and Lemma 3.2. Also, $H$ can be constructed in $O(n^{2})$ time directly. By
Equation 2, we can solve the 4-clique problem by first running $\mathcal{A}$
on graph $H$ and then checking for each adjacent $a\in A,d\in D$ whether
$\kappa_{H}(a,d)$ reaches the threshold value
$4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap N_{G}(d)|+1$. This takes
$O(n^{4-\epsilon})+O(n^{3})=O(n^{4-\epsilon})$ time because computing the
threshold value for each $a\in A,d\in D$ takes totally $O(n^{3})$ extra time.
This contradicts 2.1.
∎
###### Remark 3.8.
The proof of Theorem 3.1 basically shows that if the APVC problem can be
solved in time $T_{\mathrm{APVC}}(n)$, then the 4-clique problem can be solved
in time $T_{\mathrm{4clique}}(n)=O(T_{\mathrm{APVC}}(n)+n^{3})$. We note that
this relation can be improved to
$T_{\mathrm{4clique}}(n)=O(T_{\mathrm{APVC}}(n)+n^{\omega})$ if we use fast
matrix multiplication to speed up the reduction. For general algorithms,
another version of the 4-clique conjecture (see e.g. [DW22]) states that
solving the 4-clique problem requires $n^{\omega(1,2,1)-o(1)}$ time.
Therefore, assuming this conjecture, solving the APVC problem also requires
$n^{\omega(1,2,1)-o(1)}$ time for general algorithms.
### 3.4 Further Results
In this section, we will show several corollaries from the lower bound of the
APVC problem.
The first corollary is a conditional lower bound of the SSVC problem.
###### Corollary 3.9.
Assuming 2.1, for $n$-vertex undirected unweighted graphs, there is no
combinatorial algorithm that solves the SSVC problem in $O(n^{3-\epsilon})$
time for any constant $\epsilon>0$.
###### Proof.
Assume for the contradiction that the SSVC problem can be solved in
$O(n^{3-\epsilon})$ time. Then for an APVC instance $G$, we may treat every
vertex in $G$ as a source to get the correct output. It takes $O(n)$ SSVC
calls and therefore the complexity is $O(n\cdot
n^{3-\epsilon})=O(n^{4-\epsilon})$, which contradicts Theorem 3.1 assuming
2.1. ∎
The second corollary is a conditional lower bound of the APVC problem for
graphs with general density.
###### Corollary 3.10.
Given any constant $\delta\in[0,1]$, assuming 2.1, there is no combinatorial
algorithm that solves the APVC problem for $n$-vertex $m$-edge unweighted
graphs, where $m=\Theta(n^{1+\delta})$ with running time $O(m^{2-\epsilon})$
for any constant $\epsilon>0$.
###### Proof.
Assume that for some constants $\delta$ and $\epsilon$, such
$O(m^{2-\epsilon})$-time algorithm $\mathcal{A}$ exists. Let $H$ be an
$\hat{n}$-vertex $\hat{m}$-edge APVC hard instance with
$\hat{m}=\Theta(\hat{n}^{2})$, constructed as above for some 4-clique
instance. Let $G$ be the union of $H$ and $\Theta(\hat{m}^{1/(1+\delta)})$
isolated vertices. Observe that $G$ now has
$n=\hat{n}+\Theta(\hat{m}^{1/(1+\delta)})$ vertices and $m=\hat{m}$ edges,
i.e. $m=\Theta(n^{1+\delta})$. By applying algorithm $\mathcal{A}$ on $G$, the
all-pairs vertex connectivity of $H$ can be computed in $O(m^{2-\epsilon})$,
i.e. $O(\hat{n}^{4-2\epsilon})$ time, contradicting 2.1 by the argument in the
proof of Theorem 3.1. ∎
The last corollary is a conditional lower bound of the SSVC problem for graphs
with general density. The proof is omitted since it is analogous to Corollary
3.10.
###### Corollary 3.11.
Given any constant $\delta\in[0,1]$, assuming 2.1, there is no combinatorial
algorithm that solves the SSVC problem for $n$-vertex $m$-edge unweighted
graphs, where $m=\Theta(n^{1+\delta})$, with running time
$O(m^{3/2-\epsilon})$ for any constant $\epsilon>0$.
## 4 The Lower Bound for Steiner Vertex Connectivity Problem
In this section, we will prove Theorem 4.1, a conditional lower bound of the
Steiner vertex connectivity problem in undirected unweighted graphs,
conditioning on the edge-universal 4-clique problem.
###### Theorem 4.1.
For $n$-vertex undirected unweighted graphs, assuming 2.3, there is no
combinatorial algorithm that solves the Steiner vertex connectivity problem in
$O(n^{4-\epsilon})$ time for any constant $\epsilon>0$.
Given an $n$-vertex edge-universal 4-clique instance $G$ with a set
$E_{\mathrm{dem}}$ of _demand_ edges, let $H$ be the hard APVC instance we
construct in Section 3. We will strengthen $H$ to another graph $J$ such that
the edge-universal 4-clique problem in $G$ can be reduced to a Steiner vertex
connectivity problem in $J$ of the terminal set $A\cup D$. As mentioned in
Section 1.2, the main idea is to add extra “flow” paths from $A$ to $D$ to
make the additive deviations between $\kappa_{J}(a,d)$ and
$\kappa_{\hat{H}_{ad}}(a,d)$ uniform for all pairs of $a\in A$ and $d\in D$.
Furthermore, to exclude the interference of vertex connectivity of pairs
$(a_{1},a_{2})$ s.t. $a_{1},a_{2}\in A$ or $(d_{1},d_{2})$ s.t.
$d_{1},d_{2}\in D$, we artificially add large additive deviations for these
pairs.
The construction of $J$ is as follows and see Figure 4 for an illustration.
The vertex set $V(J)=V(H)\cup Z\cup W\cup A^{\prime}\cup D^{\prime}$, where
$Z,W,A^{\prime},D^{\prime}$ are disjoint groups of additional vertices to
create extra “flow” paths. Concretely, $Z$ and $W$ will be copies of original
vertex set $V(G)$, and $A^{\prime}$ and $D^{\prime}$ are additional sets of
vertices of size $|A^{\prime}|=|D^{\prime}|=10n$. Let
$\displaystyle E_{Z}=$ $\displaystyle\\{(a,z)|a\in A,z\in Z,(a,z)\in
E(G)\\}\cup$ $\displaystyle\\{(d,z)|d\in D,z\in Z,(d,z)\in E(G)\\}$
and
$\displaystyle E_{W}=$ $\displaystyle\\{(a,w)|a\in A,w\in W,(a,w)\not\in
E(G)\\}\cup$ $\displaystyle\\{(d,w)|d\in D,w\in W,(d,w)\not\in E(G)\\}$
be the extra edges to “equalize the deviation” of all pairs $(a,d)$ between
$A$ and $D$. Let
$E_{A^{\prime}}=\\{(a,a^{\prime})\mid a\in A,a^{\prime}\in A^{\prime}\\}$
and
$E_{D^{\prime}}=\\{(d,d^{\prime})\mid d\in D,d^{\prime}\in D^{\prime}\\}$
be extra edges that bring large deviations to pairs inside $A$ or $D$.
Finally, we construct a set of extra edges
$E_{AD}=\\{(a,d)\mid a\in A,d\in D,(a,d)\notin E_{\mathrm{dem}}\\}$
to prevent non-demand pairs $(V(G)\times V(G))\setminus E_{\mathrm{dem}}$ from
affecting the Steiner connectivity value. The whole edge set $E(J)=V(H)\cup
E_{Z}\cup E_{W}\cup E_{A^{\prime}}\cup E_{D^{\prime}}\cup E_{AD}$.
Figure 4: The graph $J$. There are bipartite cliques between $A$ and
$A^{\prime}$, $D$ and $D^{\prime}$, and the edges between $A,D$ and $Z,W$ are
linked based on $E_{Z},E_{W}$.
The correctness of the reduction will be established by the following lemmas.
###### Lemma 4.2.
In graph $J$, for each $a_{1},a_{2}\in A$ with $a_{1}\neq a_{2}$ and
$d_{1},d_{2}\in D$ with $d_{1}\neq d_{2}$, we have
$\kappa_{J}(a_{1},a_{2})\geq 10n$ and $\kappa_{J}(d_{1},d_{2})\geq 10n$.
###### Proof.
For each pair $a_{1},a_{2}\in A$ with $a_{1}\neq a_{2}$, we can construct at
least $10n$ internally vertex-disjoint paths from $a_{1}$ to $a_{2}$ via
vertices in $A^{\prime}$ and edges in $E_{A^{\prime}}$, so
$\kappa_{J}(a_{1},a_{2})\geq 10n$. Similarly, $\kappa_{J}(d_{1},d_{2})\geq
10n$.
∎
###### Lemma 4.3.
In graph $J$, for each $a\in A$, $d\in D$ with $(a,d)\not\in
E_{\mathrm{dem}}$, $\kappa_{J}(a,d)\geq 5n+1$.
###### Proof.
From our construction, the number of nodes in $Z$ adjacent to both $a$ and $d$
is
$|\\{z\in Z|(a,z)\in E_{Z},(d,z)\in E_{Z}\\}|=|N_{G}(a)\cap N_{G}(d)|,$
and the number of nodes in $W$ adjacent to both $a$ and $d$ is
$|\\{w\in W|(a,w)\in E_{Z},(d,w)\in
E_{Z}\\}|=|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|.$
Furthermore, there is an individual edge in $E_{AD}$ directly connecting $a$
and $d$ for each such $(a,d)\notin E_{\mathrm{dem}}$.
Therefore, we have $|N_{G}(a)\cap
N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|+1$ extra vertex-disjoint paths
from $a$ to $d$. Furthermore, Inequality (1) from Equation 2 shows that
$\kappa_{H}(a,d)\geq 4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap
N_{G}(d)|$, so we can construct a vertex-disjoint path set with size at least
$\displaystyle 4n$
$\displaystyle+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap N_{G}(d)|$
$\displaystyle+|N_{G}(a)\cap
N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|+1=5n+1,$
which implies $\kappa_{J}(a,d)\geq 5n+1$.
∎
###### Lemma 4.4.
In graph $J$, for each $a\in A$, $d\in D$ with $(a,d)\in E_{\mathrm{dem}}$,
there is a 4-clique containing $a$ and $d$ in $G$ if and only if
$\kappa_{J}(a,d)\geq 5n+1$.
###### Proof.
First, assume there is a 4-clique containing $a$ and $d$ in $G$. From Equation
2, we can construct $4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap
N_{G}(d)|+1$ internally vertex-disjoint paths connecting $a$ and $d$ in $H$.
An argument similar to the proof of Lemma 4.3 shows that we have extra
$|N_{G}(a)\cap N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|$ internally
vertex-disjoint paths via vertices in $Z,W$ and edges in $E_{Z},E_{W}$.
Therefore, in graph $J$, we can find a set of internally vertex-disjoint paths
from $a$ to $d$ with size
$\displaystyle 4n$
$\displaystyle+|N_{G}(d)\cap\bar{N_{G}}(a)|+|N_{G}(a)\cap\bar{N_{G}}(d)|+1$
$\displaystyle+|N_{G}(a)\cap
N_{G}(d)|+|\bar{N_{G}}(a)\cap\bar{N_{G}}(d)|=5n+1.$
Now assume there is no 4-clique containing $a$ and $d$ in $G$. From Lemma 3.2,
we define $J_{ad}=J\setminus((A\cup D)\setminus\\{a,d\\})$ by removing
vertices in $A$ and $D$ other than $a$ and $d$, and then we have
$\kappa_{J}(a,d)=\kappa_{J_{ad}}(a,d)+|A|+|D|.$ (8)
Note that although the source-sink isolating gadget $Q(A,D)$ is constructed by
applying Lemma 3.2 on graph $H$ and vertex sets $A,D$, the construction of
$Q(A,D)$ is independent from $H$, so we can also apply Lemma 3.2 on graph $J$
and vertex sets $A,D$.
Next, we will show
$\kappa_{J_{ad}}(a,d)\leq\kappa_{H_{ad}}(a,d)+|N_{G}(a)\cap
N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|,$ (9)
where $H_{ad}=H\setminus((A\cup D)\setminus\\{a,d\\})$ as defined in the proof
of Equation 2. The reason is that, compared to $H_{ad}$, the extra vertices in
$J_{ad}$ are in $A^{\prime},D^{\prime},Z,W$, which are only directly connected
to $a$ or $d$. Let $C_{H}$ be the minimum vertex cut of size
$|C_{H_{ad}}|=\kappa_{H_{ad}}(a,d)$ whose removal disconnects $a$ and $d$ in
$H_{ad}$. Let
$\displaystyle C_{J_{ad}}=$ $\displaystyle C_{H_{ad}}\cup\\{z\in Z\mid(a,z)\in
E_{Z},(d,z)\in E_{Z}\\}\cup$ $\displaystyle\\{w\in W\mid(a,w)\in
E_{Z},(d,w)\in E_{Z}\\}.$
We can easily observe that removing $C_{J_{ad}}$ will disconnect $a$ and $d$
in $J_{ad}$, which immediately implies Inequality (9).
Finally, combining Equation (8) and Inequality (9), we have
$\displaystyle\kappa_{J}(a,d)\leq$
$\displaystyle\kappa_{H_{ad}}(a,d)+|A|+|D|+|N_{G}(a)\cap N_{G}(d)|$
$\displaystyle+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|.$
From Equation 2, we know that when there is no 4-clique containing $a$ and $d$
in $G$, $\kappa_{H}(a,d)=4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap
N_{G}(d)|$. Combining $\kappa_{H}(a,d)=\kappa_{H_{ad}}(a,d)+|A|+|D|$ (by Lemma
3.2 again),
$\displaystyle\kappa_{J}(a,d)\leq$ $\displaystyle\kappa_{H_{ad}}(a,d)+|A|+|D|$
$\displaystyle+|N_{G}(a)\cap N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|$
$\displaystyle=$ $\displaystyle\kappa_{H}(a,d)+|N_{G}(a)\cap
N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|$ $\displaystyle=$ $\displaystyle
4n+|N_{G}(a)\cap\bar{N}_{G}(d)|+|\bar{N}_{G}(a)\cap N_{G}(d)|$
$\displaystyle+|N_{G}(a)\cap N_{G}(d)|+|\bar{N}_{G}(a)\cap\bar{N}_{G}(d)|$
$\displaystyle=$ $\displaystyle 5n<5n+1.$
∎
We are now ready to prove Theorem 4.1.
###### Proof of Theorem 4.1.
Given an edge-universal 4-clique instance $G$ with a set $E_{\mathrm{dem}}$ of
demand edges. We first construct the Steiner vertex connectivity instance $J$
with terminal set $A\cup D$ as shown above. Then $G$ with $E_{\mathrm{dem}}$
is a yes instance of the edge-universal 4-clique problem if and only if the
Steiner vertex connectivity of $A\cup D$ in $J$ is at least $5n+1$, by Lemmas
4.2, 4.3 and 4.4.
For time analysis, note that $V(J)=O(n)$ by construction. Therefore, an
$O(n^{4-\epsilon})$-time combinatorial algorithm for the Steiner vertex
connectivity problem will imply an $O(n^{4-\epsilon})$ time combinatorial
algorithm for the edge-universal 4-clique problem, which contradicts 2.3.
∎
## 5 The Upper Bounds
In this section, we will show the upper bounds of the APVC problem and SSVC
problem in undirected unweighted sparse graphs (see Theorem 5.1 and Theorem
5.2 respectively). We only prove Theorem 5.1 in detail and briefly mention the
proof of Theorem 5.2 since they are analogous.
###### Theorem 5.1.
Given an undirected unweighted graph $G$ with $n$ vertices and $m$ edges,
there is a randomized algorithm that solves the APVC problem in
$\hat{O}(m^{\frac{11}{5}})$ time with high probability. Furthermore, assuming
2.5, the running time of this algorithm becomes $\hat{O}(m^{2})$.
###### Theorem 5.2.
Given an undirected unweighted graph $G$ with $n$ vertices, $m$ edges and a
source vertex, there is a randomized algorithm to solve the SSVC problem in
$\hat{O}(m^{\frac{5}{3}})$ time with high probability. Furthermore, assuming
2.5, the running time becomes $\hat{O}(m^{1.5})$.
A key subroutine to obtain the algorithm in Theorem 5.1 is the subroutine
shown in Lemma 5.3, which can capture all vertex connectivity bounded by $k$.
The proof of Lemma 5.3 has been shown in [IN12, PSY22] and we defer it to
Section 5.1 for completeness.
###### Lemma 5.3.
Given an $n$-vertex $m$-edge undirected unweighted graph $G$ and a threshold
$k$, there is a randomized algorithm that computes the value of
$\min\\{\kappa_{G}(u,v),k\\}$ for all vertex pairs $(u,v)$ in
$\hat{O}(mk^{3}+n^{2})$ time with high probability. Furthermore, assuming 2.5,
the running time of this algorithm becomes $\hat{O}(mk^{2}+n^{2})$.
We now complete the proof of Theorem 5.1 using Lemma 5.3.
###### Proof of Theorem 5.1.
As discussed in Section 1.2, our APVC algorithm will handle vertex
connectivity between high-degree vertices and vertex connectivity involving
low-degree vertices separately. Let $k$ be a degree threshold which will be
chosen later. Given an input graph $G$, let $V_{h}=\\{v\in
V(G)\mid\deg_{G}(v)>k\\}$ and $V_{l}=\\{v\in V(G)\mid\deg_{G}(v)\leq k\\}$.
Note that for each vertex pair $(u,v)$ such that $u\in V_{l}$ or $v\in V_{l}$,
the vertex connectivity $\kappa_{G}(u,v)$ is at most $k$. By Lemma 5.3, the
vertex connectivity of all such pairs can be computed exactly in time
$\hat{O}(mk^{3}+n^{2})$. Now consider all remaining pairs $(u,v)$ with $u\in
V_{h}$ and $v\in V_{h}$. Since $|V_{h}|\leq O(m/k)$ by definition, there are
$O(m^{2}/k^{2})$ remaining pairs. The vertex connectivity of each pair can be
computed in $\hat{O}(m)$ time using one max flow call [CKL+22], so totally
$\hat{O}(m^{3}/k^{2})$ time suffices for remaining pairs.
The total running time of the above algorithm is then
$\hat{O}(mk^{3}+n^{2}+m^{3}/k^{2})$, by choosing $k=\Theta(m^{2/5})$, the
running time will be $\hat{O}(m^{11/5})$. Assuming 2.5, the running time will
be improved to $\hat{O}(mk^{2}+n^{2}+m^{3}/k^{2})$, which is $\hat{O}(m^{2})$
by choosing $k=\Theta(m^{1/2})$. ∎
The proof of Theorem 5.2 is analogous and we sketch it below.
###### Proof of Theorem 5.2.
Let $s\in V(G)$ be the source. Similarly, we partition $V(G)$ into two set
$V_{h}=\\{v\in V(G)\mid\deg_{G}(v)>k\\}$ and $V_{l}=\\{v\in
V(G)\mid\deg_{G}(v)\leq k\\}$ where $k$ is the degree threshold. The vertex
connectivity $\kappa_{G}(s,v)$ for all pairs $(s,v)$ such that $v\in V_{l}$
can be computed in $\hat{O}(mk^{2}+n)$ time, using a subroutine analogous to
Lemma 5.3. The vertex connectivity of remaining pairs can be computed in
$\hat{O}(m^{2}/k)$ by trivially calling max flows.
The total running time is $\hat{O}(mk^{2}+n+m^{2}/k)$, which will be
$\hat{O}(m^{5/3})$ by choosing $k=\Theta(m^{1/3})$. Assuming 2.5, the running
time is $\hat{O}(mk+n+m^{2}/k)$, which will be $\hat{O}(m^{3/2})$ by choosing
$k=\Theta(m^{1/2})$.
∎
### 5.1 Proof of Lemma 5.3
The algorithm and analysis follow the ideas in [IN12].
The algorithm is as follows. First, we create $t=O(k^{2}\log n)$ sample sets
$U_{1},...,U_{t}$, each of which is generated by sampling each vertex in
$V(G)$ independently with probability $1/k$. Moreover, for each set $U_{i}$,
we construct a $k$-Gomory-Hu tree for element connectivity using Theorem 2.4.
From Theorem 2.4, for each set $U_{i}$ and each pair $u,v\in U_{i}$, we can
query $a_{i}(u,v)=\min\\{\kappa^{\prime}_{G,U_{i}}(u,v),k\\}$ in nearly
constant time. Then for each $u,v\in V(G)$, we let the final output be
$a(u,v)=\min\\{a_{i}(u,v)\mid u,v\in U_{i}\\}$.
We then analyze the running time. The time to construct all $k$-Gomory-Hu
trees is $\hat{O}(t\cdot mk)=\hat{O}(mk^{3})$. To compute all $a_{i}(u,v)$ and
final output $a(u,v)$, each set $U_{i}$ will have size $\tilde{O}(n/k)$ w.h.p.
by Chernoff bound, so the time is
$\tilde{O}(t\cdot(n/t)^{2})=\tilde{O}(n^{2})$ (by aborting the algorithm when
some $U_{i}$ has size not bounded by $\tilde{O}(n/k)$). Therefore, the total
running time is $\hat{O}(mk^{3}+n^{2})$.
The correctness is shown as follows. Consider a fixed $u,v\in V(G)$. First,
$a(u,v)$ is well-defined with high probability, because for one sample set
$U_{i}$, $u$ and $v$ are inside $U_{i}$ and $a_{i}(u,v)$ is well-defined with
probability $1/k^{2}$ and there are $O(k^{2}\log n)$ sample sets. Given that
$a(u,v)$ is well-defined, we know $a(u,v)\geq\min\\{\kappa_{G}(u,v),k\\}$
since $\kappa^{\prime}_{G,U_{i}}(u,v)\geq\kappa_{G}(u,v)$ for all $U_{i}$ by
the definition of element connectivity. By the same reason, if
$\kappa_{G}(u,v)\geq k$, we must have $a(u,v)=k=\min\\{\kappa_{G}(u,v),k\\}$
which is the correct answer. From now we suppose $\kappa_{G}(u,v)<k$ and we
are going to show there exists $U_{i}$ such that $u,v\in U_{i}$ and
$a_{i}(u,v)=\kappa_{G}(u,v)$ with high probability. Note that
$\kappa^{\prime}_{G,U_{i}}(u,v)=\kappa$ if $U_{i}$ is disjoint with some
minimum $u$-$v$ vertex cut $C_{u,v}$. From our sampling strategy, $U_{i}$
contains $u,v$ and is disjoint with $C_{u,v}$ with probability
$\frac{1}{k^{2}}(1-\frac{1}{k})^{k}=\Omega(1/k^{2})$. Because there are
$O(k^{2}\log n)$ sample set, $U_{i}$ exists with high probability.
If we assuming 2.5, we simply subsitute $k$-Gomory-Hu tree for element
connectivity with element connectivity Gomory-Hu tree and follow the same
algorithm and analysis. We will obtain an algorithm with running time
$\hat{O}(mk^{2}+n^{2})$.
## References
* [ABHS22] Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. Scheduling lower bounds via AND subset sum. J. Comput. Syst. Sci., 127:29–40, 2022.
* [AGI+19] Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemyslaw Uznanski, and Daniel Wolleb-Graf. Faster algorithms for all-pairs bounded min-cuts. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 7:1–7:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
* [AKL+22] Amir Abboud, Robert Krauthgamer, Jason Li, Debmalya Panigrahi, Thatchaphol Saranurak, and Ohad Trabelsi. Breaking the cubic barrier for all-pairs max-flow: Gomory-hu tree in nearly quadratic time. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 884–895. IEEE, 2022.
* [AKT20] Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. New algorithms and lower bounds for all-pairs max-flow in undirected graphs. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 48–61. SIAM, 2020.
* [AKT21a] Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. APMF < apsp? gomory-hu tree for unweighted graphs in almost-quadratic time. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1135–1146. IEEE, 2021.
* [AKT21b] Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. Subcubic algorithms for gomory-hu tree in unweighted graphs. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1725–1737. ACM, 2021.
* [AWW16] Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377–391. SIAM, 2016.
* [Ben95] András A. Benczúr. Counterexamples for directed and node capacitated cut-trees. SIAM J. Comput., 24(3):505–510, 1995.
* [BGL17] Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A dichotomy for regular expression membership testing. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 307–318. IEEE Computer Society, 2017.
* [CKL+22] Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 612–623. IEEE, 2022.
* [CKM+11] Paul F. Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 273–282. ACM, 2011.
* [CRX15] Chandra Chekuri, Thapanapong Rukkanchanunt, and Chao Xu. On element-connectivity preserving graph simplification. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, volume 9294 of Lecture Notes in Computer Science, pages 313–324. Springer, 2015.
* [DW22] Mina Dalirrooyfard and Virginia Vassilevska Williams. Induced cycles and paths are harder than you think. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 531–542. IEEE, 2022.
* [FF56] L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399–404, 1956.
* [GH61] R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551–570, 1961.
* [GIKW19] Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and Ryan Williams. Completeness for first-order properties on sparse structures with algorithmic applications. ACM Trans. Algorithms, 15(2):23:1–23:35, 2019.
* [GR98] Andrew V. Goldberg and Satish Rao. Beyond the flow decomposition barrier. J. ACM, 45(5):783–797, 1998.
* [HRG00] Monika Rauch Henzinger, Satish Rao, and Harold N. Gabow. Computing vertex connectivity: New bounds from old techniques. J. Algorithms, 34(2):222–250, 2000.
* [IN12] Rani Izsak and Zeev Nutov. A note on labeling schemes for graph connectivity. Inf. Process. Lett., 112(1-2):39–43, 2012.
* [Kar00] David R. Karger. Minimum cuts in near-linear time. J. ACM, 47(1):46–76, 2000.
* [Kle69] D. Kleitman. Methods for investigating connectivity of large graphs. IEEE Transactions on Circuit Theory, 16(2):232–233, 1969.
* [KLOS14] Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 217–226. SIAM, 2014.
* [LLW88] Nathan Linial, László Lovász, and Avi Wigderson. Rubber bands, convex embeddings and graph connectivity. Comb., 8(1):91–102, 1988.
* [LNP+21] Jason Li, Danupon Nanongkai, Debmalya Panigrahi, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Vertex connectivity in poly-logarithmic max-flows. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 317–329. ACM, 2021.
* [LP20] Jason Li and Debmalya Panigrahi. Deterministic min-cut in poly-logarithmic max-flows. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 85–92. IEEE, 2020.
* [LP21] Jason Li and Debmalya Panigrahi. Approximate gomory-hu tree is faster than _n_ \- 1 max-flows. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1738–1748. ACM, 2021.
* [LPS21] Jason Li, Debmalya Panigrahi, and Thatchaphol Saranurak. A nearly optimal all-pairs min-cuts algorithm in simple graphs. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1124–1134. IEEE, 2021.
* [Mad16] Aleksander Madry. Computing maximum flow with augmenting electrical flows. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 593–602. IEEE Computer Society, 2016.
* [PSY22] Seth Pettie, Thatchaphol Saranurak, and Longhui Yin. Optimal vertex connectivity oracles. In Stefano Leonardi and Anupam Gupta, editors, STOC ’22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 \- 24, 2022, pages 151–161. ACM, 2022.
* [She13] Jonah Sherman. Nearly maximum flows in nearly linear time. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 263–269. IEEE Computer Society, 2013.
* [Vas09] Virginia Vassilevska. Efficient algorithms for clique problems. Inf. Process. Lett., 109(4):254–257, 2009.
* [vdBLN+20] Jan van den Brand, Yin Tat Lee, Danupon Nanongkai, Richard Peng, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. Bipartite matching in nearly-linear time on moderately dense graphs. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 919–930. IEEE, 2020.
* [WW18] Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1–27:38, 2018.
|
# Extrinsic to intrinsic mechanism crossover of anomalous Hall effect in the
Ir-doped MnPtSn Heusler system
Sk Jamaluddin School of Physical Sciences, National Institute of Science
Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India Roumita Roy School of Physical Sciences, Indian
Institute of Technology Goa, Ponda-403401, Goa, India Amitabh Das Solid
State Physics Division, Bhabha Atomic Research Centre, Trombay Mumbai 400085,
India Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India
Sudipta Kanungo<EMAIL_ADDRESS>School of Physical Sciences, Indian
Institute of Technology Goa, Ponda-403401, Goa, India Ajaya K. Nayak
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science
Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India
###### Abstract
Recent findings of large anomalous Hall signal in nonferromagnetic and
nonferrimagnetic materials suggest that the magnetization of the system is not
a critical component for the realization of the anomalous Hall effect (AHE).
Here, we present a combined theoretical and experimental study demonstrating
the evolution of different mechanisms of AHE in a cubic Heusler system
MnPt1-xIrxSn. With the help of magnetization and neutron diffraction studies,
we show that the substitution of nonmagnetic Ir in place of Pt significantly
reduces the net magnetic moment from 4.17 $\mu_{B}$/f.u. in MnPtSn to 2.78
$\mu_{B}$/f.u. for MnPt0.5Ir0.5Sn. In contrast, the anomalous Hall resistivity
is enhanced by nearly three times from 1.6 $\mu\Omega$ cm in MnPtSn to about 5
$\mu\Omega$ cm for MnPt0.5Ir0.5Sn. The power law analysis of the Hall
resistivity data suggests that the extrinsic contribution of AHE that
dominates in the case of the parent MnPtSn almost vanishes for MnPt0.5Ir0.5Sn,
where the intrinsic mechanism plays the major role. The experimental results
are well supported by our theoretical study, which shows a considerable
enhancement of the spin-orbit coupling when Ir is introduced into the system.
Our finding of a crossover of the anomalous Hall effect with chemical
engineering is a major contribution toward the recent interest in controlling
the band topology of topological materials, both in bulk and thin-film forms.
Anomaous Hall Effect, Heusler compounds
## I INTRODUCTION
In general, the anomalous Hall effect (AHE) in ferromagnetic
(FM)/ferrimagnetic (FiM) materials scales with the magnetization of the
samples. However, the recent observation of extremely large anomalous Hall
conductivity in some of the moderate magnetization based FM materials [2, 1,
3] and nonmagnetic materials [4, 5] suggests that the spin-orbit coupling
driven band topology is one of the primary microscopic mechanisms that governs
the AHE. In the case of FM materials, the total Hall resistivity can be
expressed as $\rho_{xy}=\rho^{o}_{xy}+\rho^{A}_{xy}=R_{0}H_{z}+R_{s}M_{z}$,
where $R_{0}$ and $R_{S}$ are the ordinary and anomalous Hall coefficients,
respectively. It is well established that mainly two different microscopic
mechanisms, such as a scattering-dependent extrinsic contribution and
scattering-independent intrinsic process, dictate the AHE in FM materials [6].
The extrinsic mechanism of AHE involves two types of scattering processes, the
skew scattering [7] and the side jump [8]. Both of them are related to the
scattering of charge carriers by the impurity sites and are the consequences
of spin-orbit interaction. In contrast, the scattering-independent intrinsic
mechanism proposed by Karplus and Luttinger arises due to the anomalous
velocity of the charge carriers caused by the electronic band structure in the
presence of spin-orbit coupling (SOC) [9]. Furthermore, the inclusion of the
Berry phase to understand the intrinsic mechanism of AHE throws a deep insight
regarding the band topology of the system [10, 12, 11].
In real systems, different microscopic mechanisms of AHE have been understood
in terms of a power law relation between anomalous Hall resistivity
($\rho^{A}_{xy}$) and longitudinal resistivity ($\rho_{xx}$). For the
intrinsic contribution, $\rho^{A}_{xy}$ scales quadratically with the
longitudinal resistivity $\rho_{xx}$, i.e.,
$\rho^{A}_{xy}\propto\rho^{2}_{xx}$ [13, 14]. In the case of extrinsic skew
scattering, the $\rho^{A}_{xy}$ linearly scales with $\rho_{xx}$, i.e.,
$\rho^{A}_{xy}\propto\rho_{xx}$, while for the side jump the relation is
quadratic in nature, i.e., $\rho^{A}_{xy}$ $\propto$ $\rho^{2}_{xx}$. These
relations can be further generalized to a scaling law
$\rho^{A}_{xy}\propto\rho^{\alpha}_{xx}$ to identify the dominant mechanism of
AHE [15], where $\alpha$ = 2 and 1 correspond to intrinsic / side jump and
extrinsic skew scattering contributions, respectively [16, 17, 18]. Since the
intrinsic and side jump mechanisms exhibit a similar relationship with the
longitudinal resistivity, it is difficult to differentiate them
experimentally. Recently, Tian et al [19] introduced a new scaling relation
called the Tian-Ye-Jin ($TYJ$) model to separate the different contributions
of AHE experimentally [20, 21, 22, 23]. It has been shown that the side jump
contribution is negligibly small in most of the metallic ferromagnetic systems
[19, 20], except in some of the multilayer systems where interfacial
scattering plays an important role [24, 25]. The new scaling relation without
considering the side jump mechanism can be expressed as
$\rho^{A}_{xy}=a\rho_{xx0}+b\rho^{2}_{xx}$ (1)
where $\rho_{xx0}$ is the residual resistivity. The first term corresponds to
the extrinsic skew scattering and the second one deals with the intrinsic
mechanism, where $b$ is the intrinsic parameter.
Figure 1: (a) Temperature-dependent zero-field-cooled (ZFC; open symbols) and
field cooled (FC; closed symbols) magnetization [M(T)] curves for MnPt1-xIrxSn
(x=0, 0.1, 0.2, 0.3, 0.5). (b) Field-dependent isothermal magnetization loops
[$M(H)$] measured at 5 K for the MnPt1-xIrxSn samples. (c) Valence electron
count (e/a) dependent Curie temperature and saturation magnetization for these
samples.
Although AHE is a well studied phenomenon in several materials, a systematic
manipulation of different mechanisms as discussed above is extremely important
for its future implementation in real devices. In this regard, Heusler
materials lay a great platform to willfully tune the material property by
chemical doping [26, 27, 28, 29]. In some of the recent studies, it is
demonstrated that these materials can exhibit extremely large AHE due to the
presence of Weyl nodes near the Fermi energy [30, 31]. Therefore, expanding
the material base in the Heusler family with respect to the AHE and
understanding its governing mechanism is an important step to move forward. In
this report, we carry out a combined theoretical and experimental study to
show a systematic change in the governing mechanism of AHE in cubic Heusler
compounds MnPt1-xIrxSn. First, we show that the magnetic moment of the system
can be greatly modified by changing only the nonmagnetic element Pt and Ir.
Then with the help of the scaling relation for the AHE and theoretical study,
we present a detailed study on the changeover of the AHE mechanism from
extrinsic to intrinsic by replacing Pt with Ir.
## II Methods
Polycrystalline samples of MnPt1-xIrxSn (x=0, 0.1, 0.2, 0.3, 0.5) are prepared
by arc-melting of ultrahighly pure constituent elements Mn, Pt, Ir, and Sn in
argon atmosphere. For better chemical homogeneity, the ingots are melted four
to five times by flipping the sides. As-prepared samples are annealed in an
evacuated quartz tube at 850∘C for seven days, followed by quenching in ice
water mixture. To check the phase purity, powder x-ray diffraction (XRD)
measurement is performed on all the samples using a Rigaku SmartLab x-ray
diffractometer with a Cu-Kα source. The neutron diffraction patterns are
recorded using the PD2 powder diffractometer ($\lambda$ = 1.2443 ${\AA}$) at
the Dhruva reactor, Bhabha Atomic Research Centre, Mumbai, India, at selected
temperatures between 1.5 K and 300 K. The magnetic characterizations are
carried out with the help of a Quantum Design MPMS3 (SQUID-VSM). The ac
transport measurements are performed using a physical property measurement
system (PPMS; Quantum Design).
The density-functional theory calculations are performed using the plane-wave
basis and pseudopotential framework as implemented in the Vienna ${ab-initio}$
simulation package (VASP) [32, 33]. The exchange-correlation functional is
employed following the Perdew-Burke-Ernzerhof (PBE) prescription [34]. The
experimentally determined lattice parameters are used in the calculations,
while relaxing the atomic positions toward equilibrium until the Hellmann-
Feynman force becomes less than 0.001 eV/$\AA$. In order to incorporate
correlations beyond the scope of mean-field PBE, the Hubbard on-site $U$ is
introduced by performing GGA+$U$ calculations [35, 36] with suitable values of
Ueff (U-JH) of 5 eV at the Mn site and 2 eV at the Pt and Ir sites,
respectively. The effect of SOC is introduced as a fully relativistic approach
in the self-consistent calculations. The self-consistent electronic structure
calculations are performed with a plane-wave cutoff of 500 eV, and an 8$\times
8\times 8$ k-mesh is used for the Brillouin zone integration.
## III RESULTS AND DISCUSSION
Figure 2: Rietveld refinement of powder neutron diffraction (ND) patterns. (a)
Room-temperature Rietveld refinement of the powder ND pattern with 10% atomic
disorders between Mn/Ir site of MnPt0.5Ir0.5Sn. The region between 34 o and 36
o has been excluded due to contributions from the cryostat. (b) Low-
temperature Rietveld refinement of powder ND pattern with 10 % atomic disorder
for MnPt0.5Ir0.5Sn. (c) Low-temperature Rietveld refinement of the powder ND
pattern for MnPtSn.
### III.1 Structural and Magnetic Characterization
The structural characterization for all the samples is carried out by room-
temperature XRD measurements combined with Rietveld analysis using the
FullProf suite (see Fig. S1 in the Supplemental Material) [37]. All the
samples crystallize in a single cubic structure with space group $F\bar{4}3m$.
A negligible change in the lattice parameter from $a=$ 6.25 ${\AA}$ in MnPtIn
to 6.23 ${\AA}$ is found with Ir doping for MnPt0.5Ir0.5Sn. The temperature-
dependent magnetization, $M(T)$, curves measured in zero-field-cooled (ZFC)
and field-cooled (FC) modes for all the samples are shown in Fig. 1(a). The
magnetic ordering temperature, $T_{C}$, for the parent MnPtSn sample is found
to be 326 K. The $T_{C}$ decreases substantially with increasing Ir
concentration and falls to 226 K for MnPt0.5Ir0.5Sn. The ZFC and FC $M(T)$
curves do not display any noticeable difference, signifying the absence of any
spin-glass-like phase or considerable magnetic anisotropy in the system. The
isothermal magnetization, $M(H)$, loops measured at 5 K for all the samples
are depicted in Fig. 1(b). As can be seen, the saturation magnetization,
$M_{S}$, significantly reduces from 4.17 $\mu_{B}$/f.u. in the case of MnPtSn
to about 2.78 $\mu_{B}$/f.u. for MnPt0.5Ir0.5Sn. It is well known that the
cubic $L2_{1}$ based Heusler materials show a linear variation of the magnetic
moment with the valence electrons concentration [38, 39, 40]. In the present
case, the substitution of Ir in place of Pt decreases the number of valence
electron in the system. This is expected to greatly affect the electronic and
magnetic properties of the system. Figure 1(c) shows the variation of total
magnetic moment and $T_{C}$ with the valence electron concentration (e/a). As
can be seen, both $M_{S}$ and $T_{C}$ exhibit a nearly linear dependency with
the e/a ratio.
To further understand the microscopic origin of the reduction in the total
magnetic moment with Ir doping, we carry out a neutron diffraction (ND) study
on our well-characterized polycrystalline samples MnPtSn and MnPt0.5Ir0.5Sn.
In the case of MnPt0.5Ir0.5Sn, the ND measurements are performed at 2 K and
300 K. Since the $T_{C}$ of MnPt0.5Ir0.5Sn is 226 K, we use the ND pattern at
300 K for this sample to extract the nuclear parameters. For this purpose, the
Rietveld refinement using the FullProf suite [41] is performed with the
Wyckoff positions of Sn, Mn, and Pt/Ir as 4a (0, 0, 0), 4b ($\frac{1}{2}$,
$\frac{1}{2}$, $\frac{1}{2}$), and 4c ($\frac{1}{4}$, $\frac{1}{4}$,
$\frac{1}{4}$), respectively. However, we are unable to achieve a good fitting
for the 300 K ND using the above mentioned Wyckoff positions. It is worth
mentioning that the atomic disorder is frequently encountered in Heusler
compounds with different transition metals of comparable ionic radii. Although
Ir is substituted in place of Pt, the higher electronegativity difference
between Ir and Sn in comparison to that of Pt-Sn may result in some of the Ir
atoms sitting along with the Sn atoms [26, 42]. To check this, we replace
about 10% of Mn atoms with Ir in the Mn Wyckoff position (4b) and put these Mn
atoms in the Ir position (4c), as shown in Fig. 2(a). By doing so, we achieve
a better fitting in comparison to that of without disorder [see Supplemental
Material Fig. S2 (a)] [37]. In the case of 0% disorder, the values of the
$\chi^{2}$ and $R_{Bragg}$ factors are 6.40 and 13.74, respectively, whereas
for 10% atomic disorder the values of the $\chi^{2}$ and $R_{Bragg}$ factors
are reduced to be 5.96 and 11.63, respectively. By further increasing the
disorder, both the $\chi^{2}$ and $R_{Bragg}$ factors increase [see the
Supplemental Material table S1]. Hence, it can be concluded from our Rietveld
refinement that MnPt0.5Ir0.5Sn consists of about 10% Mn-Ir atomic disorder.
Figure 3: (a) Field-dependent Hall resistivity [$\rho_{xy}(H)$] measured at 5
K for different samples. (b) Variation of anomalous Hall resistivity
[$\rho_{xy}^{A}(H)$; red curve] and saturation magnetization ($M_{S}$; blue
curve) with iridium composition at 5 K for MnPt1-xIrxSn. (c) Compositional-
dependent anomalous Hall conductivity ($\sigma^{A}_{xy}$) and anomalous Hall
angle (AHA) at 5 K for MnPt1-xIrxSn.
Next, we concentrate on the low-temperature powder ND data to find out the
magnetic ground state of our samples. In the case of MnPt0.5Ir0.5Sn, the Mn
atoms occupy both 4b and 4c sites due to the atomic disorder. First, we
perform the refinement considering the magnetic moment at the 4b and 4c sites.
The Mn moment at the 4b site comes to be 3.14 $\mu_{B}$, whereas the 4c site
displays a very small moment. This may be due to the presence of a very small
percentage of Mn atoms at the 4c site that makes it difficult to get the
magnetic information. Then we carry out the refinement by giving the magnetic
moment only at the 4b position. This scenario does not affect the fitting
parameter much and we obtain a magnetic moment of 3.17 $\mu_{B}$ for the Mn
atoms. This results in a total moment of 2.85 $\mu_{B}/f.u.$ which is in good
agreement with the experimental magnetic moment observed from the
magnetization data. In the case of the MnPtSn sample, the $T_{C}$ is about 326
K. We are unable to perform the ND experiment above room temperature; only 2 K
ND data are collected for this sample. Here, we consider the regular structure
without disorder and the refined magnetic moment of Mn at the 4b site comes
out to be 3.52 $\mu_{B}$. In general for Heusler compounds, the magnetic atoms
sitting at the 4b and 4c sites align antiferromagnetically. Hence, the
reduction in the magnetic moment for MnPt0.5Ir0.5Sn might be arising from the
lower Mn concentration at the 4b site as well as the antiferromagnetic
alignment between Mn at the 4b and Mn at the 4c site. Hence, our ND study
categorically establishes that the reduction of magnetic moment in the case of
the Ir-doped samples arises mainly from the induced atomic disorder in the
system.
### III.2 Anomalous Hall Effect
In order to explore the effect of Ir substitution on the electrical transport,
we focus on a detailed Hall resistivity measurement on our well-characterized
samples. The field-dependent Hall resistivity [$\rho_{xy}(H)$] measured at 5 K
for different Ir-doped samples is shown in Fig. 3(a). As is found,
$\rho_{xy}(H)$ exhibits a monotonic increment from about 1.6 $\mu\Omega$ cm
for MnPtSn to about 5 $\mu\Omega$ cm for MnPt0.5Ir0.5Sn. It is worth noting
here that the enhancement of the $\rho_{xy}(H)$ is observed despite a
significant reduction in the magnetic moment from about 4.2 $\mu_{B}$/f.u. for
MnPtSn to 2.8 $\mu_{B}$/f.u. for MnPt0.5Ir0.5Sn [Fig. 3(b)]. Although there is
a three fold increase in the Hall resistivity, we observe a small decrease in
the anomalous Hall conductivity $\sigma^{A}_{xy}$ with Ir doping [Fig. 3(c),
left y-axis]. The decrease in the $\sigma^{A}_{xy}$ might be arising due to
the increase in residual resistivity as a result of atomic disorder due to Ir
doping. In contrast, the anomalous Hall angle (AHE) considerably increases
with increasing the Ir concentration. This suggests that the Ir doping
actually helps in the overall conversion of the longitudinal resistivity to
Hall resistivity.
Figure 4: (a-e) Anomalous Hall resistivity ($\rho^{A}_{xy}$) as a function of
square of longitudinal resistivity ($\rho^{2}_{xx}$) for MnPt1-xIrxSn (x= 0,
0.1, 0.2, 0.3, 0.5). Solid lines are the linear fitting to the equation
$\rho^{A}_{xy}=a\rho_{xx0}+b\rho^{2}_{xx}$. Top axis indicates the
corresponding measurement temperatures. (f) The extrinsic parameter ($a$; open
squares) and intrinsic parameter ($b$; closed squares) as a function of
iridium concentration $x$. The lines are guides to the eye.
To understand the mechanism that governs the observed change in the AHE with
Ir doping, we carry out a thorough analysis of the $\rho^{A}_{xy}$ data using
$TYJ$ scaling relations [19] and the power law relation. In order to map out
the magnitude of different contributions of AHE in the Ir-doped samples, we
employ the scaling relation given in Eq. (1). For this purpose, we plot
$\rho^{A}_{xy}$ vs $\rho^{2}_{xx}$ for different samples, as depicted in Figs.
4(a)-4(e). The $\rho_{xx}$ data are obtained from the temperature-dependent
longitudinal resistivity measurements and $\rho^{A}_{xy}$ points are taken
from the field-dependent Hall resistivity measurements at different
temperatures as shown in the Supplemental Materials [37]. The fitting
parameters obtained from the linear fit [as shown by black solid lines in
Figs. 4(a)-4(e)] of the experimental data can be used to extract the
information about different contributions of AHE. The slope of the linear
fitting gives the intrinsic parameter $b$, whereas the extrinsic parameter $a$
can be calculated from the intercept. Figure 4(f) shows the variation of $a$
and $b$ with the iridium concentration. The extrinsic parameter $a$ decreases
and the intrinsic parameter $b$ increases with increasing iridium
concentration. This illustrates that the Ir doping enhances the intrinsic
contribution to the AHE in the present system.
It has been shown that the side jump contribution ($\sigma_{xx}^{SJ}$) is
related to $\frac{e^{2}}{ha}(\frac{\epsilon_{so}}{E_{F}})$ [43, 44], where
$\epsilon_{so}$ is the spin-orbit interaction energy, $E_{F}$ is the Fermi
energy, and $a$ is the lattice constant. Using this expression, we have
calculated the approximate side jump contribution using
$\frac{\epsilon_{so}}{E_{F}}\sim 0.01$ and the experimental lattice constant
$a$. We find that the side jump contribution for MnPtSn is almost negligible
when compared with the extrinsic and intrinsic parameters [37]. For
MnPt0.5Ir0.5Sn, the most dominant contribution arises from the intrinsic part,
which is much larger than the side jump effect. However, the magnitudes of
skew scattering and side jump contributions for this sample are comparable as
both components are very small. Hence the extrinsic parameter ‘a’ mainly
represents the skew scattering. We summarize the values of $a$, $b$, anomalous
Hall resistivity arising from the extrinsic contribution $\rho^{ext}_{xy}$,
the intrinsic anomalous Hall resistivity $\rho^{int}_{xy}$, and the total
$\rho^{tot}_{xy}$ for all the compounds in table 1.
Figure 5: Power law representation. (a-e) Plot of log ($\rho^{A}_{xy}$) vs
log ($\rho_{xx}$) for MnPt1-xIrxSn (x= 0, 0.1, 0.2, 0.3, 0.5). Black solid
lines are the fit using the relation
($\rho^{A}_{xy}\propto\rho^{\alpha}_{xx}$). Top axis indicates the
corresponding measurement temperatures. (f) Scaling factor $\alpha$ as a
function of iridium concentration $x$. Table 1: Values of extrinsic and
intrinsic parameters and different contribution of AHE.
—————————————————————————— x a b $\rho^{ext}_{xy}$ $\rho^{int}_{xy}$
$\rho^{tot}_{xy}$ $\times 10^{-3}$ $(\Omega$ $cm)^{-1}$ $(\mu\Omega$ $cm)$ 0.0
10.07 73.14 0.97 0.68 1.65 0.1 9.22 82.60 1.07 1.13 2.20 0.2 7.89 101.9 1.06
1.84 2.90 0.3 5.79 119.3 0.93 3.09 4.05 0.5 0.856 131.1 0.16 4.86 5.02
For the power law analysis, we plot log $\rho^{A}_{xy}$ vs log $\rho_{xx}$ as
shown in Figs. 5(a)-5(e). The solid lines represent the linear fitting to the
data and the slope gives the scaling factor $\alpha$. For each sample a good
fit to the data can be obtained up to a certain temperature below which the
magnetization is almost constant. As is found, $\alpha$ increases linearly
with Ir doping from 0.96 for MnPtSn to 1.93 in MnPt0.5Ir0.5Sn [Fig. 5(f)]. The
scaling factor $\alpha$ close to 1 suggests that the extrinsic contribution is
the dominant mechanism that governs the AHE in MnPtSn, whereas $\alpha$ of
nearly 2 in the case of MnPt0.5Ir0.5Sn points toward its intrinsic nature. The
change of $\alpha$ from 0.96 to 1.93 implies a changeover in the mechanism of
AHE from extrinsic to intrinsic with Ir doping.
Figure 6: Projected density of states (DOS) with UMn = 5eV and UPt/Ir = 2eV
for (a) MnPtSn, (b)MnPt0.5Ir0.5Sn, and (c) MnIrSn. The Mn-d, Pt-d, Ir-d, and
Sn-p states are represented by red, green, blue, and cyan, respectively.
Projected density of states with the variation of spin-orbit coupling (SOC)
strength. (d), (e), and (f) represent the variation of DOS for Pt-d in MnPtSn,
Pt-d in MnPt0.5Ir0.5Sn, and Ir-d in MnPt0.5Ir0.5Sn, respectively. The color
scheme for various values of $\alpha$ is mentioned in the right panel. The
Fermi level in the energy scale is set at zero.
### III.3 Electronic Band Structure and Effect of the Spin-Orbit Coupling
(SOC)
The electronic structure calculations are carried out for three compounds in
the MnPt1-xIrxSn series with x=0.0, 0.5 and 1.0. Although MnIrSn could not be
synthesized experimentally, the crystal structure is obtained by the
structural optimization starting from the MnPtSn and included in the study for
the sake of complete understanding of the role played by each component.
Figure 6 shows the calculated GGA+$U$ comparative density of states (DOS) for
the above-mentioned compounds. The DOS shows that the Mn-3d states are fully
filled in the majority spin channel and completely empty in the minority spin
channel, with a very small contribution at the Fermi energy. The major
contribution at the Fermi level arises from the Pt-$5d$ and the Sn-$5p$ states
as shown in Fig. 6(a).
The DOS clearly shows the metallic character of the electronic structure,
where the Pt-$5d$ states contribute at the Fermi energy for MnPtSn. In the
case of x=0.5, the hybridized Ir-$5d$ and Pt-$5d$ states mostly appear at the
Fermi energy. The calculated magnetic moment at the Mn site comes out to be
about 4.52 $\mu_{B}$, while the induced moments at the Sn and Pt sites appear
to be -0.12 $\mu_{B}$ and negligibly small, respectively. This results in a
total moment of 4.53 $\mu_{B}$/f.u. After 50% Ir doping at the Pt site the DOS
changes significantly as shown in Fig. 6(b). The Mn-$3d$ states remain almost
unchanged, while the major renormalization happened at the Pt states due to
incorporation of the Ir doping. Doped Ir-$5d$ states are pushed toward the
Fermi level in MnPt0.5Ir0.5Sn. The magnetic moment at the Mn site is slightly
reduced to 4.50 $\mu_{B}$, while the induced moment at the Ir site is around
-0.05 $\mu_{B}$. The difference in the theoretical magnetic moment compared to
the experimental moment obtained for the Ir doped sample might be arising due
to the fact that the present theoretical calculations are performed without
considering any disorder. For MnIrSn, the Mn-$3d$ states remain unchanged
except slightly narrowing down the band width, and the Ir-$5d$ states are
further pushed toward the Fermi level as shown in Fig. 6(c). Moreover, we
calculate the magnetic transition temperatures in a mean-field method for the
MnPtSn and MnPt0.5Ir0.5Sn. The calculated ratio of the transition temperature
($\frac{T^{MPS}_{c}}{T^{MPIS}_{c}}$) is around 1.52, which is in good
agreement with the experimental value of 1.44 (see the Supplemental Material
sec VII) [37].
Figure 7: Variation of magnetic moment with respect to (a)-(d) iridium doping
concentration (x) and (e)-(h) scale factor of SOC strength $\alpha$.
(a),(b),(e),(f) represent the variation of spin magnetic moment and
(c),(d),(g),(h) represent the variation of orbital magnetic moment. Mn-d,
Pt-d, Ir-d states are shown by circle, square, and triangle symbols,
respectively. In (e)-(h) closed symbols represent MnPtSn and open symbols
MnPt0.5Ir0.5Sn samples.
In addition to the above-discussed effects, Ir doping may also modify the
effective SOC strength in the material. To capture the effect of SOC-induced
modifications in the system, we have done electronic structure calculations
for MnPtSn and MnPt0.5Ir0.5Sn by varying the SOC strength manually. The
calculated GGA+$U$+SOC electronic DOSs with varying SOC strength ($\alpha$)
for Pt/Ir states are shown in Figs. 6(d)-6(f). As expected, the Mn-$3d$ state
does not exhibit any substantial variation with the SOC strength [37];
however, the Pt/Ir-$5d$ states change significantly and move toward the Fermi
energy with increasing SOC strength as shown in Figs. 6(d)-6(f). As discussed
above, a similar kind of modification of the DOS is also observed with Ir
doping in Figs. 6(a)-6(c). From these DOS calculations, it is very clear that
the introduction of Ir into the system substantially changes the effective
strength of the SOC as well as the Pt/Ir states.
To understand the variation of the magnetic moment with the Ir doping, we also
perform the calculations for intermediate concentrations (x= 0.25, 0.5, 0.75).
The variation of spin and orbital magnetic moments with Ir concentration is
shown in Figs. 7(a)-7(d). It is evident that the spin magnetic moment at the
Mn sites gradually decrease with increasing Ir concentration, whereas the
Pt/Ir spin moments, although very small, remain almost constant. The point to
be noted here is that the sign of the induced moments at the Ir site is
opposite to that of the Mn site. As expected, the Mn atom exhibits a very
small orbital moment in comparison to its spin magnetic moment and the orbital
moment at the Pt/Ir site. Interestingly, the orbital magnetic moment at the Ir
site is larger than that of the Pt site with opposite sign. However, the spin
and orbital magnetic moments at the Ir sites follow almost the same trend with
the Ir doping , as expected due to the modification of the effective strength
of the SOC in the system.
Figure 8: Magnetization density for the MnPtSn and 50% Ir doped material for
different SOC strength $\alpha$. (a) and (b) are the magnetization density of
MnPtSn for $\alpha$=1.0 and 4.0 respectively, whereas (c) represents the same
for MnPt0.5Ir0.5Sn with $\alpha$=1.0. Yellow and green represent the
isosurfaces with + and - signs respectively.
From our experimental and theoretical studies, it is very clear that Ir doping
plays a crucial role in deciding the magnetic state of the material. Naively,
the physical consequence of the Ir doping at the place of Pt can be mapped as
reducing one electron per atom in the material. As we can see from the DOS
calculation, Ir doping changes the SOC which influences the resultant magnetic
moment of the system. Figures 7(e)-7(h) show the variation of spin and orbital
magnetic moment as a function of the scale factor ($\alpha$) of the SOC
strength. As the $\alpha$ increases, the spin magnetic moment at the Mn sites
[Fig. 7(e)] decreases, whereas the orbital magnetic moment [Fig. 7(g)]
increases. This trend can be understood from the transfer of spin contribution
to the orbital contribution of magnetic moment with increasing SOC. In
contrast, the variation of the spin and orbital moments as a function of
$\alpha$ are very different for Pt and Ir. There is a marginal change in the
spin and orbital magnetic moment for Pt as a function of $\alpha$, whereas the
variation is quite substantial for Ir. The effect of SOC is more prominent in
the case of Ir than Pt in both the MnPtSn and 50$\%$ Ir doped cases.
Interestingly, the variations of magnetic moment at the Mn site as a function
of the Ir doping and SOC strength ($\alpha$) are very similar, which suggests
that the Ir doping at the Pt site plays a crucial role in dictating the
effective SOC of the materials, apart from injecting an electron in the
system.
### III.4 Magnetization Density
The effect of the different roles played by the Pt and Ir atoms in terms of
SOC strength can be conclusively visualized by the magnetization density plot
shown in Fig. 8. Figures 8(a) and 8(b) show the magnetization density for
MnPtSn with $\alpha$=1.0 and 4.0, respectively, whereas Fig. 8(c) represents
the magnetization density for MnPt0.5Ir0.5Sn with $\alpha$=1.0. Since the Mn
magnetic moment is much larger, the spin isosurface sitting at the Mn site is
very large compared to the Pt/Ir sites. However, the small isosurface of the
Pt or Ir carries very important information about the effective strength of
the SOC. The Pt isosurface in Figs. 8(a) and 8(b), i.e. for $\alpha$=1.0 and
4.0, are very different, which is quite expected as the SOC strength modifies
the shape of the magnetization density. A very interesting fact is that the
shape of the Ir isosurface in Fig. 8(c) (i.e, for MnPt0.5Ir0.5Sn with
$\alpha$=1.0) is similar to that of the Pt isosurface in Fig. 8(b) [i.e, for
MnPtSn with $\alpha$=4.0]. This observation suggests that the effective
strength of the SOC gets modified due to the Ir doping in the MnPtSn.
Qualitatively, it can be said that the effective strength of the SOC in the
case of the 50$\%$ Ir doped sample is greater than the parent MnPtSn. Hence,
it is expected that the change in the SOC with Ir doping may significantly
affect the electron transport in the system. The above discussed theoretical
result nicely corresponds with the experimental findings. It is well
established that the intrinsic contribution of the AHE is directly
proportional to the relevant local Berry curvature of the material. To realize
a non-vanishing Berry curvature by the breaking of the symmetry of the system,
the SOC in the material plays a very crucial role [45]. From the electronic
structure calculations, we show that Ir doping enhances the effective strength
of SOC; as a result the intrinsic contribution to the AHE should also
increase.
The theoretical findings corroborate our experimental observation of enhanced
AHE in case of the Ir-doped samples. It may be noted that the Ir-doped samples
possess some amount of disorder as well as higher residual resistivity
compared to the parent MnPtSn compound. Although the skew scattering
contribution is sensitive to the impurity, it is larger in a clean regime [low
resistivity, $\sigma_{xx}>10^{6}$ $(\Omega$ $cm)^{-1}$] and decays in a bad
metal regime [high resistivity, $\sigma_{xx}<10^{4}$ $(\Omega$ $cm)^{-1})$].
This suggests that as the conductivity $(\sigma_{xx})$ decreases, the skew
scattering contribution starts diminishing and the intrinsic mechanism begins
to dominate [6, 46]. In the present case all the samples fall into moderate to
bad metal regimes. Hence, our experimental and theoretical findings suggest
that one can tune different contributions of AHE with changing the overall SOC
strength in the system. In this regard, AHE driven by different underlying
mechanisms is reported recently in several materials. In some cases, a large
intrinsic anomalous Hall signal has been observed due to the presence of Weyl
nodes [47, 48]. Additionally, nontrivial scalar spin chirality can also induce
a large extrinsic anomalous Hall effect as recently found in MnGe [49].
Besides this, there are reports of modification of intrinsic and extrinsic AHE
by tuning the Fermi level with the help of chemical doping [20, 50, 51].
Although SOC is the primary mechanism responsible for the AHE, it is important
to understand the correlation among other mechanisms contributing its cause.
In the present study, we demonstrate the manipulation of different
contributions of AHE by introducing chemical doping.
## IV CONCLUSION
In conclusion, we present a detailed study on the magnetic and electronic
transport properties of the half-Heusler system MnPt1-xIrxSn. We find that the
saturation magnetization systematically decreases while the anomalous Hall
resistivity increases with increasing iridium concentration. The experimental
Hall signal is analyzed using the scaling relation between anomalous Hall
resistivity and longitudinal resistivity. We find that the scaling factor
changes from close to 1 for the parent MnPtSn sample to quadratic for
MnPt0.5Ir0.5Sn, signifying that Ir doping enhances the intrinsic contribution
by suppressing the extrinsic mechanism. Our experimental results are well
supported by the theoretical study that showed that the Ir doping
significantly enhances the spin-orbit coupling in the system. The present
study is an important contribution toward the basic understanding of different
mechanisms of AHE, and thereby possesses a great importance in designing
anomalous Hall sensor based spintronic devices.
## V Acknowledgment
A.K.N. acknowledges support from the Department of Atomic Energy (DAE), the
Department of Science and Technology (DST) Ramanujan Research Grant No.
SB/S2/RJN-081/2016. R.R. acknowledges IIT Goa for her research fellowship and
S.K. acknowledges DST INSPIRE for research funding.
## References
* [1] I. Belopolski1, K. Manna, D. S. Sanchez, G.Chang,B. Ernst, J. Yin1, S. S. Zhang1, T. Cochran1, N. Shumiya1,H. Zheng1, B. Singh, G. Bian, D. Multer1, M. Litskevich1, X. , Shin-Ming Huang, B Wang, Tay-Rong Chang, Su-Yang Xu1,A. Bansi, C. Felser, H. Lin, M. Zahid Hasan, Science 365, 1278 (2019).
* [2] Q. Wang, S. Sun, X. Zhang, F. Pang, and H. Lei, Phys. Rev. B 94, 075135 (2016).
* [3] P. Li, J. Koo, W. Ning, J. Li, L. Miao, L. Min, Y. Zhu, Y. Wang, N. Alem, C.-X. Liu, Z. Mao, and B. Yan, Nat. Commun. 11, 3476 (2020).
* [4] T. Liang, J. Lin, Q. Gibson, S. Kushwaha, M. Liu, W. Wang, H. Xiong, J. A. Sobota, M. Hashimoto, P. S. Kirchmann, Z.-X. Shen, R. J. Cava, N. P. Ong, Nat. Phys. 14, 451 (2018).
* [5] S.-Y. Yang, Y. Wang, B. R. Ortiz, D. Liu, J. Gayles, E. Derunova, R. Gonzalez-Hernandez, L. Šmejkal, Y. Chen, S. S. P. Parkin, S. D. Wilson, E. S. Toberer, T. McQueen, and M. N. Ali, Sci. Adv. 6, eabb6003 (2020).
* [6] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
* [7] J. Smit, Physica 21, 877 (1955); 24, 39 (1958).
* [8] L. Berger, Phys. Rev. B 2, 4559 (1970).
* [9] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954).
* [10] T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 88, 207208 (2002).
* [11] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature (London) 527, 212 (2015).
* [12] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C.Komarek, C. Shekhar, N. Kumar, W. Schnelle, J. Kübler, C.Felser, and S. S. P. Parkin, Sci. Adv. 2, e1501870 (2016).
* [13] W. L. Lee, S. Watauchi2, V. L. Miller, R. J. Cava, N. P. Ong, Science 303, 1647 (2004).
* [14] Y. Pu, D. Chiba, F. Matsukura, H. Ohno, and J. Shi, Phys. Rev. Lett. 101, 117208 (2008).
* [15] J. Shen, Q. Zeng, S. Zhang, H. Sun, Q. Yao, X. Xi, W. Wang, G. Wu, B. Shen, Q. Liu, and E. Liu, Adv. Funct. Mater. 30, 2000830, 2020.
* [16] Q. Wang, Y. Xu, R. Lou, Z Liu, M. Li1, Y. Huang, D. Shen ,H. Weng, S. Wang1 and H. Lei, Nat. Commun. 9, 3681 (2018).
* [17] H. Li, B. Zhang, J. Liang, B. Ding, J. Chen, J. Shen, Z. Li, E. Liu, X. Xi, G. Wu, Y. Yao, H. Yang, and W. W. Phys. Rev. B 101, 140409(R) (2020).
* [18] Yu Liu , H. Tan, Z. Hu, B. Yan, and C. Petrovic, Phys. Rev. B 103, 045106 (2021).
* [19] Y. Tian, L. Ye, and X. Jin, Phys. Rev. Lett. 103, 087206 (2009).
* [20] D. Yu. Karpenkov, K. P. Skokov, I. A. Radulov, O. Gutfleisch, J. Weischenberg, and H. Zhang, Phys. Rev. B 100, 094445 (2019).
* [21] K. K. Meng , L. J. Zhu, Z. H. Jin, E. K. Liu, X. P. Zhao, I. A. Malik, Z. G. Fu, Y. Wu, J. Miao, X. G. Xu, J. X. Zhang, J. H. Zhao, and Y. Jiang, Phys. Rev. B 100, 184410 (2019).
* [22] L. J. Zhu, D. Pan and J. H. Zhao, Phys. Rev. B 89, 220406(R) (2014).
* [23] J. Shen, Q. Yao, Q. Zeng, H. Sun, X. Xi, G. Wu, W. Wang, B. Shen, Q. Liu, and E. Liu, Phys. Rev. Lett. 125, 086602 (2020).
* [24] Q. Zhang, P. Li, Y. Wen, C. Zhao, J. W. Zhang, A. Manchon, W. B. Mi, Y. Peng, and X. X. Zhang, Phys. Rev. B 94, 024428 (2016).
* [25] Y. Zhang, Y. Yin, G. Dubuis, T. Butler, N. V. Medhekar and S. Granville, npj Quantum Mater, 6, 17 (2021).
* [26] T. Graf, C. Felser, and S. S. P. Parkin, Prog. Solid State Chem. 39, 1 (2011).
* [27] J. Kübler, G. H. Fecher, and C. Felser, Phys. Rev. B 76, 024414 (2007).
* [28] E. Decolvenaere, M. Gordon, R. Seshadri, and Anton V. der Ven, Phys. Rev. B 96, 165109 (2017).
* [29] S. D. Harrington, A. D. Rice, T. L. Brown-Heft, B. Bonef, A. Sharan, A. P. McFadden, J. A. Logan, M. Pendharkar, M. M. Feldman, O. Mercan, A. G. Petukhov, A. Janotti, L. Colakerol Arslan, and C. J. Palmstrøm, Phys. Rev. Mater. 2, 014406 (2018).
* [30] A.A. Burkov, Phys. Rev. Lett. 113, 187202, (2014).
* [31] J. Chen, X. Xu, H. Li, T. Guo, B. Ding, P. Chen, H. Zhang, X. Xi, and W. Wang, Phys. Rev. B 103, 144425,(2021).
* [32] G. Kresse and J. Hafner, Phys. Rev. B $\bf{47}$, 558(R) (1993).
* [33] G. Kresse and J. Furthmüller, Phys. Rev. B $\bf{54}$, 11169 (1996).
* [34] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. $\bf{77}$ ,3865 (1996).
* [35] V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czyzyk, and G. A. Sawatzky, Phys. Rev. B $\bf{48}$, 16929 (1993).
* [36] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B $\bf{57}$, 1505 (1998).
* [37] See Supplemental Material for details of structural, transport measurements, additional theoretical calculations and reference no. 50 to 51.
* [38] S. Skaftouros, K. Özdoğan, E. Şaşıoğlu, and I. Galanakis, Phys. Rev. B 87, 024420 (2013).
* [39] E. Şaşıoğlu, L. M. Sandratskii, and P. Bruno, Phys. Rev. B 77, 064417 (2008).
* [40] L. Wollmann, S. Chadov, J. Kubler, and C Felser, Phys. Rev. B 90,214420 (2014).
* [41] Juan Rodríguez-Carvajal, Physica B 192, 55 (1993).
* [42] A. K. Nayak, C. Salazar Mejia, S. W. D’Souza, S. Chadov, Y. Skourski, C. Felser, and M. Nicklas, Phys. Rev. B 90, 220408(R) (2014).
* [43] S. Onoda, N. Sugimoto, and N. Nagaosa, Phys. Rev. Lett. 97, 126602 (2006).
* [44] P. Nozières. And C. Lewiner, J. Phys. France 34, 901 (1973).
* [45] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. $\bf{82}$, 1959 (2010).
* [46] Y. Shiomi, Y. Onose, and Y. Tokura, Phys. Rev. B 79, 100404 (2009).
* [47] B. Ernst, R. Sahoo, Y. Sun, J. Nayak, L. Muchler, A K. Nayak, N. Kumar,J. Gayles, A. Markou, G. H. Fecher, and C. Felser, Phys. Rev. B 100, 054445 (2019).
* [48] T. Suzuki, R. Chisnell, A. Devarakonda, Y.-T. Liu, W. Feng, D. Xiao, J. W. Lynn and J. G. Checkelsky, Nat. Phys. 12, 1119 (2016).
* [49] Y. Fujishiro, N. Kanazawa, R. Kurihara, H. Ishizuka, T. Hori, F. S. Yasin, X. Yu, A. Tsukazaki, M. Ichikawa, M. Kawasaki, N. Nagaosa, M. Tokunaga and Y. Tokura. Nat. Commun. 12, 317, (2021).
* [50] H.-Yu Yang, B. Singh, B. Lu, C.-Yi Huang, F. Bahrami, W.-Chi Chiu, D. Graf, S.M. Huang, B. Wang, H. Lin, D. Torchinsky, A. Bansil, and F. Tafti, APL Materials 8, 011111 (2020).
* [51] K. M. Seemann, Y. Mokrousov, A. Aziz, J. Miguel, F. Kronast, W. Kuch, M. G. Blamire, A. T. Hindmarch, B. J. Hickey, I. Souza, and C. H. Marrows, Phys. Rev. Lett. 104, 076402, (2010).
## SUPPLEMENTARY INFORMATION
## VI X-ray Diffraction
X-ray diffraction (XRD) data are collected at room temperature with a Cu-K
source on a Rigaku SmartLab x-ray diffractometer to check the phase purity of
all the samples. The Rietveld refinement of the room temperature powder XRD
patterns for MnPt0.5Ir0.5Sn are shown in FIG. 9. The Rietveld refinement for
(x=0, 0.1, 0.2, 0.3) is performed using the Wyckoff positions as Sn at (0, 0,
0), Mn at (1/2, 1/2, 1/2), and Pt/Ir at (1/4, 1/4, 1/4). For x=0.5, same
Wyckoff positions are used with 10% atomic disorder between Mn/Ir sites (see
section VII). The refinement confirms that all the samples are in single-phase
and exhibit XRD patterns of cubic crystal structure of space group
F$\bar{4}$3m.
Figure 9: Room temperature powder XRD patterns and Rietveld refinements for
MnPt1-xIrxSn (x=0, 0.1, 0.2, 0.3, 0.5
## VII Neutron Diffraction (ND)
The room-temperature Rietveld refinement of the powder ND data for
MnPt0.5Ir0.5Sn is carried out using the FullProf software with different
percentages of atomic disorder between Mn and Ir. Figs. 10 (a)-10 (d) shows
the Rietveld refinement with 0%, 5%, 15%, 20% atomic disorders between the Mn
and Ir atoms, respectively. The refined parameters with different percentages
of atomic disorders are given in table II. We observe that 10% atomic disorder
between Mn/Ir results in a minimum value of $\chi^{2}$ and $R_{Bragg}$. The
magnitude of $\chi^{2}$ and $R_{Bragg}$ increases as the disorder increase or
decreases. Our findings indicate that the best fit for MnPt0.5Ir0.5Sn is
obtained with 10% atomic disorder between Mn/Ir sites.
Figure 10: Rietveld refinement of the room-temperature ND pattern for MnPt0.5Ir0.5Sn with different atomic disorder between the Mn and Ir sites. The region between $34^{o}$ and $36^{o}$ is excluded due to contribution from the cryostat. Table 2: The percentage of disorder, site occupations, and other refined parameters for the room temperature powder neutron diffraction data of MnPt0.5Ir0.5Sn. Sl. No | | Wyckoff
---
Position
Site Occupations (%) | $\chi^{2}$ | $R_{Bragg}$
0% Disorder | | 4b
---
4c
| 100 Mn + 0 Ir
---
50 Pt +50 Ir
6.4 | 13.71
5% Disorder | | 4b
---
4c
| 95 Mn + 5 Ir
---
50 Pt +45 Ir + 5 Mn
5.96 | 12.06
10% Disorder | | 4b
---
4c
| 90 Mn + 10 Ir
---
50 Pt + 40 Ir + 10 Ir
5.96 | 11.62
15% Disorder | | 4b
---
4c
| 85 Mn + 15 Ir
---
50 Pt +35 Ir + 15Ir
6.42 | 14.38
20% Disorder | | 4b
---
4c
| 80 Mn + 20 Ir
---
50 Pt +30 +Ir +20 Ir
7.45 | 21.08
## VIII Temperature dependence of longitudinal resistivity
Longitudinal resistivity $\rho_{xx}$ as a function of temperature (T) for all
the samples is shown in Fig. 11. Below the Curie temperature (Tc) resistivity
decreases as the temperature decreases indicating that all the samples are in
a metallic regime. At low temperature, $\rho_{xx}$ approaches the residual
resistivity $\rho_{xx0}$
Figure 11: Temperature variation of longitudinal resistivity $\rho_{xx}$ (T)
for the MnPt1-xIrxSn (with x= 0, 0.1, 0.2, 0.3, 0.5) samples.
## IX Field dependent Hall resistivity
The Hall resistivity measurements for all the samples are performed at
different temperatures with field sweep $\pm$ 5T, as shown in Fig. 12. For all
the samples, the Hall resistivity increases with increasing temperature before
it starts to decrease as the temperature approaches the Curie temperature.
Figure 12: Field variation of Hall resistivity $\rho_{xy}(H)$ at different
temperatures for the MnPt1-xIrxSn (with x= 0, 0.1, 0.2, 0.3, 0.5) samples.
## X Density of states with the variation of spin-orbit coupling (SOC)
strength
The density of states for Mn-d states of MnPtSn and MnPt0.5Ir0.5Sn with the
variation of spin orbit-coupling (SOC) strength are calculated and shown in
Figs. 13 (a)-13 (b), respectively. The calculated density of states with
different SOC strengths does not show any significant changes for both the
samples.
Figure 13: Projected density of states of Mn-d with the variation of spin
orbit coupling (SOC) strength for (a) MnPtSn and (b) MnPt0.5Ir0.5Sn
## XI Magnetic Exchange Interactions
Figure 14: (a) Magnetic exchange interactions between different Mn sites as a
function of the Mn-Mn distances for the MnPtSn and MnPt0.5Ir0.5Sn. Positive
and negative signs represent ferromagnetic and antiferromagnetic type of the
interactions, respectively. (b) Density of states (DOS) for MnPtSn. The Mn-3d,
Pt-5d, Sn-5p, Sn-5s are represented by gray, green, blue and red respectively.
Inset shows the Wannier function plot.
With the Ir doping, substantial changes are observed in the magnetic
properties of the MnPtSn half Heusler compound. In the experiments, it is
observed that Mn spins are ferromagnetically ordered in the case of MnPtSn,
however, there is some sort of antiferromagnetic correlation in the Ir doped
cases. Through the electronic structure total energy method, we calculate the
exchange interactions between different Mn sites as a function of the Mn-Mn
distances as shown in Fig. 14 (a). We find that the magnetic exchange
interactions of the MnPt0.5Ir0.5Sn are weaker than the MnPtSn, which is
consistent with the smaller value of the TC for the Ir doped samples compared
to that of the parent MnPtSn. We also find that the type of the exchange
interactions are oscillatory in nature which ensures the conduction electron-
driven RKKY type of ferromagnetic exchange interactions are dominating in
these materials. In the case of MnPtSn, the Mn-Mn distance is large; thus the
Mn-3d state does not overlap directly, and the direct exchange interactions
can be neglected. On the contrary, the RKKY type of exchange interaction
between two magnetic Mn ions is also mediated by the itinerant conduction
electrons. The itinerant electrons close to the Fermi energy and having
substantial hybridization with the Mn-d states will act as a mediator in the
RKKY interactions. For the MnPtSn, the density of states (Fig Fig. 14 (b))
clearly shows that Sn-sp mixed states are taking part in the conduction
electron near the Fermi energy and acting as a mediator for the RKKY type
exchange interactions between two Mn-d ions. We also found a small presence of
Pt/Ir-5d states in the Mn-Mn exchange interactions. The below DOS and Wannier
function (inset figure Fig. 14 (b)) plot clearly shows that the finite tail
(marked by red dotted circle in the inset) situated at the Sn site with a
shape of sp hybridized state as a mediator between Mn-Mn interactions. We also
found a very tiny tail at the Pt site, which also has little contribution as a
mediator.
## XII Calculation of Magnetic transition temperature
The magnetic transition temperatures in a mean-field approach for both the
systems are calculated using the equation
$T_{C}=\frac{2}{3}S(S+1)\sum_{i}J_{i}Z_{i}$ (2)
The experimental ratio of $T_{C}$ is $\frac{T_{C}^{MPS}}{T_{C}^{MPIS}}$ =1.44
(MPS represents for MnPtSn and MPIS represents for MnPt0.5Ir0.5Sn). The
calculated ratio of $T_{C}$ is $\frac{T_{C}^{MPS}}{T_{C}^{MPIS}}$ =1.52.
## XIII Side Jump Contribution
Figure 15: Skew scattering $\rho_{xy}^{Skew}$, intrinsic $\rho_{xy}^{Int}$ and
side-jump $\rho_{xy}^{SJ}$ contributions with respect to residual resistivity
$\rho_{xx0}$ for MnPt$1-x$IrxSn (for x = 0, 0.1, 0.2, 0.3, 0.5).
According to the existing theoretical model, the extrinsic side-jump
contribution is in the order of
$\sigma_{xy}^{SJ}\approx\frac{e^{2}}{ha}(\frac{\epsilon_{SO}}{E_{F}})$ , where
$\epsilon_{SO}$ is the spin-orbit interaction energy, $E_{F}$ is the Fermi
energy, a is the lattice constant. For metallic ferromagnet
$\frac{\epsilon_{SO}}{E_{F}}\approx 0.01$. With the help of this expression,
one can have the order of side jump contribution and compares it with the
other mechanism. We calculated the approximate side jump contribution for our
system using the above expression and taking the experimental lattice
parameter. We plotted the different contributions with residual resistivity
$\rho_{xx0}$ . It is clearly seen that the SJ contribution is minimal compared
to other dominant contributions.
|
# Additive structure of non-monogenic simplest cubic fields
Daniel Gil-Muñoz Department of Algebra, Faculty of Mathematics and Physics,
Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic
Magdaléna Tinková Faculty of Information Technology, Czech Technical
University in Prague, Thákurova 9, 160 00 Praha 6, Czech Republic
###### Abstract
We consider non-monogenic simplest cubic fields $K=\mathbb{Q}(\rho)$ in the
family introduced by Shanks, and among these, we focus in the fields whose
generalized module index $[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$ is a prime
number $p$. We prove that these fields arise exactly for $p=3$ or $p\equiv
1\,(\mathrm{mod}\,6)$ and we use the method in [16] to find the additive
indecomposables of $\mathcal{O}_{K}$. We determine the whole structure of
indecomposables for the family with $p=3$ and obtain that the behaviour is not
uniform with respect to the indecomposables of $\mathbb{Z}[\rho]$. From the
knowledge of the indecomposables we derive some arithmetical information on
$K$, namely: the smallest and largest norms of indecomposables, the Pythagoras
number of $\mathcal{O}_{K}$ and bounds for the minimal rank of universal
quadratic forms over $K$.
## 1 Introduction
The main motivation for the study of the additive structure of a totally real
number field $K$ comes from the topic of universal quadratic forms. A
quadratic form with coefficients in the ring of integers $\mathcal{O}_{K}$ is
said to be universal (or universal over $K$) if it represents all totally
positive elements in $\mathcal{O}_{K}$ (that is, those whose image by any
embedding of $K$ is a positive real number). Some classical problems in
arithmetic may be formulated in terms of these objects; for instance the well
known Lagrange’s four-square theorem asserts that the quadratic form
$x^{2}+y^{2}+z^{2}+w^{2}$ is universal over $\mathbb{Q}$. Although the sum of
three squares $x^{2}+y^{2}+z^{2}$ is not universal over $\mathbb{Q}$, Maaß
[24] proved in 1941 that it is universal over $\mathbb{Q}(\sqrt{5})$. Shortly
after, in 1945, Siegel [31] proved that these are the only fields under which
sums of squares can be universal. In his proof, indecomposables, which we
study in this paper, appear under the name of extremal elements.
All the examples of universal quadratic forms above are totally positive
definite, that is, they only represent totally positive integers. In this
paper we will restrict exclusively to such quadratic forms. From a result by
Hsia, Kitaoka and Kneser [10] it follows that for every totally real number
field there is some totally positive definite quadratic form which is
universal over $K$. Let $m(K)$ be the minimal integer among all ranks of
universal quadratic forms over $K$. The number $m(K)$ turns out to be rather
unpredictable. In fact, Kitaoka conjectured that $m(K)=3$ only for finitely
many totally real number fields $K$. Although a general solution is unknown,
there are some partial results [6, 8, 22, 18].
The connection of $m(K)$ with additive indecomposables of $K$ came from the
works by Blomer and Kala [1, 14], which is based on the suitability of these
elements to construct quadratic lattices of large rank. For a totally real
quadratic field $K=\mathbb{Q}(\sqrt{D})$, Perron [27], Dress and Scharlau [7]
found a complete list of indecomposable integers in $K$ in terms of the
continued fraction expansion of $\frac{-1+\sqrt{D}}{2}$ if $D\equiv
1\,(\mathrm{mod}\,4)$ and of $\sqrt{D}$ otherwise. Blomer and Kala used this
description to show that for every positive integer $n$ there are infinitely
many totally real quadratic fields $K$ such that $m(K)\geq n$. In other words,
there are infinitely many quadratic fields without universal quadratic forms
of rank $n$ arbitrarily large. In the following years an analogous result was
proved for the families of multiquadratic and cubic fields [15, 36]. In fact,
Kala [13] proved that it holds for each degree $d$ divisible by $2$ or $3$. On
the other hand, Blomer and Kala [2] found concrete bounds for the number
$m_{\mathrm{diag}}({\mathbb{Q}(\sqrt{D})})$, the minimal rank of diagonal
universal quadratic forms over $\mathbb{Q}(\sqrt{D})$.
This substantial progress on quadratic fields motivated the search of
indecomposables in families of totally real number fields with higher degree.
In this direction, Kala and the second author [16] proved that all
indecomposables in a totally real number field $K$ are, up to multiplication
by totally positive units, lattice points in a finite disjoint union of
simplicial cones in the Minkowski space $\mathbb{R}^{d}$, where
$d=[K:\mathbb{Q}]$. In the case that $d=3$, this description reduces to two
specific parallelepipeds, whose nodes depend on a proper pair of totally
positive units (according to the terminology in [32]). These considerations
led to a geometric method consisting in finding all lattice points within
those two parallelepipeds and identifying the indecomposables among them.
A tool to identify the indecomposables among lattice points relies on their
traces. The minimal trace of an algebraic integer $\alpha\in\mathcal{O}_{K}$
is the minimum among all numbers of the form $\mathrm{Tr}(\delta\alpha)$, for
totally positive elements $\delta$ in the codifferent
$\mathcal{O}_{K}^{\vee}=\\{\delta\in
K\,|\,\mathrm{Tr}(\delta\alpha)\in\mathbb{Z}\text{ for all
}\alpha\in{\mathcal{O}}_{K}\\}$
of $K$. Its importance in the search of indecomposables relies in the fact
that every totally positive algebraic integer of minimal trace $1$ is
indecomposable. While in the case of quadratic fields the converse holds, in
the case of monogenic simplest cubic fields there is an indecomposable with
minimal trace $2$. In fact, the second author [34, Theorem 1.1] proved that
the minimal trace of indecomposables in totally real cubic orders can be
arbitrarily large.
The focus in [16] was in the family of simplest cubic fields. Those are the
totally real cubic fields generated by a polynomial of the form
$f(x)=x^{3}-ax^{2}-(a+3)x-1,\quad a\in\mathbb{Z}_{\geq-1},$
meaning that $K=\mathbb{Q}(\rho)$ for a root $\rho$ of $f$. These fields were
originally studied in detail by Shanks [30], and among their nice arithmetic
properties, there is the one that $\mathrm{disc}(f)=\Delta^{2}$, where
$\Delta=a^{2}+3a+9$. Moreover, for a positive density of $a$, this is just the
discriminant of $K$, and those are the cases in which
$\mathcal{O}_{K}=\mathbb{Z}[\rho]$. In [16, Theorem 1.2] the complete list of
the indecomposables in $\mathbb{Z}[\rho]$ is provided, which led to some
bounds on the minimal rank of universal quadratic forms (see [16, Theorem
1.1]). Full lists of indecomposables of an analogous order of fields in some
other families are given in [16, Section 8] and [34]. For some partial results
on indecomposable integers in real biquadratic fields, see also [5, 22].
In this paper we shall consider the whole ring of integers $\mathcal{O}_{K}$
of the simplest cubic fields $K$ such that
$\mathcal{O}_{K}\neq\mathbb{Z}[\rho]$. In Section 3 we shall review the
characterization of the monogenity of $K$ found by Kashio and Sekigawa [19],
and some consequences that will be needed later on. We have that $K$ is
monogenic if and only if the generalized module index
$\delta\coloneqq[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$ is the cube of some
integer number, unless $K$ is defined by more than one parameter $a$. In
Section 4 we will focus on the simplest cubic fields $K$ with integral basis
$B_{p}(k,l)\coloneqq\Big{\\{}1,\rho,\frac{k+l\rho+\rho^{2}}{p}\Big{\\}},$
for a prime number $p$ and integers $1\leq k,l\leq p-1$. Our main results
concerning these can be summarized in:
###### Theorem 1.1.
Let $K=\mathbb{Q}(\rho)$ be a simplest cubic field and call
$\delta=[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$.
* 1.
$\delta=3$ if and only if $K$ has integral basis $B_{3}(1,1)$. These simplest
cubic fields arise exactly when $a\equiv 3$ or $21\,(\mathrm{mod}\,27)$,
$a>12$ and $\frac{\Delta}{27}$ is square-free.
* 2.
$\delta=p$, $p>3$ prime, if and only if $K$ has integral basis $B_{p}(k,l)$
for some $1\leq k,l\leq p-1$. These simplest cubic fields arise exactly when
$p\equiv 1\,(\mathrm{mod}\,6)$. In that case, the parameter $a$ may take two
possible values $a_{1}$, $a_{2}$ modulo $p^{2}$, and when $a\equiv
a_{i}\,(\mathrm{mod}\,p^{2})$ for $i\in\\{1,2\\}$, there are unique integers
$k_{i}$, $l_{i}$ modulo $p$ such that $k\equiv k_{i}\,(\mathrm{mod}\,p)$ and
$l\equiv l_{i}\,(\mathrm{mod}\,p)$.
In order to find the elements of $K$ that may possibly be indecomposable, we
will essentially follow the aforementioned method introduced in [16]. It turns
out that they are contained in the same two parallelepipeds found by Kala and
the second author for the case that $\mathcal{O}_{K}=\mathbb{Z}[\rho]$. While
the first parallelepiped can be fully described without difficulty (see
Proposition 4.10), the second one is much trickier. In Section 5, we will
focus in the case that $p=3$, that is, the simplest cubic fields with integral
basis
$g_{1}=1,\quad g_{2}=\rho,\quad g_{3}=\frac{1+\rho+\rho^{2}}{3}.$
This family had already been considered in a different (but equivalent) form
in [4, Theorem 3]. For these, we describe the lattice points in the second
parallelepiped. The main result in this paper is a complete list of
indecomposables for fields in this family, which we prove in Section 6.
###### Theorem 1.2.
Let $K$ be a simplest cubic field with $a\equiv 3$ or
$21\,(\mathrm{mod}\,27)$, $a>12$ and $\frac{\Delta}{27}$ square-free. Up to
multiplication by totally positive units, the indecomposable integers in $K$
are 1,
* (i)
$g_{3}$,
* (ii)
$-g_{1}-(r+1)g_{2}+3g_{3}=-r\rho+\rho^{2}$ where $1\leq r\leq\frac{a}{3}$,
* (iii)
$-(2v+1)g_{1}-(v(a+3)+2)g_{2}+3(v+1)g_{3}=-v-(v(a+2)+1)\rho+(v+1)\rho^{2}$
where $\frac{2a}{3}+1\leq v\leq a$,
* (iv)
$-(2v+1)g_{1}-(v+1)(a+2)g_{2}+3(v+1)g_{3}=-v-(v(a+2)+a-v+1)\rho+(v+1)\rho^{2}$
where $0\leq v\leq\frac{a}{3}-1$,
* (v)
$-(2v+1)g_{1}-(v(a+3)+r+1)g_{2}+(3v+2)g_{3}=\\\
-v-(v(a+2)+r)\rho+(v+1)\rho^{2}-g_{3}$ where $0\leq v\leq\frac{a}{3}-1$ and
$\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$,
* (vi)
$-(r+1)g_{2}+g_{3}=1-r\rho+\rho^{2}-2g_{3}$ where $0\leq r\leq\frac{a}{3}-1$,
* (vii)
$-(2v+2)g_{1}-(v(a+3)+\frac{2a}{3}+3)g_{2}+(3v+4)g_{3}=\\\
-v-(v(a+2)+\frac{2a}{3}+1)\rho+(v+2)\rho^{2}-2g_{3}$ where $0\leq
v\leq\frac{a}{3}-1$,
* (viii)
$-(2v+2)g_{1}-(v(a+3)+\frac{4a}{3}-v+3)g_{2}+(3v+4)g_{3}=\\\
-v-(v(a+2)+\frac{4a}{3}-v+1)\rho+(v+2)\rho^{2}-2g_{3}$ where $\frac{a}{3}\leq
v\leq\frac{2a}{3}-1$.
The points from (ii) to (iv) have minimal trace $2$, while the remaining ones
have minimal trace $1$. The total number of indecomposables in $K$ (up to
multiplication by totally positive units) is $\frac{a^{2}+3a}{18}+2a+2$.
The ordering of the indecomposables in the statement above will follow from
the classification of lattice points carried out in Section 5.
In Section 7, we shall study some applications of Theorem 1.2 for simplest
cubic fields with integral basis $B_{3}(1,1)$. In [23], Lemmermeyer and Pethö
proved that every $\gamma\in\mathbb{Z}[\rho]$ not associated with a rational
integer, the norm $N(\gamma)$ of $\gamma$ satisfies $|N(\gamma)|\geq 2a+3$. In
Proposition 7.1 we will see that except for some small values of $a$, this is
also true for our subfamily with $p=3$ and $(k,l)=(1,1)$. On the other hand,
we will also study the largest possible norm of the indecomposables, and
concretely we will prove that $|N(\alpha)|\leq\frac{(a^{2}+3a+9)^{2}}{729}$,
again with some exceptions (see Proposition 7.4). We will also show that the
Pythagoras number of $K$ is $6$, just as in the monogenic case studied by Kala
and the second author. We shall use these informations to find some bounds for
$m(K)$, in the style of the stated ones in [16, Theorem 1.1]. For more
details, see Section 7.
Our results are in the line of the existing works for the additive structure
of the totally real fields, which show evidence of its unpredictability.
However, this work is pioneer in determining the whole structure of
indecomposables in the ring of integers in a family of non-monogenic totally
real fields. Remarkably, we have found this absence of pattern even within the
same family of simplest cubic fields, for which the value of generalized
module index $[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$ turns out to be extremely
influential.
## 2 Preliminaries
Let $K$ be a totally real number field of degree $d$ and let
$\sigma_{1},\dots,\sigma_{d}\colon K\hookrightarrow\mathbb{R}$ be its
embeddings. By trace $\text{Tr}(\alpha)$ and norm $N(\alpha)$ of $\alpha\in
K$, we will mean
$\text{Tr}(\alpha)=\sum_{i=1}^{d}\sigma_{i}(\alpha)\hskip 28.45274pt\text{ and
}\hskip 28.45274ptN(\alpha)=\prod_{i=1}^{d}\sigma_{i}(\alpha).$
An element $\alpha\in K$ is totally positive if $\sigma_{i}(\alpha)>0$ for
every $1\leq i\leq d$, and we will denote it by $\alpha\succ 0$. The subset of
the totally positive integers in $K$ is denoted by $\mathcal{O}_{K}^{+}$.
Moreover, by $\alpha\succ\beta$, we will mean $\alpha-\beta\succ 0$. We will
also use symbol $\succeq$ to include equality between our elements.
A totally positive integer $\alpha\in\mathcal{O}_{K}$ is said to be
indecomposable if it cannot be written as the sum of other two totally
positive integers in $\mathcal{O}_{K}$. The codifferent of $\mathcal{O}_{K}$
is
$\mathcal{O}_{K}^{\vee}=\\{\delta\in
K\,|\,\text{Tr}(\delta\alpha)\in\mathbb{Z}\hbox{ for every
}\alpha\in\mathcal{O}_{K}\\}.$
The subset of the totally positive integers in $\mathcal{O}_{K}^{\vee}$ is
denoted by $\mathcal{O}_{K}^{\vee,+}$. For $\alpha\in\mathcal{O}_{K}^{+}$, the
number
$\mathrm{min}_{\delta\in\mathcal{O}_{K}^{\vee,+}}\mathrm{Tr}(\delta\alpha)$
will be referred to as the minimal trace of $\alpha$. It is immediate that the
totally positive algebraic integers with minimal trace $1$ are indecomposable.
We proceed to summarize the method introduced in [16] to find the
indecomposables in $K$. Let us embed $K$ on the Minkowski space
$\mathbb{R}^{d}$ by means of an integral basis of $K$. Following [32, Section
1, p. 2], we introduce the following terminology:
###### Definition 2.1.
A pair $(\varepsilon_{1},\varepsilon_{2})$ of units in $K$ is proper if the
following statements hold:
* (1)
The subgroup of $K^{*}$ generated by $\\{\varepsilon_{1},\varepsilon_{2}\\}$
has rank $2$.
* (2)
The set $\\{1,\varepsilon_{1},\varepsilon_{2}\\}$ is a $\mathbb{Q}$-basis of
$K$.
* (3)
In the linear combination of $\varepsilon_{1}\varepsilon_{2}$ with respect to
the above basis, the coefficient before $1$ is negative.
On the other hand, the parallelepiped with nodes
$\ell_{1},\dots,\ell_{e}\in\mathbb{Z}^{d}$ is
$\mathcal{D}(\ell_{1},\dots,\ell_{e})=[0,1]\ell_{1}+\dots+[0,1]\ell_{e}.$
The key result is the following refinement of [32, Theorem 1] (see [16,
Section 4, p. 11-12]):
###### Proposition 2.2.
Let $K$ be a totally real cubic field and let
$\\{\varepsilon_{1},\varepsilon_{2}\\}$ be a proper pair of totally positive
units of $K$. The indecomposables of $K$ lie, up to multiplication by totally
positive units in $K$, in either of the parallelepipeds
$\mathcal{D}(1,\varepsilon_{1},\varepsilon_{2}),\quad\mathcal{D}(1,\varepsilon_{1},\varepsilon_{1}\varepsilon_{2}^{-1}).$
This method was applied successfully to the case that $K$ is a simplest cubic
field (see Section 3 below) with $\mathcal{O}_{K}=\mathbb{Z}[\rho]$, leading
to a list of lattice points among which the indecomposables were identified.
###### Theorem 2.3.
[16, Theorem 1.2] Let $\rho$ be a root of the polynomial
$x^{3}-ax^{2}-(a+3)x-1$ where $a\in{\mathbb{Z}}_{\geq-1}$. Up to
multiplication by totally positive units, the indecomposables in the order
$\mathbb{Z}[\rho]$ of ${\mathbb{Q}}(\rho)$ are $1$,
* •
the element $1+\rho+\rho^{2}$ with minimal trace $2$,
* •
the elements in the set
$\blacktriangle=\\{-v-w\rho+(v+1)\rho^{2}\,|\,0\leq v\leq a,\,v(a+2)+1\leq
w\leq(v+1)(a+1)\\},$
all of which have minimal trace $1$.
## 3 Simplest cubic fields and their monogenity
A simplest cubic field is a number field $K=\mathbb{Q}(\rho)$, where $\rho$ is
a root of the polynomial
$f(x)=x^{3}-ax^{2}-(a+3)x-1,\quad a\in\mathbb{Z}_{\geq-1}.$
It is a cyclic cubic extension of $\mathbb{Q}$ (and then totally real).
Let $\rho^{\prime}$ and $\rho^{\prime\prime}$ be the conjugates of $\rho$,
that is, the other roots of $f$. When $a\geq 7$, reordering these roots if
necessary, we can assume that [23]
$a+1<\rho<a+1+\frac{2}{a},\quad-1-\frac{1}{a}<\rho^{\prime}<-1-\frac{1}{2a},\quad-\frac{1}{a+2}<\rho^{\prime\prime}<-\frac{1}{a+3}.$
We also have the explicit expressions
$\rho^{\prime}=-1-\frac{1}{\rho}=a+2+a\rho-\rho^{2}\quad\text{and}\quad\rho^{\prime\prime}=-\frac{1}{1+\rho}=-2-(a+1)\rho+\rho^{2}.$
Let us denote $\Delta\coloneqq a^{2}+3a+9$. We know that
$\mathrm{disc}(f)=\Delta^{2}$. If $\Delta$ is square-free, then
$\mathcal{O}_{K}=\mathbb{Z}[\rho]$. The converse does not hold in general.
We say that $K$ is monogenic if there is an element $\gamma\in\mathcal{O}_{K}$
such that $\mathcal{O}_{K}=\mathbb{Z}[\gamma]$. In that case,
$\\{1,\gamma,\gamma^{2}\\}$ is a power integral basis of $K$. Denote by
$\mathfrak{c}$ the conductor of $K$. The monogenity of $K$ is characterized in
[19] in the following way:
###### Theorem 3.1.
[19, Corollary 1.6] The following are equivalent:
* 1.
The field $K$ is monogenic.
* 2.
We have $a\in\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$ or
$\frac{\Delta}{\mathfrak{c}}$ is a cube.
* 3.
We have $a\in\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$ or $a\not\equiv
3,21\,(\mathrm{mod}\,27)$ and $v_{p}(\Delta)\not\equiv 2\,(\mathrm{mod}\,3)$
for all $p\neq 3$.
In such a case, a power integral basis of $K$ is generated by
$\gamma=\frac{\rho-b}{n}$, where $n=\sqrt[3]{\frac{\Delta}{27}}$ and
$b\in\mathbb{Z}$ satisfies $b\equiv\frac{a}{3}\,(\mathrm{mod}\,n)$ if $3\mid
n$ and $3a\equiv b\,(\mathrm{mod}\,n)$ otherwise.
###### Remark 3.2.
The list of possible values for $a$ in the second and the third statement in
Theorem 3.1 are the ones for which the corresponding simplest cubic field $K$
is also defined by some other value $a^{\prime}$ in the same list. Hence, its
presence is due to the fact that for such a simplest cubic field $K$, one of
the values satisfies the conditions of monogenity and the other one does not,
resulting that $K$ is monogenic.
We proceed to reinterpret the relevant number $\frac{\Delta}{\mathfrak{c}}$ in
the second statement of Theorem 3.1. Since $K$ is an abelian extension of
$\mathbb{Q}$, from Kronecker-Weber theorem we know that it is contained in
some cyclotomic field. Since in addition $K$ is totally real, $\mathfrak{c}$
is the smallest integer $n\in\mathbb{Z}_{>0}$ such that $K$ is contained in
the $n$-th cyclotomic field. In particular, $\mathfrak{c}$ is positive.
Moreover, it is related to the discriminant by the so called conductor-
discriminant formula, $\mathfrak{c}^{2}=\mathrm{disc}(K)$. Together with the
well known formula
$\mathrm{disc}(f)=[\mathcal{O}_{K}:\mathbb{Z}[\rho]]^{2}\mathrm{disc}(K)$, we
deduce that $\mathfrak{c}$ divides $\Delta$ and
$[\mathcal{O}_{K}:\mathbb{Z}[\rho]]=\frac{\Delta}{\mathfrak{c}}.$
###### Remark 3.3.
Note that $K$ may be monogenic even if $\mathcal{O}_{K}\neq\mathbb{Z}[\rho]$,
namely whenever $[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$ is a non-trivial cube.
This means that $\mathcal{O}_{K}=\mathbb{Z}[\gamma]$ for some
$\gamma\neq\rho$. However, we will usually refer to the case
$\mathcal{O}_{K}=\mathbb{Z}[\rho]$ as the monogenic case.
In order to compute effectively the generalized module index
$[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$, we denote $\Delta=bc^{3}$ with $b,c>0$
coprime integers and $b$ cube-free. Then:
###### Proposition 3.4.
[19, Remark 1.5] The conductor can be described as follows:
$\mathfrak{c}=\begin{cases}\prod_{p\mid b}p&\hbox{if }3\nmid a\hbox{ or
}a\equiv 12\,(\mathrm{mod}\,27),\\\ 3^{2}\prod_{p\mid b,p\neq
3}p&\hbox{otherwise}.\end{cases}$
###### Remark 3.5.
If $3\nmid a$, we have that $\Delta=a^{2}+3a+9\equiv 1\,(\mathrm{mod}\,3)$,
and if $a\equiv 12\,(\mathrm{mod}\,27)$, then $\Delta\equiv
0\,(\mathrm{mod}\,27)$. In both cases, we see that $3\nmid\mathfrak{c}$. We
deduce that the conductor of a simplest cubic field $K$ satisfies either that
$3\nmid\mathfrak{c}$ or $v_{3}(\mathfrak{c})=2$.
###### Example 3.6.
Let $K$ be the simplest cubic field with $a=21$. Then $\Delta=3^{3}\cdot 19$.
Therefore, $c=3$ and $b=19$. Since $3$ divides $a$ and $a\not\equiv
12\,(\mathrm{mod}\,27)$, we find that $\mathfrak{c}=19\cdot 3^{2}$. Then
$\frac{\Delta}{\mathfrak{c}}=3$, which is not a cube, and then $K$ is not
monogenic. Actually, the integral basis of $K$ is
$\Big{\\{}1,\rho,\frac{1+\rho+\rho^{2}}{3}\Big{\\}}.$
###### Example 3.7.
Let $K$ be the simplest cubic field with $a=41$. Then $\Delta=7^{2}\cdot 39$
and $\mathfrak{c}=7\cdot 39$, so $\frac{\Delta}{\mathfrak{c}}=7$ is not a cube
and $K$ is not monogenic. In this case, the integral basis of $K$ is
$\Big{\\{}1,\rho,\frac{4+3\rho+\rho^{2}}{7}\Big{\\}}.$
### 3.1 The parallelepipeds of lattice points
We focus on the problem of finding the indecomposable integers in a simplest
cubic field $K$ for the case when $\mathcal{O}_{K}\neq\mathbb{Z}[\rho]$. That
is, calling $\delta=[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$, we remove the
restriction that $\delta=1$ and we aim to find all indecomposables in $K$. In
order to find candidates, we will follow a slight variation of the method in
[16], based on the well known fact that
$\delta\mathcal{O}_{K}\subseteq\mathbb{Z}[\rho]$.
Let us fix the power basis $\\{1,\rho,\rho^{2}\\}$, which is not integral
unless $\mathcal{O}_{K}=\mathbb{Z}[\rho]$. We have that $\mathcal{O}_{K}$
embeds in $\frac{1}{\delta}\mathbb{Z}^{3}$ through
$\begin{array}[]{rccl}\tau\colon&K&\longrightarrow&\mathbb{R}^{3},\\\
&\alpha_{1}+\alpha_{2}\rho+\alpha_{3}\rho^{2}&\longmapsto&(\alpha_{1},\alpha_{2},\alpha_{3}).\end{array}$
Now, let us call $\varepsilon_{1}=\rho^{2}$ and
$\varepsilon_{2}=(\rho^{\prime\prime})^{-2}$. From [32, Corollary 2] we know
that the pairs $(1,\varepsilon_{1},\varepsilon_{2})$ and
$(1,\varepsilon_{1},\varepsilon_{1}\varepsilon_{2}^{-1})$ are proper (note
that conditions (1)-(3) in Definition 2.1 do not depend on whether we have
$\mathcal{O}_{K}=\mathbb{Z}[\rho]$). Then, the problem now turns to find the
intersection of $\frac{1}{\delta}\mathbb{Z}^{3}$ with the parallelepipeds
$\mathcal{D}(1,\rho^{2},1+2\rho+\rho^{2}),\quad\mathcal{D}(1,\rho^{2},-1-a-(a^{2}+3a+3)\rho+(a+2)\rho^{2}).$
(1)
From now on, these will be referred to as the first parallelepiped and the
second parallelepiped, respectively.
###### Remark 3.8.
There might be elements in $\frac{1}{\delta}\mathbb{Z}[\rho]$ that do not lie
in $\mathcal{O}_{K}$. In other words, there might be some lattice points that
are not algebraic integers.
We shall find the lattice points in the first of the above parallelepipeds.
The second one will be completely determined for a specific family of simplest
cubic fields $K$ in Section 5, by using an integral basis of $K$.
We look for $t_{1},t_{2},t_{3}\in[0,1]$ such that for some
$(m,n,o)\in\mathbb{Z}^{3}$,
$t_{1}+t_{2}\rho^{2}+t_{3}(1+2\rho+\rho^{2})=\frac{1}{\delta}(m+n\rho+o\rho^{2}).$
This gives rise to the system of equations
$\displaystyle t_{1}+t_{3}$ $\displaystyle=\frac{1}{\delta}m,$ $\displaystyle
2t_{3}$ $\displaystyle=\frac{1}{\delta}n,$ $\displaystyle t_{2}+t_{3}$
$\displaystyle=\frac{1}{\delta}o.$
The second equation gives that $t_{3}=\frac{n}{2\delta}$ with $0\leq n\leq
2\delta$. Carrying this to the first equation, $t_{1}=\frac{2m-n}{2\delta}$,
with $\frac{n}{2}\leq m\leq\frac{n}{2}+\delta$. As for the third one,
$t_{2}=\frac{2o-n}{2\delta}$, with $\frac{n}{2}\leq o\leq\frac{n}{2}+\delta$.
We plug these values in the parametric equation, obtaining the points
$\frac{1}{\delta}(m+n\rho+o\rho^{2}),\quad 0\leq n\leq
2\delta,\quad\frac{n}{2}\leq m,o\leq\frac{n}{2}+\delta,$
which are candidates to be indecomposables of $K$. Note that choosing
$m=n=o=\delta$ yields the point $1+\rho+\rho^{2}$, which was identified as
indecomposable of the $\mathbb{Z}$-order $\mathbb{Z}[\rho]$ in [16]. On the
other hand, unlike in that case, the trivial choices $n=0$ or $n=2\delta$ give
non-trivial candidates for $\frac{n}{2}<m,o<\frac{n}{2}+\delta$.
###### Proposition 3.9.
Let $K$ be a simplest cubic field and let
$\delta=[\mathcal{O}_{K}:\mathbb{Z}[\rho]]$. The candidates (i.e, lattice
points that are possibly indecomposables of $K$) from the first parallelepiped
in (1) are the points in the set
$\left\\{\frac{m}{\delta}+\frac{n}{\delta}\rho+\frac{o}{\delta}\rho^{2};\;0\leq
n\leq 2\delta,\;\frac{n}{2}\leq
m,o\leq\frac{n}{2}+\delta\right\\}\cap{\mathcal{O}}_{K}^{+}.$
## 4 Non-monogenic simplest cubic fields with prime index
In this section we introduce the simplest cubic fields $K$ with an integral
basis of the form
$B_{p}(k,l)\coloneqq\Big{\\{}1,\rho,\frac{k+l\rho+\rho^{2}}{p}\Big{\\}},$ (2)
where $p$ is an odd prime, and $k,l\in\mathbb{Z}$ are not divisible by $p$.
Note that we may assume without loss of generality that $1\leq k,l\leq p-1$.
When $(k,l)$ is implicit in the context, we just denote $B_{p}=B_{p}(k,l)$.
The simplest cubic fields in Examples 3.6 and 3.7 are of this form: in the
former we have $(k,l)=(1,1)$ and $p=3$, while in the latter it is
$(k,l)=(4,3)$ and $p=7$.
We begin with the following important lemma.
###### Lemma 4.1.
Let $K$ be a simplest cubic field with integral basis $B_{p}(k,l)$ for $1\leq
k,l\leq p-1$. Then, $K$ does not admit an integral basis of the same form for
another pair $(k^{\prime},l^{\prime})\neq(k,l)$, $1\leq
k^{\prime},l^{\prime}\leq p-1$.
###### Proof.
We start with the assumption that $K$ has integral basis as in the statement,
so that $\frac{k+l\rho+\rho^{2}}{p}\in\mathcal{O}_{K}$. Let
$(k^{\prime},l^{\prime})$ be a pair of integer numbers such that
$\frac{k^{\prime}+l^{\prime}\rho+\rho^{2}}{p}$ belongs to $\mathcal{O}_{K}$ as
well. Then, their difference
$\frac{k^{\prime}-k+(l^{\prime}-l)\rho}{p}\in\mathcal{O}_{K}.$
Let us call $c=k^{\prime}-k$ and $d=l^{\prime}-l$. Then there are unique
$x_{1},x_{2},x_{3}\in\mathbb{Z}$ such that
$\frac{c+d\rho}{p}=x_{1}+x_{2}\rho+x_{3}\frac{k+l\rho+\rho^{2}}{p}.$
Using the uniqueness of coordinates with respect to the power basis, we obtain
that $x_{1}=\frac{c}{p}$, $x_{2}=\frac{d}{p}$ and $x_{3}=0$. Hence $c$ and $d$
are multiples of $p$, meaning that $k\equiv k^{\prime}\,(\mathrm{mod}\,p)$ and
$l\equiv l^{\prime}\,(\mathrm{mod}\,p)$. ∎
In order to find the pair $(k,l)$ for which $\frac{k+l\rho+\rho^{2}}{p}$ is
the third element of the integral basis, we follow this procedure. The minimal
polynomial of $\frac{k+l\rho+\rho^{2}}{p}$ is
$x^{3}-\frac{h_{1}}{p}x^{2}+\frac{h_{2}}{p^{2}}x-\frac{h_{3}}{p^{3}},$
where
$\begin{split}h_{1}=h_{1}(k,l)&=a^{2}+(l+2)a+3k+6,\\\
h_{2}=h_{2}(k,l)&=(2k-l+1)a^{2}+(-l^{2}+2kl+4k-3l+4)a+3k^{2}-3l^{2}+12k-3l+9,\\\
h_{3}=h_{3}(k,l)&=(k^{2}-kl+k)a^{2}+(k^{2}l-kl^{2}+2k^{2}+l^{2}-3kl+4k-l)a+k^{3}+l^{3}\\\
&\hskip 28.45274pt-3kl^{2}+6k^{2}-3kl+9k-3l+1.\end{split}$
Then, we are just looking for the pair $(k,l)$, $1\leq k,l\leq p-1$, such that
$h_{i}(k,l)\equiv 0\,(\mathrm{mod}\,p^{i}),\quad i\in\\{1,2,3\\}.$ (3)
Eventually we will prove that such a pair actually exists.
Let $K$ be a simplest cubic field with integral basis $B_{p}(k,l)$ and let us
compute the module index $\delta=\frac{\Delta}{\mathfrak{c}}$. It may be
checked, for instance using mathematical software, that
$\mathrm{disc}\Big{(}1,\rho,\frac{k+l\rho+\rho^{2}}{p}\Big{)}=\mathrm{det}\begin{pmatrix}1&\rho&\frac{k+l\rho+\rho^{2}}{p}\\\
1&\rho^{\prime}&\frac{k+l\rho^{\prime}+\rho^{\prime 2}}{p}\\\
1&\rho^{\prime\prime}&\frac{k+l\rho^{\prime\prime}+\rho^{\prime\prime
2}}{p}\end{pmatrix}^{2}=\Big{(}\frac{a^{2}+3a+9}{p}\Big{)}^{2},$
and recall that the conductor $\mathfrak{c}$ must be positive, so
$\mathfrak{c}=\frac{a^{2}+3a+9}{p}$. It follows that $\delta=p$, so that
$\Delta=p\mathfrak{c}$. In particular, $K$ is not monogenic.
Recall from Remark 3.2 that the simplest cubic fields defined by more than one
value of $a$ (i.e. $a\in\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$) are monogenic.
Since $K$ is not monogenic, this is not the case. Then, Theorem 3.1 (3) gives
that $a\equiv 3$ or $21$ $(\mathrm{mod}\,27)$ or $v_{q}(\Delta)\equiv
2\,(\mathrm{mod}\,3)$ for some prime $q\neq 3$. Keeping the assumption that
$K$ has integral basis $B_{p}(k,l)$, we explore each of these two
possibilities.
#### Case $a\equiv 3$ or $21$ $(\mathrm{mod}\,27)$
Suppose that $a\equiv 3$ or $21$ $(\mathrm{mod}\,27)$. A direct computation
shows that $\Delta=a^{2}+3a+9$ is divisible by $27$. But since $K$ has
integral basis $B_{p}$, we have that $\Delta=p\mathfrak{c}$, and from
Proposition 3.4 we see that $v_{3}(\mathfrak{c})=2$, so it must be $p=3$. That
is, $K$ has integral basis $B_{3}(k,l)$ for some $1\leq k,l\leq 2$ (which are
unique by Lemma 4.1). If we choose $(k,l)=(1,1)$, we have
$\begin{split}h_{1}&=a^{2}+3a+9=\Delta\equiv 0\,(\mathrm{mod}\,3),\\\
h_{2}&=2a^{2}+6a+18=2\Delta\equiv 0\,(\mathrm{mod}\,9),\\\
h_{3}&=a^{2}+3a+9=\Delta\equiv 0\,(\mathrm{mod}\,27).\end{split}$
Then the only pair $(k,l)$ with $1\leq k,l\leq 2$ for which
$\frac{k+l\rho+\rho^{2}}{3}$ is the third element of $B_{3}$ is $(k,l)=(1,1)$.
In fact, $a\equiv 3$ or $21$ $(\mathrm{mod}\,27)$ only when $K$ has integral
basis $B_{3}$. In the next section, we will completely characterize the
simplest cubic fields with integral basis of this form.
#### Case $v_{q}(\Delta)\equiv 2\,(\mathrm{mod}\,3)$, $q\neq 3$ prime
Next, assume that $K$ satisfies that $v_{q}(\Delta)\equiv
2\,(\mathrm{mod}\,3)$ for some prime $q\neq 3$, which is equivalent to
$q^{2}\mid\Delta$. Since $\Delta=p\mathfrak{c}$ and $q\neq 3$, from
Proposition 3.4 it follows that $q=p$. Hence we obtain that $p\neq 3$ and
$p^{2}\mid\Delta$. In particular, if $K$ has integral basis $B_{p}$ for $p>3$,
then $p^{2}\mid\Delta$. There is a partial converse for this implication.
###### Proposition 4.2.
Let $K=\mathbb{Q}(\rho)$ be a simplest cubic field such that $p^{2}\mid\Delta$
for a prime $p>3$. Then there are integers $1\leq k,l\leq p-1$ such that
$\frac{k+l\rho+\rho^{2}}{p}\in\mathcal{O}_{K}$.
###### Proof.
We will check that under the hypothesis that $p^{2}\mid\Delta$, there exist
$1\leq k,l\leq p-1$ such that $h_{i}(k,l)\equiv 0\,(\mathrm{mod}\,p^{i})$ for
all $i\in\\{1,2,3\\}$.
Since $p>3$, $3$ has a modular inverse $3^{-1}$ modulo $p$. If the equations
(3) have some solution $(k,l)$ as stated, from the first one we obtain that
$k=-3^{-1}(al+a^{2}+2a+6)+Up,$
where $U\in\mathbb{Z}$ is such that $1\leq k\leq p-1$. We will check that this
is satisfied for some value of $l$ modulo $p$.
Let us take $U\in\mathbb{Z}$. Since $p^{2}\mid\Delta$, we have that
$3h_{2}(-3^{-1}(al+a^{2}+2a+6)+Up,l)=-(a^{2}+a+1+(2a+1)l+l^{2})\Delta+9p^{2}U^{2}\equiv
0\,(\mathrm{mod}\,p^{2}).$
Thus, if $p\mid h_{1}(k,l)$ for some $(k,l)\in\mathbb{Z}^{2}$, then $p^{2}\mid
h_{2}(k,l)$. On the other hand,
$27h_{3}(-3^{-1}(al+a^{2}+2a+6)+Up,l)=g(l)-9p\Delta
U(a^{2}+a+1+(2a+1)l+l^{2})+27p^{3}U^{3},$
where
$g(l)=\Delta((2a^{2}-3)\Delta+8(2a+3)+((6a-3)\Delta-12a+36)l+(6\Delta-6a-36)l^{2}+(2a+3)l^{3}).$
Now, using again that $p^{2}\mid\Delta$ gives
$27h_{3}(-3^{-1}(al+a^{2}+2a+6)+Up,l)\equiv g(l)\,(\mathrm{mod}\,p^{3})$. The
proof will be finished as soon as we check that $g(l)$ has some root modulo
$p^{3}$, and since $\Delta|g(l)$ for every $l$, it is enough to check that
$\frac{g(l)}{\Delta}$ has a root modulo $p$. Let
$q(l)=\frac{g(l)}{\Delta}\in\mathbb{Z}[l]$. We have that
$\begin{split}q^{\prime}(l)&=(6a-3)\Delta-12a+36+2(6\Delta-6a-36)l+3(2a+3)l^{2}\\\
&=3(2a^{3}+5a^{2}+11a+3)+6(2a^{2}+4a+6)l+3(2a+3)l^{2}.\end{split}$
If $2a+3\equiv 0\,(\mathrm{mod}\,p)$, then $a\equiv-3\cdot
2^{-1}\,(\mathrm{mod}\,p)$, and replacing in the congruence $a^{2}+3a+9\equiv
0\,(\mathrm{mod}\,p)$ gives that $27\equiv 0\,(\mathrm{mod}\,p)$, which is a
contradiction because $p>3$. Then $2a+3\not\equiv 0\,(\mathrm{mod}\,p)$; let
$(2a+3)^{-1}$ be its modular inverse modulo $p$. Since
$\mathrm{disc}(q^{\prime})=2^{2}3^{3}\Delta\equiv 0\,(\mathrm{mod}\,p)$,
$q(l)$ and $q^{\prime}(l)$ share a root modulo $p$, which is
$l\equiv-(2a^{2}+4a+6)(2a+3)^{-1}\,(\mathrm{mod}\,p)$. In particular this is a
root of $q(l)$ modulo $p$, and since $p^{2}\mid\Delta$, it is a root of $g(l)$
modulo $p^{3}$. ∎
From the previous proof we can extract a simple relation between $k$ and $l$.
###### Corollary 4.3.
Let $K=\mathbb{Q}(\rho)$ be a simplest cubic field such that
$p^{2}\mid\Delta$. The integers $k$, $l$ in the proof of Proposition 4.2
satisfy $2k-l\equiv-2\,(\mathrm{mod}\,p)$. In particular, this holds if $K$
has integral basis $B_{p}(k,l)$ with $p>3$.
###### Proof.
We have that $l\equiv-(2a^{2}+4a+6)(2a+3)^{-1}\,(\mathrm{mod}\,p)$ and
$k\equiv-3^{-1}(al+a^{2}+2a+6)\,(\mathrm{mod}\,p)$. Then
$\begin{split}2k-l&\equiv-2\cdot 3^{-1}(al+a^{2}+2a+6)-l\\\
&\equiv-3^{-1}((2a+3)l+2(a^{2}+2a+6))\\\
&\equiv-3^{-1}(-2a^{2}-4a-6+2a^{2}+4a+12)\equiv-2\,(\mathrm{mod}\,p)\end{split}$
The last part of the statement holds because of the uniqueness of $k$ and $l$
given by Lemma 4.1. ∎
###### Corollary 4.4.
Let $p>3$ be a prime number and let $K=\mathbb{Q}(\rho)$ be a simplest cubic
field. Then $K$ has integral basis $B_{p}$ if and only if $p^{2}\mid\Delta$
and $\Delta=p^{2}d$ or $\Delta=9p^{2}d$ for some $d\in\mathbb{Z}$ square-free
and coprime to $3$.
###### Proof.
We already know that if $K$ has integral basis $B_{p}$, then $p^{2}\mid\Delta$
and $\Delta=p\mathfrak{c}$. Looking at the description of the conductor in
Proposition 3.4 and Remark 3.5, we see that $\Delta=p^{2}d$ or $9p^{2}d$ for
some square-free integer $d$ coprime to $3$, depending on which of the two
possible description of $\mathfrak{c}$ occurs. Conversely, if
$p^{2}\mid\Delta$ and $\Delta=p^{2}d$ or $9p^{2}d$ with $d$ square-free and
coprime to $3$, by Proposition 4.2 we have that
$\frac{k+l\rho+\rho^{2}}{p}\in\mathcal{O}_{K}$ for unique integers $1\leq
k,l\leq p-1$. Now,
$\mathrm{disc}(1,\rho,\frac{k+l\rho+\rho^{2}}{p})=\Big{(}\frac{\Delta}{p}\Big{)}^{2}$.
Recall from Remark 3.5 that either $3\nmid\mathfrak{c}$ or
$v_{3}(\mathfrak{c})=2$. Thus the hypothesis gives that
$\frac{\Delta}{p}=\mathfrak{c}$, so the discriminant of $B_{p}$ is just the
discriminant of $K$. Then $B_{p}$ is an integral basis of $K$. ∎
###### Remark 4.5.
If $K$ is a simplest cubic field with $[\mathcal{O}_{K}:\mathbb{Z}[\rho]]=p$
for $p>3$, then $\Delta=p\mathfrak{c}$, so $p^{2}\mid\Delta$ and
$\Delta=p^{2}d$ or $9p^{2}d$ for some square-free integer $d$. Then, we can
argue as in the second part of the previous proof to conclude that $B_{p}$ is
an integral basis of $K$.
###### Example 4.6.
The simplest cubic field $K$ defined by $a=90$ has integral basis $B_{7}(4,3)$
and $\Delta=9\cdot 7^{2}\cdot 19$. In this case, we have that $3\mid a$ and
$a\equiv 41\,(\mathrm{mod}\,49)$.
We may interpret the condition that $p^{2}\mid\Delta$ as an equation
$a^{2}+3a+9\equiv 0\,(\mathrm{mod}\,p^{2})$ in $a$. Then, in order to have a
simplest cubic field with integral basis $B_{p}$ we need this equation to be
solvable. If we call $h(x)=x^{2}+3x+9$, we know by Hensel’s lemma that any
factorization of $h$ modulo $p$ lifts to a factorization modulo $p^{2}$. Now,
if the equation is solvable, $h$ has two simple roots modulo $p$, so they lift
to two simple roots modulo $p^{2}$. We conclude that for a prime $p$ such that
the equation above is solvable, there is a simplest cubic field with integral
basis $B_{p}$ for exactly two values of $a$ modulo $p^{2}$ (and each such an
integer $a$ produces a simplest cubic field of this type unless
$a\in\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$). In the next result, we find a
nice characterization of the condition that $p^{2}\mid\Delta$.
###### Lemma 4.7.
Let $p>3$ be a prime number. The equation $a^{2}+3a+9\equiv
0\,(\mathrm{mod}\,p^{2})$ is solvable if and only if $p\equiv
1\,(\mathrm{mod}\,6)$.
###### Proof.
Let us call $h(x)=x^{2}+3x+9$. We have that $\mathrm{disc}(h)=-27$, so the
equation in the statement is solvable only if $-27$ is a square modulo $p$. If
we consider the Legendre symbol, we obtain
$\genfrac{(}{)}{}{0}{-27}{p}=\genfrac{(}{)}{}{0}{-1}{p}\genfrac{(}{)}{}{0}{3}{p}^{3}.$
Then the result follows easily by determining the value of each factor in
terms of $p$. ∎
###### Remark 4.8.
If $a\in\mathbb{Z}$ is such that $a^{2}+3a+9\equiv 0\,(\mathrm{mod}\,p^{2})$,
the other solution of this equation in congruences is the class of
$a^{2}+2a+6$ modulo $p^{2}$, that is, $(a^{2}+2a+6)^{2}+3(a^{2}+2a+6)+9\equiv
0\,(\mathrm{mod}\,p^{2})$.
###### Proposition 4.9.
For each prime number $p>3$, there is a simplest cubic field with integral
basis $B_{p}$ if and only if $p\equiv 1\,(\mathrm{mod}\,6)$.
###### Proof.
Assume that $K$ is a simplest cubic field with integral basis $B_{p}$. Then we
know that $p^{2}\mid\Delta$, so that the equation $a^{2}+3a+9\equiv
0\,(\mathrm{mod}\,p^{2})$ is solvable. By Lemma 4.7, we deduce that $p\equiv
1\,(\mathrm{mod}\,6)$. Conversely, assume that $p\equiv 1\,(\mathrm{mod}\,6)$.
Then the equation $a^{2}+3a+9\equiv 0\,(\mathrm{mod}\,p^{2})$ has two
solutions, namely it is satisfied for two values of $a$ modulo $p^{2}$. Among
the values of $a\in\mathbb{Z}$ satisfying the equation above modulo $p^{2}$,
we claim that there is some for which $\frac{a^{2}+3a+9}{p^{2}}$ is square-
free and $a\notin\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$. Indeed, if for
$a_{0}\in\mathbb{Z}$ we have $\Delta_{0}\coloneqq a_{0}^{2}+3a_{0}+9\equiv
0\,(\mathrm{mod}\,p^{2})$, the integers $a_{k}=a_{0}+kp^{2}$,
$k\in\mathbb{Z}$, also satisfy this equation. Now, for $\Delta_{k}\coloneqq
a_{k}^{2}+3a_{k}+9$, we have
$\frac{\Delta_{k}}{p^{2}}=p^{2}k^{2}+(2a_{0}+3)k+\frac{\Delta_{0}}{p^{2}}.$
This is a quadratic polynomial in $k$, so it is well known that it takes
infinitely many square-free values, see [25]. By Corollary 4.4, the simplest
cubic field $K$ with some such a parameter $a$ has integral basis $B_{p}$. ∎
Thus, given a prime $p>3$ with $p\equiv 1\,(\mathrm{mod}\,6)$, we can
determine the family of simplest cubic fields with integral basis $B_{p}$ as
follows:
* •
We find the two values of $a$ modulo $p^{2}$ for which $a^{2}+3a+9\equiv
0\,(\mathrm{mod}\,p^{2})$.
* •
For each of those values, we find the pair $(k,l)\in{\mathbb{Z}}_{p}^{2}$ so
that $\frac{k+l\rho+\rho^{2}}{p}\in{\mathcal{O}}_{K}$.
In Table 1, we show the two values of $a$ for which there is a simplest cubic
field with integral basis $B_{p}$, for some low values of $p>3$, $p\equiv
1\,(\mathrm{mod}\,6)$. It is implicitly assumed that
$a\notin\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$ and $\Delta=p^{2}d$ or
$9p^{2}d$ for $d\in\mathbb{Z}$ square-free.
$p$ | Values of $a$ mod $p^{2}$ | $(k,l)$
---|---|---
$7$ | $5$ | $(2,6)$
$41$ | $(4,3)$
$13$ | $66$ | $(3,8)$
$100$ | $(9,7)$
$19$ | $154$ | $(11,5)$
$204$ | $(7,16)$
$31$ | $356$ | $(25,21)$
$602$ | $(5,12)$
$37$ | $374$ | $(10,22)$
$992$ | $(26,17)$
$43$ | $577$ | $(36,31)$
$1269$ | $(6,14)$
$p$ | Values of $a$ mod $p^{2}$ | $(k,l)$
---|---|---
$61$ | $1259$ | $(47,35)$
$2459$ | $(13,28)$
$67$ | $2097$ | $(37,9)$
$2389$ | $(29,60)$
$73$ | $1265$ | $(64,57)$
$4061$ | $(8,18)$
$79$ | $1096$ | $(55,33)$
$5142$ | $(23,48)$
$97$ | $4451$ | $(35,72)$
$4955$ | $(61,27)$
$103$ | $271$ | $(46,94)$
$10335$ | $(56,11)$
Table 1: Families of simplest cubic fields with integral basis $B_{p}$ for
$7\leq p\leq 103$
### 4.1 Lattice points in the first parallelepiped
Let $K$ be a simplest cubic field with integral basis $B_{p}(k,l)$ for $1\leq
k,l\leq p-1$. We describe the lattice points in the first parallelepiped
following the description obtained in the general case.
Namely, we have the points
$\frac{1}{p}(m+n\rho+o\rho^{2}),\quad 0\leq n\leq 2p,\quad\frac{n}{2}\leq
m,o\leq\frac{n}{2}+p.$
Let us assume that $p$ divides $n$. Then $n=n^{\prime}p$, $0\leq
n^{\prime}\leq 2$. Therefore, the lattice point above is
$\frac{m}{p}+n^{\prime}\rho+\frac{o}{p}\rho^{2}=\frac{m-ko}{p}+\Big{(}n^{\prime}-\frac{lo}{p}\Big{)}\rho+o\frac{k+l\rho+\rho^{2}}{p}.$
Since $k$ and $l$ are not divisible by $p$, this belongs to $\mathcal{O}_{K}$
if and only if $p\mid m$ and $p\mid o$. From this, we obtain the candidate
$1+\rho+\rho^{2}$; the other choices give sums of totally positive elements.
Now assume that $p$ does not divide $n$, so that
$n\in\\{1,\dots,p-1\\}\cup\\{p+1,\dots,2p-1\\}$. Then, we have
$\frac{m}{p}+\frac{n}{p}\rho+\frac{o}{p}\rho^{2}=\frac{m-ko}{p}+\frac{n-lo}{p}\rho+o\frac{k+l\rho+\rho^{2}}{p},$
which belongs to $\mathcal{O}_{K}$ if and only if $m\equiv
ko\,(\mathrm{mod}\,p)$ and $n\equiv lo\,(\mathrm{mod}\,p)$.
Let $n\in\\{1,\dots,p-1\\}\cup\\{p+1,\dots,2p-1\\}$. We claim that there are
unique $\frac{n}{2}\leq m,o\leq\frac{n}{2}+p$ such that the congruences above
are satisfied. Since $K$ has integral basis $B_{p}(k,l)$, Corollary 4.3 gives
that $2k-l\equiv-2\,(\mathrm{mod}\,p)$. Now, since the interval depending on
$n$ has length $p$, the only exception to uniqueness would be that $m$ or
$o\equiv\frac{n}{2}\,(\mathrm{mod}\,p)$. If
$m\equiv\frac{n}{2}\,(\mathrm{mod}\,p)$, since also $n\equiv
lo\,(\mathrm{mod}\,p)$, we obtain that $2ko\equiv lo\,(\mathrm{mod}\,p)$. If
it was $o\equiv 0\,(\mathrm{mod}\,p)$, then $p$ divides $n$, which is against
the assumption on $n$. Then $o\not\equiv 0\,(\mathrm{mod}\,p)$ and $2k\equiv
l\,(\mathrm{mod}\,p)$, which contradicts that
$2k-l\equiv-2\,(\mathrm{mod}\,p)$. If $o\equiv\frac{n}{2}\,(\mathrm{mod}\,p)$,
the congruence $n\equiv lo\,(\mathrm{mod}\,p)$ gives that $l\equiv
2\,(\mathrm{mod}\,p)$. Now, since $2k-l\equiv-2\,(\mathrm{mod}\,p)$, we have
that $k$ is divisible by $p$, which is again a contradiction. This proves the
claim. Therefore, we have $2p-2$ lattice points arising from these choices.
###### Proposition 4.10.
Let $K$ be a simplest cubic field with integral basis of the form (2). Then
the lattice points in the first parallelepiped which are algebraic integers
are
$\frac{1}{p}(m+n\rho+o\rho^{2}),$
where, for each $1\leq n\leq 2p-1$, $m$ and $o$ are the unique integer numbers
in the interval $[\frac{n}{2},\frac{n}{2}+p]$ satisfying the congruences
$m\equiv ko\,(\mathrm{mod}\,p)$ and $n\equiv lo\,(\mathrm{mod}\,p)$.
## 5 The family with $p=3$
From now on, we focus on the simplest cubic fields $K$ with integral basis
$B_{p}$ for $p=3$ and $(k,l)=(1,1)$, that is,
$B_{3}=\Big{\\{}1,\rho,\frac{1+\rho+\rho^{2}}{3}\Big{\\}}.$
We first describe them in terms of the parameter $a$.
###### Proposition 5.1.
The simplest cubic fields $K$ with integral basis
$\Big{\\{}1,\rho,\frac{1+\rho+\rho^{2}}{3}\Big{\\}}$ are the ones for which
$a\equiv 3,21\,(\mathrm{mod}\,27)$, $a>12$ and $\frac{\Delta}{27}$ is square-
free.
###### Proof.
Assume that $K$ has an integral basis as in the statement. Then
$\mathrm{disc}(K)=\mathrm{disc}(1,\rho,\frac{1+\rho+\rho^{2}}{3})=\Big{(}\frac{\Delta}{3}\Big{)}^{2}$,
so $\Delta=3\mathfrak{c}$ and $K$ is not monogenic. In particular, $3$ divides
$\Delta=a^{2}+3a+9$, so $3\mid a$ and $9\mid\Delta$. Thus $3\mid\mathfrak{c}$,
so Remark 3.5 gives that $9\mid\mathfrak{c}$, whence $27\mid\Delta$. Since
$p^{2}$ does not divide $\mathfrak{c}$ for every prime $p\neq 3$, we obtain
that $\frac{\Delta}{27}$ is square-free. On the other hand, by Theorem 3.1
(3), the non-monogenity of $K$ gives that $a\equiv 3$ or $21$
$(\mathrm{mod}\,27)$. Now, since $K$ is not monogenic, $K$ is defined by a
single value of $a$. That is, $a\notin\\{-1,0,1,2,3,5,12,54,66,1259,2389\\}$.
But in this list no element greater than $12$ is congruent to $3$ or $21$ mod
$27$, so $a>12$.
Conversely, let $K$ be a simplest cubic field defined by $a$ such that
$a\equiv 3,21\,(\mathrm{mod}\,27)$ (so that $\Delta$ is divisible by $27$),
$a>12$ and $\frac{\Delta}{27}$ is square-free. Then, we can write $\Delta=27d$
with $d$ square-free and from Proposition 3.4 we see that $\mathfrak{c}=9d$,
so $\delta=3$. In particular, $K$ is not monogenic. On the other hand,
$\frac{1+\rho+\rho^{2}}{3}$ is an algebraic integer because the coefficients
of its minimal polynomial over $\mathbb{Q}$ are integers. Now,
$\mathrm{disc}(1,\rho,\frac{1+\rho+\rho^{2}}{3})=\Big{(}\frac{\Delta}{3}\Big{)}^{2}=\mathfrak{c}^{2}$,
so $K$ has integral basis
$\Big{\\{}1,\rho,\frac{1+\rho+\rho^{2}}{3}\Big{\\}}$. ∎
###### Remark 5.2.
In the first part of the previous proof, we only have used that
$\Delta=3\mathfrak{c}$. We deduce that the only simplest cubic fields with
$[\mathcal{O}_{K}:\mathbb{Z}[\rho]]=3$ are the ones with integral basis
$B_{3}$.
###### Remark 5.3.
Not all simplest cubic fields with $a\equiv 3,21\,(\mathrm{mod}\,27)$ satisfy
that $\frac{\Delta}{27}$ is square-free. For instance, if $a=678\equiv
3\,(\mathrm{mod}\,27)$, then $v_{7}(\Delta)=2$. In that case, the integral
basis of $K$ is
$\Big{\\{}1,\rho,\frac{4+10\rho+\rho^{2}}{21}\Big{\\}}.$
Let us write $a=27t+r$, where $r\in\\{3,21\\}$. Then,
$\frac{\Delta}{27}=\begin{cases}27t^{2}+9t+1&\hbox{ if }a\equiv
3\,(\mathrm{mod}\,27),\\\ 27t^{2}+45t+19&\hbox{ if }a\equiv
21\,(\mathrm{mod}\,27).\end{cases}$
These are quadratic polynomials in $t$, so they take infinitely many square-
free values. Then, we have infinitely many simplest cubic fields within this
family.
Our family of simplest cubic fields is just the family considered by Cánovas
Orvay in [4, Theorem 3].
###### Lemma 5.4.
The simplest cubic fields $K$ with integral basis $B_{3}(1,1)$ are just the
fields of the form $K=\mathbb{Q}(\theta)$ with $\theta^{3}-p\theta+pq=0$,
where $p=9d$ with $d\in\mathbb{Z}$ square-free and $q\in\mathbb{Z}$ is such
that $q>2$, $3\nmid q$ and $4p-27q^{2}=9$.
###### Proof.
Let $K=\mathbb{Q}(\rho)$ be a simplest cubic field as in the statement.
Setting $\theta=-\rho+\frac{a}{3}$, we have $K=\mathbb{Q}(\theta)$. Let
$p=\frac{a^{2}+3a+9}{3}$ and $q=\frac{2a+3}{9}$. Since $\Delta=27d$, we have
$\begin{split}\theta^{3}-p\theta+pq&=-\rho^{3}+\rho^{2}a-\rho\frac{a^{2}}{3}+\frac{a^{3}}{27}+\Big{(}\frac{a^{2}}{3}+a+3\Big{)}\rho-\frac{(a^{2}+3a+9)(a-3)}{27}\\\
&=-\rho^{3}+a\rho^{2}+(a+3)\rho+\frac{a^{3}}{27}-\frac{a^{3}-27}{27}\\\
&=-\rho^{3}+a\rho^{2}+(a+3)\rho+1=0,\end{split}$
proving that $\theta$ has minimal polynomial $x^{3}-px+pq$ over $\mathbb{Q}$.
Moreover, we have that
$4p-27q^{2}=4\Big{(}\frac{a^{2}}{3}+a+3\Big{)}-\frac{1}{3}\Big{(}4a^{2}+12a+9\Big{)}=9.$
Conversely, let $p=9d$ with $d\in\mathbb{Z}$ square-free, $q\in\mathbb{Z}$
with $q>2$ and $3\nmid q$ and $4p-27q^{2}=9$. Let $K=\mathbb{Q}(\theta)$ with
$\theta^{3}-p\theta+pq=0$, and define $a=\frac{9q-3}{2}$. From the equality
with $p$ and $q$ we deduce that $q\equiv 1$ or $5$ $(\mathrm{mod}\,6)$, so
$a\in\mathbb{Z}$ and $a\equiv 3$ or $21$ $(\mathrm{mod}\,27)$. Let us call
$\rho=-\theta+\frac{a}{3}$. Going backwards in the chain of equalities above,
we find that $\rho$ has irreducible polynomial $x^{3}-ax^{2}-(a+3)x-1$ over
$\mathbb{Q}$. ∎
###### Corollary 5.5.
Let $K=\mathbb{Q}(\rho)$ be a simplest cubic field with integral basis
$B_{3}(1,1)$. Then $\\{\rho,\rho^{\prime}\\}$ is a system of fundamental units
of $K$.
###### Proof.
Let $\theta_{1}=4\frac{p}{3}-3q\theta-2\theta^{2}$,
$\sigma=\frac{\theta_{1}}{3}$,
$\tau=\frac{1}{q}\Big{(}\frac{q-1}{2}+\frac{q-1}{2}\sigma+\sigma^{2}\Big{)}$
and $\mu=\sigma+\tau$. In [4, Theorem 3 (iii)], it is stated that
$\\{\mu,\mu^{\prime}\\}$ is a system of fundamental units of $K$. Now, we may
check directly that $\mu=\rho^{\prime}+1$. Since Galois automorphisms preserve
systems of fundamental units, we deduce that $\\{\rho+1,\rho^{\prime}+1\\}$ is
a system of fundamental units. Finally, since $\rho+1=-\rho\rho^{\prime}$ and
$\rho^{\prime}+1=-\frac{1}{\rho}$, $\\{\rho,\rho^{\prime}\\}$ is a system of
fundamental units of $K$. ∎
### 5.1 Algebraic integers in parallelepipeds
Following Proposition 4.10, in the first parallelepiped we have the lattice
points
$\frac{1}{3}(m+n\rho+o\rho^{2}),\quad 1\leq n\leq 5,\quad\frac{n}{2}\leq
m,o\leq\frac{n}{2}+3.$
In this case, the congruences therein become $m\equiv n\equiv
o\,(\mathrm{mod}\,3)$, and since $\frac{n}{2}\leq m,n,o\leq\frac{n}{2}+3$, we
have $m=n=o$. Then we obtain the points $n\frac{1+\rho+\rho^{2}}{3}$ for
$1\leq n\leq 5$. The choices $n>1$ give sums of the choice for $n=1$, so the
former gives the unique candidate of this parallelepiped
$\frac{1+\rho+\rho^{2}}{3}.$
For the second parallelepiped, we change the strategy and work with the
integral basis of a simplest cubic field in this family
$\Big{\\{}1,\rho,\frac{1+\rho+\rho^{2}}{3}\Big{\\}}.$
We just rewrite the nodes of the second parallelepiped with respect to this
integral basis. Call $g_{1}=1$, $g_{2}=\rho$,
$g_{3}=\frac{1+\rho+\rho^{2}}{3}$. Then, the second parallelepiped becomes
$\mathcal{D}(g_{1},-g_{1}-g_{2}+3g_{3},-(2a+3)g_{1}-(a^{2}+4a+5)g_{2}+3(a+2)g_{3}),$
and we want to determine its lattice points. That is, we look for
$t_{1},t_{2},t_{3}\in[0,1]$ and integer numbers $m,n,o\in\mathbb{Z}$ such that
$t_{1}g_{1}+t_{2}(-g_{1}-g_{2}+3g_{3})+t_{3}(-(2a+3)g_{1}-(a^{2}+4a+5)g_{2}+3(a+2)g_{3})=mg_{1}+ng_{2}+og_{3}.$
This is equivalent to the following system of equations
$\displaystyle t_{1}-t_{2}-(2a+3)t_{3}$ $\displaystyle=m,$ $\displaystyle-
t_{2}-(a^{2}+4a+5)t_{3}$ $\displaystyle=n,$ (4) $\displaystyle
3t_{2}+3(a+2)t_{3}$ $\displaystyle=o.$
From the second and the third equation we obtain $-3(a^{2}+3a+3)t_{3}=o+3n$.
As in the monogenic case, the choices $t_{3}=0$ or $t_{3}=3(a^{2}+3a+3)$ give
either totally positive units or their sums. Therefore,
$t_{3}=\frac{u}{3(a^{2}+3a+3)},\quad 1\leq u\leq 3(a^{2}+3a+3)-1.$ (5)
From the first and the second equation we obtain that
$t_{1}+(a^{2}+2a+2)t_{3}=m-n$, that is,
$t_{1}=m-n-u\Big{(}\frac{1}{3}-\frac{a+1}{3(a^{2}+3a+3)}\Big{)}.$
Write $u=3w+s$, where $0\leq w\leq a^{2}+3a+2$, $0\leq s\leq 2$ and $s>0$ if
$w=0$. Then
$t_{1}=m-n-w+\frac{w(a+1)}{a^{2}+3a+3}-s\Big{(}\frac{1}{3}-\frac{a+1}{3(a^{2}+3a+3)}\Big{)}.$
Write $w=v(a+2)+r$, where $0\leq v,r\leq a+1$ and $r=0$ if $v=a+1$. Then
$t_{1}=m-n-w+v+\frac{r(a+1)-v}{a^{2}+3a+3}-s\frac{a^{2}+2a+2}{3(a^{2}+3a+3)}.$
Let us define
$e_{1}=\Big{\lceil}-\Big{(}\frac{r(a+1)-v}{a^{2}+3a+3}-s\frac{a^{2}+2a+2}{3(a^{2}+3a+3)}\Big{)}\Big{\rceil}.$
Hence, $t_{1}\in[0,1]$ if and only if $m-n=w-v+e_{1}$. We can easily write
$e_{1}=\Big{\lceil}\frac{s(a^{2}+2a+2)-3(r(a+1)-v)}{3(a^{2}+3a+3)}\Big{\rceil}.$
Then, we have that
* •
For $s=0$, $e_{1}=1$ if $r=0$, and $e_{1}=0$ otherwise.
* •
For $s=1$, $e_{1}=1$ if $3(r(a+1)-v)\leq a^{2}+2a$, and $e_{1}=0$ otherwise.
* •
For $s=2$, $e_{1}=1$ if $3(r(a+1)-v)\leq 2a^{2}+4a+3=2a(a+2)+3$, and $e_{1}=0$
otherwise.
###### Remark 5.6.
In the case $s=0$, $e_{1}=0$ would be also possible if $v=0$ and $r=0$, but
this corresponds to the zero point, which is not totally positive.
On the other hand, replacing the expression (5) of $t_{3}$ in the second
equation of (4) gives
$t_{2}=-n-u\Big{(}\frac{1}{3}+\frac{a+2}{3(a^{2}+3a+3)}\Big{)}.$
Now, replacing $u=3w+s$,
$t_{2}=-n-w-\frac{w(a+2)}{a^{2}+3a+3}-s\Big{(}\frac{1}{3}+\frac{a+2}{3(a^{2}+3a+3)}\Big{)}.$
Write $w=t(a+1)+l$, with $0\leq t\leq a+2$, $0\leq l\leq a$ and $l=0$ if
$t=a+2$. Now,
$t_{2}=-n-w-t+\frac{t-l(a+2)}{a^{2}+3a+3}-s\frac{a^{2}+4a+5}{3(a^{2}+3a+3)}.$
Let us call
$e_{2}=\Big{\lceil}-\Big{(}\frac{t-l(a+2)}{a^{2}+3a+3}-s\frac{a^{2}+4a+5}{3(a^{2}+3a+3)}\Big{)}\Big{\rceil}.$
Then $t_{2}\in[0,1]$ if and only if $n=-w-t-e_{2}$. An easy calculation shows
that
$e_{2}=\Big{\lceil}\frac{s(a^{2}+4a+5)+3(l(a+2)-t)}{3(a^{2}+3a+3)}\Big{\rceil}.$
Thus, in this case:
* •
For $s=0$, $e_{2}=0$ if $l=0$, and $e_{2}=1$ otherwise.
* •
For $s=1$, $e_{2}=1$ if $3(l(a+2)-t)\leq 2a^{2}+5a+3=(2a+3)(a+1)$, and
$e_{2}=2$ otherwise.
* •
For $s=2$, $e_{2}=1$ if $3(l(a+2)-t)\leq a^{2}+a-3=a(a+1)-3$, and $e_{2}=2$
otherwise.
###### Remark 5.7.
In the case $s=0$, $e_{2}=0$ would be also possible if $l=1$ and $t=a+2$, but
we have excluded that combination by taking $l=0$ if $t=a+2$.
Moreover, since $-u=o+3n$, we have that $o=-u+3(w+t+e_{2})$.
This gives the lattice points
$-(v+t-e_{1}+e_{2})g_{1}-(w+t+e_{2})g_{2}+(-u+3(w+t+e_{2}))g_{3}.$
Since $u=3w+s$, we may rewrite them as
$-(v+t-e_{1}+e_{2})g_{1}-(w+t+e_{2})g_{2}+(-s+3(t+e_{2}))g_{3}.$
With respect to the power (non-integral) basis, these become
$-v-\frac{s}{3}+e_{1}-\Big{(}w+\frac{s}{3}\Big{)}\rho+\Big{(}t-\frac{s}{3}+e_{2}\Big{)}\rho^{2}.$
###### Proposition 5.8.
Let $K$ be a simplest cubic field with $a\equiv 3$ or
$21\,(\mathrm{mod}\,27)$, $a>12$ and $\frac{\Delta}{27}$ square-free. The
lattice points of $K$ (and therefore non-unit candidates to be indecomposables
of $K$; up to multiplication by totally positive units) are the point
$\frac{1+\rho+\rho^{2}}{3}$ from the first parallelepiped, and the points
$-(v+t-e_{1}+e_{2})g_{1}-(w+t+e_{2})g_{2}+(-s+3(t+e_{2}))g_{3}$
$=-(v-e_{1})-w\rho+(t+e_{2})\rho^{2}-sg_{3}$
from the second one, where:
* •
$0\leq w\leq a^{2}+3a+2$,
* •
$0\leq s\leq 2$, and $s>0$ if $w=0$,
* •
$v$ and $r$ are defined from $w=v(a+2)+r$, $0\leq v,r\leq a+1$ and $r=0$ if
$v=a+1$,
* •
$t$ and $l$ are defined from $w=t(a+1)+l$, $0\leq t\leq a+2$, $0\leq l\leq a$,
and $l=0$ if $t=a+2$,
* •
$e_{1}=\Big{\lceil}\frac{s(a^{2}+2a+2)+3(v-r(a+1))}{3(a^{2}+3a+3)}\Big{\rceil}$,
* •
$e_{2}=\Big{\lceil}\frac{s(a^{2}+4a+5)+3(l(a+2)-t)}{3(a^{2}+3a+3)}\Big{\rceil}$.
Note that each lattice point $\alpha$ in the second parallelepiped is
completely determined from $s$ and the integers $v$ and $r$ as in Proposition
5.8 such that $\alpha=-(v-e_{1})-(v(a+2)+r)\rho+(t+e_{2})\rho^{2}-sg_{3}$,
because such a choice determines completely the pair $(t,l)$. From now on, we
will denote $\alpha=\alpha_{s}(v,r)$ or, for $w=v(a+2)+r$,
$\alpha=\alpha_{s}(w)$.
### 5.2 Codifferent
In order to decide whether a given candidate is indecomposable or not, the
codifferent appears as a very useful tool. Recall that
$\mathcal{O}_{K}^{\vee}=\\{\delta\in
K\,|\,\mathrm{Tr}(\alpha\delta)\in\mathbb{Z}\,\forall\alpha\in\mathcal{O}_{K}\\}.$
To determine an ${\mathbb{Z}}$-basis of $\mathcal{O}_{K}^{\vee}$, we use [26,
Proposition 4.14]. Recall the notation $g_{1}=1$, $g_{2}=\rho$,
$g_{3}=\frac{1+\rho+\rho^{2}}{3}$. Since $\\{g_{i}\\}_{i=1}^{3}$ is an
integral basis of $K$, a collection $\\{\varphi_{i}\\}_{i=1}^{3}$ of elements
in $K$ satisfying
$\mathrm{Tr}(g_{i}\varphi_{j})=\delta_{ij},\quad i,j=1,2,3,$
is an ${\mathbb{Z}}$-basis of $\mathcal{O}_{K}^{\vee}$. At the same time, we
may write $\varphi_{j}=\sum_{k=1}^{3}a_{jk}g_{k}$, $a_{jk}\in\mathbb{Q}$. Then
$\delta_{ij}=\mathrm{Tr}(g_{i}\varphi_{j})=\sum_{k=1}^{3}a_{jk}\mathrm{Tr}(g_{i}g_{k}).$
We can express these equalities as
$\begin{pmatrix}\mathrm{Tr}(g_{1}^{2})&\mathrm{Tr}(g_{1}g_{2})&\mathrm{Tr}(g_{1}g_{3})\\\
\mathrm{Tr}(g_{1}g_{2})&\mathrm{Tr}(g_{2}^{2})&\mathrm{Tr}(g_{2}g_{3})\\\
\mathrm{Tr}(g_{1}g_{3})&\mathrm{Tr}(g_{2}g_{3})&\mathrm{Tr}(g_{3}^{2})\end{pmatrix}\begin{pmatrix}a_{11}&a_{21}&a_{31}\\\
a_{12}&a_{22}&a_{32}\\\
a_{13}&a_{23}&a_{33}\end{pmatrix}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}.$
Let $M=(\mathrm{Tr}(g_{i}g_{k}))_{i,k=1}^{3}$. Then the coordinates of
$\varphi_{j}$ with respect to the basis $\\{g_{i}\\}_{i=1}^{3}$ is given by
the $j$-th column of the matrix $M^{-1}$. Carrying out the product traces, we
have
$M=\begin{pmatrix}3&a&\frac{a^{2}+3a+9}{3}\\\
a&a^{2}+2a+6&\frac{(a+1)(a^{2}+3a+9)}{3}\\\
\frac{a^{2}+3a+9}{3}&\frac{(a+1)(a^{2}+3a+9)}{3}&\frac{(a^{2}+3a+5)(a^{2}+3a+9)}{9}\end{pmatrix}.$
From this, we find
$M^{-1}=\frac{1}{a^{2}+3a+9}\begin{pmatrix}a^{2}+7a+21&a^{2}+7a+9&-3(a+6)\\\
a^{2}+7a+9&2(a^{2}+3a+3)&-3(2a+3)\\\ -3(a+6)&-3(2a+3)&18\end{pmatrix}.$
Then, we conclude that
$\begin{split}\varphi_{1}&=\frac{1}{a^{2}+3a+9}\Big{(}a^{2}+7a+21+(a^{2}+7a+9)\rho-3(a+6)\frac{1+\rho+\rho^{2}}{3}\Big{)},\\\
\varphi_{2}&=\frac{1}{a^{2}+3a+9}\Big{(}a^{2}+7a+9+2(a^{2}+3a+3)\rho-3(2a+3)\frac{1+\rho+\rho^{2}}{3}\Big{)},\\\
\varphi_{3}&=\frac{1}{a^{2}+3a+9}\Big{(}-3(a+6)-3(2a+3)\rho+18\frac{1+\rho+\rho^{2}}{3}\Big{)}.\end{split}$
When computing codifferent traces of specific elements, we have the following
important remark: If $\alpha=\sum_{i=1}^{3}a_{i}g_{i}\in\mathcal{O}_{K}$ and
$\varphi=\sum_{j=1}^{3}b_{j}\varphi_{j}\in\mathcal{O}_{K}^{\vee}$, then
$\mathrm{Tr}(\varphi\alpha)=\sum_{i,j=1}^{3}a_{i}b_{j}\mathrm{Tr}(g_{i}\varphi_{j})=\sum_{i,j=1}^{3}a_{i}b_{j}\delta_{ij}=\sum_{i=1}^{3}a_{i}b_{i}.$
## 6 Indecomposables
In this part we will prove Theorem 1.2. In the previous sections, we found all
the lattice points in the two parallelepipeds, and now we determine which ones
are indecomposable, either using the codifferent or other means.
We identify the lattice points for the case $w=0$. In this case, $s=0$ would
give the zero point, which is excluded. Otherwise, since we have $v=t=r=l=0$,
then $e_{1}=e_{2}=1$. We obtain the points
$\alpha_{2}(0,0)=-g_{2}+g_{3},\quad\alpha_{1}(0,0)=-g_{2}+2g_{3}.$ (6)
The case $w=(a+1)(a+2)$ is easy as well. This value of $w$ corresponds to
$t=a+2$, $v=a+1$ and $r=l=0$. For $s=0$, we see that $e_{1}=1$ and $e_{2}=0$.
We obtain the point
$\alpha_{0}(a+1,0)=-(2a+2)-(a+2)^{2}g_{2}+3(a+2)g_{3}=-a-(a+1)(a+2)\rho+(a+2)\rho^{2}.$
For $s\neq 0$, we have $e_{1}=e_{2}=1$, giving the point
$\begin{split}\alpha_{s}(a+1,0)&=-(2a+3)-((a+2)^{2}+1)g_{2}+(-s+3(a+3))g_{3}\\\
&=-a-(a+1)(a+2)\rho+(a+3)\rho^{2}-sg_{3}.\end{split}$ (7)
Fixed a value of $s$, we will distribute the lattice points $\alpha_{s}(v,r)$
in four cases depending on the value of $w=v(a+2)+r$.
* (a)
$x(a+2)<w<(x+1)(a+1)$, $0\leq x\leq a-1$, which corresponds to $1\leq r\leq
a-v$. In this case, $v=t=x$ and $l=r+v$.
* (b)
$w=(x+1)(a+1)$, $0\leq x\leq a$. In this case, $v=x$, $t=v+1$, $r=a-v+1$ and
$l=0$.
* (c)
$(x+1)(a+1)<w<(x+1)(a+2)$, $1\leq x\leq a$, which corresponds to $v=x$,
$a-v+2\leq r\leq a+1$, $t=v+1$ and $l=r-(a-v+1)$.
* (d)
$w=(x+1)(a+2)$, $0\leq x\leq a-1$. Then $v=x+1$, $r=0$ and $t=l=v$.
It may be checked easily that these cases, together with $\alpha_{s}(0,0)$,
$s\in\\{1,2\\}$ and $\alpha_{s}(a+1,0)$, $s\in\\{0,1,2\\}$, cover exactly once
all points in Proposition 5.8.
### 6.1 The case $s=0$
If in Proposition 5.8 we take $s=0$, this corresponds to the intersection of
the second parallelepiped with $\mathbb{Z}[\rho]$. Then, we are left just with
the lattice points in $\mathbb{Z}[\rho]$, which have been already determined
in [16]. In other words, the points $\alpha_{s}(v,r)$ with $s=0$ are exactly
the ones that have integer coordinates with respect to the power basis
$\\{1,\rho,\rho^{2}\\}$. With our notation, these are
$\begin{array}[]{ll}-v-w\rho+(v+1)\rho^{2},&\hbox{if
}v(a+2)<w\leq(v+1)(a+1)\hbox{ and }0\leq v\leq a,\\\
-v-w\rho+(v+2)\rho^{2},&\hbox{if }(v+1)(a+1)<w<(v+1)(a+2)\hbox{ and }0\leq
v\leq a,\\\ -(v-1)-w\rho+(v+1)\rho^{2},&\hbox{if }w=v(a+2)\hbox{ and }0\leq
v\leq a+1.\\\ \end{array}$
While the first line corresponds to situations (a) and (b), the second (resp.
third) one corresponds to situation (c) (resp. (d)).
### 6.2 The case $s=1$
Assume that $s=1$. Recall that $w=v(a+2)+r=t(a+1)+l$. Now, we have that
$e_{1}=\begin{cases}1&\hbox{if }3(r(a+1)-v)\leq a(a+2),\\\
0&\hbox{otherwise},\end{cases}$ $e_{2}=\begin{cases}1&\hbox{if
}3(l(a+2)-t)\leq(2a+3)(a+1),\\\ 0&\hbox{otherwise}.\end{cases}$
#### Situation (a)
We first take $x(a+2)<w<(x+1)(a+1)$, $0\leq x\leq a-1$. We know that $x=v$.
Then:
$\begin{split}&3(r(a+1)-v)\leq a(a+2)\\\
\Longleftrightarrow\;&r(a+1)\leq\frac{a}{3}(a+2)+v\\\
\Longleftrightarrow\;&r\leq\Big{\lfloor}\frac{a(a+2)}{3(a+1)}+\frac{v}{a+1}\Big{\rfloor}=\Big{\lfloor}\frac{a}{3}\Big{(}1+\frac{1}{a+1}\Big{)}+\frac{v}{a+1}\Big{\rfloor}=\frac{a}{3}+\Big{\lfloor}\frac{3v+a}{3(a+1)}\Big{\rfloor}.\end{split}$
Now, $3v+a<3(a+1)$ if and only if $v<\frac{2a}{3}+1$. We deduce that
$e_{1}=\begin{cases}1&\hbox{if }0\leq v\leq\frac{2a}{3}\hbox{ and }1\leq
r\leq\frac{a}{3}\\\ &\hbox{or }\frac{2a}{3}+1\leq v\leq a-1,\\\ 0&\hbox{if
}0\leq v\leq\frac{2a}{3}-1\hbox{ and }\frac{a}{3}+1\leq r\leq a-v.\end{cases}$
On the other hand,
$\begin{split}&3(l(a+2)-t)=3(v(a+1)+r(a+2))\leq(2a+3)(a+1)\\\
\Longleftrightarrow\;&r(a+2)\leq\Big{(}\frac{2a}{3}+1-v\Big{)}(a+1)\\\
\Longleftrightarrow\;&r\leq\Big{\lfloor}\Big{(}\frac{2a}{3}+1-v\Big{)}\frac{a+1}{a+2}\Big{\rfloor}=\frac{2a}{3}+1-v+\Big{\lfloor}\frac{1}{a+2}\Big{(}v-\Big{(}\frac{2a}{3}+1\Big{)}\Big{)}\Big{\rfloor}.\end{split}$
Note that the right-side member needs to be positive, i.e. $v<\frac{2a}{3}+1$.
Under this assumption, we have that $e_{2}=1$ if and only if $1\leq
r\leq\frac{2a}{3}-v$ (since $\frac{2a}{3}+1-v<a+2$). We deduce that
$\begin{cases}e_{2}=1&\hbox{if }0\leq v\leq\frac{2a}{3}\hbox{ and }1\leq
r\leq\frac{2a}{3}-v,\\\ e_{2}=2&\hbox{if }0\leq v\leq\frac{2a}{3}\hbox{ and
}\frac{2a}{3}+1-v\leq r\leq a-v\\\ &\hbox{or }\frac{2a}{3}+1\leq v\leq a.\\\
\end{cases}$
Joining all the information together, we have that:
* •
$e_{1}=e_{2}=1$ exactly in the cases:
* –
$0\leq v\leq\frac{a}{3}\hbox{ and }1\leq r\leq\frac{a}{3}$,
* –
$\frac{a}{3}+1\leq v\leq\frac{2a}{3}-1\hbox{ and }1\leq r\leq\frac{2a}{3}-v$.
These correspond to the points
$-2vg_{1}-(w+v+1)g_{2}+(3v+2)g_{3}=-(v-1)-w\rho+(v+1)\rho^{2}-g_{3}.$
* •
$e_{1}=0$ and $e_{2}=1$ if and only if $0\leq v\leq\frac{a}{3}-1$ and
$\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$. These correspond to the points
$-(2v+1)g_{1}-(w+v+1)+(3v+2)g_{3}=-v-w\rho+(v+1)\rho^{2}-g_{3}.$
* •
$e_{1}=1$ and $e_{2}=2$ exactly in the cases:
* –
$\frac{a}{3}+1\leq v\leq\frac{2a}{3}$ and $\frac{2a}{3}-v+1\leq
r\leq\frac{a}{3}$,
* –
$\frac{2a}{3}+1\leq v\leq a-1$.
These correspond to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-g_{3}.$
* •
$e_{1}=0$ and $e_{2}=2$ exactly in the cases:
* –
$0\leq v\leq\frac{a}{3}$ and $\frac{2a}{3}-v+1\leq r\leq a-v$,
* –
$\frac{a}{3}+1\leq v\leq\frac{2a}{3}-1$ and $\frac{a}{3}+1\leq r\leq a-v$.
These correspond to the points
$-(2v+2)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-v-w\rho+(v+2)\rho^{2}-g_{3}.$
#### Situation (b)
Suppose that $w=(x+1)(a+1)$ with $0\leq x\leq a$ and $x=v$. Now, we have
$\begin{split}&3(r(a+1)-v)=3((a-v)(a+2)+1)\leq a(a+2)\\\
\Longleftrightarrow\;&(a-v)(a+2)+1\leq\frac{a}{3}(a+2)\\\
\Longleftrightarrow\;&a-v+\frac{1}{a+2}\leq\frac{a}{3}\\\
\Longleftrightarrow\;&a-v+1\leq\frac{a}{3}\\\
\Longleftrightarrow\;&v\geq\frac{2a}{3}+1.\end{split}$
Therefore, $e_{1}=1$ if $v\geq\frac{2a}{3}+1$, and $e_{1}=0$ otherwise. On the
other hand, since $l=0$ we always have that $e_{2}=1$. Then:
* •
$e_{1}=1$ and $e_{2}=1$ if and only if $\frac{2a}{3}+1\leq v\leq a$. This
corresponds to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-g_{3}.$
* •
$e_{1}=0$ and $e_{2}=1$ if and only if $0\leq v\leq\frac{2a}{3}$. This
corresponds to the points
$-(2v+2)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-v-w\rho+(v+2)\rho^{2}-g_{3}.$
#### Situation (c)
Next, assume that $(x+1)(a+1)+1\leq w<(x+1)(a+2)$ with $1\leq x\leq a$ and
$x=v$. Arguing as in the situation (a), we see that $3(r(a+1)-v)\leq a(a+2)$
if and only if
$r\leq\frac{a}{3}+\Big{\lfloor}\frac{3v+a}{3(a+1)}\Big{\rfloor}$. Since
$\frac{a}{3}+1\geq a-v+2$ if and only if $v\geq\frac{2a}{3}+1$, we deduce that
$e_{1}=\begin{cases}1&\hbox{if }\frac{2a}{3}+1\leq v\leq a\hbox{ and
}a-v+2\leq r\leq\frac{a}{3}+1,\\\ 0&\hbox{if }\frac{2a}{3}+1\leq v\leq a\hbox{
and }\frac{a}{3}+2\leq r\leq a+1\\\ &\hbox{or }1\leq v\leq\frac{2a}{3}.\\\
\end{cases}$
On the other hand,
$\begin{split}&3(l(a+2)-t)=3(l(a+2)-(v+1))\leq(2a+3)(a+1)\\\
\Longleftrightarrow\;&l(a+2)\leq\Big{(}\frac{2a}{3}+1\Big{)}(a+1)+v+1\\\
\Longleftrightarrow\;&l\leq\Big{(}\frac{2a}{3}+1\Big{)}\Big{(}1-\frac{1}{a+2}\Big{)}+\frac{v+1}{a+2}=\frac{2a}{3}+1+\frac{v-\frac{2a}{3}}{a+2}\\\
\Longleftrightarrow\;&r\leq\frac{5a}{3}-v+2+\frac{v-\frac{2a}{3}}{a+2},\end{split}$
as $l=r-(a-v+1)$. Now, we see that
$\Big{\lfloor}\frac{v-\frac{2a}{3}}{a+2}\Big{\rfloor}=\begin{cases}-1&\hbox{if
}v<\frac{2a}{3},\\\ 0&\hbox{if }v\geq\frac{2a}{3}.\end{cases}$
In the first case, we have that $e_{2}=1$ when $a-v+2\leq
r\leq\mathrm{min}(a+1,\frac{5a}{3}-v+1)=a+1$, so there are no further
restrictions on $r$. As for the second one, we have that $e_{2}=1$ when
$a-v+2\leq r\leq\mathrm{min}(a+1,\frac{5a}{3}-v+2)$. This equals
$\frac{5a}{3}-v+2$ unless $v=\frac{2a}{3}$, in which case the minimum is
$a+1$. We conclude that
$e_{2}=\begin{cases}1&\hbox{if }1\leq v\leq\frac{2a}{3},\\\ &\hbox{or
}\frac{2a}{3}+1\leq v\leq a\hbox{ and }a-v+2\leq r\leq\frac{5a}{3}-v+2,\\\
2&\hbox{if }\frac{2a}{3}+1\leq v\leq a\hbox{ and }\frac{5a}{3}-v+3\leq r\leq
a+1.\end{cases}$
We note that $e_{1}=1$ implies that $e_{2}=1$, and consequently the case that
$e_{1}=1$ and $e_{2}=2$ is not possible.
* •
$e_{1}=e_{2}=1$ if and only if $\frac{2a}{3}+1\leq v\leq a\hbox{ and
}a-v+2\leq r\leq\frac{a}{3}+1$. This corresponds to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-g_{3}.$
* •
$e_{1}=0$ and $e_{2}=1$ exactly in the cases:
* –
$1\leq v\leq\frac{2a}{3}$.
* –
$\frac{2a}{3}+1\leq v\leq a$ and $\frac{a}{3}+2\leq r\leq\frac{5a}{3}-v+2$.
This corresponds to the points
$-(2v+2)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-v-w\rho+(v+2)\rho^{2}-g_{3}.$
* •
$e_{1}=0$ and $e_{2}=2$ if and only if $\frac{2a}{3}+2\leq v\leq a$ and
$\frac{5a}{3}-v+3\leq r\leq a+1$. This corresponds to the points
$-(2v+3)g_{1}-(w+v+3)g_{2}+(3v+8)g_{3}=-v-w\rho+(v+3)\rho^{2}-g_{3}.$
#### Situation (d)
Let us take $w=(x+1)(a+2)$ with $0\leq x\leq a-1$. Now, we have $v=x+1$. Since
$r=0$, we always have $3(r(a+1)-v)<a(a+2)$, so $e_{1}=1$. Moreover,
$3(l(a+2)-t)=3v(a+1)\leq(2a+3)(a+1)$ if and only if $v\leq\frac{2a}{3}+1$.
Therefore, $e_{2}=1$ if $v\leq\frac{2a}{3}+1$ and $e_{2}=2$ otherwise. Then:
* •
$e_{1}=1$ and $e_{2}=1$ if and only if $1\leq v\leq\frac{2a}{3}+1$. Together
with the point $-g_{2}+2g_{3}$ from (6) (so that $0\leq v\leq\frac{2a}{3}+1$),
this corresponds to the points
$-2vg_{1}-(w+v+1)g_{2}+(3v+2)g_{3}=-(v-1)-w\rho+(v+1)\rho^{2}-g_{3}.$
* •
$e_{1}=1$ and $e_{2}=2$ if and only if $\frac{2a}{3}+2\leq v\leq a$. Together
with the point from (7) with $s=1$ (so that $\frac{2a}{3}+2\leq v\leq a+1$),
this corresponds to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+5)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-g_{3}.$
### 6.3 The case $s=2$
Assume that $s=2$. In this case,
$e_{1}=\begin{cases}1&\hbox{if }3(r(a+1)-v)\leq 2a(a+2)+3,\\\
0&\hbox{otherwise},\end{cases}$ $e_{2}=\begin{cases}1&\hbox{if
}3(l(a+2)-t)\leq a(a+1)-3,\\\ 2&\hbox{otherwise}.\end{cases}$
#### Situation (a)
We first take $x(a+2)<w<(x+1)(a+1)$, where $x=v$. Then:
$\begin{split}&3(r(a+1)-v)\leq 2a(a+2)+3\\\
\Longleftrightarrow\;&r(a+1)\leq\frac{2a}{3}(a+2)+v+1\\\
\Longleftrightarrow\;&r\leq\frac{2a}{3}\Big{(}1+\frac{1}{a+1}\Big{)}+\frac{v+1}{a+1}=\frac{2a}{3}+\frac{v+1+\frac{2a}{3}}{a+1}.\end{split}$
We deduce that
$e_{1}=\begin{cases}1&\hbox{if }0\leq v\leq\frac{a}{3}-1\hbox{ and }1\leq
r\leq\frac{2a}{3}\\\ &\hbox{or }\frac{a}{3}\leq v\leq a-1,\\\ 0&\hbox{if
}0\leq v\leq\frac{a}{3}-1\hbox{ and }\frac{2a}{3}+1\leq r\leq a-v.\end{cases}$
On the other hand,
$\begin{split}&3(l(a+2)-t)=3(v(a+1)+r(a+2))\leq a(a+1)-3\\\
\Longleftrightarrow\;&r(a+2)\leq\frac{a}{3}(a+1)-1-v(a+1)=\Big{(}\frac{a}{3}-v\Big{)}(a+1)-1\\\
\Longleftrightarrow;&r\leq\Big{(}\frac{a}{3}-v\Big{)}\Big{(}1-\frac{1}{a+2}\Big{)}-\frac{1}{a+2}=\frac{a}{3}-v+\frac{v-\frac{a}{3}-1}{a+2}.\end{split}$
Then,
$e_{2}=\begin{cases}1&\hbox{if }0\leq v\leq\frac{a}{3}-2\hbox{ and }1\leq
r\leq\frac{a}{3}-v-1,\\\ 2&\hbox{if }0\leq v\leq\frac{a}{3}-2\hbox{ and
}\frac{a}{3}-v\leq r\leq a-v\\\ &\hbox{or }\frac{a}{3}-1\leq v\leq
a-1.\end{cases}$
* •
$e_{1}=e_{2}=1$ if and only if $0\leq v\leq\frac{a}{3}-2$ and $1\leq
r\leq\frac{a}{3}-v-1$. This corresponds to the points
$-2vg_{1}-(w+v+1)g_{2}+(3v+1)g_{3}=-(v-1)-w\rho+(v+1)\rho^{2}-2g_{3}.$
* •
Since $e_{2}=1$ implies that $e_{1}=1$, the case that $e_{1}=0$ and $e_{2}=1$
is not possible.
* •
$e_{1}=1$ and $e_{2}=2$ exactly in the cases:
* –
$0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}-v\leq r\leq\frac{2a}{3}$.
* –
$\frac{a}{3}\leq v\leq a-1$.
These correspond to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-2g_{3}.$
* •
$e_{1}=0$ and $e_{2}=2$ if and only if $0\leq v\leq\frac{a}{3}-1$ and
$\frac{2a}{3}+1\leq r\leq a-v$. In this case, we have the points
$-(2v+2)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-v-w\rho+(v+2)\rho^{2}-2g_{3}.$
#### Situation (b)
Suppose that $w=(x+1)(a+1)$ with $0\leq x\leq a$ and $x=v$. Now, we have
$\begin{split}&3(r(a+1)-v)=3((a-v)(a+2)+1)\leq 2a(a+2)+3\\\
\Longleftrightarrow\;&a(a+2)-v(a+2)+1\leq\frac{2a}{3}(a+2)+1\\\
\Longleftrightarrow\;&v(a+2)\geq\frac{a}{3}(a+2)\\\
\Longleftrightarrow\;&v\geq\frac{a}{3}.\end{split}$
We deduce that $e_{1}=1$ if $v\geq\frac{a}{3}$ and $e_{1}=0$ otherwise. On the
other hand, we always have $e_{2}=1$ because $l=0$.
* •
$e_{1}=e_{2}=1$ if and only if $\frac{a}{3}\leq v\leq a$, giving the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-2g_{3}.$
* •
$e_{1}=0$ and $e_{2}=1$ if and only if $0\leq v\leq\frac{a}{3}-1$, which
corresponds to
$-(2v+2)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-v-w\rho+(v+2)\rho^{2}-2g_{3}.$
#### Situation (c)
Next, assume that $(x+1)(a+1)+1\leq w<(x+1)(a+2)$ with $1\leq x\leq a$ and
$x=v$. Arguing as in the situation (a), $3(r(a+1)-v)\leq 2a(a+2)+3$ if and
only if $r\leq\frac{2a}{3}+\frac{v+1+\frac{2a}{3}}{a+1}$. Assume that $0\leq
v\leq\frac{a}{3}-1$. If it was $e_{1}=1$, we would have that
$r\leq\frac{2a}{3}$ but $a-v+2>\frac{2a}{3}$, which is a contradiction. For
$v=\frac{a}{3}$, if it was $e_{1}=1$ we would have that $r\leq\frac{2a}{3}+1$,
but $a-v+2=\frac{2a}{3}+2$, giving again a contradiction. Thus, when
$v\leq\frac{a}{3}$ we always have $e_{1}=0$. Now, suppose that
$v\geq\frac{a}{3}+1$. Again, $e_{1}=1$ if and only if $r\leq\frac{2a}{3}+1$,
and in this case $a-v+2\geq\frac{2a}{3}+1$. Thus,
$e_{1}=\begin{cases}1&\hbox{if }\frac{a}{3}+1\leq v\leq a\hbox{ and }a-v+2\leq
r\leq\frac{2a}{3}+1,\\\ 0&\hbox{if }\frac{a}{3}+1\leq v\leq a\hbox{ and
}\frac{2a}{3}+2\leq r\leq a+1\\\ &\hbox{or }1\leq
v\leq\frac{a}{3}.\end{cases}$
On the other hand,
$\begin{split}&3(l(a+2)-t)=3(l(a+2)-(v+1))\leq a(a+1)-3\\\
\Longleftrightarrow\;&l(a+2)\leq\frac{a}{3}(a+1)+v\\\
\Longleftrightarrow\;&l\leq\frac{a}{3}\frac{a+1}{a+2}+\frac{v}{a+2}\\\
\Longleftrightarrow\;&l\leq\frac{a}{3}+\frac{v-\frac{a}{3}}{a+2}\\\
\Longleftrightarrow\;&r\leq\frac{4a}{3}-v+1+\frac{v-\frac{a}{3}}{a+2}.\end{split}$
If $1\leq v\leq\frac{a}{3}-1$, we have that $e_{2}=1$ if and only if
$a-v+2\leq r\leq\mathrm{min}(a+1,\frac{4a}{3}-v)=a+1$, so there are no further
restrictions on $r$. Otherwise, if $\frac{a}{3}\leq v\leq a$, $e_{2}=1$ if and
only if $a-v+2\leq r\leq\min(a+1,\frac{4a}{3}-v+1)=\frac{4a}{3}-v+1$. We
conclude that
$e_{2}=\begin{cases}1&\hbox{if }1\leq v\leq\frac{a}{3}-1\\\ &\hbox{or
}\frac{a}{3}\leq v\leq a\hbox{ and }a-v+2\leq r\leq\frac{4a}{3}-v+1,\\\
2&\hbox{if }\frac{a}{3}+1\leq v\leq a\hbox{ and }\frac{4a}{3}-v+2\leq r\leq
a+1.\end{cases}$
* •
$e_{1}=e_{2}=1$ exactly in the cases:
* –
$\frac{a}{3}+1\leq v\leq\frac{2a}{3}$ and $a-v+2\leq r\leq\frac{2a}{3}+1$,
* –
$\frac{2a}{3}+1\leq v\leq a$ and $a-v+2\leq r\leq\frac{4a}{3}-v+1$.
These correspond to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-2g_{3}.$
* •
$e_{1}=0$ and $e_{2}=1$ exactly in the cases:
* –
$1\leq v\leq\frac{a}{3}-1$,
* –
$\frac{a}{3}\leq v\leq\frac{2a}{3}-1$ and $\frac{2a}{3}+2\leq
r\leq\frac{4a}{3}-v+1$.
These correspond to the points
$-(2v+2)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-v-w\rho+(v+2)\rho^{2}-2g_{3}.$
* •
$e_{1}=1$ and $e_{2}=2$ if and only if $\frac{2a}{3}+1\leq v\leq a$ and
$\frac{4a}{3}-v+2\leq r\leq\frac{2a}{3}+1$. This corresponds to the points
$-(2v+2)g_{1}-(w+v+3)g_{2}+(3v+7)g_{3}=-(v-1)-w\rho+(v+3)\rho^{2}-2g_{3}.$
* •
$e_{1}=0$ and $e_{2}=2$ exactly in the cases:
* –
$\frac{a}{3}+1\leq v\leq\frac{2a}{3}$ and $\frac{4a}{3}-v+2\leq r\leq a+1$,
* –
$\frac{2a}{3}+1\leq v\leq a$ and $\frac{2a}{3}+2\leq r\leq a+1$.
These correspond to the points
$-(2v+3)g_{1}-(w+v+3)g_{2}+(3v+7)g_{3}=-v-w\rho+(v+3)\rho^{2}-2g_{3}.$
#### Situation (d)
Let us take $w=(x+1)(a+2)$ with $0\leq x\leq a-1$. In this case, $x=v+1$.
First, we always have $3(r(a+1)-v)<2a(a+2)+3$, so $e_{1}=1$. Moreover,
$3(l(a+2)-t)=3v(a+1)\leq a(a+1)-3$ if and only if $v\leq\frac{a}{3}-1$.
Therefore, $e_{2}=1$ if $v\leq\frac{a}{3}-1$ and $e_{2}=2$ otherwise. Then:
* •
$e_{1}=1$ and $e_{2}=1$ if and only if $1\leq v\leq\frac{a}{3}-1$. Together
with the point $-g_{2}+g_{3}$ from (6) (so that $0\leq v\leq\frac{a}{3}-1$),
this corresponds to the points
$-2vg_{1}-(w+v+1)g_{2}+(3v+1)g_{3}=-(v-1)-w\rho+(v+1)\rho^{2}-2g_{3}.$
* •
$e_{1}=1$ and $e_{2}=2$ if and only if $\frac{a}{3}\leq v\leq a$. Together
with the point in (7) with $s=2$ (so that $\frac{a}{3}\leq v\leq a+1$), this
corresponds to the points
$-(2v+1)g_{1}-(w+v+2)g_{2}+(3v+4)g_{3}=-(v-1)-w\rho+(v+2)\rho^{2}-2g_{3}.$
### 6.4 Representing the lattice points
We list all lattice points in the second parallelepiped in Tables 2, 3 and 4
for the cases $s=0$, $s=1$ and $s=2$, respectively. We call $P_{i}$ (resp.
$R_{i}$, resp. $S_{i}$) the set of points in the $i$-th row of Table 2 (resp.
3, resp. 4).
$P_{i}$ | Point | Situation | Region
---|---|---|---
$P_{1}$ | $-v-w\rho+(v+1)\rho^{2}$ | (a) | $0\leq v\leq a-1$, $1\leq r\leq a-v$
$P_{2}$ | (b) | $0\leq v\leq a$, $r=a-v+1$
$P_{3}$ | $-v-w\rho+(v+2)\rho^{2}$ | (c) | $0\leq v\leq a$, $a-v+2\leq r\leq a+1$
$P_{4}$ | $-(v-1)-w\rho+(v+1)\rho^{2}$ | (d) | $0\leq v\leq a+1$, $r=0$
Table 2: Lattice points in the second parallelepiped with $s=0$ $R_{i}$ | Point | Situation | Region
---|---|---|---
$R_{1}$ | $-(v-1)-w\rho+(v+1)\rho^{2}-g_{3}$ | (a) | $0\leq v\leq\frac{a}{3}$, $1\leq r\leq\frac{a}{3}$
$R_{2}$ | (a) | $\frac{a}{3}+1\leq v\leq\frac{2a}{3}-1$, $1\leq r\leq\frac{2a}{3}-v$
$R_{3}$ | (d) | $0\leq v\leq\frac{2a}{3}+1$, $r=0$
$R_{4}$ | $-v-w\rho+(v+1)\rho^{2}-g_{3}$ | (a) | $0\leq v\leq\frac{a}{3}-1$, $\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$
$R_{5}$ | $-(v-1)-w\rho+(v+2)\rho^{2}-g_{3}$ | (a) | $\frac{a}{3}+1\leq v\leq\frac{2a}{3}$, $\frac{2a}{3}-v+1\leq r\leq\frac{a}{3}$
$R_{6}$ | (a) | $\frac{2a}{3}+1\leq v\leq a-1$, $1\leq r\leq a-v$
$R_{7}$ | (b) | $\frac{2a}{3}+1\leq v\leq a$, $r=a-v+1$
$R_{8}$ | (c) | $\frac{2a}{3}+1\leq v\leq a$, $a-v+2\leq r\leq\frac{a}{3}+1$
$R_{9}$ | (d) | $\frac{2a}{3}+2\leq v\leq a+1$, $r=0$
$R_{10}$ | $-v-w\rho+(v+2)\rho^{2}-g_{3}$ | (a) | $0\leq v\leq\frac{a}{3}$, $\frac{2a}{3}-v+1\leq r\leq a-v$
$R_{11}$ | (a) | $\frac{a}{3}+1\leq v\leq\frac{2a}{3}-1$, $\frac{a}{3}+1\leq r\leq a-v$
$R_{12}$ | (b) | $0\leq v\leq\frac{2a}{3}$, $r=a-v+1$
$R_{13}$ | (c) | $1\leq v\leq\frac{2a}{3}$, $a-v+2\leq r\leq a+1$
$R_{14}$ | (c) | $\frac{2a}{3}+1\leq v\leq a$, $\frac{a}{3}+2\leq r\leq\frac{5a}{3}-v+2$
$R_{15}$ | $-v-w\rho+(v+3)\rho^{2}-g_{3}$ | (c) | $\frac{2a}{3}+2\leq v\leq a$, $\frac{5a}{3}-v+3\leq r\leq a+1$
Table 3: Lattice points in the second parallelepiped with $s=1$ $S_{i}$ | Point | Situation | Region
---|---|---|---
$S_{1}$ | $-(v-1)-w\rho+(v+1)\rho^{2}-2g_{3}$ | (a) | $0\leq v\leq\frac{a}{3}-2$, $1\leq r\leq\frac{a}{3}-v-1$
$S_{2}$ | (d) | $0\leq v\leq\frac{a}{3}-1$, $r=0$
$S_{3}$ | $-(v-1)-w\rho+(v+2)\rho^{2}-2g_{3}$ | (a) | $0\leq v\leq\frac{a}{3}-1$, $\frac{a}{3}-v\leq r\leq\frac{2a}{3}$
$S_{4}$ | (a) | $\frac{a}{3}\leq v\leq a-1$, $1\leq r\leq a-v$
$S_{5}$ | (b) | $\frac{a}{3}\leq v\leq a$, $r=a-v+1$
$S_{6}$ | (c) | $\frac{a}{3}+1\leq v\leq\frac{2a}{3}$, $a-v+2\leq r\leq\frac{2a}{3}+1$
$S_{7}$ | (c) | $\frac{2a}{3}+1\leq v\leq a$, $a-v+2\leq r\leq\frac{4a}{3}-v+1$
$S_{8}$ | (d) | $\frac{a}{3}\leq v\leq a+1$, $r=0$
$S_{9}$ | $-v-w\rho+(v+2)\rho^{2}-2g_{3}$ | (a) | $0\leq v\leq\frac{a}{3}-1$, $\frac{2a}{3}+1\leq r\leq a-v$
$S_{10}$ | (b) | $0\leq v\leq\frac{a}{3}-1$, $r=a-v+1$
$S_{11}$ | (c) | $1\leq v\leq\frac{a}{3}-1$, $a-v+2\leq r\leq a+1$
$S_{12}$ | (c) | $\frac{a}{3}\leq v\leq\frac{2a}{3}-1$, $\frac{2a}{3}+2\leq r\leq\frac{4a}{3}-v+1$
$S_{13}$ | $-(v-1)-w\rho+(v+3)\rho^{2}-2g_{3}$ | (c) | $\frac{2a}{3}+1\leq v\leq a$, $\frac{4a}{3}-v+2\leq r\leq\frac{2a}{3}+1$
$S_{14}$ | $-v-w\rho+(v+3)\rho^{2}-2g_{3}$ | (c) | $\frac{a}{3}+1\leq v\leq\frac{2a}{3}$, $\frac{4a}{3}-v+2\leq r\leq a+1$
$S_{15}$ | (c) | $\frac{2a}{3}+1\leq v\leq a$, $\frac{2a}{3}+2\leq r\leq a+1$
Table 4: Lattice points in the second parallelepiped with $s=2$
For a fixed value of $s$, we identify a lattice point $\alpha_{s}(v,r)$ with a
point $(v,r)\in\mathbb{Z}^{2}$. In Figures 1, 3 and 3, we represent the
previous regions in two dimensional planes, where the X-axis (resp. Y-axis)
corresponds to the values of $v$ (resp. $r$).
In the statement of Theorem 1.2, the element $g_{3}$ in (i) is the only
candidate from the first parallelepiped. The elements in (ii), (iii) and (iv)
come from the case $s=0$, the ones in (v) have $s=1$, and the remaining ones
correspond to $s=2$. In the already mentioned figures, we highlight them in
red colour.
$0$$v$$r$$a+1$$\frac{2a}{3}+1$$\frac{a}{3}+1$$a+1$$\frac{2a}{3}+1$$\frac{a}{3}+1$$1$$P_{1}$$P_{2}$$P_{3}$$P_{4}$
Figure 1: Lattice points with $s=0$
$0$$v$$r$$a+1$$\frac{2a}{3}+1$$\frac{a}{3}+1$$a+1$$\frac{2a}{3}+1$$\frac{a}{3}+1$$1$$R_{1}$$R_{2}$$R_{3}$$R_{4}$$R_{5}$$R_{6}$$R_{7}$$R_{8}$$R_{9}$$R_{10}$$R_{11}$$R_{12}$$R_{13}$$R_{14}$$R_{15}$
Figure 2: Lattice points with $s=1$
$0$$v$$r$$a+1$$\frac{2a}{3}+1$$\frac{a}{3}+1$$a+1$$\frac{2a}{3}+1$$\frac{a}{3}+1$$1$$S_{1}$$S_{2}$$S_{3}$$S_{4}$$S_{5}$$S_{6}$$S_{7}$$S_{8}$$S_{9}$$S_{10}$$S_{11}$$S_{12}$$S_{13}$$S_{14}$$S_{15}$
Figure 3: Lattice points with $s=2$
### 6.5 Proof of the main theorem
We proceed to validate Theorem 1.2. We need to prove that the elements listed
therein are indecomposable and that all the other lattice points in the
parallelepipeds are decomposable.
#### 6.5.1 Transformations of lattice points
For later use, we consider the transformations $T_{1}$ and $T_{2}$ introduced
in [16, Section 5.1] and extend them to all elements of $K$. Namely, for
$\alpha\in K$, we define
$T_{1}(\alpha)=\alpha^{\prime}(-1-a-(a^{2}+3a+3)\rho+(a+2)\rho^{2}),\quad
T_{2}(\alpha)=\alpha^{\prime\prime}\rho^{2}.$
The choice $s=0$ corresponds to the ones studied in [16]. Our points
$\alpha_{0}(v,r)$ are represented therein as $\alpha(v,W)$ with $W=r-1$. With
our notation, the definitions of $T_{1}$ and $T_{2}$ in $\blacktriangle$ are:
$\begin{split}&T_{1}(\alpha_{0}(v,r))=(\alpha_{0}(v,r))^{\prime}(-1-a-(a^{2}+3a+3)\rho+(a+2)\rho^{2})=\alpha_{0}(r-1,a+2-v-r),\\\
&T_{2}(\alpha_{0}(v,r))=(\alpha_{0}(v,r))^{\prime\prime}\rho^{2}=\alpha_{0}(a+1-v-r,v+1).\end{split}$
Let us study the behaviour of transformations $T_{1}$ and $T_{2}$ over lattice
points with $s\neq 0$. It turns out that their behaviour over these points is
far more unpredictable than in the case $s=0$. For the sake of simplicity, we
will just work with the transformation $T_{1}$.
Let us assume that $s=1$ and let us identify a lattice point $\alpha_{1}(v,r)$
with a point $(v,r)\in\mathbb{Z}^{2}$. Using some mathematical software, we
can easily check the following:
* •
If $(v,r)\in\bigcup_{i=1}^{3}R_{i}$,
$T_{1}(v,r)=\begin{cases}\Big{(}\frac{2a}{3}+1,0\Big{)}&\hbox{if
}(v,r)=(0,0),\\\
\Big{(}r+\frac{2a}{3},a+2-v-r\Big{)}&\hbox{otherwise}.\end{cases}$
* •
If $(v,r)\in R_{4}$,
$T_{1}(v,r)=\Big{(}r-\frac{a}{3}-1,a+1-v-r\Big{)}.$
* •
If $(v,r)\in\bigcup_{i=5}^{9}R_{i}$,
$T_{1}(v,r)=\begin{cases}\Big{(}r+\frac{2a}{3},a+2-v-r\Big{)}&\hbox{if
}v+r\leq a+2,\\\ \Big{(}r+\frac{2a}{3}-1,2a+4-v-r\Big{)}&\hbox{if
}v+r>a+2.\end{cases}$
* •
If $(v,r)\in\bigcup_{i=10}^{14}R_{i}$,
$T_{1}(v,r)=\begin{cases}\Big{(}r-\frac{a}{3}-1,a+1-v-r\Big{)}&\hbox{if
}v+r\leq a+1,\\\ \Big{(}r-\frac{a}{3}-2,2a+3-v-r\Big{)}&\hbox{if
}v+r>a+1.\end{cases}$
* •
If $(v,r)\in R_{15}$,
$T_{1}(v,r)=\Big{(}r-\frac{a}{3}-2,2a+3-v-r\Big{)}.$
We see that at each case $T_{1}$ behaves as a composition of symmetries and
translations. In the following, we list the images of some regions $R_{i}$ by
$T_{1}$.
###### Lemma 6.1.
Under the previous notations, we have:
* •
$T_{1}(R_{1}\cup R_{2})=R_{14}$,
* •
$T_{1}(R_{3}-\\{(0,0)\\})=\\{v=\frac{2a}{3},\,\frac{a}{3}+1\leq r\leq
a+1\\}\subset R_{12}\cup R_{13}$,
* •
$T_{1}(0,0)=(\frac{2a}{3}+1,0)$,
* •
$T_{1}(R_{10}\cup R_{11})=R_{1}\cup R_{2}$,
* •
$T_{1}(R_{12})=R_{3}-\\{(\frac{2a}{3}+1,0)\\}$,
* •
$T_{1}(R_{5})=R_{8}$,
* •
$T_{1}(R_{9}\cup R_{15})=R_{5}$.
###### Proof.
We may check directly each inequality. Let us do it explicitly for the first
one. Given $(v,r)\in R_{1}\cup R_{2}$, we know that
$T_{1}(v,r)=(v^{\prime},r^{\prime})$, where
$\begin{cases}v^{\prime}=r+\frac{2a}{3},\\\ r^{\prime}=a+2-v-r.\end{cases}$
If $(v,r)\in R_{1}$, we have that $0\leq v\leq\frac{a}{3}$ and $1\leq
r\leq\frac{a}{3}$. We see directly that $\frac{2a}{3}+1\leq v^{\prime}\leq a$.
On the other hand, $\frac{2a}{3}+2-r\leq r^{\prime}\leq a+2-r$. Now,
$\frac{2a}{3}+2-r=\frac{4a}{3}+2-v^{\prime}$ and
$a+2-r=\frac{5a}{3}-v^{\prime}+2$, so $\frac{4a}{3}-v^{\prime}+2\leq
r^{\prime}\leq\frac{5a}{3}-v^{\prime}+2$. If $(v,r)\in R_{2}$, we have that
$\frac{a}{3}+1\leq v\leq\frac{2a}{3}-1$ and $1\leq r\leq\frac{2a}{3}-v$. It is
direct that $\frac{2a}{3}+1\leq v^{\prime}\leq\frac{4a}{3}-v+1\leq a$. On the
other hand, since $v+r\leq\frac{2a}{3}$, we have that
$r^{\prime}\geq\frac{2a}{3}+2$. Finally,
$v^{\prime}+r^{\prime}=\frac{5a}{3}+2-v\leq\frac{4a}{3}+1$, so
$r^{\prime}\leq\frac{4a}{3}-v^{\prime}+1$. Joining the two cases, we see that
$\frac{2a}{3}+1\leq v^{\prime}\leq a$ and $\frac{2a}{3}+2\leq
r^{\prime}\leq\frac{5a}{3}-v^{\prime}+2$. Since all these steps are
reversible, we conclude that $T_{1}(R_{1}\cup R_{2})=R_{14}$. ∎
Now, let us assume that $s=2$ and study the behaviour of $T_{1}$ over the
points $\alpha_{2}(v,r)$. As in the previous case, we identify a lattice point
$\alpha_{2}(v,r)$ with a point $(v,r)\in\mathbb{Z}^{2}$.
* •
If $(v,r)\in S_{1}\cup S_{2}$,
$T_{1}(v,r)=\Big{(}r+\frac{a}{3},a+1-r-v\Big{)}.$
* •
If $(v,r)\in S_{3}\cup S_{4}\cup S_{5}\cup S_{6}\cup S_{7}\cup S_{8}$,
$T_{1}(v,r)=\begin{cases}\Big{(}r+\frac{a}{3},a+1-v-r\Big{)}&\hbox{if }v+r\leq
a+1,\\\ \Big{(}r+\frac{a}{3}-1,2a+3-v-r\Big{)}&\hbox{if }v+r>a+1.\end{cases}$
* •
If $(v,r)\in S_{9}\cup S_{10}\cup S_{11}\cup S_{12}$,
$T_{1}(v,r)=\begin{cases}\Big{(}r-\frac{2a}{3}-1,a-r-v\Big{)}&\hbox{if
}v+r\leq a,\\\ \Big{(}r-\frac{2a}{3}-2,2a-r-v+2\Big{)}&\hbox{if
}v+r>a.\end{cases}$
* •
If $(v,r)\in S_{13}$,
$T_{1}(v,r)=(r+\frac{a}{3}-1,2a+3-v-r).$
* •
If $(v,r)\in S_{14}\cup S_{15}$,
$T_{1}(v,r)=\Big{(}r-\frac{2a}{3}-2,2a+2-v-r\Big{)}.$
Again, we can use these expressions to identify the images of regions $S_{i}$
by $T_{1}$.
###### Lemma 6.2.
Let us identify a lattice point $\alpha_{2}(v,r)$ with a point
$(v,r)\in\mathbb{Z}^{2}$. We have:
* •
$T_{1}(S_{1}\cup S_{2})=S_{12}$,
* •
$T_{1}(S_{3})=S_{6}\cup S_{7}$,
* •
$T_{1}(S_{6}\cup S_{7})=S_{14}\cup S_{15}$,
* •
$T_{1}(S_{12})=S_{9}$.
#### 6.5.2 Proving the indecomposability
First, we prove the indecomposability of the elements in the statement of
Theorem 1.2. We begin with the elements of minimal trace 1.
###### Proposition 6.3.
The elements in Theorem 1.2 (i), (v), (vi), (vii) and (viii) have minimal
trace $1$, and consequently they are indecomposables.
###### Proof.
For (i), i.e. the point $g_{3}$, we just note that
$\varphi_{1}+\varphi_{3}\in\mathcal{O}_{K}^{\vee}$ is a root of the polynomial
$x^{3}-x^{2}+\frac{a+3}{a^{2}+3a+9}x-\frac{2a+3}{(a^{2}+3a+9)^{2}},$
so it is totally positive, and that
$\mathrm{Tr}((\varphi_{1}+\varphi_{3})g_{3})=1.$
For (v), let $0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}+1\leq
r\leq\frac{2a}{3}-v$. The element
$3\varphi_{1}+2\varphi_{3}\in\mathcal{O}_{K}^{\vee}$ is a root of the
polynomial
$x^{3}-3x^{2}+\frac{18}{a^{2}+3a+9}x-\frac{27}{(a^{2}+3a+9)^{2}},$
so it is totally positive. Now, we have
$\begin{split}&\mathrm{Tr}((3\varphi_{1}+2\varphi_{3})(-(2v+1)g_{1}-(v(a+3)+r+1)g_{2}+(3v+2)g_{3})=\\\
&-6v-3+6v+4=1.\end{split}$
Since the elements in (vii) and (viii) are associated with conjugates of
elements in (vi), it is enough to work with the last ones. Let $0\leq
r\leq\frac{a}{3}-1$. We already know that
$\varphi_{1}+\varphi_{3}\in\mathcal{O}_{K}^{\vee,+}$. Now, we have
$\mathrm{Tr}((\varphi_{1}+\varphi_{3})(-(r+1)g_{2}+g_{3}))=1.$
∎
On the other hand, the remaining indecomposables have minimal trace $2$.
###### Proposition 6.4.
The elements in Theorem 1.2 (ii), (iii) and (iv) have minimal trace $2$.
###### Proof.
Let us recover the notation in Section 6.1. We have that $T_{1}$ maps
bijectively the points from (ii) to (iv), from (iv) to (iii), and from (iii)
to (ii). This means that the elements in (iii) and (iv) are associated with
conjugates of elements in (ii). Moreover, the points in (ii) lie in
$\blacktriangle_{0}(a)$. Therefore, it is enough to prove the statement of the
points in (ii).
Let $\alpha=-g_{1}-(r+1)g_{2}+3g_{3}$, $1\leq r\leq\frac{a}{3}$, be a lattice
point as in (ii). Since $\mathrm{Tr}((\varphi_{1}+\varphi_{3})\alpha)=2$, the
minimal trace of $\alpha$ is upper bounded by $2$. Let us assume that there
exists a totally positive element of the codifferent
$\delta=u_{1}\varphi_{1}+u_{2}\varphi_{2}+u_{3}\varphi_{3}$ such that
$\mathrm{Tr}(\alpha\delta)=1$, i.e.,
$\mathrm{Tr}(\alpha\delta)=-u_{1}-(r+1)u_{2}+3u_{3}=1.$
It implies that $u_{1}=-(r+1)u_{2}+3u_{3}-1$. Now we will consider several
totally positive elements, whose product with $\delta$ gives a totally
positive element with a positive trace. We will show that these traces cannot
be all positive at the same time.
First of all, let us consider the element $-g_{1}-rg_{2}+3g_{3}$. This element
is totally positive for $1\leq r\leq\frac{a}{3}$. Note that it is also true
for $r=1$ as $-g_{1}-g_{2}+3g_{3}=\rho^{2}$. Thus, we obtain
$\mathrm{Tr}(\delta(-g_{1}-rg_{2}+3g_{3}))=(r+1)u_{2}-3u_{3}+1-ru_{2}+3u_{3}=u_{2}+1.$
This trace is positive only if $u_{2}\geq 0$. Similarly, when we consider the
totally positive element $-g_{1}-(r+2)g_{2}+3g_{3}$, we get
$\mathrm{Tr}(\delta(-g_{1}-(r+2)g_{2}+3g_{3}))=(r+1)u_{2}-3u_{3}+1-(r+2)u_{2}+3u_{3}=-u_{2}+1.$
This trace is positive only if $u_{2}\leq 0$, which, together with the
previous part, gives $u_{2}=0$.
When we consider the element $g_{3}$, we can conclude that
$\mathrm{Tr}(\delta g_{3})=u_{3},$
which implies $u_{3}>0$. On the other hand, for
$-g_{1}-(\frac{a}{3}+2)g_{2}+2g_{3}$, we see that
$\mathrm{Tr}\Big{(}\delta\Big{(}-g_{1}-\Big{(}\frac{a}{3}+2\Big{)}g_{2}+2g_{3}\Big{)}\Big{)}=-3u_{3}+1+2u_{3}=-u_{3}+1,$
which leads to $u_{3}\geq 0$. Thus, there is no $u_{3}\in\mathbb{Z}$ for which
$\delta$ would be totally positive. ∎
Next, we prove that these are indecomposable.
###### Proposition 6.5.
The elements in Theorem 1.2 (ii), (iii) and (iv) are indecomposable.
###### Proof.
As in Proposition 6.4, it is enough to prove the indecomposability for points
in (ii).
Let $\alpha=-g_{1}-(r+1)g_{2}+3g_{3}$, $1\leq r\leq\frac{a}{3}$, be a lattice
point as in (ii). Since it has minimal trace $2$, its only possible
decomposition is as a sum of elements whose trace after multiplication by
$\varphi_{1}+\varphi_{3}$ is equal to $1$, i.e.
$\alpha=(-v_{1}g_{1}-w_{1}g_{2}+(v_{1}+1)g_{3})+(-v_{2}g_{1}-w_{2}g_{2}+(v_{2}+1)g_{3}).$
First of all, let us discuss the case when $v_{1}<0$, and firstly assume
$v_{1}\leq-2$. If $w_{1}>0$, then
$-v_{1}g_{1}-w_{1}g_{2}+(v_{1}+1)g_{3}<0,$
since $a+1<\rho<a+1+\frac{2}{a}$ and $g_{3}\geq 3$ for $a\geq 21$. Thus, in
this case we must have $w_{1}\leq 0$. Moreover, we can conclude that
$\begin{split}\mathrm{Tr}(-v_{1}g_{1}-w_{1}g_{2}+(v_{1}+1)g_{3})&=-3v_{1}-aw_{1}+(v_{1}+1)\frac{a^{2}+3a+9}{3}\\\
&=\frac{a^{2}+3a}{3}v_{1}-aw_{1}+\frac{a^{2}+3a+9}{3}.\end{split}$
This trace is negative or zero if
$\frac{(a+3)(v_{1}+1)}{3}+\frac{3}{a}\leq w_{1},$
in which case our element is not totally positive. Thus, let us assume that
$\frac{(a+3)(v_{1}+1)}{3}+\frac{3}{a}>w_{1}$. We have
$-v_{1}g_{1}-w_{1}g_{2}+(v_{1}+1)g_{3}=\frac{-2v_{1}+1}{3}-\Big{(}w_{1}-\frac{v_{1}+1}{3}\Big{)}\rho+\frac{v_{1}+1}{3}\rho^{2}.$
Under our assumption on $v_{1}$ and $w_{1}$, we see that the coefficient
before $\rho$ is positive, and the coefficient before $\rho^{2}$ is negative.
Since $-1-\frac{1}{a}<\rho^{\prime}<-1-\frac{1}{2a}$, we can conclude that
$\begin{split}\frac{-2v_{1}+1}{3}-\Big{(}w_{1}-\frac{v_{1}+1}{3}\Big{)}\rho^{\prime}+\frac{v_{1}+1}{3}\rho^{\prime
2}<\frac{-2v_{1}+1}{3}+\Big{(}w_{1}-\frac{v_{1}+1}{3}\Big{)}+\frac{v_{1}+1}{3}\\\
\leq\frac{-2v_{1}+1}{3}+\frac{(a+3)(v_{1}+1)}{3}+\frac{3}{a}-\frac{v_{1}+1}{3}+\frac{v_{1}+1}{3}=\frac{(a+1)(v_{1}+1)}{3}+1+\frac{3}{a}\leq
0\end{split}$
for $a\geq 21$. It implies that if $v_{1}\leq-2$, then the element of this
form is never totally positive.
If $v_{1}=-1$, we obtain the element $1-w_{1}\rho$. If $w_{1}>0$, then
$1-w_{1}\rho<0$, and if $w_{1}<0$, then $1-w_{1}\rho^{\prime}<0$. We obtain a
totally positive element only if $w_{1}=0$, i.e., the element $1$. However, we
know that $\alpha$ is smaller than $1$ for some embedding, so $1$ cannot
appear in its decomposition.
Therefore, we must have $v_{1}\geq 0$, and this is also true for $v_{2}$.
Without loss of generality, we thus must have $v_{1}=1$ and $v_{2}=0$. If
$w_{2}<0$, we get
$-w_{2}\rho^{\prime}+\frac{1+\rho^{\prime}+\rho^{\prime
2}}{3}=\frac{1}{3}-\Big{(}w_{2}-\frac{1}{3}\Big{)}\rho^{\prime}+\frac{1}{3}\rho^{\prime
2}<\frac{1}{3}+w_{2}-\frac{1}{3}+\frac{\Big{(}1+\frac{1}{a}\Big{)}^{2}}{3}=w_{2}+\frac{\Big{(}1+\frac{1}{a}\Big{)}^{2}}{3}<0$
as, clearly, $\frac{\Big{(}1+\frac{1}{a}\Big{)}^{2}}{3}<1$ for $a\geq 21$.
Therefore, $w_{2}\geq 0$.
Similarly, if $w_{1}<0$, we obtain
$-1+\frac{2}{3}-\Big{(}w_{1}-\frac{2}{3}\Big{)}\rho^{\prime}+\frac{2}{3}\rho^{\prime
2}<-1+\frac{2}{3}+w_{1}-\frac{2}{3}+\frac{2}{3}\Big{(}1+\frac{1}{a}\Big{)}^{2}<0.$
Thus, we have $0\leq w_{1},w_{2}\leq r+1\leq\frac{a}{3}+1$.
Now, we will use the remaining root
$-\frac{1}{a+2}<\rho^{\prime\prime}<-\frac{1}{a+3}$. If $w_{1}=0$, we have
$-1+2\frac{1+\rho^{\prime\prime}+\rho^{\prime\prime
2}}{3}<-\frac{1}{3}-\frac{2}{3}\frac{1}{a+3}+\frac{2}{3}\frac{1}{(a+2)^{2}}<0$
for $a\geq 21$. Similarly, if $w_{1}>0$, we obtain
$-1-w_{1}\rho^{\prime\prime}+2\frac{1+\rho^{\prime\prime}+\rho^{\prime\prime
2}}{3}=-\frac{1}{3}-\left(w_{1}-\frac{2}{3}\right)\rho^{\prime\prime}+\frac{2}{3}\rho^{\prime\prime}<-\frac{1}{3}+\left(w_{1}-\frac{2}{3}\right)\frac{1}{a+2}+\frac{2}{3}\frac{1}{(a+2)^{2}}$
The expression on the right side is positive only if
$w_{1}>\frac{a}{3}+\frac{4}{3}-\frac{2}{3(a+2)}>\frac{a}{3}+1$
for $a\geq 21$. This is impossible as we must have $w_{1}\leq\frac{a}{3}+1$.
Therefore, $\alpha$ is indecomposable. ∎
#### 6.5.3 Finding decompositions of lattice points
We have proved that the elements in Theorem 1.2 are indecomposable, and we
have given their minimal traces. Now, we prove that all the other lattice
points in the second parallelepiped are decomposable. Again, we have to
distinguish different cases depending on the value of $s$.
Let us assume that $s=0$. We already know that the points from $P_{3}$ and
$P_{4}$ are decomposable (because they are indecomposables in
$\mathbb{Z}[\rho]$, see [16, Lemma 4.2]). Thus, we are left with the regions
$P_{1}$ and $P_{2}$. Their union correspond to the triangle $\blacktriangle$
(note that since the points in situation (d) are discarded, we can denote
$v=x$).
Recall the definition of $T_{1}$ and $T_{2}$ in Section 6.5.1. Since the
factors multiplying the conjugate of $\alpha_{0}(v,r)$ at each line are
totally positive units, we have that for each $i\in\\{1,2\\}$,
$\alpha_{0}(v,r)$ is indecomposable if and only if so is
$T_{i}(\alpha_{0}(v,r))$. Moreover, for each $\alpha\in\blacktriangle$, there
is $i\in\\{1,2\\}$ such that $T_{i}(\alpha)$ lies in the region
$\blacktriangle_{0}(a)=\Big{\\{}\alpha_{0}(v,r)\,|\,0\leq
v\leq\frac{a}{3}-1,\,v+1\leq r\leq
a-2v\Big{\\}}\cup\Big{\\{}\alpha_{0}\Big{(}\frac{a}{3},\frac{a}{3}+1\Big{)}\Big{\\}}.$
Therefore, it is enough to study the decomposability of the lattice points in
$\blacktriangle_{0}(a)$.
###### Proposition 6.6.
The elements with $s=0$ other than the ones in Theorem 1.2 (ii), (iii) and
(iv) are decomposable.
###### Proof.
We know that it is enough to look at the points in $\blacktriangle_{0}(a)$.
Thus, let us consider $\alpha\in\blacktriangle_{0}(a)$, i.e. we have
$\begin{split}\alpha=-v-(v(a+2)+r)\rho+(v+1)\rho^{2},\end{split}$ (8)
where $0\leq v\leq\frac{a}{3}-1$ and $v+1\leq r\leq a-2v$, or
$\alpha=\alpha_{0}(\frac{a}{3},\frac{a}{3}+1)$. Among these, the points that
are known to be indecomposable (from Proposition 6.5) are the ones in Theorem
1.2 (ii), i.e. for $v=0$ and $1\leq r\leq\frac{a}{3}$.
Let us assume that $0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}+1\leq
r\leq\frac{2a}{3}-v$. Then $\alpha-g_{3}$ is one of the points in Theorem 1.2
(v), so $\alpha$ is decomposable.
If $v=0$ and $\frac{2a}{3}+1\leq r\leq a$, we have that
$\begin{split}\alpha&=\Big{[}-\frac{2a}{3}\rho+\rho^{2}-g_{3}\Big{]}\\\
&+\Big{[}1-\Big{(}r-\frac{2a}{3}-1\Big{)}\rho+\rho^{2}-2g_{3}\Big{]}.\end{split}$
The first summand is $\alpha_{1}(0,\frac{2a}{3})\in R_{4}$. On the other hand,
$0\leq r^{\prime}\leq\frac{a}{3}-1$ for $r^{\prime}=r-\frac{2a}{3}-1$, so the
second summand is the point $\alpha_{2}(0,r^{\prime})$ that belongs to $S_{1}$
if $1\leq r^{\prime}\leq\frac{a}{3}-1$ and to $S_{2}$ if $r^{\prime}=0$.
Therefore, $\alpha$ is decomposable for this case.
Assume that $1\leq v\leq\frac{a}{3}-1$ and $v+1\leq r\leq\frac{a}{3}$. We have
$\begin{split}\alpha&=\Big{[}-\Big{(}\frac{a}{3}+1\Big{)}\rho+\rho^{2}-g_{3}\Big{]}\\\
&+\Big{[}-(v-1)-(v(a+2)+r-\frac{a}{3}-2)\rho+(v+1)\rho^{2}-2g_{3}\Big{]}.\end{split}$
The first summand is $\alpha_{1}(0,\frac{a}{3}+1)\in R_{4}$. The second one
can be rewritten as
$-(v-1)-\Big{(}(v-1)(a+2)+r+\frac{2a}{3}\Big{)}\rho+(v+1)\rho^{2}-2g_{3}.$
For $v^{\prime}=v-1$ and $r^{\prime}=r+\frac{2a}{3}$, this is of the form
$-v^{\prime}-(v^{\prime}(a+2)+r^{\prime})\rho+(v^{\prime}+2)\rho^{2}-2g_{3},$
with $0\leq v^{\prime}\leq\frac{a}{3}-2$.
* •
If $r\leq\frac{a}{3}-v+1$, then $\frac{2a}{3}+1+v\leq r^{\prime}\leq
a-v+1=a-v^{\prime}$ and the point above is
$\alpha_{2}(v^{\prime},r^{\prime})\in S_{9}$.
* •
If $r=\frac{a}{3}-v+2$, then $r^{\prime}=a-v^{\prime}+1$ and we obtain
$\alpha_{2}(v^{\prime},r^{\prime})\in S_{10}$.
* •
In the remaining cases, $a-v^{\prime}+2\leq r^{\prime}\leq a$ and we obtain
$\alpha_{2}(v^{\prime},r^{\prime})\in S_{11}$.
Now, consider the same decomposition for
$\alpha=\alpha_{0}(\frac{a}{3},\frac{a}{3}+1)$, i.e. $v=\frac{a}{3}$ and
$r=\frac{a}{3}+1$. Then $v^{\prime}=\frac{a}{3}-1$ and $r^{\prime}=a+1$, so
that $\alpha_{2}(v^{\prime},r^{\prime})\in S_{11}$.
Assume that $1\leq v\leq\frac{a}{3}-1$ and $\frac{2a}{3}-v+1\leq r\leq a-2v$.
We have
$\begin{split}\alpha&=\Big{[}-\frac{2a}{3}\rho+\rho^{2}-g_{3}\Big{]}\\\
&+\Big{[}-(v-1)-\Big{(}v(a+2)+r-\frac{2a}{3}-1\Big{)}\rho+(v+1)\rho^{2}-2g_{3}\Big{]}.\end{split}$
We already know that the first summand is totally positive. Let us assume that
$\frac{2a}{3}-v+1\leq r\leq\frac{2a}{3}$ (for simplicity, even when
$a-2v\leq\frac{2a}{3}$, which happens if and only if
$v\geq\lceil\frac{a}{6}\rceil$ and in which case $\alpha$ is as in (8)). We
rewrite the second summand as
$-(v-1)-\Big{(}(v-1)(a+2)+r+\frac{a}{3}+1\Big{)}\rho+(v+1)\rho^{2}-2g_{3},$
which is of the form
$-v^{\prime}-(v^{\prime}(a+2)+r^{\prime})\rho+(v^{\prime}+2)\rho^{2}-2g_{3}$
for $v^{\prime}=v-1$ and $r^{\prime}=r+\frac{a}{3}+1$, so that
$a-v^{\prime}+1\leq r^{\prime}\leq a+1$. If $v=1$, then $v^{\prime}=0$ and
$r^{\prime}=a+1$, and we obtain the lattice point $\alpha_{2}(0,a+1)\in
S_{10}$. Otherwise, if $2\leq v\leq\frac{a}{3}-1$, then $1\leq
v^{\prime}\leq\frac{a}{3}-2$, and we obtain a lattice point
$\alpha_{2}(v^{\prime},r^{\prime})$ in $S_{10}$ if $r^{\prime}=a-v^{\prime}+1$
and in $S_{11}$ if $a-v^{\prime}+2\leq r^{\prime}\leq a+1$. Finally, let us
assume that $\frac{2a}{3}+1\leq r\leq a-2v$, which in particular implies that
$1\leq v\leq\lfloor\frac{a-3}{6}\rfloor<\frac{a}{3}-1$. Let
$r^{\prime}=r-\frac{2a}{3}-1$. If $r=\frac{2a}{3}+1$, then $r^{\prime}=0$ and
the second summand above equals $\alpha_{2}(v,0)\in S_{2}$. Otherwise, if
$\frac{2a}{3}+2\leq r\leq a-2v$, then $1\leq r^{\prime}\leq\frac{a}{3}-2v-1$,
and we obtain $\alpha_{2}(v,r^{\prime})\in S_{1}$.
We have seen that all points in $\blacktriangle_{0}(a)$ other than the ones in
Theorem 1.2 (ii) can be written as a sum of totally positive elements, so they
are decomposable. ∎
In the cases with $s\neq 0$, $T_{1}$ also preserves decompositions in both
directions. Hence, we can use it to reduce the list of lattice points for
which we have to find a decomposition.
###### Proposition 6.7.
The elements with $s=1$ other than the ones in Theorem 1.2 (v) are
decomposable.
###### Proof.
Note that the points in Theorem 1.2 (v) are just the ones in $R_{4}$. By Lemma
6.1, it is enough to prove the decomposability of points in $R_{1}$, $R_{2}$,
$R_{3}$, $R_{5}$, $R_{6}$, $R_{7}$ and
$R_{13}-\\{v=\frac{2a}{3},\,\frac{a}{3}+2\leq r\leq a+1\\}$.
We first consider the points of the form
$\alpha=-(v-1)-(v(a+2)+r)\rho+(v+1)\rho^{2}-g_{3},$
which correspond to the regions $R_{1}$, $R_{2}$ and $R_{3}$. We have the
decompositions
$\begin{split}\alpha&=[1+\rho^{2}-2g_{3}]+[-(v-1)-(v(a+2)+r-1)\rho+(v+1)\rho^{2}-2g_{3}]=\alpha_{2}(0,0)+P,\\\
\alpha&=\Big{[}-\frac{2a}{3}\rho+\rho^{2}-g_{3}\Big{]}+\Big{[}-(v-1)-\Big{(}(v-1)(a+2)+r+\frac{a}{3}+2\Big{)}\rho+v\rho^{2}\Big{]}\\\
&=\alpha_{1}\Big{(}0,\frac{2a}{3}\Big{)}+Q.\end{split}$
We will prove that either $P$ or $Q$ are totally positive depending on the
values of $v$ and $r$. Let us suppose that $\alpha\in R_{1}\cup R_{2}$, so
$0\leq v\leq\frac{2a}{3}-1$ and $1\leq
r\leq\mathrm{min}(\frac{a}{3},\frac{2a}{3}-v)$. If $v=0$ and we call
$r^{\prime}=r-1$, then $P=1-r^{\prime}\rho+\rho^{2}-2g_{3}$ with $0\leq
r^{\prime}\leq\frac{a}{3}-1$. For $r=1$, we obtain $P=\alpha_{2}(0,0)\in
S_{2}$, with $(e_{1},e_{2})=(1,1)$, while for $2\leq r\leq\frac{a}{3}$, we
obtain $P=\alpha_{2}(0,r^{\prime})\in S_{1}$. Next, suppose that $1\leq
v\leq\frac{2a}{3}-1$. Let us call $v^{\prime}=v-1$ and
$r^{\prime}=r+\frac{a}{3}+2$. Then
$Q=-v^{\prime}-(v^{\prime}(a+2)+r^{\prime})\rho+(v^{\prime}+1)\rho^{2}$, with
$0\leq v^{\prime}\leq\frac{2a}{3}-2$ and $\frac{a}{3}+3\leq
r^{\prime}\leq\mathrm{min}(\frac{2a}{3},a-v^{\prime}+1)$. Then
$Q=\alpha_{0}(v^{\prime},r^{\prime})\in\blacktriangle$ is totally positive.
Finally, we assume that $\alpha\in R_{3}$, so that $0\leq v\leq\frac{2a}{3}+1$
and $r=0$. For $v=0$, we have $\alpha=1+\rho^{2}-g_{3}=\alpha_{1}(0,0)$, which
we already know is decomposable, and
$T_{1}(\alpha_{1}(0,0))=\alpha_{1}(\frac{2a}{3}+1,0)$, so we may assume that
$1\leq v\leq\frac{2a}{3}$. For $v^{\prime}=v-1$ and
$r^{\prime}=\frac{a}{3}+2$, we have
$Q=-v^{\prime}-(v^{\prime}(a+2)+r^{\prime})\rho+(v^{\prime}+1)\rho^{2}$, with
$0\leq v^{\prime}\leq\frac{2a}{3}-1$ and $0<r^{\prime}<a-v^{\prime}+1$. Then
$Q=\alpha_{0}(v^{\prime},r^{\prime})\in\blacktriangle$ is totally positive.
Next, let us take $\alpha\in R_{5}\cup R_{6}\cup R_{7}$, so that
$\alpha=-(v-1)-(v(a+2)+r)\rho+(v+2)\rho^{2}-g_{3}$
with $\frac{a}{3}+1\leq v\leq\frac{2a}{3}$ and $\frac{2a}{3}-v+1\leq
r\leq\frac{a}{3}$ or $\frac{2a}{3}+1\leq v\leq a$ and $1\leq r\leq a-v+1$. Let
us write
$\alpha=\alpha_{2}(0,0)+[-(v-1)-(v(a+2)+r-1)\rho+(v+2)\rho^{2}-2g_{3}].$
On the other hand, our hypotheses on $v$ and $r$ imply that in particular
$\frac{a}{3}\leq v\leq a$ and $1\leq r\leq a-v+1$, so the second summand is
just $\alpha_{2}(v,r)\in S_{4}\cup S_{5}$, and hence it is totally positive.
Therefore, $\alpha$ is decomposable for this case.
It remains to study the points in the region
$R_{13}-\\{v=\frac{2a}{3},\,\frac{a}{3}+2\leq r\leq a+1\\}$. That is, we
consider
$\alpha=-v-(v(a+2)+r)\rho+(v+2)\rho^{2}-g_{3},$
with $1\leq v\leq\frac{2a}{3}-1$ and $a-v+2\leq r\leq a+1$. Let us write
$\begin{split}\alpha&=\Big{[}-\Big{(}\frac{a}{3}+1\Big{)}\rho+\rho^{2}-g_{3}\Big{]}+[-v-\Big{(}v(a+2)+r-\frac{a}{3}-1\Big{)}\rho+(v+1)\rho^{2}]\end{split}$
The first summand is $\alpha_{1}(0,\frac{a}{3}+1)$, which is totally positive.
As for the second one, let $r^{\prime}=r-\frac{a}{3}-1$ and assume that
$r\leq\frac{4a}{3}-v+1$ (note that this does not impose any restriction if
$1\leq v\leq\frac{a}{3}$). Then $\frac{2a}{3}-v+1\leq
r^{\prime}\leq\mathrm{min}(\frac{2a}{3},a-v)$, so we obtain
$\alpha_{0}(v,r^{\prime})\in\blacktriangle$, which is totally positive. Hence,
$\alpha$ is decomposable. The remaining case corresponds to the points for
which $\frac{a}{3}+1\leq v\leq\frac{2a}{3}-1$ and $\frac{4a}{3}-v+2\leq r\leq
a+1$. But those are the images by $T_{1}$ of the points in $R_{13}$ for which
$1\leq v\leq\frac{a}{3}$, so they are decomposable. ∎
###### Proposition 6.8.
The elements with $s=2$ other than the ones in Theorem 1.2 (vi), (vii) and
(viii) are decomposable.
###### Proof.
Among these lattice points, the indecomposables correspond to:
* •
the line $\\{v=0,\,0\leq r\leq\frac{a}{3}-1\\}$, contained in $S_{1}\cup
S_{2}$,
* •
the line $\\{0\leq v\leq\frac{a}{3}-1,\,r=\frac{2a}{3}+1\\}$, contained in
$S_{9}$,
* •
the line $\\{\frac{a}{3}\leq v\leq\frac{2a}{3}-1,\,r=\frac{4a}{3}-v+1\\}$,
contained in $S_{12}$.
and we know that they are indecomposable from Proposition 6.3. Moreover, by
Lemma 6.2, it is enough to find decompositions for all the other points in the
regions $S_{1}$, $S_{2}$, $S_{3}$, $S_{4}$, $S_{5}$, $S_{8}$, $S_{10}$,
$S_{11}$ and $S_{13}$.
We start with the regions $S_{1}$ and $S_{2}$. Among their points, the line
$\\{\alpha_{2}(0,r)\,|\,0\leq r\leq\frac{a}{3}-1\\}$ are the elements in
Theorem 1.2 (vi), and we know that they are indecomposable from Proposition
6.3. The remaining points are of the form
$\begin{split}\alpha&=-(v-1)-(v(a+2)+r)\rho+(v+1)\rho^{2}-2g_{3}\\\
&=\alpha_{1}\Big{(}0,\frac{2a}{3}\Big{)}+\Big{[}-(v-1)-\Big{(}v(a+2)+r-\frac{2a}{3}\Big{)}\rho+v\rho^{2}-g_{3}\Big{]}\end{split}$
with $1\leq v\leq\frac{a}{3}-1$ and $0\leq r\leq\frac{a}{3}-v-1$. We rewrite
the second summand as
$-v^{\prime}-\Big{(}v^{\prime}(a+2)+r^{\prime}\Big{)}\rho+(v^{\prime}+1)\rho^{2}-g_{3},$
where $v^{\prime}=v-1$ and $r^{\prime}=r+\frac{a}{3}+2$. Since $0\leq
v^{\prime}\leq\frac{a}{3}-2$ and $\frac{a}{3}+2\leq
r^{\prime}\leq\frac{2a}{3}-v^{\prime}$, this is one of the indecomposables in
Theorem 1.2 (v).
Next, we consider the points from the regions $S_{3}$, $S_{4}$, $S_{5}$ and
$S_{8}$, which are of the form
$\alpha=-(v-1)-(v(a+2)+r)\rho+(v+2)\rho^{2}-2g_{3}$
with $0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}-v\leq r\leq\frac{2a}{3}$ or
$\frac{a}{3}\leq v\leq a+1$ and $0\leq r\leq a+1-v$. First we assume that
$r>0$, so that $v\leq a$. Let us write
$\alpha=[1+\rho^{2}-2g_{3}]+[-v-(v(a+2)+r)\rho+(v+1)\rho^{2}].$
Under these hypotheses we have $1\leq r\leq a-v+1$, so the second summand
belongs to $\blacktriangle$ and hence it is totally positive. Now, let us
choose $r=0$ and write
$\alpha=\Big{[}-\Big{(}\frac{a}{3}+1\Big{)}\rho+\rho^{2}-g_{3}\Big{]}+\Big{[}-(v-1)-\Big{(}(v-1)(a+2)+\frac{2a}{3}+1\Big{)}\rho+(v+1)\rho^{2}-g_{3}\Big{]}.$
The first summand is $\alpha_{1}(0,\frac{a}{3}+1)$. The second one can be
rewritten as
$-v^{\prime}-\Big{(}v^{\prime}(a+2)+r^{\prime}\Big{)}\rho+(v^{\prime}+2)\rho^{2}-g_{3},$
where $v^{\prime}=v-1$ and $r^{\prime}=\frac{2a}{3}+1$. This belongs to:
* •
$R_{10}$, if $v=\frac{a}{3}$;
* •
$R_{12}$, if $v=\frac{a}{3}+1$;
* •
$R_{13}$, if $\frac{a}{3}+2\leq v\leq\frac{2a}{3}+1$;
* •
$R_{14}$, if $\frac{2a}{3}+2\leq v\leq a+1$.
Hence, the points in these regions are decomposable.
Now, we consider the points from the regions $S_{10}$ and $S_{11}$, which are
of the form
$\alpha=-v-(v(a+2)+r)\rho+(v+2)\rho^{2}-2g_{3}$
with $0\leq v\leq\frac{a}{3}-1$ and $a-v+1\leq r\leq a+1$. Let us write
$\alpha=\Big{[}-\Big{(}v+r-\frac{2a}{3}\Big{)}\rho+\rho^{2}-g_{3}\Big{]}+\Big{[}-v-\Big{(}v(a+2)+\frac{2a}{3}-v\Big{)}\rho+(v+1)\rho^{2}-g_{3}\Big{]}.$
The second summand is clearly the point $\alpha_{1}(v,\frac{2a}{3}-v)\in
R_{4}$, and consequently totally positive. As for the first one, if we define
$r^{\prime}=v+r-\frac{2a}{3}$, from $a-v+1\leq r\leq a+1$ we obtain that
$\frac{a}{3}+1\leq r^{\prime}\leq\frac{a}{3}+1+v$, and since
$v\leq\frac{a}{3}-1$ we have actually that $\frac{a}{3}+1\leq
r^{\prime}\leq\frac{2a}{3}$. Therefore the first summand is
$\alpha_{1}(0,r^{\prime})\in R_{4}$. Hence it is totally positive, and
$\alpha$ is decomposable.
Finally, we consider the points of the region $S_{13}$, which are of the form
$\alpha=-(v-1)-(v(a+2)+r)\rho+(v+3)\rho^{2}-2g_{3},$
where $\frac{2a}{3}+1\leq v\leq a$ and $\frac{4a}{3}-v+2\leq
r\leq\frac{2a}{3}+1$. We have that
$\alpha=[1+\rho^{2}-2g_{3}]+[-v-(v(a+2)+r)\rho+(v+2)\rho^{2}].$
Since $r>a-v+2$, the second summand is a lattice point $\alpha_{0}(v,r)\in
P_{3}$, so it is totally positive, and $\alpha$ is decomposable. ∎
We conclude that Theorem 1.2 is established from Propositions 6.3, 6.4, 6.5,
6.6, 6.7 and 6.8.
### 6.6 Other families
$p$ | $a$ mod $p^{2}$ | $(k,l)$ | Indecomposables in the first parallelepiped
---|---|---|---
$7$ | $5$ | $(2,6)$ | none
$41$ | $(4,3)$ | none
$13$ | $66$ | $(3,8)$ | $-g_{2}+2g_{3}$, $-g_{1}-3g_{2}+6g_{3}$, $-g_{1}-3g_{2}+5g_{3}$
$100$ | $(9,7)$ | $-g_{1}-g_{2}+2g_{3}$, $-3g_{1}-2g_{2}+5g_{3}$, $-4g_{1}-3g_{2}+6g_{3}$
$19$ | $154$ | $(11,5)$ | $-2g_{1}-g_{2}+4g_{3}$, $-3g_{1}-g_{2}+6g_{3}$, $-5g_{1}-2g_{2}+9g_{3}$
$204$ | $(7,16)$ | $-g_{1}-3g_{2}+4g_{3}$, $-2g_{1}-5g_{2}+6g_{3}$, $-3g_{1}-7g_{2}+9g_{3}$
$31$ | $356$ | $(25,21)$ | $-2g_{1}-2g_{2}+3g_{3}$, $-10g_{1}-8g_{2}+13g_{3}$, $-12g_{1}-10g_{2}+15g_{3}$,
$-6g_{1}-5g_{2}+8g_{3}$, $-7g_{1}-6g_{2}+9g_{3}$, $-11g_{1}-9g_{2}+14g_{3}$
$602$ | $(5,12)$ | $-g_{2}+3g_{3}$, $-2g_{1}-5g_{2}+15g_{3}$, $-2g_{1}-5g_{2}+13g_{3}$,
$-g_{1}-3g_{2}+9g_{3}$, $-g_{1}-3g_{2}+8g_{3}$, $-2g_{1}-5g_{2}+14g_{3}$
Table 5: Indecomposables in the first parallelepiped for low values of $p>3$
Now we will briefly discuss indecomposables for families for other primes $p$.
It seems that the behaviour of indecomposables is different in comparison to
$p=3$, which can be seen even from some partial results. For example, in Table
5, we can find all indecomposable integers in the first parallelepiped formed
by units $1$, $\rho^{2}$ and $\rho^{2}\rho^{\prime 2}$. Recall that for $p=3$,
we have one indecomposable in this parallelepiped, namely
$g_{3}=\frac{1+\rho+\rho^{2}}{3}$. For $p=7$, we do not obtain any
indecomposable integer here. On the other hand, for families with $p=13,19$,
we get three indecomposables, and for $p=31$, we obtain six such elements.
Note that each of these indecomposables has minimal trace $1$. Moreover,
similarly as in the second parallelepiped, these elements form triples
$\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ so that $\alpha_{1}=\alpha$,
$\alpha_{2}=\alpha^{\prime}\varepsilon_{1}$ and
$\alpha_{3}=\alpha^{\prime\prime}\varepsilon_{2}$ where $\varepsilon_{1}$ and
$\varepsilon_{2}$ are concrete units in ${\mathcal{O}}_{K}$. In the following,
similarly as for $p=3$, we use the notation $g_{1}=1$, $g_{2}=\rho$ and
$g_{3}=\frac{k+l\rho+\rho^{2}}{p}$.
Moreover, the fact that the situation significantly differs for primes $p>3$
can also be seen in the following proposition, which gives the complete
characterization of indecomposable integers in ${\mathcal{O}}_{K}$ for $a=41$.
###### Proposition 6.9.
Let $K={\mathbb{Q}}(\rho)$ where $\rho$ is a root of the polynomial
$x^{3}-41x^{2}-44x-1$. Then, up to multiplication of totally positive units,
all the indecomposable integers in ${\mathcal{O}}_{K}$ are $1$,
1. 1.
$-(5v+2)g_{1}-wg_{2}+(7v+3)g_{3}$ where $0\leq v\leq 5$ and $46v+14\leq w\leq
45v+19$,
2. 2.
$-vg_{2}+g_{3}$ where $1\leq v\leq 6$,
3. 3.
$-g_{1}-vg_{2}+2g_{3}$ where $1\leq v\leq 12$,
4. 4.
$-(5v+125)g_{1}-(45v+1132)g_{2}+(7v+176)g_{3}$ where $0\leq v\leq 5$,
5. 5.
$-(5v+31)g_{1}-(45v+283)g_{2}+(7v+44)g_{3}$ where $0\leq v\leq 11$,
6. 6.
$-(5v+5)g_{1}-(46v+22)g_{2}+(7v+8)g_{3}$ where $0\leq v\leq 5$,
7. 7.
$-(5v+6)g_{1}-(46v+41)g_{2}+(7v+9)g_{3}$ where $0\leq v\leq 11$,
which are of minimal trace $1$,
1. 8.
$-3g_{1}-vg_{2}+5g_{3}$ where $9\leq v\leq 14$,
2. 9.
$-(5v+3)g_{1}-(45v+32)g_{2}+(7v+5)g_{3}$ where $0\leq v\leq 5$,
3. 10.
$-(5v+93)g_{1}-(46v+837)g_{2}+(7v+131)g_{3}$ where $0\leq v\leq 5$,
4. 11.
$-4g_{1}-4g_{2}+7g_{3}$, $-4g_{1}-45g_{2}+7g_{3}$ and
$-209g_{1}-1890g_{2}+294g_{3}$,
which are of minimal trace $2$,
1. 12.
$-4g_{1}-vg_{2}+7g_{3}$ where $5\leq v\leq 9$,
2. 13.
$-(5v+4)g_{1}-(45v+45)g_{2}+(7v+7)g_{3}$ where $1\leq v\leq 5$,
3. 14.
$-(5v+184)g_{1}-(46v+1660)g_{2}+(7v+259)g_{3}$ where $0\leq v\leq 4$,
which are of minimal trace $3$.
The distribution of elements in Proposition 6.9 is not random, and it is
highly influenced by elements of the codifferent, which we use in the
determination of the minimal trace. In particular, for almost all elements, we
can use one of four concrete elements of the codifferent as follows: first one
for the triangle of elements in (1), second one for (2), (3), (8) and (12),
third one for (4), (5), (9) and (13), and fourth one for (6), (7), (10) and
(14). Only elements from (11) need different elements of the codifferent to
get the minimal trace $2$ (and not $3$, which we would get by applying one of
four elements of the codifferent mentioned above). We can also highlight that
in comparison to the family with $p=3$, we obtain indecomposable integers with
minimal trace $3$. Note that these elements are also indecomposable in the
order ${\mathbb{Z}}[\rho]$.
## 7 Consequences
In this section, we will use our knowledge of indecomposable integers for
$p=3$ to derive several other results for these fields. First, we will focus
on the minimal norm of algebraic integers in $K$ not associated with rational
integers. Such a minimal norm for monogenic simplest cubic fields is $2a+3$ as
was shown by Lemmermeyer and Pethö [23]. We will prove that except for a few
cases of $a$, $2a+3$ is still minimal for our family with $p=3$.
Then, we will find an upper bound on the norm of indecomposable integers in
${\mathcal{O}}_{K}$. Recall that in any totally real number field, the norm of
indecomposable integers is bounded [3, 18]. So far, this upper bound was
intensively studied for real quadratic fields [7, 11, 12, 35] and several
families of monogenic totally real cubic number fields [34]. Let us highlight
that our bound differs from the bound given in [34] for the simplest cubic
fields with ${\mathcal{O}}_{K}={\mathbb{Z}}[\rho]$.
Moreover, we will use indecomposable integers to show that the Pythagoras
number of ${\mathcal{O}}_{K}$ is always $6$ in our subfamily. A similar result
was derived by Tinková [33], who proved that the Pythagoras number of
${\mathbb{Z}}[\rho]$ is $6$ whenever $a\geq 3$. Moreover, in the final
subsection, we will find both upper and lower bounds on the minimal number of
variables of universal quadratic forms over ${\mathcal{O}}_{K}$. For that, we
will follow the procedure developed in [16], which is based on the knowledge
of indecomposable integers and their minimal traces after multiplication by
elements of the codifferent.
### 7.1 The smallest norm
Now, we will find the smallest norm of elements which are not associated with
rational integers. To do that, we use the knowledge of the structure of
indecomposables stated in Theorem 1.2.
###### Proposition 7.1.
Let $\alpha\in{\mathcal{O}}_{K}$. Then either
$|N(\alpha)|\geq\left\\{\begin{array}[]{ll}\frac{\Delta}{27}&\text{ if
}a=21,30,48,\\\ 2a+3&\text{ if }a>48,\end{array}\right.$
or $\alpha$ is associated with a rational integer. Moreover, this lower bound
is attained by some $\alpha\in{\mathcal{O}}_{K}$.
###### Proof.
The smallest such norm can be attained either by an indecomposable integer or
by a sum of two totally positive units. Let us first discuss the second case.
Recall that even in this case, a system of fundamental units is formed by the
pair $\rho$ and $\rho^{\prime}$, see Corollary 5.5. Every sum of two units is
associated with an element of the form $1+\varepsilon$ where $\varepsilon$ is
a totally positive unit. Moreover, we have $\varepsilon\neq 1$ since
otherwise, $1+\varepsilon$ would be associated with the rational integer $2$.
Therefore, by [16, Lemma 6.2], $\varepsilon$ is greater than $a^{2}$ in some
embedding. Thus,
$N(1+\varepsilon)=N(1)+N(\varepsilon)+\text{Tr}(\varepsilon)+\text{Tr}(\varepsilon\varepsilon^{\prime})>a^{2}.$
However, $a^{2}>2a+3$ for $a\geq 21$, and $2a+3$ is the norm of one of the
indecomposable integers in ${\mathcal{O}}_{K}$. Thus, a sum of two units
cannot have the smallest norm.
Since the elements in Theorem 1.2 (iii) and (iv) are associated with
conjugates of elements in 1.2 (ii), and the same is true for (vi)–(viii), it
is enough to consider elements in (i), (ii), (v) and (vi).
The norm of $\frac{1+\rho+\rho^{2}}{3}$ is equal to $\frac{a^{2}+3a+9}{27}$.
The smallest norm among elements in (ii) is $2a+3$.
Regarding elements in (vi), we have
$N\left(-(r+1)g_{2}+g_{3}\right)=-r^{3}-3r^{2}+\frac{a^{2}+3a-18}{9}r+\frac{4a^{2}+12a+9}{27}$
where $0\leq r\leq\frac{a}{3}-1$. This norm is a cubic polynomial in $r$ with
a negative leading coefficient. Moreover, for $r=-2$, it is equal to
$-\frac{2a^{2}+6a-9}{27}<0$ for $a\geq 21$. It means that in the interval
$\left[0,\frac{a}{3}-1\right]$, it attains its smallest value at $0$ or at
$\frac{a}{3}-1$. For $r=0$, we obtain $\frac{4a^{2}+12a+9}{27}>2a+3$ for
$a\geq 21$. Similarly, for $r=\frac{a}{3}-1$, we again obtain the norm
$\frac{4a^{2}+12a+9}{27}$. Thus, the smallest norm is not attained by elements
in (vi).
Let us now focus on elements in (v). We can deduce that
$N\left(-(2v+1)g_{1}-(v(a+3)+r+1)g_{2}+(3v+2)g_{3}\right)\\\
=-r^{3}-(av+3v-a)r^{2}-\left(av^{2}-a^{2}v-3av-3v+\frac{2a^{2}}{9}-\frac{a}{3}-1\right)r\\\
+v^{3}+\frac{a^{2}}{3}v^{2}-\frac{2a^{3}+7a^{2}+12a+9}{9}v-\frac{4a^{2}+12a+9}{27}$
where $0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$.
This is again a cubic polynomial in $r$ with a negative leading coefficient.
If $r=\frac{a}{3}$, we obtain
$v^{3}-\frac{a^{3}+3a+9}{9}v-\frac{a^{2}+3a+9}{27},$
which is a cubic polynomial in $v$ with a positive leading coefficient.
Moreover, for $v=-1$, it is $\frac{2a^{2}+6a-9}{27}>0$; for $v=0$, it is equal
to $-\frac{a^{2}+3a+9}{27}<0$, and for $v=\frac{a}{3}-1$, it equals
$-\frac{10a^{2}-24a+9}{27}<0$ for $a\geq 21$. It gives that whenever
$r=\frac{a}{3}$ and $0\leq v\leq\frac{a}{3}-1$, then our norm is negative. It
implies that for a fixed $v$, the smallest norm is attained either for
$r=\frac{a}{3}+1$, or for $r=\frac{2a}{3}-v$.
Let us now discuss the case when $r=\frac{a}{3}+1$. Here, we get the norm
$v^{3}-av^{2}+\frac{2a^{2}-3a-9}{9}v+\frac{2a^{2}+6a-9}{27}.$
For $v=\frac{a}{3}$, it is equal to $-\frac{a^{2}+3a+9}{27}<0$. Thus, it again
attains its smallest value in one of the border points of our interval. For
both of them, it is equal to $\frac{2a^{2}+6a-9}{27}>2a+3$ for $a\geq 26$. If
$a=21$, then $\frac{2a^{2}+6a-9}{27}=37>19=\frac{a^{2}+3a+9}{27}$, so we can
exclude this norm even in this case.
Similarly, for $r=\frac{2a}{3}-v$, we obtain
$-v^{3}-3v^{2}+\frac{a^{2}+3a-18}{9}v+\frac{2a^{2}+6a-9}{27}.$
For $v=-1$, it is equal to $-\frac{a^{2}+3a+9}{27}$; thus, the smallest norm
is attained in a border point. For both of them, we again obtain the value
$\frac{2a^{2}+6a-9}{27}$ which we have excluded before.
Thus, we are left with the norms $\frac{a^{2}+3a+9}{27}$ and $2a+3$, and
$\frac{a^{2}+3a+9}{27}>2a+3$ if $a\geq 53$. Between $21$ and $52$, only the
cases $a=21,30,48$ belong to our family, and for them, the norm
$\frac{a^{2}+3a+9}{27}$ is the smallest one. ∎
### 7.2 The largest norm
In this part, we will find a sharp upper bound on the norm of indecomposable
integers in ${\mathcal{O}}_{K}$. As in the case of the smallest norm, it is
enough to discuss elements in (i), (ii), (v) and (vi). We will start with
elements in (ii) and (vi).
###### Lemma 7.2.
Let $\alpha\in{\mathcal{O}}_{K}$ be an indecomposable integer. Then:
1. 1.
If $\alpha$ is as in (ii), then
$N(\alpha)\leq\frac{2a^{3}+9a^{2}+27a+27}{27}=\frac{(2a+3)\Delta}{27}$.
2. 2.
If $\alpha$ is as in (vi), then $N(\alpha)<\frac{(2a+3)\Delta}{27}$.
###### Proof.
To prove (1), it is not difficult to see that the largest norm among these
elements is attained when $r=\frac{a}{3}$, for which
$N(\alpha)=\frac{2a^{3}+9a^{2}+27a+27}{27}$.
On the other hand, if $\alpha=-(r+1)g_{2}+g_{3}$, we have
$g(r)=\frac{2a^{3}+9a^{2}+27a+27}{27}-N(\alpha)=r^{3}+3r^{2}-\frac{a^{2}+3a-18}{9}r+\frac{2a^{3}+5a^{2}+15a+18}{27}.$
This is a cubic polynomial in $r$ with a positive leading coefficient.
Moreover, the discriminant of $g$ is negative, in which case $g$ has only one
real root. This together with $g(0)=\frac{2a^{3}+5a^{2}+15a+18}{27}>0$ implies
that $g(r)>0$ for all $0\leq r\leq\frac{a}{3}-1$, completing the proof. ∎
Note that
$N\left(\frac{1+\rho+\rho^{2}}{3}\right)=\frac{a^{2}+3a+9}{27}<\frac{2a^{3}+9a^{2}+27a+27}{27}$
for $a\geq 21$. Therefore, we can also exclude the element
$\frac{1+\rho+\rho^{2}}{3}$ from the consideration.
It remains to discuss the elements in (v), which form the triangle $R_{4}$ in
Section 6.4. Let us denote these integers as
$\alpha(v,r)=-(2v+1)g_{1}-(v(a+3)+r+1)g_{2}+(3v+2)g_{3}$
where $0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$.
Furthermore, for the transformation $T_{1}$ in Section 6.5.3 we have
$T_{1}(R_{4})=R_{4}$. In other words, $R_{4}$ also contains
$\alpha^{\prime}\varepsilon_{1}$ and $\alpha^{\prime\prime}\varepsilon_{2}$
where $\varepsilon_{1}$ and $\varepsilon_{2}$ are concrete totally positive
units. Thus, it suffices to consider only one third of this triangle, in
particular,
$R_{4}^{0}=\left\\{\alpha(v,r);0\leq v\leq\frac{a-3}{9}-1,\frac{a}{3}+1+v\leq
r\leq\frac{2a}{3}-2v-1\right\\}\cup\left\\{\alpha\left(\frac{a-3}{9},\frac{a}{3}+1+\frac{a-3}{9}\right)\right\\}.$
The following lemma compares norms of elements contained in $R_{4}^{0}$. Note
that the proof is almost the same as in [34, Lemma 3.2].
###### Lemma 7.3.
We have
1. 1.
$N(\alpha(v,r))<N(\alpha(v+1,r))$ for all $0\leq v\leq\frac{a-3}{9}-1$ and
$\frac{a}{3}+2+v\leq r\leq\frac{2a}{3}-2v-3$,
2. 2.
$N\big{(}\alpha\big{(}v,\frac{a}{3}+1+v\big{)}\big{)}<N\big{(}\alpha\big{(}v+1,\frac{a}{3}+1+v+1\big{)}\big{)}$
for all $0\leq v\leq\frac{a-3}{9}-1$,
3. 3.
$N\big{(}\alpha\big{(}v,\frac{2a}{3}-2v-1\big{)}\big{)}<N\big{(}\alpha\big{(}v+1,\frac{2a}{3}-2(v+1)-1\big{)}\big{)}$
for all $0\leq v\leq\frac{a-3}{9}-2$,
4. 4.
$N\big{(}\alpha\big{(}v,\frac{2a}{3}-2v-2\big{)}\big{)}<N\big{(}\alpha\big{(}v+1,\frac{2a}{3}-2(v+1)-2\big{)}\big{)}$
for all $0\leq v\leq\frac{a-3}{9}-2$.
Now we have everything we need to determine the upper bound on the norm of
indecomposable integers in ${\mathcal{O}}_{K}$.
###### Proposition 7.4.
If $\alpha$ is indecomposable in ${\mathcal{O}}_{K}$, then
$|N(\alpha)|\leq\left\\{\begin{array}[]{ll}\frac{2a^{3}+9a^{2}+27a+27}{27}\text{
if }a=21,30,48,\\\ \frac{(a^{2}+3a+9)^{2}}{729}\text{ if
}a>48,\end{array}\right.$
###### Proof.
As we have seen, the largest norm is either $\frac{2a^{3}+9a^{2}+27a+27}{27}$
found in Proposition 7.2 or is attained by some elements from $R_{4}^{0}$.
Moreover, applying Lemma 7.3, we are left only with two candidates from
$R_{4}^{0}$: $\alpha\left(\frac{a-3}{9}-1,\frac{a}{3}+2+\frac{a-3}{9}\right)$
and $\alpha\left(\frac{a-3}{9},\frac{a}{3}+1+\frac{a-3}{9}\right)$.
Furthermore,
$N\left(\alpha\left(\frac{a-3}{9}-1,\frac{a}{3}+2+\frac{a-3}{9}\right)\right)=\frac{a^{4}+6a^{3}-54a^{2}-189a+81}{729}\\\
<\frac{(a^{2}+3a+9)^{2}}{729}=N\left(\alpha\left(\frac{a-3}{9},\frac{a}{3}+1+\frac{a-3}{9}\right)\right).$
Moreover,
$\frac{2a^{3}+9a^{2}+27a+27}{27}<\frac{(a^{2}+3a+9)^{2}}{729}$
for $a\geq 53$, which gives a few exceptional cases listed in the statement of
the proposition. ∎
### 7.3 Pythagoras number
In this part, we will show that the Pythagoras number of ${\mathcal{O}}_{K}$
is $6$. Before that, let us briefly overview basic facts and results on this
number. Consider a commutative ring ${\mathcal{O}}$, for which we define the
following two sets:
$\sum{\mathcal{O}}_{K}^{2}=\Bigg{\\{}\sum_{i=1}^{n}\alpha_{i}^{2};\alpha_{i}\in{\mathcal{O}}_{K}\text{
and }n\in{\mathbb{N}}\Bigg{\\}}$
and for each $m\in\mathbb{N}$
$\sum^{m}{\mathcal{O}}_{K}^{2}=\Bigg{\\{}\sum_{i=1}^{m}\alpha_{i}^{2};\alpha_{i}\in{\mathcal{O}}_{K}\Bigg{\\}}.$
Then by the Pythagoras number ${\mathcal{P}}({\mathcal{O}})$ of
${\mathcal{O}}$, we will mean the following infimum:
${\mathcal{P}}({\mathcal{O}})=\inf\Bigg{\\{}m\in{\mathbb{N}}\cup\\{\infty\\};\sum{\mathcal{O}}_{K}^{2}=\sum^{m}{\mathcal{O}}_{K}^{2}\Bigg{\\}}.$
For example, ${\mathcal{P}}({\mathbb{C}})={\mathcal{P}}({\mathbb{R}})=1$ and
${\mathcal{P}}({\mathbb{Z}})={\mathcal{P}}({\mathbb{Q}})=4$. Although there
are many results on the Pythagoras number of fields, we will only summarize
results in the case when ${\mathcal{O}}$ is an order of a totally real number
field $K$.
For them, we know that the value of ${\mathcal{P}}({\mathcal{O}})$ is finite
but can attain arbitrarily large values [29]. Moreover, in this case, the
Pythagoras number can be bounded by a function depending on the degree $d$ of
a field $K$, and, furthermore, ${\mathcal{P}}({\mathcal{O}})\leq d+3$ for
$2\leq d\leq 5$ [17]. So far, the Pythagoras number was fully determined for
orders in real quadratic fields [28], and we have some partial results for
real biquadratic fields [21, 9] and totally real cubic fields [20, 33]. In
particular, we closely follow [33], where the second author showed that
${\mathcal{P}}({\mathbb{Z}}[\rho])=6$ for $\rho$ being a root of the
polynomial $x^{3}-ax^{2}-(a+3)x-1$ with $a\geq 3$, i.e., for a concrete order
of the simplest cubic fields.
In cubic fields, we know that ${\mathcal{P}}({\mathcal{O}}_{K})\leq 6$ by the
result of Kala and Yatsyna [17], and we will prove that
${\mathcal{P}}({\mathcal{O}}_{K})\geq 6$ for our subfamily of non-monogenic
simplest cubic fields with the base $B_{3}(1,1)$. For that, we will use the
method developed in [33], and our determination of the structure of
indecomposable integers in ${\mathcal{O}}_{K}$. In particular, we will do the
following:
1. 1.
We will choose a suitable element $\gamma\in{\mathcal{O}}_{K}$ such that
$\gamma$ can be written as a sum of six non-zero squares of elements in
${\mathcal{O}}_{K}$.
2. 2.
We will determine all elements $\alpha$ from Theorem 1.2 and all units
$\varepsilon\in{\mathcal{O}}_{K}$ such that
$\gamma\succeq(\varepsilon\alpha)^{2}$.
3. 3.
Using elements in (2), we will find all elements $\omega\in{\mathcal{O}}_{K}$
such that $\gamma\succeq\omega^{2}$.
4. 4.
Considering elements in (3), we will discuss all possible decompositions of
$\gamma$ as a sum of squares and show that for every such decomposition, we
need at least $6$ squares.
Note that in fact, elements $\varepsilon\alpha$ in (2) are indecomposable
integers in other signatures. Since the simplest cubic fields have units of
all signatures, they can be expressed as presented here, i.e., as
$\varepsilon\alpha$ where $\varepsilon$ is a (not necessarily totally
positive) unit and $\alpha$ is one of totally positive indecomposables listed
in Theorem 1.2. For more details, see, for example, [33].
In our proof, we will fix $\gamma$ as
$\displaystyle\gamma$
$\displaystyle=1+1+1+4+\left(\frac{a+6}{3}g_{1}+\frac{a}{3}g_{2}-g_{3}\right)^{2}+\left(\frac{5a+3}{9}g_{1}+\frac{2a+3}{3}g_{2}-2g_{3}\right)^{2}$
$\displaystyle=\frac{34a^{2}+15a+783}{81}+\frac{11a^{2}-29a-39}{27}\rho-\frac{11a-33}{27}\rho^{2}.$
Using estimates on $\rho,\rho^{\prime}$, and $\rho^{\prime\prime}$ (see
Section 2; note that for $\rho^{\prime}$, we use bounds
$-1-\frac{1}{a+1}<\rho^{\prime}<-1-\frac{1}{a+2}$ found in [33]), we obtain
$\displaystyle\gamma$
$\displaystyle<\frac{13a^{2}}{81}+\frac{14a}{27}+\frac{197}{27}-\frac{26}{9a}<a^{2},$
$\displaystyle\gamma^{\prime}$
$\displaystyle<\frac{a^{4}+40a^{3}+1234a^{2}+4596a+4725}{81(a+2)^{2}}<a^{2},$
$\displaystyle\gamma^{\prime\prime}$
$\displaystyle<\frac{34a^{4}+186a^{3}+1167a^{2}+5178a+7497}{81(a+3)^{2}}<a^{2}$
for $a\geq 21$. We see that if $\omega\in{\mathbb{Z}}$ is such that
$\gamma\succeq\omega^{2}$, then $|\omega|<a$. In contrast with [33], we will
not determine the precise set of elements $\omega$ satisfying
$\gamma\succeq\omega^{2}$, but we will significantly restrict the set of such
elements. As we will see, this restriction will be enough to show that we need
at least $6$ squares to express $\gamma$.
First of all, in a series of lemmas, we will find all
$\omega\in{\mathcal{O}}_{K}$ such that $\gamma\succeq\omega^{2}$. We will
start with units and the indecomposable integer
$g_{3}=\frac{1+\rho+\rho^{2}}{3}$. Recall that in the simplest cubic fields,
all totally positive units are squares. Moreover, conjugates of $g_{3}$ are
associated with $g_{3}$.
###### Lemma 7.5.
We have
1. 1.
if $\gamma\succeq\varepsilon^{2}$ where $\varepsilon$ is a unit in
${\mathcal{O}}_{K}$, then $\varepsilon^{2}=1$,
2. 2.
if $\gamma\succeq\left(\varepsilon\frac{1+\rho+\rho^{2}}{3}\right)^{2}$ where
$\varepsilon$ is a unit in ${\mathcal{O}}_{K}$, then
$\varepsilon^{2}=\rho^{\prime 2}\rho^{\prime\prime 2}$.
###### Proof.
Recall from Corollary 5.5 that a system of fundamental units is formed by the
pair $\rho$ and $\rho^{\prime}$. Thus, we can use [16, Lemma 6.2], which says
that if $\varepsilon^{2}\neq 1$ is a unit in ${\mathcal{O}}_{K}$, then
$\varepsilon^{2}$ has a conjugate greater than $a^{2}$. Since $\gamma$ has all
conjugates smaller than $a^{2}$, no totally positive unit except for $1$ can
be totally smaller than $\gamma$.
Considering the second part of the statement, we see that
$\displaystyle\left(\frac{1+\rho+\rho^{2}}{3}\right)^{2}$
$\displaystyle>\frac{(a^{2}+3a+3)^{2}}{9}>a^{2},$
$\displaystyle\left(\frac{1+\rho^{\prime}+\rho^{\prime 2}}{3}\right)^{2}$
$\displaystyle>\frac{1}{9}\left(\frac{(a+3)^{2}}{(a+2)^{2}}-\frac{1}{a+1}\right)^{2},$
$\displaystyle\left(\frac{1+\rho^{\prime\prime}+\rho^{\prime\prime
2}}{3}\right)^{2}$
$\displaystyle>\frac{1}{9}\left(1-\frac{1}{a+2}+\frac{1}{(a+3)^{2}}\right)^{2}.$
Thus, we can immediately exclude that $\varepsilon^{2}=1$. Moreover, it can be
easily shown that
$a^{2}\left(\frac{1+\rho^{\prime}+\rho^{\prime
2}}{3}\right)^{2}>\gamma^{\prime}\qquad\text{ and }\qquad
a^{4}\left(\frac{1+\rho^{\prime\prime}+\rho^{\prime\prime
2}}{3}\right)^{2}>\gamma^{\prime\prime}.$
Here, we use [16, Lemma 6.3]. It says that if $\varepsilon>a^{2}$ is a totally
positive unit such that $\varepsilon\neq\rho^{2},\rho^{2}\rho^{\prime 2}$,
then either $\varepsilon>a^{4}$, or one of $\varepsilon^{\prime}$ and
$\varepsilon^{\prime\prime}$ is greater than $a^{2}$. Together with previous
estimates, it implies $\varepsilon^{\prime\prime
2}=\rho^{2},\rho^{2}\rho^{\prime 2}$ in our case, from which we can exclude
$\varepsilon^{\prime\prime 2}=\rho^{2}$, which does not have
$\varepsilon^{2}<1$. ∎
We will proceed with indecomposables from ${\mathbb{Z}}[\rho]$.
###### Lemma 7.6.
Let $\alpha$ be as in (ii), (iii) or (iv) of Theorem 1.2. Then there is no
unit $\varepsilon\in{\mathcal{O}}_{K}$ such that
$\gamma\succeq(\varepsilon\alpha)^{2}$.
###### Proof.
Let $\alpha=-r\rho+\rho^{2}$ where $1\leq r\leq\frac{a}{3}$, i.e., $\alpha$ is
one of elements in (ii). In this case, we have
$\displaystyle(-r\rho+\rho^{2})^{2}$
$\displaystyle>((a+1)^{2}-r(a+2))^{2}\geq\frac{(2a^{2}+4a+3)^{2}}{9}>a^{2},$
$\displaystyle(-r\rho^{\prime}+\rho^{\prime 2})^{2}$
$\displaystyle>(r+1)^{2},$
$\displaystyle(-r\rho^{\prime\prime}+\rho^{\prime\prime 2})^{2}$
$\displaystyle>\frac{r^{2}}{(a+3)^{2}}.$
Therefore, we must have $\varepsilon^{2}<1$, $\varepsilon^{\prime 2}<a^{2}$
and $\varepsilon^{\prime\prime 2}>a^{2}$, which again follows from [16, Lemma
6.2]. Moreover, easily,
$\frac{r^{2}a^{4}}{(a+3)^{2}}>\gamma,\gamma^{\prime},\gamma^{\prime\prime}$.
Therefore, again by [16, Lemma 6.3], we can conclude that
$\varepsilon^{\prime\prime 2}=\rho^{2},\rho^{2}\rho^{\prime 2}$. However, for
both of these units, we have
$(\varepsilon\alpha)^{2}>\gamma,\gamma^{\prime},\gamma^{\prime\prime}$.
Note that by this, we have also excluded Cases (iii) and (iv). It follows from
the fact that elements in (iii) and (iv) are associated with conjugates in
(ii), and, thus,
$(\varepsilon\alpha)^{2}>\gamma,\gamma^{\prime},\gamma^{\prime\prime}$ also
covers elements originating from indecomposables in (iii) and (iv). ∎
Now we will discuss the other three lines of indecomposables.
###### Lemma 7.7.
Let $a\geq 55$. Let $\alpha$ be as in (vi), (vii) or (viii) of Theorem 1.2,
and let $\gamma\succeq(\varepsilon\alpha)^{2}$ for some unit $\varepsilon$.
Then $\alpha$ is as in (vi) and $\varepsilon^{2}=\rho^{\prime
2}\rho^{\prime\prime 2}$.
###### Proof.
Let $\alpha=-(r+1)g_{2}+g_{3}$ where $0\leq r\leq\frac{a}{3}-1$, i.e.,
$\alpha$ is one of elements in (vi). Then we have
$\displaystyle\alpha^{2}$
$\displaystyle>\left(\frac{a^{2}}{3}-\frac{4}{3a}-r\Big{(}a+1+\frac{2}{a}\Big{)}\right)^{2}>\left(\frac{2a}{3}+\frac{1}{3}+\frac{2}{3a}\right)^{2}>\gamma,\gamma^{\prime},\gamma^{\prime\prime},$
$\displaystyle\alpha^{\prime 2}$
$\displaystyle>\left(\frac{4}{3}+\frac{4}{3(a+2)}+\frac{1}{3(a+2)^{2}}+r\Big{(}1+\frac{1}{a+2}\Big{)}\right)^{2}>1,$
$\displaystyle\alpha^{\prime\prime 2}$
$\displaystyle>\left(\frac{1}{3}+\frac{2}{3(a+3)}+\frac{1}{3(a+3)^{2}}+\frac{r}{a+3}\right)^{2}.$
Moreover, $a^{4}\alpha^{\prime\prime
2}>\gamma,\gamma^{\prime},\gamma^{\prime\prime}$. It implies that again
$\varepsilon^{\prime\prime 2}=\rho^{2},\rho^{2}\rho^{\prime 2}$ by [16, Lemma
6.3]. We can exclude $\varepsilon^{\prime\prime 2}=\rho^{2}$ as
$\varepsilon^{2}>1$.
Now we will discuss all three conjugates of
$(\rho^{\prime}\rho^{\prime\prime}\alpha)^{2}$. Since
$(\rho\rho^{\prime}\alpha^{\prime\prime})^{2}>\gamma^{\prime}$, the conjugate
$(\rho^{\prime\prime}\rho\alpha^{\prime})^{2}$ cannot be totally smaller than
$\gamma$. If $r\geq\frac{a}{6}$, then
$(\rho\rho^{\prime}\alpha^{\prime\prime})^{2}>\gamma$, and if $r<\frac{a}{6}$,
then $(\rho^{\prime}\rho^{\prime\prime}\alpha)^{2}>\gamma^{\prime}$ for $a\geq
55$. This excludes the conjugate
$(\rho\rho^{\prime}\alpha^{\prime\prime})^{2}$. Thus, we are left only with
$(\rho^{\prime}\rho^{\prime\prime}\alpha)^{2}$, which states the lemma.
Note that similarly as in the proof of Lemma 7.6, conjugates of elements in
(vi) are associated with elements in (vii) and (viii), and in this proof, we
have discussed all conjugates of $\alpha$ at the same time. ∎
In the end, we will look at the triangle of indecomposables in (v). For that,
let us denote
$\alpha(v,r)=-(2v+1)g_{1}-(v(a+3)+r+1)g_{2}+(3v+2)g_{3}$
where $0\leq v\leq\frac{a}{3}-1$ and $\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$.
###### Lemma 7.8.
Let $a\geq 63$. Moreover, let $\gamma\succeq(\varepsilon\alpha(v,r))^{2}$ for
some unit $\varepsilon\in{\mathcal{O}}_{K}$, $0\leq v\leq\frac{a}{3}-1$ and
$\frac{a}{3}+1\leq r\leq\frac{2a}{3}-v$. Then either
1. 1.
$(\varepsilon\alpha(v,r))^{2}=\rho^{\prime\prime 2}\rho^{2}\alpha(0,r)^{\prime
2}$ for some $\frac{a}{3}+1\leq r<\frac{5a-6}{12}$, or
2. 2.
$(\varepsilon\alpha(v,r))^{2}=\rho^{2}\rho^{\prime 2}\alpha(0,r)^{\prime\prime
2}$ for some $\frac{5(a-3)}{9}\leq r\leq\frac{2a}{3}-1$.
###### Proof.
As before, it suffices to restrict to a subset of the triangle of
indecomposables in (v). In particular, we can discuss only elements in
$R_{4}^{0}$. Thus, let $\alpha(v,r)\in R_{4}^{0}$, i.e., $0\leq
v\leq\frac{a-3}{9}-1$ and $\frac{a}{3}+1+v\leq r\leq\frac{2a}{3}-2v-1$, or
$v=\frac{a-3}{9}$ and $r=\frac{a}{3}+1+\frac{a-3}{9}$. Then
$\displaystyle\alpha(v,r)^{2}$
$\displaystyle>\left(\frac{2a^{2}}{3}+a-\frac{2}{3a}-v\Big{(}a+4+\frac{4}{a}\Big{)}-r\Big{(}a+1+\frac{2}{a}\Big{)}\right)^{2}\geq\left(\frac{4a}{3}-\frac{1}{3}+\frac{4}{3a}\right)^{2}>a^{2},$
$\displaystyle\alpha(v,r)^{\prime 2}$
$\displaystyle>\left(\frac{2}{3}+\frac{5}{3(a+2)}+\frac{2}{3(a+2)^{2}}+v\Bigg{(}a+2+\frac{a+4}{a+2}+\frac{1}{(a+2)^{2}}\Bigg{)}+r\Bigg{(}1+\frac{1}{a+2}\Bigg{)}\right)^{2}$
$\displaystyle\hskip
142.26378pt\geq\left(\frac{a+5}{3}+\frac{a+8}{3(a+2)}+\frac{2}{3(a+2)^{2}}\right)^{2}>1,$
$\displaystyle\alpha(v,r)^{\prime\prime 2}$
$\displaystyle>\left(-\frac{1}{3}+\frac{1}{3(a+3)}+\frac{2}{3(a+3)^{2}}+v\Bigg{(}-1+\frac{a+2}{a+3}+\frac{1}{(a+3)^{2}}\Bigg{)}+\frac{r}{a+3}\right)^{2}\geq\frac{(a+5)^{2}}{9(a+3)^{4}}.$
Furthermore, $a^{6}\alpha(v,r)^{\prime\prime
2}>\gamma,\gamma^{\prime},\gamma^{\prime\prime}$. That means that our unit has
to satisfy $\varepsilon^{2}<1$, $\varepsilon^{\prime 2}<a^{2}$ and
$a^{2}<\varepsilon^{\prime\prime 2}<a^{6}$. This is fulfilled by units
$\varepsilon^{\prime\prime 2}=\rho^{2}\rho^{\prime 2},\rho^{2}\rho^{\prime
4},\rho^{4}\rho^{\prime 2},\rho^{4}\rho^{\prime 4},\rho^{4}\rho^{\prime 6}$.
Now, we will use the fact that if $\gamma\succeq\omega^{2}$ for some
$\omega\in{\mathcal{O}}_{K}$, then necessarily
$\text{Tr}(\gamma)\geq\text{Tr}(\omega^{2})$. It can be computed that
$\text{Tr}(\gamma)=\frac{16a^{2}-24a+981}{27}$. Moreover, we have
$\displaystyle\text{Tr}(\rho^{\prime 2}\rho^{\prime\prime 4}\alpha(v,r)^{2})$
$\displaystyle\geq\frac{a^{4}+12a^{3}+56a^{2}+132a+153}{9}>\text{Tr}(\gamma),$
$\displaystyle\text{Tr}(\rho^{\prime 4}\rho^{\prime\prime 6}\alpha(v,r)^{2})$
$\displaystyle\geq\frac{a^{4}+10a^{3}+41a^{2}+96a+126}{9}>\text{Tr}(\gamma),$
so we can exclude $\varepsilon^{2}=\rho^{\prime 2}\rho^{\prime\prime
4},\rho^{\prime 4}\rho^{\prime\prime 6}$. Moreover, if $v\geq 1$, then
$\displaystyle\text{Tr}(\rho^{\prime 2}\rho^{\prime\prime 2}\alpha(v,r)^{2})$
$\displaystyle\geq\frac{17a^{2}+96a+369}{9}>\text{Tr}(\gamma),$
$\displaystyle\text{Tr}(\rho^{\prime 4}\rho^{\prime\prime 2}\alpha(v,r)^{2})$
$\displaystyle\geq\frac{26a^{2}+6a+153}{9}>\text{Tr}(\gamma),$
$\displaystyle\text{Tr}(\rho^{\prime 4}\rho^{\prime\prime 4}\alpha(v,r)^{2})$
$\displaystyle\geq\frac{41a^{2}+150a+234}{9}>\text{Tr}(\gamma),$
by which we have excluded all cases with $v\geq 1$.
Let us now focus on the case when $v=0$. If $r\geq\frac{a}{3}+2$, then
$\displaystyle\text{Tr}(\rho^{\prime 4}\rho^{\prime\prime 2}\alpha(0,r)^{2})$
$\displaystyle\geq\frac{26a^{2}+42a+126}{9}>\text{Tr}(\gamma),$
$\displaystyle\text{Tr}(\rho^{\prime 4}\rho^{\prime\prime 4}\alpha(0,r)^{2})$
$\displaystyle\geq\frac{26a^{2}+114a+234}{9}>\text{Tr}(\gamma).$
If $r=\frac{a}{3}+1$, we obtain
$(\gamma-\beta)(\gamma^{\prime}-\beta^{\prime})+(\gamma-\beta)(\gamma^{\prime\prime}-\beta^{\prime\prime})+(\gamma^{\prime}-\beta^{\prime})(\gamma^{\prime\prime}-\beta^{\prime\prime})<0$
(9)
where $\beta$ is any conjugate of $\rho^{\prime 4}\rho^{\prime\prime
2}\alpha\big{(}0,\frac{a}{3}+1\big{)}^{2}$ or $\rho^{\prime
4}\rho^{\prime\prime 4}\alpha\big{(}0,\frac{a}{3}+1\big{)}^{2}$. Note that (9)
gives the second coefficient in the minimal polynomial of $\gamma-\beta$,
which has to be nonnegative if $\gamma-\beta\succeq 0$. By this, we have
excluded the units $\varepsilon^{2}=\rho^{\prime 4}\rho^{\prime\prime
2},\rho^{\prime 4}\rho^{\prime\prime 4}$.
Thus, it remains to discuss the unit $\varepsilon^{2}=\rho^{\prime
2}\rho^{\prime\prime 2}$. We have $\rho^{\prime\prime
2}\rho^{2}\alpha(0,r)^{\prime 2}>\gamma^{\prime}$ for all $\frac{a}{3}+1\leq
r\leq\frac{2a}{3}-1$, which excludes the conjugate $\rho^{\prime
2}\rho^{\prime\prime 2}\alpha(0,r)^{2}$.
Let us now focus on the conjugate $\rho^{\prime\prime
2}\rho^{2}\alpha(0,r)^{\prime 2}$. If $r\geq\frac{5a-6}{12}$, we have
$\rho^{\prime\prime 2}\rho^{2}\alpha(0,r)^{\prime
2}>\frac{1}{(a+3)^{2}}(a+1)^{2}\frac{(5a^{3}+27a^{2}+52a+44)^{2}}{144(a+2)^{4}}>\gamma$
for $a\geq 63$. Therefore, we must have $\frac{a}{3}+1\leq r<\frac{5a-6}{12}$,
which gives the first case in our statement.
For the conjugate $\rho^{2}\rho^{\prime 2}\alpha(0,r)^{\prime\prime 2}$, we
similarly see that for $\frac{a}{3}+1\leq r\leq\frac{5(a-3)}{9}-1$, we obtain
$\rho^{\prime 2}\rho^{\prime\prime
2}\alpha(0,r)^{2}>\left(1+\frac{1}{a+2}\right)^{2}\frac{1}{(a+3)^{2}}\frac{(a^{3}+28a^{2}+14a+42)^{2}}{81a^{2}}>\gamma^{\prime}$
for $a\geq 34$. This implies $\frac{5(a-3)}{9}\leq r\leq\frac{2a}{3}-1$, i.e.,
the second case in our statement. ∎
In Lemmas 7.5, 7.6, 7.7 and 7.8, we have found some of the squares totally
smaller than $\gamma$. There can exist other squares
$\omega^{2}\in{\mathcal{O}}_{K}$ satisfying $\gamma\succeq\omega^{2}$;
however, these squares can be determined from elements derived so far. In
particular, every such $\omega$ can be written as
$\omega=\sum_{i=1}^{n}\beta_{i}$ where elements $\beta_{i}$ are square roots
of squares found in above lemmas. Moreover, we can assume that all $\beta_{i}$
in this decomposition have the same signature. By the signature of an element
$\alpha\in K$, we mean the triple
$(\text{sgn}(\alpha),\text{sgn}(\alpha^{\prime}),\text{sgn}(\alpha^{\prime\prime}))$,
where sgn is the signum function. In the following, we will also replace $1$
and $-1$ in this triple by symbols $+$ and $-$.
Therefore,
$\gamma\succeq\omega^{2}=\left(\sum_{i=1}^{n}\beta_{i}\right)^{2}\succeq(\beta_{i}+\beta_{j})^{2}\succeq\beta_{i}^{2}$
for every $1\leq i,j\leq n$. Thus, to determine all the remaining squares
totally smaller than $\gamma$, it is enough for each possible signature
1. (1)
to take all square roots of squares from Lemmas 7.5, 7.6, 7.7 and 7.8 with
this signature,
2. (2)
to study the element $(\beta_{i}+\beta_{j})^{2}$ for all pairs of elements
$\beta_{i}$ and $\beta_{j}$ from 1.
* •
If $(\beta_{i}+\beta_{j})^{2}\succ\gamma$ for all such pairs, then we do not
get any additional square totally smaller than $\gamma$. In the following
proof, we will sometimes show that
$\beta_{i}^{2},\beta_{j}^{2}\succ\frac{1}{2}\gamma$, from which
$(\beta_{i}+\beta_{j})^{2}\succ\gamma$ follows.
* •
Otherwise, we can get more squares. However, we will see that for $\gamma$,
this can occur only in one trivial case.
Moreover, $\alpha$ and $-\alpha$ produce the same square, so it suffices to
study only one of every pair of (mutually opposite) signatures; e.g., instead
of studying both elements with signatures $(+,+,+)$ and $(-,-,-)$, it is
enough to examine those with the signature $(+,+,+)$.
This problem is related to the study of indecomposability of algebraic
integers which are not totally positive. The above summary is not detailed,
and the full description can be found in [33].
###### Lemma 7.9.
Let $a\geq 63$ and let $\gamma\succeq\omega^{2}$ where
$\omega^{2}\in{\mathcal{O}}_{K}$ is not as in Lemmas 7.5, 7.6, 7.7 and 7.8.
Then $\omega\in{\mathbb{Z}}$.
###### Proof.
Recall that the unit $\rho$ has the signature $(+,-,-)$. Thus, the so far
found elements, whose squares are totally smaller than $\gamma$, have the
following signatures:
1. 1.
$1$ has the signature $(+,+,+)$,
2. 2.
$\rho^{\prime}\rho^{\prime\prime}\frac{1+\rho+\rho^{2}}{3}$ has the signature
$(+,-,-)$,
3. 3.
$\rho^{\prime}\rho^{\prime\prime}\left(-(r+1)g_{2}+g_{3}\right)$ where $0\leq
r\leq\frac{a}{3}-1$ have the signature $(+,-,-)$,
4. 4.
$\rho^{\prime\prime}\rho\alpha(0,r)^{\prime}$ where $\frac{a}{3}+1\leq
r<\frac{5a-6}{12}$ have the signature $(-,-,+)$,
5. 5.
$\rho\rho^{\prime}\alpha(0,r)^{\prime\prime}$ where $\frac{5(a-3)}{9}\leq
r\leq\frac{2a}{3}-1$ have the signature $(-,+,-)$.
From elements in (1), we can get only squares $\omega^{2}$ such that
$\omega\in{\mathbb{Z}}$, which are included in the statement of the lemma.
We will proceed with elements in (2) and (3), which have the same signature.
Let $\alpha_{1}=-(r_{1}+1)g_{2}+g_{3}$ and $\alpha_{2}=-(r_{2}+1)g_{2}+g_{3}$
where $-1\leq r_{1},r_{2}\leq\frac{a}{3}-1$. Note for $r_{i}=-1$, we obtain
the element $\frac{1+\rho+\rho^{2}}{3}$ from (2). Then
$(\rho\rho^{\prime}\alpha_{1}^{\prime\prime}+\rho\rho^{\prime}\alpha_{2}^{\prime\prime})^{2}=\rho^{2}\rho^{\prime
2}(\alpha_{1}^{\prime\prime
2}+2\alpha_{1}^{\prime\prime}\alpha_{2}^{\prime\prime}+\alpha_{2}^{\prime\prime
2})\\\
>4(a+1)^{2}\left(1+\frac{1}{a+2}\right)^{2}\frac{1}{9}\left(1-\frac{1}{a+2}+\frac{1}{(a+3)^{2}}\right)^{2}>\gamma^{\prime\prime}.$
Thus, the square of a sum of elements in (2) and (3) cannot be totally smaller
than $\gamma$.
Let us now focus on elements in (4). For them, we have
$\rho^{\prime\prime 2}\rho^{2}\alpha(0,r)^{\prime
2}>\frac{1}{(a+3)^{2}}(a+1)^{2}\left(\frac{a+5}{3}+\frac{a+8}{3(a+2)}+\frac{2}{3(a+2)^{2}}\right)^{2}>\frac{1}{2}\gamma$
for $\frac{a}{3}+1\leq r<\frac{5a-6}{12}$. That means that the square of a sum
of two elements in (4) cannot be totally smaller than $\gamma$.
Likewise, if we consider elements in (5), we obtain
$\rho^{\prime\prime 2}\rho^{2}\alpha(0,r)^{\prime
2}>\frac{1}{(a+3)^{2}}(a+1)^{2}\left(\frac{5a}{9}-1+\frac{5a}{9(a+2)}+\frac{2}{3(a+2)^{2}}\right)^{2}>\frac{1}{2}\gamma^{\prime\prime}$
for $\frac{5(a-3)}{9}\leq r\leq\frac{2a}{3}-1$. That completes the proof. ∎
Now, we are able to prove that the Pythagoras number of ${\mathcal{O}}_{K}$ is
$6$ as for ${\mathbb{Z}}[\rho]$.
###### Proposition 7.10.
We have ${\mathcal{P}}({\mathcal{O}}_{K})=6$.
###### Proof.
If $a<63$, i.e, for $a=21,30,48,57$, we can use a computer program to check
that we need at least $6$ squares to express $\gamma$. Mathematica notebook
with our code is available at
https://sites.google.com/view/tinkovamagdalena/research. Thus, let us assume
that $a\geq 63$. In Lemmas 7.5, 7.6, 7.7, 7.8 and 7.9, we have determined all
squares which can be totally smaller than $\gamma$. We have obtained the
following elements:
1. 1.
$n^{2}\in{\mathbb{Z}}$ where $n^{2}<a^{2}$,
2. 2.
$\rho^{\prime 2}\rho^{\prime\prime
2}\left(\frac{1+\rho+\rho^{2}}{3}\right)^{2}=\frac{a^{2}+4a+6}{9}g_{1}+\frac{a^{2}+2a+3}{9}g_{2}-\frac{a}{3}g_{3}$,
3. 3.
$\rho^{\prime 2}\rho^{\prime\prime
2}\left(-(r+1)g_{2}+g_{3}\right)^{2}=\frac{a^{2}+10a+33+(6a+36)r+9r^{2}}{9}g_{1}+\frac{a^{2}+8a+3+6ar}{9}g_{2}-\frac{a+6+6r}{3}g_{3}$
where $0\leq r\leq\frac{a}{3}-1$,
4. 4.
$\rho^{\prime\prime
2}\rho^{2}\left(-g_{1}^{\prime}-(r+1)g_{2}^{\prime}+2g_{3}^{\prime}\right)^{2}=\frac{4a^{2}+10a+6-(12a+18)r+9r^{2}}{9}g_{1}+\frac{3a^{2}+2a+3-6ar}{9}g_{2}-(a-2r)g_{3}$
where $\frac{a}{3}+1\leq r<\frac{5a-6}{12}$,
5. 5.
$\rho^{2}\rho^{\prime
2}\left(-g_{1}^{\prime\prime}-(r+1)g_{2}^{\prime\prime}+2g_{3}^{\prime\prime}\right)^{2}=\frac{15-8a+36r+9r^{2}}{9}g_{1}+\frac{3-4a-4a^{2}+(12a+18)r}{9}g_{2}+\frac{4a-3-12r}{3}g_{3}$
where $\frac{5(a-3)}{9}\leq r\leq\frac{2a}{3}-1$.
Recall that only these elements can appear in a decomposition of $\gamma$ to a
sum of squares.
First of all, let us note that for all these elements, the coefficients before
$g_{2}$ are nonnegative, and the coefficients before $g_{3}$ are negative or
zero. This is clear for Cases (1), (2) and (3). For (4), we see that
$\frac{3a^{2}+2a+3-6ar}{9}>\frac{a^{2}+10a+6}{18}>0$
and
$-(a-2r)<-1-\frac{a}{6}<0.$
Similarly, for (5), we obtain
$\frac{3-4a-4a^{2}+(12a+18)r}{9}\geq\frac{8a^{2}-42a-81}{27}>0$
and
$\frac{4a-3-12r}{3}\leq-\frac{8a-51}{9}<0.$
Recall that
$\displaystyle\gamma$
$\displaystyle=\frac{34a^{2}+15a+783}{81}+\frac{11a^{2}-29a-39}{27}\rho-\frac{11a-33}{27}\rho^{2}$
$\displaystyle=\frac{2(17a^{2}+24a+342)}{81}g_{1}+\frac{11a^{2}-18a-72}{27}g_{2}-\frac{11(a-3)}{9}g_{3}.$
If $a$ is even, then $\frac{11(a-3)}{9}$ is odd. If we consider the
coefficient before $g_{3}$ for elements in (1)–(5), we see that this
coefficient is odd only for squares in (5) under the assumption that $a$ is
odd. Thus, at least one element from (5) must appear in every decomposition of
$\gamma$ to a sum of squares. Similarly, let $a$ be odd. Then
$\frac{11a^{2}-18a-72}{27}$, i.e., the coefficient before $g_{2}$ of $\gamma$,
is odd. If we again consider the squares in (1)–(5), only elements in (5) has
an odd coefficient before $g_{2}$ for odd $a$. That implies that for all
possible $a\geq 63$, at least one element from (5) appear in every
decomposition of $\gamma$ to a sum of squares.
Considering elements from (5), we can conclude that
$-\frac{11(a-3)}{9}=\frac{4a-3-12r}{3}$ only if $r=\frac{23a-42}{36}$. For
this concrete $r$, the coefficient before $g_{2}$ is
$\frac{3-4a-4a^{2}+(12a+18)r}{9}=\frac{11a^{2}}{27}-\frac{13a}{18}-2.$
Moreover,
$\frac{11a^{2}}{27}-\frac{13a}{18}-2=\frac{11a^{2}-18a-72}{27}$
only if $a=12$, which does not give a simplest cubic field from our subfamily.
Thus, if this concrete element $r=\frac{23a-42}{36}$ were in the decomposition
of $\gamma$, then some other summand would have to have
* •
a non-zero coefficient before $g_{2}$,
* •
a zero coefficient before $g_{3}$.
However, these conditions are not both satisfied for any element from (1)–(5).
Thus, the element from (5), which appears in the decomposition of $\gamma$,
satisfies
$-\frac{11(a-3)}{9}\neq\frac{4a-3-12r}{3}.$
That means that we need at least one more element from (2)–(5) to express
$\gamma$.
We know that for all elements from (5), the coefficient before $g_{3}$
satisfies
$\frac{4a-3-12r}{3}\leq-\frac{8a-51}{9}.$
Thus, the other summand from (2)–(5) has this coefficient between zero and
$-\frac{11(a-3)}{9}+\frac{8a-51}{9}=-\frac{a}{3}-2.$
That is true for
1. (a)
the element
$\frac{a^{2}+4a+6}{9}g_{1}+\frac{a^{2}+2a+3}{9}g_{2}-\frac{a}{3}g_{3}$ from
(2),
2. (b)
the element with $r=0$ from (3),
3. (c)
elements from (4).
Let us now discuss the case when at least two elements from a–c appeared in
the decomposition of $\gamma$. In this case, it suffices to realize that a sum
of two elements from a–c never has the coefficient before $g_{3}$ between
$-\frac{a}{3}-2$ and $0$, which we have derived to be a necessary condition in
this part of the proof. This follows from the fact that the largest
coefficient before $g_{3}$ is achieved by the element in c with the largest
$r$, where we have
$-(a-2r)<-1-\frac{a}{6}.$
So its twice is
$-2(a-2r)<-\frac{a}{3}-2.$
It implies that in every decomposition of $\gamma$, there must appear exactly
one element from (5), and exactly one element from a–c. The remaining summands
belong to (1).
Let us now focus on elements from c, i.e., elements from (4), and let us
assume that in a decomposition of $\gamma$, there is one element from (4) and
one from (5). If we consider coefficients before $g_{2}$ and $g_{3}$, we can
obtain the following system of equations:
$\frac{11a^{2}-18a-72}{27}=\frac{3a^{2}+2a+3-6ar_{1}}{9}+\frac{3-4a-4a^{2}+(12a+18)r_{2}}{9}$
and
$-\frac{11(a-3)}{9}=-(a-2r_{1})+\frac{4a-3-12r_{2}}{3}$
where $\frac{a}{3}+1\leq r_{1}<\frac{5a-6}{12}$ and $\frac{5(a-3)}{9}\leq
r_{2}\leq\frac{2a}{3}-1$. This system has the solution $r_{1}=\frac{a}{3}-1$
and $r_{2}=\frac{5(a-3)}{9}$. However, $r_{1}=\frac{a}{3}-1$ does not belong
to our interval. Thus, the elements from c cannot appear in any decomposition
of $\gamma$ to a sum of squares.
We will proceed with a. In this case, the coefficient before $g_{3}$ leads to
the equation
$-\frac{11(a-3)}{9}=-\frac{a}{3}+\frac{4a-3-12r}{3}$
with the solution $r=\frac{10a-21}{18}$. However, regarding the coefficient
before $g_{2}$, we obtain
$\frac{a^{2}+2a+3}{9}+\frac{3-4a-4a^{2}+(12a+18)r}{9}=\frac{11a^{2}-18a-45}{27}\neq\frac{11a^{2}-18a-72}{27}$
for every $a$.
Thus, only the element in b can appear in every decomposition of $\gamma$.
Moreover, $r=\frac{5(a-3)}{9}$ for the element from (5) in this decomposition,
and, thus,
$\gamma=\left(\frac{a^{2}+10a+33}{9}g_{1}+\frac{a^{2}+8a+3}{9}g_{2}-\frac{a+6}{3}g_{3}\right)\\\
+\left(\frac{25a^{2}-42a-180}{81}g_{1}+\frac{8a^{2}-42a-81}{27}g_{2}-\frac{8a-51}{9}g_{3}\right)+7.$
To express $7$, we need at least $4$ squares. Therefore, we need at least $6$
squares to express $\gamma$, which gives ${\mathcal{P}}({\mathcal{O}}_{K})=6$
as desired. ∎
### 7.4 Universal quadratic forms
In this part, we will use the results on indecomposables in
${\mathcal{O}}_{K}$ to get information on universal quadratic forms over
${\mathcal{O}}_{K}$. Before that, we will summarize several basic facts on
quadratic forms over number fields.
Let us consider quadratic form of the form
$Q(x_{1},x_{2},\ldots,x_{n})=\sum_{1\leq i\leq j\leq n}a_{ij}x_{i}x_{j}$ where
$a_{ij}\in{\mathcal{O}}_{K}$. We say that $Q$ is totally positive definite if
$Q(\gamma_{1},\gamma_{2},\ldots,\gamma_{n})\in{\mathcal{O}}_{K}^{+}$ for every
$n$-tuple $\gamma_{1},\ldots,\gamma_{n}\in{\mathcal{O}}_{K}$ such that
elements $\gamma_{i}$ are not all equal to zero. The quadratic form $Q$ is
classical if $2$ divides $a_{ij}$ for all $i\neq j$. The quadratic form $Q$ is
called diagonal if $a_{ij}=0$ whenever $i\neq j$. And finally, $Q$ is
universal if it represents all totally positive algebraic integers in $K$. In
this part, we will assume that all quadratic forms are totally positive
definite.
To bound the minimal number of variables of universal quadratic forms over our
subfamily of the non-monogenic simplest cubic fields, we will use the method
developed in [16, Section 7]. This approach is based on the knowledge of
indecomposable integers in $K$. In this part, we will use the following
notation. Let $S$ denote the set of indecomposable integers in
${\mathcal{O}}_{K}$ up to multiplication by squares of units in
${\mathcal{O}}_{K}$. Moreover, let us assume that there exist
$\delta\in{\mathcal{O}}_{K}^{\vee,+}$ and
$\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in{\mathcal{O}}_{K}^{+}$ such that
$\text{Tr}(\alpha_{i}\delta)=1$ for all $1\leq i\leq n$. Then, a more general
result for totally real number fields derived in [16, Section 7] implies the
following theorem for cubic fields:
###### Theorem 7.11 ([16, Section 7]).
Let $K$ be a totally real cubic field.
1. 1.
There exist a diagonal universal quadratic form over ${\mathcal{O}}_{K}$ with
${\mathcal{P}}({\mathcal{O}}_{K})\\#S$ variables.
2. 2.
Every classical universal quadratic form over ${\mathcal{O}}_{K}$ has at least
$\frac{n}{3}$ variables.
3. 3.
If $n\geq 240$, then every (non-classical) universal quadratic form over
${\mathcal{O}}_{K}$ has at least $\frac{\sqrt{n}}{3}$ variables.
Now we will use Theorem 7.11 and results on indecomposable integers to derive
bounds on the minimal number of variables of universal quadratic forms over
our nice subfamily of the simplest cubic fields.
###### Proposition 7.12.
Let $K$ be a simplest cubic field such that $a\equiv 3,21\;(\textup{mod 27})$,
$a>12$ and $\frac{\Delta}{27}$ is square-free. Then:
1. 1.
There exists a diagonal universal quadratic form over ${\mathcal{O}}_{K}$ with
$\frac{a^{2}+3a}{3}+12a+12$ variables.
2. 2.
Every classical universal quadratic form over ${\mathcal{O}}_{K}$ has at least
$\frac{a^{2}+3a}{54}$ variables.
3. 3.
If $a>64$, then every (non-classical) universal quadratic form over
${\mathcal{O}}_{K}$ has at least $\frac{\sqrt{a^{2}+3a}}{9\sqrt{2}}$
variables.
###### Proof.
Regarding the first part of the statement, we know that
${\mathcal{P}}({\mathcal{O}}_{K})=6$ by Proposition 7.10. Moreover, in $K$,
every totally positive unit is a square. Thus $\\#S=\frac{a^{2}+3a}{18}+2a+2$,
which is the number of indecomposable integers up to multiplication by totally
positive units.
To get the largest possible value in (2) and (3), it is convenient to consider
the triangle of indecomposables $\alpha$ in (v) of Theorem 1.2, for which we
have found one totally positive element $\delta$ of the codifferent satisfying
$\text{Tr}(\alpha\delta)=1$. In this case, $n=\frac{a^{2}+3a}{18}$, which
gives parts (2) and (3) of the proposition. ∎
## Acknowledgements
The authors want to thank Vítězslav Kala and Giacomo Cherubini for enriching
discussions and helpful suggestions on the contents of this work. The authors
also wish to express their thanks to László Remete and Stéphane Louboutin for
their quick answers. The first author was supported by Czech Science
Foundation GAČR, grant 21-00420M, and by Charles University Research Centre
program UNCE/SCI/022. The second author was supported by Czech Science
Foundation GAČR, grant 22-11563O.
## References
* [1] V. Blomer and V. Kala “Number fields without universal n-ary quadratic forms” In _Math. Proc. Cambridge Philos. Soc._ 159, 2015, pp. 239–252
* [2] V. Blomer and V. Kala “On the rank of universal quadratic forms over real quadratic fields” In _Doc. Math._ 23, 2018, pp. 15–34
* [3] H. Brunotte “Zur Zerlegung totalpositiver Zahlen in Ordnungen totalreeller algebraischer Zahlkörper” In _Arch. Math. (Basel)_ 41 (6), 1983, pp. 502–503
* [4] F. Cánovas Orvay “On cyclic cubic fields.” In _Extracta Mathematicae_ 6.1, 1991, pp. 28–30 URL: http://eudml.org/doc/39912
* [5] M. Čech et al. “Universal quadratic forms and indecomposables over biquadratic fields” In _Math. Nachr._ 292, 2019, pp. 540–555
* [6] W.. Chan, M.-H. Kim and S. Raghavan “Ternary universal integral quadratic forms over real quadratic fields” In _Japan. J. Math._ 22, 1996, pp. 263–273
* [7] A. Dress and R. Scharlau “Indecomposable Totally Positive Numbers in Real Quadratic Orders” In _J. Number Theory_ 14, 1982, pp. 292–306
* [8] A.. Earnest and A. Khosravani “Universal positive quaternary quadratic lattices over totally real number fields” In _Mathematika_ 44, 1997, pp. 342–347
* [9] Z. He and Y. Hu “Pythagoras number of quartic orders containing $\sqrt{2}$” Preprint., 2022
* [10] J.. Hsia, Y. Kitaoka and M. Kneser “Representations of positive definite quadratic forms” In _J. Reine Angew. Math._ 301, 1978, pp. 132–141
* [11] S.. Jang and B.. Kim “A refinement of the Dress-Scharlau theorem” In _J. Number Theory_ 158, 2016, pp. 234–243
* [12] V. Kala “Norms of indecomposable integers in real quadratic fields” In _J. Number Theory_ 166, 2016, pp. 193–207
* [13] V. Kala “Number fields without universal quadratic forms of small rank exist in most degrees” To appear in Math. Proc. Cambridge Philos. Soc., 2022
* [14] V. Kala “Universal quadratic forms and elements of small norm in real quadratic fields” In _Bull. Aust. Math. Soc._ 94, 2016, pp. 7–14
* [15] V. Kala and J. Svoboda “Universal quadratic forms over multiquadratic fields” In _Ramanujan J._ 48, 2019, pp. 151–157
* [16] V. Kala and M. Tinková “Universal quadratic forms, small norms and traces in families of number fields” To appear in Int. Math. Res. Not. IMRN, 2022
* [17] V. Kala and P. Yatsyna “Lifting problem for universal quadratic forms” In _Adv. Math._ 337, 2021, pp. 107497
* [18] V. Kala and P. Yatsyna “On Kitaoka’s conjecture and lifting problem for universal quadratic forms” To appear in Bull. Lond. Math. Soc., 2022
* [19] T. Kashio and R. Sekigawa “The characterization of cyclic cubic fields with power integral bases” In _Kodai Mathematical Journal_ 44, 2021, pp. 290–306
* [20] J. Krásenský “A cubic ring of integers with the smallest Pythagoras number” In _Arch. Math (Basel)_ 118, 2022, pp. 39–48
* [21] J. Krásenský, M. Raška and E. Sgallová “Pythagoras numbers of orders in biquadratic fields” To appear in Expo. Math., 2022
* [22] J. Krásenský, M. Tinková and K. Zemková “There are no universal ternary quadratic forms over biquadratic fields” In _Proc. Edinb. Math. Soc._ 63 (3), 2020, pp. 861–912
* [23] F. Lemmermeyer and A. Pethö “Simplest Cubic Fields” In _Manuscripta Math._ 88, 1995, pp. 53–58
* [24] H. Maaß “Über die Darstellung total positiver Zahlen des Körpers $R(\sqrt{5})$ als Summe von drei Quadraten” In _Abh. Math. Sem. Univ. Hamburg_ 14, 1941, pp. 185–191
* [25] T. Nagell “Zur Arithmetik der Polynome” In _Abhandl. Math. Sem. Hamburg_ 1, 1922, pp. 178–193
* [26] W. Narkiewicz “Elementary and Analytic Theory of Algebraic Numbers” PWN/POLISH SCIENTIFIC PUBLISHE, 1974
* [27] O. Perron “Die Lehre von den Kettenbrüchen” B. G. Teubner, 1913
* [28] M. Peters “Quadratische Formen über Zahlringen” In _Acta Arith._ 24, 1973, pp. 157–164
* [29] R. Scharlau “On the Pythagoras number of orders in totally real number fields” In _J. Reine Angew. Math._ 316, 1980, pp. 208–210
* [30] D. Shanks “The Simplest Cubic Fields” In _Math. Comput._ 28, 1974, pp. 1137–1152
* [31] C.. Siegel “Sums of $m$-th powers of algebraic integers” In _Annals of Mathematics_ 46, 1945, pp. 313–339
* [32] E. Thomas and A.. Vasquez “On the Resolution of Cusp Singularities and the Shintani Decomposition in Totally Real Cubic Number Fields” In _Math. Ann._ 247, 1980, pp. 1–20
* [33] M. Tinková “On the Pythagoras number of the simplest cubic fields” Preprint, 2022
* [34] M. Tinková “Trace and norm of indecomposable integers in cubic orders” To appear in Ramanujan J., 2022
* [35] M. Tinková and P. Voutier “Indecomposable integers in real quadratic fields” In _J. Number Theory_ 212, 2020, pp. 458–482
* [36] P. Yatsyna “A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real field” In _Comment. Math. Helvet._ 94, 2019, pp. 221–239
|
# Electronic spectrum and optical properties of Y-shaped Kekulé-patterned
graphene: Band nesting resonance as an optical signature
Yawar Mohammadi E-mail address<EMAIL_ADDRESS>
###### Abstract
Employing tight-binding model we investigate the effects of a uniform Y-shaped
Kekulé lattice distortion on the electronic spectrum and optical conductivity
of graphene. We derive a low-energy effective Hamiltonian which is found to be
in excellent agreement with one calculated from a diagonalization of the full
tight-binding Hamiltonian. Then using the low-energy Hamiltonian and Kubo
formula we obtain an analytical expression for the real part of the optical
conductivity used to explore the effects of chemical potential, temperature
and on-site and hopping energy deviations in details. In particular we find
that Y-shaped Kekué-patterned graphene at finite chemical potential displays a
large optical response called band nesting resonance. This effect is shown to
be robust against increasing temperature, facilitating its detection as an
optical signature for the Y-shaped Kekulé distortion even at room temperature.
Department of Physics, Farhangian University, Tehran, Iran
_Keywords_ : Kekulé-patterned graphene; Kubo Formula; Optical conductivity;
Band nesting resonance.
## 1 Introduction
Recently Gutiérrez et. al [1] have realized experimentally a Y-shaped periodic
alternation of weak and strong C-C bonds in a superlattice of graphene grown
epitaxially onto copper substrate, called Y-shaped Kekulé lattice distortion
with reference to the Kekulé dimerization in a benzene ring. Using density
functional theory calculations they showed [1] that removing copper atoms from
the topmost surface allows the system to relax created an alternating network
of C-C bonds similar to the Y-shaped Kekulé lattice distortion. Gamayun et.al
introduced a tight-binding Hamiltonian to explore the effects of a uniform
Y-shaped Kekulé lattice on the properties of graphene, in which this effect
has been taken into account by adding a nearest-neighbor hopping energy
deviation to the minimal tight-binding Hamiltonian of graphene. According to
this Hamiltonian the uniform Y-shaped Kekulé distortion locks the valley
degree of freedom of the charge carriers to their direction of motion,
resulting in the breaking of the valley degeneracy of graphene and emerging
two species of massless Dirac fermions[2]. Some researchers, using this
Hamiltonian, have investigated different properties of Y-shaped Kekulé-
patterned graphene. Wu et. al[3] proposed a type of valley field-effect
transistors for Y-shaped Kekulé-patterned graphene and explored tuning valley
pseudomagnetoresistance via an electric field. Andrade et. al[4] studied the
effects of uniaxial strain on the band structure of all types of Kekulé-
distorted graphene. In other works the effect of the Y-shaped Kekulé
distortion on the electronic transport properties[5, 6], dynamical
polarization[7], magneto-optical conductivity[8] and quantum Hall effect[9] in
graphene has been investigated.
In this paper we consider the electronic spectrum and the optical conductivity
of such a system with particular emphasis on the effects of the on-site energy
deviation, which in the previous researches has been neglected[3, 4, 5, 6, 7,
8, 9, 10, 11] while as shown here it leads to fascinating optical properties.
The structure of our paper is organized as follows. In section 2, we introduce
our tight-binding model in which the effects of the Y-shaped Kekulé distortion
is taken into account by including both on-site and hopping energy deviations
in the minimal tight-binding Hamiltonian of graphene. Then we derive a low-
energy effective Hamiltonian in subsection 2.1 which is found to be in
excellent agreement with one calculated from a diagonalization of the full
tight-binding Hamiltonian. The main effect of the on-site energy deviation on
the band structure is that a set of bands gains an effective mass and a shift
in energy, thus lifting the degeneracy of the conduction bands at the Dirac
point. In the next subsection we obtain an analytical expression for the real
part of the optical conductivity using Kubo formula, which is used to explore
the effects of temperature, chemical potential and on-site and nearest-
neighbor hopping energy deviations on the optical conductivity. Then we
present our results for the optical conductivity as a function of the photon
energy in next section. In particular we find that in the limit of zero
chemical potential the optical conductivity displays a dip-peak structure
located at the photon energy corresponding to 2 times the effective on-site
energy deviation. Furthermore, it is shown that at finite chemical potential
Y-shaped Kekué-patterned graphene exhibits a large optical response caused by
nesting of the conduction or valance bands. This effect is shown to be robust
with respect to increasing temperature, facilitating its detection even at
room temperature. In this section we also discuss the effects of next-nearest-
neighbor hopping energy deviation. Finally we end the paper by presenting
summary and conclusions in section 4.
## 2 Model and Formulation
A schematic representation of Y-shaped Kekulé-patterned graphene superlattice
has been displayed in Fig. 1(a), in which the three thick lines in each unit
cell of graphene superlattice denote the bonds that connect the carbon atom
located on the copper-atom vacancies in substrate to its nearest neighbors.
These bonds acquire a shorter nearest-neighbor bond [1], leading to elongation
of the other nearest-neighbor bonds, which is shown by thin lines. Also the
on-site energy of carbon atoms located on the cooper-atom vacancies is
expected to be changed with respect to other carbon atoms. Therefore, by
taking into account deviations in the hopping and on-site energies [12], the
tight-binding Hamiltonian for Y-shaped Kekulé-patterned graphene in the
nearest-neighbor approximation is given by
$\displaystyle H$ $\displaystyle=$ $\displaystyle
u\sum_{i=1}^{N/3}a_{\vec{\mathbf{R}}_{i}}^{{\dagger}}a_{\vec{\mathbf{R}}_{i}}-(t_{0}+\delta
t)\sum_{i=1}^{N/3}\sum_{n=1}^{3}(b_{\vec{\mathbf{R}}_{i}+\vec{\mathbf{\delta}}_{n}}^{{\dagger}}a_{\vec{\mathbf{R}}_{i}}+h.c.)$
(1) $\displaystyle-$ $\displaystyle(t_{0}-\delta
t)\sum_{i=1}^{N/3}\sum_{n=1}^{3}(b_{\vec{\mathbf{R}}_{i}+\vec{\mathbf{\delta}}_{3}-\vec{\mathbf{\delta}}_{1}+\vec{\mathbf{\delta}}_{n}}^{{\dagger}}a_{\vec{\mathbf{R}}_{i}+\vec{\mathbf{\delta}}_{3}-\vec{\mathbf{\delta}}_{1}}+b_{\vec{\mathbf{R}}_{i}+\vec{\mathbf{\delta}}_{3}-\vec{\mathbf{\delta}}_{2}+\vec{\mathbf{\delta}}_{n}}^{{\dagger}}a_{\vec{\mathbf{R}}_{i}+\vec{\mathbf{\delta}}_{3}-\vec{\mathbf{\delta}}_{2}}+h.c.)$
where $t_{0}$ and $N$ are the nearest-neighbor hopping energy and the number
of the unit cells in pristine graphene respectively (The number of the nuit
cells of the Y-shaped Kekulé-patterned graphene is $N/3$.). The operator
$a_{\vec{\mathbf{R}}_{i}}^{{\dagger}}$
($b_{\vec{\mathbf{R}}_{i}}^{{\dagger}}$) creates an electron in the carbon
atoms on sublattice $A$ ($B$) located at $i^{th}$ unit cell.
$\vec{\mathbf{R}}_{i}=n_{i}\vec{\mathbf{A}}_{1}+m_{i}\vec{\mathbf{A}}_{2}$ are
the translation vectors of Y-shaped Kekulé-patterned graphene with
$\vec{\mathbf{A}}_{1}=(3\sqrt{3}a/2,3a/2)$ and
$\vec{\mathbf{A}}_{2}=(-3\sqrt{3}a/2,3a/2)$ its primitive translation vectors,
and $n_{i}$ and $m_{i}$ are integer numbers. The vectors
$\vec{\mathbf{\delta}}_{1}=(\sqrt{3}a/2,a/2)$,
$\vec{\mathbf{\delta}}_{2}=(\sqrt{3}a/2,a/2)$ and
$\vec{\mathbf{\delta}}_{3}=(0,-a)$ are drown from each A carbon atom to its
nearest neighbors and $a$ is the shortest carbon-carbon distance. The on-site
energy deviation, $u$, is assumed to be nonzero only for the carbon atoms
located on the cooper-atom vacancies in the substrate. The hopping energy
deviation, which is due to the change in the bond length, is assumed to be
$+\delta t$ and $-\delta t$ for carbon-carbon bonds shown by thick and thin
lines, respectively, in Fig. 1(a). This Hamiltonian can be diagonalized by
selecting the appropriate unit cell, red hexagonal in 1(a), in space k. The
corresponding band structure for different values of the hopping and on-site
energy deviations, (a) $\delta t=0.1t_{0}$ and $u=0$, (b) $\delta t=0$ and
$u=0.1t_{0}$, (c) $\delta t=0.1t_{0}$ and $u=0.1t_{0}$ has been shown in
Figure 1 by solid black lines.
Figure 1: (a) A schematic representation of a single layer graphene with a
uniform Y-shaped Kekulé lattice distortion in which primitive cells of
pristine and Y-shaped Kekulé-patterned has been shown by green and red
hexagonal respectively. $\vec{a}_{i}$ ($\vec{A}_{i}$) indicates primitive
translation vectors of pristine (Y-shaped Kekulé-patterned) graphene. (b)
Reciprocal lattices for pristine(in green) and Y-shaped Kekulé-patterned (in
black) graphene.
### 2.1 Low-energy effective Hamiltonian
To build a low-energy effective Hamiltonian, we must first express the
Hamiltonian of Eq. 1 in the momentum space. As it is clear from Fig. 1(b), the
first Brillouin zone of pristine graphene can be represented through three
copies of that of graphene superlattice which are set to be centered at
$\mathbf{K}_{j}$ (j=1,2 and 3),
$\mathbf{K}_{1}=\mathbf{K}=(+4\pi/3\sqrt{3},0)$,
$\mathbf{K}_{2}=\mathbf{\Gamma}=(0,0)$ and
$\mathbf{K}_{2}=\mathbf{K}^{{}^{\prime}}=(-4\pi/3\sqrt{3},0)$. Therefore, the
annihilation (creation) operators can be expanded in the momentum space as
[13]
$a_{i}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}\sum_{j}e^{i(\mathbf{K}_{j}+\mathbf{k}).\mathbf{R}_{i}}a_{\mathbf{K}_{j}+\mathbf{k}}$
($a_{i}^{{\dagger}}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}\sum_{j}e^{-i(\mathbf{K}_{j}+\mathbf{k}).\mathbf{R}_{i}}a^{{\dagger}}_{\mathbf{K}_{j}+\mathbf{k}}$)
where N is the number of unit cells of pristine graphene, and $\mathbf{k}$
runs over the first Brillouin of graphene superlattice, red hexagonal in 1(b).
The Fourier transform of Eq. (1) is then,
$\displaystyle H(\mathbf{k})$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}}\sum_{j}(\epsilon^{\ast}(\mathbf{K}_{j}+\mathbf{k})b_{\mathbf{K}_{j}+\mathbf{k}}^{{\dagger}}a_{\mathbf{K}_{j}+\mathbf{k}}+h.c.)+\frac{u}{3}\sum_{\mathbf{k}}\sum_{j,j^{{}^{\prime}}}a_{\mathbf{K}_{j}+\mathbf{k}}^{{\dagger}}a_{\mathbf{K}_{j^{{}^{\prime}}}+\mathbf{k}}$
(2) $\displaystyle+$
$\displaystyle\Delta\sum_{\mathbf{k}}\sum_{j,j^{{}^{\prime}}}([1-2\cos(2(j-j^{{}^{\prime}})\pi/3)]\epsilon^{\ast}(\mathbf{K}_{j}+\mathbf{k})b_{\mathbf{K}_{j}+\mathbf{k}}^{{\dagger}}a_{\mathbf{K}_{j^{{}^{\prime}}}+\mathbf{k}}+h.c.),$
where
$\epsilon^{\ast}(\mathbf{K}_{j}+\mathbf{k})=-t_{0}\sum_{n=1}^{3}e^{-i(\mathbf{K}_{j}+\mathbf{k}).\mathbf{\delta}_{n}}$
and $\Delta=\frac{\delta t}{3t_{0}}$.
By defining
$\hat{\psi}_{\mathbf{k}}=(a_{\mathbf{\Gamma}+\mathbf{k}},b_{\mathbf{\Gamma}+\mathbf{k}},a_{\mathbf{K}+\mathbf{k}},b_{\mathbf{K}+\mathbf{k}},a_{\mathbf{K}^{{}^{\prime}}+\mathbf{k}},b_{\mathbf{K}^{{}^{\prime}}+\mathbf{k}})^{T}$
we can rewrite the Hamiltonian as
$\displaystyle
H(\mathbf{k})=\hat{\psi}^{{\dagger}}_{\mathbf{k}}\left(\begin{array}[]{cc}\hat{H}_{H}&\hat{T}^{{\dagger}}\\\
\hat{T}&\hat{H}_{L}\end{array}\right)\hat{\psi}_{\mathbf{k}},$ (5)
where
$\displaystyle\hat{H}_{H}=\left(\begin{array}[]{cc}\frac{u}{3}&(1-\Delta)\epsilon(\mathbf{\Gamma}+\mathbf{k})\\\
(1-\Delta)\epsilon^{\ast}(\mathbf{\Gamma}+\mathbf{k})&0\end{array}\right),$
(8)
and
$\displaystyle\hat{H}_{L}=\left(\begin{array}[]{cccc}\frac{u}{3}&(1-\Delta)\epsilon(\mathbf{K}+\mathbf{k})&\frac{u}{3}&2\Delta\epsilon(\mathbf{K}^{{}^{\prime}}+\mathbf{k})\\\
(1-\Delta)\epsilon^{\ast}(\mathbf{K}+\mathbf{k})&0&2\Delta\epsilon^{\ast}(\mathbf{K}+\mathbf{k})&0\\\
\frac{u}{3}&2\Delta\epsilon(\mathbf{K}+\mathbf{k})&\frac{u}{3}&(1-\Delta)\epsilon(\mathbf{K}^{{}^{\prime}}+\mathbf{k})\\\
2\Delta\epsilon^{\ast}(\mathbf{K}^{{}^{\prime}}+\mathbf{k})&0&(1-\Delta)\epsilon^{\ast}(\mathbf{K}^{{}^{\prime}}+\mathbf{k})&0\end{array}\right),$
(13)
are the high and low energy sectors of $\hat{H}(\mathbf{k})$, respectively,
and
$\displaystyle\hat{T}=\left(\begin{array}[]{cccc}\frac{u}{3}&2\Delta\epsilon(\mathbf{K}+\mathbf{k})&\frac{u}{3}&2\Delta\epsilon(\mathbf{K}^{{}^{\prime}}+\mathbf{k})\\\
2\Delta\epsilon^{\ast}(\mathbf{\Gamma}+\mathbf{k})&0&2\Delta\epsilon^{\ast}(\mathbf{\Gamma}+\mathbf{k})&0\end{array}\right),$
(16)
is the coupling between them. Then, we consider the Schrodinger equations
applied to Eq. (5),
$\displaystyle\hat{H}_{H}\hat{\Psi}_{H}+\hat{T}\hat{\Psi}_{L}=E\hat{\Psi}_{H},$
(17)
and
$\displaystyle\hat{T}^{{\dagger}}\hat{\Psi}_{H}+\hat{H}_{L}\hat{\Psi}_{L}=E\hat{\Psi}_{L},$
(18)
on each space, where $\hat{\Psi}_{H}$ and $\hat{\Psi}_{L}$ are the components
of the eigenfunction, $\hat{\Psi}=(\hat{\Psi}_{H},\hat{\Psi}_{L})$. One can
obtain $\hat{\Psi}_{H}$ from Eq. (17) and use it on Eq. (18) to obtain
$\displaystyle\hat{H}_{eff}=\hat{H}_{L}+\hat{T}^{{\dagger}}(E\hat{1}-\hat{H}_{L})^{-1}\hat{T},$
(19)
which is the effective Hamiltonian for the $\hat{\Psi}_{L}$ component. Then,
we expand $\epsilon(\mathbf{K}_{j}+\mathbf{k})$ and their complex conjugates
in Eqs. (8-16) up to the first order in $\mathbf{k}$,
$\epsilon(\mathbf{\Gamma}+\mathbf{k})\approx-3t_{0}$,
$\epsilon(\mathbf{K}+\mathbf{k})\approx\frac{3}{2}t_{0}a(k_{x}+ik_{y})$ and
$\epsilon(\mathbf{K}^{{}^{\prime}}+\mathbf{k})\approx-\frac{3}{2}t_{0}a(k_{x}-ik_{y})$,
and insert them into Eq. (19). Therefore, we arrive at
$\displaystyle\hat{H}_{eff}=\left(\begin{array}[]{cccc}U&v_{B}(k_{x}+ik_{y})&U&-v_{C}(k_{x}-ik_{y})\\\
v_{B}(k_{x}-ik_{y})&0&v_{C}(k_{x}-ik_{y})&0\\\
U&v_{C}(k_{x}+ik_{y})&U&-v_{B}(k_{x}-ik_{y})\\\
-v_{C}(k_{x}+ik_{y})&0&-v_{B}(k_{x}+ik_{y})&0\end{array}\right),$ (24)
where
$U=\frac{u}{3}(1-\frac{4\Delta}{1-\Delta}+\frac{4\Delta^{2}}{(1-\Delta)^{2}})$,
$v_{B}=\frac{3}{2}t_{0}a(1-\Delta-\frac{4\Delta^{2}}{1-\Delta})$, and
$v_{C}=\frac{3}{2}t_{0}a(2\Delta-\frac{4\Delta^{2}}{1-\Delta})$ (Notice that
although Eq. (24) is convenient to obtain the paramagnetic current operator,
to get the diamagnetic part of the current operator we must keep the terms up
to second order in $\mathbf{k}$). The energy bands of the low-energy effective
Hamiltonian are
$\displaystyle E_{1\mathbf{k}}$ $\displaystyle=$ $\displaystyle-
E_{2\mathbf{k}}=-(v_{B}-v_{C})k,$ $\displaystyle E_{3\mathbf{k}}$
$\displaystyle=$ $\displaystyle U-\sqrt{(v_{B}+v_{C})^{2}k^{2}+U^{2}},$
$\displaystyle E_{4\mathbf{k}}$ $\displaystyle=$ $\displaystyle
U+\sqrt{(v_{B}+v_{C})^{2}k^{2}+U^{2}},$ (25)
where $k=\sqrt{k^{2}_{x}+k^{2}_{y}}$. Figure 2 displays the exact band
structure resulting from Eq. (1) or (2) in comparison with that obtained from
our low-energy effective Hamiltonian for different values of $u$ and $\delta
t$, (a) $u=0$ and $\delta t=0.1t_{0}$, (b) $u=0.1t_{0}$ and $\delta t=0$, (c)
$u=0.1t_{0}$ and $\delta t=0.1t_{0}$. An excellent agreement between the
energy bands of the effective Hamiltonian and the exact ones in the low-energy
regime is clear. One can see that, the Y-shaped Kekulé distortion breaks the
valley degeneracy and couples the energy bands at $\mathbf{K}$ and
$\mathbf{K}^{{}^{\prime}}$ in the firs brillouin zone, resulting in two
concentric energy bands with different velocities at $\mathbf{\Gamma}$ point,
but contrary to what has been reported in previous works [2, 14], their
deviation from initial velocity is not symmetric. According to our results,
different velocities are $(1-3\Delta)v_{F}$ and
$(1+\Delta-\frac{8\Delta^{2}}{1-\Delta})v_{F}$ with $v_{F}=\frac{3}{2}t_{0}a$
being the initial Fermi velocity. On the other hand, the on-site potential
deviation, in addition two breaking valley symmetry, has two other effects.
First, it causes two energy bands gain effective mass. Second, it shifts the
massive energy bands, resulting in a particle-hole symmetry breaking and
lifting the four fold degeneracy at Dirac point. These effects lead to the
appearance of attractive optical signatures, which are discussed in the next
section.
Figure 2: Comparison of the exact energy bands resulting from Eq. (1) or (2)
with the effective low-energy bands, Eq. (2.1), for different values of the
hopping and on-site energy deviations, (a) $\delta t=0.1t_{0}$ and $u=0$, (b)
$\delta t=0$ and $u=0.1t_{0}$ and (c) $\delta t=0.1t_{0}$ and $u=0.1t_{0}$.
$\mathbf{M^{\ast}}=(+2\pi/3\sqrt{3},0)$, $\mathbf{\Gamma}=(0,0)$ and
$\mathbf{K^{\ast}}=(+2\pi/3\sqrt{3},+2\pi/9)$ are high-symmetry points in the
first brillouin zone of graphene superlattice. See red hexagonal in Fig. 1(b)
.
### 2.2 Current operator and Kubo Formula
The finite frequency conductivity calculated by Kubo Formula is given by[15]
$\displaystyle\sigma_{\alpha\alpha}(\Omega)=\frac{\Pi_{\alpha\alpha}(\Omega+i0^{+})}{i\hbar(\Omega+i0^{+})S}+\frac{<j_{\alpha}^{D}>}{i(\Omega+i0^{+})S},$
(26)
where $S$ is the area of the sample, $\alpha$ denotes $x$ or $y$, $\Omega$ is
the frequency, $\Pi_{\alpha\alpha}(\Omega+i0^{+})$ is the correlation function
of the paramagnetic current operator and $<j_{\alpha}^{D}>$ is the expectation
value of the diamagnetic part of current operator. The function
$\Pi_{\alpha\alpha}(\Omega+i0^{+})$ is obtained by analytical continuum of the
corresponding Matsubara current-current correlation function defined as[16,
17, 18]
$\displaystyle\Pi_{\alpha\alpha}(i\omega_{n})=\int_{0}^{\hbar\beta}d\tau
e^{i\omega_{n}\tau}<T_{\tau}j_{\alpha}^{P}(\tau)j_{\alpha}^{P}(0)>,$ (27)
where $i\omega_{n}$ are the bosonic Matsubara frequencies, $\beta=1/k_{B}T$
with $k_{B}$ the Boltzmann constant and T the temperature, and $T_{\tau}$ is
time ordering operator. One can show that the paramagnetic current operator
can be written as [17]
$j_{\alpha}^{P}=-g_{s}e\Sigma_{\mathbf{k}}\hat{\psi}^{{\dagger}}_{\mathbf{k},L}\hat{v}_{\alpha}\hat{\psi}_{\mathbf{k},L}$
where $g_{s}=2$ is the spin degeneracy, $e$ is the electron charge,
$\hat{\psi}_{\mathbf{k},L}$ is the low-energy component of
$\hat{\psi}_{\mathbf{k}}=(\hat{\psi}_{\mathbf{k},H},\hat{\psi}_{\mathbf{k},L})^{T}$,
and $\hat{v}_{\alpha}=\partial\hat{H}_{eff}/\hbar\partial k_{\alpha}$.
Consequently, the current-current correlation function can then be written in
the usual bubble approximation as[16, 17]
$\displaystyle\Pi_{\alpha\alpha}(i\nu_{m})=\frac{g_{s}e^{2}}{\beta\hbar}\sum_{i\nu_{m}}\sum_{\mathbf{k}}Tr[\hat{v}_{\alpha}\hat{G}(i\nu_{m}+i\omega_{n},\mathbf{k})\hat{v}_{\alpha}\hat{G}(i\nu_{m},\mathbf{k})],$
(28)
where
$\hat{G}(i\nu_{m},\mathbf{k})=(i\nu_{m}\hat{1}-\hat{H}_{eff}/\hbar)^{-1}$,
$Tr[\hat{A}]$ denotes the trace of $\hat{A}$ matrix, and $i\nu_{m}$ are the
fermionic Matsubara frequencies. Using the spectral representation of the
Green’s function matrix,[17, 19]
$\displaystyle\hat{G}(z,\mathbf{k})=\int_{-\infty}^{+\infty}\frac{d\omega^{{}^{\prime}}}{2\pi}\frac{\hat{A}(\omega^{{}^{\prime}},\mathbf{k})}{z-\omega^{{}^{\prime}}},$
(29)
the imaginary and real part of the current-current correlation function can be
written as
$\displaystyle\Im\Pi_{xx}(\Omega+i0^{+})=-\frac{\pi
g_{s}e^{2}}{\hbar^{2}}\sum_{\mathbf{k}}\sum_{i,j}\chi(E_{j\mathbf{k}},E_{i\mathbf{k}})[n_{F}(E_{j\mathbf{k}})-n_{F}(E_{i\mathbf{k}})]\delta(\Omega+E_{j\mathbf{k}}/\hbar-
E_{i\mathbf{k}}/\hbar),$ (30)
and
$\displaystyle\Re\Pi_{xx}(\Omega+i0^{+})=\frac{g_{s}e^{2}}{\hbar^{2}}\sum_{\mathbf{k}}\sum_{i,j}\chi(E_{i\mathbf{k}},E_{j\mathbf{k}})\frac{n_{F}(E_{j\mathbf{k}})-n_{F}(E_{i\mathbf{k}})}{\Omega+E_{j\mathbf{k}}/\hbar-
E_{i\mathbf{k}}/\hbar},$ (31)
where $n_{F}(x)=1/(1+exp[(x-\mu)/k_{B}T])$ is Fermi-Dirac distribution
function and
$\displaystyle\chi(E_{i\mathbf{k}},E_{j\mathbf{k}})$ $\displaystyle=$
$\displaystyle
4(v^{2}_{B}\cos(2\varphi_{\mathbf{k}})+v^{2}_{C})\mathcal{M}_{12,E_{i\mathbf{k}}}\mathcal{M}_{12,E_{j\mathbf{k}}}+4(v^{2}_{B}+v^{2}_{C}\cos(2\varphi_{\mathbf{k}}))\mathcal{M}_{12,E_{i\mathbf{k}}}\mathcal{M}_{12,E_{j\mathbf{k}}}$
(32) $\displaystyle+$ $\displaystyle
4v_{B}v_{C}(1+\cos(2\varphi_{\mathbf{k}}))[\mathcal{M}_{12,E_{i\mathbf{k}}}\mathcal{M}_{13,E_{j\mathbf{k}}}+\mathcal{M}_{13,E_{i\mathbf{k}}}\mathcal{M}_{12,E_{j\mathbf{k}}}]$
$\displaystyle+$ $\displaystyle
2(v^{2}_{B}+v^{2}_{C})[\mathcal{M}_{11,E_{i\mathbf{k}}}\mathcal{M}_{22,E_{j\mathbf{k}}}+\mathcal{M}_{22,E_{i\mathbf{k}}}\mathcal{M}_{11,E_{j\mathbf{k}}}]$
$\displaystyle+$ $\displaystyle
4v_{B}v_{C}[\mathcal{M}_{11,E_{i\mathbf{k}}}\mathcal{M}_{23,E_{j\mathbf{k}}}+\mathcal{M}_{23,E_{i\mathbf{k}}}\mathcal{M}_{11,E_{j\mathbf{k}}}]$
$\displaystyle-$ $\displaystyle
4v_{B}v_{C}\cos(2\varphi_{\mathbf{k}})[\mathcal{M}_{14,E_{i\mathbf{k}}}\mathcal{M}_{22,E_{j\mathbf{k}}}+\mathcal{M}_{22,E_{i\mathbf{k}}}\mathcal{M}_{14,E_{j\mathbf{k}}}]$
$\displaystyle-$ $\displaystyle
2(v^{2}_{B}+v^{2}_{C})[\mathcal{M}_{14,E_{i\mathbf{k}}}\mathcal{M}_{23,E_{j\mathbf{k}}}+\mathcal{M}_{23,E_{i\mathbf{k}}}\mathcal{M}_{14,E_{j\mathbf{k}}}],$
in which $\varphi_{\mathbf{k}}=\tan^{-1}(k_{y}/k_{x})$ and
$\mathcal{M}_{mn,E_{i\mathbf{k}}}$ are the coefficients of delta functions
$\delta(\omega-E_{i\mathbf{k}}/\hbar)$ in the matrix components of the
spectral function,
$A_{mn}(\omega,\mathbf{k})=2\pi\sum_{i}\mathcal{M}_{mn,E_{i\mathbf{k}}}\delta(\omega-
E_{i\mathbf{k}}/\hbar)$.
To obtain $<j_{\alpha}^{D}>$ we keep the terms up to second order in
$\mathbf{k}$ in the expansion of $\epsilon(\mathbf{K}_{j}+\mathbf{k})$ and its
complex conjugate in Eq. (19), so we can write the diamagnetic current
operator as[21]
$j^{D}_{\alpha}=-g_{s}e^{2}\Sigma_{\mathbf{k}}\hat{\psi}^{{\dagger}}_{\mathbf{k},L}\hat{w}_{\alpha}\hat{\psi}_{\mathbf{k},L}$,
where
$\displaystyle\hat{w}_{x}=\frac{1}{\hbar^{2}}\frac{\partial^{2}\hat{H}_{eff}}{\partial
k^{2}_{x}}=\frac{a}{2\hbar^{2}}\left(\begin{array}[]{cccc}0&-v_{B}&0&v_{C}\\\
-v_{B}&0&-v_{C}&0\\\ 0&-v_{C}&0&v_{B}\\\ v_{C}&0&v_{B}&0\end{array}\right).$
(37)
After inserting above equations into $<j_{\alpha}^{D}>$ and performing some
straightforward calculations[20], we reach the following statement
$\displaystyle<j_{x}^{D}>=\frac{g_{s}ae^{2}}{2\hbar^{{}^{2}}}\sum_{\mathbf{k}}\sum_{i}\Lambda(E_{i\mathbf{k}})n_{F}(E_{i\mathbf{k}}),$
(38)
where
$\displaystyle\Lambda(E_{i\mathbf{k}})=4v_{B}\mathcal{M}_{12,E_{i\mathbf{k}}}\cos(2\varphi_{\mathbf{k}})+2v_{C}\mathcal{M}_{14,E_{i\mathbf{k}}}\cos(4\varphi_{\mathbf{k}})+2v_{C}\mathcal{M}_{23,E_{i\mathbf{k}}}.$
(39)
The real part of the finite frequency conductivity, which is related to the
optical quantities such optical absorption and reflectivity, is given by
$\displaystyle\Re\sigma_{\alpha\alpha}(\Omega)=D\delta(\Omega)+\frac{\Im\Pi_{\alpha\alpha}(\Omega+i0^{+})}{\hbar\Omega
S},$ (40)
with
$\displaystyle
D=-\pi\frac{<j_{\alpha}^{D}>}{S}-\pi\frac{\Re\Pi_{\alpha\alpha}(\Omega+i0^{+})}{\hbar
S}.$ (41)
called Drude weight. Finally, by inserting Eqs. (30), (31) and (38) into (40),
we arrive at the following equation for the real part of the optical
conductivity
$\displaystyle\Re\sigma_{\alpha\alpha}(\Omega)/\sigma_{0}=$ $\displaystyle-$
$\displaystyle\frac{2\pi
ag_{s}}{\hbar}\frac{1}{S}\sum_{\mathbf{k}}\sum_{i}\Lambda(E_{i\mathbf{k}})n_{F}(E_{i\mathbf{k}})\delta(\Omega)$
$\displaystyle-$ $\displaystyle\frac{4\pi
g_{s}}{\hbar^{2}}\frac{1}{S}\sum_{\mathbf{k}}\sum_{i,j}\chi(E_{i\mathbf{k}},E_{j\mathbf{k}})\frac{n_{F}(E_{j\mathbf{k}})-n_{F}(E_{i\mathbf{k}})}{\Omega+E_{j\mathbf{k}}/\hbar-
E_{i\mathbf{k}}/\hbar}\delta(\Omega)$ $\displaystyle-$
$\displaystyle\frac{4\pi
g_{s}}{\hbar^{2}}\frac{1}{S}\sum_{\mathbf{k}}\sum_{i,j}\chi(E_{i\mathbf{k}},E_{j\mathbf{k}})\frac{[n_{F}(E_{j\mathbf{k}})-n_{F}(E_{i\mathbf{k}})]}{\Omega}\delta(\Omega+E_{j\mathbf{k}}/\hbar-
E_{i\mathbf{k}}/\hbar),$
where $\sigma_{0}=\frac{e^{2}}{4\hbar}$ is the conductivity of the pristine
graphene.
Eqs. (32), (39), and (2.2) along with the low-energy effective Hamiltonian and
its energy bands are our main results, which are solved numerically in the
next section to investigate the effects of the kekulé distortion on the
optical conductivity of graphene at low energy regime.
## 3 Results and discussion
Here we present our results exploring the effects of Y-shaped Kekulé
distortion on the optical conductivity of graphene, which is obtained by
evaluating Eq. (2.2) numerically. Although it is expected that the Y-shaped
Kekulé distortion leads to a deviation in both hopping and on-site energies,
but in order to clarify the role of each of them separately, in addition to
the $\delta t\neq 0$ and $u\neq 0$ case which is expected to be more
realistic, we also investigate the cases in which $\delta t\neq 0$ or $u\neq
0$. For the numerical evaluation of Eq. (2.2), we use the Lorentzian
representation of the delta function, $\delta(x)=(\eta/\pi)/(x^{2}+\eta^{2})$
with broadening $\eta$ reflecting the effect of electron scattering from
disorder, $1/\tau_{imp}=2\eta$, phenomenologically. In all calculations we set
$\eta=0.00002t_{0}$. Furthermore, we put $t_{0}$ as the energy scale and the
optical conductivity curves are scaled by $\sigma_{0}$. So,
$\sigma_{xx}/\sigma_{0}$ is calculated and plotted as a function of
$\hbar\Omega/t_{0}$ with $\hbar\Omega$ being the photon energy.
Zero chemical potential: Our results for the optical conductivity at zero
chemical potential and temperature are shown in Fig. 3, in which the hopping
and on-site energy deviations are (a) $\delta t=0.10t_{0}$ and $u=0$, (b)
$\delta t=0$ and $u=0.10t_{0}$ and (c) $\delta t=0.10t_{0}$ and $u=0.10t_{0}$.
The results for Kekulé-patterned (pristine) graphene are shown by solid
magenta (dashed black) curves. All possible optical transitions are displayed
in the lower insets of Fig. 3 and the resulting structures in the optical
conductivity are identified by arrows with the same color on the curves. The
upper insets exhibit the contribution of each transition in optical
conductivity separately identified by the curve with same color. In the case
of $\delta t=0.10t_{0}$ and $u=0$, the optical conductivity displays a flat
interband response at all frequencies which also is found in undoped pristine
graphene[22, 23], reflecting the linear band structure of Kekulé-patterned
graphene near $\mathbf{\Gamma}$ point. This is to be contrast with the case of
$u\neq 0$ shown in the middle panels Fig. 3, in which a dip-peak structure is
also seen around $2U$. The appearance of this structure in the optical
conductivity, which can be detected as a optical signature for the Y-shaped
Kekulé distortion in single layer graphene at the charge neutrality, is
explained by examining the contribution of each of the transitions in the
optical conductivity. In the presence of the on-site energy deviation
$E_{3\mathbf{k}}\rightarrow E_{2\mathbf{k}}$ transition is possible for all
$\hbar\Omega>0$, but $E_{1\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ transition,
due to the presence of the energy gap, is restricted to $\hbar\Omega>2U$. The
$E_{3\mathbf{k}}\rightarrow E_{2\mathbf{k}}$ ($E_{1\mathbf{k}}\rightarrow
E_{4\mathbf{k}}$) transition, as shown in the upper inset of the middle frame
of Fig. 3, starts at zero ($2U$) photon energy with a spectral weight that
decreases (increases) by increasing the photon energy and finally approaches
$\sigma_{0}/2$ at high photon energies. These facts manifest themselves as a
dip in the curve of the total optical conductivity. This spectral weight lost,
for $\hbar\Omega>2U$, is compensate by the appearance of a new transition
which starts at $\hbar\Omega=2U$, $E_{3\mathbf{k}}\rightarrow
E_{4\mathbf{k}}$, and its large spectral weight also leads to the appearance
of the peak structure. At high photon energies, where the effect of the energy
gap becomes insignificant, $E_{2\mathbf{k}}$ and $E_{4\mathbf{k}}$ bands
inherit the chirality of the energy bands in $\mathbf{K}$ and
$\mathbf{K}^{{}^{\prime}}$ valleys in pristine graphene and become completely
chiral and anti-chiral respectively. So, the optical transitions between them
becomes forbidden[24]. Therefore, at high photon energies only
$E_{1\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ and $E_{2\mathbf{k}}\rightarrow
E_{4\mathbf{k}}$ transitions contribute to the optical conductivity and the
usual flat interband response is recovered. Returning to the case of $\delta
t=0.10t_{0}$ and $u=0.10t_{0}$ (Fig. 3(c)), one can see that the only
additional change caused by the simultaneous presence of the hopping and the
on-site energy deviations is that the dip-peak structure of the optical
conductivity is shifted to lower energies originating from the dependence of
$U$ on the hopping energy deviation,
$U=\frac{u}{3}(1-\frac{4\Delta}{1-\Delta}+\frac{4\Delta^{2}}{(1-\Delta)^{2}})$.
Figure 3: The normalized optical conductivity, $\sigma_{xx}/\sigma_{0}$, of
undoped Kekulé-patterned graphene (solid magenta curves) as function of
$\hbar\Omega/t_{0}$ at zero temperature for (a) $\delta t=0.1t_{0}$ and $u=0$,
(b) $\delta t=0$ and $u=0.1t_{0}$, and (c) $\delta t=0.1t_{0}$ and
$u=0.1t_{0}$, compared with that of pristine graphene (dashed black curves).
The lower insets show the band structures around $\mathbf{\Gamma}$ point for
each case, and allowed optical transitions indicated by arrows which give rise
to structure in the conductivity curves shown by arrows with same color on
plots. The upper insets in each panel displays the contribution of each of the
optical transitions in the optical conductivity separately.
Finite chemical potential: Figure 4 shows our numerical results for the
optical conductivity of Y-shaped Kekulé-patterned graphene in comparison with
that of pristine graphene for $\mu>0$. The upper (lower) panels display the
result for $\mu=+0.05t_{0}$ ($\mu=+0.10t_{0}$), in which the left, middle and
panels is for $\delta t=0.10t_{0}\leavevmode\nobreak\ \leavevmode\nobreak\
u=0$, $\delta t=0.0\leavevmode\nobreak\ \leavevmode\nobreak\ u=0.10t_{0}$ and
$\delta t=0.10t_{0}\leavevmode\nobreak\ \leavevmode\nobreak\ u=0.10t_{0}$
respectively. As in Fig. 3, the solid magenta (dashed black) curves display
the optical conductivity of Kekulé-patterned (pristine) graphene, and in each
panel, the inset shows the corresponding band structure and the allowed inter-
band optical transitions. The colored arrows in the band structure show the
allowed optical transitions which give rise to some structures in the optical
conductivity identified by arrows with same color on the plots. Figure 5 shows
the same plots as Fig. 4, but for other different values of the chemical
potentials, $\mu=-0.05t_{0}$ (upper panels) and $\mu=-0.10t_{0}$ (lower
panels). For $\mu\neq 0$, as can be seen in all panels of Figs. 4 and 5, there
is the usual Drude conductivity arising from the intra-band transitions.
Starting from the case of $\delta t=0.10t_{0}$ and $u=0$, shown in Figs. 4(a)
and 3(d), one can see that the interband absorption edge that is at $2\mu$ for
pristine graphene[21] is now splitting into two edges moving to lower and
higher photon energies. So, the optical conductivity exhibits a two-steps
absorption which onset at $2\mu\frac{v_{B}}{v_{B}+v_{C}}$ and
$2\mu\frac{v_{B}}{v_{B}-v_{C}}$ photon energies coming from the interband
transitions between different valleys, $E_{1\mathbf{k}}\rightarrow
E_{4\mathbf{k}}$ and $E_{3\mathbf{k}}\rightarrow E_{2\mathbf{k}}$. The
presence of the Kekule distortion, which couples different valleys, also leads
to the appearance of a sharp peak in the optical conductivity which occurs in
the range of $2\mu\frac{v_{C}}{v_{B}+v_{C}}$ to
$2\mu\frac{v_{C}}{v_{B}-v_{C}}$ and is due to the inter-valley transitions in
the conduction band, shown by red arrow in the insets. See also (a), (d), (g)
and (j) panels of Fig. 6 in which the contribution of all transitions of Figs.
4 and 5 has been displayed. From Fig. 4 (d) one can also see that increasing
the chemical potential enhances the splitting of the interband absorption
edges and moves them further to higher and lower energies. It also shifts the
sharp peak to higher energies and increase its spectral weight.
The effects of nonzero on-site energy deviation on the optical conductivity
has been addressed in (b) and (e) panels of Fig. 4. One can see that due to
the gapped nature of $E_{4\mathbf{k}}$ band, which increases the energy
required for $E_{3\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ transition, the
sharp peak in optical conductivity is enhanced and moved to higher energies,
with a nonzero spectral weight in the energy range
$U+\sqrt{U^{2}+[(v_{B}+v_{C})/(v_{B}-v_{C})]^{2}\mu^{2}}<\hbar\Omega<2U$. If
the chemical potential increases and cuts $E_{4\mathbf{k}}$ band, due to the
occurrence of a phenomenon called band nesting resonance[25, 26, 27], the
spectral weight of the peak, as can be seen in Fig. 4(b), increases sharply
and its broadening becomes limited to a small range of photon energies,
facilitating its observation as an optical signature for the Y-shaped Kekulé
distortion with nonzero on-site energy deviation.
In order to understand the origin of band nesting resonance, we rewrite the
contribution of $E_{i\mathbf{k}}\rightarrow E_{j\mathbf{k}}$ transition in the
optical conductivity in Eq. 2.2, $-\frac{4\pi
g_{s}}{\hbar^{2}}\frac{1}{S}\sum_{\mathbf{k}}\chi(E_{j\mathbf{k}},E_{i\mathbf{k}})\frac{[n_{F}(E_{i\mathbf{k}})-n_{F}(E_{j\mathbf{k}})]}{\Omega}\delta(\Omega+E_{i\mathbf{k}}/\hbar-
E_{j\mathbf{k}}/\hbar)$, in a more illustrative form. If we consider cuts
$S(E)$ of constant energy $E$, $E=E_{j\mathbf{k}}-E_{i\mathbf{k}}$, in the
band structure, we can write
$d^{2}\mathbf{k}=dS[d(E_{j\mathbf{k}}-E_{i\mathbf{k}})/|\nabla_{\mathbf{k}}(E_{j\mathbf{k}}-E_{i\mathbf{k}})|]$.
So, we arrive at
$\displaystyle\Re\sigma_{xx}^{E_{i}\rightarrow
E_{j}}(\Omega)/\sigma_{0}=-\frac{g_{s}}{\pi\hbar\Omega}\int_{S(E)}\frac{\chi(E_{j\mathbf{k}},E_{i\mathbf{k}})}{|\nabla_{\mathbf{k}}(E_{j\mathbf{k}}-E_{i\mathbf{k}})|}dS,$
(43)
for the contribution of $E_{i\mathbf{k}}\rightarrow E_{j\mathbf{k}}$
transition in optical conductivity at zero temperature indicating that strong
peaks in the optical conductivity will come from regions in the spectrum where
$|\nabla_{\mathbf{k}}(E_{j\mathbf{k}}-E_{i\mathbf{k}})|\approx 0$. One of the
ways to fulfill the condition
$|\nabla_{\mathbf{k}}(E_{j\mathbf{k}}-E_{i\mathbf{k}})|\approx 0$ is that the
condition
$|\nabla_{\mathbf{k}}E_{i\mathbf{k}}|\approx|\nabla_{\mathbf{k}}E_{j\mathbf{k}}|>0$,
which occurs when two energy bands are approximately equispaced over regions
in the Brillouin zone and called band nesting, is satisfied. Band nesting
results in a strong peak in the optical conductivity which is called band
nesting resonance. One can easily check that the band nesting condition is
satisfied for $E_{2\mathbf{k}}$ and $E_{4\mathbf{k}}$ bands in the case of
$u\neq 0$ when the chemical potential become larger than $2U$, as can be seen
in Fig. 4(e), resulting in the appearance of band nesting resonance in case of
$\delta t=0$ and $u=0.1t_{0}$. Notice that the band nesting resonance also
takes place for $E_{1\mathbf{k}}\rightarrow E_{3\mathbf{k}}$ transition but
with a spectral weight more dependent on the chemical potential and restricted
to limited ranges of the chemical potential.
The simultaneous presence of the hopping and on-site energy deviations makes
the band nesting condition better fulfilled. Therefore, as Fig. 4(f) shows,
for the case of $\delta t=0.1t_{0}$ and $u=0.1t_{0}$ the intensity of the peak
increases. Furthermore, since the nonzero hopping and on-site energy
deviations decreases $U$ and shifts $E_{4\mathbf{k}}$ band to lower energies,
the band nesting condition is also satisfied for a wide range of $\mathbf{k}$
in $E_{2\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ transition and as a result, we
see a large increase in the intensity of the peak in Fig. 4(c) compared to
that in Fig. 4(b).
Figure 4: The normalized optical conductivity, $\sigma_{xx}/\sigma_{0}$, of
undoped Kekulé-patterned graphene (solid magenta curves) as function of
$\hbar\Omega/t_{0}$ at zero temperature for different values of the chemical
potential, $\mu=0.05t_{0}$ (upper panels) and $\mu=0.10t_{0}$ (lower panels),
and different values of the on-site and hopping energy deviations, $\delta
t=0.1t_{0}$ and $u=0$ (left panels), $\delta t=0$ and $u=0.1t_{0}$ (middle
panels) and $\delta t=0.1t_{0}$ and $u=0.1t_{0}$ (right panels), compared with
that of pristine graphehe. The insets show the corresponding band structures
around $\mathbf{\Gamma}$ point. The colored arrows in the band structure show
the allowed optical transitions which give rise to some structures in the
optical conductivity curves identified by arrows with same color on the plots.
Returning to the cases of $u\neq 0$ and $\mu=0.05t_{0}<2U$, shown in Figs.
4(b) and 4(c), one can see that the optical conductivity displays a similar
two-steps interband absorbtion but with a interband absorbtion edges that,
with respect to that in case of $\delta t=0.1t_{0}$ and $u=0$, moves to lower
energies and onsets at $2U$ and
$\mu+\sqrt{U^{2}+[(v_{B}+v_{C})/(v_{B}-v_{C})]^{2}\mu^{2}}-U$. A nonzero
hopping energy deviation, as shown in Fig. 4(c), enhances the splitting of the
interband absorbtion edge. Moreover, the shape of the steps, in cases of
$u\neq 0$ and $\mu<2U$, which is typical of a gapped band structure [28, 29]
shows a peak at the absorbtion edge arising from a discontinuity in the
electronic density of states. As the photon energy increases the usual flat
interband response of pristine single layer graphene is recovered. By
increasing the chemical potential, as can be seen in Figs. 4(e) and 4(f),
$E_{2\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ transitions become similar to
that between linear energy bands leading to disappearance of the peak at the
absorbtion edge. One can also see that in this case the optical conductivity
displays a three-steps absorbtion arising from the splitting of the absorbtion
edge of $E_{1\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ and
$E_{3\mathbf{k}}\rightarrow E_{4\mathbf{k}}$ transitions.
Figure 5 shows the same plots as Fig. 4, but for $\mu=-0.05t_{0}$ (upper
panels) and $\mu=-0.10t_{0}$ (lower panels). As expected, due to the presence
of particle-hole symmetry, the optical conductivity in the case of $u=0$ does
not change with changing the sign of the chemical potential, leading to
similar plots as Figs. 4(a) and 4(d). This is in contrast to the case of
nonzero on-site energy deviation, middle and right panels of Fig. 5, in which
the particle-hole symmetry is absent and the allowed optical transition due to
the gapped nature of $E_{4\mathbf{k}}$ is shifted to higher energies. In these
cases the optical conductivity, similar to that in the middle and right panels
of Fig. 4, displays a three-steps interband absorbtion but with the edges that
move to higher energies. For $\mu<0$ the band nesting condition is satisfied
for the case of $\delta t=0$ and $u=0.1t_{0}$ leading to appearance of band
nesting resonance, but when both hopping and on-site energy deviation are
nonzero the energy difference between $E_{1\mathbf{k}}$ and $E_{3\mathbf{k}}$
which determine the location and intensity of the intra-valley transition in
the valance band (that turns into in band nesting resonance in the appropriate
conditions[25]) is strongly dependent on the chemical potentials.
Figure 5: Same plots as Fig. 4, but for $\mu=-0.05t_{0}$ (upper panels) and
$\mu=-0.10t_{0}$ (lower panels). Figure 6: Calculated normalized optical
conductivity, $\sigma_{xx}/\sigma_{0}$, as function of $\hbar\Omega/t_{0}$ for
each of the optical transitions shown in the insets of Figs. 4 and 4
separately.
The main optical feature of the case of $\delta t\neq 0$ and $u\neq 0$, which
is expected to be more realistic, is the occurrence of band nesting resonance.
To show that the band nesting resonance is robust with respect to increasing
temperature, the optical conductivity of Y-shaped Kekul-patterned graphene at
different temperatures, $T=0$ (solid magenta curves) and
$T=300\leavevmode\nobreak\ K$ (dashed black curves), has been displayed in
Figure 7. One can see that increasing temperature only broadens the interband
absorbtion edge. It is also clear that the band nesting resonance which occurs
in both electron- and hole-doped cases is resistant against increasing
temperature except when the chemical potential approaches the intersection of
$E_{1\mathbf{k}}$ and $E_{3\mathbf{k}}$ energy bands ((d) panel). Therefore
the occurrence of band nesting resonance is a significant optical signature to
detect the Y-shaped Kekulé distortion in single layer graphene.
Figure 7: The optical conductivity of Kekulé-patterned graphene for the case
of $\delta t=0.1t_{0}$ and $u=0.1t_{0}$ at different temperatures,
$T=0\leavevmode\nobreak\ K$ (magenta solid curves) and
$T=300\leavevmode\nobreak\ K$ (black dashed-dotted curves), and for different
chemical potentials (a) $\mu=+0.05\leavevmode\nobreak\ t_{0}$, (b)
$\mu=+0.05\leavevmode\nobreak\ t_{0}$, (c) $\mu=+0.05\leavevmode\nobreak\
t_{0}$ and (d) $\mu=+0.05\leavevmode\nobreak\ t_{0}$. It is evident that the
band nesting resonance identicated by red arrow, except when the chemical
potential approaches the intersection of $E_{1\mathbf{k}}$ and
$E_{3\mathbf{k}}$ energy bands ((d) panel), is robust with respect to
increasing temperature.
Finally it is desirable to mention to the effect of the next-nearest hopping
energy deviations on the optical conductivity of Y-shaped Kekulé-patterned
graphene. In Ref. [21] the effect of next-nearest hopping energy on the
optical conductivity of pristine graphene was investigated and it was found
that its effect is negligible at low energies. Moreover, recently the effect
of the next nearest neighbor hopping energy deviation on the electronic
structure of kekule-patterned graphene has been investigated analytically[14].
Comparing equation 14c of this article with our result for the effective low
energy Hamiltonian, Eq. 24, shows that including this effect in our
calculations only leads to the normalization of $U$. Therefore, it can be
expected that including the next nearest hopping energy deviation does not
change our results for the optical conductivity in general.
## 4 Summary and conclusions
In summary, we investigated the effects of a uniform Y-shaped Kekulé
distortion of C-C bonds on the electronic band structure and optical
conductivity of graphene. First we introduce our tight-binding model in which
the effects of the Y-shaped Kekulé distortion was taken into account by
including both on-site and hopping energy deviations in the minimal tight-
binding Hamiltonian of graphene. Then by projecting the high energy bands onto
the subspace defined by the low energy bands we derived a low-energy effective
Hamiltonian which was found to be in excellent agreement with one calculated
from a diagonalization of the full tight-binding Hamiltonian. It was shown
that in addition the coupling of energy bands in different valleys caused by
both on-site and nearest-neighbor hopping energy deviations, the other main
effect of the on-site energy deviation on the low-energy band structure is
that a set of bands gains an effective mass and a shift in energy, thus
lifting the degeneracy of the conduction bands at the Dirac point. Then, using
Kubo formula, we obtained an analytical expression for the real part of the
optical conductivity. In the next section we presented our results for the
optical conductivity of as a function of the photon energy. We found that in
the zero limit of the chemical potential and the simultaneous presence of on-
site and nearest-neighbor hopping energy deviations, the optical conductivity
displays a dip-peak structure located at the photon energy corresponding to 2
times the effective hopping energy whose occurrence was explained by
considering the allowed optical transitions. This is in contrast to the case
of zero hopping energy deviation which similar to pristine graphene leads to a
flat interband response at all frequencies. Furthermore it was shown that at
finite chemical potential the interband absorbtion edge, which for pristine
graphene is at $2\mu$, is splitted into two edges whose splitting depends on
the amounts of the on-site and nearest-neighbor hopping energy deviations and
also the chemical potential. As a notable result it was found that the
Y-shaped Kekulé-patterned graphene at finite chemical potential also displays
a large optical response called band nesting resonance. We showed that this
effect is robust with respect to increasing temperature except when the
chemical potential approaches the intersection of $E_{1\mathbf{k}}$ and
$E_{3\mathbf{k}}$ energy bands. Therefore the occurrence of band nesting
resonance is a significant optical signature to detect the Y-shaped Kekulé
distortion in single layer graphene. We also discussed the effect of the next-
nearest hopping energy deviations which is found to only lead to
renormalization of the effective on-site energy deviation preserving our
general results for the optical conductivity.
## 5 acknowledgment
This work has been supported by Farhangian University.
## References
* [1] C. Gutiérrez, C.-J. Kim, L. Brown, T. Schiros, D. Nordlund, E. B. Lochocki, K. M. Shen, J. Park and A. N. Pasupathy, Nat. Phys. 12 (2016) 950.
* [2] O. V. Gamayun,V. P. Ostroukh, N. V. Gnezdilov, İ. Adagideli and C. W. J. Beenakker, New J. Phys. 20 (2018) 023016.
* [3] Q.-P. Wu, L.-L. Chang, Y.-Z. Li, Z.-F. Liu, and X.-B. Xiao, Nanoscale Research Letters 15 (2020) 1.
* [4] E. Andrade, R. Carrillo-Bastos, and G. G. Naumis, Phys. Rev. B 99 (2019) 035411.
* [5] E. Andrade, R. Carrillo-Bastos, P. A. Pantaleón, and F. Mireles, J. Appl. Phys. 127 (2020) 054304.
* [6] J. J. Wang, S. Liu, J. Wang, and J.-F. Liu, Phys. Rev. B 98 (2018) 195436.
* [7] S. A. Herrera and G. G. Naumis, Phys. Rev. B 102 (2020) 205429.
* [8] Y. Mohammadi, ECS J. Solid State Sci. Technol. 10(6) (2021) 061011.
* [9] Y. Mohammadi and S. Bahrami, Chin. Phys. B 31 (1) (2022) 017305.
* [10] M. A. Mojarro, V. G. Ibarra-Sierra, J. C. Sandoval-Santana, R. Carrillo-Bastos, and Gerardo G. Naumis, Phys. Rev. B 102 (2020) 165301.
* [11] S. A. Herrera and Gerardo G. Naumis, Phys. Rev. B 104 (2021) 115424.
* [12] W. A. Harrison, Solid State Theory, Dover Publications, 1970.
* [13] Y. Ren, X. Deng, Z. Qiao, C. Li, J. Jung, C. Zeng, Z. Zhang, and Q. Niu, Phys. Rev. B 91 (2015) 245415.
* [14] E. Andrade, G. G. Naumis and R Carrillo-Bastos, J. Phys.: Condens. Matter 33 (2021) 225301
* [15] T. Stauber, N. M. R. Peres, and A. K. Geim, Phys. Rev. B 78 (2008) 085432.
* [16] G. D. Mahan, Many-Particle Physics, Plenum, New York (1990).
* [17] E. J. Nicol and J. P. Carbotte, Phys. Rev. B 77 (2008) 155409.
* [18] Y Mohammadi, R Moradian, F Shirzadi Tabar, Solid State Commun. 193 (2014) 1.
* [19] Y. Mohammadi, Superlattices Microstruct. 156 (2021) 106955.
* [20] H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction, Oxford University Press, 2002.
* [21] N. M. R. Peres, T. Stauber, Int. J. Mod. Phys. B 22 (2008) 2529.
* [22] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005); Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature 438, (2005) 201.
* [23] N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73 (2006) 125411.
* [24] S. A. Herrera and G. G. Naumis, Phys. Rev. B 101 (2020) 205413.
* [25] A. Carvalho, R. M. Ribeiro, and A. H. Castro Neto, Phys. Rev. B 88 (2013) 115205.
* [26] Z. Rashidian, Y. V. Bludov, R. M. Ribeiro, N. M. R. Peres and M. I Vasilevskiy, J. Phys.: Condens. Matter 26 (2014) 395301.
* [27] L. Mennel, V. Smejkal, L. Linhart, J. Burgdörfer, F. Libisch and T. Mueller, Nano Lett. 20(6) (2020) 4242.
* [28] C. J. Tabert and E. J. Nicol, Phys. Rev. B 86 (2012) 075439.
* [29] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys. Rev. Lett. 96 (2006) 256802.
|
[orcid=0000-0003-3204-238X<EMAIL_ADDRESS>]
[orcid=0000-0001-8116-9446<EMAIL_ADDRESS>]
# Ontomathedu Ontology Enrichment Method
Konstantin S. Nikolaev Federal State Institution of the Federal Research
Center NIISI RAS Kazan Federal University Olga A. Nevzorova
###### Abstract
Nowadays, distance learning technologies have become very popular. The recent
pandemic has had a particularly strong impact on the development of distance
education technologies. Kazan Federal University has a distance learning
system based on LMS Moodle. This article describes the structure of the
OntoMathEdu ecosystem aimed at improving the process of teaching school
mathematics courses, and also provides a method for improving the OntoMathEdu
ontology structure based on identifying new connections between contextually
related concepts.
###### keywords:
OntoMathEdu ontologysoftware serviceeducational applicationontology enrichment
## 1 Introduction
The ontological approach is often used to organize training in a more
structured format, as well as to help the student form a full picture of the
current discipline, to more clearly understand the existing patterns and
relationships between concepts in the problem area under consideration. The
basis of such an educational infrastructure can serve as a qualitative
ontology. Currently, there are many training systems that use various
ontologies to organize subject data. For example, Lutoshkina et al. in the
work [1], the ontology of the subject area is used to systematize the
multimedia content of electronic courses (the connections between the concepts
of ontology are used to construct an integral structure of the lesson). In
addition, Stancin et al. [2] provide a detailed overview of the use of
ontologies in education, among which the most interesting for us are
ontologies describing the subject under study and e-learning services. Here
are some examples of such solutions:
1\. LiReWiapproach (Conde et al. [3]). Educational ontology is used to
describe the topics to be mastered by students and the pedagogical relations
between the topics.
2\. MOOC-KG (Dang et al. [4]). To build a knowledge graph, an ontology is used
that models knowledge about online learning resources.
3\. MKMSE (Martinez-Ramirez et al. [5]). Ontology is used to store
mathematical knowledge.
4\. Ontology is used to represent the semantic description of document
resources and the relationships between document resources and other types of
resources (Hai [6]).
5\. CALMS (Erazo-Garzon et al. [7]). Ontologies are used to define concepts
and semantic relations between academic information and conceptual aspects.
6\. Ontology is used as an expression of knowledge to support the ends of the
content of education. (Kubekov et al. [8])
## 2 Structure of the OntoMathEdu ecosystem
The article discusses the structure and main tasks of the infrastructure of
software components built around the ontology of school mathematics
OntoMathEdu [9, 10]. Ontology OntoMathEdu is a knowledge system that describes
a set of concepts, relationships between them and statements within the
framework of a school course in planimetry. Figure 1 shows the composition of
the OntoMathEdu digital ecosystem. The ecosystem consists of the following
components.
Figure 1: Components of the OntoMathEdu ecosystem
1. 1.
Intelligent digital educational platform for school mathematics. This
component occupies a central place in the OntoMathEdu ecosystem. Its main
purpose is the application of semantic and ontological technologies in
teaching school mathematics.
2. 2.
Collection of questions. This collection is used as an input data set for
various components of the ecosystem, in particular, for the module "Ontology
Enrichment Services", as well as in the central component "Intelligent Digital
Educational Platform for School Mathematics". The collection is constantly
updated.
3. 3.
Formula storage. This component contains formulas extracted from school
geometry textbooks. Formulas are presented in various formats (plain text,
LaTeX, OpenMath). We plan to use this repository as an auxiliary data set for
building test tasks and other information components.
4. 4.
Digital educational resources. This component combines all auxiliary data
sources located on the Internet (for example, open educational databases,
resources for the construction of geometric shapes, etc.)
5. 5.
OntoMathEdu ontology. The OntoMathEdu ontology is a reflection of the level of
knowledge corresponding to the level of school mathematics. It serves as the
main repository of concepts and their relationships that are used by other
services.
6. 6.
Ontology enrichment service. This component includes a set of methods that are
used to clarify the relationships between the concepts of ontologies and
improve the horizontal connectivity of ontologies.
7. 7.
Test generator. This component is used to automatically create new test tasks
based on the analysis of the structure and concepts of existing tasks.
8. 8.
Search by semantic formulas. A search module that performs semantic search of
mathematical texts present in the ecosystem. As a result of the search, it
returns a set of texts containing the requested concept and offers a list of
related concepts for further navigation through the text database.
9. 9.
Recommendation system. This module helps the user to study various sections of
the training course, providing an individual learning trajectory.
## 3 Ontology enrichment service
The ontology OntoMathEdu is currently in the active stage, and contains more
than 900 concepts and more than 20 relationships. The task of introducing new
relations between concepts is urgent. We have developed a method for
introducing new relations between concepts based on the use of a collection of
questions. To find new connections between concepts, we use a collection of
test questions on school geometry. Our main idea is as follows. Each question
contains some basic concepts of ontology, about which the question is asked.
We assume that the answer to this question can be found with the help of
ontology, in which there is a chain of concepts of limited length connecting
the original concepts. In addition, the search for such connections allows us
to evaluate the structural properties of the ontology, namely the horizontal
connectivity of the resource.
The source of the texts is a collection of school test tasks on geometry with
the markup of nominal groups. These nominal groups will act as candidates to
search for the corresponding concept in the OntoMathEdu ontology. A noun group
of the first type (Noun phrase, NP) is understood as a syntactic construction
in which there is a vertex (noun) and dependent words. An example of NP is a
"finite set of points", where the real "set" is the vertex of a given phrase
with two dependent words. A nominal group of the second type (Prepositional
phrase, PP) is a phrase in which the main word is a preposition. An example of
PP is the phrase "up to two characters after the dot", where the preposition
"to" is the vertex of the nominal group.
Then, for each pair of concepts, all existing relationships are searched, and
the search is performed twice: in the first case, only hierarchical
relationships (rdfs:subClassOf and ome:hasChild) are used as relationships
between concepts, and in the second case, all possible relationships are used.
We call the first type of relationship as a "hierarchical relationship", and
the second as a "complete relationship". The presence of an optimal
relationship is an event in which the shortest complete link is shorter than
the hierarchical one. This connection is optimal, because any pair of concepts
is somehow connected by a hierarchical connection (we can always climb from
the first concept to the node shared with the second concept and descend to
the second concept), but this connection will not be the shortest if there are
direct connections between the concepts under consideration.
The resulting set of shortest links with an indication of optimality is sent
for analysis to a group of experts for manual updating of the ontology. It can
also be used as auxiliary information when checking test tasks.
Figure 2: Block diagram of the ontology enrichment method
Figure 2 shows a block diagram of the task of enriching ontology through
method names. The following sections will describe the main stages of the
algorithm for finding optimal connections between concepts in the texts of
questions with the name of the method. Here are some comments on Figure 2.
1. 1.
Extraction of NP and PP. At this stage, we need to select a set of NP and PP
in the texts of questions and text variants of answers (this means that we do
not consider answer options for computational problems and for problems with
the choice of an answer presented as a number or a set of characters that are
not a dictionary unit). Thanks to the already preliminary markup of the text
by the nominal groups NP (TERM1 tag) and PP (TERM2 tag), this stage is reduced
to highlighting the contents of these tags using the standard xml library for
Python 3.
2. 2.
Tokenization and normalization of phrases from nominal groups (NP and PP):
TokenizeAndLemmatizeTexts() function. Tokenization of phrases received at the
last stage is performed using the nltk library. Normalization of the resulting
set of tokens is carried out using the pymorphy2 library, which has the
necessary functionality to bring words to a normal form. For example, for the
word "triangles", the normal form is "triangle".
3. 3.
Tokenization and normalization of ontology concepts:
TokenizeAndLemmatizeOntology() function. Tokenization of phrases received at
the last stage is performed using the nltk library. Tokens are normalized
using the pymorphy2 library.
4. 4.
Match search: Match() function. Next, we need to find similar sequences of
tokens in the question text and names of ontological concepts, using a
modified Jacquard measure to work with a sequence of strings. The sequence
element in our case is a lemma from the phrases NP or PP. However, since the
words in the compared sequences may not completely match, we also apply the
Jacquard measure when searching for common words between sequences. In this
case, the elements of the sequence will be individual letters.
When performing this method, we will introduce two thresholds: a lower
accuracy threshold at which two words are considered the same, and a lower
accuracy threshold at which two sequences of words are considered similar.
Their values were obtained experimentally on a specific set of questions.
5. 5.
Search for connections through hierarchical relations: the
SearchHierarchicalConnections() function. In this method, we look for
connections between pairs of concepts located in the same question, using only
hierarchical relationships (rdfs:subClassOf and ome:hasChild) as relationships
between neighboring concepts. This type of connection is guaranteed to be
present between any two concepts in the ontology, but it is not necessarily
optimal. We select the shortest connection from the resulting set of this
method.
6. 6.
Search for connections through any relations: Search All Connections(). In
this method, we look for connections between pairs of concepts located in the
same question, using all types of relations as relations between neighboring
concepts. We select the shortest connection from the resulting set of this
method.
7. 7.
Comparison of shortest distances with Previous methods: CompareDistances()
function. In this method, we check a link using all types of links, which is
shorter than the corresponding link using only hierarchical links. If such a
connection is found, then the optimal connection between a pair of concepts is
confirmed. Table 3 shows examples of optimal connections found in ontology,
indicating the lengths of these connections.
## 4 Conclusion
This article examines the structure of the OntoMathEdu ecosystem, designed to
support the process of personalized teaching of school mathematics, and also
provides an algorithm for enriching ontology, based on the contextual
proximity of concepts in geometry tasks, which contributes to improving the
horizontal connectivity of ontology.
Future work is related to the implementation of the personalization component
in all implemented test generators, as well as the overall development of
these components. In addition, it is planned to develop a component for the
automatic analysis of a detailed response by drawing up a framework (scheme)
of the solution, and filtering tasks by types of these frameworks.
Table 1: Examples of optimal connections found Source theorem | First concept | Second concept | Link length
---|---|---|---
If the diagonals of a given quadrilateral are perpendicular, then this quadrilateral has: 1) equal middle lines, 2)<perpendicular> <middle lines> 3) equal opposite angles | Perpendicular | Triangle middle line | 3
If a circle can be described around a quadrilateral, then 1) the center of the circle is equidistant from the sides of the quadrilateral 2) the bisectors of the angles of the quadrilateral intersect at one point 3) the sum of <opposite angles> is equal to <right angle> | Opposite corners of a quadrilateral | Right angle | 4
<Segments> connecting the midpoints of the sides of any quadrilateral and the segment connecting the midpoints of <diagonals> | The middle of the segment | Diagonal of the polygon | 3
The sum of two axial symmetries with intersecting axes can be replaced by <rotation> around the <point> at which the ax-es intersect at an angle equal to the angle between the axes | Point | Rotation angle | 4
###### Acknowledgements.
The article was prepared within the framework of the government task of the
Federal State Institution of the Federal Research Center NIISI RAS for
2022-2024 (topic No. 0580-2022-0014 (FNEF-2022-0014)).
## References
* Lutoshkina and Vysotin [2011] N. Lutoshkina, A. Vysotin, Use of ontologies in interactive courses, Info-Strategiya 2011: Obshchestvo. Gosudarstvo. Obrazovanie: Sbornik materialov konferencii (2011) 56–59.
* Stancin et al. [2020] K. Stancin, P. Poscic, D. Jaksic, Ontologies in education – state of the art, Education and Information Technologies 25 (2020) 5301–5320. doi:10.1007/s10639-020-10226-z.
* Conde et al. [2019] A. Conde, M. Larranaga, A. Arruarte, J. A. Elorriaga, A Combined Approach for Eliciting Relationships for Educational Ontologies Using General-Purpose Knowledge Bases, IEEE Access 7 (2019) 48339–48355. doi:10.1109/ACCESS.2019.2910079.
* Dang et al. [2019] F. Dang, J. Tang, S. Li, MOOC-KG: A MOOC knowledge graph for cross-platform online learning resources, ICEIEC 2019 - Proceedings of 2019 IEEE 9th International Conference on Electronics Information and Emergency Communication (2019) 596–603. URL: https://ieeexplore.ieee.org/abstract/document/8784572?casa_token=Hn9FqPbXI1AAAAAA:X0MhRkIRwXQjleGUG0HWTmaPOj0D2jsvR9ATrUG1_wWOWHSlSrVkv400fbFbsYX_iKBDYP7vYyuv. doi:10.1109/ICEIEC.2019.8784572.
* Martinez-Ramirez et al. [2018] Y. Martinez-Ramirez, A. Ramirez-Noriega, M. Zayas-Esquer, S. Miranda-Mondaca, J. Armenta-Bojorquez, M. Quintero-Fonseca, M. Gonzialez-Videgaray, C. Cortes-Velazquez, Architecture of mathematical knowledge management system in education: Ontology-based and case-based, in: 2018 International Symposium on Computers in Education, Proceedings, 2018, pp. 1–5. doi:10.1109/SIIE.2018.8586771.
* Hai [2019] J. Hai, An efficient semantic retrieval method for network education information resources, Proceedings - 2019 11th International Conference on Measuring Technology and Mechatronics Automation, ICMTMA 2019 (2019) 522–526. doi:10.1109/ICMTMA.2019.00121.
* Erazo-Garzon et al. [2019] L. Erazo-Garzon, A. Patino, P. Cedillo, A. Bermeo, CALMS: A context-aware learning mobile system based on ontologies, in: 2019 6th International Conference on eDemocracy and eGovernment, ICEDEG 2019, 2019, pp. 84–91. doi:10.1109/ICEDEG.2019.8734423.
* Kubekov et al. [2018] B. Kubekov, N. Zhaksybaeva, V. Naumenko, A. Utegenova, Methodology of formation of educational resources on the basis of ontology, IEEE 12th International Conference on Application of Information and Communication Technologies, AICT 2018 - Proceedings (2018). doi:10.1109/ICAICT.2018.8747069.
* Kirillovich et al. [2020] A. Kirillovich, O. Nevzorova, M. Falileeva, E. Lipachev, L. Shakirova, OntomathEdu: A linguistically grounded educational mathematical ontology 12236 LNAI (2020) 157–172. doi:10.1007/978-3-030-53518-6_10.
* Kirillovich et al. [2019] A. Kirillovich, L. Shakirova, M. Falileeva, E. Lipachev, Towards an Educational Mathematical Ontology, INTED2019 Proceedings 1 (2019) 6823–6829. doi:10.21125/inted.2019.1657.
|
# Deep Neural Network Techniques for Monaural Speech Enhancement and
Separation: State of the art Analysis
Peter Ochieng
Department of Computer Science.
University of Cambridge.
<EMAIL_ADDRESS>
###### Abstract
Deep neural networks (DNN) techniques have become pervasive in domains such as
natural language processing and computer vision. They have achieved great
success in tasks such as machine translation and image generation. Due to
their success, these data driven techniques have been applied in audio domain.
More specifically, DNN models have been applied in speech enhancement and
separation to achieve denoising, dereverberation, speaker extraction and
speaker separation in monaural speech intelligibility improvement. In this
paper, we review some dominant DNN techniques being employed to achieve speech
enhancement and separation. The review looks at the pipeline of speech
enhancement and separation techniques from feature extraction, how DNN-based
tools models both global and local features of speech and model training
(supervised and unsupervised). The review also covers the use of domain
adaptation techniques and pre-trained models to boost speech enhancement
process.
## 1 Introduction
Techniques for monaural speech intelligibility improvement can be categorised
either as speech enhancement or separation. Speech enhancement involves
isolating a target speech either from noise [1] or a mixed speech [2]. Speech
enhancement involves tasks such as dereverberation, denoising and speaker
extraction. Speaker separation on the other hand seeks to estimate independent
speeches composed in a mixed speech [3]. Speech enhancement and separation
have applications in multiple domains such as automatic speech recognition,
mobile speech communication and designing of hearing aids [4]. Initial
research on speech enhancement and separation exploited techniques such as
non-negative matrix factorization [5] [6] [7] probabilistic models [8] and
computational auditory scene analysis (CASA)[9]. However, these techniques are
tailored for closed-set speakers (i.e., do not work well with mixtures with
unknown speakers) which significantly restricts their applicability in real
environments. Due to the recent success of deep learning models in different
domains such natural language processing and computer vision, these data
driven techniques have been introduced to process audio dataset. In
particular, DNN models have become popular in speech enhancement and
separation and have achieved great performance in terms of boosting speech
intelligibility and their ability to enhance speech with unknown speakers [10]
[11]. In order to be effective in speech enhancement and separation, DNN
models must extract important features of speech, maintain order of audio
frames, exploit both local and global contextual information to achieve
coherent separation of speech data. This necessitates that DNN models should
include techniques tailored to meet these requirements. Discussion of these
techniques is the core subject of this review. Further, in computer vision and
text domain, large pre-trained models are used to extract universal
representations that are beneficial to downstream tasks. The review discusses
the impact of pre-trained models to the speech enhancement and separation
domain. It also discusses DNN techniques being adopted by speech enhancement
and separation tools to reduce computation complexity to enable them work in
low latency and resource constrained environments. The review therefore
focuses on the whole pipeline of DNN application to speech enhancement and
separation, i.e., from feature extraction, model implementation, training and
evaluation. Our goal is to uncover the dominant techniques at each level of
DNN implementation. In each section, we highlight key emerging features and
challenges that exist. A recent review [12] only looked at supervised
techniques of performing speech separation and in this review, we discuss both
supervised and unsupervised methods. Moreover, with the fast-growing field of
deep learning, new techniques have emerged that necessitates a new look into
how these techniques have been implemented in speech enhancement and
separation. The review is constrained to discussing how DNN techniques are
being applied to monaural speech enhancement and so we do not focus on multi-
channel speech separation (which has been covered in [13]).
The paper first explains the types of speech enhancement and separation
(section 2) by highlighting their key elements and the tools that focus on
each type. It discusses the key speech features that are being used by speech
enhancement and separation tools in section 3. This section looks at how the
features are derived and how they are used to train the DNN models in
supervised learning technique. Section 5 discusses the techniques the tools
use to model long dependencies that exist in speech. The paper discusses model
size compression techniques in section 6. In section 7, the paper discusses
some of the popular objective functions used in speech enhancement and
separation. Section 8 discusses how some tools are implementing unsupervised
techniques to achieve speech enhancement and separation. Section 9 discusses
how the speech separation and enhancement tools are being adapted to the
target environment. In section 10 the paper looks at how pre-trained models
are being utilized in the speech enhancement and separation pipeline. Finally,
section 11 looks at future direction. Figure 1 gives an overall organization
and topics covered by the paper.
forked edges, for tree= edge+=thick, inner color=gray!5, outer color=gray!20,
rounded corners=2pt, draw, thick, tier/.option=level, align=center, font=,
blur shadow, , where level<=1 rotate=90, anchor=east, , [Deep Neural Network
Techniques for Monaural Speech Enhancement and Separation [, coordinate,
[Types of SE and separation [Speech separation] [Dereverberation] [Denoising]
[Speech extraction]]] [Features [Fourier spectrum [Log-power spectrum
features][Mel-frequency spectrum features][DFT magnitude features][Complex DFT
features][Complementary features][Supervised Training [Spectral Mapping
Techniques][Spectral Masking Techniques][Generative Modelling]] [Phase
handling]] [Time domain [Supervised training[Adaptive front-end
method][Waveform mapping][Generative modelling]]] ] [Long term
dependency modelling [RNN] [TCN] [Use of transformers]] [Model size
reduction techniques, calign with current [Use of dilated convolution]
[Parameter quantization] [Use of depthwise separable convolution] [Knowledge
distillation] [Parameter pruning] [Weight sharing:] ] [Objective functions
[SSTOI] [SI-SDR, calign with current] [PMSQR] ] [Unsupervised
Techniques] [Domain
Adaptation
Techniques] [Use of
Pre-trained
Models] [Future
directions]] ] [expatriate, inner color=green!10, outer color=green!25, child
anchor=west, edge path’=(!u.parent anchor) -| (.child anchor), before drawing
tree=y’+=7.5pt ] ]
Figure 1: Overall structure of the topics covered by this review.
## 2 Types of speech separation and enhancement
### 2.1 Speech separation
Scenarios arise where more than one target speech signals are composed in a
given speech mixture and the goal is to isolate each independent speech
composed in a mixture. This problem is known as speech separation. For a
mixture that is composed of $C$ independent speech signals $x_{c}(n)$ with
$c=1,\cdots,C$, a recording $y(n)$ composed of the $C$ speech signals can be
represented as:
$y(n)=\sum_{c=1}^{C}x_{c}(n)$ (1)
Here, $n$ indexes time. The goal of speech separation is to estimate each
independent $x_{c}$ speech signal composed in $y(n)$. Separating speech from
another speech is a daunting task by the virtue that all speakers belong to
the same class and share similar characteristics[14]. Some models such as [15]
and [16] lessen this by performing speech separation on a mixed speech signal
based on gender voices present. They exploit the fact that there is large
discrepancy between male and female voices in terms of vocal track,
fundamental frequency contour, timing, rhythm, dynamic range etc. This results
in a large spectral distance between male and female speakers in most cases to
facilitate a good gender segregation. For speech separation that the mixture
involves speakers of the same gender, the separation task is much difficult
since the pitch of the voice is in the same range [14]. Most speech separation
tools that solve this task such as [17] [18] [19] [20] [14] and [10] cast the
problem as a multi-class regression. In that case, training a DNN model
involves comparing its output to a source speaker. DNN models always output a
dimension for each target class and when multiple sources of the same type
exist, the system needs to select arbitrarily which output dimension to map to
each output and this raises a permutation problem (permutation ambiguity)
[14]. Taking a case of a two speaker separation, if the model estimates
$\hat{a_{1}}$ and $\hat{a_{2}}$ as the magnitude of the reference speech
magnitudes $a_{1}$ and $a_{2}$ respectively, it is unclear the order in which
the model will output the estimates i.e. the order of output can either be
$\\{\hat{a}_{1},\hat{a}_{2}\\}$ or $\\{\hat{a}_{2},\hat{a}_{1}\\}$. A naive
approach shown in figure 2 [21] is to present the reference speech magnitudes
in a fixed order and hope that it is the same order in which the system will
output its estimation.
Figure 2: Naive approach of solving label matching problem for a two-talker
speech separation model
In case of a mismatch, the loss computation will be based on the wrong
comparison resulting in low quality of separated speeches. Therefore, systems
that perform speaker separation have an extra burden of designing mechanisms
that are geared towards handling the permutation problem. There are several
strategies that are being implemented by speech separation tools to tackle
permutation problem. In [19], a number of DNN techniques are implemented that
estimates two clean speeches contained in a two-talker mixed speech. They
employ supervised training to train DNN models to discriminate the two
speeches based on average energy, pitch and instantaneous energy of a frame.
Work in [22] and [21] introduce permutation invariant training (PIT) technique
of computing permutation loss such that permutations of reference labels are
presented as a set to be compared with the output of the system. The
permutation with the lowest loss is adopted as the correct order. For a a two-
speaker separation system introduced earlier, the reference sources
permutation will be $\\{a_{1},a_{2}\\}$ and $\\{a_{2},a_{1}\\}$ such that the
possible permutation losses are computed as:
$loss_{1}=D([a_{1},a_{2}],[\hat{a}_{1},\hat{a}_{2}])=D[a_{1},\hat{a}_{1}]+D[a_{2},\hat{a}_{2}]$
$loss_{2}=D([a_{1},a_{2}],[\hat{a}_{2},\hat{a}_{1}])=D[a_{1},\hat{a}_{2}]+D[a_{2},\hat{a}_{1}]$
The one that returns the lowest loss between the two is selected as the
permutation loss to be minimized (see figure 3). For an $S$ speaker separation
system a total of $S!$ permutations are generated.
Figure 3: Permutation invariant training implementation of label matching for
a two-talker speech separation model
For a system that performs $S$ speaker separation and $S$ is high (e.g. 10),
implementation of PIT which has a computation complexity of $O(S!)$ is
computationally expensive [23] [24]. Due to this, [24] casts the permutation
problem as a linear sum problem where Hungarian algorithm is exploited to find
the permutation which minimizes the loss at computation complexity of
$O(S^{3})$. Work in [23] proposes SinkPIT loss which is based on Sinkhorn’s
matrix balancing algorithm. They utilize the loss to reduce the complexity of
PIT loss from $O(C!)$ to $O(kC^{2})$. Work in [17] employs minimum loss
permutation computation at each time step $t$. The best permutation (argmin)
at each time-step is exploited to re-order the embedding vectors to be
consistent with the training labels. To evade the permutation problem, they
train two separate DNN models for each of the two speakers to be identified.
Another prominent technique of handling permutation problem is to employ a DNN
clustering technique [25] [26] [20] [27] [28] to identify the multiple
speakers present in a mixed speech signal. The DNN $f_{\theta}$ accepts as its
input the whole spectrogram $X$ and generates a $D$ dimension embedding vector
V i.e., $V=f_{\theta}(X)\in R^{N\times D}$. Here, the embedding $V$ learns the
features of the spectrogram $X$ and is considered a permutation- and
cardinality-independent encoding of the network’s estimate of the signal
partition. For the network $f_{\theta}$ to be learn how to generate an
embedding vector $V$ given the input $X$, it is trained to minimize the cost
function.
$C_{Y}(V)=||VV^{T}-YY^{T}||_{F}^{2}=\sum_{ij}(<v_{i},v_{j}>-<y_{i},y_{j}>)^{2}$
(2)
Here, $Y=\\{y_{i,c}\\}$ represents the target partition that maps the
spectrogram $S_{i}$ to each of the $C$ clusters such that $y_{i,c=1}$ if
element $i$ is in cluster $c$ . $YY^{T}$ is taken here as a binary affinity
matrix that represents the cluster assignment in a partition-independent way.
The goal in equation 2 is to minimise the distance between the network
estimated affinity matrix $VV^{T}$ and the true affinity matrix $YY^{T}$. The
minimization is done over the training examples. $||A||_{F}^{2}$ is the
squared Frobenius norm. Once $V$ has been established, its rows are clustered
into partitions that will represent the binary masks. To cluster the rows
$v_{i}$ of $V$, K-means clustering algorithm is used. The resulting clusters
of $V$ are then used as binary masks to separate the sources by applying the
masks on mixed spectrogram $X$. The separated sources are then reconstructed
separated by using inverse STFT. Even though PIT is popular in speech
separation models, it is unable to handle the output dimension mismatch
problem where there is a mismatch on the number of speakers between training
and inference [29]. For example, training a speech separation model on $n$
speaker mixtures but testing it on $t\neq n$ speaker mixtures. The PIT-based
methods cannot directly deal with this problem due to their fixed output
dimension. Most speech separation models such as [30] [21] [31] [32] deal with
the problem by setting a maximum number of sources $C$ that the model should
output from any given mixture. If an inference mixture has $K$ sources, where
$C>K$, $C-K$ outputs are invalid, and the model needs to have techniques to
handle the invalid sources. In case of invalid sources, some models such as
[31], [30], [21] design the model to output silences for invalid sources while
[32] outputs the mixture itself which are then discarded by comparing the
energy level of the outputs relative to the mixture. The challenge with models
that output silences for invalid sources is that they rely on a pre-defined
energy threshold, which may be problematic if the mixture also has a very low
energy [32]. Some models handle the output dimension mismatch problem by
generating a single speech in each iteration and subtracting it from the
mixture until no speech is left [33], [34], [35][36], [37]. The iterative
technique despite being trained with a mixture with low number of sources can
generalize to mixtures with a higher number of sources [35]. It however faces
criticism that setting iteration termination criteria is difficult and the
separation performance decreases in later iterations due degradations
introduced in prior iterations [35]. Other speech separation models include
[38] [39][40] [31] [41] [19] [20] [42].
### 2.2 Speaker extraction
Some speech enhancement DNN models have been developed where in a mixed speech
such as an equation 1, they design methods to extract a single target speech.
These models focus only on a single target speech $x_{t}$ and treat all other
speeches as interfering signals, therefore they modify equation 1 as shown in
3.
$y(n)=\sum_{c=1}^{C}x_{c}(n)=x_{t}(n)+\sum_{c\not=t}^{C}x_{c}(n)$ (3)
where $x_{t}(n)$ is the target speech. By focusing on only a single target
speech, the permutation ambiguity problem is avoided. They formulate the
speech extraction task into a binary classification problem, where the
positive class is the target speech, and the negative class is formed by the
combination of all other speakers. A popular technique of speaker extraction
is to give as input to the DNN models additional speaker dependent information
that can be used to isolate a target speaker [43]. Speaker dependent
information can be injected into the DNN models by either concatenating
speaker dependent auxiliary clues with the input features or adapting part of
the DNN model parameters for each speaker [44]. This addition information
about a speaker injects a bias that is necessary to differentiate the target
speaker from the rest in the mixture [2]. Several auxiliary clues have been
exploited by DNN models which include pre-recorded enrolment utterances of the
target speaker [44] [2] [45] [46] [47] [48], electroglottographs (EGGs) of the
target speaker [49] and i-vectors extracted at speaker level [50] [51]. Tool
in [52] adapt parameters for each speaker by allocating a speaker dependent
module to a selected intermediate layer of DNN. Speech extraction tool in [53]
does not use auxiliary clues of the target speaker but design attractor points
that are compared with the mixed speech embeddings to generate the mask used
to extract the target speech.
### 2.3 Dereverberation
This is a speech enhancement technique that seeks to eliminate the effect of
reverberation contained in speech. When speech is captured in an enclosed
space by a microphone that is at distance $d$ from the talker, the observed
signal consists of a superposition of many delayed and attenuated copies of
the speech resulting from reflections of the enclosed space walls and existing
objects within the space (see figure 4) [54]. The signal received by the
microphone consists of direct sound, reflections that arrive shortly after
direct sound ( within approximately 50ms) i.e., early reverberation and
reflections that arrive after early reverberation i.e., late reverberation
[55]. Normally, early reverberation does not affect speech intelligibility
much [56] and much of perceptual degradation of speech is attributed to late
reverberation. Speech degradation due to reverberation can be attributed to
two types of masking [57], overlap masking- where the energy of a preceding
phoneme overlaps with the one following or self-masking-where internal
temporal which refers to the time and frequency alterations of an individual
phoneme. Reverberation therefore can be viewed as the convolution of the
direct sound and the room impulse response (RIR). A reverberant speech can be
formally represented according to equation 4:
Figure 4: How Reverberation happens.
$y(t)=h(t)*s(t)$ (4)
Here, $*$ represents convolution, $s(t)$ is the clean anechoic speech. $h(t)$
represents room impulse response i.e., direct speech $h_{d}(t)$, early
reverberation $h_{e}(t)$and late reverberation $h_{l}(t)$. Hence $h(t)$ can be
represented as
$h(t)=h_{d}(t)+h_{e}(t)+h_{l}(t)$ (5)
Using the distributive property of convolution [58] equation 4 becomes:
$y(t)=h_{d}(t)*s(t)+h_{e}(t)*s(t)+h_{l}(t)*s(t)$ (6)
The goal of dereverberation is therefore to establish $s(t)$ from $y(t)$.
Hence it can be viewed as a deconvolution between the speech signal and RIR
[59]. Dereverberation is considered a more challenging task than denoising for
a number of reasons. First, it is difficult to pinpoint direct speech from its
copies especially when the reverberation is strong. Secondly, the key
underlying assumption of sparsity and orthogonality of speech representations
in the feature domain that is commonly used in monaural mask-based speech
separation does not hold for speech under reverberation [60]. Due to these
unique features of reverberation, most tools designed for denoising, or
speaker separation are ill poised to perform dereverberation [60]. The DNN
tools for speaker separation and denoising mostly make assumption that they
are working on reverberation free speech hence do not make special
consideration for eliminating reverberation (with exception of a few such as
[61] [62]). For instance, in [60] they demonstrate that SepFormer [11]
performance can significantly improve by making adjustments to include
techniques that handle reverberation. Several deep learning models have been
designed with a goal to estimate clean speech from a reverberant one. Similar
to speech denoising and speech separation, DNN models performing
dereverberation exploit these models to fit a nonlinear function to map
features of a reverberant speech to features of clean anechoic speech either
directly [63] [64] [65] [66] [67] [68] or by use of mask [18] [55] [69] [70]
[65] [55] [70]. Therefore, one way of categorising the existing
dereverberation DNN tools is based on the type of target (spectrogram or ratio
mask) they employ. Another way in which dereverberation tools can be
categorised is based on whether a tool performs general dereverberation ( i.e
suppress $h(t)$ see equation 4) or focus only on eliminating late
reverberation ($h_{l}*s(t)$ see equation 6). Tools such as [71] [59] [72] [73]
[74] [75] explore elimination of late reverberation. This is because early
reverberation does not affect speech intelligibility much. Finally, the DNN
dereverberation tools can be categorised based on the type of training
technique used (supervised or unsupervised). Tools such as [64] and [18]
perform speech dereverberation by implementing supervised training where the
DNN model is trained to directly estimate features clean speech when given
features from a reverberant speech.
$D(k,f)=M(k,f)\times Y(k,f)$ (7)
Here $D(k,f)$, $M(k,f)$and $Y(k,f)$ are the STFTs of the clean speech, the
ideal ratio mask, and the reverberant speech at time frame k and frequency
channel f respectively. Work in [76] exploits conditional GAN to perform
unsupervised dereverberation of a reverberant speech.
Dereverberation in DTF magnitude domain: When dereverberation is to be
performed in DFT magnitude domain (see section 3.1), a DFT has to be applied
to equation 4 such that,
$DFT(y(t))=Y(t,f)=H(t,f)\times S(t,f)$ (8)
the assumption in equation 8 is that the convolution of the clean signal
$s(t)$ with RIR $h(t)$ corresponds to the multiplications of their Fourier
transform in the T-F domain. However, this is only true if the extent of
$H(t,f)$ is smaller than the analysis window [60]. Therefore, when performing
dereverberation in the TF domain the selection of the window is crucial on the
performance of the DNN model [60].
Target selection in dereverberation:In dereverberation training, most tools
use direct speech as the target. This therefore means that the estimated
speech will have to be compared with the direct path speech via a selected
loss function. This has the potential of resulting in large prediction errors
which can cause speech distortion [59]. Due to this, recent work [75] proposes
the use of a target that has early reverberation. By doing this, they suggest
it will improve the quality of enhanced speech. In fact, experiments in [75]
demonstrate that allowing early reverberation in the target speech improves
the quality of enhanced speech.
### 2.4 Speech denoising
This is a speech enhancement technique of separating background noise from the
target speech. Formally, the noisy speech is represented as:
$y_{t}=s_{t}+n_{t}$ (9)
where $y_{t}$ is the noisy speech, $s_{t}$ is the target speech and $n_{t}$ is
the noise. Speech denoising seeks to isolate a single target speech from
noise. Hence data driven DNN models are optimized to predict $s_{t}$ from
$y_{t}$. Since speech denoising has only a single target speech it does not
suffer from global permutation ambiguity problem. Some DNN tools that perform
speech denoising include [77] [78] [1] [79] [80] [81] [82] [83] [84] [85].
## 3 Speech separation and enhancement features
Speech enhancement and separation tools’ input features can be categorised
into two:
1. 1.
Fourier spectrum features.
2. 2.
Time domain features.
### 3.1 Fourier spectrum features
Speech enhancement and separation tools that use these features do not work
directly on the raw signal (i.e., signal in the time domain) rather they
incorporate the discrete Fourier transform (DFT) in their signal processing
pipeline mostly as the first step to transform a time domain signal into
frequency domain. These models recognise that speech signals are highly non-
stationary, and their features vary in both time and frequency. Therefore,
extracting their time-frequency features using DFT will better capture the
representation of speech signal [86]. To demonstrate the DFT process we
exploit a noisy speech signal shown in equation 10. The same process can be
applied in speech separation. A noisy raw waveform signal of speech, $y(t)$,
can be represented as in equation 10.
$y(t)=x(t)+n(t)$ (10)
where $x(t)$ and $n(t)$ represent discrete clean speech and noise
respectively. Since speech is assumed to be statistically static for a short
period of time, it is analysed frame-wise using DFT as shown in equation 11
[86] [87] [88].
$Y[t,k]=\sum_{m=\infty}^{-\infty}y(m)w(t-m)\exp^{-j2\pi km/L}$ (11)
Here, $k$ represents the index ( frequency bin) of the discrete frequency, $L$
is the length of the frequency analysis and $w(n)$ is the analysis window. In
speech analysis, the Hamming window is mostly used as $w(n)$ [89]. Once the
DFT has been applied to the signal $y(t)$, it is transformed into time-
frequency domain represented as:
$Y[t,k]=X[t,k]+N[t,k]$ (12)
$Y[t,k],X[t,k]$ and $N[t,k]$ are the DFT representations of the noisy speech,
clean speech and noise respectively. Each term in equation 12 can be expressed
in terms of DFT magnitude and phase spectrum. For example, the polar form
(including magnitude and phase) of the noisy signal $Y[t,k]$ is:
$Y[t,k]=|Y[t,k]|\exp^{j\angle{Y[t,k]}}$ (13)
$|Y[t,k]|$ and $\angle{Y[t,k]}$ are the magnitude and phase spectra of
$Y[t,k]$ respectively. Equation 13 can be written in Cartesian coordinates as
shown in equation 14.
$Y[t,k]=|Y[t,k]|(\cos{\theta}+i\sin{\theta})=|Y[t,k]|\cos{\theta}+i|Y[t,k]|\cos{\theta}$
(14)
Both the phase and the magnitude are computed from the real and the imaginary
part of $Y[t,k]$ i.e.
$|Y[t,k]|=\sqrt{\mathbb{R}(Y[t,k])^{2}+\Im{(Y[t,k]^{2})}}$ (15)
$\angle{Y[t,k]}=\tan^{-1}\frac{\Im{(Y[t,k])}}{\mathbb{R}(Y[t,k])}$ (16)
All models that work with the Fourier spectrum features either use the DFT
representations directly as the input of the model or further modify the DFT
features. The features based on Fourier spectrum include:
1. 1.
Log-power spectrum features.
2. 2.
Mel-frequency spectrum features.
3. 3.
DFT magnitude features.
4. 4.
Complex DFT features.
5. 5.
Complementary features.
DFT magnitude features: These are features where the mixed raw waveform $y(t)$
is first converted into time-frequency (TF) representation (spectrogram) using
DFT ( specifically, short-time Fourier transform (STFT) (equation 11) [90].
The magnitude of the time-frequency representation (equation 13) acts as the
input to a deep neural network (DNN) model for speech separation. The DNN
model is then trained to learn how to separate the TF-bins such that those
that comprise each source speech are grouped together. DNN speech enhancements
and separation models that exploit DFT features include systems such as [91]
[83] [92] [93] [94] [95]. The use of DFT magnitude as features work with high
frequency resolution hence necessitating the use of larger time window which
is typically more that 32ms [20] [21] for speech and more than 90ms for music
separation [96]. Due to this, these models must handle increased computational
complexity [97]. This has motivated other speech separation models to work
with lower dimensional features as compared to those of DFT magnitude.
DFT complex features: Unlike the DFT magnitude features that only use the
magnitude of T-F representations, tools that use DFT complex features include
both the magnitude and the phase of the noisy (mixed) speech signal in the
estimation of the enhanced or separated speech. Therefore, each T-F unit of a
complex features is a complex number with a real and imaginary component (see
equation 13). The magnitude and phase of a signal is computed according to
equation 15 and 16 respectively. Tools that use DFT complex features include
[98] [55] [99] [100].
Mel-frequency cepstral coefficients (MFCC) features: Given the mixed speech
signal such as in equation 10, to extract Mel frequency cepstral features, the
following steps are executed:
1. 1.
Perform DFT of the input noisy signal $DFT(y(t))=Y[t,k]=X[t,k]+N[t,k]$
2. 2.
Given the DFT features $Y[n,k]$ of the input signal, a filterbank with M
filters i.e. a $1\leq m\leq M$ is defined where $m$ is a triangular filter
given by:
$\displaystyle H_{m}[k]=\begin{cases}0&k<f[m-1]\\\
\frac{(2(k-f[k-m])}{f[m+1]-f[m-1])(f[m]-f[m-1])}&f[m-1]\leq k\leq f[m]\\\
\frac{2(f[m+1]-1)}{(f[m+1]-f[m-1])(f[m+1]-f[m])}&f[m]\leq k\leq f[m+1]\\\
0&k>f[m+1]\end{cases}$ (17)
The filters are used to compute the average spectrum around centre frequencies
with increasing bandwidths as shown in figure 1. Here, $f[m]$ are uniformly
spaced boundary points in the Mel-scale which is computed according to
equation 18. The Mel-scale B is given by equation 19 and $B^{-1}$ which is its
inverse is computed as shown in equation 20.
$f[m]=\frac{N}{F_{s}}B^{-1}(B(f)+m\frac{B(f_{h})-B(f_{l})}{M+1})$ (18)
$F_{s}$ is the sampling frequency, $f_{l}$ and $f_{h}$ represent the lowest
and the highest frequencies of the filter bank in Hz. N is the size of DFT and
M is the number of filters.
$B(f)=1125\ln{(1+\frac{f}{700})}$ (19)
$B^{-1}(b)=700(\exp{(\frac{b}{1125})-1})$ (20)
Figure 5: Triangular filters used in the computation of the Mel-cepstrum using
equation 18
3. 3.
Scale the magnitude spectrum $|Y[t,k]|$ of the noisy signal in both frequency
and magnitude using mel-filter bank $H(k,m)$ and then take the logarithm of
the scaled frequency.
$X^{\prime}(m)=\log(\sum_{k=0}^{N-1}|Y[t,k]|^{2}H_{m}[k]$ (21)
for $m=0,\cdots,M$where $M$ is the number of filter banks.
4. 4.
Compute the Mel frequency by computing the discrete cosine transform of the
$m$ filter outputs as shown in equation 22.
$c[n]=\sum_{m=0}^{m-1}X^{\prime}(m)\cos{(\pi n(m+1/2)/M)}$ (22)
where $0\leq n<M$
The motivation for working with MFCC is that it results in reduced resolution
space as compared to DFT features. Fewer parameters are easier to learn and
may generalise better to unseen speakers and noise [97]. The challenge however
with working on a reduced resolution such as MFCC is that the DNN estimated
features must be extrapolated to the DFT feature space. Due to working on a
reduced resolution, MFCC degree-of-freedom will be restricted by the
dimensionality of the reduced resolution feature space which is much less than
that of the DFT space. The low-rank approximation generates a sub-optimal
Wiener filter which cannot account for all the added noise content and yields
reduced SDR [97]. MFCC features have been exploited in tools such as [101]
[68] [102] [83][103] [104].
Log-power spectra features: To compute these features, a short-time Fourier
analysis is applied to the raw signal computing the DFT of each overlapping
waveform (see equation 11). The log-power spectra are then computed from the
output of the DFT. Consider a noisy speech signal in the time-frequency domain
i.e., where DFT has been applied to the signal (see equation 12). From
equation 14, the power spectrum of the noisy signal can be represented as in
equation 23.
$|Y[k]|^{2}=|X[k]|^{2}+N[k]|^{2}=|X[k]|^{2}+|N[k]|^{2}+2|X[k]||N[k]|\cos{\theta}$
(23)
Here, $\theta$ represents the angle between the two complex variables $|X[k]|$
and $|N[k]|$. Most models that exploit log-power spectra features ignore the
last term ( assume the value to be zero) and employ equation 24.
$Y^{l}=log(|Y[k]|^{2})=log(|X[K]|^{2})+log(|N[k]|^{2})$ (24)
Figure 6 [105] summarises the process of log-power feature extraction.
Figure 6: Demonstrating feature extraction. Here, $Y^{t}$ represents the noisy
signal in time domain, $Y^{f}$ represents the transformed signal in the
frequency domain. $Y^{l}$ is the log power features of the input signal
Examples of models that use Log-power spectra features include [98] [105]
[106] [107].
Complementary features: Since different features strongly capture different
acoustic features which characterise different properties of the speech
signal, some DNN models exploit a combination of the features to perform
speech separation. This is based on works such as [108] and [109] which
demonstrated that complementary features significantly improve performance in
speech recognition. The complementary features used in [109] [110]
[111]include perceptual linear prediction, amplitude modulation spectrogram
(AMS), relative spectral transform and perceptual linear product (RASTA-PLP),
Gammatone frequency cepstral coefficient, MFCC, pitch-based features. The
complementary features are combined by concatenation. Research in [111]
reports that the use of complementary features registered better results as
compared to those of DFT magnitude. The challenge with using complementary
features is how to effectively combine the different features, such that those
complementing each other are retained while redundant ones are eliminated
[110].
#### 3.1.1 Supervised speech enhancement and separation training with Fourier
spectrum features
DNN models that are trained via supervised learning using Fourier spectrum
features employ several strategies to learn how to generate estimated clean
signal from a noisy (mixed) signal. These strategies can be classified into
three categories based on the target of the model.
1. 1.
Spectral mapping techniques.
2. 2.
Spectral masking techniques.
3. 3.
Generative modelling.
Spectral mapping techniques
These models fit a nonlinear function to learn a mapping from a mixed signal
feature to an estimated clean signal feature (see figure 7).
Figure 7: Supervised training of speech enhancement model using spectrogram as
input and spectrogram as output.
The training dataset of these models consist of a noisy speech signal (source)
and clean speech (target) features. The process of training these models can
be generalised in the following steps:
1. 1.
Given $N$ raw waveforms of mixed (noisy) speech, convert the $N$ raw waveform
of noisy speech to the desired representation (such as spectrogram).
2. 2.
Convert the respective $N$ clean speech waveform in time domain to the same
representation as that of the noisy speech.
3. 3.
Create an annotated dataset consisting of a pair of noisy speech features and
that of clean speech i.e.,
$<noisy\\_speech\\_features_{i},clean\\_speech\\_features_{i}>$ with $1\leq
i\leq N$
4. 4.
Train a deep learning model $g_{\theta}$ to learn how to estimate clean
features $clean\\_speech\\_features_{i}$ given a noisy speech feature as input
$noisy\\_speech\\_features_{i}$ by minimizing an objective function.
5. 5.
Given new a noisy speech features $x_{j}$ the trained model $g_{\theta}$
should estimate a clean speech feature $y_{j}$.
6. 6.
Using the estimated clean speech features $y_{j}$, reconstruct its raw
waveform by performing the inverse of the feature generation process (such as
using the inverse short-time Fourier transform if the features are in time-
frequency domain).
The above generalisation has been exploited in [83] [92] [95] [106] [112]
[113] [84] [85] to achieve speech enhancement and in [94]and [103] to perform
speech separation and enhancement. Figure 8 gives a summary of the steps when
time-frequency(spectrogram) is exploited as the input of the speech
enhancement model.
Figure 8: Showing steps involves to train speech enhancement model to fit a
regression function from noisy spectrogram to an estimated clean speech
spectrogram
Spectral masking techniques
Here, the task of estimating clean speech features from a noisy (mixed) speech
input features is formulated as that of predicting real-valued or complex-
valued masks [55]. The mask function is usually constrained to be in range the
[0,1] even though different types of soft masks have been proposed (see [21]
[114] [115]). Source separation based on masks is predicated on the assumption
of sparsity and orthogonality of the sources in the domain in which the masks
are computed [60]. Due to the sparsity assumption, the dominant signal at a
given range (such as time-frequency bin) is taken to be the only signal at
that range (i.e. all other signals are ignored except the dominant signal). In
that case, the role of DNN estimated mask is to estimate the dominant source
at a given range. To do this, the mask is applied on the input features such
that it eliminates portion of the signal( where the mask has a value of 0)
while allowing others (mask value of 1)[116] [117]. The masks are always
established by computing the signal-to-noise (SNR) within each TF bin against
a threshold or a local criterion [116]. It has been demonstrated
experimentally that the use of masks significantly improves speech
intelligibility when an original speech is composed of noise or a masker
speech signal [118] [119]. For deep learning models working on the time-
frequency domain, a model $g_{\theta}$ is designed such that given a noisy or
mixed speech spectrogram $Y[t,n]$ at time frame $t$, it estimates the mask
$m_{t}$ at that time frame. The established mask $m_{t}$ is then applied to
the input spectrogram to estimate target or denoised spectrogram i.e.,
$\hat{S}_{t}=m_{t}\otimes Y[t,n]$ (see figure 9). Here, $\hat{S}_{t}$ is the
spectrogram estimate of the clean speech at time frame $t$ and $\otimes$
denotes element wise multiplication. To train the model $g_{\theta}$, there
are two key objective variants. The first type minimizes an objective function
$D$ such as mean squared error (MSE) between the model estimated mask
$\hat{m}_{m}$ and the target mask ($tm$).
$\mathcal{L}=\sum_{u,t,f}D|tm_{u,t,f},\hat{m}_{u,t,f}|$
This approach however cannot effectively handle silences where $|Y[t,n]|=0$
and $|X[t,n]|=0,$ because the target masks $tm$ will be undefined at the
silence bins. Note that target masks such ideal amplitude mask (IRM) that is
defined as $IRM(t,f)=\frac{|X_{s}(t,f)|}{\sum_{i=1}^{s}|Y_{s}(t,f)|}$ involves
division of $|X[t,n]|$ by $|Y[t,n]|$ hence silence regions will make the
target mask undefined [21]. This cost function also focuses on minimizing the
disparity between the masks instead of the features of estimated signal and
the target clean signal [21]. The second type of cost function seeks to
minimize the features of estimated signal $\hat{S}_{t}=m_{t}\otimes Y[t,n]$
and those of target clean signal $S$ directly as shown equation 25.
$\mathcal{L}=\sum_{u,t,f}D|\hat{m}_{u,t}Y_{u,t,f},S_{u,t,f}|$ (25)
The sum is over all the speech $u$ and time-frequency bin $(t,f)$. Here, $Y$
and $S$ represents noisy (mixed) and clean (target) speech respectively. So,
for DNN tools using indirect estimation of clean signal features, instead of
them estimating the clean features directly from the noisy features input, the
models first estimate binary masks. The binary masks are then applied to the
noisy features to separate the sources (see figure 5, here, the features are
the TF spectrogram). This technique has been applied in [3] [20] [103] [84]
[114] [120] [121] [25] [122] [123] [124] [125] [18] [123] [31].
Figure 9: DNN model for mask estimation from a noisy spectrogram.
Generative modelling
Given an observed sample $x$, the goal of a generative model is to model its
true distribution $p(x)$. The established model can then be used to generate
new samples that are similar to the observed samples $x$. In speech separation
and enhancement, these models have been exploited almost exclusively to
perform speech denoising. Several generative models have been employed in a
supervised manner to generate clean speech from a noisy one. Generative
adversarial network (GAN) [126] is an unsupervised model that constitutes two
key parts: the generator $\mathcal{G}$ and the discriminator $\mathcal{D}$,
where $\mathcal{G}$ generates samples which are then judged by the
$\mathcal{D}$. The generator $\mathcal{G}$ generates the synthetic data by
sampling from a simple prior $z\sim p(z)$ and the outputs a final sample
$g_{\theta}(z)$ where $g_{\theta}$ is non-linear function more specifically a
DNN. The discriminator $\mathcal{D}$ on the other hand must be able to catch
synthetic data as fake and real data from $p(x)$ as real. The training
objective is shown in equation 26.
$\min_{\mathcal{G}}\max_{\mathcal{D}}=\mathbb{E}_{x\sim p(x)}[\log
D(x)]-\mathbb{E}_{z\sim p(z)}[\log(1-D(G(z)))]$ (26)
The objective function in equation 26 is maximised w.r.t to $D(.)$ and
minimised w.r.t $G(.)$. In speech enhancement, GAN was first introduced by
SEGAN( see section 3.2.3). SEGAN which works in time-domain, uses a
conditioned version of the objective function of equation 26. In conditioned
GAN, both the generator and the discriminator are given extra information.
This allows GAN to perform classification and mapping. SEGAN uses the least-
squares GAN loss as opposed to sigmoid cross-entropy loss used for training.
CGAN [104] just like SEGAN uses conditioned GAN but works in T-F domain to
generate a denoised speech. Since most automatic speech recognition (ASR)
tools work in T-F domain, CGAN hypothesises that the generative model working
in T-F domain will be more robust for ASR as compared to those working in raw
waveform. Therefore, CGAN can be seen as version of SEGAN that accepts input
in T-F domain. To address the problem of mismatch between the training
objective used in CGAN and the evaluation metrics, MetricGAN [93] proposes to
integrate evaluation metric in the discriminator. By doing this, instead of
the generator giving a false (0) or true (1) discrete values, it will generate
continuous values based on the evaluation metric. MetricGAN can therefore be
trained to generate data according to the the selected metric score. Through
this modification, MetricGAN produces more robust enhanced speech. Another
common generative group of generative models is variational auto-encoder (VAE)
technique [127]. Like GAN, VAE is mainly used for denoising i.e. where the
mixture is modelled as:
$x_{fn}=\sqrt{g_{n}}s_{fn}+b_{fn}$ (27)
Here, $x_{fn}$ denotes the mixture at the frequency index $f$ and the time-
frame index $n$, $g_{n}\in R_{+}$ is a frequency independent but frame
dependent gain while $s_{fn}$ and $b_{fn}$ represent the clean speech and the
noise respectively at the frequency index $f$ and the time-frame index $n$. We
first give a brief overview of VAE before we discuss how it is adapted for
speech enhancement. Mathematically, given an observable sample $s$, the goal
of a generative VAE model is to model true data distribution $p(s)$. To do
this, VAE assumes that the observed sample $s$ are generated by associated
latent variable $z$ and their joint distribution is $p(s,z)$. The model
therefore seeks to learn how to maximize the likelihood $p(s)$ over all
observed data.
$p(s)=\int p(s,z)dz$
Integrating out all the latent variables $z$ in the above equation is
intractable. However, using Evidence lower bound (ELBO) which quantifies the
log-likelihood of observed data $p(s)$ can be estimated. ELBO is given in
equation 28 (refer to [81] to see derivation of relationship between $p(s)$
and ELBO).
$\log p(s)\geq\mathbb{E}_{q_{\phi}(z|s)}[\log\frac{p(s,z)}{q_{\phi}(z\mid
s)}]$ (28)
Here, $q_{\phi}(z\mid s)$ is a flexible variational distribution with
parameters $\phi$ that the model seeks to maximize. Equation 28 can be written
as equation 29 using Bayes theorem.
$\log p(s)\geq\mathbb{E}_{q_{\phi}(z|s)}[\log\frac{p_{\theta}(s\mid
z)p(z)}{q_{\phi}(z\mid s)}]$ (29)
Equation 29 can be expanded as:
$\log p(s)\geq\mathbb{E}_{q_{\phi}(z\mid s)}[\log p_{\theta}(s\mid
z)]+\mathbb{E}_{q_{\phi}(z\mid s)}[\frac{\log p(z)}{q_{\phi}(z\mid s)}]$ (30)
Equation 30 can be expanded as:
$\log p(s)\geq\mathbb{E}_{q_{\phi}(z\mid s)}[\log p_{\theta}(s\mid
z)]+D_{KL}(q_{\phi}(z\mid s)\mid\mid p(z))$ (31)
The second term on the right of equation 31 seeks to learn the prior $p(z)$
via $q_{\phi}(z\mid s)$ while the first term reconstructs data based on the
learned latent variable $z$. $q_{\phi}(z\mid s)$ is always modelled by a DNN
and referred to as encoder and the reconstruction term is another DNN referred
to as decoder. Both the encoder and decoder are trained simultaneously. The
encoder is normally chosen to model a multivariate Gaussian with diagonal
covariance and the prior is often selected to be a standard multivariate
Gaussian:
$q_{\phi}(z\mid x)=\mathcal{N}(z;\mu_{\theta}(x),\delta^{2}_{\theta}(x)I)$
$p(z)=\mathcal{N}(z;0,I)$
To estimate clean speech based on variational-autoencoder pre-training, the
tools execute several techniques that can be generalised into the following
steps:
1. 1.
Train a model such that it can maximise the likelihood $p_{\theta}(s\mid z)$.
Here, $s$ denotes the clean speech dataset that is composed of F-dimensional
samples i.e $s_{t}\in R^{F},1\leq t\leq T$. The variational autoencoder
assumes a D-dimensional latent variable $z_{t}\in R^{D}$. The latent variable
$z_{t}$ and the clean speech $s_{t}$ have the following distribution:
$z_{t}\sim\mathcal{N}(0,I_{D})$ $s_{t}\sim p(s_{t}|z_{t})$
Here, $\mathcal{N}(\mu,\delta)$ denotes a Gaussian distribution with mean
$\mu$ and variance $\delta$. Basically, a decoder $p_{\theta}(s_{t}\mid
z_{t})$ is trained to generate clean speech $s_{t}$ when given the latent
variable $z_{t}$, the decoder is parameterized by $\theta$. The decoder
$p_{\theta}(s_{t}\mid z_{t})$ is learned by deep learning model during
training. The encoder is trained to estimate the posterior
$q_{\phi}(z_{t}|s_{t})$ using a DNN. The overall objective of the variational
auto-encoder training is to maximise equation 32.
$p(s)=\operatorname*{argmin}_{\theta,\phi}\sum_{i=1}^{L}\log p_{\theta}(s\mid
z^{i})+D_{KL}(q_{\phi}(z\mid s)\mid\mid p(z))$ (32)
The posterior estimator $q_{\phi}(z\mid s)$ is a Gaussian distribution with
parameters $\mu_{d}$ and $\delta_{d}$. These parameters are to be established
by the encoder deep neural network such that $\mu_{d}:R^{F}\rightarrow R$ and
$\delta_{d}:R^{F}\rightarrow R_{+}$.
2. 2.
Set up a noise model using unsupervised techniques such as NMF [128]. For
example, in case of NMF the noise $b_{fn}$ in equation 27 can be modelled as
$b_{bf};w_{b,f},h_{b,n}\sim\mathcal{N}(0,(W_{b},H_{b})_{f,n})$ (33)
where $\mathcal{N}(0,\delta)$ is a Gaussian distribution with zero mean and
variance of $\delta$.
3. 3.
Set up a mixture model such that $p(x\mid z,\theta_{s},\theta_{u})$ is
maximised. Here $x$ is the noisy speech signal, $\theta_{s}$ are parameters
from the pre-trained model in step 1 i.e $\phi$ and $\theta$.
$\theta_{u}={g_{n},(W_{b},H_{b})_{f,n}}$ represents the parameters to be
optimised. The parameters are $\theta_{u}$ are optimised by appropriate
Bayesian inference technique.
4. 4.
Reconstruct the clean speech $\hat{s}$ such that
$p(\hat{s}|\theta_{u},\theta_{s},x)$ is maximised based on the parameters
$\theta_{u},\theta_{s}$ from step 1 and 3 respectively and the observed mixed
speech $x$.
Works that exploit different versions of variational auto-encoder technique
include [77] [78] [1] [79] [129]. Another generative modelling technique that
has been used in speech enhancement is the variational diffusion model
(VDM)[130]. VDM is composed of two processes i.e., diffusion and reverse
process. The diffusion process perturbs data to noise and the reverse process
seeks to recover data from noise. The goal of diffusion therefore is to
transform a given data distribution into a simple prior distribution mostly
standard Gaussian while the reverse process recovers data by learning a
decoder parameterised by DNN. Formally, representing true data samples and
latent variables as $x_{t}$ where $t=0$ represents true data and $1\leq t\leq
T$ represents a sequence of latent variables, the VDM posterior is represented
as:
$q(x_{1:T}\mid x_{0})=\prod_{t=1}^{T}q(x_{t}\mid x_{t-1)}$ (34)
The VDM encoder $q(x_{t}\mid x_{t-1})$ unlike that of VAE, is not learned
rather it is a predefined linear Gaussian model. The Gaussian encoder is
parameterized with mean $u_{t}(x_{t})=\sqrt{\alpha_{t}}x_{t-1}$ and variance
$\varepsilon_{t}=(1-\alpha_{t})I$. Therefore, the encoder $q(x_{t}\mid
x_{t-1})$ can mathematically be represented as
$q(x_{t}\mid
x_{t-1})=\mathcal{N}(x_{t};\sqrt{\alpha_{t}}x_{t-1},(1-\alpha_{t})I).$ (35)
$\alpha_{t}$ evolves over time such that the final distribution of the latent
$p(x_{T})$ is a standard Gaussian. The reverse process seeks to train a
decoder that starts from the standard Gaussian distribution $p(x_{T})$.
Formally the reverse process can be represented as:
$p(x_{0:T})=p(x_{T})\prod_{i=1}^{T}p_{\theta}(x_{t-1}\mid x_{t})$ (36)
Here $p(x_{T})=\mathcal{N}(x_{T};0,I)$. The reverse process seeks to set up a
decoder $p_{\theta}(x_{t-1}\mid x_{t})$ that optimizes the parameter $\theta$
such that: the conditionals $p_{\theta}(x_{t-1}\mid x_{t})$ are established.
Once the VDM is optimized, a sample from the Gaussian noise $p(x_{T})$ can
iteratively be denoised through transitions $p_{\theta}(x_{t-1}\mid x_{t})$
for T steps to generate a simulated $x_{0}$. Using reparameterization trick,
$x_{t}$ in equation 34 can be rewritten as:
$x_{t}=\sqrt{\alpha_{t}}x_{t-1}+\sqrt{1-\alpha_{t}}\epsilon$ (37)
where $\epsilon\sim\mathcal{N}(\epsilon,O,I)$ Similarly,
$x_{t-1}=\sqrt{\alpha_{t-1}}x_{t-2}+\sqrt{1-\alpha_{t-1}}\epsilon$ (38)
Based on this and through iterative derivation of equation 34, it can be shown
that:
$x_{t}=\sqrt{\bar{\alpha_{t}}}x_{0}+\sqrt{1-\bar{\alpha_{t}}}\epsilon_{0}$
(39)
In the reverse process in equation 36, the transition probability
$p_{\theta}(x_{t-1}\mid x_{t})$ can be represented by two parameters
$\mu_{\theta}$ and $\delta_{\theta}$ as
$\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\delta_{\theta}(x_{t},t)^{2}I)$
with $\theta$ being the learnable parameters. It has been shown in [131] that
$\mu_{\theta}(x_{t},t)$ can be established as:
$\mu_{\theta}(x_{t},t)=\frac{1}{\sqrt{\alpha_{t}}}x_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha_{t}}}\sqrt{\alpha_{t}}}\epsilon_{\theta}(x_{t},t)$
(40)
Based on equation 40, to estimate $\mu_{\theta}(x_{t},t)$ the DNN
$\epsilon_{\theta}(x_{t},t)$ needs to estimate the Gaussian noise $\epsilon$
in $x_{t}$ which was injected during the diffusion process. Like VAE, VDM uses
ELBO objective for optimization. Please see [131] for a thorough discussion on
VDM. In speech denoising, work in [81] uses conditional diffusion process to
model the encoder $q(x_{t}\mid x_{t-1})$. In conditional encoder, instead of
$q(x_{t}\mid x_{t-1})$, they define it as $q(x_{t}\mid x_{0},y)$ i.e.,
$q(x_{t}\mid
x_{0},y)=\mathcal{N}(x_{t};(1-m_{t})\sqrt{\bar{\alpha}}x_{0}+m_{t}\sqrt{\bar{\alpha}}y,\delta_{t}I)$.
Here $x_{0}$,$y$ represents the clean speech and noisy speech respectively.
The encoder is modeled as a linear interpolation between clean speech $x_{0}$
and the noise speech $y$ with interpolation ratio $m_{t}$. The reverse process
$p_{\theta}(x_{t-1}\mid x_{t})$ is also modified to $p_{\theta}(x_{t-1}\mid
x_{t},y)=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},y,t),\delta I)$. Here,
$\mu_{\theta}(x_{t},y,t)$ is the mean of the conditional reverse process.
similar to equation 40, $\mu_{\theta}(x_{t},y,t)$ is estimated as
$\mu_{\theta}(x_{t},y,t)=c_{xt}x_{t}+c_{yt}y-c_{\epsilon
t}\epsilon_{\theta}(x_{t},y,t)$ (41)
where $\epsilon_{\theta}(x_{t},y,t)$ is a DNN model to estimate the
combination of Gaussian and non-Gaussian noise. The coefficients $c_{xt}$,
$c_{yt}$ and $c_{\epsilon t}$ are established via the ELBO optimization. Other
generative modelling techniques for speech enhancement(denoising).
#### 3.1.2 Highlights on Fourier spectrum features
1. 1.
When performing a DFT on the input signal, an optimum window length must be
selected. The choice of the window has a direct impact on the frequency
resolution and the latency of the system. To achieve good performance, most
systems use 32ms. This may limit the use of the DFT based models in
environments which require short latency [132].
2. 2.
DFT is a generic method for signal transformation that may not be optimised
for waveform transformation in speech separation. It is therefore important to
know to what extent does it place an upper bound on the performance level of
speech enhancement techniques.
3. 3.
Accurate reconstruction of estimated clean speech from the estimated features
is not easy and the erroneous reconstruction of clean speech places an upper
bound on the accuracy of the reconstructed audio.
4. 4.
Perhaps the biggest challenge when working in the frequency domain is how to
handle the phase. Most DNN models only use the magnitude spectrum of the noisy
signal to train the DNN then factor in the phase of the noisy signal during
reconstruction. Recent works such as [89] have shown that this technique does
not generate optimum results.
5. 5.
While working in the frequency domain, experimental research has demonstrated
that spectral masking generates better results in terms of enhanced speech
quality as compared to the spectral mapping method [119].
#### 3.1.3 Handling of phase in frequency domain
The assumption made by most DNN models that use Fourier spectrum features is
that phase information is not crucial for human auditory. Therefore, they
exploit only the magnitude or power of the input speech to train the DNN
models to learn the magnitude spectrum of the clean signal and factor in the
phase during the reconstruction of the signal( see figure 10)[113] [133] [105]
[134] [135]. The use of the phase from the noisy signal to estimate the clean
signal is based on works such as [136] that demonstrated that the optimal
estimator of the clean signal is the phase of the noisy signal. Further, most
speech separation models work on frames that are of size between 20-40 ms and
believe that the short-time phase contain low information [137] [138] [139]
[140] and therefore not crucial when estimating clean speech. However, recent
research [89] have demonstrated through experiments that further improvements
in quality of estimated clean speech can be attained by processing both the
short-time phase and magnitude spectra. Further, the factoring in of the noisy
input phase during reconstruction has been noted to be a problem since the
phase errors in the input interact with the amplitude of the estimated clean
signal hence causing the amplitude of the estimated clean signal to differ
with the amplitude of the actual clean signal being estimated [115], [64].
Figure 10: Showing how DNN models exploit the phase of the noisy signal during
reconstruction of the estimated signal
Based on the realisation of the importance of phase, some studies have avoided
factoring in the phase of the noisy signal but rather exploit a modified phase
to estimate the clean signal. The existing techniques of modifying the phase
by the DNN models working on Fourier spectrum features can be categorised into
two:
Phase learning: These models make the phase part of the objective function
i.e., they learn the phase of the estimated clean signal during training. To
integrate the phase in the learning process works such as [115] [21] use a
phase sensitive objective by replacing equation 42 with 43. It essentially
exploits a phase sensitive spectrum approximation objective by minimising the
distance between the raw waveform of the estimated speech and that of the
target clean speech.
$\mathcal{L}=\sum_{u,t,f}D|\hat{m}_{u,t}Y_{u,t,f},S_{u,t,f}|$ (42)
$\mathcal{L}=\sum_{u,t,f}D|\hat{m}_{u,t}Y_{u,t,f},S_{u,t,f}(\cos{\theta}_{y}-\cos{\theta}_{s})|$
(43)
Here $D$, is a selected objective function such as MSE, $\theta_{y}$ and
$\theta_{s}$ represent the phase of the noisy and clean (target) speech
respectively. The sum is over all the speech $u$ and time-frequency bin
$(t,f)$. Experiments conducted based on the objective function in equation 43
show superior results in terms of signal-to-distortion ratio (SDR)[115]. Work
in [111] trains a DNN model to generate masks that are composed of both the
real and imaginary part (see equation 44). The complex mask will then be
applied to a complex representation of the noisy signal to generate the
estimated clean signal. By learning a mask that has both the real and
imaginary part, they integrate the phase as part of the learning.
$L=\frac{1}{2N}\sum_{t}\sum_{f}[(O_{r}(t,f)-M_{r}(t,f))^{2}+(O_{i}(t,f)-M_{i}(t,f))^{2}]$
(44)
$O_{r}$ is the real part of the mask estimated by the DNN model while $O_{i}$
is the imaginary part. $M_{r}$ is the real part of the target mask while
$M_{i}$ is the imaginary part. $N$ is the number of frames and (t,f) is a
given TF bin. The complex mask implementation has been exploited in [55]
[115][141] where the targets are formulated in the complex coordinate system
i.e. the magnitude and phase are composed as part of the learning process.
Work in [42] proposes a model that learns the phase during training via input
spectrogram inversion (MISI) algorithm [142]. Work in [143] proposes a
generative adversarial network (GAN) [126] based technique of learning the
phase during training. Other works that learn the phase during training
include [63] and [144]. Techniques that include phase as part of the training
face the difficulty of processing a phase spectrogram which is randomly
distributed and highly unstructured [145]. To mitigate this problem and derive
a highly structured phase-aware target masks, [145] employs instantaneous
frequency (IF)[146] to extract structured patterns from phase spectrograms.
Post-processing phase update: The models that use this technique, train the
DNN models using only the magnitude spectrum. Once the model has been trained
to estimate the magnitude spectrum of the clean signal, they iteratively
update the phase of the noisy signal to be as close as possible to that of the
target clean signal. The algorithm being exploited by the models performing
post-processing phase update is based on the Griffin-Lim algorithm proposed in
[147]. For example, in [64], they exploit the magnitude $X^{0}$ of the target
clean signal to iteratively obtain an optimal phase $\phi$ from the phase of
the noisy signal ( see algorithm 1). The obtained phase is then used in the
reconstruction of the estimated clean signal together with the magnitude
$\hat{X}$ estimated by the DNN. The technique is also used in [148].
Techniques that implement Griffin-Lim algorithm such as in algorithm 1 perform
iterative phase reconstruction of each source independently and may not be
effective for multiple source separation where the sources must sum up to the
mixture [42]. Work in [42] proposes to jointly reconstruct the phase of all
sources in a given mixture by exploiting their estimated magnitudes and the
noisy phase using the multiple input spectrogram inversion (MISI) algorithm
[142]. They ensure that the sum of the reconstructed time-domain signals after
each iteration must sum to the mixture signal. Work [149] and [62] also uses
post-processing to update the phase of the noisy signal.
Algorithm 1 Iteratively updating the phase of a noisy signal
Target clean magnitude $X^{0}$, noisy phase $\phi^{0}$, iteration N.
$X\leftarrow X^{0},\phi\leftarrow\phi^{0},n\leftarrow 1$
while $n\leq N$ do
$s^{n}\leftarrow iSTFT(X,\phi)$
$(X^{n},\phi^{n})\leftarrow STFT(s^{n})$
$X\leftarrow X^{0}$
$\phi\leftarrow\phi^{n}$
$n\leftarrow n+1$
end while
$s\leftarrow s^{n}$
### 3.2 Time-domain features
Due to the challenges highlighted in section 3.1.2 of working in the time-
frequency domain, different models such as [132] [150] [10] [151] [152] [11]
[153] [154][155] [61] [156] [157] explore the idea of designing a deep
learning model for speech separation that accepts speech signal in the time-
domain. The fundamental concept for these models is to replace the DFT based
input with a data-driven representation that is jointly learned during model
training. The models therefore accept as their input the mixed raw waveform
sound and then generates either the estimated clean sources or masks that are
applied on the noisy waveform to generate clean sources. By working on the raw
waveform, these models address two key limitations of DFT based models. First,
the models are designed to fully learn the magnitude and phase information of
the input signal during training [150]. Secondly, they avoid reconstruction
challenges faced when working with DFT features. The time domain methods can
broadly be classified into two categories [150].
#### 3.2.1 Adaptive front-end based method
The models in this category can roughly be discussed as composed of three key
modules i.e., the encoder, separation and decoder modules ( see figure 11).
1. 1.
Encoder: The encoder can be regarded as an adaptive front-end which seeks to
replace STFT with a differentiable transform that is jointly trained (learned)
with the separation model. It accepts as its input a time-domain mixture
signal then learns STFT-like representation [11] [155]. By working directly
with the time-domain signal, these models avoid the decoupling of the
magnitude and phase of the input signal [10]. Most systems employ
1-dimensional convolution as the encoder to learn the features of the input
signal. The transform generated by the encoder is then passed to the
separation module. Work in [158] demonstrates that learned bases from raw data
produce better results for speech/non-speech separation.
2. 2.
Separation module: This module is fed by the output of the encoder. It
implements techniques to identify the different sources present in the input
signal.
3. 3.
Decoder: It accepts input from the separation module and sometimes from the
encoder(for residual implementation ). It is mostly implemented as an inverse
of the encoder in order to reconstruct the separated signals [132] [10] [11].
Figure 11: Showing adaptive front-end implementation.
#### 3.2.2 Waveform Mapping
The second category of systems implement end-to-end systems where they utilise
deep learning models to fit a regression function that maps an input mixed
signal to its constituent estimated clean signal without an explicit front-end
encoder (see figure 12). The models are trained using a pair of mixed(noisy)
and clean speech. The model is fed with features of mixed signal for it to
estimate clean speech. The training involves minimising an objective function
such as minimum mean square error(MMSE) between the features of the clean
signal and the estimated clean signal generated by the model. This approach
has been implemented in [159] [160] [161].
Figure 12: Direct approach training of DNN models using raw waveform.
#### 3.2.3 Generative modelling
SEGAN [162] is GAN based model for speech denoising that conditions both $G$
and $D$ of equation 26 on extra information $z$ representing latent
representation of the input. To solve the problem of vanishing gradient
associated with optimizing objective in equation 26, they replace the cross-
entropy loss by a least square function in equation 45.
$\min_{\mathcal{G}}=\mathbb{E}_{z\sim p(z),\bar{x}\sim
p(\bar{x})}[(D(G(z,\bar{x}),\bar{x})-1)^{2})]+\lambda||G(z,\bar{x})-x)||_{1}]$
(45)
Here, $\bar{x}$ is the noisy speech, $x$ is the clean speech, $z$ is the extra
input latent representation and $||.||_{1}$ is the $l_{1}$ norm distance
between the clean sample x and the generated sample $|G(z,\bar{x})$ to
encourage the generator G to generate more realistic audio. Work in [163]
improves SEGAN to handle a more generalised speech signal distortion case
which involves distortions such as chunk removal, band reduction, clipping and
whispered speech. Work [164] improves SEGAN by implementing multiple
generators as opposed to one and demonstrates that by doing so the speech
quality of the enhanced speech is better than when a single generator is used.
Work in [165] proposes a variation of SEGAN that is more tailored towards
speech synthesis and not ASR. They replace the original loss function used in
SEGAN with Wasserstein distance with gradient penalty(WGAN) [166]. They also
exploit gated linear unit as activation function which has been shown in [167]
to be more robust in generating realistic speech. Other GAN based models for
speech enhancement in the time domain include [168]. Other tools that
implement supervised conditional GAN include [104] [169] [170] [93]. In [171],
Bayesian network is exploited to generate estimated clean speech from a noisy
one.
### 3.3 Challenges of working with time-domain features
1. 1.
Time domain features lack direct frequency representation; this hinders the
features from capturing speech phonetics that are present in the frequency
domain. Due to this, artefacts are always introduced in the reconstructed
speech in the time domain [172].
2. 2.
The time domain waveform has a large input space. Based on this, models
working with raw waveforms are often deep and complex in order to effectively
model the dependencies in the waveform. This is computationally expensive [72]
[162] [11] [173].
.
## 4 Which feature produces superior quality of enhanced speech?
We performed analysis of 500 papers that exploit DNN to perform speech
enhancement(i.e., multi-talker speech separation or denoising or
dereverberation). We selected papers published from 2018 to 2022. We were
interested to answer the question, which features are more popular with these
tools? The summary is presented in figure 13. Based on the analysis, time-
domain features popularity has grown rapidly from 2018 to 2022. The use of DFT
features has slightly dropped, however remains popular over the five years.
The popularity of MFCC and LPS has diminished. The popularity of features that
are computationally expensive such as time-domain and DFT features may be
attributed to the improved computation power of computers and efficient
sequence modelling techniques such as transformers and temporal convolutional
networks (see section 5 for discussion). Features such as MFCC are becoming
less popular due to their reduced resolution, which must be extrapolated
during reconstruction hence placing an upper bound on the quality of enhanced
speech.
Figure 13: Feature popularity between 2018 and 2022
We also investigated whether DFT or time-domain features produced the highest
quality enhanced speech. Several works have conducted experiments with the
goal to answer this question. Notable works include[174] and [175]. For
example, [174] investigates Conv-TasNet’s [10] performance under different
input types in the encoder and decoder. Conv-TasNet uses a frame length of
4ms, stride of 2ms and overlap of 2ms. Sample results presented in [174] are
presented in table 1 where evaluation parameters include scale-invariant
signal-to-distortion (si_SDR), signal-to-distortion (SDR), word error rate
(WER).
Table 1: Comparison of different encoder and decoder combination using Lw = 4 ms and Ls = 2ms on the test set of the WSJ0-2mix database loss | Encoder | Decoder | si_SDR dB | SDB dB | WER %
---|---|---|---|---|---
$L^{SI\\_SDR}$ | Learned | Learned | 14.4 | 14.7 | 21.71
$L^{SI\\_SDR}$ | STFT | Learned | 13.9 | 14.3 | 21.92
$L^{SI\\_SDR}$ | Learned | iSTFT | 14.1 | 14.5 | 21.87
$L^{SI\\_SDR}$ | STFT | iSTFT | 12.4 | 12.8 | 24.69
The results in table 1 show that the Conv-TasNet model gives marginally better
results in terms of $si\\_SDR$, $SDBdB$ and $WER$ when the input is in time
domain where the signal representation is learned by the encoder and output is
learned by the decoder. The results are significantly reduced in all the three
parameters if STFT is used as the input and its inverse used in the decoder.
For instance, Conv-TasNet model achieves a SDR of 14.7 when time-domain
features are used. This drops to 12.8 when DFT features are used. This shows
that working in the time domain may be better for this setting as compared to
the frequency domain. Work in [175] also shows the same trend where working in
time-domain provides better results as compared to frequency domain. However,
for mixed speech with reverberation, the use of a time domain signal does not
improve the same results as compared to the frequency domain and further
investigation on behaviour of both time and frequency features in the presence
of reverberation is needed[175].
## 5 Long term dependencies modelling
To effectively perform speech separation, the speech separation tools need to
model both long and short sequences within the audio signal. To do this,
existing tools have employed several techniques:
### 5.1 Use of RNN
The initial speech separation models such as [85] [106] [176] relied on a
feedforward DNN to estimate clean speech from a noisy one. However,
feedforward DNN models are ill poised for speech data since they are unable to
effectively model long dependencies across time that are present in the speech
data. Due to this, researchers progressively introduced recurrent neural
networks (RNN) which have a feedback structure such that the representations
at given time step $t$ is a function of the data at time $t$, the hidden state
and memory at time $t-1$. One such RNN that has been exploited in speech
separation is long-short-term memory (LSTM) [177]. LSTM has memory blocks that
are composed of a memory cell to remember the temporary state and several
gates to control the information and gradient flow. LSTM structures can be
used to model sequential prediction networks which can exploit long-term
contextual information [177]. Works in [103] [120][178] exploit LSTM to
perform speech separation while [115] uses bidirectional long short-term
memory (BLSTM) networks to make use of contextual information from both sides
in the sequence. Due to their inherently sequential nature, RNN models are
unable to support parallelization of computation. This limits their use when
working with large datasets with long sequences due to slow training [11].
Moreover, in speech separation, a typical frame(input features) is usually
25ms which corresponds to 400 samples at a 16kHz sampling rate, for LSTM to
work directly on the raw waveform, it would require unrolling the LSTM for an
unrealistic large number of time steps to cover an audio of modest length
[179]. Other models that use different versions of RNN include [180]. Models
such as [181] use the gated recurrent unit (GRU)[182] to perform speech
denoising.
### 5.2 Use of temporal convolution network
Conventional convolution neural networks(CNN)have been used to design speech
separation models[94] [183]. However, CNNs are limited in their ability to
model long-range dependencies due to limited receptive fields [184]. They are
therefore mainly tailored to learn local features. They exploit local window
which maintain translation equivariance to learn a shared position-based
kernel[185]. For CNN to capture long range dependencies ( i.e., to enlarge the
receptive field), there is a need to stack many layers. This increases
computation cost due to the large number of parameters. These shortcomings of
the CNN and RNN, have motivated the use of dilated temporal convolution
network (TCN) in speech separation to encode long-range dependencies using
hierarchical convolutional layers [186] [187] [188] [17] [189]. TCN is
composed of two key distinguishing characteristics: the convolution in the
model must be causal i.e., a given activation of a certain layer $l$ at time
$t$ is only influenced by activations of the previous layer $l-1$ from time
steps that are less that $t$, 2) the model takes the sequence of any length
and maps it into an output sequence of the same length. To achieve the second
characteristic, TCN models are implemented using a 1-dimensional convolutional
network such that each hidden layer is the same length as the input layer. To
ensure same length, a zero padding of length $filtersize-1$ is added to keep
subsequent layers the same length as previous ones [190] (see figure 14).The
first property is achieved through the use of causal convolutions i.e. where
an output at time $t$ is convolved only with elements from time t and earlier
in the previous layer. To increase the receptive fields, models implement
dilated TCN. Dilated convolution is where the filter is applied to a region
larger than its size [191]. This is achieved by skipping input with certain
specified steps (see figure 10). More formally, for 1D sequence such as speech
signal, the input $x\in R^{n}$ and the kernel $f:\\{0,\cdots,k-1\\}\rightarrow
R$, the dilated convolution operation $F$ on an element $s$ of a given
sequence is defined according to equation 20 [190].
$F(s)=(x\ast_{d}f)(s)=\sum_{i=0}^{k-1}f(i)x_{s-di}$ (46)
where $x$ is the $1D$ input signal, $k$ is the kernel and $d$ is the dilation
factor. The effect of this is to expand the receptive field without loss of
resolution and drastically increase the number of parameters. Stacked dilated
convolution expands the receptive field with only a few layers. The expanded
receptive field allows the network to capture temporal dependence of various
resolutions with the input sequences [152]. In effect, TCN introduces the idea
of time-hierarchy where the upper layers of the network model longer input
sequences on larger timescales while local information are modelled by lower
layers and are mainly maintained in the network through residuals and skip
connections [152]. TCN also uses causal convolution where a given output at
layer $l$ in time step $t$ is computed only based on time steps up to time
step $t-1$ in the previous layer. The dilated TCN is exploited by [10] to
model sequences that exist within the input speech signal. They implement TCN
such that each layer is composed of 1-D dilated convolution blocks. The layers
have 1-D CNN blocks with increasing dilation factors. This is to uncover long
range dependencies that exist in the audio input. The dilation factors
increase exponentially over the layers in order to cover a large temporal
context window to exploit the long-range dependencies that exist within a
speech signal.
$y(m,n)=\mathop{\sum_{i=1}^{M}\sum_{j=1}^{N}}x(m+r\times i,n+r\times j)w(i,j)$
(47)
Here, $y(m,n)$ is the output of a given layer of dilated convolution, $x(m,n)$
is the input and $w(i,j)$ is the filter with the length and the width of $M$
and $N$ respectively. The parameter $r$ is the dilation rate. Note that if
$r=1$, the dilated convolution becomes the normal convolution convolution.
Figure 14: Dilated TCN with four layers.
.
### 5.3 Use of transformers
A transformer[192] is an attention-based deep learning technique that has been
successful in modelling sequences and allows uncovering of dependencies that
exist within an input without regard to the distance between any two values of
the input. Transformers consist only of feed-forward layers which allows them
to exploit the parallel processing capabilities of GPUs leading to fast
training [192]. In speech separation, [11] introduces a speech separation
system that fully relies on transformers to model the dependencies that exist
in the mixed audio signal. This is used to extract a mask for each of the
speakers in the audio mixture. The transformer is used to uncover both the
short-term dependencies (within a frame) and long-term dependencies (between
frames). Work in [184] also exploits transformers in the encoder to model the
dependencies that exist in the mixed audio while [67] uses transformers to
perform speech dereverberation. Despite their ability to model long-range
dependencies and ability to work well with parallelization, the attention
mechanism of transformers, has $O(N^{2})$ complexity that brings a major
memory bottleneck [193]. For a sequence of length $N$, the transformer needs
to compare $N^{2}$ elements which results in a computational bottleneck
especially for long signals such as speech. Transformers also use many
parameters aggravating the memory problem further. Several versions of
transformers such as Longformer[194], LinFormer[195] and Reformer[196] have
been proposed with a goal to reduce the computation complexity of the
transformers. Work in [197] investigates the performance of the three versions
of transformers in speech separation and concludes that they are suitable for
speech separation applications since they achieve a highly favourable trade-
off between performance and computational requirements. Work in [198] proposes
a technique of parameter sharing to reduce the computation complexity of the
transformer while [193] reduces complexity by avoiding frame overlap. In
[199], a teacher-student speech separation model based on transformer is
proposed. The student model which is much smaller than the teacher model is
used to reduce computation complexity. Other transformer-based speech
enhancement tools include [173] [200]. Another key limitation of a transformer
is that while it can model long-range global context, they do not extract
fine-grained local features patterns well. Based on this, transformer-based
speech separation tools apply attention within a frame (chunk) to capture
local features and between frames(chunks) to capture global features [11]
[201].
## 6 Model size reduction techniques
To achieve high performance i.e., generate speech with high intelligibility,
DNN models for speech enhancements are becoming large by exploiting millions
of parameters [202]. High number of parameters increase the memory
requirements, computation complexity and latency. To reduce these parameters
significantly without compromising quality and make speech enhancement tools
to work in resource constrained platform, several techniques are being
exploited. The techniques include:
Use of dilated convolution: To increase the receptive field of 1D CNN and
subsequently increase the temporal window and model long range dependencies
within a speech, speech separation such as [132] and [186] implement dilated
CNN. Dilated convolution initially introduced by [167] involves a convolution
where a kernel is applied to an area that is larger that it. This is achieved
by skipping input values by a defined step. It is like implementing a sparse
kernel ( i.e., dilating the kernel with zeros). When dilated convolution is
applied in a stacked network, it enables the network to increase its receptive
field with few layers hence minimizing parameters and reducing computation
[188]( see figure 14). This ensures that the models can capture long range
dependencies while keeping the number of parameters at minimum. The dilating
factors are made to increase exponentially per layer( see figure 10).
Parameter quantization: To reduce computation, inference complexity of DNN
models and to scale down the number of parameters, models such as [203] [204]
[205] [206] [207] use parameter quantization. In quantization, the objective
is to reduce the precision of model parameters and activation values to a low
precision with minimal effects on the generalization capability of the DNN
model. To achieve this, a quantization operator $Q$ is defined that maps a
floating value to a quantized one [208].
Use of depthwise separable convolution: This type of convolution, decouples
the convolution process into two i.e. depthwise convolution where a single
filter is applied to each input channel and pointwise convolution which is
applied to the output of depthwise convolution to achieve a linear convolution
of the depthwise layer. Depthwise separable convolution has been shown to
reduce the number of parameters as compared to the convectional one [209]
[210]. Speech enhancement tools that exploit depthwise separable convolution
include [10] [26] [67].
Knowledge distillation: Knowledge distillation involves training a large
teacher model which can easily extract the structure of data then the
knowledge learned by the teacher is distilled down to a smaller model called
the student. Here, the student is trained under the supervision of the
teacher[211] [212] [213]. The student model must mimic the teacher and by
doing so achieve superior or similar performance but at reduced computation
cost due to reduced parameters. Knowledge distillation technique has been
exploited to reduce latency in speech enhancement tool [199] [214].
Parameter pruning: In order to reduce the number of parameters and hence speed
up computation, some speech enhancement tools use parameter pruning [206]
[215]. Pruning involves converting a dense DNN model into a sparse one by
significantly scaling down the number of parameters without compromising
model’s output’s quality. In [216] they train a speech enhancement DNN model
to obtain an initial parameter set $\Theta$, they then prune the parameters by
dropping the weights whose absolute values are below a set pruning threshold.
The sparse network is again re-trained to obtain final parameters. Work in
[203] estimates the sparsity $S(k)$ of a given channel $F_{jk}$. If the
sparsity $S(k)>\theta$ where $\theta$ is a predefined threshold, the weights
within $F_{jk}$ is set to zero and the model is retrained. After several
iterations, the channel $F_{jk}$ is dropped.
Weight sharing: This involves identifying clusters of weights that have a
common value. The clusters are normally identified using K-means algorithm. So
instead of storing each weight value, only the indexes of the shared values
are stored. Through this memory requirements of the model is reduced [217].
Speech enhancement tools that use weight sharing include [204], [218].
## 7 Objective functions for speech enhancement and separation
Most DNN monaural speech enhancement and separation models especially those
working on features in the frequency domain exploit mean-square-error (MSE) as
the training objective [21] [38] [136] [151]. The DNN models that have the
mask as the target use the training objective to minimise the MSE between the
estimated mask and the ideal mask target. For models that predict estimated
features (such as T-F spectrogram) of the clean source speech, MSE is used to
minimise the difference between target features and the estimated features by
the model. Despite the dominance of MSE as an objective function in the speech
enhancement tools, it has been criticised since it is not closely related to
human auditory perception [93]. Its major weakness is that it treats
estimation elements independently and equally. For instance, it treats each
time-frequency unit separately rather than whole spectral [106]. This leads to
muffled sound quality and compromises intelligibility [106]. MSE also treats
every estimation element with equal importance which is not the case [219]. It
also does not discriminate between the positive or negative differences
between the clean and estimated spectra. A positive difference between the
clean and estimated spectra represents attenuation distortion, while a
negative spectral difference represents amplification distortion. MSE treats
the effects of these two distortions on speech intelligibility as equivalent
which is problematic [220] [221]. Moreover, the MSE is usually defined in the
linear frequency scale while the human auditory perception is on the Mel-
frequency scale. To avoid the problem of treating every estimation element
with equal importance [222], [223] propose a weighted MSE. Due to the
shortcomings of MSE, objective functions that are closely related to the human
auditory perception have been introduced to train the DNN [219] [224] [225]
[226] [120]. Some of the human auditory perception training objectives being
used by speech enhancement tools are also used as metrics for perceptual
evaluation. They include:
1. 1.
Short-time objective intelligibility (STOI)[227].
2. 2.
Scale invariant signal-to-distortion ratio(SI-SDR)[228].
3. 3.
Perceptual metric for speech quality evaluation(PMSQE).
Scale invariant signal-to-distortion ratio Work in [228] proposes an
intelligibility measure such that given the target signal $s$ and the model
estimated signal $\hat{s}$, they re-scale either $s$ or $\hat{s}$ such that
the residual $(s-\beta\hat{s})$ after scaling $\hat{s}$ or $(\alpha
s-\hat{s})$ after scaling the target $s$ is orthogonal to the target as:
$(s-\beta\hat{s}).s=0$ or $(\hat{s}-\alpha s).s=0$ based on this,$\alpha$ can
be computed as:
$(\hat{s}-\alpha s).s=0$ $\hat{s}.s-\alpha s.s=0\\\ $
$\alpha=\frac{\hat{s}^{T}s}{s.s}$
based on scaling of the target $s$, the signal-to-noise ratio(SNR) equation
$SDR=10\log_{10}\frac{||s||^{2}}{||s-\hat{s}||^{2}}$ (48)
is transformed to:
$SDR=10\log_{10}\frac{||\alpha s||^{2}}{||s-\hat{\alpha s}||^{2}}$ (49)
replacing the $\alpha$ we get the SI-SDR:
$SI-
SDR=10\log_{10}\frac{||\frac{\hat{s}^{T}s}{||s^{2}||}s||^{2}}{||\frac{\hat{s}^{T}s}{||s^{2}||}s-\hat{s}||^{2}}$
(50)
This objective function has been used in [11] [10] [175] [30][193] [229],
[230] [26] [28] [202].
Short-time objective intelligibility: This objective has been used in [225]
[226] [219] [93]. STOI [227][231] is a speech intelligibility measure that is
achieved by executing the following steps:
1. 1.
Given discrete time signals of clean speech signal $x(n)$ and enhanced speech
$y(n)$, perform a DFT on both $x(n)$ and $y(n)$ i.e $X(n,k)=DFT(y(n))$ and
$Y(n,k)=DFT(y(n))$. Here, $k$ refers to the index of the discrete frequency.
2. 2.
Remove silences in both the clean signal and the enhanced signals. Silences
are removed by first identifying the frame with maximum energy($max_{energy}$)
in the clean signal. All frames with energy of 40 dB less than $max_{energy}$
are dropped.
3. 3.
Reconstruct both the clean and enhanced speech signals.
4. 4.
Perform a one-third band octave analysis on both clean and enhanced speech by
grouping DFT bins i.e the complex-valued STFT coefficients, $X(n,k)$, are
combined into J third-octave bands by computing the TF units.
$X_{j}(m)=\sqrt{\sum_{k=k_{1}(j)}^{k_{2}(j)-1}|X(n,k)|^{2}}j=1,\cdots,J$ (51)
Here, $k_{1}$ and $k_{2}$ represent the one-third octave band edges. The same
octave analysis is performed on the enhanced speech. The one-third octave of
the enhanced speech is defined in a similar manner.
5. 5.
Define a short temporal envelope of both enhanced and clean speech as:
$Y_{j,m}=[Y_{j}(m-N+1),Y_{j}(m-N+2),\cdots,X_{j}(m)]^{T}$ and
$X_{j,m}=[X_{j}(m-N+1),X_{j}(m-N+2),\cdots,X_{j}(m)]^{T}$ respectively. STOI
exploits correlation coefficients to compare the temporal envelopes of clean
and enhanced speech for a short time region. Note that N=30.
6. 6.
Normalise the short temporal envelopes of the enhanced speech. Let
$y_{j,m}(n)$ denote the $n^{th}$ envelope of enhanced speech. The normalised
enhanced speech $y^{\prime}_{j,m}(n)$ of $Y_{j,m}(n)$ is given by
$y^{\prime}_{j,m}(n)=\frac{|X_{j,m}|}{||Y_{j,m}||}Y_{j,m}(n)$. $||.||$ is the
$l_{2}$ norm. The intuition behind normalisation of the enhanced speech is to
reduce global level differences between clean and enhanced speech. These
global level differences should not have a strong effect on speech
intelligibility.
7. 7.
Clip the normalised enhanced speech as
$\bar{y}_{j,m}(n)=\min(y^{\prime}_{j,m},(1+10^{\frac{\beta}{20}})x_{j,m}(n))$.
Clipping is done to ensure the effects of severely degraded frames of the
enhanced speech on the model is upper bounded. Here, $\beta=-15dB$ is the
lower signal-to-distortion(SDR) bound.
8. 8.
Compute intermediate intelligibility measure as
$d_{j,m}=\frac{(x_{j,m}-\mu_{x_{j,m}})^{T}(y_{j,m}-\mu_{y_{j,m}})}{||x_{j,m}-\mu_{x_{j,m}}||y_{j,m}-\mu_{y_{j,m}}||}$
(52)
Here,$\mu_{(.)}$ refers to the sample mean of the corresponding vector.
9. 9.
Compute the average intermediate intelligibility of all frames as
$d=\frac{1}{JM}\sum_{j,m}d_{j,m}$ (53)
where $M$ represents the total number of frames and $J$ the number of one-
third octave band.
Short-time spectral amplitude mean square error. Let $X[n,k]$ with $1\leq
n\leq N$ and $1\leq k\leq K$ be an $N$ point DFT of $x$ and $K$ is the number
of frames.Let $A[n,k]=X[n,k]$ with $k,\cdots,\frac{N}{2}+1$ and $k=1,\cdots,K$
denote the single sided amplitude spectra of $X[n,k]$. Let $\hat{A}[n,k]$ be
an estimate of $A[n,k]$, the short-time spectral amplitude mean square
error(STSA-MSE) is given by
$\mathcal{L}_{STSA-
MSE}=\frac{1}{(N/2+1)K}\sum_{n=1}^{N=K/2+1}\sum_{k=1}^{k=K}(\hat{A}[n,k]-A[n,k])^{2}$
(54)
Equation 36 represents the mean square error between single-sided amplitude
spectra of the clean speech $x$ and the DNN estimated speech. Equation 36 is
not sensitive to the phase spectrum of the two signals $\hat{x}$
Perceptual metric for speech quality evaluation(PMSQE). This is an objective
function that is based on the adaptation of perceptual evaluation of speech
quality (PESQ) algorithm [232]. Given the MSE loss in the log-power spectrum
with mean and variance normalisation i.e.
$MSE_{t}=\frac{1}{k}\sum_{n=1}^{K}(\frac{\log|x[n,k]|^{2}-\mu_{k}}{\delta_{k}}-\frac{\log|\hat{X}[n,k]|^{2}-\mu_{k}}{\delta_{k}})^{2}=\frac{1}{K}\sum_{k=1}^{k=K}\frac{1}{\delta_{k}^{2}}(\log\frac{|X[n,k]|^{2}}{|\hat{X}[n,k]|^{2}})^{2}$
(55)
Here, $X[n,k]|^{2}$ and $\hat{X}[n,k]|^{2}$ represent the power spectra of the
clean and enhanced speech respectively. $\mu_{k}$ is the mean log-power
spectrum and $\delta_{k}$ is its standard deviation. The indices n and k
represent the frame and frequency, while $K$ is the number of frequency bins.
From equation 55, the MSE is entirely dependent on power spectra across
frequency bands hence not factoring in the perceptual factors such as loudness
difference, masking and threshold effects[233]. To factor in the perceptual
factors in the MSE, PMSQE modifies the MSE loss by incorporating two
disturbance terms (symmetrical disturbance and asymmetrical disturbance) which
are based on the PESQ algorithm both computed in a frame-by-frame basis[233].
$MSE_{t}=\sum_{t}{}MSE_{t}+\alpha D_{t}^{s}+\beta D_{t}^{a}$ (56)
Here $D_{t}^{s}$ and $D_{t}^{a}$ represent symmetrical and asymmetrical
disturbances respectively. The parameters $\alpha$ and $\beta$ are weighting
factors which are determined experimentally. Work in [233] describes how to
arrive at the values of $D_{t}^{s}$ and $D_{t}^{a}$. Since PESQ is non-
differentiable, the PMSQE objective function provides a way of estimating it.
PMSQE objective function is designed to be inversely proportional to PESQ,
such that a low PMSQE value corresponds to a high PESQ value and vice versa.
The key question here is: Which objective function is superior? work in [234]
tries to answer this question where they evaluate six objective functions.
Their conclusion is that the evaluation metric should be a major factor in
deciding on the objective function to use in the speech enhancement model. In
case a given model targets to improve a specific evaluation metric, then
selection of an objective function related to that metric is advantageous.
## 8 Unsupervised techniques for speech enhancement
Although supervised techniques of speech enhancement and separation have
achieved great success towards improving speech intelligibility, the inherent
problems associated with supervised learning still prohibits their
applications in all scenarios. First, collecting parallel data of clean and
noisy (mixed) data remains costly and time consuming. This limits the amount
of data that can be used to train these models. Consequently, the models are
not exposed to enough variations of the recording during training hence
affecting their generalizability to noise types and acoustic conditions that
were not seen during training [235] [236]. Collecting clean audio is always
difficult and requires a well-controlled studio exacerbating the already high
cost of data annotation [236]. Unsupervised learning offers an alternative to
solving these problems. The existing unsupervised techniques for speech
enhancement and separation can roughly be categorised into three: MixIT based
techniques, generative modelling technique and teacher-student based
techniques. Few novel techniques have also been proposed that fall outside
these three dominant categories. Work in [237] proposes mixture invariant
training (MixIT) to perform unsupervised speech separation. Given a set of $X$
that is composed of mixed speech i.e. $X=\\{x_{1},x_{2},\cdots,x_{n}\\}$ where
each mixture $x_{i}$ is composed of up to $N$ sources, mixtures are drawn at
random from the set $X$ without replacement and a mixture of mixture (MoM)
created by adding the drawn mixtures, for example if two mixtures $x_{1}$ and
$x_{2}$ are drawn from the set $X$, $MoM$ $\bar{x}$ is created by adding
$x_{1}$ and $x_{2}$ i.e $\bar{x}=x_{1}+x_{2}$. The MoM $\bar{x}$ is the input
to a DNN model which is trained to estimate sources $\hat{s}$ composed in
$x_{1}$ and $x_{2}$. The DNN model is trained to minimize the loss function in
equation 57.
$L_{MixIT}=\min_{A}\sum_{i=1}^{2}L(x_{i},[A\hat{s}]_{i})$ (57)
For a case where MoM is composed of only two mixtures, $A\in B^{2\times M}$ is
a set of binary matrices where each column sums to 1. The loss function is
trained to minimize the loss between mixtures $x_{i}$ and the remixed
separated sources $A\hat{s}$. MixIT has been criticised for over-separation
where it outputs estimated sources greater than the actual number of
underlying sources in the mixtures $x_{i}$ [238]. Further, MixIT does not work
well for speech enhancement ( i.e., denoising) [239]. MixIT teacher-student
unsupervised model has been proposed in [238] to tackle the problem of over-
separation in MixIT. It trains a student model such that its output matches
the number of sources in the mixed speech $x$. Another MixIT based technique
for solving over-separation problem is discussed in MixCycle [240]. Work in
[241] proposes to improve MixIT to make it more tailored for denoising by
exploiting an ASR pre-trained model to modify MixIT’s loss function. Work in
[239] also seeks to improve MixIT for denoising by improving loss function and
noise augmentation scheme. RemixIT [242] is an unsupervised speech denoising
tool that exploits teacher-student DNN model. Given a batch of noisy speeches
of size $b$, the teacher estimates the clean speech sources $\hat{s}_{i}$ and
noises $\hat{n}_{i}$ where $1\leq i\leq b$. The teacher estimated noises
$\hat{n}_{i}$ are mixed at random to generate $n^{p}$. The mixed noise $n^{p}$
together with the teacher estimated sources are used to generate new mixtures
$\hat{m}_{i}=\hat{s}_{i}+n^{p}$. The new mixtures $\hat{m}_{i}$ are used as
input to the student. The student is optimised to generate $\hat{s}_{i}$ and
noise $n^{p}$ i.e., $\hat{s}_{i}$ and $n^{p}$ are the targets. Through this
the teacher-student model is trained to denoise the speech. In RemixIT, a pre-
trained speech enhancement model is used as the teacher model. Motivated by
RemixIT, [243] also proposes a speech denoising unsupervised tool using
teacher-student DNN model. They propose various techniques of student
training. MetricGAN-U [76] is an unsupervised GAN based speech enhancement
tool that trains a conditioned GAN discriminator without a reference clean
speech, MetricGAN-U employs objective in equation 58 to train the speech
enhancement model.
$L=\mathbb{E}_{x}[(D(G(x))-Q\prime(G(x)))^{2}+(D(x)-Q\prime(G(x))^{2}]$ (58)
In equation 58, $Q\prime$ is a non-intrusive metric (i.e does not require
reference clean speech) that is used to score the enhanced speech from the
generator. The scores obtained by $Q\prime$ are used to optimize the model. In
MetricGAN-U, DNSMOS [244] is used as $Q\prime$. Another GAN based technique
for unsupervised learning is [245] which exploits CycleGAN [246] multi-
objective learning to perform parallel-data-free speech enhancement. Tools in
[235] and [247] propose unsupervised speech denoising technique based on
variations of VAE. Work in [236] proposes a speech denoising technique that
uses only the noisy speech. It exploits the idea that was first proposed in
[248] where they demonstrated that it is possible to recover signals under
corruptions without observing clean signals. Predicating their work on these
findings, given a noisy speech signal $x$, and noise $n$, [236] creates a more
noisy speech $y=x+n$. They then train a DNN model to predict an enhanced
speech $\hat{s}$ by having the more noisy input $y$ as the input and noisy
speech $x$ as the target. Consequently, the DNN is trained by minimizing the
loss in equation 59.
$L=\frac{1}{M}\sum_{i=1}^{M}D(\hat{s}_{m},x_{m})$ (59)
Here, $D$ is the objective function and $M$ is the sample size. This technique
works on the basis that DNN cannot predict random noise hence the noise
component in the training data is mapped to their expected values. Therefore,
by assuming the noise as zero mean random variable, the objective function
eliminates the noise [236]. Work in [16] proposes unsupervised techniques to
perform speech separation based on gender. They exploit i-vectors to model
large discrepancy in vocal tract, fundamental frequency contour, timing,
rhythm, dynamic range, etc between speakers of different genders. In this case
DNN model can be viewed as gender separator.
## 9 Domain adaptive techniques for speech enhancement and Separation
Training data used to train speech enhancement and separation tools mostly
have acoustic features that are significantly different from the acoustic
features of the speech signals where the tools are deployed. This mismatch
between the training data and target data leads to degradation in the tool’s
performance in their deployed environment [249].The target environment
dataset’s acoustic features may vary from the training data in noise type,
speaker and signal-noise-ratio [247]. One potential way of tackling this
problem is to collect massive training data that covers different variation of
deployment environment. However, this is mostly not possible due to
prohibitive cost. Due to this, some tools are proposing DNN based techniques
for domain adaptation. Domain adaptation seeks to exploit either labelled or
unlabelled target domain data to transfer a given tool from training data
domain to the target data domain. Basically, domain adaptation seeks to reduce
the covariance shift between the source and target domains data. The domain
adaptation techniques in literature for speech separation and enhancement
tools can be categorised into two: Unsupervised domain adaptation techniques
such as [250] [251] which use unlabelled target domain dataset to adapt a DNN
model and supervised domain adaptation techniques such as [252] [247] [253]
which exploit limited labelled target domain dataset to perform domain
adaptation of a DNN model for speech enhancement or separation. To make speech
enhancement tools portable to new a new language, [252] proposes to use
transfer learning. Transfer learning entails tailoring trained DNN models to
apply knowledge acquired during training to a new domain where there is some
commonality in type of task. The tool fine-tunes the top layers of a trained
DNN model for speech enhancement by using labelled data of a new language
while freezing the lower layer which are composed of parameters acquired
during training of the original language. Work in [253] also uses transfer
learning to show that pre-trained SEGAN can achieve high performance in new
languages with unseen speakers and noise with just short training time. To
make it more adaptable to different types of noise, tool in [247] proposes to
employ multiple encoders where each encoder is trained in a supervised manner
to focus only on given acoustic feature. The features are categorized into two
i.e., utterance-level features such as gender, age, ascent of the speaker,
signal-to-noise ratio and noise type and the signal-level features such high
and low frequency of the speech parts. Feature focused encoders are trained to
learn how to extract a given feature representation such as gender
representation composed in the speech. Through the feature focused encoders,
the experimental results show that the tool can adapt more to unseen noise
types as compared to using a single global encoder. To adapt the DNN speech
enhancement model to unseen noise type, work in [250] utilizes domain
adversarial training (DAT)[254] to train an encoder to extract noise-invariant
features. To do this, it utilizes the labelled source data and unlabelled
target data. Through the feedback from the discriminator which gives the
probability distribution over multiple noise types, the encoder is trained to
produce noise-invariant features, hence reducing the mismatch problem. Work in
[251] also exploits unsupervised DAT for speaker mismatch resolution. Work in
[247] exploits importance-weighting (IW) using the classifiers of the networks
to classify the source domain samples from the outlier weights and hence
reduces the shift between the source and target domain.
## 10 Use of pre-trained models in speech separation and enhancement
Pre-trained models have become popular especially in Natural language
processing(NLP) and Computer vision. In NLP, for example, large corpus of text
can be used to learn universal language representations which are beneficial
for downstream NLP tasks. Due to their success in domains such as NLP and
computer vision, pre-trained models based on unsupervised learning have been
introduced in audio data [255] [256] [257] [258][259][260]. Such pre-trained
models are beneficial in several ways:
1. 1.
Pre-trained models are trained in large speech dataset hence can learn
universal speech representations which can be beneficial to speech separation
by boosting the quality of enhanced speech generated by these models.
2. 2.
Pre-trained models provide models with better initialization which can result
in better generalization and speed up convergence during training of speech
enhancement models.
3. 3.
Pre-trained speech models can act as regularizers to help speech enhancements
models to avoid over fitting.
Work in [261] seeks to establish if pre-trained speech models will help
generate more robust features for downstream speech denoising and separation
task as compared to features established without pre-trained models. To do
this they use 13 speech pre-trained models to generate features of a noisy
speech which are then passed through a three-layer BLSTM network which
generates speech denoising or separation mask. They compare the performance of
these features with those of baseline STFT and mel filerbank (FBANK) features.
Their experiments establish that the 13 pre-trained models used do not
significantly improve feature representations as compared to those of
baselines. Hence the quality of enhanced and separated speech generated by
features of pre-trained models are only slightly better or worse in some cases
as compared to those generated based on the baseline features. They attribute
this to domain mismatch and information loss. Since most of the pre-trained
models were trained with clean speech, they are not portable to a noisy speech
domain. Pre-trained models are usually trained to model global features and
long-term dependencies hence some local features of the noisy or mixed speech
signal may be lost due to this during feature extraction. Using HuBERT Large
model [262], they demonstrate that the last layer of the model does not
produce the best feature representation for speech enhancement and separation.
In fact, for speech separation, the higher layers features are of low quality
as compared to lower layers. They show that the weighted-sum representations
of the representations from the different layers of pre-trained models where
lower layers are given more weight generate better speech the enhancement and
separation results as compared to isolated layers representations. They
hypothesise that this could be due loss of some local signal information
necessary for speech reconstruction tasks in deeper layers. To address the
problem of information loss in pre-trained model, [263] proposes two
solutions, first they utilize cross-domain features as model inputs to
compensate lost information and secondly, they fine-tune a pre-trained model
by using a speech enhancement model such that the extracted features are more
tailored towards speech enhancement. Research in [264] seeks to synthesise
clean speech directly from a noisy one using pre-trained model and HiFiGAN
[265] speech synthesis model. It exploits the pre-trained model to extract
features of the noisy model. The features are then used as input of HiFiGAN
which generates estimated clean speech from these features. Based on the
results reported in [261] that demonstrated that the final layer of pre-
trained model does not give optimized representation for speech enhancement,
they exploit weighted average of the representations of all the layers of the
pre-trained model to generate representations of the noisy speech. The novelty
of this work is that they do not use a model dedicated for speech denoising
rather show that given features of a noisy speech, speech synthesis model can
perform denoising. In [266], a pre-trained model has been exploited to design
the loss function. Given a pre-trained model $\Phi$ with $m$ layers, the tool
uses weighted $L^{1}$ loss to compute the difference between the feature
activations of clean and noisy speech generated by different layers of the
pre-trained model according to equation 60.
$L=\sum_{i=1}^{m}\lambda_{w}||\Phi_{m}(s)-\Phi_{m}(g_{\theta}(x))||$ (60)
Here, $s$ and $x$ are the clean and noisy speech respectively, $g_{\theta}$ is
the denoising DNN model and $\lambda_{m}$ are the weights of contribution of
each pre-trained model layer features to the loss function. Work in [267]
proposes a two-stage speech enhancement framework where they first pre-train a
model using unpaired noisy and clean data and utilize the pre-train model to
perform speech enhancement. Unlike the previous works that use general pre-
trained models for audio, the pre-trained model in [267] is trained using
speech enhancement dataset. They report state of the art results in speech
enhancement and ability of the tool to generalize to unseen noise.
## 11 Future direction for research in speech enhancement
Unsupervised techniques for speech separation: Majority of speech enhancements
tools use supervised learning technique. For those that use unsupervised
learning discussed in section 8, they almost entirely focus on speech
denoising and not speaker separation and dereverberation. There is therefore a
gap of extending the unsupervised DNN techniques to perform multi-speaker
speech separation and dereverberation.
Dimension mismatch problem: Most speech separation tools set a fixed number
$C$ of speakers therefore cannot deal with an inference mixture with $K$
sources, where $C\neq K$. Currents tools deal with the dimension mismatch
problem by either outputting silences if $C>K$ or performing speech separation
through iteration. However, both techniques have been found to be inefficient
(see discussion in section 2.1). Therefore, there need to explore on
developing DNN techniques for speech separation which are dynamic to the
number of speakers present in inference mixture and adapt appropriately.
Focus on data: Most model compression techniques for speech enhancement speed
up the model performance by reducing or optimizing the model parameters. They
don’t focus on the data-side impact on the model performance. For instance,
what is the ideal sequence length and chunk overlap when working in time-
domain that can speed up the speech enhancement process without compromising
the quality of enhancement. More focus needs to turn towards exploring the
data modifications that can speech up the speech enhancement process.
Dataset modification: In dereverberation, tools are beginning to explore the
use of speech with early reverberation [75] as the target as opposed to using
anechoic target. Experiments in [75] demonstrate that allowing early
reverberation in the target speech improves the quality of enhanced speech.
There is need to develop a standardized dataset where the target is composed
of early reverberation to allow for standardized evaluation of the tools on
this dataset.
Pre-trained Model: The pre-trained models that have been utilized for speech
enhancement or separation have been trained on clean dataset hence failing
potability test when used to generate features of a noisy speech signal [261].
There is need for development of a pre-trained model tailored for speech
separation and enhancement.
## 12 Conclusion
This review gives a discussion on how DNN techniques are being exploited by
monaural speech enhancement tools. The objective was to uncover the key trends
and dominant techniques being used by DNN tools at each stage of speech
enhancement process. The review therefore discusses the type of features being
exploited by these tools, the modelling of speech contextual information, how
the models are trained (both supervised and unsupervised), key objective
functions, how pre-trained speech models are being utilized and dominant
dataset for evaluating the performance of the speech enhancement tools. In
each section we highlight the standout challenges and how tools are dealing
with these challenges. Our target is to give an entry into the speech
enhancement domain by getting a thorough overview of the concepts and the
trends of the domain. The hope is that the review gives a snapshot of the
current research on DNN application to speech enhancement.
## References
* Bando et al. [2018] Y. Bando, M. Mimura, K. Itoyama, K. Yoshii, and T. Kawahara, “Statistical Speech Enhancement Based on Probabilistic Integration of Variational Autoencoder and Non-Negative Matrix Factorization,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, no. Mcmc, pp. 716–720, 2018.
* Xiao et al. [2019] X. Xiao, Z. Chen, T. Yoshioka, H. Erdogan, C. Liu, D. Dimitriadis, J. Droppo, and Y. Gong, “Single-channel speech extraction using speaker inventory and attention network,” in _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 86–90.
* Wang and Wang [2013] Y. Wang and D. L. Wang, “Towards scaling up classification-based speech separation,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 21, no. 7, pp. 1381–1390, 2013.
* Wang et al. [2014] Y. Wang, A. Narayanan, and D. L. Wang, “On training targets for supervised speech separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 22, no. 12, pp. 1849–1858, 2014.
* Schmidt and Olsson [2006] M. N. Schmidt and R. K. Olsson, “Single-channel speech separation using sparse non-negative matrix factorization,” _INTERSPEECH 2006 and 9th International Conference on Spoken Language Processing, INTERSPEECH 2006 - ICSLP_ , vol. 5, pp. 2614–2617, 2006.
* Wang and Sha [2014] Z. Wang and F. Sha, “Discriminative Non-Negative Matrix Factorization For Single-Channel Speech Separation,” _2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)_ , pp. 3777–3781, 2014. [Online]. Available: https://pdfs.semanticscholar.org/854a/454106bd42a8bca158426d8b12b48ba0cae8.pdf
* Virtanen and Cemgil [2009] T. Virtanen and A. T. Cemgil, “Mixtures of gamma priors for non-negative matrix factorization based speech separation,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 5441, no. 3, pp. 646–653, 2009.
* Virtanen [2006] T. Virtanen, “Speech recognition using factorial hidden Markov models for separation in the feature space,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 1, pp. 89–92, 2006.
* Shao and Wang [2006] Y. Shao and D. Wang, “Model-based sequential organization in cochannel speech,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 14, no. 1, pp. 289–298, 2006.
* Luo and Mesgarani [2019] Y. Luo and N. Mesgarani, “Conv-TasNet: Surpassing Ideal Time-Frequency Magnitude Masking for Speech Separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 27, no. 8, pp. 1256–1266, 2019.
* Subakan et al. [2021] C. Subakan, M. Ravanelli, S. Cornell, M. Bronzi, and J. Zhong, “Attention is all you need in speech separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2021-June, pp. 21–25, 2021.
* Wang and Chen [2018] D. Wang and J. Chen, “Supervised speech separation based on deep learning: An overview,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 26, no. 10, pp. 1702–1726, 2018.
* Gannot et al. [2017] S. Gannot, E. Vincent, S. Markovich-Golan, and A. Ozerov, “A Consolidated Perspective on Multimicrophone Speech Enhancement and Source Separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 25, no. 4, pp. 692–730, 2017.
* Hershey et al. [2016a] J. R. Hershey, Z. Chen, J. Le Roux, and S. Watanabe, “Deep clustering: Discriminative embeddings for segmentation and separation,” in _2016 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2016, pp. 31–35.
* Wang et al. [2017] Y. Wang, J. Du, L. R. Dai, and C. H. Lee, “A gender mixture detection approach to unsupervised single-channel speech separation based on deep neural networks,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 25, no. 7, pp. 1535–1546, 2017.
* Wang et al. [2016] Y. Wang, J. Du, L.-R. Dai, and C.-H. Lee, “Unsupervised single-channel speech separation via deep neural network for different gender mixtures,” in _2016 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA)_. IEEE, 2016, pp. 1–4.
* Zeghidour and Grangier [2021] N. Zeghidour and D. Grangier, “Wavesplit: End-to-End Speech Separation by Speaker Clustering,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 29, no. iv, pp. 2840–2849, 2021.
* Huang et al. [2011] P.-S. Huang, M. Kim, M. Hasegawa-Johnson, and P. Smaragdis, “Deep Learning For Monaural Speech Separation,” _Acta Physica Polonica B_ , vol. 42, no. 1, pp. 33–44, 2011.
* Weng et al. [2015] C. Weng, D. Yu, M. L. Seltzer, and J. Droppo, “Deep neural networks for single-channel multi-talker speech recognition,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 23, no. 10, pp. 1670–1679, 2015.
* Isik et al. [2016] Y. Isik, J. Le Roux, Z. Chen, S. Watanabe, and J. R. Hershey, “Single-channel multi-speaker separation using deep clustering,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-Sept, pp. 545–549, 2016.
* Kolbæk et al. [2017a] M. Kolbæk, D. Yu, Z.-H. Tan, and J. Jensen, “Multitalker speech separation with utterance-level permutation invariant training of deep recurrent neural networks,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 25, no. 10, pp. 1901–1913, 2017.
* Yu et al. [2017] D. Yu, M. Kolbaek, Z. H. Tan, and J. Jensen, “Permutation invariant training of deep models for speaker-independent multi-talker speech separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 241–245, 2017.
* Tachibana [2021] H. Tachibana, “Towards listening to 10 people simultaneously: An efficient permutation invariant training of audio source separation using Sinkhorn’s algorithm,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2021-June, pp. 491–495, 2021\.
* Dovrat et al. [2021] S. Dovrat, E. Nachmani, and L. Wolf, “Many-speakers single channel speech separation with optimal permutation training,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 4, pp. 2408–2412, 2021.
* Hershey et al. [2016b] J. R. Hershey, Z. Chen, J. L. Roux, and S. Watanabe, “Deep Clustering : Discriminative Embeddings For Segmentation And Separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 31–35, 2016.
* Byun and Shin [2021] J. Byun and J. W. Shin, “Monaural speech separation using speaker embedding from preliminary separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 29, pp. 2753–2763, 2021.
* Qin et al. [2020] S. Qin, T. Jiang, S. Wu, N. Wang, and X. Zhao, “Graph convolution-based deep clustering for speech separation,” _IEEE Access_ , vol. 8, pp. 82 571–82 580, 2020.
* Lee et al. [2022] J. H. Lee, J. H. Chang, J. M. Yang, and H. G. Moon, “NAS-TasNet: Neural Architecture Search for Time-Domain Speech Separation,” _IEEE Access_ , vol. 10, pp. 56 031–56 043, 2022.
* Jiang and Duan [2020] F. Jiang and Z. Duan, “Speaker attractor network: Generalizing speech separation to unseen numbers of sources,” _IEEE Signal Processing Letters_ , vol. 27, pp. 1859–1863, 2020.
* Nachmani et al. [2020] E. Nachmani, Y. Adi, and L. Wolf, “Voice separation with an unknown number of multiple speakers,” _37th International Conference on Machine Learning, ICML 2020_ , vol. PartF16814, pp. 7121–7132, 2020.
* Liu and Wang [2019] Y. Liu and D. Wang, “Divide and Conquer: A Deep CASA Approach to Talker-Independent Monaural Speaker Separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 27, no. 12, pp. 2092–2102, 2019.
* Luo and Mesgarani [2020] Y. Luo and N. Mesgarani, “Separating varying numbers of sources with auxiliary autoencoding loss,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, pp. 2622–2626, 2020.
* Shi et al. [2018] J. Shi, J. Xu, G. Liu, and B. Xu, “Listen, think and listen again: Capturing top-down auditory attention for speaker-independent speech separation,” _IJCAI International Joint Conference on Artificial Intelligence_ , vol. 2018-July, pp. 4353–4360, 2018.
* Kinoshita et al. [2018] K. Kinoshita, L. Drude, M. Delcroix, and T. Nakatani, “Listening to each speaker one by one with recurrent selective hearing networks,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, pp. 5064–5068, 2018.
* Takahashi et al. [2019] N. Takahashi, S. Parthasaarathy, N. Goswami, and Y. Mitsufuji, “Recursive speech separation for unknown number of speakers,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2019-September, pp. 1348–1352, 2019.
* Neumann et al. [2019] T. V. Neumann, K. Kinoshita, M. Delcroix, S. Araki, T. Nakatani, and R. Haeb-Umbach, “All-neural Online Source Separation, Counting, and Diarization for Meeting Analysis,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2019-May, pp. 91–95, 2019.
* von Neumann et al. [2020] T. von Neumann, C. Boeddeker, L. Drude, K. Kinoshita, M. Delcroix, T. Nakatani, and R. Haeb-Umbach, “Multi-talker ASR for an unknown number of sources: Joint training of source counting, separation and ASR,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, pp. 3097–3101, 2020.
* Luo et al. [2018] Y. Luo, Z. Chen, and N. Mesgarani, “Speaker-Independent Speech Separation with Deep Attractor Network,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 26, no. 4, pp. 787–796, 2018.
* Yul et al. [2017] D. Yul, M. Kalbcek, Z.-h. Tan, and J. Jensen, “SPEAKER-INDEPENDENT MULTI-TALKER SPEECH SEPARATION,” in _IEEE International Conference on Acoustics, Speech and Signal Processing_ , 2017, pp. 241–245.
* Chang et al. [2020a] X. Chang, W. Zhang, Y. Qian, J. Le Roux, and S. Watanabe, “End-to-end multi-speaker speech recognition with transformer,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 6134–6138.
* Chang et al. [2020b] X. Chang, W. Zhang, Y. Qian, J. L. Roux, and S. Watanabe, “End-To-End Multi-Speaker Speech Recognition with Transformer,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2020-May, pp. 6134–6138, 2020.
* Wang et al. [2018a] Z. Q. Wang, J. L. Roux, and J. R. Hershey, “Alternative objective functions for deep clustering,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, pp. 686–690, 2018.
* Veselỳ et al. [2016] K. Veselỳ, S. Watanabe, K. Žmolíková, M. Karafiát, L. Burget, and J. H. Černockỳ, “Sequence summarizing neural network for speaker adaptation,” in _2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2016, pp. 5315–5319.
* Wang et al. [2018b] J. Wang, J. Chen, D. Su, L. Chen, M. Yu, Y. Qian, and D. Yu, “Deep extractor network for target speaker recovery from single channel speech mixtures,” _arXiv preprint arXiv:1807.08974_ , 2018.
* Wang et al. [2018c] Q. Wang, H. Muckenhirn, K. Wilson, P. Sridhar, Z. Wu, J. Hershey, R. A. Saurous, R. J. Weiss, Y. Jia, and I. L. Moreno, “Voicefilter: Targeted voice separation by speaker-conditioned spectrogram masking,” _arXiv preprint arXiv:1810.04826_ , 2018.
* Ji et al. [2020] X. Ji, M. Yu, C. Zhang, D. Su, T. Yu, X. Liu, and D. Yu, “Speaker-aware target speaker enhancement by jointly learning with speaker embedding extraction,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 7294–7298.
* Zhang et al. [2021a] C. Zhang, M. Yu, C. Weng, and D. Yu, “Towards robust speaker verification with target speaker enhancement,” in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 6693–6697.
* Delcroix et al. [2018] M. Delcroix, K. Zmolikova, K. Kinoshita, A. Ogawa, and T. Nakatani, “Single channel target speaker extraction and recognition with speaker beam,” in _2018 IEEE international conference on acoustics, speech and signal processing (ICASSP)_. IEEE, 2018, pp. 5554–5558.
* Chen et al. [2023a] L. Chen, Z. Mo, J. Ren, C. Cui, and Q. Zhao, “An electroglottograph auxiliary neural network for target speaker extraction,” _Applied Sciences_ , vol. 13, no. 1, p. 469, 2023.
* Miao et al. [2015] Y. Miao, H. Zhang, and F. Metze, “Speaker adaptive training of deep neural network acoustic models using i-vectors,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 23, no. 11, pp. 1938–1949, 2015\.
* Senior and Lopez-Moreno [2014] A. Senior and I. Lopez-Moreno, “Improving dnn speaker independence with i-vector inputs,” in _2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2014, pp. 225–229.
* Ochiai et al. [2014] T. Ochiai, S. Matsuda, X. Lu, C. Hori, and S. Katagiri, “Speaker adaptive training using deep neural networks,” in _2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2014, pp. 6349–6353.
* Chen et al. [2017] Z. Chen, Y. Luo, and N. Mesgarani, “Deep attractor network for single-microphone speaker separation,” in _2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2017, pp. 246–250.
* Patrick A. Naylor [2010] N. D. G. Patrick A. Naylor, _Speech Dereverberation_ , N. D. G. Patrick A. Naylor, Ed. London: Springer-Verlag London, 2010, vol. 53, no. 1.
* Williamson and Wang [2017a] D. S. Williamson and D. Wang, “Time-frequency masking in the complex domain for speech dereverberation and denoising,” _IEEE/ACM transactions on audio, speech, and language processing_ , vol. 25, no. 7, pp. 1492–1501, 2017\.
* Arweiler and Buchholz [2011] I. Arweiler and J. M. Buchholz, “The influence of spectral characteristics of early reflections on speech intelligibility,” _The Journal of the Acoustical Society of America_ , vol. 130, no. 2, pp. 996–1005, 2011.
* Nábělek et al. [1989] A. K. Nábělek, T. R. Letowski, and F. M. Tucker, “Reverberant overlap- and self-masking in consonant identification,” _Journal of the Acoustical Society of America_ , vol. 86, no. 4, pp. 1259–1265, 1989.
* Oppenheim [1999] A. V. Oppenheim, _Discrete-Time Signal Processing_ , 2nd ed. upper Saddle River, NJ,USA:: Prentice-Hall, 1999\.
* Zhou et al. [2022] R. Zhou, W. Zhu, and X. Li, “Single-Channel Speech Dereverberation using Subband Network with A Reverberation Time Shortening Target,” _arXiv preprint arXiv:2210.11089_ , 2022. [Online]. Available: http://arxiv.org/abs/2204.08765
* Cord-Landwehr et al. [2021] T. Cord-Landwehr, C. Boeddeker, T. von Neumann, C. Zorila, R. Doddipatla, and R. Haeb-Umbach, “Monaural source separation: From anechoic to reverberant environments,” in _In 2022 International Workshop on Acoustic Signal Enhancement (IWAENC)_ , 2021, pp. 1–5. [Online]. Available: http://arxiv.org/abs/2111.07578
* Su et al. [2020] J. Su, Z. Jin, and A. Finkelstein, “HiFi-GAN: High-fidelity denoising and dereverberation based on speech deep features in adversarial networks,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, no. 3, pp. 4506–4510, 2020.
* Choi et al. [2020] H.-S. Choi, H. Heo, J. H. Lee, and K. Lee, “Phase-aware Single-stage Speech Denoising and Dereverberation with U-Net,” _arXiv preprint arXiv:2006.00687_ , 2020. [Online]. Available: http://arxiv.org/abs/2006.00687
* Wang et al. [2019] Z. Q. Wang, K. Tan, and D. Wang, “Deep Learning Based Phase Reconstruction for Speaker Separation: A Trigonometric Perspective,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2019-May, pp. 71–75, 2019.
* Han et al. [2015] K. Han, Y. Wang, D. L. Wang, W. S. Woods, I. Merks, and T. Zhang, “Learning Spectral Mapping for Speech Dereverberation and Denoising,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 23, no. 6, pp. 982–992, 2015.
* Jiang et al. [2014] Y. Jiang, D. L. Wang, R. S. Liu, and Z. M. Feng, “Binaural classification for reverberant speech segregation using deep neural networks,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 22, no. 12, pp. 2112–2121, 2014.
* Gamper and Tashev [2018] H. Gamper and I. J. Tashev, “Blind reverberation time estimation using a convolutional neural network,” _16th International Workshop on Acoustic Signal Enhancement, IWAENC 2018 - Proceedings_ , pp. 136–140, 2018.
* Zhao et al. [2020] Y. Zhao, D. Wang, B. Xu, and T. Zhang, “Monaural Speech Dereverberation Using Temporal Convolutional Networks with Self Attention,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 28, pp. 1598–1607, 2020.
* Ueda et al. [2016] Y. Ueda, L. Wang, A. Kai, X. Xiao, E. S. Chng, and H. Li, “Single-channel Dereverberation for Distant-Talking Speech Recognition by Combining Denoising Autoencoder and Temporal Structure Normalization,” _Journal of Signal Processing Systems_ , vol. 82, no. 2, pp. 151–161, 2016.
* Williamson and Wang [2017b] D. S. Williamson and D. Wang, “Speech Dereverberation And Denoising Using Complex Ratio Masks ,” _IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 2017_ , pp. 5590–5594, 2017.
* Jin and Wang [2009] Z. Jin and D. Wang, “A supervised learning approach to monaural segregation of reverberant speech,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 17, no. 4, pp. 625–638, 2009.
* León and Tobar [2021] D. León and F. Tobar, “Late reverberation suppression using U-nets,” _arXiv preprint arXiv:2110.02144._ , no. 1, 2021. [Online]. Available: http://arxiv.org/abs/2110.02144
* Défossez et al. [2020] A. Défossez, G. Synnaeve, and Y. Adi, “Real time speech enhancement in the waveform domain,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, pp. 3291–3295, 2020.
* Isik et al. [2020] U. Isik, R. Giri, N. Phansalkar, J. M. Valin, K. Helwani, and A. Krishnaswamy, “PoCoNet: Better speech enhancement with frequency-positional embeddings, semi-supervised conversational data, and biased loss,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, pp. 2487–2491, 2020.
* Li et al. [2021a] A. Li, W. Liu, X. Luo, G. Yu, C. Zheng, and X. Li, “A simultaneous denoising and dereverberation framework with target decoupling,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2, pp. 796–800, 2021.
* Valin et al. [2022] J.-M. Valin, R. Giri, S. Venkataramani, U. Isik, and A. Krishnaswamy, “To Dereverb Or Not to Dereverb? Perceptual Studies On Real-Time Dereverberation Targets,” _arXiv:2206.07917_ , 2022. [Online]. Available: http://arxiv.org/abs/2206.07917
* Fu et al. [2022] S.-W. Fu, C. Yu, K.-H. Hung, M. Ravanelli, and Y. Tsao, “Metricgan-u: Unsupervised speech enhancement/dereverberation based only on noisy/reverberated speech,” in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 7412–7416.
* Leglaive et al. [2019] S. Leglaive, U. Simsekli, A. Liutkus, L. Girin, and R. Horaud, “Speech Enhancement with Variational Autoencoders and Alpha-stable Distributions,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2019-May, pp. 541–545, 2019.
* Leglaive et al. [2018] S. Leglaive, L. Girin, and R. Horaud, “A variance modeling framework based on variational autoencoders for speech enhancement,” _IEEE International Workshop on Machine Learning for Signal Processing, MLSP_ , vol. 2018-September, 2018.
* Leglaive et al. [2020a] S. Leglaive, X. Alameda-Pineda, L. Girin, and R. Horaud, “A Recurrent Variational Autoencoder for Speech Enhancement,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2020-May, pp. 371–375, 2020.
* Kolbæk et al. [2017b] M. Kolbæk, Z. H. Tan, and J. Jensen, “Speech Intelligibility Potential of General and Specialized Deep Neural Network Based Speech Enhancement Systems,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 25, no. 1, pp. 149–163, 2017.
* Lu et al. [2022] Y.-J. Lu, Z.-Q. Wang, S. Watanabe, A. Richard, C. Yu, and Y. Tsao, “Conditional diffusion probabilistic model for speech enhancement,” in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 7402–7406.
* Lu et al. [2021] Y.-J. Lu, Y. Tsao, and S. Watanabe, “A study on speech enhancement based on diffusion probabilistic model,” in _2021 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC)_. IEEE, 2021, pp. 659–666.
* Fu et al. [2018a] S. W. Fu, T. W. Wang, Y. Tsao, X. Lu, H. Kawai, D. Stoller, S. Ewert, S. Dixon, X. Lu, Y. Tsao, S. Matsuda, C. Hori, Y. Xu, J. Du, L. R. Dai, C. H. Lee, T. Gao, J. Du, L. R. Dai, C. H. Lee, S. W. Fu, Y. Tsao, X. Lu, F. Weninger, J. R. Hershey, J. Le Roux, B. Schuller, Y. Xu, J. Du, L. R. Dai, C. H. Lee, F. Lluís, J. Pons, and X. Serra, “Speech enhancement based on deep denoising autoencoder,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-Sept, no. 1, pp. 7–19, 2018.
* Fu et al. [2016] S. W. Fu, Y. Tsao, and X. Lu, “SNR-aware convolutional neural network modeling for speech enhancement,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-Sept, pp. 3768–3772, 2016.
* Gao et al. [2016] T. Gao, J. Du, L. R. Dai, and C. H. Lee, “SNR-based progressive learning of deep neural network for speech enhancement,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-Sept, pp. 3713–3717, 2016.
* Portnoff [1980] M. R. Portnoff, “Time-Frequency Representation of . Digital Signals,” _IEEE Transactions on Acoustics, Speech and Signal Processing ASSP_ , vol. 28, no. 1, pp. 55–69, 1980.
* Allen [1982] J. B. Allen, “Applications of the short time Fourier transform to speech processing and spectral analysis,” in _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 1982-May, 1982, pp. 1012–1015.
* Allen and Rabiner [1977] J. B. Allen and L. R. Rabiner, “A Unified Approach to Short-Time Fourier Analysis and Synthesis,” _Proceedings of the IEEE_ , vol. 65, no. 11, pp. 1558–1564, 1977.
* Paliwal et al. [2011] K. Paliwal, K. Wójcicki, and B. Shannon, “The importance of phase in speech enhancement,” _Speech Communication_ , vol. 53, no. 4, pp. 465–494, 2011.
* Natsiou and O’Leary [2021] A. Natsiou and S. O’Leary, “Audio representations for deep learning in sound synthesis: A review,” _Proceedings of IEEE/ACS International Conference on Computer Systems and Applications, AICCSA_ , vol. 2021-Decem, 2021\.
* Nossier et al. [2020a] S. A. Nossier, J. Wall, M. Moniri, C. Glackin, and N. Cannings, “A Comparative Study of Time and Frequency Domain Approaches to Deep Learning based Speech Enhancement,” in _Proceedings of the International Joint Conference on Neural Networks_ , 2020.
* Grais and Plumbley [2018] E. M. Grais and M. D. Plumbley, “Single channel audio source separation using convolutional denoising autoencoders,” _2017 IEEE Global Conference on Signal and Information Processing, GlobalSIP 2017 - Proceedings_ , vol. 2018-Janua, pp. 1265–1269, 2018.
* Fu et al. [2019] S. W. Fu, C. F. Liao, Y. Tsao, and S. D. Lin, “MetricGAN: Generative adversarial networks based black-box metric scores optimization for speech enhancement,” _36th International Conference on Machine Learning, ICML 2019_ , vol. 2019-June, pp. 3566–3576, 2019.
* Jansson et al. [2017] A. Jansson, E. Humphrey, N. Montecchio, R. Bittner, A. Kumar, and T. Weyde, “Singing voice separation with deep U-Net convolutional networks,” _Proceedings of the 18th International Society for Music Information Retrieval Conference, ISMIR 2017_ , pp. 745–751, 2017.
* Kim and Smaragdis [2015] M. Kim and P. Smaragdis, “Adaptive denoising autoencoders: A fine-tuning scheme to learn from test mixtures,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 9237, pp. 100–107, 2015.
* Luo et al. [2017] Y. Luo, Z. Chen, J. R. Hershey, J. Le Roux, and N. Mesgarani, “Deep clustering and conventional networks for music separation: Stronger together,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 61–65, 2017.
* Baby et al. [2014] D. Baby, T. Virtanen, T. Barker, and H. Van Hamme, “Coupled dictionary training for exemplar-based speech enhancement,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 2883–2887, 2014.
* Fu et al. [2017] S. W. Fu, T. Y. Hu, Y. Tsao, and X. Lu, “Complex spectrogram enhancement by convolutional neural network with multi-metrics learning,” _IEEE International Workshop on Machine Learning for Signal Processing, MLSP_ , vol. 2017-September, pp. 1–6, 2017.
* Kothapally and Hansen [2022a] V. Kothapally and J. H. Hansen, “Complex-valued time-frequency self-attention for speech dereverberation,” _arXiv preprint arXiv:2211.12632_ , 2022.
* Kothapally and Hansen [2022b] ——, “Skipconvgan: Monaural speech dereverberation using generative adversarial networks via complex time-frequency masking,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 30, pp. 1600–1613, 2022.
* Liu et al. [2022] H. Liu, X. Liu, Q. Kong, Q. Tian, Y. Zhao, D. Wang, C. Huang, and Y. Wang, “VoiceFixer: A Unified Framework for High-Fidelity Speech Restoration,” _arXiv preprint arXiv:2204.05841._ , no. September, pp. 4232–4236, 2022.
* Du et al. [2020] Z. Du, X. Zhang, and J. Han, “A Joint Framework of Denoising Autoencoder and Generative Vocoder for Monaural Speech Enhancement,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 28, pp. 1493–1505, 2020.
* Weninger et al. [2014] F. Weninger, J. R. Hershey, J. Le Roux, and B. Schuller, “Discriminatively trained recurrent neural networks for single-channel speech separation,” _2014 IEEE Global Conference on Signal and Information Processing, GlobalSIP 2014_ , pp. 577–581, 2014.
* Donahue et al. [2018] C. Donahue, B. Li, and R. Prabhavalkar, “Exploring speech enhancement with generative adversarial networks for robust speech recognition,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, no. Figure 1, pp. 5024–5028, 2018\.
* Du and Huo [2008] J. Du and Q. Huo, “A speech enhancement approach using piecewise linear approximation of an explicit model of environmental distortions,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , pp. 569–572, 2008.
* Xu et al. [2015] Y. Xu, J. Du, L. R. Dai, and C. H. Lee, “A regression approach to speech enhancement based on deep neural networks,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 23, no. 1, pp. 7–19, 2015.
* Du et al. [2014] J. Du, Y. Tu, Y. Xu, L. Dai, and C. H. Lee, “Speech separation of a target speaker based on deep neural networks,” _International Conference on Signal Processing Proceedings, ICSP_ , vol. 2015-January, no. October, pp. 473–477, 2014.
* Garau and Renals [2008] G. Garau and S. Renals, “Combining spectral representations for large-vocabulary continuous speech recognition,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 16, no. 3, pp. 508–518, 2008.
* Zolnay et al. [2007] A. Zolnay, D. Kocharov, R. Schlüter, and H. Ney, “Using multiple acoustic feature sets for speech recognition,” _Speech Communication_ , vol. 49, no. 6, pp. 514–525, 2007.
* Wang et al. [2013] Y. Wang, K. Han, and D. Wang, “Exploring monaural features for classification-based speech segregation,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 21, no. 2, pp. 270–279, 2013.
* Williamson et al. [2016] D. S. Williamson, Y. Wang, and D. L. Wang, “Complex ratio masking for monaural speech separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 24, no. 3, pp. 483–492, 2016.
* Lu et al. [2013] X. Lu, Y. Tsao, S. Matsuda, and C. Hori, “Speech enhancement based on deep denoising autoencoder,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , no. August, pp. 436–440, 2013.
* Xu et al. [2014a] Y. Xu, J. Du, L. R. Dai, and C. H. Lee, “An experimental study on speech enhancement based on deep neural networks,” _IEEE Signal Processing Letters_ , vol. 21, no. 1, pp. 65–68, 2014.
* Narayanan and Wang [2013] A. Narayanan and D. Wang, “Ideal Ratio Mask Estimation Using Deep Neural Networks For Robust Speech Recognition,” _2013 IEEE International Conference on Acoustics, Speech and Signal Processing_ , pp. 7092–7096, 2013.
* Erdogan et al. [2015] H. Erdogan, J. R. Hershey, S. Watanabe, and J. Le Roux, “Phase-sensitive and recognition-boosted speech separation using deep recurrent neural networks,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2015-Augus, pp. 708–712, 2015.
* Kjems et al. [2009] U. Kjems, J. B. Boldt, M. S. Pedersen, T. Lunner, and D. Wang, “Role of mask pattern in intelligibility of ideal binary-masked noisy speech,” _The Journal of the Acoustical Society of America_ , vol. 126, no. 3, pp. 1415–1426, 2009.
* Wang [2008] D. Wang, “Time – Frequency Masking for Speech Hearing Aid Design,” _Trends In Amplification_ , vol. 12, pp. 332–353, 2008. [Online]. Available: http://www.ncbi.nlm.nih.gov/pubmed/18974204
* Brungart et al. [2006] D. S. Brungart, P. S. Chang, B. D. Simpson, and D. Wang, “Isolating the energetic component of speech-on-speech masking with ideal time-frequency segregation,” pp. 4007–4018, 2006.
* Nossier et al. [2020b] S. A. Nossier, J. Wall, M. Moniri, C. Glackin, and N. Cannings, “A Comparative Study of Time and Frequency Domain Approaches to Deep Learning based Speech Enhancement,” _Proceedings of the International Joint Conference on Neural Networks_ , 2020.
* Chen et al. [2015] Z. Chen, S. Watanabe, H. Erdogan, and J. R. Hershey, “Speech enhancement and recognition using multi-task learning of long short-term memory recurrent neural networks,” in _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2015-Janua, 2015, pp. 3274–3278.
* Huang et al. [2015] P. S. Huang, M. Kim, M. Hasegawa-Johnson, and P. Smaragdis, “Joint Optimization of Masks and Deep Recurrent Neural Networks for Monaural Source Separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 23, no. 12, pp. 2136–2147, 2015.
* Grais et al. [2014] E. M. Grais, M. U. Sen, and H. Erdogan, “Deep neural networks for single channel source separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 3734–3738, 2014.
* Zhang and Wang [2016] X. L. Zhang and D. Wang, “A deep ensemble learning method for monaural speech separation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 24, no. 5, pp. 967–977, 2016.
* Narayanan and Wang [2015] A. Narayanan and D. Wang, “Improving robustness of deep neural network acoustic models via speech separation and joint adaptive training,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 23, no. 1, pp. 92–101, 2015.
* Weninger et al. [2015] F. Weninger, H. Erdogan, S. Watanabe, E. Vincent, J. Le Roux, J. R. Hershey, and B. Schuller, “Speech enhancement with LSTM recurrent neural networks and its application to noise-robust ASR,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 9237, pp. 91–99, 2015.
* Goodfellow [2016] I. Goodfellow, “NIPS 2016 Tutorial: Generative Adversarial Networks,” _arXiv preprint arXiv_ , 2016. [Online]. Available: http://arxiv.org/abs/1701.00160
* Kingma and Welling [2014] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” _2nd International Conference on Learning Representations, ICLR 2014 - Conference Track Proceedings_ , no. Ml, pp. 1–14, 2014.
* Hien et al. [2015] T. D. Hien, D. V. Tuan, P. V. At, and L. H. Son, “Novel algorithm for non-negative matrix factorization,” _New Mathematics and Natural Computation_ , vol. 11, no. 2, pp. 121–133, 2015.
* Leglaive et al. [2020b] S. Leglaive, X. Alameda-Pineda, L. Girin, and R. Horaud, “A recurrent variational autoencoder for speech enhancement,” in _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 371–375.
* Sohl-Dickstein et al. [2015] J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, “Deep unsupervised learning using nonequilibrium thermodynamics,” in _International Conference on Machine Learning_. PMLR, 2015, pp. 2256–2265.
* Luo [2022] C. Luo, “Understanding diffusion models: A unified perspective,” _arXiv preprint arXiv:2208.11970_ , 2022.
* Luo and Mesgarani [2018] Y. Luo and N. Mesgarani, “TaSNet: Time-Domain Audio Separation Network for Real-Time, Single-Channel Speech Separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, pp. 696–700, 2018.
* Kumar and Florencio [2016] A. Kumar and D. Florencio, “Speech enhancement in multiple-noise conditions using deep neural networks,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-September-2016, pp. 3738–3742, 2016.
* Tu et al. [2014] Y. Tu, J. Du, Y. Xu, L. Dai, and C. H. Lee, “Deep neural network based speech separation for robust speech recognition,” _International Conference on Signal Processing Proceedings, ICSP_ , vol. 2015-January, no. October, pp. 532–536, 2014.
* Li et al. [2017] H. Li, X. Zhang, H. Zhang, and G. Gao, “Integrated Speech Enhancement Method Based on Weighted Prediction Error and DNN for Dereverberation and Denoising,” _arXiv preprint arXiv:1708.08251_ , no. 2, 2017. [Online]. Available: http://arxiv.org/abs/1708.08251
* Ephraim and Malah [1984] Y. Ephraim and D. Malah, “Speech Enhancement Using a Minimum Mean-Square Error Short-Time Spectral Amplitude Estimator,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 32, no. 6, pp. 1109–1121, 1984\.
* Lim and Oppenheim [1979] J. S. Lim and A. V. Oppenheim, “Enhancement and Bandwidth Compression of Noisy Speech,” _Proceedings of the IEEE_ , vol. 67, no. 12, pp. 1586–1604, 1979.
* Oppenheim and Lim [1981] A. V. Oppenheim and J. S. Lim, “The Importance of Phase in Signals,” _Proceedings of the IEEE_ , vol. 69, no. 5, pp. 529–541, 1981.
* Vary and Eurasip [1985] P. Vary and M. Eurasip, “Noise suppression by spectral magnitude estimation -mechanism and theoretical limits-,” _Signal Processing_ , vol. 8, no. 4, pp. 387–400, 1985.
* Wang and Lim [1982] D. L. Wang and J. S. Lim, “The Unimportance of Phase in Speech Enhancement,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 30, no. 4, pp. 679–681, 1982.
* Lee et al. [2017] Y.-S. Lee, C.-Y. Wang, S.-F. Wang, J.-C. Wang, and C.-H. Wu, “Fully complex deep neural network for phase-incorporating monaural source separation,” in _2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2017, pp. 281–285.
* Gunawan and Sen [2010] D. Gunawan and D. Sen, “Iterative phase estimation for the synthesis of separated sources from single-channel mixtures,” _IEEE Signal Processing Letters_ , vol. 17, no. 5, pp. 421–424, 2010.
* Ai et al. [2021] Y. Ai, H. Li, X. Wang, J. Yamagishi, and Z. Ling, “Denoising-and-Dereverberation Hierarchical Neural Vocoder for Robust Waveform Generation,” _2021 IEEE Spoken Language Technology Workshop, SLT 2021 - Proceedings_ , pp. 477–484, 2021.
* Le Roux et al. [2019] J. Le Roux, G. Wichern, S. Watanabe, A. Sarroff, and J. R. Hershey, “Phasebook and friends: Leveraging discrete representations for source separation,” _IEEE Journal on Selected Topics in Signal Processing_ , vol. 13, no. 2, pp. 370–382, 2019.
* Zheng and Zhang [2019] N. Zheng and X. L. Zhang, “Phase-aware speech enhancement based on deep neural networks,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 27, no. 1, pp. 63–76, 2019.
* Friedman [1985] D. H. Friedman, “Instantaneous-Frequency Distribution Vs. Time: An Interpretation Of The Phase Structure Of Speech.” in _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , 1985, pp. 1121–1124.
* Griffin and Lim [1984] D. W. Griffin and J. S. Lim, “Signal Estimation from Modified Short-Time Fourier Transform,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 32, no. 2, pp. 236–243, 1984.
* Zhao et al. [2019] Y. Zhao, Z. Q. Wang, and D. Wang, “Two-stage deep learning for noisy-reverberant speech enhancement,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 27, no. 1, pp. 53–62, 2019.
* Li et al. [2016] K. Li, B. Wu, and C. H. Lee, “An iterative phase recovery framework with phase mask for spectral mapping with an application to speech enhancement,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-Sept, pp. 3773–3777, 2016\.
* Luo et al. [2020] Y. Luo, Z. Chen, and T. Yoshioka, “Dual-Path RNN: Efficient Long Sequence Modeling for Time-Domain Single-Channel Speech Separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2020-May, pp. 46–50, 2020.
* Venkataramani et al. [2018] S. Venkataramani, J. Casebeer, and P. Smaragdis, “End-To-End Source Separation With Adaptive Front-Ends,” _2018 52nd Asilomar Conference on Signals, Systems, and Computers_ , no. 1, pp. 684–688, 2018.
* Zhang et al. [2020a] L. Zhang, Z. Shi, J. Han, A. Shi, and D. Ma, “FurcaNeXt: End-to-end monaural speech separation with dynamic gated dilated temporal convolutional networks,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 11961 LNCS, pp. 653–665, 2020.
* Tzinis et al. [2020a] E. Tzinis, S. Venkataramani, Z. Wang, C. Subakan, and P. Smaragdis, “Two-Step Sound Source Separation: Training on Learned Latent Targets,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2020-May, pp. 31–35, 2020.
* Tzinis et al. [2020b] E. Tzinis, Z. Wang, and P. Smaragdis, “Sudo RM -RF: Efficient networks for universal audio source separation,” _IEEE International Workshop on Machine Learning for Signal Processing, MLSP_ , vol. 2020-Septe, 2020.
* Kong et al. [2022] Z. Kong, W. Ping, A. Dantrey, and B. Catanzaro, “Speech Denoising in the Waveform Domain With Self-Attention,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2022-May, pp. 7867–7871, 2022.
* Lam et al. [2021a] M. W. Lam, J. Wang, D. Su, and D. Yuy, “Sandglasset: A Light Multi-Granularity Self-Attentive Network for Time-Domain Speech Separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2021-June, pp. 5759–5763, 2021\.
* Lam et al. [2021b] M. W. Lam, J. Wang, D. Su, and D. Yu, “Effective Low-Cost Time-Domain Audio Separation Using Globally Attentive Locally Recurrent Networks,” _2021 IEEE Spoken Language Technology Workshop, SLT 2021 - Proceedings_ , pp. 801–808, 2021.
* Kavalerov et al. [2019] I. Kavalerov, S. Wisdom, H. Erdogan, B. Patton, K. Wilson, J. Le Roux, and J. R. Hershey, “Universal sound separation,” _IEEE Workshop on Applications of Signal Processing to Audio and Acoustics_ , vol. 2019-October, pp. 175–179, 2019.
* Stoller et al. [2018] D. Stoller, S. Ewert, and S. Dixon, “Wave-u-net: A multi-scale neural network for end-to-end audio source separation,” _arXiv preprint arXiv:1806.03185_ , 2018.
* Fu et al. [2018b] S. W. Fu, T. W. Wang, Y. Tsao, X. Lu, and H. Kawai, “End-to-End Waveform Utterance Enhancement for Direct Evaluation Metrics Optimization by Fully Convolutional Neural Networks,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 26, no. 9, pp. 1570–1584, 2018.
* Lluís et al. [2019] F. Lluís, J. Pons, and X. Serra, “End-to-end music source separation: Is it possible in the waveform domain?” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2019-Septe, pp. 4619–4623, 2019.
* Pascual et al. [2017] S. Pascual, A. Bonafonte, and J. Serra, “SEGAN: Speech enhancement generative adversarial network,” in _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2017-Augus, no. D, 2017, pp. 3642–3646.
* Pascual et al. [2019] S. Pascual, J. Serrà, and A. Bonafonte, “Towards generalized speech enhancement with generative adversarial networks,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2019-September, pp. 1791–1795, 2019.
* Phan et al. [2020] H. Phan, I. V. McLoughlin, L. Pham, O. Y. Chen, P. Koch, M. De Vos, and A. Mertins, “Improving GANs for Speech Enhancement,” _IEEE Signal Processing Letters_ , vol. 27, pp. 1700–1704, 2020.
* Adiga et al. [2019] N. Adiga, Y. Pantazis, V. Tsiaras, and Y. Stylianou, “Speech enhancement for noise-robust speech synthesis using wasserstein gan.” in _INTERSPEECH_ , 2019, pp. 1821–1825.
* Gulrajani et al. [2017] I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, and A. C. Courville, “Improved training of wasserstein gans,” _Advances in neural information processing systems_ , vol. 30, 2017.
* van den Oord et al. [2016] A. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu, “Wavenet: A generative model for raw audio,” _arXiv preprint arXiv:1609.03499_ , pp. 1–15, 2016. [Online]. Available: http://arxiv.org/abs/1609.03499
* Xiao et al. [2021] F. Xiao, J. Guan, Q. Kong, and W. Wang, “Time-domain speech enhancement with generative adversarial learning,” _arXiv preprint arXiv:2103.16149_ , 2021\.
* Li et al. [2018a] Z. X. Li, L. R. Dai, Y. Song, and I. McLoughlin, “A Conditional Generative Model for Speech Enhancement,” _Circuits, Systems, and Signal Processing_ , vol. 37, no. 11, pp. 5005–5022, 2018. [Online]. Available: https://doi.org/10.1007/s00034-018-0798-4
* Qin and Jiang [2018] S. Qin and T. Jiang, “Improved Wasserstein conditional generative adversarial network speech enhancement,” _Eurasip Journal on Wireless Communications and Networking_ , vol. 2018, no. 1, 2018.
* Qian et al. [2017] K. Qian, Y. Zhang, S. Chang, X. Yang, D. Florêncio, and M. Hasegawa-Johnson, “Speech enhancement using bayesian wavenet.” in _Interspeech_ , 2017, pp. 2013–2017.
* Cao et al. [2022] R. Cao, S. Abdulatif, and B. Yang, “CMGAN: Conformer-based Metric GAN for Speech Enhancement,” _arXiv preprint arXiv:2209.11112_ , pp. 936–940, 2022\.
* Wang et al. [2021] K. Wang, B. He, and W. P. Zhu, “Tstnn: Two-Stage Transformer Based Neural Network for Speech Enhancement in the Time Domain,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2021-June, pp. 7098–7102, 2021.
* Heitkaemper et al. [2020] J. Heitkaemper, D. Jakobeit, C. Boeddeker, L. Drude, and R. Haeb-Umbach, “Demystifying TasNet: A Dissecting Approach,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2020-May, pp. 6359–6363, 2020.
* Bahmaninezhad et al. [2019] F. Bahmaninezhad, J. Wu, R. Gu, S. X. Zhang, Y. Xu, M. Yu, and D. Yu, “A comprehensive study of speech separation: Spectrogram vs waveform separation,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2019-September, pp. 4574–4578, 2019.
* Xu et al. [2014b] Y. Xu, J. Du, L.-R. Dai, and C.-H. Lee, “Global variance equalization for improving deep neural network based speech enhancement,” in _2014 IEEE China Summit & International Conference on Signal and Information Processing (ChinaSIP)_. IEEE, 2014, pp. 71–75.
* Hochreiter and Schmidhuber [1997] S. Hochreiter and J. Schmidhuber, “Long short-term memory,” _Neural computation_ , vol. 9, no. 8, pp. 1735–1780, 1997.
* Han et al. [2019] C. Han, J. O’Sullivan, Y. Luo, J. Herrero, A. D. Mehta, and N. Mesgarani, “Speaker-independent auditory attention decoding without access to clean speech sources,” _Science Advances_ , vol. 5, no. 5, pp. 1–12, 2019.
* Sainath et al. [2015] T. N. Sainath, R. J. Weiss, A. Senior, K. W. Wilson, and O. Vinyals, “Learning the speech front-end with raw waveform CLDNNs,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2015-Janua, pp. 1–5, 2015.
* Parveen and Green [2004] S. Parveen and P. Green, “Speech enhancement with missing data techniques using recurrent neural networks,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 1, no. Figure 1, pp. 13–16, 2004.
* Wichern and Lukin [2017] G. Wichern and A. Lukin, “Low-Latency approximation of bidirectional recurrent networks for speech denoising,” _IEEE Workshop on Applications of Signal Processing to Audio and Acoustics_ , vol. 2017-October, pp. 66–70, 2017.
* Cho et al. [2014] K. Cho, B. Van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio, “Learning phrase representations using RNN encoder-decoder for statistical machine translation,” _EMNLP 2014 - 2014 Conference on Empirical Methods in Natural Language Processing, Proceedings of the Conference_ , pp. 1724–1734, 2014.
* Chandna et al. [2017] P. Chandna, M. Miron, J. Janer, and E. Gómez, “Monoaural audio source separation using deep convolutional neural networks,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 10169 LNCS, pp. 258–266, 2017.
* Chen et al. [2020] J. Chen, Q. Mao, and D. Liu, “Dual-path transformer network: Direct context-aware modeling for end-to-end monaural speech separation,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-Octob, pp. 2642–2646, 2020\.
* Gulati et al. [2020] A. Gulati, J. Qin, C. C. Chiu, N. Parmar, Y. Zhang, J. Yu, W. Han, S. Wang, Z. Zhang, Y. Wu, and R. Pang, “Conformer: Convolution-augmented transformer for speech recognition,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, pp. 5036–5040, 2020.
* Rethage et al. [2018] D. Rethage, J. Pons, and X. Serra, “A wavenet for speech denoising,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, pp. 5069–5073, 2018.
* Li et al. [2018b] Y. Li, X. Zhang, and D. Chen, “CSRNet: Dilated Convolutional Neural Networks for Understanding the Highly Congested Scenes,” _Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition_ , pp. 1091–1100, 2018.
* Lea et al. [2017] C. Lea, M. D. Flynn, R. Vidal, A. Reiter, and G. D. Hager, “Temporal convolutional networks for action segmentation and detection,” in _proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2017, pp. 156–165.
* Zhang et al. [2020b] Q. Zhang, A. Nicolson, M. Wang, K. K. Paliwal, and C. Wang, “DeepMMSE: A Deep Learning Approach to MMSE-Based Noise Power Spectral Density Estimation,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 28, pp. 1404–1415, 2020.
* Bai et al. [2018] S. Bai, J. Z. Kolter, and V. Koltun, “An empirical evaluation of generic convolutional and recurrent networks for sequence modeling,” _arXiv preprint arXiv:1803.01271_ , 2018.
* He and Zhao [2019] Y. He and J. Zhao, “Temporal Convolutional Networks for Anomaly Detection in Time Series,” _Journal of Physics: Conference Series_ , vol. 1213, no. 4, 2019.
* Vaswani et al. [2017] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” _Advances in Neural Information Processing Systems_ , vol. 2017-Decem, no. Nips, pp. 5999–6009, 2017.
* Subakan et al. [2022a] C. Subakan, M. Ravanelli, S. Cornell, F. Lepoutre, and F. Grondin, “Resource-Efficient Separation Transformer,” _arXiv preprint arXiv:2206.09507_ , pp. 1–5, 2022. [Online]. Available: http://arxiv.org/abs/2206.09507
* Beltagy et al. [2020] I. Beltagy, M. E. Peters, and A. Cohan, “Longformer: The Long-Document Transformer,” _arXiv preprint arXiv:_ , 2020. [Online]. Available: http://arxiv.org/abs/2004.05150
* Wang et al. [2020] S. Wang, B. Z. Li, M. Khabsa, H. Fang, and H. Ma, “Linformer: Self-Attention with Linear Complexity,” _arXiv preprint arXiv_ , vol. 2048, no. 2019, 2020\. [Online]. Available: http://arxiv.org/abs/2006.04768
* Kitaev et al. [2020] N. Kitaev, Ł. Kaiser, and A. Levskaya, “Reformer: The Efficient Transformer,” in _International Conference on Learning Representations_ , 2020, pp. 1–12. [Online]. Available: http://arxiv.org/abs/2001.04451
* Subakan et al. [2022b] C. Subakan, M. Ravanelli, S. Cornell, F. Grondin, and M. Bronzi, “On Using Transformers for Speech-Separation,” in _International Workshop on Acoustic Signal Enhancement_ , vol. 14, no. 8, 2022, pp. 1–10. [Online]. Available: http://arxiv.org/abs/2202.02884
* Luo et al. [2022] J. Luo, J. Wang, N. Cheng, E. Xiao, X. Zhang, and J. Xiao, “Tiny-Sepformer: A Tiny Time-Domain Transformer Network For Speech Separation,” _arXiv preprint arXiv:2206.13689_ , no. 1, pp. 5313–5317, 2022.
* Chen et al. [2021] S. Chen, Y. Wu, Z. Chen, J. Wu, T. Yoshioka, S. Liu, J. Li, and X. Yu, “Ultra fast speech separation model with teacher student learning,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 3, pp. 2298–2302, 2021.
* de Oliveira et al. [2022] D. de Oliveira, T. Peer, and T. Gerkmann, “Efficient Transformer-based Speech Enhancement Using Long Frames and STFT Magnitudes,” _arXiv preprint arXiv:2206.11703._ , no. 1, pp. 2948–2952, 2022.
* Qiu and Hu [2022] W. Qiu and Y. Hu, “Dual-Path Hybrid Attention Network for Monaural Speech Separation,” _IEEE Access_ , vol. 10, pp. 78 754–78 763, 2022.
* Lutati et al. [2022] S. Lutati, E. Nachmani, and L. Wolf, “SepIt: Approaching a Single Channel Speech Separation Bound,” in _arXiv preprint arXiv:2205.11801_ , 2022, pp. 5323–5327.
* Wu et al. [2019] J. Y. Wu, C. Yu, S. W. Fu, C. T. Liu, S. Y. Chien, and Y. Tsao, “Increasing Compactness of Deep Learning Based Speech Enhancement Models with Parameter Pruning and Quantization Techniques,” _IEEE Signal Processing Letters_ , vol. 26, no. 12, pp. 1887–1891, 2019.
* Sun and Li [2017] H. Sun and S. Li, “An optimization method for speech enhancement based on deep neural network,” _IOP Conference Series: Earth and Environmental Science_ , vol. 69, no. 1, 2017.
* Lin et al. [2019] Y. C. Lin, Y. T. Hsu, S. W. Fu, Y. Tsao, and T. W. Kuo, “IA-Net: Acceleration and Compression of Speech Enhancement using Integer-adder Deep Neural Network,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2019-September, pp. 1801–1805, 2019.
* Fedorov et al. [2020] I. Fedorov, M. Stamenovic, C. Jensen, L. C. Yang, A. Mandell, Y. Gan, M. Mattina, and P. N. Whatmough, “TinyLSTMs: Efficient neural speech enhancement for hearing aids,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, pp. 4054–4058, 2020.
* Hsu et al. [2019] Y. T. Hsu, Y. C. Lin, S. W. Fu, Y. Tsao, and T. W. Kuo, “A Study on Speech Enhancement Using Exponent-Only Floating Point Quantized Neural Network (EOFP-QNN),” _2018 IEEE Spoken Language Technology Workshop, SLT 2018 \- Proceedings_ , pp. 566–573, 2019.
* Gholami et al. [2022] A. Gholami, S. Kim, Z. Dong, Z. Yao, M. W. Mahoney, and K. Keutzer, “A Survey of Quantization Methods for Efficient Neural Network Inference,” _Low-Power Computer Vision_ , pp. 291–326, 2022.
* Avery et al. [2014] K. R. Avery, J. Pan, C. C. Engler-Pinto, Z. Wei, F. Yang, S. Lin, L. Luo, and D. Konson, “Fatigue Behavior of Stainless Steel Sheet Specimens at Extremely High Temperatures,” _SAE International Journal of Materials and Manufacturing_ , vol. 7, no. 3, pp. 560–566, 2014.
* Howard et al. [2017] A. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, T. Weyand, M. Andreetto, and H. Adam, “MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications,” _arXiv preprint arXiv:1704.04861._ , 2017. [Online]. Available: http://arxiv.org/abs/1704.04861
* Hinton et al. [2015] G. Hinton, O. Vinyals, and J. Dean, “Distilling the knowledge in a neural network,” _arXiv preprint arXiv:1503.02531 2.7_ , pp. 1–9, 2015. [Online]. Available: http://arxiv.org/abs/1503.02531
* Gou et al. [2021] J. Gou, B. Yu, S. J. Maybank, and D. Tao, “Knowledge distillation: A survey,” _International Journal of Computer Vision_ , vol. 129, pp. 1789–1819, 2021\.
* Wang and Yoon [2022] L. Wang and K. J. Yoon, “Knowledge Distillation and Student-Teacher Learning for Visual Intelligence: A Review and New Outlooks,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 44, no. 6, pp. 3048–3068, 2022.
* Aihara et al. [2019] R. Aihara, T. Hanazawa, Y. Okato, G. Wichern, and J. L. Roux, “Teacher-student Deep Clustering for Low-delay Single Channel Speech Separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2019-May, pp. 690–694, 2019\.
* Tan and Wang [2021] K. Tan and D. Wang, “Towards Model Compression for Deep Learning Based Speech Enhancement,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 29, pp. 1785–1794, 2021.
* Ye et al. [2019] F. Ye, Y. Tsao, and F. Chen, “Subjective feedback-based neural network pruning for speech enhancement,” _2019 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference, APSIPA ASC 2019_ , no. November, pp. 673–677, 2019.
* Dupuis et al. [2020] E. Dupuis, D. Novo, I. O’Connor, and A. Bosio, “Sensitivity Analysis and Compression Opportunities in DNNs Using Weight Sharing,” in _Proceedings - 2020 23rd International Symposium on Design and Diagnostics of Electronic Circuits and Systems, DDECS 2020_ , 2020.
* Hu et al. [2021] X. Hu, K. Li, W. Zhang, Y. Luo, J. M. Lemercier, and T. Gerkmann, “Speech Separation Using an Asynchronous Fully Recurrent Convolutional Neural Network,” _Advances in Neural Information Processing Systems_ , vol. 27, no. NeurIPS, pp. 22 509–22 522, 2021.
* Zhang et al. [2018] H. Zhang, X. Zhang, and G. Gao, “Training Supervised Speech Separation System To Improve Stoi And Pesq Directly,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 5374–5378, 2018.
* Loizou and Kim [2011] P. C. Loizou and G. Kim, “Reasons why current speech-enhancement algorithms do not improve speech intelligibility and suggested solutions,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 19, no. 1, pp. 47–56, 2011.
* P. C. Loizou [2013] P. C. Loizou, _Speech enhancement: theory and practice_. CRC press, 2013.
* Xia and Bao [2014] B. Xia and C. Bao, “Wiener filtering based speech enhancement with Weighted Denoising Auto-encoder and noise classification,” pp. 13–29, 2014.
* Shivakumar and Georgiou [2016] P. G. Shivakumar and P. Georgiou, “Perception optimized deep denoising autoencoders for speech enhancement,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 08-12-September-2016, pp. 3743–3747, 2016.
* Koizumi et al. [2017] Y. Koizumi, K. Niwa, Y. Hioka, K. Kobayashi, and Y. Haneda, “DNN-based source enhancement self-optimized by reinforcement learning using sound quality measurements,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 81–85, 2017.
* Kolbcek et al. [2018] M. Kolbcek, Z. H. Tan, and J. Jensen, “Monaural Speech Enhancement Using Deep Neural Networks by Maximizing a Short-Time Objective Intelligibility Measure,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2018-April, pp. 5059–5063, 2018.
* Yan et al. [2018] Z. Yan, X. Buye, G. Ritwik, and Z. Tao, “Perceptually Guided Speech Enhancement Using Deep Neural Networks,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , pp. 5074–5078, 2018. [Online]. Available: https://cliffzhao.github.io/Publications/ZXGZ.icassp18.pdf%0Ahttp://arxiv.org/abs/1312.6114%0Ahttps://arxiv.org/pdf/1312.6114.pdf
* Taal et al. [2011] C. H. Taal, R. C. Hendriks, R. Heusdens, and J. Jensen, “An algorithm for intelligibility prediction of time-frequency weighted noisy speech,” _IEEE Transactions on Audio, Speech and Language Processing_ , vol. 19, no. 7, pp. 2125–2136, 2011.
* Roux et al. [2019] J. L. Roux, S. Wisdom, H. Erdogan, and J. R. Hershey, “SDR - Half-baked or Well Done?” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2019-May, pp. 626–630, 2019\.
* Li et al. [2022] H. Li, K. Chen, L. Wang, J. Liu, B. Wan, and B. Zhou, “Sound source separation mechanisms of different deep networks explained from the perspective of auditory perception,” _Applied Sciences_ , vol. 12, no. 2, p. 832, 2022.
* Fan et al. [2020] C. Fan, J. Tao, B. Liu, J. Yi, Z. Wen, and X. Liu, “End-to-End Post-Filter for Speech Separation with Deep Attention Fusion Features,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 28, pp. 1303–1314, 2020.
* Taal et al. [2010] C. H. Taal, R. C. Hendriks, R. Heusdens, and J. Jensen, “A Short-Time Objective Intelligibility Measure For Time-Frequency Weighted Noisy Speech,” _IEEE International Conference on Acoustics, Speech, and Signal Processing_ , pp. 4214–4217, 2010. [Online]. Available: http://cas.et.tudelft.nl/pubs/Taal2010.pdf
* Rix et al. [2001] A. W. Rix, J. G. Beerends, M. P. Hollier, and A. P. Hekstra, “Perceptual evaluation of speech quality (PESQ) - A new method for speech quality assessment of telephone networks and codecs,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2, pp. 749–752, 2001.
* Martin-Donas et al. [2018] J. M. Martin-Donas, A. M. Gomez, J. A. Gonzalez, and A. M. Peinado, “A deep learning loss function based on the perceptual evaluation of the speech quality,” _IEEE Signal Processing Letters_ , vol. 25, no. 11, pp. 1680–1684, 2018.
* Kolbaek et al. [2020] M. Kolbaek, Z. H. Tan, S. H. Jensen, and J. Jensen, “On Loss Functions for Supervised Monaural Time-Domain Speech Enhancement,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 28, pp. 825–838, 2020\.
* Bie et al. [2022] X. Bie, S. Leglaive, X. Alameda-Pineda, and L. Girin, “Unsupervised speech enhancement using dynamical variational autoencoders,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 30, pp. 2993–3007, 2022.
* Fujimura et al. [2021] T. Fujimura, Y. Koizumi, K. Yatabe, and R. Miyazaki, “Noisy-target training: A training strategy for dnn-based speech enhancement without clean speech,” in _2021 29th European Signal Processing Conference (EUSIPCO)_. IEEE, 2021, pp. 436–440.
* Wisdom et al. [2020] S. Wisdom, E. Tzinis, H. Erdogan, R. J. Weiss, K. Wilson, and J. R. Hershey, “Unsupervised sound separation using mixture invariant training,” in _Advances in Neural Information Processing Systems_ , vol. 2020-Decem, jun 2020. [Online]. Available: http://arxiv.org/abs/2006.12701
* Zhang et al. [2021b] J. Zhang, C. Zorila, R. Doddipatla, and J. Barker, “Teacher-student mixit for unsupervised and semi-supervised speech separation,” _arXiv preprint arXiv:2106.07843_ , 2021.
* Saito et al. [2021] K. Saito, S. Uhlich, G. Fabbro, and Y. Mitsufuji, “Training speech enhancement systems with noisy speech datasets,” _arXiv preprint arXiv:2105.12315_ , 2021\.
* Karamatlı and Kırbız [2022] E. Karamatlı and S. Kırbız, “Mixcycle: Unsupervised speech separation via cyclic mixture permutation invariant training,” _IEEE Signal Processing Letters_ , 2022.
* Trinh and Braun [2022] V. A. Trinh and S. Braun, “Unsupervised speech enhancement with speech recognition embedding and disentanglement losses,” in _ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2022, pp. 391–395.
* Tzinis et al. [2022] E. Tzinis, Y. Adi, V. K. Ithapu, B. Xu, P. Smaragdis, and A. Kumar, “Remixit: Continual self-training of speech enhancement models via bootstrapped remixing,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 16, no. 6, pp. 1329–1341, 2022.
* Chen et al. [2023b] L.-W. Chen, Y.-F. Cheng, H.-S. Lee, Y. Tsao, and H.-M. Wang, “A training and inference strategy using noisy and enhanced speech as target for speech enhancement without clean speech,” in _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , 2023, pp. 5315–5319.
* Reddy et al. [2021] C. K. Reddy, V. Gopal, and R. Cutler, “Dnsmos: A non-intrusive perceptual objective speech quality metric to evaluate noise suppressors,” in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 6493–6497.
* Xiang and Bao [2020] Y. Xiang and C. Bao, “A parallel-data-free speech enhancement method using multi-objective learning cycle-consistent generative adversarial network,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 28, pp. 1826–1838, 2020.
* Zhu et al. [2017] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, “Unpaired image-to-image translation using cycle-consistent adversarial networks,” in _Proceedings of the IEEE international conference on computer vision_ , 2017, pp. 2223–2232.
* Li et al. [2021b] Y. Li, Y. Sun, K. Horoshenkov, and S. M. Naqvi, “Domain adaptation and autoencoder-based unsupervised speech enhancement,” _IEEE Transactions on Artificial Intelligence_ , vol. 3, no. 1, pp. 43–52, 2021.
* Lehtinen et al. [2018] J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” _arXiv preprint arXiv:1803.04189_ , 2018.
* Pan et al. [2010] S. J. Pan, I. W. Tsang, J. T. Kwok, and Q. Yang, “Domain adaptation via transfer component analysis,” _IEEE transactions on neural networks_ , vol. 22, no. 2, pp. 199–210, 2010.
* Liao et al. [2018] C.-F. Liao, Y. Tsao, H.-Y. Lee, and H.-M. Wang, “Noise adaptive speech enhancement using domain adversarial training,” _arXiv preprint arXiv:1807.07501_ , 2018.
* Wang et al. [2018d] Q. Wang, W. Rao, S. Sun, L. Xie, E. S. Chng, and H. Li, “Unsupervised domain adaptation via domain adversarial training for speaker recognition,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 4889–4893.
* Xu et al. [2014c] Y. Xu, J. Du, L.-R. Dai, and C.-H. Lee, “Cross-language transfer learning for deep neural network based speech enhancement,” in _The 9th International Symposium on Chinese Spoken Language Processing_. IEEE, 2014, pp. 336–340.
* Pascual et al. [2018] S. Pascual, M. Park, J. Serrà, A. Bonafonte, and K.-H. Ahn, “Language and noise transfer in speech enhancement generative adversarial network,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 5019–5023.
* Ganin et al. [2016] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky, “Domain-adversarial training of neural networks,” _The journal of machine learning research_ , vol. 17, no. 1, pp. 2096–2030, 2016.
* Chung et al. [2019] Y. A. Chung, W. N. Hsu, H. Tang, and J. Glass, “An unsupervised autoregressive model for speech representation learning,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2019-September, pp. 146–150, 2019.
* Chung et al. [2020] Y. A. Chung, H. Tang, and J. Glass, “Vector-quantized autoregressive predictive coding,” _Proceedings of the Annual Conference of the International Speech Communication Association, INTERSPEECH_ , vol. 2020-October, no. 1, pp. 3760–3764, 2020.
* Liu et al. [2020] A. T. Liu, S. W. Yang, P. H. Chi, P. C. Hsu, and H. Y. Lee, “Mockingjay: Unsupervised speech representation learning with deep bidirectional transformer encoders,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2020-May, pp. 6419–6423, 2020.
* Liu et al. [2021] A. T. Liu, S. W. Li, and H. Y. Lee, “TERA: Self-Supervised Learning of Transformer Encoder Representation for Speech,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 29, pp. 2351–2366, 2021.
* Baevski et al. [2020] A. Baevski, H. Zhou, A. Mohamed, and M. Auli, “wav2vec 2.0: A framework for self-supervised learning of speech representations,” _Advances in Neural Information Processing Systems_ , vol. 2020-Decem, no. Figure 1, pp. 1–19, 2020.
* Hsu et al. [2021a] W. N. Hsu, B. Bolte, Y. H. H. Tsai, K. Lakhotia, R. Salakhutdinov, and A. Mohamed, “HuBERT: Self-Supervised Speech Representation Learning by Masked Prediction of Hidden Units,” _IEEE/ACM Transactions on Audio Speech and Language Processing_ , vol. 29, no. Cv, pp. 3451–3460, 2021.
* Huang et al. [2022] Z. Huang, S. Watanabe, S. W. Yang, P. García, and S. Khudanpur, “Investigating Self-Supervised Learning for Speech Enhancement and Separation,” _ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings_ , vol. 2022-May, pp. 6837–6841, 2022\.
* Hsu et al. [2021b] W.-N. Hsu, B. Bolte, Y.-H. H. Tsai, K. Lakhotia, R. Salakhutdinov, and A. Mohamed, “Hubert: Self-supervised speech representation learning by masked prediction of hidden units,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 29, pp. 3451–3460, 2021.
* Hung et al. [2022] K.-H. Hung, S.-w. Fu, H.-H. Tseng, H.-T. Chiang, Y. Tsao, and C.-W. Lin, “Boosting self-supervised embeddings for speech enhancement,” _arXiv preprint arXiv:2204.03339_ , 2022.
* Irvin et al. [2023] B. Irvin, M. Stamenovic, M. Kegler, and L.-C. Yang, “Self-supervised learning for speech enhancement through synthesis,” in _ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2023, pp. 1–5.
* Kong et al. [2020] J. Kong, J. Kim, and J. Bae, “Hifi-gan: Generative adversarial networks for efficient and high fidelity speech synthesis,” _Advances in Neural Information Processing Systems_ , vol. 33, pp. 17 022–17 033, 2020.
* Germain et al. [2018] F. G. Germain, Q. Chen, and V. Koltun, “Speech denoising with deep feature losses,” _arXiv preprint arXiv:1806.10522_ , 2018.
* Hao et al. [2023] X. Hao, C. Xu, and L. Xie, “Neural speech enhancement with unsupervised pre-training and mixture training,” _Neural Networks_ , vol. 158, pp. 216–227, 2023.
|
Dedicated to the 70th anniversary of Peter Olver
# Natural differential invariants and equivalence of third order nonlinear
differential operators
Valentin Lychagin Institute of Control Sciences of RAS, Moscow, Russia
<EMAIL_ADDRESS>and Valeriy Yumaguzhin Program Systems Institute of
RAS, Pereslavl’-Zalesskiy, Russia<EMAIL_ADDRESS>
###### Abstract.
We give a description of the field of rational natural differential invariants
for a class of nonlinear differential operators of the third order on a two
dimensional manifold and show their application to the equivalence problem of
such operators.
###### Key words and phrases:
3rd order linear partial differential operator, jet bundle, differential
invariant, equivalence problem
###### 2010 Mathematics Subject Classification:
Primary: 58J70,53C05,35A3; Secondary: 35G05,53A55
V. Yumaguzhin is the corresponding author
## 1\. Introduction
In this paper, we continue to study rational differential invariants of
differential operators on two-dimensional manifolds. Here we consider a class
of nonlinear operators that in local coordinates $x_{1},x_{2}$ have the
following form:
$A_{w}\colon
f\longmapsto\sum\limits_{i,j,k}a_{ijk}(x,f)\frac{\partial^{3}f}{\partial
x_{i}\partial x_{j}\partial
x_{k}}+\sum\limits_{i,j}a_{ij}\left(x,f\right)\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}}\\\ +\sum\limits_{i,j}a_{i}\left(x,f\right)\frac{\partial
f}{\partial x_{i}}+a_{0}\left(x,f\right)f,$ (1)
where $x=(x_{1},x_{2})$, $f=f(x)$.
We call such operators as weakly nonlinear operators.
In paper [5], we found rational differential invariants of the 2nd order
weakly nonlinear operators and used them to solve the local equivalence
problem.
Here we use the methods of [5] to find rational differential invariants for
the 3rd order weakly nonlinear operators and apply them to solve the local as
well as the global equivalence problem.
Additionally we will assume that all coefficients $a(x,y)$ of these operators
are smooth in $x$ and rational in $y$.
It is easy to see that the pseudogroup of local diffeomorphisms (in variables
$x$) naturally acts in this class of operators.
The main step in studies of such operators is a representation of operator
$A_{w}$ as a pair $(A,f)$ (we call it as a (related pair), where
$A=\sum\limits_{i,j,k}a_{ijk}\left(x,y\right)\frac{\partial^{3}}{\partial
x_{i}\partial x_{j}\partial
x_{k}}+\sum\limits_{i,j}a_{ij}\left(x,y\right)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\\\
+\sum\limits_{i}a_{i}\left(x,y\right)\frac{\partial}{\partial
x_{i}}+a_{0}(x,y)$ (2)
is a linear third order linear differential operator on the extended space
$(x,y)$.
The procedure of descent allows us to get differential invariants of weakly
nonlinear operators from invariants of related pairs and linear operators of
the form
$A_{f}=\sum\limits_{i,j,k}a_{ijk}\left(x,f\right)\frac{\partial^{3}}{\partial
x_{i}\partial x_{j}\partial
x_{k}}+\sum\limits_{i,j}a_{ij}\left(x,f\right)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\\\
+\sum\limits_{i}a_{i}\left(x,f\right)\frac{\partial}{\partial
x_{i}}+a_{0}\left(x,f\right).$ (3)
Using this procedure and a machinery of reduced Gröbner bases, we propose a
method to find invariants of the 3rd order weakly nonlinear operators.
## 2\. Notations
In this paper, we use the same notations as in [3, 4].
Let $M$ be $n$-dimensional manifold and let $\tau\colon TM\to M$ and
$\tau^{*}\colon T^{*}M\to M$ be respectively tangent and cotangent bundles
over $M$.
We will denote by $\mathit{Diff}_{k}(M)$ the module of linear $k$-th order
operators, acting in $C^{\infty}(M)$. The corresponding vector bundle of such
operators we denote by $\chi_{k}\colon\mathbf{Diff}_{k}\to M$, thus
$\mathit{Diff}_{k}(M)$ is also the module of smooth sections of $\chi_{k}$,
$\mathit{Diff}_{k}(M)=C^{\infty}(\chi_{k})$.
The group of diffeomorphisms of $M$ will be denoted by $\mathcal{G}(M)$ and
the multiplicative group of nowhere vanishing functions on $M$ will be denoted
as $\mathcal{F}(M)$.
We denote by $\Sigma^{k}(M)$ and $\Sigma_{k}(M)$ the modules of symmetric
$k$-forms and $k$-vectors respectively.
Also, we denote by $\Omega^{k}$ the modules of exterior $k$-forms.
## 3\. Linear differential operators on $M$
In this section, we recall the necessary facts from [3] on scalar linear third
order differential operators on two-dimensional manifolds.
Let $M$ be 2-dimensional smooth oriented connected manifold and let
$\pi\colon M\times\mathbb{R}\longrightarrow M$
be trivial line bundle. Denote by
$\pi_{k}\colon\mathbf{J}^{k}(\pi)\longrightarrow M$
the bundles of $k$-jets of smooth sections of $\pi$ (i.e. $k$-jets of smooth
functions on $M$).
Let $\mathcal{G}(M)$ be the group of diffeomorphisms of the manifold $M$.
Then, together with the action of $\mathcal{G}\left(M\right)$ on $M$, we have
also the actions $\mathcal{G}\left(M\right)$ in the bundles $\pi_{k}$ by
prolongations of diffeomorphisms.
The prolongations of diffeomorphisms $\phi\in\mathcal{G}(M)$ in the bundles
$\pi_{k}$ will be denoted by $\phi^{(k)}$.
### 3.1.
Let
$\chi_{3}\colon\mathbf{Diff}_{3}\left(M\right)\longrightarrow M$
be the bundle of the third order linear differential operators on $M$ and
$\chi_{3,k}\colon\mathbf{J}^{k}(\chi_{3})\longrightarrow M$
be bundles of their $k$-jets of sections of $\chi_{3}$.
The group $\mathcal{G}\left(M\right)$ acts on operators
$A\in\mathit{Diff}_{3}(M)$ in natural way:
$\phi_{\ast}\colon A\rightarrow\phi_{\ast}(A)=\phi_{*}\circ
A\circ\phi_{*}^{-1},$
where the morphism $\phi_{\ast}\colon C^{\infty}(M)\longrightarrow
C^{\infty}(M)$ is defined by the formula
$\phi_{\ast}=(\phi^{-1})^{\ast}\colon h\mapsto h\circ\phi^{-1}.$
### 3.2. Symbols
By the symbol $\sigma_{3,A}$ of the operator $A$, we means the equivalence
class
$\sigma_{3,A}=A\\!\\!\\!\mod\\!\mathit{Diff}_{2}(M).$
Let an operator $A\in\mathit{Diff}_{3}(M)$ be represented in local coordinates
$x_{1},x_{2}$ of $M$ in the form
$A=a_{1}\partial^{3}_{1}+3a_{2}\partial^{2}_{1}\partial_{2}+3a_{3}\partial_{1}\partial^{2}_{2}+a_{4}\partial^{3}_{2}\\\
+b_{1}\partial^{2}_{1}+2b_{2}\partial_{1}\partial_{2}+b_{3}\partial^{2}_{2}+c_{1}\partial_{1}+c_{2}\partial_{2}+a_{0},$
(4)
where the all coefficients are smooth functions of $x=(x_{1},x_{2})$,
$\partial_{1}$ and $\partial_{2}$ are $\partial_{x_{1}}$ and
$\partial_{x_{2}}$ respectively.
Then the symbol $\sigma_{3,A}$ is identified with the symmetric 3-vector
$\sigma_{3,A}=a_{1}\partial^{3}_{1}+3a_{2}\partial^{2}_{1}\partial_{2}+3a_{3}\partial_{1}\partial^{2}_{2}+a_{4}\partial^{3}_{2}\in\Sigma_{3}(M),$
(5)
where dots and degrees are symmetric products of the vector fields
$\partial_{1},\;\partial_{2}$.
The symbol $\sigma_{3,A}$ is regular (at point or in domain) if it as a
homogeneous cubic polynomial on the cotangent bundle $T^{*}(M)$ has distinct
roots.
Denote by $\Delta(\sigma_{3,A})$ the discriminant of $\sigma_{3,A}$, then
$\Delta(\sigma_{3,A})=6a_{1}a_{2}a_{3}a_{4}-4(a_{1}a_{3}^{3}+a_{4}a_{2}^{3})+3a_{2}^{2}a_{3}^{2}-a_{1}^{2}a_{4}^{2}.$
Recall that the symbol $\sigma_{3,A}$ has three distinct real roots if
$\Delta(\sigma_{3,A})>0$ and one real and two complex roots if
$\Delta(\sigma_{3,A})<0$.
Thus, we say that the operator $A$ is regular if its symbol $\sigma_{3,A}$ is
regular or $\Delta(\sigma_{3,A})\neq 0$
More over, we say that an operator $A$ is hyperbolic if
$\Delta(\sigma_{3,A})>0$ and ultrahyperbolic if $\Delta(\sigma_{3,A})<0$.
Locally, the symbol of a hyperbolic operator can be presented as a symmetric
product of pair wise linear independent vector fields $\sigma_{3,A}=X_{1}\cdot
X_{2}\cdot X_{3}$, and therefore there are local coordinates $x_{1},x_{2}$
such that
$\sigma_{3,A}=(a\partial_{1}+b\partial_{2})\cdot\partial_{1}\cdot\partial_{2},$
(6)
where $a$ and $b$ are smooth functions and $ab\neq 0$.
For the case of ultrahyperbolic operators we have $\sigma_{3,A}=X\cdot q$,
where $X$ is a nonzero vector field and $q\in\Sigma_{2}$ is a positive
symmetric 2-vector. Therefore, there are local coordinates $x_{1},x_{2}$ such
that
$\sigma_{3,A}=(a\partial_{1}+b\partial_{2})\cdot(\partial^{2}_{1}+\partial^{2}_{2}),$
(7)
where $a$ and $b$ are smooth functions and $a^{2}+b^{2}>0$.
### 3.3. Wagner connections
The following result due to Wagner [7, 3].
###### Theorem 1.
1. (1)
Let $A\in{\it Diff}_{3}(M)$ be a regular differential operator. Then there
exist a unique linear connection $\nabla$ such that the symbol $\sigma_{3,A}$
is parallel with respect to $\nabla$.
2. (2)
The curvature tensor of $\nabla$ is zero.
We call this connection Wagner’s connection.
In the case of hyperbolic symbol, we choose local coordinates in $M$ such that
$\sigma$ has form (6). Then the non zero Christoffel coefficients
$\Gamma^{i}_{jk}$ of the Wagner connection are the following:
$\displaystyle\Gamma^{1}_{11}=\frac{1}{3}\big{(}\ln\frac{b}{a^{2}}\big{)}_{x_{1}},\quad\Gamma^{2}_{22}=\frac{1}{3}\big{(}\ln\frac{a}{b^{2}}\big{)}_{x_{2}},$
$\displaystyle\Gamma^{1}_{12}=\frac{1}{3}\big{(}\ln\frac{b}{a^{2}}\big{)}_{x_{2}},\quad\Gamma^{2}_{21}=\frac{1}{3}\big{(}\ln\frac{a}{b^{2}}\big{)}_{x_{1}}.$
In the ultrahyperbolic case (7) we have the following nonzero Chritoffel
coefficients:
$\displaystyle\Gamma^{1}_{12}=\Gamma^{2}_{22}=-\frac{1}{6}\big{(}\ln(a^{2}+b^{2})\big{)}_{x_{2}},$
$\displaystyle\Gamma^{1}_{21}=-\Gamma^{2}_{11}=\frac{a_{x_{1}}b-ab_{x_{1}}}{a^{2}+b^{2}},\quad\Gamma^{1}_{22}=-\Gamma^{2}_{12}=\frac{ab_{x_{2}}-a_{x_{2}}b}{a^{2}+b^{2}}.$
###### Corollary 2.
1 The torsion form $\theta$ of the Wagner connection is
$\theta=\frac{1}{3}\big{(}\ln\frac{b^{2}}{a}\big{)}_{x_{1}}dx_{1}+\frac{1}{3}\big{(}\ln\frac{a^{2}}{b}\big{)}_{x_{2}}dx_{2}$
for the hyperbolic case and
$\theta=\frac{ab_{x_{2}}-a_{x_{2}}b}{a^{2}+b^{2}}dx_{1}+\frac{ab_{x_{1}}-a_{x_{1}}b}{a^{2}+b^{2}}dx_{2}-\frac{1}{6}d\big{(}\ln(a^{2}+b^{2})\big{)}$
for the ultrahyperbolic case.
### 3.4. Symols and quantization
Let $\Sigma^{\cdot}=\oplus_{k\geq 0}\Sigma^{k}(M)$ be the graded algebra of
symmetric differential forms and let $\nabla$ be the Wagner connection
associated with a regular symbol from $\Sigma_{3}(M)$. Then the covariant
differential
$d_{\nabla}:\Omega^{1}(M)\longrightarrow\Omega^{1}(M)\otimes\Omega^{1}(M)$
define derivation
$d_{\nabla}^{s}\colon\Sigma^{\cdot}\longrightarrow\Sigma^{\cdot+1}$
of degree one in graded symmetric algebra $\Sigma^{\cdot}\oplus_{k\geq
0}\Sigma^{k}(M)$. Namely, this derivation is defined by its action on
generators, and we have
$\displaystyle
d^{s}_{\nabla}=d:C^{\infty}(M)\longrightarrow\Omega^{1}(M)=\Sigma^{1},$
$\displaystyle d^{s}_{\nabla}:\Omega^{1}(M)=\Sigma^{1}\stackrel{{\scriptstyle
d_{\nabla}}}{{\longrightarrow}}\Omega^{1}(M)\otimes\Omega^{1}(M)\stackrel{{\scriptstyle\mathrm{Sym}}}{{\longrightarrow}}\Sigma^{2}.$
Let now $\alpha_{k}\in\Sigma_{k}(M)$. We define a differential operator
$\mathcal{Q}(\alpha_{k})\in\mathit{Diff}_{k}(M)$ as follows:
$\mathcal{Q}(\alpha_{k})(h)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\frac{1}{k!}\left\langle\,\alpha_{k},\,\big{(}d^{s}_{\nabla}\big{)}^{k}(h)\,\right\rangle,$
where $h\in C^{\infty}(M)$,
$\big{(}d^{s}_{\nabla}\big{)}^{k}(h)\in\Sigma^{k}(M)$, and
$\langle\cdot\,,\cdot\rangle$ is the standard convolution
$\Sigma_{k}(M)\otimes\Sigma^{k}(M)\longrightarrow C^{\infty}(M).$
Remark, that the value of the symbol of the derivation $d^{s}_{\nabla}$ at a
covector $\theta$ equals to the symmetric product by $\theta$ into the module
$\Sigma^{\cdot}$. Therefore, the symbol of operator $\mathcal{Q}(\alpha_{k})$
equals $\alpha_{k}$ as the symbol of a composition of operators equals the
composition of symbols.
We call differential operator $\mathcal{Q}(\alpha_{k})$ a quantization of
symbol $\alpha_{k}$.
Let now $A\in\mathit{Diff}_{3}(M)$ and $\sigma_{3,A}$ be its symbol. Then
operator
$A-\mathcal{Q}\big{(}\sigma_{3,A}\big{)}$
has order $2$, and let $\sigma_{2,A}$ be its symbol.
Then operator
$A-\mathcal{Q}\big{(}\sigma_{3,A}\big{)}-\mathcal{Q}\big{(}\sigma_{2,A}\big{)}$
has order $1$ and let $\sigma_{1,A}$ be its symbol. Thus we get subsymbols
$\sigma_{i,A}\in\Sigma_{i}(M)$, $0\leq i\leq 2$, such that
$A=\mathcal{Q}\big{(}\sigma_{(3)}(A)\big{)},$
where
$\sigma_{(3)}(A)=\sigma_{3,A}+\sigma_{2,A}+\sigma_{1,A}+\sigma_{0,A}$ (8)
is the total symbol and
$\mathcal{Q}\big{(}\sigma_{(3)}(A)\big{)}=\mathcal{Q}\big{(}\sigma_{3,A}\big{)}+\mathcal{Q}\big{(}\sigma_{2,A}\big{)}+\ldots+\mathcal{Q}\big{(}\sigma_{0,A}\big{)}.$
#### 3.4.1. Coordinates
Let $x_{1},x_{2}$ be local coordinates in a neighborhood $\mathcal{O}\subset
M$, where the symbol $\sigma_{3,A}$ is regular. Denote by
$x_{1},x_{2},w_{1},w_{2}$ induced standard coordinates in the tangent bundle
over $\mathcal{O}$.
Then $d_{\nabla}(dx_{k})=-\sum\Gamma^{k}_{ij}dx_{i}\otimes dx_{j}$, where
$\Gamma^{k}_{ij}$ are the Christoffel symbols of the Wagner connection
$\nabla$.
Thus, in coordinates $x,w$ we have
$d^{s}_{\nabla}(w_{k})=-\sum\Gamma^{k}_{ij}w_{i}w_{j}$ and the derivation
$d^{s}_{\nabla}$ has the form:
$d^{s}_{\nabla}=\sum
w_{i}\partial_{x_{i}}-\sum\Gamma^{k}_{ij}w_{i}w_{j}\partial_{w_{k}}.$
### 3.5. Universal differential operator
We define a total operator of third order
$\square_{3}\colon C^{\infty}(J^{k}\chi_{3})\longrightarrow
C^{\infty}(J^{k+3}\chi_{3})$
as it was done in [3].
In local coordinates this operator has the form
$\square_{3}=6\sum_{\alpha,0\leq|\alpha|\leq
3}\frac{u^{\alpha}}{\alpha!}\Big{(}\frac{d}{dx}\Big{)}^{\alpha},$ (9)
where $\alpha=(\alpha_{1},\alpha_{2})$, $|\alpha|=\alpha_{1}+\alpha_{2}$,
$(d/dx)^{\alpha}=(d/dx_{1})^{\alpha_{1}}(d/dx_{2})^{\alpha_{2}}$, and
$d/dx_{1},d/dx_{2}$ are total derivatives.
###### Theorem 3.
1 The operator $\square_{3}$ commutes with the action of the group
$\mathcal{G}(M)$ on the jet bundles.
### 3.6. Natural differential invariants of regular operators
By natural differential invariants of order k we mean a function on
$J^{k}(\chi_{3})$ which are $\mathcal{G}(M)$-invariant and rational along
fibers of the projection $\chi_{3,k}$.
###### Theorem 4.
If $I$ is a natural differential invariant of order $\leq k$ for differential
operators of the third order, then $\square_{3}(I)$ is a natural differential
invariant of the order $\leq(k+3)$ for these operators.
We say that two natural differential invariants $I_{1},I_{2}$ are in general
position if
$\hat{d}I_{1}\wedge\hat{d}I_{2}\neq 0,$ (10)
where $\hat{d}$ is the total differential, and denote by
$\mathcal{O}(I_{1},I_{2})\subset J^{\infty}(\chi_{3})$ the open domain, where
condition (10) holds.
###### Theorem 5.
Let natural differential invariants $I_{1},I_{2}$ are in general position and
let
$J^{\alpha}=\square_{3}(I^{\alpha}),$
where $\alpha=(\alpha_{1},\alpha_{2})$ and $0\leq|\alpha|\leq 3$.
Then the field of natural differential invariants for differential operators
of the third order in the domain $\mathcal{O}(I_{1},I_{2})$ is generated by
invariants $I_{1},I_{2},J^{\alpha}$ and all their Tresse derivatives
$\frac{d^{l}J^{\alpha}}{dI_{1}^{l_{1}}dI_{2}^{l_{2}}},$
where $l=l_{1}+l_{2}$.
Thus, to obtain the field of natural differential invariants of linear regular
differential operators of order 3, it is enough to have two natural
differential invariants $I_{1}$, $I_{2}$ in general position.
The invariants can be obtained by various methods.
For example, as the invariant $I_{1}$ one can take the free term $u_{0}$ of
the universal operator $\square_{3}$, or $I_{1}$ is natural invariant such
that $I_{1}(A)$ is the natural convolution of the torsion form $\theta$ of the
Wagner connection and the subsymbol $\sigma_{1,A}$.
As the natural invariant $I_{2}$ one can take, for example, the invariant
$\square_{3}(I_{1})$.
#### 3.6.1.
Let $\mathbf{F}_{k}^{(3)}$ be the field of all natural differential invariants
of order $\leq k$ of linear scalar differential operators of order $\leq 3$ on
$M$.
The $\mathcal{G}(M)$-action on $M$ is transitive and, therefore, the set of
all such invariants forms an $\mathbb{R}$-field $\mathbf{F}_{k}^{(3)}$.
The natural projections
$\chi_{k,l}\colon\mathbf{J}^{k}(\chi)\rightarrow\mathbf{J}^{l}(\chi),\;k>l$,
define the embeddings
$\mathbf{F}_{l}^{(3)}\hookrightarrow\mathbf{F}_{k}^{(3)}$ and their inductive
limit $\mathbf{F}_{*}^{(3)}$ is the field of natural differential invariants
of linear differential operators on the manifold $M$.
Remark that the $\mathcal{G}(M)$-action on differential operators satisfies
the conditions of the Lie-Tresse theorem (see [2]) and, therefore, the field
$\mathbf{F}_{*}^{(3)}$ separates regular $\mathcal{G}(M)$-orbits.
## 4\. Linear differential operators on $M\times\mathbb{R}$
Consider now linear differential operators $A$ of the third order on the
manifold $\mathbf{J}^{0}\left(\pi\right)=M\times\mathbb{R}$, that satisfy the
following condition:
$\big{(}A-A(1)\big{)}(y)=0,$
where $y$ is the fibrewise coordinate on $\mathbf{J}^{0}(\pi)$.
In local coordinates $(x_{1},x_{2})$ on $M,$ we have representation (2) of
operators of such type:
$A=\sum\limits_{i,j,k}a_{ijk}\left(x,y\right)\frac{\partial^{3}}{\partial
x_{i}\partial x_{j}\partial
x_{k}}+\sum\limits_{i,j}a_{ij}\left(x,y\right)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\\\
+\sum\limits_{i}a_{i}\left(x,y\right)\frac{\partial}{\partial
x_{i}}+a_{0}\left(x,y\right).$
The module of these operators we will denote by
$\mathit{Diff}_{3}\left(M,\mathbb{R}\right)$ and by
$\mathbb{\zeta}\colon\mathbf{Diff}_{3}\left(M,\mathbb{R}\right)\longrightarrow
M\times\mathbb{R}$
we will denote the corresponding bundle of these operators.
As before, elements of the module of $\mathit{Diff}_{3}(M,\mathbb{R})$ are
just sections of the bundle $\mathbb{\zeta}$ with the correspondence
$\mathit{Diff}_{3}\left(M,\mathbb{R}\right)\equiv
C^{\infty}(\mathbb{\zeta}),\quad A\mapsto s_{A},$
where $s_{A}\left(a,y\right)=A_{\left(a,y\right)}$ and $\left(a,y\right)\in
M\times\mathbb{R}$.
The diffeomorphism group $\mathcal{G}(M)$ acts by prolongation
$\phi\rightarrow\phi^{(0)}$ on the manifold
$\mathbf{J}^{0}(\pi)=M\times\mathbb{R}$, preserves the function $y$, and
therefore acts in the bundle $\mathbb{\zeta}$ as well as in the $k$-jet
bundles
$\mathbb{\zeta}_{k}\colon\mathbf{J}^{k}\left(\mathbb{\zeta}\right)\rightarrow
M\times\mathbb{R}$.
### 4.1. Linear differential operators $\mathbf{A_{f}}$
Given an operator $A\in\mathit{Diff}_{3}(M,\mathbb{R})$ and a function $f\in
C^{\infty}(M)$ we define the operator
$A_{f}=s_{f}^{\ast}\circ A\circ\pi^{\ast}\in\mathit{Diff}_{3}(M)$
as the operator that corresponds to the restriction of the section $s_{A}$ to
the graph of the function $f$. Here $s_{f}\colon M\rightarrow
M\times\mathbb{R}$ is the section of the bundle $\pi$ that corresponds to the
function $f$.
In local coordinates, we get the above representation (3) for this type of
operators:
$A_{f}=\sum\limits_{i,j,k}a_{ijk}\left(x,f\right)\frac{\partial^{3}}{\partial
x_{i}\partial x_{j}\partial
x_{k}}+\sum\limits_{i,j}a_{ij}\left(x,f\right)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\\\
+\sum\limits_{i}a_{i}\left(x,f\right)\frac{\partial}{\partial
x_{i}}+a_{0}\left(x,f\right).$
### 4.2. Weakly nonlinear differential operators
Define now thespace of weakly nonlinear operators of the third order
$\mathit{Diff}_{3}^{w}(M)$ as the space of differential operators $A_{w}$ on
$C^{\infty}\left(M\right)$ of the form
$A_{w}\left(f\right)=A_{f}\left(f\right),$ (11)
where $A\in\mathit{Diff}_{3}\left(M,\mathbb{R}\right)$ and $f\in
C^{\infty}(M)$.
###### Proposition 6.
The mappings
$\Theta\colon\mathit{Diff}_{3}(M,\mathbb{R})\times
C^{\infty}(M)\longrightarrow\mathit{Diff}_{3}(M),\quad\Theta\colon(A,f)\mapsto
A_{f},$
and
$\Theta_{w}\colon\mathit{Diff}_{3}(M,\mathbb{R})\longrightarrow\mathit{Diff}_{3}^{w}(M),\quad\Theta_{w}\colon
A\mapsto A_{w},$
are natural in the following sense :
$\displaystyle\phi_{\ast}\left(A,f\right)$ $\displaystyle=$
$\displaystyle\left(\phi_{\ast}^{\left(0\right)}\left(A\right),\phi_{\ast}\left(f\right)\right),$
$\displaystyle\phi_{\ast}\left(A_{f}\right)$ $\displaystyle=$
$\displaystyle\big{(}\phi_{\ast}^{(0)}(A)\big{)}_{\phi_{\ast}(f)},$
and
$\phi_{\ast}(A_{w})=\big{(}\phi_{\ast}^{(0)}(A)\big{)}_{w},$
for all diffeomorphisms $\phi\in\mathcal{G}\left(M\right)$.
###### Proof.
The proof is almost verbatim repetition of the proof of Proposition 1 in [5].
∎
## 5\. Related pairs and their invariants
The above proposition has the following consequences:
* •
By differential $\mathcal{G}(M)$-invariants of weakly nonlinear operators we
will mean differential $\mathcal{G}(M)$-invariants of operators
in$\mathit{Diff}_{3}(M,\mathbb{R})$, that are $\mathcal{G}(M)$-invariant
functions on jet spaces $\mathbf{J}^{k}(\mathbb{\zeta})$.
* •
In what follows, we will require that operators
$A\in\mathit{Diff}_{3}(M,\mathbb{R})$ under consideration are rational in $y$
(as sections of the bundle $\mathbb{\zeta}$). Also, by differential invariants
of rational weakly nonlinear operators we mean $\mathcal{G}(M)$-invariant
functions on jet spaces $\mathbf{J}^{k}(\mathbb{\zeta})$ that are rational
along fibres of the projections
$\pi\circ\mathbb{\zeta}_{k}\colon\mathbf{J}^{k}(\mathbb{\zeta})\rightarrow M$.
* •
The group $\mathcal{G}\left(M\right)$ acts transitively on the manifold $M$
and therefore rational differential invariants of order $\leq k$ for rational
weakly nonlinear operators form a field $\mathbf{F}_{k}^{w}$. We have the
embedding
$\mathbb{\zeta}_{k,l}^{\ast}:\mathbf{F}_{l}^{w}\hookrightarrow\mathbf{F}_{k}^{w},$
if $k\geq l,$ where
$\mathbb{\zeta}_{k,l}:\mathbf{J}^{k}\left(\mathbb{\zeta}\right)\rightarrow\mathbf{J}^{l}\left(\mathbb{\zeta}\right)$
are the natural projections, and $\mathbf{F}^{w}=\bigcup\limits_{k\geq
0}\mathbf{F}_{k}^{w}$ is the field of all rational differential invariants of
rational weakly nonlinear operators.
We consider the following vector bundles over manifold
$\mathbf{J}^{0}\left(\pi\right)=M\times\mathbb{R}$:
1. (1)
jet bundles of functions on manifold:
$\pi_{l,0}\colon\mathbf{J}^{l}(\pi)\longrightarrow\mathbf{J}^{0}(\pi),$
2. (2)
jet bundles of operators:
$\mathbb{\zeta}_{k}\colon\mathbf{J}^{k}(\mathbb{\zeta})\longrightarrow
M\times\mathbb{R},$
3. (3)
the Whitney sum of vector bundles $\mathbb{\zeta}_{k}$ and $\pi_{l,0}$
$r_{k,l}\colon\mathbf{RP}^{k,l}=\mathbb{\zeta}_{k}\oplus_{M\times\mathbb{R}}\pi_{l,0}\longrightarrow
M\times\mathbb{R}.$
We call the last bundle as the bundle of related pairs.
Elements of the total space of this bundle are related pairs
$\left([A]_{\left(x,y\right)}^{k},[f]_{x}^{l}\right)$ consisting of $k$-jet
$[A]_{\left(x,y\right)}^{k}$ of operator
$A\in\mathit{Diff}_{3}\left(M,\mathbb{R}\right)$ at the point
$\left(x,y\right)\in M\times\mathbb{R}$ and $l$ -jet $[f]_{x}^{l}$ of function
$f\in C^{\infty}(M)$ at the point $x\in M$ under condition that
$f\left(x\right)=y$.
The group $\mathcal{G}(M)$ acts by prolongations in the bundles $r_{k,l}$ and
by invariants of this action (or invariants of related pairs) we mean
functions on the total space
$\mathbf{CP}^{k,l}=\mathbf{J}^{k}(\zeta)\oplus_{M\times\mathbb{R}}\mathbf{J}^{l}(\pi)$.
that are $\mathcal{G}\left(M\right)$-invariant and rational along fibers of
the projection $\pi\circ r_{k,l}:\mathbf{RP}^{k,l}\rightarrow M$.
Because of transitivity of $\mathcal{G}(M)$-action on $M,$ all such functions
are completely determined by their values on fibre $\left(\pi\circ
r_{k,l}\right)^{-1}(a)$ at a base point $a\in M$. Therefore,
$\mathcal{G}\left(M\right)$-invariants of related pairs form an
$\mathbb{R}$-field $\mathbf{F}_{k,l},$ that is a subfield of the filed
$\mathbf{Q}_{k,l}$ of all rational functions on the fibre $\left(\pi\circ
r_{k,l}\right)^{-1}\left(a\right)$.
The natural projections
$\mathbf{CP}^{k^{\prime},l^{\prime}}\rightarrow\mathbf{CP}^{k,l},$ where
$k\leq k^{\prime},\,l\leq l^{\prime}$, give us embeddings of fields
$\mathbf{F}_{k,l}\subset\mathbf{F}_{k^{\prime},l^{\prime}}$,
$\mathbf{Q}_{k,l}\subset\mathbf{Q}_{k^{\prime},l^{\prime}}$ and we define the
fields $\mathbf{F}_{l},\;\mathbf{Q}_{l}$ by the inductive limits:
$\mathbf{F}_{l}=\bigcup\limits_{k\geq
0}\mathbf{F}_{k,l},\quad\mathbf{Q}_{l}=\bigcup\limits_{k\geq
0}\mathbf{Q}_{k,l}.$
Remark that $\mathbf{F}_{0}=\mathbf{F}^{w}$ is just the field of rational
differential invariants of rational weakly nonlinear differential operators.
Below we will discuss various methods of finding invariants, but first of all
we remark that the vector field $\partial_{y}$ on
$\mathbf{J}^{0}(\pi)=M\times\mathbb{R}$ is an invariant of the
$\mathcal{G}(M)$-action. Therefore its $l$-th prolongation
$\partial_{y}^{(l)}$ on $\mathbf{J}^{l}(\pi)$ is also
$\mathcal{G}(M)$-invariant. The same is valid for the total derivation
$\displaystyle\frac{d}{dy}$ that acts in $\mathbf{J}^{\infty}(\zeta)$. All
together they define $\mathcal{G}\left(M\right)$-invariant derivation $\nabla$
in the fields $\mathbf{Q}_{l}$, as well as in $\mathbf{F}_{l}$, where
$\nabla(\alpha\beta)=\frac{d\alpha}{dy}\beta+\alpha\partial_{y}^{(l)}(\beta),$
$\alpha\in\mathbf{Q}_{0}$ and $\beta$ is a function on $\mathbf{J}^{l}(\pi)$.
## 6\. Construction of invariants
At first we consider $y$ as a parameter and identify operators
$A\in\mathit{Diff}_{3}(M,\mathbb{R})$ with 1-parametric family $A_{y}$ of
operators in $\mathrm{Diff}_{3}(M)$.
Remark that $y$ is a $\mathcal{G}(M)$-invariant. Therefore, for any invariant
$I\in\mathbf{F}_{k}^{(3)}$ of linear differential operators on manifold $M$,
the function $\widehat{I}\colon J^{k}(\zeta)\rightarrow\mathbb{R}$, where
$\widehat{I}\big{(}[A]^{k}_{(x,y)}\big{)}=I\big{(}[A_{y}]^{k}_{x}\big{)}$
is a $\mathcal{G}(M)$-invariant too.
Thus we get a mapping
$\mathbf{F}^{(3)}_{*}\longrightarrow\mathbf{F}_{0}=\mathbf{F}^{w},\quad
I\mapsto\widehat{I},$
that immediately gives us invariants of weakly nonlinear operators.
Moreover, application of the invariant derivation $\nabla$ essentially
increases their amount.
As we have seen, Proposition 6, the mapping
$\Theta_{l}\colon\mathbf{RP}^{l,l}\longrightarrow\mathbf{J}^{l}(\chi),\quad\Theta_{l}\colon([A]_{(x,y)}^{l},[f]_{x}^{l})\mapsto[A_{f}]_{x}^{l},$
commutes with the $\mathcal{G}(M)$-action. Therefore,
$\Theta_{l}^{*}(I)\in\mathbf{F}_{l},$
i.e., it is an invariant of related pairs for any invariant
$I\in\mathbf{F}_{l}^{(3)}$.
#### 6.0.1. Descent procedure
1
To get invariants of weakly nonlinear operators from invariants of related
pairs we consider the following descent procedure for invariants
$\mathbf{F}_{l}\longrightarrow\mathbf{F}^{w}.$
Let $I_{0}\in\mathbf{F}_{l}$ be an invariant of related pairs and let
$I_{1}=\nabla(I_{0})\in\mathbf{F}_{l},\ldots,I_{i+1}=\nabla(I_{i})\in\mathbf{F}_{l}$
be its invariant derivatives. Remark that the transcendence degree of the
field $\mathbf{Q}_{l}$ over $\mathbf{Q}_{0}$ equals $N=\dim\pi_{l,0}$.
Therefore, ([8]), there are polynomial relations between rational functions
${I_{0},...,I_{N}}$. Denote by
$J\left(I\right)\subset\mathbf{Q}_{0}[X_{0},...,X_{N}]$ the ideal of these
relations.
###### Theorem 7.
Let $b_{1},..,b_{r}\in\mathbf{Q}_{0}[X_{0},...,X_{N}]$ be the reduced Gröbner
basis in the ideal $J(I)$ with respect to the standard lexicographic order.
Then the coefficients of polynomials $b_{i}$ are natural invariants of weakly
nonlinear operators.
###### Proof.
The action of the diffeomorphism group preserves the ideal $J\left(I\right)$
as well as the lexicographic order. The reduced Gröbner basis in an ideal with
respect to the lexicographic order is unique ([1]), and therefore, the action
preserves elements of the basis and their coefficients. ∎
## 7\. Example $n=1,k=3$
Let $M=\mathbb{R}$.
Then in coordinate $x$ on M, an operator $A\in\mathrm{Diff}_{3}(M)$ has the
form
$A=a_{3}\partial^{3}+a_{2}\partial^{2}+a_{1}\partial+a_{0},$
where the all coefficients are smooth functions on $x$ and
$\partial=\partial_{x}$.
The symbol $a_{3}\partial^{3}$ of $A$ defines an invariant connection on the
line $\mathbb{R}$ with the Christoffel coefficient $\Gamma$:
$\Gamma=-\frac{a^{\prime}_{3}}{3a_{3}},$
the total symbol $\sigma_{(3)}$ of $A$ with respect to the Wagner connection
$\nabla$ is the following:
$\sigma_{(3)}=\sigma_{3}+\sigma_{2}+\sigma_{1}+\sigma_{0},$
where
$\displaystyle\sigma_{3}=$ $\displaystyle a_{3}\partial^{3},$ (12)
$\displaystyle\sigma_{2}=$ $\displaystyle(a_{2}-a^{\prime}_{3})\partial^{2},$
$\displaystyle\sigma_{1}=$
$\displaystyle\big{(}a_{1}-\frac{2(a^{\prime}_{3})^{2}a_{3}+3a^{\prime}_{3}-a^{\prime\prime}_{3}a_{3}}{9a_{3}}-(a_{2}-a^{\prime}_{3})\frac{a^{\prime}_{3}}{3a_{3}}\big{)}\partial,$
$\displaystyle\sigma_{0}=$ $\displaystyle a_{0}$
and the natural splitting of the operator $A$ is the following:
$A=\widehat{\sigma_{(3)}}=\widehat{\sigma_{3}}+\widehat{\sigma_{2}}+\widehat{\sigma_{1}}+\widehat{\sigma_{0}},$
(13)
where the differential operators $\widehat{\sigma_{k}}$ are the quantizations
of the symbols $\sigma_{k}$, see [3]:
$\displaystyle\widehat{\sigma_{3}}$
$\displaystyle=a_{3}\partial^{3}+a^{\prime}_{3}\partial^{2}+\frac{2(a^{\prime}_{3})^{2}a_{3}+3a^{\prime}_{3}-a^{\prime\prime}_{3}a_{3}}{9a_{3}}\,\partial,$
$\displaystyle\widehat{\sigma_{2}}$
$\displaystyle=(a_{2}-a^{\prime}_{3})\big{(}\partial^{2}+\frac{a^{\prime}_{3}}{3a_{3}}\,\partial\big{)},$
$\displaystyle\widehat{\sigma_{1}}$
$\displaystyle=\big{(}\,a_{1}-\frac{2(a^{\prime}_{3})^{2}a_{3}+3a^{\prime}_{3}-a^{\prime\prime}_{3}a_{3}}{9a_{3}}-(a_{2}-a^{\prime}_{3})\frac{a^{\prime}_{3}}{3a_{3}}\,\big{)}\partial,$
$\displaystyle\widehat{\sigma_{0}}$ $\displaystyle=a_{0}.$
Therefore, we get from (12) the following $\mathcal{G}(M)$-invariants of
operators $A$:
$\displaystyle I_{0}$ $\displaystyle=a_{0},$ $\displaystyle I_{1}$
$\displaystyle=\big{\langle}\,\sigma_{1},\,da_{0}\,\big{\rangle}=\big{(}\,a_{1}-\frac{2(a^{\prime}_{3})^{2}a_{3}+3a^{\prime}_{3}-a^{\prime\prime}_{3}a_{3}}{9a_{3}}-(a_{2}-a^{\prime}_{3})\frac{a^{\prime}_{3}}{3a_{3}}\,\big{)}a^{\prime}_{0},$
$\displaystyle I_{2}$
$\displaystyle=\big{\langle}\,\sigma_{2},\,da_{0}^{2}\,\big{\rangle}=(a_{2}-a^{\prime}_{3})(a_{0}^{\prime})^{2},$
$\displaystyle I_{3}$
$\displaystyle=\big{\langle}\,\sigma_{3},\,da_{0}^{3}\,\big{\rangle}=a_{3}(a_{0}^{\prime})^{3}.$
Now take an operator $A\in\mathrm{Diff}_{3}(M,\mathbb{R})$.
$\displaystyle A$
$\displaystyle=a_{3}(x,y)\partial_{x}^{3}+a_{2}(x,y)\partial_{x}^{2}+a_{1}(x,y)\partial_{x}+a_{0}(x,y),$
$\displaystyle A_{f}$
$\displaystyle=a_{3}(x,f)\partial_{x}^{3}+a_{2}(x,f)\partial_{x}^{2}+a_{1}(x,f)\partial_{x}+a_{0}(x,f),\quad
f\in C^{\infty}(M).$
Then, the corresponding invariants of relates pairs are the following:
$\displaystyle I_{0}=$ $\displaystyle a^{0},$ $\displaystyle I_{1}=$
$\displaystyle\Big{(}\,a_{1}-\frac{2(a_{3,x}+a{3,y}f^{\prime})^{2}a_{3}+3(a_{3,x}+a_{3,y}f^{\prime})}{9a_{3}}$
$\displaystyle-\frac{\big{(}a_{3,xx}+2a_{3,xy}f^{\prime}+a_{3,yy}(f^{\prime})^{2}+(a_{3,x}+a_{3,y})f^{\prime\prime}\big{)}a_{3}}{9a_{3}}$
$\displaystyle-\big{(}a_{2}-(a_{3,x}+a_{3,y}f^{\prime})\big{)}\frac{(a_{3,x}+a_{3,y}f^{\prime})}{3a_{3}}\,\Big{)}(a_{0,x}+a_{0,y}f^{\prime}),$
$\displaystyle I_{2}=$
$\displaystyle\big{(}a_{2}-a_{3,x}-a_{3,y}f^{\prime}\big{)}(a_{0,x}+a_{0,y}f^{\prime})^{2},$
$\displaystyle I_{3}=$ $\displaystyle a_{3}(a_{0,x}+a_{0,y}f^{\prime})^{3}.$
From the expression of $I_{3}$, we get
$f^{\prime}=\big{(}(\,I_{3}/a_{3}\,)^{1/3}-a_{0,x}\,\big{)}/a_{0,y}.$
Substituting it into $I_{2}$, we get a relation of the form
$KI_{2}+K_{1}I_{3}^{1/3}+K_{2}I_{3}^{2/3}+K_{3}I_{3}+K_{0}=0,$
where
$\frac{K_{1}}{K},\quad\frac{K_{2}}{K},\quad\frac{K_{3}}{K},\quad\frac{K_{0}}{K}$
are invariants of operators $A\in\mathrm{Diff}_{3}(M,\mathbb{\mathbb{R}})$,
i.e. invariants of weakly nonlinear operators.
## 8\. Equivalence of weakly nonlinear operators
Let $z$ be natural differential invariant of weakly nonlinear operators and
$A\in\mathit{Diff}_{3}(M,\mathbb{R})$.
Then the values $z(A,y_{0})=s_{A}^{\ast}(z)$ is function rational in $y$ with
coefficients in $C^{\infty}(M)$. Values $z(A,y_{0})$ of this function for a
given value $y_{0}$ is a smooth function on $M$.
We say that the operator $A$ is in general position if for any point $a\in M$
there are: natural invariants $z_{1},z_{2}$, a value $y_{0}$ of $y$, and a
neighborhood $U\subset M$, $a\in U$, such that the mapping
$Z_{A,y_{0}}\colon
U\mathcal{\longrightarrow}\,\mathbf{D}\subset\mathbb{R}^{2},\quad
Z_{A,y_{0}}\colon x\mapsto\big{(}z_{1}(A,y_{0})(x),z_{2}(A,y_{0})(x)\big{)}$
is a local diffeomorphism.
We say, that the mapping $Z_{A,y_{0}}$ is a natural chart of the operator $A$
and the functions $z_{1}(A,y_{0}),z_{2}(A,y_{0})$ are natural coordinates on
$U$.
We call the atlas of these charts $\\{U_{\alpha},\phi_{\alpha}\colon
U_{\alpha}\rightarrow\mathbf{D}_{\alpha}\subset\mathbb{R}^{2}\\}$ natural if
coordinates
$\phi_{\alpha}=\big{(}z_{1}^{\alpha}(A,y_{0}),z_{2}^{\alpha}(A,y_{0})\big{)}$
are given by distinct invariants
$\big{(}z_{1}^{\alpha}(A,y_{0}),z_{2}^{\alpha}(A,y_{0})\big{)}\neq\big{(}z_{1}^{\beta}(A,y_{0}),z_{2}^{\beta}(A,y_{0})\big{)}$,
when $\alpha\neq\beta$.
We denote by $\mathbf{D}_{\alpha\beta}=\phi_{\alpha}(U_{\alpha}\cap
U_{\beta})$ and we assume that domains $\mathbf{D_{\alpha}}$ and
$\mathbf{D_{\alpha\beta}}$ are connected and simply connected.
Let $A_{\alpha}=\phi_{\alpha*}(A|_{U_{\alpha}})$,
$A_{\alpha\beta}=\phi_{\alpha*}(A|_{U_{\alpha}\cap U_{\beta}})$ be the images
of the operator $A$ in these coordinates.
Then $\phi_{\alpha\beta*}(A_{\alpha\beta})=A_{\beta\alpha}$, where
$\phi_{\alpha\beta}=\phi_{\beta}\circ\phi_{\alpha}^{-1}\colon\mathbf{D_{\alpha\beta}}\rightarrow\mathbf{D_{\beta\alpha}}$
are the transition mapping.
Take now two operators $A,A^{\prime}\in\mathit{Diff}_{3}(M,\mathbb{R})$ and
consider their natural charts $Z_{A,y_{0}}$ and
$Z_{A^{\prime},y_{0}^{\prime}}$.
The $\mathcal{G}(M)$-equivalence of these operators means that the local
diffeomorphism $Z_{A^{\prime},y_{0}^{\prime}}^{-1}\circ Z_{A,y_{0}}$ does not
depend on choice of $y_{0}$ and $y_{0}^{\prime}$. In this case we will say
that these charts are coordinated.
Summarizing we get the following result.
###### Theorem 8.
Let operators $A,A^{\prime}\in\mathit{Diff}_{3}(M,\mathbb{R})$ be in general
position, then they are $\mathcal{G}(M)$-equivalent if and only if the
following conditions hold:
1. (1)
the mappings $\\{\phi^{\prime}_{\alpha}\colon
U^{\prime}_{\alpha}\rightarrow\mathbf{D}_{\alpha}\\}$,
where
$\phi^{\prime}_{\alpha}=\big{(}z_{1}^{\alpha}(A^{\prime},y^{\prime}_{0}),z_{2}^{\alpha}(A^{\prime},y^{\prime}_{0})\big{)}$
and $U^{\prime}_{\alpha}=(\phi^{\prime}_{\alpha})^{-1}(\mathbf{D}_{\alpha})$,
constitute a natural atlas for the operator $A^{\prime}$,
2. (2)
the charts $\phi_{\alpha}\in\\{\phi_{\alpha}\colon
U_{\alpha}\rightarrow\mathbf{D}_{\alpha}\\}$ and
$\phi^{\prime}_{\alpha}\in\\{\phi^{\prime}_{\alpha}\colon
U^{\prime}_{\alpha}\rightarrow\mathbf{D}_{\alpha}\\}$ are coordinated,
3. (3)
$\phi^{\prime}_{\alpha\beta}=\phi_{\alpha\beta}$,
4. (4)
$A_{\alpha}=\phi^{\prime}_{\alpha*}\big{(}A^{\prime}\big{|}_{U^{\prime}_{\alpha}}\big{)}$
and
$A_{\alpha\beta}=\phi^{\prime}_{\alpha*}\big{(}A^{\prime}\big{|}_{U^{\prime}_{\alpha\cap\beta}}\big{)}$.
###### Proof.
Any diffeomorphism $\psi\colon M\rightarrow M$ such that
$\psi_{*}(A)=A^{\prime}$ transform natural atlas to the natural one and
because of $\psi^{*-1}\big{(}z(A)\big{)}=z\big{(}\psi_{*}(A)\big{)}$, for any
natural invariant $z$, this diffeomorphism has the form of the identity map in
the natural coordinates. ∎
###### Corollary 9.
Let operators $A$ and $A^{\prime}\in\mathit{Diff}_{3}(M,\mathbb{R})$ be in
general position. Then the weakly nonlinear differential operators $A_{w}$ and
$A^{\prime}_{w}$ are $\mathcal{G}(M)$\- equivalent if and only if the
operators $A$ and $A^{\prime}$ are $\mathcal{G}(M)$-equivalent.
## References
* [1] Cox, D.; Little, J.; O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, (1997), ISBN 0-387-94680-2.
* [2] Kruglikov, Boris, Lychagin, Valentin, Global Lie-Tresse theorem, Selecta Math. (N.S.) 22 (2016), no. 3, 1357–1411.
* [3] Lychagin, V.V., Yumaguzhin, V.A., On equivalence of third order linear differential operators on two-dimensional manifolds, Journal of geometry and physics, Vol. 146, December 2019, 103507.
* [4] Valentin Lychagin, Valeriy Yumaguzhin, On structure of linear differential operators, acting in line bundles, Journal of geometry and physics, Volume 148, February 2020, 103549.
* [5] Valentin Lychagin, Valeriy Yumaguzhin, Natural differential invariants and equivalence of nonlinear second order differential operators, Journal of geometry and physics, 178(2022)104549.
* [6] Peter J. Olver, Applications of Lie groups to differential equations, Springer-Verlag
* [7] Wagner,V.V., Two dimensional space with cubic metric, Scientific notes of Saratov State University, Vol. 1(XIV), Ser. FMI, No. 1, 1938. (in Russian).
* [8] Zariski O., Samuel P.:Commutative Algebra, vol. 1, 2, Van Nostrand, 1960.
|
# Universal scaling regimes in rotating fluid turbulence
Abhik Basu abhik.basu<EMAIL_ADDRESS>Theory Division, Saha
Institute of Nuclear Physics, Calcutta 700064, West Bengal, India Jayanta K
Bhattacharjee<EMAIL_ADDRESS>Department of Theoretical
Physics, Indian Association for the Cultivation of Science, 2A and 2B Raja S C
Mullick Road, Calcutta 700032, West Bengal, India
###### Abstract
We analyse the scaling properties of the energy spectra in fully developed
incompressible turbulence in forced, rotating fluids in three dimensions (3D),
which are believed to be characterised by universal scaling exponents in the
inertial range. To elucidate the scaling regimes, we set up a scaling analysis
of the 3D Navier-Stokes equation for a rotating fluid that is driven by large-
scale external forces. We use scaling arguments to extract the scaling
exponents, which characterise the different scaling regimes of the energy
spectra. We speculate on the intriguing possibility of two-dimensionalisation
of 3D rotating turbulence within our scaling theory. Our results can be tested
in large scale simulations and relevant laboratory-based experiments.
## I Introduction
Nonequilibrium systems are described by the appropriate equations of motion
for the relevant dynamical variables and exhibit much richer universal
behavior than usually observed in equilibrium critical dynamics hal .
Hydrodynamic turbulence in fluids, described by the Navier-Stokes equation
land ; frish for the evolution of the velocity field $\bf v$, is a prime
example of an out of equilibrium system, due to the external drive acting on
the fluids. Interestingly, fully developed fluid turbulence in three- (3D) and
two- (2D) dimensions show markedly different behavior: In 3D, the energy
spectra follow the well-known K41 result for homogeneous and isotropic 3D
hydrodynamics turbulence where the one-dimensional energy spectra $E(k)\sim
k^{-5/3}$ (hereafter K41) in the inertial range, where $k$ is a wavevector k41
. This K41 result is quite robust and universal, and found in wide-ranging
natural systems, e.g., shear flows shear , viscoelastic fluids visco and jet
flows jet . In contrast, 2D turbulence is characterised by an inverse cascade
of energy at very large scales with $E(k)\sim k^{-5/3}$, and forward cascade
of enstrophy with $E(k)\sim k^{-3}$ at intermediate scales 2d1 ; 2d2 ; 2d3 ;
2d4 ; 2d5 .
Rotating turbulence, i.e., turbulence in a rotating fluid, is a naturally
occurring phenomenon in many astrophysical and geophysical flows, as well as
in laboratory-based engineering fluid flows. The presence of the Coriolis
forces is the distinctive feature of rotating turbulence, which should affect
the large-scale scaling properties of rotating turbulence. In spite of
extensive studies, there is still no good agreement on the scaling of the
energy spectra in rotating turbulence, in particular how the Coriolis forces
affect the scaling of the spectra at very large scales. Varieties of
analytical, numerical, or experimental investigations of either the forced or
the decaying rotating turbulent fluid suggest that the kinetic energy spectra
in the rotation-dominated small-$k$ regions should scale as $E(k)\sim
k^{-m},\,m\in(2,3)$ 2ds1 ; 2ds2 ; 2ds3 ; 2ds4 ; 2ds5 ; 2ds6 ; 2ds7 ; 2ds8 ;
2ds9 ; 2ds10 ; 2ds11 ; 2ds12 ; 2ds13 ; 2ds14 ; 2ds15 . Recent perturbative
studies indicate that the one-dimensional kinetic energy spectra made out of
the velocity component parallel to the rotation axis scales as $k^{-5/3}$,
indistinguishable from the K41 prediction. In contrast, the one-dimensional
kinetic energy spectra made out of the velocity components lying in a plane
normal to the rotation axis scales as $k^{-3}$, different from the K41 scaling
ab-jkb-rot . The precise forms of the scaling of the energy spectra in
rotating turbulence however is still not well-settled.
There is a degree of formal similarity between the (linearised) equations of
motion of rotating turbulence, which is nothing but the Navier-Stokes equation
in a rotating frame (see below) and the equations for magnetohydrodynamic
turbulence (MHD) in the presence of a mean magnetic field $B_{0}$ jackson ;
arnab . In the former case, the Coriolis forces lead to oscillatory modes,
whereas in the MHD case, a non-zero $B_{0}$ gives rise to propagating Alfv́en
waves. Strong Alfv́en waves are known to make the energy spectra in MHD
anisotropic, and change the scaling as well jkb-mhd . In the same vein, strong
Coriolis forces should make the scaling of energy spectra in rotating
turbulence anisotropic and also different from its isotropic counterpart
(i.e., the K41 scaling).
In this work, we revisit the universal scaling of energy spectra in forced,
statistically steady rotating turbulence in its inertial range. To this end,
we have set up a scaling theory to study the scaling of the energy spectra in
the inertial range. We cover both the weak and strong rotation limits. In the
former case, unsurprisingly, the K41 result is found. With stronger rotation,
anisotropic scaling with different exponents ensues. In particular, in the
wavevector region $k_{\perp}\gg k_{\parallel}$, our scaling theory gives the
scaling of the 2D spectra $E(k_{\perp},k_{\parallel})$, where ${\bf
k}_{\perp}$ and $k_{\parallel}$ are the components of the wavevector $\bf k$
in the plane perpendicular to the rotation axis (here $\hat{z}$-axis) and
along the rotation axis, respectively. We find $E(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-5/2}k_{\parallel}^{-1/2}$ for $k_{\perp}\gg k_{\parallel}$, which
agrees with the Kuznetsov–Zakharov–Kolmogorov spectra predicted by the weak
inertial-wave turbulence theory for the rotating fluids 2ds7 . We show that
this result is unaffected by nonlinear fluctuation corrections at the one-loop
order. We further demonstrate that this result could be obtained by demanding
that the cascade of the kinetic energy flux is hindered by a non-zero
helicity, which is naturally present in a rotating fluid. In the opposite
limit of $k_{\perp}\ll k_{\parallel}$, we get $E(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-1}k_{\parallel}^{-2}$. We also show perturbatively in the rotation
$\Omega$ that the kinetic energy flux is indeed reduced by it. The remainder
of the article is organised as follows. In Sec. II, we set up the forced
Navier-Stokes equation in a rotating fluid. Then in Sec. III we set up the
scaling arguments. Then next in Sec. III.1, we revisit the K41 scaling scaling
in an isotropic, nonrotating fluid turbulence, and show how are scaling theory
reproduces it. Next, in Sec. III.2, we show that for weak rotation, the energy
spectra again show the K41 scaling. Then in Sec. III.3 we study the 2D
anisotropic energy spectra in the opposite limit of large $\Omega$. In Sec. IV
we discuss and summarise our results. We provide some technical results,
including a perturbative demostration of the reduction of the kinetic energy
flux by helicity, in the Appendix for interested readers.
## II Turbulence in a rotating fluid
The Navier-Stokes equation for the velocity field ${\bf v}({\bf r},t)$ in a
rotating frame with rotation ${\boldsymbol{\Omega}}=\omega\hat{z}$ is given by
$\frac{\partial\bf v}{\partial t}+2({\boldsymbol{\Omega}}\times{\bf
v})+\lambda({\bf v}\cdot{\boldsymbol{\nabla}}){\bf
v}=-\frac{{\boldsymbol{\nabla}}p^{*}}{\rho}+\nu\nabla^{2}{\bf v}+{\bf f},$ (1)
where $p^{*}=p+\frac{1}{2}|{\boldsymbol{\Omega}}\times{\bf v}|^{2}$ is the
effective pressure. We assume ${\boldsymbol{\Omega}}=\Omega\hat{z}$, i.e., the
rotation is about the $z$-axis; see Fig. 1.
Figure 1: Geometry of the rotating fluid. We assume the rotation to be about
the $z$-axis. In this coordinate system, ${\bf k}_{\perp}=(k_{x},k_{y})$ and
$k_{\parallel}=k_{z}$.
In this case (1) may be written in terms of components as
$\displaystyle\frac{\partial v_{z}}{\partial t}+\lambda({\bf
v}\cdot{\boldsymbol{\nabla}})v_{z}=-\frac{\partial_{z}p^{*}}{\rho}+\nu\nabla^{2}v_{z}+f_{z},$
(2) $\displaystyle\frac{\partial v_{x}}{\partial t}-2\Omega v_{y}+\lambda({\bf
v}\cdot{\boldsymbol{\nabla}})v_{x}=-\frac{\partial_{x}p^{*}}{\rho}+\nu\nabla^{2}v_{x}+f_{x},$
(3) $\displaystyle\frac{\partial v_{y}}{\partial t}+2\Omega v_{x}+\lambda({\bf
v}\cdot{\boldsymbol{\nabla}})v_{y}=-\frac{\partial_{y}p^{*}}{\rho}+\nu\nabla^{2}v_{y}+f_{y}.$
(4)
Here, $\lambda=1$. These equations in 3D admit two conserved quantities in the
inviscid limit: (i) kinetic energy $E=\int d^{3}x\rho\,v^{2}/2$ and (ii)
helicity $H=\int d^{3}x\,{\bf v}\cdot{\boldsymbol{\nabla}}\times{\bf v}={\bf
v}\cdot{\boldsymbol{\omega}}$, where
$\boldsymbol{\omega}\equiv{\boldsymbol{\nabla}}\times{\bf v}$ is the local
vorticity. In the viscous steady states, in a Kolmogorov-like picture
neglecting intermittency, $E$ and $H$ should have constant (i.e., scale
independent) fluxes. Clearly, $E/H$ has the dimension of a length, which
allows us to define a length-scale $l^{*}=E/H$. We assume the external forces
to be non-helical, i.e., no helicity injection by the forces. Thus helicity is
generated in the bulk only by the global rotation. We consider the
incompressible limit, i.e., the mass density $\rho=const.$, or equivalently
${\boldsymbol{\nabla}}\cdot{\bf v}=0$. At this stage, it is useful to set up
the notations. Below we use $\tilde{\omega}$ and $\tilde{\Omega}$ to denote
Fourier frequencies, while $\Omega$ and $\boldsymbol{\omega}$ represent the
global rotation frequency and vorticity, respectively.
## III Scaling analysis
To classify the scaling regimes, we first define the following dimensionless
numbers
(i) Rossby number $R_{o}=U/(2\Omega L)$,
(ii) Reynolds number $R_{e}=\frac{LU}{\nu}$ and
(iii) Ekman number $Ek=R_{o}/R_{e}=\nu/(2\Omega L^{2})$,
where $L$ is the linear system size, and $U$ is a typical velocity. We expect
to find two distinct scaling regimes as characterised by $R_{o}$ (or
$\Omega$):
(i) Weak rotation $\Omega\rightarrow 0$, or $R_{o}\rightarrow\infty$
(iii) Large rotation $\Omega\rightarrow\infty$, or $R_{o}\rightarrow 0$.
Since the rotation picks up a direction (the axis of rotation, here the
$z$-axis), system is generally anisotropic. We therefore construct an
anisotropic scaling theory of the system: we assume scaling under the
transformations
${\bf r}_{\perp}\rightarrow l_{\perp}{\bf r}_{\perp},\,z\rightarrow
l_{\perp}^{\xi}z,\,{\bf v}_{\perp}\rightarrow l_{\perp}^{a_{\perp}}{\bf
v}_{\perp},\,v_{z}\rightarrow l_{\perp}^{a_{z}}v_{z}.$ (5)
Here, ${\bf r}_{\perp}\equiv(x,\,y)$, ${\bf v}_{\perp}\equiv(v_{x},\,v_{y})$.
In a general anisotropic situation, $\xi\neq 1$. We also allow for the
possibility $a_{\perp}\neq a_{z}$, i.e., ${\bf v}_{\perp}$ and $v_{z}$ may not
scale in the same way under spatial rescaling. We further define time-scale
$t$ to scale as
$t\sim l_{\perp}^{\tilde{z}},$ (6)
where $\tilde{z}$ is a dynamic exponent. Furthermore, we define a
phenomenological dimensionless constant $\tilde{\Omega}$ by
$\tilde{\Omega}=\frac{[{\boldsymbol{\Omega}}]}{[{\boldsymbol{\omega}}]},$ (7)
Here, […] implies “in a dimensional sense” jkb-mhd . Clearly, the two limiting
cases $\tilde{\Omega}\rightarrow 0$ and $\tilde{\Omega}\rightarrow\infty$
phenomenologically correspond to $R_{o}\rightarrow\infty$ and
$R_{o}\rightarrow 0$.
We note that by balancing the Coriolis force terms against the advective
nonlinear terms, we can extract a length-scale $L_{o}$. In a scaling sense, we
set
$\Omega v\sim\frac{v^{2}}{L_{0}},$ (8)
giving $v\sim\Omega L_{0}$. Dimensionally speaking, energy dissipation
$\epsilon\sim\frac{v^{3}}{L_{0}}\sim\Omega^{3}L_{0}^{2}.$ (9)
This gives
$L_{0}\sim\sqrt{\frac{\epsilon}{\Omega^{3}}}.$ (10)
The corresponding wavevector $k_{0}\equiv 2\pi/L_{0}$ is the Zeeman
wavevector. For length-scales $L\gg L_{0}$ (or equivalently, for wavevector
$k\ll k_{0}$), we expect Coriolis force terms to be important, effects of the
rotation should be strong, and hence non-K41 spectra should follow. In the
opposite limit of $L\ll L_{0}$ (or $k\gg k_{0}$), Coriolis forces should be
irrelevant, and hence K41 scaling should follow. Thus a dual scaling is
believed to exist 2ds12 ; dual1 ; dual2 ; dual3 . In terms of the
dimensionless numbers, we are interested in $R_{e}\rightarrow\infty$ for fully
developed turbulence. Together with $R_{e}\rightarrow\infty$ (implying fully
developed turbulence) and $Ek\rightarrow 0$ (implying Coriolis forces
dominating over the viscous damping at large scales), we can have two
situations: (i) $R_{o}\rightarrow\infty$ for weak rotation, and (ii)
$R_{o}\rightarrow 0$ for strong rotation. Lastly, one has the dissipation
scale $\eta_{d}$, such that for length scales smaller than $\eta_{d}$, the
dissipation range ensues. Then in terms of the length scales defined above, we
can have the following scenarios. In a sufficiently large system, there should
be adequate scale separations, such that $\eta_{d}\ll L_{0}$, i.e., $L_{0}$
should belong to the inertial range. This should allow for both the scaling
regimes, viz. K41 and non-Kolmogorov scaling regimes to be observed.
### III.1 Nonrotating isotropic case
For a nonrotating, isotropic fluid, $\Omega=0$ in (3)-(4) gives the usual 3D
isotropic Navier-Stokes equation. Let us briefly revisit the extraction of the
K41 scaling by applying the scaling arguments on the usual 3D Navier-Stokes
equation first. Due to the isotropy of the system, we expect $\chi=1$
strictly, and make no distinction between $l_{\perp}$ and $l_{\parallel}$, the
rescaling factors of ${\bf r}_{\perp}$ and $z$ respectively: $l_{\perp}\sim
l_{\parallel}\sim l$. Demanding scale invariance jkb-mhd , we find
$\frac{\partial{\bf v}}{\partial t}\sim{\bf v}\cdot{\boldsymbol{\nabla}}{\bf
v}\implies l^{a-\tilde{z}}=l^{2a-1}\implies a=1-\tilde{z}.$ (11)
Next, in a mean-field like approach, we assume the kinetic energy flux or the
kinetic energy dissipation per unit mass is scale invariant in the inertial
range. This gives
$\frac{\partial v^{2}}{\partial t}\sim l^{0}\implies 2a=\tilde{z}.$ (12)
Combining then, we get $a=1/3,\,\tilde{z}=2/3$. This corresponds to a 1D
kinetic energy spectra $E(k)\sim k^{-5/3}$, the expected K41 result.
### III.2 Weak rotation effects: $k\gg 2\pi/L_{0}$
In order to study the effects of weak rotation on the scaling of the energy
spectra, we consider the limit $R_{o}\rightarrow\infty$, or $\Omega\rightarrow
0$. Equivalently, we consider length scales $L\ll L_{0}$, with the
understanding that $L\gg\eta_{r}$, the dissipation scale. In this case, the
Coriolis force is unimportant. Hence $\langle H\rangle\approx 0$, where
$<...>$ implies averages over the statistical steady states. Thus, the flux of
$E$ is the relevant (in the Kolmogorov sense) flux. Then, proceeding as in
Ref. jkb-mhd , we unsurprisingly recover the K41 scaling:
$a_{\perp}=a_{z}=1/3,\,\tilde{z}=2/3,\,\xi=1.$ (13)
The last of the above naturally means isotropic scaling (although geometry
remains anisotropic). Furthermore, if we let $R_{o}\sim l^{\eta}$ and demand
scale invariance of all the terms (including the Coriolis force terms), we
find
$\eta=\tilde{z}=2/3.$ (14)
Thus $R_{o}\rightarrow\infty$ as $l\rightarrow\infty$, which corresponds to
nonrotating and isotropic fully developed turbulence. We have assumed that the
nonlinear coupling constant does not scale under spatial rescaling, which is
consistent with the nonrenormalisation of $\lambda$ due to the Galilean
invariance of the Navier-Stokes equation. The scaling of the viscosity is
controlled by the dynamic exponent $\tilde{z}$.
### III.3 Strong rotation effects: $k\ll 2\pi/L_{0}$
We next consider the large rotation case, i.e., when $R_{o}\geq{\cal O}(1)$,
or $\Omega\geq{\cal O}(1)$. It is now expected that anisotropy is significant.
Below we analyse the scaling in several equivalent ways.
Balancing different terms of (3) and (4) we find
$\tilde{z}=0,\,a_{\perp}=1,\,a_{z}=\xi.$ (15)
To proceed further, we allow for the possibility that not only the spatial
scaling may be anisotropic, there may be different dynamic exponents for ${\bf
v}_{\perp}$ and $v_{z}$, with $\tilde{z}$ being identified as the dynamic
exponent $\tilde{z}_{\perp}$ of ${\bf v}_{\perp}$. To study this, we
separately consider the contribution to the kinetic energy from the in-plane
velocity ${\bf v}_{\perp}$ and normal component of the velocity $v_{z}$.
Interestingly, the exponents in (15) mean that the flux of the “in-plane
kinetic energy” $E_{\perp}\equiv\int d^{3}x\,v_{\perp}^{2}$ cannot be scale-
independent! Let us now consider the kinetic energy $E_{z}\equiv\int
d^{3}x\,v_{z}^{2}$ flux of $v_{z}$. If we assume $\tilde{z}=0$ is the dynamic
exponent of $v_{z}$ also, then $a_{z}=0=\xi$ can actually keep the flux of
$E_{z}$ scale-independent. However, $a_{z}=0$ is unexpected, as it means
$v_{z}$ does not scale with $l$ at all. Assume the dynamics $z_{\parallel}$ of
$v_{z}$ be non-zero: $z_{\parallel}>0$. Now consider Eq. (2) and balance
$\frac{\partial v_{z}}{\partial t}\sim v_{z}\partial_{z}v_{z}\implies
a_{z}=1-z_{\parallel}.$ (16)
Next, demanding scale-invariance of the flux of $E_{z}$ gives
$2a_{z}=\tilde{z}_{\parallel}\implies
a_{z}=\frac{1}{3},\,\tilde{z}_{\parallel}=\frac{2}{3}.$ (17)
This further means $\xi=a_{z}=1/3$. Notice that with $a_{\perp}=1,\,\xi=1/3$,
$({\bf v}_{\perp}\cdot{\boldsymbol{\nabla}}_{\perp})v_{z}$ and
$v_{z}\partial_{z}v_{z}$ scale the same way. Since we get $a_{\perp}>a_{z}$,
we should have $\langle v_{\perp}^{2}\rangle\gg\langle v_{z}^{2}\rangle$ in
the long wavelength limit, suggesting concentration of the kinetic energy in a
plane normal to the rotation axis gode . This implies an effective two-
dimensionalisation. On the other hand, $\tilde{z}<\tilde{z}_{\parallel}$
implies ${\bf v}_{\perp}\ll v_{z}$ in the long time limit, a conclusion
contradictory to our above inference. In fact, this alternative scenario
implies a type of dimensional reduction, where most of the energy is confined
to the $z$-direction. We are unable to conclusively predict which of these two
scenarios actually holds. Numerical studies should be useful in this regard.
Is the ensuing flow field in the limit $R_{o}\rightarrow\infty$ truly 2D? In
our opinion, the answer is no. First of all, the flow remains overall 3D
incompressible. This means the effective 2D flow field might be 2D
compressible, which is an interesting possibility. Secondly, it is not whether
the direction of the kinetic energy cascade becomes backward, a hallmark of
pure 2D turbulence. Thirdly, enstrophy is a conserved quantity in the inviscid
limit of pure 2D turbulence, whereas it is not expected to be so in the 3D
rotating case even in the limit of high rotation. Therefore, notwithstanding
the dominance of $v_{\perp}$ over $v_{z}$, the resulting flow field should be
fundamentally different from pure 2D nonrotating turbulence. Lastly, since
$z_{\perp}\neq z_{\parallel}$, we find weak dynamic scaling jkb-mhd ; dib . We
now calculate the scaling of the two-dimensional kinetic energy spectra
$E_{\perp}(k_{\perp},k_{\parallel})$ and $E_{z}(k_{\perp},k_{z})$, such that
total kinetic energy $E_{\text{tot}}=\int
dk_{\perp}dk_{z}[E_{\perp}(k_{\perp},k_{z})+E_{z}(k_{\perp},k_{z})]$. The
scaling of $E_{\perp}(k_{\perp},k_{z})$ and $E_{z}(k_{\perp},k_{z})$ can be
obtained as follows. We use the general definition to write in a
dimensional/scaling sense
$\displaystyle v_{m}({\bf k},\tilde{\omega})\sim\int v_{m}({\bf
x},t)\exp(i{\bf k_{\perp}}\cdot{\bf x_{\perp}})$ (18)
$\displaystyle\times\exp(ik_{\parallel}z)\exp(i\tilde{\omega}t)d^{2}x_{\perp}dzdt$
$\displaystyle\sim$ $\displaystyle
l_{\perp}^{a_{m}}l_{\perp}^{2}l_{\parallel}\sim\frac{1}{k_{\perp}^{a_{m}+2}}\frac{1}{k_{\parallel}}.$
Here, $m=\perp,\,\parallel$. Furthermore,
$\displaystyle\langle{\bf v}_{\perp}({\bf k}_{1})\cdot{\bf v}_{\perp}({\bf
k_{2}})\rangle=F_{\perp}(k_{1})\delta({\bf k}_{1}+{\bf k}_{2}),$ (19)
$\displaystyle\langle{v}_{z}({\bf k}_{1}){v}_{z}({\bf
k_{2}})\rangle=F_{\parallel}(k_{1})\delta({\bf k}_{1}+{\bf k}_{2}).$ (20)
Dimensionally then,
$F_{\perp}\sim\frac{1}{k_{\perp}^{a_{\perp}+2}k_{\parallel}},\,F_{\parallel}\sim\frac{1}{k_{\perp}^{a_{\parallel}+2}k_{\parallel}}.$
(21)
This gives for the two-dimensional energy spectra
$E_{\perp}(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-3},\,E_{z}(k_{\perp},k_{z})\sim k_{\perp}^{-5/3}.$ (22)
If we ignore anisotropy, we can define two corresponding one dimensional
energy spectra $E_{\perp}(k)$ and $E_{z}(k)$ from (22) by $E_{\text{tot}}=\int
dk[E_{\perp}(k)+E_{z}(k)]$. Notice that, neglecting anisotropy, the one-
dimensional spectra corresponding to $E_{\perp}(k)\sim k_{\perp}^{-3}$ should
scale as $k^{-2}$ as argued above, in agreement with Refs. dual1 ; dual2 ;
dual3 . Nonetheless, in spite of this agreement, we notice that our results
(22) appear to suggest that $E_{\perp}(k_{\perp},k_{z})$ and
$E_{z}(k_{\perp},k_{z})$ have no $k_{z}$-dependence, which should be
unphysical. We try to rectify this below.
First of all, for large $\Omega$, the scaling should be dominated by the
Coriolis forces. The vorticity ${\boldsymbol{\omega}}$ satisfies 2ds7
$\partial_{t}{\boldsymbol{\omega}}({\bf
k})=-2\Omega\frac{k_{\parallel}\hat{e}_{\bf k}\times{\boldsymbol{\omega}}_{\bf
k}}{k},$ (23)
giving time-scale $\tau\sim k/k_{\parallel}\sim k_{\perp}/k_{\parallel}$ for
$k_{\perp}\gg k_{\parallel}$.
It is thus reasonable to assume that $\tau$ as defined above is the relevant
time-scale when $R_{o}\rightarrow 0$. In what follows below, we do not make
any distinction between $v_{\perp}$ and $v_{z}$. We now impose the scale-
independence of the kinetic energy flux $\Pi$. The energy flux may be
calculated from the Navier-Stokes equations (1). We find
$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle-2\lambda^{2}\int_{{\bf
k,q},\tilde{\omega},\tilde{\Omega}}\bigg{[}M_{imn}({\bf k})M_{ijp}(-{\bf
k})\langle v_{m}({-k+q},-\tilde{\omega}+\tilde{\Omega})v_{p}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})\rangle$ (24) $\displaystyle+$
$\displaystyle M_{jmn}({\bf k})M_{ijp}({\bf q})\langle
v_{i}({-k},-\tilde{\omega})v_{n}({\bf k},\tilde{\omega})\rangle\langle
v_{m}({\bf-k+q},-\tilde{\omega}+\tilde{\Omega})v_{p}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})\rangle$ (25) $\displaystyle+$
$\displaystyle M_{jmn}({\bf k-q})M_{ijp}({\bf k})\langle v_{i}({-\bf
k},-\tilde{\omega})v_{n}({\bf k},\tilde{\omega})\rangle\langle v_{j}({\bf
q},\tilde{\Omega})v_{m}({\bf-q},-\tilde{\Omega})\rangle\bigg{]}.$ (26)
Here, $M_{ijp}({\bf k})=P_{ij}({\bf k})k_{p}+P_{ip}({\bf k})k_{j}$. Since we
are interested in the scaling, it suffices to consider the scaling of $\Pi$,
suppressing indices and wavevector labels. At this one-loop order, suppressing
indices and wavevector labels, and assuming $k_{\perp}\gg k_{\parallel}$
$\Pi\sim\int d^{2}k_{\perp}d^{2}p_{\perp}dk_{\parallel}dp_{\parallel}\int
dt\int d^{2}k_{\perp}\,k_{\perp}^{2}C^{2}\sim const.,$ (27)
where $C\equiv\langle v\,v\rangle$ is the correlation function, again
suppressing indices and wavevector labels. This gives
$C\sim k_{\perp}^{-7/2}k_{\parallel}^{-1/2},$ (28)
where we have used $\int dt\sim\tau$. This implies for the two-dimensional
kinetic energy spectra
$E_{\perp}(k_{\perp},k_{\parallel})\sim Ck_{\perp}\sim
k_{\perp}^{-5/2}k_{\parallel}^{-1/2},$ (29)
as in the wave turbulence theory. It now behooves us to show that the scaling
of $\tau$ with wavevector does not get renormalised at the one-loop order. We
restrict ourselves here to a scaling-level demonstration. As shown in
Appendix, the one-loop self-energy $\Sigma_{ik}({\bf k},\tilde{\omega})$ has
the form
$\displaystyle\Sigma_{ij}({\bf k},\tilde{\omega})\sim\int
d^{3}qd\tilde{\Omega}M_{imn}({\bf k})\langle v_{m}({\bf
q},\tilde{\Omega})v_{r}({-\bf q},-\tilde{\Omega})\rangle$ (30)
$\displaystyle\times$ $\displaystyle G_{ns}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})M_{srj}({\bf k-q}).$
Here, $G_{ns}({\bf k},\tilde{\omega})$ is the propagator defined via
$G_{ns}({\bf k},\tilde{\omega})\equiv\bigg{\langle}\frac{\delta v_{n}({\bf
k},\tilde{\omega})}{\delta f_{s}({\bf k},\tilde{\omega})}\bigg{\rangle}.$ (31)
Considering the one-loop self-energy $\Sigma$ and suppressing indices and
wavevector labels [see also Appendix], we obtain
$\Sigma\sim k_{\perp}^{2}\int d^{2}q_{\perp}dq_{z}\int dtG\,C,$ (32)
where $G$ is a propagator. Assuming the dominant time-scale in the above time-
integral is given by $\tau$, we get (in a scaling sense)
$k_{\perp}^{2}\int
d^{2}q_{\perp}dq_{z}\frac{q_{\perp}}{q_{z}}q_{\perp}^{-7/2}k_{z}^{-1/2}\sim[k_{\perp}]^{3/2}[k_{\parallel}]^{1/2},$
(33)
which is less singular than the the bare form of $\tau$. Thus our scaling
results on the 2D kinetic energy spectra remain unaffected by the advective
nonlinearities in the asymptotic long wavelength limit.
Interestingly, we can also derive the above results by using phenomenological
arguments of suppression of the kinetic energy flux by the helicity generated
by the rotation, which are similar to the arguments set up in Ref. jkb-mhd
for scaling of the energy spectra in the presence of a strong mean magnetic
field in magnetohydrodynamic turbulence. It is known that the predominant role
of a (large) non-zero helicity flux is to hinder the cascade of the kinetic
energy flux kraich . In fact, it is easy to see from (3) and (4) that a large
$\Omega$ should suppress the nonlinear effects, relative to the Coriolis force
terms. Since the nonlinear terms are responsible for the cascade phenomena, we
expect the flux of $E_{\perp}$ to be suppressed by a large $\Omega$. This is
similar to the suppression of the energy flux by a strong mean magnetic field
in fully developed magnetohydrodynamic turbulence jkb-mhd ; see also similar
treatment for turbulence in a stably stratified fluid ab-jkb-bolgiano . We
write
$\bigg{[}\frac{\partial v_{\perp}^{2}}{\partial
t}\bigg{]}\bigg{[}\frac{\omega^{2}}{\Omega^{2}}\bigg{]}\sim l^{0}$ (34)
as the condition of the flux being scale independent. Since dimensionally,
$[\omega^{2}]\sim[v_{\perp}^{2}/l^{2}]$, we get
$v_{\perp}\sim l_{\perp}^{3/4}l_{\parallel}^{-1/4}.$ (35)
This gives
$E_{\perp}(k_{\perp},k_{\parallel})\sim k_{\perp}^{-5/2}k_{\parallel}^{-1/2},$
(36)
in agreement with the conclusion from the wave turbulence theory approaches
2ds7 . It is easy to get the spectra in the opposite limit $k_{\parallel}\gg
k_{\perp}$. In this limit, $\tau\sim k_{\perp}^{0}k_{\parallel}^{0}$. At this
one-loop order, suppressing indices and wavevector labels, and assuming
$k_{\perp}\ll k_{z}$.
$\Pi\sim\int d^{2}k_{\perp}d^{2}p_{\perp}dk_{z}dp_{\parallel}\int
d^{2}k_{\perp}\,k_{z}^{2}C^{2}\sim const.,$ (37)
Proceeding as before, we find
$E_{\perp}(k_{\perp},k_{\parallel})\sim k_{\perp}^{-1}k_{\parallel}^{-2}.$
(38)
In each of these cases, the corresponding one-dimensional spectra, without
making any distinction between $k_{\perp}$ and $k_{\parallel}$ scale as
$k^{-2}$, consistent with the recent shell-model studies on rotating
turbulence new-shell .
A pictorial summary of the scaling regimes are shown in Fig. 2.
Figure 2: Schematic diagram illustrating the scaling regimes at different
length scales. Dissipation range for small scales, K41 scaling regime at the
intermediate scales and rotation dominated scaling regimes at the largest
scales are shown.
In the above, we have implicitly assumed that the kinetic energy flux $E$ is
the relevant flux (in the Kolmogorov sense). As we have discussed above this
holds for length scales $\gg l^{*}$. An interesting case may arise if
$l^{*}\gg L_{0}$, in which case in the window between $L_{0}$ and $l^{*}$,
helicity flux $H$ dominates over $E$. If this indeed holds, the scaling of the
energy spectra might change within this window. We do not discuss this further
here.
## IV Summary and outlook
In this work we have developed a scaling theory for fully developed
incompressible hydrodynamic turbulence in a rotating fluid in the inertial
range. We have studied the scaling of the energy spectra in the inertial range
for weak and strong rotations, i.e., for small and large Rossby number
$R_{o}$. We argue that for wavevectors smaller than the Zeeman wavevector
rotation is important, whereas in the opposite limit, rotation is unimportant.
It is therefore expected that in the former regime the scaling may be
different from the K41 scaling, but in the other regime K41 scaling should
ensue. The scaling theory that we developed here bears this out.
Our scaling theory reveals that in the rotation dominated regime, not only the
scaling itself is anisotropic (i.e., different dependence on $k_{\perp}$ and
$k_{\parallel}$), the scaling of $v_{\perp}$ and $v_{z}$ are different. Even
the dynamic exponents of $v_{\perp}$ and $v_{z}$ are different, indicating
weak dynamic scaling. This suggests that in the limit of a large rotation, the
flow fields are dominated by only some of the velocity components. However,
our theory cannot conclusively predict whether $v_{\perp}$ or $v_{z}$ will be
the dominant part. Numerical simulations should be useful to make further
conclusions.
We have throughout assumed that the kinetic energy flux is the relevant flux
(in the Kolmogorov sense), neglecting the helicity flux. However, for
sufficiently large rotation, there may be a window of length scales where the
helicity flux is the dominant flux. In this regime, the scaling of the energy
spectra should be different. This will be discussed elsewhere.
## V Acknowledgement
AB thanks the SERB, DST (India) for partial financial support through the
MATRICS scheme [file no.: MTR/2020/000406].
## Appendix A Reduction of the kinetic energy flux in rotating turbulence:
perturbation theory
We now calculate the kinetic energy flux $\Pi$ in the perturbation theory. We
derive (26) and calculate $\Pi$ to lowest non-trivial order in $\Omega$. We
start by eliminating pressure from the Navier-Stokes equation (1) in a
rotating frame. We obtain
$\displaystyle(-i\tilde{\omega}+\nu k^{2})v_{i}+2P_{im}({\bf
k})\Omega\epsilon_{mzp}v_{p}({\bf k},\tilde{\omega})$
$\displaystyle+i\frac{\lambda}{2}M_{ijp}({\bf k})\sum_{{\bf
q},\tilde{\omega}}v_{j}({\bf q},\tilde{\omega})v_{p}({\bf
k-q},\,\tilde{\omega}-\tilde{\Omega})=f_{i}.$ (39)
To the lowest order in $\Omega$, we expect $\Pi$ to depend on $\Omega$
quadratically, since the energy cascade should be independent of the sense of
rotation around the $z$-axis, i.e., should be the same for clockwise and
anticlockwise rotations. To this order in $\Omega$, it suffices to expand (39)
to ${\cal O}(\Omega)$ and construct an effective equation:
$\displaystyle(-i\tilde{\omega}+\nu k^{2})v_{i}+\frac{i\lambda}{2}M_{ijp}({\bf
k})\sum_{{\bf q},\tilde{\omega}}v_{j}({\bf q},\tilde{\Omega})v_{p}({\bf
k-q},\tilde{\omega}-\tilde{\Omega})$ (40) $\displaystyle=$ $\displaystyle
f_{i}-2P_{im}({\bf k})\epsilon_{mzp}\Omega\frac{f_{p}}{-i\tilde{\omega}+\nu
k^{2}}.$
Clearly, the last term on the rhs of (40), an effective noise, is the dominant
noise in the hydrodynamic limit. We use (40) to calculate the kinetic energy
flux $\Pi$ to ${\cal O}(\Omega^{2})$. The flux $\Pi$ follows
$\displaystyle\frac{d}{dt}\langle v^{2}\rangle$ $\displaystyle=$
$\displaystyle-\frac{i\lambda}{2}\int_{{\bf
k},\tilde{\omega}}\langle\bigg{[}M_{ijp}({\bf k})\sum_{{\bf
q},\tilde{\omega}}v_{i}({\bf-k},-\tilde{\omega})v_{j}({\bf
q},\tilde{\Omega})v_{p}({\bf k-q},\tilde{\omega}-\tilde{\Omega})$ (41)
$\displaystyle+$ $\displaystyle M_{ijp}({\bf-k})\sum_{{\bf
q},\tilde{\omega}}v_{i}({\bf k},\tilde{\omega})v_{j}({\bf
q},\tilde{\Omega})v_{p}({\bf-
k-q},-\tilde{\omega}-\tilde{\Omega})\bigg{]}\rangle.$
Next we iterate and expand the rhs of (41) up to the one-loop order, which
gives (26) above. Now, to the linear order in $\Omega$,
$\displaystyle\langle v_{j}({\bf
q},\tilde{\Omega})v_{n}({\bf-q},-\tilde{\Omega})\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle f_{j}({\bf q},\tilde{\Omega})f_{n}({-\bf
q},-\tilde{\Omega})\rangle}{\tilde{\Omega}^{2}+\nu^{2}q^{4}}-2P_{js}({\bf
q})\epsilon_{szp}G_{0}^{2}(q,\tilde{\Omega})\Omega\frac{\langle f_{p}({\bf
q},\tilde{\Omega})f_{n}({\bf-q},-\tilde{\Omega})\rangle}{i\tilde{\Omega}+\nu
q^{2}}$ (42) $\displaystyle-$ $\displaystyle 2P_{ns}({\bf
q})\epsilon_{szp}G_{0}^{2}(-q,-\tilde{\Omega})\tilde{\Omega}\frac{\langle
f_{p}({\bf-q},-\tilde{\Omega})f_{j}({\bf
q},\tilde{\Omega})\rangle}{-i\tilde{\Omega}+\nu q^{2}}.$
Substituting in (26) and evaluating, we find $\Pi(\Omega)<\Pi(\Omega=0)$,
indicating reduction of the kinetic energy flux by rotation. Since the
helicity generated by the rotation scales with $\Omega$, we conclude that with
a rising helicity, the kinetic energy flux is suppressed.
## References
* (1) P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
* (2) L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Butterworth and Heinemann, Oxford (1998).
* (3) U Frisch, Turbulence: the legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.
* (4) A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 301–305 (1941); A. N. Kolmogorov, “Dissipation of energy in locally isotropic turbulence,” Dokl. Akad. Nauk SSSR 32, 16–18 (1941).
* (5) G. E. Elsinga and I. Marusic, “The anisotropic structure of turbulence and its energy spectrum,” Phys. Fluids 28, 011701 (2016).
* (6) P. C. Valente, C. B. da Silva, and F. T. Pinho, “Energy spectra in elasto-inertial turbulence,” Phys. Fluids 28, 075108 (2016).
* (7) P. Buchhave and C. M. Velte, “Measurement of turbulent spatial structure and kinetic energy spectrum by exact temporal-to-spatial mapping,” Phys. Fluids 29, 085109 (2017).
* (8) R. H. Kraichnan, “Inertial ranges in two-dimensional turbulence,” Phys. Fluids 10, 1417 (1967).
* (9) J. Paret and P. Tabeling, “Experimental observation of the two-dimensional inverse energy cascade,” Phys. Rev. Lett. 79, 4162 (1997).
* (10) S. Chen, R. E. Ecke, G. L. Eyink, M. Rivera, M. Wan, and Z. Xiao, “Physical mechanism of the two-dimensional inverse energy cascade,” Phys. Rev. Lett. 96, 084502 (2006).
* (11) A. Vallgren and E. Lindborg, “The enstrophy cascade in forced two- dimensional turbulence,” J. Fluid Mech. 671, 168–183 (2011).
* (12) G. Boffetta and R. E. Ecke, “Two-dimensional turbulence,” Annu. Rev. Fluid. Mech. 44, 427–451 (2012).
* (13) J. Bardina, J. H. Ferziger, and R. S. Rogallo, “Effect of rotation on isotropic turbulence: Computation and modelling,” J. Fluid Mech. 154, 321–336 (1985).
* (14) Y. Zhou, “A phenomenological treatment of rotating turbulence,” Phys. Fluids 7, 2092 (1995).
* (15) V. M. Canuto and M. S. Dubovikov, “Physical regimes and dimensional structure of rotating turbulence,” Phys. Rev. Lett. 78, 666–669 (1997).
* (16) V. M. Canuto and M. S. Dubovikov, “A dynamical model for turbulence. V. The effect of rotation,” Phys. Fluids 9, 2132–2140 (1997).
* (17) L. M. Smith and F. Waleffe, “Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence,” Phys. Fluids 11, 1608 (1999).
* (18) C. N. Baroud, B. B. Plapp, Z.-S. She, and H. L. Swinney, “Anomalous self- similarity in a turbulent rapidly rotating fluid,” Phys. Rev. Lett. 88, 114501 (2002).
* (19) S. Galtier, “Weak inertial-wave turbulence theory,” Phys. Rev. E 68, 015301 (2003).
* (20) W.-C. Müller and M. Thiele, “Scaling and energy transfer in rotating turbulence,” Europhys. Lett. 77, 34003 (2007).
* (21) S. Chakraborty and J. K. Bhattacharjee, “Third-order structure function for rotating three-dimensional homogeneous turbulent flow,” Phys. Rev. E 76, 036304 (2007).
* (22) S. Chakraborty, “Signatures of two-dimensionalisation of 3D turbulence in the presence of rotation,” Europhys. Lett. 79, 14002 (2007).
* (23) D. Mininni, A. Alexakis, and A. Pouquet, “Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers,” Phys. Fluids 21, 015108 (2009).
* (24) S. Chakraborty, M. H. Jensen, and A. Sarkar, “On two-dimensionalization of three-dimensional turbulence in shell models,” Eur. Phys. J. B 73, 447–453 (2010).
* (25) M. K. Sharma, M. K. Verma and S. Chakraborty, On the energy spectrum of rapidly rotating forced turbulence, Physics of Fluids 30, 115102 (2018).
* (26) M. K. Sharma, M. K. Verma and S. Chakraborty, Anisotropic energy transfers in rapidly rotating turbulence, Phys. Fluids 31, 085117 (2019).
* (27) S. K. Rathor, S. Chakraborty and S. S. Ray, Dynamic scaling in rotating turbulence: A shell model study, Phys. Rev. E 105, L063102 (2022).
* (28) A. Basu and J. K. Bhattacharjee, Fluctuating hydrodynamics and turbulence in a rotating fluid: Universal properties, Phys. Rev. E 85, 026311 (2012).
* (29) J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley Eastern, New Delhi, 1975).
* (30) A. Raichaudhuri, The Physics of Fluids and Plasmas (Cambridge University Press, Cambridge, 1998).
* (31) O. Zeman, A note on the spectra and decay of rotating homogeneous turbulence, Phys. Fluids 6, 3221 (1994).
* (32) Y. Zhou, A phenomenological treatment of rotating turbulence, Phys. Fluids 7, 2092 (1995).
* (33) S. Chakraborty, M. H. Jensen, and A. Sarkar, On two-dimensionalization of three-dimensional turbulence in shell models, Eur. Phys. J. B 73, 447 (2010).
* (34) A. Basu and J. K. Bhattacharjee, Varieties of scaling regimes in hydromagnetic turbulence, Phys. Rev. E 98, 062143 (2018).
* (35) C. Cambon, N. N. Mansour, and F. S. Godefard, Energy transfer in rotating turbulence, J. Fluid Mech. 337, 303 (1997).
* (36) D. Das, A. Basu, M. Barma, and S. Ramaswamy, Phys. Rev. E 64, 021402 (2001).
* (37) R. H. Kraichnan, Helical turbulence and absolute equilibrium, J. Fluid Mech. 59, 745 (1973).
* (38) A. Basu and J. K. Bhattacharjee, Kolmogorov or Bolgiano-Obukhov scaling: Universal energy spectra in stably stratified turbulent fluids, Phys. Rev. E 100, 033117 (2019).
* (39) S. K. Rathor, M. K. Sharma, S. S. Ray and S. Chakraborty, Bridging inertial and dissipation range statistics in rotating turbulence, Phys. Fluids 32, 095104 (2020).
|
# A Noise-tolerant Differentiable Learning Approach for Single Occurrence
Regular Expression with Interleaving
Rongzhen Ye, 1 Tianqu Zhuang, 1 Hai Wan, 1* Jianfeng Du, 2 Weilin Luo, 1
Pingjia Liang, 1 Both Hai Wan and Jianfeng Du are corresponding authors.
###### Abstract
We study the problem of learning a _single occurrence regular expression with
interleaving_ (SOIRE) from a set of text strings possibly with _noise_. SOIRE
fully supports interleaving and covers a large portion of regular expressions
used in practice. Learning SOIREs is challenging because it requires heavy
computation and text strings usually contain noise in practice. Most of the
previous studies only learn restricted SOIREs and are not robust on noisy
data. To tackle these issues, we propose a noise-tolerant differentiable
learning approach SOIREDL for SOIRE. We design a neural network to simulate
SOIRE matching and theoretically prove that certain assignments of the set of
parameters learnt by the neural network, called _faithful encodings_ , are
one-to-one corresponding to SOIREs for a bounded size. Based on this
correspondence, we interpret the target SOIRE from an assignment of the set of
parameters of the neural network by exploring the nearest faithful encodings.
Experimental results show that SOIREDL outperforms the state-of-the-art
approaches, especially on noisy data.
## Introduction
Learning regular expressions (REs) is a fundamental task in Machine Learning.
For example, REs are the target in EXtensible Markup Language (XML) schema
inference, for covering a set of text strings about an XML element. The
regular expression with interleaving, denoted as RE(&), is an extension of
regular expressions, where the operator interleaving (&) is added to
interleave two strings. RE(&) has been widely used in various areas, ranging
from XML database system (Makoto and Clark 2003; Colazzo et al. 2013; Martens
et al. 2017) to system verification (Gischer 1981; Bojanczyk et al. 2006) and
natural language processing (Kuhlmann and Satta 2009; Nivre 2009), etc.
We focus on a subclass of RE(&), _single occurrence regular expression with
interleaving_ (SOIRE), where each symbol occurs at most once in a SOIRE.
Learning SOIREs is still meaningful, as SOIREs have the second highest
coverage rate of REs on the schema database Relax NG among all well-known
subclasses (Li et al. 2019b). Although a subclass is generally easier to learn
than the full class, it is still challenging to learn SOIREs. On one hand,
learning SOIREs is a search problem requiring heavy computation. On the other
hand, real-life text strings usually contain noise (Kearns and Li 1988;
Galassi and Giordana 2005; Bex et al. 2006). For example, in XML schema
inference the XML data may contain incorrect symbols (Bex et al. 2006). The
presence of noise makes the problem of learning SOIREs more challenging.
There have been a number of proposals for learning either the full class or
its subclasses of SOIRE, from a set of text strings (Freydenberger and Kötzing
2015; Zhang et al. 2018; Li et al. 2019a, 2020b). However, they are hard to
guarantee that the learnt REs reach the full declared expressive power. For
example, Li et al. (2019a) claimed to learn SOIREs, but Wang and Zhang (2021)
showed that they just learn special cases of SOIRE. Besides, existing
approaches are not robust on noisy data because any modification to given
strings will alter the patterns in learnt REs. As far as we know, there is no
approach to learning SOIREs that works well with both noise-free data and
noisy data.
In this paper, we propose a noise-tolerant differentiable learning approach
SOIREDL for SOIREs. Specifically, we design a neural network to simulate SOIRE
matching for text strings. Since existing work for SOIRE matching (Wang 2021b,
2022) is not suitable for differentiable learning, we propose a new SOIRE
matching algorithm SOIRETM based on the syntax tree of SOIRE, and accordingly,
an algorithm for converting SOIRETM to a neural network. We theoretically
prove that certain assignments of the set of parameters learnt by the neural
network, called _faithful encodings_ , one-to-one correspond to SOIREs for a
bounded size. This correspondence allows us to interpret the target SOIRE from
an assignment of the set of learnt parameters of the neural network by
exploring the nearest faithful encodings.
To evaluate the performance of SOIREDL on noisy data, we extract $30$ SOIREs
from the RE database built by Li et al. (2018) to make a group of datasets
covering five domains with different noise levels. Experimental results show
that at all noise levels, the average accuracy of SOIREDL is higher than
state-of-the-art (SOTA) approaches and the faithfulness of SOIREDL is always
beyond $80\%$. In particular, the average accuracy of SOIREDL only decreases
slightly with increasing noise levels, which suggests that SOIREDL is robust
on noisy data.
## Related Work
Matching algorithms for regular expressions. It has been shown (Mayer and
Stockmeyer 1994) that the matching problem of RE(&) is NP-hard. For SOIREs,
Wang (2021b) proposed a finite automata with interleaving, written FA(&), to
solve the matching problem. Wang (2022) proposed a single occurrence finite
automata, written SFA(&, #), for matching single occurrence regular
expressions with interleaving and counting. All above studies do not consider
converting the matching algorithm into a neural network. In contrast, we
present not only a matching algorithm for SOIREs but also the way to convert
it to a neural network.
Learning approaches for regular expressions. There also exists work for
learning different subclasses of REs from a set of text strings, such as the
deterministic regular expressions with counting (Wang and Chen 2018), the
deterministic regular expressions with unorder (Wang and Chen 2020), and the
deterministic regular expression with counting and unorder (Wang 2021a). Some
subclasses of the extension RE(&) have also been explored, such as chain
regular expression with interleaving (ICHARE) (Zhang et al. 2018), restricted
SOIRE (RSOIRE) (Li et al. 2019a), and k-occurrence regular expression with
interleaving (kOIRE) (Li et al. 2020a). All of them are learnt from positive
strings only. Li et al. (2020b) learnt a subclass of ICHARE, called SIRE, from
both positive and negative strings based on a genetic algorithm. The relation
of expressive powers supported by these classes is SIRE $\subset$ ICHARE
$\subset$ RSOIRE $\subset$ SOIRE $\subset$ kOIRE. Li et al. (2021) proposed a
natural language processing based RE synthesizer to learn REs from natural
language descriptions together with positive and negative strings. This
problem setting is different from ours.
Differentiable learning. Differentiable learning has attracted much research
interest recently. Most studies focus on learning logical rules from knowledge
bases (Yang, Yang, and Cohen 2017; Sadeghian et al. 2019; Cohen, Yang, and
Mazaitis 2020; Huang et al. 2021) or on neural logic programming (Yang and
Song 2020; Gao et al. 2022; Wang et al. 2020). In particular, some studies
(Rocktäschel and Riedel 2017; Minervini et al. 2020b, a) focus on neural
theorem proving. They convert the symbolic operations into differentiable
modules to enhance the reasoning ability of the neural network. Mensch and
Blondel (2018) utilized a strongly convex regularizer to smooth the max
operator and convert a broad class of dynamic programming (DP) algorithms into
differentiable operators. Wang et al. (2019) proposed a differentiable MaxSAT
solver integrated into the deep learning networks to solve the problems like
visual Sudoku, which has implicit satisfiability constraints. For REs, Jiang
et al. (2020) injected a weighted finite-state automaton (FSA) of REs into the
recurrent neural network (RNN) to improve the performance of text
classification. Further, Jiang, Jin, and Tu (2021) injected a finite-state
transducer of REs into RNN for slot filling. Both of the above methods fine-
tune initial regular expressions given from the expert knowledge to obtain
better results. Different from all above studies, we further study the one-to-
one correspondence between parameters of a neural network and SOIREs.
## Preliminaries
A regular expression with interleaving, written RE(&), over an alphabet
$\Sigma$ is defined recursively as follows (Mayer and Stockmeyer 1994):
$r:=\epsilon\big{|}a\big{|}r_{1}^{*}\big{|}r_{1}\cdot
r_{2}\big{|}r_{1}\&r_{2}\big{|}r_{1}|r_{2}$
where $\epsilon$ is empty string, the symbol $a\in\Sigma$, and $r_{1},r_{2}$
are RE(&). The operator $*$ denotes Kleene-Star, $\cdot$ denotes concatenation
(it can be omitted if there is no ambiguity), $\&$ denotes interleaving, $|$
denotes disjunction. The operators $?$ and $+$ are commonly used for
repetition. They are defined as $r^{?}:=r\big{|}\epsilon$ and $r^{+}:=r\cdot
r^{*}$, respectively. The operator $\&$ for two strings $s_{1},s_{2}$ is
defined as follows:
$s_{1}\&s_{2}:=\left\\{\begin{array}[]{cc}s_{2}&\text{if }s_{1}=\epsilon\\\
s_{1}&\text{if }s_{2}=\epsilon\\\
a(s_{1}^{\prime}\&s_{2})|b(s_{1}\&s_{2}^{\prime})&otherwise\\\
\end{array}\right.$
where $s_{1}=as_{1}^{\prime}$, $s_{2}=bs_{2}^{\prime}$, $a,b\in\Sigma$.
###### Definition 1 (Single occurrence regular expression with interleaving
(SOIRE) (Li et al. 2019a)).
SOIRE is a RE(&) where each symbol occurs at most once in the expression.
###### Example 1.
$(a\&b)c^{*}$ is a SOIRE. $(a\&b)a^{*}$ is a RE(&) but not a SOIRE, because
the symbol $a$ occurs twice.
Each SOIRE can be expressed by its prefix notation where operators are written
in front of operands rather than written in the middle as the infix notation.
By $\texttt{PreForm}(r)$ we denote the prefix notation of a SOIRE $r$. For the
SOIRE $r=(a\&b)c^{*}$ given in Example 1, $\texttt{PreForm}(r)$ is
$\cdot\&ab*c$.
Syntax tree. Each SOIRE can also be represented as a binary tree, called
_syntax tree_ , where each inner vertex in the tree represents an operator and
each leaf represents a symbol. By $\texttt{RE2Tree}(r)$ we denote the syntax
tree of a SOIRE $r$. In a syntax tree, each leaf represents a symbol in
$\Sigma$ and each symbol occurs at most once, while each inner vertex
represents an operator and the number of children of it is equal to the number
of its operands. Figure 1 shows the syntax tree of $(a\&b)c^{*}$ in Example 1.
The size of a SOIRE $r$, denoted by $|r|$, is defined as the number of
vertices in the syntax tree of $r$. For example, the size of $(a\&b)c^{*}$ is
$6$. Obviously, the preorder traversal sequence of the syntax tree of $r$ is
$\texttt{PreForm}(r)$ and each subtree represents a subexpression. In this
preorder traversal sequence, vertex $t+1$ is the left child for any inner
vertex $t$. We use $r^{t}(1\leq t\leq|r|)$ to denote the corresponding SOIRE
of the subtree of $\texttt{RE2Tree}(r)$ whose root is vertex $t$. Further, if
vertex $t$ represents a binary operator, we use $\eta^{t}$ to denote the
sequential number of its right child. For $r=(a\&b)c^{*}$ in Example 1,
$r^{2}=a\&b$ and $\eta^{2}=4$.
From prefix notation to syntax tree. We show a way to realize
$\texttt{RE2Tree}(r)$. It scans $\texttt{PreForm}(r)$ from back to front and
maintains a stack of syntax trees. Take Figure 1 for instance,
$\texttt{PreForm}(r)=\cdot\&ab*c$. $\texttt{RE2Tree}(r)$ scans
$\texttt{PreForm}(r)$ from $c$ to $\cdot$. When scanning $c$, push $c$
($v_{6}$) into the stack. When scanning $*$, pop $c$ ($v_{6}$) from the stack
and make $c$ the left child of $*$, then push $*c$ ($v_{5}$) into the stack.
When scanning $b$ and $a$, push $b$ ($v_{4}$) and $a$ ($v_{3}$) into the
stack. When scanning $\&$, pop $a$ ($v_{3}$) and $b$ ($v_{4}$) from the stack
and make $a$ the left child of $\&$ and $b$ the right child, then push $\&ab$
($v_{2}$) into the stack. In this way, the syntax tree in Figure 1 is built.
We use $\texttt{Tree2RE}(\xi)$ to denote the inverse function of
$\texttt{RE2Tree}(r)$, which returns $\texttt{PreForm}(r)$ from the syntax
tree $\xi=\texttt{RE2Tree}(r)$ by preorder traversal.
Figure 1: The syntax tree of SOIRE $(a\&b)c^{*}$. $v_{1}$, …, $v_{6}$
represent $\cdot$, $\&$, $a$, $b$, $*$ and $c$, respectively.
SOIRE matching. By $r\models s$ we denote that a SOIRE $r$ matches a string
$s$. Then the problem of SOIRE matching is to check whether $r\models s$,
which can be decided in the following way:
$\displaystyle r\models s:=$
$\displaystyle\left\\{\begin{array}[]{ll}s=a&\text{if }r=a\\\ s=\epsilon\vee
r_{1}\models s&\text{if }r=r_{1}^{?}\\\ s=\epsilon\vee\exists s_{1}\exists
s_{2},(s=s_{1}s_{2}\wedge s_{2}\neq\epsilon\\\ \wedge r_{1}^{*}\models
s_{1}\wedge r_{1}\models s_{2})&\text{if }r=r_{1}^{*}\\\ \exists s_{1}\exists
s_{2},(s=s_{1}s_{2}\wedge r_{1}^{*}\models s_{1}\wedge r_{1}\models
s_{2})&\text{if }r=r_{1}^{+}\\\ \exists s_{1}\exists s_{2},(s=s_{1}s_{2}\wedge
r_{1}\models s_{1}\wedge r_{2}\models s_{2})&\text{if }r=r_{1}\cdot r_{2}\\\
\exists s_{1}\exists s_{2},(s=s_{1}\&s_{2}\wedge r_{1}\models s_{1}\wedge
r_{2}\models s_{2})&\text{if }r=r_{1}\&r_{2}\\\ r_{1}\models s\vee
r_{2}\models s&\text{if }r=r_{1}|r_{2}\\\ \end{array}\right.$ (9)
where $r_{1},r_{2}$ are SOIREs and $a\in\\{\epsilon\\}\cup\Sigma$.
SOIRE learning. Given a set of strings $\Pi=\Pi^{+}\cup\Pi^{-}$, where
$\Pi^{+}$ (resp. $\Pi^{-}$) denotes the set of strings with the positive
(resp. negative) label, the problem of SOIRE learning is to find a SOIRE $r$
that maximizes $\mathrm{acc}(r)$ defined below.
$\displaystyle\mathrm{acc}(r)=\frac{|\\{s|r\models
s,s\in\Pi^{+}\\}|+|\\{s|r\not\models s,s\in\Pi^{-}\\}|}{|\Pi|}$ (10)
## The Proposed SOIREDL Approach
In this section, we first describe a new matching algorithm SOIRETM, then show
how to convert SOIRETM to a neural network. Afterwards, we show correspondence
between parameters of the neural network and SOIREs. Finally, we show how to
interpret the target SOIRE from the parameters.
First of all, we introduce a variant problem of SOIRE matching, called _filter
matching_. Filter matching for a SOIRE $r$ and a string $s$ is to check if $r$
matches $filter(s,\alpha(r))$, where $\alpha(r)$ denotes the set of symbol in
a SOIRE $r$, and the filter function $filter(s,V)$ returns a string that only
retains symbols in $V$, where $V\subseteq\Sigma$. For Example 1 and $s=dbac$,
the corresponding problem of filter matching is to check if $(a\&b)c^{*}$
matches $filter(dbac,\\{a,b,c\\})=bac$, as $\alpha((a\&b)c^{*})=\\{a,b,c\\}$.
Given a SOIRE $r$ and a string $s$, we use $g_{i,j}^{t}\in\\{0,1\\}$ ($1\leq
t\leq|r|$, $1\leq i,j\leq|s|$) to denote whether $r^{t}$ matches
$filter(s_{i,j},\alpha(r^{t}))$, where $s_{i,j}$ denotes the substring of $s$
from $i$ to $j$, and where $s_{1,0}=\epsilon$ specially. If $r^{t}$ matches
$filter(s_{i,j},\alpha(r^{t}))$, then $g_{i,j}^{t}=1$, otherwise
$g_{i,j}^{t}=0$. In particular, $g_{1,0}^{t}$ denotes if $r^{t}$ matches
$\epsilon$ since for all $t$ from $1$ to $|r|$,
$filter(s_{1,0},\alpha(r^{t}))=\epsilon$. For Example 1 and $s=dbac$,
$g_{1,2}^{2}$ denotes if $a\&b$ matches $ba$ and $g_{1,2}^{2}=1$.
SOIRE | Semantics of $r\models filter(s,\alpha(r))$
---|---
$r=a\in\Sigma$ | 1\. $filter(s,{a})=a$.
$r=r_{1}^{?}$ | 1\. $filter(s,\alpha(r_{1}^{?}))=\epsilon$.
$=\epsilon|r_{1}$ | 2\. $r_{1}\models filter(s,\alpha(r_{1}))$.
$r=r_{1}^{*}$ | 1\. $filter(s,\alpha(r_{1}^{*}))=\epsilon$.
$=\epsilon|r_{1}|r_{1}^{*}\cdot r_{1}$ | 2\. $r_{1}\models filter(s,\alpha(r_{1}))$.
| 3\. $r_{1}^{*}\models filter(s_{1},\alpha(r_{1}^{*}))$ and $r_{1}\models
filter(s_{2},\alpha(r_{1}))$,
| where $s=s_{1}s_{2}(s_{1},s_{2}\neq\epsilon)$.
$r=r_{1}^{+}$ | 1\. $r_{1}\models filter(s,\alpha(r_{1}))$.
$=r_{1}|r_{1}^{+}\cdot r_{1}$ | 2\. $r_{1}^{+}\models filter(s_{1},\alpha(r_{1}^{+}))$ and $r_{1}\models filter(s_{2},\alpha(r_{1}))$,
| where $s=s_{1}s_{2}(s_{1},s_{2}\neq\epsilon)$.
$r=r_{1}\cdot r_{2}$ | 1\. $r_{1}\models filter(s,\alpha(r_{1}\cdot r_{2}))$ and $r_{2}\models\epsilon$.
| 2\. $r_{2}\models filter(s,\alpha(r_{1}\cdot r_{2}))$ and
$r_{1}\models\epsilon$.
| 3\. $r_{1}\models filter(s_{1},\alpha(r_{1}\cdot r_{2}))$ and
| $r_{2}\models filter(s_{2},\alpha(r_{1}\cdot r_{2}))$, where
$s=s_{1}s_{2}(s_{1},s_{2}\neq\epsilon)$.
$r=r_{1}\&r_{2}$ | $r_{1}\models filter(s,\alpha(r_{1}))$ and $r_{2}\models filter(s,\alpha(r_{2}))$.
$r=r_{1}|r_{2}$ | 1\. $r_{1}\models filter(s,\alpha(r_{1}|r_{2}))$.
| 2\. $r_{2}\models filter(s,\alpha(r_{1}|r_{2}))$.
Table 1: The semantics of filter matching, where $r,r_{1},r_{2}$ are SOIREs
and $s,s_{1},s_{2}$ are strings. $r\models filter(s,\alpha(r))$ if and only if
at least one condition on the right is satisfied.
### SOIRE Matching by SOIRETM
We propose a new matching algorithm for SOIRE, named SOIRETM, based on dynamic
programming. Generally, SOIRETM divides the original matching problem into
smaller ones to conquer. We observe that SOIRE matching can be simplified to
filter matching, as shown in the following theorem.
###### Theorem 1.
Given a SOIRE $r$ and a string $s$, $r\models s$ iff $filter(s,\alpha(r))=s$
and $r\models filter(s,\alpha(r))$.111All the proofs of
theorems/lemmas/propositions are provided in appendix B.
Theorem 1 presents a necessary and sufficient condition for SOIRE matching.
The condition $filter(s,\alpha(r))=s$ guarantees that $\alpha(r)$ contains all
symbols in $s$, which is easy to check. Therefore, the primary problem is to
check if $r$ matches $filter(s,\alpha(r))$.
We build the syntax tree of $r$ and compute the result of filter matching of
$s$ and $r$, namely $g_{1,|s|}^{1}$. The proposed algorithm SOIRETM is
detailed in Algorithm 1. Initially, line 1 checks if $filter(s,\alpha(r))=s$.
Then we calculate $g_{i,j}^{t}$ from shorter substrings to longer ones and
from bottom to top of the syntax tree. The statements in Line 7-20 conform to
the semantics of each operator, given by Table 1 and Lemma 2.
###### Lemma 2.
Given a SOIRE $r$ and a string $s$. If $r=r_{1}\cdot r_{2}$ or
$r=r_{1}\&r_{2}$ or $r=r_{1}|r_{2}$, then for all $i\in\\{1,2\\}$,
$r_{i}\models filter(s,\alpha(r))$ iff
$filter(s,\alpha(r_{i}))=filter(s,\alpha(r))$ and $r_{i}\models
filter(s,\alpha(r_{i}))$.
The time complexity of Algorithm 1 is $O(|s|^{3}|r|)$. It is sound and
complete according to the following theorem.
###### Theorem 3.
Given a SOIRE $r$ and a string $s$, $r\models s$ iff
$\texttt{SOIRETM}(r,s)=1$.
Algorithm 1 Matching Algorithm for SOIRE, SOIRETM.
Input: A SOIRE $r$ and a string $s$.
Output: The answer of whether $r$ matches $s$.
1:if $filter(s,\alpha(r))\neq s$ then
2: Return $0$;
3:Build the syntax tree of $r$ by $\texttt{RE2Tree}(r)$;
4:Let $flag_{i,j}^{t,t^{\prime}}$ denote
$1[filter(s_{i,j},\alpha(r^{t}))=filter(s_{i,j},\alpha(r^{t^{\prime}}))]$,
where $1[\mu]=1$ iff $\mu$ is true;
5:for all substring $s_{i,j}$ from $\epsilon$ to $s$ (shorter to longer) do
6: for $t\leftarrow|r|$ downto $1$ do
7: if $r^{t}=a\in\alpha(r)$ then
8: $g_{i,j}^{t}\leftarrow 1[filter(s_{i,j},\\{a\\})=a]$;
9: else if $r^{t}=(r^{t+1})^{?}$ then
10: $g_{i,j}^{t}\leftarrow 1[filter(s_{i,j},\alpha(r^{t}))=\epsilon]\vee
g_{i,j}^{t+1}$;
11: else if $r^{t}=(r^{t+1})^{*}$ then
12: $g_{i,j}^{t}\leftarrow 1[filter(s_{i,j},\alpha(r^{t}))=\epsilon]\vee
g_{i,j}^{t+1}\vee\bigvee_{k=i}^{j-1}(g_{i,k}^{t}\wedge g_{k+1,j}^{t+1})$;
13: else if $r^{t}=(r^{t+1})^{+}$ then
14: $g_{i,j}^{t}\leftarrow
g_{i,j}^{t+1}\vee\bigvee_{k=i}^{j-1}(g_{i,k}^{t}\wedge g_{k+1,j}^{t+1})$;
15: else if $r^{t}=r^{t+1}\cdot r^{\eta^{t}}$ then
16: $g_{i,j}^{t}\leftarrow(flag_{i,j}^{t,t+1}\wedge g_{i,j}^{t+1}\wedge
g_{1,0}^{\eta^{t}})\vee(flag_{i,j}^{t,\eta^{t}}\wedge g_{i,j}^{\eta^{t}}\wedge
g_{1,0}^{t+1})\vee\bigvee_{k=i}^{j-1}(flag_{i,k}^{t,t+1}\wedge
g_{i,k}^{t+1}\wedge flag_{k+1,j}^{t,\eta^{t}}\wedge g_{k+1,j}^{\eta^{t}})$;
17: else if $r^{t}=r^{t+1}\&r^{\eta^{t}}$ then
18: $g_{i,j}^{t}\leftarrow g_{i,j}^{t+1}\wedge g_{i,j}^{\eta^{t}}$;
19: else if $r^{t}=r^{t+1}|r^{\eta^{t}}$ then
20: $g_{i,j}^{t}\leftarrow(flag_{i,j}^{t,t+1}\wedge
g_{i,j}^{t+1})\vee(flag_{i,j}^{t,\eta^{t}}\wedge g_{i,j}^{\eta^{t}})$;
21:Return $g_{1,|s|}^{1}$;
### From SOIRETM to Neural Network
We now detail how to convert SOIRETM to a trainable neural network to simulate
SOIRE matching.
SOIRETM uses the syntax tree for SOIRE matching, so the trainable parameters
of an expected neural network can be defined by constructs of the syntax tree.
There are two parts of parameters used in an expected neural network,
$\theta=(w,u)$, where $w\in[0,1]^{T\times|\mathbb{B}|}$, $u\in[0,1]^{T\times
T}$ and $\mathbb{B}=\Sigma\cup\\{?,*,+,\cdot,\&,|,\text{none}\\}$, and where
$T$ is the bounded size of the target SOIRE. For $1\leq t\leq
T,a\in\mathbb{B}$, $w^{t}_{a}$ denotes the probability of vertex $t$
representing a symbol in $\Sigma$ or an ordinary operator or the none
operator. For $1\leq t\leq T$ and $t+2\leq t^{\prime}\leq T$,
$u^{t}_{t^{\prime}}$ denotes the probability of vertex $t$ choosing vertex
$t^{\prime}$ as its right child. The total number of the parameters to be
learnt is $T|\mathbb{B}|+\frac{(T-1)(T-2)}{2}$. Example 2 shows the parameters
of the neural network $\theta=(w,u)$ corresponding to the syntax tree in
Figure 1.
###### Example 2.
When $T=6$, $w^{1}_{\cdot}$, $w^{2}_{\&}$, $w^{3}_{a}$, $w^{4}_{b}$,
$w^{5}_{*}$, $w^{6}_{c}$, $u^{1}_{5}$, $u^{2}_{4}$ are $1$s, whereas other
parameters are $0$s. When $T=8$, $w^{7}_{\text{none}}$, $w^{8}_{\text{none}}$
are $1$s in addition to the above parameters.
There are four parts that should be considered during the conversion of
SOIRETM to neural network: $\alpha(r^{t})$,
$flag_{i,j}^{t,t^{\prime}},g^{t}_{i,j}$, and the return value of Algorithm 1.
Recall that $\alpha(r^{t})$ represents the set of symbols in $r^{t}$, which is
also the set of symbols occurring in the subtree whose root is $t$. We use
$\rho^{t}_{a}$ to denote the probability of symbol $a\in\Sigma$ that occurs in
the subtree whose root is $t$. We calculate $\rho^{t}_{a}$ from bottom to top
of the syntax tree by Equation 11, where $\sigma_{01}(x)=\min(\max(x,0),1)$.
For all $t>T$ and $a\in\Sigma$, $\rho^{t}_{a}$ is set to $0$. Note that
$\min(x)=-\max(-x)$, and the $\max$ function amounts to ReLU and can be
approximated by a more differentiable LeakyReLU function.
$\rho^{t}_{a}=\sigma_{01}(w^{t}_{a}+\sum_{\begin{subarray}{c}o\in\\{?,*,+,\\\
\cdot,\&,|\\}\end{subarray}}w^{t}_{o}\rho^{t+1}_{a}+\sum_{\begin{subarray}{c}o\in\\\
\\{\cdot,\&,|\\}\end{subarray}}w^{t}_{o}\sum_{t^{\prime}=t+2}^{T}u^{t}_{t^{\prime}}\rho^{t^{\prime}}_{a})$
(11)
For converting $flag_{i,j}^{t,t^{\prime}}$, we treat it as the probability
that there does not exist a symbol occurring in both $s_{i,j}$ and
$\alpha(r^{t})$ but not occurring in $\alpha(r^{t^{\prime}})$, as defined in
Equation 12. Note that $t^{\prime}$ can only be either $t+1$ or $\eta^{t}$.
$flag_{i,j}^{t,t^{\prime}}=1-\sigma_{01}(\sum_{a\in\Sigma}\sigma_{01}(1[a\in
s_{i,j}]+(\rho^{t}_{a}-\rho^{t^{\prime}}_{a})-1))$ (12)
For converting $g_{i,j}^{t}$, we introduce $p_{i,j}^{t}(?)$ (resp.
$p_{i,j}^{t}(*)$ or $p_{i,j}^{t}(+)$) to denote the probability that the
right-hand side of Line 10 (resp. 12 or 14) in Algorithm 1 evaluates to $1$,
as well as $p_{i,j}^{t}(\cdot,t^{\prime})$ (resp. $p_{i,j}^{t}(\&,t^{\prime})$
or $p_{i,j}^{t}(|,t^{\prime})$) the probability that the right-hand side of
Line 16 (resp. 18 or 20) with $\eta^{t}$ substituted by $t^{\prime}$ evaluates
to $1$. For example, $p^{8}_{1,3}(\&,10)$ denotes the probability that
$g_{1,3}^{9}\wedge g_{1,3}^{10}$ evaluates to $1$. Since a right-hand side may
contain logical connectives $\wedge$ or $\vee$, we apply the transformations
given in Table 2 to estimate the ultimate probability that the right-hand side
evaluates to $1$, where the special transformation is only used in the third
term of Line 12 and Line 16 as well as the second term of Line 14 since these
terms may have a large number of operands, while anywhere else the general
transformations are used. With the probabilities that the right-hand sides
evaluate to $1$, the probability that $g_{i,j}^{t}$ evaluates to $1$ can be
defined recursively by Equation From SOIRETM to Neural Network, where we reuse
$g_{i,j}^{t}$ to represent such a probability.
$\displaystyle g_{i,j}^{t}=\sum_{a\in\Sigma}w^{t}_{a}\cdot
1[filter(s_{i,j},{a})=a]$
$\displaystyle+\sum_{o\in\\{?,*,+\\}}w^{t}_{o}p^{t}_{i,j}(o)+\sum_{o\in\\{\cdot,\&,|\\}}w^{t}_{o}\sum_{t^{\prime}=t+2}^{T}u^{t}_{t^{\prime}}p^{t}_{i,j}(o,t^{\prime})$
(13)
Logical operation | General transformation | Special transformation
---|---|---
$A\wedge B$ | $\min(p_{A},p_{B})$ | -
$A\vee B$ | $\sigma_{01}(p_{A}+p_{B})$ | $\max(p_{A},p_{B})$
Table 2: The transformation from logical operations to numerical computations,
where $A$ and $B$ are formulae, and $p_{A}$ (resp. $p_{B}$) is the probability
that $A$ (resp. $B$) evaluates to $1$.
For converting the return value of Algorithm 1, we consider Line 1 and Line 21
in Algorithm 1 and compute the return value by Equation 14, where the second
term in Equation 14 ensures that all symbols occurring in $s$ appear in the
target SOIRE too.
$\hat{y}=g^{1}_{1,|s|}-\max_{a\in\Sigma}\sigma_{01}(1[a\in s]-\rho^{1}_{a})$
(14)
The converted neural network is trained to minimize the objective function
$\frac{1}{2}(\hat{y}-y)^{2}$, where $y\in\\{0,1\\}$ is the ground-truth label
for $r$ matching $s$.
### Faithful Encoding
We simply refer to an assignment of the set of trainable parameters of the
converted neural network as an _encoding_ of SOIREs. We find that encodings
can one-to-one correspond to prefix notations of SOIREs for a bounded size,
when it satisfies certain conditions given by Definition 2.
###### Definition 2 (Faithful encoding).
An encoding $\theta=(w,u)$ of SOIREs with length $T$ is said to be _faithful_
if it satisfies all the following conditions:
1. 1.
$\forall 1\leq t\leq T,w^{t}$ is a one-hot vector.
2. 2.
$\forall 1\leq t\leq T,u^{t}$ is either a one-hot vector or an all-zero
vector.
3. 3.
$\forall 1\leq t\leq
T,\sum_{t^{\prime}=t+2}^{T}u^{t}_{t^{\prime}}+\sum_{a\in\Sigma\cup\\{?,*,+,\text{none}\\}}w^{t}_{a}=1$.
4. 4.
$\forall 1\leq t\leq T-1,w^{t+1}_{\text{none}}-w^{t}_{\text{none}}\geq 0$.
5. 5.
$\forall 2\leq t\leq
T,\sum_{a\in\\{?,+,*,\cdot,\&,|\\}}w^{t-1}_{a}+\sum_{t^{\prime}=1}^{t-2}u^{t^{\prime}}_{t}+w^{t}_{\text{none}}=1$.
6. 6.
$\forall 3\leq t\leq T,\forall 1\leq p\leq
t-2,(t-1-p)u^{p}_{t}+\sum_{p^{\prime}=p+1}^{t-1}\sum_{t^{\prime}=t+1}^{T}u^{p^{\prime}}_{t^{\prime}}\leq
t-1-p$.
7. 7.
$\forall a\in\Sigma,\sum_{t=1}^{T}w^{t}_{a}\leq 1$.
All conditions in a faithful encoding are translated from the construction
constraints of a syntax tree. Condition 1 guarantees that each vertex in the
syntax tree represents a symbol, an ordinary operator or the none operator.
Condition 2 guarantees that each vertex has at most one right child. Condition
3 guarantees that each vertex either has a right child, or represents a
symbol, a unary operator or the none operator. Condition 4 guarantees that if
vertex $t$ represents the none operator, then vertex $t+1$ is also the none
operator. Condition 5 guarantees that if vertex $t$ represents the none
operator, then vertex $t$ is not the child of any vertex; otherwise, vertex
$t$ is the child of exactly one vertex. Condition 6 guarantees that the
vertices are numbered in the order of preorder traversal. Condition 7
guarantees that each symbol in $\Sigma$ occurs at most once in the syntax
tree. Example 2 shows two faithful encodings with lengths $6$ and $8$,
respectively.
By $\texttt{Enc2Pre}(\theta)$ we denote the prefix notation of the SOIRE
interpreted from a faithful encoding $\theta$. $\texttt{Enc2Pre}(\theta)$
decodes $w^{t}$ and $u^{t^{\prime}}_{t}$ ($1\leq t^{\prime}\leq t-2$) into the
syntax tree of $r$ from $t=1$ to $T$ progressively until
$w^{t}_{\text{none}}=1$, and then translates the constructed syntax tree of
$r$ to the prefix notation of $r$. Take Example 2 for instance,
$\texttt{Enc2Pre}(\theta)=\cdot\&ab*c$ is decoded from the faithful encoding
$\theta$ with length $T=8$.
Proposition 4 shows that $\texttt{Enc2Pre}(\theta)$ always yields the prefix
notation of a SOIRE.
###### Proposition 4.
For any faithful encoding $\theta$, there exists a SOIRE $r$ such that
$\texttt{Enc2Pre}(\theta)=\texttt{PreForm}(r)$.
Proposition 5 and Proposition 6 show that $\texttt{Enc2Pre}(\theta)$ is
surjective and injective, respectively, for a bounded size.
###### Proposition 5.
Given a bounded size $T\in\mathbb{Z}^{+}$, for any SOIRE $r$ such that
$|r|\leq T$, there exists a faithful encoding $\theta$ with length $T$ such
that $\texttt{Enc2Pre}(\theta)=\texttt{PreForm}(r)$.
###### Proposition 6.
Given a bounded size $T\in\mathbb{Z}^{+}$, for any two different faithful
encodings $\theta_{1}$ and $\theta_{2}$ with length $T$, we have
$\texttt{Enc2Pre}(\theta_{1})\neq\texttt{Enc2Pre}(\theta_{2})$.
Since $\texttt{Enc2Pre}(\theta)$ is both injective and surjective, faithful
encodings are one-to-one corresponding to the prefix notations of SOIREs for a
bounded size.
###### Theorem 7.
Given a bounded size $T\in\mathbb{Z}^{+}$, prefix notations of SOIREs $r$ with
$|r|\leq T$ and faithful encodings $\theta$ with length $T$ have a one-to-one
correspondence, i.e., $\texttt{Enc2Pre}(\theta)=\texttt{PreForm}(r)$.
To make an encoding more faithful, we add one regularization for each
condition to the objective function222Details of regularizations are provided
in Appendix C..
Algorithm 2 SOIRE Interpretation.
Input: A training set $(\Pi_{\text{train}}^{+},\Pi_{\text{train}}^{-})$ on the
alphabet $\Sigma$, an encoding $\theta=(w,u)$ and the the beam width $\beta$.
Output: The infix notation of the target SOIRE.
1:Let $T$ be the first dimension of $w$;
2:Let $C^{t}$ be the set of candidate solutions $(r,e)$ of the subtree with
the root $t$, where $r$ is the infix notation of a SOIRE and $e$ is its score;
3:for $t$ from $T$ down to $1$ do
4: for all $a\in\Sigma$ do
5: Add $(a,w^{t}_{a})$ into $C^{t}$;
6: for all $(r_{i},e_{i})\in C^{t+1}$ do
7: Add $(r_{i}^{?},e_{i}w^{t}_{?})$ into $C^{t}$;
8: Add $(r_{i}^{*},e_{i}w^{t}_{*})$ into $C^{t}$;
9: Add $(r_{i}^{+},e_{i}w^{t}_{+})$ into $C^{t}$;
10: for $t^{\prime}$ from $t+2$ to $T$ do
11: for all $((r_{i},e_{i}),(r_{j},e_{j}))\in C^{t+1}\times C^{t^{\prime}}$ do
12: if $\alpha(r_{i})\cap\alpha(r_{j})=\emptyset$ then
13: Add $((r_{i})\cdot(r_{j}),e_{i}e_{j}w^{t}_{\cdot})$ into $C^{t}$;
14: Add $((r_{i})\&(r_{j}),e_{i}e_{j}w^{t}_{\&})$ into $C^{t}$;
15: Add $((r_{i})|(r_{j}),e_{i}e_{j}w^{t}_{|})$ into $C^{t}$;
16: Sort all $(r,e)$ in $C^{t}$ according to the descending order of
${e}^{\frac{1}{|r|}}$ and keep only the top-$\beta$ elements;
17:Get the accuracy of $r$ on
$(\Pi_{\text{train}}^{+},\Pi_{\text{train}}^{-})$ for all $r$ in $C^{1}$;
18:Return $r$ in $C^{1}$ that has the highest accuracy;
### SOIRE Interpretation
We apply beam search to find a faithful encoding nearby the learnt encoding
and then interpret it to the target SOIRE. The algorithm is shown in Algorithm
2. The interpretation steps are conducted from bottom to top of the syntax
tree. We keep $\beta$ candidate SOIREs for each subtree according to the score
of each candidate SOIRE, which is defined as the geometric mean of the
probabilities of all operators and symbols. At each step, we select different
operators and candidate SOIREs from the left child and right child (if any)
and merge them to generate new candidates. At last, we calculate the accuracy
of each SOIRE in the last step on the training set and pick out the SOIRE with
the highest accuracy.
## Evaluation
We conduct experiments to evaluate the performance of SOIREDL on both noise-
free data and noisy data.
Datasets. We extract $30$ SOIREs from the RE database built by Li et al.
(2018) and generate datasets with noise from them. The SOIREs are randomly
chosen from different classes: SIRE, ICHARE, RSOIRE, SOIRE. We set $\Sigma$ as
$10$ letters. For each SOIRE $r$, we generate a dataset $(\Pi^{+},\Pi^{-})$
randomly, making sure that for all $s\in\Pi^{-}$, there exists $s^{\prime}$
such that $r\models s^{\prime}$ and $s$ can be modified to $s^{\prime}$ by
deleting a character, inserting a character at any position, replacing a
character with another one, or moving a character to any other position. For
example, string $abc$ can be modified to $ac$, $abac$, $acc$ or $bca$. We set
the maximum length of strings to $20$ in the dataset. For training sets and
test sets, we set $|\Pi^{+}|=|\Pi^{-}|=250$, whereas for validation sets, we
set $|\Pi^{+}|=|\Pi^{-}|=50$. The noise levels are set to
$\delta=\\{0,0.05,0.1,0.15,0.2\\}$, where for each $\delta$, we reverse the
labels for $|\Pi^{+}|\delta$ positive strings and $|\Pi^{-}|\delta$ negative
strings in the training and validation sets.
Competitors. We choose approaches iSOIRE (Li et al. 2019a), GenICHARE (Zhang
et al. 2018), iSIRE (Li et al. 2020b) and RE2RNN (Jiang et al. 2020) as
competitors. iSOIRE learns RSOIREs and GenICHARE learns ICHAREs from positive
strings only. iSIRE learns SIREs from both positive and negative strings.
RE2RNN embeds a weighted FSA to improve the performance on text
classification. It can also learn an automaton if we randomly initialize the
parameters. Thus we also compare the performance between SOIREDL and RE2RNN.
Settings. We implement iSOIRE, GenICHARE, iSIRE according to their papers, and
reuse the source code of RE2RNN. The hyper-parameters of RE2RNN are set as
default, except that the number of states is set to $100$ and the threshold in
interpretation to $0.12$ for achieving the best accuracy. We train SOIREDL
with the AdamW optimizer. The hyper-parameters of SOIREDL are set as follows:
the bounded size $T$ is $4|\Sigma|-2$ according to Proposition 8, the batch
size is $64$, the regularization coefficient $\lambda$ is $0$, and the beam
width $\beta$ is $500$. The optimal $\lambda$ is selected from
$\\{0,10^{-3},10^{-2},10^{-1},1,10\\}$ and $\beta$ from
$\\{10,50,100,300,500,1000\\}$ for maximizing the accuracy on the validation
set. iSOIRE, GenICHARE and iSIRE use the union of the training and validation
sets for learning.
All experiments were conducted on a Linux machine equipped with an Intel Xeon
Gold 6248R processor with $126$ GB RAM and a single NVIDIA A$100$. We train
SOIREDL with five learning rates $0.01,0.05,0.1,0.15,0.2$ and select the SOIRE
achieving the best accuracy on the validation set. Therefore, the running time
of SOIREDL is the sum of training time and interpretation time with five
learning rates. The time limit for each approach is set to $5000$ seconds.
###### Proposition 8.
For any SOIRE $r$ over $\Sigma$, there exists another SOIRE $r^{\prime}$ over
$\Sigma$ and a faithful encoding $\theta$ with length $4|\Sigma|-2$ such that
$\texttt{PreForm}(r^{\prime})=\texttt{Enc2Pre}(\theta)$ and
$\\{s|r^{\prime}\models s\\}=\\{s|r\models s\\}$.
Evaluate metrics. We use accuracy on the test set to evaluate the performance
of all approaches. For SOIREDL and RE2RNN, We introduce faithfulness defined
as $\frac{N_{=}}{\left|\Pi^{+}\right|+\left|\Pi^{-}\right|}$ to evaluate the
consistency between the neural network and the interpreted SOIRE (SOIREDL) or
automata (RE2RNN), where $N_{=}$ is the number of test strings that the neural
network and the SOIRE or automata predicts the same label.
| Positive and negative strings | Positive strings only
---|---|---
Data | iSI | RE2RNN | SOIREDL | iSO | GenIC | iSI | RE2RNN | SOIREDL
set | RE | IRE | HARE | RE
1 | 86.0 | 52.4 (94.4) | 100.0 (100.0) | 89.4 | 89.4 | 83.8 | 48.0 (50.0) | 57.0 (57.0)
2 | 77.4 | 48.8 (91.4) | 100.0 (100.0) | 100.0 | 100.0 | 100.0 | 50.4 (50.0) | 100.0 (100.0)
3 | 90.8 | 50.6 (77.4) | 100.0 (100.0) | 100.0 | 97.2 | 95.6 | 50.6 (49.6) | 65.4 (65.4)
4 | 72.4 | 49.0 (80.2) | 99.6 (73.4) | 73.8 | 72.6 | 73.4 | 49.6 (49.8) | 61.4 (61.4)
5 | 90.8 | 52.6 (91.4) | 58.2 (87.2) | 100.0 | 100.0 | 86.2 | 49.6 (50.2) | 52.8 (52.8)
6 | 77.8 | 51.6 (62.2) | 93.4 (93.0) | 100.0 | 100.0 | 70.4 | 50.2 (50.0) | 52.6 (52.6)
7 | 81.2 | 52.2 (95.8) | 99.2 (99.2) | 100.0 | 96.4 | 89.0 | 49.2 (49.8) | 69.4 (69.4)
8 | 76.2 | 49.8 (88.0) | 100.0 (100.0) | 100.0 | 100.0 | 100.0 | 49.8 (50.0) | 100.0 (100.0)
9 | 93.8 | 44.2 (48.4) | 98.4 (98.4) | 99.8 | 98.8 | 98.0 | 47.2 (49.6) | 81.6 (81.6)
10 | 94.8 | 50.2 (89.8) | 99.8 (100.0) | 99.8 | 99.8 | 82.8 | 44.4 (50.2) | 61.2 (61.2)
11 | 91.0 | 52.4 (88.6) | 89.2 (91.0) | 100.0 | 100.0 | 91.4 | 47.6 (50.2) | 57.4 (57.4)
12 | 78.6 | 51.2 (98.4) | 100.0 (100.0) | 100.0 | 100.0 | 100.0 | 50.8 (49.4) | 100.0 (100.0)
13 | 96.6 | 52.8 (50.2) | 100.0 (100.0) | 100.0 | 100.0 | 83.6 | 51.2 (49.6) | 67.0 (67.0)
14 | 74.4 | 50.0 (79.0) | 84.8 (71.6) | 70.0 | 74.4 | 68.8 | 49.0 (50.0) | 54.4 (54.4)
15 | 95.2 | 54.0 (49.8) | 96.2 (100.0) | 100.0 | 100.0 | 100.0 | 47.8 (50.2) | 66.2 (66.2)
16 | 96.8 | 49.0 (71.4) | 94.8 (100.0) | 100.0 | 100.0 | 75.4 | 49.4 (50.0) | 75.4 (75.4)
17 | 91.0 | 46.2 (87.2) | 100.0 (100.0) | 100.0 | 100.0 | 100.0 | 50.6 (50.0) | 100.0 (100.0)
18 | 81.0 | 42.8 (78.8) | 87.4 (99.2) | 87.4 | 100.0 | 82.0 | 40.2 (50.2) | 53.0 (53.0)
19 | 88.2 | 50.0 (63.8) | 100.0 (93.4) | 100.0 | 100.0 | 88.0 | 50.4 (50.0) | 54.6 (54.6)
20 | 93.4 | 45.2 (54.4) | 83.4 (90.0) | 100.0 | 99.2 | 95.8 | 47.4 (50.0) | 56.0 (51.2)
21 | 69.8 | 48.8 (96.2) | 100.0 (100.0) | 71.2 | 71.2 | 71.2 | 49.8 (50.4) | 71.2 (71.2)
22 | 90.6 | 47.6 (50.6) | 100.0 (100.0) | 100.0 | 100.0 | 91.4 | 50.0 (50.2) | 58.2 (58.2)
23 | 85.6 | 32.8 (84.4) | 86.8 (65.2) | 90.0 | 90.0 | 88.0 | 50.0 (48.8) | 56.8 (56.8)
24 | 69.4 | 49.0 (57.2) | 74.6 (76.8) | 77.6 | 76.0 | 71.4 | 47.6 (50.0) | 54.2 (54.2)
25 | 67.8 | 54.2 (93.4) | 99.8 (74.6) | 70.0 | 70.0 | 70.0 | 50.2 (50.0) | 69.6 (69.6)
26 | 80.2 | 50.4 (50.4) | 63.8 (98.2) | 100.0 | 100.0 | 85.2 | 51.0 (49.6) | 63.8 (63.8)
27 | 92.0 | 52.0 (91.6) | 96.4 (96.4) | 100.0 | 100.0 | 97.4 | 52.2 (50.2) | 67.6 (67.6)
28 | 65.0 | 54.8 (95.2) | 100.0 (98.6) | 65.6 | 65.6 | 65.6 | 50.0 (50.0) | 65.6 (65.6)
29 | 93.4 | 56.2 (75.8) | 97.6 (97.2) | 100.0 | 99.8 | 81.2 | 49.2 (49.8) | 54.2 (54.2)
30 | 60.4 | 41.4 (61.6) | 78.8 (79.6) | 61.0 | 61.0 | 60.4 | 49.8 (49.8) | 54.8 (54.8)
Avg. | 83.4 | 49.4 (76.6) | 92.7 (92.8) | 91.9 | 92.0 | 84.9 | 49.1 (49.9) | 66.7 (66.6)
Table 3: Accuracy (%) on noise-free data with best results in bold. For X(Y),
X denotes the accuracy of the learnt SOIRE or automaton, and Y the accuracy of
the neural network.
Comparison on noise-free data. The results of different approaches on noise-
free data are shown in Table 3. For learning from both positive and negative
strings, SOIREDL outperforms iSIRE and RE2RNN on almost all datasets, achieves
comparable performance with iSOIRE and GenICHARE (both of which learn from
positive strings only), and achieves the highest average accuracy among all
approaches. Regarding the accuracy of the intermediate neural network, SOIREDL
is also superior to RE2RNN. These results show that SOIREDL achieves the SOTA
performance on noise-free data.
Comparison on noisy data. The average accuracy of different approaches on
noisy data is shown in Figure 2 (a). The performance of iSOIRE and GenICHARE
drops sharply when noise is present, suggesting that they are not robust on
noisy data. The average accuracy of RE2RNN also decreases quickly when the
noise level increases, and so does the neural network of it. Both SOIREDL and
iSIRE perform well on noisy data. The average accuracy of SOIREDL slightly
decreases when the noise level increases, but it still keeps higher than
others at all noise levels. This suggests that SOIREDL is the most robust on
noisy data.
Figure 2: Average accuracy (%) on test sets at different noise levels
$\delta$. SOIREDL-ipt and RE2RNN-ipt represent the learnt SOIREs or automata,
whereas SOIREDL-net and RE2RNN-net represent the neural networks. (a) Positive
and negative strings. (b) Positive strings only.
Comparison in terms of faithfulness. The average faithfulness of SOIREDL and
RE2RNN is shown in Figure 3. Obviously, the faithfulness of SOIREDL is much
higher than that of RE2RNN and keeps over $80\%$ at all noise levels,
suggesting that the neural network of SOIREDL and its interpreted SOIRE are
more consistent in performing SOIRE matching.
Figure 3: Average faithfulness (%) of SOIREDL and RE2RNN on test sets at
different noise levels $\delta$.
Case study. We pick one RE from each subclass of SOIREs considered in our
experiments to show their learnt results, reported in Table 4. Although the
ground-truth REs and the SOIREs learnt by SOIREDL are not exactly the same in
the subclasses ICHARE and RSOIRE, they still belong to the same subclass.
These results show that SOIREDL is able to learn different subclasses of
SOIREs.
Subclass | Dataset | Ground-truth SOIREs | SOIREDL
---|---|---|---
SIRE | 13 | $a^{?}\&b^{*}\&c^{?}$ | $a^{?}\&c^{?}\&b^{*}$
ICHARE | 22 | $\left(a|b\right)^{*}c^{*}d^{*}$ | $\left(a^{*}\&b^{*}\right)c^{*}d^{*}$
RSOIRE | 3 | $a^{+}|\left(b|c\right)^{*}|d^{+}$ | $\left(b^{+}|c\right)^{*}|a^{*}|d^{*}$
SOIRE | 1 | $\left(\left(a|b\right)c^{*}\right)^{+}d$ | $\left(\left(a|b\right)c^{*}\right)^{+}d$
Table 4: The ground-truth SOIREs and the SOIREs learnt by SOIREDL on different
subclasses of SOIREs.
We also analyse the relation between the accuracy of the SOIRE learnt by
SOIREDL and the size of the ground-truth SOIRE. Figure 4 shows that the
accuracy decreases when the size of the ground-truth SOIRE increases. This may
be due to the difficulty for a neural network to capture the long-distance
dependency in SOIRE matching.
Figure 4: The relation of the accuracy (%) of a SOIRE learnt by SOIREDL and
the size $|r|$ of the ground-truth SOIRE $r$.
The necessity for using negative strings. The results of learning from
positive strings only are shown in Table 3 and Figure 2 (b). Neural networks
do not perform well because they prefer to classify unseen strings as positive
ones when training on positive strings only. Even worse, when noise is
present, the accuracy of any competitor drops sharply to no more than $60\%$
as shown in Figure 2 (b). This suggests that negative strings are crucial in
effectively learning with noisy data, possibly because they infer what kinds
of strings that the target SOIRE cannot match. By learning from positive and
negative strings, both SOIREDL and RE2RNN achieve significantly better
performance; in particular, SOIREDL outperforms iSOIRE and GenICHARE in terms
of average accuracy, as shown in Table 3 and Figure 2 (a).
## Conclusion and Future Work
In this paper, we have proposed a noise-tolerant differentiable learning
approach SOIREDL and a matching algorithm SOIRETM based on the syntax tree for
SOIREs. The neural network introduced in SOIREDL simulates the matching
process of SOIRETM. Theoretically, the faithful encodings learnt by SOIREDL
one-to-one correspond to SOIREs for a bounded size. Experimental results have
demonstrated higher performance compared with the SOTA approaches. In the
future work, we will tackle the problem of long dependency in SOIRE matching
and extend our approach to other subclasses of REs.
## Acknowledgments
We thank Kunxun Qi for discussion on the paper and anonymous referees for
helpful comments.
This paper was supported by the National Natural Science Foundation of China
(No. 62276284, 61976232, 61876204), the National Key Research and Development
Program of China (No. 2021YFA1000504), Guangdong Basic and Applied Basic
Research Foundation (No. 2022A1515011355), Guangzhou Science and Technology
Project (No. 202201011699), Guizhou Science Support Project (No. 2022-259), as
well as Humanities and Social Science Research Project of Ministry of
Education (No. 18YJCZH006).
## References
* Bex et al. (2006) Bex, G. J.; Neven, F.; Schwentick, T.; and Tuyls, K. 2006. Inference of Concise DTDs from XML Data. In _VLDB_ , 115–126.
* Bojanczyk et al. (2006) Bojanczyk, M.; Muscholl, A.; Schwentick, T.; Segoufin, L.; and David, C. 2006. Two-Variable Logic on Words with Data. In _LICS_ , 7–16.
* Cohen, Yang, and Mazaitis (2020) Cohen, W. W.; Yang, F.; and Mazaitis, K. 2020. TensorLog: A Probabilistic Database Implemented Using Deep-Learning Infrastructure. _J. Artif. Intell. Res._ , 67: 285–325.
* Colazzo et al. (2013) Colazzo, D.; Ghelli, G.; Pardini, L.; and Sartiani, C. 2013. Efficient asymmetric inclusion of regular expressions with interleaving and counting for XML type-checking. _Theor. Comput. Sci._ , 492: 88–116.
* Freydenberger and Kötzing (2015) Freydenberger, D. D.; and Kötzing, T. 2015. Fast Learning of Restricted Regular Expressions and DTDs. _Theory Comput. Syst._ , 57(4): 1114–1158.
* Galassi and Giordana (2005) Galassi, U.; and Giordana, A. 2005. Learning Regular Expressions from Noisy Sequences. In _SARA_ , volume 3607, 92–106.
* Gao et al. (2022) Gao, K.; Inoue, K.; Cao, Y.; and Wang, H. 2022. Learning First-Order Rules with Differentiable Logic Program Semantics. In _IJCAI_ , 3008–3014.
* Gischer (1981) Gischer, J. L. 1981. Shuffle Languages, Petri Nets, and Context-Sensitive Grammars. _Commun. ACM_ , 24(9): 597–605.
* Huang et al. (2021) Huang, J.; Li, Z.; Chen, B.; Samel, K.; Naik, M.; Song, L.; and Si, X. 2021. Scallop: From Probabilistic Deductive Databases to Scalable Differentiable Reasoning. In _NeurIPS_ , 25134–25145.
* Jiang, Jin, and Tu (2021) Jiang, C.; Jin, Z.; and Tu, K. 2021. Neuralizing Regular Expressions for Slot Filling. In _EMNLP_ , 9481–9498.
* Jiang et al. (2020) Jiang, C.; Zhao, Y.; Chu, S.; Shen, L.; and Tu, K. 2020. Cold-Start and Interpretability: Turning Regular Expressions into Trainable Recurrent Neural Networks. In _EMNLP_ , 3193–3207.
* Kearns and Li (1988) Kearns, M. J.; and Li, M. 1988. Learning in the Presence of Malicious Errors (Extended Abstract). In _STOC_ , 267–280.
* Kuhlmann and Satta (2009) Kuhlmann, M.; and Satta, G. 2009. Treebank Grammar Techniques for Non-Projective Dependency Parsing. In _EACL_ , 478–486.
* Li et al. (2020a) Li, Y.; Cao, J.; Chen, H.; Ge, T.; Xu, Z.; and Peng, Q. 2020a. FlashSchema: Achieving High Quality XML Schemas with Powerful Inference Algorithms and Large-scale Schema Data. In _ICDE_ , 1962–1965.
* Li et al. (2020b) Li, Y.; Chen, H.; Zhang, L.; Huang, B.; and Zhang, J. 2020b. Inferring Restricted Regular Expressions with Interleaving from Positive and Negative Samples. In _PAKDD_ , volume 12085, 769–781.
* Li et al. (2019a) Li, Y.; Chen, H.; Zhang, X.; and Zhang, L. 2019a. An effective algorithm for learning single occurrence regular expressions with interleaving. In _IDEAS_ , 24:1–24:10.
* Li et al. (2018) Li, Y.; Chu, X.; Mou, X.; Dong, C.; and Chen, H. 2018. Practical Study of Deterministic Regular Expressions from Large-scale XML and Schema Data. In _IDEAS_ , 45–53.
* Li et al. (2021) Li, Y.; Li, S.; Xu, Z.; Cao, J.; Chen, Z.; Hu, Y.; Chen, H.; and Cheung, S. 2021\. TRANSREGEX: Multi-modal Regular Expression Synthesis by Generate-and-Repair. In _ICSE_ , 1210–1222.
* Li et al. (2019b) Li, Y.; Zhang, X.; Cao, J.; Chen, H.; and Gao, C. 2019b. Learning k-Occurrence Regular Expressions with Interleaving. In _DASFAA_ , volume 11447, 70–85.
* Makoto and Clark (2003) Makoto, M.; and Clark, J. 2003. RELAX NG. https://relaxng.org/.
* Martens et al. (2017) Martens, W.; Neven, F.; Niewerth, M.; and Schwentick, T. 2017. BonXai: Combining the Simplicity of DTD with the Expressiveness of XML Schema. _ACM Trans. Database Syst._ , 42(3): 15:1–15:42.
* Mayer and Stockmeyer (1994) Mayer, A. J.; and Stockmeyer, L. J. 1994. The Complexity of Word Problems - This Time with Interleaving. _Inf. Comput._ , 115(2): 293–311.
* Mensch and Blondel (2018) Mensch, A.; and Blondel, M. 2018. Differentiable Dynamic Programming for Structured Prediction and Attention. In _ICML_ , volume 80, 3459–3468.
* Minervini et al. (2020a) Minervini, P.; Bosnjak, M.; Rocktäschel, T.; Riedel, S.; and Grefenstette, E. 2020a. Differentiable Reasoning on Large Knowledge Bases and Natural Language. In _AAAI_ , 5182–5190.
* Minervini et al. (2020b) Minervini, P.; Riedel, S.; Stenetorp, P.; Grefenstette, E.; and Rocktäschel, T. 2020b. Learning Reasoning Strategies in End-to-End Differentiable Proving. In _ICML_ , volume 119, 6938–6949.
* Nivre (2009) Nivre, J. 2009. Non-Projective Dependency Parsing in Expected Linear Time. In _ACL_ , 351–359.
* Rocktäschel and Riedel (2017) Rocktäschel, T.; and Riedel, S. 2017. End-to-end Differentiable Proving. In _NeurIPS_ , 3788–3800.
* Sadeghian et al. (2019) Sadeghian, A.; Armandpour, M.; Ding, P.; and Wang, D. Z. 2019. DRUM: End-To-End Differentiable Rule Mining On Knowledge Graphs. In _NeurIPS_ , 15321–15331.
* Wang et al. (2019) Wang, P.; Donti, P. L.; Wilder, B.; and Kolter, J. Z. 2019. SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver. In _ICML_ , volume 97, 6545–6554.
* Wang et al. (2020) Wang, P.; Stepanova, D.; Domokos, C.; and Kolter, J. Z. 2020. Differentiable learning of numerical rules in knowledge graphs. In _ICLR_.
* Wang (2021a) Wang, X. 2021a. Inferring Deterministic Regular Expression with Unorder and Counting. In _DASFAA_ , volume 12682, 235–252.
* Wang (2021b) Wang, X. 2021b. Learning Finite Automata with Shuffle. In _PAKDD_ , volume 12713, 308–320.
* Wang (2022) Wang, X. 2022. Membership Algorithm for Single-Occurrence Regular Expressions with Shuffle and Counting. In _DASFAA_ , volume 13245, 526–542.
* Wang and Chen (2018) Wang, X.; and Chen, H. 2018. Inferring Deterministic Regular Expression with Counting. In _Conceptual Modeling - 37th International Conference, ER_ , volume 11157, 184–199.
* Wang and Chen (2020) Wang, X.; and Chen, H. 2020. Inferring Deterministic Regular Expression with Unorder. In _SOFSEM_ , volume 12011, 325–337.
* Wang and Zhang (2021) Wang, X.; and Zhang, X. 2021. Discovering an Algorithm Actually Learning Restricted Single Occurrence Regular Expression with Interleaving. _CoRR_ , abs/2103.10546.
* Yang, Yang, and Cohen (2017) Yang, F.; Yang, Z.; and Cohen, W. W. 2017. Differentiable Learning of Logical Rules for Knowledge Base Reasoning. In _NeurIPS_ , 2319–2328.
* Yang and Song (2020) Yang, Y.; and Song, L. 2020. Learn to Explain Efficiently via Neural Logic Inductive Learning. In _ICLR_.
* Zhang et al. (2018) Zhang, X.; Li, Y.; Cui, F.; Dong, C.; and Chen, H. 2018. Inference of a Concise Regular Expression Considering Interleaving from XML Documents. In _PAKDD_ , volume 10938, 389–401.
## Appendix A
### From prefix notation to the infix notation of a SOIRE
Algorithm 3 is a function to obtain the infix notaiton of a SOIRE from the
prefix notaiton.
Algorithm 3 From prefix notation to the infix notation of a SOIRE.
Input: The prefix notation of the SOIRE $r_{p}$.
Output: The infix notation of the SOIRE $r_{i}$.
1:Let $\Gamma$ be an empty stack;
2:for $i\leftarrow|r_{p}|$ downto $1$ do
3: Let $o^{i}$ be the $i$ th character of $r_{p}$;
4: if $o^{i}\in\Sigma$ then
5: Push $(o^{i})$ into stack $\Gamma$;
6: if $o^{i}\in\\{?,*,+\\}$ then
7: Pop an element from stack $\Gamma$, denoted as $r$;
8: Push $\text{concat}(`(^{\prime},r,o^{i},`)^{\prime})$ into stack $\Gamma$;
9: if $o^{i}\in\\{\cdot,\&,|\\}$ then
10: Pop an element from stack $\Gamma$, denoted as $r_{2}$;
11: Pop an element from stack $\Gamma$, denoted as $r_{1}$;
12: Push $\text{concat}(`(^{\prime},r_{1},o^{i},r_{2},`)^{\prime})$ into stack
$\Gamma$;
13:Pop an element from stack $\Gamma$, denoted as $r_{i}$;
14:Return $r_{i}$;
### Faithful encoding interpreter
Algorithm 4 Faithful encoding interpreter Enc2Pre.
Input: The faithful encoding of an SOIRE, $\theta=(w,u)$.
Output: The prefix notation of an SOIRE $r$.
1:Let $r$ be a empty string;
2:Let $T$ be bounded size of $r$;
3:for $t\leftarrow 1$ to $T$ do
4: if $w^{t}_{\text{none}}=1$ then
5: Break;
6: for all $a\in\mathbb{B}\setminus\\{\text{none}\\}$ do
7: if $w^{t}_{a}=1$ then
8: $r\leftarrow\text{concat}(a,r)$;
9:Return $r$;
## Appendix B
### Proof of Theorem 1
Proof:
1. 1.
If $r\models s$, then $filter(s,\alpha(r))=s$ and $r\models
filter(s,\alpha(r))$.
Because $r\models s$, $\alpha(r)$ contains all the symbols in $s$ and
$filter(s,\alpha(r))=s$ (Definition of SOIRE matching). Because $r\models s$
and $filter(s,\alpha(r))=s$, $r\models filter(s,\alpha(r))$.
2. 2.
If $filter(s,\alpha(r))=s$ and $r\models filter(s,\alpha(r))$, than $r\models
s$.
Because $r\models filter(s,\alpha(r))$ and $filter(s,\alpha(r))=s$, $r\models
s$.
Theorem 1 has been proved.
### Proof of Lemma 2
Proof:
We just prove that is correct for $r_{1}$ and the proof for $r_{2}$ is nearly
the same.
1. 1.
If $r_{1}\models filter(s,\alpha(r))$, then
$filter(s,\alpha(r_{1}))=filter(s,\alpha(r))$ and $r_{1}\models
filter(s,\alpha(r_{1}))$.
Because $r_{1}\models filter(s,\alpha(r))$, $\alpha(r_{1})$ contains all the
symbols in $filter(s,\alpha(r))$ (Definition of SOIRE matching). Because
$\alpha(r_{1})\subseteq\alpha(r)$,
$filter(s,\alpha(r_{1}))=filter(s,\alpha(r))$. Therefore, $r_{1}\models
filter(s,\alpha(r_{1}))$.
2. 2.
If $filter(s,\alpha(r_{1}))=filter(s,\alpha(r))$ and $r_{1}\models
filter(s,\alpha(r_{1}))$, then $r_{1}\models filter(s,\alpha(r))$.
Because $filter(s,\alpha(r_{1}))=filter(s,\alpha(r))$ and $r_{1}\models
filter(s,\alpha(r_{1}))$, $r_{1}\models filter(s,\alpha(r))$.
Lemma 2 has been proved.
### Proof of Theorem 3
Theorem 1 shows that $r\models s$ iff $filter(s,\alpha(r))=s$ and $r\models
filter(s,\alpha(r))$. In Algorithm 1, Line 1 checks if
$filter(s,\alpha(r))=s$. Therefore, we need to prove that $g_{1,|s|}^{1}=1$
iff $r\models filter(s,\alpha(r))$. Algorithm 1 calculate $g_{i,j}^{t}$ for
$s_{i,j}$ from short to long and for $r^{t}$ from bottom to top of the syntax
tree. We apply mathematical induction to the length of the substring
$|s_{i,j}|$ and structural induction to $r^{t}$.
Proof:
Mathematical induction base: $|s_{i,j}|=l=0$, i.e. $s_{i,j}=\epsilon$ and
$i=1,j=0$.
Structural induction: Each case is corresponding to one statement in Line 8-20
in Algorithm 1.
1. 1.
$r^{t}=a\in\alpha(r)$.
$r^{t}=a$ does not match $filter(s_{1,0},\\{a\\})=\epsilon$, so
$g_{1,0}^{t}=1[filter(s_{1,0},\\{a\\})=a]=0$ is correct.
2. 2.
$r^{t}=(r^{t+1})^{?}$.
$(r^{t+1})^{?}\models filter(s_{1,0},\alpha((r^{t+1})^{?}))=\epsilon$, so
$g_{1,0}^{t}=g_{1,0}^{t+1}\ \vee\
1[filter(s_{1,0},\alpha((r^{t+1})^{?}))=\epsilon]=1$ is correct.
3. 3.
$r^{t}=(r^{t+1})^{*}$.
$(r^{t+1})^{*}\models filter(s_{1,0},\alpha((r^{t+1})^{*}))=\epsilon$, so
$g_{1,0}^{t}=1[filter(s_{1,0},\alpha(r^{t}))=\epsilon]\vee
g_{1,0}^{t+1}\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\ g_{k+1,j}^{t+1})=1$ is
correct.
4. 4.
$r^{t}=(r^{t+1})^{+}$.
Because $i=1>j-1=-1$, $\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\
g_{k+1,j}^{t+1})=0$. Therefore
$g_{1,0}^{t}=g_{1,0}^{t+1}\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\
g_{k+1,j}^{t+1})=g_{1,0}^{t+1}$. Because $g_{1,0}^{t}=1$ iff
$(r^{t+1})^{+}\models\epsilon$, and $(r^{t+1})^{+}\models\epsilon$ iff
$r^{t+1}\models\epsilon$, $g_{1,0}^{t}=1$ iff $r^{t+1}\models\epsilon$.
Because $r^{t+1}\models\epsilon$ iff $g_{1,0}^{t+1}=1$ (Structural inductive
hypothesis), $g_{1,0}^{t}=1$ iff $g_{1,0}^{t+1}=1$. Therefore,
$g_{1,0}^{t}=g_{1,0}^{t+1}\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\
g_{k+1,j}^{t+1})$ is correct.
5. 5.
$r^{t}=r^{t+1}\cdot r^{\eta^{t}}$.
Because $i=1>j-1=-1$, $\vee_{k=i}^{j-1}(1[filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,k},\alpha(r^{t+1}))]\ \wedge\ g_{i,k}^{t+1}\
\wedge\ 1[filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{k+1,j}^{\eta^{t}})=0$. In addition,
$filter(s_{1,0},\alpha(r^{t+1}))=\epsilon,filter(s_{1,0},\alpha(r^{\eta^{t}}))=\epsilon$,
i.e. $filter(s_{1,0},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{t+1}))=filter(s_{1,0},\alpha(r^{\eta^{t}}))=\epsilon$.
Therefore, $g_{1,0}^{t}=(1[filter(s_{1,0},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{t+1}))]\ \wedge\ g_{1,0}^{t+1}\
\wedge\ g_{1,0}^{\eta^{t}})\ \vee\ (1[filter(s_{1,0},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{1,0}^{\eta^{t}}\ \wedge\
g_{1,0}^{t+1})\vee\vee_{k=i}^{j-1}(1[filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,k},\alpha(r^{t+1}))]\ \wedge\ g_{i,k}^{t+1}\
\wedge\ 1[filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{k+1,j}^{\eta^{t}})=g_{1,0}^{t+1}\ \wedge\ g_{1,0}^{\eta^{t}}$. Because
$g_{1,0}^{t}=1$ iff $r^{t+1}\cdot r^{\eta^{t}}\models\epsilon$, and
$r^{t+1}\cdot r^{\eta^{t}}\models\epsilon$ iff both
$r^{t+1},r^{\eta^{t}}\models\epsilon$, $g_{1,0}^{t}=1$ iff both
$r^{t+1},r^{\eta^{t}}\models\epsilon$. Because $r^{t+1}$ (reps.
$r^{\eta^{t}}$) matches $\epsilon$ iff $g_{1,0}^{t+1}=1$ (reps.
$g_{1,0}^{\eta^{t}}=1$) (Structural inductive hypothesis), $g_{1,0}^{t}=1$ iff
$(g_{1,0}^{t+1}\wedge g_{1,0}^{\eta^{t}})=1$. Therefore,
$g_{1,0}^{t}=(1[filter(s_{1,0},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{t+1}))]\ \wedge\ g_{1,0}^{t+1}\
\wedge\ g_{1,0}^{\eta^{t}})\ \vee\ (1[filter(s_{1,0},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{1,0}^{\eta^{t}}\ \wedge\
g_{1,0}^{t+1})\vee\vee_{k=i}^{j-1}(1[filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,k},\alpha(r^{t+1}))]\ \wedge\ g_{i,k}^{t+1}\
\wedge\ 1[filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{k+1,j}^{\eta^{t}})$ is correct.
6. 6.
$r^{t}=r^{t+1}\&r^{\eta^{t}}$.
Because $g_{1,0}^{t}=1$ iff $r^{t+1}\&r^{\eta^{t}}\models\epsilon$, and
$r^{t+1}\&r^{\eta^{t}}\models\epsilon$ iff both
$r^{t+1},r^{\eta^{t}}\models\epsilon$, $g_{1,0}^{t}=1$ iff both
$r^{t+1},r^{\eta^{t}}\models\epsilon$. Because $r^{t+1}$ (reps.
$r^{\eta^{t}}$) matches $\epsilon$ iff $g_{1,0}^{t+1}=1$ (reps.
$g_{1,0}^{\eta^{t}}=1$) (Structural inductive hypothesis), $g_{1,0}^{t}=1$ iff
$(g_{1,0}^{t+1}\wedge g_{1,0}^{\eta^{t}})=1$. Therefore,
$g_{1,0}^{t}=g_{1,0}^{t+1}\ \wedge\ g_{1,0}^{\eta^{t}}$ is correct.
7. 7.
$r^{t}=r^{t+1}|r^{\eta^{t}}$.
Because
$filter(s_{1,0},\alpha(r^{t+1}))=\epsilon,filter(s_{1,0},\alpha(r^{\eta^{t}}))=\epsilon$,
i.e.
$filter(s_{1,0},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{t+1}))=filter(s_{1,0},\alpha(r^{\eta^{t}}))=\epsilon$,
$g_{1,0}^{t}=(1[filter(s_{1,0},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{t+1}))]\
\wedge\ g_{1,0}^{t+1})\ \vee\
(1[filter(s_{1,0},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{\eta^{t}}))]\
\wedge\ g_{1,0}^{\eta^{t}})=g_{1,0}^{t+1}\ \vee\ g_{1,0}^{\eta^{t}}$. Because
$g_{1,0}^{t}=1$ iff $r^{t+1}|r^{\eta^{t}}\models\epsilon$, and
$r^{t+1}|r^{\eta^{t}}\models\epsilon$ iff $r^{t+1}$ or
$r^{\eta^{t}}\models\epsilon$, $g_{1,0}^{t}=1$ iff $r^{t+1}$ or
$r^{\eta^{t}}\models\epsilon$. Because $r^{t+1}$ (reps. $r^{\eta^{t}}$)
matches $\epsilon$ iff $g_{1,0}^{t+1}=1$ (reps. $g_{1,0}^{\eta^{t}}=1$)
(Structural inductive hypothesis), $g_{1,0}^{t}=1$ iff $(g_{1,0}^{t+1}\vee
g_{1,0}^{\eta^{t}})=1$. Therefore,
$g_{1,0}^{t}=(1[filter(s_{1,0},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{t+1}))]\
\wedge\ g_{1,0}^{t+1})\ \vee\
(1[filter(s_{1,0},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{1,0},\alpha(r^{\eta^{t}}))]\
\wedge\ g_{1,0}^{\eta^{t}})$ is correct.
Mathematical induction step: $|s_{i,j}|=l>0$. Suppose that for all
$s_{i^{\prime},j^{\prime}}$ ($|s_{i^{\prime},j^{\prime}}|=l^{\prime}<l$) and
$1\leq t\leq|r|$, $g_{i^{\prime},j^{\prime}}^{t}=1$ iff $r^{t}\models
s_{i^{\prime},j^{\prime}}$.
Structural induction: Each case is corresponding to one statement in Line 8-20
in Algorithm 1.
1. 1.
$r^{t}=a\in\alpha(r)$.
Because $g_{i,j}^{t}=1$ iff $r^{t}=a\in\alpha(r)\models
filter(s_{i,j},\\{a\\})$, i.e. $filter(s_{i,j},\\{a\\})=a$, so $g_{i,j}^{t}=1$
iff $filter(s_{i,j},\\{a\\})=a$. Therefore,
$g_{i,j}^{t}=1[filter(s_{i,j},\\{a\\})=a]$ is correct.
2. 2.
$r^{t}=(r^{t+1})^{?}$.
$g_{i,j}^{t}=1$ iff $(r^{t+1})^{?}\models
filter(s_{i,j},\alpha((r^{t+1})^{?}))$, which is also equivalent to that
$r^{t+1}\models filter(s_{i,j},\alpha((r^{t+1})^{?}))$ or
$filter(s_{i,j},\alpha((r^{t+1})^{?}))=\epsilon$. Because
$\alpha((r^{t+1})^{?})=\alpha(r^{t+1})$,
$filter(s_{i,j},\alpha((r^{t+1})^{?}))=filter(s_{i,j},\alpha(r^{t+1}))$.
Therefore, $(r^{t+1})^{?}\models filter(s_{i,j},\alpha((r^{t+1})^{?}))$ iff
$r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$ or
$filter(s_{i,j},\alpha((r^{t+1})^{?}))=\epsilon$. $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$ iff $g_{i,j}^{t+1}=1$ (Structural inductive
hypothesis), so $g_{i,j}^{t}=1$ iff $g_{i,j}^{t+1}=1$ or
$1[filter(s_{i,j},\alpha((r^{t+1})^{?}))=\epsilon]$. Therefore,
$g_{i,j}^{t}=g_{i,j}^{t+1}\vee
1[filter(s_{i,j},\alpha((r^{t+1})^{?}))=\epsilon]$ is correct.
3. 3.
$r^{t}=(r^{t+1})^{*}$.
$g_{i,j}^{t}=1$ iff $(r^{t+1})^{*}\models
filter(s_{i,j},\alpha((r^{t+1})^{*}))$, which is also equivalent to that
$filter(s_{i,j},\alpha((r^{t+1})^{*}))=\epsilon$ or $r^{t+1}\models
filter(s_{i,j},\alpha((r^{t+1})^{*}))$, or $\exists i\leq
k<j,(r^{t+1})^{*}\models filter(s_{i,k},\alpha((r^{t+1})^{*}))$ and
$r^{t+1}\models filter(s_{k+1,j},\alpha((r^{t+1})^{*}))$. Because
$\alpha((r^{t+1})^{*})=\alpha(r^{t+1})$,
$filter(s_{i,j},\alpha((r^{t+1})^{*}))=filter(s_{i,j},\alpha(r^{t+1}))$ and
$filter(s_{k+1,j},\alpha((r^{t+1})^{*}))=filter(s_{k+1,j},\alpha(r^{t+1}))$.
Therefore, $(r^{t+1})^{*}\models filter(s_{i,j},\alpha((r^{t+1})^{*}))$ iff
$filter(s_{i,j},\alpha((r^{t+1})^{*}))=\epsilon$ or $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$, or $\exists i\leq k<j,(r^{t+1})^{*}\models
filter(s_{i,k},\alpha((r^{t+1})^{*}))$ and $r^{t+1}\models
filter(s_{k+1,j},\alpha(r^{t+1}))$. $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$ iff $g_{i,j}^{t+1}=1$ (Structural inductive
hypothesis). Because $|s_{i,k}|=k-i+1<l$, $(r^{t+1})^{*}\models
filter(s_{i,k},\alpha((r^{t+1})^{*}))$ iff $g_{i,k}^{t}=1$ (Mathematical
inductive hypothesis). Because $|s_{k+1,j}|=j-(k+1)+1<l$, $r^{t+1}\models
filter(s_{k+1,j},\alpha(r^{t+1}))$ iff $g_{k+1,j}^{t+1}=1$ (Mathematical
inductive hypothesis). Therefore, $\exists i\leq k<j,(r^{t+1})^{*}\models
filter(s_{i,k},\alpha((r^{t+1})^{*}))$ and $r^{t+1}\models
filter(s_{k+1,j},\alpha(r^{t+1}))$ iff $\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\
g_{k+1,j}^{t+1})=1$. Therefore, $g_{i,j}^{t}=1$ iff
$1[filter(s_{i,j},\alpha(r^{t}))=\epsilon]\vee
g_{i,j}^{t+1}=1\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\ g_{k+1,j}^{t+1})=1$.
Therefore, $g_{i,j}^{t}=1[filter(s_{i,j},\alpha(r^{t}))=\epsilon]\vee
g_{i,j}^{t+1}\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\ g_{k+1,j}^{t+1})$ is
correct.
4. 4.
$r^{t}=(r^{t+1})^{+}$.
$g_{i,j}^{t}=1$ iff $(r^{t+1})^{+}\models
filter(s_{i,j},\alpha((r^{t+1})^{+}))$, which is also equivalent to that
$r^{t+1}\models filter(s_{i,j},\alpha((r^{t+1})^{+}))$, or $\exists i\leq
k<j,(r^{t+1})^{+}\models filter(s_{i,k},\alpha((r^{t+1})^{+}))$ and
$r^{t+1}\models filter(s_{k+1,j},\alpha((r^{t+1})^{+}))$. Because
$\alpha((r^{t+1})^{+})=\alpha(r^{t+1})$,
$filter(s_{i,j},\alpha((r^{t+1})^{+}))=filter(s_{i,j},\alpha(r^{t+1}))$ and
$filter(s_{k+1,j},\alpha((r^{t+1})^{+}))=filter(s_{k+1,j},\alpha(r^{t+1}))$.
Therefore, $(r^{t+1})^{+}\models filter(s_{i,j},\alpha((r^{t+1})^{+}))$ iff
$r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$, or $\exists i\leq
k<j,(r^{t+1})^{+}\models filter(s_{i,k},\alpha((r^{t+1})^{+}))$ and
$r^{t+1}\models filter(s_{k+1,j},\alpha(r^{t+1}))$. $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$ iff $g_{i,j}^{t+1}=1$ (Structural inductive
hypothesis). Because $|s_{i,k}|=k-i+1<l$, $(r^{t+1})^{+}\models
filter(s_{i,k},\alpha((r^{t+1})^{+}))$ iff $g_{i,k}^{t}=1$ (Mathematical
inductive hypothesis). Because $|s_{k+1,j}|=j-(k+1)+1<l$, $r^{t+1}\models
filter(s_{k+1,j},\alpha(r^{t+1}))$ iff $g_{k+1,j}^{t+1}=1$ (Mathematical
inductive hypothesis). Therefore, $\exists i\leq k<j,(r^{t+1})^{+}\models
filter(s_{i,k},\alpha((r^{t+1})^{+}))$ and $r^{t+1}\models
filter(s_{k+1,j},\alpha(r^{t+1}))$ iff $\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\
g_{k+1,j}^{t+1})=1$. Therefore, $g_{i,j}^{t}=1$ iff
$(g_{i,j}^{t+1}\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\ g_{k+1,j}^{t+1}))=1$.
Therefore, $g_{i,j}^{t}=g_{i,j}^{t+1}\vee\vee_{k=i}^{j-1}(g_{i,k}^{t}\ \wedge\
g_{k+1,j}^{t+1})$ is correct.
5. 5.
$r^{t}=r^{t+1}\cdot r^{\eta^{t}}$.
$g_{i,j}^{t}=1$ iff $r^{t+1}\cdot r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$, which is also equivalent
to that $r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$ and
$r^{\eta^{t}}\models\epsilon$, or $r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$ and
$r^{t+1}\models\epsilon$, or $\exists i\leq k<j,r^{t+1}\models
filter(s_{i,k},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$ and $r^{\eta^{t}}\models
filter(s_{k+1,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$. $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$ iff
$filter(s_{i,j},\alpha(r^{t+1}))=filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$ (Lemma
2). $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$
iff $filter(s_{i,j},\alpha(r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{\eta^{t}}))$
(Lemma 2). Therefore, $r^{t+1}\cdot r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}\cdot r^{\eta^{t}}))$ iff
$filter(s_{i,j},\alpha(r^{t+1}))=filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$ and
$r^{\eta^{t}}\models\epsilon$, or
$filter(s_{i,j},\alpha(r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{\eta^{t}}))$
and $r^{t+1}\models\epsilon$, or $\exists i\leq k<j,\
filter(s_{i,k},\alpha(r^{t+1}))=filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{t+1}\models filter(s_{i,k},\alpha(r^{t+1}))$ and
$filter(s_{k+1,j},\alpha(r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{\eta^{t}}\models
filter(s_{k+1,j},\alpha(r^{\eta^{t}}))$. $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$ iff $g_{i,j}^{t+1}=1$ (Structural inductive
hypothesis). $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{\eta^{t}}))$ iff
$g_{i,j}^{\eta^{t}}=1$ (Structural inductive hypothesis). $r^{t+1}$ (reps.
$r^{\eta^{t}}$) matches $\epsilon$ iff $g_{1,0}^{t+1}=1$ (reps.
$g_{1,0}^{\eta^{t}}=1$) (Mathematical inductive hypothesis). Because
$|s_{i,k}|=k-i+1<l$, $r^{t+1}\models filter(s_{i,k},\alpha(r^{t+1}))$ iff
$g_{i,k}^{t+1}=1$ (Mathematical inductive hypothesis). Because
$|s_{k+1,j}|=j-(k+1)+1<l$, $r^{\eta^{t}}\models
filter(s_{k+1,j},\alpha(r^{\eta^{t}}))$ iff $g_{k+1,j}^{\eta^{t}}=1$
(Mathematical inductive hypothesis). Therefore, $\exists i\leq k<j,\
filter(s_{i,k},\alpha(r^{t+1}))=filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{t+1}\models filter(s_{i,k},\alpha(r^{t+1}))$ and
$filter(s_{k+1,j},\alpha(r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))$ and $r^{\eta^{t}}\models
filter(s_{k+1,j},\alpha(r^{\eta^{t}}))$ iff
$\vee_{k=i}^{j-1}(1[filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,k},\alpha(r^{t+1}))]\ \wedge\ g_{i,k}^{t+1}\
\wedge\ 1[filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{k+1,j}^{\eta^{t}})=1$. Therefore, $g_{i,j}^{t}=1$ iff
$(1[filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}))]\ \wedge\ g_{i,j}^{t+1}\
\wedge\ g_{1,0}^{\eta^{t}})\ \vee\ (1[filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{i,j}^{\eta^{t}}\ \wedge\
g_{1,0}^{t+1})\vee\vee_{k=i}^{j-1}(1[filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,k},\alpha(r^{t+1}))]\ \wedge\ g_{i,k}^{t+1}\
\wedge\ 1[filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{k+1,j}^{\eta^{t}})=1$. Therefore,
$g_{i,j}^{t}=(1[filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}))]\ \wedge\ g_{i,j}^{t+1}\
\wedge\ g_{1,0}^{\eta^{t}})\ \vee\ (1[filter(s_{i,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{i,j}^{\eta^{t}}\ \wedge\
g_{1,0}^{t+1})\vee\vee_{k=i}^{j-1}(1[filter(s_{i,k},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{i,k},\alpha(r^{t+1}))]\ \wedge\ g_{i,k}^{t+1}\
\wedge\ 1[filter(s_{k+1,j},\alpha(r^{t+1}\cdot
r^{\eta^{t}}))=filter(s_{k+1,j},\alpha(r^{\eta^{t}}))]\ \wedge\
g_{k+1,j}^{\eta^{t}})$ is correct.
6. 6.
$r^{t}=r^{t+1}\&r^{\eta^{t}}$.
$g_{i,j}^{t}=1$ iff $r^{t+1}\&r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}\&r^{\eta^{t}}))$. Because each symbol only
occurs once in an SOIRE and $r^{t},r^{t+1},r^{\eta^{t}}$ are SOIREs,
$\alpha(r^{t})=\alpha(r^{t+1})\cup\alpha(r^{\eta^{t}}),\alpha(r^{t+1})\cap\alpha(r^{\eta^{t}})=\emptyset$.
Therefore, each symbol in $filter(s_{i,j},\alpha(r^{t+1}\&r^{\eta^{t}}))$ is
either in $\alpha(r^{t+1})$ or $\alpha(r^{\eta^{t}})$.
$r^{t+1}\&r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{t+1}\&r^{\eta^{t}}))$
iff $filter(s_{i,j},\alpha(r^{t+1}\&r^{\eta^{t}}))$ can be devided into two
subsequences and $r^{t+1}$ matches one subsequence and $r^{\eta^{t}}$ matches
the other. Because each symbol in
$filter(s_{i,j},\alpha(r^{t+1}\&r^{\eta^{t}}))$ is either in $\alpha(r^{t+1})$
or $\alpha(r^{\eta^{t}})$, one subsequence consists of all the symbols in
$\alpha(r^{t+1})$, i.e. $filter(s_{i,j},\alpha(r^{t+1}))$, which is matched by
$r^{t+1}$, and the other subsequence consists of all the symbols in
$\alpha(r^{\eta^{t}})$, i.e. $filter(s_{i,j},\alpha(r^{\eta^{t}}))$, which is
matched by $r^{\eta^{t}}$. Therefore, $r^{t+1}\&r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}\&r^{\eta^{t}}))$ iff $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$ and $r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{\eta^{t}}))$. $r^{t+1}\models
filter(s_{i,j},\alpha(r^{t+1}))$ iff $g_{i,j}^{t+1}=1$ (Structural inductive
hypothesis). $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{\eta^{t}}))$ iff
$g_{i,j}^{\eta^{t}}=1$ (Structural inductive hypothesis). Therefore
$g_{i,j}^{t}=1$ iff $(g_{i,j}^{t+1}\ \wedge\ g_{i,j}^{\eta^{t}})=1$. Therefore
$g_{i,j}^{t}=g_{i,j}^{t+1}\ \wedge\ g_{i,j}^{\eta^{t}}$ is correct.
7. 7.
$r^{t}=r^{t+1}|r^{\eta^{t}}$.
$g_{i,j}^{t}=1$ iff $r^{t+1}|r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$, which is equivalent to that
$r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$ or
$r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$.
$r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$ iff
$filter(s_{i,j},\alpha(r^{t+1}))=filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$
and $r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$ (Lemma 2).
$r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$ iff
$filter(s_{i,j},\alpha(r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$
and $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{\eta^{t}}))$ (Lemma 2).
Therefore, $r^{t+1}|r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$ iff
$filter(s_{i,j},\alpha(r^{t+1}))=filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$
and $r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$, or
$filter(s_{i,j},\alpha(r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))$
and $r^{\eta^{t}}\models filter(s_{i,j},\alpha(r^{\eta^{t}}))$.
$r^{t+1}\models filter(s_{i,j},\alpha(r^{t+1}))$ iff $g_{i,j}^{t+1}=1$
(Structural inductive hypothesis). $r^{\eta^{t}}\models
filter(s_{i,j},\alpha(r^{\eta^{t}}))$ iff $g_{i,j}^{\eta^{t}}=1$ (Structural
inductive hypothesis). Therefore, $g_{i,j}^{t}=1$ iff
$(1[filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}))]\
\wedge\ g_{i,j}^{t+1})\ \vee\
(1[filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{\eta^{t}}))]\
\wedge\ g_{i,j}^{\eta^{t}})=1$. Therefore,
$g_{i,j}^{t}=(1[filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{t+1}))]\
\wedge\ g_{i,j}^{t+1})\ \vee\
(1[filter(s_{i,j},\alpha(r^{t+1}|r^{\eta^{t}}))=filter(s_{i,j},\alpha(r^{\eta^{t}}))]\
\wedge\ g_{i,j}^{\eta^{t}})$ is correct.
Theorem 3 has been proved.
### Proof of Proposition 4
Proof:
Algorithm 3 shows the properties of the prefix notation of an SOIRE. Consider
the number of elements of the stack, denoted as $|\Gamma|$. If
$o^{i}\in\Sigma$, $|\Gamma|$ adds $1$. If $o^{i}\in\\{?,*,+\\}$, $|\Gamma|\geq
1$ and after that $|\Gamma|$ stays unchanged. If $o^{i}\in\\{\cdot,\&,|\\}$,
$|\Gamma|\geq 2$ and after that $|\Gamma|$ minus $1$. Finally, $|\Gamma|=1$.
Therefore,
$\displaystyle\forall 1\leq
i\leq|r_{p}|,(\sum_{j=i}^{|r_{p}|}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq
1$
$\displaystyle\wedge(\sum_{i=1}^{|r_{p}|}1[o^{i}\in\Sigma]-1[o^{i}\in\\{\cdot,\&,|\\}])=1$
(15)
We shows that $\texttt{Enc2Pre}(\theta)$ satisfies Equation Proof of
Proposition 4, given the faithful encoding $\theta=(w,u)$. Let $last$ be the
biggest number that $w^{last}_{\text{none}}=0$ ($|r_{p}|=last$) and $T$ be the
bounded size. We first use mathematical induction to prove $\forall 1\leq
i\leq|r_{p}|,(\sum_{j=i}^{|r_{p}|}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq
1$.
Mathematical induction base: $i=last$.
If $i=last=T$, then $\sum_{a\in\Sigma}w^{i}_{a}=1$ (the last vertex can only
select a symbol), i.e. $o^{i}\in\Sigma$. Therefore,
$(\sum_{j=i}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq 1$ is
correct. If $i=last<T$, because $w^{i+1}_{\text{none}}=1$, $\forall
a\in\\{?,*,+,\cdot,\&,|\\},w^{i}_{a}=0$ (Definition 2 Condition 5). Therefore,
$\sum_{a\in\Sigma}w^{i}_{a}=1$ (Definition 2 Condition 1), i.e.
$o^{i}\in\Sigma$. Therefore,
$(\sum_{j=i}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq 1$ is
correct.
Mathematical induction step: $1\leq i<last$. Suppose for $i+1$,
$(\sum_{j=i+1}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq 1$.
$(\sum_{j=i+1}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq 1$
(Mathematical inductive hypothesis). If $o^{i}\in\Sigma\cup\\{?,*,+\\}$,
$(\sum_{j=i}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq 1$ is
correct. If $o^{i}\in\\{\cdot,\&,|\\}$, consider two cases:
1. 1.
$(\sum_{j=i+1}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])>1$
$(\sum_{j=i+1}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])+1[o^{i}\in\Sigma]-1[o^{i}\in\\{\cdot,\&,|\\}]>1-1$.
Therefore,
$(\sum_{j=i}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])>1$ is
correct.
2. 2.
$(\sum_{j=i+1}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])=1$
Let $B=\\{j|i\leq j\leq last\wedge o^{j}\in\\{\cdot,\&,|\\}\\}$. For all $j\in
B$, $\exists j+2\leq k\leq last,u^{j}_{k}=1$, denoted as $u^{j}=k$ (Definition
2 Condition 3). For any $j,k\in B$, if $j\neq k$, $u^{j}\neq u^{k}$
(Definition 2 Condition 2 and 5). For all $j\in B$, $o^{u^{j}-1}\in\Sigma$
(Definition 2 Condition 5 and 1). Because $o^{i}\in\\{\cdot,\&,|\\}$,
$(\sum_{j=i}^{last}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])=0$.
Therefore, $\\{u^{j}-1|j\in B\\}$ is the set of all $o\in\Sigma$. Therefore,
$o^{last}\notin\Sigma$, which is a contradiction. Therefore, this case does
not exist.
Therefore, $\texttt{Enc2Pre}(\theta)$ satisfies $\forall 1\leq
i\leq|r_{p}|,(\sum_{j=i}^{|r_{p}|}1[o^{j}\in\Sigma]-1[o^{j}\in\\{\cdot,\&,|\\}])\geq
1$.
Let $B=\\{i|1\leq i\leq last\wedge o^{i}\in\\{\cdot,\&,|\\}\\}$. For all $i\in
B$, $\exists i+2\leq j\leq last,u^{i}_{j}=1$, denoted as $u^{i}=j$ (Definition
2 Condition 3). For any $i,j\in B$, if $i\neq j$, $u^{i}\neq u^{j}$
(Definition 2 Condition 2 and 5). For all $i\in B$, $o^{u^{i}-1}\in\Sigma$
(Definition 2 Condition 5 and 1). In addition, $last\notin\\{u^{i}-1|i\in
B\\}$ and $o^{last}\in\Sigma$. Therefore, $\texttt{Enc2Pre}(\theta)$ satisfies
$(\sum_{i=1}^{|r_{p}|}1[o^{i}\in\Sigma]-1[o^{i}\in\\{\cdot,\&,|\\}])=1$.
Therefore, $\texttt{Enc2Pre}(\theta)$ satisfies Equation Proof of Proposition
4 and $\texttt{Enc2Pre}(\theta)$ returns the prefix notation of an SOIRE.
Proposition 4 has been proved.
### Proof of Proposition 5
Proof:
We construct the faithful encoding $\theta=(w,u)$, given $T\in\mathbb{Z}^{+}$,
and an SOIRE $r(|r|\leq T)$ written in prefix notation. Construct the syntax
tree of $r$, and $o^{t}(1\leq t\leq|r|)$ denotes the symbol or operator
represented by vectex $t$ and vertex $v^{t}$ is the right son of vertex $t$.
Let $w=\mathbf{0}$ and $u=\mathbf{0}$ initially. $\forall|r|<t\leq
T,w^{t}_{\text{none}}\leftarrow 1$. $\forall 1\leq
t\leq|r|,w^{t}_{o^{t}}\leftarrow 1$. $\forall 1\leq t\leq|r|\wedge
o^{t}\in\\{\cdot,\&,|\\},u^{t}_{v^{t}}\leftarrow 1$. Because $\theta=(w,u)$ is
constructed based on the syntax tree of $r$, $\theta=(w,u)$ are faithful
encoding and $\texttt{Enc2Pre}(\theta)=\texttt{PreForm}(r)$.
Proposition 5 has been proved.
### Proof of Proposition 6
Proof:
We apply proof by contradiction to prove this theorem. Given the bouned size
of SOIREs $T\in\mathbb{Z}^{+}$, for any two different faithful encodings
$\theta_{1},\theta_{2}(\theta_{1}\neq\theta_{2})$,
$\texttt{Enc2Pre}(\theta_{1})=\texttt{Enc2Pre}(\theta_{2})$. Let
$\theta_{1}=(w^{(1)},u^{(1)}),\theta_{2}=(w^{(2)},u^{(2)})$. Because
$\texttt{Enc2Pre}(\theta_{1})=\texttt{Enc2Pre}(\theta_{2})$, $w^{(1)}=w^{(2)}$
(Algorithm 4 and Definition 2 Condition 4). Let $p$ be the biggest number that
$u^{(1)p}\neq u^{(2)p}$. Because $w^{(1)}=w^{(2)}$,
$u^{(1)p}=u^{(2)p}=\mathbf{0}$ or $\exists p+2\leq t_{1},t_{2}\leq
T,u^{(1)p}_{t_{1}}=1\wedge u^{(2)p}_{t_{2}}=1$. In addition, $u^{(1)p}\neq
u^{(2)p}$. Therefore, $\exists p+2\leq t_{1},t_{2}\leq
T,u^{(1)p}_{t_{1}}=1\wedge u^{(2)p}_{t_{2}}=1$. Without loss of generality,
let $t_{1}<t_{2}$. Because $u^{(1)p}_{t_{1}}=1$, $\forall
a\in\\{?,*,+,\cdot,\&,|\\},w^{(1)t_{1}-1}_{a}=0$ (Definition 2 Condition 5).
Because $w^{(1)}=w^{(2)}$, $\forall
a\in\\{?,*,+,\cdot,\&,|\\},w^{(2)t_{1}-1}_{a}=0$ (Definition 2 Condition 5).
Therefore, $\exists 1\leq p_{2}\leq t_{1}-2,u^{(2)p_{2}}_{t_{1}}=1$
(Definition 2 Condition 5). Because $p$ be the biggest number that
$u^{(1)p}\neq u^{(2)p}$, $1\leq p_{2}<p$. For $\theta_{2}$, because vertex
$p_{2}$ has a right son, vertex $t_{1}$, then for all vertex
$p^{\prime}(p_{2}<p^{\prime}<t_{1})$, if vertex $p^{\prime}$ has a right son,
vertex $t^{\prime}$, then $t^{\prime}<t_{1}$ (Definition 2 Condition 6).
However, $p_{2}<p<t_{1}$ and vertex $p$ has a right son, vertex $t_{2}$, and
$t_{2}>t_{1}$, which is a contradiction. Therefore, there does not exist two
different faithful encoding $\theta_{1},\theta_{2}(\theta_{1}\neq\theta_{2})$,
where the size of $\theta_{1}$ and $\theta_{2}$ is $T$,
$\texttt{Enc2Pre}(\theta_{1})=\texttt{Enc2Pre}(\theta_{2})$.
Proposition 6 has been proved.
### Proof of Theorem 7
Proof:
Theorem 7 can be proved by combining Proposition 4, Proposition 6 and
Proposition 5. Therefore, Faithful encoding is one-to-one corresponding to the
SOIRE in prefix notation for a certain size of the parameters.
### Proof of Proposition 8
Proof:
Because
$(r^{?})^{?}=r^{?},(r^{?})^{*}=r^{*},(r^{?})^{+}=r^{*},(r^{*})^{*}=r^{*},(r^{*})^{?}=r^{*},(r^{*})^{+}=r^{*},(r^{+})^{+}=r^{+},(r^{+})^{?}=r^{*},(r^{+})^{*}=r^{*}$,
the consecutive unary operators can be replaced by one. For an SOIRE $r$ over
$\Sigma$, we use $r^{\prime}$ ($r^{\prime}\equiv r$) to denote the SOIRE
without consecutive unary operators. For the syntax tree of $r^{\prime}$,
there are no consecutive unary operators in $r^{\prime}$, so only the parent
of a vertex representing a symbol or a binary operator can represent a unary
operator. The Equation Proof of Proposition 4 shows that the number of the
binary operators is $|\alpha(r^{\prime})|-1$ for $r^{\prime}$. Therefore, the
size of the syntax tree of $r^{\prime}$ is
$(|\alpha(r^{\prime})|+|\alpha(r^{\prime})|-1)*2=4|\alpha(r^{\prime})|-2\leq
4|\Sigma|-2$. Proposition 5 shows that given $T=4|\Sigma|-2$, there exists a
faithful encoding $\theta$ that
$\texttt{Enc2Pre}(\theta)=\texttt{PreForm}(r^{\prime})$, because
$r^{\prime}\leq T$. Therefore, given $T=4|\Sigma|-2$, for each SOIRE $r$ over
$\Sigma$, there exists a faithful encoding $\theta$ that
$\texttt{Enc2Pre}(\theta)=\texttt{PreForm}(r^{\prime})$ and
$\\{s|r^{\prime}\models s\\}=\\{s|r\models s\\}$.
Proposition 8 has been proved.
## Appendix C
### Regularization
To make the set of parameters more faithful, we can add one regularization for
each condition when training the neural network. We use Equation 16-22 as
regularization for each condition.
$\text{Onehotloss}(x)=\text{Mean}_{i}(1-x_{i})x_{i}+(1-\sum_{i}x_{i})^{2}$ for
a vector $x$. $\text{ReLU}(x)=\max(x,0)$ for a scalar $x$.
$\displaystyle\text{Mean}_{t=1}^{T}(\text{Onehotloss}(w^{t}))$ (16)
$\displaystyle\text{Mean}_{t=1}^{T}(\text{Onehotloss}(u^{t}))$ (17)
$\displaystyle\text{Mean}_{t=1}^{T}(\text{Onehotloss}([u^{t}_{t+2},u^{t}_{t+3},\dots,u^{t}_{T},$
$\displaystyle
w^{t}_{a\in\Sigma},w^{t}_{?},w^{t}_{*},w^{t}_{+},w^{t}_{\text{none}}]))$ (18)
$\displaystyle\text{Mean}_{t=1}^{T-1}(\text{ReLU}(w^{t}_{\text{none}}-w^{t+1}_{\text{none}}))$
(19)
$\displaystyle\text{Mean}_{t=2}^{T}(\text{Onehotloss}([w^{t-1}_{?},w^{t-1}_{+},w^{t-1}_{*},w^{t-1}_{\cdot},$
$\displaystyle
w^{t-1}_{\&},w^{t-1}_{|},u^{1}_{t},u^{2}_{t},\dots,u^{t-2}_{t},w^{t}_{\text{none}}]))$
(20)
$\displaystyle\text{Mean}_{t=3}^{T}(\text{Mean}_{p=1}^{t-2}(\text{ReLU}((t-1-p)u^{p}_{t}+$
$\displaystyle\sum_{p^{\prime}=p+1}^{t-1}\sum_{t^{\prime}=t+1}^{T}u^{p^{\prime}}_{t^{\prime}}-(t-1-p))))$
(21)
$\displaystyle\text{Mean}_{a\in\Sigma}(\text{ReLU}(\sum_{t=1}^{T}w^{t}_{a}-1))$
(22)
### Results on noise-free data
The SOIREs learnt by SOIREDL on each dataset with noise-free data are show in
Table 5.
Dataset | Ground-truth SOIREs | SOIREDL
---|---|---
1 | $((a|b)c^{*})^{+}d$ | $((a|b)c^{*})^{+}d$
2 | $(a|b|c|d|e^{?})^{*}$ | $(b^{*}e^{*}a^{*}c^{*}d^{*})^{*}$
3 | $a^{+}|(b|c)^{*}|d^{+}$ | $(b^{+}|c)^{*}|a^{*}|d^{*}$
4 | $(ab^{*})^{+}|c^{+}$ | $(a^{+}b^{*})^{*}$
5 | $ab(c|d|e|f|g)^{*}$ | $((c|g^{*}|a\&b|d^{*})e^{*})^{*}$
6 | $a(b|c|d|e)^{+}f^{*}$ | $(b^{*}\&a\&d^{*}\&e^{*}\&c^{*})f^{*}$
7 | $a^{*}|(b|c|d|e)^{*}$ | $(c^{*}d^{*}e^{*})^{*}\&b^{*}$
8 | $(a|b|c|d|e)^{+}\&f^{*}\&g^{*}$ | $(f^{*}g^{*}d^{*}a^{*}e^{*}c^{*}b^{*})^{+}$
9 | $a^{+}|b|(c|d)^{+}$ | $(c^{+}|d^{+})^{+}$
10 | $(a|b)^{+}c$ | $((a^{*}\&b^{*})^{+}c)^{?}$
11 | $a^{?}(b|c|d)^{*}e$ | $(((b^{*}|d^{*})a^{*}c^{*})^{+}e)^{?}$
12 | $(a|b|c|d^{?})^{*}$ | $(a^{*}b^{*}d^{*}c^{*})^{*}$
13 | $a^{?}\&b^{*}\&c^{?}$ | $a^{?}\&c^{?}\&b^{*}$
14 | $(a(bc^{?})^{*})^{*}d^{?}e^{?}$ | $(a(b^{+}\&c)^{*}d^{*}e^{?})^{*}$
15 | $a^{?}\&b^{*}\&c^{*}\&d^{?}$ | $c^{*}\&a^{?}\&d\&b^{*}$
16 | $a^{*}\&b^{?}\&c^{*}$ | $a^{*}\&c^{*}\&b$
17 | $a^{+}\&(b|c|d|e)^{+}$ | $(c^{*}|e^{*}\&(a^{*}|b^{+})|d^{*})^{*}$
18 | $a^{?}(b|c)^{*}(d|e)$ | $a^{*}(b|c)^{*}d^{?}e^{?}$
19 | $a(b|c|d)^{*}e^{*}$ | $a((c|d)^{?}|b)^{*}e^{*}$
20 | $(a^{+}|b^{?}|c^{?}|(d|e)^{*})fgh^{*}$ | $(d^{*}\&f\&g^{*}\&e^{*})^{?}h^{*}$
21 | $(a^{*}b)^{+}$ | $(a^{*}b)^{*}$
22 | $(a|b)^{*}c^{*}d^{*}$ | $(a^{*}\&b^{*})c^{*}d^{*}$
23 | $((ab^{*})^{+}|c^{+})d$ | $(ab^{*}d^{?})^{+}$
24 | $(a|b|c)^{+}|(de^{*}f^{*})^{+}$ | $((c^{*}a^{*})^{*}b^{*})^{+}$
25 | $a|(b^{?}c)^{+}$ | $(b^{?}c^{+})^{+}$
26 | $a^{*}(b|c|d)^{*}$ | $((b^{*}|a^{*}d)c^{*})^{*}$
27 | $a^{?}(b^{+}\&c^{*}\&d^{*}\&e^{*}\&f^{*})$ | $((c^{*}\&d^{*}\&e^{*})f^{*}b^{*})^{*}$
28 | $(a^{?}b)^{+}$ | $(a^{?}b)^{*}$
29 | $(a^{+}|b^{+}|(c|d)^{*})e$ | $(d^{*}c^{*})^{*}e$
30 | $a^{?}(b^{?}c^{+}d)^{+}$ | $((a|b)^{*}cd^{?})^{*}$
Table 5: The ground-truth SOIREs and the SOIREs learnt by SOIREDL on each
dataset with noise-free data.
### Hyperparameters
All hyperparameters are tuned on the first noise-free dataset.
Hyperparameters of RE2RNN
Max-state and $\gamma$ are two important hyperparameters of RE2RNN. We use
grid search to choose max-state from $\\{$$20$, $40$, $60$, $80$, $100$,
$120$, $140$, $160$, $180$, $200$$\\}$ and choose $\gamma$ from $\\{$$0$,
$0.02$, $0.04$, $0.06$, $0.08$, $0.1$, $0.12$, $0.14$, $0.16$, $0.18$,
$0.2$$\\}$.
Max-state is the number of state in the automaton and Figure 5 shows that max-
state should be set as $100$ to get the best performance.
Figure 5: Accuracy(%) of RE2RNN on the first dataset with different number of
states max-state.
$\gamma$ is the threshold in the interpretation. The values above $\gamma$ are
set as $1$s and the values below $\gamma$ are set as $0$s. Figure 6 shows that
$\gamma$ should be set as $0.12$ to get the best performance.
Figure 6: Accuracy(%) of RE2RNN on the first dataset with different thresholds
in the interpretation $\gamma$.
Hyperparameters of SOIREDL
The beam width $\beta$ and the coefficient of regularization $\lambda$ are two
important hyperparameters of SOIREDL. We use grid search to choose $\beta$
from $\\{$$10$, $50$, $100$, $300$, $500$, $1000$$\\}$ and choose $\lambda$
from $\\{$$0$, $10^{-3}$, $10^{-2}$, $10^{-1}$, $1$, $10$$\\}$.
Figure 7 shows that SOIREDL gets the best result when $\beta\geq 500$, so
$\beta$ should be set as $500$ to make a balance between accuracy and running
time.
Figure 7: Accuracy(%) of SOIREDL on the first dataset with different beam
width $\beta$.
Figure 8 and 9 show that the accuracy and faithfulness decrease when
$\lambda>10^{-3}$. The larger coefficient of regularization makes the neural
network more difficult to train and the results show that SOIREDL can also
gets a well performance without regularization, so we set $\lambda$ as $0$.
Figure 8: Accuracy(%) of SOIREDL on the first dataset with different
coefficient of regularization $\lambda$. Figure 9: Faithfulness(%) of SOIREDL
on the first dataset with different coefficient of regularization $\lambda$.
|
# Emerging Diversity in a Population of Evolving Intransitive Dice
Julius B. Kirkegaard Kim Sneppen Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark
###### Abstract
Exploiting the mathematical curiosity of intransitive dice, we present a
simple theoretical model for co-evolution that captures scales ranging from
the genome of the individual to the system-wide emergence of species
diversity. We study a set of evolving agents that interact competitively in a
closed system, in which both the dynamics of mutations and competitive
advantage emerge directly from interpreting a genome as the sides of a die.
The model demonstrates sympatric speciation where new species evolve from
existing ones while in contact with the entire ecosystem. Allowing free
mutations both in the genomes and the mutation rates, we find, in contrast to
hierarchical models of fitness, the emergence of a metastable state of finite
mutation rate and diversity.
## I Introduction
Evolution is the optimization scheme of the biological realm: with the correct
initial conditions and ample time, random mutations and natural selection are
sufficient to ensure the emergence of highly complex organisms. But the nature
of what is being optimized is context-dependent. Even for a fixed environment,
evolution does not necessarily have a single goal. If we were to rerun the
“experiment” of evolution, it is virtually guaranteed this would result in
species distinct from those that are alive today.
Evolution of specific biological features of a species depend on properties of
the other species in its environment ehrlich1964butterflies ;
woolhouse2002biological . Such co-evolution is believed to be central to
describe evolution on the large scale KaufffmanBook ; Bak1992 ; Sneppen1993 ;
Weitz2005 ; Xue2017 as implied by the Red Queen hypothesis of Van Valen
van1973new ; liow2011red . Further, species interactions are not necessarily
ranked, as observed among corals, plants and microbes jackson1975alleopathy ;
taylor1990complex ; cameron2009parasite ; kirkup2004antibiotic ; Kerr2002 .
Many studies have been devoted to understanding and evolving such intransitive
interactions, ranging from molecular scale autocatalytic network
eigen1978hypercycle ; jain1998autocatalytic ; segre2000composing to
extensions of the competitive game of rock, paper and scissors Kerr2002 ;
Laird2006 ; Reichenbach2006 ; Reichenbach2007 ; Mathiesen2011 ; Mitarai2012 ;
Dobrinevski2012 ; Ulrich2014 ; szolnoki2014cyclic ; Levine2017 ; Soliveres2018
.
Non-hierarchical species dynamics can readily be studied for a set of species
whose interactions are fixed. Here, in contrast, we are interested in evolving
systems where intransitive interactions emerge ex nihilo. We suggest a minimal
model for such a system consisting of individuals that interact by rolling
dice.
Figure 1: The intransitive interaction of five 9-sided proper dice. Direction
of the arrows indicate domination. The graph formed from the interactions has
two Hamiltonian paths, one of which is indicated in the background. In other
words, out of the $4!=24$ orderings of the dice, there are two that form an
intransitive loop of dominance.
## II Model
We define the characteristics of an individual in our system by its genome as
given by a list of $n$ integers. We only consider competitive interactions
between species and settle these by interpreting the genome of integers as the
sides of dice that are rolled. The outcome of a fight is stochastic, but
certain dice will tend to out-compete other dice. For instance, a fight
between $A=(3,3,3,3,3,6)$ and $B=(2,2,2,5,5,5)$ will typically be won by $A$
despite $\sum_{i}A_{i}=\sum_{i}B_{i}$. The probability of $A$ winning in the
present example is $n^{-2}\sum_{i}\sum_{j}[A_{i}>B_{j}]=7/12$. What makes this
particular interaction interesting in the context of competing species is the
fact that we can introduce a species $C$ such that both $B\succ C$ and $C\succ
A$, or succinctly: $A\succ B\succ C\succ A$. This is for instance the case for
$C=(1,4,4,4,4,4)$. This intransitive behavior of dice is well-known
Finkelstein2006 ; Conrey2016a , but its applicability and simplicity for
modeling co-evolution are unexplored.
A plethora of systems could be designed around the above interaction rule. We
choose to consider one of the simplest and study $k$ individuals that interact
in a well-mixed scenario. At each time step of our simulation, we let each
individual randomly attack another. Two individuals, $X$ and $Y$, are
considered to belong to the same species if $\sum_{i}|X_{i}-Y_{i}|<\delta$, in
which case they will not fight. Otherwise, the losing individual of the
competition will be replaced by a copy of the winner. On ties, a random winner
is chosen.
Figure 2: Probability of intransitive (Hamiltonian) $k$-loops in sets of $k$
proper dice as a function of the number of sides $n$ on the dice. Here, proper
dice are sorted, has $\sum_{i}X_{i}=n(n+1)/2$ and $X_{i}\leq n$. A set of $k$
dice $\\{X_{i}\\}$ has a $k$-loop if an ordering $\sigma$ exists such that
$X_{\sigma_{1}}\prec X_{\sigma_{2}}\prec\cdots\prec X_{\sigma_{k}}\prec
X_{\sigma_{1}}$. Each point is the result of averaging over $100,\\!000$ Monte
Carlo samples with uniform probability for each valid die. Figure 3:
Speciation. (a–b) show $1,\\!500$ individuals of the system with links between
them if they consider one another the same species. (a) has $n=9$ and (b)
$n=14$, and in both cases $\mu=0.1$. Nodes are colored, for visualization
purposes only, using the Leiden community detection algorithm Traag2019 on
the undirected, unweighted graph. (c–e) show species richness $S$ (solid
lines) and diversity $D$ (dashed) stabilizing in a simulations of varying
genomic complexity $n$ and mutation rate $\mu$ as indicated by the legend.
Curves are averages over simulations with initial conditions of all dice
equaling the standard die. All simulation were run with $\delta=3$.
$k=10,\\!000$.
A crucial novelty of our model is that the dice interpretation not only sets
the rules for interactions but also naturally provides a genome space in which
mutations may occur. In our model, at each time step, each individual mutates
with probability $\mu$. A mutation event is the random change ($\pm 1$) of one
of its genome digits. As dice that are permutations of one another have
identical competitive advantages, we restrict our genome space to that of
ordered dice. Thus we disallow mutations that break the sorted nature of a
genome. For instance, $A$ may mutate to $(2,3,3,3,3,6)$ but not to
$(3,2,3,3,3,6)$. In effect this accelerates the dynamics of our system, as
most neutral competitions are avoided that would otherwise have to be settled
by stochastic extinction. Further, naturally, it is universally better for a
species to mutate up in sum rather than down. We set a fitness ceiling by only
allowing mutations that keep $\sum_{i}X_{i}\leq n\,(n+1)/2$ (the sum of the
standard die) and any $X_{i}\leq n$; the latter of which excludes a large of
set of dice that are typically less competitive and allows efficient
enumeration of the set of allowed dice.
In total, we have thus created a mutation model where universal fitness can be
measured, but where, occasionally, universally unfavorable mutations are
preferred to adapt to coexistent competitors.
Instead of defining the interaction as a single roll of the dice, one might
consider competitions of $r$ rounds. As $r\rightarrow\infty$, any slight
competitive advantage will result in certain overall wins. In this way, $r$
can be chosen to control the ruggedness of the fitness landscape KaufffmanBook
. Here, we limit our attention to $r=1$.
## III Probability of intransitivity
Before delving into the dynamics of the model, it is useful to have a feeling
for the prevalence of intransitivity in random dice. Consider all dice such
that $\sum X_{i}=n(n+1)/2$ and $1\leq X_{i}\leq n$. Fig. 2 shows the
probability that $k$ such dice contains at least one intransitive
(Hamiltonian) $k$-loop as function of the number of sides of the dice $n$. The
plot shows the fact that if you choose $k=3$ random dice (with $n<15$ sides),
the probability that these dice interact intransitively is less than $\sim
20\%$. With a larger set of dice, not only do the probability of $k$-loops
increase, but so does probability of smaller sub-cycles [see SI paperSI for a
version of Fig. 2 for loops of any size]. In Fig. 2 we only consider a small
number of dice, but very long intransitive cycles can also be found. For
instance, in the set of $k=910$ allowed $9$-sided dice, the longest possible
cycle is at least $891$ (finding the precise length is NP-hard).
Thus, intransitivity is by no means rare, but (short) loops are not the norm
either. However, in the dynamics of our model, it is much more unlikely for a
species to go extinct in an intransitive loop than when species interact in a
dominant manner. Thus we expect one of two things to happen in the long run:
the system will be taken over by one species or will be inhabited by a number
of species that interact intransitively and show oscillations.
There is also a large heterogeneity in the advantages of the different dice.
For instance, for $n=6$, the die that has an advantage over most other dice is
$(1,3,3,4,5,5)$, beating on average $\sim 60\,\%$ of the other dice. The worst
is $(1,2,2,4,6,6)$, which is better than only $\sim 40\,\%$. In contrast, the
standard die $(1,2,3,4,5,6)$ has precisely a $50\,\%$ chance of beating any
other proper die. Similar conclusions can be made for all $n$.
We note that the precise statistics of Fig. 2 would be different if we
considered not only proper dice, as some dice have many more permutations of
their sides than others. The qualitative conclusions drawn would remain
similar, nonetheless.
## IV Finite mutation rate upholds species diversity
Random, well-mixed ecosystems of many competing species interacting under
demographic noise are unstable may1972will , and competitive exclusion often
leads to a collapse to only a single or a few surviving species
hardin1960competitive ; levin1970community ; haerter2016food . This,
naturally, also applies to the present dynamical system. However, since the
genome space of our model is not hierarchically organized, a finite mutation
rate can induce a perpetual co-evolutionary arms race.
One complication in counting the number of species in a system is due to the
fact that the network of individuals is very unlikely to organize into fully
connected components. This is a complication that is not unique to our system,
but indeed any problem related to speciation tucker2017guide . For each pair
of individuals, our genome distance rule specifies if they belong to the same
species. Denote by $C_{i}$ the number of individuals that individual $i$ is
considered the same species as (including $i$ itself). An effective measure
for species richness is then given by $S=\sum_{i}C_{i}^{-1}$. In the case of
fully connected species, with no overlap between them, this measure coincides
with simply counting the number of species. Likewise, we can define a species
diversity measure that also accounts for evenness as
$D=k^{2}\left(\sum_{i}C_{i}\right)^{-1}$. This is equal to the number of
species only if each species occupy the same fraction of the entire system and
thus small species contribute only negligibly to its value.
Fig. 3 shows that both the mutation rate $\mu$ and the genome complexity $n$
(dice size) set the number of species that a system of a certain size can
maintain. At low mutation rates, the system is dominated by a small cloud of
individuals that form a quasispecies eigen1989molecular , since, in this case,
a single species can be locally dominant and no individuals can escape this
local optimum at the low mutation rate.
At higher mutation rates, however, intransitive interactions appear and
oscillatory dynamics of a high diversity system emerges. The systems have
intransitive loops of many lengths, but the dynamics are dominated by short
cycles ($\lesssim 5$ in the systems studied here). This is demonstrated and
studied in the SI by considering the mean-field Lotka–Volterra equations of
the system. In detail, the system behaves oscillatory with a frequency that
remain relatively constant for multiple oscillation periods. On long time
scales, however, stochastic events can change the dynamics, such as when a
species in an intransitive loop stochastically goes extinct or when an
individual suddenly mutates to dominate the existent intransitive interactions
initiating a “punctuated equilibrium” event causing a sudden shift in
oscillation frequency.
Fig. 3(a–b) visualizes species connectivity in the steady-state ecosystems
that evolve from the dynamics of the model. In these graphs, an edge is drawn
between two dice if they consider each other to belong to the same species.
Despite starting with a single species, the system can evolve to one that has
many species that are genomically disconnected. While no single measure can
capture the complexity of these inter-species connections, running community
detection algorithms on these graphs tend to find a number of clusters in the
same order of magnitude as our richness $S$ and diversity $D$ measures. We
note that for comparisons with other system sizes $k$, the values of the
mutation rates should be rescaled accordingly, as the number of attempted
mutation events per step is $\sim\mu k$. Thus systems with a higher number of
individuals can support high diversity in spite of having a low mutation rate
per individual.
## V Heterogeneous mutation rates
Since a finite mutation rate is needed to maintain a finite diversity, we have
an ecosystem collapse if the mutation rate is taken to zero. For instance, in
a system of purely hierarchically interacting species, the dominant species
will prefer a low mutation rate thus leading to a collapse of ecosystem
diversity. In the present system, however, there is no global optimum, and a
high mutations rate means quick adaptability and increases the chance of an
individual to out-compete others by an evolutionary advantage. A high mutation
rate is not strictly an advantage though, since it also means a high rate of
bad mutation events towards either locally or globally worse genomes.
In Fig. 4(inset), we show the competition between two populations with
$\mu_{A}=0.2$ and $\mu_{B}=0.001$, respectively. For early times, a low
mutation rate gives an advantage because there is a low rate of genomic decay
and thus we see population $B$ winning initially. This reflects the advantage
of localizing a population around a local fitness maximum over more diffuse
quasispecies at higher mutation rates eigen1989molecular . However, at some
point population $A$ finds a competitive advantage over the slowly adapting
population $B$ and annihilates the latter completely. Varying $\mu_{A}$ and
$\mu_{B}$, the average outcome of these scenarios is shown in the main part of
Fig. 4. The exact results depend on the initialization of the dice, but in
this case, we see that $\mu\approx 0.1$ is generally advantageous.
Figure 4: Annihilation statistics of two populations with different mutation
rates. Inset shows a simulation of $2\times 10,\\!000$ dice with one half
having $\mu_{A}=0.2$ and the other $\mu_{B}=0.001$. The time of annihilation
of population $B$ is $T=130$ time steps. Main plot shows the average of $1/T$
measured with a negative sign if $B$ wins. High values thus indicate that $B$
tends to be annihilated and negative values that $A$ tends to be annihilated.
Values near zero indicate a system dominated by stochastic extinction or one
where annihilation takes a very long time. All dice were initialized to
$(1,1,1,6,6,6)$ at the beginning of the simulations. Figure 5: Mutating the
mutation rate. Simulations for which each individual mutates its mutation rate
with a 1 % chance of a 1 % change per time step (Gaussian multiplicative
noise). (abc) Simulations for $n=6,\,9,\,14$ showing the system average
mutation rate $\langle\mu\rangle$ as a function of time steps. Orange lines
show average over all simulations, whereas pink line shows an average over
only those simulations that at $t=10^{7}$ have $\langle\mu\rangle>10^{-3}$.
The full distribution of $\mu$ is narrowly peaked around $\langle\mu\rangle$
(see SI paperSI ). Inset in (c) shows, for each realization, the average
mutation rate versus the system diversity $D$ (dark curves) for $t>5\cdot
10^{5}$, with two trajectories emphasized, one of which collapses towards
$\langle\mu\rangle=0$. All simulations use $\delta=3$, but obtain similar
results for other $\delta$. The fraction of systems in the high mutation state
depends greatly on initial mutation rate, even for the $n=6$ die (see SI
paperSI ).
We complete the design of our model by, finally, also permitting mutations in
the mutation rate itself. In a hierarchical setting, this would lead to the
immediate collapse of both the mutation rates and diversity. Fig. 4 indicates,
however, that in the present system there is also an advantage to having a
finite mutation rate. Each panel in Fig. 5 show realizations of a system of
individuals, all initialized with the same genome and an initial mutation rate
of $\mu=0.01$. We observe two distinct outcomes: most trajectories reach a
metastable state with a high mutations rate (die size $n=6$ never reaches this
state, but would do so if we instead started with initial $\mu=0.1$, see SI
paperSI ), and some that decay towards zero mutation rates.
A characteristic of the metastable state is the rare but sudden decay events
of both mutation rate and diversity of the entire system [Fig. 5(b–c)]. As the
mutation rate decreases, it becomes less and less likely to escape the low
diversity situation thereby creating a positive feedback loop for decreasing
the mutation rate even further. Once collapsed, very large perturbations are
needed to bring the system out of this situation. Even changing the mutation
rate of half of the collapsed system to $\mu=0.1$ is typically not enough to
return to the high diversity state [see SI paperSI ]. In contrast, for the
surviving high diversity ecosystems the average number of mutations that
separate two random individuals is large. For instance,
$\langle\sum_{i}|X_{i}-Y_{i}|\rangle_{X,Y}\approx 12$ for $n=14$, which is
about $2/3$ of the average obtained between random dice.
## VI Perspective
We have presented a theoretical model that, on one hand, is exceedingly simple
to define, and at the same time successfully describes a host of complex
phenomena related to co-evolution. At fixed, finite mutation rates, the model
permits a state of finite diversity in co-evolutionary balance. For
hierarchically interacting systems, allowing mutations in the mutation rates
themselves, will lead to an ecosystem collapse. In contrast, we find a
metastable state of finite diversity, whose stability increases quickly with
genomic complexity, measured by the number of sides of the dice $n$.
We only considered a well-mixed system, meaning that at all times each
individual could meet any other individual. Introducing space to the model,
e.g. putting the agents on a lattice, will most likely stabilize the observed
effects even further; with spatial dynamics, intransitive relations will decay
very slowly Mitarai2012 , and thus the rate of species extinctions decreases.
Furthermore, speciation events should increase in frequency as space allows
for transient allopatric speciation.
Precise quantification of intransitivity in the dynamical system is another
interesting avenue for further research: despite being dominated by few
intransitive cycles, a static view of the systems at any given time will not
reveal the dominance of these cycles as at least one species will have a low
population count due to the oscillatory Lotka–Volterra-like dynamics imposed
by the dynamics of the intransitive loops.
In conclusion, from the simple rules of competing dice emerge a natural
balance of mutation rates and diversity. Too high a mutation rate risks
genomic decay and the disintegration of quasispecies: “mutate and die”. Too
few mutations are disfavoured in analogy to the Red Queen hypothesis: “mutate
or die”.
###### Acknowledgements.
This project has received funding from the Novo Nordisk Foundation Grant
Agreement NNF20OC0062047 and from the European Research Council (ERC) under
the European Union’s Horizon 2020 Research and Innovation Programme, Grant
Agreement No. 740704.
## References
* (1) Paul R Ehrlich and Peter H Raven. Butterflies and plants: a study in coevolution. Evolution, pages 586–608, 1964.
* (2) Mark EJ Woolhouse, Joanne P Webster, Esteban Domingo, Brian Charlesworth, and Bruce R Levin. Biological and biomedical implications of the co-evolution of pathogens and their hosts. Nature genetics, 32(4):569–577, 2002.
* (3) Stuart A Kauffmann. The Origins of Order. Oxford University Press, 1993.
* (4) Per Bak, Henrik Flyvbjerg, and Benny Lautrup. Coevolution in a rugged fitness landscape. Physical Review A, 46(10):6724–6730, 1992.
* (5) Per Bak and Kim Sneppen. Punctuated Equilibribum and Criticality in a simple model of evolution. Physical Review Letters, 71(24):4083–4086, 1993.
* (6) J S Weitz, H Hartman, and S A Levin. Coevolutionary arms races between bacteria and bacteriophage. Proceedings of the National Academy of Sciences of the United States of America, 102(27):9535–9540, 2005.
* (7) Chi Xue and Nigel Goldenfeld. Coevolution Maintains Diversity in the Stochastic ”kill the Winner” Model. Physical Review Letters, 119(26):1–5, 2017.
* (8) Leigh Van Valen. A new evolutionary law. Evol theory, 1:1–30, 1973.
* (9) Lee Hsiang Liow, Leigh Van Valen, and Nils Chr Stenseth. Red queen: from populations to taxa and communities. Trends in ecology & evolution, 26(7):349–358, 2011.
* (10) JBC Jackson and LEO Buss. Alleopathy and spatial competition among coral reef invertebrates. Proceedings of the National Academy of Sciences, 72(12):5160–5163, 1975.
* (11) Douglas R Taylor and Lonnie W Aarssen. Complex competitive relationships among genotypes of three perennial grasses: implications for species coexistence. The American Naturalist, 136(3):305–327, 1990.
* (12) Duncan D Cameron, Andy White, and Janis Antonovics. Parasite–grass–forb interactions and rock–paper–scissor dynamics: predicting the effects of the parasitic plant rhinanthus minor on host plant communities. Journal of Ecology, 97(6):1311–1319, 2009.
* (13) Benjamin C Kirkup and Margaret A Riley. Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo. Nature, 428(6981):412–414, 2004.
* (14) Benjamin Kerr, Margaret A. Riley, Marcus W. Feldman, and Brendan J.M. Bohannan. Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature, 418(6894):171–174, 2002.
* (15) Manfred Eigen and Peter Schuster. The hypercycle. Naturwissenschaften, 65(1):7–41, 1978.
* (16) Sanjay Jain and Sandeep Krishna. Autocatalytic sets and the growth of complexity in an evolutionary model. Physical Review Letters, 81(25):5684, 1998.
* (17) Daniel Segré and Doron Lancet. Composing life. EMBO reports, 1(3):217–222, 2000.
* (18) Robert A. Laird and Brandon S. Schamp. Competitive intransitivity promotes species coexistence. American Naturalist, 168(2):182–193, 2006.
* (19) Tobias Reichenbach, Mauro Mobilia, and Erwin Frey. Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 74(5):1–11, 2006.
* (20) Tobias Reichenbach, Mauro Mobilia, and Erwin Frey. Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games. Nature, 448(7157):1046–1049, 2007.
* (21) Joachim Mathiesen, Namiko Mitarai, Kim Sneppen, and Ala Trusina. Ecosystems with mutually exclusive interactions self-organize to a state of high diversity. Physical Review Letters, 107(18):1–5, 2011.
* (22) Namiko Mitarai, Joachim Mathiesen, and Kim Sneppen. Emergence of diversity in a model ecosystem. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 86(1):1–6, 2012.
* (23) Alexander Dobrinevski and Erwin Frey. Extinction in neutrally stable stochastic Lotka-Volterra models. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 85(5):1–12, 2012.
* (24) Werner Ulrich, Santiago Soliveres, Wojciech Kryszewski, Fernando T. Maestre, and Nicholas J. Gotelli. Matrix models for quantifying competitive intransitivity from species abundance data. Oikos, 123(9):1057–1070, 2014.
* (25) Attila Szolnoki, Mauro Mobilia, Luo-Luo Jiang, Bartosz Szczesny, Alastair M Rucklidge, and Matjaž Perc. Cyclic dominance in evolutionary games: a review. Journal of the Royal Society Interface, 11(100):20140735, 2014.
* (26) Jonathan M. Levine, Jordi Bascompte, Peter B. Adler, and Stefano Allesina. Beyond pairwise mechanisms of species coexistence in complex communities. Nature, 546(7656):56–64, 2017.
* (27) Santiago Soliveres and Eric Allan. Everything you always wanted to know about intransitive competition but were afraid to ask. Journal of Ecology, 106(3):807–814, 2018.
* (28) Mark Finkelstein and Edward O Thorp. Nontransitive Dice With Equal Means *. pages 1–24, 2006.
* (29) Brian Conrey, James Gabbard, Katie Grant, Andrew Liu, and Kent E. Morrison. Intransitive Dice. Mathematics Magazine, 89(2):133–143, 2016.
* (30) V. A. Traag, L. Waltman, and N. J. van Eck. From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9(1):1–12, 2019.
* (31) See supplemental material at [url] for more details.
* (32) Robert M May. Will a large complex system be stable? Nature, 238(5364):413–414, 1972.
* (33) Garrett Hardin. The competitive exclusion principle: An idea that took a century to be born has implications in ecology, economics, and genetics. science, 131(3409):1292–1297, 1960.
* (34) Simon A Levin. Community equilibria and stability, and an extension of the competitive exclusion principle. The American Naturalist, 104(939):413–423, 1970.
* (35) Jan O Haerter, Namiko Mitarai, and Kim Sneppen. Food web assembly rules for generalized lotka-volterra equations. PLoS computational biology, 12(2):e1004727, 2016.
* (36) Caroline M Tucker, Marc W Cadotte, Silvia B Carvalho, T Jonathan Davies, Simon Ferrier, Susanne A Fritz, Rich Grenyer, Matthew R Helmus, Lanna S Jin, Arne O Mooers, et al. A guide to phylogenetic metrics for conservation, community ecology and macroecology. Biological Reviews, 92(2):698–715, 2017.
* (37) Manfred Eigen, John McCaskill, and Peter Schuster. The molecular quasi-species. Advances in chemical physics, 75:149–263, 1989.
|
# Domain Adaptive Scene Text Detection via Subcategorization
Zichen Tian1, Chuhui Xue2, Jingyi Zhang1, Shijian Lu1
1Nanyang Technological University, 2ByteDance Inc.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Most existing scene text detectors require large-scale training data which
cannot scale well due to two major factors: 1) scene text images often have
domain-specific distributions; 2) collecting large-scale annotated scene text
images is laborious. We study domain adaptive scene text detection, a largely
neglected yet very meaningful task that aims for optimal transfer of labelled
scene text images while handling unlabelled images in various new domains.
Specifically, we design SCAST, a subcategory-aware self-training technique
that mitigates the network overfitting and noisy pseudo labels in domain
adaptive scene text detection effectively. SCAST consists of two novel
designs. For labelled source data, it introduces pseudo subcategories for both
foreground texts and background stuff which helps train more generalizable
source models with multi-class detection objectives. For unlabelled target
data, it mitigates the network overfitting by co-regularizing the binary and
subcategory classifiers trained in the source domain. Extensive experiments
show that SCAST achieves superior detection performance consistently across
multiple public benchmarks, and it also generalizes well to other domain
adaptive detection tasks such as vehicle detection. Codes are available at
https://github.com/doem97/SCAST.
## 1 Introduction
Figure 1: The proposed subcategorization technique mitigates the overfitting
in domain adaptive scene text detection: Existing domain adaptive scene text
detector suffers from clear overfitting with over-confident predictions with
most prediction probabilities lying around $0$ or $1$ (blue curves). The
proposed subcategorization mitigates the overfitting effectively (orange
curves), leading to higher entropy and lower pseudo-labeling error. The
studies adopt scene text detector EAST [71] over domain adaptive scene text
detection task SynthText $\to$ IC15.
Scene text detection has been studied intensely for years, largely due to its
wide applications in many real-world tasks such as scene understanding [1,
46], image retrieval [55, 17], autonomous indoor and outdoor navigation [39],
etc. With the recent advances of deep neural networks, it has achieved
significant progress under the presence of large-scale annotated training
images [29, 50, 71, 31, 40, 68, 56, 9, 30, 67]. However, collecting large-
scale annotated scene text images is laborious which has become a bottleneck
while facing various scene text detection tasks that often have different
camera setups, environmental parameters, etc.
Unsupervised domain adaptation (UDA), which aims to learn from a labeled
source domain for a well-performing model in an unlabelled target domain, has
achieved great success in different computer vision tasks such as semantic
segmentation [18, 64, 54], person re-identification [14, 34, 66], object
detection [19, 6, 58, 70, 69], etc. However, existing UDA methods often suffer
from a clear overfitting problem while applied to the scene text detection
task. Take the prevalent self-training [57, 18, 72, 73] as an example. It
tackles UDA by pseudo-labeling target samples for network retraining and has
demonstrated superb performance while handling multi-class data as in general
semantic segmentation and object detection tasks. Scene text detection instead
involves a bi-class pseudo-label prediction task (i.e., foreground text and
background stuff) where the prediction scores often have an extreme bimodal
pattern with two sharp peaks at probabilities 0 and 1 as illustrated in Fig.
1. As a result, the predictions tend to be over-confidently wrong which
introduces clear overfitting together with very high model likelihood and
severe noises in pseudo-labelling.
We design SCAST, a subcategory-aware self-training technique that introduces
subcategorization for robust UDA for the scene text detection task. For the
labelled source-data, SCAST identifies multiple pseudo subcategories for both
foreground texts and background stuff via clustering which converts a bi-class
prediction task to a multi-class prediction task, leading to more
generalizable source models via regularization with a multi-class learning
objective. For unlabelled target data, SCAST predicts pseudo labels with both
bi-class and subcategory-aware classifiers which mitigates overfitting and
allows transferring more diverse and informative source knowledge to the
target domain. In addition, it exploits the prediction consistency between the
bi-class and subcategory-aware classifiers over target samples which leads to
better domain adaptation and more accurate pseudo-labelling of target images.
The contributions of this work are threefold. First, we identify the
overfitting issue in domain adaptive scene text detection and propose a
subcategorization method that mitigates the overfitting effectively. To the
best of our knowledge, this is the first work that explores subcategorization
in domain adaptive scene text detection tasks. Second, we design SCAST that
can learn more generalizable source models by identifying subcategories for
model regularization. SCAST can transfer more diverse source knowledge and
improve pseudo-labeling of target samples with prediction consistency between
the bi-class and subcategory-aware classifiers learnt from source data. Third,
extensive experiments show that SCAST achieve superior detection performance
and the subcategorization idea can generalize to other bi-class detection
tasks with little adaptation.
Figure 2: The framework of the proposed SCAST: SCAST mitigates the overfitting
of text detectors via subcategorization in the source and the target domains:
1) In the labeled source domain, we conduct feature clustering to discover
fine-grained subcategories learning with which relieves the overfitting in bi-
class classification greatly; 2) In the unlabeled target domain, we co-
regularize pseudo-labelling with the bi-class and multi-class classifiers
learnt from source samples, which reduces pseudo labelling noises in self-
training effectively.
## 2 Related Work
Scene text detection aims to locate texts in scene images and it has been
widely explored via box regression and image segmentation. The regression
approach exploits different object detection frameworks and text-specific
shapes and orientations [71, 28, 27, 41, 68, 49, 60, 62, 61, 63]. For example,
RRD [31] extracts rotation-sensitive features with SSD [35] for rotation-
invariant detection. Textboxes++ [28] modifies convolutional kernels and
anchor boxes to capture various text shapes effectively. EAST [71] directly
infers quadrangles without proposal mechanism. However, the regression
approach often struggle while handling scene texts with irregular shapes. The
segmentation approach addresses this issue effectively by predicting a
semantic label for each image pixel [10, 56, 9, 51, 30]. For example,
PixelLink [10] locates text regions by associating neighborhood image pixels.
PSENet [56] generates text bounding boxes from multi-scale segmentation maps.
DB [30] introduces differentiable binarization for adaptive thresholding while
predicting bounding regions from segmentation maps.
Many existing methods can achieve very impressive detection performance under
the presence of large-scale labelled training images that are often laborious
to collect. Domain adaptive scene text detection, which aims to exploit
previous collection of annotated scene text images for handling various new
data, is largely neglected. Several recent studies [4, 57, 7, 66] attempt to
tackle this challenge but they focus on adapting existing UDA methods without
addressing text-specific problems.
Unsupervised domain adaptation (UDA) has been studied extensively via three
typical approaches, namely, adversarial learning, image translation, and self-
training. Adversarial learning aligns source and target domains by minimizing
certain distribution discrepancy in feature spaces [3, 37, 38, 15, 16]. Image
translation works by translating source images to have similar target styles
[48, 22, 2]. The two approaches cannot handle scene text detection well as
they focus on adapting low-frequency appearance information. Self-training
works by pseudo-labelling target data, which has been widely explored for
various multi-class recognition tasks such as sign recognition [44], semantic
segmentation [72, 73] and panoptic segmentation [24, 33].
We adopt self-training that pseudo-labels target images with a source-trained
model. However, we focus on a more challenging bi-class self-training task
that suffers from severe overfitting. We address the challenge by introducing
subcategory pseudo-labeling that mitigates the overfitting by learning from
multiple subcategories beyond the original background and foreground texts.
## 3 Method
The proposed subcategory-aware self-training explores subcategorization to
mitigate the overfitting in domain adaptive scene text detection task as
illustrated in Fig. 2. In the source domain, it aims for Subcategory Discovery
and Learning that first performs feature clustering to determine multiple
subcategories beyond the original two classes and then employs the subcategory
labels as pseudo labels to learn multi-class classification model. In the
target domain, it aims for Subcategory Regularized Self-Training that employs
the learnt multi-class classifier to label target samples and retrain the
network iteratively, more details to be described in the ensuing subsections.
### 3.1 Problem Definition
Given scene text images in source and target domains
$\left\\{X^{s},X^{t}\right\\}\subset\mathbb{R}^{H\times W\times 3}$ where only
the source data $X^{s}$ are labelled, we aim to learn a scene text detection
model that performs well over the target samples $X^{t}$. The baseline model
$\mathbf{G}$ is trained on labeled source data with loss:
$\mathcal{L}_{\mathrm{bi}}\left(X^{s},Y^{s};\mathbf{G}=\mathbf{C}(\mathbf{E})\right),$
(1)
where $Y^{s}$ is bi-class annotations of the source data and
$\mathcal{L}_{\mathrm{bi}}$ denotes a binary classification loss (e.g. dice
loss [56] or binary cross-entropy loss [30, 71]). The baseline model
$\mathbf{G}$ consists of a bi-class classifier $\mathbf{C}$ and a feature
extractor $\mathbf{E}$.
### 3.2 Subcategory Discovery and Learning
In the source domain, we perform feature clustering to discover multiple
subcategories on top of the original two-class labels to address the
overfitting problem. Besides a bi-class classifier trained with the original
scene text bounding boxes, a multi-class classifier is trained by using the
discovered subcategories as pseudo labels as illustrated in Fig. 2. The multi-
class classifier mitigates the overconfident predictions effectively which can
be observed in the Histograms of Prediction Scores that are produced by the
bi-class and multi-class classifiers.
Subcategory Discovery in Source Domain. SCAST discovers subcategories by
clustering features instead of spatial relationships among image patches or
pixels as in [20, 12, 65, 52, 23, 5]. Feature clustering can produce fine-
grained sub-clusters of scene texts that are distinguishable by their high-
level features such as textures, strokes, etc.
Given $X^{s}$, we first obtain their feature maps $f^{s}$ with a pre-trained
feature extractor $\mathbf{E}$ of the baseline model $\mathbf{G}$. DBSCAN [12]
clustering is then applied to $f^{s}$ to discover sub-clusters for texts and
image background (obtained with the source annotations) separately. The
indexes of the identified sub-clusters are then used as pseudo labels
$\hat{Y}^{s}_{\mathrm{sub}}$ to train a multi-class classifier. The
subcategory pseudo-labeling process can be formulated by:
$\hat{Y}_{\mathrm{sub}}^{s}=\mathbf{\Gamma}\left(f^{s}\right),f^{s}=\mathbf{E}\left(X^{s}\right),$
(2)
where $\mathbf{\Gamma}$ denotes the subcategory pseudo labelling operation.
For efficiency, we perform clustering on down-sampled feature maps with a
down-sample ratio $d$, where each feature point corresponds to a text-height
area.
Figure 3: The framework of the proposed SCAST: SCAST consists of a multi-
class subcategory classifier $\mathbf{C}_{\mathrm{sub}}$, a source-domain
subcategorizer $\mathbf{\Gamma}$, and a target-domain co-regularizer
$\mathbf{\Phi}_{\mathrm{reg}}$. In the source domain, $\mathbf{\Gamma}(\cdot)$
clusters features of source samples to produce multi-class subcategory pseudo
labels $\hat{Y}_{\mathrm{sub}}^{s}$ and applies them to re-train a network
model that suffers from much less overfitting than that trained with the
original bi-class labels $Y^{s}$. In the target domain,
$\mathbf{\Phi}_{\mathrm{reg}}(\cdot)$ co-regularizes between the bi-class and
multi-class subcategory pseudo labels $\hat{Y}_{\mathrm{bi}}^{t}$ and
$\hat{Y}_{\mathrm{sub}}^{t}$ to filter out noisy pseudo labels which further
mitigates overfitting and improves domain adaptation effectively.
Subcategory Learning in Source Domain. To learn from the pseudo-labeled
subcategories in the source domain, we include a multi-class classifier
$\mathbf{C}_{\mathrm{sub}}$ on top of the feature extractor $\mathbf{E}$. The
multi-class subcategory classifier is learnt with a cross-entropy loss:
$\mathcal{L}_{\mathrm{sub}}\left(P^{s}_{\mathrm{sub}},\hat{Y}^{s}_{\mathrm{sub}}\right)=\\\
-\sum_{h,w}\sum_{k\in
K}\left(\hat{Y}_{\mathrm{sub}}^{s}\right)^{(h,w,k)}\log\left(\left(P^{s}_{\mathrm{sub}}\right)^{(h,w,k)}\right),$
(3)
where $P^{s}_{\mathrm{sub}}$ is predictions of multi-class classifier and
$\hat{Y}_{\mathrm{sub}}^{s}$ refers to multi-class pseudo labels (Eq. 2). We
denote the subcategory class number as $K$, which is decided automatically by
DBSCAN algorithm. The learned model $\mathbf{G}^{s}$ is then applied to target
domain through a subcategory regularized self-training, introduced next.
### 3.3 Subcategory Regularized Self-Training
In the target domain, we employ bi-class and subcategory classifiers to
predict pseudo labels and co-regularize them for effective knowledge transfer.
As illustrated in Fig. 2, the subcategory predictions with less overfitting
co-regularize the bi-class predictions, which help filter out noisy pseudo
labels as ‘Untrain’ effectively.
Subcategory learning in target domain. We select the most confident
predictions by $\mathbf{G}^{s}$ as the pseudo labels for target images.
Algorithm 1 Determination of threshold $\theta$
0: Predictions $P_{\mathrm{sub}}^{t}$; Selection portion $\rho\%$
1: for $k=1$ to $K$ do
2: $M^{k}$ = sort($(P^{t}_{\mathrm{sub}})^{k}$, order = descending)
3: $l^{k}$ = length$(M^{k})\times\rho\%$
4: $\theta^{k}$ = $M^{k}[l^{k}]$
5: end for
6: return $\theta=\\{\theta^{k}|k\in K\\}$
As summarized in Algorithm. 1, for each subcategory, we employ a selection
proportion $\rho$ [72, 73] to select the top $\rho\%$ most confident
predictions as pseudo labels. With the threshold value $\theta$ at the top
$\rho\%$-th position, the pseudo-labeling process can be formulated as:
$\hat{Y}^{t}_{\mathrm{sub}}=\mathbf{\Theta}\left(\theta,P^{t}_{\mathrm{sub}}\right)=\begin{cases}1,&\text{if}\
(P^{t}_{\mathrm{sub}})^{k}>\theta^{k},\\\ 0,&\text{otherwise},\\\ \end{cases}$
(4)
where $k\in K$ is the subcategory index, and $\hat{Y}^{t}_{\mathrm{sub}}$ is
the subcategory pseudo labels. The bi-class pseudo labelling follows the same
formula:
$\hat{Y}^{t}_{\mathrm{bi}}=\mathbf{\Theta}\left(\theta,P^{t}_{\mathrm{bi}}\right)$
where $k=0,1$ refers to texts or image background, respectively.
The network can then be re-trained with the target data with the obtained
pseudo labels
$\left\\{\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}}\right\\}$. The
training losses are the same as the supervised losses
$\mathcal{L}_{\mathrm{bi}}$ and $\mathcal{L}_{\mathrm{sub}}$ (in Eq. 1 and Eq.
3) in the source domain. The learning objective in the target domain is thus a
weighted combination of bi-class loss and subcategory loss:
$\displaystyle\begin{split}\mathcal{L}^{t}=&\lambda_{\mathrm{sub}}\mathcal{L}_{\mathrm{sub}}\left(P_{\mathrm{sub}}^{t},\hat{Y}_{\mathrm{sub}}^{t};\mathbf{C}_{\mathrm{sub}},\mathbf{E}\right)\
+\\\
&\lambda_{\mathrm{bi}}\mathcal{L}_{\mathrm{bi}}\left(P_{\mathrm{bi}}^{t},\hat{Y}_{\mathrm{bi}}^{t};\mathbf{C}_{\mathrm{bi}},\mathbf{E}\right).\end{split}$
(5)
where $\lambda_{\mathrm{bi}}$ is a weight parameter and we set it empirically
at 1 in our implementation. Eq. 4 and Eq. 5 define a single iteration of self-
training that generates pseudo labels and retrains the network in single-
round. In practice, we implement multi-round optimization as described in Sec
3.4.
Subcategory regularization in target domain. As shown in Fig. 2, we discard
inconsistent predictions between the bi-class and subcategory classifiers,
which effectively reduces false positives and pseudo-label noises and leads to
more robust pseudo-labeling. Specifically, we measure the distance between bi-
class predictions and subcategory predictions by cross-entropy and drop out
predictions with large distances (Eq. 7). To control the proportion of dropped
predictions, we set a ratio $\rho_{\mathrm{reg}}$ to drop out positions with
top $\rho_{\mathrm{reg}}\%$ distance (we set $\rho_{\mathrm{reg}}\%=10\%$
empirically). We denote such co-regularization process as
$\mathbf{\Phi}_{\mathrm{reg}}$ and formulate it as a loss minimization
problem:
$\displaystyle\mathbf{\Phi}_{\mathrm{reg}}=\operatorname*{arg\,min}_{\left\\{P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}}\right\\}}\mathcal{L}^{t}_{\mathrm{reg}}\left(P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}};\rho_{\mathrm{reg}}\right),$
(6)
where
$\mathcal{L}^{t}_{\mathrm{reg}}\left(P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}}\right)=-\left(P^{t}_{\mathrm{sub}}\right)\log\left(P^{t}_{\mathrm{bi}}\right)$
is the cross-entropy distance between bi- and multi-class predictions. We can
get regularized pseudo labels by solving Eq. 6:
$\left\\{\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}}\right\\}=\begin{cases}1,&\text{if}\
\mathcal{L}^{t}_{\mathrm{reg}}\left(P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}}\right)<\theta_{\mathrm{reg}},\\\
0,&\text{otherwise},\\\ \end{cases}$ (7)
where $\theta_{\mathrm{reg}}$ is the threshold value at top
$\rho_{\mathrm{reg}}\%$-th position.
### 3.4 Network Optimization
Method | Arch. | ICDAR13 [26] | ICDAR15 [25] | COCO-Text [53] | Total-Text (Reg) [45] | Mean
---|---|---|---|---|---|---
Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score
Supervised | - | 92.64 | 82.67 | 87.37 | 84.36 | 81.27 | 82.79 | 50.39 | 32.40 | 39.45 | 50.00 | 36.20 | 42.00 | 69.35 | 58.14 | 62.90
Baseline (EAST [71]) | - | 65.33 | 68.85 | 67.05 | 69.63 | 53.44 | 60.47 | 38.23 | 20.73 | 26.89 | 40.35 | 36.61 | 38.39 | 53.38 | 44.90 | 48.20
TST [57] | ST | 71.50 | 70.70 | 71.10 | 69.30 | 60.50 | 64.60 | 53.00 | 22.76 | 31.85 | 43.48 | 37.56 | 40.30 | 59.32 | 47.88 | 51.96
EntMin [18] | ST | 68.29 | 67.81 | 68.05 | 70.41 | 53.16 | 60.58 | 57.12 | 18.52 | 27.97 | 43.99 | 34.50 | 38.67 | 59.95 | 43.50 | 48.82
CBST [72] | ST | 70.28 | 71.08 | 70.68 | 71.36 | 58.40 | 64.23 | 55.17 | 20.96 | 30.38 | 43.17 | 38.23 | 40.55 | 60.00 | 47.17 | 51.46
CRST [73] | ST | 72.21 | 73.05 | 72.63 | 71.85 | 62.19 | 66.67 | 51.44 | 22.28 | 31.09 | 46.31 | 39.13 | 42.42 | 60.45 | 49.16 | 53.20
FDA [64] | Tran. | 47.28 | 73.88 | 57.66 | 59.13 | 55.18 | 57.09 | 44.69 | 19.79 | 27.43 | 37.24 | 40.15 | 38.64 | 43.86 | 47.02 | 43.35
ADVENT [54] | Adv. | 62.15 | 66.30 | 64.16 | 63.00 | 61.00 | 61.99 | 38.60 | 21.06 | 27.25 | 38.44 | 37.51 | 37.97 | 50.55 | 46.47 | 47.84
Ours (SCAST) | ST | 79.88 | 76.65 | 78.23 | 75.80 | 65.27 | 70.14 | 49.28 | 26.07 | 34.10 | 52.86 | 42.97 | 47.40 | 64.46 | 52.74 | 57.47
Table 1: Regular scene text detection: The experiments are conducted over
domain adaptive scene text detection tasks SynthText $\to$ {ICDAR13, ICDAR15,
COCO-Text and Total-Text (Reg)} using EAST [71]. ST stands for Self-training,
Adv. for Adversarial training, and Tran. for Image-to image translation.
Supervised refers to supervised learning from the labeled target data.
Method | Arch. | Total-Text (Curve) [45] | CTW1500 [36] | TextSeg [59] | Mean
---|---|---|---|---|---
Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score
Supervised | - | 81.77 | 75.11 | 78.30 | 80.57 | 75.55 | 78.00 | 89.02 | 81.55 | 85.12 | 83.79 | 77.40 | 80.47
Baseline (PSENet [56]) | - | 72.41 | 46.12 | 56.35 | 44.03 | 44.26 | 44.15 | 77.24 | 49.33 | 60.20 | 64.56 | 46.57 | 53.57
TST [57] | ST | 68.94 | 51.03 | 58.65 | 42.07 | 50.98 | 46.09 | 82.91 | 51.03 | 63.17 | 64.64 | 51.01 | 55.97
EntMin [18] | ST | 72.20 | 46.33 | 56.44 | 43.38 | 48.50 | 45.80 | 82.23 | 48.34 | 60.88 | 65.94 | 47.72 | 54.37
CBST [72] | ST | 71.39 | 47.57 | 57.09 | 44.58 | 49.19 | 46.77 | 79.17 | 52.15 | 62.88 | 65.05 | 49.64 | 55.58
CRST [73] | ST | 73.64 | 50.31 | 59.78 | 46.35 | 47.75 | 47.04 | 82.30 | 51.41 | 63.29 | 67.43 | 49.82 | 56.70
FDA [64] | Tran. | 71.60 | 43.50 | 54.12 | 45.02 | 44.38 | 44.70 | 78.21 | 50.02 | 61.02 | 64.94 | 45.97 | 53.28
ADVENT [54] | Adv. | 76.27 | 34.34 | 47.36 | 41.37 | 46.97 | 43.99 | 80.83 | 50.04 | 61.81 | 66.16 | 43.78 | 51.05
Ours (SCAST) | ST | 71.82 | 56.73 | 63.39 | 51.02 | 56.10 | 53.44 | 81.90 | 58.78 | 68.44 | 68.25 | 57.20 | 61.76
Table 2: Irregular scene text detection: The experiments are conducted over
domain adaptive scene text detection tasks SynthText $\to$ {Total-Text
(Curve), CTW1500, TextSeg} using PSENet [56]. ST stands for Self-training,
Adv. for Adversarial training, and Tran. for Image translation architecture.
Supervised refers to supervised learning from the labeled target data. Figure
4: Samples of clustered subcategories: Our method captures high-level features
($e.g.$ curvature, texture, stroke shapes, hue, etc.) and clusters text
instances and background into different subcategories with different features.
The experiments were conducted on ICDAR13 with EAST detector. $k$ is the
subcategory index and we show five subcategories ($k=1$, 12 and 19 for text,
37 and 40 for background) for illustration.
Method | Arch. | ICDAR13 [26] | ICDAR15 [53] | Total-Text (Reg) [45] | Mean
---|---|---|---|---|---
Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score | Precision | Recall | F-Score
Supervised | - | 92.64 | 82.67 | 87.37 | 84.36 | 81.27 | 82.79 | 50.00 | 36.20 | 42.00 | 75.67 | 66.71 | 70.72
Baseline (EAST [71]) | - | 62.31 | 70.31 | 66.07 | 71.29 | 55.20 | 62.22 | 41.03 | 35.28 | 37.94 | 58.21 | 53.60 | 55.41
TST [57] | ST | 66.27 | 72.05 | 69.04 | 73.46 | 57.21 | 64.32 | 42.23 | 38.10 | 40.06 | 60.65 | 55.79 | 57.81
EntMin [18] | ST | 64.13 | 72.98 | 68.27 | 70.68 | 56.08 | 62.54 | 44.06 | 36.19 | 39.74 | 59.62 | 55.08 | 56.85
CBST [72] | ST | 68.68 | 71.36 | 69.99 | 69.05 | 59.30 | 63.80 | 42.28 | 37.92 | 39.98 | 60.00 | 56.19 | 57.93
CRST [73] | ST | 71.33 | 74.42 | 72.84 | 73.89 | 63.09 | 68.06 | 43.60 | 39.33 | 41.36 | 62.94 | 58.95 | 60.75
FDA [64] | Tran. | 43.62 | 61.50 | 51.04 | 62.07 | 52.75 | 57.03 | 37.92 | 33.40 | 35.52 | 47.87 | 49.22 | 47.86
ADVENT [54] | Adv. | 57.42 | 61.96 | 59.60 | 65.83 | 51.47 | 57.77 | 43.15 | 33.29 | 37.58 | 55.47 | 48.91 | 51.65
Ours (SCAST) | ST | 76.48 | 78.26 | 77.36 | 77.24 | 67.40 | 71.99 | 47.06 | 41.75 | 44.25 | 66.93 | 62.47 | 64.53
Table 3: Real to real adaptation: The experiments are conducted over real-to-
real domain adaptive scene text detection tasks COCO-Text $\to$ {ICDAR13,
ICDAR15 and Total-Text (Reg)} using EAST [71]. ST stands for Self-training,
Adv. for Adversarial training, and Tran. for Image translation architecture.
Supervised refers to supervised learning from the labeled target data.
We optimize SCAST by multiple rounds of self-training. Each round consists of
two alternate steps including updating of pseudo labels
$\\{\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}}\\}$ and retraining
network $\mathbf{G}$ with the updated pseudo labels. The optimization can be
formulated as a unified loss minimization problem:
$\operatorname*{arg\,min}_{\mathbf{G},\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}}}\mathcal{L}^{t}\left(P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}},\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}};\mathbf{G},\theta\right),$
(8)
where
$\mathbf{G}(\cdot)=\left\\{\mathbf{C}_{\mathrm{bi}}\left(\mathbf{E}(\cdot)\right),\mathbf{C}_{\mathrm{sub}}\left(\mathbf{E}(\cdot)\right)\right\\}$
and $\mathcal{L}^{t}$ is the weighted cross-entropy loss (Eq. 5).
Pseudo label prediction. We fix network $\mathbf{G}$ while predicting pseudo
labels $\hat{Y}^{t}_{\mathrm{bi}}$ and $\hat{Y}^{t}_{\mathrm{sub}}$ with
thresholds $\theta$ in each round. We predict pseudo labels in an “easy-to-
hard” manner so that the network can learn from confident pseudo labels first.
This strategy helps reduce the impact of noisy pseudo labels for both bi-class
multi-class objectives effectively. In implementation, we increase the
selection proportion $\rho\%$ gradually after each training round which
reduces the corresponding thresholds $\theta$ accordingly. The optimization in
this step can be formulated by:
$\operatorname*{arg\,min}_{\hat{Y}_{\mathrm{sub}}^{t},\hat{Y}_{\mathrm{bi}}^{t}}\mathcal{L}^{t}\left(P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}},\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}};\mathbf{G},\theta_{(i)}\right),$
(9)
where $i$ is the optimization round.
Update the network. We fix pseudo labels while retraining $\mathbf{G}$. As
more pseudo labels are selected round by round, model $\mathbf{G}$ learns and
mitigates the domain gap gradually. The optimization can be formulated by:
$\operatorname*{arg\,min}_{\mathbf{G}}\mathcal{L}^{t}\left(P^{t}_{\mathrm{bi}},P^{t}_{\mathrm{sub}},\hat{Y}^{t}_{\mathrm{bi}},\hat{Y}^{t}_{\mathrm{sub}};\mathbf{G},\theta_{(i)}\right),$
(10)
where $i$ is the optimization round.
## 4 Experiments
### 4.1 Datasets and Evaluation
Datasets. Our experiments involve several domain adaptive scene text detection
tasks including SynthText $\to$ {ICDAR13, ICDAR15, COCO-Text17, Total-Text
(Reg)} for regular scene text and SynthText $\to$ {Total-Text (Curve),
CTW1500, TextSeg} for irregular scene text. More dataset details are provided
in the appendix.
Evaluation. We evaluate using Precision, Recall and F-Score as in [25, 26].
The evaluations are based on the Intersection-over-Union (IoU) criterion in
PASCAL [13], with a widely adopted threshold of $50\%$. During training and
evaluations, we ignore unreadable text regions that are labeled by either “do
not care” or “illegal” in all datasets. Following previous UDA methods [72,
73, 54], we adopt dense evaluation in the experiments.
### 4.2 Implementation Details
Subcategorizor. We perform DBSCAN clustering on feature maps to discover
subcategories. Specifically, we down-sample the feature maps to make each
feature point correspond to a text-height region. We conduct DBSCAN clustering
on these feature maps for both text and background (the distance $\epsilon$ in
DBSCAN is discussed in Sec. 4.3).
Network architectures. We evaluate our method with regular text detector EAST
[71] and irregular text detector PSENet [56]. The two detectors have simple
detection strategies without complex post-processing, which reduce
interference in evaluations. For EAST, we adopt the settings of the officially
released code, which use VGG-16 pre-trained model on FPN [32] as the backbone,
and dice-loss [47] as classification loss. During post-processing, we use
Local-Aware NMS with a threshold $0.5$, same as the original paper. For
PSENet, we follow PSENet-1s from the original paper, which uses ResNet [21]
pre-trained on ImageNet [11] as backbone. In addition, we train the network
from scratch on SynthText without pretraining on the ICDAR17-MLT [42] to avoid
introducing extra data.
Training details. We use SGD optimizer with momentum $0.9$ and a weight decay
$5e-4$ in training. The initial learning rate is $1e-3$ and then decays by a
polynomial policy of power $0.9$. For training with source data, we adopted
all data augmentation strategies as used in the original papers. For training
with target data, we remove all data augmentations and resize training images
to $512\times 512$. We set the training batch-size at $12$, including $6$
source images and $6$ target images during domain adaption.
Self-Training details. The pseudo label proportion $\rho\%$ and related
threshold $\theta$ are important for pseudo label prediction. To avoid
heuristic setups, we adopt linearly increased $\rho\in\\{20,40,60,80,100\\}$
to select the top $\\{20\%,40\%,60\%,80\%,100\%\\}$ confident predictions as
pseudo labels.
### 4.3 Ablation Study and Analysis
We study the contributions of each design in the proposed SCAST, including a
$K$-Subcategorizor $SC_{\mathrm{K}}$ for source data, a subcategory self-
training $ST_{\mathrm{K}}$ and a subcategory regularized self-training
$ST_{\mathrm{K,reg}}$ over the target data.
As Table 4 shows, including subcategorization in $+SC_{\mathrm{K}}$ in the
source domain outperforms the Baseline by a large margin, showing that the
proposed subcategorization mitigates the overfitting effectively. For target
data, including the bi-class self-training with $+ST_{\mathrm{2}}$ improves
the scene text detection greatly, and combining the source-domain
subcategorization with the target-domain bi-class self-training in
$+ST_{\mathrm{2}}+SC_{\mathrm{K}}$ further improves the detection by a large
margin. In addition, including the multi-class subcategory self-training on
target data in $+ST_{\mathrm{2}}+SC_{\mathrm{K}}+ST_{\mathrm{K}}$ also
improves clearly. Finally, the inclusion of co-regularization between bi-class
and multi-class pseudo labels of target data in
$+ST_{\mathrm{2}}+SC_{\mathrm{K}}+ST_{K,reg}$ produces the best detection. The
ablation studies show the effectiveness of the proposed subcategorization and
subcategory regularized self-training.
### 4.4 Comparison with SOTA
Method | $\mathcal{L}^{s}_{\mathrm{bi}}$ | $\mathcal{L}^{s}_{\mathrm{sub}}$ | $\mathcal{L}^{t}_{\mathrm{bi}}$ | $\mathcal{L}^{t}_{\mathrm{sub}}$ | $\mathcal{L}^{t}_{\mathrm{reg}}$ | Prec. | Rec. | F-sco.
---|---|---|---|---|---|---|---|---
Baseline | ✓ | | | | | 69.63 | 53.44 | 60.47
\+ $SC_{\mathrm{K}}$ | ✓ | ✓ | | | | 67.94 | 62.93 | 65.34
\+ $ST_{\mathrm{2}}$ | ✓ | | ✓ | | | 71.36 | 58.40 | 64.23
\+ $ST_{\mathrm{2}}$ \+ $SC_{\mathrm{K}}$ | ✓ | ✓ | ✓ | | | 75.12 | 63.01 | 68.53
\+ $ST_{\mathrm{2}}$ \+ $SC_{\mathrm{K}}$ \+ $ST_{\mathrm{K}}$ | ✓ | ✓ | ✓ | ✓ | | 73.09 | 65.98 | 69.35
\+ $ST_{\mathrm{2}}$ \+ $SC_{\mathrm{K}}$ \+ $ST_{K,reg}$ | ✓ | ✓ | ✓ | ✓ | ✓ | 75.80 | 65.27 | 70.14
Table 4: Ablation study of SCAST over domain adaptive scene text detection
task SynthText $\to$ ICDAR15 using EAST. $SC_{\mathrm{K}}$, $ST_{\mathrm{2}}$,
$ST_{\mathrm{K}}$, and $ST_{K,reg}$ refer to subcategorization (via
$\mathcal{L}^{s}_{\mathrm{sub}}$), bi-class CBST [72] (via
$\mathcal{L}^{t}_{\mathrm{bi}}$), subcategory self-training (via
$\mathcal{L}^{t}_{\mathrm{sub}}$), and subcategory regularized self-training
(via $\mathcal{L}^{t}_{\mathrm{reg}}$), respectively.
Method | $\epsilon$ | Subcategory Number | Prec. | Rec. | F-sc.
---|---|---|---|---|---
Text | Background
Baseline | - | - | - | 69.63 | 53.44 | 60.47
$+SC_{K=15}$ | $0.5$ | 6 | 9 | 64.28 | 60.82 | 62.50
$+SC_{K=21}$ | $0.1$ | 10 | 11 | 69.48 | 59.30 | 63.99
$+SC_{K=33}$ | $0.05$ | 19 | 14 | 65.66 | 62.27 | 63.92
$+SC_{K=54}$ | $0.01$ | 35 | 19 | 67.94 | 62.93 | 65.34
Table 5: Maximum cluster distance $\epsilon$. With the decrease of the maximum
cluster distance $\epsilon$, DBSCAN clustering discovers more subcategories.
$SC_{\mathrm{K}}$ stands for $K$-classes SubCategorization in the source
domain, where $K$ equals to the number of subcategories in both Text and
Background. The experiments were performed over task SynthText $\to$ ICDAR15
with EAST backbone.
Network | ICDAR13 | ICDAR15 | COCO-Text | Total-Text
---|---|---|---|---
Entropy | Err. Rate | Likelihood | Entropy | Err. Rate | Likelihood | Entropy | Err. Rate | Likelihood | Entropy | Err. Rate | Likelihood
Baseline (Text) | 0.0092 | 38.49% | 844.79 | 0.0099 | 63.35% | 833.52 | 0.0101 | 67.83% | 832.02 | 0.0091 | 68.49% | 844.80
\+ $SC_{\mathrm{K}}$ | 0.1553 | 33.16% | 587.70 | 0.2305 | 58.90% | 537.10 | 0.2724 | 67.92% | 511.69 | 0.2222 | 63.65% | 587.70
Baseline (Back) | 0.0007 | 0.78% | 1101.30 | 0.0001 | 0.15% | 1262.55 | 0.0002 | 0.21% | 1101.30 | 0.0006 | 0.83% | 1116.67
\+ $SC_{\mathrm{K}}$ | 0.1186 | 0.70% | 653.80 | 0.1013 | 0.11% | 770.34 | 0.1451 | 0.15% | 653.80 | 0.1290 | 0.96% | 648.00
Table 6: Overfitting mitigation. Experiments are conducted on SynthText $\to$
ICDAR15 with EAST [71]. Baseline (Text) and Baseline (Back) measure
overfitting of text and background with the baseline model. $SC_{\mathrm{K}}$
denotes the proposed $K$-class subcategorization. ‘Likelihood’ is described in
page 2 footnote and ‘Err. Rate’ is classification error rate. Figure 5:
Comparison with the state-of-the-art. The Baseline detector and the compared
methods produce over-confident predictions with an extreme bimodal
distribution (i.e., most prediction probabilities are around $0$ or $1$),
indicating overfitting with degraded detection (with more false negatives).
The proposed subcategory-aware detector instead generates prediction
probabilities that smoothly distribute between $0$ and $1$, indicating less
overfitting. The evaluation is performed on task SynthText $\to$ TotalText
with EAST detector.
We compare SCAST with several state-of-the-art UDA methods that adopt self-
training with EntMin [18], TST [57], CBST [72], and CRST [73]), image
translation with FDA [64] and adversarial with ADVENT [54].
Tables 1 and 2 show experimental results on regular and irregular scene text
under the synthetic-to-real setup, respectively. We also evaluate real-to-real
adaptation to show the robustness of our method in Table 3. We can see that
SCAST outperforms all compared methods in F-score. Specifically, methods using
image translation and adversarial learning do not perform well as image
translation tends to smooth text strokes while adversarial learning focuses on
aligning image background. For UDA using self-training, TST [57] and CRST [73]
outperform the Baseline clearly as retraining using pseudo-labeled target data
often produces stronger models. However, the improvements are limited as their
predicted pseudo labels are noisy due to the overfitted classifiers as shown
in Table 6. As a comparison, SCAST mitigates the overfitting and reduces
pseudo label errors and produces clearly better detection as illustrated in
Fig. 5 (more samples available in appendix). Note TST combines adversarial
learning with self-training while SCAST performs self-training alone.
Note SCAST even outperforms supervised models over the Total-Text (Reg) as
shown in Tables 1 and 3. The outstanding performance is largely because the
subcategory-aware classifier predicts text regions as multiple fragments
instead of a single quadrangle. Such predictions are much more compatible for
the detection of curved text instances which cannot be located well with
quadrangles.
### 4.5 Discussion
Maximum cluster distance $\epsilon$. Table 5 shows the impact of the maximum
cluster distance $\epsilon$ in DBSCAN clustering, where the baseline is EAST
detector. We can observe a trade-off between the recall and precision, showing
that our method greatly improves the recall ($53.44$ to $62.93$) while
introducing marginal decrease in precision ($69.63$ to $67.94$) which can be
compensated with self-training over target data. We adopt $\epsilon=0.01$ in
our implemented system.
Subcategorization. Fig. 4 shows subcategorized foreground texts and image
backgrounds. We can see that DBSCAN clustering can group image patches into
subcategories with similar features in polygonal lines, traffic signs, curved
corners and thin strokes (e.g., subcategories 1, 12, 19, 37, and 40 for
illustration). Fig. 5 further shows subcategory predictions where over-
confident predictions are mitigated and False Negatives are reduced
effectively.
Overfitting mitigation. We adopt entropy, error rate, and likelihood metrics
to evaluate how the proposed subcategorization helps mitigate the overfitting
in domain adaptive scene text detection. Table 6 shows experimental results
over four widely adopted datasets. It can be observed that including the
proposed subcategorization (i.e., via +$SC_{k}$) helps mitigate the
overfitting effectively with higher entropy, fewer pseudo-labeling errors, and
lower likelihood.
Other bi-class tasks The proposed SCAST can generalize well to other bi-class
detection tasks. We evaluate another two bi-class detection tasks on vehicle
detection and pedestrian detection. Experiments over the adaptation task GTA5
[43] $\rightarrow$ Cityscapes [8] show that SCAST outperforms the state-of-
the-art by 2.9% and 3.1% (in foreground IoU), respectively. Please refer to
the appendix for details.
## 5 Conclusion
This paper presents SCAST, a subcategory-aware self-training technique for
mitigating overfitting in domain adaptive scene text detection. We
subcategorize source data by feature clustering which mitigates the
overfitting substantially. For target data, we design a subcategory
regularized self-training that employs the source-learned subcategory priors
to co-regularizes bi-class and multi-class subcategory pseudo labels which
further mitigates the overfitting effectively. Extensive experiments show that
SCAST mitigates the overfitting in scene text detection task with superior
detection performance. The idea of subcategorization is generic and can adapt
to other bi-class detection and recognition tasks. We will explore it in our
future research.
## References
* [1] Ali Furkan Biten, Ruben Tito, Andres Mafla, Lluis Gomez, Marçal Rusinol, Ernest Valveny, CV Jawahar, and Dimosthenis Karatzas. Scene text visual question answering. In Proceedings of the IEEE/CVF international conference on computer vision, pages 4291–4301, 2019.
* [2] Konstantinos Bousmalis, Nathan Silberman, David Dohan, Dumitru Erhan, and Dilip Krishnan. Unsupervised pixel-level domain adaptation with generative adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3722–3731, 2017.
* [3] Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip Krishnan, and Dumitru Erhan. Domain separation networks. Advances in neural information processing systems, 29:343–351, 2016\.
* [4] Danlu Chen, Lihua Lu, Yao Lu, Ruizhe Yu, Shunzhou Wang, Lin Zhang, and Tingxi Liu. Cross-domain scene text detection via pixel and image-level adaptation. In International Conference on Neural Information Processing, pages 135–143. Springer, 2019.
* [5] Tao Chen, Shijian Lu, and Jiayuan Fan. S-cnn: Subcategory-aware convolutional networks for object detection. IEEE transactions on pattern analysis and machine intelligence, 40(10):2522–2528, 2017.
* [6] Yuhua Chen, Wen Li, Christos Sakaridis, Dengxin Dai, and Luc Van Gool. Domain adaptive faster r-cnn for object detection in the wild. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3339–3348, 2018.
* [7] Yudi Chen, Wei Wang, Yu Zhou, Fei Yang, Dongbao Yang, and Weiping Wang. Self-training for domain adaptive scene text detection. In 2020 25th International Conference on Pattern Recognition (ICPR), pages 850–857. IEEE, 2021.
* [8] Marius Cordts, Mohamed Omran, Sebastian Ramos, Timo Rehfeld, Markus Enzweiler, Rodrigo Benenson, Uwe Franke, Stefan Roth, and Bernt Schiele. The cityscapes dataset for semantic urban scene understanding. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
* [9] Pengwen Dai, Sanyi Zhang, Hua Zhang, and Xiaochun Cao. Progressive contour regression for arbitrary-shape scene text detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7393–7402, 2021.
* [10] Dan Deng, Haifeng Liu, Xuelong Li, and Deng Cai. Pixellink: Detecting scene text via instance segmentation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
* [11] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
* [12] Martin Ester, Hans-Peter Kriegel, Jörg Sander, Xiaowei Xu, et al. A density-based algorithm for discovering clusters in large spatial databases with noise. In kdd, volume 96, pages 226–231, 1996.
* [13] Mark Everingham, SM Ali Eslami, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes challenge: A retrospective. International journal of computer vision, 111(1):98–136, 2015.
* [14] Yang Fu, Yunchao Wei, Guanshuo Wang, Yuqian Zhou, Honghui Shi, and Thomas S Huang. Self-similarity grouping: A simple unsupervised cross domain adaptation approach for person re-identification. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6112–6121, 2019.
* [15] Yaroslav Ganin and Victor Lempitsky. Unsupervised domain adaptation by backpropagation. In International conference on machine learning, pages 1180–1189. PMLR, 2015.
* [16] Muhammad Ghifary, W Bastiaan Kleijn, Mengjie Zhang, David Balduzzi, and Wen Li. Deep reconstruction-classification networks for unsupervised domain adaptation. In European conference on computer vision, pages 597–613. Springer, 2016.
* [17] Lluis Gomez, Andrés Mafla, Marçal Rusinol, and Dimosthenis Karatzas. Single shot scene text retrieval. In Proceedings of the European Conference on Computer Vision (ECCV), pages 700–715, 2018.
* [18] Yves Grandvalet, Yoshua Bengio, et al. Semi-supervised learning by entropy minimization. CAP, 367:281–296, 2005.
* [19] Dayan Guan, Jiaxing Huang, Aoran Xiao, Shijian Lu, and Yanpeng Cao. Uncertainty-aware unsupervised domain adaptation in object detection. IEEE Transactions on Multimedia, 2021.
* [20] John A Hartigan and Manchek A Wong. Algorithm as 136: A k-means clustering algorithm. Journal of the royal statistical society. series c (applied statistics), 28(1):100–108, 1979.
* [21] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
* [22] Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. Cycada: Cycle-consistent adversarial domain adaptation. In International conference on machine learning, pages 1989–1998. PMLR, 2018.
* [23] Jiaxing Huang, Dayan Guan, Shijian Lu, and Aoran Xiao. Mlan: Multi-level adversarial network for domain adaptive semantic segmentation. arXiv preprint arXiv:2103.12991, 2021.
* [24] Jiaxing Huang, Dayan Guan, Aoran Xiao, and Shijian Lu. Cross-view regularization for domain adaptive panoptic segmentation. arXiv preprint arXiv:2103.02584, 2021.
* [25] Dimosthenis Karatzas, Lluis Gomez-Bigorda, Anguelos Nicolaou, Suman Ghosh, Andrew Bagdanov, Masakazu Iwamura, Jiri Matas, Lukas Neumann, Vijay Ramaseshan Chandrasekhar, Shijian Lu, et al. Icdar 2015 competition on robust reading. In 2015 13th International Conference on Document Analysis and Recognition (ICDAR), pages 1156–1160. IEEE, 2015.
* [26] Dimosthenis Karatzas, Faisal Shafait, Seiichi Uchida, Masakazu Iwamura, Lluis Gomez i Bigorda, Sergi Robles Mestre, Joan Mas, David Fernandez Mota, Jon Almazan Almazan, and Lluis Pere De Las Heras. Icdar 2013 robust reading competition. In 2013 12th International Conference on Document Analysis and Recognition, pages 1484–1493. IEEE, 2013.
* [27] Minghui Liao, Guan Pang, Jing Huang, Tal Hassner, and Xiang Bai. Mask textspotter v3: Segmentation proposal network for robust scene text spotting. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XI 16, pages 706–722. Springer, 2020.
* [28] Minghui Liao, Baoguang Shi, and Xiang Bai. Textboxes++: A single-shot oriented scene text detector. IEEE transactions on image processing, 27(8):3676–3690, 2018.
* [29] Minghui Liao, Baoguang Shi, Xiang Bai, Xinggang Wang, and Wenyu Liu. Textboxes: A fast text detector with a single deep neural network. In Thirty-first AAAI conference on artificial intelligence, 2017\.
* [30] Minghui Liao, Zhaoyi Wan, Cong Yao, Kai Chen, and Xiang Bai. Real-time scene text detection with differentiable binarization. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 11474–11481, 2020.
* [31] Minghui Liao, Zhen Zhu, Baoguang Shi, Gui-song Xia, and Xiang Bai. Rotation-sensitive regression for oriented scene text detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5909–5918, 2018.
* [32] Tsung-Yi Lin, Piotr Dollár, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie. Feature pyramid networks for object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2117–2125, 2017.
* [33] Dongnan Liu, Donghao Zhang, Yang Song, Fan Zhang, Lauren O’Donnell, Heng Huang, Mei Chen, and Weidong Cai. Unsupervised instance segmentation in microscopy images via panoptic domain adaptation and task re-weighting. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 4243–4252, 2020.
* [34] Jiawei Liu, Zheng-Jun Zha, Di Chen, Richang Hong, and Meng Wang. Adaptive transfer network for cross-domain person re-identification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7202–7211, 2019.
* [35] Wei Liu, Dragomir Anguelov, Dumitru Erhan, Christian Szegedy, Scott Reed, Cheng-Yang Fu, and Alexander C Berg. Ssd: Single shot multibox detector. In European conference on computer vision, pages 21–37. Springer, 2016.
* [36] Yuliang Liu, Lianwen Jin, Shuaitao Zhang, Canjie Luo, and Sheng Zhang. Curved scene text detection via transverse and longitudinal sequence connection. Pattern Recognition, 90:337–345, 2019.
* [37] Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan. Learning transferable features with deep adaptation networks. In International conference on machine learning, pages 97–105. PMLR, 2015.
* [38] Mingsheng Long, Han Zhu, Jianmin Wang, and Michael I Jordan. Deep transfer learning with joint adaptation networks. In International conference on machine learning, pages 2208–2217. PMLR, 2017.
* [39] Shangbang Long, Xin He, and Cong Yao. Scene text detection and recognition: The deep learning era. International Journal of Computer Vision, 129(1):161–184, 2021\.
* [40] Pengyuan Lyu, Minghui Liao, Cong Yao, Wenhao Wu, and Xiang Bai. Mask textspotter: An end-to-end trainable neural network for spotting text with arbitrary shapes. In Proceedings of the European Conference on Computer Vision (ECCV), pages 67–83, 2018.
* [41] Pengyuan Lyu, Cong Yao, Wenhao Wu, Shuicheng Yan, and Xiang Bai. Multi-oriented scene text detection via corner localization and region segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7553–7563, 2018.
* [42] Nibal Nayef, Fei Yin, Imen Bizid, Hyunsoo Choi, Yuan Feng, Dimosthenis Karatzas, Zhenbo Luo, Umapada Pal, Christophe Rigaud, Joseph Chazalon, et al. Icdar2017 robust reading challenge on multi-lingual scene text detection and script identification-rrc-mlt. In 2017 14th IAPR International Conference on Document Analysis and Recognition (ICDAR), volume 1, pages 1454–1459. IEEE, 2017.
* [43] Stephan R. Richter, Vibhav Vineet, Stefan Roth, and Vladlen Koltun. Playing for data: Ground truth from computer games. In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, European Conference on Computer Vision (ECCV), volume 9906 of LNCS, pages 102–118. Springer International Publishing, 2016.
* [44] Kuniaki Saito, Yoshitaka Ushiku, and Tatsuya Harada. Asymmetric tri-training for unsupervised domain adaptation. In International Conference on Machine Learning, pages 2988–2997. PMLR, 2017.
* [45] Baoguang Shi, Cong Yao, Minghui Liao, Mingkun Yang, Pei Xu, Linyan Cui, Serge Belongie, Shijian Lu, and Xiang Bai. Icdar2017 competition on reading chinese text in the wild (rctw-17). In 2017 14th IAPR International Conference on Document Analysis and Recognition (ICDAR), volume 1, pages 1429–1434. IEEE, 2017.
* [46] Amanpreet Singh, Vivek Natarajan, Meet Shah, Yu Jiang, Xinlei Chen, Dhruv Batra, Devi Parikh, and Marcus Rohrbach. Towards vqa models that can read. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8317–8326, 2019.
* [47] Carole H Sudre, Wenqi Li, Tom Vercauteren, Sebastien Ourselin, and M Jorge Cardoso. Generalised dice overlap as a deep learning loss function for highly unbalanced segmentations. In Deep learning in medical image analysis and multimodal learning for clinical decision support, pages 240–248. Springer, 2017.
* [48] Yaniv Taigman, Adam Polyak, and Lior Wolf. Unsupervised cross-domain image generation. arXiv preprint arXiv:1611.02200, 2016.
* [49] Shangxuan Tian, Shijian Lu, and Chongshou Li. Wetext: Scene text detection under weak supervision. In Proceedings of the IEEE International Conference on Computer Vision, pages 1492–1500, 2017.
* [50] Zhi Tian, Weilin Huang, Tong He, Pan He, and Yu Qiao. Detecting text in natural image with connectionist text proposal network. In European conference on computer vision, pages 56–72. Springer, 2016.
* [51] Zhuotao Tian, Michelle Shu, Pengyuan Lyu, Ruiyu Li, Chao Zhou, Xiaoyong Shen, and Jiaya Jia. Learning shape-aware embedding for scene text detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4234–4243, 2019.
* [52] Yi-Hsuan Tsai, Kihyuk Sohn, Samuel Schulter, and Manmohan Chandraker. Domain adaptation for structured output via discriminative patch representations. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 1456–1465, 2019.
* [53] Andreas Veit, Tomas Matera, Lukas Neumann, Jiri Matas, and Serge Belongie. Coco-text: Dataset and benchmark for text detection and recognition in natural images. arXiv preprint arXiv:1601.07140, 2016.
* [54] Tuan-Hung Vu, Himalaya Jain, Maxime Bucher, Matthieu Cord, and Patrick Pérez. Advent: Adversarial entropy minimization for domain adaptation in semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2517–2526, 2019.
* [55] Hao Wang, Xiang Bai, Mingkun Yang, Shenggao Zhu, Jing Wang, and Wenyu Liu. Scene text retrieval via joint text detection and similarity learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4558–4567, 2021.
* [56] Wenhai Wang, Enze Xie, Xiang Li, Wenbo Hou, Tong Lu, Gang Yu, and Shuai Shao. Shape robust text detection with progressive scale expansion network. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9336–9345, 2019.
* [57] Weijia Wu, Ning Lu, Enze Xie, Yuxing Wang, Wenwen Yu, Cheng Yang, and Hong Zhou. Synthetic-to-real unsupervised domain adaptation for scene text detection in the wild. In Proceedings of the Asian Conference on Computer Vision, 2020\.
* [58] Chang-Dong Xu, Xing-Ran Zhao, Xin Jin, and Xiu-Shen Wei. Exploring categorical regularization for domain adaptive object detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11724–11733, 2020.
* [59] Xingqian Xu, Zhifei Zhang, Zhaowen Wang, Brian Price, Zhonghao Wang, and Humphrey Shi. Rethinking text segmentation: A novel dataset and a text-specific refinement approach. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12045–12055, 2021.
* [60] Chuhui Xue, Shijian Lu, and Steven Hoi. Detection and rectification of arbitrary shaped scene texts by using text keypoints and links. Pattern Recognition, 124:108494, 2022.
* [61] Chuhui Xue, Shijian Lu, and Fangneng Zhan. Accurate scene text detection through border semantics awareness and bootstrapping. In Proceedings of the European conference on computer vision (ECCV), pages 355–372, 2018.
* [62] Chuhui Xue, Shijian Lu, and Wei Zhang. Msr: multi-scale shape regression for scene text detection. arXiv preprint arXiv:1901.02596, 2019.
* [63] Chuhui Xue, Wenqing Zhang, Yu Hao, Shijian Lu, Philip HS Torr, and Song Bai. Language matters: A weakly supervised vision-language pre-training approach for scene text detection and spotting. In European Conference on Computer Vision, pages 284–302. Springer, 2022.
* [64] Yanchao Yang and Stefano Soatto. Fda: Fourier domain adaptation for semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4085–4095, 2020.
* [65] Xiangyu Yue, Yang Zhang, Sicheng Zhao, Alberto Sangiovanni-Vincentelli, Kurt Keutzer, and Boqing Gong. Domain randomization and pyramid consistency: Simulation-to-real generalization without accessing target domain data. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2100–2110, 2019.
* [66] Kaiwei Zeng, Munan Ning, Yaohua Wang, and Yang Guo. Hierarchical clustering with hard-batch triplet loss for person re-identification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 13657–13665, 2020.
* [67] Fangneng Zhan, Chuhui Xue, and Shijian Lu. Ga-dan: Geometry-aware domain adaptation network for scene text detection and recognition. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9105–9115, 2019.
* [68] Chengquan Zhang, Borong Liang, Zuming Huang, Mengyi En, Junyu Han, Errui Ding, and Xinghao Ding. Look more than once: An accurate detector for text of arbitrary shapes. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10552–10561, 2019.
* [69] Gongjie Zhang, Kaiwen Cui, Rongliang Wu, Shijian Lu, and Yonghong Tian. PNPDet: Efficient few-shot detection without forgetting via plug-and-play sub-networks. In WACV, 2021.
* [70] Gongjie Zhang, Zhipeng Luo, Kaiwen Cui, Shijian Lu, and Eric P. Xing. Meta-DETR: Image-level few-shot detection with inter-class correlation exploitation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022\.
* [71] Xinyu Zhou, Cong Yao, He Wen, Yuzhi Wang, Shuchang Zhou, Weiran He, and Jiajun Liang. East: an efficient and accurate scene text detector. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 5551–5560, 2017.
* [72] Yang Zou, Zhiding Yu, BVK Kumar, and Jinsong Wang. Unsupervised domain adaptation for semantic segmentation via class-balanced self-training. In Proceedings of the European conference on computer vision (ECCV), pages 289–305, 2018.
* [73] Yang Zou, Zhiding Yu, Xiaofeng Liu, BVK Kumar, and Jinsong Wang. Confidence regularized self-training. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 5982–5991, 2019.
|
> Proceedings of the
> 3rd International Workshop on
> Reading Music Systems
>
> 23rd July, 2021
> Alicante, Spain
## Organization
General Chairs
Jorge Calvo-Zaragoza | | University of Alicante, Spain
---|---|---
Alexander Pacha | | TU Wien, Austria
Program Committee
Ichiro Fujinaga | | McGill University, Canada
---|---|---
Jose M. Iñesta | | University of Alicante, Spain
Heinz Roggenkemper | | Canamus, United States of America
George Fazekas | | Queen Mary University of London, United Kingdom
Gabriel Vigliensoni | | Goldsmiths University of London, United Kingdom
Elona Shatri | | Queen Mary University of London, United Kingdom
Alicia Fornés | | Computer Vision Center, Spain
Proceedings of the 3rd International Workshop on Reading Music Systems,
Alicante, 2021
Edited by Jorge Calvo-Zaragoza and Alexander Pacha
© The respective authors.
Licensed under a Creative Commons Attribution 4.0 International License (CC-
BY-4.0).
Logo made by Freepik from www.flaticon.com. Adapted by Alexander Pacha.
## Preface
Dear colleagues!
We are more than pleased to present to you the proceedings of the 3rd
International Workshop on Reading Music Systems (WoRMS). With the pandemic
hopefully slowly fading, we can finally resume our tradition of having an
annual workshop that brings together researchers and practitioners that work
on music reading systems.
For us, it was always important to create an interactive workshop that brings
together people that share a common interest in music reading systems,
allowing them to exchange ideas and form relationships with one another. 2020
was a difficult year for many of us. Personal situations changed rapidly,
research projects were canceled, and not knowing whether an online-only
edition of WoRMS would be desirable under these circumstances, we decided,
heavy-hearted, to skip WoRMS 2020.
Nevertheless, we are looking forward to this year’s edition, which will take
place in a hybrid mode with some participants being on-site, while others
joining remotely via Zoom. We believe that this way, we can find the balance
between enabling interaction and keeping everyone safe. Nothing can replace
in-person communication, so we hope that future editions will be fully in-
person again. However, we also want to highlight the benefits of this format:
offering an online option allows people to join the workshop that could not
participate otherwise.
This year’s edition features 11 contributions, reaching from exciting new
datasets to multi-modal methods that might change the way how we think about
processing written music. We noted that machine-learning remains a common
theme throughout most papers—a trend that we expect to resume in the future.
Finally, we want to thank University of Alicante Polytechnic School for
providing the room and the TU Wien for providing Zoom conferencing facilities.
Jorge Calvo-Zaragoza and Alexander Pacha
###### Contents
1. Preface
See pages - of papers/WoRMS_2021_paper_1.pdf See pages - of
papers/WoRMS_2021_paper_2.pdf See pages - of papers/WoRMS_2021_paper_3.pdf See
pages - of papers/WoRMS_2021_paper_4.pdf See pages - of
papers/WoRMS_2021_paper_5.pdf See pages - of papers/WoRMS_2021_paper_6.pdf See
pages - of papers/WoRMS_2021_paper_7.pdf See pages - of
papers/WoRMS_2021_paper_8.pdf See pages - of papers/WoRMS_2021_paper_11.pdf
See pages - of papers/WoRMS_2021_paper_12.pdf See pages - of
papers/WoRMS_2021_paper_13.pdf
|
# The effect of linear background rotational flows on magnetoacoustic modes of
a photospheric magnetic flux tube
S. J. Skirvin1,2, V. Fedun2, S. S. A. Silva2, T. Van Doorsselaere1, N. Claes1,
M. Goossens1 and G. Verth3
1Centre for mathematical Plasma Astrophysics, Mathematics Department, KU
Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium.
2Plasma Dynamics Group, Department of Automatic Control & Systems Engineering,
The University of Sheffield, Sheffield, S3 7RH, UK
3Plasma Dynamics Group, Department of Mathematics and Statistics, The
University of Sheffield, Sheffield, S3 7RH, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Magnetoacoustic waves in solar magnetic flux tubes may be affected by the
presence of background rotational flows. Here, we investigate the behaviour of
$m=0$ and $m=\pm 1$ modes of a magnetic flux tube in the presence of linear
background rotational flows embedded in a photospheric environment. We show
that the inclusion of a background rotational flow is found to have little
effect on the obtained eigensolutions for the axisymmetric $m=0$ sausage mode.
However, solutions for the kink mode are dependent on the location of the flow
resonance modified by the slow frequency. A background rotational flow causes
the modified flow resonances to possess faster phase speeds in the thin-tube
(TT) limit for the case $m=1$. This results in solutions for the slow body and
slow surface kink modes to follow this trajectory, changing their dispersive
behaviour. For a photospheric flux tube in the TT limit, we show that it
becomes difficult to distinguish between the slow surface and fast surface
kink ($m=1$) modes upon comparison of their eigenfunctions. 2D velocity field
plots demonstrate how these waves, in the presence of background rotational
flows, may appear in observational data. For slow body kink modes, a swirling
pattern can be seen in the total pressure perturbation. Furthermore, the tube
boundary undergoes a helical motion from the breaking of azimuthal symmetry,
where the $m=1$ and $m=-1$ modes become out of phase, suggesting the resulting
kink wave is circularly polarised. These results may have implications for
seismology of magnetohydrodynamic waves in solar magnetic vortices.
###### keywords:
magnetohydrodynamics (MHD) – waves
††pubyear: 2022††pagerange: The effect of linear background rotational flows
on magnetoacoustic modes of a photospheric magnetic flux tube–The effect of
linear background rotational flows on magnetoacoustic modes of a photospheric
magnetic flux tube
## 1 Introduction
High resolution observations have revealed, over the last few decades, that
magnetohydrodynamic (MHD) waves are ubiquitous throughout the Sun’s atmosphere
(Nakariakov et al., 1999; Aschwanden et al., 1999; De Pontieu et al., 2007;
Tomczyk et al., 2007; Morton et al., 2015; Grant et al., 2015; Keys et al.,
2018; Stangalini et al., 2022; Bate et al., 2022). Understanding MHD wave
properties from a theoretical point of view is of the utmost importance in
solar physics as these waves could contribute to the heating of local plasma
and may also be used as a proxy to determine sub-resolution plasma properties.
Additionally, observed MHD wave behaviour can be used to provide an estimate
of the properties of local plasma that cannot be measured directly, for
example the magnetic field in the corona. Furthermore, MHD waves may also be
responsible in the formation of some observed phenomena, e.g. jets, in the
solar atmosphere (De Pontieu et al., 2004; Rouppe van der Voort et al., 2007;
Scullion et al., 2011).
The uniform cylindrical waveguide model (see e.g. Wilson, 1979; Spruit &
Zweibel, 1979; Edwin & Roberts, 1983) has provided a foundation for analytical
studies into MHD wave investigations under solar atmospheric conditions. These
studies have shown that the trapped wave modes of a magnetic flux tube can be
described by the number of nodes present in each geometrical direction. The
axisymmetric sausage mode has zero nodes in the azimuthal direction $m=0$,
whereas the (linearly polarised) non-axisymmetric kink mode has a single
azimuthal node $m=1$ or $m=-1$. Whilst this uniform model is simplistic in
nature, due to the uniform plasma considered inside and outside the waveguide,
it has been modified over recent decades to model specific configurations
which better match observational data (see e.g. Van Doorsselaere et al., 2004;
Verth et al., 2007; Erdélyi & Fedun, 2010; Aldhafeeri et al., 2021; Ruderman &
Petrukhin, 2022). The cylinder model can be further extended to incorporate
additional physical environments that may be common configurations in magnetic
flux tubes, such as including a background rotational flow. Magnetic flux
tubes with background rotational flows are a common configuration observed in,
e.g., flux tubes rooted in intergranular lanes, solar tornadoes and spicules
(Bonet et al., 2010; Wedemeyer-Böhm et al., 2012; Tziotziou et al., 2018;
Shetye et al., 2019). Such structures display dynamic characteristics and
excite a wide range of MHD waves which couple different layers of the solar
atmosphere. As a result, these structures act as a natural conduit for the
transfer of mass, momentum and energy throughout the solar atmosphere.
Furthermore, rotational flows naturally appear in numerical MHD simulations of
regions in the solar atmosphere with vortex drivers (see, e.g. Fedun et al.,
2011a; Fedun et al., 2011b; Shelyag et al., 2011, 2012, 2013; González-Avilés
et al., 2017; González-Avilés et al., 2018; Snow et al., 2018) and also in
magnetoconvection simulations (Yadav et al., 2020, 2021; Silva et al., 2021).
The presence of a background rotational flow would manifest itself as a non-
zero azimuthal component of the background velocity field vector. Considering
$m=\pm 1$ transverse kink modes, the effect of introducing a background
rotational flow into the model would break the symmetry of the system with
respect to the direction of azimuthal wave propagation. Similarly, it has been
shown that the inclusion of a steady vertical background plasma flow, aligned
with the magnetic field, can obtain solutions which also break the symmetry of
the forward and backward propagating waves (Nakariakov & Roberts, 1995; Terra-
Homem et al., 2003; Soler et al., 2009; Skirvin et al., 2022). The presence of
the longitudinal flow results in a Doppler-shifted wave frequency, depending
upon the amplitude of the flow and the vertical wavenumber, which may shift
certain wave modes into different physical regimes. Furthermore, a background
longitudinal flow may also affect the wavelength and damping length of
resonant absorption for kink modes, causing more efficient damping of backward
propagating modes compared to forward propagating modes (Soler et al., 2011).
Circularly polarised kink modes, previously observed in chromospheric magnetic
elements (Stangalini et al., 2017) and sunspots (Jess et al., 2017), have also
been studied recently by Magyar et al. (2022) where the authors discuss the
differences that resonant absorption and phase mixing have on linearly and
circularly polarised kink waves. Magyar et al. (2022) also conclude, upon
analysis of the Doppler signatures for both polarisation states, that there is
very little difference between the two polarisation’s in Doppler observations.
Circularly polarised kink waves have been shown to develop in coronal loops
with twisted magnetic fields (Terradas & Goossens, 2012; Ruderman & Terradas,
2015).
Previous analytical studies have investigated the stability status of rotating
flux tubes, as an azimuthal velocity shear across the waveguide boundary may
be susceptible to the Kelvin-Helmholtz instability (KHI) (Soler et al., 2010;
Zaqarashvili et al., 2015; Zhelyazkov & Chandra, 2019). However, these studies
assume zero plasma-$\beta$ and focus on coronal conditions only, ignoring the
slow magnetoacoustic modes entirely. The stability of a magnetic flux tube
with a linear background magnetic twist and rotational flow component was
studied by Cheremnykh et al. (2018) who found that the $m=0$ sausage mode
becomes unstable for azimuthal flow speeds that create a centrifugal force
which can overcome the magnetic tension, whereas, the $m=1$ kink mode can only
become unstable for sufficiently large values of longitudinal (axial) flow
speed. Whilst the stability studies mentioned above investigate the
susceptibility of the KHI to various wave modes in magnetic flux tubes with
background flows, we stress that it is not the goal of this study. The primary
aim of the present work, however, is to provide a description of how $m=0$ and
$m=\pm 1$ wave modes would manifest themselves in solar magnetic flux tubes,
under photospheric conditions, in the presence of background rotational flows
and to aid mode interpretations of observational and numerical data.
This paper is presented as follows, in Section 2 the governing equations
describing a magnetic flux tube in the presence of a background rotational
flow is presented, along with a brief description of the numerical eigensolver
implemented in this work. Section 3 presents the results of an investigation
into the effect of a linear profile of background rotational flow on the
properties of magnetoacoutsic waves under photospheric conditions. Section 3
looks closely at the obtained eigenvalues, one dimensional eigenfunctions and
two/three dimensional visualisations of the kink mode in the presence of a
background rotational flow. Finally, in Section 4, we summarise the findings
presented in this work and discuss avenues of future research.
## 2 Method
The ideal MHD equations adopted in this study are:
$\displaystyle\frac{d\rho}{dt}+\rho\nabla\cdot\textbf{v}$ $\displaystyle=0,$
(1) $\displaystyle\rho\left(\frac{d\textbf{v}}{dt}\right)$
$\displaystyle=-\nabla
p+\frac{1}{\mu_{0}}\left(\nabla\times\textbf{B}\right)\times\textbf{B},$ (2)
$\displaystyle\frac{d}{dt}\left(\frac{p}{\rho^{\gamma}}\right)$
$\displaystyle=0,$ (3) $\displaystyle\frac{\partial\textbf{B}}{\partial t}$
$\displaystyle=\nabla\times\left(\textbf{v}\times\textbf{B}\right),$ (4)
$\displaystyle\nabla\cdot\textbf{B}$ $\displaystyle=0,$ (5)
where $\rho$, v, $p$, B, $\gamma$ and $\mu$ denote plasma density, plasma
velocity, plasma pressure, magnetic field, ratio of specific heats (taken
$\gamma=5/3$) and the magnetic permeability respectively. We consider a
cylindrical geometry $(r,\varphi,z)$ where the initial equilibrium has a
radially spatially dependent form of the velocity field vector
$\textbf{v}_{0}=(0,v_{\varphi}(r),0)$. In our model the magnetic field is
taken to be straight and uniform with no background axial plasma flow. In the
lower solar atmosphere it should be noted that magnetic flux tubes possess
significant non-vertical magnetic field as a result of flux tube expansion to
maintain pressure balance, however, this effect can be considered to be
negligible in the current study due to the analysis in the local plasma
environment. Since the equilibrium quantities depend on $r$ only, the
perturbed quantities can be Fourier-analysed with respect to the ignorable
coordinates $\varphi,z$ and time $t$ and put proportional to:
$\text{exp}\left[i\left(m\varphi+kz-\omega t\right)\right],$
where $m$ is the azimuthal wave number, $k$ the vertical wavenumber and
$\omega$ the wave frequency.
After linearising Equations (1)-(5), we arrive at a system of two differential
equations containing the total pressure perturbation $\hat{P}_{T}$ and the
radial displacement perturbation $r\hat{\xi}_{r}$ (see e.g. Sakurai et al.,
1991; Goossens et al., 1992), which can be written as:
$\displaystyle D\frac{d}{dr}\left(r\hat{\xi}_{r}\right)$
$\displaystyle=C_{1}r\hat{\xi}_{r}-C_{2}r\hat{P}_{T},$ (6) $\displaystyle
D\frac{d\hat{P}_{T}}{dr}$ $\displaystyle=C_{3}\hat{\xi}_{r}-C_{1}\hat{P}_{T},$
(7)
where,
$\displaystyle D$
$\displaystyle=\rho\left(c^{2}+v_{A}^{2}\right)\left(\Omega^{2}-k^{2}v_{A}^{2}\right)\left(\Omega^{2}-k^{2}c_{T}^{2}\right),$
(8) $\displaystyle\Omega$ $\displaystyle=\omega-\frac{m}{r}v_{\varphi},$ (9)
$\displaystyle c^{2}$ $\displaystyle=\frac{\gamma p}{\rho},\ \ \ \
v_{A}^{2}=\frac{B_{z}^{2}}{\mu\rho},\ \ \ \
c_{T}^{2}=\frac{v_{A}^{2}c^{2}}{\left(c^{2}+v_{A}^{2}\right)},$ (10)
$\displaystyle C_{1}$
$\displaystyle=Q\Omega^{2}-2m\left(c^{2}+v_{A}^{2}\right)\left(\Omega^{2}-k^{2}c_{T}^{2}\right)\frac{T^{2}}{r^{2}},$
(11) $\displaystyle C_{2}$
$\displaystyle=\Omega^{4}-\left(c^{2}+v_{A}^{2}\right)\left(\frac{m^{2}}{r^{2}}+k^{2}\right)\left(\Omega^{2}-k^{2}c_{T}^{2}\right),$
(12) $\displaystyle C_{3}$
$\displaystyle=D\left\\{\rho\left(\Omega^{2}-k^{2}v_{A}^{2}\right)+r\frac{d}{dr}\left[-\rho\left(\frac{v_{\varphi}}{r}\right)^{2}\right]\right\\}+$
(13)
$\displaystyle+Q^{2}-4\left(c^{2}+v_{A}^{2}\right)\left(\Omega^{2}-k^{2}c_{T}^{2}\right)\frac{T^{2}}{r^{2}},$
$\displaystyle Q$
$\displaystyle=-\left(\Omega^{2}-k^{2}v_{A}^{2}\right)\frac{\rho
v_{\varphi}^{2}}{r},$ (14) $\displaystyle T$ $\displaystyle=\rho\Omega
v_{\varphi}.$ (15)
It should be noted here that we assume $B_{\varphi}=v_{z}=0$, which simplifies
the fully inclusive
($B_{\varphi}=B_{\varphi}(r),B_{z}=B_{z}(r),v_{z}=v_{z}(r),\rho=\rho(r)$) set
of equations previously noted in literature by Goossens et al. (1992).
Quantities $c^{2}$, $v_{A}^{2}$, and $c_{T}^{2}$ define the squares of the
local sound speed, Alfvén speed and cusp (tube) speed respectively. The
quantity $\Omega$ represents the Doppler shifted frequency as a result of the
background rotational plasma flow.
Equations (6)-(7) can be combined to create a single differential equation in
either $r\hat{\xi}_{r}$:
$\frac{d}{dr}\left[f(r)\frac{d}{dr}\left(r\hat{\xi}_{r}\right)\right]-g(r)\left(r\hat{\xi}_{r}\right)=0,$
(16)
where,
$f(r)=\frac{D}{rC_{2}},$ (17)
$g(r)=\frac{d}{dr}\left(\frac{C_{1}}{rC_{2}}\right)-\frac{1}{rD}\left(C_{3}-\frac{C_{1}^{2}}{C_{2}}\right),$
(18)
or $\hat{P}_{T}$:
$\frac{d}{dr}\left[\tilde{f}(r)\frac{d\hat{P}_{T}}{dr}\right]-\tilde{g}(r)\hat{P}_{T}=0,$
(19)
where,
$\tilde{f}(r)=\frac{rD}{C_{3}},$ (20)
$\tilde{g}(r)=-\frac{d}{dr}\left(\frac{rC_{1}}{C_{3}}\right)-\frac{r}{D}\left(C_{2}-\frac{C_{1}^{2}}{C_{3}}\right).$
(21)
For a non-uniform plasma, the governing Equations (6)-(7) possess regular
singularities where the wave frequency matches the local characteristic
frequencies at:
$\omega=\frac{m}{r}v_{\varphi}(r)\pm kv_{A},$ (22)
$\omega=\frac{m}{r}v_{\varphi}(r)\pm kc_{T}.$ (23)
Equations (22) and (23) define the flow continua modified by the local Alfvén
($kv_{A}$) and slow ($kc_{T}$) frequencies, respectively. In ideal MHD, the
wave solutions existing inside the continua, with positions given by Equations
(22) and (23), are known as ‘quasi-modes’ where the wave frequency becomes a
complex quantity (De Groof & Goossens, 2000; Goedbloed & Poedts, 2004;
Geeraerts et al., 2022). The nature of the solutions lying inside the continua
is not discussed in the present study and instead will be the focus of future
work.
Both Equations (16) and (19) have no known closed form analytical solutions,
without making assumptions that somehow reduce the mathematical complexity.
Therefore, investigating the properties of wave modes propagating within an
equilibrium which is non-uniform must be done numerically. The numerical
approach used in this study is based on the eigensolver applied in Skirvin et
al. (2021); Skirvin et al. (2022) for non-uniform magnetic slabs and non-
uniform flux tubes, respectively. The eigensolver implements the numerical
shooting and bisection methods whilst also relying on fundamental properties
of the sausage and kink modes. The numerical shooting method solves Equations
(16) and (19) ensuring continuity of $\hat{P}_{T}$ and $\hat{\xi}_{r}$ across
the boundary of the flux tube. This technique has been applied before in solar
physics by, e.g. Tirry & Goossens (1996); Pinter et al. (1998); Andries et al.
(2000); Taroyan & Erdélyi (2002, 2003). These studies also utilise the jump
conditions introduced by Sakurai et al. (1991); Goossens et al. (1992) to deal
with the regular singularities that appear in the governing equations,
allowing the authors to study the resulting quasi-modes. However, following
the primary objective of the present study, we only consider eigenmodes with
real valued wave frequency and wavenumber.
## 3 Magnetic flux tube in the presence of a linear background rotational
flow
In this section, a magnetic flux tube in the presence of a linear rotational
background flow is investigated. For all cases the magnetic flux tube is
otherwise uniform such that the equilibrium plasma density and magnetic field
is constant across the flux tube. A profile comparable to the magnetic twist
profile incorporated by Erdélyi & Fedun (2007) and Erdélyi & Fedun (2010) is
chosen but applied to the azimuthal velocity field component $v_{\varphi}$
instead. A rotational flow can be either clockwise or counter-clockwise in the
reference frame relative to the observer. The only difference between a
clockwise rotational flow and an anti-clockwise rotational flow will be the
sign in front of $v_{\varphi}$ and the direction of shifted wave frequency
relative to the flow. In the following sections, a magnetic flux tube is
presented with an equilibrium azimuthal flow component, acting in a counter-
clockwise direction, which takes the form:
$v_{\varphi}(r)=A\left(\frac{r}{a}\right)^{\alpha},$ (24)
where $A$ is the amplitude of the rotational flow, $\alpha$ is the parameter
(exponent) dictating the radial profile of the rotational flow and $a$
indicates the location of the boundary of the flux tube, taken to be $a=1$ in
our study. The case when $\alpha=1$ for example corresponds to a linear
rotational flow, which will be the focus of this study, (see e.g. Figure 1).
It can be shown that when $v_{\varphi}$ is linear, Equations (6)-(15)
simplify, and in many cases the dependence on the radial coordinate is
removed. The rotational flow is constant with height $z$ in all cases
considered in this work. Obtaining an equilibrium in a magnetic cylinder with
a background rotational velocity component is not as mathematically simple as
the scenario of a uniform magnetic cylinder. In order to maintain total
pressure balance across the waveguide the following expression must be
satisfied (Goossens et al., 2011):
$\frac{d}{dr}\left(p+\frac{B_{0z}^{2}}{2\mu}\right)=\frac{\rho
v_{\varphi}^{2}(r)}{r}=\rho A^{2}r.$ (25)
Integration of Equation (25) yields:
$p+\frac{B_{0z}^{2}}{2\mu}=\rho A^{2}\frac{r^{2}}{2},$ (26)
where the constant of integration is absorbed into the gas pressure term, $p$,
and corresponds to the plasma pressure on the axis of the cylinder where the
amplitude of the flow is zero (see e.g. Cheremnykh et al., 2018). Under the
photospheric conditions considered in this work, the total pressure balance is
achieved by an increase in temperature to balance the increase in azimuthal
flow amplitude towards the boundary of the flux tube. For configurations where
the amplitude of the rotational flow is weak (e.g. $A<0.5c_{i}$), then the
change in spatial behaviour of the plasma pressure and temperature is small,
but must not be dismissed.
The presence of a background rotational flow not only modifies the equilibrium
pressure balance relationship, but also affects the continuity conditions on
the boundary of the waveguide. Considering a magnetic flux tube in the
presence of a background rotational flow, the resulting boundary continuity
conditions state:
$\displaystyle\hat{\xi}_{re}\Bigr{\rvert}_{r=a}$
$\displaystyle=\hat{\xi}_{ri}\Bigr{\rvert}_{r=a},$ (27)
$\displaystyle\hat{P}_{Te}\Bigr{\rvert}_{r=a}$
$\displaystyle=\left(\hat{P}_{Ti}+\frac{\rho_{0i}v_{\varphi}^{2}}{a}\hat{\xi}_{ri}\right)\Bigr{\rvert}_{r=a}.$
(28)
The change in boundary conditions are accounted for in the numerical
eigensolver, and a pair of eigenvalues will only be retrieved for values
satisfying the above conditions for each respective case study.
Finally, when the background rotational flow is taken to be linear with
respect to the radial direction, Equations (22) and (23) become:
$\displaystyle\omega$ $\displaystyle=Am\pm kv_{A},$ (29) $\displaystyle\omega$
$\displaystyle=Am\pm kc_{T}.$ (30)
Equations (29) and (30) no longer define continuum regions as the resonant
positions no longer cover a range of frequencies, rather they define singular
resonant locations which correspond to the flow resonance position modified by
the Alfvén and slow frequencies respectively. The locations of these resonant
positions depend heavily on the amplitude of the rotational flow in the setup
presented in this work.
### 3.1 Linear rotating magnetic flux tube under photospheric conditions
Figure 1: Equilibrium background rotational flow profiles for cases with
increasing amplitude for a photospheric cylinder. In all cases the profiles
are linear with respect to spatial coordinate $r$ (i.e. $\alpha=1$). The
amplitude of the rotational flow increases linearly up to a boundary value of
$A=0.01$ (green line), $A=0.05$ (black line), $A=0.1$ (yellow line), $A=0.15$
(red line) and $A=0.25$ (blue line). The boundary of the flux tube is located
at $r=1$.
In this section, a magnetic flux tube under photospheric conditions
($v_{Ae}<c_{i}<c_{e}<v_{Ai}$) in the presence of a linear background
rotational flow is investigated. For all photospheric cases in this work, the
numerical values correspond to $c_{i}=1$, $c_{e}=1.5c_{i}$, $v_{Ai}=2c_{i}$,
$v_{Ae}=0.5c_{i}$ and $\rho_{i}=1$. This choice of equilibrium parameters
results in a density contrast between the internal and external plasma to be
roughly $\rho_{i}/\rho_{e}=0.567$. The background velocity vector inside the
waveguide can be written as $\mathbf{v}_{0i}=(0,Ar,0)$. The flow outside the
cylinder is zero which results in a velocity shear across the cylinder
boundary at $r=a$, however the value of $A$ is chosen to be small and both
sub-Alfvénic, sub-sonic and below the threshold for the Kelvin-Helmholtz
instability (Soler et al., 2010). This choice of amplitude $A$ also agrees
with observed values of photospheric flows when compared with the local
sound/Alfvén speed (Bonet et al., 2008). Shown in Figure 1 are the linear
profiles of background rotational flow considered in this section. In all
cases the flow amplitude is proportional to the radial distance from the
center of the flux tube, at $r=0$, up to the boundary at $r=1$, however the
amplitude is allowed to vary.
#### 3.1.1 Eigenvalues
(a)
(b)
(c)
(d)
Figure 2: Dispersion diagrams for a photospheric cylinder with a linear
background rotational flow of varying flow amplitudes. The different cases
with varying amplitude, displayed on the top of each panel, are shown
corresponding to those in Figure 1. The red curves indicate solutions for the
$m=0$ sausage mode and the blue curves show the $m=+1$ kink mode solutions.
The green curve in all cases shows the flow resonance locations modified by
the slow frequency as given by Equation (30). Dash-dotted lines indicate the
equilibrium characteristic kink speed, $c_{k}$ sound speed, $c$, and tube
speed, $c_{T}$, internal (subscript ‘i’) and external (subscript ‘e’) to the
flux tube. The dashed box region shown in Figure 2(c) is investigated in more
detail in Figure 5.
Figure 2 highlights the change in eigenvalues for the different cases of flow
profiles considered in Figure 1. For now, we focus on the forward propagating
wave modes ($\omega/k>0$). The axisymmetric $m=0$ sausage mode appears
unaffected, or at least not significantly affected, by the presence of the
background flow, which is a result of the $m=0$ mode being the axisymmetric
mode in the azimuthal direction. As a result of this, the rotational flow does
not break the symmetry of this mode. This can also be seen analytically, by
setting $m=0$ in Equations (6)-(15), the majority of terms containing the
presence of the (linear) background flow are removed when $m=0$. Furthermore,
when $m=0$, the resonant locations described by Equations (29) and (30) reduce
to the resonant positions corresponding to a uniform magnetic flux tube (Edwin
& Roberts, 1983). Whilst Equations (6)-(15) do contain some terms, for example
in the variables $Q$ and $T$, which are dependent on the background rotational
flow and that remain for the $m=0$ solution, we do not observe any significant
modification to the obtained eigenvalues for the sausage mode in Figure 2 when
compared to the analytical solutions for the uniform magnetic flux tube,
although very minor effects can be seen when the amplitude of the rotational
flow is sufficiently large ($A>0.25$). Investigating this result both
analytically and numerically may be a focus of future work.
Conversely, there is a considerable effect on the $m=1$ kink mode solutions
due to the presence of a background rotational flow. In the long wavelength
(thin-tube) limit, the phase speeds of the slow body and slow surface kink
modes tend to an infinite phase speed, similar to the case study of a linear
background magnetic twist (Erdélyi & Fedun, 2010), where they may even enter
the leaky regime. As the amplitude of the azimuthal flow is increased, the
corresponding phase speeds of the slow kink modes also increases for all
values of wavenumber. The flow resonance locations modified by the slow
frequency given by Equation (30) is shown by the green line in Figure 2. The
slow body and slow surface kink modes follow this resonance curve in the long
wavelength limit and even undergo an avoided crossing (Abdelatif, 1990; Mather
& Erdélyi, 2016; Allcock & Erdélyi, 2017) where the slow surface mode
approaches the fast surface mode. This avoided crossing implies a transfer of
properties between the fast and slow surface modes. In the long wavelength
limit it may not be appropriate to refer to the slow surface mode as a slow
mode anymore as, in the reference frame of the observer, it possesses phase
speeds similar to that of the fast surface mode. Therefore, using the observed
phase speeds alone may introduce some difficulty when distinguishing between
the fast and slow surface kink modes in rotating photospheric flux tubes. It
may be more appropriate to differentiate between the two modes by comparing
their eigenfunctions both parallel and perpendicular to the magnetic field,
which is discussed in more detail in Section 3.1.2. In the case of a uniform
photospheric cylinder, the slow body modes tend toward $c_{Ti}$ in the long
wavelength limit (Edwin & Roberts, 1983; Priest, 2014), however with the
inclusion of a background rotational flow, these modes now encounter the
modified flow resonance point where they may become resonantly damped.
Furthermore, the dispersive nature of the fast surface kink mode is also
modified by the presence of a background rotational flow. In a uniform
photospheric flux tube, the phase speed of the fast surface mode tends to the
kink speed in the long wavelength limit (Edwin & Roberts, 1983). However,
similar to the slow modes, the fast surface mode encounters the modified flow
resonance in the long-wavelength limit, where the phase speed of this mode
also appears to increase sharply depending on the amplitude of the flow. The
fast mode may also enter the leaky regime in the long-wavelength limit under
the assumed equilibrium configuration, which may be important in the context
of solar observations.
Of course, the kink mode possesses an azimuthal wavenumber which can be either
positive or negative. Granular buffeting in the lower solar photosphere due to
convective motions beneath the solar surface, may excite both $m=1$ and $m=-1$
kink modes. In the uniform cylinder model of a solar waveguide, the
‘traditional’ kink mode is considered to display signatures that resemble a
periodic transverse displacement of the waveguide. This is because, due to the
symmetry of the model, the opposite rates of rotation set up a standing wave
in the azimuthal direction. However, introducing a background rotational flow
breaks this azimuthal symmetry and, as a result, the $m=1$ and $m=-1$ modes
are expected to behave differently and the resulting observed mode will no
longer be a standing wave in the azimuthal direction. In Figure 3, we show the
obtained wave solutions for the $m=1$ and $m=-1$ modes in a photospheric flux
tube with a linear background rotational flow of amplitude $A=0.1$. For the
case when the kink wave is rotating with the flow, the phase speed of the fast
surface kink mode increases in the long-wavelength limit and enters the leaky
regime for small $ka$. This suggests that the fast kink surface mode, when
propagating in a thin waveguide with the background rotational flow, should
not be seen in observations of rotating photospheric structures, for example
in magnetic bright points or flux bundles rooted within intergranular lanes.
However, the obtained solutions for the fast surface mode are different for
the two cases when $m=1$ and $m=-1$. This is due to the $m=1$ mode rotating in
the same direction as the background flow, in our specific configuration.
Therefore, the $m=1$ mode constructively interferes with the background
rotational flow, increasing its phase velocity, whereas the $m=-1$ mode,
rotating against the flow, destructively superimposes with the flow, hence
lowering its phase velocity. The most notable difference between the case when
$m=1$ and $m=-1$ can be seen by the dispersive behaviour of the slow surface
and body modes. In both cases, the slow modes follow the trajectory of the
flow resonance locations modified by the slow frequency as given by Equation
(30), however for the case when $m=-1$, this curve decreases with decreasing
$ka$. As a result, the difference in phase speed for a given wavenumber
between $m=1$ and $m=-1$ is greater for the slow modes, with a more dramatic
difference seen as $ka$ approaches zero. This example of the phase speed
difference between the forward propagating slow body and surface modes, for
$m=1$ and $m=-1$, demonstrates how the background flow breaks the symmetry of
the kink mode.
Figure 3: The dispersion diagrams showing the obtained solutions for the $m=1$
and $m=-1$ modes denoted by the blue curves. The green curve shows the
behaviour of Equation (30) for both cases of $m=\pm 1$, respectively. The
respective zoom in plots highlight the difference in the behaviour of the
various category of modes for different values of $m$.
#### 3.1.2 Eigenfunctions
Figure 4: The resulting eigenfunctions for the slow surface kink mode ($m=1$)
for all linear cases of rotational flow with the colour scheme consistent with
Figure 1. A wavenumber value of $k=0.6$ was chosen for all plots.
Shown in Figure 4 are the spatial eigenfunctions for the slow surface kink
mode at a fixed wavenumber $ka=0.6$ for varying rotational flow amplitudes
with $m=1$. The colour scheme shown in the eigenfunctions is consistent with
that for the rotational flow profiles shown in Figure 1. It can be seen that
increasing the amplitude of the equilibrium linear rotational flow, changes
the spatial behaviour of the observable eigenfunctions. For the case of
$A=0.01$ which corresponds to a very small rotational flow parameter, the
eigenfunctions still obey a ‘surface-like’ structure, that is, the amplitude
of the radial displacement and velocity perturbations possesses a maximum at
the boundary of the flux tube and decays away from the boundary. However,
increasing the amplitude of the background rotational flow causes the radial
displacement perturbation to increase towards the centre of the flux tube,
such that the maximum displacement perturbation is no longer at the point
where $r=a$. This results in an eigenfunction that shares striking
similarities to that of the fundamental kink mode, and may therefore be
misinterpreted in observational data.
To further emphasise this point, it is possible to plot the eigenfunctions of
$\hat{P}_{T}$ and $\hat{\xi}_{r}$ for eigenvalues of a similar phase speed on
either side of the modified flow resonance point. One of these solutions
corresponds to the slow surface kink mode and the other is the fast surface
kink mode.
(a)
(b)
Figure 5: Panels showing the obtained wave solutions in a zoomed region
denoted by the dashed box in Figure 2(c). The eigenfunctions $\hat{P}_{T}$,
$\hat{\xi}_{r}$, $\hat{\xi}_{z}$ and $\hat{\xi}_{\varphi}$ for the (a) fast
and (b) slow magnetoacoustic surface kink mode solutions in a photospheric
flux tube with a background rotational flow given by $v_{\varphi}=0.1r$. Both
panels (a) and (b) are for the same $v_{ph}=\omega/k=1.3$ indicated by the
blue dot on the dispersion diagram. All plots are normalised such that the
external value of each eigenfunction is equal to unity at the boundary.
Shown in Figure 5 are these eigenfunctions for the slow and fast
magnetoacoustic kink modes at a similar phase speed. It can be seen that the
normalised eigenfunctions for $\hat{P}_{T}$, $\hat{\xi}_{r}$ and
$\hat{\xi}_{\varphi}$ are difficult to distinguish between the slow and the
fast surface modes. Furthermore, the modes no longer display the typical
characteristics of the surface mode anymore. In particular, the main
characteristic of a surface mode from the uniform cylinder model is that it
possesses a maximum amplitude of radial displacement perturbation at the
boundary of the waveguide, which is no longer the case when a rotational
background plasma flow is present. Investigating the nature of perturbations
parallel and perpendicular to the magnetic field may aid in distinguishing
between the slow and fast surface kink modes. Slow modes tend to propagate
(mainly) along the magnetic field lines, which in our study are straight and
vertical, so we should expect $\hat{\xi}_{z}$ to dominate for the slow mode
when compared to the fast mode, which may propagate at an angle across the
magnetic field. In Figure 5, we also show the normalised $z$ component of the
displacement perturbation for both the fast and slow surface kink modes. As
expected, $\hat{\xi}_{z}$ is dominant over $\hat{\xi}_{r}$ for the slow
surface mode (see e.g. Moreels & Van Doorsselaere, 2013), however, the same is
also true for the fast surface mode. Although, the absolute magnitude of the
normalised $z$ component is greater for the slow surface mode compared to the
fast surface modes, by roughly a factor of $2$, suggesting that the component
of displacement is still more dominant for the slow surface kink mode in a
rotating photospheric flux tube. It should be noted that, displayed in some
plots of the eigenfunctions, the amplitude of the resulting eigenfunctions
increases as we approach the center of the cylinder. This is a feature which
relates to the fact that there is a singularity in the set of Equations (16)
and (19) at $r=0$. However, this is purely an artifact of the plotting
technique and does not affect the eigensolver obtaining the solutions.
#### 3.1.3 2D and 3D velocity fields
Following the analysis from Section 2, the wave solutions are put proportional
to $\text{exp}\left[i\left(m\varphi+kz-\omega t\right)\right]$, therefore, it
is possible to convert the one dimensional radial eigenfunctions, for example
those shown in Figures 4 and 5, into two and three dimensional plots, to
better represent an observer’s perspective.
Figure 6: Snapshots of the $2$D velocity field for the scenario of a slow body
mode in a photospheric flux in the presence of a background flow given by
$v_{\varphi}=0.1r$. The plots are arranged in a $3\times 2$ configuration
where the left hand column shows the velocity perturbation only, whereas the
right hand column shows the perturbed velocity field plus the background
velocity field. The top row corresponds to the solution for the $m=1$ mode,
the middle row shows the solution for the $m=-1$ mode and the bottom row shows
the resulting velocity field and total pressure perturbation for the sum of
the $m=1$ and $m=-1$ modes. In all panels the velocity vectors are normalised
by their maximum values. The colour contour denotes the normalised total
pressure perturbation, $\hat{P}_{T}$, which is the same for both the left and
right columns. The boundary of the flux tube is highlighted by the solid blue
line. An animated movie of this Figure is available online.
In Figure 6, we show a snapshot of the two dimensional (converted into
Cartesian $x$ and $y$) velocity field for the perturbations alongside the
addition of the background flow onto the perturbations. This snapshot is
chosen specifically at the time when the $m=1$ and the $m=-1$ perturbations
effectively cancel each other such that the sum of the perturbed velocity
field is zero. Of course, due to the breaking of symmetry from the presence of
the background rotational flow, the boundary of the flux tube is no longer at
equilibrium, as the $m=1$ and $m=-1$ modes are slightly out of phase with one
another. In this snapshot, the perturbed total pressure, displayed by the
colour contour, can be seen to display a swirling characteristic, albeit with
a small magnitude, and is clearly visible in the bottom panels of Figure 6.
This swirling behaviour is present purely in the perturbations, signifying the
effect that a rotating waveguide has on the perturbed eigenfunctions. This
swirling behaviour of the total pressure perturbation may represent an
observational signature of the slow body kink mode propagating in a rotating
photospheric flux tube, although detecting this may be a challenge for
observers due to the small absolute value of magnitude. In the plots in the
right hand column of Figure 6, we add the background velocity field to the
perturbed velocity field. The background velocity field is given by
$v_{\varphi}=0.1r$ and acts counter-clockwise in the $xy$ plane. The addition
of the background velocity field completely changes the observed distribution
of the velocity field for the slow body kink mode. In the case of a uniform
cylinder, the velocity field for the slow body kink mode can be seen emanating
from two ‘islands’, which correspond to two anti-nodes in the total pressure
eigenfunction. This can be seen in the left hand column showing the
perturbations only for the case of $m=1$ and $m=-1$, however, when the
background flow is added, the velocity field no longer displays these typical
characteristics. It should be noted that both the perturbed velocity field and
the background velocity field are normalised separately to their respective
maximum values to aid these visualisations, as it is to be expected that the
perturbed components will be significantly smaller in absolute value than the
background quantities.
Figure 7: Same as Figure 6 but a snapshot at a later time. An animated movie
of this Figure is available online.
In Figure 7, we show the same quantities as those displayed in Figure 6 at a
later time. As we have shown in Figure 3, there is a significant difference in
the phase speed for the $m=1$ and $m=-1$ forward propagating slow body waves.
Therefore, we would expect these perturbations to become increasingly out of
phase with one another as time progresses, a behaviour which is displayed in
Figure 7. The two modes becoming out of phase with one another can be seen by
comparing the top row and the middle rows of Figure 7 (which display the $m=1$
and $m=-1$ modes respectively). For the case of a uniform cylinder with no
background rotational flow, we would expect that the $m=1$ and $m=-1$ modes
would rotate in opposite directions but perfectly in phase with one another,
however, this is no longer true when a rotational background flow is added. As
a result of one mode rotating with the background flow (in our case this is
the $m=1$ mode), and the other mode propagating against the background flow
($m=-1$ mode), the sum of the two modes results in a total pressure
perturbation which is also propagating (rotating) in the azimuthal direction.
Furthermore, the boundary of the flux tube can be seen to rotate (see
associated online movie). The observed rotating motion when combining the
individual $m=1$ and $m=-1$ wave modes presents another observational
signature that can be sought to identify the kink mode in the presence of a
background rotational flow.
(a)
(b)
Figure 8: This figure shows the three dimensional structure of the slow body
kink mode for (a) a uniform magnetic flux tube without the presence of a
background flow and (b) a magnetic flux tube in the presence of a background
rotational flow, by visualising the normalised total pressure perturbation
$\hat{P}_{T}$. On the left panels, the magnetic flux tube is immersed in the
volume rendering of $\hat{P}_{T}$. The three cross-sectional cuts (shown as
coloured rings at three different heights, $z=0.0,2.5$ and $5.0$) correspond
to the corresponding right subplots. These three subplots show the LIC
visualisation at the same heights. The white rings represent the boundary of
the flux tube. An animated version of this Figure is available online.
Figure 8 shows a three dimensional visualisation of how the sum of the $m=\pm
1$ slow body kink mode manifests itself in a uniform magnetic flux tube and
also for the case of a magnetic flux tube in the presence of a background
rotational flow. Figure 8 also displays a Line Integral Contour (LIC) plot at
three different heights in the flux tube which are denoted by white rings in
the 3D plot. A LIC visualisation aids imaging the velocity field of the plasma
(see, e.g. Cabral & Leedom, 1993). In other words, LIC visualisations display
the path lines that an object would follow if it were placed into the fluid
with a stationary velocity field. It can be seen that the LIC for the slow
body kink mode in a uniform magnetic flux tube (Figure 8(a)) remains unchanged
with height, as the wave propagates and the structure oscillates in one plane,
in other words it is linearly polarised. In addition, the LIC also displays a
‘double X-point’ close to $x=0$ and $y=\pm 1$, near the boundary in the plane
in which the structure does not oscillate. This behaviour is due to the
rotational motions that are inherently associated with the non-axisymmetric
kink mode (Goossens et al., 2014). However, once a rotational flow is present,
the observed evolution of the flux tube motion is affected. Firstly, we can
observe that the three dimensional behaviour of the total pressure (density)
perturbation begins to represent a helical structure, in contrast to the
sinusoidal oscillation in the uniform case, as the perturbation now also
displays a rotational behaviour. This suggests that the resulting kink wave is
circularly polarised, as a result of the $\pm 1$ modes becoming out of phase
with one another. Therefore, we show that, in the presence of a background
rotational flow, the kink mode may manifest itself as a circularly polarised
mode, similar to a twisted magnetic flux tube (Terradas & Goossens, 2012;
Ruderman & Terradas, 2015).
Furthermore, the LIC visualisations show that the velocity field evolution is
modified in the presence of a background rotational flow. Firstly, the ‘double
X-point’ feature seen in the LIC for the uniform flux tube no longer displays
the same characteristics in the scenario including a background rotational
flow. Instead, the ‘X-points’ can be seen at different locations at certain
times as the wave propagates upwards. Secondly, the evolution of the velocity
field is no longer constant and does not appear to oscillate in a plane on a
single axis anymore. Similar to the behaviour seen in Figures 6 and 7, the LIC
visualisation shows that the velocity field also rotates as it evolves, and
the traditional signatures of the kink mode in a uniform flux tube are no
longer present. These features could possibly be retrieved from numerical
simulations of kink waves in rotating structures.
## 4 Conclusions
In the present study, we have extended previous studies investigating the
properties of MHD waves in photospheric waveguides, by introducing a linear
azimuthal component to the background velocity field. In order to conduct this
investigation, we utilised a previously developed numerical eigensolver
(Skirvin et al., 2021; Skirvin et al., 2022) to model a rotating flux tube in
a photospheric environment with a background $v_{\varphi}$ component. For the
inclusion of a linear rotational flow, very similar results to those of the
linear magnetic twist were recovered (Erdélyi & Fedun, 2010). We find that the
obtained kink mode solutions possess phase speeds along the magnetic field
which tend to infinite values in the long wavelength limit where they may
become leaky. We have shown this to be a result of the modes being guided by
the trajectory of the flow resonance location modified by the slow frequency.
As a result of considering a linear background flow, the equations describing
the continua regions reduce to single point locations. Therefore, there
becomes a point where, under photospheric conditions, the slow surface kink
mode and the slow body kink mode approach the same solution at the resonant
point and undergo an avoided crossing where their properties become mixed.
Comparison of the eigenfunctions for the slow surface kink mode and fast
surface kink mode, in the presence of a linear background rotational flow at
similar phase speeds, indicated that identification of the two modes becomes
extremely difficult in the long wavelength limit using the total pressure and
radial displacement perturbations. However, comparison of $\hat{\xi}_{z}$
shows that this component is still dominant for slow modes, comparable to
uniform theory. Furthermore, we find that the axisymmetric $m=0$ sausage mode
remains unaffected by any background azimuthal component. Analytically this
can be understood by examining the governing set of Equations (6)-(15) when
setting $m=0$ and noticing many terms either simplifying or disappearing
altogether. In addition, it suggests that sausage mode observations in the
lower solar atmosphere in e.g. pores and sunspots, may not be a suitable wave
mode to conduct atmospheric-seismology, if the structure is in the presence of
any magnetic twist or background rotational flows.
We have also presented 2D plots showing the velocity field with and without
the inclusion of the background flow (Figures 6-7) and 3D plots showing the
perturbation of the normalised total pressure $\hat{P}_{T}$ (Figure 8). These
plots clearly show the breaking of symmetry between the $m=1$ and $m=-1$ kink
modes when a flux tube is in the presence of a rotational background flow.
Furthermore, inspection of the total pressure perturbations shows that it
displays signatures of a swirling motion when the flux tube experiences its
maximum displacement. It is hoped that these visualisations will aid future
interpretations of observational data from high resolution instruments on
state of the art telescopes such as DKIST. Finally, we have also produced
three dimensional visualisations of the kink mode propagating in a
photospheric flux tube with a linear background rotational flow incorporated.
We have also shown, in Figure 8, the LIC visualisations at different heights
displaying the evolution of the velocity field as the wave propagates along
the magnetic field, which may be retrieved from numerical simulations.
In this study, we have suggested potential observational signatures of
magnetoacoustic kink modes in photospheric flux tubes with background
rotational flows. These signatures share striking characteristics with
previous observations of circularly polarised kink modes in the lower solar
atmosphere (Jess et al., 2017; Stangalini et al., 2017). However, naturally,
the magnitude of the background flows are much greater than those of the
perturbations arising from the waves, which then raises the important question
of how to actually observe this phenomena in the solar atmosphere. Both the
rotational motion of the waveguide combined with the swirling pattern seen in
the total pressure perturbation are observational signatures that may be
detected with current and future telescopes, however distinguishing between
the contribution of the background and perturbed components may be
challenging.
Furthermore, we have focused only on a magnetic flux tube in a photospheric
environment. Under the photospheric conditions adopted in this work, the
Alfvén continuum given by Equation (22) exists in the leaky regime at phase
speeds above the cut-off speed at $v_{ph}=c_{e}$. Therefore this resonant
region is not discussed in detail in this work, however, it will become
important for studies under coronal conditions, where contributions from both
Equations (22) and (23) become important. In the solar atmosphere, vortices
can be buffeted from multiple directions, for example convective motion due to
convection cells. In this study, we have considered just one fixed driver that
oscillates from side to side, however in reality the picture will be more
complicated due to drivers in additional directions. These perturbations along
different axis will manifest themselves as apparent rotation, regardless of a
background rotational flow, this is work to be investigated in the future.
Additional future work includes a closer inspection on the effect that
rotational flows have on the sausage mode. We have shown that the sausage mode
may be affected by the presence of background rotational flow in magnetic flux
tubes, however, more insight is required to quantify exactly how the sausage
mode is affected and how it manifests itself in rotating flux tubes. Finally,
future work to be investigated includes modelling a background rotational flow
with a realistic radial profile. For example Silva et al. (2020) have shown,
using information from MURAM simulations, that the radial profile of the
azimuthal velocity component in solar vortex tubes can be accurately modelled
with a cubic polynomial. In this case, the background rotational flow becomes
non-linear, such that the value of parameter $\alpha$ in Equation (24) no
longer equals $1$. We expect that in this example, the MHD spectrum is densely
occupied by the (modified) flow continua, consequently, trapped modes may find
it difficult to exist. Obtaining the eigenvalues that lie inside the continua
can be achieved by modifying the numerical eigensolver employed in this work
or by using an alternative 1D eigensolver such as Legolas (Claes et al.,
2020). Modelling a non-linear background rotational flow such as this is an
objective for future studies.
## Acknowledgements
This work has been supported by STFC (UK). SJS is grateful to STFC for the PhD
studentship project reference (2135820). VF, GV, and SSAS are grateful to
Science and Technology Facilities Council (STFC) grant ST/V000977/1, and The
Royal Society, International Exchanges Scheme, collaboration with Brazil
(IES191114) and Chile (IE170301). TVD was supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 724326) and the C1 grant TRACEspace of Internal
Funds KU Leuven. The results received support from the FWO senior research
project with number G088021N. SJS and VF would like to thank the International
Space Science Institute (ISSI) in Bern, Switzerland, for the hospitality
provided to the members of the team on ‘The Nature and Physics of Vortex Flows
in Solar Plasmas’. SJS and GV wish to acknowledge scientific discussions with
the Waves in the Lower Solar Atmosphere (WaLSA; https://www.WaLSA.team) team,
which is supported by the Research Council of Norway (project no. 262622).
This research has also received financial support from the ISEE, International
Joint Research Program (Nagoya University, Japan) and the European Union’s
Horizon 2020 research and innovation program under grant agreement No. 824135
(SOLARNET).
Software: Numpy (Harris et al., 2020), matplotlib (Hunter, 2007), SciPy
(Virtanen et al., 2020). The numerical code, Sheffield Dispersion Diagram Code
(SDDC) introduced and applied in this work is available on the Plasma Dynamics
Group (PDG) website 111https://sites.google.com/sheffield.ac.uk/pdg/solar-
codes?authuser=0 along with the user manual which explains some cases shown in
this work. This code and the accompanying tools have been developed using
Python an open-source and community-developed programming language.
## Data Availability
Accessibility of data used in this research is available upon request from the
authors.
## References
* Abdelatif (1990) Abdelatif T. E., 1990, Sol. Phys., 129, 201
* Aldhafeeri et al. (2021) Aldhafeeri A. A., Verth G., Brevis W., Jess D. B., McMurdo M., Fedun V., 2021, ApJ, 912, 50
* Allcock & Erdélyi (2017) Allcock M., Erdélyi R., 2017, Sol. Phys., 292, 35
* Andries et al. (2000) Andries J., Tirry W. J., Goossens M., 2000, ApJ, 531, 561
* Aschwanden et al. (1999) Aschwanden M. J., Fletcher L., Schrijver C. J., Alexander D., 1999, ApJ, 520, 880
* Bate et al. (2022) Bate W., et al., 2022, ApJ, 930, 129
* Bonet et al. (2008) Bonet J. A., Márquez I., Sánchez Almeida J., Cabello I., Domingo V., 2008, ApJ, 687, L131
* Bonet et al. (2010) Bonet J. A., et al., 2010, ApJ, 723, L139
* Cabral & Leedom (1993) Cabral B., Leedom L. C., 1993, in Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH ’93. Association for Computing Machinery, New York, NY, USA, p. 263–270, doi:10.1145/166117.166151, https://doi.org/10.1145/166117.166151
* Cheremnykh et al. (2018) Cheremnykh O., Fedun V., Ladikov-Roev Y., Verth G., 2018, ApJ, 866, 86
* Claes et al. (2020) Claes N., De Jonghe J., Keppens R., 2020, ApJS, 251, 25
* De Groof & Goossens (2000) De Groof A., Goossens M., 2000, A&A, 356, 724
* De Pontieu et al. (2004) De Pontieu B., Erdélyi R., James S. P., 2004, Nature, 430, 536
* De Pontieu et al. (2007) De Pontieu B., et al., 2007, Science, 318, 1574
* Edwin & Roberts (1983) Edwin P. M., Roberts B., 1983, Sol. Phys., 88, 179
* Erdélyi & Fedun (2007) Erdélyi R., Fedun V., 2007, Sol. Phys., 246, 101
* Erdélyi & Fedun (2010) Erdélyi R., Fedun V., 2010, Sol. Phys., 263, 63
* Fedun et al. (2011a) Fedun V., Shelyag S., Verth G., Mathioudakis M., Erdélyi R., 2011a, Annales Geophysicae, 29, 1029
* Fedun et al. (2011b) Fedun V., Shelyag S., Erdélyi R., 2011b, ApJ, 727, 17
* Geeraerts et al. (2022) Geeraerts M., Vanmechelen P., Van Doorsselaere T., Soler R., 2022, arXiv e-prints, p. arXiv:2203.02359
* Goedbloed & Poedts (2004) Goedbloed J. P. H., Poedts S., 2004, Principles of Magnetohydrodynamics
* González-Avilés et al. (2017) González-Avilés J. J., Guzmán F. S., Fedun V., 2017, ApJ, 836, 24
* González-Avilés et al. (2018) González-Avilés J. J., Guzmán F. S., Fedun V., Verth G., Shelyag S., Regnier S., 2018, ApJ, 856, 176
* Goossens et al. (1992) Goossens M., Hollweg J. V., Sakurai T., 1992, Sol. Phys., 138, 233
* Goossens et al. (2011) Goossens M., Erdélyi R., Ruderman M. S., 2011, Space Sci. Rev., 158, 289
* Goossens et al. (2014) Goossens M., Soler R., Terradas J., Van Doorsselaere T., Verth G., 2014, ApJ, 788, 9
* Grant et al. (2015) Grant S. D. T., et al., 2015, ApJ, 806, 132
* Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
* Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
* Jess et al. (2017) Jess D. B., et al., 2017, ApJ, 842, 59
* Keys et al. (2018) Keys P. H., et al., 2018, ApJ, 857, 28
* Magyar et al. (2022) Magyar N., Duckenfield T., Van Doorsselaere T., Nakariakov V. M., 2022, A&A, 659, A73
* Mather & Erdélyi (2016) Mather J. F., Erdélyi R., 2016, ApJ, 822, 116
* Moreels & Van Doorsselaere (2013) Moreels M. G., Van Doorsselaere T., 2013, A&A, 551, A137
* Morton et al. (2015) Morton R. J., Tomczyk S., Pinto R., 2015, Nature Communications, 6, 7813
* Nakariakov & Roberts (1995) Nakariakov V. M., Roberts B., 1995, Sol. Phys., 159, 213
* Nakariakov et al. (1999) Nakariakov V. M., Ofman L., Deluca E. E., Roberts B., Davila J. M., 1999, Science, 285, 862
* Pinter et al. (1998) Pinter B., Cadez V. M., Goossens M., 1998, A&A, 332, 775
* Priest (2014) Priest E. R., 2014, Magnetohydrodynamics of the Sun
* Rouppe van der Voort et al. (2007) Rouppe van der Voort L. H. M., De Pontieu B., Hansteen V. H., Carlsson M., van Noort M., 2007, ApJ, 660, L169
* Ruderman & Petrukhin (2022) Ruderman M. S., Petrukhin N. S., 2022, Sol. Phys., 297, 72
* Ruderman & Terradas (2015) Ruderman M. S., Terradas J., 2015, A&A, 580, A57
* Sakurai et al. (1991) Sakurai T., Goossens M., Hollweg J. V., 1991, Sol. Phys., 133, 227
* Scullion et al. (2011) Scullion E., Erdélyi R., Fedun V., Doyle J. G., 2011, ApJ, 743, 14
* Shelyag et al. (2011) Shelyag S., Fedun V., Keenan F. P., Erdélyi R., Mathioudakis M., 2011, Annales Geophysicae, 29, 883
* Shelyag et al. (2012) Shelyag S., Mathioudakis M., Keenan F. P., 2012, ApJ, 753, L22
* Shelyag et al. (2013) Shelyag S., Cally P. S., Reid A., Mathioudakis M., 2013, ApJ, 776, L4
* Shetye et al. (2019) Shetye J., Verwichte E., Stangalini M., Judge P. G., Doyle J. G., Arber T., Scullion E., Wedemeyer S., 2019, ApJ, 881, 83
* Silva et al. (2020) Silva S. S. A., Fedun V., Verth G., Rempel E. L., Shelyag S., 2020, ApJ, 898, 137
* Silva et al. (2021) Silva S. S. A., Verth G., Rempel E. L., Shelyag S., Schiavo L. A. C. A., Fedun V., 2021, ApJ, 915, 24
* Skirvin et al. (2021) Skirvin S. J., Fedun V., Verth G., 2021, MNRAS, 504, 4077
* Skirvin et al. (2022) Skirvin S. J., Fedun V., Silva S. S. A., Verth G., 2022, MNRAS, 510, 2689
* Snow et al. (2018) Snow B., Fedun V., Gent F. A., Verth G., Erdélyi R., 2018, ApJ, 857, 125
* Soler et al. (2009) Soler R., Oliver R., Ballester J. L., 2009, ApJ, 693, 1601
* Soler et al. (2010) Soler R., Terradas J., Oliver R., Ballester J. L., Goossens M., 2010, ApJ, 712, 875
* Soler et al. (2011) Soler R., Terradas J., Goossens M., 2011, ApJ, 734, 80
* Spruit & Zweibel (1979) Spruit H. C., Zweibel E. G., 1979, Sol. Phys., 62, 15
* Stangalini et al. (2017) Stangalini M., et al., 2017, ApJ, 840, 19
* Stangalini et al. (2022) Stangalini M., et al., 2022, Nature Communications, 13, 479
* Taroyan & Erdélyi (2002) Taroyan Y., Erdélyi R., 2002, Physics of Plasmas, 9, 3121
* Taroyan & Erdélyi (2003) Taroyan Y., Erdélyi R., 2003, Physics of Plasmas, 10, 266
* Terra-Homem et al. (2003) Terra-Homem M., Erdélyi R., Ballai I., 2003, Sol. Phys., 217, 199
* Terradas & Goossens (2012) Terradas J., Goossens M., 2012, A&A, 548, A112
* Tirry & Goossens (1996) Tirry W. J., Goossens M., 1996, ApJ, 471, 501
* Tomczyk et al. (2007) Tomczyk S., McIntosh S. W., Keil S. L., Judge P. G., Schad T., Seeley D. H., Edmondson J., 2007, Science, 317, 1192
* Tziotziou et al. (2018) Tziotziou K., Tsiropoula G., Kontogiannis I., Scullion E., Doyle J. G., 2018, A&A, 618, A51
* Van Doorsselaere et al. (2004) Van Doorsselaere T., Debosscher A., Andries J., Poedts S., 2004, A&A, 424, 1065
* Verth et al. (2007) Verth G., Van Doorsselaere T., Erdélyi R., Goossens M., 2007, A&A, 475, 341
* Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
* Wedemeyer-Böhm et al. (2012) Wedemeyer-Böhm S., Scullion E., Steiner O., Rouppe van der Voort L., de La Cruz Rodriguez J., Fedun V., Erdélyi R., 2012, Nature, 486, 505
* Wilson (1979) Wilson P. R., 1979, A&A, 71, 9
* Yadav et al. (2020) Yadav N., Cameron R. H., Solanki S. K., 2020, ApJ, 894, L17
* Yadav et al. (2021) Yadav N., Cameron R. H., Solanki S. K., 2021, A&A, 645, A3
* Zaqarashvili et al. (2015) Zaqarashvili T. V., Zhelyazkov I., Ofman L., 2015, ApJ, 813, 123
* Zhelyazkov & Chandra (2019) Zhelyazkov I., Chandra R., 2019, Sol. Phys., 294, 20
|
> Proceedings of the
> 2nd International Workshop on
> Reading Music Systems
>
> 2nd November, 2019
> Delft, The Netherlands
## Organization
General Chairs
Jorge Calvo-Zaragoza | | University of Alicante, Spain
---|---|---
Alexander Pacha | | TU Wien, Austria
Heinz Roggenkemper | | Canemus, United States
Program Committee
Andreas Arzt | | Johannes Kepler Universität, Austria
---|---|---
Jürgen Diet | | Bavarian State Library, Germany
Ichiro Fujinaga | | McGill University, Canada
Jose M. Iñesta | | University of Alicante, Spain
Gabriel Vigliensoni | | McGill University, Canada
Proceedings of the 2nd International Workshop on Reading Music Systems, Delft,
2019
Edited by Jorge Calvo-Zaragoza and Alexander Pacha
© The respective authors.
Licensed under a Creative Commons Attribution 4.0 International License (CC-
BY-4.0).
Logo made by Freepik from www.flaticon.com. Adapted by Alexander Pacha.
## Preface
Dear colleagues,
it is our greatest pleasure to introduce the proceedings of the 2nd
International Workshop on Music Reading Systems (WoRMS). After the first
successful installation last year, we are more than happy to have another
edition of a workshop that brings together researchers and practitioners that
work on reading music systems.
This workshop has gained global interest with submissions from all around the
world. Similar to last year, we have sessions on both technical approaches as
well as applications that can be used by end-users or other researchers to
build upon. What is even more pleasant is the increase in open-source software
that seems to become the scientific norm.
Following the suggestions from last year, the workshop will provide the
opportunity for discussions right after each presentation, as well as in a
panel after each session. This way we hope to drive the interactive character
of WoRMS, while giving sufficient time to present ongoing research.
Finally, we want to thank Jaehun Kim and Ginny Ruiter who assisted us in
organizing the venue.
Jorge Calvo-Zaragoza and Alexander Pacha
###### Contents
1. Preface
See pages - of papers/WoRMS_2019_paper_7.pdf
See pages - of papers/WoRMS_2019_paper_1.pdf
See pages - of papers/WoRMS_2019_paper_8.pdf
See pages - of papers/WoRMS_2019_paper_9.pdf
See pages - of papers/WoRMS_2019_paper_6.pdf
See pages - of papers/WoRMS_2019_paper_4.pdf
See pages - of papers/WoRMS_2019_paper_3.pdf
|
# SPOT: Secure and Privacy-preserving prOximiTy protocol for e-healthcare
systems
Souha Masmoudi Samovar, Telecom SudParis,
Institut Polytechnique de Paris
Palaiseau, France
Member of the Chair Values and
Policies of Personal Information
ORCID: 0000-0002-7194-8240 Nesrine Kaaniche Samovar, Telecom SudParis,
Institut Polytechnique de Paris
Palaiseau, France
Member of the Chair Values and
Policies of Personal Information
ORCID: 0000-0002-1045-6445 Maryline Laurent Samovar, Telecom SudParis,
Institut Polytechnique de Paris
Palaiseau, France
Member of the Chair Values and
Policies of Personal Information
ORCID: 0000-0002-7256-3721
###### Abstract
This paper introduces SPOT, a Secure and Privacy-preserving prOximity based
protocol for e-healthcare systems. It relies on a distributed proxy-based
approach to preserve users’ privacy and a semi-trusted computing server to
ensure data consistency and integrity. The proposed protocol ensures a balance
between security, privacy and scalability. As far as we know, in terms of
security, SPOT is the first one to prevent malicious users from colluding and
generating false positives. In terms of privacy, SPOT supports both anonymity
of users being in proximity of infected people and unlinkability of contact
information issued by the same user. A concrete construction based on
structure-preserving signatures and NIWI proofs is proposed and a detailed
security and privacy analysis proves that SPOT is secure under standard
assumptions. In terms of scalability, SPOT’s procedures and algorithms are
implemented to show its efficiency and practical usability with acceptable
computation and communication overhead.
###### Index Terms:
Anonymity, e-healthcare, NIWI proofs, Privacy, Structure-preserving signature,
Unlinkability.
## I Introduction
The recent health crisis has led governments to apply different tracing
solutions to control the contamination chain among the population. These
solutions are aimed at sharing valuable data while preserving users’ privacy.
However, there are still privacy threats and robustness challenges as long as
users are required to disclose and share correct sensitive and personal
information with different third parties with various levels of trust.
Most of the solutions rely on the Bluetooth technology, namely Bluetooth Low
Energy (BLE), to exchange contact information, thanks to its efficiency in
active communications [17]. Among these governmental solutions, the
TraceTogether application [11] has been launched by Singapore. TraceTogether
enables to collect, via the Bluetooth technology, temporary IDs (generated by
a central trusted server) of users in close proximity. Collected IDs are
stored in an encrypted form using the server’s public key, at users’ devices,
and in case of infection, they are shared with the server. The COVIDSafe
application [6] from the Australian government is another Bluetooth-based
solution. It also logs encrypted users’ contact information, and share them
once an infection is detected. The server is required to alert users at risk
without revealing the identity of the infected user. Both TraceTogether and
COVIDSafe applications are set upon a centralized architecture. Many other
applications like Stop COVID-19 (Croatia) [3], CA Notify (California) [1] rely
on the Google and Apple Exposure Notification (GAEN) service [5], which is set
upon a decentralized architecture. Although contact tracing applications have
helped governments to alleviate the widespread of the pandemic by automating
the manual contact tracing done by health authorities, they raise critical
privacy concerns, namely users tracking and identification [15].
Academic solutions have been also proposed to support both centralized [13]
and decentralized [7, 8, 18, 14, 16] architectures. However, each architecture
has his merits and limits in terms of security and privacy. Indeed, using
centralized solutions, users guarantee the reception of correct alerts as long
as the generation of users’ contact tokens and the verification in case of
infection are performed by a centralized server. This guarantee is compensated
with threats to users’ privacy, i.e., users are exposed to tracking and
identification of their contact lists. Decentralized solutions have been
proposed to mitigate these privacy threats. Users are responsible for
generating their contact tokens in order to ensure their privacy and
anonymity, but, they are not prevented from forging contact information, which
results in high level of false alerts. To get the best of both architectures,
hybrid architecture based solutions [9, 12] have been proposed. However,
security and privacy requirements, like the correctness of contact information
and the anonymity of contacted users, are not yet handled and ensured
together.
In this paper, we present SPOT, a novel hybrid Secure and Privacy-preserving
prOximity-based protocol for e-healthcare systems. It combines a decentralized
proxy front-end architecture, ensuring both users’ anonymity and contact
information integrity, and a centralized back-end computing server,
guaranteeing a real time verification of contact information integrity. SPOT
assumes that two users in close proximity rely on their Ephemeral Bluetooth
IDentifiers (EBID) to compute a common contact message. This message is
relayed to a central server through a group of proxies. With the help of the
computing server and relying on a proof-based group signature, SPOT prevents
users from forging their contact lists. The signed contact messages are given
to the user to be locally stored. In case of a detected infection, the user
consents to share his contact list, i.e., a set of signed contact messages,
with the health authority. This latter checks the correctness of the received
list and shared it back with all the involved users, if the verification
holds. The contributions of this paper are summarized as follows:
* •
we design a proximity protocol for e-health services that prevents the
injection of false positives, i.e., alert users to be at risk when they are
not. SPOT enforces the verification of the correctness and the integrity of
users’ contact information by health authorities, thanks to the support of
both a computing server and a group of proxies.
* •
we guarantee strong privacy properties namely the anonymity of users being in
contact with infected people, and the unlinkability of users’ transactions
when relying on random EBIDs that can neither be linked to each other nor be
linked to their issuer.
* •
we propose a concrete construction of the SPOT protocol relying on a
structure-preserving signature scheme [4] that supports securely signing
group’s elements, i.e., contacts’ information. This signature scheme is
coupled with Groth-Sahai Non-Interactive Witness-Indistinguishable (NIWI)
proof [10] in order to ensure integrity of proxies’ keys. Indeed, NIWI proofs
guarantee the anonymity of proxies while the health authority is still able to
verify that proxies are trustful by verifying the validity of their keys
without having access to them.
* •
we evaluate the performances of SPOT through the full implementation of
different procedures and algorithms. The conducted experiments have shown
acceptable computation times proving the efficiency and practical usability of
the proposed solution.
This paper is organized as follows. Section II introduces and compares most
closely-related proximity tracing algorithms and solutions to SPOT. Section
III gives an overview of SPOT. After introducing the underlying building
blocks in Section IV The concrete construction of the proposed SPOT protocol
is presented in Section V. Section VI and Section VII provide security and
privacy properties and a detailed discussion of SPOT’s conducted experiments,
before concluding in Section VIII.
## II Related Work
Recently, several industrial and research contact tracing solutions have been
proposed for e-health applications [19, 17]. These solutions aim at ensuring
security properties, namely anti-replay mitigating the multi-submission of the
same contact information, and unforgeability preventing malicious
entities111Malicious entities involve either a single malicious adversary or
colluding adversaries. from threatening data integrity. Privacy properties
have been significantly addressed, including the anonymity of end-users and
the unlinkability between their different transactions (i.e., a formal
definition of security and privacy requirements is given in Section III-D).
Indeed, researchers from Inria, France, and Fraunhofer, Germany proposed
Robert [13], a contact tracing protocol that relies on a centralized
architecture, where a central server delivers pseudonyms to users. Each user
collects pseudonyms of users in close proximity and shares them with the
server when being infected. In such centralized solution, users are sure that
they receive correct alerts (i.e., collected pseudonyms are neither replayed
nor falsified by malicious entities), however, their privacy is threatened as
long as the server is able to identify users’ contacts and to track them. In
[7], Troncoso _et al._ introduced the Decentralized Privacy-Preserving
Proximity Tracing (DP-3T) solution which is one of the most popular contact-
tracing protocols. DP-3T has been proved to mitigate the privacy threats of
centralized solutions as there is no need for a central entity which collects
users’ contact information with the risk of tracking them. However, it does
not prevent relay and replay attacks and gives no mean to verify the
correctness of contact information. Thus, users are exposed to false alerts
from malicious entities either by creating falsified information or replaying
information of previous sessions. Afterwards, Castelluccia _et al._ proposed
Desire [9], a proximity tracing protocol that leverages the advantages of the
centralized and decentralized solutions. However, some security and privacy
issues have not been considered in this solution. First malicious users are
able to collude and merge their contact lists, which leads to false positive
injection attacks. Second, the server requested to compute the exposure status
and risk, is able to link users’ requests, and to de-anonymize them. Two very
similar proposals called PACTs are also introduced. The east coast PACT [18]
and the west coast PACT [8] are very close to DP-3T. The two solutions rely on
random pseudonyms derived from a private seed, that are broadcasted to users
in proximity via Bluetooth. The pseudonyms are generated using cryptographic
pseudorandom generators and pseudorandom functions. Apart from the non-
resistance against replay attacks, these two proposals give no mean to check
the correctness of the contact information before being broadcasted. Pietrzak
[16] proposed a decentralized contact-tracing solution to mitigate replay
attacks against DP-3T. However, privacy concerns are raised, namely tracking
users, as geo-location and time of contacts are shared within the Bluetooth
message. In [12], Hoepman proposed two tracing protocols, the first one relies
on an interactive session between two users in proximity to register contact
information. If the interaction fails, the contact is not registered. Thus,
the second protocol comes to mitigate this risk of failure and relies on an
authority that relays information between users. In both protocols, the
identities of users who have been in contact with an infected person, are
revealed to a central entity, which contradicts the defined anonymity
requirement. Liu et al. [14] use zero-knowledge proofs and group signatures in
order to preserve users’ privacy for their proposed tracing protocol. Zero-
knowledge proofs are generated by users over the contact information they
collected. Indeed, users prove the contacts to their doctors without revealing
the information. Afterwards, doctors, being members of a group, sign the
proofs and publish them in a public board. Then, relying on their secrets,
users can check if they were in contact with infected people. As such, no
entity can identify the contacts of an infected user. However, based on a long
interactive protocol between two devices, the collection of contacts’
information may result in a failed interaction, thus causing the non-
registration of the contact. Furthermore, the authors only consider honest but
curious adversaries, which leads to possible false alerts due to malicious
colluding users. It is also worth noticing that, unlike SPOT and other related
work [9, 7, 16], [14] assume that all the computations (handshake, zero-
knowledge, verification) are performed by the user’s device, which leads to
device’s battery depletion. A new contact tracing strategy based on online
social networking is proposed in [20] but does not provide privacy guarantees.
Table I provides a comparative summary between SPOT and related works in terms
of architecture settings and properties. As shown, SPOT is the only solution
which supports strong security and privacy requirements.
TABLE I: Comparison between SPOT and related works
| | SPOT | [14] | [12] | [7] | [18] [8] | [13] | [9] | [16]
---|---|---|---|---|---|---|---|---|---
Architecture | Centralized | - | - | - | - | - | ✓ | - | -
Decentralized | - | ✓ | - | ✓ | ✓ | - | - | ✓
Semi-centralized | ✓ | - | ✓ | - | - | - | ✓ | -
Properties | Unforgeability a | ✓ | ✓b | ✓b | ✗ | ✗ | N.A. | ✗ | ✗
Anti-replay | ✓ | ✓ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓
Unlinkability | ✓ | N.A. | ✗ | ✓ | ✓ | ✗ | ✓ | ✗
Anonymity | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗
NOTE: N.A. is the abbreviation for Not Applicable; a indicates that the
unforgeability implies the integrity of users’ contact information and the
prevention of false positives injection; a indicates that unforgeability is
partially satisfied while not considering malicious colluding entities.
## III SPOT Overview
This section first presents the involved entities and gives an overview of
SPOT. Then, it details the system model with its procedures and algorithms and
defines the threat model through formal security games.
### III-A Entities
Figure 1: Overview of the SPOT protocol
Figure 1 depicts the four main actors involved in SPOT with their interactions
according to the different phases. Four actors are defined as follows:
* •
The user ($\mathcal{U}$) represents the entity that owns the device where the
proximity-tracing application is installed. During the Generation phase,
$\mathcal{U}$ broadcasts his EBID (Ephemeral Bluetooth IDentifier), collects
the EBIDs of other users in proximity, and computes a common contact message
shared between each two users being in contact. $\mathcal{U}$ wants to receive
alerts if he was in contact with confirmed cases.
* •
The Health Authority ($\mathcal{HA}$) is responsible for issuing users’
identifiers during the Sys_Init phase, and for checking the correctness of the
contact messages provided by an infected user during the Verification phase.
* •
The Server ($\mathcal{S}$) is responsible for anonymously collecting and
storing users’ contact messages relayed by proxies during the Generation
phase222The Server can be distributed by considering one or several servers
per geographical area, each server participating in locally storing part of
users’ contact messages databases. All the parts are then collected on offline
in a centralized server. Thus, for ease of presentation, we consider only one
server.. $\mathcal{S}$ performs a real-time verification of the received
contacts during the Generation phase, in order to help $\mathcal{HA}$ to
verify the correctness and integrity of the contact messages.
* •
The Proxy ($\mathcal{P}$) is considered as a member of a group of proxies
managed by the group manager ($\mathcal{GM}$)333Proxies are distributed over
several geographical areas. We assume that a load-balancing is established
between at least two proxies in the same geographical area to ensure the
system availability in case of failure or overload. More precisely, proxies in
a same geographical area are separated into two subsets - a primary and a
secondary - and two users in a contact interaction must contact proxies
belonging to different subsets in order to prevent a proxy from gaining too
much knowledge about users’ interactions.. Proxies form an intermediate layer
by relaying the common contact messages of users to $\mathcal{S}$ in order to
ensure the anonymity of involved users towards the server during the
Generation phase. Proxies also play an important role in ensuring data
integrity and user geolocation privacy thanks to group signatures.
### III-B Overview
SPOT is set upon an hybrid architecture that leverages the best of the
centralized and decentralized settings in proximity-tracing protocols. It
relies on a proxy-based solution to preserve users’ privacy (i.e., users
remain anonymous towards the server, thus preventing users’ tracking) and is
based on a semi-trusted computing server to ensure data consistency and
integrity (i.e., users are ensured that the received alerts are correct). The
architecture of the proposed protocol is depicted in Figure 1. SPOT involves
three main phases: Sys_Init, Generation and Verification presented hereafter.
The Sys_Init phase consists of initializing the whole system. It relies on
seven algorithms, referred to as $\mathsf{Set\\_params}$,
$\mathsf{HA\\_keygen}$, $\mathsf{S\\_keygen}$,
$\mathsf{Setup\\_ProxyGr}_{\mathcal{GM}}$ and
$\mathsf{Join\\_ProxyGr}_{\mathcal{P}/\mathcal{GM}}$,
$\mathsf{Set\\_UserID}_{\mathcal{HA}}$ and
$\mathsf{Userkeygen}_{\mathcal{U}}$. During the Sys_Init phase, a trusted
authority444For ease of presentation, the trusted authority is neither
presented in Figure 1 nor in the system’s model entities. generates the system
public parameters published to all involved entities and the pair of keys of
both $\mathcal{HA}$ and $\mathcal{S}$, relying on $\mathsf{Set\\_params}$,
$\mathsf{HA\\_keygen}$ and $\mathsf{S\\_keygen}$ algorithms. During this
phase, the group manager defines the group of proxies. It generates the group
signature parameters using the $\mathsf{Setup\\_ProxyGr}_{\mathcal{GM}}$
algorithm and it interacts with each group member to derive the associated
keys relying on the $\mathsf{Join\\_ProxyGr}_{\mathcal{P}/\mathcal{GM}}$
algorithm. The Health Authority is also involved in this phase to register a
user when installing the proximity-tracing application. $\mathcal{HA}$
generates a specific secret value $t_{\mathcal{U}}$ (only known by
$\mathcal{HA}$) and a unique identifier $\mathtt{ID}_{\mathcal{U}}$ for each
user ($\mathcal{U}$), using the $\mathsf{Set\\_UserID}_{\mathcal{HA}}$
algorithm. Finally, $\mathcal{U}$ uses his identifier to generate his pair of
keys relying on the $\mathsf{Userkeygen_{U}}$ algorithm. The user’s identifier
$\mathtt{ID}_{\mathcal{U}}$, secret value $t_{\mathcal{U}}$ and public key are
stored in a database $DB_{USER}$ owned by $\mathcal{HA}$. We note that the
trusted authority, the group manager and proxies are involved only once in the
Sys_Init phase, while the health authority must intervene every time a user
wants to register.
The Generation phase occurs when two users $\mathcal{U}_{A}$ and
$\mathcal{U}_{B}$ are in contact. It represents the process of generating
contact messages and contact lists for users. Three main entities participate
in this phase relying on three different algorithms, referred to as
$\mathsf{Set\\_CCM}_{\mathcal{U}}$, $\mathsf{S\\_PSign}_{\mathcal{S}}$ and
$\mathsf{P\\_Sign}_{\mathcal{P}}$. At first, $\mathcal{U}_{A}$ and
$\mathcal{U}_{B}$ execute the $\mathsf{Set\\_CCM}_{\mathcal{U}}$ algorithm to
generate a common contact message relying on their random $EBID$s (denoted by
$\mathtt{D}^{e}_{A}$ and $\mathtt{D}^{e}_{B}$) for an epoch $e$555An epoch $e$
denotes a period of time in which the Bluetooth identifier (EBID) remains
unchanged.. $\mathcal{U}_{A}$ and $\mathcal{U}_{B}$ choose two different
proxies to relay their common contact message to the server. For this purpose,
they compare their $EBID$s, i.e., if $\mathtt{D}^{e}_{A}$ $>$
$\mathtt{D}^{e}_{B}$, $\mathcal{U}_{A}$ chooses the first proxy and
$\mathcal{U}_{B}$ selects the second one, and vice versa. Each of the two
proxies relays the common contact message to the server. $\mathcal{S}$ checks
if the two copies are similar. If so, $\mathcal{S}$ executes the
$\mathsf{S\\_PSign}_{\mathcal{S}}$ algorithm to partially sign the common
contact message, thus proving that the contact message correctly reached the
server. Afterwards, given back only a correct message, each proxy executes the
$\mathsf{P\\_Sign}_{\mathcal{P}}$ algorithm. Indeed, each proxy extends the
message, given by $\mathcal{S}$, with the corresponding user’s identifier and
it signs the resulting message on behalf of the group. He, finally, sends back
the message and the corresponding group signature to the user and closes the
communication session, while removing all the exchanged and generated contact
information. The user adds the group signature, along with the common contact
message, the date, time and duration of contact, in his contact list
$CL_{\mathcal{U}}$. Note that each contact information is stored for $\Delta$
days.
The Verification phase is run by the health authority to check the correctness
of a contact list $CL_{\mathcal{U}}$ provided by $\mathcal{U}$ during a period
of time $t$. To this end, $\mathcal{HA}$ performs three successive
verifications relying on two main algorithms, referred to as
$\mathsf{Sig\\_Verify}_{\mathcal{HA}}$ and
$\mathsf{CCM\\_Verify}_{\mathcal{HA}}$. (i) $\mathcal{HA}$ checks if, in his
$DB_{USER}$ database, $\mathcal{U}$ is infected666We suppose that the health
status of a user is updated when being tested. Indeed, to be tested,
$\mathcal{U}$ has to provide an encrypted form of his identifier
$\mathtt{ID}_{\mathcal{U}}$ (i.e., $\mathtt{ID}_{\mathcal{U}}$ is encrypted
meaning the $\mathcal{HA}$ public key). Afterwards, the analysis’ result is
sent with the encrypted identifier to $\mathcal{HA}$, that extracts the
identifier and updates the user’s health status in the $DB_{USER}$ database..
(ii) $\mathcal{HA}$ checks the validity of the group signatures relying on the
$\mathsf{Sig\\_Verify}_{\mathcal{HA}}$ algorithm, w.r.t. the messages
contained in the contact list $CL_{\mathcal{U}}$. (iii) $\mathcal{HA}$
verifies that the contact messages have been correctly generated and have
successfully reached $\mathcal{S}$, using the
$\mathsf{CCM\\_Verify}_{\mathcal{HA}}$ algorithm.
It is worth mentioning that if one of the verifications given above fails, the
contact message is rejected. Otherwise, $\mathcal{HA}$ collects all verified
messages of all infected users in a set $S_{\mathtt{CCM}}$ that she signs.
Note that for each period of time $t$, $\mathcal{HA}$ removes users’ contact
lists after verifications. $S_{\mathtt{CCM}}$ and the corresponding signature
are sent to the server that shares them with all users. To compute the risk
score, each user compares the set $S_{\mathtt{CCM}}$ with his contact list,
taking into account the number of infected users being contacted and the
contact duration.
For ease of presentation, the different notations used in this paper are
depicted in Table II.
TABLE II: Notations used in this paper
Notation | Description
---|---
$\mathcal{U}$ | User
$\mathcal{HA}$ | Health Authority
$\mathcal{S}$ | Server
$\mathcal{P}$ | Proxy
$\mathcal{GM}$ | Group Manager
$\mathcal{TA}$ | Trusted Authority
$\mathtt{ID}_{\mathcal{U}}$ | An identifier of a user $\mathcal{U}$
$t_{\mathcal{U}}$ | A secret value associated to $\mathcal{U}$
$DB_{USER}$ | The users’ database at $\mathcal{HA}$
$\mathtt{D}^{e}$ | An ephemeral Bluetooth Identifier during an epoch $e$
$CL_{\mathcal{U}}$ | A user’s contact list
$\lambda$ | A security parameter
$pp$ | The system’s public parameters
$\mathtt{sk}$ | A private key
$\mathtt{pk}$ | A public key
$\mathtt{vk}_{g}$ | The group public parameters
$\mathtt{CCM}$ | A common contact message
$(\mathtt{PS},\mathtt{PS}^{\prime})$ | A partial signature
$\sigma$ | A signature
$\mathtt{M}$ | A message derived from $\mathtt{PS}$ and $\mathtt{ID}_{\mathcal{U}}$
$\pi$ | A NIWI proof
$S_{\mathtt{CCM}}$ | A set of verified $\mathtt{CMM}$s of infected users
### III-C System Model
Based on the three phases, Figure 2 presents the chronological sequence of
twelve PPT algorithms, defined below. For ease of presentation, we consider
only one user and one proxy in the sequence diagram. For the Generation phase,
we suppose that two users have been in contact and exchanged their EBIDs. The
Verification phase occurs only if the user receives a negative analysis’
result.
Figure 2: Workflow of the SPOT protocol
* •
Sys_Init phase:
$\mathsf{Set\\_params}(\lambda)\rightarrow pp$ – run by a trusted authority.
Given the security parameter $\lambda$, this algorithm generates the system
public parameters $pp$ that will be considered as a default input for all the
following algorithms.
$\mathsf{K\\_keygen}()\rightarrow(\mathtt{sk}_{j},\mathtt{pk}_{j})$ –
performed by a trusted authority. It returns the pair of keys
$(\mathtt{sk}_{j},\mathtt{pk}_{j})$ of $j$ where
$j=\\{\mathcal{HA},\mathcal{S}\\}$.
$\mathsf{Setup\\_ProxyGr}_{\mathcal{GM}}()\rightarrow(\mathtt{sk}_{g},\mathtt{vk}_{g})$
– this algorithm is performed by the group manager to set up the group
signature. It returns the proxies’ group verification key $\mathtt{vk}_{g}$
represented as $(\mathtt{pk}_{g},\Sigma_{\mathsf{NIWI}})$, where
$\mathtt{pk}_{g}$ is the public key of the group manager and
$\Sigma_{\mathsf{NIWI}}$ is the Common Reference String CRS of a NIWI proof
[10]. The $\mathsf{Setup\\_ProxyGr}_{\mathcal{GM}}$ algorithm also outputs the
secret key $\mathtt{sk}_{g}$ that is privately stored by $\mathcal{GM}$.
$\mathsf{Join\\_ProxyGr_{\mathcal{P}/\mathcal{GM}}}(\mathtt{sk}_{g})\rightarrow(\mathtt{sk}_{p},\mathtt{pk}_{p},\sigma_{p})$
– this algorithm is performed through an interactive session between the proxy
and the group manager. It takes as input the secret key $\mathtt{sk}_{g}$, and
outputs the pair of keys $(\mathtt{sk}_{p},\mathtt{pk}_{p})$ of $\mathcal{P}$
belonging to the group (i.e., $\mathcal{P}$ is responsible for generating his
pair of keys), and a signature $\sigma_{p}$ over $\mathcal{P}$’s public key
$\mathtt{pk}_{p}$ (i.e., $\sigma_{p}$ is generated by $\mathcal{GM}$).
$\mathsf{Set\\_UserID_{\mathcal{HA}}}()\rightarrow(t_{\mathcal{U}},\mathtt{ID}_{\mathcal{U}})$
– this algorithm is run by $\mathcal{HA}$ and returns a secret value
$t_{\mathcal{U}}$ specific for $\mathcal{U}$ and the identifier
$\mathtt{ID}_{\mathcal{U}}$ of $\mathcal{U}$.
$\mathsf{Userkeygen_{U}}(\mathtt{ID}_{\mathcal{U}})\rightarrow(\mathtt{sk}_{\mathcal{U}},\mathtt{pk}_{\mathcal{U}})$
– performed by $\mathcal{U}$ to set his pair of keys
$(\mathtt{sk}_{\mathcal{U}},\mathtt{pk}_{\mathcal{U}})$ relying on the
identifier $\mathtt{ID}_{\mathcal{U}}$.
* •
Generation phase:
$\mathsf{Set\\_CCM_{\mathcal{U}}}(\mathtt{D}^{e}_{A},\mathtt{D}^{e}_{B})\rightarrow\mathtt{CCM}^{e}_{AB}$
– run by each of two users $\mathcal{U}_{A}$ and $\mathcal{U}_{B}$ being in
contact during an epoch $e$. Given two Bluetooth identifiers
$\mathtt{D}^{e}_{A}$ and $\mathtt{D}^{e}_{B}$, this algorithm generates a
common contact message $\mathtt{CCM}^{e}_{AB}$.
$\mathsf{S\\_PSign_{\mathcal{S}}}(\mathtt{CCM}^{e}_{AB},\mathtt{sk}_{\mathcal{S}})\rightarrow(\mathtt{PS}^{e}_{AB},\mathtt{PS^{\prime}}^{e}_{AB})$
– run by $\mathcal{S}$. Given a common contact message $\mathtt{CCM}^{e}_{AB}$
sent by $\mathcal{U}_{A}$ and $\mathcal{U}_{B}$ through two different proxies
$\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, this algorithm outputs the couple
($\mathtt{PS}^{e}_{AB}$, $\mathtt{PS^{\prime}}^{e}_{AB}$) that is stored with
$\mathtt{CCM}^{e}_{AB}$ at $\mathcal{S}$, for $\Delta$ days. Note that only
$\mathtt{PS}^{e}_{AB}$ is given back to $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$ to prove that $\mathtt{CCM}^{e}_{AB}$ has been successfully
received and verified by $\mathcal{S}$ (i.e., a real contact took place),
while $\mathtt{PS^{\prime}}^{e}_{AB}$ is kept secret at $\mathcal{S}$ and is
sent only to $\mathcal{HA}$ to check the correctness of a contact message
provided by a infected user.
$\mathsf{P\\_Sign}_{\mathcal{P}}(\mathtt{vk}_{g},\mathtt{sk}_{p},\mathtt{pk}_{p},\sigma_{p},\mathtt{ID}_{\mathcal{U}_{A}},\mathtt{PS}^{e}_{AB})\rightarrow(\mathtt{M}^{e}_{AB}$,
$\sigma_{m}$, $\pi)$777In this algorithm, we only consider user
$\mathcal{U}_{A}$ with $\mathtt{ID}_{\mathcal{U}_{A}}$. The same operations
are performed for user $\mathcal{U}_{B}$ with $\mathtt{ID}_{\mathcal{U}_{B}}$.
– performed by the proxy $\mathcal{P}$ ($\mathcal{P}_{1}$ or
$\mathcal{P}_{2}$). This algorithm takes as input the proxies’ group public
parameters $\mathtt{vk}_{g}$, the pair of keys
$(\mathtt{sk}_{p},\mathtt{pk}_{p})$ of $\mathcal{P}$, the signature
$\sigma_{p}$ over $\mathcal{P}$’s public key, the identifier
$\mathtt{ID}_{\mathcal{U}_{A}}$ of user $\mathcal{U}_{A}$ and the message
$\mathtt{PS}^{e}_{AB}$. It returns a signature $\sigma_{m}$ over a new message
$\mathtt{M}^{e}_{AB}$ and a group signature represented by a NIWI proof $\pi$
over the two signatures $\sigma_{p}$ and $\sigma_{m}$. The couple
($\mathtt{M}^{e}_{AB}$, $\pi$) is sent to user $\mathcal{U}_{A}$ to be stored
with the contact message $\mathtt{CCM}^{e}_{AB}$ in his contact list. Note
that each input of the contact list is stored for $\Delta$ days.
* •
Verification phase:
$\mathsf{Sig\\_Verify_{\mathcal{HA}}}(\mathtt{vk}_{g},\mathtt{M}^{e}_{AB},\pi)\rightarrow
b$ – performed by $\mathcal{HA}$. Given the public parameters
$\mathtt{vk}_{g}$, a message $\mathtt{M}^{e}_{AB}$ from the contact list of an
infected user, and the corresponding NIWI proof $\pi$, the
$\mathsf{Sig\\_Verify_{\mathcal{HA}}}$ algorithm returns a bit $b\in\\{0,1\\}$
stating whether the proof is valid.
$\mathsf{CCM\\_Verify_{\mathcal{HA}}}(\mathtt{M}^{e}_{AB},\mathtt{PS^{\prime}}^{e}_{AB},\mathtt{pk}_{\mathcal{S}},t_{\mathcal{U}_{A}})\rightarrow
b$ – run by $\mathcal{HA}$. This algorithm takes as input the message
$\mathtt{M}^{e}_{AB}$, the message $\mathtt{PS^{\prime}}^{e}_{AB}$ requested
from $\mathcal{S}$, the server’s public key $\mathtt{pk}_{\mathcal{S}}$ and
the secret value $t_{\mathcal{U}_{A}}$, and outputs a bit $b\in\\{0,1\\}$,
i.e., $\mathtt{CCM}^{e}_{AB}$ is correctly generated or not.
### III-D Threat Model
In this section, we first present the adversaries considered in SPOT, and
then, the formal definitions of the different security and privacy properties.
* •
A malicious user ($\mathcal{U}$): this adversary attempts to inject false
contact messages or contact messages of other users in his contact list. He
may also collude with corrupted proxies or malicious users.
* •
A honest but curious health authority ($\mathcal{HA}$): given a valid group
signature, $\mathcal{HA}$ tries to identify the signer (i.e., proxy) of a
particular message, hence for identifying the appropriate geographical area
and for tracking the user’s movements. She may also attempt to link two
signatures issued by the same group member. A curious $\mathcal{HA}$ may also
try to identify, from a contact list of a particular user, the list of users
he had been in contact with.
* •
A honest but curious server ($\mathcal{S}$): he attempts to link several
common contact messages generated by the same user, to trace users’ movements.
* •
A malicious proxy ($\mathcal{P}$): this adversary, either colluding with a
malicious user or with $n$ other proxies, attempts to forge the partial
signature of the server and to generate a valid signature over a false contact
message.
Both malicious users and malicious proxies are considered against security
properties, i.e., unforgeability, anti-replay, while the curious health
authority and server are considered against privacy requirements, i.e.,
unlinkability and anonymity. The different adversaries are involved in
different phases.
Note that the anti-replay property which aims at mitigating the multi-
submission of the same contact information is not formally presented below,
but is informally discussed in Section VI. The following properties are
defined w.r.t the corresponding phases and the involved adversaries.
###### Remark 1.
We do not deeply analyze the case of a malicious $\mathcal{GM}$ although our
scheme is resistant against this adversary. Indeed, proxies are responsible
for generating their key pair and only their public keys are shared with
$\mathcal{GM}$. Thus, unless holding a proxy’s secret key, $\mathcal{GM}$ is
not able to generate a valid signature on behalf of $\mathcal{P}$ thanks to
the unforgeability of the signature scheme.
#### III-D1 Unforgeability
The unforgeability property ensures the security of SPOT for the different
phases. It states that a malicious user is not able to forge his contact list
(i.e., forging either the group signature or the server’s partial signature
when colluding with a malicious proxy)888We assume that (i) malicious user
refers to either a single user or colluding users, and (ii) the group
signature scheme used in SPOT is unforgeable as proven in [4], thus in the
security model and analysis, we will only consider the unforgeability of the
server’s partial signature.. Formally, this is defined in a game
$\textbf{Exp}_{\mathcal{A}}^{unforg}$ where an adversary $\mathcal{A}$,
playing the role of a corrupted proxy colluding with a malicious user, has
access to a $\mathsf{S\\_PSign}$ oracle. We note that, for each session $i$,
$\mathcal{A}$ only gets $\mathtt{PS}^{i}$ from the $\mathsf{S\\_PSign}$
oracle, while $\mathtt{PS^{\prime}}^{i}$ is kept secret from the adversary.
Then, given a valid message $\mathtt{PS^{\prime}}$ that cannot be obtained by
combining either a part of or all messages $\mathtt{PS}^{i}$, $\mathcal{A}$
succeeds if it outputs a valid message $\mathtt{PS}^{*}$ to be signed using
$\mathsf{P\\_Sign}$, such that the $\mathsf{CCM\\_Verify}$ verification holds.
###### Definition 1.
Unforgeability – We say that SPOT satisfies the unforgeability property, if
for every _PPT_ adversary $\mathcal{A}$, there exists a negligible function
$\kappa$ such that:
$Pr[\textbf{Exp}_{\mathcal{A}}^{unforg}(1^{\lambda})=1]\leq\kappa(\lambda)$,
where $\textbf{Exp}_{\mathcal{A}}^{unforg}$ is given below.
$\textbf{{Exp}}_{\mathcal{A}}^{unforg}(\lambda)$
$pp\leftarrow\mathsf{Set\\_params}(\lambda)$
$(\mathtt{sk_{\mathcal{HA}}},\mathtt{pk_{\mathcal{HA}}})\leftarrow\mathsf{HA\\_keygen}(pp)$
$(\mathtt{sk_{\mathcal{S}}},\mathtt{pk_{\mathcal{S}}})\leftarrow\mathsf{S\\_keygen}(pp)$
$(\mathtt{sk}_{g},\mathtt{vk}_{g})\leftarrow\mathsf{Setup\\_ProxyGr}(pp)$
$(\mathtt{sk}_{p},\mathtt{pk}_{p},\sigma_{p})\leftarrow\mathsf{Join\\_ProxyGr}(pp,\mathtt{sk}_{g})$
$\mathtt{ID_{\mathcal{U}}}\leftarrow\mathsf{Set\\_UserID}(pp)$
$(\mathtt{sk}_{\mathcal{U}},\mathtt{pk}_{\mathcal{U}})\leftarrow\mathsf{Userkeygen}(pp,\mathtt{ID_{\mathcal{U}}})$
$\mathtt{CCM}\leftarrow\mathsf{Set\\_CCM}(\mathtt{D}_{1},\mathtt{D}_{2})$
$(\mathtt{PS},\mathtt{PS^{\prime}})$ $\leftarrow$
$\\{\mathsf{S\\_PSign}(\mathtt{CCM},\mathtt{sk_{\mathcal{S}}})\\}$
$\mathcal{O}$ $\leftarrow$
$\\{\mathsf{S\\_PSign}(\cdot,\mathtt{sk_{\mathcal{S}}})\\}$
$\mathtt{PS}^{*}\leftarrow\mathcal{A}^{\mathcal{O}}(\mathtt{vk_{g}},\mathtt{sk_{p}},\mathtt{pk_{p}},\sigma_{p},ID_{\mathcal{U}},pp,\mathtt{PS^{\prime}})$
letting $\mathtt{CCM}$ and $\mathtt{PS}^{i}$ denote the queries and answers to
and from oracle $\mathsf{S\\_PSign}$
$(\mathtt{M}^{*},\sigma^{*},\pi^{*})\leftarrow\mathsf{P\\_Sign}(\mathtt{vk}_{g},\mathtt{sk}_{p},\mathtt{pk}_{p},\sigma_{p},ID_{\mathcal{U}},\mathtt{PS}^{*})$
If
$\mathsf{CCM\\_Verify}(\mathtt{M}^{*},\mathtt{PS^{\prime}}^{*},\mathtt{pk_{\mathcal{S}}},t_{\mathcal{U}})=1$
return 1 Else return 0
#### III-D2 Unlinkability
The unlinkability property can be divided into two sub-properties. The first
one constitutes the _group-signature unlinkability_ stating that a curious
health authority is not able to link two or several group signatures issued by
the same proxy during the Verification phase. The second sub-property _multi-
CCM unlinkability_ ensures that a curious server is not able to link two or
several common contact messages to the same user during the Generation phase
999The collusion between the health authority and the server does not pose
additional and plausible threats to the different procedures of the whole
framework. Indeed, during the Generation phase, contact messages are anonymous
to the server (and a possible colluding health authority); during the
Verification phase, the health authority knows the true identity of the
confirmed cases with their contact information; as such, a collusion between
the server and the authority does not bring extra knowledge..
We note that the _multi-CCM unlinkability_ property will be informally
discussed in Section VI. In this section, we only focus on the _group-
signature unlinkability_. Formally, this property is defined in a game
$\textbf{Exp}_{\mathcal{A}}^{unlink}$ where an adversary $\mathcal{A}$ acting
as a curious $\mathcal{HA}$ has access to a $\mathsf{P\\_Sign}$ oracle. The
adversary may query this oracle on the same message $\mathtt{PS}^{*}$ and on a
tuple $((\mathtt{sk}_{p_{j}},\mathtt{pk}_{p_{j}},\sigma_{p_{j}})$, where
$j\in\\{0,1\\}$ (i.e., the tuple belongs either to proxy $\mathcal{P}_{0}$ or
proxy $\mathcal{P}_{1}$). A left-or-right oracle $\mathsf{LoRSig}$ is
initialized with a secret random bit $b$ and returns to $\mathcal{A}$
$\mathsf{P\\_Sign}$ on message $\mathtt{PS}^{*}$ and respectively on tuples
$(\mathtt{sk}_{p_{0}},\mathtt{pk}_{p_{0}},\sigma_{p_{0}})$ and
$(\mathtt{sk}_{p_{b}},\mathtt{pk}_{p_{b}},\sigma_{p_{b}})$. The adversary wins
the game if he successfully predicts the bit $b$ (i.e., the guessing
probability should be greater than $\frac{1}{2}$).
###### Definition 2.
Unlinkability – We say that SPOT satisfies the unlinkability property, if for
every _PPT_ adversary $\mathcal{A}$, there exists a negligible function
$\kappa$ such that:
$Pr[\textbf{Exp}_{\mathcal{A}}^{unlink}(\lambda)=1]=\frac{1}{2}\pm\kappa(\lambda)$,
where $\textbf{Exp}_{\mathcal{A}}^{unlink}$ is defined below.
$\textbf{{Exp}}_{\mathcal{A}}^{unlink}(\lambda)$
$pp\leftarrow\mathsf{Set\\_params}(\lambda)$
$(\mathtt{sk_{\mathcal{HA}}},\mathtt{pk_{\mathcal{HA}}})\leftarrow\mathsf{HA\\_keygen}(pp)$
$(\mathtt{sk_{\mathcal{S}}},\mathtt{pk_{\mathcal{S}}})\leftarrow\mathsf{S\\_keygen}(pp)$
$(\mathtt{sk}_{g},\mathtt{vk}_{g})\leftarrow\mathsf{Setup\\_ProxyGr}(pp)$
$(\mathtt{sk}_{p_{i}}$, $\mathtt{pk}_{p_{i}}$, $\sigma_{p_{i}})$ $\leftarrow$
$\mathsf{Join\\_ProxyGr}$ ($pp$, $\mathtt{sk}_{g}$), $i\in\\{0,1\\}$
$\mathtt{ID_{\mathcal{U}}}\leftarrow\mathsf{Set\\_UserID}(pp)$
$(\mathtt{sk}_{\mathcal{U}},\mathtt{pk}_{\mathcal{U}})\leftarrow\mathsf{Userkeygen}(pp,\mathtt{ID_{\mathcal{U}}})$
$m^{*}\leftarrow\mathsf{Set\\_CCM}(\mathtt{D}_{1},\mathtt{D}_{2})$
$(\mathtt{PS}^{*},\mathtt{PS^{\prime}}^{*})\leftarrow\mathsf{S\\_PSign}(m^{*},\mathtt{sk_{S}})$
$b\leftarrow\\{0,1\\}$ $\mathcal{O}$ $\leftarrow$
$\\{\mathsf{P\\_Sign}$($\cdot$,$\mathtt{sk}_{p_{j}}$,$\mathtt{pk}_{p_{j}}$,$\sigma_{p_{j}}$,$\cdot$,$\cdot$),
$\mathsf{LoRSig}$($\cdot$,$\cdot$,$b)\\}$ $b^{\prime}$ $\leftarrow$
$\mathcal{A}^{\mathcal{O}}$
$(\mathtt{sk_{\mathcal{HA}}}$,$\mathtt{pk_{\mathcal{HA}}}$,
$\mathtt{vk}_{g}$,$\mathtt{pk}_{p_{0}}$,$\mathtt{pk}_{p_{1}}$,$pp$,$\mathtt{PS}^{*})$
If $b=b^{\prime}$ return 1 Else return 0
$\mathsf{LoRSig}$ $(\mathtt{vk}_{g}$, (($\mathtt{sk}_{p_{0}}$,
$\mathtt{pk}_{p_{0}}$, $\sigma_{p_{0}})$, ($\mathtt{sk}_{p_{1}}$,
$\mathtt{pk}_{p_{1}}$, $\sigma_{p_{1}}$)), $\mathtt{ID_{\mathcal{U}}}$,
$\mathtt{PS}^{*},b)$ $({\mathtt{M}}^{*},\sigma^{*},\pi^{*})$ $\leftarrow$
$\mathsf{P\\_Sign}$($\mathtt{vk}_{g}$, $\mathtt{sk}_{p_{0}}$,
$\mathtt{pk}_{p_{0}}$, $\sigma_{p_{0}}$, $\mathtt{ID_{\mathcal{U}}}$ ,
$\mathtt{PS}^{*})$ $({\mathtt{M}}^{*},\sigma^{*}_{b},\pi^{*}_{b})$
$\leftarrow$ $\mathsf{P\\_Sign}$($\mathtt{vk}_{g}$, $\mathtt{sk}_{p_{b}}$,
$\mathtt{pk}_{p_{b}}$, $\sigma_{p_{b}}$, $\mathtt{ID_{\mathcal{U}}}$ ,
$\mathtt{PS}^{*})$ return
$(({\mathtt{M}}^{*},\pi^{*}),({\mathtt{M}}^{*},\pi^{*}_{b}))$
#### III-D3 Anonymity
This property guarantees that no entity is able to identify users involved in
a contact list (i.e., the owner and the contacted users), during the
Verification phase, and is described through the game
$\textbf{Exp}_{\mathcal{A}}^{anon}$. The anonymity property implies that even
if $\mathcal{HA}$ knows that a contact list belongs to a user ($\mathcal{U}$),
$\mathcal{HA}$ is not able to identify users being in contact with
$\mathcal{U}$ 101010We assume that the probability of two confirmed users
being in contact and submitting their respective contact lists to
$\mathcal{HA}$ at the same period, is low.. This should hold even if an
efficient adversary, playing the role of the curious health authority, is
given access to $\mathsf{Set\\_CCM}$, $\mathsf{S\\_PSign}$,
$\mathsf{P\\_Sign}$ oracles. $\mathcal{A}$ can learn contact messages and
signatures associated to the selected users’ identifiers. $\mathcal{A}$ also
gets access to a left-or-right oracle $\mathsf{LoRCU}$ which is initialized
with a secret random bit $b\in\\{0,1\\}$. $\mathcal{A}$ may query this oracle
on $\mathtt{ID_{\mathcal{U}_{0}}}$ and $\mathtt{ID_{\mathcal{U}_{1}}}$
referred to as the identifiers of respectively user $\mathcal{U}_{0}$ and user
$\mathcal{U}_{1}$. Observe that user $\mathcal{U}_{A}$ is involved in all
queries. $\mathtt{D}^{*}_{\mathcal{U}_{A}}$ and
$\mathtt{D}^{*}_{\mathcal{U}_{b}}$, respectively belonging to user
$\mathcal{U}_{A}$ and user $\mathcal{U}_{b}$, are randomly selected in order
to execute the $\mathsf{LoRCU}$ oracle. To win the proposed anonymity game,
the adversary should predict the bit $b$ (i.e., which one of users
$\mathcal{U}_{0}$ and $\mathcal{U}_{1}$ is involved in the contact with user
$\mathcal{U}_{A}$) with a probability greater than $\frac{1}{2}$.
###### Definition 3.
Anonymity – We say that SPOT fulfills the anonymity requirement, if for every
_PPT_ adversary $\mathcal{A}$, there exists a negligible function $\kappa$
such that:
$Pr[\textbf{Exp}_{\mathcal{A}}^{anon}(1^{\lambda})=1]=\frac{1}{2}\pm\kappa(\lambda)$,
where $\textbf{Exp}_{\mathcal{A}}^{anon}$ is defined as follows.
$\textbf{{Exp}}_{\mathcal{A}}^{anon}(\lambda)$
$pp\leftarrow\mathsf{Set\\_params}(\lambda)$
$(\mathtt{sk_{\mathcal{HA}}},\mathtt{pk_{\mathcal{HA}}})$ $\leftarrow$
$\mathsf{HA\\_keygen}(pp)$
$(\mathtt{sk_{\mathcal{S}}},\mathtt{pk_{\mathcal{S}}})$ $\leftarrow$
$\mathsf{S\\_keygen}(pp)$ $(\mathtt{sk}_{g},\mathtt{vk}_{g})$ $\leftarrow$
$\mathsf{Setup\\_ProxyGr}(pp)$ $(\mathtt{sk}_{p},\mathtt{pk}_{p},\sigma_{p})$
$\leftarrow$ $\mathsf{Join\\_ProxyGr}(pp,\mathtt{sk}_{g})$
$\mathtt{ID}_{\mathcal{U}_{A}}$ $\leftarrow$ $\mathsf{Set\\_UserID}(pp)$
$(\mathtt{sk}_{\mathcal{U}_{A}},\mathtt{pk}_{\mathcal{U}_{A}})$ $\leftarrow$
$\mathsf{Userkeygen}(pp,\mathtt{ID}_{\mathcal{U}_{A}})$
$\mathtt{ID}_{\mathcal{U}_{i}}$ $\leftarrow$ $\mathsf{Set\\_UserID}(pp)$,
$i\in\\{0..N\\}$
$(\mathtt{sk}_{\mathcal{U}_{i}},\mathtt{pk}_{\mathcal{U}_{i}})$ $\leftarrow$
$\mathsf{Userkeygen}(pp,\mathtt{ID}_{\mathcal{U}_{i}})$, $i\in\\{0..N\\}$
$b\leftarrow\\{0,1\\}$ $\mathcal{O}$ $\leftarrow$
$\\{\mathsf{Set\\_CCM}$($\cdot$,$\cdot$),
$\mathsf{S\\_PSign}$($\cdot$,$\mathtt{sk_{\mathcal{S}}}$),
$\mathsf{P\\_Sign}$($\cdot$,$\mathtt{sk_{p}}$,
$\cdot$,$\sigma_{p}$,$\cdot$,$\cdot$),
$\mathsf{LoRCU}$($\cdot$,$\cdot$,$b$,$\cdot)$ $b^{\prime}$ $\leftarrow$
$\mathcal{A}^{\mathcal{O}}$ $(\mathtt{sk_{\mathcal{HA}}}$,
$\mathtt{pk_{\mathcal{HA}}}$, $pp$, $\mathtt{ID}_{\mathcal{U}_{A}}$,
$\\{\mathtt{ID}_{\mathcal{U}_{i}}\\}_{i=0}^{N})$ If $b=b^{\prime}$ return 1
Else return 0
$\mathsf{LoRCU}$ $(\mathtt{D}^{*}_{\mathcal{U}_{A}}$,
$\mathtt{D}^{*}_{\mathcal{U}_{b}}$, $b$, $\mathtt{vk}_{g}$,
$\mathtt{sk_{\mathcal{S}}}$, $\mathtt{sk}_{p}$, $\mathtt{pk}_{p}$,
$\sigma_{p}$, $\mathtt{ID}_{\mathcal{U}_{A}}$,
$\mathtt{ID}_{\mathcal{U}_{b}})$ ${\mathsf{CCM}}^{*}_{b}$ $\leftarrow$
$\mathsf{Set\\_CCM_{\mathcal{U}_{A}}}$ $(\mathtt{D}^{*}_{\mathcal{U}_{A}}$,
$\mathtt{D}^{*}_{\mathcal{U}_{b}})$
$({\mathsf{PS}}^{*}_{b},{\mathsf{PS}^{\prime}}^{*}_{b})$ $\leftarrow$
$\mathsf{S\\_PSign}$ $({\mathsf{CCM}}^{*}_{b}$, $\mathtt{sk_{\mathcal{S}}})$
$(\mathtt{M}^{*}_{b},\sigma^{*}_{b},\pi^{*}_{b})$ $\leftarrow$
$\mathsf{P\\_Sign}$ $(\mathtt{vk}_{g}$, $\mathtt{sk}_{p}$, $\mathtt{pk}_{p}$,
$\sigma_{p}$, $\mathtt{ID}_{\mathcal{U}_{A}}$, ${\mathsf{PS}}^{*}_{b})$ return
$({\mathsf{CCM}}^{*}_{b},\mathtt{M}^{*}_{b},\pi^{*}_{b})$
## IV Building Blocks
After introducing bilinear maps and security standard assumptions in Section
IV-A, the section next presents structure-preserving signatures [4] with their
different variants as main building blocks of the SPOT protocol. Sections IV-B
and IV-C describe respectively constant-size signatures and signatures on
mixed-group messages that are instantiated in Appendix IV-D to build a group
signature scheme on group element messages.
### IV-A Mathematical Background and Cryptographic Assumptions
Hereafter, we define bilinear maps and we present the computational
indistinguishability property and the CDH assumption.
#### IV-A1 Bilinear Maps
Let $\mathbb{G}_{1}$=$\langle{g_{1}}\rangle$ and
$\mathbb{G}_{2}$=$\langle{g_{2}}\rangle$ be two cyclic groups of order $n$ so
there exists a bilinear map
$e:\mathbb{G}_{1}\times\mathbb{G}_{2}\rightarrow\mathbb{G}_{3}$ that satisfies
the following properties: (i) bilinearity for all $g_{1}\in\mathbb{G}_{1}$,
$g_{2}\in\mathbb{G}_{2}$, (ii) non-degeneracy: $e(g_{1},g_{2})\neq 1$ and
(iii) $e(g_{1},g_{2})$ is efficiently computable for any
$g_{1}\in\mathbb{G}_{1}$ and $g_{2}\in\mathbb{G}_{2}$.
#### IV-A2 Computational Witness-Indistinguishability
The Computational Witness-Indistinguishability property is defined as follows:
Let $L\in\mathcal{NP}$ be a language and let $(\mathcal{P},\mathcal{V})$ be an
interactive proof system for $L$. We say that $(\mathcal{P},\mathcal{V})$ is
_witness-indistinguishable_ (WI) if for every _PPT_ algorithm
$\mathcal{V}^{*}$ and every two sequences $\\{w^{1}_{x}\\}_{x\in L}$ and
$\\{w^{2}_{x}\\}_{x\in L}$ such that $w^{1}_{x}$ and $w^{2}_{x}$ are both
witnesses for $x$, the following ensembles are computationally
indistinguishable, where $z$ is an auxiliary input to $\mathcal{V}^{*}$:
1. 1.
$\\{\langle\mathcal{P}(w^{1}_{x}),\mathcal{V}^{*}\rangle(x)\\}_{x\in
L,z\in\\{0,1\\}^{*}}$
2. 2.
$\\{\langle\mathcal{P}(w^{2}_{x}),\mathcal{V}^{*}\rangle(x)\\}_{x\in
L,z\in\\{0,1\\}^{*}}$
#### IV-A3 Computational Diffie Hellman Assumption (CDH)
The CDH assumption is defined as follows: Let $\mathbb{G}$ be a group of prime
order $n$, and $g$ is a generator of $\mathbb{G}$. The CDH problem is defined
as: Given the tuple of elements $(g,g^{x},g^{y})$, where
$\\{x,y\\}\leftarrow\mathbb{Z}_{n}$, there is no efficient algorithm
$\mathcal{A}_{CDH}$ that can compute $g^{xy}$.
### IV-B Structure-preserving Constant-size Signature
Structure-preserving constant-size signature was defined by Abe _et al._ [4]
as the main scheme of structure-preserving signatures used to sign a message
$\vec{m}=(m_{1},...,m_{k})\in{\mathbb{G}_{2}}^{k}$, considering an asymmetric
bilinear group $(n,\mathbb{G}_{1},\mathbb{G}_{2},$
$\mathbb{G}_{3},g_{1},g_{2},e)$. A constant-size signature scheme
$\mathsf{CSIG}$ [4] relies on the following three PPT algorithms
($\mathsf{CSIG}.\mathsf{Key}$, $\mathsf{CSIG}.\mathsf{Sign}$,
$\mathsf{CSIG}.\mathsf{Verify}$):
${\mathsf{CSIG}}.\mathsf{Key}(1^{\lambda})$: This algorithm takes as input the
security parameter $(1^{\lambda})$ and outputs the pair of public and secret
keys $(\mathtt{sk},\mathtt{pk})$ of the signer. It chooses two random
generators $g_{r},h_{u}\leftarrow\mathbb{G}^{*}_{1}$ and random values
$\gamma_{i},\delta_{i}\leftarrow\mathbb{Z}^{*}_{n}$ and computes
$g_{i}={g_{r}}^{\gamma_{i}}$ and $h_{i}={h_{u}}^{\delta_{i}}$, for
$i=1,...,k$. It then selects
$\gamma_{z},\delta_{z}\leftarrow\mathbb{Z}^{*}_{n}$ and computes
$g_{z}={g_{r}}^{\gamma_{z}}$ and $h_{z}={h_{u}}^{\delta_{z}}$. It also chooses
$\alpha,\beta\leftarrow\mathbb{Z}^{*}_{n}$ and sets the couples
$(g_{r},g_{2}^{\alpha})$ and $(h_{u},g_{2}^{\beta})$. The public key is set as
$\mathtt{pk}=(g_{z},h_{z},g_{r},h_{u},g_{2}^{\alpha},g_{2}^{\beta},\\{g_{i},h_{i}\\}^{k}_{i=1})$
and the secret key is set as
$\mathtt{sk}=(\mathtt{pk},\alpha,\beta,\gamma_{z},\delta_{z},\\{\gamma_{i},\delta_{i}\\}^{k}_{i=1})$.
$\mathsf{CSIG}.\mathsf{Sign}(\mathtt{sk},\vec{m})$: This algorithm generates a
signature $\sigma$ over a message $\vec{m}$ using the secret key
$\mathtt{sk}$. That is, the signer randomly selects
$\zeta,\rho,\tau,\varphi,\omega\leftarrow\mathbb{Z}^{*}_{n}$ and computes
$z=g_{2}^{\zeta},r={g_{2}}^{\alpha-\rho\tau-{\gamma_{z}}\zeta}{\prod}^{k}_{i=1}{m_{i}}^{-\gamma_{i}},s={g_{r}}^{\rho},t={g_{2}}^{\tau},$
$u={g_{2}}^{\beta-\varphi\omega-{\delta_{z}}\zeta}{\prod}^{k}_{i=1}{m_{i}}^{-\delta_{i}},v={h_{u}}^{\varphi},w={g_{2}}^{\omega}$
The signature is set as $\sigma=(z,r,s,t,u,v,w)$.
$\mathsf{CSIG}.\mathsf{Verify}(\mathtt{pk},\vec{m},\sigma)$: This algorithm
checks the validity of the signature $\sigma$ on the message $m$ relying on
the signer’s public key $\mathtt{pk}$. It outputs 1 if the signature is valid
and 0 otherwise. The verifier checks if the following equations hold:
$A=e(g_{z},z)e(g_{r},r)e(s,t){\prod}^{k}_{i=1}e(g_{i},m_{i})$ (1)
$B=e(h_{z},z)e(h_{u},u)e(v,w){\prod}^{k}_{i=1}e(h_{i},m_{i})$ (2)
where $A=e(g_{r},g_{2}^{\alpha})$ and $B=e(h_{u},g_{2}^{\beta})$
### IV-C Structure-preserving signature on mixed-group messages
A structure-preserving signature on mixed-group messages $\mathsf{XSIG}$ [4]
represents a signature scheme where the message space is a mixture of the two
groups $\mathbb{G}_{1}$ and $\mathbb{G}_{2}$. We consider two constant-size
signature schemes $\mathsf{CSIG}1$ and $\mathsf{CSIG}2$. $\mathsf{CSIG}2$ is
the same scheme as in Section IV-B where the message space is
${\mathbb{G}_{2}}^{k_{2}}$, while $\mathsf{CSIG}1$ is a ’dual’ scheme obtained
by exchanging $\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ in the same scheme, where
the message space is ${\mathbb{G}_{1}}^{k_{1}}$. The message space for the
$\mathsf{XSIG}$ is then
${\mathbb{G}_{1}}^{k_{1}}\times{\mathbb{G}_{2}}^{k_{2}}$. Let
$(\vec{m},\vec{\tilde{m}})$ be a message in
${\mathbb{G}_{1}}^{k_{1}}\times{\mathbb{G}_{2}}^{k_{2}}$. For a vector
$\vec{\tilde{m}}\in{\mathbb{G}_{1}}^{k_{1}}$ and a single element
$s\in\mathbb{G}_{1}$, let $\vec{m}||s$ denote a vector in
${\mathbb{G}_{1}}^{k_{1}+1}$ obtained by appending $s$ to the end of
$\vec{m}$.
A mixed-group messages signature scheme $\mathsf{XSIG}$ relies on the
following three PPT algorithms ($\mathsf{XSIG}.\mathsf{Key}$,
$\mathsf{XSIG}.\mathsf{Sign}$, $\mathsf{XSIG}.\mathsf{Verify}$):
$\mathsf{XSIG}.\mathsf{Key}(1^{\lambda})$: This algorithm runs
$(\mathtt{sk}_{1}$, $\mathtt{pk}_{1})$ $\leftarrow$
$\mathsf{CSIG}1.\mathsf{Key}(1^{\lambda})$ and $(\mathtt{sk}_{2}$,
$\mathtt{pk}_{2})$ $\leftarrow$ $\mathsf{CSIG}2.\mathsf{Key}(1^{\lambda})$ and
sets
$(\mathtt{sk},\mathtt{pk})=((\mathtt{sk}_{1},\mathtt{sk}_{2}),(\mathtt{pk}_{1},\mathtt{pk}_{2}))$.
$\mathsf{XSIG}.\mathsf{Sign}(\mathtt{sk},(\vec{m},\vec{\tilde{m}}))$: This
algorithm runs $\sigma_{2}$ = $(z,r,s,t,u,v,w)$ $\leftarrow$
$\mathsf{CSIG}2.\mathsf{Si}$\- $\mathsf{gn}(\mathtt{sk_{2}}$,
$\vec{\tilde{m}})$ and
$\sigma_{1}=(z^{\prime},r^{\prime},s^{\prime},t^{\prime},u^{\prime},v^{\prime},w^{\prime})\leftarrow\mathsf{CSIG}1.\mathsf{Sign}(\mathtt{sk_{1}},\vec{m}||s)$,
and outputs $\sigma=(\sigma_{1},\sigma_{2})$.
$\mathsf{XSIG}.\mathsf{Verify}(\mathtt{pk},(\vec{m},\vec{\tilde{m}}),(\sigma_{1},\sigma_{2}))$:
This algorithm takes $s\in\mathbb{G}_{1}$ from $\sigma_{2}$, runs
$b_{2}=\mathsf{CSIG}2.\mathsf{Verify}(\mathtt{pk_{2}},\vec{\tilde{m}},\sigma_{2})$
and
$b_{1}=\mathsf{CSIG}1.\mathsf{Verify}(\mathtt{pk_{1}},\vec{m}||s,\sigma_{1})$.
If $b_{1}=b_{2}=1$, the algorithm outputs 1, otherwise it outputs 0.
### IV-D Group signatures drawn from structure-preserving signatures
We present hereafter an instantiation of a group signature scheme that allows
to sign a group element message relying on a constant-size signature scheme
$\mathsf{CSIG}$, a mixed-group messages signature scheme $\mathsf{XSIG}$ and a
witness indistinguishable proof of knowledge system $\mathsf{NIWI}$ [10] (cf.
Appendix A).
A group signature scheme $\mathsf{GSIG}$ relies on the four following
algorithms ($\mathsf{GSIG}.\mathsf{Setup}$, $\mathsf{GSIG}.\mathsf{Join}$,
$\mathsf{GSIG}.\mathsf{Sign}$, $\mathsf{GSIG}.\mathsf{Verify}$):
$\mathsf{GSIG}.\mathsf{Setup}$ : represents the setup algorithm. It runs
$\mathsf{XSIG}.\mathsf{Key}$ algorithm that generates the key pair
$(\mathtt{sk_{g}},\mathtt{pk}_{g})$ of the group manager and sets up a CRS
$\Sigma_{\mathsf{NIWI}}$ for the $\mathsf{NIWI}$ proof. The group verification
key is set as $\mathtt{vk}_{g}=(\mathtt{pk}_{g},\Sigma_{\mathsf{NIWI}})$,
while the certification secret key $\mathtt{sk_{g}}$ is privately stored by
the group manager.
$\mathsf{GSIG}.\mathsf{Join}$: represents the join algorithm. It is composed
of two steps. In the first one, the group member generates his key-pair
$(\mathtt{sk_{p}},\mathtt{pk}_{p})$ while running the
$\mathsf{CSIG}.\mathsf{Key}$ algorithm. Only the public key $\mathtt{pk}_{p}$
is sent to the group manager. This latter generates a signature $\sigma_{p}$
over $\mathtt{pk}_{p}$, using the $\mathsf{XSIG}.\mathsf{Sign}$ algorithm, and
sends it to the group member.
$\mathsf{GSIG}.\mathsf{Sign}$: represents the signing algorithm run by a group
member on a message $m\in\mathbb{G}_{2}$. The group member generates, over the
message $m$, a signature
$\sigma_{m}\leftarrow\mathsf{CSIG}.\mathsf{Sign}(\mathtt{sk}_{p},m)$ and a
non-interactive witness indistinguishable proof of knowledge
$\pi\leftarrow\mathsf{NIWI}.\mathsf{Proof}(\Sigma_{\mathsf{NIWI}},pub,wit)$
that proves
$1=\mathsf{XSIG}.\mathsf{Verify}(\mathtt{pk}_{g},\mathtt{pk}_{p},\sigma_{p})$
and $1=\mathsf{CSIG}.\mathsf{Verify}(\mathtt{pk}_{p},m,\sigma_{m})$ with
respect to the witness $wit=(\mathtt{pk}_{p},\sigma_{p},\sigma_{m})$ and the
public information $pub=(\mathtt{pk}_{g},m)$. The signing algorithm outputs
the group signature $\pi$.
$\mathsf{GSIG}.\mathsf{Verify}$: represents the group signature verification
algorithm run by a verifier. It takes $(\mathtt{vk}_{g},m,\pi)$ as input and
verifies the correctness of the $\mathsf{NIWI}$ proof $\pi$ w.r.t.
$pub=(\mathtt{pk}_{g},m)$ and the CRS $\Sigma_{\mathsf{NIWI}}$.
## V SPOT Algorithms
This section gives a concrete construction of the different phases and
algorithms of SPOT, introduced in Section III-A. SPOT relies on the different
variants of structure-preserving signatures represented in Appendix IV.
### V-A Sys_Init phase
* •
$\mathsf{Set\\_params}$ – a trusted authority sets an asymmetric bilinear
group $(n$, $\mathbb{G}_{1}$, $\mathbb{G}_{2}$, $\mathbb{G}_{3}$, $g_{1}$,
$g_{2}$, $e)$ relying on the security parameter $\lambda$, where
$\mathbb{G}_{1}$ and $\mathbb{G}_{2}$ are two cyclic groups of prime order
$n$, $g_{1}$ and $g_{2}$ are generators of respectively $\mathbb{G}_{1}$ and
$\mathbb{G}_{2}$ and $e$ is a bilinear map such that
$e:\mathbb{G}_{1}\times\mathbb{G}_{2}\rightarrow\mathbb{G}_{3}$. The trusted
authority also considers a cryptographic hash function
$\mathbf{H}:\\{0,1\\}^{*}\rightarrow\mathbb{Z}_{n}$. The output of the
$\mathsf{Set\\_params}$ algorithm represents the system global parameters that
are known by all the system entities. The tuple
$(n,\mathbb{G}_{1},\mathbb{G}_{2},\mathbb{G}_{3},g_{1},g_{2},e,\mathbf{H})$ is
denoted by $pp$, and is considered as a default input of all algorithms.
* •
$\mathsf{HA\\_keygen}$ – a trusted authority takes as input the public
parameters $pp$, selects a random $x\in\mathbb{Z}^{*}_{n}$ and generates the
pair of secret and public keys
$(\mathtt{sk}_{\mathcal{HA}},\mathtt{pk}_{\mathcal{HA}})$ of the health
authority as follows:
$\mathtt{sk}_{\mathcal{HA}}=x\quad;\quad\mathtt{pk}_{\mathcal{HA}}=g_{2}^{x}$
* •
$\mathsf{S\\_keygen}$ – a trusted authority generates the pair of secret and
public keys $(\mathtt{sk}_{\mathcal{S}},\mathtt{pk}_{\mathcal{S}})$ of the
server as given below, relying on the system public parameters $pp$ and two
selected randoms $y_{1},y_{2}\in\mathbb{Z}^{*}_{n}$.
$\mathtt{sk}_{\mathcal{S}}=(y_{1},y_{2})\quad;\quad\mathtt{pk}_{\mathcal{S}}=(Y_{1},Y_{2})=(g_{2}^{y_{1}},g_{2}^{y_{2}})$
* •
$\mathsf{Setup\\_ProxyGr_{\mathcal{GM}}}$ – $\mathcal{GM}$ sets up the group
of proxies by generating a group public key $\mathtt{vk}_{g}$ and a
certification secret key $\mathtt{sk}_{g}$ as shown in Algorithm 1.
* •
$\mathsf{Join\\_ProxyGr_{\mathcal{P}/\mathcal{GM}}}$ – $\mathcal{P}$ first
generates his pair of keys $(\mathtt{sk}_{p},\mathtt{pk}_{p})$ w.r.t. the
${\mathsf{CSIG}}.\mathsf{Key}$ algorithm (cf. Section IV-B). Afterwards,
$\mathcal{GM}$ generates a signature $\sigma_{p}$ over the public key
$\mathtt{pk}_{p}$ w.r.t. the ${\mathsf{XSIG}}.\mathsf{Sign}$ algorithm (cf.
Section IV-C). The $\mathsf{Join\\_ProxyGr_{\mathcal{P}/\mathcal{GM}}}$
algorithm is detailed in Algorithm 2.
Algorithm 1 $\mathsf{Setup\\_ProxyGr_{\mathcal{GM}}}$ algorithm
1:Input: the system public parameters $pp$
2:Output: the public parameters $\mathtt{vk}_{g}$ of the proxies’ group and
the secret key $\mathtt{sk}_{g}$
3:// The next iterations are executed to generate the pair of keys of
$\mathcal{GM}$
4:pick at random $g_{r1},h_{u1}\leftarrow\mathbb{G}^{*}_{1}$,
$g_{r2},h_{u2}\leftarrow\mathbb{G}^{*}_{2}$
5:for $i=1$ to $2$ do pick at random
$\gamma_{1i},\delta_{1i}\leftarrow\mathbb{Z}^{*}_{n}$ compute
$g_{1i}\leftarrow{g_{r1}}^{\gamma_{1i}}$,
$h_{1i}\leftarrow{h_{u1}}^{\delta_{1i}}$
6:end for
7:for $j=1$ to $7$ do pick at random
$\gamma_{2j},\delta_{2j}\leftarrow\mathbb{Z}^{*}_{n}$ compute
$g_{2i}\leftarrow{g_{r2}}^{\gamma_{2j}}$ and
$h_{2j}\leftarrow{h_{u2}}^{\delta_{2j}}$
8:end for
9:pick at random
$\gamma_{1z},\delta_{1z},\gamma_{2z},\delta_{2z}\leftarrow\mathbb{Z}^{*}_{n}$
;
10:compute $g_{1z}\leftarrow{g_{r1}}^{\gamma_{1z}}$,
$h_{1z}\leftarrow{h_{u1}}^{\delta_{1z}}$,
$g_{2z}\leftarrow{g_{r2}}^{\gamma_{2z}}$ and
$h_{2z}\leftarrow{h_{u2}}^{\delta_{2z}}$ ;
11:pick at random
$\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\leftarrow\mathbb{Z}^{*}_{n}$ ;
12:$\mathtt{pk}_{1}\leftarrow(g_{2z},h_{2z},g_{2r},h_{2u},g_{1}^{\alpha_{2}},g_{1}^{\beta_{2}},\\{g_{2j},h_{2j}\\}^{7}_{j=1})$
and
$\mathtt{sk}_{1}\leftarrow(\mathtt{pk}_{1},\alpha_{2},\beta_{2},\gamma_{2z},\delta_{2z},\\{\gamma_{2j},\delta_{2j}\\}^{7}_{j=1})$
;
13:$\mathtt{pk}_{2}\leftarrow(g_{1z},h_{1z},g_{1r},h_{1u},g_{2}^{\alpha_{1}},g_{2}^{\beta_{1}},\\{g_{1i},h_{1i}\\}^{2}_{i=1})$
and
$\mathtt{sk}_{2}\leftarrow(\mathtt{pk}_{2},\alpha_{1},\beta_{1},\gamma_{1z},\delta_{1z},\\{\gamma_{1i},\delta_{1i}\\}^{2}_{i=1})$
;
14:set $\mathtt{pk}_{g}\leftarrow(\mathtt{pk}_{1},\mathtt{pk}_{2})$ and
$\mathtt{sk}_{g}\leftarrow(\mathtt{sk}_{1},\mathtt{sk}_{2})$ ;
15:// The next iterations are executed to generate the CRS
$\Sigma_{\mathsf{NIWI}}$
16:pick at random $r,s\leftarrow\mathbb{Z}^{*}_{n}$ and set
$\mathcal{U}=rg_{1}$ and $\mathcal{V}=sg_{2}$ ;
17:set
$\Sigma_{\mathsf{NIWI}}=(\mathbb{G}_{1},\mathbb{G}_{2},\mathbb{G}_{3},e,\iota_{1},p_{1},\iota_{2},p_{2},\iota_{3},\mathcal{U},\mathcal{V})$
;
18:$\mathtt{vk}_{g}\leftarrow(\mathtt{pk}_{g},\Sigma_{\mathsf{NIWI}})$ ;
19:return $(\mathtt{sk}_{g},\mathtt{vk}_{g})$
* •
$\mathsf{Set\\_UserID_{\mathcal{HA}}}$ – every time, a user ($\mathcal{U}$)
installs the application and wants to register, $\mathcal{HA}$ picks a secret
$t_{\mathcal{U}}\in\mathbb{Z}^{*}_{n}$ and sets the user’s identifier
$\mathtt{ID}_{\mathcal{U}}$ as
$\mathtt{ID}_{\mathcal{U}}=h_{\mathcal{U}}=g_{2}^{t_{\mathcal{U}}}$
* •
$\mathsf{Userkeygen_{\mathcal{U}}}$ – After receiving his identifier
$\mathtt{ID}_{\mathcal{U}}$, a user generates his pair of secret and private
keys $(\mathtt{sk}_{\mathcal{U}},\mathtt{pk}_{\mathcal{U}})$. Indeed,
$\mathcal{U}$ randomly selects $q_{\mathcal{U}}\in\mathbb{Z}^{*}_{n}$ and sets
$(\mathtt{sk}_{\mathcal{U}},\mathtt{pk}_{\mathcal{U}})$ as
$\mathtt{sk}_{\mathcal{U}}=q_{\mathcal{U}}\quad;\quad\mathtt{pk}_{\mathcal{U}}={h_{\mathcal{U}}}^{q_{\mathcal{U}}}$
### V-B Generation phase
* •
$\mathsf{Set\\_CCM_{\mathcal{U}}}$ – For each epoch $e$, $\mathcal{U}_{A}$ and
$\mathcal{U}_{B}$ generate random EBIDs $\mathtt{D}^{e}_{\mathcal{U}_{A}}$ and
$\mathtt{D}^{e}_{\mathcal{U}_{B}}$, respectively. $\mathcal{U}_{A}$ and
$\mathcal{U}_{B}$ exchange their EBIDs and each of them executes the
$\mathsf{Set\\_CCM}$ algorithm. $\mathcal{U}_{A}$ (resp. $\mathcal{U}_{B}$)
computes
$m^{e}_{AB}=\mathtt{D}^{e}_{\mathcal{U}_{A}}*\mathtt{D}^{e}_{\mathcal{U}_{B}}$
and sets the common contact element between $\mathcal{U}_{A}$ and
$\mathcal{U}_{B}$ as $\mathtt{CCM}^{e}_{AB}=\mathbf{H}(m^{e}_{AB})$.
* •
$\mathsf{S\\_PSign_{\mathcal{S}}}$ – After checking that he receives two
copies of $\mathtt{CCM}^{e}_{AB}$, the server picks at random
$r_{s}\leftarrow\mathbb{Z}^{*}_{n}$ and, relying on his secret key
$\mathtt{sk}_{\mathcal{S}}$, he computes the two messages
$\mathtt{PS}^{e}_{AB}$ and $\mathtt{PS^{\prime}}^{e}_{AB}$ such that
$\mathtt{PS}^{e}_{AB}=\mathtt{CCM}^{e}_{AB}y_{1}r_{s}+y_{2}\quad
and\quad\mathtt{PS^{\prime}}^{e}_{AB}=\mathtt{CCM}^{e}_{AB}r_{s}$
* •
$\mathsf{P\\_Sign_{\mathcal{P}}}$ – We consider that when being requested by a
user $\mathcal{U}_{A}$, the proxy opens a session and saves the user’s
identifier $\mathtt{ID}_{\mathcal{U}_{A}}$. This latter is used when executing
the $\mathsf{P\\_Sign_{\mathcal{P}}}$ algorithm (c.f. Algorithm 3) to generate
a new message $\mathtt{M}^{e}_{AB}$ (Line 4). The proxy then signs
$\mathtt{M}^{e}_{AB}$ (Line 6 – Line 8) following the
${\mathsf{CSIG}}.\mathsf{Sign}$ algorithm and finally generates a proof $\pi$
(Line 10 – Line 16) w.r.t. the ${\mathsf{GSIG}}.\mathsf{Sign}$ algorithm.
Algorithm 2 $\mathsf{Join\\_ProxyGr_{\mathcal{P}/\mathcal{GM}}}$ algorithm
1:Input: the security parameter $\lambda$ and the secret key of the group
manager $\mathtt{sk}_{g}$
2:Output: the pair of keys of a proxy group member
$(\mathtt{sk}_{p},\mathtt{pk}_{p})$ and the signature $\sigma_{p}$ over the
public key the public $\mathtt{pk}_{p}$
3:// The next is set by $\mathcal{P}$
4:pick at random $g_{r},h_{u}\leftarrow\mathbb{G}^{*}_{1}$,
$\gamma,\delta\leftarrow\mathbb{Z}^{*}_{n}$ ;
5:compute $g_{\gamma}\leftarrow{g_{r}}^{\gamma}$ and
$h_{\delta}\leftarrow{h_{u}}^{\delta}$ ;
6:pick at random $\gamma_{z},\delta_{z}\leftarrow\mathbb{Z}^{*}_{n}$ ;
7:compute $g_{z}\leftarrow{g_{r}}^{\gamma_{z}}$ and
$h_{z}\leftarrow{h_{u}}^{\delta_{z}}$ ;
8:pick at random $\alpha,\beta\leftarrow\mathbb{Z}^{*}_{n}$ ;
9:set
$\mathtt{pk}_{p}=(g_{z},h_{z},g_{r},h_{u},g_{2}^{\alpha},g_{2}^{\beta},g_{\gamma},h_{\delta})$
and
$\mathtt{sk}_{p}=(\mathtt{pk}_{p},\alpha,\beta,\gamma_{z},\delta_{z},\gamma,\delta)$
;
10:// The next is set by $\mathcal{GM}$
11:$\sigma_{p}\leftarrow\mathsf{XSIG}.\mathsf{Sign}(\mathtt{sk}_{g},\mathtt{pk}_{p})$
;
12:return $(\mathtt{sk}_{p},\mathtt{pk}_{p},\sigma_{p})$
Algorithm 3 $\mathsf{P\\_Sign_{\mathcal{P}}}$ algorithm
1:Input: the public parameters of the proxies’ group $\mathtt{vk}_{g}$, the
secret key $\mathtt{sk}_{p}$, the signature $\sigma_{p}$ over the proxy’s
public key, the identifier $\mathtt{ID}_{\mathcal{U}_{A}}$ of user
$\mathcal{U}_{A}$ and the message $\mathtt{PS}$
2:Output: a message $\mathtt{M}$, the corresponding signature $\sigma_{m}$ and
a proof $\pi$
3:// The next is executed by $\mathcal{P}$ to generate $\mathtt{M}$
4:compute $\mathtt{M}={\mathtt{ID}_{\mathcal{U}_{A}}}^{\mathtt{PS}}$;
5:// The next is executed by $\mathcal{P}$ to sign $\mathtt{M}$
6:pick at random $\zeta,\rho,\tau,\varphi,\omega\leftarrow\mathbb{Z}^{*}_{n}$
;
7:run $z=g_{2}^{\zeta}$,
$r={g_{2}}^{\alpha-\rho\tau-{\gamma_{z}}\zeta}{\mathtt{M}}^{-\gamma}$,
$s={g_{r}}^{\rho}$, $t={g_{2}}^{\tau}$,
$u={g_{2}}^{\beta-\varphi\omega-{\delta_{z}}\zeta}{\mathtt{M}}^{-\delta}$,
$v={h_{u}}^{\varphi}$, $w={g_{2}}^{\omega}$ ;
8:set $\sigma_{m}=(z,r,s,t,u,v,w)$ ;
9:// The next is set to generate a proof on equations
$\\{(\vec{\mathcal{A}_{im}},\vec{\mathcal{B}_{im}},\Gamma_{im},t_{im})\\}^{2}_{i=1}$
where $\vec{\mathcal{A}_{im}}$ = $\vec{\mathcal{B}_{im}}$ = $\vec{0}$,
$\Gamma_{im}$ = $\mathcal{MAT}_{3\times 3}(1)$ for $i=1,2$, $t_{1m}$ =
$t_{2m}$ = $1_{\mathbb{G}_{3}}$
10:$\vec{\mathcal{X}}_{1m}=(g_{z},g_{r},s)$,
$\vec{\mathcal{X}}_{2m}=(h_{z},h_{u},v)$,
$\vec{\mathcal{Y}}_{1m}=(z,{g_{2}}^{\alpha-\rho\tau-{\gamma_{z}}\zeta},t)$ and
$\vec{\mathcal{Y}}_{2m}=(z,{g_{2}}^{\beta-\rho\tau-{\delta{z}}\zeta},w)$ ;
11:$\pi_{m}=\\{(\vec{\mathcal{C}_{im}},\vec{\mathcal{D}_{im}},\pi_{im},\theta_{im})\\}^{2}_{i=1}\leftarrow\mathsf{NIWI}.\mathsf{Proof}(\mathtt{vk}_{g}$,
$\\{(\vec{\mathcal{A}_{im}},\vec{\mathcal{B}_{im}},\Gamma_{im},t_{im})\\}^{2}_{i=1},$
$\\{(\vec{\mathcal{X}_{im}},\vec{\mathcal{Y}_{im}})\\}^{2}_{i=1})$ ;
12:// The next is set to generate a proof on equations
$\\{(\vec{\mathcal{A}_{ip}},\vec{\mathcal{B}_{ip}},\Gamma_{ip},t_{ip})\\}^{4}_{i=1}$
where $\vec{\mathcal{A}}_{1p}=(g_{1}^{\alpha_{2}})$,
$\vec{\mathcal{A}}_{2p}=(g_{1}^{\beta_{2}})$,
$\vec{\mathcal{A}}_{3p}=(g_{1z},g_{1r})$,
$\vec{\mathcal{A}}_{4p}=(h_{1z},h_{1u})$,
$\vec{\mathcal{B}}_{1p}=(g_{2z},g_{2r})$,
$\vec{\mathcal{B}}_{2p}=(h_{2z},h_{2u})$,
$\vec{\mathcal{B}}_{3p}=(g_{2}^{\alpha_{1}})$,
$\vec{\mathcal{B}}_{4p}=(g_{2}^{\beta_{1}})$, $\Gamma_{1p}=(\gamma_{2z},-1)$,
$\Gamma_{2p}=(\delta_{2z},-1)$, $\Gamma_{3p}=(\gamma_{1z},-1)$,
$\Gamma_{4p}=(\delta_{1z},-1)$, $t_{1p}=e(g_{1}^{\alpha_{2}},g_{2r})$,
$t_{2p}=e(g_{1}^{\beta_{2}},h_{2u})$, $t_{3p}=e(g_{1r},g_{2}^{\alpha_{1}})$
and $t_{4p}=e(h_{1u},g_{2}^{\beta_{1}})$
13:$\vec{\mathcal{X}}_{1p}=(z_{1},{g_{1}}^{\alpha_{2}-\rho_{1}\tau_{1}-{\gamma_{2z}}\zeta_{1}})$,
$\vec{\mathcal{X}}_{2p}=(z_{1},{g_{1}}^{\beta_{2}-\rho_{1}\tau_{1}-{\delta_{2z}}\zeta_{1}})$,
$\vec{\mathcal{X}}_{3p}=(g_{1r})$, $\vec{\mathcal{X}}_{4p}=(h_{1u})$,
$\vec{\mathcal{Y}}_{1p}=(g_{2r})$, $\vec{\mathcal{Y}}_{2p}=(h_{2u})$,
$\vec{\mathcal{Y}}_{3p}=(z_{2},{g_{2}}^{\alpha_{1}-\rho_{2}\tau_{2}-{\gamma_{1z}}\zeta_{2}})$,
and
$\vec{\mathcal{Y}}_{4p}=(z_{2},{g_{2}}^{\beta_{1}-\rho_{2}\tau_{2}-{\delta_{1z}}\zeta_{2}})$
;
14:$\pi_{p}=\\{(\vec{\mathcal{C}_{ip}},\vec{\mathcal{D}_{ip}},\pi_{ip},\theta_{ip})\\}^{4}_{i=1}\leftarrow\mathsf{NIWI}.\mathsf{Proof}(\mathtt{vk}_{g},\\{(\vec{\mathcal{A}_{ip}},\vec{\mathcal{B}_{ip}},\Gamma_{ip},t_{ip})\\}^{4}_{i=1},$
$\\{(\vec{\mathcal{X}_{ip}},\vec{\mathcal{Y}_{ip}})\\}^{4}_{i=1})$ ;
15:set $\pi_{p}=((\pi_{ip},\theta_{ip})_{i=1}^{4})$ ;
16:set $\pi=(\pi_{p},\pi_{m})$ ;
17:return $(\mathtt{M},\sigma_{m},\pi)$
### V-C Verification phase
* •
$\mathsf{Sig\\_Verify_{\mathcal{HA}}}$ – Given a contact list of user
$\mathcal{U}_{A}$ (a list of tuples $(\mathtt{CCM}$, $\mathtt{M}$, $\pi)$ such
that $\pi$ can be parsed as
$\\{(\vec{\mathcal{A}_{i}},\vec{\mathcal{B}_{i}},\Gamma_{i},t_{i})\\}^{N}_{i=1}$,
$\\{(\vec{\mathcal{C}_{i}},\vec{\mathcal{D}_{i}},\pi_{i},\theta_{i})\\}^{N}_{i=1}$),
$\mathcal{HA}$ verifies the validity of the group signature of each message,
w.r.t. ${\mathsf{GSIG}}.\mathsf{Verify}$ algorithm (cf. Appendix IV-D).
* •
$\mathsf{CCM\\_Verify_{\mathcal{HA}}}$ – We consider that $\mathcal{HA}$
requests from $\mathcal{S}$ the message $\mathtt{PS^{\prime}}$ corresponding
to a contact message $\mathtt{CCM}$ contained in the contact list of user
$\mathcal{U}_{A}$. The message $\mathtt{PS^{\prime}}$ is taken as input with
the message $\mathtt{M}$ (corresponding to $\mathtt{CCM}$), the server’s
public key $\mathtt{pk}_{\mathcal{S}}$ and the secret value
$t_{\mathcal{U}_{A}}$ specific to user $\mathcal{U}_{A}$, to the
$\mathsf{CCM\\_Verify_{HA}}$ algorithm that checks if the equation 3 holds:
$\mathtt{M}={Y_{1}}^{t_{\mathcal{U}_{A}}\mathtt{PS^{\prime}}}{Y_{2}}^{t_{\mathcal{U}_{A}}}$
(3)
## VI Security and Privacy Analysis
In this section, we prove that SPOT achieves the defined security and privacy
requirements with respect to the threat models defined in Section III-D, by
relying on the following theorems and lemmas.
###### Theorem 1 (Unforgeability).
If a probabilistic-polynomial time (PPT) adversary $\mathcal{A}$ wins
$\textbf{Exp}_{\mathcal{A}}^{unforg}$, as defined in Section III-D1, with a
non-negligible advantage $\epsilon$, then a PPT simulator $\mathcal{B}$ can be
constructed to break the CDH assumption with a non-negligible advantage
$\epsilon$.
###### Proof.
In this proof, we show that a simulator $\mathcal{B}$ can be constructed with
the help of an adversary $\mathcal{A}$ having advantage $\epsilon$ against
SPOT scheme.
The CDH challenger $\mathcal{C}$ sends to $\mathcal{B}$ the tuple
$(g_{2},g_{2}^{a},g_{2}^{b})$, where $a,b\leftarrow\mathbb{Z}^{*}_{n}$ are
randomly selected. $\mathcal{C}$ asks $\mathcal{B}$ to compute $g_{2}^{ab}$.
Then, $\mathcal{B}$ sets $g_{2}^{t_{\mathcal{U}}}$ to $g_{2}^{a}$ and
$g_{2}^{y_{2}}$ to $g_{2}^{b}$. During the challenge phase, $\mathcal{B}$
randomly selects $y_{1}\in\mathbb{Z}^{*}_{n}$ and sends $g_{2}^{y_{1}}$ to
$\mathcal{A}$ as part of the server’s public key. $\mathcal{A}$ forges the
partial signature over the message $\mathtt{PS^{\prime}}$ and generates the
message $\mathtt{M}^{*}$ with advantage $\epsilon$: $\mathtt{M}^{*}$ =
${\mathtt{ID}_{\mathcal{U}}}^{\mathtt{PS}^{*}}$ =
${g_{2}}^{t_{\mathcal{U}}(\mathtt{PS^{\prime}}y_{1}+y_{2})}$. The tuple
$({g_{2}}^{t_{\mathcal{U}}},\mathtt{PS^{\prime}},\mathtt{M}^{*})$ is sent back
to $\mathcal{B}$. Upon receiving this tuple and knowing $y_{1}$, $\mathcal{B}$
can compute the value of $g_{2}^{t_{\mathcal{U}}y_{2}}$ which is the same as
$g_{2}^{ab}$ and can then send the result to the CDH challenger. As such,
$\mathcal{B}$ succeeds the forgery against the CDH assumption with advantage
$\epsilon$. ∎
###### Theorem 2 (Unlinkability).
Our SPOT system achieves the unlinkability requirement with respect to the
_group-signature unlinkability_ and _multi-CCM unlinkability_ properties.
We prove Theorem 2 through Lemma 3 and Lemma 4 with respect to _group-
signature unlinkability_ and _multi-CCM unlinkability_ properties,
respectively.
###### Lemma 3 (Group-signature unlinkability).
SPOT satisfies the group signature unlinkability requirement with respect to
the computational witness indistinguishability property of the NIWI proof.
###### Proof.
In this proof, the objective is to show that the adversary is not able to
distinguish group signatures issued by the same proxy. For this purpose, we
suppose that, for each session $i$, the adversary receives the message $M^{*}$
(i.e., the same message $M^{*}$ is returned by each oracle) and the NIWI proof
$\pi^{i}$ = $(\pi_{m}^{i},\pi_{p}^{i})$ =
$((\pi^{i}_{jm},\theta^{i}_{jm})_{j=1}^{2}$,
$(\pi^{i}_{jp},\theta^{i}_{jp})_{j=1}^{4})$.
To simplify the proof, we will only consider the NIWI proof $\pi_{m}^{i}$, as
the statements used to generate the proofs $\pi_{p}^{i}$ do not give any
information about the proxy generating the proof (i.e., statements do not
include the proxy’s public key). Thus, for each session $i$, the adversary is
given the tuples
$(\vec{\mathcal{C}^{i}_{k1}},\vec{\mathcal{D}^{i}_{k1}},\pi^{i}_{k1},\theta^{i}_{k1})$
and
$(\vec{\mathcal{C}^{i}_{k2}},\vec{\mathcal{D}^{i}_{k2}},\pi^{i}_{k2},\theta^{i}_{k2})$
referred to as the group signature generated by a proxy $P_{k}$, where
$k\in\\{0,1\\}$. During the challenge phase, the adversary is also given two
group signatures. The first signature is represented by the tuples
$(\vec{\mathcal{C}^{*}_{1}},\vec{\mathcal{D}^{*}_{1}},\pi^{*}_{1},\theta^{*}_{1})$
and
$(\vec{\mathcal{C}^{*}_{2}},\vec{\mathcal{D}^{*}_{2}},\pi^{*}_{2},\theta^{*}_{2})$
generated by proxy $\mathcal{P}_{0}$, while the second one is represented by
the tuples
$(\vec{\mathcal{C}^{*}_{b1}},\vec{\mathcal{D}^{*}_{b1}},\pi^{*}_{b1},\theta^{*}_{b1})$
and
$(\vec{\mathcal{C}^{*}_{b2}},\vec{\mathcal{D}^{*}_{b2}},\pi^{*}_{b2},\theta^{*}_{b2})$
and is generated by a proxy $\mathcal{P}_{b}$ ($b\in\\{0,1\\}).$
Let us consider a simulator $\mathcal{B}$ that can be constructed with the
help of an adversary $\mathcal{A}$ having advantage $\epsilon$ against SPOT
scheme. A challenger $\mathcal{C}$ selects two couples of witnesses
$(X_{0},Y_{0})$ and $(X_{1},Y_{1})$. $\mathcal{C}$ computes a commitment
$(C,D)$ over $(X_{0},Y_{0})$, and then selects a bit $b\in\\{0,1\\}$ and
computes a commitment $(C^{\prime}_{b},D^{\prime}_{b})$ over $(X_{b},Y_{b})$.
$\mathcal{C}$ asks $\mathcal{B}$ to guess the bit $b$. Then, $\mathcal{B}$
selects the tuples $(A,B,\Gamma,t)$ and
$(A^{\prime}_{b},B^{\prime}_{b},\Gamma^{\prime}_{b},t^{\prime}_{b})$ and
computes the proofs $(\pi,\theta)$ and
$(\pi^{\prime}_{b},\theta^{\prime}_{b})$. $\mathcal{B}$ returns the two proofs
to $\mathcal{A}$. Finally, $\mathcal{A}$ outputs a bit $b^{\prime}$ that it
sends to $\mathcal{B}$. This latter outputs the same bit $b^{\prime}$ to its
own challenger $\mathcal{C}$. As such, $\mathcal{A}$ succeeds in breaking the
group-signature unlinkability with advantage $\epsilon$, which is the same as
breaking the computational witness-indistinguishability property.
∎
###### Corollary 3.1.
If SPOT satisfies the unlinkability property, then the proxies’ group
signature (i.e., NIWI proof) fulfills the anonymity requirement stating that
it is not possible to identify the proxy that issued a particular group
signature.
###### Lemma 4 (Multi-CCM unlinkability).
SPOT satisfies the multi-CCM unlinkability requirement with respect to the
common contact message structure.
###### Sketch of proof.
Let $\mathcal{A}$ be a successful adversary against the _multi-CCM
unlinkability_ property. Assume that $\mathcal{A}$ receives two messages
$\mathtt{CCM_{1}}=\mathbf{H}(\mathtt{D}_{i}*\mathtt{D}_{j})$ and
$\mathtt{CCM_{2}}=\mathbf{H}(\mathtt{D}_{i}*\mathtt{D}_{k})$ (with $j\neq k$)
meaning that user $U_{i}$ met $U_{j}$ and $U_{k}$, then $\mathcal{A}$ is not
able to link $\mathtt{CCM_{1}}$ and $\mathtt{CCM_{2}}$ to the same user
$\mathcal{U}_{i}$ as all EBIDs are randomly generated in each epoch $e$, and
the hashing function $\mathbf{H}$ behaves as a pseudo-random function. ∎
###### Theorem 5 (Anonymity).
SPOT satisfies the anonymity property, in the sense of Definition 3, if and
only if, the CCM-unlinkability requirement is fulfilled.
###### Sketch of proof.
We prove that our proximity-based protocol SPOT satisfies the anonymity
property using an _absurdum_ reasoning. We suppose that an adversary
$\mathcal{A}$ can break the anonymity of SPOT, in the sense of Definition 3,
by reaching the advantage
$Pr[\textbf{Exp}_{\mathcal{A}}^{anon}(1^{\lambda})=1]\geq\frac{1}{2}\pm\kappa(\lambda)$.
$\mathcal{A}$ is given the pair of public-private keys
$(\mathtt{pk}_{\mathcal{HA}},\mathtt{sk}_{\mathcal{HA}})$ of the health
authority, the identifiers $\mathtt{ID}_{\mathcal{U}}$ of all users and a
contact list of a particular user $\mathcal{U}_{A}$, obtained when relying on
several sessions. Then, relying on the _left-or-right_ $\mathsf{LoRCU}$
oracle, $\mathcal{A}$ tries to distinguish the user $\mathcal{U}_{b}$ being in
contact with $\mathcal{U}_{A}$, better than a flipping coin. That is, given
the tuple $({\mathsf{CCM}}^{*}_{b},\mathtt{M}^{*}_{b},\pi^{*}_{b})$,
$\mathcal{A}$ successfully predicts the identifier
$\mathtt{ID}_{\mathcal{U}_{b}}$. Obviously, $\mathcal{A}$ tries to identify
user $\mathcal{U}_{b}$ relying on the message ${\mathsf{CCM}}^{*}_{b}$, since
both $\mathtt{M}^{*}_{b}$ and $\pi^{*}_{b}$ are generated based on
${\mathsf{CCM}}^{*}_{b}$ and give no further information about the user
$\mathcal{U}_{b}$. This refers to link the message ${\mathsf{CCM}}^{*}_{b}$ to
its issuer $\mathcal{U}_{b}$. Thus, if $\mathcal{A}$ succeeds, this means that
$\mathcal{A}$ is able to link two or several common contact messages to the
same user, which contradicts the _multi-CCM unlinkability_ property previously
discussed. As such, we prove that the adversary succeeds
$\textbf{Exp}_{\mathcal{A}}^{anon}(1^{\lambda})$ with a probability
$Pr=\frac{1}{2}\pm\kappa(\lambda)$, where $\kappa(\lambda)$ is negligible.
Thus, SPOT satisfies anonymity.
∎
###### Theorem 6 (Anti-replay).
SPOT satisfies the anti-replay requirement and supports false positive
hindrance, if the proposed scheme is unforgeable.
###### Sketch of proof.
To successfully replay a common contact message generated in an epoch $e$, in
another epoch $e^{\prime}\neq e$, a malicious user can perform in two ways.
(i) The user reinserts, in his contact list, the tuple $(\mathtt{CCM}^{e}$,
$\mathtt{M}^{e}$, $\pi^{e})$ generated in an epoch $e$. The reinsertion is
performed in an epoch $e^{\prime}>e+\Delta$111111It makes no sense to reinsert
an element in an epoch $e^{\prime}<e+\Delta$, as duplicated messages will be
deleted either by the server or at the user’s end-device.. Afterwards, the
contact list is sent to $\mathcal{HA}$ when the user is infected.
$\mathcal{HA}$ asks the server to provide the message $\mathtt{PS}^{\prime}$
corresponding to $\mathtt{CCM}^{e}$. As the server has no entry corresponding
to $\mathtt{CCM}^{e}$ in the last $\Delta$ days, the second verification
performed by $\mathcal{HA}$ does not hold and the tuple is rejected. (ii) We
assume that, in an epoch $e^{\prime}>e+\Delta$, the user is able to replay a
message $\mathtt{CCM}^{e}$ with two different proxies and he successfully
receives the corresponding message $\mathtt{M}$ and the group signature $\pi$.
Thus, when the user is infected, the health authority validates false
positives, but this has no impact on the computation of the risk score, as no
user has the same entry in his contact list. As such, we can prove the
resistance of SPOT against replay attacks. ∎
## VII Performance Analysis
This section introduces SPOT test-bed, discusses the experimental results,
presented in Table III, and demonstrates the usability of the proposed
construction for real world scenarios.
### VII-A SPOT Test-bed
For our experiments, we developed a prototype of the SPOT protocol that
implements the three phases Sys_Init, Generation and Verification including
the twelve algorithms121212The source code is available at
https://github.com/privteam/SPOT. The tests are made on an Ubuntu $18.04.3$
machine - with an _Intel_ Core<EMAIL_ADDRESS>\- 4 cores processor and $8GB$
memory. The twelve algorithms were implemented based on JAVA version $11$, and
the cryptographic library JPBC131313http://gas.dia.unisa.it/projects/jpbc/. We
evaluate the computation time of each algorithm relying on two types of
bilinear pairings, i.e., type A and type F. The pairing type A is the fastest
symmetric pairing type in the JPBC library constructed on the curve
$y^{2}=x^{3}+x$ with an embedding degree equal to 2. The pairing type F is an
asymmetric pairing type introduced by Barreto and Naehrig [2]. It has an
embedding degree equal to 12. For the two types of pairing, we consider two
different levels of security i.e., 112-bits and 128-bits security levels
recommended by the US National Institute of Standards and
Technology141414http://keylength.com (NIST).
Based on the selected cryptographic library and the implementation of Groth-
Sahai proofs151515https://github.com/gijsvl/groth-sahai, the SPOT test-bed is
built with six main java classes, w.r.t. to the different entities of SPOT,
referred to as _TrustedAuthority.java_ , _GroupManager.java_ , _Proxy.java_ ,
_HealthAuthority.java_ , _User.java_ and _Server.java_. Each class encompasses
the algorithms that are performed by the relevant entity as described in
Section III-A. In order to obtain accurate measurements of the computation
time, each algorithm is run 100 times. Thus, the computation times represent
the mean of the 100 runs while considering a standard deviation of an order
$10^{-2}$.
### VII-B Communication and Computation Performances of SPOT
This section first proposes a theoretical analysis of the communication cost.
Then, it presents the experimental results of the implementation of SPOT
algorithms.
TABLE III: Computation time and communication overhead of SPOT algorithms
Algorithm | Entity | Synch/Asynch | Communication cost | Computation time (ms)
---|---|---|---|---
A/112-bits | A/128-bits | F/112-bits | F/128-bits
$\mathsf{Set\\_params}$ | $\mathcal{TA}$ | Asynch. | $|\mathbb{Z}_{n}|+|\mathbb{G}_{1}|+|\mathbb{G}_{2}|+|\mathbb{G}_{3}|$ | 874 | 2521 | 1230 | 1364
$\mathsf{HA\\_Keygen}$ | $\mathcal{TA}$ | Asynch. | $|\mathbb{G}_{1}|$ | 59 | 123 | 12 | 16
$\mathsf{S\\_Keygen}$ | $\mathcal{TA}$ | Asynch. | $2|\mathbb{G}_{2}|$ | 119 | 244 | 24 | 31
$\mathsf{Setup\\_ProxyGr}$ | $\mathcal{GM}$ | Asynch. | $21(|\mathbb{G}_{1}|+|\mathbb{G}_{2}|)$ | 1955 | 4075 | 346 | 451
$\mathsf{Join\\_ProxyGr}$ | $\mathcal{P}$/$\mathcal{GM}$ | Synch. | $\mathcal{P}:8|\mathbb{G}_{1}|+2|\mathbb{G}_{2}|$ / $\mathcal{GM}:7(|\mathbb{G}_{1}|+|\mathbb{G}_{2}|)$ | 2861 | 6014 | 1159 | 1409
$\mathsf{Set\\_UserID}$ | $\mathcal{HA}$ | Synch. | $|\mathbb{G}_{2}|$ | 58 | 121 | 12 | 16
$\mathsf{Userkeygen}$ | $\mathcal{U}$ | Asynch. | $|\mathbb{G}_{2}|$ | 117 | 242 | 24 | 31
$\mathsf{Set\\_CCM}$ | $\mathcal{U}$ | Synch. | $|\mathbb{Z}_{n}|$ | 0.1 | 0.1 | 0.1 | 0.1
$\mathsf{S\\_PSign}$ a | $\mathcal{S}$ | Synch. | $|\mathbb{Z}_{n}|$ | 0.1 | 0.08 | 0.02 | 0.02
$\mathsf{P\\_Sign}$ a | $\mathcal{P}$ | Synch. | $6|\mathbb{G}_{1}|+7|\mathbb{G}_{2}|$ | 19353 | 40371 | 3164 | 4170
$\mathsf{Sig\\_Verify}$ a | $\mathcal{HA}$ | Asynch. | N.A. | 6541 | 15406 | 31637 | 36892
$\mathsf{CCM\\_Verify}$ a | $\mathcal{HA}$ | Asynch. | N.A. | 174 | 360 | 148 | 190
NOTE: Synch./Asynch. indicates whether the algorithm must be run online (i.e.,
in real time) or offline (i.e., later); a indicates that the algorithm is
performed on a single contact message that is generated by the
$\mathsf{Set\\_CCM}$ algorithm; $|\mathbb{G}_{1}|$ (resp. $|\mathbb{G}_{2}|$,
$|\mathbb{G}_{3}|$ and $|\mathbb{Z}_{n}|$) indicates the size of an element in
${\mathbb{G}_{1}}$ (resp. ${\mathbb{G}_{2}}$, ${\mathbb{G}_{3}}$ and
${\mathbb{Z}_{n}}$); N.A. is the abbreviation for Not Applicable.
As presented in Table III, the communication cost is measured according to the
size of the elements ${\mathbb{G}_{1}}$, ${\mathbb{G}_{2}}$,
${\mathbb{G}_{3}}$ and ${\mathbb{Z}_{n}}$ exchanged between entities.
$\mathsf{Setup\\_ProxyGr}$ and $\mathsf{Join\\_ProxyGr}$ are the most
bandwidth consuming algorithms, however this result must be put into
perspective as both algorithms are performed once. Other algorithms have
acceptable communication overhead, in particular those performed repeatedly by
the user, which proves the efficiency of SPOT.
From Table III, it is worth noticing that the computation time depends on the
selected pairings types and is strongly related to the security level. Some
algorithms of the Sys_Init phase are consuming but they are limited to only
one execution from a powerful trusted authority. For the Generation phase, the
most consuming algorithm is $\mathsf{P\\_Sign}$ which requires 19 seconds
(resp. 40 seconds) for pairing type A and 3 seconds (resp. 4 seconds) for
pairing type F. The computation time of $\mathsf{Set\\_CCM}$ and
$\mathsf{S\\_PSign}$ are negligible, which means that the user and the server
are not required to have important computation capacities. Finally, to verify
the correctness of a single contact message, the Verification phase requires
approximately 7 seconds (resp. 15 seconds) for pairing type A and 32 seconds
(resp. 37 seconds) for pairing type F.
It is clear that $\mathsf{P\\_Sign}$ and $\mathsf{Sig\\_Verify}$ are the most
consuming algorithms in terms of computation time as they include a large
number of exponentiations and pairing functions, however this result must be
put into perspective as both the proxy and the health authority are assumed to
have advanced hardware features. Table III also shows that the
$\mathsf{Sig\\_Verify}$ algorithm run with pairing type A is faster than with
pairing type F, as this latter requires excessive memory allocation and
deallocation. The $\mathsf{P\\_Sign}$ algorithm has an opposite behavior where
the execution with pairing type F is faster than pairing type A, which is
compliant to the JPBC library
benchmark161616http://gas.dia.unisa.it/projects/jpbc/benchmark.html showing
that elementary functions of multiplication and exponentiation require less
computation time for pairing type F.
From Table III, we can deduce that the algorithms executed at the user’s side,
have very low computation and communication overhead, which confirms the
usability of SPOT, even when being run on a smartphone with low capacities.
For both consuming algorithms ($\mathsf{P\\_Sign}$ and
$\mathsf{Sig\\_Verify}$) that are repeatedly run, some performance improvement
means are proposed in the next subsection.
### VII-C Improved Performances with Multithreading and Preprocessing
For both computation consuming algorithms $\mathsf{P\\_Sign}$ and
$\mathsf{Sig\\_Verify}$ (see Section VII-B), in an effort to make the
computation time as efficient as possible, although they run on powerful SPOT
entities, we rely on a two step improvement:
* •
Multithreading: applied to both $\mathsf{P\\_Sign}$ and
$\mathsf{Sig\\_Verify}$ algorithms. It enables simultaneous multiple threads
execution (e.g., the computation of the different parts of the NIWI proof for
the $\mathsf{P\\_Sign}$ algorithm, the computation of either the different
verification equations of the NIWI proof, or the two sides of each equation,
for the $\mathsf{Sig\\_Verify}$ algorithm).
* •
Preprocessing: applied only to $\mathsf{Sig\\_Verify}$ algorithm. It enables
to prepare in advance a value to be later paired several times, like the
variables $\mathcal{U}$ and $\mathcal{V}$ which are used as input to pairing
functions for each verification equation.
(a) Influence of multithreading on $\mathsf{P\\_Sign}$ algorithm
(b) Influence of preprocessing or/and multithreading on
$\mathsf{Sig\\_Verify}$ algorithm
Figure 3: Influence of improvements on $\mathsf{P\\_Sign}$ and
$\mathsf{Sig\\_Verify}$ algorithms
Figure 3 exposes the impacts of one or two combined improvements applied to
$\mathsf{P\\_Sign}$ and $\mathsf{Sig\\_Verify}$. From Figure 3(a), we notice
that multithreading reduces the computation time for $\mathsf{P\\_Sign}$ of
approximately $35\%$, for the two types of pairing and the two levels of
security. For $\mathsf{Sig\\_Verify}$, Figure 3(b) shows that multithreading
has a greater impact on the computation time (i.e., approximately $40\%$ for
pairing type A and $28\%$ for pairing type F) than preprocessing (i.e.,
approximately $10\%$ for pairing type A and $5\%$ for pairing type F 171717For
type F - 128 bits, the preprocessing decreases the performances. This is due
to the excessive memory allocation and deallocation required by the pairing
type F.). The two combined improvements ensure a gain of almost $50\%$ for
pairing type A and $30\%$ for pairing type F.
## VIII Conclusion
In this paper, a novel secure and privacy-preserving proximity-based SPOT
protocol for e-healthcare systems is introduced. The objective of SPOT is to
help governments and healthcare systems to deal with pandemics by automating
the process of contact tracing, with security guarantees against fake contacts
injection and privacy preservation for users. Thanks to the underlying network
architecture relying on a centralized computing server and decentralized
proxies, SPOT enables users to determine whether they were in close proximity
with infected people, with no risk of false positive alerts. The strength of
the paper is to provide a full concrete construction of SPOT which is proven
to be secure and to support several privacy properties under standard
assumptions. Another strength of the contribution is a PoC of SPOT including a
full implementation of the different algorithms, where practical computation
costs measurements demonstrate the feasibility of our proposed protocol.
Further research will consider aggregating the verification of multiple
contact messages in an effort to improve verification performances.
## References
* [1] Ca notify. https://canotify.uchealth.edu/ucberkeley/.
* [2] Jpbc library: Bilinear pairing parameters generators. http://gas.dia.unisa.it/projects/jpbc/docs/ecpg.html.
* [3] Stop covid-19 app. https://github.com/Stop-COVID-19-Croatia/stopcovid19-docs, [Online accessed June 2022].
* [4] Masayuki Abe, Kristiyan Haralambiev, and Miyako Ohkubo. Signing on elements in bilinear groups for modular protocol design. IACR Cryptology ePrint Archive, 2010.
* [5] Apple Inc. and Google Inc. Contact tracing, bluetooth specification. https://www.blog.google/documents/58/Contact_Tracing_-_Bluetooth_, 2020\.
* [6] Australian Government Department of Health. Covidsafe app. https://www.health.gov.au/resources/apps-and-tools/covidsafe-app, 2020.
* [7] Troncoso Carmela, Payer Mathias, Hubaux Jean-Pierre, Salathé Marcel, Larus James, Bugnion Edouard, Lueks Wouter, Stadler Theresa, Pyrgelis Apostolos, Antonioli Daniele, Barman Ludovic, Chatel Sylvain, Paterson Kenneth, Capkun Srdjan, Basin David, Beutel Jan, Jackson Dennis, Roeschlin Marc, Leu Patrick, Preneel Bart, Smart Nigel, Abidin Aysajan, and et al. Decentralized privacy-preserving proximity tracing. https://github.com/DP-3T/documents/blob/master/DP3T%20White%20Paper.pdf, 2020\.
* [8] Justin Chan, D. Foster, Shyam Gollakota, E. Horvitz, J. Jaeger, Sham M. Kakade, T. Kohno, J. Langford, J. Larson, S. Singanamalla, J. Sunshine, and S. Tessaro. Pact: Privacy-sensitive protocols and mechanisms for mobile contact tracing. 2020\.
* [9] Castelluccia Claude, Bielova Nataliia, Boutet Antoine, Cunche Mathieu, Lauradoux Cedric, Le Metayer Daniel, and Roca Vincent. Desire: A third way for a european exposure notification system. https://github.com/3rd-ways-for-EU-exposure-notification/project-DESIRE/blob/master/DESIRE-specification-EN-v1_0.pdf, 2020\.
* [10] Jens Groth and Amit Sahai. Efficient non-interactive proof systems for bilinear groups. In Nigel Smart, editor, Advances in Cryptology – EUROCRYPT 2008, pages 415–432, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg.
* [11] Asghar Hassan, Farokhi Farhad, Kaafar Dali, and Rubinstein Ben. On the privacy of tracetogether, the singaporean covid-19 contact tracing app and recommendations for australia, 2020.
* [12] Jaap-Henk Hoepman. Hansel and gretel and the virus: Privacy conscious contact tracing. arXiv preprint arXiv:2101.03241, 2021.
* [13] Inria and Fraunhofer AISEC. Robert: Robust and privacy-preserving proximity tracing. https://github.com/ROBERT-proximity-tracing/documents/blob/master/ROBERT-specification-EN-v1_1.pdf, 2020, [Online accessed June 2022].
* [14] K. Liu Joseph, Au Man Ho, Yuen Tsz Hon, Zuo Cong, Wang Jiawei, Sakzad Amin, Luo Xiapu, and Li Li. Privacy-preserving covid-19 contact tracing app: A zero-knowledge proof approach. IACR Cryptol. ePrint Arch, 2020, (528), 2020.
* [15] Douglas J Leith and Stephen Farrell. Contact tracing app privacy: What data is shared by europe’s gaen contact tracing apps. In IEEE INFOCOM 2021, pages 1–10. IEEE, 2021.
* [16] Krzysztof Pietrzak. Delayed authentication: Preventing replay and relay attacks in private contact tracing. In Progress in Cryptology – INDOCRYPT 2020, pages 3–15, Cham, 2020\. Springer International Publishing.
* [17] Leonie Reichert, Samuel Brack, and BjÖRN Scheuermann. A survey of automatic contact tracing approaches using bluetooth low energy. ACM Trans. Comput. Healthcare, 2021.
* [18] Rivest Ronald L., Callas Jon, Canetti Ran, Esvelt Kevin, Gillmor Daniel Kahn, Kalai Yael Tauman, Lysyanskaya Anna, Norige Adam, Raskar Ramesh, Shamir Adi, Shen Emily, Soibelman Israel, Specter Michael, Teague Vanessa, Trachtenberg Ari, Varia Mayank, Viera Marc, Weitzner Daniel, Wilkinson John, and Zissman Marc. The pact protocol specification, 2020.
* [19] Viktoriia Shubina, Sylvia Holcer, Michael Gould, and Elena Simona Lohan. Survey of decentralized solutions with mobile devices for user location tracking, proximity detection, and contact tracing in the covid-19 era. Data, 5(4), 2020.
* [20] Sahraoui Yesin, Ludovica Lucia, Anna Maria Vegni, Chaker Kerrache, Marica Amadeo, and Ahmed Korichi. Traceme: Real-time contact tracing and early prevention of covid-19 based on online social networks. pages 893–896, 2022.
## Appendix A Non-Interactive Witness Indistinguishable Proof
In this section we represent the Groth-Sahai NIWI proof scheme applied on
pairing product equations with an asymmetric bilinear map. Witness-
indistinguishability implies that the verifier of a group signature is not
able to find the group member that has generated the signature. The
$\mathsf{NIWI}$ scheme we consider, involves four PPT algorithms
($\mathsf{NIWI}.\mathsf{Setup}$, $\mathsf{NIWI}.\mathsf{CRS}$,
$\mathsf{NIWI}.\mathsf{Proof}$, $\mathsf{NIWI}.\mathsf{Verify}$):
$\mathsf{NIWI}.\mathsf{Setup}$: This algorithm outputs a setup
$(\mathtt{gk},\mathtt{sk})$ such that $\mathtt{gk}=(n$, $\mathbb{G}_{1}$,
$\mathbb{G}_{2}$, $\mathbb{G}_{3}$, $g_{1}$, $g_{2}$, $e)$ and
$\mathtt{sk}=(p,q)$ where $n=pq$.
$\mathsf{NIWI}.\mathsf{CRS}$: This algorithm generates a common reference
string $\mathtt{CRS}$. It takes $(\mathtt{gk},\mathtt{sk})$ as inputs and
produces $\mathtt{CRS}$ =
$(\mathbb{G}_{1},\mathbb{G}_{2},\mathbb{G}_{3},e,\iota_{1},p_{1},\iota_{2},p_{2},\iota_{3},p_{3},\mathcal{U},\mathcal{V})$,
where $\mathcal{U}=rg_{1}$, $\mathcal{V}=sg_{2}$ ; $r,s\in\mathbb{Z}^{*}_{n}$
and
$\iota_{1}$: | $\mathbb{G}_{1}\longrightarrow\mathbb{G}_{1}$ | $\iota_{2}$: | $\mathbb{G}_{2}\longrightarrow\mathbb{G}_{2}$ | $\iota_{3}$: | $\mathbb{G}_{3}\longrightarrow\mathbb{G}_{3}$
---|---|---|---|---|---
| $x\longmapsto x$ | | $y\longmapsto y$ | | $z\longmapsto z$
$p_{1}$: | $\mathbb{G}_{1}\longrightarrow\mathbb{G}_{1}$ | $p_{2}$: | $\mathbb{G}_{2}\longrightarrow\mathbb{G}_{2}$ | $p_{3}$: | $\mathbb{G}_{3}\longrightarrow\mathbb{G}_{3}$
| $x\longmapsto\lambda x$ | | $y\longmapsto\lambda y$ | | $z\longmapsto z^{\lambda}$
$\mathsf{NIWI}.\mathsf{Proof}$: This algorithm generates a NIWI proof for
satisfiability of a set of pairing product equations of the form of
${\prod}^{l}_{i=1}e(\mathcal{A}_{i},\mathcal{Y}_{i}){\prod}^{k}_{i=1}e(\mathcal{X}_{i},\mathcal{B}_{i}){\prod}^{k}_{i=1}{\prod}^{l}_{j=1}e(\mathcal{X}_{i},\mathcal{Y}_{j})^{\gamma_{ij}}=t$
also written as
$(\vec{\mathcal{A}}\cdot\vec{\mathcal{Y}})(\vec{\mathcal{X}}\cdot\vec{\mathcal{B}})(\vec{\mathcal{X}}\cdot\Gamma\vec{\mathcal{Y}})=t$
It takes as input $\mathtt{gk}$, $\mathtt{CRS}$ and a list of pairing product
equations
$\\{(\vec{\mathcal{A}_{i}},\vec{\mathcal{B}_{i}},\Gamma_{i},t_{i})\\}^{N}_{i=1}$
and a satisfying witness $\vec{\mathcal{X}}\in\mathbb{G}_{1}^{k}$,
$\vec{\mathcal{Y}}\in\mathbb{G}_{2}^{l}$. To generate a proof over a pairing
product equation, the algorithm, first, picks at random $\mathcal{R}\leftarrow
Vec_{k}(\mathbb{Z}_{n})$ and $\mathcal{S}\leftarrow Vec_{l}(\mathbb{Z}_{n})$,
commits to all variables as
$\vec{\mathcal{C}}:=\vec{\mathcal{X}}+\mathcal{R}\mathcal{U}$ and
$\vec{\mathcal{D}}:=\vec{\mathcal{Y}}+\mathcal{S}\mathcal{V}$, and computes
${\pi}=\mathcal{R}^{\top}\iota_{2}(\vec{\mathcal{B}})+\mathcal{R}^{\top}\Gamma\iota_{2}(\vec{\mathcal{Y}})+\mathcal{R}^{\top}\Gamma\mathcal{S}\mathcal{V}$
${\theta}=\mathcal{S}^{\top}\iota_{1}(\vec{\mathcal{A}})+\mathcal{S}^{\top}\Gamma^{\top}\iota_{1}(\vec{\mathcal{X}})$
The algorithm outputs the proof $(\pi,\theta)$.
$\mathsf{NIWI}.\mathsf{Verify}$: This algorithm checks if the proof is valid.
It takes $\mathtt{gk}$, $\mathtt{CRS}$,
$\\{(\vec{\mathcal{A}_{i}},\vec{\mathcal{B}_{i}},\Gamma_{i},t_{i})\\}^{N}_{i=1}$
and
$(\vec{\mathcal{C}_{i}},\vec{\mathcal{D}_{i}},\\{(\pi_{i},\theta_{i})\\}^{N}_{i=1})$
as inputs and for each equation, checks the following equation:
$e(\iota_{1}(\vec{\mathcal{A}_{i}}),\vec{\mathcal{D}_{i}})e(\vec{\mathcal{C}_{i}},\iota_{2}(\vec{\mathcal{B}_{i}}))e(\vec{\mathcal{C}_{i}},\Gamma_{i}\vec{\mathcal{D}_{i}})=\iota_{3}(t_{i})e(\mathcal{U},\pi_{i})e(\theta_{i},\mathcal{V})$
(4)
The algorithm outputs 1 if the equation holds, else it outputs 0.
|
# Tetrahedron of flavors:
One Three to rule them all
Aharon Davidson<EMAIL_ADDRESS>https://physics.bgu.ac.il/~davidson
Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105,
Israel
(October10, 2022)
###### Abstract
Under the conception that the total number three of fermion families must have
the one and the same gauge theoretical origin as all other threes which
accompany the single family grand unifiable group structure, we trade the
trinification $SU(3)$ symmetry building block for its semi-simple Vertical-
Horizontal symmetric $SU(3)_{V}\otimes SU(3)_{H}$ extension. The anomaly free
flavor-chiral fermion representation is then constructed, solely out of threes
and without any superfluous replication, by treating each standard $V$-triplet
(and anti triplet) as an $H$-singlet, and conversely, by letting each standard
$V$-singlet transform as an $H$-triplet (or anti triplet). The model can be
schematically described by a tetrahedron of flavors. In its bi-trinification
phase, the model exhibits two coupled trinification cycles, sharing the same
color group, and is furthermore accompanied by a built-in dark sector. In its
isomorphic quartification phase, the model presents a novel non-Pati-Salam
version of quark/lepton correspondence, where quarks are paired with anti-
leptons to cancel horizontal anomalies, and all fermion masses stem from one
and the same Yukawa source.
## I Introduction
Decoding the mass spectrum of quarks and leptons is without any doubt the holy
grail of theoretical particle physics. While the mass generating Higgs
mechanism, mediated by Yukawa couplings, is rooted in the heart of the
standard electro/nuclear theory, we still do not have the slightest idea what
physics actually determines the eigenmasses and mixings. At this stage, even
an empirical formula to account for the observed mass hierarchy is certainly
welcome. The structure of the Fermi mass matrix is however just the tip of the
flavor puzzle iceberg.
The various fermionic pieces which make a single family (or a generation) are
well organized, up to some electro/nuclear neutral members, within the
framework of the by now standard $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$
gauge theory. Anomaly cancellation is then a necessary self consistency
condition, but obviously not a sufficient one for shedding light on the full
picture. The consequent grand unification procedure to combine the otherwise
separated fermionic pieces into a single representation has been exclusively
realized for the $SU(5)\subset SO(10)\subset E(6)$ group sequence, and is most
welcome, but leaves a fundamental question unanswered. Namely, why can/must
grand unification be realized (at least partially) already at the single
family level, while apparently leaving the observed multiplicity of fermion
families field/group theoretically out of the game.
At the time, embedding the single family GUT within a larger multi family GUT
seemed to be a logical step forwards. Unfortunately, such a potentially
promising direction has been practically exhausted, and so far failed to
deliver. On the short list of theoretical (on top of experimental) obstacles
encountered one can find:
(i) The magic yet theoretically challenging number of exactly _three_ standard
fermion families.
(ii) The restrictive chiral flavor structure required at the low energy group
theoretical reduction level.
(iii) The ever growing sizes (counting generators) of the candidate GUT groups
involved.
(iv) Heavy field theoretical artillery, such as super-symmetry and dimensional
reduction, while being attractive from various other reasons, has not
contributed so far its part to the flavor puzzle. The more so superstring
theory, including its string phenomenology outer branch, has not been proven
too useful either.
It may well be that the tempting strategy of passing through the single family
grand unification stage may have counter intuitively blocked us from unveiling
the full flavor picture.
## II Vertical $\leftrightarrow$ Horizontal symmetry
Equipped with no compelling answers, the flavor puzzle has sourced plenty of
imaginative ideas, ranging from atomic style isotopes all the way to
compositeness, involving a variety of field theoretical techniques. In this
paper, however, with the focus solely on group theory, we humbly follow the
hypothesis that the total number _three_ of fermion families shares the one
and the same gauge theoretical origin as all other _threes_ which govern the
single family unifiable (not grand unified) group structure. Adopting this
line of thought, let our starting point be trinification 333 ; trinification ,
based on the semi-simple gauge group
$G=SU(3)_{C}\otimes SU(3)_{L}\otimes SU(3)_{R}~{}.$ (1)
An accompanying $C_{3}$ discrete symmetry is optional, and becomes mandatory
if a common gauge coupling constant is in order. Associated with is the
familiar anomaly free flavor chiral representation of left handed fermions
$\psi_{L}=\begin{array}[]{|c ||c|c|c|}\hline\cr q&3&3^{\ast}&1\\\ \hline\cr
q^{c}&3^{\ast}&1&3\\\ \hline\cr\ell+\ell^{c}&1&3&3^{\ast}\\\
\hline\cr\end{array}$ (2)
which can always be supplemented, if so required, by a bunch of dark
$G$-singlets
$\begin{array}[]{|c ||c|c|c|}\hline\cr~{}~{}\chi{}{}&~{}1{}&~{}1{}&~{}1{}\\\
\hline\cr\end{array}$ (3)
Schematically, the model can be neatly represented by a triangle, as depicted
in Fig.1, where the vertices represent fermions and the edges stand for the
various $SU(3)$ group factors involved.
Figure 1: The trinification model: The vertices are associated with the
various fermions involved., and the three edges stand for the three $SU(3)$
group factors involved. An incoming/outgoing arrow signals a triplet/anti-
triplet of the respective $SU(3)$. Schematically, blue and red circles
represent triplets and anti-triplets respectively
Already at the trinification level, prior to $E(6)$ unification E6 , there is
a heavy price (not everyone is willing) to pay, namely extending the standard
single fermion family, composed of 16 members, to include extra 27-16=11 non-
standard members. The latter supplement, easily classified by means of the
electric charge operator
$Q=T_{3L}+T_{3R}+\frac{1}{2}(Y_{L}+Y_{R})~{},$ (4)
decouples however once $SU(3)_{L}\otimes SU(3)_{R}$ parent symmetry breaks
down to its left-right symmetric sub-group $SU(2)_{L}\otimes SU(2)_{R}\otimes
U(1)_{B-L}$. Furthermore, notice that the full quark/lepton correspondence,
i.e. treating lepton number as the forth color, PS is not restored until (and
if) $E(6)$ embedding is realized.
Multiplying the trinification group $G$ by extra $SU(3)$ factors is then quite
a conservative step forwards. $SU(3)^{N}$ models, with $N>3$, mostly single
family models, have been extensively studied by Ma SUnk and by others SU34 .
Horizontal symmetries horizontal , primarily invoked to classify the otherwise
electro/nuclear degenerate fermionic families, start from minimal $U(1)$,
local or else global (incorporating the Peccei-Quinn mechanism familon ) and
in particular include an $SU(3)$ example HSU3 . Here, we revive the idea of
Vertical $\leftrightarrow$ Horizontal symmetry VH , and trade the single
family group, now referred to as $G_{V}$, for its $G_{V}\otimes G_{H}$ semi-
simple extension. Note that models incorporating semi-simple group structure,
for example Pati-Salam’s $SU(4)^{4}$ PS , first ever unification scheme
(originally designed for two families), the hybrid single family left-right
symmetric $SU(5)_{L}\otimes SU(5)_{R}$ model 5x5 hosting chiral color
chiralcolor , and multi family models of the $SO(10)\otimes SO(10)$ type 10x10
, have already been discussed in the literature.
To construct the tenable $G_{V}\otimes G_{H}$ fermion representation, we adopt
the following prescription:
_Treat each standard $V$-triplet (and anti-triplet) as an $H$-singlet, and
conversely, let each standard $V$-singlet transform as an $H$-triplet (or
anti-triplet)._
This prescription generalizes Eq.(2) into
$\begin{array}[]{ccc}&~{}~{}\quad\quad V\quad\quad\quad\quad H&\\\
&\psi_{L}=\begin{array}[]{|c |c |c ||c |c |c |}\hline\cr 3&3^{\ast}&1&1&1&x\\\
\hline\cr 3^{\ast}&1&3&1&y&1\\\ \hline\cr 1&3&3^{\ast}&z&1&1\\\
\hline\cr\hline\cr 1&1&1&z^{\ast}&y^{\ast}&x^{\ast}\\\
\hline\cr\end{array}&\end{array}$ (5)
The cancelation of ABJ anomalies is done pairwise, while maintaining overall
flavor chirality, so that each of the $SU(3)$ representations $x,y,z$ can be
either a $3$ or else a $3^{\ast}$. While the particular choice seems
irrelevant from any individual $SU(3)$ point of view, a matter of definition,
it does become relevant in the presence of an accompanying discrete symmetry.
Altogether, with only $SU(3)$ triplets and anti-triplets at our disposal, only
two independent configurations exist.
The manifestly symmetric configuration
$x=y=z\quad\textrm{}~{},$ (6)
could have been our naive preference. But unfortunately, as demonstrated in
Fig.(2), its associated discrete symmetry does not really go much beyond the
original trinity model, offering the same three ways to identify
$\\{q,q^{c},\ell+\ell^{c}\\}$, and this without involving $V\leftrightarrow H$
interplay, and without reviving quark/lepton correspondence.
Figure 2: With blue and red circles representing triplets and anti-triplets
respectively, the three trinification ways to identify
$\\{q,q^{c},\ell+\ell^{c}\\}$ are illustrated by $\\{a,b,c\\}$ cyclic
permutations accompanied by suitable re-arrangements of the six $SU(3)$ group
factors.
## III Tetrahedron of flavors
In this paper, however, from reasons to be specified, we construct the so-
called bi-trinification model (along with its isomorphic quartification phase)
based on the alternative configuration
$x=y=z^{\ast}\quad\textrm{up to permutations}~{}.$ (7)
It offers six variant ways to identify $\\{q,q^{c},\ell+\ell^{c}\\}$, each of
which comes with its own (practically equivalent) permuted and/or conjugate
horizontal assignments. To fully appreciate this point, as demonstrated in
Fig.3, one may rearrange raws as well as columns, identify an alternative
trinification V-sector (three left columns), and then verify how one H-sector
(three right columns) gets systematically replaced by an equivalent one, such
that $(x,y,z)\rightarrow(x^{\prime},y^{\prime},z^{\prime})$, where the latter
is a permuted version of the former, with or without conjugation.
Figure 3: Horizontal bi-trinification: By rearranging raws (fermion
assignments) and columns ($SU(3)$) group factors), while recasting the exact
structure of the trinification V-sector (marked with lines), one can easily
switch from one H-sector (three right columns) to another. The six variants
are thus equivalent to each other. Note that triplets and anti-triplets are
represented here by blue and red circles, respectively. Figure 4: The
Tetrahedron model (Bi-Trinification phases): The two coupled 3-cycles (red
arrows), namely $231$ (left plot) and $246$ (right plot), describe two
different trinification V-sectors. Attached to each one of them is a
corresponding H-sector (blue arrows). In this phase, $\ell$ and $\ell^{c}$
conventionally share a common leptonic vertex, and associated with the fourth
vertex $\chi$ is a novel dark (= electro/nuclear neutral) sector.
The fermion representation furnishes a tetrahedron. While the edges, numbered
$k=1,2,...,6$, correspond to the individual $SU(3)_{k}$ group factors, the
vertices are associated with the four fermion classes
$q,q^{c},\ell+\ell^{c},\chi$. The various arrows involved, making the edges
directed, stand for triplets (outgoing arrows) and anti-triplets (ingoing
arrows). Notice that there are in fact two different ways to consistently
identify the V-sector. Sticking to Fig.(3), it is either the 231-cycle (three
upper configurations) or alternatively the 246-cycle (three lower
configurations). It is this characteristic feature which justifies using the
terminology bi-trinification. The two cycles correspond to the trinifications
$SU(3)_{C}\otimes\left[SU(3)_{L}\otimes SU(3)_{R}\right]_{V,H}$, respectively.
This in turn gives rise to two electric charges
$Q_{V,H}=\left[T_{3L}+T_{3R}+\frac{1}{2}Y_{L+R}\right]_{V,H}~{},$ (8)
one of which, that is $Q_{H}$, must be spontaneously broken, thereby opening
the door for the Holdom effect Holdom , also known as photon mixing.
The $\chi$-fermions are required on anomaly cancelation grounds. They are
$V$-sector singlets by construction, and as such, have no standard
electro/nuclear interactions. This makes them, by definition, candidates for
dark matter particles of the WIMP kind. They do interact with ordinary matter
though, with the $H$-sector serving as the tenable portal. The mechanism
involved, as demonstrated in Fig.(5) for the special lepto/dark case, is the
exchange of super heavy horizontal gauge bosons.
Figure 5: Dark matter portal: Dark matter particles ($V$-sector singlets)
interact with standard model particles ($V$-sector non singlets) by exchanging
heavy horizontal gauge bosons. Depicted in this figure is the lepto/dark case.
## IV Bi-Tri/Quart isomorphism
Alternative rearrangements of raws and columns can make the single family
group flavor assignments depart from their built-in trinification (and thus
bi-trinification) construction, and at the expense of a smaller H-sector,
extend the V-sectior into a quartification phase. Depicted in Fig.(6) are the
four associated quartification variant configurations. Indeed, they share a
common extended V-sector (left four columns connected with lines) and admit
shrank (only two, rather than three, right columns) horizontal structures.
They are equivalent to each other. And most importantly, the existence of
these four variants serves to establish that, as far as our model is
concerned, bi-trinification and quartification are two faces of the same coin.
Figure 6: Horizontal quartification: Alternative rearrangements of particle
assignments (raws) and $SU(3)$ group factors (columns) depart from
trinification, and extend the V-sectior (marked with lines) into
quartification at the expense of a smaller (two right columns) H-sector. The
equivalence of these four variants to each other is interpreted as a novel
version of quark/lepton correspondence. As before, triplets and anti-triplets
are represented by blue and red circles, respectively.
Back to the tetrahedron of flavors, the quartification phase calls for some
reshuffling of fermion assignments. To be specific, the leptonic sub-
representation bifurcates, such that $\ell$ and $\ell^{c}$ belong now to two
different vertices. To classify the fermions under the extended single family
group $SU(3)^{4}$, with the individual subscripts being $C,L,R,N$ (with $N$
denoting the new comer), we introduce the fully symmetric electric charge
formula
$Q=T_{3L}+T_{3R}+T_{3N}+\frac{1}{2}\left(Y_{L}+Y_{R}+Y_{N}\right)~{}.$ (9)
Apart from the manifest $LRN$ symmetry, it has the advantage that all
individual electric charges involved are standard. The electric charge $Q$,
which generalizes Eq.(4), should be contrasted with Eq.(8), and with a
different electric charge formula recently advocated for a similar $SU(3)^{4}$
model. To be more explicit, and expose the extra nine leptons per family,
introduced on top of the trinification scheme, we piecewise specify the
$SU(3)_{C}\otimes SU(2)_{L}\otimes SU(2)_{R}\otimes U(1)_{B-L}$ content of the
fermion representation, namely
$\begin{array}[]{ccl}(3,3^{\ast},1,1)&=&(3,2,1)_{\frac{1}{3}}+(3,1,1)_{-\frac{2}{3}}~{},\\\
(3^{\ast},1,3,1)&=&(3^{\ast},1,2)_{-\frac{1}{3}}+(3^{\ast},1,1)_{\frac{2}{3}}~{},\\\
(1,3,1,3^{\ast})&=&(1,2,1)_{-1,1}+(1,2,1)_{-1}~{}+\\\
&+&(1,1,1)_{0,2}+(1,1,1)_{0}~{},\\\
(1,1,3^{\ast},3)&=&(1,1,2)_{-1,1}+(1,1,2)_{1}~{}+\\\
&+&(1,1,1)_{-2,0}+(1,1,1)_{0}~{}.\end{array}$ (10)
Of special interest are those vector-like fermions which carry non-standard
$B-L$ charges. From a group theory point of view, they are primarily
responsible for the small value
$\sin^{2}\theta_{W}\rightarrow\frac{\sum T_{L3}^{2}}{\sum Q^{2}}=\frac{1}{4}$
(11)
of the Weinberg angle at the symmetry limit, which is smaller than the
conventional value of $\frac{3}{8}$. However, the vector-like surplus of
quarks and leptons is not protected by the $SU(3)_{C}\otimes SU(2)_{L}\otimes
SU(2)_{R}\otimes U(1)_{B-L}$ symmetry subgroup, and decouples at some
presumably very heavy mass scale. Under the latter residual symmetry, we are
left with the exact flavor chiral family of the Left-Right symmetric model,
that is
$\psi=(3,2,1)_{\frac{1}{3}}+(3^{\ast},1,2)_{-\frac{1}{3}}+(1,2,1)_{-1}+(1,1,2)_{1}~{}.$
(12)
Figure 7: The Tetrahedron model (quartification phase): The focus now is on
the 4-cycle $2451$ (red arrows) which describes a single family $SU(3)^{4}$
fermionic representation. It accounts for the revival of a full
$q,q^{c}\leftrightarrow\ell,\ell^{c}$ quark/lepton correspondence, as
expressed by the four variants shown in Fig.(6). The blue arrows account for
the threefold family structure under horizontal $SU(3)_{I}\otimes SU(3)_{II}$.
## V Quark/Lepton correspondence resurrection
$B-L$ does not serve in our model as the forth color. The reason is obvious:
There is no Pati-Salam $SU(4)$ to the rescue. Quark/lepton correspondence is
absent from the trinification model as well. Unlike $q$ and $q^{c}$ which are
assigned to two different vertices, see Fig.(1), $\ell$ and $\ell^{c}$ share a
common third vertex. It is only in Ma’s $SU(3)^{4}$ model that quark/lepton
correspondence has been revived, at least in the sense that $\ell$ and
$\ell^{c}$ split vertices, see Fig.(7). But there is more to it.
The H-sector is primarily in charge of the threefold family replication.
Denoting the quartification horizontal group by $SU(3)_{I}\otimes SU(3)_{II}$,
and choosing for definiteness one of the four variants depicted in Fig.(6), we
face the horizontal assignments
$q\sim(1,3)~{},~{}q^{c}\sim(3^{\ast},1)~{},~{}\ell\sim(3,1)~{},~{}\ell^{c}\sim(1,3^{\ast})~{}.$
(13)
It is the pair $\\{q^{c},\ell\\}$ which cancels the $SU(3)_{I}$ anomalies,
while $\\{q,\ell^{c}\\}$ takes care of the $SU(3)_{II}$ anomalies. Such an
unprecedented horizontal pairing, which seems as a unified generalization of
Foot-Lew quark/lepton symmetry FL , is fully dictated by the group theoretical
structure of our tetrahedron model, and gives a novel perspective to the inter
relations between quarks and leptons.
Another attractive aspect of quartification has to do with the Higgs sector.
The standard model has already taught us that quarks and leptons alike should
acquire their masses via Yukawa couplings with the one and the same Higgs
scalar. One complex doublet suffices to govern all masses (and may even be the
only source of spontaneous symmetry breaking). Truly, this feature is
forcefully shared by the original $SU(3)^{3}$ trinification model as well.
Starting from the fermionic representation Eq.(2), the mass generating job is
carried out by the scalar $\phi$ which exhibits the fermion bilinear quantum
numbers
$\phi\sim qq^{c}\sim\ell\ell^{c}\sim(1,3^{\ast},3)~{}.$ (14)
This is strikingly not the case for the tetrahedron model in the bi-
trinification phase, for which
$\begin{array}[]{rcc}&qq^{c}\sim(1,3^{\ast},3||1,3,3^{\ast})&\\\
&\ell\ell^{c}\sim(1,3^{\ast},3||3^{\ast},1,1)&\end{array}$ (15)
The more so, once $\chi$ has entered the game, we cannot avoid having
$(\ell+\ell^{c})\chi\sim(1,3,3^{\ast}||1,3^{\ast},3)$ as well. This seems to
suggest that $\ell$ and $\ell^{c}$ better not share a common tetrahedron
vertex. Indeed, one is back on safe (and quite attractive) grounds provided
the tetrahedron model in adopted in its the quartification phase.
Choosing for the sake of definiteness a particular horizontal variant, say
$\begin{array}[]{|c ||c|c|c|c||c|c|}\hline\cr q&3&3^{\ast}&1&1&1&3\\\
\hline\cr q^{c}&3^{\ast}&1&3&1&3^{\ast}&1\\\
\hline\cr\ell&1&3&1&3^{\ast}&3&1\\\
\hline\cr\ell^{c}&1&1&3^{\ast}&3&1&3^{\ast}\\\ \hline\cr\end{array}~{},$ (16)
quartification naturally offers the conjugate fermion bilinear quantum numbers
$\begin{array}[]{rcc}&qq^{c}\sim\phi\sim(1,3^{\ast},3,1||3^{\ast},3)&\\\
&\ell\ell^{c}\sim\phi^{\ast}\sim(1,3,3^{\ast},1||3,3^{\ast})&\end{array}$ (17)
In turn, as dictated by their revived correspondence, quarks and leptons do
share now a common Yukawa coupling origin. The associated VEV pattern needs
not be simple, to say the least, reflecting the double spinorial structure
(expressed by $i,j=1,2,3$) of $\langle\phi\rangle=v_{L,R}^{ij}$ under
$SU(3)_{I,II}$ of each of the two (LR symmetric) Weinberg-Salam doublets. To
add a complication to the list, notice that generically the two VEV matrices
$v_{L}^{ij}$ and $v_{R}^{ij}$ cannot be diagonalized simultaneously. This is
apparently the reason underlying the fermion mixing phenomenon, but
unfortunately not even a tentative mixing formula can be derived at this
preliminary stage. It is highly suggestive that $SU(3)_{I}\otimes SU(3)_{II}$
gets broken at a certain stage down to $SU(3)_{I+II}$, under which $\phi$
transforms as a singlet, with all horizontal fine details ripped off.
## VI Epilogue
In this paper, unfortunately and admittedly, we have offered no detailed
insight into the structure of the Fermi mass matrix. While this is undoubtedly
a drawback, it should be gracefully criticized, recalling that after forty
years of flavor puzzle frustration, and after so many imaginative yet unripe
theoretical ideas per year, no one can so far claim even the slightest of
victories. The major clues, so we would like to believe, rowing against the
conventional stream, are already with us. Namely, the challenging total number
three of fermion families, and the striking failure of all grand unification
trails (string theory and extra dimensions included) to account for it. The
naive yet powerful idea that all threes which accompany flavor physics must
have the one and the same gauge theoretical origin then simply paves the way
for Vertical-Horizontal symmetric trinification. We can only hope that the
resulting hereby presented tetrahedron of flavors, stemming from trinification
but eventually focused on its quartification (rather than bi-trinification)
phase, would shed some light on the quark/lepton correspondence and elegantly
account for the threefold family replication, at least at the group
theoretical level.
###### Acknowledgements.
## References
* (1) A. de Rujula, H. Georgi and S.L. Glashow, in Fifth Workshop on Grand Unification, edited by K. Kang, H. Fried and P. Frampton, World Scientific (1984).
* (2) K.S. Babu, X.G. He and S. Pakvasa, Phys. Rev. D33, 763 (1986); J.E. Kim, Phys. Lett. B591, 119 (2004); J. Sayre, S. Wiesenfeldt and S. Willenbrock, Phys. Rev. D73, 035013 (2006); J. Hetzel and B. Stech, Phys. Rev. D91, 055026 (2015); G.M. Pelaggi, A. Strumia and S. Vignali, JHEP 08, 130 (2015); K.S. Babu, B. Bajc, M. Nemevšek and Z. Tavartkiladze, AIP Conf. Proc. 1900, 020002 (2017); Z.W. Wang, A. Al Balushi, R. Mann and H.M. Jiang, Phys. Rev. D99, 115017 (2019); K.S. Babu, S. Jana and A. Thapa, JHEP 02, 051 (2022).
* (3) F. Gursey, P. Ramond and P. Sikivie, Phys. Lett. 60B, 177 (1976); Y. Achiman and B. Stech, Phys. Lett. 77B, 389 (1978); Q. Shafi, Phys. Lett. B79, 301 (1978); R. Barbieri and D.V. Nanopoulos, Phys. Lett. 91B, 369 (1980).
* (4) J.C. Pati and A. Salam, Phys. Rev. D10, 275 (1974).
* (5) E. Ma, Phys. Lett. B593, 198 (2004); E. Ma, Mod. Phys. Lett. A20, 1953 (2005); E. Ma, M. Mondragon and G. Zoupanos, JHEP 12, 026 (2005); Q.H. Cao, S.L. Chen and E. Ma, Phys. Rev. D73, 015009 (2006).
* (6) K.S. Babu, 1, Thomas, W. Kephart and H. Pas, Phys. Rev. D77,116006 (2008); T. Banks and J.F. Fortin, JHEP 07, 046 (2009); C. Kownacki, E. Ma, N. Pollard, O. Popov and M. Zakeri, Phys. Lett. B769, 267 (2017); ibid Nucl. Phys. B928, 520 (2018); ibid Phys. Lett. B777, 121 (2018); E. Ma, Phys. Lett. B820, 136561 (2021).
* (7) F. Wilczek and A. Zee, Phys. Rev. Lett. 42, 421 (1979); A. Davidson, Phys. Rev. Lett. 43, 92 (1979).
* (8) A. Davidson and K.C.Wali, Phys. Rev. Lett. 48, 11 (1982); F. Wilczek and A. Zee, Phys. Rev. Lett., 49:1549 (1982).
* (9) T. Yanagida, Phys. Rev. D20, 2986 (1979); Z.G. Berezhiani, Phys. Lett. B150, 177 (1985).
* (10) A. Davidson and K.C. Wali, Phys. Lett. B94, 359 (1980).
* (11) A. Davidson and K.C. Wali, Phys. Rev. Lett. 58, 2623 (1987).
* (12) P.H. Frampton and S.L. Glashow, Phys. Lett. B190, 157(1987); A. Davidson and K.C. Wali, Phys. Lett. B194, 247 (1987).
* (13) A. Davidson, P.D. Mannheim and K.C. Wali, Phys. Rev. Lett. 47, 149 (1981).
* (14) B. Holdom, Phys. Lett. 166B, 196 (1986).
* (15) R. Foot, and H. Lew, Phys. Rev. D41,3502 (1990); R. Foot, H. Lew and R.R. Volkas, Phys. Rev. D44, 1531 (1991).
|
Date: ]
# Kaon condensation in skyrmion matter and compact stars
Christoph Adam<EMAIL_ADDRESS>Departamento de Física de Partículas,
Universidad de Santiago de Compostela Instituto Galego de Física de Altas
Enerxias (IGFAE) E-15782 Santiago de Compostela, Spain Alberto García Martín-
Caro<EMAIL_ADDRESS>Instituto Galego de Física de Altas Enerxias
(IGFAE) E-15782 Santiago de Compostela, Spain Physics Dept., Brookhaven
National Laboratory, Bldg. 510A, Upton, NY 11973, USA Miguel Huidobro
<EMAIL_ADDRESS>Ricardo Vázquez<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad de Santiago de Compostela
Instituto Galego de Física de Altas Enerxias (IGFAE) E-15782 Santiago de
Compostela, Spain Andrzej Wereszczynski<EMAIL_ADDRESS>Institute of Physics, Jagiellonian University, Lojasiewicza 11, Kraków, Poland
([)
###### Abstract
We address the possibility of the appearance of a charged kaon condensate in
neutron star cores described within a generalized Skyrme model. Our treatment
of strange degrees of freedom is based on the Bound State Approach by Callan
and Klebanov, which allows to obtain an in-medium effective potential for the
$s$-wave kaon condensate. We predict the onset of kaon condensation at a
certain threshold density—whose value depends on the parameters of the model,
and ranges between $1.5$ and $2.5$ times saturation density—, and obtain both
the particle fractions and equation of state for dense matter in the kaon
condensed phase. Finally, we discuss the effect of such condensates on the
mass-radius curves and other observable properties of neutron stars with kaon
condensed cores.
###### Contents
1. I Introduction
2. II Generalized Skyrme model and skyrmion crystals
3. III Kaon condensate in skyrmion crystals
1. III.1 Kaon fluctuations in the Skyrme model
2. III.2 The kaon condensate on classical crystals
1. III.3 Effect of kaon condensation on the quantum corrections to Skyrme crystals
1. IV Results
1. V Neutron stars with kaon condensed cores within the Skyrme model.
1. V.1 The Skyrme crystal EoS for (a-)symmetric nuclear matter.
2. V.2 Maxwell construction versus Gibbs construction
3. V.3 The TOV system and NS properties
1. VI Conclusions
1. A Derivation of the WZW and sextic terms contribution to $V_{K}$
2. B Explicit expressions
## I Introduction
While heavy ion collision experiments and lattice QCD simulations provide
insight into the properties of hot and dense QCD, neutron stars (NS) are the
only known objects in the universe that may allow us to deepen our
understanding of the rich structure of cold, ultra-dense strongly interacting
nuclear matter. The astrophysical inference of NS masses, radii and moments of
inertia in low mass x-ray binaries and isolated NS from the NICER experiment
as well as in binary NS inspirals and subsequent mergers from Gravitational-
Wave (GW) observatories have helped to significantly constrain the NS equation
of state (EoS) at supra-saturation densities, which cannot be reached in
laboratory experiments.
These constraints, however, do not yield any information on the microscopic
structure of dense matter. Indeed, owing to the non-perturbative nature of QCD
at energies below the confinement scale, the precise phase structure of cold,
strongly interacting matter at both finite baryon and isospin chemical
potential is still very speculative. Novel phases of dense baryonic matter are
expected to occur in the inner core of NS, containing additional particle
species such as $\Delta$ isobar resonances Li _et al._ (2018), hyperons
Glendenning (1985), or pion or kaon condensates Hartle _et al._ (1975);
Celenza _et al._ (1977); Kaplan and Nelson (1988); Glendenning and Schaffner-
Bielich (1999); Pal _et al._ (2000). There have been theoretical proposals of
even more exotic scenarios, where a transition to deconfined quark matter
takes place inside the core Heiselberg _et al._ (1993); Benvenuto and Lugones
(1999); Annala _et al._ (2020), or a new state of matter in which both
hadronic and quark degrees of freedom coexist, the so-called quarkionic matter
McLerran and Pisarski (2007); McLerran and Reddy (2019). Studies of the
dynamical features of compact stars, such as the occurrence of phase
transitions during mergers or the cooling rate of proto-neutron stars may
produce complementary data, as they may strongly depend on the specific
microscopic degrees of freedom as well as on the EoS.
Many calculations for the EoS in dense matter predict that strangeness degrees
of freedom may become important in the interior of compact stars, in the form
of hyperons (strange baryons) or a Bose-Einstein condensate of negatively
charged kaons, for densities just a few times nuclear saturation. For a recent
review, see Tolos and Fabbietti (2020). Indeed, hyperons may become stable at
sufficiently high isospin chemical potential, where the decay of neutrons
relieve the Fermi pressure exerted by the nucleons. On the other hand, the
strong attraction between $K^{-}$ mesons and baryons increases with density
and lowers the energy of the zero momentum state. A condensate is formed when
this energy equals the kaon chemical potential, since kaons are favored over
negatively charged fermions for achieving charge neutrality, as they are
bosons and can condense in the lowest energy state.
It is generally assumed that hyperons should appear at densities above $\sim
2-3$ times the nuclear saturation density $n_{0}$, whereas the critical
density for kaon condensation is usually predicted to be a bit larger, around
$\sim(3-4)n_{0}$ (although the specific values are of course model and
parameter dependent). A density of this order is smaller than the central
density of a typical NS, so a kaon condensate could be present in its core.
The possibility of kaon condensates in the core of neutron stars has been
extensively investigated in the literature, using different approaches. Its
appearance tends to soften the EoS, producing smaller values for the allowed
maximum masses. Therefore, the presence of hyperons at too low densities is
not compatible with the stiffness required by the existence of such massive
stars. This is the so-called _Hyperon puzzle_ , presently a subject of very
active research Bombaci (2017); Vidaña (2016).
In this work we will make use of a Generalized Skyrme model, a
phenomenological, nonlinear chiral model that, due to its nonperturbative
nature, can in principle be used as a simple model to study strongly
interacting matter at all scales, from single baryons and nuclei to nuclear
matter in neutron stars. The model contains a rather limited number of
fundamental degrees of freedom, which in the simplest version are just pions,
encoded into an $SU(2)$ valued field $U$, as chiral symmetry is nonlinearly
realized. Nucleons and atomic nuclei emerge as collective, topologically
nontrivial excitations of the mesonic fields. Mathematically they are
described by topological solitons, called skyrmions, whose topological degree
can be identified with the baryon number. The most attractive feature of the
model is the small number of free parameters, which implies a rather strong
predictive power.
The Skyrme model Skyrme (1962) and its generalizations have been applied
successfully to the description of nucleon properties Adkins _et al._ (1983),
Ding and Yan (2007), nuclear interaction potentials Halcrow and Harland
(2020), ground and excited states of atomic nuclei Lau and Manton (2014),
Halcrow _et al._ (2017), or the problem of nuclear binding energies Adam _et
al._ (2013); Gillard _et al._ (2015); Gudnason (2016). Typically, the best
fit to phenomenological observations requires the extension of the original
Skyrme Lagrangian Skyrme (1962), either by the addition of new degrees of
freedom, e.g., vector mesons Naya and Sutcliffe (2018), or additional,
physically-motivated higher derivative terms, like the so-called sextic term
Adam _et al._ (2010), an effective term related to two-body interactions
mediated by $\omega$ mesons.
Simultaneously, in the last years there has been a significant progress in the
application of the Skyrme model to investigate properties of dense nuclear
matter and neutron stars Naya (2019); Adam _et al._ (2020, 2022a, 2022b),
where the sextic term is especially important, as this part of the action
provides the leading contribution in the regime of high pressure and density
Adam _et al._ (2015). Indeed, it makes the skyrmionic matter much stiffer at
extreme conditions which results in physically acceptable values of the
maximal mass of neutron stars.
The extension of the Skyrme model for a larger number of flavors has also been
discussed in the literature. In particular, for $N_{F}=3$, it has been used to
describe pentaquarks and strange hyperons, both in the flavor symmetric limit,
in which the full $SU(3)_{F}$ group is quantized, and in the flavor symmetry
breaking limit, the so called Bound State Approach Callan and Klebanov (1985);
Klebanov (1990), in which oscillations into the strange sector are treated as
perturbations of the $SU(2)$ valued classical soliton. Since kaon degrees of
freedom are best described in the latter approach, in this paper we will take
this path and extend its application to the crystalline phases of Skyrmion
matter, in order to be able to describe the phenomenon of kaon condensation in
dense matter as predicted by the Skyrme model.
The paper is organized as follows: in section II, we introduce the model and
review the classical and quantum properties of crystal solutions. In section
III we review the bound state approach to kaons in the Skyrme model, and
compute the contribution to the total energy from the kaon condensate. In
section IV, we obtain the system of equations from minimizing the free energy
of $npe\mu\bar{K}$ matter, and solve it to find the onset of kaon condensation
for different sets of parameters, and finally in section V we calculate the
Equation of State of skyrmion matter including a kaon condensate, and compute
the corresponding mass-radius curves for NS.
## II Generalized Skyrme model and skyrmion crystals
The generalized Skyrme model we will consider is given by the following
Lagrangian density
$\displaystyle\mathcal{L}=-\frac{f^{2}_{\pi}}{16}$ $\displaystyle\Tr
L_{\mu}L^{\mu}+\frac{1}{32e^{2}}\Tr\left[L_{\mu},L_{\nu}\right]^{2}$
$\displaystyle-\lambda^{2}\pi^{4}\mathcal{B}_{\mu}\mathcal{B}^{\mu}+\frac{m^{2}_{\pi}f^{2}_{\pi}}{8}\Tr\left(U-I\right),$
(1)
where $L_{\mu}=U^{\dagger}\partial_{\mu}U$ is the left invariant Maurer-Cartan
current and the Skyrme field can be written as
$U=\sigma+i\pi_{a}\tau_{a}.$ (2)
Here, $\pi_{a}$ ($a$ = 1, 2, 3) are the pions and $\tau_{a}$ are the Pauli
matrices. The unitarity of the matrix field implies
$\sigma^{2}+\pi_{a}\pi_{a}=1$. Furthermore, $\mathcal{B}^{\mu}$ is the
conserved topological current which, in the standard manner, defines the
topological index of maps $U$, i.e., the baryon charge $B$
$B=\int d^{3}x\mathcal{B}^{0},\hskip
5.69054pt\mathcal{B}^{\mu}=\frac{1}{24\pi^{2}}\epsilon^{\mu\nu\alpha\beta}\Tr\left\\{L_{\nu}L_{\alpha}L_{\beta}\right\\}.$
(3)
The generalized Skyrme effective model contains only four terms and,
therefore, four coupling constants, $f_{\pi},m_{\pi},e,\lambda$, two of which
have a direct phenomenological interpretation as the pion decay constant and
the pion mass. In addition, $\lambda$ can be related to a ratio between the
mass and the coupling constant of the $\omega$ meson. From the very beginning,
we assume the physical mass of the pions, $m_{\pi}=140$ MeV. The remaining
constants are fitted to some properties of infinite nuclear matter.
We remark that the third term in the action, although often omitted in the
context of light nuclei, is obligatory when one studies the properties of
nuclear matter at high density, which is a natural environment in the core of
neutron stars. Indeed, this sextic term governs the equation of state at this
regime and asymptotically leads to the maximally stiff EoS.
The canonical description of NS is provided by the relativistic TOV approach
where a particular model of infinite nuclear matter gives a source term of the
Einstein equations. Effectively, it enters via an EoS, that is, a relation
between e.g., pressure and density. In the Skyrme model, infinite skyrmionic
matter is described by a periodic minimizer of the static energy ($E=-\int
d^{3}x\mathcal{L}$) and, therefore, it is usually referred to as the Skyrme
crystal. Obviously, while the total energy of the crystal is infinite, the
energy per baryon number remains finite
$\frac{E}{B}=\frac{N_{\text{cells}}\>E_{\text{cell}}}{N_{\text{cells}}\>B_{\text{cell}}}=\frac{E_{\text{cell}}}{B_{\text{cell}}}.$
(4)
Here, $N_{\text{cells}}$ is the number of cells and $E_{\text{cell}}$,
$B_{\text{cell}}$ are the energy and baryon charge in a single, periodic cell.
As mentioned before, skyrmion crystals minimize the static energy functional,
$\displaystyle E$ $\displaystyle=$ $\displaystyle\int
d^{3}x\left(\mathcal{E}_{2}+\mathcal{E}_{4}+\mathcal{E}_{6}+\mathcal{E}_{0}\right)$
(5) $\displaystyle=$ $\displaystyle\frac{1}{24\pi^{2}}\int
d^{3}x\left[-\frac{1}{2}\Tr\\{L_{i}L_{i}\\}-\frac{1}{4}\Tr\\{\left[L_{i},L_{j}\right]^{2}\\}\right.$
$\displaystyle+\left.4\pi^{4}c_{6}(\mathcal{B}^{0})^{2}+\frac{c_{0}}{2}\Tr\left(I-U\right)\right]$
over a finite region of space with periodic boundary conditions. Here $U$ is
the SU(2) valued Skyrme field and $L_{\mu}=U^{\dagger}\partial_{\mu}U$.
Further, $\mathcal{E}_{2}$ and $\mathcal{E}_{4}$ are the standard terms of the
Skyrme model quadratic and quartic in derivatives, and $\mathcal{E}_{6}$ is
the sextic term mentioned above. Finally, $\mathcal{E}_{0}$ is the pion mass
potential. We have defined the dimensionless constants
$c_{6}=2\lambda^{2}\frac{f^{2}_{\pi}e^{4}}{\hbar^{3}}$,
${c_{0}=2\frac{m^{2}_{\pi}}{f^{2}_{\pi}e^{2}}}$ and use the so-called Skyrme
model units, so that our energy and length units are
$E_{s}=3\pi^{2}f_{\pi}/e,\quad l_{s}=\hbar/(f_{\pi}e),$ (6)
respectively. Both the size of the unit cell (characterized by the unit cell
length parameter $L$) and its geometry will affect $E_{\rm cell}$. It turns
out that, for our purposes, the ground state of skyrmion crystals is well
described by a cubic unit cell with side length $2L$ composed of skyrmions in
a face-centered cubic (FCC) arrangement, but with an additional symmetry.
Concretely, it respects the following symmetries,
$\displaystyle\text{S}_{1}$ $\displaystyle:(x,y,z)\rightarrow(-x,y,z),$
$\displaystyle(\sigma,\pi_{1},\pi_{2},\pi_{3})\rightarrow(\sigma,-\pi_{1},\pi_{2},\pi_{3}),$
(7) $\displaystyle\text{S}_{2}$ $\displaystyle:(x,y,z)\rightarrow(y,z,x),$
$\displaystyle(\sigma,\pi_{1},\pi_{2},\pi_{3})\rightarrow(\sigma,\pi_{2},\pi_{3},\pi_{1}),$
(8) $\displaystyle\text{S}_{3}$ $\displaystyle:(x,y,z)\rightarrow(x,z,-y),$
$\displaystyle(\sigma,\pi_{1},\pi_{2},\pi_{3})\rightarrow(\sigma,-\pi_{1},\pi_{3},-\pi_{2}),$
(9) $\displaystyle\text{S}_{4}$ $\displaystyle:(x,y,z)\rightarrow(x+L,y,z),$
$\displaystyle(\sigma,\pi_{1},\pi_{2},\pi_{3})\rightarrow(-\sigma,-\pi_{1},\pi_{2},\pi_{3}).$
(10)
A detailed description of the construction of the Skyrme crystal and the
comparison of different symmetries can be found in Adam _et al._ (2022a). As
in that previous work, the unit cell that we will consider has size $2L$ and a
baryon content of $B_{\text{cell}}=4$. Because of the additional symmetry for
this crystal, the unit cell of size $2L$ decomposes into 8 cubes of side
length $L$, each forming a simple cubic arrangement of half-skyrmions, where
half-skyrmions are located in the corners of the cube and lead to a baryon
content of $1/2$. The fields, however, are periodic only in $2L$, hence the
unit cell has side length $2L$. We obtain the value of the energy for each
value of $L$ and, as explained in Adam _et al._ (2022a), the energy-size
curve, $E_{\text{cell}}(L)$, is a convex function which has a minimum at a
certain $L_{0}$. We identify this point with the nuclear saturation point of
infinite, symmetric nuclear matter, which also presents a minimum in the
energy per baryon number curve as a function of the baryon density
$n_{B}=(4/(2L)^{3})=(1/2)L^{-3}$.
Up to now, we have considered the classical skyrmion crystal, which
corresponds to symmetric nuclear matter. However, for a realistic description
of nuclear matter inside a NS we need to consider an almost completely
isospin-asymmetric state, where only a small amount of protons is allowed. We
have already considered this scenario in Adam _et al._ (2022b) via a
semiclassical quantization of the isospin degrees of freedom (DOF) of the
skyrmion crystal. Indeed, it is standard in nuclear physics to define the
binding energy of a nuclear system in the following way,
$\frac{E}{B}(n_{B},\delta)=E_{N}(n_{B})+S_{N}(n_{B})\delta^{2}+\order{\delta^{3}},$
(11)
where $\delta$ is the isospin asymmetry parameter, defined in terms of the
proton fraction $\gamma$ of the system, $\delta=(1-2\gamma)$. $E_{N}(n_{B})$
denotes the binding energy of isospin-symmetric matter, and $S_{N}(n_{B})$
represents the so-called _symmetry energy_ , which is responsible for the
change in the binding energy when the neutron-to-proton ratio changes for a
fixed value of the baryon number. The knowledge of the symmetry energy at high
densities is fundamental for a correct description of NS interiors. However,
although the values of the symmetry energy at saturation is well known
($S_{0}\sim 30$ MeV) Fiorella Burgio and Fantina (2018), the difficulty in
experimentally measuring its behavior at high densities forces us to express
it as an expansion in powers of the baryon density around $n_{0}$,
$S_{N}(n_{B})=S_{0}+\frac{n_{B}-n_{0}}{3n_{0}}L_{\rm
sym}+\frac{(n_{B}-n_{0})^{2}}{18n_{0}^{2}}K_{\rm sym}+\cdots$ (12)
where
$L_{\rm
sym}=3n_{0}\partialderivative{S_{N}}{n_{B}}\evaluated{}_{n_{B}=n_{0}},\quad
K_{\rm
sym}=9n_{0}^{2}\partialderivative[2]{S_{N}}{n_{B}}\evaluated{}_{n_{B}=n_{0}}$
(13)
denote the slope and curvature of the symmetry energy at saturation,
respectively. The values of these coefficients are still very uncertain, but
recent analysis of combined astrophysical and nuclear observations made
possible to constrain the symmetry energy above $n_{0}$ Essick _et al._
(2021); Tang _et al._ (2021); de Tovar _et al._ (2021); Gil _et al._
(2021); Li _et al._ (2021).
Let us now review the procedure for calculating the symmetry energy of an
$SU(2)$ skyrmion crystal, as it will be generalized to the 3 flavor case in
the next section. First, let us rewrite the Skyrme Lagrangian (1) as
$\mathcal{L}=a\Tr\left\\{L_{\mu}L^{\mu}\right\\}+b\Tr\left\\{\left[L_{\mu},L_{\nu}\right]^{2}\right\\}+c\,\mathcal{B}_{\mu}\mathcal{B}^{\mu}+d\Tr\left(U-I\right).$
(14)
In our dimensionless units, we have
$a=-\frac{1}{2},\quad b=\frac{1}{4},\quad
c=-8\lambda^{2}\pi^{4}f_{\pi}^{2}e^{4},\quad
d=\frac{m^{2}_{\pi}}{f^{2}_{\pi}e^{2}}.$ (15)
and consider a (time-dependent) isospin transformation of a static Skyrme
field configuration:
$U(\vec{x})\rightarrow\tilde{U}(\vec{x},t)\equiv
g(t)U(\vec{x})g^{\dagger}(t).$ (16)
The time dependent isospin matrices $g(t)$ are collective coordinates, whose
dynamics is given by a kinetic term in the energy functional,
$T=\frac{1}{2}\omega_{i}\Lambda_{ij}\omega_{j}$ (17)
where $\Lambda_{ij}$ is the isospin inertia tensor, given by
$\displaystyle\Lambda_{ij}$ $\displaystyle=$
$\displaystyle\int\\{2a\Tr\\{T_{i}T_{j}\\}-4b\Tr\\{[T_{i},L_{k}][T_{j},L_{k}]\\}-$
(18)
$\displaystyle-\frac{c}{32\pi^{4}}\varepsilon^{abc}\Tr\\{T_{i}L_{b}L_{c}\\}\varepsilon_{ars}\Tr\\{T_{j}L_{r}L_{s}\\}\\}\,d^{3}x$
$\displaystyle=$ $\displaystyle\;\Lambda\,\delta_{ij}$
being $T_{a}$ the $\mathfrak{su}(2)$-valued current
$T_{a}=\frac{i}{2}U^{\dagger}[\tau_{a},U]$ and $\vec{\omega}$ the associated
isospin angular velocity, defined by
$g^{\dagger}\dot{g}=\tfrac{i}{2}\omega_{a}\tau_{a}$.
As shown in Adam _et al._ (2022b), we may canonically quantize the isospin
collective degrees of freedom and obtain a Hamiltonian, which for a cubic
crystal with a number $N$ of unit cells is given by
$H=\frac{\hbar^{2}}{2N\Lambda_{\rm cell}}I^{\rm{tot}}(I^{\rm{tot}}+1)$ (19)
in terms of the isospin moment of inertia $\Lambda=N\Lambda_{\rm cell}$, and
the total isospin angular momentum eigenvalue $I^{\rm tot}$, given by the
product of the total number of unit cells times the total isospin of each unit
cell, which can be obtained by composing the isospins of each of the cells. In
the charge neutral case, all cells will have the highest possible value of
isospin angular momentum, so that on each unit cell with baryon number $B$,
the total isospin will be $\frac{1}{2}B$, and hence for the full crystal will
be $I^{\rm{tot}}=\frac{1}{2}NB$.
Thus, the quantum correction to the energy (per unit cell) due to the isospin
degrees of freedom in the neutral (i.e. purely neutronic) limit would be
(assuming $N\rightarrow\infty$):
$E^{\rm{iso}}=\frac{\hbar^{2}}{8\Lambda_{\rm cell}}B^{2},$ (20)
where the value of $\hbar$ is related to the value of $e$ through:
$\hbar=\frac{e^{2}}{3\pi^{2}}.$ (21)
The classical skyrmion crystal configurations can therefore be understood as
models for isospin-symmetric nuclear matter, i.e. nuclear matter with zero
total isospin. Deviations from the exact isospin symmetric case yield quantum
isospin corrections to the crystal energy per baryon, which depend on the
difference between protons and neutrons through the total isospin number per
unit cell. Hence, by considering the effect of iso-rotations over classical
solutions we are effectively breaking the isospin symmetry of the static
energy functional by adding a correction of quantum origin that explicitly
breaks it. Moreover, knowing the energy correction due to isospin, it is
straightforward to obtain the associated isospin chemical potential for the
skyrmion crystal using its thermodynamical definition:
$\mu_{I}=-\partialderivative{E}{N_{I}}$, where $N_{I}$ is the (third component
of) the isospin number per unit cell. Given that
$(I^{\rm{tot}})^{2}=I_{1}^{2}+I_{2}^{2}+I_{3}^{2}$ and $N_{I}=I_{3}/N$, we may
rewrite eq. 20 as
$E^{\rm{iso}}=\frac{1}{2\Lambda_{\rm
cell}}\quantity(N_{I}^{2}+\frac{I_{2}^{2}}{N^{2}}+\frac{I_{1}^{2}}{N^{2}})$
(22)
and then
$\mu_{I}=-\partialderivative{E^{\rm{iso}}}{N_{I}}=-\frac{N_{I}}{\Lambda_{\rm
cell}}.$ (23)
Let us now consider a finite chunk of the Skyrme crystal of $N$ unit cells,
and let $A=N\times B_{\rm cell}$, where $B_{\rm cell}$ is the baryon number of
a unit cell. We do not enforce charge neutrality at this step, and further
leave unknown the quantum state of the crystal. As in Adam _et al._ (2022b),
we take a mean field approximation and consider that the isospin density in an
arbitrary skyrmion crystal quantum state is approximately uniform so that
$\expectationvalue{I^{0}_{3}}=\frac{\expectationvalue{I_{3}}}{\int
d^{3}x}=\frac{\expectationvalue{I_{3}}}{NV_{\rm
cell}}\doteq\frac{N_{I}}{V_{\rm cell}}$ (24)
where $N_{I}$ is the isospin charge per unit cell in this arbitrary quantum
state. The effective proton fraction that would yield such an isospin charge
per unit cell is $\gamma$, so we write
$N_{I}=-\frac{1}{2}(1-2\gamma)B_{\rm cell}=-\frac{B_{\rm cell}}{2}\delta.$
(25)
Hence, the isospin energy per unit cell of the skyrmion crystal in such a
state can be written in terms of the asymmetry parameter
$E^{\rm iso}=\frac{\hbar^{2}}{8\Lambda}B_{\rm cell}^{2}\delta^{2},$ (26)
and thus the symmetry energy for Skyrme crystals is given by
$S_{N}(n_{B})=\frac{L^{3}}{8\Lambda}n_{B}.$ (27)
As argued in Klebanov (1985), any quantum state different from purely neutron
matter leads to a divergent Coulomb energy term for the skyrmion crystal.
Therefore, in order to allow for a nonzero positive electric charge within the
unit cell we consider the existence of a neutralizing background of negatively
charged leptons, namely, electrons and muons. In this scenario, the effects of
the positive charge become almost completely screened, and the residual
Coulomb energy is negligible, so we do not take it into account.
Apart from electromagnetic forces, nuclear matter interacts with leptons via
the weak force. Indeed, the exchange between leptons and nucleons inside NS is
completely described by the $\beta$-decay and electron capture processes,
$n\rightarrow p+l+\bar{\nu}_{l}\quad,\quad p+l\rightarrow n+\nu_{l},$ (28)
which take place simultaneously, as long as the charge neutrality and
$\beta$-equilibrium conditions are satisfied,
$\displaystyle n_{p}$ $\displaystyle=\frac{Z}{V}=n_{e}+n_{\mu},$ (29)
$\displaystyle\mu_{n}=\mu_{p}+\mu_{l}$
$\displaystyle\implies\mu_{I}=\mu_{l},\quad l=e,\mu.$ (30)
Leptons inside a NS can be described as a non-interacting relativistic Fermi
gas. Then the chemical potential for a given kind of lepton is,
$\mu_{l}=\sqrt{(\hbar k_{F})^{2}+m_{l}^{2}},$ (31)
where $k_{F}=(3\pi^{2}n_{l})^{1/3}$ is the corresponding Fermi momentum, and
$m_{l}$ is the mass of the corresponding lepton. Considering the most general
case, in which we include muons, we combine this last expression with the
above equilibrium conditions and obtain the following system of equations,
$\displaystyle n_{\mu}=\frac{1}{3\pi^{2}}$
$\displaystyle\quantity[\quantity(\frac{\hbar B_{\rm
cell}(1-2\gamma)}{2\Lambda})^{2}-\left(\frac{m_{\mu}}{\hbar}\right)^{2}]^{\tfrac{3}{2}},$
(32) $\displaystyle\frac{\hbar B_{\rm cell}}{2\Lambda}$
$\displaystyle(1-2\gamma)=\quantity[3\pi^{2}\quantity(\frac{\gamma B_{\rm
cell}}{8L^{3}}-n_{\mu})]^{\tfrac{1}{3}},$ (33)
In order to solve the system for $\gamma$, we take the ultrarelativistic
approximation $\mu_{e}\approx\hbar k_{F,e}$ for electrons. Besides we start
solving the system at low densities considering only electrons, hence we drop
the first equation and set $n_{\mu}=0$ until the condition $\mu_{e}=m_{\mu}$
is reached. Then, muons start to appear and we solve both equations. For each
length of the unit cell we obtain the value of the proton fraction, hence we
reconstruct the curve $\gamma(L)$.
The total energy per unit cell in a $\beta$-equilibrated skyrmion crystal is
therefore given by
$E=E_{\rm class}+E_{\rm iso}(\gamma)+E_{e}(\gamma)+E_{\mu}(\gamma),$ (34)
where $E_{\rm class}$ correspond to the classical energy of the Skyrme
crystal, $E_{\rm iso}$ is calculated from eq. 26 and the energies of the
leptons are the usual energy of a relativistic Fermi gas with mass $m_{l}$ at
zero temperature,
$\displaystyle E_{\rm lep}$
$\displaystyle=\int_{0}^{k_{f}}\frac{k^{2}dk}{\pi^{2}}\sqrt{k^{2}+m_{l}^{2}}=$
(35)
$\displaystyle=\frac{m^{4}_{l}}{8\pi^{2}}\quantity[x_{r}(1+2x_{r}^{2})\sqrt{1+x_{r}^{2}}-\ln{x_{r}+\sqrt{1+x_{r}^{2}}}],$
where $x_{r}=k_{F}/m_{l}$. Recall that for electrons we take the approximation
$x^{-1}_{r}\rightarrow 0$, which is justified for densities $n\geq n_{0}$.
## III Kaon condensate in skyrmion crystals
Having obtained the proton fraction (hence the electron chemical potential) in
$npe\mu$-matter as a function of density, we can now turn to the question of
whether kaon fields may condense inside a Skyrme crystal for a sufficiently
high density, and, if so, whether this critical density value is relevant for
the description of matter inside compact stars.
### III.1 Kaon fluctuations in the Skyrme model
Following the bound-state approach first proposed in Callan and Klebanov
(1985) we may include strange degrees of freedom in the Skyrme model by
extending the skyrmion field to a $SU(3)$-valued field $U$ through modelling
kaon fluctuations on top of a $SU(2)$ skyrmion-like background $u$. With the
only requirement that unitarity must be preserved, different ansätze have been
proposed in the literature for the total $SU(3)$ field describing both pions
and kaons. In this work, we choose the ansatz proposed by Blom et al in Blom
_et al._ (1989):
$U=\sqrt{U_{K}}U_{\pi}\sqrt{U_{K}}.$ (36)
In this ansatz $U_{\pi}$ represents the $SU(3)$ embedding of the purely pionic
part $u$, and the field $U_{K}$ are the fluctuations in the strange
directions. It can be shown that this ansatz is equivalent to the one first
proposed by Callan and Klebanov in Callan and Klebanov (1985) when computing
static properties of hyperons, although both may differ in other predictions
of the model Nyman and Riska (1990).
In the simplest $SU(3)$ embedding, the $SU(2)$ field $u$ is extended to
$U_{\pi}$ by filling the rest of entries with ones in the diagonal and zeros
outside. On the other hand, the kaon ansatz is modelled by a
$\mathfrak{su}$(3) -valued matrix $\mathcal{D}$ which is non trivial in the
off-diagonal elements:
$\begin{split}U_{\pi}&=\begin{pmatrix}u&0\\\ 0&1\end{pmatrix},\hskip
8.53581ptU_{K}=e^{i\frac{2\sqrt{2}}{f_{\pi}}\mathcal{D}},\\\
&u=\sigma+i\pi_{a}\tau_{a},\quad\mathcal{D}=\matrixquantity(0\hfil&K\\\
K^{\dagger}&0)\end{split}$ (37)
where $K$ consists of a scalar doublet of complex fields representing charged
and neutral kaons:
$K=\matrixquantity(K^{+}\\\ K^{0}),\quad K^{\dagger}=(K^{-},\bar{K}^{0}).$
(38)
The extension of the Generalized Skyrme Lagrangian from (1) to include strange
degrees of freedom consists in the replacement of the $\mathcal{L}_{0}$ term
by Nyman and Riska (1990):
$\displaystyle\mathcal{L}_{0}^{\rm new}=$
$\displaystyle\frac{f^{2}_{\pi}}{48}\left(m^{2}_{\pi}+2m^{2}_{K}\right)\Tr{U+U^{\dagger}-2}+$
$\displaystyle+\frac{\sqrt{3}}{24}f^{2}_{\pi}\left(m^{2}_{\pi}-m^{2}_{K}\right)\Tr{\lambda_{8}\left(U+U^{\dagger}\right)},$
(39)
where $\lambda_{8}$ is the eighth Gell-Mann matrix and $m_{K}$ is the vacuum
kaon mass, and the addition of the Wess-Zumino-Witten (WZW) term, which can be
expressed in terms of a 5-dimensional action:
$S_{WZ}=-i\frac{N_{c}}{240\pi^{2}}\int
d^{5}x\>\epsilon^{\mu\nu\alpha\beta\gamma}\Tr{L_{\mu}L_{\nu}L_{\alpha}L_{\beta}L_{\gamma}}.$
(40)
### III.2 The kaon condensate on classical crystals
The onset of kaon condensation in the Skyrme model takes place at a critical
density $n_{\rm cond}$ at which $\mu_{e}$ becomes greater than the energy of
the kaon zero-momentum mode (s-wave condensate). Thus, for baryon densities
$n\geq n_{\rm cond}$, the macroscopic contribution of the kaon condensate to
the energy must be taken into account when obtaining the EoS of dense matter.
To do so, we follow the standard procedure to describe Bose-Einstein
condensation of a (complex) scalar field (see eg Schmitt (2010)) in which the
field condensates correspond to the non-zero vacuum expectation values (vev),
$\expectationvalue{K^{\pm}}$, which are assumed to be constant in space and
whose time dependence is given by:
$\expectationvalue{K^{\mp}}=\phi e^{\mp i\mu_{K}t}$ (41)
The real constant $\phi$ corresponds to the zero-momentum component of the
fields, which acquires a nonvanishing, macroscopic value after the
condensation. Its exact value is determined from the minimization of the
corresponding effective potential, to whose calculation we will dedicate the
rest of this section. On the other hand, the phase $\mu_{K}$ is nothing but
the corresponding kaon chemical potential. First, we will need an explicit
form of the $SU(3)$ Skyrme field in the kaon condensed phase. Assuming the
charged kaons will be the first mesons to condense 111Actually, that the
charged (in particular, the negatively charged) kaons will condense first is
true in our approach (whenever $\mu_{e}>0$), since the chemical potential
associated to neutral kaons is zero, so that the onset of neutral kaon
condensation is given by $m^{*}_{K}=0$., we can safely drop the neutral kaon
contribution, and define the following matrix
$\tilde{\mathcal{D}}=\matrixquantity(0&0&\phi e^{i\mu_{K}t}\\\ 0&0&0\\\ \phi
e^{-i\mu_{k}t}&0&0)$ (42)
which results from substituting the kaon fields in $\mathcal{D}$ as defined in
(37) by their corresponding vev in the kaon condensed phase. Also, taking
advantage of the property $\mathcal{D}^{3}=\phi^{2}\mathcal{D}$, we may write
the $SU(3)$ element generated by $\tilde{\mathcal{D}}$ explicitly in matrix
form:
$\Sigma=e^{i\tfrac{\sqrt{2}}{f_{\pi}}\tilde{\mathcal{D}}}=\matrixquantity(\cos\tilde{\phi}&0&ie^{i\mu_{K}t}\sin\tilde{\phi}\\\
0&1&0\\\ ie^{-i\mu_{K}t}\sin\tilde{\phi}&0&\cos\tilde{\phi})$ (43)
where $\tilde{\phi}=\tfrac{\sqrt{2}}{f_{\pi}}\phi$ is the dimensionless
condensate amplitude.
Furthermore, assuming the backreaction from the kaon condensate to the
skyrmion crystal is negligible, and thus the classically obtained crystal
configuration will be the physically correct background even in the kaon
condensed phase, we may write the $SU(3)$ field in this phase as $U=\Sigma
U_{\pi}\Sigma$, where $U_{\pi}$ is the $SU(3)$ embedding of the $SU(2)$
skyrmion background as in (37). Introducing this $U$ in the total action
yields the standard Skyrme action for the $SU(2)$ field plus an effective
potential term for the kaon condensate:
$S_{Sk}(U)+S_{\rm WZW}(U)=S_{Sk}(U_{\pi})-\int dtV_{K}(\tilde{\phi}),$ (44)
where
$V_{K}=\frac{1}{24\pi^{2}}\int
d^{3}x\Big{[}V^{(2)}_{K}+V^{(4)}_{K}+V^{(6)}_{K}+V^{(0)}_{K}\Big{]}+V^{(WZW)}_{K}.$
(45)
Let us now calculate the contribution to the effective potential $V_{K}$ of
each term in the action:
* •
Quadratic term: Given that the crystal background is static and the kaon
condensate does not depend on spatial coordinates, the kaon part of the
quadratic term may be written as
$\Tr{L_{0}^{2}}=-[\Tr\\{\partial_{t}\Sigma^{\dagger}\partial_{t}\Sigma\\}+\Tr\\{\Sigma^{\dagger}\partial_{t}\Sigma
U_{\pi}^{\dagger}\Sigma\partial_{t}\Sigma^{\dagger}U_{\pi}\\}].$ (46)
Introducing the explicit expression for $\Sigma$, (43), yields
$V_{K}^{(2)}=\mu_{K}^{2}\sin^{2}\tilde{\phi}[(1+\sigma^{2}+\pi^{2}_{3})\sin^{2}\tilde{\phi}-2(1+\sigma\cos^{2}\tilde{\phi})].$
(47)
* •
Quartic term: In the quartic term, the kaon effective potential comes from the
terms with time derivatives of the total field,
$\displaystyle\Tr{[L_{0},L_{i}]^{2}}=2[\Tr\\{\partial_{t}U^{\dagger}\partial_{i}U\partial_{t}U^{\dagger}\partial_{i}U\\}-$
(48)
$\displaystyle-\Tr\\{\partial_{i}U^{\dagger}\partial_{t}U\partial_{i}U^{\dagger}\partial_{t}U\\}],$
which, after substitution of the expression for $\Sigma$, gives
$\displaystyle V_{K}^{(4)}$
$\displaystyle=-2\mu_{K}^{2}\sin^{2}{\tilde{\phi}}\big{\\{}(1+\sigma)\partial_{i}n^{2}\cos^{2}{\tilde{\phi}}+$
(49) $\displaystyle+$ $\displaystyle
2[\partial_{i}\sigma^{2}(1-\pi_{3}^{2})+\partial_{i}\pi_{3}^{2}(1-\sigma^{2})+2\sigma\pi_{3}\partial_{i}\sigma\partial_{i}\pi_{3}]\sin^{2}\tilde{\phi}\big{\\}}$
* •
Mass term: The kaon part associated to the mass term gives the following
contribution,
$V_{K}^{(0)}(\tilde{\phi})=2\frac{m^{2}_{K}}{f^{2}_{\pi}e^{2}}(1+\sigma)\sin^{2}\tilde{\phi}$
(50)
* •
Wess-Zumino-Witten term: The WZW term is written as a 5-form integrated over
an auxiliar 5-dimensional disk $D$ whose boundary is the spacetime manifold
$M$, but in appendix A we show that the variation after the kaon fluctuations
of the pion background yields a local term which may be written as an
effective four-dimensional lagrangian. Indeed, we show that
(51)
* •
Sextic term: The contribution from the sextic term is also obtained in
appendix A to be
$V_{K}^{(6)}=-\frac{\lambda^{2}f^{2}_{\pi}e^{4}}{16}\Tr\\{[R_{j},R_{k}]\xi_{0}\\}^{2}$
(52)
where
$\xi_{\mu}=U_{\pi}\Sigma\partial_{\mu}\Sigma^{\dagger}U_{\pi}^{\dagger}-\Sigma^{\dagger}\partial_{\mu}\Sigma$.
Once the traces are evaluated, we end up with
$\displaystyle V_{K}^{(6)}=$
$\displaystyle-\lambda^{2}f^{2}_{\pi}e^{4}\mu_{K}^{2}\sin^{4}(\tilde{\phi})(\partial_{i}\pi_{3}\partial_{j}\sigma-\partial_{i}\sigma\partial_{j}\pi_{3})^{2}.$
(53)
### III.3 Effect of kaon condensation on the quantum corrections to Skyrme
crystals
In the above calculations, we have taken separately the contributions of a
kaon condensate and an isospin angular momentum of the skyrmion crystal, and
the kaon condensate interacts with the skyrmion isospin only indirectly via
the charge neutrality and $\beta$ equilibrium conditions, which relate their
corresponding chemical potentials. However, since kaons possess an isospin
quantum number, we should consider a (time-dependent) isospin transformation
of the full Skyrme field + kaon condensate configuration $U=\Sigma
U_{\pi}\Sigma$:
$U\rightarrow\tilde{U}\equiv A(t)UA^{\dagger}(t),$ (54)
where $A$ is an element of $SU(3)$ modelling an isospin rotation,
$A=\matrixquantity(a&0\\\ 0&1),\quad a\in SU(2).$ (55)
The Maurer-Cartan form transforms as ($\dot{A}=dA/dt$)
$\tilde{U}^{\dagger}\partial_{\mu}\tilde{U}=\left\\{\begin{array}[]{ll}AU^{\dagger}\partial_{i}UA^{\dagger},\quad(\mu=i=1,2,3),&\\\\[5.69054pt]
AU^{\dagger}\partial_{0}UA^{\dagger}+A(U^{\dagger}[A^{\dagger}\dot{A},U])A^{\dagger},&(\mu=0).\end{array}\right.$
(58)
We now define the isospin angular velocity $\vec{\omega}$ as
$A^{\dagger}\dot{A}=\tfrac{i}{2}\omega_{a}\lambda_{a}$ ($a=1,2,3$), with
$\lambda_{\textsc{a}}$ the Gell-Mann matrices generating $SU(3)$ for
$\textsc{a}=1,\cdots 8$. Notice that $\vec{\omega}$ is a three-vector, since
$A^{\dagger}\dot{A}$ belongs to the isospin $\mathfrak{su}(2)$ subalgebra of
$\mathfrak{su}(3)$ Then, we may write the time component of the Maurer-Cartan
current as
$\tilde{U}^{\dagger}\partial_{0}\tilde{U}=AL_{0}A^{\dagger}+AT_{a}A^{\dagger}\omega_{a}$,
where $T_{a}$ is the $\mathfrak{su}(3)$-valued current:
$T_{a}=\frac{i}{2}U^{\dagger}[\lambda_{a},U]\equiv
iT_{a}^{\textsc{a}}\lambda_{\textsc{a}},$ (59)
where we have made use of the parametrization (37). The time dependence of the
new Skyrme field induces the existence of a kinetic term in the energy
functional, given by 222Remember that we are using the mostly minus convention
for the metric signature.
$\begin{split}T=\int\\{a&\quantity(\Tr\\{L_{0}L_{0}\\}+2\Tr\\{L_{0}T_{a}\\}\omega_{a}+\Tr\\{T_{a}T_{b}\\}\omega_{a}\omega_{b})\\\
-2b\big{(}\Tr\\{[&(L_{0}+T_{a}\omega_{a}),L_{k}][(L_{0}+T_{b}\omega_{b}),L_{k}]\\}\big{)}-c\,\,\mathcal{B}^{i}\mathcal{B}_{i}\\}d^{3}x,\end{split}$
(60)
with $\mathcal{B}^{i}$ the spatial components of the topological current (3):
$\mathcal{B}^{i}=\frac{3}{24\pi^{2}}\varepsilon^{ijk}\Tr\\{(L_{0}+T_{a}\omega_{a})L_{j}L_{k}\\}.$
(61)
We may rewrite the kinetic isorotational energy in the standard way as a
quadratic form acting on the components of the isospin angular velocity,
$T=\frac{1}{2}\omega_{a}\Lambda_{ab}\omega_{b}+\Delta_{a}\omega_{a}-V_{K}$
(62)
where $\Lambda_{ab}$ is the isospin inertia tensor and $\Delta_{a}$ is the
kaon condensate isospin current, given by
$\displaystyle\Lambda_{ab}$
$\displaystyle=\int\left\\{2a\Tr\\{T_{a}T_{b}\\}-4b\Tr\\{[T_{a},L_{k}][T_{b},L_{k}]\\}-\frac{c}{32\pi^{4}}\varepsilon^{lmn}\Tr\\{T_{a}L_{m}L_{n}\\}\varepsilon_{lrs}\Tr\\{T_{j}L_{r}L_{s}\\}\right\\}\,d^{3}x,$
(63) $\displaystyle\Delta_{a}$
$\displaystyle=\int\left\\{2a\Tr\\{L_{0}T_{a}\\}-4b\Tr\\{[T_{a},L_{k}][L_{0},L_{k}]\\}-\frac{c}{32\pi^{4}}\varepsilon^{lmn}\Tr\\{L_{0}L_{m}L_{n}\\}\varepsilon_{lrs}\Tr\\{T_{a}L_{r}L_{s}\\}\right\\}\,d^{3}x,$
(64)
where $a,b$ and $c$ are those in eq. 15.
The symmetries of the crystalline configuration that we consider in this work,
concretely the $S_{1}$ and $S_{2}$ transformations, imply that the isospin
inertia tensor becomes proportional to the identity, i.e.
$\Lambda^{\rm{crystal}}_{ab}=\Lambda\delta_{ab}$. However, the presence of a
kaon condensate breaks this symmetry to a $U(1)$ subgroup, so that
$\Lambda_{ab}$ presents two different eigenvalues in the condensate phase,
$\Lambda_{\rm cond}=\text{diag}(\Lambda,\Lambda,\Lambda_{3})$. Similarly,
$\Delta_{a}=0$ in the purely barionic phase, and its third component acquires
a non-zero value in the condensate phase, $\Delta_{\rm cond}=(0,0,\Delta)$.
The explicit expressions for $\Lambda_{3}$ and $\Delta$ in the condensed phase
are written in appendix B. One can easily check that in the non-condensed
phase, $\phi=0$ and the results of the previous section are recovered, namely,
$\Lambda_{3}=\Lambda$, $\Delta=0$.
The quantization procedure now goes along the same lines as in the first
section. However, the isospin breaking due to the kaon condensate implies that
the canonical momentum associated to the third component of the isospin
angular velocity will now be different, and given by
$I_{3}=\Lambda_{3}\omega_{3}+\Delta$.
Thus, after a Legendre transformation to rewrite (62) in Hamiltonian form, and
making the $N\rightarrow\infty$ approximation, one can write the quantum
energy correction per unit cell of the crystal in the kaon condensed phase as
$E_{\rm{quant}}=\frac{1}{2\Lambda_{3}}(I_{3}^{2}-\Delta^{2}).$ (65)
The first term on the rhs is just the isospin correction, while now there is
an additional second term due to the isospin of the kaons. Indeed, since the
kaon field enters also in the expression of the isospin moment of inertia
$\Lambda_{3}$, both terms will depend nontrivially on the kaon vev field.
## IV Results
When the kaon field develops a nonzero vev, apart from the neutron decay and
lepton capture processes of eq. 28, additional processes involving kaons may
occur:
$n\leftrightarrow p+K^{-},\qquad l\leftrightarrow K^{-}+\nu_{l}$ (66)
such that the chemical equilibrium conditions
$\mu_{n}=\mu_{p}+\mu_{K},\qquad\mu_{l}=\mu_{K}$ (67)
are satisfied. These are the extension of eq. 30 to the condensate phase.
The total energy within the unit cell may be obtained as the sum of the
baryon, lepton and kaon contributions:
$\begin{split}E=E_{\rm class}+&E_{\rm
iso}(\gamma,\tilde{\phi})+E_{K}(\mu_{e},\tilde{\phi})+\\\
+&E_{e}(\mu_{e})+\Theta(\mu_{e}^{2}-m_{\mu}^{2})E_{\mu}(\mu_{e})\end{split}$
(68)
The kaon contribution is the effective potential energy
$E_{K}(\mu_{e},\tilde{\phi})=V_{K}-\frac{\Delta^{2}}{2\Lambda_{3}},$ (69)
which depends on the condensate $\tilde{\phi}$ and on the lepton chemical
potential through the explicit dependence on $\mu_{K}$ of both $V_{K}$ and
$\Delta$, and $\mu_{K}=\mu_{e}$ due to the equilibrium conditions (67).
Therefore, the energy of the full system depends on the proton fraction, the
kaon vev field and the electron chemical potential. Their respective values
can be obtained, for fixed $n_{B}$ (or equivalently, fixed $L$) by minimizing
the free energy
$\Omega=E-\mu_{e}(N_{e}+\Theta(\mu_{e}^{2}-m_{\mu}^{2})N_{\mu}-\gamma B)$ (70)
with respect to $\gamma$, $\tilde{\phi}$ and $\mu_{e}$, i.e.
$\partialderivative{\Omega}{\gamma}\evaluated{}_{n_{B}}\\!\\!\\!(\gamma,\tilde{\phi},\mu_{e})=\partialderivative{\Omega}{\tilde{\phi}}\evaluated{}_{n_{B}}\\!\\!\\!(\gamma,\tilde{\phi},\mu_{e})=\partialderivative{\Omega}{\mu_{e}}\evaluated{}_{n_{B}}\\!\\!\\!(\gamma,\tilde{\phi},\mu_{e})=0.$
(71)
The first equation imposes the expected condition
$\mu_{e}=\mu_{I}=2\hbar^{2}(1-2\gamma)/\Lambda_{3}$. Then, after substituting
into the other two conditions we get:
$\displaystyle\gamma n_{B}-$
$\displaystyle\frac{(\mu^{2}_{I}-m^{2}_{e})^{3/2}+(\mu^{2}_{I}-m^{2}_{\mu})^{3/2}}{3\pi^{2}\hbar^{3}}+\frac{n_{B}}{4}\partialderivative{E_{K}}{\mu_{e}}\evaluated{}_{\mu_{e}=\mu_{I}}=0,$
(72)
$\displaystyle\partialderivative{V_{K}}{\tilde{\phi}}-\frac{\Delta}{\Lambda_{3}}\partialderivative{\Delta}{\tilde{\phi}}+\partialderivative{\Lambda_{3}}{\tilde{\phi}}\left(\frac{\Delta^{2}}{2\Lambda_{3}^{2}}-\frac{\mu_{I}^{2}}{2\hbar^{2}}\right)=0,$
(73)
which are precisely the charge neutrality condition, and the minimization of
the grand canonical potential with respect to the kaon field. We note here
that we drop the ultrarrelativistic consideration for electrons since the
appearance of kaons may decrease hugely the electron fraction. By solving the
system of equations 72 and(73) for $\gamma$ and $\tilde{\phi}$ we obtain all
the needed information for the new kaon condensed phase. Then we may compare
the particle fractions and energies between both phases, which we will call
$npe\mu$ and $npe\mu\overline{K}$.
Before solving the full system for different values of the lattice length $L$,
we may try to obtain the value of the length at which kaons condense, $L_{\rm
cond}$. This value is indeed important since it will determine wether or not a
condensate of kaons will appear at some point in the interior of NS This is
accomplished with the same system of eqs. 72 and 73 by factoring the
$\sin\tilde{\phi}$ from the second equation and setting $\tilde{\phi}=0$. Then
we may see the system as a pair of equations to obtain the values of
$\gamma_{\text{cond}}$ and $L_{\text{cond}}$, the values of the proton
fraction and the length parameter for which the kaons condense.
We show in the table below the density at which kaons condense for different
values of the parameters as well as the values of some nuclear observables
they yield. All the values are given in units of MeV or fm, respectively.
label | $f_{\pi}$ | $e$ | $\lambda^{2}$ | $E_{0}$ | $n_{0}$ | $S_{0}$ | $L_{\text{sym}}$ | $n_{\text{cond}}/n_{0}$
---|---|---|---|---|---|---|---|---
set 1 | 133.71 | 5.72 | 5 | 920 | 0.165 | 23.5 | 29.1 | 2.3
set 2 | 138.11 | 6.34 | 5.78 | 915 | 0.175 | 24.5 | 28.3 | 2.2
set 3 | 120.96 | 5.64 | 2.68 | 783 | 0.175 | 28.7 | 38.7 | 1.6
set 4 | 139.26 | 5.61 | 2.74 | 912 | 0.22 | 28.6 | 38.9 | 1.6
Table 1: Sets of parameter values and observables at nuclear saturation
Parameter sets 1 and 2 are chosen so that the energy per baryon and baryon
density at saturation are fitted to experimental values, whereas the sets 3
and 4 correctly fit the symmetry energy and slope at saturation.
In fig. 1, we show the $E(L)$ curves both without and with kaon condensation,
in dimensionless Skyrme units. It is clearly visible that for sufficiently
small $L$ a nonzero kaon condensate is preferred. In fig. 2 we show the
resulting particle fractions. In fig. 3 we plot the symmetry energy as a
function of both $n_{B}$ and the kaon condensate $\phi$.
Figure 1: Energy vs lattice length in dimensionless Skyrme units, for the set
1 of parameters. The energy is shown for the classical crystal without isopin
contributions (green), isospin asymmetric ($npe\mu$) matter with (black) and
without (blue) kaons. We also plot the completely asymmetric neutron matter
(magenta) which lies slightly above the blue curve. Figure 2: Particle
fractions as a function of baryon density for the set 1 of parameters, both
with (solid lines) and without (discontinuous lines) kaon condensate. For the
case with kaon condensate, the contribution of muons is negligible. Figure 3:
Symmetry energy of nuclear matter as a function of baryon density and kaon
$vev$, for the parameter set 1. The surface is obtained by treating $n_{B}$
and $\phi$ as independent variables, whereas the red curve corresponds to the
energy-minimizing solution for each $L$, i.e., $n_{B}$.
## V Neutron stars with kaon condensed cores within the Skyrme model.
In this section we briefly recapitulate how to obtain the EoS from the Skyrme
crystal solution. Then we calculate the full $npe\mu\bar{K}$ matter EoS, and
finally solve the TOV system and compare the NS properties with and without
kaons.
### V.1 The Skyrme crystal EoS for (a-)symmetric nuclear matter.
The energy per baryon as a function of the lattice length, $E(L)$, has a
minimum at a certain value $L_{0}$. The density $n_{0}=(1/2)L_{0}^{-3}$ at
which this minimum is achieved is the so called nuclear saturation density,
which has been experimentally found to be $n_{0}\simeq 0.16$ fm-3. Further,
the energy per baryon at nuclear saturation is $E_{0}=923\,{\rm MeV}$. On the
other hand, for the crystal solutions one can define the energy density
$\rho$, pressure $p$ and baryon density $n$ as
$\displaystyle\rho$
$\displaystyle=\frac{E}{V}=\frac{E_{\text{cell}}}{V_{\text{cell}}},$ (79)
$\displaystyle p$ $\displaystyle=-\frac{\partial E}{\partial
V}=-\frac{\partial E_{\text{cell}}}{\partial V_{\text{cell}}},$ (80)
$\displaystyle n_{B}$
$\displaystyle=\frac{B}{V}=\frac{B_{\text{cell}}}{V_{\text{cell}}}=\frac{1}{2L^{3}}.$
(81)
We can understand the cell length parameter $L$ as labelling the different
energy-minimizing configurations at different densities. Therefore, the three
quantities above are related as functions of $L$. This relation is precisely
the equation of state (EoS) for skyrmion crystals. It has been recently shown
in Adam _et al._ (2020) that this EoS stiffens in the generalized Skyrme
model, i.e. when the sextic term in (5) is included. This stiffening
significantly rises the maximal NS masses that can be reached, which is a
strong motivation for the inclusion of the sextic term, as it is necessary in
order to reach the mass range of massive pulsars, around $2-2.5M_{\odot}$
according to recent observations.
The global energy minimum of the crystal is reached at $L=L_{0}$, at which
skyrmionic matter remains in equilibrium, i.e. at zero pressure. For $L<L_{0}$
the crystal is squeezed, which translates into larger values of pressure and
density. In the opposite region, $L>L_{0}$, the matter content inside the unit
cell spreads and we enter the unstable branch. Indeed, the pressure in this
region is negative, hence we conclude that this (low-density) regime is not
well described by the crystal solution. We actually expect that matter inside
the unit cell will rearrange in a kind of inhomogeneous central lump
surrounded by vacuum Adam _et al._ (2022a). More exotic configurations
presenting non-homogeneous structures have also been constructed in the Skyrme
model, Park _et al._ (2019); Canfora _et al._ (2020). However, the physical
relevance of these configurations as true energy minimizers in the model still
remains unclear.
Hence, the main ingredient to obtain the EoS for the Skyrme crystal is the
energy dependence on the unit cell size. The $npe\mu$ matter case is easy to
obtain using eq. 34 for different values of $L$ after solving the
$\beta-$equilibrium and charge neutrality conditions for $\gamma$. However,
once we include kaons, the change in the energy curve fig. 1 may lead to a
first or second order phase transition. To distinguish the order of the phase
transition in our case, we need to know accurately the pressure near the
condensation point. Therefore, we computed more points for the energy near the
condensation value with higher accuracies, and we obtained the pressure using
a numerical derivative. We conclude that the kaon condensation produces a
first order phase transition for our choices of parameters in the Skyrme
model. This can be seen in the right plot of fig. 4, where we show the EoS for
our best accuracy and, clearly, there is a non-physical region which must be
bridged by a first order phase transition.
Figure 4: Left plot: energy against the side length of the crystal, calculated
with more points near the condensation values for both branches and their
interpolations. Right plot: pressure against the energy density (both in
Skyrme units) from which we conclude that there is a first order phase
transition.
### V.2 Maxwell construction versus Gibbs construction
The Maxwell construction (MC) is typically used to obtain a physical equation
of state when a first order transition is present. Indeed, the MC has been
already studied in the Skyrme crystals context to describe the transition
between crytals with different symmetries Adam _et al._ (2022a). This
construction is based on a mixed phase of constant pressure which connects the
two solutions. However, the MC is only correct when there is a single
conserved charge (in this case, the baryon number) for which the associated
chemical potential is enforced to be common for both phases in the mixed phase
Glendenning (1992). If, instead, an additional charge is conserved, like the
electric charge in the case of $npe\mu$ matter, the Gibbs conditions for the
phase equilibrium,
$p^{\rm I}=p^{\rm II},\hskip 5.69054pt\mu^{\rm I}_{i}=\mu^{\rm II}_{i},\hskip
4.2679pti=B,q$ (82)
cannot be both satisfied in a standard MC. In the last expression $\mu_{B}$
and $\mu_{q}$ represent the chemical potentials associated to the conserved
baryon and electric charges, respectively. Instead, one should perform a Gibbs
construction (GC) Glendenning (1992); Glendenning and Schaffner-Bielich
(1999). Indeed, the GC has also been proven useful in the context of a hadron-
to-quark phase transition inside NS Bhattacharyya _et al._ (2010).
We may write the chemical potential of each particle species as a linear
combination of the chemical potentials associated to the conserved charges of
our system:
$\mu_{i}=B_{i}\mu_{B}+q_{i}\mu_{q},$ (83)
where $B_{i}$ and $q_{i}$ are the baryon number and electric charge of the
particle species $i$. Then we might identify the baryon and electric charge
chemical potentials with the neutron and electron chemical potentials
respectively. The main difference between MC and GC is that, in the mixed
phase, the first one imposes charge neutrality locally, i.e. both phases are
neutral independently, however in the GC it is imposed globally in the mixed
phase. Considering a volume fraction $\chi$ of the kaon condensed phase,
charge neutrality is imposed in the GC as:
$n^{MP}_{q}=(1-\chi)n^{\rm I}_{q}+\chi n^{\rm II}_{q}=0.$ (84)
The mixed phase in the GC is calculated by identifying first the contributions
to the pressure and charge densities in each phase separately. Then we have to
solve the system of equations composed by eqs. 82, 84 and 73. We use the unit
cell length parameter of the first ($npe\mu$) phase $L_{\rm I}$ as the
variable defining our position in the phase diagram, then the unknowns are the
length in the second ($npe\mu\bar{K}$) phase $L_{\rm II}$, the proton
fractions $\gamma_{\rm I}$, $\gamma_{\rm II}$, the kaon field $\tilde{\phi}$
and the volume fraction $\chi$.
Figure 5: $E(L)$ curves for the two phases. The different slopes at the point
of phase separation indicate a first-order phase transition. We also show the
curves resulting from a Maxwell construction (MC) and a Gibbs construction
(GC).
We remark that we assumed in our calculations of the kaon condensate in
section III that the backreaction of the condensate on the crystal is
negligible, such that our two phases are always considered in the same
classical crystal background, and the energies per baryon of the two phases
are compared for the same length $L$. As a result, we always should have
$L_{\rm I}=L_{\rm II}$ and, consequently, $n_{B,{\rm I}}=n_{B,{\rm II}}$ by
construction. On the other hand, the relation between $L$ and the
thermodynamical variables $p$, $\mu_{i}$ and $\phi$ used in eqs. 82, 84 and 73
is quite nontrivial in both phases. We, therefore, treat $L_{\rm II}$ as an
independent variable in our numerical calculations. We find that always
$L_{\rm I}=L_{\rm II}$ within our numerical precision, which provides us with
an additional consistency check both for our numerics and for the
thermodynamical transformations we used.
We show our results in fig. 5 and in fig. 6. We find that the mixed phase of
the GC, and hence the values at which the kaon field becomes non-zero, starts
at a smaller density than the value obtained in table 1. This is also found in
Glendenning and Schaffner-Bielich (1999), for which the GC mixed phase extends
to a larger region than the one obtained from the MC, because the mixed phase
in the GC no longer is for constant pressure. In our case, even the minimum of
$E(L)$ is shifted to slightly lower values, see the insert in fig. 5 and,
hence, the use of the GC affects the low density regime of the EoS, as can be
seen in fig. 6.
Figure 6: EoS for the three different cases that we have constructed. The jump
in the MC due to the first order transition and the different behaviour of the
GC at low densities are clearly visible. We also show the standard nuclear
physics EoS of Sharma _et al._ (2015) (BCPM) and a hybrid EoS obtained by
joining the BCPM EoS at low pressure with the GC EoS at high pressure.
We may also calculate the particle fractions in the mixed phase of the GC
using an expression equivalent to eq. 84 for each particle. We show the new
particle fractions in fig. 7. Besides, during the mixed phase, we find that
there are more protons in the second phase than in the first one. However, the
presence of more kaons than protons in the second phase results in a partial
negative charge density. That negative charge is compensated by the overall
positive charge density of the first phase. In both phases the number of
electrons is much less than that of protons and kaons.
Figure 7: Particle fractions for the GC. The main difference with respect to
fig. 2 is the earlier appearance of kaons.
### V.3 The TOV system and NS properties
In order to calculate the mass and radius for a non-rotating NS we have to
solve the standard TOV (Tolman-Oppenheimer-Volkoff) system of ODEs. It is
obtained inserting a spherically symmetric ansatz of the spacetime metric,
$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (85)
in the Einstein equations,
$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}.$ (86)
To describe matter inside the star, in the right-hand side of the equation, we
use the stress-energy tensor of a perfect fluid,
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (87)
where the pressure $p$ and the energy density $\rho$ are not independent but
related by the EoS. Hence the EoS describes the nuclear interactions inside
the NS and different EoS lead to different observables.
The resulting TOV system involves 3 differential equations for $A,B$ and $p$,
which must be solved for a given value of the pressure in the centre of the NS
($p(r=0)=p_{0}$) until the condition $p(r=R)=0$ is achieved.
We use a 4th order Runge-Kutta method of step $\Delta r=1$ m to solve the
system and to obtain the main observables from the solutions. The radial point
at which the pressure vanishes defines the radius of the NS, and the mass $M$
is obtained from the Schwarzschild metric definition outside the star,
$B(r=R)=\frac{1}{(1-\frac{2GM}{R})}.$ (88)
The results of this section are plotted in fig. 8 for the 4 sets of
parameters. We compare the results between the MC and GC as well as with the
EoS without kaons. The first observation is that the addition of kaons to the
EoS agrees with the expectation, reducing the achievable maximum mass. This
represents the so-called hyperon puzzle in which the appearance of new strange
degrees of freedom softens the EoS such that it may not lead to sufficiently
massive NS ($\sim 2M_{\odot}$). As can be seen, this is not the case in the
generalized Skyrme model since we may obtain very high masses easily due to
the contribution of the sextic term. Furthermore the radii of NS are also
reduced, which benefits our concrete model since the radii for skyrmion
crystals are in some cases too large.
The main difference between the two different constructions is that the MC
starts at a given density, hence it deviates from the $npe\mu$ EoS at a
certain mass. On the other hand, since the GC changes the location of the
minimum, it leads to different results also in the low mass region. However,
both constructions practically merge in the high masses region, in which they
follow the same $npe\mu\overline{K}$ EoS.
As already explained, the thermodynamically stable region of the $E(L)$ curves
and the corresponding EoS based on the Skyrme crystal is $L\leq L_{0}$ or,
equivalently, $n_{B}\geq n_{0}$. As a consequence, NS based on the Skyrme
crystal have $n_{B}=n_{0}$ at the NS surface or, in other words, Skyrme
crystal NS have no crust. In the right panel of fig. 8, therefore, we show the
result of adding a crust to the NS by joining the GC equations of state of the
Skyrme crystal with a standard nuclear physics EoS for low densities (the
corresponding EoS are shown in fig. 6). Concretely, we use the BCPM EoS Sharma
_et al._ (2015) and joint the two EoS at the pressure $p=p_{*}$ where the two
EoS coincide, i.e., $\rho_{\rm BCPM}(p_{*})=\rho_{\rm,crystal}(p_{*})$,
exactly as we did in Adam _et al._ (2020). In terms of the baryon density,
the joining occurs at $n_{B,*}\sim 1.1n_{0}$ for the parameter sets 1-3, and
for $n_{B,*}\sim 1.2n_{0}$ for the set 4. Again as in Adam _et al._ (2020),
we assume a smooth joining between the two EoS (concretely, described by a
quadratic interpolation) in order to avoid an artificial phase transition at
$p_{*}$. We also plot in fig. 8 the most likely mass-radius relations for the
NS corresponding to GW170817 Abbott _et al._ (2017) and GW190425 Abbott _et
al._ (2020) events (orange and blue regions). The green regions represent the
estimations for the mass and radius values of PSR J0740+6620 (top) Miller _et
al._ (2019) and J0030+0451 (bottom) Riley _et al._ (2021). The purple region
constraints the mass-radius curves from the statistical analysis done in
Altiparmak _et al._ (2022). We find that the NS resulting from the addition
of a crust to the Skyrme crystal EoS with a nonzero kaon condensate agree very
well with these recent constraints. Further, the softening of the EoS due to
the presence of kaons and the resulting smaller NS radii are important for
this agreement.
Figure 8: Mass-Radius curves of NS with a kaon condensed core. The different
sets of parameters that we consider are shown with different colors. Left
panel: Solid lines represent $npe\mu$ matter, dashed-dotted lines are obtained
with a MC and the dashed with the GC. Right panel: The effect of adding a
standard nuclear physics crust to the Skyrme crystal EoS with kaon condensate
obtained from the Gibbs construction (GC).
## VI Conclusions
The quantization of the isospin degrees of freedom in Adam _et al._ (2022b)
allowed us to find the dependence of the isospin contribution to the energy
(hence the isospin chemical potential) on the lattice length of the skyrmion
crystals, which in turn determines the proton (and electron) fraction in
$\beta$-equilibrated skyrmion crystals. In this paper, we have used this
information to precisely determine the critical density at which charged kaons
will condense inside neutron stars described by the Skyrme crystal. Although
the prediction of the condensation of charged mesons at high (isospin)
chemical potential is common in nuclear matter literature, and there have been
some partial results within the Skyrme model Westerberg (1995); Park _et al._
(2010), we have, to our knowledge, for the first time provided a framework
which allows to precisely calculate the value of the density where kaon
condensation sets in. This value turns out to be around twice nuclear
saturation for physically relevant sets of parameters, so the prediction from
the Skyrme model is that strangeness will be present at the core of neutron
stars in the form of an (anti)kaon condensate. Further, we have computed the
EoS for skyrmion matter in the condensate phase, which becomes softer due to
the additional degrees of freedom. This has appreciable effects on the Mass-
Radius curves for physically relevant parameters in the model, reducing the
maximum radii by about 0.5-1 km. We found that a correct treatment of the
resulting first-order phase transition between the phase with and without kaon
condensate by a Gibbs construction, as originally advocated in Glendenning
(1992), is important for these results.
Despite the prediction of a very early onset of kaon condensation as compared
with other nuclear EoS, the maximum mass limit for neutron stars doesn’t get
significantly reduced, due to the sextic term being dominant at such
densities. The fact that the presence of a kaon condensate does not pose a
problem for reaching high masses in the Skyrme model, however, does not imply
that the hyperon problem is completely solved in Skyrmion-based EoS. Indeed,
in addition to kaons, one should take into account also hyperon degrees of
freedom within the Skyrme model. Whilst hyperons can be succesfully described
within this model Klebanov (1990), and even a proposal to study modifications
of hyperon properties in dense matter has bee provided in Hong _et al._
(2019), it is not clear for us how to apply these ideas to Skyrmion crystals.
A possibility would be to extend the isospin quantization scheme proposed in
Adam _et al._ (2022b) to the three flavor case, and quantize the whole
$SU(3)$ rotation group. This method, however, is much more technically
involved, and relies on the assumption of an approximate $SU(3)$ symmetry of
the Hamiltonian, which only holds for sufficiently large densities.
###### Acknowledgements.
The authors acknowledge financial support from the Ministry of Education,
Culture, and Sports, Spain (Grant No. PID2020-119632GB-I00), the Xunta de
Galicia (Grant No. INCITE09.296.035PR and Centro singular de investigación de
Galicia accreditation 2019-2022), the Spanish Consolider-Ingenio 2010
Programme CPAN (CSD2007-00042), and the European Union ERDF. AW is supported
by the Polish National Science Centre, grant NCN 2020/39/B/ST2/01553. AGMC is
grateful to the Spanish Ministry of Science, Innovation and Universities, and
the European Social Fund for the funding of his predoctoral research activity
(_Ayuda para contratos predoctorales para la formación de doctores_ 2019). MHG
is also grateful to the Xunta de Galicia (Consellería de Cultura, Educación y
Universidad) for the funding of his predoctoral activity through _Programa de
ayudas a la etapa predoctoral_ 2021\.
## Appendix A Derivation of the WZW and sextic terms contribution to $V_{K}$
To work with the WZW and sextic terms, it is useful to employ the formalism of
Lie algebra-valued differential forms, which we will extensively do in this
appendix. Let us first review the basic properties of such objects and
establish the notation that we will follow. A $\mathfrak{g}$-valued
differential form $\alpha$ can be written in terms of the Lie algebra
generators $T_{a}$, as $\alpha=\alpha^{a}\otimes T_{a}$. The exterior
derivative is then simply obtained as $d\alpha=d\alpha^{a}\otimes T_{a}$.
Furthermore, the wedge product on $\mathfrak{g}$-valued forms is defined as
$\alpha\wedge\beta=\alpha^{a}\wedge\beta^{b}\otimes T_{a}T_{b}.$ (89)
So that the following useful properties hold:
$\displaystyle d\alpha\wedge\beta$
$\displaystyle=(d\alpha)\wedge\beta+(-1)^{\absolutevalue{\alpha}}\alpha\wedge(d\beta)$
(90) $\displaystyle\Tr{\alpha\wedge\beta}$
$\displaystyle=(-1)^{\absolutevalue{\alpha}\absolutevalue{\beta}}\Tr{\beta\wedge\alpha},$
(91)
where $\absolutevalue{\alpha}$ denotes the degree of $\alpha$. Also, by
linearity of the trace, both the trace and the exterior derivative commute,
i.e.
$d\Tr{\alpha}=\Tr{d\alpha}.$ (92)
To alleviate the notation, in the following we will drop the wedge product
symbol and denote the product (89) simply by $\alpha\beta$. Then, for
instance, if $\alpha$ and $\beta$ denote two 1-forms, we have
$\Tr{\alpha\beta}=-\Tr{\beta\alpha}$, $d(\alpha\beta)=d\alpha\beta-\alpha
d\beta.$
Let us now perform the most general chiral transformation to the Skyrme field
$U_{\pi}$, given by $U=g_{l}U_{\pi}g_{r}^{\dagger}$, with $(g_{l},g_{r})\in
SU(3)_{L}\times SU(3)_{R}$ and define the following $\mathfrak{su}(3)$-valued
differential forms,
$\begin{split}V=U^{\dagger}&dU,\,\,L=U_{\pi}^{\dagger}dU_{\pi},\,\,\alpha=g_{l}^{\dagger}dg_{l},\,\beta=g_{r}dg_{r}^{\dagger}.\end{split}$
(93)
By definition, we have the following relation between the forms above:
$V=(U_{\pi}g_{r})^{\dagger}[\alpha+U_{\pi}(L-\beta)U^{\dagger}_{\pi}]U_{\pi}g_{r}.$
(94)
On the other hand, the WZW action is then given by the pullback of a volume
5-form $\Omega_{5}$ by an extended Skyrme field $U:D^{5}\rightarrow SU(3)$
333The result is of course independent of such extension, because
$\pi_{4}(SU(3))$ vanishes. integrated over an auxiliar 5-dimensional disk $D$
whose boundary is the spacetime manifold $M$,
$S_{\rm WZW}=-i\frac{N_{C}}{240\pi^{2}}\int_{D}U^{*}(\Omega_{5})$ (95)
The form $U^{*}(\Omega_{5})$ can be expressed in terms of $L$ as
$\displaystyle S_{\rm WZW}(L)$
$\displaystyle=-i\frac{N_{C}}{240\pi^{2}}\int_{D}\Tr\\{V^{5}\\}=$
$\displaystyle=$
$\displaystyle-\frac{iN_{C}}{240\pi^{2}}\int_{D}\Tr\\{[\alpha+U_{\pi}(L-\beta)U^{\dagger}_{\pi}]^{5}\\}.$
(96)
Let us denote the 1-form $U_{\pi}(L-\beta)U^{\dagger}_{\pi}$ by $\omega$. The
exterior derivative of this form is:
$\displaystyle d\omega$
$\displaystyle=dU_{\pi}(L-\beta)U^{\dagger}_{\pi}+U_{\pi}(dL-d\beta)U^{\dagger}_{\pi}-U_{\pi}(L-\beta)dU^{\dagger}_{\pi}=$
$\displaystyle=-U^{\dagger}_{\pi}(dL+d\beta+L\beta+\beta L)U_{\pi},$ (97)
where we have used the fact that both $\beta$ and $L$ satisfy the Maurer-
Cartan equation $d\beta=-\beta^{2}$. Moreover, one can straightforwardly see
that
$\omega^{2}=U^{\dagger}_{\pi}(L-\beta)^{2}U_{\pi}=U^{\dagger}_{\pi}(-d\beta-
dv-\beta v-v\beta)U_{\pi}=d\omega$ (98)
Knowing this, we have
$\begin{split}S_{\rm
WZW}(L)=&-i\frac{N_{C}}{240\pi^{2}}\int_{\mathcal{M}}\Tr\\{[\alpha+\omega]^{5}\\}\\\
=&S_{\rm WZW}(\alpha)+S_{\rm
WZW}(\omega)-\frac{iN_{C}}{48\pi^{2}}\int_{\mathcal{M}}\Tr\\{\alpha^{4}\omega+\omega^{4}\alpha+\alpha^{2}\omega^{3}+\omega^{2}\alpha^{3}+\alpha\omega\alpha\omega^{2}+\omega\alpha\omega\alpha^{2}\\}\\\
=&S_{\rm WZW}(\alpha)+S_{\rm
WZW}(\omega)-\frac{iN_{C}}{48\pi^{2}}\int_{\partial\mathcal{M}}\Tr\\{\omega^{3}\alpha-\alpha^{3}\omega-\frac{1}{2}(\alpha\omega)^{2}\\},\end{split}$
(99)
where we have used eqs. 90, 91 and 92, the relation (98) for $\omega$, the M-C
equation for $\alpha$ and Stokes’ theorem in the last step. Repeating the same
calculation for $S_{\rm WZW}(\omega)$ yields:
$\displaystyle S_{\rm WZW}(\omega)=$
$\displaystyle-i\frac{N_{C}}{240\pi^{2}}\int_{\mathcal{M}}\Tr\\{[L-\beta]^{5}\\}=$
(100) $\displaystyle=$ $\displaystyle-S_{\rm WZW}(\beta)+S_{\rm WZW}(L)-$
$\displaystyle-\frac{iN_{C}}{48\pi^{2}}\int_{\partial\mathcal{M}}\\!\\!\Tr\\{L^{3}\beta-\beta^{3}L-\frac{1}{2}(L\beta)^{2}\\},$
so that
$\displaystyle S_{\rm WZW}$ $\displaystyle(V)=S_{\rm WZW}(L)+S_{\rm
WZW}(\alpha)-S_{\rm WZW}(\beta)-$ $\displaystyle-\frac{iN_{C}}{48\pi^{2}}$
$\displaystyle\int_{\partial\mathcal{M}}\Tr\\{L^{3}\beta-\beta^{3}L-\frac{1}{2}(L\beta)^{2}+\omega^{3}\alpha-\alpha^{3}\omega-\frac{1}{2}(\alpha\omega)^{2}\\}.$
(101)
Eq.(101) shows that a chiral transformation of the $SU(3)$ Skyrme field
induces an additional local term in the action due to the nontrivial
transformation of the (nonlocal) WZW term. Furthermore, if we fix the chiral
transformation fields to only depend on one spacetime coordinate,
$g_{l/r}(x)\equiv g_{l/r}(t)$, any power of $\alpha$ and $\beta$ will vanish
in the local, $4$-dimensional effective term. Taking this into account, we
arrive to the final result:
$S_{\rm WZW}(V)=S_{\rm
WZW}(L)-\frac{iN_{C}}{48\pi^{2}}\int_{\partial\mathcal{M}}\Tr\\{L^{3}(\beta+U^{\dagger}_{\pi}\alpha
U_{\pi})\\},$ (102)
from where eq. 51 is readily obtained.
Let us now turn to the sextic term. The coordinate free version of
$\mathcal{L}_{6}$ is given by
$\mathcal{L}_{6}=\lambda^{2}\pi^{4}\mathcal{B}\wedge\star\mathcal{B}$ (103)
where $\star$ denotes the Hodge star operator, and $\mathcal{B}$ the 1-form in
spacetime whose coordinates in a local chart coincide with the Baryon current,
$\mathcal{B}=B_{\mu}dx^{\mu}$. We can construct such form as the Hodge dual of
the baryon number density three-form,
$\displaystyle b$
$\displaystyle=(24\pi^{2})^{-1}U^{*}(\Omega_{3})=\frac{1}{24\pi^{2}}\Tr\\{L_{\mu}L_{\nu}L_{\rho}\\}dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\rho}$
$\displaystyle=\frac{1}{3!}b_{\mu\nu\rho}dx^{\mu}\wedge dx^{\nu}\wedge
dx^{\rho},$ (104)
i.e.
$\displaystyle\mathcal{B}=\star b=$
$\displaystyle\frac{1}{3!}b^{\nu\rho\sigma}\varepsilon_{\nu\rho\sigma\mu}dx^{\mu}=$
$\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}}\varepsilon_{\mu}^{\,\,\,\nu\rho\sigma}\Tr\\{L_{\nu}L_{\rho}L_{\sigma}\\}=B_{\mu}dx^{\mu}.$
(105)
Thus $\mathcal{L}_{6}=\lambda^{2}\pi^{4}\star b\wedge b$. Expressing the
sextic term in such form is most useful for calculating its contribution to
the kaon potential employing the formalism of differential forms in the same
way as for the WZW term. Indeed, we see that
$b=\frac{1}{24\pi^{2}}\Tr\\{L^{3}\\}=\frac{1}{24\pi^{2}}\Tr\\{\omega^{3}+3w\omega^{2}\\},$
(106)
where we used the fact that $w^{n}=0$ for $n\geq 2$ because the kaon field
$\Sigma$ only depends on time. Hence, also $v^{n}=0$ for $n\geq 2$ and $vw=0$
hold, and, given that $\omega=U_{\pi}^{\dagger}(\beta-v)U_{\pi}$, we may write
$\displaystyle b=$
$\displaystyle\frac{1}{24\pi^{2}}\Tr\\{\beta^{3}+3(wU_{\pi}^{\dagger}(\beta-v)^{2}U_{\pi}-\beta^{2}v)\\}=$
$\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}}\Tr\\{\beta^{3}+3\beta^{2}(U_{\pi}wU_{\pi}^{\dagger}-v)\\}.$
(107)
Thus, the baryon current density in the kaon condensed phase will be modified
by $B^{\mu}=B^{\mu}_{\pi}+C^{\mu}$, where $B^{\mu}_{\pi}$ is the baryon
current due to the pionic background and
$C^{\mu}=\frac{1}{8\pi^{2}}\varepsilon^{\mu\nu\rho\sigma}\Tr\\{R_{\nu}R_{\rho}(U_{\pi}\Sigma\partial_{\sigma}\Sigma^{\dagger}U_{\pi}^{\dagger}-\Sigma^{\dagger}\partial_{\sigma}\Sigma).\\}$
(108)
For a time independent pion background and a homogeneous kaon condensate, we
have $B_{\pi}^{\mu}C_{\mu}=0$, and hence the only constribution from the
sextic term ($\propto B_{\mu}B^{\mu}$) to $V_{K}$ comes from the additional
term:
$\displaystyle C_{\mu}C^{\mu}$
$\displaystyle=\frac{1}{64\pi^{4}}\varepsilon^{\mu\nu\rho\sigma}\epsilon_{\mu\alpha\beta}\Tr\\{R_{\nu}RE_{\rho}\xi_{\sigma}\\}\Tr\\{R_{\alpha}R_{\beta}\xi_{\gamma}\\}=$
$\displaystyle=\frac{-1}{64\pi^{4}}\varepsilon_{ijk}\varepsilon_{ilm}\Tr\\{R_{j}R_{k}\xi_{0}\\}Tr\\{R_{l}R_{m}\xi_{0}\\}$
(109)
where
$\xi_{\mu}=U_{\pi}\Sigma\partial_{\mu}\Sigma^{\dagger}U_{\pi}^{\dagger}-\Sigma^{\dagger}\partial_{\mu}\Sigma$.
## Appendix B Explicit expressions
We show here the explicit expressions of the (third component of the) inertia
tensor and the kaon isospin current, divided into separated contributions
coming from the quadratic, quartic, sextic and Wess-Zumino-Witten terms. A
superindex labels the origin of each contribution.
$\displaystyle\Lambda_{33}$
$\displaystyle\equiv\Lambda_{3}=\Lambda_{3}^{(2)}+\Lambda_{3}^{(4)}+\Lambda_{3}^{(6)}$
(110) $\displaystyle\Delta_{3}$
$\displaystyle\equiv\Delta=\Delta^{(2)}+\Delta^{(4)}+\Delta^{(6)}+\Delta^{(\rm
WZW)}$ (111)
with
$\displaystyle\Lambda^{(2)}_{3}$ $\displaystyle=2a\Tr\\{T_{3}T_{3}\\}=$ (112)
$\displaystyle=\frac{\pi_{1}^{2}+\pi_{2}^{2}}{2}(1+\cos^{2}\phi)^{2}+(1+\sigma)\sin^{2}(2\phi)/4,$
$\displaystyle\Lambda^{(4)}_{3}$
$\displaystyle=-4b\Tr\\{[T_{3},L_{k}][T_{3},L_{k}]\\}=2(1+\cos^{2}\phi)\times$
$\displaystyle\times\big{[}(1-\pi^{2}_{3})\partial_{i}\sigma^{2}+\sigma\pi_{3}\partial_{i}\sigma\partial_{i}\pi_{3})+(\sigma\leftrightarrow\pi_{3})\big{]}+$
$\displaystyle+\partial_{i}n^{2}(1+\sigma)/4\sin^{2}(2\phi),$ (113)
$\displaystyle\Lambda^{(6)}_{3}$
$\displaystyle=-\frac{c}{32\pi^{4}}\varepsilon^{lmn}\Tr\\{T_{3}L_{m}L_{n}\\}\varepsilon_{lrs}\Tr\\{T_{3}L_{r}L_{s}\\}=$
$\displaystyle=$
$\displaystyle\frac{\lambda^{2}f^{2}_{\pi}e^{4}\mu_{K}^{2}}{2}(1+\cos^{2}(\tilde{\phi}))^{2}(\partial_{i}\pi_{3}\partial_{j}\sigma-\partial_{i}\sigma\partial_{j}\pi_{3})^{2}$
(114) $\displaystyle\Delta^{(2)}$ $\displaystyle=2a\Tr\\{T_{3}L_{0}\\}=$ (115)
$\displaystyle=-i\mu_{K}[(\pi_{1}^{2}+\pi_{2}^{2})(\cos^{4}\phi-1)+(1+\sigma)\sin^{2}(2\phi)/2],$
$\displaystyle\Delta^{(4)}$
$\displaystyle=-4b\Tr\\{[T_{a},L_{k}][T_{b},L_{k}]\\}=-2i\mu_{K}\big{[}2(1-\cos^{4}\phi)\times$
$\displaystyle\times(\pi_{1}^{2}\partial_{i}\pi_{2}^{2}+\pi_{2}^{2}\partial_{i}\pi_{1}^{2}-2\pi_{1}\pi_{2}\partial_{i}\pi_{1}\partial_{i}\pi_{2}-\partial_{i}n^{2}(\pi_{1}^{2}+\pi^{2}_{2}))+$
$\displaystyle+\partial_{i}n^{2}/4(1+\sigma)\sin^{2}(2\phi)\big{]},$ (116)
$\displaystyle\Delta^{(6)}$
$\displaystyle=-\frac{c}{32\pi^{4}}\varepsilon^{lmn}\Tr\\{T_{3}L_{m}L_{n}\\}\varepsilon_{lrs}\Tr\\{L_{0}L_{r}L_{s}\\}=$
$\displaystyle=i\mu_{K}\lambda^{2}f^{2}_{\pi}e^{4}(1+\cos^{2}\phi)\sin^{2}\phi(\partial_{i}\pi_{3}\partial_{j}\sigma-\partial_{i}\sigma\partial_{j}\pi_{3})^{2}$
(117) $\displaystyle\Delta$ ${}^{(\rm WZW)}=-\frac{N_{C}B_{\rm
cell}}{2}\sin^{2}{\left(\phi\right)}$ (118)
## References
* Li _et al._ (2018) J. J. Li, A. Sedrakian, and F. Weber, Phys. Lett. B 783, 234 (2018), arXiv:1803.03661 [nucl-th] .
* Glendenning (1985) N. K. Glendenning, Astrophys. J. 293, 470 (1985).
* Hartle _et al._ (1975) J. B. Hartle, R. F. Sawyer, and D. J. Scalapino, Astrophys. J. 199, 471 (1975).
* Celenza _et al._ (1977) L. S. Celenza, W. Nutt, and C. Shakin, Physics Letters B 72, 23 (1977).
* Kaplan and Nelson (1988) D. B. Kaplan and A. E. Nelson, Nucl. Phys. A 479, 273c (1988).
* Glendenning and Schaffner-Bielich (1999) N. K. Glendenning and J. Schaffner-Bielich, Phys. Rev. C 60, 025803 (1999), arXiv:astro-ph/9810290 .
* Pal _et al._ (2000) S. Pal, D. Bandyopadhyay, and W. Greiner, Nucl. Phys. A 674, 553 (2000), arXiv:astro-ph/0001039 .
* Heiselberg _et al._ (1993) H. Heiselberg, C. J. Pethick, and E. F. Staubo, Phys. Rev. Lett. 70, 1355 (1993).
* Benvenuto and Lugones (1999) O. G. Benvenuto and G. Lugones, Mon. Not. Roy. Astron. Soc. 304, L25 (1999).
* Annala _et al._ (2020) E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, Nature Physics 16, 907 (2020).
* McLerran and Pisarski (2007) L. McLerran and R. D. Pisarski, Nucl. Phys. A 796, 83 (2007), arXiv:0706.2191 [hep-ph] .
* McLerran and Reddy (2019) L. McLerran and S. Reddy, Phys. Rev. Lett. 122, 122701 (2019), arXiv:1811.12503 [nucl-th] .
* Tolos and Fabbietti (2020) L. Tolos and L. Fabbietti, Prog. Part. Nucl. Phys. 112, 103770 (2020), arXiv:2002.09223 [nucl-ex] .
* Bombaci (2017) I. Bombaci, JPS Conf. Proc. 17, 101002 (2017), arXiv:1601.05339 [nucl-th] .
* Vidaña (2016) I. Vidaña, J. Phys. Conf. Ser. 668, 012031 (2016), arXiv:1509.03587 [nucl-th] .
* Skyrme (1962) T. H. R. Skyrme, Nuclear Physics 31, 556 (1962).
* Adkins _et al._ (1983) G. S. Adkins, C. R. Nappi, and E. Witten, Nuclear Physics B 228, 552 (1983).
* Ding and Yan (2007) G.-J. Ding and M.-L. Yan, Phys. Rev. C 75, 034004 (2007), arXiv:nucl-th/0702037 .
* Halcrow and Harland (2020) C. Halcrow and D. Harland, Phys. Rev. Lett. 125, 042501 (2020).
* Lau and Manton (2014) P. H. C. Lau and N. S. Manton, Phys. Rev. Lett. 113, 232503 (2014).
* Halcrow _et al._ (2017) C. J. Halcrow, C. King, and N. S. Manton, Phys. Rev. C 95, 031303 (2017).
* Adam _et al._ (2013) C. Adam, C. Naya, J. Sanchez-Guillen, and A. Wereszczynski, Phys. Rev. Lett. 111, 232501 (2013), arXiv:1312.2960 [nucl-th] .
* Gillard _et al._ (2015) M. Gillard, D. Harland, and M. Speight, Nucl. Phys. B 895, 272 (2015).
* Gudnason (2016) S. B. Gudnason, Phys. Rev. D 93, 065048 (2016).
* Naya and Sutcliffe (2018) C. Naya and P. Sutcliffe, Phys. Rev. Lett. 121, 232002 (2018).
* Adam _et al._ (2010) C. Adam, J. Sánchez-Guillén, and A. Wereszczyński, Physics Letters B 691, 105–110 (2010).
* Naya (2019) C. Naya, International Journal of Modern Physics E 28, 1930006 (2019).
* Adam _et al._ (2020) C. Adam, A. G. Martín-Caro, M. Huidobro, R. Vázquez, and A. Wereszczynski, Phys. Lett. B 811, 135928 (2020).
* Adam _et al._ (2022a) C. Adam, A. G. Martin-Caro, M. Huidobro, R. Vazquez, and A. Wereszczynski, Phys. Rev. D 105, 074019 (2022a), arXiv:2109.13946 [hep-th] .
* Adam _et al._ (2022b) C. Adam, A. G. Martín-Caro, M. Huidobro, R. Vázquez, and A. Wereszczynski, (2022b), arXiv:2202.00953 [nucl-th] .
* Adam _et al._ (2015) C. Adam, C. Naya, J. Sanchez-Guillen, R. Vazquez, and A. Wereszczynski, Physics Letters B 742, 136–142 (2015).
* Callan and Klebanov (1985) C. G. Callan and I. Klebanov, Nuclear Physics B 262, 365 (1985).
* Klebanov (1990) I. Klebanov, “Strangeness in the skyrme model,” in _Hadrons and Hadronic Matter_, edited by D. Vautherin, F. Lenz, and J. W. Negele (Springer US, Boston, MA, 1990) pp. 223–262.
* Fiorella Burgio and Fantina (2018) G. Fiorella Burgio and A. F. Fantina, Astrophys. Space Sci. Libr. 457, 255 (2018).
* Essick _et al._ (2021) R. Essick, P. Landry, A. Schwenk, and I. Tews, Phys. Rev. C 104, 065804 (2021).
* Tang _et al._ (2021) S.-P. Tang, J.-L. Jiang, M.-Z. Han, Y.-Z. Fan, and D.-M. Wei, Phys. Rev. D 104, 063032 (2021).
* de Tovar _et al._ (2021) P. B. de Tovar, M. Ferreira, and C. Providência, Phys. Rev. D 104, 123036 (2021).
* Gil _et al._ (2021) H. Gil, P. Papakonstantinou, and C. H. Hyun, (2021), arXiv:2110.09802 [nucl-th] .
* Li _et al._ (2021) B.-A. Li, B.-J. Cai, W.-J. Xie, and N.-B. Zhang, Universe 7, 182 (2021).
* Klebanov (1985) I. Klebanov, Nuclear Physics B 262, 133 (1985).
* Blom _et al._ (1989) U. Blom, K. Dannbom, and D. Riska, Nuclear Physics 493, 384 (1989).
* Nyman and Riska (1990) E. M. Nyman and D. Riska, Reports on Progress in Physics 53, 1137 (1990).
* Schmitt (2010) A. Schmitt, _Dense matter in compact stars: A pedagogical introduction_, Vol. 811 (2010) arXiv:1001.3294 [astro-ph.SR] .
* Note (1) Actually, that the charged (in particular, the negatively charged) kaons will condense first is true in our approach (whenever $\mu_{e}>0$), since the chemical potential associated to neutral kaons is zero, so that the onset of neutral kaon condensation is given by $m^{*}_{K}=0$.
* Note (2) Remember that we are using the mostly minus convention for the metric signature.
* Park _et al._ (2019) B.-Y. Park, W.-G. Paeng, and V. Vento, Nucl. Phys. A 989, 231 (2019), arXiv:1904.04483 [hep-ph] .
* Canfora _et al._ (2020) F. Canfora, M. Lagos, and A. Vera, Eur. Phys. J. C 80, 697 (2020), arXiv:2007.11543 [hep-th] .
* Glendenning (1992) N. K. Glendenning, Phys. Rev. D 46, 1274 (1992).
* Bhattacharyya _et al._ (2010) A. Bhattacharyya, I. N. Mishustin, and W. Greiner, J. Phys. G 37, 025201 (2010), arXiv:0905.0352 [nucl-th] .
* Sharma _et al._ (2015) B. K. Sharma, M. Centelles, X. Viñas, M. Baldo, and G. F. Burgio, Astron. Astrophys. 584, A103 (2015).
* Abbott _et al._ (2017) B. P. Abbott _et al._ (LIGO , VIRGO), Phys. Rev. Lett. 119, 161101 (2017).
* Abbott _et al._ (2020) B. Abbott _et al._ (LIGO Scientific, Virgo), Astrophys. J. Lett. 892, L3 (2020).
* Miller _et al._ (2019) M. C. Miller _et al._ , Astrophys. J. Lett. 887, L24 (2019), arXiv:1912.05705 [astro-ph.HE] .
* Riley _et al._ (2021) T. E. Riley _et al._ , Astrophys. J. Lett. 918, L27 (2021).
* Altiparmak _et al._ (2022) S. Altiparmak, C. Ecker, and L. Rezzolla, (2022), arXiv:2203.14974 [astro-ph.HE] .
* Westerberg (1995) K. M. Westerberg, Phys. Rev. D 51, 5030 (1995), arXiv:hep-ph/9411430 .
* Park _et al._ (2010) B.-Y. Park, J.-I. Kim, and M. Rho, Phys. Rev. C 81, 035203 (2010), arXiv:0912.3213 [hep-ph] .
* Hong _et al._ (2019) K.-H. Hong, U. Yakhshiev, and H.-C. Kim, Phys. Rev. C 99, 035212 (2019), arXiv:1806.06504 [nucl-th] .
* Note (3) The result is of course independent of such extension, because $\pi_{4}(SU(3))$ vanishes.
|
# Manipulating non-reciprocity in a two-dimensional magnetic quantum walk
Quan Lin Beijing Computational Science Research Center, Beijing 100084, China
Wei Yi<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University
of Science and Technology of China, Hefei 230026, China CAS Center For
Excellence in Quantum Information and Quantum Physics, Hefei 230026, China
Peng Xue<EMAIL_ADDRESS>Beijing Computational Science Research Center,
Beijing 100084, China
###### Abstract
Non-reciprocity is an important topic in fundamental physics and quantum-
device design, as much effort has been devoted to its engineering and
manipulation. Here we experimentally demonstrate non-reciprocal transport in a
two-dimensional quantum walk of photons, where the directional propagation is
highly tunable through dissipation and synthetic magnetic flux. The non-
reciprocal dynamics hereof is a manifestation of the non-Hermitian skin
effect, with its direction continuously adjustable through the photon-loss
parameters. By contrast, the synthetic flux originates from an engineered
geometric phase, which competes with the non-Hermitian skin effect through
magnetic confinement. We further demonstrate how the non-reciprocity and
synthetic flux impact the dynamics of the Floquet topological edge modes along
an engineered boundary. Our results exemplify an intriguing strategy for
achieving tunable non-reciprocal transport, highlighting the interplay of non-
Hermiticity and gauge fields in quantum systems of higher dimensions.
Open quantum systems are ubiquitous in nature, and exhibit rich and complex
behaviors unknown to their closed counterparts openbook . The recent
progresses in non-Hermitian physics offer fresh insights into open systems
from a unique perspective, giving rise to exotic symmetries and new paradigms
of topology benderreview ; review2 ; photonpt1 ; review3 ; nonHtopo1 ;
nonHtopo2 ; Lee ; WZ1 . A much studied non-Hermitian phenomenon of late is the
non-Hermitian skin effect (NHSE) WZ1 ; WZ2 ; murakami ; ThomalePRB ; stefano ;
tianshu ; Budich ; mcdonald ; alvarez ; lli ; yzsgbz ; stefano2 ; lyc ;
fangchenskin2 ; teskin ; photonskin ; metaskin ; scienceskin ; teskin2d ; dzou
; fangchenskin ; kawabataskin , whereby a macroscopic number of eigenstates
become exponentially localized toward the boundaries. The NHSE has significant
impact on the band topology WZ1 ; WZ2 ; murakami , the spectral symmetry
nonblochpt1 ; nonblochpt2 ; XDW+21 , and dynamics quench1 ; quench2 ; coldatom
. One of the most salient dynamic signatures of the NHSE is the directional
bulk flow stefano ; coldatom ; ql ; ql2 , which is closely connected to the
global topology of the spectrum on the complex plane fangchenskin . Such non-
reciprocal dynamics can have potential applications in topological transport
and quantum-device design, but the generation and control of this peculiar
form of non-reciprocity, particularly in higher dimensions, remain
experimentally unexplored.
In this work, we experimentally demonstrate the tuning of non-reciprocal
transport in photonic quantum walks on a synthetic two-dimensional square
lattice. The non-reciprocal dynamics underlies the NHSE of the two-dimensional
quantum walk—the unidirectional flow leads to the accumulation of eigenstates
toward boundaries in the corresponding direction. By tuning the photon-loss
parameters, we show how the direction of the flow (hence the direction of the
NHSE) can be continuously adjusted. The much discussed corner skin effect in
two dimensions is but a special case here, corresponding to a directional flow
along the diagonal of the square lattice. By engineering the quantum-walk
setup, we also introduce a synthetic flux to the lattice hof ; sai , which we
observe to suppress the non-reciprocal dynamics. Such a suppression is the
result of the competition between two localization mechanisms: magnetic
confinement and NHSE chenwei . We quantitatively characterize the tunability
of the non-reciprocity through loss and flux, and further demonstrate their
impact on the dynamics of topological edge modes along the boundary. Our
experiment is the first observation of the magnetic suppression of NHSE, and
further illustrates the flexible control over the NHSE-induced non-reciprocity
in higher dimensions.
Figure 1: Two-dimensional non-Hermitian quantum walk with a synthetic gauge
field. a Schematics for possible movements of a walker at spatial position
$(x,y)$ during each time step. b A time-multiplexed implementation of the two-
dimensional photonic quantum walk. The photons are initialized at position
$(0,0)$ in the superposition of the polarizations
$(\ket{H}+i\ket{V})/\sqrt{2}$. Once coupled into the setup through a low-
reflectivity beam splitter (BS, reflectivity $3\%$), their polarization state
is manipulated by a half-wave plate (HWP). The photonic wave packets are split
by a polarizing beam splitter (PBS) and routed through a pair of single-mode
fibers (SMF) of length $287.03$m and $270$m, respectively, implementing a
temporal step in the $x$ direction. A temporal step in the $y$ direction is
implemented by another two-PBS loop based on the same principle, but in the
free space instead of fibers. At each step, photons are partially coupled out
to a polarization resolving detection of the arrival time via avalanche
photodiodes (APDs). ND: neutral density filter; AOM: optical switch acousto-
optic modulator; EOM: electro-optic modulator. c Illustration of the operation
sequence of the time-multiplexed quantum walk. Here $V_{\text{EOM}}$ is the
control voltage applied to the EOMs. Figure 2: Tunable non-reciprocity and
NHSE. The walker with the polarization $(\ket{H}+i\ket{V})/\sqrt{2}$ starts
from the lattice site $(0,0)$ with $\alpha=0$. Probability distributions are
measured after $16$ time steps. a Probability distribution for a Hermitian
two-dimensional quantum walk with $\gamma_{x}=\gamma_{y}=0$. b Probability
distribution following a non-Hermitian quantum walk with
$\gamma_{x}=\gamma_{y}=0.125$. c Directional displacements after the final
time step ($t=16$) for quantum walks with varying $\gamma_{x}$ and
$\gamma_{y}$. (Left) Color contour of the azimuthal angle of the displacement
$\bm{d}$ on the $x$–$y$ plane. (Right) Measured (blue arrows) and simulated
(gray arrows) of the displacement along the red vertical line of the color
contour (left panel). d Probability distribution following a non-Hermitian
quantum walk in the presence of domain walls (marked by red dashed lines),
with $\gamma_{x}=0.125$ for $x\geq-6$, $\gamma_{x}=-0.125$ for $x<-6$,
$\gamma_{y}=0.125$ for $y\geq-6$, and $\gamma_{y}=-0.125$ for $y<-6$. Figure
3: Magnetic suppression of the non-reciprocal transport. The walker is
initialized in the state
$\frac{1}{\sqrt{2}}(\ket{H}+i\ket{V})\otimes\ket{x=0}\ket{y=0}$. Measured
probability distributions of $16$-time-step quantum walks with the tuning
parameter $\alpha=0.05$, and the loss parameter $\gamma_{x}=0$,
$\gamma_{y}=0.1$ in a, with $\alpha=0.15$, $\gamma_{x}=0$, and
$\gamma_{y}=0.1$ in b, with $\alpha=0.05$ and $\gamma_{x}=\gamma_{y}=0.1$ in
c, and with $\alpha=0.15$ and $\gamma_{x}=\gamma_{y}=0.1$ in d. e Norm of the
directional displacement $\bm{d}$ for $16$-time-step quantum walks under
different flux $\alpha$. Symbols represent the experimental data, and curves
are the corresponding numerical simulations. Error bars are due to the
statistical uncertainty in photon-number-counting. f Measured (blue) and
simulated (gray) $\bm{d}$. Blue arrows represent the experimental results, and
gray ones indicate the corresponding numerical simulations.
Results
Time-multiplexed two-dimensional quantum walk. In discrete-time quantum walks,
the walker state $|\psi(t)\rangle$ evolves according to
$|\psi(t)\rangle=U^{t}|\psi(0)\rangle$, where $t$ indicates the discrete time
steps, and $U$ is thus identified as the Floquet operator that periodically
drives the system. We consider such a quantum walk on a two-dimensional square
lattice, with the Floquet operator
$U=M_{y}S_{y}PCM_{x}S_{x}C.$ (1)
Here the shift operators are defined as
$S_{j}=\sum_{\bm{r}}\ket{0}\bra{0}\otimes\ket{\bm{r}-\bm{e}_{j}}\bra{\bm{r}}+\ket{1}\bra{1}\otimes\ket{\bm{r}+\bm{e}_{j}}\bra{\bm{r}}$,
with $\bm{r}=(x,y)\in\mathbb{Z}^{2}$ labeling the coordinates of the lattice
sites, $j\in\\{x,y\\}$, and $\bm{e}_{x}=(1,0)$ and $\bm{e}_{y}=(0,1)$. The
shift operators move the walker in the corresponding directions, depending on
the walker’s internal degrees of freedom in the basis of
$\\{|0\rangle,|1\rangle\\}$ (dubbed the coin states). These coin states are
subject to rotations under the coin operator
$C=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}\otimes\mathds{1}_{\bm{r}}$, where
$\mathds{1}_{\bm{r}}=\sum_{\bm{r}}\ket{\bm{r}}\bra{\bm{r}}$. The gain-loss
operators are given by (here $j\in\\{x,y\\}$)
$\displaystyle M_{j}(\gamma_{j})=\begin{pmatrix}e^{\gamma_{j}}&0\\\
0&e^{-\gamma_{j}}\end{pmatrix}\otimes\mathds{1}_{\bm{r}},$ (2)
which make the quantum walk non-unitary for finite $\gamma_{x}$ or
$\gamma_{y}$.
A key ingredient to our scheme is the phase-shift operator, defined as
$P=\sum_{\bm{r}}\begin{pmatrix}e^{i2\pi\alpha x}&0\\\ 0&e^{-i2\pi\alpha
x}\end{pmatrix}\otimes\ket{\bm{r}}\bra{\bm{r}},$ (3)
which enforces a position-dependent geometric phase on the walker, so that the
latter acquires a phase $2\pi\alpha$ when going around any single plaquette of
the square lattice (see Fig. 1a). Similar to that of the Hofstadter model hof
, the accumulated phase shift of the walker on the lattice is equal to the
Aharonov-Bohm phase of a charged particle in a uniform magnetic field, with a
magnetic flux $\alpha$ threaded through each plaquette. We therefore regard
$\alpha$ as the synthetic flux, which takes value in the range
$\left[0,1\right)$.
We experimentally implement the two-dimensional quantum walk above using
photons. As illustrated in Fig. 1, the overall architecture is that of a fiber
network sch ; ql ; ql2 , through which attenuated single-photon pulses are
sent, with each full cycle around the network representing a discrete time
step. The coin states $\\{|0\rangle,|1\rangle\\}$ are encoded in the photon
polarizations $\\{|H\rangle,|V\rangle\\}$. The spatial degrees of freedom of
the square lattice are encoded in the time domain, following a time-
multiplexed scheme. This is achieved by building path-dependent time delays
into the four different paths (labeled $x\pm 1$ and $y\pm 1$ in Fig. 1a)
within the network (see Methods for details). The superpositions of multiple
well-resolved pulses within the same discrete time step thus represent those
of multiple spatial positions at the given time step (see Fig. 1b).
The shift and coin operators are implemented with beam splitters (BSs) and
wave plates (WPs), and the phase operator with one of the electro-optical
modulators (EOM1 in Fig. 1). We further implement polarization-dependent loss
operators $M^{\prime}_{j}=e^{-\gamma_{j}}M_{j}$ in each path, using a
combination of the WPs and the EOMs. The time-evolved state driven by $U$ is
then related to that in the experiment by adding a factor
$e^{(\gamma_{x}+\gamma_{y})t}$ to the latter.
For all experiments, avalanche photo-diodes (APDs) with temporal and
polarization resolutions are employed to record the probability distribution
of the walker states. This enables us to construct the site-resolved
population of the synthetic lattice, with
$\displaystyle P_{\text{exp}}(x,y,t)=\frac{N(x,y,t)}{\sum_{x,y}N(x,y,t)},$ (4)
where $N(x,y,t)$ is the total photon number on site $(x,y)$ at time $t$.
Figure 4: Dynamic detection of topological edge modes in $16$-time-step two-
dimensional magnetic quantum walks. The walker is initialized in the state
$\frac{1}{\sqrt{2}}(\ket{H}+i\ket{V})\otimes\ket{x=0}\ket{y=0}$ under a
domain-wall geometry. Specifically, in a, b, c, we set $\alpha=0.05$ for
$x\leq 0$, and $\alpha=-0.05$ for $x>0$. In d, e, f, we set $\alpha=1/3$ for
$x\leq 0$, and $\alpha=-1/3$ for $x>0$. The left, middle, and right columns
show the measured probability distribution with the parameters
$\gamma_{x}=\gamma_{y}=0$, $\gamma_{x}=0.1,\gamma_{y}=0$, and
$\gamma_{x}=\gamma_{y}=0.1$, respectively.
NHSE and tunable non-reciprocity. In the absence of flux, quantum walks driven
by $U$ already show non-reciprocal transport under finite photon losses. In
Fig. 2a, we show the measured populations of the synthetic lattice sites after
$t=16$ time steps. Starting from a local initial state at $\bm{r}=(0,0)$, the
propagation in the synthetic spatial dimensions is symmetric along the four
lattice directions (Fig. 2a). However, under finite photon-loss parameters,
the final-time photon distribution becomes asymmetric with a preferred
direction. Such a directional flow is the signature of the non-reciprocal
transport. For instance, when $\gamma_{x}=\gamma_{y}\neq 0$, as shown in Fig.
2b, the flow is diagonal to the square lattice. By tuning the ratio of
$\gamma_{y}/\gamma_{x}$, we can continuously adjust the direction of the
asymmetric pattern. This is explicitly shown in Fig. 2c, where we define the
directional displacement
$\displaystyle\bm{d}(t)=\sum_{x,y}\bm{r}P_{\text{exp}}(x,y,t).$ (5)
As $\gamma_{y}/\gamma_{x}$ varies, the direction of the displacement at the
final time step can be continuously tuned (see the left panel of Fig. 2c). In
our experiment, we adjust $\gamma_{y}/\gamma_{x}$ in the range of $[-1,1]$ for
$16$-time-step quantum walks. Correspondingly, the measured polar angle of
$\bm{d}$ changes from $3\pi/4$ to $-3\pi/4$ (the right panel of Fig. 2c).
Underlying this loss-induced directional flow is the NHSE in two dimensions.
While it is straightforward to show that the direction of the non-reciprocal
transport also indicates the direction of the eigenstate accumulation under
the open boundary condition supp , from an experimental perspective, we
observe the dynamic localization of the walker toward the boundary when a
domain-wall boundary condition is imposed (see Fig. 2d). Combined with the
theoretical spectral analysis that there are no topological edge states
present under the parameters of Fig. 2d, it is clear that the localization is
due to the NHSE.
Magnetic suppression of the NHSE. In Figs. 3a-d, we show the final population
distribution with the synthetic flux switched on, following $16$-time-step
quantum walks. Compared to Fig. 2, the directional flow appears to be
increasingly suppressed under larger $\alpha$, regardless of its direction. In
Figs. 3e and f, we show the absolute values of the directional displacement
$\bm{d}$ as functions of $\alpha$, for various loss parameters. The
suppression is the largest when $\alpha$ is tuned in between $0$ and $0.5$.
Such a suppression reflects the competition between the magnetic confinement
and the NHSE, and can be used for the manipulation of the non-reciprocal
transport.
Impact on topological edge states. In the absence of loss, the Floquet
operator $U$ describes an anomalous Floquet Chern insulator, characterized by
the Floquet topological invariant sai ; asb ; rud , which can be calculated
for each quasienergy gap and is fully responsible for the topological edge
states. Here we experimentally investigate how the NHSE under loss and
magnetic confinement impact the topological edge states. For this purpose, we
engineer a domain-wall configuration by choosing different values of $\alpha$
on either side of $x=0$.
As shown in Fig. 4a, for lossless quantum walks, a pair of topological edge
modes emerge, moving in opposite directions along the boundary. This is
consistent with the prediction of the Floquet topological invariant (see
Methods and supp ). When only the loss parameter $\gamma_{x}$ is turned on,
the NHSE induces a horizontal directional flow toward the region with $x<0$.
From the measured population following a $16$-time-step quantum walk (see Fig.
4b), both the bulk flow and the topological edge states are clearly visible.
Since the directional flow is perpendicular to the boundary, it has no direct
impact on the motion of the topological edge states. This is no longer the
case when both $\gamma_{x}$ and $\gamma_{y}$ become finite, as in Fig. 4c.
Here, besides a diagonal bulk flow induced by the corner skin effect, the
topological edge modes moving in the negative (positive) $y$ direction are
enhanced (suppressed) by the NHSE.
In Figs. 4d-f, we show the final probability distribution under a larger
synthetic flux $\alpha$. Compared to Figs. 4a-c, the bulk propagation is
significantly suppressed, whereas the topological edge modes are largely
unaffected by flux. This suggests that magnetic confinement is helpful for the
dynamics detection of topological edge modes in systems with the NHSE.
Discussion. We have experimentally demonstrated how the interplay of synthetic
flux and dissipation enables the full control over the non-reciprocal
transport underlying the NHSE. Since the quantum walk simulates an anomalous
Floquet Chern insulator, we further illustrate how the motion of topological
edge modes on the boundary is affected by the tuning parameters. While the
high tunability can be exploited for topological device design, our
implementation of a dissipative anomalous Floquet Chern insulator further
raises theoretical questions as to how the NHSE affects the bulk-boundary
correspondence herein. Our experiment also paves the way for engineering more
exotic forms of the non-Hermitian skin effect in higher dimensions
fangchenskin2 using quantum-walk dynamics.
## References
* (1) Breuer, H. P. & Petruccione, F. The Theory of Open Quantum Systems. (Oxford University Press, 2007).
* (2) Bender, C. M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007).
* (3) El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11-19 (2018).
* (4) Miri, M.-A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).
* (5) Ashida, Y., Gong, Z. & Ueda, M. Non-Hermitian physics. Advances in Physics 69, 3 (2020).
* (6) Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and topology in non-Hermitian physics. Phys. Rev. X 9, 041015 (2019).
* (7) Zhou, H. & Lee, J. Y. Periodic table for topological bands with non-Hermitian symmetries. Phys. Rev. B 99, 235112 (2019).
* (8) Lee, T. E. Anomalous edge state in a non-Hermitian lattice. Phy. Rev. Lett. 16, 133903 (2016).
* (9) Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phy. Rev. Lett. 121, 086803 (2018).
* (10) Yao, S., Song, F. & Wang, Z. Non-Hermitian Chern bands. Phy. Rev. Lett. 121, 136802 (2018).
* (11) Yokomizo, K. & Murakami, S. Non-Bloch band theory of non-Hermitian systems. Phy. Rev. Lett. 123, 066404 (2019).
* (12) Lee, C. H. & Thomale, R. Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B 99, 201103 (2019).
* (13) Longhi, S. Probing non-Hermitian skin effect and non-Bloch phase transitions. Phys. Rev. Research 1, 023013 (2019).
* (14) Deng, T.-S. & Yi, W. Non-Bloch topological invariants in a non-Hermitian domain wall system. Phys. Rev. B 100, 035102 (2019).
* (15) Kunst, F. K., Edvardsson, E., Budich, J. C. & Bergholtz, E. J. Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Phy. Rev. Lett. 121, 026808 (2018).
* (16) McDonald, A., Pereg-Barnea, T. & Clerk, A. Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev-Majorana chain. Phys. Rev. X 8, 041031 (2018).
* (17) Alvarez, V. M., Vargas, J. B. & Torres, L. F. Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points. Phys. Rev. B 97, 121401 (2018).
* (18) Li, L., Lee, C. H., Mu, S. & Gong, J. Critical non-Hermitian skin effect. Nat. Commun. 11, 5491 (2020).
* (19) Yang, Z., Zhang, K., Fang, C. & Hu, J. Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory. Phys. Rev. Lett. 125, 226402 (2020).
* (20) Longhi, S. Self-healing of non-Hermitian topological skin modes. Phy. Rev. Lett. 128, 157601 (2022).
* (21) Li, Y., Liang, C., Wang, C., Lu, C. & Liu, Y.-C. Gain-loss-induced hybrid skin-topological effect. Phy. Rev. Lett. 128, 223903 (2022).
* (22) Zhang, K., Yang, Z. & Fang, C. Universal non-Hermitian skin effect in two and higher dimensions. Nat. Commun. 13, 2496 (2022).
* (23) Helbig, T. et al. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747 (2020).
* (24) Xiao, L. et al. Non-Hermitian bulk–boundary correspondence in quantum dynamics. Nat. Phys. 16, 761 (2020).
* (25) Ghatak, A., Brandenbourger, M., Van Wezel, J. & Coulais, C. Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial. Proc. Natl. Acad. Sci. 117, 29561 (2020).
* (26) Weidemann, S. et al. Topological funneling of light. Science 368, 311-314 (2020).
* (27) Hofmann, T. et al. Reciprocal skin effect and its realization in a topolectrical circuit. Phys. Rev. Research 2, 023265 (2020).
* (28) D. Zou. et al. Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits. Nat. Commun. 12, 7201 (2021).
* (29) Zhang, K., Yang, Z. & Fang, C. Correspondence between winding numbers and skin modes in non-Hermitian systems. Phys. Rev. Lett. 125, 126402 (2020).
* (30) Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020).
* (31) Longhi, S. Non-Bloch PT symmetry breaking in non-Hermitian photonic quantum walks. Opt. Lett. 44, 5804 (2019).
* (32) Hu, Y.-M., Wang, H.-Y., Wang, Z. & Song, F. Geometric Origin of non-Bloch PT symmetry breaking. arXiv:2210.13491.
* (33) Xiao, L. et al. Observation of non-Bloch parity-time symmetry and exceptional points. Phys. Rev. Lett. 126, 230402 (2021).
* (34) Li, T., Sun, J.-Z., Zhang, Y.-S & Yi, W. Non-Bloch quench dynamics. Phys. Rev. Research 3, 023022 (2021).
* (35) Wang, K. et al. Detecting non-Bloch topological invariants in quantum dynamics. Phys. Rev. Lett. 127, 270602 (2021).
* (36) Liang, Q. et al. Dynamic signatures of non-Hermitian skin effect and topology in ultracold atoms. Phys. Rev. Lett. 129, 070401 (2022).
* (37) Lin, Q. et al. Observation of non-Hermitian topological Anderson insulator in quantum dynamics. Nat. Commun. 13, 3229 (2022).
* (38) Lin, Q. et al. Topological phase transitions and mobility edges in non-Hermitian quasicrystals. Phys. Rev. Lett. 129, 113601 (2022).
* (39) Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239 (1976).
* (40) Sajid, M., Asbóth, J. K., Meschede, D., Werner, R. F. & Alberti, A. Creating anomalous Floquet Chern insulators with magnetic quantum walks. Phys. Rev. B 99, 214303 (2019).
* (41) Shao, K. et al. Cyclotron quantization and mirror-time transition on nonreciprocal lattices. Phys. Rev. B 106, L081402 (2022).
* (42) Schreiber, A. et al. A 2D quantum walk simulation of two-particle dynamics. Science 336, 55-58 (2012).
* (43) see Supplemental Material.
* (44) Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).
* (45) Asbóth, J. K. & Alberti, A. Spectral flow and global topology of the Hofstadter butterfly. Phys. Rev. Lett. 118, 216801 (2017).
Methods
Experimental setup. We adopt a time-multiplexed scheme for the experimental
realization of photonic quantum walks sch ; ql ; ql2 . As illustrated in Fig.
1, the photon source is provided by a pulsed laser with a central wavelength
of $808$nm, a pulse width of $88$ps, and a repetition rate of $15.625$kHz. The
pulses are attenuated by a neutral density filter, such that an average photon
number per pulse is less than $2.4\times 10^{-4}$, which ensures a negligible
probability of multi-photon events. The photons are coupled in and out of a
time-multiplexed setup through a BS with a reflectivity of $3\%$,
corresponding to a low coupling rate of photons into the network. Such a low-
reflectivity BS also enables the out-coupling of photons for measurement. A
HWP with the setting angle $\pi/8$ is used to implement the coin operator $C$.
Four different paths in a fiber network correspond to the four different
directions a walker can take in one step on a two-dimensional lattice. Two-PBS
loops are used to realize polarization-dependent optical delays. The shift
operator $S_{x}$ is implemented by separating photons corresponding to their
two polarization components and routing them through the fibre loops,
respectively. Polarization-dependent time delay is then introduced. Since the
lengths of the two fiber loops are $287.03$m and $270.00$m, respectively, the
time difference of photons traveling through two fiber loops is $80$ns. The
shift operator $S_{y}$ is implemented by another two-PBS loop based on the
same principle, where the vertical component of photons is delayed relative to
the horizontal component by a $1.61$m free space path difference. The
corresponding time difference in the $y$ direction is then $4.83$ns.
The position-dependent phase operator $P$ is implemented using the first
electro-optical modulator (EOM1). The rise/fall times of EOM ($4$ns) are much
shorter than the time difference between adjacent positions ($80$ns and
$4.83$ns for $x$ and $y$ directions, respectively), which enable us to control
the parameter $\phi$ precisely.
To realize a polarization-dependent loss operation
$M^{\prime}_{x}(\gamma_{x})=e^{-\gamma_{x}}M_{x}(\gamma_{x})$, two HWPs and an
EOM are introduced into each fiber loop. Here HWPs are used to keep the
polarizations of photons unchanged before and after they pass through the
fiber loops. For $\gamma_{x}>0$, for the short loop, the voltage of EOM3 is
tuned to $0$. Thus, after passing through the first PBS, horizontally
polarized photons are all transmitted by the second PBS and are subject to
further time evolution. Whereas for the long loop, by controlling the voltage
of EOM2 to satisfy $\cos\theta/2=e^{-2\gamma_{x}}$, we flip part of the
photons $(1-e^{4\gamma_{x}})$ with vertical polarization into horizontal ones.
They are subsequently transmitted by the second PBS, and leak out of the
setup. Otherwise, for $\gamma_{x}<0$, horizontally polarized photons are all
transmitted by the second PBS and are subject to further time evolution in the
long loop. By contrast, for the short loop, part of the photons
$(1-e^{4\gamma_{x}})$ with vertical polarization are flipped by EOM3,
transmitted by the second PBS and subsequently leak out of the setup. We use
the same method to realize $M^{\prime}_{y}(\gamma_{y})$.
We compare the ideal theoretical distribution with the measured distribution
via the similarity,
$S(t)=\sum_{x,y}\sqrt{P_{\text{th}}(x,y,t)P_{\text{exp}}(x,y,t)},$ (6)
which quantifies the equality of two probability distributions. Here $S=0$
stands for completely orthogonal distributions, and $S=1$ for identical
distributions. We observe $S\geq 0.914$ in Fig. 2, $S\geq 0.922$ in Fig. 3,
and $S\geq 0.930$ in Fig. 4, respectively. Here the theoretical value
$P_{\text{th}}(x,y,t)$ is given by
$\displaystyle
P_{\text{th}}(x,y,t)=\sum_{m=H,V}\frac{|(\langle\bm{r}|\otimes\langle
m|)e^{-(\gamma_{x}+\gamma_{y})t}|\psi(t)\rangle|^{2}}{\sum_{\bm{r},m}|(\langle\bm{r}|\otimes\langle
m|)e^{-(\gamma_{x}+\gamma_{y})t}|\psi(t)\rangle|^{2}},$ (7)
where $|\psi(t)\rangle$ is the time-evolved walker state under $U$.
In our experiment, photon loss is caused by the loss of photons through an
optical element. Our round-trip single-loop efficiency is about $0.66$ even
for a unitary quantum walk. This is calculated by multiplying the transmission
rates of each optical component used in the round trip, including the
transmission rates of the BS ($\sim 0.97$), the collection efficiency from
free space to fiber ($\sim 0.78$), the EOM ($\sim 0.96$), and all other
optical components ($\sim 0.91$). We therefore estimate the single-loop
efficiency as $0.78\times 0.97\times 0.96\times 0.91\simeq 0.66$.
Floquet topological invariant. The walker state evolves according to
$\displaystyle|\psi(t)\rangle=U^{t}|\psi(0)\rangle=e^{-iH_{\rm{eff}}t}|\psi(0)\rangle,$
(8)
where $H_{\rm{eff}}=i\ln U$ is defined as the effective Hamiltonian. While the
quantum walk is identified as the periodically-driven Floquet dynamics, the
eigenenergies of $H_{\rm{eff}}$ constitute the quasienergy spectrum of the
Floquet system. We fix the branch cut of the logarithm such that the
quasienergy spectrum lies within the range $[-\pi,\pi)$.
To calculate the Floquet topological invariant, we follow Refs. sai ; asb ;
rud , and define
$U^{\prime}=e^{i\tilde{E}}M_{y}S_{y}P^{\prime}CM_{x}S_{x}C,$ (9)
where $e^{i\tilde{E}}$ shifts the quasienergy spectrum by $-\tilde{E}$. The
modified phase-shift operator is
$P^{\prime}(\beta,\alpha)=\sum_{\bm{r}}\exp\left[i\sigma_{z}(\beta\lfloor
x/q\rfloor+2\pi\alpha x)\right]\otimes\ket{\bm{r}}\bra{\bm{r}},$ (10)
where $\alpha=p/q$, $\beta=2\pi/s$, and $\lfloor x/q\rfloor$ is the greatest
integer less than or equal to $x/q$. Here $p$ and $q$ are coprime integers,
and $s$ is a sufficiently large integer (in our case, for $\alpha=1/3$, $s=15$
is sufficient).
We denote the eigenvalues of $U^{\prime}$ as $e^{-iE_{j}}$, and the
topological invariant for the quasienergy gap (corresponding to $U$)
comprising $\tilde{E}$ can be calculated through
$R=\frac{1}{2\pi}\left(\sum_{j=1}^{2sq}E_{j}(1/s,\alpha,\tilde{E})-\sum_{j=1}^{2sq}E_{j}(0,\alpha,\tilde{E})\right).$
(11)
Under an open boundary condition, the value of $R$ indicates the number of
anomalous Floquet edge states emerging within the quasienergy gap.
In Fig. 2d, all quasienergy gaps are closed, hence there are no topological
edge states along the boundaries, and the gap topological invariants are ill-
defined. In Fig. 4, for the $x\leq 0$ ($x>0$) region of the domain-wall
configurations, we have $p=1,q=20$ ($p=-1,q=20$) in Figs. 4a, b, c, and
$p=1,q=3$ ($p=-1,q=3$) in Figs. 4d, e, f, respectively. While there are now a
host of quasienergy gaps, for any given gap, the topological invariants $R$ of
the two regions are always finite and differ by a sign supp . As a
consequence, Floquet topological edge modes emerge along the domain-wall
boundary.
We find that the Floquet topological invariant $R$ is capable of predicting
the anomalous topological edge states under all our experimental parameters,
despite the presence of the NHSE. Whether the NHSE can have significant impact
on $R$ beyond our experimental parameters (particularly when the photon loss
is further increased) is an interesting theoretical question that we leave to
future studies.
Acknowledgments
We thank Chen Fang for helpful discussions. This work has been supported by
the National Natural Science Foundation of China (Grant Nos. 92265209,
12025401 and 11974331). W. Y. acknowledges support from the National Key
Research and Development Program of China (Grant Nos. 2016YFA0301700 and
2017YFA0304100).
## A Supplemental Material for “Manipulating non-reciprocity in a two-
dimensional magnetic quantum walk”
In this Supplemental Material, we provide numerical results characterizing the
non-Hermitian skin effect (NHSE) and the Floquet topological invariants.
### A.1 Quasienergy spectra
The presence of NHSE can be confirmed by examining the quasienergy spectra of
the effective Hamiltonian under different boundary conditions, and the spatial
distribution of eigenstates under the open boundary condition. As discussed in
the main text (Methods section), we define the effective Hamiltonian
$H_{\rm{eff}}=i\ln U$, where $U$ is the Floquet operator in Eq. (1) of the
main text, and the branch cut of the logarithm is taken to be the negative
real axis. The quasienergy spectrum of $H_{\rm{eff}}$ thus lies within the
range $[-\pi,\pi)$.
In Figs. S1a and c, we show the quasienergy spectra under both the periodic
(grey) and open (red) boundary conditions. The collapse of the spectra, when
the boundary condition is changed from periodic to open, strongly suggests the
presence of the NHSE. This is explicitly confirmed in Figs. S1b and d, where
we show the spatial distribution of the eigenstates under the open boundary
condition. They accumulate to one edge or one corner, depending on the loss
parameters. Note that $H_{\rm{eff}}$ and $U$ share the same set of
eigenstates.
Figure S1: a Numerically calculated quasienergy spectra of $H_{\rm{eff}}$ on
the complex plane, with $\gamma_{x}=\gamma_{y}=0.125$, under either the
periodic boundary condition (grey) or the open boundary condition (red). b
Spatial distribution of the eigenstates under the open boundary condition,
with the same parameters used in a. c Numerically calculated quasienergy
spectra of $H_{\rm{eff}}$ with $\gamma_{x}=0.1$, $\gamma_{y}=0$, under the
periodic boundary condition (grey) and the open boundary condition (red),
respectively. d Spatial distribution of the eigenstates under the open
boundary condition, with the same parameters used in c. For numerical
calculations, a lattice size of $45\times 45$ is taken as an example, with
$\alpha=0$.
### A.2 Floquet topological invariants and edge states
In Figs. S2a and b, we show the real components of the quasienergy spectra
under the domain-wall geometry, respectively under the parameters of Fig. 4d
and Fig. 4f of the main text. While the Floquet topological edge states are
visible within each quasienergy gap, the quasienergy spectra are close to each
other, because of the smallness of the loss parameters in Fig. S2b.
The Floquet topological edge states can be characterized by the gap invariant
$R$ defined in the Methods section of the main text. In Fig. S2c, we show the
calculated gap invariants $R$ for the left (red) and right (black) regions of
the domain-wall configuration in Fig. S2b. The topological invariants of the
two regions are always finite but differ by their signs. Importantly, within
each quasienergy gap, the difference in the gap topological invariants between
the two regions is $2$. This corresponds to the number of Floquet topological
edge states within each quasienergy gap, as illustrated in Fig. S2b.
Figure S2: a, b The real component of the quasienergy spectra under the
domain-wall geometry of Figs. 4d and f in the main text, with
$\gamma_{x}=\gamma_{y}=0$ in a, and $\gamma_{x}=\gamma_{y}=0.1$ in b. Floquet
topological edge states (red) are seen to emerge within each quasienergy gap.
c Floquet topological invariants $R$ as functions of the real component of the
quasienergy. Within each quasienergy gap, the red (black) line indicates $R$
of the left (right) region. The shaded areas are the density of states
normalized to the total number of states, indicating the regions of
quasienergy bands. For numerical calculations, we take a lattice with
$45\times 45$ sites, and fix $\alpha=1/3$ and $\alpha=-1/3$ for the left and
right regions, respectively.
|
# Kick-motion Training with DQN in AI Soccer Environment
††thanks: This work was supported by the Institute for Information
communications Technology Promotion (IITP) grant funded by the Korean
government (MSIT) (No.2020-0-00440, Development of Artificial Intelligence
Technology that continuously improves itself as the situation changes in the
real world).
Bumgeun Park Cho Chun Shik Graduate School of Mobility
Korea Advanced Institute of Science and Technology (KAIST)
Daejeon, South Korea
<EMAIL_ADDRESS>Jihui Lee Division of Future Vehicle
Korea Advanced Institute of Science and Technology (KAIST)
Daejeon, South Korea
<EMAIL_ADDRESS>Taeyoung Kim Cho Chun Shik Graduate School of Mobility
Korea Advanced Institute of Science and Technology (KAIST)
Daejeon, South Korea
<EMAIL_ADDRESS>Dongsoo Har Cho Chun Shik Graduate School of Mobility
Korea Advanced Institute of Science and Technology (KAIST)
Daejeon, South Korea
<EMAIL_ADDRESS>
###### Abstract
This paper presents a technique to train a robot to perform kick-motion in AI
soccer by using reinforcement learning (RL). In RL, an agent interacts with an
environment and learns to choose an action in a state at each step. When
training RL algorithms, a problem called the curse of dimensionality (COD) can
occur if the dimension of the state is high and the number of training data is
low. The COD often causes degraded performance of RL models. In the situation
of the robot kicking the ball, as the ball approaches the robot, the robot
chooses the action based on the information obtained from the soccer field. In
order not to suffer COD, the training data, which are experiences in the case
of RL, should be collected evenly from all areas of the soccer field over
(theoretically infinite) time. In this paper, we attempt to use the relative
coordinate system (RCS) as the state for training kick-motion of robot agent,
instead of using the absolute coordinate system (ACS). Using the RCS
eliminates the necessity for the agent to know all the (state) information of
entire soccer field and reduces the dimension of the state that the agent
needs to know to perform kick-motion, and consequently alleviates COD. The
training based on the RCS is performed with the widely used Deep Q-network
(DQN) and tested in the AI Soccer environment implemented with Webots
simulation software.
###### Index Terms:
Reinforcement learning (RL), Deep Q-Network (DQN), AI soccer, Curse of
dimensionality (COD), Coordinate transformation matrix (CTM)
## I Introduction
Reinforcement Learning (RL), which is one of the widely used machine learning
techniques, makes an agent maximize the accumulated rewards in given
environment [1]. To do so requires the use of experiences, e.g., experience
replay, during training. The agent selects an action based on the current
state and receives the reward from the environment. Through this interaction,
the agent can learn which action is more useful to achieve the goal. Recently,
a deep neural network (DNN) for nonlinear approximation enables the RL to
handle more complicated tasks such as playing a range of Atari games [2],
playing the game of GO [3, 4], controlling the robot arm [5, 6, 7], and
planning path for mobile robots [8, 9].
AI soccer environment, which is one of the environments for training multi-
agent RL algorithms, is introduced in the previous work [10]. The main target
of AI Soccer is winning the soccer game by controlling each robot. Each
robot’s movement consists of many small motions such as ”going to a target
spot”, ”blocking robot or ball”, ”positioning”, ”heading”, and ”kicking”.
Therefore, the entire AI soccer training can be divided into various tasks of
training motions. The state, action, and reward, which are three basic
elements for RL, should be designed depending on the motion to be trained.
The curse of dimensionality(COD) occurs when the dimension of the state
increases and the amount of training data is inappropriately small. The
inappropriately small data for training causes the RL models to be overfitted
and consequently leads to deteriorated performance.
In this paper, we consider training kick-motion with a deep Q-network (DQN).
To train kick-motion successfully, the state should include various
information about the robot and ball on the soccer field, which is described
by their current position, speed, and the direction of the movement of ball or
robot. However, designing the state to include large information can cause the
COD, unless experiences are gained evenly over all areas of the soccer field.
To eliminate the need for the agent to know all the game information over
entire soccer field and to reduce the dimension of the state, we attempt to
design the state for efficient kick-motion training by using a coordinate
transformation matrix (CTM) used to transform coordinates in robotics or
computer vision [11]. The main contributions of this paper are as follows:
1. 1.
The attempted designing of the state avoids deterioration arising from the COD
and improve the performance of the RL algorithm.
2. 2.
The RCS taken for AI Soccer eliminates the need for excessive exploration and
reduces the dimension of the state.
## II Background
### II-A AI soccer environment
In this paper, experiments are conducted on the AI soccer environment employed
for AI World Cup event [12]. The environment consists of a ball, two teams of
robots and a soccer field. Each team consists of five robots each driven by
two wheels. Like the human soccer game, each team tries to beat the other team
by controlling their robots to take the best action for each game state. The
robots are controlled at each timestep by setting wheel speeds. The two
separately controllable wheels extend the range of actions and increase the
degree of freedom. Value of the best action can be measured by corresponding
reward. Each robot, as an agent, takes an action based on the situation of the
field to get a higher accumulated reward. This is a very challenging task,
because various strategies are needed for the agents to adapt to the vast
state space which is observed in dynamic game situation.
### II-B Curse of dimensionality (COD)
Traditional RL algorithms suffer from the COD when the dimension of the space
in the environment is excessively high [13]. As the dimension of the state
increases, the number of experiences needed to train an agent grows
exponentially. The lack of experiences makes the RL model overfitted. Although
using function approximation such as DNN alleviates this problem, the
performance of the algorithm is still deteriorated. In the AI soccer
environment, large information describing the (game) state and two-wheeled
actions with a high degree of freedom increase the dimension of the state
space and action space, which readily causes the COD.
### II-C Deep Q-network (DQN)
The DQN algorithm, which is a widely used RL algorithm, is a deep RL method
that approximates an action-value function, called Q-function by using a DNN.
The Q-function with parameters $\theta$ is defined as follows:
$Q^{\theta}(s,a)=\mathbb{E}_{s,a,\theta}[\sum_{t=1}^{\infty}\gamma^{t-1}r_{t}]$
(1)
where a, s, r, and $\gamma$ are action, state, reward, and discount factor
respectively. DQN is trained to minimize the loss function, which can be
presented as follows:
$L(\theta)=\mathbb{E}[(y_{t}-Q^{\theta}(s_{t},a_{t}))^{2}]$ (2)
The target Q function $y_{t}$ is calculated by reward and next state
$s^{\prime}$ as follows:
$y_{t}=r(s_{t},a_{t})+\gamma
max_{a^{\prime}}Q^{\theta^{-}}(s^{\prime},a^{\prime})$ (3)
where $\theta^{-}$ represents the parameters of a fixed and separate target
network.
Figure 1: Illustration of the absolute coordinate system (ACS). The origin of
the ACS is placed in the center of the field. The orange dotted line presents
the direction of ball movement. The yellow arrow presents the direction the
robot movement.
Figure 2: Illustration of the relative coordinate system (RCS). The origin of
the RCS is placed in the center of the robot, and the direction the robot is
aligned with the $X^{\prime}$-axis.
## III Relative coordinate system for description of game state
In the task of training the robot to perform kick-motion, the robot chooses at
each timestep the best action out of a set of actions, taking into account the
information about the ball and the robot such as the current coordinates, the
velocity of the ball and the robot, and the approaching angle of the ball to
the robot. To avoid the COD and train the robot to kick the ball at any
location, it is required to use evenly distributed experiences over entire
field. For this purpose, we adopt the relative coordinate system (RCS) to
represent the state, instead of the absolute coordinate system (ACS). The RCS
can alleviate the COD and make training process straightforward. Most
important benefit obtained from the use of the RCS is that the robot can
achieve the task at any location after training at only one location.
In the AI soccer environment, all raw information given to the robot from the
environment is presented in the ACS format. The origin of the ACS is defined
as the center of the field with $X$-axis and $Y$-axis as shown in Fig. 1. In
this case, the state given to the robot is defined as follows:
$S_{ACS}=[R_{x},R_{y},R_{\theta},B_{x},B_{y},V_{R_{x}},V_{R_{y}},V_{B_{x}},V_{B_{y}},\theta]$
(4)
Each element in (4) respectively represents $X$ and $Y$ coordinates of the
robot, the heading angle of the robot, $X$ and $Y$ coordinates of the ball,
$X$ and $Y$ direction velocity of the robot, $X$ and $Y$ direction velocity of
the ball, and the angle of the direction in which the ball moves. The high
dimensional state space formulated in (4) can deteriorate the performance of
RL algorithms.
With the RCS, the dimensional space is reduced and training becomes efficient.
In the RCS, the origin is defined at the center of the robot with
$X^{\prime}$-axis and $Y^{\prime}$-axis and the direction the robot movement
is the same as the direction of $X^{\prime}$-axis shown in Fig. 2. With the
RCS, the values of $R_{x}$, $R_{y}$, and $R_{\theta}$ are always zero since
the origin of the RCS is the center of the robot. The value of $V_{R}{}_{y}$
is also always zero because the robot can take only one of forward, backward,
or stop, during the task of the kick-motion. As a result, the state can be
defined as follows:
$S_{RCS}=[B_{x}^{RCS},B_{y}^{RCS},V_{R_{x}}^{RCS},V_{B_{x}}^{RCS},V_{B_{y}}^{RCS},\theta^{RCS}]$
(5)
where $(\cdot)^{RCS}$ means $(\cdot)$ in the RCS. The relationship between the
RCS and the ACS can be presented using the CTM as shown in (6).
$M=\begin{bmatrix}\cos(R_{\theta})&-\sin(R_{\theta})&R_{x}\\\
\sin(R_{\theta})&\cos(R_{\theta})&R_{y}\\\ 0&0&1\end{bmatrix}$ (6)
The position, velocity, and angle in the RCS can be computed with the CTM and
those in the ACS, and can be shown as (7), (8), and (9).
$\begin{bmatrix}B_{x}^{RCS}&V_{B_{x}}^{RCS}\\\ B_{y}^{RCS}&V_{B_{y}}^{RCS}\\\
1&1\end{bmatrix}=M^{-1}\begin{bmatrix}B_{x}&V_{B_{x}}\\\ B_{y}&V_{B_{y}}\\\
1&1\end{bmatrix}$ (7)
$V_{R_{x}}^{RCS}={V_{R_{x}}}^{2}+{V_{R_{y}}}^{2}$ (8)
$\theta^{RCS}=\theta-R_{\theta}$ (9)
The change of the coordinate system from the ACS to the RCS reduces the
dimension of the state by 60%, which means significantly easier training
process.
Figure 3: Illustration of a situation in which the ball and the robot are in
contact. The blue line connects from the center of the robot to the center of
the ball and the yellow arrow presents the direction toward the target point.
## IV Experiments
This section describes AI Soccer environment and presents the result of
experiments conducted to show the effect of the COD.
### IV-A Experimental Environment
Experiments are conducted for the kick-motion task in the AI soccer
environment developed with Webot physics simulator [14]. The goal of the kick-
motion task is that robot kicks the ball to a target point. The robot is
facing the target point and the ball is located at a random point with a
distance of 2m from the robot. With the start of an episode of training, the
ball starts to move at a random speed to a random point near the robot. At
each timestep, the robot chooses an action based on the state. Each episode
consist of 40 experiences over 40 timesteps. The RL framework for this task
can be described as follows:
* States:
State consists of 6 values shown as (5) in Section III.
* Rewards:
Reward is given as a sum of 3 types of reward,
$r_{contact}+r_{theta}+r_{velocity}$. The $r_{contact}$ is a reward of
contact, which is -0.1 before the robot contact the ball and 0 after the
contact. The $r_{Theta}$ is a reward according to distance of the contact
point from the center of the robot, which is represented as
$3\cos{\theta_{contact}}$, where $\theta_{contact}$ is the angle between the
blue and the yellow lines in Fig. 3. The $r_{velocity}$ is a reward according
to the speed of the contact with the ball, which is represented as
$3.92V_{contact}$, where $V_{contact}$ is the velocity of the robot at the
contact.
* Actions:
List of available actions: going straight forward with steady-state speeds of
2.55 (m/s), stop, going straight backward with steady-state speeds of -2.55
(m/s).
### IV-B Experimental Results
The performances with the kick-motion task are evaluated in terms of the sum
of the reward for each training episode. To reduce granularity of the reward
variation, the moving average of the sum of the reward across 250 episodes is
taken.
Figure 4: Total reward according to the number of dummy variables. It shows
that increasing the dimension of state space slows down the training speed and
as a result degrades performance. The blue line represents the model with the
RCS and the others represent comparative models. Especially, the purple line
is equivalent to the model with the ACS.
To present the intuitive understanding of the COD, models used to compare with
the proposed method are trained with states including ’dummy variables’, which
are some meaningless random variables. The states for the comparison models
are the state in the RCS with ’dummy variables’. In Fig. 4, the name of ’$+n$’
means that $n$ dummy variables are added to the state.
Figure 4 shows the results of the experiments. The (total accumulated) reward
of the action for RCS based state exceeds -4 from 1500-th episode, which is
about 500 episodes faster than by ACS based method. The RCS based method
achieves the highest reward value of 6.8, although the reward value of the RCS
based method is similar to the others in the latter part of the training,
e.g., 3600-4700 episodes.
## V Conclusion
Efficient training of the RL algorithms with high dimensional state space is
hard to achieve, due to the COD. In this paper, a method to reduce the
dimension of the state space for training the kick-motion task in the AI
soccer environment is attempted. The attempted method uses the CTM to convert
the state in the ACS to the state in the RCS. The dimension of the state space
required for the agent to achieve the kick-motion is reduced from ten to six
since 4 values in the state are always zero by the RCS based method. The
experimental result shows that the kick-motion can be achieved with the
reduced state by the RCS based method. The result obtained along with the
comparative models with different number of variables, which are trained with
the states including dummy variables, shows that the lower dimension of the
state space can avoid deterioration due to the COD.
## Acknowledgment
This work was supported by the Institute for Information communications
Technology Promotion (IITP) grant funded by the Korean government (MSIT)
(No.2020-0-00440, Development of Artificial Intelligence Technology that
continuously improves itself as the situation changes in the real world).
## References
* [1] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.
* [2] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013.
* [3] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529(7587):484–489, 2016.
* [4] David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. nature, 550(7676):354–359, 2017.
* [5] Minah Seo, Luiz Felipe Vecchietti, Sangkeum Lee, and Dongsoo Har. Rewards prediction-based credit assignment for reinforcement learning with sparse binary rewards. IEEE Access, 7:118776–118791, 2019.
* [6] Luiz Felipe Vecchietti, Taeyoung Kim, Kyujin Choi, Junhee Hong, and Dongsoo Har. Batch prioritization in multigoal reinforcement learning. IEEE Access, 8:137449–137461, 2020.
* [7] Luiz Felipe Vecchietti, Minah Seo, and Dongsoo Har. Sampling rate decay in hindsight experience replay for robot control. IEEE Transactions on Cybernetics, 2020.
* [8] Inhwan Kim, Sarvar Hussain Nengroo, and Dongsoo Har. Reinforcement learning for navigation of mobile robot with lidar. In 2021 5th International Conference on Electronics, Communication and Aerospace Technology (ICECA), pages 148–154. IEEE, 2021.
* [9] Woohyeon Moon, Bumgeun Park, Sarvar Hussain Nengroo, Taeyoung Kim, and Dongsoo Har. Path planning of cleaning robot with reinforcement learning. arXiv preprint arXiv:2208.08211, 2022.
* [10] Taeyoung Kim, Luiz Felipe Vecchietti, Kyujin Choi, Sanem Sariel, and Dongsoo Har. Two-stage training algorithm for ai robot soccer. PeerJ Computer Science, 7:e718, 2021.
* [11] Richard Hartley and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, 2003.
* [12] Chansol Hong, Inbae Jeong, Luiz Felipe Vecchietti, Dongsoo Har, and Jong-Hwan Kim. Ai world cup: Robot-soccer-based competitions. IEEE Transactions on Games, 13(4):330–341, 2021.
* [13] Abhijit Gosavi. Reinforcement learning: A tutorial survey and recent advances. INFORMS Journal on Computing, 21(2):178–192, 2009.
* [14] Olivier Michel. Cyberbotics ltd. webots™: professional mobile robot simulation. International Journal of Advanced Robotic Systems, 1(1):5, 2004\.
|
# Statistical Chronometry of Meteorites. I. A Test of ${}^{26}{\rm Al}$
homogeneity and the Pb-Pb age of the Solar System’s $t\\!\\!=\\!\\!0$.
Steven J. Desch Daniel R. Dunlap Emilie T. Dunham Curtis D. Williams
Prajkta Mane
###### Abstract
We use rapidly cooled achondrites to test the assumption of ${}^{26}{\rm Al}$
homogeneity in the solar nebula, by checking if there is a single value of
$t_{\rm SS}$, the absolute “Pb-Pb” age of the Solar System’s
$t\\!\\!=\\!\\!0$, that makes concordant their ages from the Al-Mg and Pb-Pb
systems. We find that values $t_{\rm SS}=4568.42\pm 0.24$ Myr do make these
ages concordant, and therefore the hypothesis of homogeneous ${}^{26}{\rm Al}$
is not falsified. This age, defined to be when the solar nebula had
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})=5.23\times 10^{-5}$, is
significantly older than the $\approx$ 4567.3 Myr inferred from direct
measurements of Pb-Pb ages in CAIs. Discrepancies between the Al-Mg and Pb-Pb
chronometers in chondrules and CAIs have previously been interpreted as
arising from heterogeneities in ${}^{26}{\rm Al}$, under the presumption that
the Al-Mg and Pb-Pb systems in CAIs closed simultaneously. We examine this
assumption and show that resetting is to be expected in CAIs. In particular,
we quantitatively demonstrate that it is plausible that Pb-Pb ages of CAIs
were reset at late times, without resetting the earlier Al-Mg ages, if they
were transiently heated in the same manner as chondrules. We critically
examine Pb-Pb isochrons, refining data and suggesting best practices for their
calculation and reporting. We advocate reporting chronometry as times of
formation after $t\\!\\!=\\!\\!0$ rather than absolute ages, as only the
former is useful for astrophysical models of the solar nebula. We advocate
averaging of multiple samples, rather than anchoring to individual meteorites,
to improve precision.
###### keywords:
Solar System formation 1530 , Planet formation 1241 , Meteorites 1038 ,
Achondrites 15 , Chondrites 228
††journal: Icarus
[inst1]organization=School of Earth and Space Exploration, Arizona State
University,addressline=PO Box 871404, city=Tempe, postcode=85287-1404,
state=Arizona, country=USA
[inst2]organization=Oak Ridge National Laboratory, addressline=1 Bethel Valley
Rd, city=Oak Ridge, postcode=37830, state=Tennessee, country=USA
[inst6]organization=Department of Earth, Planetary and Space Sciences,
University of California, Los Angeles, addressline=PO Box 951567, city=Los
Angeles, postcode=90095-1567, state=California, country=USA
[inst3]organization=Earth and Planetary Sciences Department, University of
California, Davis,addressline=One Shields Ave., city=Davis, postcode=95616,
state=California, country=USA
[inst4]organization=Lunar and Planetary Institute, USRA, addressline=3600 Bay
Area Blvd., city=Houston, postcode=77058, state=Texas, country=USA
We re-evaluate and average data from the Al-Mg and Pb-Pb radiometric dating
systems, for achondrites and chondrules, to attain better accuracy and
precision in the time of formation of meteorites and meteoritic inclusions.
We find remarkable concordancy between the systems, provided the Pb-Pb age of
the Solar System is $4568.4\pm 0.2$ Myr, older than previous estimates; this
suggests late resetting of the Pb-Pb system in Ca-rich, Al-rich inclusions
(CAIs), and supports homogeneity of ${}^{26}{\rm Al}$ in the solar nebula.
## 1 Introduction
### 1.1 The Al-Mg and Pb-Pb Chronometers
To learn about the birth of planets and the story of our Solar System’s first
few million years, we study meteorites that bear witness to this era. It is
especially crucial to constrain the times at which their constituent
components formed, the times at which their parent planetesimals accreted and
melted, when collisions occurred, and to constrain the relative order of these
events in the context of the solar nebula. The need of astrophysical models is
to know the time $\Delta t$ after $t\\!\\!=\\!\\!0$ that an event occurred,
where $t\\!\\!=\\!\\!0$ is a defined event or time in Solar System history.
To obtain these times $\Delta t$, radiometric dating systems are employed. The
Al-Mg system is most commonly used. Calcium-rich, aluminum-rich inclusions
(CAIs), incorporated live ${}^{26}{\rm Al}$, a short-lived radionuclide (SLR)
that decays to ${}^{26}{\rm Mg}$ with a half-life of 0.717 Myr, or mean-life
$\tau_{26}=1.034$ Myr (Auer et al., 2009; Kondev, 2021). CAIs are among the
first meteoritic components to have formed in the Solar System, apparently
during a short time interval that can be associated with $t\\!\\!=\\!\\!0$.
Most CAIs incorporated Al with an isotopic ratio $\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$}\approx(5.23\pm 0.13)\times 10^{-5}$ (Jacobsen et al.,
2008). A useful working hypothesis is that this was the widespread and
spatially homogeneous abundance of ${}^{26}{\rm Al}$ in the solar nebula when
CAIs formed. We define $t\\!\\!=\\!\\!0$ to be the time in the solar nebula
when ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ equalled $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{\rm SS}\equiv 5.23\times 10^{-5}$ exactly. Regardless
of the “true” initial value, most CAIs appear to have formed in a short time
interval around this time (Liu et al., 2019), so this is a convenient
definition. Assuming homogeneity of ${}^{26}{\rm Al}$, if a different object
formed with a lower initial ratio $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$, it must have formed later, at a time
$\Delta t_{26}=\tau_{26}\,\ln\left(\frac{(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{\rm SS}}{(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}}\right).$ (1)
This measures a time of formation relative to $t\\!\\!=\\!\\!0$. In practice,
the Al-Mg chronometer often has uncertainty of less than $\pm 0.1$ Myr (§2).
None of the original ${}^{26}{\rm Al}$ in CAIs exists today, and so its one-
time existence must be inferred. Its initial abundance relative to stable Al
is determined from a correlation between the isotopic ratio $({}^{26}{\rm
Mg}/{}^{24}{\rm Mg})$ and the ratio $({}^{27}{\rm Al}/{}^{24}{\rm Mg})$, as
measured in different minerals that condensed or crystallized from the same
isotopic reservoir within the inclusion (or igneous achondrite). If the
minerals have not been disturbed, e.g., not heated or altered in a way that
would lead to mobilization of Mg isotopes, then the correlation must be linear
and form an “internal isochron” in a plot of $({}^{26}{\rm Mg}/{}^{24}{\rm
Mg})$ versus $({}^{27}{\rm Al}/{}^{24}{\rm Mg})$ ratios of the inclusion’s
minerals. The slope of the isochron is the inferred initial
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratio at the time the
inclusion achieved isotopic closure.
Because ${}^{26}{\rm Al}$ must be inferred from an isochron, there are caveats
associated with its determination. First, determination of
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ is only possible if the data
form a linear isochron. This requires a statistical test: the mean squares
weighted deviation (MSWD)] of the linear correlation must not exceed a maximum
value $\approx 1+2(2/(N-2))^{1/2}$, where $N$ is the number of data points
(Wendt and Carl, 1991). A non-linear fit suggests disturbance or an additional
process; without a model of this process, $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ cannot be inferred. Second, it must be remembered that the initial
ratio $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ records not the
formation of the CAI, but the time at which the Al and Mg isotopes no longer
moved through the inclusion (i.e., when it reached isotopic closure). On the
parent body, aqueous alteration is a likely cause of disturbance. On the
parent body or in the solar nebula, a common cause of disturbance is heat-
related diffusion of isotopes. Often $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}$ records the time at which the CAI remained below a critical
temperature, the “closure temperature,” such that diffusion of Mg isotopes is
slower than continued cooling. This is discussed further in (§ 2.2).
The Pb-Pb system provides a second way to measure times of formation. In this
system, ${}^{235}{\rm U}$ essentially decays to ${}^{207}{\rm Pb}$ with half-
life $t_{1/2}=703.81\pm 0.96(1\sigma)$ Myr, and ${}^{238}{\rm U}$ essentially
decays to ${}^{206}{\rm Pb}$ with half-life $t_{1/2}=4468.3\pm 4.8(1\sigma)$
Myr (Jaffey et al., 1971; Villa et al., 2016), but no natural isotopes decay
to ${}^{204}{\rm Pb}$. The ratio of radiogenic ${}^{207}{\rm Pb}$ to
radiogenic ${}^{206}{\rm Pb}$ is therefore
$\left(\frac{{}^{207}{\rm Pb}}{{}^{206}{\rm Pb}}\right)_{\rm
r}=\left(\frac{{}^{235}{\rm U}}{{}^{238}{\rm U}}\right)_{\rm
t}\,\frac{\exp(+t/\tau_{235})-1}{\exp(+t/\tau_{238})-1},$ (2)
where $({}^{235}{\rm U}/{}^{238}{\rm U})_{\rm t}$ is the isotope ratio
measured in the sample today, and $t$ is the time that has elapsed since the
inclusion formed. The radiogenic Pb isotopic ratio is found by creating
different washes and leachates and residues by dissolution of a sample by
different acids, and then measuring Pb isotopes in each. A regression of
$y={}^{207}{\rm Pb}/{}^{206}{\rm Pb}$ vs. $x={}^{204}{\rm Pb}/{}^{206}{\rm
Pb}$ in each fraction yields a line (sometimes called the “inverse isochron”)
with $y$-intercept (limit of zero non-radiogenic component) equal to the left-
hand side of Equation 2. Different washes or leachates may incorporate
variable amounts of Pb from terrestrial contamination or initial non-
radiogenic Pb, either of which may cause a particular data point to fall off
an otherwise reasonable regression. As with Al-Mg isochrons, adherence to a
line can be tested using the MSWD of the linear regression. Before about 2010,
it was assumed that all samples were characterized by $\mbox{${}^{238}{\rm
U}/{}^{235}{\rm U}$}=137.88$; but it has become recognized that CAIs,
achondrites, etc., vary significantly in this ratio. This is attributable to
variable amounts of radiogenic ${}^{235}{\rm U}$ from decay of ${}^{247}{\rm
Cm}$ (Brennecka et al., 2010; Tissot et al., 2016), but also evaporation and
other processes (Tissot and Dauphas, 2015; Tissot et al., 2017, 2019). A
fractional change of $10^{-3}$ in the U isotope ratio corresponds to a shift
in the age of 1.45 Myr, and typical variations in CAIs can lead to
inaccuracies in their ages $>1$ Myr. Therefore it is now standard to report
“U-corrected” Pb-Pb ages, in which ${}^{238}{\rm U}/{}^{235}{\rm U}$ is
measured in the sample.
An additional issue is that not all uranium isotopic ratios are reported using
the same assumed values for standards. This may lead to variations in
${}^{238}{\rm U}/{}^{235}{\rm U}$ on the order of
$0.05\mbox{\text{\textperthousand}}$, leading to variations in inferred Pb-Pb
ages of 0.07 Myr (F. Tissot, personal communication). Future renormalization
of the data appears warranted.
The uncertainty in Pb-Pb absolute ages is often misunderstood to be $<1$ Myr.
(i.e., $\pm y$ in the notation of Tissot et al. (2017)), but ignores other
sources of uncertainty, the largest of which is the half-lives of U. Although
both are known at the 0.1% level, the uncertainty in absolute Pb-Pb ages is
$\pm 9$ Myr, due to uncertainties in the ${}^{238}{\rm U}$ and especially
${}^{235}{\rm U}$ half-lives (Ludwig, 2000; Tissot et al., 2017). However,
when two Pb-Pb ages are computed using the same half-lives, the systematic
uncertainties cancel, and the ($2\sigma$) precision can approach $0.3-0.5$ Myr
(Tissot et al., 2017). It is essential that the half-lives be identical. Other
half-lives are sometimes used—e.g., the ${}^{235}{\rm U}$ half-life of 703.20
Myr (Mattinson, 2010) used by Palk et al. (2018) in their Pb-Pb dating of
enstatite chondrites—but fortunately almost all absolute ages in the
literature are computed using the half-lives quoted above from Jaffey et al.
(1971) and Villa et al. (2016). The Pb-Pb system is the most precise and
versatile absolute chronometer, but its greater utility and precision comes
from using it as a relative chronometer to determine the sequence of events in
the solar nebula.
To be effective as a relative chronometer, absolute ages measured by the Pb-Pb
system, $t_{\rm Pb}$, must be converted into times after $t\\!\\!=\\!\\!0$,
$\Delta t_{\rm Pb}$, by subtracting them from the absolute age of
$t\\!\\!=\\!\\!0$, which we denote $t_{\rm SS}$:
$\Delta t_{\rm Pb}=t_{\rm SS}-t_{\rm Pb}.$ (3)
Here $t_{\rm SS}$ should be interpreted as the Pb-Pb age that would be
obtained by measuring the age of a sample that achieved isotopic closure at
$t\\!\\!=\\!\\!0$, using the same uranium half-lives as assumed for other
samples. Determination of $t_{\rm SS}$, the “age of the Solar System,” is one
focus of this paper.
### 1.2 Absolute Age of CAIs and the Solar System
To date, most determinations of $t_{\rm SS}$ and the time elapsed since
$t\\!\\!=\\!\\!0$ have been made by radiometrically dating CAIs. As described
above, an inverse isochron is made by measuring Pb isotopes in different
washes, leachates and residues (“fractions”) obtained by reacting a sample
with various acids, and for each fraction obtaining isotopic fractionation-
corrected $y={}^{207}{\rm Pb}/{}^{206}{\rm Pb}$ and $x={}^{204}{\rm
Pb}/{}^{206}{\rm Pb}$ ratios. Assuming a linear correlation, the $y$-intercept
of the line is the purely radiogenic end-member $({}^{207}{\rm
Pb}/{}^{206}{\rm Pb})$, and this plus a measurement of ${}^{235}{\rm
U}/{}^{238}{\rm U}$ allows one to solve for $t$ in Equation 2. The uncertainty
in the age comes from adding in quadrature the uncertainties in the
${}^{235}{\rm U}/{}^{238}{\rm U}$ ratio and the uncertainties in the
$y$-intercept from the linear regression. The latter derives from the
measurement uncertainties in the $x$ and $y$ data points.
Perhaps because of the difficulty of the measurement (wet chemistry of small
samples while avoiding terrestrial contamination) and the newness of the
uranium correction, there are only four CAIs with peer-reviewed, U-corrected
Pb-Pb ages. Amelin et al. (2010) dated Allende CAI SJ101 at $4567.18\pm 0.50$
Myr. MacPherson et al. (2017) reported an internal isochron with
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(5.20\pm 0.53)\times 10^{-5}$
for this CAI. Connelly et al. (2012) found ages for Efremovka CAIs 22E, 31E
and 32E of $4567.35\pm 0.28$ Myr, $4567.23\pm 0.29$ Myr, and $4567.38\pm 0.21$
Myr. Larsen et al. (2011) found excesses of ${}^{26}{\rm Mg}$ in 22E and 31E
that are consistent with formation with $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}\approx 5.23\times 10^{-5}$. Assuming that all of these CAIs
experienced isotopic closure in Pb-Pb at identical times, that time is the
weighted average of the Pb-Pb ages, given by Connelly et al. (2012) as
$4567.30\pm 0.16$ Myr. Further assuming that they all closed in Al-Mg at this
same time, this time is taken to be $t\\!\\!=\\!\\!0$. These four CAIs are the
only ones with peer-reviewed, U-corrected Pb-Pb ages, and all appear
consistent with this average age, so this value has been widely adopted as the
time elapsed since $t\\!\\!=\\!\\!0$.
There are strong hints of older ages of CAIs, however. Bouvier et al. (2011a)
reported a U-corrected Pb-Pb age for NWA 6991 CAI B4 of $4567.94\pm 0.31$ Myr.
This CAI also has canonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
(Wadhwa et al., 2014). In addition, Bouvier and Wadhwa (2010) inferred an age
$4568.22\pm 0.18$ Myr in NWA 2364 CAI B1. These ages have not been widely
accepted, perhaps because the former has not been published in the refereed
literature, and the latter was not U-corrected using a direct measurement of U
isotopes, but rather a correlation between Th/U and ${}^{238}{\rm
U}/{}^{235}{\rm U}$ found by Brennecka et al. (2010); this relation is not
universal (Tissot et al., 2016), and so might not yield the correct age
adjustment for this CAI. The two latter ages do not agree with $4567.30\pm
0.16$ Myr to within the uncertainties. If accepted or confirmed, CAIs B1 and
SJ101 would be seen to have formed at times that differ from $4567.3$ Myr by
over 1.0 Myr (at the $4\sigma$ level).
More Pb-Pb dates of CAIs would be enormously helpful, to determine whether
CAIs reached isotopic closure in the Pb-Pb system at the same time or across a
range of times. Because the four CAIs averaged by Connelly et al. (2012) seem
to have formed at a single time, that time is taken to be $t\\!\\!=\\!\\!0$.
However, as is well-known and as is discussed below, if CAIs actually formed
with canonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx
5.23\times 10^{-5}$ at 4567.30 Myr ago, then almost all other chronometers
would be left discordant. This has led to the proposal that ${}^{26}{\rm Al}$
must have been spatially or temporally heterogeneous in the solar nebula, with
the CAI-forming region holding nearly four times as much ${}^{26}{\rm Al}$ as
the rest of the solar nebula (Larsen et al., 2011; Bollard et al., 2019).
In this paper, we explore other reasons why Al-Mg and Pb-Pb ages might not
match. One important factor we identify is that the practice of building
isochrons from a subset of fractions means that some Pb-Pb ages are inaccurate
and their uncertainties have been considerably underestimated. Another
pertinent fact is that CAIs were likely transiently heated at random times up
to 3 Myr after $t\\!\\!=\\!\\!0$; we demonstrate that the Pb-Pb system could
have been isotopically reset, without disturbing the Al-Mg system. As these
are plausible scenarios, it is a distinct possibility that the reported Pb-Pb
ages of CAIs might not actually record the time of $t\\!\\!=\\!\\!0$.
An alternative to using radiometric dating of CAIs to determine $t_{\rm SS}$
is to mathematically find values that reduce the discrepancies between the Pb-
Pb and other chronometers to levels consistent with measurement errors. Many
other groups have sought to correlate Al-Mg and other chronometers. For
example, Lugmair and Shukolyukov (1998) correlated Mn-Cr ages against Pb-Pb
ages in many achondrites, finding a range of ages 4568 to 4571 Myr would
minimize the discrepancies between the chronometers. These ages overlapped
with the then-accepted age of CAIs, $4566\pm 2$ Myr (Göpel et al., 1991), so
it was possible to conclude the chronometers were concordant. Of course, none
of these ages was U-corrected. Nyquist et al. (2009) correlated Al-Mg and Mn-
Cr ages to show that their ages were linearly correlated and thereby constrain
the half-life of ${}^{53}{\rm Mn}$ and the Solar System initial value
$(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$. They then used the
initial $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{0}$ inferred for the
achondrite LEW 86010 to determine the time of its formation after
$t\\!\\!=\\!\\!0$. They added this to the Pb-Pb age of LEW 86010 to derive
$t_{\rm SS}=4568.2$ Myr. This Pb-Pb age was not U-corrected, either. Sanborn
et al. (2019) directly correlated U-corrected Pb-Pb ages against Al-Mg
formation times for several achondrites and CAIs. Although not explicitly
stated, their Figure 6 makes clear that they would infer a value of $t_{\rm
SS}\approx 4567.8$ Myr, with an uncertainty that we estimate at $\sim 0.5$
Myr. Recently, Piralla et al. (2023) also directly correlated U-corrected Pb-
Pb ages against Al-Mg formation times of several achondrites and concluded
$t_{\rm SS}=4568.7$ Myr. We further discuss these treatments below. One key
aspect missing from these treatments is that they did not check whether the
values of $t_{\rm SS}$ they derived led to a statistically valid fit between
the Al-Mg and other chronometers, i.e., whether the discrepancies between
their inferred $\Delta t_{26}$ and other formation times are attributable
solely to measurement uncertainties. Without this assessment of concordancy,
it is impossible to test the underlying assumption of ${}^{26}{\rm Al}$
homogeneity.
### 1.3 Outline
In this work we test the hypothesis that ${}^{26}{\rm Al}$ was distributed
homogeneously in the solar nebula. We do this by comparing Al-Mg and Pb-Pb
ages in the samples above, as in the work by Sanborn et al. (2019). A
prediction of the hypothesis is that in samples in which the systems must have
closed simultaneously, a single value $t_{\rm SS}$ must reconcile the
chronometers, in a statistical sense. If a single value of $t_{\rm SS}$ does
not reconcile the Al-Mg and Pb-Pb ages, then the hypothesis of ${}^{26}{\rm
Al}$ homogeneity is falsified. If the chronometers are reconciled, then that
supports the homogeneous hypothesis, and futher suggests that the range of
values of $t_{\rm SS}$ that reconciles them is the Pb-Pb age of the Solar
System at $t\\!\\!=\\!\\!0$.
In §2 we compile and discuss the meteoritic data, which comprises five
achondrites from the non-carbonaceous chondrite (NC) isotopic reservoir, two
achondrites from the carbonaceous chondrite (CC) isotopic reservoir, four
chondrules, and one CAI. We review closure temperatures and why there can be
no discrepancies between Al-Mg and Pb-Pb ages for the rapidly cooled
achondrites, and therefore why these can falsify the homogeneous hypothesis.
We discuss why the other samples, especially the CAI, do not necessarily test
the hypothesis. We re-analyze the Pb-Pb isochrons from all samples, to
determine the Pb-Pb age and the uncertainty for each.
In §3 we take weighted means of the data to derive the value of $t_{\rm SS}$
that minimizes the discrepancies between the Al-Mg and Pb-Pb chronometers in
the appropriate samples. We calculate the goodness-of-fit metric
$\chi_{\nu}^{2}$ (also known as MSWD) to test whether the samples are
adequately fit by a single value of $t_{\rm SS}$. We find the the Pb-Pb and
Al-Mg ages are indeed reconciled in a statistical sense for the seven
achondrites, and even the 4 chondrules as well, if $t_{\rm SS}=4568.42\pm
0.24$ Myr. Had they not been, this would have falsified the hypothesis of
homogeneity, but instead this finding supports it. We find that the same
$t_{\rm SS}$ that minimizes the discrepancies between the Al-Mg and Pb-Pb
systems among the NC achondrites also makes the chondrules concordant, but
cannot reconcile the CAI SJ101, which appears reset.
In §4 we compare our approach to previous similar approaches. We discuss how
the data support homogeneity of ${}^{26}{\rm Al}$ and discuss why
astrophysical models strongly support such an interpretation. One of our
primary conclusions is that the Pb-Pb age of CAI SJ101 and other CAIs may have
been reset 1.2 Myr after it formed and recorded an Al-Mg age, which was not
reset. We strongly support reporting times of formation after
$t\\!\\!=\\!\\!0$ rather than absolute ages. Astrophysical models are unable
to use absolute ages, only relative formation times, and relative times of
formation are far more precise anyway.
In §5 we draw conclusions. It does not appear that ${}^{26}{\rm Al}$ was
heterogeneous in the solar nebula. We argue that statistical approaches, like
those of Nyquist et al. (2009), Sanborn et al. (2019), Piralla et al. (2023),
and presented here, are more reliable determinants of $t_{\rm SS}$ than direct
measurements of CAIs. We advocate for reporting dates as times after
$t\\!\\!=\\!\\!0$, not absolute ages, and we advocate against use of
individual anchors.
## 2 Meteoritic Data
### 2.1 Which Samples test Homogeneity?
The hypothesis of homogeneous ${}^{26}{\rm Al}$ makes the testable prediction
that times of formation of appropriate samples found using the Al-Mg
chronometer should match times of formation found using the Pb-Pb chronometer.
Several conditions must be met for a sample to be an appropriate test. The
most important is that the $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
and Pb-Pb age actually can be found in an object; the former requires a valid
Al-Mg isochron, and the latter a valid Pb-Pb isochron and measurement of the
${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio in the sample. It also must be certain
that both the Al-Mg and Pb-Pb systems date the same event. The Al-Mg ages can
be defined as a time of formation after $t\\!\\!=\\!\\!0$, $\Delta t_{26}$,
using Equation 1. The Pb-Pb absolute ages must be converted to a time of
formation after $t\\!\\!=\\!\\!0$, $\Delta t_{\rm Pb}$, using Equation 3.
These requirements drive our choice of samples. To our knowledge, the objects
in which both Al-Mg and U-corrected Pb-Pb ages have been determined includes
seven bulk achondrites, four chondrules, and one CAI. The achondrites include
five with the signatures of having formed in the NC isotopic reservoir: the
quenched (volcanic) angrites D’Orbigny, SAH 99555, and NWA 1670; the eucrite-
like achondrite Asuka 881394; and the ungrouped achondrite NWA 7325 (paired
with NWA 8486). Two of the achondrites have signatures of having formed in the
CC isotopic reservoir: the ungrouped achondrites NWA 2976 (paired with NWA
011) and NWA 6704 (paired with NWA 6693). In some samples, we are confident
that the Al-Mg and Pb-Pb systems probably date the same event; in others we
have no evidence to support that.
There are many reasons why the two isotopic systems might not date the same
event. As discussed below, bulk achondrites generally record the time at which
the temperature dropped below a “closure temperature”. This can be hundreds of
K different for different minerals and isotopic systems, so Al-Mg and Pb-Pb
can record different times. If the cooling rate is slow, only $\sim
10^{2}\,{\rm K}\,{\rm Myr}^{-1}$, then those times can be $\sim 10^{6}$ years
apart. Alternatively, chondrules notoriously were exposed to transient heating
events with peak temperatures $>1800$ K, followed by cooling rates $\sim
10-10^{3}\,{\rm K}\,{\rm hr}^{-1}$ (Desch et al., 2012). Being located in the
nebula at the same time as chondrules before parent-body accretion, CAIs also
could have been exposed to these same heating events. Below we explore how
such transient heating events could have reset the Pb-Pb isochron at a late
time, and even do so without resetting the Al-Mg chronometer, which would
record a much earlier event.
### 2.2 Closure Temperatues of the Al-Mg and Pb-Pb Systems
To quantify the largest possible difference between the times recorded by the
Al-Mg and Pb-Pb systems, we calculate their closure temperatures, the
temperatures below which isotopic redistribution within the sample (here, by
thermal diffusion) is slower than further cooling. Closure temperatures are
found by comparing the cooling rate of the sample against the diffusion rate
of key elements in specific minerals, and depend on the grain size of the
minerals (more precisely, the lengthscales over which isotopic gradients must
be preserved). It is assumed the diffusion coefficient of the element obeys an
Arrhenius relationship:
$D(T)=D_{0}\,\exp\left(-E/RT\right),$ (4)
where $D_{0}$ is the pre-factor and $E$ the activation energy, both specific
to the element and mineral, and $R$ the gas constant. In that case the closure
temperature is given by
$T_{\rm c}=\frac{E}{R}\,\left[\ln\left(\frac{A\,T_{\rm
c}^{2}\,D_{0}}{(E/R)\,(dT/dt)\,a^{2}}\right)\right]^{-1}$ (5)
(Dodson, 1973), where $A=55$ (for a sphere; $A=27$ for a cylinder, $A=8.7$ for
a plane), $a$ is the grain radius, $dT/dt$ is the cooling rate, and it is
assumed that the cooling is from a peak temperature above $T_{\rm c}$. The
closure temperature must be estimated, substituted into the right-hand side,
and the solution iterated to convergence. The majority of Pb-Pb dates are
based on data from pyroxene grains. We have calculated the closure temperature
for Pb in clinopyroxene using $D_{0}=44\,{\rm m}^{2}\,{\rm s}^{-1}$, and
$E/R=62,400$ K (Cherniak, 1998). Al-Mg measurements have been taken in a
variety of minerals, including anorthite, melilite, pyroxene and spinel. For
Mg diffusion in anorthite, we assume $D_{0}=1.2\times 10^{-6}\,{\rm
m}^{2}\,{\rm s}^{-1}$ and $E/R=33440$ K (LaTourrette and Wasserburg, 1998).
For Mg diffusion in melilite, we assume $D_{0}=3.02\times 10^{-9}\,{\rm
m}^{2}\,{\rm s}^{-1}$ and $E/R=28990$ K (Ito and Ganguly, 2009). For Mg
diffusion in pyroxene, we assume $D_{0}=1.9\times 10^{-9}\,{\rm m}^{2}\,{\rm
s}^{-1}$ and $E/R=24300$ K (Müller et al., 2013). For Mg diffusion in spinel,
we assume $D_{0}=2.77\times 10^{-7}\,{\rm m}^{2}\,{\rm s}^{-1}$ and
$E/R=38560$ K (Liermann and Ganguly, 2002).
In general, during transient heating events, Pb-Pb is easier to reset than Al-
Mg. In Figure 1 we plot the calculated closure temperatures $T_{\rm c}$ of Pb
and Mg in clinopyroxene, as functions of the cooling rate for grain sizes of
20 and $200\,\mu{\rm m}$. At slow cooling rates characteristic of secular
cooling of asteroids, $\sim 100\,{\rm K}\,{\rm Myr}^{-1}$ ($3\times
10^{-12}\,{\rm K}\,{\rm s}^{-1}$), $T_{\rm c}\approx 1040$ K for Pb, but lower
($\approx 670$ K) for Mg. The order of isotopic closure tends to reverse for
small inclusions in transient heating events with cooling rates $\sim
1-10^{3}\,{\rm K}\,{\rm hr}^{-1}$ ($3\times 10^{-4}-0.3\,{\rm K}\,{\rm
s}^{-1}$).
Figure 1: Closure temperatures of Pb in clinopyroxene (blue curves) and Mg in
clinopyroxene (red curves), as a function of cooling rate, assuming grain
sizes (diameters) of $200\,\mu{\rm m}$ (top), and $20\,\mu{\rm m}$ (bottom).
For slow cooling rates typical of secular cooling on asteroids (e.g., 100
K/Myr), Al-Mg has a lower closure temperature than the Pb-Pb system and is
more easily reset. For a cooling rate 500 K/hr, associated with transient
heating events experienced by CAIs or chondrules, the closure temperatures are
reversed. At a hypothetical peak temperature of $\approx 1730$ K and cooling
rate 500 K/hr, the Al-Mg system would remain closed in clinopyroxene grains
$100\,\mu{\rm m}$ in size (closure temperature 1764 K, red diamond), but the
Pb-Pb system in pyroxene grains $100\,\mu{\rm m}$ in size would be disturbed
(closure temperature 1704 K, blue diamond).
The quenched or “volcanic” angrites (D’Orbigny, SAH 99555, NWA 1670) are
widely accepted, based on petrologic features suggesting a lava erupted onto
an asteroid surface, to have cooled very rapidly, at rates $\approx 300\,{\rm
K}\,{\rm hr}^{-1}$ in the 1273 - 1573 K range (Keil, 2012). Likewise, NWA 7325
shares these petrologic features, and is inferred from Mg diffusion in
plagioclase to have cooled at rates $\sim 500-650\,{\rm K}\,{\rm hr}^{-1}$, at
high temperatures (Yang et al., 2019). The achondrite NWA 6704 was inferred by
Hibiya et al. (2019) from textures to have cooled at rates $1-10^{2}\,{\rm
K}\,{\rm hr}^{-1}$ at high temperatures, and from olivine-spinel
geospeedometry to have cooled at rates $<10^{-2}\,{\rm K}\,{\rm hr}^{-1}$ at
1123 - 1243 K. The cooling rate of Asuka 881394 has not been constrained
petrologically; but given its early time of formation in common with the
achondrites above, we assume it also cooled rapidly. It is very likely that
these achondrites formed from complete melts at high temperatures and
continued to cool below the closure temperatures of Al-Mg and Pb-Pb in a
matter of years, so that these two systems closed effectively simultaneously.
Al-Mg and (properly determined) Pb-Pb ages must be concordant in these
achondrites.
In contrast to the achondrites, the Al-Mg and Pb-Pb systems will not
necessarily have closed at the same time in chondrules and CAIs. This is not
because they were not rapidly cooled, but because their last heating event did
not reset both chronometers. Chondrules are silicate melts that crystallized
following a transient heating event in the solar nebula, typified by peak
temperatures anywhere from subsolidus or barely above the solidus (Wasson and
Rubin, 2003; Ruzicka et al., 2008), to $>1800$ K or more, and cooling rates
$\sim 10-10^{3}\,{\rm K}\,{\rm hr}^{-1}$ (Desch et al., 2012). Many chondrules
appear to have experienced multiple such events. A representative thermal
history might be heating to a peak temperature $\approx 1700-1800$ K, followed
by cooling at $500\,{\rm K}\,{\rm hr}^{-1}$ (a cooling rate consistent with
all chondrule textures).
A majority of CAIs also were transiently heated in the solar nebula. Compact
type A CAIs were at least partially melted or annealed, and type B and C CAIs
have igneous textures showing they were partially or completely melted at
least once. During their last heating event, type B CAIs appear from zoning in
melilite to have cooled at rates of $0.5-50\,{\rm K}\,{\rm hr}^{-1}$ (Stolper
and Paque, 1986; Jones et al., 2000), or (from the need to limit diffusion of
oxygen isotopes) $>6\times 10^{4}\,{\rm K}\,{\rm hr}^{-1}$ (Kawasaki et al.,
2021). These hours-long cooling times are of the same order as the cooling
times experienced by chondrules.
Both chondrules and CAIs resided in the same regions of the solar nebula, for
the same $\sim 2-4$ Myr it took before chondrites formed (Desch et al., 2018).
Given that a large fraction of the material in chondrites is chondrules, all
of which were transiently heated by some event, it would be remarkable if CAIs
avoided being transiently heated as well. It seems very likely that CAIs must
have been transiently heated, some to the melting point and some to only
subsolidus temperatures, over the same period of time as chondrules (1-3 Myr
after $t\\!\\!=\\!\\!0$; Villeneuve et al. 2009). In fact, evidence exists of
this late heating of CAIs (Mane et al., 2022).
These transient heating events could have reset isotopic systems in chondrules
and CAIs, depending on the peak temperatures and the cooling rates. If a CAI
were to be heated to peak temperatures of about 1750 K, and cool at $500\,{\rm
K}\,{\rm hr}^{-1}$, pre-existing minerals would survive. Pyroxene (enstatite)
grains survive to temperatures of 1830 K (Greenwood and Hess, 1996), although
the more Ti-rich pyroxene more typically found in CAIs may partially melt at
lower temperatures $\approx 1500$ K (A. Davis, personal communication).
Spinel grains survive to similar temperatures $>1800$ K (Whattam et al.,
2022). For $20\,\mu{\rm m}$ grains ($a=10\mu{\rm m}$) and cooling rates of 500
K/hr, we calculate closure temperatures of 2071 K in spinel, 2022 K in
melilite, 1764 K in pyroxene, and 1687 K in anorthite. With the possible
exception of anorthite, which would be slightly disturbed, a heating event
with peak temperatures just above 1700 K would not reset the Al-Mg
chronometer. In contrast, the closure temperature of the Pb-Pb system in
$20\,\mu{\rm m}$ pyroxene grains would be no higher than 1703 K, so Pb-Pb ages
would be reset, especially if the more Ti-rich pyroxene grains were to
partially melt. The types of transient heating events experienced by
chondrules and assuredly also CAIs are capable of resetting the Pb-Pb
chronometer without necessarily resetting the Al-Mg chronometer, at least if
it Al-Mg ages are determined using isochrons based on pyroxene and spinel.
Chondrules and CAIs may have formed at $t\\!\\!=\\!\\!0$, or any time over the
next 2-3 Myr, before a transient heating event reset their Pb-Pb ages without
necessarily resetting their Al-Mg ages. Therefore chondrules and CAIs are not
samples that reliably or necessarily test the homogeneity hypothesis.
It has been noted before that the discrepancies between the Al-Mg and Pb-Pb
systems imply a late-stage resetting of the Pb-Pb system. Bouvier and Wadhwa
(2010) suggested that diffusion of Pb in pyroxene might be faster than
diffusion of Mg in melilite or anorthite, although they did not outline a
scenario in which this would occur, identify a nebular or parent-body setting,
or quantify the thermal histories that would be required.
With these caveats, we examine the ages recorded by each sample.
### 2.3 Achondrites
#### 2.3.1 D’Orbigny (quenched angrite)
D’Orbigny is a quenched angrite with an unshocked and unbrecciated texture
consisting of large laths of anorthite intergrown with Al-Ti-bearing diopside-
hedenbergite, Ca-rich olivine and kirschsteinite as well as abundant glasses
and round vugs (Keil, 2012). The glass is not of shock origin or externally
introduced at a later time (Glavin et al., 2004). The glass and vugs speak to
a rapid cooling rate $\sim 300\,{\rm K}\,{\rm hr}^{-1}$ (Keil, 2012).
D’Orbigny has long been considered an anchor in which the different isotopic
systems likely closed simultaneously.
The initial $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratio in
D’Orbigny has been determined by multiple groups: Spivak-Birndorf et al.
(2009) found $(5.06\pm 0.92)\times 10^{-7}$, MSWD $=2.5$; Schiller et al.
(2010) found $(3.88\pm 0.27)\times 10^{-7}$, MSWD $=1.9$; and Schiller et al.
(2015) found $(3.98\pm 0.15)\times 10^{-7}$, MSWD $=1.9$. Combining these data
and rejecting one outlier, Sanborn et al. (2019) found $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}=(3.93\pm 0.39)\times 10^{-7}$, the value we adopt.
It has also been reported to have been $(3.97\pm 0.21)\times 10^{-7}$, but
from an isochron with MSWD above the threshold for acceptance (Kleine and
Wadhwa, 2017). For our adopted initial ratio of $5.23\times 10^{-5}$, we find
a time of formation $\Delta t_{26}=5.06\pm 0.10\,{\rm Myr}$ (Equation 1).
Amelin (2008b) measured Pb isotopic ratios using whole-rock and pyroxene
fractions in D’Orbigny. Excluding two outliers with excess initial Pb, and one
with low Pb concentration, and using 9 of the 13 fractions, their regression
yields an intercept corresponding to an age $4564.42\pm 0.12$ Myr, and MSWD
$=1.18$ (assuming ${}^{238}{\rm U}/{}^{235}{\rm U}=137.88$), We concur with
their selection of points and confirm their regression, finding $4564.40\pm
0.13$ Myr, with MSWD $=1.14$. The ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio in
whole-rock samples of D’Orbigny has been measured by Brennecka and Wadhwa
(2012), who found $137.780\pm 0.021$, and Tissot et al. (2017), who found
$137.793\pm 0.025$. We take the weighted mean of these, or $137.785\pm 0.016$,
and use this to update the age and its uncertainty according to equations 3
and 4. We find $4563.42\pm 0.21$ Myr.
An important caveat is that the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio was
measured in whole-rock samples, whereas the Pb-Pb ages were determined using
data primarily from Pb in (clino)pyroxene (especially the points with the most
radiogenic Pb). This matters because the U in pyroxene appears to be
isotopically light, most probably because of isotopic fractionation during
magmatic differentiation; Tissot et al. (2017) model this process and conclude
that pyroxene grains have $\delta^{238}{\rm U}$ roughly
$0.25\mbox{\text{\textperthousand}}$ below the value for $\delta^{238}{\rm U}$
in the silicate melt in both D’Orbigny and Angra dos Reis. Whether the
${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio in the pyroxene grains that figure
into the isochron is well represented by the ratio in the bulk samples depends
on $f_{\rm cpx}$, the fraction of all U residing in clinopyroxene, with
$\delta^{238}{\rm U}_{\rm cpx}=\delta^{238}{\rm U}_{\rm bulk}-(1-f_{\rm
cpx})\,(0.25\mbox{\text{\textperthousand}}).$ (6)
Tissot et al. (2017) estimate $f_{\rm cpx}\approx 50\%$ in most angrites and
similar achondrites, so that using whole-rock U isotopic ratios overestimates
the ages of these achondrites, by about $0.19$ Myr, but possibly by up to
$0.38$ Myr ($f_{\rm cpx}=0$), or not at all ($f_{\rm cpx}=1$). We therefore
favor $4563.24\pm 0.21\,{\rm Myr}$.
#### 2.3.2 SAH 99555 (quenched angrite)
SAH 99555 is a quenched angrite with an unshocked, fine-grained texture
composed of anorthite, Al-Ti-bearing hedenbergite, olivine and mm-sized
vesicles (Keil, 2012).
We take the value $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(3.64\pm
0.18)\times 10^{-6}$ determined by Schiller et al. (2015). This yields a time
of formation $\Delta t_{26}=5.14\pm 0.05\,{\rm Myr}$.
The Pb isotopic ratios of SAH 99555 were determined by both Amelin (2008a) and
Connelly et al. (2008). Amelin (2008a) built an isochron using 8 out of 10
whole-rock fractions, rejecting two clear outliers. Assuming ${}^{238}{\rm
U}/{}^{235}{\rm U}=137.88$, they found an age $4564.86\pm 0.38$ Myr, MSWD
$=1.5$. We essentially reproduce this regression, finding $4564.88\pm 0.10$
Myr, MSWD $=1.5$. Connelly et al. (2008) built an isochron using 8 of 11
whole-rock leachates, plus the residue from the pyroxene samples. Assuming
${}^{238}{\rm U}/{}^{235}{\rm U}=137.88$, they found an age $4564.58\pm 0.14$
Myr, MSWD $=0.99$. We again essentially reproduce this regression, finding
$4564.58\pm 0.07$ Myr, MSWD $=1.17$. The weighted mean of the ages is
$4564.614\pm 0.131$ Myr. For the U isotopic rate, we use the weighted mean of
the values measured by Tissot et al. (2017), ${}^{238}{\rm U}/{}^{235}{\rm
U}=137.805\pm 0.029$, and by Connelly et al. (2012), ${}^{238}{\rm
U}/{}^{235}{\rm U}=137.784\pm 0.024$ (after renormalizing to the same U
standard as Tissot et al. 2017), and adopt ${}^{238}{\rm U}/{}^{235}{\rm
U}=137.793\pm 0.019$. With this correction, we find an age $4563.70\pm
0.24\,{\rm Myr}$.
Again we note the caveat that the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio is
measured in the bulk sample, not the pyroxene grains, meaning the value of
$t_{\rm Pb}$ should be reduced by about 0.19 Myr. We therefore favor
$4563.51\pm 0.24\,{\rm Myr}$.
#### 2.3.3 NWA 1670 (quenched angrite)
NWA 1670 is a quenched angrite with a porphyritic texture including large
olivine megacrysts in a fine-grained matrix of olivine, pyroxene, kirsch-
steinite and anorthite as well as other accessory minerals (Keil, 2012). These
indicate rapid cooling at $\sim 300\,{\rm K}\,{\rm hr}^{-1}$ (Mikouchi et al.,
2003).
It was analyzed by Schiller et al. (2015), who found $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}=(5.92\pm 0.59)\times 10^{-7}$, which yields a time
of formation $\Delta t_{26}=4.63\pm 0.10\,{\rm Myr}$.
Schiller et al. (2015) also measured Pb isotopic ratios to determine a Pb-Pb
age of $4565.39\pm 0.24$ Myr, assuming ${}^{238}{\rm U}/{}^{235}{\rm
U}=137.786\pm 0.013$. We note that this value appears to have been taken from
whole-rock measurements of other angrites (Connelly et al., 2012). Besides
this, we were not able to confirm this age, because it is unclear which points
to include in the regression. This isochron was based on using only six of the
13 available fractions, with no clear criteria for including or excluding
points from the regression. As a result, the uncertainty in the age has been
considerably underestimated, as we now demonstrate.
As described above, the absolute age of a sample can be found by performing a
linear regression on the data $y_{i}={}^{207}{\rm Pb}/{}^{206}{\rm Pb}$ vs.
$x_{i}={}^{204}{\rm Pb}/{}^{206}{\rm Pb}$, where each point $i$ represents an
isotopic measurement in a different wash, leachate, or residue of a sample
after it has been dissolved in various acids. Provided there is only one non-
radiogenic source of lead—either primordial lead or terrestrial
contamination—the data should array along a line, and the $y$-intercept of
that line yields the absolute age according to Equation 2. Primordial lead is
a likely component of meteorites and inclusions, and terrestrial contamination
is pervasive (Gopel et al., 1994; Palk et al., 2018), so it is likely that one
of the fractions may contain an anomalous amount of one source of Pb or the
other, and fall off the isochron. It is almost always necessary to exclude one
or more points from the Pb-Pb isochron regression, and this is indeed the case
for all the other Pb-Pb isochrons discussed in this paper. For example, in
their analyses of D’Orbigny and SAH 99555, Amelin (2008b) and Amelin (2008a),
excluded from their regressions data points with elevated common Pb, or low
total Pb. In other analyses of Asuka 881394 (Wadhwa et al., 2009; Wimpenny et
al., 2019), NWA 2976 (Bouvier et al., 2011b), and NWA 6704 (Amelin et al.,
2019), points were excluded essentially only if the ${}^{206}{\rm
Pb}/{}^{204}{\rm Pb}$ was below a threshold value (indicating too little
radiogenic Pb).
In contrast, Connelly et al. (2008) (SAH 99555), Schiller et al. (2015) (NWA
1670), and Bollard et al. (2017) (NWA 5697 chondrules) did not include or
exclude points based on stated criteria such as dissolution order or threshold
${}^{206}{\rm Pb}/{}^{204}{\rm Pb}$. We believe they instead excluded up to
half their data based on whether the data fell off a pre-determined isochron.
This procedure is prone to confirmation bias, as we illustrate using the
example of NWA 1670.
Schiller et al. (2015) obtained Pb isotopic data for NWA 1670 from the
following fractions: two acid washes of whole-rock samples (W1, W2), eight
leachates (L1-L8) and a residue (R) of whole-rock samples, and two leachates
(C-L3, C-L4) of clinopyroxene grains. In Figure 2 we plot these 13 data
(except for W1 and W2, with very high ${}^{204}{\rm Pb}/{}^{206}{\rm
Pb}\approx 0.01$). When we perform a York regression (York et al., 2004) on
the same 6 data points selected by Schiller et al. (2015) (L1, L2, L4, L5, L7,
C-L3), we infer a slope $4.093\pm 0.009$ and intercept $0.624205\pm 0.000038$,
with MSWD $=0.30$. With ${}^{238}{\rm U}/{}^{235}{\rm U}=137.786\pm 0.013$, we
infer a Pb-Pb age of $4564.40\pm 0.22$ Myr. This is almost identical to the
result obtained by Schiller et al. (2015), who found an age $4564.39\pm 0.24$
Myr, with MSWD $=0.31$.
Except perhaps for W1 and W2, the reasons for excluding data points from the
regression were not based on wash order or ${}^{204}{\rm Pb}/{}^{206}{\rm Pb}$
ratio, as this would not explain why L4 and L7 were included, but L6 and L8
were not, or why C-L3 was included but C-L4 was not. Schiller et al. (2015)
state that the reasons for excluding data points was the need to have
components that only sampled radiogenic and terrestrial Pb, not initial Pb.
They point out that the regression line is consistent with mixing between
radiogenic and terrestrial Pb; we also find that this regression line lies
only 0.0038 above the terrestrial Pb point (${}^{204}{\rm Pb}/{}^{206}{\rm
Pb}=0.0542$, ${}^{207}{\rm Pb}/{}^{206}{\rm Pb}=0.84228$; Stacey and Kramers
1975). Schiller et al. (2015) interpret points above the line (e.g., L3) to
contain minor amounts of initial Pb, and points below the line (e.g., L6, L8,
R, C-L4) to have lost radiogenic Pb. Notably, these interpretations are not
based on any physical characteristics of the samples, but based on where they
lie relative to the regression line. Thus, this interpretation is circular
logic. Points were excluded from the regression; those points that were not
included in the regression fell off the regression line; for that reason
alone, their exclusion was justified. The fits obtained by such analyses have
low MSWD, but significantly, an MSWD of only 0.30 is “too good to be true”.
The probability of six data points with random measurement errors exhibiting a
fit with such a low MSWD is $\approx 5\%$ (Wendt and Carl, 1991), which is
usually the threshold for acceptability. This entire approach is prone to
confirmation bias.
Figure 2: Pb-Pb isochron of NWA 1670, based on the six data points regressed
by Schiller et al. (2015), in black, with excluded points in red. The
intercept of this isochron, and its uncertainty, yield an age $4564.40\pm
0.21$ Myr.
Indeed, we have found a range of combinations of points that yield more
probable isochrons (in terms of MSWD) and a wide variety of ages. In Figure 3,
we show a regression using a different set of five points (W1, L1, L2, L6,
C-L3). We infer a slope $4.151\pm 0.006$ and intercept $0.623937\pm 0.000036$,
with MSWD $=1.71$. This line passes only 0.006 above terrestrial Pb, and a
regression of five points with MSWD $=1.71$ is actually just as likely as one
with 0.30. The points that lie below the line (L8, R, C-L4) can be equally
interpreted as having lost radiogenic Pb, and the points above the line (L3,
L4, L5, L7) as having incorporated initial Pb. Thus, this is an equally valid
isochron, but the age derived from it is $4563.77\pm 0.21$ Myr, 0.63 Myr
younger than the age derived in Figure 1.
Figure 3: Pb-Pb isochron of NWA 1670, based a different subset of data points.
The intercept of this isochron, and its uncertainty, yield an age $4563.77\pm
0.21$ Myr, 0.63 Myr younger than the age from Figure 1.
Likewise, in Figure 4, we show a regression using a different set of six
points (L4, L5, L6, L7, R, C-L3). We infer a slope $4.000\pm 0.018$ and
intercept $0.624309\pm 0.000051$, with MSWD $=1.12$. This line passes only
0.001 below terrestrial Pb, and a regression of six points with MSWD $=1.12$
is very probable. The points that lie below the line (L8, C-L4) can be equally
interpreted as having lost radiogenic Pb, and the points above the line (L1,
L2, L3) as having incorporated initial Pb. This isochron is as valid as the
others, perhaps even more so because it passes closest to terrestrial Pb, has
the most probable MSWD, and there is some order to which fractions are
included in the regression. The age derived from this regression is
$4564.64\pm 0.23$ Myr, 0.24 Myr older than the isochron derived in Figure 1.
Figure 4: Pb-Pb isochron of NWA 1670, based a different subset of data points.
The intercept of this isochron, and its uncertainty, yield an age $4564.64\pm
0.23$ Myr, 0.24 Myr older than the age from Figure 1.
Through these examples, we have established that two isochrons equally valid
to the one derived by Schiller et al. (2015) yield a range of ages, from 0.63
Myr younger to 0.24 Myr older. These isochrons are equally valid, because they
use equal numbers of apparently equally valid points. Without objective
criteria like ${}^{204}{\rm Pb}/{}^{206}{\rm Pb}$ ratio or Pb concentration to
select points, points included in our regressions can equally well be argued
to be valid because they lie on the regression line; points excluded from the
regression can equally well be argued to exhibit loss of radiogenic Pb or
excess of primordial Pb. Without further information, we consider the 95%
confidence interval of possible Pb-Pb ages of NWA 1670 to extend from 4563.55
to 4564.87 Myr, and we take its Pb-Pb age to be $4564.21\pm 0.66\,{\rm Myr}$.
This encompasses the age $4564.39\pm 0.24$ Myr reported by Schiller et al.
(2015), but acknowledges a larger uncertainty. This age uncertainty may be
magnified, depending on whether the isochron draws mostly from pyroxenes or
whole rock washes and leachates, because of isotopic fractionation of U
isotopes.
Again we note the caveat that the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio is
measured in the bulk sample, not the pyroxene grains, meaning the value of
$t_{\rm Pb}$ should be reduced by about 0.19 Myr. We therefore favor
$4564.02\pm 0.66\,{\rm Myr}$.
#### 2.3.4 Asuka 881394 (eucrite-like achondrite)
Asuka 881394 is a eucrite-like achondrite with a coarse-grained igneous
texture with near equal amounts of anorthite and pyroxene. The granoblastic
texture suggests post-formation, low-grade metamorphism could have affected
Asuka 881394 (Wimpenny et al., 2019). While originally classified as a
cumulate eucrite (Takeda, 1997), Asuka 881394 has geochemcial and isotopic
qualities that preclude classification as such. As discusssed by Wimpenny et
al. (2019), the major element chemistry of the primary phases do not resemble
cumulate eucrites (e.g., its plagioclase is too calcic, and the Mg-rich
pyroxenes do not show evidence of inversion textures). Additionally, the
$\Delta^{17}{\rm O}$ oxygen isotope composition of Asuka 881394 is
15${\sigma}$ above the mean value for HEDs (Scott et al., 2009).
The Al-Mg systematics of Asuka 881394 have been measured by three groups:
Nyquist et al. (2003), who found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}=(1.18\pm 0.14)\times 10^{-6}$, reanalyzed by Wimpenny et al. (2019)
as $(1.18\pm 0.31)\times 10^{-6}$; Wadhwa et al. (2009), who found
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(1.28\pm 0.07)\times 10^{-6}$,
eliminating one outlier from the regression; and Wimpenny et al. (2019), who
found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(1.48\pm 0.12)\times
10^{-6}$. We take the weighted average of these, $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}=(1.31\pm 0.06)\times 10^{-6}$. This yields a time
of formation $\Delta t_{26}=3.82\pm 0.04\,{\rm Myr}$.
Pb-Pb dating of Asuka 881394 was done by Wimpenny et al. (2019), who built an
isochron using their pyroxene and whole-rock residue data, plus one whole-rock
wash point, plus the most radiogenic residues (${}^{206}{\rm Pb}/{}^{204}{\rm
Pb}>400$) from the analysis by Wadhwa et al. (2009). They also determined
${}^{238}{\rm U}/{}^{235}{\rm U}=137.786\pm 0.038$ in the bulk rock, and found
a Pb-Pb age of $4564.95\pm 0.53\,{\rm Myr}$, based on an isochron with MSWD
$=1.4$.
Again we note the caveat that the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio is
measured in the bulk sample, not the pyroxene grains, meaning the value of
$t_{\rm Pb}$ should be reduced by about 0.19 Myr. We therefore favor
$4564.76\pm 0.53\,{\rm Myr}$.
### 2.4 NWA 7325 (ungrouped achondrite)
NWA 7325 is an ungrouped achondrite with a medium-grained cumulate texture
consisting of Mg-rich olivine, Cr-bearing diopside and Ca-rich plagioclase
(Goodrich et al., 2017). The positive Eu anomaly in NWA 7325 is a geochemical
sign of being a cumulate rock, but also can be interpreted as the result of
melting or of a basaltic or gabbroic lithology (Barrat et al., 2015).
The Al-Mg systematics of NWA 7325 were measured by Koefoed et al. (2016), who
found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(3.03\pm 0.14)\times
10^{-7}$, which yields a time of formation $\Delta t_{26}=5.33\pm 0.05\,{\rm
Myr}$.
Koefoed et al. (2016) also analyzed its Pb-Pb systematics, restricting their
regression to pyroxene residues, and building an isochron that excluded those
points with ${}^{206}{\rm Pb}/{}^{204}{\rm Pb}<50$, as they included obvious
terrestrial contamination. They did not measure the U isotopic ratio, instead
adopting a value ${}^{238}{\rm U}/{}^{235}{\rm U}=137.794$ as representative
of materials from the inner Solar System (Goldmann et al., 2015). Koefoed et
al. (2016) found a Pb-Pb age for NWA 7325 of $4563.4\pm 2.6\,{\rm Myr}$.
More recently, Cartwright et al. (2016) analyzed NWA 8486, which is paired
with NWA 7325. Combining the data for the two achondrites, and apparently
assuming the same U isotopic ratio, they found an age of $4563.9\pm 1.7$ Myr.
As it is consistent with the Koefoed et al. (2016) result but provides a more
extreme test, we adopt this value with its smaller uncertainties.
Again we note the caveat that the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio is
measured in the bulk sample, not the pyroxene grains, meaning the value of
$t_{\rm Pb}$ should be reduced by about 0.19 Myr. We therefore favor
$4563.7\pm 1.7\,{\rm Myr}$.
#### 2.4.1 NWA 2976 (ungrouped carbonaceous achondrite)
NWA 2976 (paired with NWA 011) is an unshocked, unbrecciated, ungrouped
achondrite with coarse-grained pigeonite surrounded by fine-grained
plagioclase with prevalent 120∘ triple junctions. (Yamaguchi et al., 2002).
The 54Cr, 50Ti and ${\Delta}^{17}$O isotope systematics of NWA 011 have strong
affinities to carbonaceous chondrites, and to the CR clan of chondrites in
particular (Floss et al., 2005; Sanborn et al., 2019). The major element data
presented by Floss et al. (2005) point to oxidizing conditions during
formation of NWA 011/2976.
The Al-Mg systematics of NWA 2976 were measured by Bouvier et al. (2011b), who
found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(3.94\pm 0.16)\times
10^{-7}$. Schiller et al. (2010) also reported $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}=(4.91\pm 0.46)\times 10^{-7}$ in NWA 2976. Sugiura
and Yamaguchi (2007) reported $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}=(6.93\pm 2.12)\times 10^{-7}$ in the paired achondrite NWA 011. We
adopt the weighted mean of the values from Bouvier et al. (2011b) and Schiller
et al. (2010), $(4.05\pm 0.15)\times 10^{-7}$ and find a time of formation
$\Delta t_{26}=5.03\pm 0.04\,{\rm Myr}$.
For NWA 2976, Bouvier et al. (2011b) regressed the five whole-rock and
pyroxene samples obtained from the last dissolution step, with ${}^{206}{\rm
Pb}/{}^{204}{\rm Pb}>1010$. They measured ${}^{238}{\rm U}/{}^{235}{\rm
U}=137.751\pm 0.018$, and found a Pb-Pb age for NWA 2976 of $4262.89\pm 0.59$
Myr from an isochron with MSWD $=0.02$. We agree with their choice of data to
regress and reproduce their regression. Connelly et al. (2012) measured an
isotopic ratio ${}^{238}{\rm U}/{}^{235}{\rm U}=137.787\pm 0.011$. We combine
these, finding ${}^{238}{\rm U}/{}^{235}{\rm U}=137.777\pm 0.009$, which then
yields an age $4563.16\pm 0.57\,{\rm Myr}$,
As before, it is important to note that the U isotopic ratio was measured in
whole-rock samples, and may not represent the isotopic ratio in the pyroxenes.
Here we argue, though, that for the two ‘carbonaceous’ achondrites NWA 2976
and NWA 6704, likely $f_{\rm cpx}\approx 1$, most of the U probably resides in
the pyroxene grains, and the whole-rock isotopic ratio probably is
appropriate. Both of these achondrites are more oxidized. As members of the CC
isotopic reservoir, they would be expected to contain more water and other
volatiles. Both Warren et al. (2013) and Hibiya et al. (2019) estimated that
the CC achondrite NWA 6693, paired with NWA 6704, formed under oxygen fugacity
conditions 2 log units above the iron-wüstite buffer ($\Delta{\rm IW}=+2$), as
opposed to angrites, which formed at $\Delta{\rm IW}=+1$ (Brett et al., 1977;
Jurewicz et al., 1993; McKay et al., 1994; Tissot et al., 2022). More
importantly, Izawa et al. (2022) concluded that the paired achondrite NWA 011,
paired with NWA 2976, experienced hydrous magmatism and a water-bearing melt.
Under these redox conditions, U is expected to be mostly in the U(iv) $4^{+}$
valence state (as insoluble UO2) with some in the U(vi) $6^{+}$ state as the
soluble ${\rm UO}_{2}^{2+}$ uranyl ion. (Chevreux et al., 2021). At IW+2,
compared to IW+1, slightly more U speciates as U(vi) instead of U(iv), but the
more important factor is the presence of H2O in the magma. This could
drastically change the partitioning of U due to the ability of water to
depolymerize the aluminosilicate network and produce Non-Bridging Oxygen (NBO)
sites. These are the coordination sites by which U (of either valence) can
enter a growing crystal. We expect that in a hydrous magma, U will readily
partition into the major crystallizing phases, as opposed to remaining in the
melt to partition into late-forming phosphates as in an anhydrous magma. As a
result, the pyroxene grains will take up most the U and share the same U
isotopic ratio as the whole rock. Assuming $f_{\rm cpx}\approx 1$, we do not
apply an age correction to NWA 2976 or NWA 6704. By our reasoning, it should
not be applied to any achondrite formed from hydrous magmatism.
#### 2.4.2 NWA 6704 (ungrouped carbonaceous achondrite)
NWA 6704 is an unshocked, ungrouped achondrite with a medium-grained, cumulate
texture comprised of low-Ca pyroxene along with Ni-rich olivine and sodic
plagioclase (Hibiya et al., 2019). Warren et al. (2013) argue the coarse
texture and bulk subchondritic MgO/SiO2 indicate origin as an igneous
cumulate. However, Hibiya et al. (2019) argue that these same textural
features are best explained by a rapid initial crystallization, which agrees
with the apparent absence of geochemical evidence for significant magmatic
differentiation. The Ni-rich phases along with the V/(Al+Cr) ratio in the
spinels suggest formation at relatively oxidizing oxygen fugacity $\Delta{\rm
IW}=+2$, 2 log units above the Iron-Wüstite) buffer (Warren et al., 2013).
Both Hibiya et al. (2019) and Warren et al. (2013) demonstrate the volatile-
depleted nature of NWA 6704/6693. Sanborn et al. (2019) showed the
${\Delta}^{17}$O, ${\epsilon}^{50}$Ti and ${\epsilon}^{54}$Cr isotope
composition of NWA 6704/6693 agree with the CR clan of chondrites.
The Al-Mg systematics of NWA 6704 were measured by Sanborn et al. (2019), who
found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(3.15\pm 0.38)\times
10^{-7}$, the value we adopt, which yields a time of formation $\Delta
t_{26}=5.29\pm 0.13\,{\rm Myr}$.
The Pb-Pb age of NWA 6704 was determined by Amelin et al. (2019), who
regressed 13 pyroxene fraction data with ${}^{206}{\rm Pb}/{}^{204}{\rm
Pb}>700$, excluding one outlier. Based on a measured ${}^{238}{\rm
U}/{}^{235}{\rm U}=137.7784\pm 0.0097$, their regression yielded an age
$4562.76^{+0.22}_{-0.30}\,{\rm Myr}$.
As with NWA 2976, we assume the U isotopic ratio in the pyroxenes is well
represented by the whole rock ratio, and do not apply a correction.
### 2.5 NWA 5697 (L3) Chondrules
The Al-Mg and Pb-Pb systematics were simultaneously measured by Bollard et al.
(2017) and Bollard et al. (2019) for eight chondrules from the carbonaceous
chondrite Allende (CV3) and the ordinary chondrite NWA 5697 (L3). However,
although Bollard et al. (2017) reported Pb-Pb dates for eight chondrules
analyzed for Al-Mg, only four of these (NWA 5697: 2-C1, 3-C5, 5-C2, 11-C1)
were U-corrected using a ${}^{238}{\rm U}/{}^{235}{\rm U}$ measured in the
chondrule; the rest (NWA 5697: 5-C10, C1, C3; and Allende: C30) assumed a bulk
chondrite value ${}^{238}{\rm U}/{}^{235}{\rm U}=137.786\pm 0.013$. Because
some chondrules are resolvably different from this assumed value (e.g., NWA
5697 5-C2 has ${}^{238}{\rm U}/{}^{235}{\rm U}=137.756\pm 0.029$), we do not
assume all chondrules should have this value, and we restrict our analysis
only to those four measured ordinary chondrite chondrules. As well, Bollard et
al. (2017) constructed their isochrons using a subset of the fractions. As
with the case of NWA 1670, we have found it necessary to re-analyze the Pb-Pb
isochrons of these four chondrules. Each of these recalculated Pb-Pb ages of
the chondrules encompasses the values reported by Bollard et al. (2017), but
acknowledges a much greater uncertainty.
Chondrule 2-C1: For this chondrule, Bollard et al. (2017) regressed 11 out of
20 fractions (L3, L4, L6, L8, L9, R, W11-2, L3-2, L4-2, L7-2, L8-2) and
reported an age $4567.57\pm 0.56$ Myr and MSWD $=1.20$. Performing the same
regression, we find a similar $4567.45\pm 0.48$ Myr and MSWD $=1.12$.
Regressing a different subset of 11 fractions (L3, L4, L5, L6, L8, L9, R,
L4-2, L7-2, L8-2, L9-2) yields an age $4567.33\pm 0.48$ Myr and MSWD $=1.88$.
And regressing yet another subset of 11 fractions (L1, L2, L7, L9, R, L2-2,
L3-2, L4-2, L7-2, L8-2, L9-2) yields an age $4567.85\pm 0.50$ Myr and MSWD
$=0.44$. We take the age of chondrule 2-C1 to be $4567.60\pm 0.75\,{\rm Myr}$.
Bollard et al. (2019) found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}=(7.56\pm 0.13)\times 10^{-6}$ (MSWD=1.3), which yields a time of
formation $\Delta t_{26}=2.00\pm 0.21\,{\rm Myr}$.
Chondrule 3-C5: For this chondrule, Bollard et al. (2017) regressed 10 out of
14 fractions (W11, L1, L2, L3, L4, L5, L6, L7, L8, L9) and reported an age
$4566.20\pm 0.63$ Myr and MSWD $=1.27$. Performing the same regression, we
find a similar $4566.13\pm 0.51$ Myr and MSWD $=1.29$. Simply removing point
L8 from the regression, we find an age $4565.54\pm 0.54$ Myr and MSWD $=0.46$.
Regressing instead a different set of nine points (W11, L2, L4, L5, L6, L7,
L8, L11, l12), we find instead an age $4566.57\pm 0.56$ Myr and MSWD $=0.63$.
We take the age of chondrule 3-C5 to be $4566.07\pm 1.07\,{\rm Myr}$. Bollard
et al. (2019) found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(7.04\pm
1.51)\times 10^{-6}$ (MSWD$=0.9$, which yields a time of formation $\Delta
t_{26}=1.84\pm 0.21\,{\rm Myr}$.
Chondrule 5-C2: For this chondrule, Bollard et al. (2017) regressed 8 out of
15 fractions (W11, L3, L4, L6, L7, L8, L9, L10) and reported an age
$4567.54\pm 0.52$ and MSWD $=0.66$, a result we reproduce exactly. Regressing
a subset of 9 fractions (L1, L2, L3, L4, L6, L7, L9, L10, L11) yields an age
$4566.84\pm 0.56$ Myr and a more probable MSWD $=0.94$. It is not clear why
the other points should be rejected. Regressing a subset of eight data points
(W10, W11, L2, L3, L4, L6, L8, L10, L11) yields an age $4567.70\pm 0.49$ Myr
and MSWD $=0.72$. We take the age of chondrule 5-C2 to be $4567.24\pm
0.95\,{\rm Myr}$. Bollard et al. (2019) found $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}=(8.85\pm 1.83)\times 10^{-6}$ (MSWD$=1.8$), which
yields a time of formation after $t\\!\\!=\\!\\!0$ of $\Delta t_{26}=2.07\pm
0.22\,{\rm Myr}$.
Chondrule 11-C1: Finally, for this chondrule, Bollard et al. (2017) regressed
8 out of 15 fractions (W11, L4, L5, L6, L7, L8, L9, L10) and reported an age
$4565.84\pm 0.72$ and MSWD $=1.16$. We find similar but different $4565.64\pm
0.55$ Myr, MSWD $=1.29$. Regressing a subset of eight data points (W11, L2,
L4, L5, L7, L9, L10, L12) yields an age $4565.36\pm 0.59$ Myr and MSWD
$=0.85$. Regressing a different subset of 10 fractions (W11, L2, L4, L5, L6,
L7, L8, L9, L10, L12) yields an age $4565.69\pm 0.56$ Myr and MSWD $=1.77$. We
take the age of chondrule 11-C1 to be $4565.51\pm 0.74\,{\rm Myr}$. Bollard et
al. (2019) found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(5.55\pm
1.84)\times 10^{-6}$ (MSWD$=0.7$), which yields a time of formation $\Delta
t_{26}=2.32\pm 0.34\,{\rm Myr}$.
### 2.6 CAIs
Although the age of $t\\!\\!=\\!\\!0$ is commonly taken to be the Pb-Pb age of
CAIs, the assumption that the Pb-Pb chronometer closed at the same time that
the Al-Mg did (at $t\\!\\!=\\!\\!0$) is untested. Given that CAIs almost
certainly spent millions of years in the same region of the solar nebula where
chondrules formed, it is highly probable they could have been subject to
transient heating events like those that formed chondrules. As discussed above
(§2.2), this could have reset the Pb-Pb chronometer without resetting the Al-
Mg chronometer. With this in mind, we ask whether any of the CAIs from which
Connelly et al. (2012) determined a Pb-Pb age (22E, 31E, 33E, SJ101), actually
show evidence that the Pb-Pb chronometer closed at $t\\!\\!=\\!\\!0$, when
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\approx 5\times 10^{-5}$.
To our knowledge, Al-Mg systematics have not been measured in CAI 33E. Larsen
et al. (2011) measured bulk isotopic ratios in CAIs 22E and 31E, but not an
internal isochron; if a late resetting of the Al-Mg system in these CAIs
occurred, this would not have been detected by bulk measurements.
This leaves only CAI SJ101. Amelin et al. (2010) measured ${}^{238}{\rm
U}/{}^{235}{\rm U}=137.876\pm 0.043$ and an isochron with intercept
$0.625000\pm 0.000092$ (MSWD $=1.07$) and derived a Pb-Pb age of $4567.18\pm
0.50$ Myr for this CAI. Its Al-Mg systematics were measured by MacPherson et
al. (2017), who found $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(5.2\pm
0.5)\times 10^{-5}$. It is widely assumed that the event recorded by the Pb-Pb
chronometer is the same as the one recorded by the Al-Mg chronometer, i.e.,
the CAI’s formation. This in turn rests on the assumption that any thermal
event resetting the Pb-Pb chronometer would cause diffusion of Mg and reset
the Al-Mg chronometer, as well. As discussed in §2.2, however, a typical
chondrule-forming transient heating event would have reset the Pb-Pb
chronometer, and also the Al-Mg system in anorthite, but not in melilite,
pyroxene, or spinel. It is highly significant that the isochron built by
MacPherson et al. (2017) is built from these three minerals, and that it
clearly shows that anorthite is slightly disturbed, with points in anorthite
falling below the isochron. The CAI SJ101 data admit the possibility that it
formed and its Al-Mg chronometer was set at $t\\!\\!=\\!\\!0$, but that it was
later transiently heated to peak temperatures $\approx 1750$ K and cooling
rates $\approx 500\,{\rm K}\,{\rm hr}^{-1}$ like chondrules. It cannot be
known that the Pb-Pb system closed at the same time as the Al-Mg system.
For completeness, we also examine the Pb-Pb isochrons of other CAIs. For the
CAI 22E, Connelly et al. (2012) reported a Pb-Pb age of $4567.35\pm 0.28$ Myr,
based on a regression with MSWD $=0.91$ and measured $\mbox{${}^{238}{\rm
U}/{}^{235}{\rm U}$}=137.627\pm 0.022$. They generated 20 different
leachates/washes/residues in their analysis, but based the intercept on a
regression involving only 11 of these points (L1, L3, L6, L7, L8, L9, L10,
L11, W7, W8, W9). Performing a York regression on these data points, and
propagating the uncertainties in the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratio
and intercept in quadrature, we find a similar age $4567.32\pm 0.27$ Myr, on
an isochron with MSWD $=1.08$. As was the case for NWA 1670, we were able to
find many combinations of 11 data points that yielded acceptable isochrons
(MSWD $<1.94$, the maximum for 11 points; Wendt and Carl 1991). For one set of
11 points (W1, W5, W6, W7, W8, W9, L3, L6, L9, L10, L11), we found MSWD
$=0.29$, and an age $4567.21\pm 0.29$ Myr. For a different set of 11 points
(W1, W5, W6, W7, W9, L1, L3, L4, L6, L7, L8), we found MSWD $=0.77$ and an age
$4567.54\pm 0.31$ Myr. As with the case of NWA 1670, no physical criteria were
laid out for the exclusion of data points. Connelly et al. (2008) argued that
the points they included in the regression eliminated are free of
contamination by terrestrial Pb, on the basis that the regression line extends
to primordial Pb, and they eliminated points below this line (i.e., toward
terrestrial Pb). However, their regression line falls below primordial Pb,
suggesting (by their criteria) that actually all their data have some
terrestrial contamination. Moreover, some included data (e.g., W7, L7) in fact
plot above the regression line. In the absence of independent criteria for
judging the isochrons, we consider all the isochrons described above to be
equally valid, and conclude that the 95% confidence interval for the Pb-Pb age
of CAI 22E should extend from 4566.92 to 4567.85 Myr, i.e., $4567.39\pm 0.47$
Myr. We find basically the same Pb-Pb age for 22E as Connelly et al. (2012),
but find the uncertainty has been underestimated by about a factor of almost
2. We find similar outcomes for CAIs 31E and 32E.
We find no issue with the reported Pb-Pb ages of the other CAIs SJ101, B1, or
B4. It is worth noting that CAI B1 records $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}=(5.03\pm 0.26)\times 10^{-5}$ (Bouvier and Wadhwa,
2010). We conclude that CAIs actually do record Pb-Pb ages over a range of
times from about 4568.2 Myr to 4567.2 Myr, and that their Pb-Pb ages may have
been reset about 1 Myr after they formed.
### 2.7 Summary
In Table 1 we convert the $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$
initial ratios of each sample into a time of formation after $t\\!\\!=\\!\\!0$
assuming an abundance $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm
SS}=5.23\times 10^{-5}$ at $t\\!\\!=\\!\\!0$ and mean-life 1.034 Myr, and we
compile our adopted Pb-Pb ages, $t_{\rm Pb}$, for each achondrite and
chondrule, as well as the implied value of $t_{\rm SS}=t_{\rm Pb}+\Delta
t_{26}$. For the first five achondrites we have applied the 0.19 Myr
correction of Tissot et al. (2017), making the Pb-Pb ages younger. We have not
applied this to the achondrites NWA 2976 or NWA 6704, because we assume their
whole-rock U isotopic ratios reflect the composition of the pyroxene grains in
which the Pb-Pb ages were determined. We have not applied the correction to
chondrules because ${}^{238}{\rm U}/{}^{235}{\rm U}$ was measured in them
directly. The uncertainties in $t_{\rm SS}$ are found by adding the
uncertainties in $t_{\rm Pb}$ and $\Delta t_{26}$ in quadrature.
Table 1: Adopted $\Delta t_{26}$ and Pb-Pb ages ages of seven bulk achondrites and four chondrules, plus the implied value of $t_{\rm SS}=t_{\rm Pb}+\Delta t_{26}$. Sample | $\Delta t_{26}$ (Myr) | $t_{\rm Pb}$ (Myr) | $t_{\rm SS}$ (Myr)
---|---|---|---
D’Orbigny | $5.06\pm 0.10$ | $4563.24\pm 0.21$ | $4568.30\pm 0.23$
SAH 99555 | $5.14\pm 0.05$ | $4563.51\pm 0.24$ | $4568.65\pm 0.25$
NWA 1670 | $4.64\pm 0.10$ | $4564.02\pm 0.66$ | $4568.66\pm 0.67$
Asuka 881394 | $3.82\pm 0.04$ | $4564.76\pm 0.53$ | $4568.58\pm 0.53$
NWA 7325 | $5.33\pm 0.05$ | $4563.71\pm 1.7$ | $4569.04\pm 1.7$
NWA 2796 | $5.03\pm 0.04$ | $4563.17\pm 0.57$ | $4568.20\pm 0.57$
NWA 6704 | $5.29\pm 0.13$ | $4562.76\pm 0.26$ | $4568.05\pm 0.29$
NWA 5697 2-C1 | $2.00\pm 0.21$ | $4567.60\pm 0.75$ | $4569.60\pm 0.78$
NWA 5697 3-C5 | $1.84\pm 0.21$ | $4566.07\pm 1.07$ | $4567.91\pm 1.09$
NWA 5976 5-C2 | $2.07\pm 0.22$ | $4567.24\pm 0.95$ | $4569.31\pm 0.97$
NWA 5697 11-C1 | $2.32\pm 0.34$ | $4565.51\pm 0.74$ | $4567.83\pm 0.81$
These samples all imply values of $t_{\rm SS}$ ranging from 4567.8 and 4569.6
Myr, centered around 4568.7 Myr. The average uncertainty in the Pb-Pb model
ages $t_{\rm SS}$ is 0.8 Myr, due almost entirely to the uncertainties in the
Pb-Pb ages of each sample.
## 3 Determination of $t_{\rm SS}$
### 3.1 Statistical analysis
As seen from Table 1, the majority of achondrites and chondrules imply that
$t\\!\\!=\\!\\!0$ of the Solar System occurred more than 4568 Myr ago. The
best estimate of the Pb-Pb age of $t\\!\\!=\\!\\!0$, $t_{\rm SS}^{*}$, is the
weighted average of the estimates of $t_{{\rm SS},i}$ from each sample
(indexed by $i$):
$t_{\rm
SS}^{*}=\left[\sum_{i=1}^{N}w_{i}\right]^{-1}\,\times\,\sum_{i=1}^{N}\,w_{i}\left[t_{{\rm
Pb},i}+\tau_{26}\,\ln\left(R_{26,{\rm SS}}/R_{26,i}\right)\right],$ (7)
where $R_{26,i}$ and $t_{{\rm Pb},i}$ are the initial ratio
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ and Pb-Pb age in sample $i$,
$w_{i}=1/\sigma_{i}^{2}$,
$\sigma_{t26,i}^{2}=\tau_{26}^{2}(\sigma_{R26,i}/R_{26,i})^{2}$, and
$\sigma_{i}^{2}=\sigma_{t26,i}^{2}+\sigma_{t{\rm Pb},i}^{2}$ accounts for the
uncertainties in the ages. The uncertainty in the weighted mean is
$\left[\sum_{i=1}^{N}w_{i}\right]^{-1/2}$. Although the uncertainties in each
sample’s model age are typically 0.8 Myr, by averaging several estimates of
$t_{\rm SS}$ together, the uncertainty in $t_{\rm SS}$ can be reduced. This
treatment gives the best estimate (and the uncertainty in our estimate) of
$t_{\rm SS}^{*}$.
Whether the ages actually are concordant with that estimate of $t_{\rm
SS}{{}^{*}}$ must be determined by calculating the $z$ scores of each age. We
first determine for each sample $i$ our best estimate of when that achondrite
or chondrule formed, $\Delta t_{i}$, assuming both the Al-Mg and Pb-Pb systems
closed simultaneously. We calculate $\Delta t_{{\rm Pb},i}=t_{\rm
SS}^{*}-t_{{\rm Pb},i}$ and
$\Delta t_{i}=\left(\frac{1}{\sigma_{t26,i}^{2}}+\frac{1}{\sigma_{t{\rm
Pb},i}^{2}}\right)^{-1}\,\left(\frac{\Delta
t_{26,i}}{\sigma_{t26,i}^{2}}+\frac{\Delta t_{{\rm Pb},i}}{\sigma_{t{\rm
Pb},i}^{2}}\right).$ (8)
The $z$ scores measure the departure of the measured time of formation of a
sample from this modeled time of formation, in units of $1\sigma$:
$z_{26,i}=2\frac{\Delta t_{26,i}-\Delta t_{i}}{\sigma_{t26,i}}$ (9)
and
$z_{{\rm Pb},i}=2\frac{\Delta t_{{\rm Pb},i}-\Delta t_{i}}{\sigma_{t{\rm
Pb},i}},$ (10)
where the factor of 2 recognizes that the $\sigma$ are reported as $2\sigma$
errors. If $\left|z_{26,i}\right|>2$ or $\left|z_{{\rm Pb},i}\right|>2$, then
it is improbable ($<5\%$) that that single time of formation $\Delta t_{26,i}$
or $\Delta t_{{\rm Pb},i}$ is being adequately fit by the data.
Of course, in a dataset with $>20$ ages being fit, it would be improbable not
to have at least one age with $\left|z\right|>2$. This is accounted for by
calculating the $\chi_{\nu}^{2}$ or MSWD quantity for the dataset:
$\chi_{\nu}^{2}=\frac{1}{2A-1}\,\sum_{i=1}^{A}\left(z_{26,i}^{2}+z_{{\rm
Pb},i}^{2}\right),$ (11)
where $A$ is the number of samples (achondrites, chondrules, etc.) being
considered, and it is recognized that the number of data being fit is $N=2A$
(two ages for each sample), and the number of degrees of freedom in the
dataset is 1 (only $t_{\rm SS}^{*}$ is being fit). This quantity must not
exceed a critical value $\chi_{\nu,{\rm max}}^{2}\approx 1+2(2/(N-1))^{1/2}$
or else there is a $<5\%$ probability that the dataset as a whole is being
adequately fit by the single value of $t_{\rm SS}^{*}$ (Wendt and Carl, 1991).
### 3.2 Achondrites
We first consider just the seven achondrites as a group, since we are
relatively certain that each achondrite cooled rapidly enough that the Al-Mg
and Pb-Pb systems closed simultaneously and were not reset. We find $t_{\rm
SS}^{*}=4568.39$ Myr. Among this set of 14 times of formation to be fit, we
find that two (the Pb-Pb ages of SAH 99555 and NWA 6704) are barely fit, at
the $2.0\sigma$ and $2.1\sigma$ levels; then again, in a sample of 14
formation times, we would expect 5%, or about 1, to be $>2.0\sigma$
discrepant. Overall, the data are fit very well, with
$\chi_{\nu}^{2}=0.98<\chi_{\nu,{\rm max}}^{2}=1.72$ (48% probability). These
data are concordant. This is our preferred case.
The above analysis assumes a correction of 0.19 Myr applied to Pb-Pb ages of
the five achondrites that did not involve hydrous silicate melts, as suggested
by Tissot et al. (2017) for angrites, assuming $f_{\rm cpx}=0.5$ in NC
achondrites and $f_{\rm cpx}=1$ in CC achondrites. Had we assumed $f_{\rm
cpx}=0$ in the NC achondrites and applied a correction of -0.38 Myr, the fit
would be improved: $t_{\rm SS}^{*}=4568.24$ Myr, with all 14 $z$ scores $<2$,
and $\mbox{$\chi_{\nu}^{2}$}=0.57$. Had we assumed $f_{\rm cpx}=1$ in the NC
achondrites and applied no correction, the fit would be worsened: $t_{\rm
SS}^{*}=4568.52$ Myr, with the Pb-Pb ages of SAH 99555 and NWA 6704 discordant
at the $>2.5\sigma$ and $>2.9\sigma$ levels, and
$\mbox{$\chi_{\nu}^{2}$}=1.63$ (7% probability). If $f_{\rm cpx}<1$ for the CC
achondrites, this would significantly worsen the fits. We do not treat them as
free parameters, but we consider it likely that $f_{\rm cpx}\approx 0.5$ at
most in the NC achondrites and $f_{\rm cpx}\approx 1$ in the CC achondrites,
due to the hydrous nature of the melt. We adopt these values of $f_{\rm cpx}$
and age corrections going forward.
### 3.3 Chondrules
We next consider just the four chondrules. We fit these eight formation times
with $t_{\rm SS}^{*}=4568.76$ Myr, but barely. Two of the Pb-Pb ages have
$z>2.0$, and $\mbox{$\chi_{\nu}^{2}$}=1.94$, barely below the critical value
2.07, indicating a probability $\approx 6\%$ that the chondrule data are
concordant with each other. However, the two values of $t_{\rm SS}^{*}$
implied by the achondrites (4568.39 Myr) and chondrules (4568.76 Myr) are only
0.34 Myr apart, suggesting the achondrites and chondrules might be concordant
with each other.
If we adopt the value of $t_{\rm SS}^{*}=4568.39$ Myr from the achondrites fit
(or 4568.36 Myr from Paper II), the four chondrules are fit well, except for
one anomalous $z$ score, the Pb-Pb formation time of 2-C1 being discordant at
the $3.0\sigma$ level. We find
$\mbox{$\chi_{\nu}^{2}$}=1.35<\mbox{$\chi_{\nu}^{2}$}_{\rm max}=1.56$ (13%
probability).
If we average the seven achondrites and four chondrules together, we find
$t_{\rm SS}^{*}=4568.42$ Myr. Among the 14 achondrite formation times, only
one (the Pb-Pb age of NWA 6704) is off, at the $2.3\sigma$ level. Among the
eight chondrule formation times, only the Pb-Pb age of 2-C1 is off, at the
$2.9\sigma$ level. That is, two out of 22 formation times have $z$ scores
$>2$, which is close the value (1.1) expected statistically. For this
ensemble, $\mbox{$\chi_{\nu}^{2}$}=1.35$, which does not exceed
$\mbox{$\chi_{\nu}^{2}$}_{\rm max}=1.56$ for these 22 data points, and is
significant (probability 12%). This fit is slightly better for the chondrules,
not much worse for the achondrites, and can be considered concordant, although
the Pb-Pb age of 2-C1 is a concern.
We can vary $t_{\rm SS}^{*}$ about this value and ask what range of $t_{\rm
SS}^{*}$ yields $\mbox{$\chi_{\nu}^{2}$}\leq 1.62$. We find the range is
4568.31 to 4568.54 Myr, implying the Pb-Pb age of $t\\!\\!=\\!\\!0$ is roughly
$4568.42\pm 0.12\,{\rm Myr}$.
### 3.4 CAIs
For none of the values of $t_{\rm SS}^{*}$ above is the formation times of CAI
SJ101 concordant with the others. This would require $t_{\rm
SS}^{*}=4567.17\pm 0.51$ Myr, 1.3 Myr younger than our preferred value. For
our preferred $t_{\rm SS}^{*}=4568.42$ Myr, the Pb-Pb $z$ score of SJ101 is
discordant at the $5.2\sigma$ level.
### 3.5 Effects of varying half-lives
These calculations assume a half-life of ${}^{26}{\rm Al}$ of 0.717 Myr, but
the value is not known this precisely. As reviewed by Nishiizumi (2003),
estimates had converged by the 1970s at $\approx 0.72$ Myr, but with some
range. Rightmire et al. (1958) reported $0.738\pm 0.29(1\sigma)$ Myr based on
gamma ray measurements; but this was superseded by the work of Samworth et al.
(1972), who revised the branching ratios and reported $0.716\pm
0.032(1\sigma)$ Myr. Norris et al. (1983) undertook independent gamma ray
spectrometry and found $0.705\pm 0.024(1\sigma)$ Myr. Middleton et al. (1983)
used activity accelerator mass spectrometry to find 0.699 and 0.705 Myr using
two different standards; they recommended $0.702\pm 0.056(1\sigma)$ Myr.
Finally, Thomas et al. (1984) found $0.78\pm 0.05(1\sigma)$ Myr based on a
technique involving the gamma ray cross section. Auer et al. (2009) took the
weighted mean of these values to derive $0.717\pm 0.017(1\sigma)$ Myr, and
this is commonly cited. The value listed in the chart of the nuclides (Walker
et al., 1989), 0.73 Myr, also is commonly cited, but the source of this value
is not clear. We adopt the Auer et al. (2009) value. This is also the value
listed in the compilation of Kondev (2021).
We have repeated the calculations above, varying the half-life of ${}^{26}{\rm
Al}$ across its allowed range $0.717\pm 0.034(2\sigma)$ Myr. For a half-life
of 0.683 Myr, we find $t_{\rm SS}^{*}=4568.17$ Myr,for the $A=7$ samples. For
a half-life of 0.751 Myr, we find $t_{\rm SS}^{*}=4568.63$ Myr. The $2\sigma$
uncertainty of $\pm 0.049$ Myr in the ${}^{26}{\rm Al}$ mean-life translates
into a difference of $\pm 0.22$ Myr in $t_{\rm SS}^{*}$.
Of course the uncertainties in the ${}^{238}{\rm U}$ and ${}^{235}{\rm U}$
half-lives lead to $40\times$ larger uncertainties in the Pb-Pb times of
formation, $\pm 9$ Myr (Tissot et al., 2017). It is understood that the
reported values of $t_{\rm SS}^{*}$ refer to the Pb-Pb age of a sample that
closed at $t\\!\\!=\\!\\!0$, assuming standard half-lives (§1.1).
Figure 5: Times of formation after $t\\!\\!=\\!\\!0$ implied by the Al-Mg
system, $\Delta t_{26}$, and the age determined by the Pb-Pb system, $t_{\rm
Pb}$, for the five NC achondrites (blue), two CC achondrites (purple), four
chondrules (red), and one CAI (orange) in which both are measured. The black
line denotes the locus of points yielding $t_{\rm Pb}+\Delta t_{26}=4568.42$
Myr, the apparent value of $t_{\rm SS}$. All 14 formation times of the 7
achondrites are consistent with that value, except the Pb-Pb age of NWA 6704
is inconsistent at the $2.3\sigma$ level. The achondrite Erg Chech 002 has a
Pb-Pb age (Reger et al., 2023) but two very different reported Al-Mg formation
times (Barrat et al., 2021; Reger et al., 2023); because of this discrepancy
will be the subject of a forthcoming paper. The 8 chondrule formations times
are marginally consistent with that value, except the Pb-Pb age of 2-C1 is
inconsistent at the $2.9\sigma$ level. Still, the overall goodness-of-fit
parameter for the combined 22 ages is $\mbox{$\chi_{\nu}^{2}$}=1.36$, which is
statistically significant (12% probability). The ages are concordant. The
light black line is shifted to the right by 0.19 Myr, corresponding to $t_{\rm
SS}=4568.61$ Myr if the five NC achondrites’ Pb-Pb ages had not been made 0.19
Myr younger to account for the isotopic fractionation of U in pyroxene grains
relative to whole-rock measurements, as suggested by Tissot et al. (2017). In
that scenario, the ages would be barely concordant. CAI SJ101 is not
concordant with the other samples, which we attribute to a late resetting of
the Pb-Pb chronometer without resetting the Al-Mg chronometer (§2.2).
## 4 Discussion
### 4.1 Comparison to other treatments
Our approach is similar to, but distinct from, a few other attempts to
determine the age of the Solar System’s $t\\!\\!=\\!\\!0$ by statistically
correlating Pb-Pb ages against the Al-Mg or other systems.
Lugmair and Shukolyukov (1998) correlated Mn-Cr ages against Pb-Pb ages of
several achondrites to determine $t_{\rm SS}$. They found a range of values
4568 to 4571 Myr, compared to the then-accepted age of CAIs, $4566\pm 2$ Myr
(Göpel et al., 1991), although again none of these Pb-Pb ages was U-corrected.
According to this data, an age 4568 Myr would be consistent with both CAI
formation and $t_{\rm SS}$. These Pb-Pb ages were not as precise as more
modern methods allow, and were not in any event U-corrected.
A different approach was taken by Nyquist et al. (2009) to estimate $t_{\rm
SS}$ (their “$T_{\rm SS}$”). They compiled $\log_{10}(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ and $\log_{10}(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm
Mn}$})_{0}$ ratios, as well as Pb-Pb ages, for several samples: the angrites
D’Orbigny and Sahara (SAH) 99555; the eucrite-like Asuka 881394; ureilites DaG
165 and Dag 319; an estimate for the howardite-eucrite-diogenite (HED) parent
body; and Semarkona chondrules. It is difficult to convert
$(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{0}$ ratios directly into $\Delta
t_{53}$ ages because the initial abundance $(\mbox{${}^{53}{\rm
Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$ and even the half-life of ${}^{53}{\rm Mn}$
are poorly known. Nyquist et al. (2009) did not correlate $\Delta t_{26}$ ages
against Pb-Pb ages $t_{\rm Pb}$, but instead used the Al-Mg system to
calibrate the Mn-Cr system, then found the Mn-Cr time of formation $\Delta
t_{53}$ for one achondrite, Lewis Cliff (LEW) 86010, and used its measured Pb-
Pb age $t_{\rm Pb}$ as an anchor, to derive $t_{\rm SS}=t_{\rm Pb}+\Delta
t_{53}$.
Specifically, Nyquist et al. (2009) regressed the Al-Mg and Mn-Cr data to show
that the quantities were linearly correlated as
$\log_{10}(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{0}=(-5.485\pm 0.028)$
$+(0.23\pm 0.04)\times\left[\log_{10}(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}-(-6.252\pm 0.074)\right]$ (12)
The slope in this linear correlation, $0.23\pm 0.04(2\sigma)$, which should be
the ratio of half-lives, Nyquist et al. (2009) estimated from measurements
should be $0.20\pm 0.02(1\sigma)$. This suggests a ${}^{53}{\rm Mn}$ half-life
$3.12\pm 0.65(2\sigma)$ Myr. This linear correlation was extended to the value
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=5.1\times 10^{-5}$, the value
they assigned to the solar nebula at $t\\!\\!=\\!\\!0$, to find the ratio in
the solar nebula value then, $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{\rm
SS}=9.14\times 10^{-6}$. We note that this implicitly assumes that
${}^{26}{\rm Al}$ and ${}^{53}{\rm Mn}$ were homogeneous (or identically
heterogeneous) between the regions of CAI and achondrite formation. From
there, using the inferred value $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm
Mn}$})_{0}=1.35\times 10^{-6}$ in the achondrite LEW 86010, they calculated
$\Delta t_{53}=10.23$ Myr based on a ${}^{53}{\rm Mn}$ half-life of 3.7 Myr,
with 10% uncertainty ($1\sigma$). Adopting a Pb-Pb age of $4558.55\pm 0.15$
Myr for LEW 86010, Nyquist et al. (2009) then calculated $t_{\rm SS}=4568.8\pm
1.0$ Myr. Nyquist et al. (2009) repeated this calculation using a slope
$0.20\pm 0.02$ in the correlation above, finding $t_{\rm SS}=4568.1\pm 0.6$
Myr. They then took the weighted mean of these two estimates to get their best
estimate of $t_{\rm SS}=4568.2\pm 0.5$ Myr.
It is notable that this value of $t_{\rm SS}$ is much older than the accepted
Pb-Pb ages of CAIs measured today ($4567.30\pm 0.16$ Myr), and even the
majority of Pb-Pb ages of CAIs used in the analysis of Nyquist et al. (2009).
However, there are many caveats that may have prevented adoption of this value
as the age of the Solar System. First, of course, none of the Pb-Pb ages in
their analysis was U-corrected, including those of LEW 86010 or the CAIs. Thus
the value of $t_{\rm SS}$ potentially could be shifted by about $>1$ Myr
(Brennecka and Wadhwa, 2012; Tissot et al., 2017). Second, the calculation of
$t_{\rm SS}$ was done quite indirectly. Since ${}^{26}{\rm Al}$ was implicitly
assumed to be homogeneous anyway, it might have been simpler and more precise
to correlate $\Delta t_{26}$ against Pb-Pb ages directly, as Nyquist et al.
(2009) did in their Figure 3; this, incidentally, would have yielded a value
$t_{\rm SS}\approx 4569.3$ Myr. Third, although some statistical approaches
were employed, ultimately the value of $t_{\rm SS}$ relied on a single anchor,
LEW 86010. This increases the uncertainty in $t_{\rm SS}$ due to the combined
uncertainties in $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$,
$\tau_{53}$, $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{0}$ in LEW 86010,
and the Pb-Pb age of LEW 86010. In fact, the $1\sigma$ uncertainty in the
half-life of ${}^{53}{\rm Mn}$ is $\pm 10\%$ (Honda and Imamura, 1971), and
this dominates the uncertainty in $t_{\rm SS}$ in their treatment, so Nyquist
et al. (2009) should have reported $t_{\rm SS}=4568.2\pm 1.0$ Myr.
In similar fashion, Sanborn et al. (2019), correlated times of formation
derived from Al-Mg systematics, and Pb-Pb ages, for several achondrites and
the CAIs listed above. They were focused on the slope (which yields a
${}^{26}{\rm Al}$ half-life of $0.69\pm 0.12$ Myr), and did not report the
intercept, necessary for finding the Pb-Pb age of $t\\!\\!=\\!\\!0$ when
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})=5.23\times 10^{-5}$. Nevertheless,
it is clear by inspection of their Figure 6 that they would have inferred an
age $4567.8$ Myr, with an uncertainty $\sim 0.5$ Myr, regressing the
achondrite and CAI data. Their regression would appear to be discordant or
just barely concordant with the CAI Pb-Pb ages of Connelly et al. (2012).
More recently, Piralla et al. (2023) have attempted to determine $t_{\rm SS}$
using three approaches. In the first, they averaged the model ages of eight
volcanic achondrites, ($t_{\rm Pb}+\Delta t_{26}$ or $t_{\rm Pb}+\Delta
t_{182}$), like those reported in our Table 1. Through a Monte Carlo approach,
they determined $t_{\rm SS}=4568.50_{-0.99}^{+0.91}\,{\rm Myr}$ using the Al-
Mg ages. Their second approach was to compare Pb-Pb ages and Al-Mg formation
times of their measured spinels and, essentially, anchor to their oldest
chondrule. They found $t_{\rm SS}=4568.36\pm 0.59$ Myr. Their third approach
was to combine a probabilistic approach to their use of the oldest chondrule,
to derive $t_{\rm SS}=4568.67_{-0.59}^{+0.79}\,{\rm Myr}$.
Our approach is similar in some ways to these methods, but is marked by many
innovations. First, we have selected data based on the need to have measured,
U-corrected Pb-Pb ages as well as Al-Mg ages from internal isochrons, for all
samples; we abandon CAIs in favor of achondrites. We have quantitatively
demonstrated the possibility that the Al-Mg and Pb-Pb systems did not close
simultaneously in CAIs that were transiently heated. We have carefully
compiled and vetted the Pb-Pb isochrons to make sure they are being analyzed
in the same fashion, and that they provide robust estimates of both the ages
and the uncertainties. This involved applying examining how points were
included or excluded from the isochron, properly averaging and propagating the
errors when combining multiple analyses, etc. We have applied the likely
corrections of $\delta^{238}{\rm U}\approx-0.19$ Myr to the angrite-like
achondrites, as advocated by Tissot et al. (2017), where geochemically
appropriate. We have directly correlated Pb-Pb ages against Al-Mg ages, as
others have, but most importantly, we have applied statistical techniques to
assess the goodness of fit and concordancy.
### 4.2 Heterogeneity of ${}^{26}{\rm Al}$?
We have calculated an age of $t_{\rm SS}=4568.42\pm 0.24$ Myr for the time at
which the solar nebula was characterized by the canonical ratio
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})=5.23\times 10^{-5}$ (assuming
homogeneity of ${}^{26}{\rm Al}$). In contrast, the ages of CAIs E22, E31,
SJ101, and B4, which are generally considered to record near-canonical
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ ratios (Larsen et al., 2011;
MacPherson et al., 2017; Bouvier and Wadhwa, 2010), are $4567.3\pm 0.2$ Myr
(Connelly et al., 2012) to $4568.2\pm 0.2$ Myr (Bouvier et al., 2011a).
Because a homogeneous amount of ${}^{26}{\rm Al}$ at 4568.42 Myr would have
decayed by a factor of 3.3 by 4567.18 Myr, it has been inferred that
${}^{26}{\rm Al}$ was not homogeneously distributed in the solar system (e.g.,
Larsen et al., 2011).
For example, Bollard et al. (2019) concluded that the reservoir from which
achondrites formed was depleted in ${}^{26}{\rm Al}$ by a factor of 3.8
compared to the CAI-forming region. They reached this conclusion by obtaining
the $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ in eight chondrules,
taking the Pb-Pb ages $t_{\rm Pb}$ inferred for the same chondrules by Bollard
et al. (2017), then extrapolating back to the time of $t\\!\\!=\\!\\!0$ at
$t_{\rm SS}$, to estimate the ${}^{26}{\rm Al}/{}^{27}{\rm Al}$ of the
material comprising each chondrule:
$\left(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$}\right)_{t=0}=(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm
Al}$})_{0}\,\exp\left(+(t_{\rm SS}-t_{\rm Pb})/\tau_{26}\right).$ (13)
Taking $t_{\rm SS}=4567.3$ Myr, they inferred values of $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{t=0}$ ranging from $0.4\times 10^{-5}$ to $2.7\times
10^{-5}$. The average was $1.36\times 10^{-5}$, with hints of a bimodal
distribution: the four lowest values averaged to $5\times 10^{-6}$, and the
four highest values to $1.8\times 10^{-5}$. They concluded that ${}^{26}{\rm
Al}$ was heterogeneous.
However, if one instead assumes $t_{\rm SS}=4568.42$ Myr, then the CAIs must
have taken 1.24 Myr longer to form than previously thought, and their
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{t=0}$ values should be multiplied
by $\exp(+1.24/1.034)=3.3$. In that case, the average value of
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{t=0}$ implied by the eight
chondrules would be $4.5\times 10^{-5}$, very close to the canonical value.
Conclusions of ${}^{26}{\rm Al}$ heterogeneity based on discordancy between
the Al-Mg and Pb-Pb chronometers ultimately derive from the assumption that
the Pb-Pb system closed at $t\\!\\!=\\!\\!0$, i.e., at the same time as the
Al-Mg system. As we have demonstrated, the 22 Al-Mg and Pb-Pb ages of seven
achondrites and four chondrules are made concordant in a statistical sense
with a single value of $t_{\rm SS}$. It could have been the case that no
single value of $t_{\rm SS}$ could make the samples, especially the
achondrites, concordant; this would have falsified the hypothesis of
${}^{26}{\rm Al}$ homogeneity. Instead, the concordance of the data strongly
suggest that ${}^{26}{\rm Al}$ was homogeneous and that CAI SJ101 and the
other CAIs are the anomaly. Given that transient heating events like those
experienced by chondrules can reset the Pb-Pb chronometer without resetting
the Al-Mg system, we consider CAIs to be unreliable testers of the homogeneity
hypothesis.
The consistency of the data with homogeneity of ${}^{26}{\rm Al}$ is exactly
what is expected from astrophysical models, which do not generally predict
conditions in which the CAI-forming region would have $3-4$ times the level of
${}^{26}{\rm Al}$ as other reservoirs. Such large variations in ${}^{26}{\rm
Al}/{}^{27}{\rm Al}$ would not be expected in the molecular cloud, as
molecular cloud cores take several Myr to collapse, during which time
turbulent diffusion will readily mix materials across several parsecs (Pan et
al., 2012), an expectation borne out by the surprising chemical similarity of
stars born in the same cluster (Feng and Krumholz, 2014; Armillotta et al.,
2018). On the scale of a molecular cloud core, $<1$ pc across, there should be
no differences in the composition of the accreting material. Any spatial
variations that did exist across a molecular cloud core would be further mixed
during collapse (Kuffmeier et al., 2017). Models have been proposed in which
the infalling material varied over time; for example, to explain stable
isotope anomalies, Nanne et al. (2019) proposed that early-accreted material
might be richer in supernova-derived material than later-accreted material.
But in such models, heterogeneity in the molecular cloud is merely assumed or
asserted, not demonstrated. Moreover, as in the model of (Nanne et al., 2019),
increased ${}^{26}{\rm Al}$ in the inner disk would require the cloud core
interior to contain stellar ejecta, but not its exterior, an unlikely
scenario.
Because heterogeneities of ${}^{26}{\rm Al}$ cannot be inherited from the
molecular cloud, they would have to result from late injections of
${}^{26}{\rm Al}$-bearing material from supernovas or Wolf-Rayet stars, into
the protoplanetary disk. Late injection has long been invoked as an
explanation for why some hibonite-dominated grains apparently lacked
${}^{26}{\rm Al}$: presumably they formed before ${}^{26}{\rm Al}$ was
injected (Sahijpal and Goswami, 1998). However, the fact that only certain
rare corundum- or hibonite-dominated inclusions exhibit a lack of ${}^{26}{\rm
Al}$ (e.g., Krot et al. 2012) more strongly suggests a chemical heterogeneity,
as suggested by (Larsen et al., 2020), and not spatial or temporal variations.
As demonstrated by Ouellette et al. (2007), it is possible to inject dust
grains from stellar ejecta into a protoplanetary disk; but it is highly
improbable for a disk to form close enough to a stellar source to receive
relevant amounts of ejecta (Ouellette et al., 2010). Although it has not been
quantitatively explored whether CAIs could form with $4\times$ more
${}^{26}{\rm Al}$ than the rest of the disk, a simple argument suggests CAIs
should form with less ${}^{26}{\rm Al}$, not more: over the scale of the disk,
the stellar ejecta injects a uniform mass of ${}^{26}{\rm Al}$ per area; but
that injected material is more diluted by greater surface densities in the
inner disk where CAIs form.
Approaching the problem from a different angle, the abundances of about a
dozen short-lived radionuclides in the solar nebula are very successfully
explained as being inherited from the molecular cloud (contaminated by
supernovae and Wolf-Rayet winds), and not through late injection or
irradiation in the solar nebula (Young, 2020). This demands these isotopes,
including ${}^{26}{\rm Al}$, would have been spatially well mixed in the solar
nebula from before $t\\!\\!=\\!\\!0$.
Ascribing ${}^{26}{\rm Al}$ heterogeneities to something to do with star
formation is too simplistic. Astrophysical models overall suggest ${}^{26}{\rm
Al}$ should have been homogeneous in the Sun’s protoplanetary disk, and that
the discrepancy between the Al-Mg and Pb-Pb chronometers has to do with the
Pb-Pb chronometer itself.
### 4.3 Relative vs. Absolute Ages
The determination of $t_{\rm SS}=4568.42\pm 0.24$ Myr allows us to use the Pb-
Pb data available for a number of achondrites, and unlock the potential of the
Pb-Pb system as a relative chronometer. Relative ages are much more precise
than absolute ages: absolute ages are uncertain to within $\pm 9$ Myr
($2\sigma$) due to the uncertainties in the half-lives of ${}^{235}{\rm U}$
and ${}^{238}{\rm U}$ (Amelin, 2006; Tissot et al., 2017). These systematic
uncertainties cancel when using the Pb-Pb system to calculate relative ages,
and the precision can approach $\pm 0.3$ Myr, a factor of 30 more precise.
Even ignoring the half-life uncertainties and comparing to the measurement
uncertainties, the uncertainties in times of formation after $t\\!\\!=\\!\\!0$
are typically 0.13 Myr, 5 times better than the uncertainties in the model
ages, which are typically 0.75 Myr.
Relative ages are also much more relevant to models of the protoplanetary
disk. All astrophysical models of planet formation desperately need
quantification of the order of events in the solar nebula, relative to a
commonly accepted event at a defined $t\\!\\!=\\!\\!0$. These could then be
compared to astronomical constraints on the ages of young stellar objects, for
which the timing of evolutionary stages is typically precise to within $<1$
Myr (Haisch et al., 2001; Mamajek, 2009). In contrast, there is almost no
model of stellar or planetary evolution, that uses information of how long ago
the protoplanetary disk stage was, that would be affected if it turned out the
Solar System were 4560 Myr or 4580 Myr old instead of 4569 Myr old. For
example, the tightest astrophysical constraints from helioseismology on the
Sun’s age are $4600\pm 40(1\sigma)$ Myr (Houdek and Gough, 2011), and $4587\pm
7$ Myr or $4569\pm 6(1\sigma)$ Myr, based on different databases of physical
quantities (Bonanno and Fröhlich, 2015). For these allied fields of study,
“4.57 Gyr ago” is precise enough.
In addition, the need to choose anchors to report absolute ages introduces
confusion and imprecision, especially to those not working in the meteoritics
community, as a recent example from the literature illustrates. The two
achondrites NWA 11119 and Erg Chech 002 are each among the most ancient
crustal rocks in the meteoritic record. Which formed first? Srinivasan et al.
(2018) reported $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(1.69\pm
0.09)\times 10^{-6}$ in NWA 11119, while Barrat et al. (2021) reported
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}=(5.72\pm 0.07)\times 10^{-6}$
for Erg Chech 002. Assuming they both formed from material sampling the
canonical $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})$ ratio, then NWA 11119
formed at $\Delta t_{26}=3.55\pm 0.06$ Myr, and Erg Chech 002 at $\Delta
t_{26}=2.29\pm 0.01$ Myr. If these values are correct and if these two
achondrites formed from a common reservoir, then clearly NWA 11119 formed a
time $\Delta t=\tau_{26}\,\ln\left[(5.72\pm 0.07)/(1.69\pm 0.09)\right]$
$=1.26\pm 0.08$ Myr after Erg Chech 002. However, the only dating information
reported by Srinivasan et al. (2018) in the abstract of their paper is that
NWA 11119 has an absolute age of $4564.8\pm 0.3$ Myr. Likewise, the only
dating information Barrat et al. (2021) reported in the abstract of their
paper was that Erg Chech 002 has an absolute age $4565.0$ (presumably $\pm
0.3$) Myr. Based on these headline ages, the age difference is $0.2\pm 0.4$
Myr, which is clearly incompatible with an age difference of $1.26\pm 0.08$
Myr.
Only a very careful reading of the papers will clear up the discrepancy. The
age $4564.8\pm 0.3$ Myr for NWA 11119 was derived by Srinivasan et al. (2018)
by anchoring to Al-Mg and Pb-Pb ages of D’Orbigny, although at no point did
they explicitly state the $(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}$ and
Pb-Pb age of D’Orbigny they were assuming. In contrast, the $4565.0$ Myr for
Erg Chech 002 was derived by Barrat et al. (2021) by anchoring to CAIs,
implicitly assuming their Pb-Pb age is 4567.3 Myr, although at no point did
they state the assumed Pb-Pb age of CAIs, recognize the collateral assumption
that ${}^{26}{\rm Al}$ would have to be heterogeneous, or even cite the source
(Connelly et al., 2012) of these assumptions. So many assumptions are
introduced when using anchors that the entire meaning of a sample age becomes
opaque. Chronometry papers are replete with phrases like “if anchored to
D’Orbigny, then… but if anchored to CAIs, then…”. This is confusing and not
even necessary, as the quantity of greatest interest is the time of formation
after $t\\!\\!=\\!\\!0$. Moreover, the use of absolute ages necessarily
introduces unneeded uncertainty, because of the uncertainties in even Pb-Pb
relative ages, $\pm 0.3$ Myr, or $\pm 0.7$ Myr in the example of NWA 6704
above. Had the formation times of NWA 11119 and Erg Chech 002 been reported as
times after $t\\!\\!=\\!\\!0$, the precision in the difference in their
formation times would have been $\pm 0.08$, a factor of 4 more precise than
could be obtained using the ages reported for them.
For all these reasons, we strongly advocate reporting all ages, even Pb-Pb
ages, as times of formation relative to $t\\!\\!=\\!\\!0$, avoiding the use of
anchors, and stating all assumptions clearly. This means reporting the assumed
value of $t_{\rm SS}$ for Pb-Pb dating, or the value of $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{\rm SS}$ or even $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm
Mn}$})_{\rm SS}$ that was assumed. We strongly advocate avoiding use of
anchors to report absolute ages.
### 4.4 Improved Chronometry
Fixing $t_{\rm SS}=4568.42\pm 0.24$ Myr allows us to use the Pb-Pb system as a
relative chronometer, in concert with the Al-Mg chronometer, improving the
precision of dating. In Table 2 we present for various meteoritic samples: our
calculated time of formation after $t\\!\\!=\\!\\!0$ based on the Al-Mg
system, $\Delta t_{26}$; our calculated time time of formation based on the
Pb-Pb system and $t_{\rm SS}=4568.42$ Myr, $\Delta t_{\rm Pb}$; and the
weighted mean of these, $\Delta t$. To connect to the broader literature, we
also list the implied absolute age, based on $t_{\rm SS}=4568.42$ Myr (and the
assumption of commonly used U half-lives), but we emphasize again that the
relative ages are what matter for chronometry. The uncertainties reflect only
the uncertainty in the Al-Mg ages, as is appropriate when taking the
differences in model ages. We also note that in a companion paper (Desch et
al. 2023, hereafter Paper II) we update these estimates using information from
the Mn-Cr and Hf-W systems, and many more achondrites; these shift the
inferred times of formation $\Delta t$ and model ages, by about 0.07 Myr.
Table 2: Adopted times of formation $\Delta t_{26}$ and $\Delta t_{\rm Pb}$ (assuming $t_{\rm SS}=4568.42$ Myr), and weighted mean, $\Delta t$, plus implied Pb-Pb age, of seven bulk achondrites and four chondrules. Sample | $\Delta t_{26}$ (Myr) | $\Delta t_{\rm Pb}$ (Myr) | $\Delta t$ (Myr) | Model age
---|---|---|---|---
D’Orbigny | $5.06\pm 0.10$ | $4.99\pm 0.21$ | $5.05\pm 0.09$ | $4563.37\pm 0.09$
SAH 99555 | $5.14\pm 0.05$ | $4.72\pm 0.24$ | $5.12\pm 0.05$ | $4563.30\pm 0.05$
NWA 1670 | $4.64\pm 0.10$ | $4.21\pm 0.66$ | $4.63\pm 0.10$ | $4563.79\pm 0.10$
Asuka 881394 | $3.82\pm 0.04$ | $3.47\pm 0.53$ | $3.82\pm 0.04$ | $4564.60\pm 0.04$
NWA 7325 | $5.33\pm 0.05$ | $4.52\pm 1.70$ | $5.33\pm 0.05$ | $4563.09\pm 0.05$
NWA 2796 | $5.03\pm 0.04$ | $5.26\pm 0.57$ | $5.03\pm 0.04$ | $4563.39\pm 0.04$
NWA 6704 | $5.29\pm 0.13$ | $5.66\pm 0.26$ | $5.36\pm 0.11$ | $4563.13\pm 0.13$
NWA 5697 2-C1 | $2.00\pm 0.21$ | $0.82\pm 0.75$ | $1.91\pm 0.20$ | $4566.51\pm 0.20$
NWA 5697 3-C5 | $1.84\pm 0.21$ | $2.35\pm 1.07$ | $1.86\pm 0.21$ | $4566.56\pm 0.21$
NWA 5976 5-C2 | $2.07\pm 0.22$ | $1.18\pm 0.95$ | $2.03\pm 0.21$ | $4566.39\pm 0.21$
NWA 5697 11-C1 | $2.32\pm 0.34$ | $2.91\pm 0.74$ | $2.42\pm 0.31$ | $4566.00\pm 0.31$
Having a precise and accurate value for $t_{\rm SS}$ obtained by averaging of
multiple samples allows improved chronometry across the board, not just for
the samples above with Al-Mg dates, but also for samples with Mn-Cr
measurements. For example, if a sample has a Pb-Pb age $t_{\rm Pb}$ and a
measurement of $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{0}$, one could
extrapolate backward to find the implied value of $(\mbox{${}^{53}{\rm
Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$ at $t\\!\\!=\\!\\!0$ in the solar nebula:
$(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{\rm SS}=(\mbox{${}^{53}{\rm
Mn}/{}^{55}{\rm Mn}$})_{0}\,\exp\left(+\Delta t_{Pb}/\tau_{53}\right).$ (14)
Even samples formed too late to have Al-Mg ages could be used to estimate
$(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$, as Nyquist et al.
(2009) did using LEW 86010. However, we advocate using averages of many such
samples to derive $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$. Once
it is derived, a single measurement of $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm
Mn}$})_{0}$ in a sample, even without a Al-Mg or Pb-Pb date, would suffice to
date the formation time of the sample. Deriving such quantities is one goal of
the companion Paper II (Desch et al., 2023).
## 5 Conclusions
We have correlated Pb-Pb ages and Al-Mg times of formation of seven
achondrites and four chondrules for which both $(\mbox{${}^{26}{\rm
Al}/{}^{27}{\rm Al}$})_{0}$ and U-corrected Pb-Pb ages $t_{\rm Pb}$ exist. We
define $t\\!\\!=\\!\\!0$ to be the time in the solar nebula when
$(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})=5.23\times 10^{-5}$ (assuming
homogeneity of ${}^{26}{\rm Al}$). The time of formation after
$t\\!\\!=\\!\\!0$ is then $\Delta
t_{26}=\tau_{26}\,\ln\left[(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{\rm
SS}/(\mbox{${}^{26}{\rm Al}/{}^{27}{\rm Al}$})_{0}\right]$, where
$\tau_{26}=1.034$ Myr is the mean-life of ${}^{26}{\rm Al}$. The time of
formation according to the Pb-Pb system is $\Delta t_{\rm Pb}=t_{\rm
SS}-t_{\rm Pb}$, where $t_{\rm SS}$ is the Pb-Pb age of samples that achieved
isotopic closure at $t\\!\\!=\\!\\!0$, using standard half-lives for
${}^{235}{\rm U}$ and ${}^{238}{\rm U}$. The achondrites rapidly cooled and
should have achieved simultaneous isotopic closure in both systems. In these
samples in particular, the times of formation $\Delta t_{26}$ and $\Delta
t_{\rm Pb}$ should match if ${}^{26}{\rm Al}$ was homogeneous. These samples
therefore test the hypothesis of homogeneity. Specifically, if a single value
of $t_{\rm SS}$ does not reconcile (in a statistical sense) the $\Delta
t_{26}$ and $\Delta t_{\rm Pb}$ formation times for the achondrites, this
would falsify the homogeneous hypothesis.
We find that the hypothesis of ${}^{26}{\rm Al}$ homogeneity is not falsified.
The seven achondrites’ ages are reconciled by $t_{\rm SS}=4568.42\pm 0.24$
Myr. Thirteen of their 14 formation times are concordant at the $<2\sigma$
level, and one (the Pb-Pb age of NWA 6704) is discrepant at the $2.4\sigma$
level, consistent with the scatter being due only to measurement errors.
Likewise, the goodness-of-fit parameter for the ensemble is a statistically
significant $\mbox{$\chi_{\nu}^{2}$}=0.98$ (47% probability). This supports
the assumption that ${}^{26}{\rm Al}$ was homogeneous.
Because transient heating events might have reset one chronometer in
chondrules but not the other, chondrules do not necessarily test homogeneity.
Nevertheless, they also appear concordant. Combining them with the seven
achondrites, we find that the 22 ages are reconciled for values of $t_{\rm
SS}=4568.42\pm 0.24$ Myr. Only two of the 22 ages are discordant: the Pb-Pb
ages of NWA 6704 (at the $2.3\sigma$ level) and chondrule 2-C1 (at the
$2.9\sigma$ level). Still, $\mbox{$\chi_{\nu}^{2}$}=1.36$ for this ensemble,
which is statistically significant (12% probability). Despite the spread in
the chondrule ages, they also appear concordant, further supporting the
assumption of homogeneity.
There is only a single CAI, SJ101, for which a U-corrected Pb-Pb age and an
internal Al-Mg isochron both exist. These ages are quite discrepant. Although
this has been interpreted in the past as heterogeneity of ${}^{26}{\rm Al}$,
we have demonstrated that the underlying assumption—that the Al-Mg and Pb-Pb
systems closed at the same time—is not justified. If this CAI were subjected
to a transient heating event in the solar nebula like those that chondrules
experienced, then the Pb-Pb system would be reset, the Al-Mg system in
anorthite would be marginally reset, and the Al-Mg system in pyroxene,
melilite and spinel would not be modified. The Al-Mg system could record the
time of the CAI’s formation at $t\approx 0$, while the Pb-Pb system would
record a time 1-3 Myr later. The isochron of SJ101 is exactly consistent with
this behavior, making it untenable to argue that the Al-Mg and Pb-Pb systems
had to close simultaneously, or that CAIs’ Pb-Pb chronometers record
$t\\!\\!=\\!\\!0$.
As part of our analysis, we reevaluated Pb-Pb ages of all the samples. We
performed the regressions as a check, and averaged the intercepts of the Pb-Pb
isochrons and averaged the ${}^{238}{\rm U}/{}^{235}{\rm U}$ ratios to
reevaluate the Pb-Pb ages, where appropriate. We discovered that the isochrons
built by Connelly et al. (2008) [SAH 99555], Connelly et al. (2012) [CAIs],
Schiller et al. (2015) [NWA 1670], and Bollard et al. (2017) [NWA 5697
chondrules] employed a methodology for selecting points to regress that is
prone to confirmation bias, leading to ages that are overly precise. We have
reevaluated the uncertainties in these ages. We have also applied the isotopic
correction factor $\delta^{238}{\rm
U}\approx-0.3\mbox{\text{\textperthousand}}$ advocated by Tissot et al.
(2017), to account for the fact that pyroxene grains (from which Pb-Pb
isochrons are typically built) are isotopically lighter than uranium in the
usually-measured whole rock (in which uranium isotopes are usually measured),
leading to Pb-Pb ages being overestimated by about 0.19 Myr. We argue that
this correction should not be applied to the achondrites NWA 2976 and NWA 6704
from the CC isotopic reservoir, due to them forming from hydrous magmas.
The difficulty of measurements of Pb-Pb ages in individual CAIs, and the
controversy over the results, makes clear that statistical approaches like
that of Nyquist et al. (2009), Sanborn et al. (2019), Piralla et al. (2023),
and the approach taken here are preferred. Averages of many samples are more
reliable and lead to greater precision, than use of single anchors. While we
find $t_{\rm SS}=4568.42\pm 0.24$ Myr from the samples above, we encourage
further measurements of Al-Mg and Pb-Pb dates for many more systems for which
the Al-Mg and Pb-Pb systems are likely to have achieved isotopic closure
simultaneously, especially volcanic achondrites. This will severely test the
prediction of homogeneity, but if ${}^{26}{\rm Al}$ was homogeneous then we
expect $t_{\rm SS}$ to be determined even more precisely as more measurements
are made.
We advocate a move away from reporting formation times as absolute ages
determined by anchoring to samples. Use of anchors introduces a number of
assumptions about the ages, which are rarely stated explicitly in papers. At
any rate, absolute ages are not needed for models of protoplanetary disks and
planet formation, or for most purposes; formation times $\Delta t$ relative to
$t\\!\\!=\\!\\!0$, or time differences between two events, are demanded.
Relative ages are much more precise than Pb-Pb absolute ages, which have
uncertainties of $\pm 9$ Myr due to uncertainties in the ${}^{235}{\rm U}$
half-life. The real utility of the Pb-Pb system is as a relative dating
system, with precision $\pm 0.3$ Myr. This requires fixing $t_{\rm SS}$ to
define $\Delta t_{\rm Pb}=t_{\rm SS}-t_{\rm Pb}$. We hope that our finding of
$t_{\rm SS}=4568.42\pm 0.24$ Myr, will allow precise relative dating of
samples using the Pb-Pb system.
This increased precision should open the door to more precise chronometry
using other isotopic systems. In the companion Paper II (Desch et al., 2023),
we refine $t_{\rm SS}$ and show how knowledge of $t_{\rm SS}$ allows us to use
statistical approaches to better determine $(\mbox{${}^{53}{\rm
Mn}/{}^{55}{\rm Mn}$})_{\rm SS}$ and $(\mbox{${}^{182}{\rm Hf}/{}^{180}{\rm
Hf}$})_{\rm SS}$ at $t\\!\\!=\\!\\!0$ in the solar nebula, despite the
difficulties of measuring these quantities in CAIs directly. This allows a
measurement of $(\mbox{${}^{53}{\rm Mn}/{}^{55}{\rm Mn}$})_{0}$ or
$(\mbox{${}^{182}{\rm Hf}/{}^{180}{\rm Hf}$})_{0}$ in a sample to be converted
into a time of formation directly, without the need for anchors or absolute
ages.
Acknowledgments: The authors would like to acknowledge the efforts of
cosmochemists from multiple laboratories around the world whose work makes
possible the data cited in Table 1 and throughout this paper. Statistical
chronometry necessarily distills very difficult and painstaking analytical
work into mere numbers to be crunched, but the efforts to obtain those numbers
are appreciated. We thank Zachary Torrano for useful discussions. We thank
Francois Tissot and two other anonymous reviewers, whose suggestions greatly
improved the quality of our work. The work herein benefitted from
collaborations and/or information exchange within NASA’s Nexus for
Exoplanetary System Science research coordination network sponsored by NASA’s
Space Mission Directorate (grant NNX15AD53G, PI Steve Desch). Emilie Dunham
gratefully acknolwedges support from a 51 Pegasi b Fellowship, grant
#2020-1829.
The data in Table 1 and the calculations by which we derived our results are
included as an Excel spreadsheet as Research Data.
## References
* Amelin (2006) Amelin, Y., 2006. The prospect of high-precision Pb isotopic dating of meteorites. Meteoritics and Planetary Science 41, 7–17. doi:10.1111/j.1945-5100.2006.tb00189.x.
* Amelin (2008a) Amelin, Y., 2008a. The U Pb systematics of angrite Sahara 99555. Geochimica et Cosmochimica Acta 72, 4874–4885. doi:10.1016/j.gca.2008.07.008.
* Amelin (2008b) Amelin, Y., 2008b. U Pb ages of angrites. Geochimica et Cosmochimica Acta 72, 221–232. doi:10.1016/j.gca.2007.09.034.
* Amelin et al. (2010) Amelin, Y., Kaltenbach, A., Iizuka, T., Stirling, C.H., Ireland, T.R., Petaev, M., Jacobsen, S.B., 2010. U-Pb chronology of the Solar System’s oldest solids with variable 238U/ 235U. Earth and Planetary Science Letters 300, 343–350. doi:10.1016/j.epsl.2010.10.015.
* Amelin et al. (2019) Amelin, Y., Koefoed, P., Iizuka, T., Fernandes, V.A., Huyskens, M.H., Yin, Q.Z., Irving, A.J., 2019. U-Pb, Rb-Sr and Ar-Ar systematics of the ungrouped achondrites Northwest Africa 6704 and Northwest Africa 6693. Geochimica et Cosmochimica Acta 245, 628–642. doi:10.1016/j.gca.2018.09.021.
* Armillotta et al. (2018) Armillotta, L., Krumholz, M.R., Fujimoto, Y., 2018. Mixing of metals during star cluster formation: statistics and implications for chemical tagging. Monthly Notices of the Royal Astronomical Society 481, 5000–5013. doi:10.1093/mnras/sty2625, arXiv:1807.01712.
* Auer et al. (2009) Auer, M., Wagenbach, D., Wild, E.M., Wallner, A., Priller, A., Miller, H., Schlosser, C., Kutschera, W., 2009\. Cosmogenic 26Al in the atmosphere and the prospect of a 26Al/ 10Be chronometer to date old ice. Earth and Planetary Science Letters 287, 453–462. doi:10.1016/j.epsl.2009.08.030.
* Barrat et al. (2021) Barrat, J.A., Chaussidon, M., Yamaguchi, A., Beck, P., Villeneuve, J., Byrne, D.J., Broadley, M.W., Marty, B., 2021\. A 4,565-My-old andesite from an extinct chondritic protoplanet. Proceedings of the National Academy of Science 118, 2026129118. doi:10.1073/pnas.2026129118, arXiv:2105.01911.
* Barrat et al. (2015) Barrat, J.A., Greenwood, R.C., Verchovsky, A.B., Gillet, P., Bollinger, C., Langlade, J.A., Liorzou, C., Franchi, I.A., 2015\. Crustal differentiation in the early solar system: Clues from the unique achondrite Northwest Africa 7325 (NWA 7325). Geochimica et Cosmochimica Acta 168, 280–292. doi:10.1016/j.gca.2015.07.020.
* Bollard et al. (2017) Bollard, J., Connelly, J.N., Whitehouse, M.J., Pringle, E.A., Bonal, L., Jørgensen, J.K., Nordlund, Å., Moynier, F., Bizzarro, M., 2017. Early formation of planetary building blocks inferred from Pb isotopic ages of chondrules. Science Advances 3, e1700407. doi:10.1126/sciadv.1700407, arXiv:1708.02631.
* Bollard et al. (2019) Bollard, J., Kawasaki, N., Sakamoto, N., Olsen, M., Itoh, S., Larsen, K., Wielandt, D., Schiller, M., Connelly, J.N., Yurimoto, H., Bizzarro, M., 2019. Combined U-corrected Pb-Pb dating and 26Al-26Mg systematics of individual chondrules - Evidence for a reduced initial abundance of 26Al amongst inner Solar System chondrules. Geochimica et Cosmochimica Acta 260, 62–83. doi:10.1016/j.gca.2019.06.025.
* Bonanno and Fröhlich (2015) Bonanno, A., Fröhlich, H.E., 2015\. A Bayesian estimation of the helioseismic solar age. Astronomy and Astrophysics 580, A130. doi:10.1051/0004-6361/201526419, arXiv:1507.05847.
* Bouvier et al. (2011a) Bouvier, A., Brennecka, G.A., Wadhwa, M., 2011a. Absolute Chronology of the First Solids in the Solar System, in: Workshop on Formation of the First Solids in the Solar System, p. 9054.
* Bouvier et al. (2011b) Bouvier, A., Spivak-Birndorf, L.J., Brennecka, G.A., Wadhwa, M., 2011b. New constraints on early Solar System chronology from Al-Mg and U-Pb isotope systematics in the unique basaltic achondrite Northwest Africa 2976. Geochimica et Cosmochimica Acta 75, 5310–5323. doi:10.1016/j.gca.2011.06.033.
* Bouvier and Wadhwa (2010) Bouvier, A., Wadhwa, M., 2010\. The age of the Solar System redefined by the oldest Pb-Pb age of a meteoritic inclusion. Nature Geoscience 3, 637–641. doi:10.1038/ngeo941.
* Brennecka and Wadhwa (2012) Brennecka, G.A., Wadhwa, M., 2012\. Uranium isotope compositions of the basaltic angrite meteorites and the chronological implications for the early Solar System. Proceedings of the National Academy of Science 109, 9299–9303. doi:10.1073/pnas.1114043109.
* Brennecka et al. (2010) Brennecka, G.A., Weyer, S., Wadhwa, M., Janney, P.E., Zipfel, J., Anbar, A.D., 2010\. 238U/235U Variations in Meteorites: Extant 247Cm and Implications for Pb-Pb Dating. Science 327, 449\. doi:10.1126/science.1180871.
* Brett et al. (1977) Brett, R., Huebner, J.S., Sato, M., 1977. Measured oxygen fugacities of the Angra dos Reis achondrite as a function of temperature. Earth and Planetary Science Letters 35, 363–368. doi:10.1016/0012-821X(77)90139-X.
* Cartwright et al. (2016) Cartwright, J.A., Amelin, Y., Koefoed, P., Wadhwa, M., 2016\. U-Pb Age of the Ungrouped Achondrite NWA 8486, in: LPI Editorial Board (Ed.), 79th Annual Meeting of the Meteoritical Society, p. 6231.
* Cherniak (1998) Cherniak, D.J., 1998. Pb diffusion in clinopyroxene. Chemical Geology 150, 105–117. doi:10.1016/S0009-2541(98)00056-4.
* Chevreux et al. (2021) Chevreux, P., Tissandier, L., Laplace, A., Vitova, T., Bahl, S., Guyadec, F.L., Deloule, E., 2021. Uranium solubility and speciation in reductive soda-lime aluminosilicate glass melts. Journal of Nuclear Materials 544, 152666. doi:10.1016/j.jnucmat.2020.152666.
* Connelly et al. (2012) Connelly, J.N., Bizzarro, M., Krot, A.N., Nordlund, Å., Wielandt, D., Ivanova, M.A., 2012\. The Absolute Chronology and Thermal Processing of Solids in the Solar Protoplanetary Disk. Science 338, 651\. doi:10.1126/science.1226919.
* Connelly et al. (2008) Connelly, J.N., Bizzarro, M., Thrane, K., Baker, J.A., 2008\. The Pb Pb age of Angrite SAH99555 revisited. Geochimica et Cosmochimica Acta 72, 4813–4824. doi:10.1016/j.gca.2008.06.007.
* Desch et al. (2023) Desch, S.J., Dunlap, Daniel, R., Dunham, E.T., Williams, C.D., Mane, P., 2023. Statistical Chronometry of Meteorites: II Abundances and Homogeneity of Short-Lived Radionuclides. Geochimica and Cosmochimica Acta submitted, 00. doi:10.3847/1538-4365/aad95f, arXiv:1710.03809.
* Desch et al. (2018) Desch, S.J., Kalyaan, A., O’D. Alexander, C.M., 2018. The Effect of Jupiter’s Formation on the Distribution of Refractory Elements and Inclusions in Meteorites. The Astrophysical Journal Supplement Series 238, 11. doi:10.3847/1538-4365/aad95f, arXiv:1710.03809.
* Desch et al. (2012) Desch, S.J., Morris, M.A., Connolly, H.C., Boss, A.P., 2012\. The importance of experiments: Constraints on chondrule formation models. Meteoritics and Planetary Science 47, 1139–1156. doi:10.1111/j.1945-5100.2012.01357.x.
* Dodson (1973) Dodson, M.H., 1973. Closure temperature in cooling geochronological and petrological systems. Contributions to Mineralogy and Petrology 40, 259–274. doi:10.1007/BF00373790.
* Feng and Krumholz (2014) Feng, Y., Krumholz, M.R., 2014\. Early turbulent mixing as the origin of chemical homogeneity in open star clusters. Nature 513, 523–525. doi:10.1038/nature13662, arXiv:1408.6543.
* Floss et al. (2005) Floss, C., Taylor, L.A., Promprated, P., Rumble, D., I., 2005\. Northwest Africa 011: A “eucritic” basalt from a non-eucrite parent body. Meteoritics and Planetary Science 40, 343. doi:10.1111/j.1945-5100.2005.tb00387.x.
* Glavin et al. (2004) Glavin, D.P., Kubny, A., Jagoutz, E., Lugmair, G.W., 2004\. Mn-Cr isotope systematics of the D’Orbigny angrite. Meteoritics and Planetary Science 39, 693–700. doi:10.1111/j.1945-5100.2004.tb00112.x.
* Goldmann et al. (2015) Goldmann, A., Brennecka, G., Noordmann, J., Weyer, S., Wadhwa, M., 2015. The uranium isotopic composition of the Earth and the Solar System. Geochimica et Cosmochimica Acta 148, 145–158. doi:10.1016/j.gca.2014.09.008.
* Goodrich et al. (2017) Goodrich, C.A., Kita, N.T., Yin, Q.Z., Sanborn, M.E., Williams, C.D., Nakashima, D., Lane, M.D., Boyle, S., 2017\. Petrogenesis and provenance of ungrouped achondrite Northwest Africa 7325 from petrology, trace elements, oxygen, chromium and titanium isotopes, and mid-IR spectroscopy. Geochimica et Cosmochimica Acta 203, 381–403. doi:10.1016/j.gca.2016.12.021.
* Göpel et al. (1991) Göpel, C., Manhes, G., Allegre, C.J., 1991. Constraints on the time of accretion and thermal evolution of chondrite parent bodies by precise U-Pb dating of phosphates. Meteoritics 26, 338\.
* Gopel et al. (1994) Gopel, C., Manhes, G., Allegre, C.J., 1994. U-Pb systematics of phosphates from equilibrated ordinary chondrites. Earth and Planetary Science Letters 121, 153–171. doi:10.1016/0012-821X(94)90038-8.
* Greenwood and Hess (1996) Greenwood, J.P., Hess, P.C., 1996\. Congruent melting kinetics: constraints on chondrule formation., in: Chondrules and the Protoplanetary Disk, pp. 205–211.
* Haisch et al. (2001) Haisch, Karl E., J., Lada, E.A., Lada, C.J., 2001. Disk Frequencies and Lifetimes in Young Clusters. The Astrophysical Journal 553, L153–L156. doi:10.1086/320685, arXiv:astro-ph/0104347.
* Hibiya et al. (2019) Hibiya, Y., Archer, G.J., Tanaka, R., Sanborn, M.E., Sato, Y., Iizuka, T., Ozawa, K., Walker, R.J., Yamaguchi, A., Yin, Q.Z., Nakamura, T., Irving, A.J., 2019\. The origin of the unique achondrite Northwest Africa 6704: Constraints from petrology, chemistry and Re-Os, O and Ti isotope systematics. Geochimica et Cosmochimica Acta 245, 597–627. doi:10.1016/j.gca.2018.04.031.
* Honda and Imamura (1971) Honda, M., Imamura, M., 1971\. Half-Life of Mn53. Physical Review C 4, 1182–1188. doi:10.1103/PhysRevC.4.1182.
* Houdek and Gough (2011) Houdek, G., Gough, D.O., 2011\. On the seismic age and heavy-element abundance of the Sun. Monthly Notices of the Royal Astronomical Society 418, 1217–1230. doi:10.1111/j.1365-2966.2011.19572.x, arXiv:1108.0802.
* Ito and Ganguly (2009) Ito, M., Ganguly, J., 2009\. Mg Diffusion in Minerals in CAIs: New Experimental Data for Melilites and Implications for the Al-Mg Chronometer and Thermal History of CAIs, in: 40th Annual Lunar and Planetary Science Conference, p. 1753.
* Izawa et al. (2022) Izawa, M.R.M., Yokoyama, S., Yamashita, S., Okuchi, T., Jephcoat, A.P., Cloutis, E.A., Applin, D.M., Hall, B.J., 2022\. Evidence for Hydrous Magmatism on the Northwest Africa 011 Parent Asteroid, in: 53rd Lunar and Planetary Science Conference, p. 2369.
* Jacobsen et al. (2008) Jacobsen, B., Yin, Q.z., Moynier, F., Amelin, Y., Krot, A.N., Nagashima, K., Hutcheon, I.D., Palme, H., 2008\. 26Al- 26Mg and 207Pb- 206Pb systematics of Allende CAIs: Canonical solar initial 26Al/ 27Al ratio reinstated. Earth and Planetary Science Letters 272, 353–364. doi:10.1016/j.epsl.2008.05.003.
* Jaffey et al. (1971) Jaffey, A.H., Flynn, K.F., Glendenin, L.E., Bentley, W.C., Essling, A.M., 1971. Precision Measurement of Half-Lives and Specific Activities of 235U and 238U. Physical Review C 4, 1889–1906. doi:10.1103/PhysRevC.4.1889.
* Jones et al. (2000) Jones, R.H., Lee, T., Connolly, H. C., J., Love, S.G., Shang, H., 2000. Formation of Chondrules and CAIs: Theory VS. Observation, in: Mannings, V., Boss, A.P., Russell, S.S. (Eds.), Protostars and Planets IV, p. 927.
* Jurewicz et al. (1993) Jurewicz, A.J.G., Mittlefehldt, D.W., Jones, J.H., 1993. Experimental partial melting of the Allende (CV) and Murchison (CM) chondrites and the origin of asteroidal basalts. Geochimica et Cosmochimica Acta 57, 2123–2139. doi:10.1016/0016-7037(93)90098-H.
* Kawasaki et al. (2021) Kawasaki, N., Itoh, S., Sakamoto, N., Simon, S.B., Yamamoto, D., Yurimoto, H., Marrocchi, Y., 2021. Oxygen and Al-Mg isotopic constraints on cooling rate and age of partial melting of an Allende Type B CAI, Golfball. Meteoritics and Planetary Science 56, 1224–1239. doi:10.1111/maps.13701.
* Keil (2012) Keil, K., 2012. Angrites, a small but diverse suite of ancient, silica-undersaturated volcanic-plutonic mafic meteorites, and the history of their parent asteroid. Chemie der Erde / Geochemistry 72, 191–218. doi:10.1016/j.chemer.2012.06.002.
* Kleine and Wadhwa (2017) Kleine, T., Wadhwa, M., 2017\. Chronology of Planetesimal Differentiation. pp. 224–245. doi:10.1017/9781316339794.011.
* Koefoed et al. (2016) Koefoed, P., Amelin, Y., Yin, Q.Z., Wimpenny, J., Sanborn, M.E., Iizuka, T., Irving, A.J., 2016. U-Pb and Al-Mg systematics of the ungrouped achondrite Northwest Africa 7325. Geochimica et Cosmochimica Acta 183, 31–45. doi:10.1016/j.gca.2016.03.028.
* Kondev (2021) Kondev, F.G., 2021. Nuclear Data Sheets for A=201. arXiv e-prints , arXiv:2107.07613arXiv:2107.07613.
* Krot et al. (2012) Krot, A.N., Makide, K., Nagashima, K., Huss, G.R., Ogliore, R.C., Ciesla, F.J., Yang, L., Hellebrand, E., Gaidos, E., 2012. Heterogeneous distribution of 26Al at the birth of the solar system: Evidence from refractory grains and inclusions. Meteoritics and Planetary Science 47, 1948–1979. doi:10.1111/maps.12008.
* Kuffmeier et al. (2017) Kuffmeier, M., Haugbølle, T., Nordlund, Å., 2017. Zoom-in Simulations of Protoplanetary Disks Starting from GMC Scales. The Astrophysical Journal 846, 7\. doi:10.3847/1538-4357/aa7c64, arXiv:1611.10360.
* Larsen et al. (2011) Larsen, K.K., Trinquier, A., Paton, C., Schiller, M., Wielandt, D., Ivanova, M.A., Connelly, J.N., Nordlund, Å., Krot, A.N., Bizzarro, M., 2011\. Evidence for Magnesium Isotope Heterogeneity in the Solar Protoplanetary Disk. The Astrophysical Journal 735, L37. doi:10.1088/2041-8205/735/2/L37.
* Larsen et al. (2020) Larsen, K.K., Wielandt, D., Schiller, M., Krot, A.N., Bizzarro, M., 2020. Episodic formation of refractory inclusions in the Solar System and their presolar heritage. Earth and Planetary Science Letters 535, 116088. doi:10.1016/j.epsl.2020.116088.
* LaTourrette and Wasserburg (1998) LaTourrette, T., Wasserburg, G.J., 1998\. Mg diffusion in anorthite: implications for the formation of early solar system planetesimals. Earth and Planetary Science Letters 158, 91–108. doi:10.1016/S0012-821X(98)00048-X.
* Liermann and Ganguly (2002) Liermann, H.P., Ganguly, J., 2002\. Diffusion kinetics of Fe 2+ and Mg in aluminous spinel . experimental determination and applications. Geochimica et Cosmochimica Acta 66, 2903–2913. doi:10.1016/S0016-7037(02)00875-X.
* Liu et al. (2019) Liu, M.C., Han, J., Brearley, A.J., Hertwig, A.T., 2019\. Aluminum-26 chronology of dust coagulation and early solar system evolution. Science Advances 5, eaaw3350. doi:10.1126/sciadv.aaw3350.
* Ludwig (2000) Ludwig, K.R., 2000. Decay constant errors in U-Pb concordia-intercept ages. Chemical Geology 166, 315–318. doi:10.1016/S0009-2541(99)00219-3.
* Lugmair and Shukolyukov (1998) Lugmair, G.W., Shukolyukov, A., 1998\. Early solar system timescales according to 53Mn- 53Cr systematics. Geochimica et Cosmochimica Acta 62, 2863–2886. doi:10.1016/S0016-7037(98)00189-6.
* MacPherson et al. (2017) MacPherson, G.J., Bullock, E.S., Tenner, T.J., Nakashima, D., Kita, N.T., Ivanova, M.A., Krot, A.N., Petaev, M.I., Jacobsen, S.B., 2017. High precision Al-Mg systematics of forsterite-bearing Type B CAIs from CV3 chondrites. Geochimica et Cosmochimica Acta 201, 65–82. doi:10.1016/j.gca.2016.12.006.
* Mamajek (2009) Mamajek, E.E., 2009. Initial Conditions of Planet Formation: Lifetimes of Primordial Disks, in: Usuda, T., Tamura, M., Ishii, M. (Eds.), Exoplanets and Disks: Their Formation and Diversity, pp. 3–10. doi:10.1063/1.3215910, arXiv:0906.5011.
* Mane et al. (2022) Mane, P., Wallace, S., Bose, M., Wallace, P., Wadhwa, M., Weber, J., Zega, T.J., 2022. Earliest evidence of nebular shock waves recorded in a calcium-aluminum-rich Inclusion. Geochimica et Cosmochimica Acta 332, 369–388. doi:10.1016/j.gca.2022.06.006.
* Mattinson (2010) Mattinson, J.M., 2010. Analysis of the relative decay constants of 235U and 238U by multi-step CA-TIMS measurements of closed-system natural zircon samples. Chemical Geology 275, 186–198. doi:10.1016/j.chemgeo.2010.05.007.
* McKay et al. (1994) McKay, G., Le, L., Wagstaff, J., Crozaz, G., 1994\. Experimental partitioning of rare earth elements and strontium: Constraints on petrogenesis and redox conditions during crystallization of Antarctic angrite Lewis Cliff 86010. Geochimica et Cosmochmica Acta 58, 2911–2919. doi:10.1016/0016-7037(94)90124-4.
* Middleton et al. (1983) Middleton, R., Klein, J., Raisbeck, G.M., Yiou, F., 1983\. Accelerator mass spectrometry with 26Al. Nuclear Instruments and Methods in Physics Research 218, 430–438. doi:10.1016/0167-5087(83)91017-7.
* Mikouchi et al. (2003) Mikouchi, T., McKay, G., Koizumi, E., Monkawa, A., Miyamoto, M., 2003. Northwest Africa 1670: A New Quenched Angrite. Meteoritics and Planetary Science Supplement 38, 5218.
* Müller et al. (2013) Müller, T., Dohmen, R., Becker, H.W., ter Heege, J.H., Chakraborty, S., 2013. Fe-Mg interdiffusion rates in clinopyroxene: experimental data and implications for Fe-Mg exchange geothermometers. Contributions to Mineralogy and Petrology 166, 1563–1576. doi:10.1007/s00410-013-0941-y.
* Nanne et al. (2019) Nanne, J.A.M., Nimmo, F., Cuzzi, J.N., Kleine, T., 2019\. Origin of the non-carbonaceous-carbonaceous meteorite dichotomy. Earth and Planetary Science Letters 511, 44–54. doi:10.1016/j.epsl.2019.01.027.
* Nishiizumi (2003) Nishiizumi, K., 2003. Cosmogenic nuclides studies from meteorites to the Earth and beyond. Geochimica et Cosmochimica Acta Supplement 67, 336.
* Norris et al. (1983) Norris, T.L., Gancarz, A.J., Rokop, D.J., Thomas, K.W., 1983\. Half-life of Al-26. Lunar and Planetary Science Conference Proceedings 88, B331–B333. doi:10.1029/JB088iS01p0B331.
* Nyquist et al. (2009) Nyquist, L.E., Kleine, T., Shih, C.Y., Reese, Y.D., 2009\. The distribution of short-lived radioisotopes in the early solar system and the chronology of asteroid accretion, differentiation, and secondary mineralization. Geochimica et Cosmochimica Acta 73, 5115–5136. doi:10.1016/j.gca.2008.12.031.
* Nyquist et al. (2003) Nyquist, L.E., Reese, Y., Wiesmann, H., Shih, C.Y., Takeda, H., 2003. Fossil 26Al and 53Mn in the Asuka 881394 eucrite: evidence of the earliest crust on asteroid 4 Vesta. Earth and Planetary Science Letters 214, 11–25. doi:10.1016/S0012-821X(03)00371-6.
* Ouellette et al. (2007) Ouellette, N., Desch, S.J., Hester, J.J., 2007. Interaction of Supernova Ejecta with Nearby Protoplanetary Disks. The Astrophysical Journal 662, 1268–1281. doi:10.1086/518102, arXiv:0704.1652.
* Ouellette et al. (2010) Ouellette, N., Desch, S.J., Hester, J.J., 2010. Injection of Supernova Dust in Nearby Protoplanetary Disks. The Astrophysical Journal 711, 597–612. doi:10.1088/0004-637X/711/2/597.
* Palk et al. (2018) Palk, C., Andreasen, R., Rehkämper, M., Stunt, A., Kreissig, K., Coles, B., Schönbächler, M., Smith, C., 2018\. Variable Tl, Pb, and Cd concentrations and isotope compositions of enstatite and ordinary chondrites—Evidence for volatile element mobilization and decay of extinct 205Pb. Meteoritics and Planetary Science 53, 167–186. doi:10.1111/maps.12989.
* Pan et al. (2012) Pan, L., Desch, S.J., Scannapieco, E., Timmes, F.X., 2012\. Mixing of Clumpy Supernova Ejecta into Molecular Clouds. The Astrophysical Journal 756, 102\. doi:10.1088/0004-637X/756/1/102, arXiv:1206.6516.
* Piralla et al. (2023) Piralla, M., Villeneuve, J., Schnuriger, N., Bekaert, D., Marrocchi, Y., 2023. A unified chronology of dust formation in the early solar system. Icarus 10, 230–237. doi:10.1016/j.icarus.2023.115427.
* Reger et al. (2023) Reger, P.M., Roebbert, Y., Neumann, W., Gannoun, A., Regelous, M., Schwarz, W.H., Ludwig, T., Trieloff, M., Weyer, S., Bouvier, A., 2023\. Al-Mg and U-Pb chronological records of Erg Chech 002 ungrouped achondrite meteorite. Geochimica et Cosmochimica Acta 343, 33–48. doi:10.1016/j.gca.2022.12.025.
* Rightmire et al. (1958) Rightmire, R.A., Kohman, T.P., Hinten-Berger, H., 1958. Über die Halbwertszeit des langlebigen 26Al. Zeitschrift Naturforschung Teil A 13, 847–853. doi:10.1515/zna-1958-1010.
* Ruzicka et al. (2008) Ruzicka, A., Floss, C., Hutson, M., 2008. Relict olivine grains, chondrule recycling, and implications for the chemical, thermal, and mechanical processing of nebular materials. Geochimica et Cosmochimica Acta 72, 5530–5557. doi:10.1016/j.gca.2008.08.017.
* Sahijpal and Goswami (1998) Sahijpal, S., Goswami, J.N., 1998\. Refractory Phases in Primitive Meteorites Devoid of 26Al and 41Ca:Representative Samples of First Solar System Solids? The Astrophysical Journal 509, L137–L140. doi:10.1086/311778.
* Samworth et al. (1972) Samworth, E.A., Warburton, E.K., Engelbertink, G.A., 1972. Beta Decay of the 26Al Ground State. Physical Review C 5, 138–142. doi:10.1103/PhysRevC.5.138.
* Sanborn et al. (2019) Sanborn, M.E., Wimpenny, J., Williams, C.D., Yamakawa, A., Amelin, Y., Irving, A.J., Yin, Q.Z., 2019. Carbonaceous achondrites Northwest Africa 6704/6693: Milestones for early Solar System chronology and genealogy. Geochimica et Cosmochimica Acta 245, 577–596. doi:10.1016/j.gca.2018.10.004.
* Schiller et al. (2010) Schiller, M., Baker, J.A., Bizzarro, M., 2010. 26Al- 26Mg dating of asteroidal magmatism in the young Solar System. Geochimica et Cosmochimica Acta 74, 4844–4864. doi:10.1016/j.gca.2010.05.011.
* Schiller et al. (2015) Schiller, M., Connelly, J.N., Glad, A.C., Mikouchi, T., Bizzarro, M., 2015. Early accretion of protoplanets inferred from a reduced inner solar system 26Al inventory. Earth and Planetary Science Letters 420, 45–54. doi:10.1016/j.epsl.2015.03.028.
* Scott et al. (2009) Scott, E.R.D., Greenwood, R.C., Franchi, I.A., Sanders, I.S., 2009\. Oxygen isotopic constraints on the origin and parent bodies of eucrites, diogenites, and howardites. Geochimica et Cosmochimica Acta 73, 5835–5853. doi:10.1016/j.gca.2009.06.024.
* Spivak-Birndorf et al. (2009) Spivak-Birndorf, L., Wadhwa, M., Janney, P., 2009. 26Al-26Mg systematics in D’Orbigny and Sahara 99555 angrites: Implications for high-resolution chronology using extinct chronometers. Geochimica et Cosmochimica Acta 73, 5202–5211. doi:10.1016/j.gca.2009.02.038.
* Srinivasan et al. (2018) Srinivasan, P., Dunlap, D.R., Agee, C.B., Wadhwa, M., Coleff, D., Ziegler, K., Zeigler, R., McCubbin, F.M., 2018\. Silica-rich volcanism in the early solar system dated at 4.565 Ga. Nature Communications 9, 3036\. doi:10.1038/s41467-018-05501-0.
* Stacey and Kramers (1975) Stacey, J.S., Kramers, J.D., 1975\. Approximation of terrestrial lead isotope evolution by a two-stage model. Earth and Planetary Science Letters 26, 207–221. doi:10.1016/0012-821X(75)90088-6.
* Stolper and Paque (1986) Stolper, E., Paque, J.M., 1986\. Crystallization sequences of Ca-Al-rich inclusions from Allende: The effects of cooling rate and maximum temperature. Geochimica et Cosmochimica Acta 50, 1785–1806. doi:10.1016/0016-7037(86)90139-0.
* Sugiura and Yamaguchi (2007) Sugiura, N., Yamaguchi, A., 2007\. Al-Mg and Mn-Cr Ages of Northwest Africa 011 Achondrite, in: 38th Annual Lunar and Planetary Science Conference, p. 1431.
* Takeda (1997) Takeda, H., 1997. Mineralogical records of early planetary processes on the HED parent body with reference to Vesta. Meteoritics and Planetary Science 32, 841–853. doi:10.1111/j.1945-5100.1997.tb01574.x.
* Thomas et al. (1984) Thomas, J.H., Rau, R.L., Skelton, R.T., Kavanagh, R.W., 1984\. Half-life of 26Al. Physical Review C 30, 385–387. doi:10.1103/PhysRevC.30.385.
* Tissot et al. (2022) Tissot, F.L.H., Collinet, M., Namur, O., Grove, T.L., 2022\. The case for the angrite parent body as the archetypal first-generation planetesimal: Large, reduced and Mg-enriched. Geochimica et Cosmochimica Acta 338, 278–301. doi:10.1016/j.gca.2022.09.031.
* Tissot and Dauphas (2015) Tissot, F.L.H., Dauphas, N., 2015\. Uranium isotopic compositions of the crust and ocean: Age corrections, U budget and global extent of modern anoxia. Geochimica et Cosmochimica Acta 167, 113–143. doi:10.1016/j.gca.2015.06.034.
* Tissot et al. (2016) Tissot, F.L.H., Dauphas, N., Grossman, L., 2016. Origin of uranium isotope variations in early solar nebula condensates. Science Advances 2, e1501400–e1501400. doi:10.1126/sciadv.1501400, arXiv:1603.01780.
* Tissot et al. (2017) Tissot, F.L.H., Dauphas, N., Grove, T.L., 2017. Distinct 238U/235U ratios and REE patterns in plutonic and volcanic angrites: Geochronologic implications and evidence for U isotope fractionation during magmatic processes. Geochimica et Cosmochimica Acta 213, 593–617. doi:10.1016/j.gca.2017.06.045.
* Tissot et al. (2019) Tissot, F.L.H., Ibanez-Mejia, M., Boehnke, P., Dauphas, N., McGee, D., Grove, T.L., Harrison, T.M., 2019. 238U/235U measurement in single-zircon crystals: implications fo the Hadean environment, magmatic differentiation and geochronology. Journal of Analytical Atomic Spectrometry 34, 2035–2052. doi:10.1039/C9JA00205G.
* Villa et al. (2016) Villa, I.M., Bonardi, M.L., De Bièvre, P., Holden, N.E., Renne, P.R., 2016. IUPAC-IUGS status report on the half-lives of 238U, 235U and 234U. Geochimica et Cosmochimica Acta 172, 387–392. doi:10.1016/j.gca.2015.10.011.
* Villeneuve et al. (2009) Villeneuve, J., Chaussidon, M., Libourel, G., 2009. Homogeneous Distribution of 26Al in the Solar System from the Mg Isotopic Composition of Chondrules. Science 325, 985\. doi:10.1126/science.1173907.
* Wadhwa et al. (2009) Wadhwa, M., Amelin, Y., Bogdanovski, O., Shukolyukov, A., Lugmair, G.W., Janney, P., 2009\. Ancient relative and absolute ages for a basaltic meteorite: Implications for timescales of planetesimal accretion and differentiation. Geochimica et Cosmochimica Acta 73, 5189–5201. doi:10.1016/j.gca.2009.04.043.
* Wadhwa et al. (2014) Wadhwa, M., Kita, N.T., Nakashima, D., Bullock, E.S., MacPherson, G.J., Bouvier, A., 2014\. High Precision 26Al-26Mg Systematics for an Almost Pristine Refractory Inclusion: Implications for the Absolute Age of the Solar System, in: Lunar and Planetary Science Conference, p. 2698\.
* Walker et al. (1989) Walker, F.W., Parrington, J.R., Feiner, F., 1989. Nuclides and Isotopes. General Electric Co.
* Warren et al. (2013) Warren, P.H., Rubin, A.E., Isa, J., Brittenham, S., Ahn, I., Choi, B.G., 2013\. Northwest Africa 6693: A new type of FeO-rich, low-$\Delta$17O, poikilitic cumulate achondrite. Geochimica et Cosmochimica Acta 107, 135–154. doi:10.1016/j.gca.2012.12.025.
* Wasson and Rubin (2003) Wasson, J.T., Rubin, A.E., 2003\. Ubiquitous low-FeO relict grains in type II chondrules and limited overgrowths on phenocrysts following the final melting event. Geochimica et Cosmochimica Acta 67, 2239–2250. doi:10.1016/S0016-7037(03)00023-1.
* Wendt and Carl (1991) Wendt, I., Carl, C., 1991\. The statistical distribution of the mean squared weighted deviation. Chemical Geology (Isotope Geoscience Section) 86, 275–285. doi:10.1016/0168-9622(91)90010-TGet.
* Whattam et al. (2022) Whattam, S.A., Hewins, R.H., Seo, J., Devouard, B., 2022\. Refractory inclusions as Type IA chondrule precursors: Constraints from melting experiments. Geochimica et Cosmochimica Acta 319, 30–55. doi:10.1016/j.gca.2021.12.022.
* Wimpenny et al. (2019) Wimpenny, J., Sanborn, M.E., Koefoed, P., Cooke, I.R., Stirling, C., Amelin, Y., Yin, Q.Z., 2019. Reassessing the origin and chronology of the unique achondrite Asuka 881394: Implications for distribution of 26Al in the early Solar System. Geochimica et Cosmochimica Acta 244, 478–501. doi:10.1016/j.gca.2018.10.006.
* Yamaguchi et al. (2002) Yamaguchi, A., Clayton, R., Mayeda, T., Ebihara, M., Oura, Y., Miura, Y., Haramura, H., Misawa, K., Kojima, H., Nagao, K., 2002\. A New Source of Basaltic Meteorites Inferred from Northwest Africa 011. Science 296, 334–336. doi:10.1126/science.1069408.
* Yang et al. (2019) Yang, J., Zhang, C., Miyahara, M., Tang, X., Gu, L., Lin, Y., 2019\. Evidence for early impact on a hot differentiated planetesimal from Al-rich micro-inclusions in ungrouped achondrite Northwest Africa 7325. Geochimica et Cosmochimica Acta 258, 310–335. doi:10.1016/j.gca.2019.03.010.
* York et al. (2004) York, D., Evensen, N.M., Martínez, M.L., De Basabe Delgado, J., 2004. Unified equations for the slope, intercept, and standard errors of the best straight line. American Journal of Physics 72, 367–375. doi:10.1119/1.1632486.
* Young (2020) Young, E.D., 2020. The birth environment of the solar system constrained by the relative abundances of the solar radionuclides, in: Elmegreen, B.G., Tóth, L.V., Güdel, M. (Eds.), Origins: From the Protosun to the First Steps of Life, pp. 70–77. doi:10.1017/S1743921319001777, arXiv:1909.06361.
|
# The canonical trace of determinantal rings
Antonino Ficarra, Jürgen Herzog, Dumitru I. Stamate and Vijaylaxmi Trivedi
Antonino Ficarra, Department of mathematics and computer sciences, physics and
earth sciences, University of Messina, Viale Ferdinando Stagno d’Alcontres 31,
98166 Messina, Italy<EMAIL_ADDRESS>Jürgen Herzog, Fakultät für
Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany
<EMAIL_ADDRESS>Dumitru I. Stamate, Faculty of Mathematics and
Computer Science, University of Bucharest, Str. Academiei 14, Bucharest –
010014, Romania<EMAIL_ADDRESS>Vijaylaxmi Trivedi, School of
Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,
Mumbai-400005, India<EMAIL_ADDRESS>
###### Abstract.
We compute the canonical trace of generic determinantal rings and provide a
sufficient condition for the trace to specialize. As an application we
determine the canonical trace $\operatorname{tr}(\omega_{R})$ of a
Cohen–Macaulay ring $R$ of codimension two, which is generically Gorenstein.
It is shown that if the defining ideal $I$ of $R$ is generated by $n$
elements, then $\operatorname{tr}(\omega_{R})$ is generated by the
$(n-2)$-minors of the Hilbert-Burch matrix of $I$.
###### Key words and phrases:
canonical traces, determinantal rings, Teter numbers, perfect codimension 2
ideals
###### 2010 Mathematics Subject Classification:
Primary 13H10; Secondary 13M05.
This paper was written while the first, the third and the fourth author
visited the Faculty of Mathematics of Essen. D.I. Stamate was partly supported
by a grant of the Ministry of Research, Innovation and Digitization, CNCS -
UEFISCDI, project number PN-III-P1-1.1-TE-2021-1633, within PNCDI III.
## Introduction
Let $(R,{\mathfrak{m}})$ be a Cohen–Macaulay local ring which admits a
canonical module or a finitely generated graded Cohen–Macaulay $K$-algebra
with graded maximal ideal ${\mathfrak{m}}$. In both cases we denote by
$\omega_{R}$ the canonical module of $R$.
The trace of an $R$-module $M$ is defined to be the ideal
$\operatorname{tr}_{R}(M)=\sum_{\varphi\in\operatorname{Hom}_{R}(M,R)}\varphi(M)\subseteq
R.$
We omit the index $R$ in $\operatorname{tr}_{R}(M)$, if it is clear from the
context in which ring $R$ we are computing the trace. The trace of
$\omega_{R}$ is called the canonical trace of $R$. There is a particular
interest in the canonical trace because it determines the non-Gorenstein locus
of $R$. It also allows to define nearly Gorenstein rings, which are the rings
for which ${\mathfrak{m}}\subseteq\operatorname{tr}_{R}(\omega_{R})$. This
class of rings have first been considered in [11]. The name “nearly
Gorenstein” was introduced in [9]. In 1974 William Teter [13] studied
$0$-dimensional local rings which can be represented as Gorenstein rings
modulo their socle. Such rings are nowadays called Teter rings. It has been
shown (see [13],[11] and [5]) that $R$ is a Teter ring if and only if there
exists an epimorphism $\varphi\colon\omega_{R}\rightarrow{\mathfrak{m}}$. This
result shows that a Teter ring is nearly Gorenstein. Inspired by this result,
the authors of [6] define the Teter number $\textup{teter}(R)$ of $R$ as the
smallest integer $t$ for which there exist $R$-module homomorphisms
$\varphi_{1},\ldots,\varphi_{t}\colon\omega_{R}\rightarrow\operatorname{tr}(\omega_{R})$
with $\operatorname{tr}(\omega_{R})=\sum_{i=1}^{t}\varphi_{i}(\omega_{R})$.
For further studies of the canonical trace we refer the reader to [4].
In this paper we study the canonical trace of determinantal rings. Let $K$ be
a field, and let $X$ be an $m\times n$ matrix of indeterminates with $m<n$.
Furthermore, assume that $1<r+1\leq m$ and let $R=K[X]/I_{r+1}(X)$, where
$I_{r+1}(X)$ is the ideal of $K[X]$ generated by the $(r+1)$-minors of $X$.
Section 1 of this paper is devoted to prove that
$\operatorname{tr}(\omega_{R})=I_{r}(X)^{n-m}R$, see Theorem 1.1. As a
consequence, the non-Gorenstein locus and the singular locus of $R$ coincide.
In Section 2 we determine the Teter number of $R$. It can be easily seen that
in general the Teter number of a ring is bounded above by the minimal number
of generators of the anti-canonical module $\omega_{R}^{-1}$. We show in
Theorem 2.3 that for our determinantal rings this upper bound is reached, and
hence by [3, Proposition 4.1] it is equal to the determinant of
$\big{[}\binom{2n-m-j}{n-i}\big{]}_{1\leq i,j\leq r}$.
It is natural to ask whether the canonical trace specializes. It is known and
easily seen that if ${\mathbf{x}}$ is an $R$-sequence and
$\overline{R}=R/{\mathbf{x}}R$, then
$\operatorname{tr}(\omega_{R})\overline{R}\subseteq\operatorname{tr}(\omega_{\overline{R}})$.
If equality holds, we say that canonical trace specializes. In Section 3 we
give an explicit example which shows that the trace does not always
specialize. On the other hand, we prove in Theorem 3.1 that the canonical
trace specializes if $R$ and $\overline{R}$ are generically Gorenstein and
$\omega_{R}^{2}$ is a Cohen–Macaulay $R$-module. For our determinantal rings
this condition on $\omega_{R}^{2}$ is satisfied if and only if $n\leq 2m-r$,
see [3, Theorem 4.3]. We should stress the fact that Theorem 3.1 only gives a
sufficient condition for the canonical trace to specialize. Indeed, we don’t
have any example of a determinantal ring whose canonical trace does not
specialize.
Let $I$ be graded perfect ideal of height $2$ in the polynomial ring
$S=K[x_{1},\ldots,x_{s}]$. Assume that $I$ is minimally generated by $n$
homogeneous elements and that $R=S/I$ is generically Gorenstein. Theorem 3.1
is used in Corollary 3.5 to show that
$\operatorname{tr}(\omega_{R})=I_{n-2}(A)R$, where $A$ is a Hilbert-Burch
matrix of $I$. This result is applied to rings defined by generic perfect
monomial ideals of grade $2$ and also to numerical semigroup rings generated
by $3$ elements, to recover a result in [10].
## 1\. The canonical trace of determinantal rings
The aim of this section is to prove
###### Theorem 1.1.
Let $K$ be a field, and let $X=(x_{ij})$ be an $m\times n$ matrix of
indeterminates with $m\leq n$. Let $1\leq r+1\leq m$ and $R=K[X]/I_{r+1}(X)$.
Then
$\operatorname{tr}(\omega_{R})=I_{r}(X)^{n-m}R.$
The proof of this theorem needs some preparations. Let $X=(x_{ij})$ be an
$m\times n$ matrix of indeterminates with $m\leq n$. Let
$A\subseteq[m],B\subseteq[n]$ with $|A|=|B|=s$, and assume that
$A=\\{a_{1}<\ldots<a_{s}\\}$ and $B=\\{b_{1}<\ldots<b_{s}\\}$. We denote by
$[A|B]=[a_{1},\dots,a_{s}|b_{1},\dots,b_{s}]$ the $r$-minor of $X$, whose
$i$-th row is the $a_{i}$-th row of $X$ and whose $j$-th column is the
$b_{j}$-th column of $X$.
Let $[C|D]=[c_{1},\dots,c_{t}|d_{1},\dots,d_{t}]$ be any other minors of $X$.
We set
$[A|B]\leq[C|D]\ \Longleftrightarrow\ s\geq t\ \text{and}\ a_{i}\leq c_{i},\
b_{i}\leq d_{i},\ i=1,\dots,t.$
Then $\leq$ is a partial order on the set of all minors of $X$.
Let $r+1\leq m$. It is well-known that $K[x_{ij}]/I_{r+1}(X)$ is an algebra
with straightening laws. In particular, if $u,v$ are incomparable $r$-minors,
we have a straightening relation
(1) $u\cdot v=\sum_{i}\lambda_{i}u_{i}v_{i},$
where $\lambda_{i}\in K$, $\lambda_{i}\neq 0$, and for all $i$, $u_{i},v_{i}$
are $r$-minors with $u_{i}\leq v_{i}$, $u_{i}<u$ and $u_{i}<v$.
We denote again by $X$ the image of the matrix $X$ in $R$. Let $P$ be the
ideal of $R$ generated by the $r$-minors of the first $r$ rows of $X$.
Likewise, let $Q$ be the ideal of $R$ generated by the $r$-minors of the first
$r$ columns of $X$. Furthermore, we denote by $\delta$ the minor
$[1,\dots,r|1,\dots,r]$.
If $I$ is a graded ideal in $R$, $\mu(I)$ denotes the minimal number of
generators.
###### Lemma 1.2.
With the notation introduced, we have for all $\ell\geq 1$,
$\mu((PQ)^{\ell})=\mu(P^{\ell})\mu(Q^{\ell}).$
###### Proof.
Let $Y=(y_{ij})$ be an $m\times r$ matrix and $Z=(z_{ij})$ be an $r\times n$
matrix of indeterminates. For an integer $k$, we set $[k]=\\{1,\dots,k\\}$.
Furthermore, if $I\subseteq[m]$ with $|I|=r$, we denote by $Y_{I}$ the
$r\times r$ submatrix of $Y$ whose rows are indexed by $I$. Similarly, if
$J\subseteq[n]$ with $|J|=r$, we denote by $Z_{J}$ the $r\times r$ submatrix
of $Z$ whose columns are indexed by $J$. We set $\delta_{Y}=\det(Y_{[r]})$ and
$\delta_{Z}=\det(Z_{[r]})$.
For the proof we use the isomorphism
$\displaystyle\varphi:R\rightarrow K[Y\cdot Z],\ \ x_{ij}\mapsto(Y\cdot
Z)_{ij}\ \ \textup{for all}\ \ i\in[m],\ j\in[n].$
Let $u=[1,\dots,r|J]\in P$ be an $r$-minor, with $J\subseteq[n]$, $|J|=r$.
Then
$\varphi(u)=\det((Y\cdot Z)_{i\in[r],j\in J})=\det(Y_{[r]}\cdot
Z_{J})=\delta_{Y}\det(Z_{J}).$
Thus $\varphi(P)=\delta_{Y}I_{r}(Z)$. Similarly, we have
$\varphi(Q)=\delta_{Z}I_{r}(Y)$. This implies that
$\varphi(P^{\ell}Q^{\ell})=(\delta_{Y}\delta_{Z})^{\ell}I_{r}(Y)^{\ell}I_{r}(Z)^{\ell}.$
Therefore,
$\mu((PQ)^{\ell})=\mu(I_{r}(Y)^{\ell}I_{r}(Z)^{\ell})=\mu(I_{r}(Y)^{\ell})\mu(I_{r}(Z)^{\ell})=\mu(P^{\ell})\mu(Q^{\ell}).$
The second equality follows from the fact that the generators of $I_{r}(Y)$
and $I_{r}(Z)$ are polynomials in different sets of variables. ∎
###### Corollary 1.3.
With the notation introduced, we have
$PQ=\delta I_{r}(X).$
###### Proof.
Let $u\in P$ and $v\in Q$. If $u=\delta$ or $v=\delta$, then $uv\in\delta
I_{r}(X)$. Suppose both of them are different from $\delta$. Then $u$ and $v$
are incomparable $r$-minors. The only $r$-minor which is less than $u$ and $v$
is $\delta$. Therefore, equation (1) implies that for each $i$,
$u_{i}=\delta$. This shows that $PQ\subseteq\delta I_{r}(X)$.
It is clear that $\mu(\delta
I_{r}(X))=\mu(I_{r}(X))=\binom{m}{r}\binom{n}{r}$. Hence, if we show that
$\mu(PQ)=\binom{m}{r}\binom{n}{r}$, then equality follows. This follows from
the previous lemma with $\ell=1$, because $\mu(P)=\binom{n}{r}$ and
$\mu(Q)=\binom{m}{r}$. ∎
Theorem 1.1 is now a consequence of the following more general result,
observing that $Q^{n-m}$ is the canonical ideal $\omega_{R}$ of $R$ [1,
Theorem 7.3.6].
###### Theorem 1.4.
We have $\operatorname{tr}(Q^{\ell})=I_{r}(X)^{\ell}R$, for all $\ell\geq 1$.
###### Proof.
By Corollary 1.3 we have $Q^{\ell}(\delta^{-1}P)^{\ell}=I_{r}(X)^{\ell}$. On
[1, Page 315] it is noted that $\operatorname{height}(I_{r}(X)R)>1$.
Therefore, for any prime ideal $L$ of $R$ of height one, we have
$Q^{\ell}R_{L}(\delta^{-1}P)^{\ell}R_{L}=R_{L}$. This implies that
$(Q^{\ell})^{-1}R_{L}=(\delta^{-1}P)^{\ell}R_{L}$. Since $Q^{\ell}$ and
$(\delta^{-1}P)^{\ell}$ are divisorial ideals and the previous equation holds
for all height one prime ideals $L$ of $R$, it follows that
$(Q^{\ell})^{-1}=(\delta^{-1}P)^{\ell}$. Here we use the fact that if $I$ is a
divisorial ideal, then $I=\bigcap_{\mathfrak{p}}IR_{\mathfrak{p}}$, where the
intersection is taken over all height one prime ideals $\mathfrak{p}$ of $R$.
Since $Q^{\ell}$ has positive grade, by [9, Lemma 1.1]
$\operatorname{tr}(Q^{\ell})=Q^{\ell}(Q^{\ell})^{-1}$, and hence the desired
result follows. ∎
###### Corollary 1.5.
Suppose the determinantal ring $R$ is not Gorenstein. Then the non-Gorenstein
locus and the singular locus of $R$ coincide.
###### Proof.
It is noted in [9, Lemma 2.1] that the canonical trace determines the non-
Gorenstein locus. Our determinantal ring $R$ is Gorenstein if and only if
$m=n$. Hence, if $R$ is not Gorenstein, Theorem 1.1 implies that
$\sqrt{\operatorname{tr}(\omega_{R})}=I_{r}(X)R$. By [1, Proposition 7.3.4]
$I_{r}(X)R$ determines the singular locus. This implies the assertion. ∎
## 2\. The Teter number of determinantal rings
In [6] the notion of Teter ring was introduced. Let $R$ be a Cohen–Macaulay
ring which is either a local ring with canonical module or a graded
$K$-algebra over a field $K$. Since by definition
$\operatorname{tr}(\omega_{R})=\sum_{\varphi\in\textup{Hom}_{R}(\omega_{R},R)}\varphi(\omega_{R})$,
it is natural to determine the smallest number $t$ of maps
$\varphi_{1},\dots,\varphi_{t}\in\textup{Hom}_{R}(\omega_{R},R)$ such that
$\operatorname{tr}(\omega_{R})=\sum_{i=1}^{t}\varphi_{i}(\omega_{R})$. We call
such a number, the Teter number of $R$, and denote it by $\textup{teter}(R)$.
If $\textup{teter}(R)=1$, $R$ is called a ring of Teter type.
The purpose of this section is to determine the Teter number of a
determinantal ring. For this aim, we adopt the following strategy. Recall that
a Cohen–Macaulay $R$ is _generically Gorenstein_ if $R_{\mathfrak{p}}$ is
Gorenstein for all minimal prime ideals $\mathfrak{p}$ or $R$. If this is the
case, then $\omega_{R}$ can be identified with an ideal of height one in $R$.
Hence $\operatorname{tr}(\omega_{R})=\omega_{R}\cdot\omega_{R}^{-1}$, by [9,
Lemma 1.1]. Since $\textup{Hom}_{R}(\omega_{R},R)\cong\omega_{R}^{-1}$, each
map $\varphi:\omega_{R}\rightarrow R$ is multiplication by some element
$x\in\omega_{R}^{-1}$. Hence, the Teter number is the smallest number $t$ such
that there exist $x_{1},\dots,x_{t}\in\omega_{R}^{-1}$ with
$x_{1}\omega_{R}+\dots+x_{t}\omega_{R}=\operatorname{tr}(\omega_{R}).$
Set $J=(x_{1},\dots,x_{t})$. Therefore,
$J\omega_{R}=\operatorname{tr}(\omega_{R})$. Let $\mu(\omega_{R}^{-1})$ be the
minimal number of generators of $\omega_{R}^{-1}$. Since
$\omega_{R}^{-1}\omega_{R}=\operatorname{tr}(\omega_{R})$, our discussion
yields the following upper bound:
###### Proposition 2.1.
Let $R$ be a ring as introduced above. Then
$\textup{teter}(R)\leq\mu(\omega_{R}^{-1})$.
Below we discuss a special case where this upper bound is reached.
###### Lemma 2.2.
Let $R$ be a ring as introduced above. Assume that
$\mu(\operatorname{tr}(\omega_{R}))=\mu(\omega_{R})\mu(\omega_{R}^{-1})$. Then
$\textup{teter}(R)=\mu(\omega_{R}^{-1})$.
###### Proof.
Suppose by contradiction that $\textup{teter}(R)<\mu(\omega_{R}^{-1})$. Then,
we can find $J\subseteq\omega_{R}^{-1}$ with $\mu(J)<\mu(\omega_{R}^{-1})$ and
$J\omega_{R}=\operatorname{tr}(\omega_{R})$. Therefore,
$\mu(\operatorname{tr}(\omega_{R}))\leq\mu(J)\mu(\omega_{R})<\mu(\omega_{R}^{-1})\mu(\omega_{R})=\mu(\operatorname{tr}(\omega_{R})).$
A contradiction. Since we also have
$\textup{teter}(R)\leq\mu(\omega_{R}^{-1})$, the conclusion follows. ∎
###### Theorem 2.3.
Let $K$ be a field, and let $X=(x_{ij})$ be an $m\times n$ matrix of
indeterminates with $m<n$. Let $1<r+1\leq m$ and $R=K[X]/I_{r+1}(X)$. Then
$\textup{teter}(R)=\det\bigg{[}\binom{2n-m-j}{n-i}\bigg{]}_{1\leq i,j\leq r}.$
###### Proof.
By using the notation introduced before Corollary 1.3, we know that
$\omega_{R}=Q^{n-m}$, $\omega_{R}^{-1}=(\delta^{-1}P)^{n-m}$ and
$\operatorname{tr}(\omega_{R})=I_{r}(X)^{n-m}$. Obviously,
$\mu(\omega_{R}^{-1})=\mu(P^{n-m})$ and
$\mu(\operatorname{tr}(\omega_{R}))=\mu(\delta^{n-m}I_{r}(X)^{n-m})$. By Lemma
1.2 $\mu(P^{n-m})\mu(Q^{n-m})=\mu((PQ)^{n-m})$. By Lemma 2.2 it follows that
$\textup{teter}(R)=\mu(P^{n-m})$ and using [3, Proposition 4.1] we get the
formula in the statement. ∎
We conclude this section with the following remark.
###### Remark 2.4.
Suppose $R$ is generically Gorenstein and that $\omega_{R}$ is a divisorial
ideal. Let $J\subseteq\omega_{R}^{-1}$ such that
$J\omega_{R}=\operatorname{tr}(\omega_{R})$. Then
$J\omega_{R}=\omega_{R}^{-1}\omega_{R}$. Hence,
$\omega_{R}:(\omega_{R}J)=\omega_{R}:(\omega_{R}\omega_{R}^{-1}).$
On the other hand
$\omega_{R}:(\omega_{R}J)=(\omega_{R}:\omega_{R}):J=R:J=J^{-1}$ and
$\omega_{R}:(\omega_{R}\omega_{R}^{-1})=(\omega_{R}:\omega_{R}):\omega_{R}^{-1}=R:\omega_{R}^{-1}=(\omega_{R}^{-1})^{-1}=\omega_{R}$,
because we assumed that $\omega_{R}$ is divisorial. Therefore,
$J^{-1}=\omega_{R}$. Because $J$ may not be divisorial, we only obtain the
following inclusion,
$J\subseteq(J^{-1})^{-1}=\omega_{R}^{-1}.$
Hence, if $J\omega_{R}=\operatorname{tr}(\omega_{R})$, the divisorial closure
$(J^{-1})^{-1}$ of $J$ has to be equal to the anti-canonical module
$\omega_{R}^{-1}$, but $J$ itself may be a proper sub-ideal of
$\omega_{R}^{-1}$.
## 3\. Reduction of traces
Let $R$ be a local ring or a positively graded graded $K$-algebra which is
Cohen–Macaulay and admits a canonical module $\omega_{R}$. Let
${\mathbf{x}}=x_{1},\ldots,x_{m}$ be a (graded) sequence of elements in $R$,
and set $\overline{M}=M/({\mathbf{x}})M$. It is easy to see that
$\operatorname{tr}(\omega_{R})\overline{R}\subseteq\operatorname{tr}(\omega_{\overline{R}})$,
if ${\bf x}$ is an $R$-sequence. The question is whether this inclusion may be
strict.
When $R$ is generically Gorenstein, then $\omega_{R}$ may be identified with
an ideal, and we may consider the powers of $\omega_{R}$.
###### Theorem 3.1.
Let $R$ be a ring as introduced above, and assume that $R$ is generically
Gorenstein. Then
$\operatorname{tr}(\omega_{R})\overline{R}=\operatorname{tr}(\omega_{\overline{R}}),$
if $\omega_{R}^{2}$ is Cohen–Macaulay and $\overline{R}$ is generically
Gorenstein as well.
The proof of the theorem is based on the following
###### Lemma 3.2.
Let $R$ be a Cohen–Macaulay ring as before and assume that $R$ is generically
Gorenstein. Then
$\operatorname{Hom}_{R}(\omega_{R}^{2},\omega_{R})\cong\operatorname{Hom}_{R}(\omega_{R},R).$
###### Proof.
Let $U$ be the kernel of the canonical map
$\omega_{R}\otimes_{R}\omega_{R}\rightarrow\omega_{R}^{2}$. Then we obtain the
exact sequence
$0\rightarrow\operatorname{Hom}_{R}(\omega_{R}^{2},\omega_{R})\rightarrow\operatorname{Hom}_{R}(\omega_{R}\otimes_{R}\omega_{R},\omega_{R})\rightarrow\operatorname{Hom}_{R}(U,\omega_{R})$
Since $R$ is generically Gorenstein it follows that $\omega_{R_{P}}\cong
R_{P}$ for all minimal prime ideals $P$ of $R$. This implies that
$\omega_{R}\otimes_{R}\omega_{R}\rightarrow\omega_{R}^{2}$ becomes an
isomorphism after localization at a minimal prime ideal of $R$. Hence,
$U_{P}=0$ for all minimal prime ideal $P$ of $R$. Since
$\operatorname{Ass}\operatorname{Hom}_{R}(U,\omega_{R})=\operatorname{Supp}U\cap\operatorname{Ass}\omega_{R}$
and since $\operatorname{Ass}\omega_{R}=\operatorname{Ass}R$, we conclude that
$\operatorname{Ass}\operatorname{Hom}_{R}(U,\omega_{R})=\emptyset$. This
implies that $\operatorname{Hom}_{R}(U,\omega_{R})=0$. Therefore,
$\operatorname{Hom}_{R}(\omega_{R}^{2},\omega_{R})\cong\operatorname{Hom}_{R}(\omega_{R}\otimes_{R}\omega_{R},\omega_{R}).$
By using that $\operatorname{Hom}_{R}(\omega_{R},\omega_{R})\cong R$, we see
that
$\operatorname{Hom}_{R}(\omega_{R}\otimes_{R}\omega_{R},\omega_{R})\cong\operatorname{Hom}_{R}(\omega_{R},\operatorname{Hom}_{R}(\omega_{R},\omega_{R}))\cong\operatorname{Hom}_{R}(\omega_{R},R).$
Thus, the desired conclusion follows. ∎
###### Proof of Theorem 3.1.
Consider the commutative diagram
${\operatorname{Hom}_{R}(\omega_{R},R)\times\omega_{R}}$${R}$${\overline{\operatorname{Hom}_{R}(\omega_{R},R)}\times\overline{\omega_{R}}}$${\overline{R}}$$\scriptstyle{\varphi}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\psi}$
with its natural maps. Then
$\operatorname{Im}(\varphi)=\operatorname{tr}(\omega_{R})$, and hence
$\operatorname{Im}(\beta\circ\varphi)=\operatorname{tr}(\omega_{R})\overline{R}$.
It follows that
$\operatorname{Im}(\psi)=\operatorname{tr}(\omega_{R})\overline{R}$.
Lemma 3.2, implies that
$\operatorname{Hom}_{R}(\omega_{R},R)\cong\operatorname{Hom}_{R}(\omega_{R}^{2},\omega_{R})$,
since we assume that $R$ is generically Gorenstein. By assumption,
$\omega_{R}^{2}$ is Cohen–Macaulay, and since $\dim\omega_{R}^{2}=\dim R$, it
is a maximal Cohen–Macaulay module.
Thus, [1, Theorem 3.3.10(c)(ii)] implies that
$\operatorname{Ext}^{i}_{R}(\omega_{R}^{2},\omega_{R})=0$ for $i>0$, and hence
it follows from [1, Proposition 3.3.3] that
$\overline{\operatorname{Hom}_{R}(\omega_{R}^{2},\omega_{R})}\cong\operatorname{Hom}_{\overline{R}}(\overline{\omega_{R}^{2}},\overline{\omega_{R}})\cong\operatorname{Hom}_{\overline{R}}(\omega^{2}_{\overline{R}},\omega_{\overline{R}}).$
Now we use that $\overline{R}$ is generically Gorenstein, and apply again
Lemma 3.2 to obtain that
$\operatorname{Hom}_{\overline{R}}(\omega^{2}_{\overline{R}},\omega_{\overline{R}})\cong\operatorname{Hom}_{\overline{R}}(\omega_{\overline{R}},\overline{R})$.
This shows that
$\operatorname{Im}\psi=\operatorname{tr}(\omega_{\overline{R}}),$ and
completes the proof of the theorem. ∎
In the following example, using Macaulay2 [7] we verified that the canonical
trace does not specialize in general.
###### Example 3.3.
Consider the monomial ideal
$I=(x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{1}x_{2},x_{2}y_{3},x_{1}x_{3})\subset
S=K[x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}]$. Then $R=S/I$ is a Cohen–Macaulay
$K$-algebra and the canonical trace of $R$ is
$\operatorname{tr}(\omega_{R})=(x_{1},x_{2},x_{3}^{2},x_{3}y_{1},x_{3}y_{2},y_{1}y_{2},y_{1}y_{3},y_{2}y_{3},y_{3}^{2})R.$
The element $x_{1}-y_{1}$ is regular on $R$. We set
$\overline{R}=R/(x_{1}-y_{1})R$. Then,
$\operatorname{tr}(\omega_{\overline{R}})=(x_{2},x_{3}^{2},x_{3}y_{2},y_{1},y_{3})\overline{R}.$
Hence, the canonical trace does not specialize in such a case. In this example
$R$ and $\overline{R}$ are both generically Gorenstein, but $\omega_{R}^{2}$
is not Cohen–Macaulay, because $\operatorname{depth}\omega_{R}^{2}=2$, while
$\operatorname{depth}R=3$.
Let $K$ be a field, and let $S=K[x_{1},\ldots,x_{s}]$ be the graded polynomial
ring with $\deg x_{i}=d_{i}>0$ for $i=1,\ldots,s$. Furthermore, let
$A=(f_{ij})$ be an $m\times n$-matrix of homogeneous polynomials of $S$ such
that $\deg f_{ij}=a_{i}-b_{j}$, where $a_{1},\ldots,a_{m}$ and
$b_{1},\ldots,b_{n}$ positive integers and where $f_{ij}=0$ if
$a_{i}-b_{j}\leq 0$. Under these assumptions all minors of $A$ are homogeneous
polynomials. Hence $R=S/I_{r+1}(A)$ is a graded $K$-algebra. We have
$\operatorname{height}I_{r+1}(A)\leq\operatorname{height}I_{r+1}(X)=(n-r)(m-r)$.
If equality holds, then $R$ is Cohen–Macaulay.
###### Corollary 3.4.
With the notation introduced, assume that
$\operatorname{height}I_{r+1}(A)=(n-r)(m-r)$, that $R$ is generically
Gorenstein and that $n\leq 2m-r$. Then
$\operatorname{tr}(\omega_{R})=I_{r}(A)^{n-m}R.$
###### Proof.
Let $X$ be the $m\times n$ matrix whose entries are the indeterminates
$x_{ij}$, $i=1,\ldots,m$ and $j=1,\ldots,n$, and let $S^{\prime}=S[X]$ and
$R^{\prime}=S^{\prime}/I^{\prime}$, where $I^{\prime}=I_{r+1}(X)S^{\prime}$.
We set $\deg x_{ij}=b_{j}-a_{i}$ for all $i$ and $j$. Then $I^{\prime}$ is a
graded ideal in the non-standard graded polynomial ring $S^{\prime}$, and $R$
is a specialization of $R^{\prime}$. In other words, the sequence
${\mathbf{g}}=x_{11}-f_{11},\ldots,x_{n-1,n}-f_{n-1,n}$ is a homogeneous
regular sequence on $S^{\prime}$ and $R^{\prime}$ with
$S^{\prime}/({\mathbf{g}})=S$ and $R^{\prime}/({\mathbf{g}})=R$.
The ring $R^{\prime}$ is generically Gorenstein, since it is a Cohen–Macaulay
domain, and $R$ is generically Gorenstein, by assumption. Therefore, by
Theorem 3.1,
$\operatorname{tr}(\omega_{R})=\operatorname{tr}(\omega_{R^{\prime}})R=I_{r}(A)^{n-m}R$,
if $\omega_{R^{\prime}}^{2}$ is Cohen–Macaulay. This condition for
$\omega_{R^{\prime}}$ is obviously satisfied if and only if the square of the
canonical module $Q^{n-m}$ of $K[X]/I_{r+1}(X)$ is Cohen–Macaulay. By [3,
Theorem 4.3] this is the case if and only if $2(n-m)\leq n-r$, equivalently if
$n\leq 2m-r$. This completes the proof of the corollary. ∎
As an application, we consider a graded perfect ideal $I$ of height $2$ in the
polynomial ring $S=K[x_{1},\ldots,x_{s}]$ minimally generated by $n$
homogeneous elements. Then $R=S/I$ has a graded resolution
$0\rightarrow\bigoplus_{j=1}^{n-1}S(-b_{j})\stackrel{{\scriptstyle
A}}{{\longrightarrow}}\bigoplus_{i=1}^{n}S(-a_{i})\rightarrow R\rightarrow 0$
with $A$ an $n\times(n+1)$-matrix. The Hilbert-Burch theorem [1, Theorem
1.4.17] tells us that $I$ is generated by the maximal minors of $A$, which is
the same as the maximal minors of $A^{T}$ \- the transpose of $A$. In
particular, $I$ is obtained as a specialization of the ideal of maximal minors
of the generic $(n-1)\times n$ matrix $X$.
The matrix $A$ is called the Hilbert-Burch matrix of $I$ (with respect to the
given resolution).
###### Corollary 3.5.
Let $I$ be a graded perfect ideal of height $2$ in the polynomial ring
$S=K[x_{1},\ldots,x_{s}]$ minimally generated by $n$ homogeneous elements and
with Hilbert-Burch matrix $A$. Let $R=S/I$, and assume that $R$ is generically
Gorenstein. Then
$\operatorname{tr}(\omega_{R})=I_{n-2}(A)R.$
###### Proof.
Since $R$ is the specialization of $K[X]/I_{n-1}(X)$ with $X$ an $(n-1)\times
n$-matrix of indeterminates and since $r=n-2$, Corollary 3.4 yields the
desired conclusion. ∎
As a very special case of Theorem 3.1 we recover a result in [10]. Let $H$ be
a numerical semigroup generated by 3 elements, and let $K$ be any field. The
numerical semigroup ring $K[H]$ is a $1$-dimensional (non-standard) graded
domain. This implies that the defining ideal $I_{H}$ of $K[H]$ is a graded
perfect height 2 prime ideal in the polynomial ring in 3 variables. By [8],
the ideal $I_{H}$ is generated by $3$ elements, so that its Hilbert-Burch
matrix $A_{H}$ is a $2\times 3$ matrix. Thus, Corollary 3.5 implies that the
canonical trace of $K[H]$ is generated by the entries of $A_{H}$ modulo
$I_{H}$, see [10, Proposition 3.1] for more details.
In [2, Remark 6.3] it is observed that any perfect monomial ideal of height
$2$ arises from a generic perfect monomial ideal of height 2 by
specialization. The family of generic perfect monomial ideals of height 2 are
parameterized by trees. This has been worked out in detail in [12], whose
presentation we follow here. To each tree $\Gamma$ on the vertex set
$[n]=\\{1,\ldots,n\\}$ one assigns an $(n-1)\times n$-matrix
$A(\Gamma)=(a_{ij})$, whose entries are either variables or $0$. One chooses a
total order of the edges of $\Gamma$, and assigns to the $k$th edge
$\\{i,j\\}$ of $\Gamma$, $i<j$, the $k$th row of $A(\Gamma)$ as follows:
$\displaystyle a_{k\ell}=\begin{cases}-x_{ij},&\text{if $\ell=i$}\\\
x_{ji},&\text{if $\ell=j$}\\\ 0,&\text{otherwise}.\end{cases}$
Let $I(\Gamma)=I_{n-1}(A(\Gamma))$. Then $I(\Gamma)$ is a perfect squarefree
monomial ideal of height $2$, whose Hilbert-Burch matrix is the transpose of
$A(\Gamma)$. The generators of $I(\Gamma)$ are determined as follows: let
$i,j$ be two distinct vertices of $\Gamma$. Then there exists a unique path
from $i$ to $j$, that is, a sequence of pairwise distinct numbers
$i=i_{0},i_{1},i_{2},\ldots,i_{k-1},i_{k}=j$, where
$\\{i_{\ell},i_{\ell+1}\\}$ is an edge for $\ell=1,\ldots,k-1$. We set
$b(i,j)=i_{1}$. Let $v_{j}$ be the minor of $A(\Gamma)$ which is obtained by
omitting the $j$th column of $A(\Gamma)$. Then
$v_{j}=\pm\prod_{i=1\atop i\neq j}^{n}x_{ib(i,j)}$
for $j=1,2,\ldots,n$, and $I(\Gamma)=(v_{1},\ldots,v_{n})$.
The ideal $I(\Gamma)$ is a squarefree monomial ideal, and therefore
generically Gorenstein. It follows from Corollary 3.5 that the $(n-2)$-minors
of $A(\Gamma)$ modulo $I(\Gamma)$ generate the canonical trace of
$R=S/I(\Gamma)$, where $S$ is the polynomial ring over $K$ generated by the
entries of $A(\Gamma)$. Expanding the $(n-1)$-minor $v_{j}$ along the $i$th
row, we see that the $n-2$ minors of $A(\Gamma)$ are the monomials
$v_{j}/x_{ib(i,j)}$
with $j=1,\ldots,n$, $i=1,\ldots n$ and $i\neq j$.
In general, these monomials do not form a minimal set of generators of the
canonical trace.
For a monomial ideal $I$ we denote by $I(x)$ the monomial ideal which is
obtained from $I$ by substituting $x$ by $1$ in each monomial generator of
$I$. This operation is a special case of monomial localization. The above
discussions give us the following result.
###### Corollary 3.6.
Let $\Gamma$ be a tree on the vertex set $[n]$, and let $R=S/I(\Gamma)$, where
$S$ is the polynomial ring over $K$ generated by the entries of $A(\Gamma)$.
Then
$\operatorname{tr}(\omega_{R})=\big{(}\sum
I(\Gamma)(x_{ij})\big{)}/I(\Gamma),$
where the sum is taken over all $i$ and $j$ for which $x_{ij}$ is a variable
appearing in $A(\Gamma)$.
###### Example 3.7.
Let $\Gamma$ be the tree with the edges
$\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{3,5\\}$. Then
$\displaystyle A(\Gamma)=\begin{pmatrix}-x_{12}&x_{21}&0&0&0\\\
0&-x_{23}&x_{32}&0&0\\\ 0&0&-x_{34}&x_{43}&0\\\
0&0&-x_{35}&0&x_{53}\end{pmatrix}$
and
$\displaystyle I(\Gamma)$ $\displaystyle=$
$\displaystyle(x_{21}x_{32}x_{43}x_{53},x_{12}x_{32}x_{43}x_{53},x_{12}x_{23}x_{43}x_{53},x_{12}x_{23}x_{34}x_{53},x_{12}x_{23}x_{43}x_{35}).$
We write $A(\Gamma)$ modulo $I(\Gamma)$ as
$\displaystyle\begin{pmatrix}-a&b&0&0&0\\\ 0&-c&d&0&0\\\ 0&0&-e&f&0\\\
0&0&-g&0&h\end{pmatrix}.$
Then the canonical trace of the ring $R$ defined by $I(\Gamma)$ is the ideal
of $3$-minors of this matrix, and we obtain
$\operatorname{tr}(\omega_{R})=(a\,c\,e,\,a\,c\,g,\,a\,c\,f,\,a\,d\,f,\,a\,f\,g,\,b\,d\,f,\,b\,f\,g,\,c\,f\,g,\,a\,c\,h,\,\\\
a\,d\,h,\,a\,e\,h,\,b\,d\,h,\,b\,e\,h,\,c\,e\,h,\,a\,f\,h,\,b\,f\,h,\,c\,f\,h,\,d\,f\,h).$
## References
* [1] W. Bruns and J. Herzog, Cohen–Macaulay Rings, revised ed., Cambridge Stud. Adv. Math., 39, Cambridge University Press, Cambridge, 1998
* [2] W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Camb. Phil. Soc. 118(1995), 245–257.
* [3] W. Bruns, T. Römer, A. Wiebe, Initial algebras of determinantal rings, Cohen–Macaulay and Ulrich ideals. Michigan Mathematical Journal, 53 (2005), 71–81.
* [4] H. Dao, T. Kobayashi, R. Takahashi, Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being Gorenstein. Journal of Pure and Applied Algebra, 225(2021), 106655.
* [5] J. Elias, MS, Takatuji. On Teter rings. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 147 (2017), 125–39.
* [6] O. Gasanova, J. Herzog, T. Hibi, S. Moradi. Rings of Teter type. Nagoya Mathematical Journal, 248 (2022), 1005–1033.
* [7] D. R. Grayson, M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
* [8] J. Herzog, Generators and relations of abelian semigroups and semigroup rings. manuscripta math. 3(1970), 175–193.
* [9] J. Herzog, T. Hibi and D.I. Stamate, The trace of the canonical module, Israel Journal of Mathematics 233 (2019), 133–165.
* [10] J. Herzog, J., T. Hibi, D.I. Stamate, Canonical trace ideal and residue for numerical semigroup rings. Semigroup Forum 103(2021), 550–566.
* [11] C. Huneke, A. Vraciu, Rings that are almost Gorenstein, Pacific J. Math. 225 (2006), 85–102.
* [12] M. Naeem, Cohen–Macaulay monomial ideals of codimension $2$, manuscripta math. 127(2008), 533–545.
* [13] W. Teter, Rings which Are a Factor of a Gorenstein Ring by Its Socle, Inventiones math. 23 (1974), 153–162.
|
# Dirac’s Theorem and Multigraded syzygies
Antonino Ficarra, Jürgen Herzog Antonino Ficarra, Department of mathematics
and computer sciences, physics and earth sciences, University of Messina,
Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina, Italy
<EMAIL_ADDRESS>Jürgen Herzog, Fakultät für Mathematik, Universität
Duisburg-Essen, 45117 Essen, Germany<EMAIL_ADDRESS>
###### Abstract.
Let $G$ be a simple finite graph. A famous theorem of Dirac says that $G$ is
chordal if and only if $G$ admits a perfect elimination order. It is known by
Fröberg that the edge ideal $I(G)$ of $G$ has a linear resolution if and only
if the complementary graph $G^{c}$ of $G$ is chordal. In this article, we
discuss some algebraic consequences of Dirac’s theorem in the theory of
homological shift ideals of edge ideals. Recall that if $I$ is a monomial
ideal, $\textup{HS}_{k}(I)$ is the monomial ideal generated by the $k$th
multigraded shifts of $I$. We prove that $\textup{HS}_{1}(I)$ has linear
quotients, for any monomial ideal $I$ with linear quotients generated in a
single degree. For and edge ideal $I(G)$ with linear quotients, it is not true
that $\textup{HS}_{k}(I(G))$ has linear quotients for all $k\geq 0$. On the
other hand, if $G^{c}$ is a proper interval graph or a forest, we prove that
this is the case. Finally, we discuss a conjecture of Bandari, Bayati and
Herzog that predicts that if $I$ is polymatroidal, $\textup{HS}_{k}(I)$ is
polymatroidal too, for all $k\geq 0$. We are able to prove that this
conjecture holds for all polymatroidal ideals generated in degree two.
###### Key words and phrases:
monomial ideal, multigraded shifts, homological shift ideals, edge ideals,
polymatroidal ideals
###### 2020 Mathematics Subject Classification:
Primary 13F20; Secondary 13H10
.
## Introduction
Let $S=K[x_{1},\ldots,x_{n}]$ be the standard graded polynomial ring with
coefficients in a field $K$ and $G$ be a simple graph on the vertex set
$V(G)=\\{1,\dots,n\\}$ and with edge set $E(G)$. The edge ideal of $G$ is the
ideal $I(G)$ in $S$ generated by the monomials $x_{i}x_{j}$, such that
$\\{i,j\\}\in E(G)$. The classification of all Cohen–Macaulay edge ideals and
the classification of all edge ideals with linear resolution are fundamental
problems. While the first problem is widely open and considered to be
intractable in general, for the second problem we have a complete answer. The
complementary graph $G^{c}$ of $G$ is the graph with vertex set
$V(G^{c})=V(G)$ and where $\\{i,j\\}$ is an edge of $G^{c}$ if and only if
$\\{i,j\\}\notin E(G)$. Ralph Fröberg in [9] proved that $I(G)$ has a linear
resolution if and only $G^{c}$ is chordal, that is, it has no induced cycles
of length bigger than three. In turn, the classical and fundamental Dirac’s
theorem on chordal graphs says that a graph $G$ is chordal if and only if $G$
admits a perfect elimination order [5].
Recently, a new research trend in the theory of monomial ideals has been
initiated by the second author, Moradi, Rahimbeigi and Zhu in [13], see, also,
[2, 3, 4, 6, 7, 14]. For ${\bf a}=(a_{1},\dots,a_{n})\in{\mathbb{Z}}_{\geq
0}^{n}$, we denote $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ by ${\bf x^{a}}$. Let
$I\subset S$ be a monomial ideal and let ${\mathbb{F}}$ be its minimal
multigraded free $S$-resolution. Then, the $k$th free $S$-module in
${\mathbb{F}}$ is $F_{k}=\bigoplus_{j=1}^{\beta_{k}(I)}S(-{\bf a}_{kj})$,
where ${\bf a}_{kj}\in{\mathbb{Z}}_{\geq 0}^{n}$ are the $k$th multigraded
shifts of $I$. The _$k$ th homological shift ideal_ of $I$ is the monomial
ideal generated by the monomials ${\bf x}^{{\bf a}_{kj}}$ for
$j=1,\ldots,\beta_{k}(I)$. Note that $\textup{HS}_{0}(I)=I$. It is natural to
ask what combinatorial and homological properties are satisfied by all
$\textup{HS}_{k}(I)$, $k=0,\ldots,\operatorname{pd}(I)$. Any such property is
called an homological shift property of $I$. If all $\textup{HS}_{k}(I)$ have
linear quotients, or linear resolution, we say that $I$ has homological linear
quotients or homological linear resolution, respectively.
In this article we discuss the algebraic consequences of Dirac’s theorem on
chordal graphs related to the theory of homological shift ideals of edge
ideals.
The article is structured as follows. In Section 1, we investigate arbitrary
monomial ideals with linear quotients generated in one degree. Our main
theorem states that for such an ideal $I$, $\textup{HS}_{1}(I)$ always has
linear quotients. The proof relies upon the fact that certain colon ideals are
generated by linear forms (Lemma 1.1). In particular, $\textup{HS}_{1}(I)$ has
a linear resolution. At present we are not able to generalize this result for
all monomial ideals with linear resolution. In this case, one could expect
even that $\textup{HS}_{1}(I)$ also has linear quotients, if $I$ has a linear
resolution. On the other hand, if $I$ is generated in more than one degree, in
Example 1.4 we show that Theorem 1.3 is no longer valid.
Sections 2 and 3 are devoted to homological shifts of edge ideals with linear
resolution. Let $G$ be a graph and $I(G)$ be its edge ideal. For unexplained
terminology look at Section 2. Unfortunately, even if $I(G)$ has linear
resolution, it may not have homological linear resolution in general, (Example
2.3). At present we do not have a complete classification of all edge ideals
with homological linear quotients or homological linear resolution. Thus, we
determine many classes of cochordal graphs whose edge ideals have homological
linear resolution. In particular, for proper interval graphs and forests, we
prove that the edge ideals of their complementary graphs have homological
linear quotients, (Theorems 2.4 and 3.1). For the proof of the first result we
introduce the class of reversible chordal graphs, and show that any proper
interval graph is a reversible graph, (Lemma 2.5). For the second result, we
consider two operations on chordal graphs that preserve the homological linear
quotients property. Namely, adding whiskers to a chordal graph and taking
unions of disjoint chordal graphs, (Propositions 3.2 and 3.4). Using these
results, it is easy to see that $I(G)$ has homological linear quotients, if
$G$ is a forest. Indeed, any forest is the union of pairwise disjoint trees,
and any tree can be constructed by iteratively adding whiskers to a previously
constructed tree on a smaller vertex set.
In the last section, we consider polymatroidal ideals. An equigenerated
monomial ideal $I$ is called polymatroidal if its minimal set of monomial
generators $G(I)$ corresponds to the set of bases of a discrete polymatroid,
see [12, Chapter 12]. Polymatroidal ideals are characterized by the fact that
they have linear quotients with respect to the lexicographic order induced by
any ordering of the variables. Such characterization is due to Bandari and
Rahmati-Asghar [3]. It was conjectured by Bandari, Bayati and Herzog that all
homological shift ideals of a polymatroidal ideal are polymatroidal. At
present this conjecture is widely open. On the other hand, Bayati proved that
the conjecture holds for any squarefree polymatroidal ideal [2]. The second
author of this paper, Moradi, Rahimbeigi and Zhu proved that it holds for
polymatroidal ideals that satisfy the strong exchange property [13, Corollary
3.6]; whereas the first author of this paper proved that $\textup{HS}_{1}(I)$
is again polymatroidal if $I$ is such [6], pointing towards the validity of
the conjecture in general.
We prove in Theorem 4.5 that for any polymatroidal ideal $I$ generated in
degree two, all homological shift ideals are polymatroidal. In the squarefree
case, $I$ may be seen as the edge ideal of a cochordal graph and we apply our
criterion on reversibility of perfect elimination orders. Unfortunately our
methods are very special and they can not be applied to prove that homological
shifts of polymatroidal ideals, generated in higher degree than two, are
polymatroidal.
## 1\. The first Homological shift of ideals with Linear quotients
Let $S=K[x_{1},\dots,x_{n}]$ be the standard graded polynomial ring, with $K$
a field. A monomial ideal $I\subset S$ has linear quotients if for some
ordering $u_{1},\dots,u_{m}$ of its minimal set of monomial generators $G(I)$,
all colon ideals $(u_{1},\dots,u_{i-1}):u_{i}$, $i=1,\dots,m$, are generated
by variables. We call $u_{1},\dots,u_{m}$ an admissible order of $I$. Such
order is called non-increasing if
$\deg(u_{1})\leq\deg(u_{2})\leq\dots\leq\deg(u_{m})$. By [16, Lemma 2.1], an
ideal with linear quotients always has a non-increasing admissible order. So,
from now, we consider only non-increasing admissible orders.
Let $u_{1},\dots,u_{m}$ be an admissible order of an ideal $I\subset S$ having
linear quotients. For $i\in\\{1,\dots,m\\}$, we let
$\textup{set}(u_{i})=\\{j:x_{j}\in(u_{1},\dots,u_{i-1}):u_{i}\\}.$
Given a non-empty subset $A$ of $\\{1,\dots,n\\}$, we set ${\bf
x}_{A}=\prod_{i\in A}x_{i}$ and ${\bf x}_{\emptyset}=1$. The multigraded
version of [15, Lemma 1.5] implies that
(1) $\textup{HS}_{k}(I)\ =\ (u_{i}{\bf x}_{A}\ :\ i=1,\dots,m,\
A\subseteq\textup{set}(u_{i}),\ |A|=k).$
The ideal $(u_{1},\dots,u_{i-1}):u_{i}$ is generated by the monomials
$u_{j}:u_{i}=\textup{lcm}(u_{j},u_{i})/u_{i}$. Hence, $I$ has linear quotients
if and only if for all $i=1,\dots,m$ and all $j<i$ there exists $\ell<i$ such
that $u_{\ell}:u_{i}=x_{p}$ for some $p$, and $x_{p}$ divides $u_{j}:u_{i}$.
Hereafter, we denote the set $\\{1,\dots,n\\}$ by $[n]$. For a monomial $u\in
S$ and $i\in[n]$, the $x_{i}$-degree of $u$ is the integer
$\deg_{x_{i}}(u)=\max\\{j\geq 0:x_{i}^{j}\ \textup{divides}\ u\\}$.
For the proof of our main result we need Corollary 1.2 of the following lemma.
###### Lemma 1.1.
Let $I$ be an equigenerated graded ideal with linear relations. Let
$f_{1},\dots,f_{m}$ be a minimal set of generators of $I$. Then, for any
$1\leq i\leq m$,
$(f_{1},\dots,f_{i-1},f_{i+1},\dots,f_{m}):f_{i}$
is generated by linear forms.
###### Proof.
To simplify the notation, we may assume that $i=m$, and we set
$J=(f_{1},\dots,f_{m-1}):f_{m}$. Since the $f_{i}$ are homogeneous elements,
$J$ is a graded ideal. Let $r_{m}\in J$ be an homogeneous element. Then, there
exist $r_{1},\ldots,r_{m-1}$ such that
$r_{m}f_{m}=-\sum_{i=1}^{m-1}r_{i}f_{i}$ with $\deg(r_{i})=\deg(r_{m})$ for
$i=1,\ldots,m-1$. Therefore, $r=(r_{1},\ldots,r_{m})$ is a homogeneous
relation of $I$. By assumption, the relation module of $I$ is generated by
linear relations, say $\ell_{i}=(\ell_{i1},\ldots,\ell_{im})$ for
$i=1,\ldots,t$. Therefore, there exist homogeneous elements $s_{i}\in S$ such
that $r=\sum_{i=1}^{t}s_{i}\ell_{i}$. This implies that
$r_{m}=\sum_{i=1}^{t}s_{i}\ell_{i,m}$. Since $\ell_{i,m}\in J$, the desired
conclusion follows.
###### Corollary 1.2.
Let $I$ be an equigenerated monomial ideal with linear quotients and let
$u_{1},\dots,u_{m}$ be its minimal monomial generators. Then, for any $1\leq
i\leq m$,
$(u_{1},\dots,u_{i-1},u_{i+1},\dots,u_{m}):u_{i}$
is generated by variables.
###### Theorem 1.3.
Let $I\subset S$ be an equigenerated monomial ideal having linear quotients.
Then $\textup{HS}_{1}(I)$ has linear quotients.
###### Proof.
We proceed by induction on $m\geq 1$. For $m=1$ or $m=2$ there is nothing to
prove.
Let $m>2$ and set $J=(u_{1},\dots,u_{m-1})$. Let
$L=(x_{i}:i\in\textup{set}(u_{m}),x_{i}u_{m}\notin\textup{HS}_{1}(J))$. Then,
by equation (1),
$\textup{HS}_{1}(I)=\textup{HS}_{1}(J)+u_{m}L.$
By inductive hypothesis, $\textup{HS}_{1}(J)$ has linear quotients. Let
$v_{1},\dots,v_{r}$ be an admissible order of $\textup{HS}_{1}(J)$. If
$L=(x_{j_{1}},\dots,x_{j_{s}})$, we claim that
$v_{1},\dots,v_{r},x_{j_{1}}u_{m},\dots,x_{j_{s}}u_{m}$ is an admissible order
of $\textup{HS}_{1}(I)$. We only need to show that
(2) $(v_{1},\dots,v_{r},x_{j_{1}}u_{m},\dots,x_{j_{t-1}}u_{m}):x_{j_{t}}u_{m}$
is generated by variables, for all $t=1,\dots,s$.
Note that each generator $x_{j_{\ell}}u_{m}:x_{j_{t}}u_{m}=x_{j_{\ell}}$, with
$\ell<t$ is already a variable. Consider now a generator
$v_{\ell}:x_{j_{t}}u_{m}$ for some $\ell=1,\dots,r$. Then
$v_{\ell}=x_{h}u_{j}$ for some $j<m$ and $h\in\text{set}(u_{j})$. Moreover, we
can write $x_{j_{t}}u_{m}=x_{p}u_{k}$ for some $k<m$.
If $j=k$, then
$v_{\ell}:x_{j_{t}}u_{m}=x_{h}u_{k}:x_{p}u_{k}=x_{h}$
is a variable and there is nothing to prove.
Suppose now $j\neq k$. Since $u_{1},\dots,u_{m-1}$ is an admissible order, by
Corollary 1.2
$Q=(u_{1},\dots,u_{k-1},u_{k+1},\dots,u_{m-1}):u_{k}$
is generated by variables. Since $j\neq k$ and $j<m$, $u_{j}:u_{k}$ belongs to
$Q$. Hence, we can find $b<m$, $b\neq k$ such that $u_{b}:u_{k}=x_{q}$ and
$x_{q}$ divides $u_{j}:u_{k}$. Thus $x_{q}u_{k}\in\textup{HS}_{1}(J)$.
Note that $x_{q}$ divides also $x_{h}u_{j}:x_{p}u_{k}$. Indeed $x_{q}$ divides
$u_{j}:u_{k}$. If $x_{q}$ does not divide $x_{h}u_{j}:x_{p}u_{k}$, then
necessarily $p=q$. But this would imply that
$x_{j_{t}}u_{m}=x_{q}u_{k}\in\textup{HS}_{1}(J)$, against the fact that
$x_{j_{t}}\in L$. Hence $x_{q}$ divides $x_{h}u_{j}:x_{p}u_{k}$. But
$x_{q}u_{k}:x_{j_{t}}u_{m}=x_{q}u_{k}:x_{p}u_{k}=x_{q}$
belongs to the ideal (2). Hence $x_{h}u_{j}:x_{p}u_{k}$ is divided by a
variable belonging to the ideal (2). This concludes our proof.
It is natural to ask the following question. Let $I\subset S$ be a monomial
ideal having a linear resolution. Is it true that $\textup{HS}_{1}(I)$ has a
linear resolution, too?
Theorem 1.3 is no longer valid for monomial ideals with linear quotients
generated in more than one degree, as next example of Bayati et all shows [3].
###### Example 1.4.
([3, Example 3.3]). Let
$I=\left(x_{1}^{2},\,x_{1}x_{2},\,x_{2}^{4},\,x_{1}x_{3}^{4},\,x_{1}x_{3}^{3}x_{4},\,x_{1}x_{3}^{2}x_{4}^{2}\right)$
be an ideal of $S=K[x_{1},x_{2},x_{3},x_{4}]$. $I$ is a (strongly) stable
ideal whose Borel generators are
$x_{1}x_{2},x_{2}^{4},x_{1}x_{3}^{2}x_{4}^{2}$. It is well-known that stable
ideals have linear quotients. Thus $I$ has linear quotients. Using Macaulay2
[10] the package [7], we verified that
$\displaystyle\textup{HS}_{1}(I)\ =\ \big{(}$ $\displaystyle
x_{1}^{2}x_{2},\,x_{1}x_{2}^{4},\,x_{1}x_{3}^{3}x_{4}^{2},\,x_{1}x_{2}x_{3}^{2}x_{4}^{2},\,x_{1}^{2}x_{3}^{2}x_{4}^{2},\,x_{1}x_{3}^{4}x_{4},$
$\displaystyle
x_{1}x_{2}x_{3}^{3}x_{4},\,x_{1}^{2}x_{3}^{3}x_{4},\,x_{1}x_{2}x_{3}^{4},\,x_{1}^{2}x_{3}^{4}\big{)}$
has the following Betti table
$\begin{array}[]{c|cccc}&0&1&2&3\\\ \hline\cr 3&1&.&.&.\\\ 4&.&.&.&.\\\
5&1&1&.&.\\\ 6&8&15&8&1\\\ 7&.&.&.&.\\\ 8&.&3&5&2\end{array}$
We show that $\textup{HS}_{1}(I)$ does not have linear quotients. Suppose by
contradiction that $\textup{HS}_{1}(I)$ has linear quotients. Then, since the
Betti numbers of an ideal with linear quotients do not depend upon the
characteristic of the underlying field $K$, we may assume that $K$ has
characteristic zero. Hence $\textup{HS}_{1}(I)$ would be componentwise linear,
see [12, Corollary 8.2.21]. However, this cannot be the case by virtue of [12,
Theorems 8.2.22. and 8.2.23(a)]. Indeed $\beta_{1,1+8}(\textup{HS}_{1}(I))\neq
0$, while $\beta_{0,8}(\textup{HS}_{1}(I))=0$.
## 2\. Homological shifts of proper interval graphs
Let $G$ be a finite simple graph with vertex set $V(G)=[n]$ and edge set
$E(G)$. Let $K$ be a field. The edge ideal of $G$ is the squarefree monomial
ideal $I(G)$ of $S=K[x_{1},\dots,x_{n}]$ generated by the monomials
$x_{i}x_{j}$ such that $\\{i,j\\}\in E(G)$. A graph $G$ is complete if every
$\\{i,j\\}$ with $i,j\in[n]$, $i\neq j$, is an edge of $G$. The open
neighbourhood of $i\in V(G)$ is the set
$N_{G}(i)=\big{\\{}j\in V(G):\\{i,j\\}\in E(G)\big{\\}}.$
A graph $G$ is called chordal if it has no induced cycles of length bigger
than three. Recall that a perfect elimination order of $G$ is an ordering
$v_{1},\dots,v_{n}$ of its vertex set $V(G)$ such that $N_{G_{i}}(v_{i})$
induces a complete subgraph on $G_{i}$, where $G_{i}$ is the induced subgraph
of $G$ on the vertex set $\\{i,i+1,\dots,n\\}$. Hereafter, if $1,2,\dots,n$ is
a perfect elimination order of $G$, we denote it by $x_{1}>x_{2}>\dots>x_{n}$.
###### Theorem 2.1.
(Dirac). A simple finite graph $G$ is chordal if and only if $G$ admits a
perfect elimination order.
The complementary graph $G^{c}$ of $G$ is the graph with vertex set
$V(G^{c})=V(G)$ and where $\\{i,j\\}$ is an edge of $G^{c}$ if and only if
$\\{i,j\\}\notin E(G)$. A graph $G$ is called cochordal if and only if $G^{c}$
is chordal.
###### Theorem 2.2.
(Fröberg). Let $G$ be a simple finite graph. Then, $I(G)$ has a linear
resolution if and only if $G$ is cochordal.
It is known by [12, Theorem 10.2.6] that $I(G)$ has linear resolution if and
only if it has linear quotients. The theorems of Dirac and Fröberg classify
all edge ideals with linear quotients. Furthermore if
$x_{1}>x_{2}>\dots>x_{n}$ is a perfect elimination order of $G^{c}$, then
$I(G)$ has linear quotients with respect to the lexicographic order
$>_{\textup{lex}}$ induced by $x_{1}>x_{2}>\dots>x_{n}$.
Now we turn to the homological shifts of edge ideals with linear quotients.
Unfortunately, in general an edge ideal with linear quotients does not even
has homological linear resolution as next example shows.
###### Example 2.3.
Let $G$ be the following cochordal graph on six vertices.
123456
Let $I=I(G)\subset S=K[x_{1},\dots,x_{6}]$. Using the package [7] we verified
that $\textup{HS}_{0}(I)$ and $\textup{HS}_{1}(I)$ have linear quotients.
However the last homological shift ideal
$\textup{HS}_{2}(I)=(x_{1}x_{2}x_{3}x_{4},\,x_{1}x_{4}x_{5}x_{6})$ has the
following non-linear resolution
$0\rightarrow S(-6)\rightarrow
S(-4)^{2}\rightarrow(x_{1}x_{2}x_{3}x_{4},x_{1}x_{4}x_{5}x_{6})\rightarrow 0.$
In graph theory, one distinguished class of chordal graphs is the family of
proper interval graphs. A graph $G$ is called an interval graph, if one can
label its vertices with some intervals on the real line so that two vertices
are adjacent in $G$, when the intersection of their corresponding intervals is
non-empty. A proper interval graph is an interval graph such that no interval
properly contains another.
Now we are ready to state our main result in the section.
###### Theorem 2.4.
Let $G$ be a cochordal graph on $[n]$ whose complementary graph $G^{c}$ is a
proper interval graph. Then, $I(G)$ has homological linear quotients.
In order to prove the theorem we introduce a more general class of graphs.
We call a perfect elimination order $x_{1}>x_{2}>\dots>x_{n}$ of a chordal
graph $G$ reversible if $x_{n}>x_{n-1}>\dots>x_{1}$ is also a perfect
elimination order of $G$. We call a chordal graph $G$ reversible if $G$ admits
a reversible perfect elimination order. Moreover, a cochordal graph $G$ is
called reversible if and only if $G^{c}$ is reversible.
###### Lemma 2.5.
Let $G$ be a proper interval graph. Then $G$ is reversible.
###### Proof.
By [17, Theorem 1 and Lemma 1], up to a relabeling of the vertex set of $G$,
the following property is satisfied:
1. $(*)$
for all $i<j$, $\\{i,j\\}\in E(G)$ implies that the induced subgraph of $G$ on
$\\{i,i+1\dots,j\\}$ is a clique, _i.e._ , a complete subgraph.
With such a labeling, both $x_{1}>x_{2}>\dots>x_{n}$ and
$x_{n}>x_{n-1}>\dots>x_{1}$ are perfect elimination orders of $G$. By
symmetry, it is enough to show that $x_{1}>x_{2}>\dots>x_{n}$ is a perfect
elimination order. Let $i\in[n]$, $j,k\in N_{G}(i)$ with $j,k>i$. We prove
that $\\{j,k\\}\in E(G)$. Suppose $j>k$. By $(*)$, the induced subgraph of $G$
on $\\{i,i+1\dots,j\\}$ is a clique. Since $j>k>i$, we obtain that
$\\{j,k\\}\in E(G)$, as wanted.
With this lemma at hand, Theorem 2.4 follows from the following more general
result.
###### Theorem 2.6.
Let $G$ be a cochordal graph on $[n]$, and let $x_{1}>\dots>x_{n}$ be a
reversible perfect elimination order of $G^{c}$. Then, $\textup{HS}_{k}(I(G))$
has linear quotients with respect to the lexicographic order
$>_{\textup{lex}}$ induced by $x_{1}>\dots>x_{n}$, for all $k\geq 0$.
For the proof of this theorem, we need a description of the homological shift
ideals.
###### Lemma 2.7.
Let $G$ be a cochordal graph on $[n]$, and let $x_{1}>x_{2}>\dots>x_{n}$ be a
perfect elimination order of $G^{c}$. Then, for all $\\{i,j\\}\in E(G)$, with
$i<j$,
(3) $\textup{set}(x_{i}x_{j})=\\{1,\dots,i-1\\}\cup(\\{i+1,\dots,j-1\\}\cap
N_{G}(i)).$
In particular,
$\displaystyle\textup{HS}_{k}(I(G))=\big{(}{\bf x}_{A}{\bf x}_{B}\ :$
$\displaystyle\ A,B\subseteq[n],A,B\neq\emptyset,\max(A)<\min(B),|A\cup
B|=k+2,$ $\displaystyle\ \\{\max(A),b\\}\in E(G),\ \text{for all}\ b\in
B\big{)}.$
###### Proof.
As remarked before $I(G)$ has linear quotients with respect to the
lexicographic order $>_{\textup{lex}}$ induced by $x_{1}>x_{2}>\dots>x_{n}$.
Let $\\{i,j\\}\in E(G)$ with $i<j$. Let us determine
$\textup{set}(x_{i}x_{j})$. If $k\in\textup{set}(x_{i}x_{j})$, then
$x_{k}(x_{i}x_{j})/x_{\ell}\in I(G)$ and
$x_{k}(x_{i}x_{j})/x_{\ell}>_{\textup{lex}}x_{i}x_{j}$ for some
$\ell\in\\{i,j\\}$. Note that $k<j$, indeed for $k>j$, both
$x_{i}x_{k},x_{j}x_{k}$ are smaller than $x_{i}x_{j}$ in the lexicographic
order. Thus either $k<i$ or $i<k<j$. We distinguish the two possible cases.
Case 1. Suppose $k<i$. Assume that none of $x_{k}x_{i},x_{k}x_{j}$ is in
$I(G)$. Then $\\{k,i\\},\\{k,j\\}\in E(G^{c})$. Since
$x_{1}>x_{2}>\dots>x_{n}$ is a perfect elimination order, the induced graph of
$G_{i}^{c}$ on the vertex set $N_{G_{k}^{c}}(k)$ is complete. But $i,j>k$ and
$i,j\in N_{G_{k}^{c}}(k)$. Thus we would have $\\{i,j\\}\in E(G^{c})$, that
is, $x_{i}x_{j}\notin I(G)$, absurd.
Case 2. Suppose $i<k<j$. Since $k>i$, $x_{k}x_{j}<_{\textup{lex}}x_{i}x_{j}$.
Thus $k\in\textup{set}(x_{i}x_{j})$ if and only if $x_{i}x_{k}\in E(G)$, that
is $k\in N_{G}(i)$.
The two cases above show that equation (3) holds. The formula for
$\textup{HS}_{k}(I(G))$ follows immediately by applying equations (1) and (3).
For the proof of the theorem we recall the concept of Betti splitting [8].
Let $I$, $I_{1}$, $I_{2}$ be monomial ideals of $S$ such that $G(I)$ is the
disjoint union of $G(I_{1})$ and $G(I_{2})$. We say that $I=I_{1}+I_{2}$ is a
Betti splitting if
$\beta_{i,j}(I)=\beta_{i,j}(I_{1})+\beta_{i,j}(I_{2})+\beta_{i-1,j}(I_{1}\cap
I_{2})\ \ \ \textup{for all}\ i,j.$
###### Proof of Theorem 2.6..
We proceed by induction on $n\geq 1$. Let $G^{\prime}$ be the induced subgraph
of $G$ on the vertex set $\\{2,3,\dots,n\\}$. Then $x_{2}>x_{3}>\dots>x_{n}$
is again a reversible perfect elimination order of $(G^{\prime})^{c}$ and
$G^{\prime}$ is a reversible cochordal graph.
Let $J=(x_{i}:x_{1}x_{i}\in I(G))$. Then, $I(G)=x_{1}J+I(G^{\prime})$ is a
Betti splitting, because $G(I(G))$ is the disjoint union of $G(x_{1}J)$ and
$G(I(G^{\prime}))$, and $x_{1}J$, $I(G^{\prime})$ have linear resolutions, see
[8, Corollary 2.4]. Since $I(G^{\prime})\cap x_{1}J=x_{1}I(G^{\prime})$, [4,
Proposition 1.7] gives
$\textup{HS}_{k}(I(G))=x_{1}\big{(}\textup{HS}_{k-1}(I(G^{\prime}))+\textup{HS}_{k}(J)\big{)}+\textup{HS}_{k}(I(G^{\prime})).$
We claim that $\textup{HS}_{k}(I(G))$ has linear quotients with respect to the
lexicographic order $>_{\textup{lex}}$ induced by $x_{1}>x_{2}>\dots>x_{n}$.
For $k=0$ this is true. Let $k>0$.
Let $u=x_{i_{1}}x_{j_{1}}{\bf x}_{F_{1}},v=x_{i_{2}}x_{j_{2}}{\bf
x}_{F_{2}}\in G(\textup{HS}_{k}(I(G)))$, with $u>_{\textup{lex}}v$,
$i_{1}<j_{1},i_{2}<j_{2}$, $x_{i_{1}}x_{j_{1}},\ x_{i_{2}}x_{j_{2}}\in I(G)$,
$F_{1}\subseteq\textup{set}(u)$, $F_{2}\subseteq\textup{set}(v)$. We are going
to prove that there exists $w\in G(\textup{HS}_{k}(I(G)))$ such that
$w>_{\textup{lex}}v$, $w:v=x_{p}$ and $x_{p}$ divides $u:v$.
We can write
$u=x_{p_{1}}x_{p_{2}}\cdots x_{p_{k+2}},\ \ \ \ v=x_{q_{1}}x_{q_{2}}\cdots
x_{q_{k+2}},$
with $p_{1}<p_{2}<\dots<p_{k+2}$, $q_{1}<q_{2}<\dots<q_{k+2}$. Since
$u>_{\textup{lex}}v$ then $p_{1}=q_{1}$, $p_{2}=q_{2}$, $\dots$,
$p_{s-1}=q_{s-1}$, $p_{s}<q_{s}$ for some $s\in\\{1,\dots,k+2\\}$. If $s=k+2$,
then $u:v=x_{p_{k+2}}=x_{j_{1}}$ and there is nothing to prove. Therefore, we
may assume $s<k+2$. Thus $p_{s}<q_{s}<q_{k+2}=j_{2}$. Set $p=p_{s}$ and
$q=q_{s}$, then $x_{p}$ divides $u:v$.
Suppose for the moment that $x_{1}$ divides $v$. Then by definition of
$>_{\textup{lex}}$, $p_{1}=q_{1}=1$ and $x_{1}$ divides $u$, too. There are
four cases to consider.
Case 1. Suppose $i_{1}=i_{2}=1$. Setting $u^{\prime}=u/x_{1}$ and
$v^{\prime}=v/x_{1}$, we have $u^{\prime},v^{\prime}\in G(\textup{HS}_{k}(J))$
and $u^{\prime}>_{\textup{lex}}v^{\prime}$. Since $J$ is an ideal generated by
variables, it has homological linear quotients with respect to
$>_{\textup{lex}}$. Hence, there exists $w^{\prime}\in G(\textup{HS}_{k}(J))$
with $w^{\prime}>_{\textup{lex}}v^{\prime}$ such that
$w^{\prime}:v^{\prime}=x_{\ell}$ and $x_{\ell}$ divides
$u^{\prime}:v^{\prime}$. Setting $w=x_{1}w^{\prime}$, we have that
$w>_{\textup{lex}}v$ and $w\in G(x_{1}\textup{HS}_{k}(J))\subseteq
G(\textup{HS}_{k}(I(G)))$. Hence $w:v=w^{\prime}:v^{\prime}=x_{\ell}$ and
$x_{\ell}$ divides $u:v=u^{\prime}:v^{\prime}$.
Case 2. Suppose $i_{1}>1$ and $i_{2}>1$. Setting $u^{\prime}=u/x_{1}$ and
$v^{\prime}=v/x_{1}$, we have $u^{\prime},v^{\prime}\in
G(\textup{HS}_{k-1}(I(G^{\prime})))$ and
$u^{\prime}>_{\textup{lex}}v^{\prime}$. By inductive hypothesis,
$I(G^{\prime})$ has homological linear quotients with respect to
$>_{\textup{lex}}^{\prime}$ induced by $x_{2}>x_{3}>\dots>x_{n}$. Hence, there
exists $w^{\prime}\in G(\textup{HS}_{k-1}(I(G^{\prime})))$ with
$w^{\prime}>_{\textup{lex}}^{\prime}v^{\prime}$ such that
$w^{\prime}:v^{\prime}=x_{\ell}$ and $x_{\ell}$ divides
$u^{\prime}:v^{\prime}$. Setting $w=x_{1}w^{\prime}$, we have that
$w>_{\textup{lex}}v$ and $w\in
G(x_{1}\textup{HS}_{k-1}(I(G^{\prime})))\subseteq G(\textup{HS}_{k}(I(G)))$.
Hence $w:v=w^{\prime}:v^{\prime}=x_{\ell}$ and $x_{\ell}$ divides
$u:v=u^{\prime}:v^{\prime}$.
Case 3. Suppose $i_{1}>1$ and $i_{2}=1$. Then $1=i_{2}<p<j_{2}$.
Subcase 3.1. Assume $x_{1}x_{p}\in I(G)$, then
$p\in\textup{set}(x_{i_{2}}x_{j_{2}})$. Setting $w=x_{p}(v/x_{q})$, by
equation (1) $w\in G(\textup{HS}_{k}(I(G)))$, and $w>_{\textup{lex}}v$,
because $p<q$. Moreover $w:v=x_{p}$ and $x_{p}$ divides $u:v$.
Subcase 3.2. Assume that $x_{1}x_{p}\notin I(G)$. By hypothesis,
$x_{n}>x_{n-1}>\dots>x_{1}$ is also a perfect elimination order of $G^{c}$.
Thus, by Lemma 2.7, we can write $u={\bf x}_{A}{\bf x}_{B}$ with
$A=\\{p_{k+2},p_{k+1},\dots,p_{r}\\}$, $B=\\{p_{r-1},\dots,p_{2},p_{1}\\}$ for
some $r>1$ and with $\\{p_{r},p_{\ell}\\}\in E(G)$ for all
$\ell=r-1,\dots,2,1$. Since $\\{1,p\\}=\\{p_{1},p_{s}\\}\notin E(G)$, by the
above presentation of $u$, $s>r$. Using again Lemma 2.7, but considering the
reversed perfect elimination order $x_{n}>x_{n-1}>\dots>x_{1}$, we see that
$\displaystyle w$ $\displaystyle=x_{q_{s+1}}x_{q_{s+2}}\cdots
x_{q_{k+2}}u/(x_{p_{s+1}}x_{p_{s+2}}\cdots x_{p_{k+2}})$ $\displaystyle={\bf
x}_{(A\setminus\\{p_{s+1},p_{s+2},\dots,p_{k+2}\\})\cup\\{q_{s+1},q_{s+2},\dots,q_{k+2}\\}}{\bf
x}_{B}\in G(\textup{HS}_{k}(I(G))).$
Moreover, $w>_{\textup{lex}}v$, $w:v=x_{p}$ and $x_{p}$ divides $u:v$, as
desired.
Case 4. Suppose $i_{1}=1$ and $i_{2}>1$. Recall that $p<j_{2}$. Moreover
$p\neq i_{2}$, because $x_{p}$ divides $u:v$ but $x_{i_{2}}$ divides $v$. Thus
there are two cases to consider.
Subcase 4.1. Assume $p<i_{2}$. By Lemma 2.7,
$p\in\textup{set}(x_{i_{2}}x_{j_{2}})$. If $q\neq i_{2}$, then $q<j_{2}$ and
by equation (1) $w=x_{p}(v/x_{q})$ is a minimal generator of
$\textup{HS}_{k}(I(G))$. Moreover $w>_{\textup{lex}}v$ and $w:v=x_{p}$ divides
$u:v$, as wanted. Suppose now that $q=i_{2}$. If there exists $\ell$ such that
$x_{\ell}$ divides $v$ and $i_{2}<\ell<j_{2}$, then $\ell>p$ and
$w=x_{p}(v/x_{\ell})$ is a minimal generator of $\textup{HS}_{k}(I(G))$ such
that $w>_{\textup{lex}}v$ and with $w:v=x_{p}$ dividing $u:v$, as wanted.
Otherwise, suppose no such integer $\ell$ exists. Then, $s=k+1$,
$q_{k+1}=i_{2}$ and $q_{k+2}=j_{2}$. Since
$p\in\textup{set}(x_{i_{2}}x_{j_{2}})$, then $x_{p}x_{\ell}\in I(G)$, where
$\ell\in\\{i_{2},j_{2}\\}$. Then $p<\ell$ and by Lemma 2.7 we see that
$w=x_{p}(v/x_{\ell})$ is a minimal generator of $\textup{HS}_{k}(I(G))$ such
that $w>_{\textup{lex}}v$ and with $w:v=x_{p}$ dividing $u:v$.
Subcase 4.2. Assume now $i_{2}<p<j_{2}$. If $x_{i_{2}}x_{p}\in I(G)$, by Lemma
2.7, $p\in\textup{set}(x_{i_{2}}x_{j_{2}})$. Setting $w=x_{p}(v/x_{q})$, we
have $w\in G(\textup{HS}_{k}(I(G)))$, $w>_{\textup{lex}}v$ and $w:v=x_{p}$
divides $u:v$. Suppose now that $x_{i_{2}}x_{p}\notin I(G)$. By hypothesis,
$x_{n}>x_{n-1}>\dots>x_{1}$ is also a perfect elimination order of $G^{c}$.
Thus, by Lemma 2.7, we can write $u={\bf x}_{A}{\bf x}_{B}$ with
$A=\\{p_{k+2},p_{k+1},\dots,p_{r}\\}$, $B=\\{p_{r-1},\dots,p_{2},p_{1}\\}$ for
some $r>1$ and with $\\{p_{r},p_{\ell}\\}\in E(G)$ for all
$\ell=r-1,\dots,2,1$. Note that $i_{2}<p$, so $x_{i_{2}}$ divides $u$. Since
$\\{i_{2},p\\}=\\{i_{2},p_{s}\\}\notin E(G)$, by the above presentation of
$u$, $s>r$. Using again Lemma 2.7, but considering the reversed perfect
elimination order $x_{n}>x_{n-1}>\dots>x_{1}$, we see that
$\displaystyle w$ $\displaystyle=x_{q_{s+1}}x_{q_{s+2}}\cdots
x_{q_{k+2}}u/(x_{p_{s+1}}x_{p_{s+2}}\cdots x_{p_{k+2}})$ $\displaystyle={\bf
x}_{A}{\bf
x}_{(B\setminus\\{p_{s+1},p_{s+2},\dots,p_{k+2}\\})\cup\\{q_{s+1},q_{s+2},\dots,q_{k+2}\\}}\in
G(\textup{HS}_{k}(I(G))).$
Moreover, $w>_{\textup{lex}}v$, $w:v=x_{p}$ and $x_{p}$ divides $u:v$, as
desired.
Suppose now that $x_{1}$ does not divide $v$. Then $v\in
G(\textup{HS}_{k}(I(G^{\prime})))$. If $x_{1}$ does not divide $u$, then $u\in
G(\textup{HS}_{k}(I(G^{\prime})))$, too. Let $>_{\textup{lex}}^{\prime}$ be
the lexicographic order induced by $x_{2}>x_{3}>\dots>x_{n}$. Since by
induction $I(G^{\prime})$ has homological linear quotients with respect to
$>_{\textup{lex}}^{\prime}$ and also $u>_{\textup{lex}}^{\prime}v$, there
exists $w\in G(\textup{HS}_{k}(I(G^{\prime})))$, with
$w>_{\textup{lex}}^{\prime}v$, $w:v=x_{\ell}$ and $x_{\ell}$ divides $u:v$.
But also we have $w\in G(\textup{HS}_{k}(I(G)))$ and $w>_{\textup{lex}}v$.
Otherwise if $x_{1}$ divides $u$, then $x_{1}$ divides $u:v$. Since
$\textup{HS}_{k}(I(G^{\prime}))\subseteq\textup{HS}_{k-1}(I(G^{\prime}))$ and
$k>0$, we can write $v=x_{t}w^{\prime}$ with $w^{\prime}\in
G(\textup{HS}_{k-1}(I(G^{\prime})))$. Let $w=x_{1}w^{\prime}$. Then
$w>_{\textup{lex}}v$ and $w:v=x_{1}$ divides $u:v$.
Hence, the inductive proof is complete and the theorem is proved.
###### Remark 2.8.
Let $x_{1}>x_{2}>\dots>x_{n}$ be a reversible perfect elimination order of
$G^{c}$. By symmetry, Theorem 2.6 shows also that $\textup{HS}_{k}(I(G))$ has
linear quotients with respect to the lexicographic order induced by
$x_{n}>x_{n-1}>\dots>x_{1}$.
###### Examples 2.9.
Let $n,m$ be two positive integers.
1. (a)
Let $G=K_{n,m}$ be the complete bipartite graph. That is, $V(G)=[n+m]$ and
$E(G)=\big{\\{}\\{i,j\\}:i\in[n],j\in\\{n+1,\dots,n+m\\}\big{\\}}$. For
example, for $n=3$ and $m=4$
1234567
It is easy to see that $G^{c}$ is the disjoint union of two complete graphs
$\Gamma_{1}$ and $\Gamma_{2}$ on vertex sets $[n]$ and $\\{n+1,\dots,n+m\\}$
respectively. Furthermore, any ordering of the vertices is a perfect
elimination order of $G^{c}$. Applying the previous theorem,
$I(G)=(x_{1},\dots,x_{n})(x_{n+1},\dots,x_{m})$
has homological linear quotients with respect to the lexicographic order
induced by any ordering of the variables.
2. (b)
Let $G$ be the graph with vertex set $V(G)=[n+m]$ and edge set
$E(G)=\big{\\{}\\{i,j\\}:i\in[n+m],n+1\leq j\leq n+m,i<j\big{\\}}.$
We claim that $G$ is a reversible cochordal graph. Indeed $G^{c}$ is the
disjoint union of the complete graph $K_{n}$ on the vertex set $[n]$ together
with the set of isolated vertices $\\{n+1,\dots,n+m\\}$. It easily seen that
any ordering of the vertices is a perfect elimination order of $G^{c}$.
Applying Theorem 2.6
$I(G)=(x_{1},\dots,x_{n})(x_{n+1},\dots,x_{m})+(x_{i}x_{j}:n+1\leq i<j\leq
n+m)$
has homological linear quotients with respect to the lexicographic order
induced by any ordering of the variables.
## 3\. Homological shifts of Trees
In this section we construct several classes of edge ideals with homological
linear quotients, by considering various operations on cochordal graphs that
preserve the homological linear quotients property. As a main application of
all these results we will prove the following theorem.
###### Theorem 3.1.
Let $G$ be a graph such that $G^{c}$ is a forest. Then $I(G)$ has homological
linear quotients.
The squarefree Veronese ideal $I_{n,d}$ of degree $d$ in
$S=K[x_{1},\dots,x_{n}]$ is the ideal of $S$ generated by all squarefree
monomials of degree $d$ in $S$. It is well-known that $I_{n,d}$ has
homological linear quotients, (see for instance [13, Corollary 3.2]).
The first operation we consider consists in adding whiskers. Let
$\Gamma^{\prime}$ be a graph on vertex set $[n-1]$. Let $i\in[n-1]$ and let
$\Gamma$ be the graph with vertex set $[n]$ and edge set
$V(\Gamma)=V(\Gamma^{\prime})\cup\\{\\{i,n\\}\\}$. $\Gamma$ is called the
whisker graph of $\Gamma^{\prime}$ obtained by adding the whisker $\\{i,n\\}$
to $\Gamma^{\prime}$.
###### Proposition 3.2.
Let $\Gamma^{\prime}$ be a graph on vertex set $[n-1]$ and $\Gamma$ be the
graph on vertex set $[n]$ and edge set
$V(\Gamma)=V(\Gamma^{\prime})\cup\\{\\{i,n\\}\\}$ for some $i\in[n-1]$. Set
$G=\Gamma^{c}$. Suppose $I((\Gamma^{\prime})^{c})$ has homological linear
quotients. Then $I(G)$ has homological linear quotients, too.
###### Proof.
Since $\Gamma^{\prime}$ is chordal, obviously $\Gamma$ is chordal, too. Set
$J=I((\Gamma^{\prime})^{c})$, $I=I(G)$ and
$L=(x_{j}:j\in[n-1]\setminus\\{i\\})$. Since $N_{G^{c}}(n)=\\{i\\}$, we have
the Betti splitting:
(4) $I=x_{n}L+J.$
Since $G$ is cochordal, $\textup{HS}_{0}(I)$ and $\textup{HS}_{1}(I)$ have
linear quotients. So we only have to show that $\textup{HS}_{k}(I)$ has linear
quotients for $k\geq 2$. By equation (4), for all $k\geq 2$,
$\textup{HS}_{k}(I)=x_{n}\textup{HS}_{k}(L)+x_{n}\textup{HS}_{k-1}(J)+\textup{HS}_{k}(J).$
Note that $\textup{HS}_{k}(L)$ is the squarefree Veronese ideal of degree
$k+1$ in the polynomial ring $K[x_{j}:j\in[n-1]\setminus\\{i\\}]$. Thus
$\textup{HS}_{k}(L)$ has linear quotients with admissible order, say,
$u_{1},\dots,u_{m}$. Let $v_{1},\dots,v_{r}$ and $w_{1},\dots,w_{s}$ be
admissible orders of $\textup{HS}_{k-1}(J)$ and $\textup{HS}_{k}(J)$,
respectively. Let $v_{j_{1}},\dots,v_{j_{p}}$, with $j_{1}<j_{2}<\dots<j_{p}$,
the monomials in $G(\textup{HS}_{k-1}(J))\setminus G(\textup{HS}_{k}(L))$. We
claim that
(5) $x_{n}u_{1},\dots,x_{n}u_{m},\ x_{n}v_{j_{1}},\dots,x_{n}v_{j_{p}},\
w_{1},\dots,w_{s}$
is an admissible order of $\textup{HS}_{k}(J)$.
Let $\ell\in\\{1,\dots,m\\}$. Then
$(x_{n}u_{1},\dots,x_{n}u_{\ell-1}):x_{n}u_{\ell}=(u_{1},\dots,u_{\ell-1}):u_{\ell}$
is generated by variables.
Let $\ell\in\\{1,\dots,p\\}$. We show that
$\displaystyle Q$
$\displaystyle=(x_{n}u_{1},\dots,x_{n}u_{m},x_{n}v_{j_{1}},\dots,x_{n}v_{j_{\ell-1}}):x_{n}v_{j_{\ell}}$
$\displaystyle=(u_{1},\dots,u_{m},v_{j_{1}},\dots,v_{j_{\ell-1}}):v_{j_{\ell}}$
is generated by variables. Consider $v_{j_{q}}:v_{j_{\ell}}$, then we can find
$d<j_{\ell}$ such that $v_{d}:v_{j_{\ell}}$ is a variable that divides
$v_{j_{q}}:v_{j_{\ell}}$. Either $d=j_{b}$, for some $b<\ell$, or
$v_{d}\in\textup{HS}_{k}(L)$. In any case,
$x_{n}v_{d}\in(u_{1},\dots,u_{m},v_{j_{1}},\dots,v_{j_{\ell-1}})$ and
$v_{d}:v_{j_{\ell}}\in Q$ divides $v_{j_{q}}:v_{j_{\ell}}$.
Consider now $u_{q}:v_{j_{\ell}}$, $1\leq q\leq m$. Hence $x_{i}$ divides
$v_{j_{\ell}}$, lest $v_{j_{\ell}}\in G(\textup{HS}_{k}(L))$. But then
$v_{j_{\ell}}/x_{i}\in\textup{HS}_{k-1}(L)$. Let $x_{t}$ dividing
$u_{q}:v_{j_{\ell}}$. Then $u=x_{t}v_{j_{\ell}}/x_{i}\in\textup{HS}_{k}(L)$
and $u:v_{j_{\ell}}=x_{t}\in Q$ divides $u_{q}:v_{j_{\ell}}$.
Finally, let $\ell\in\\{1,\dots,s\\}$. We show that
$\displaystyle Q$
$\displaystyle=(x_{n}u_{1},\dots,x_{n}u_{m},x_{n}v_{j_{1}},\dots,x_{n}v_{j_{p}},w_{1},\dots,w_{\ell-1}):w_{\ell}$
$\displaystyle=(x_{n}\textup{HS}_{k}(L)+x_{n}\textup{HS}_{k-1}(J)):w_{\ell}+(w_{1},\dots,w_{\ell-1}):w_{\ell}$
is generated by variables. Since $w_{1},\dots,w_{s}$ is an admissible order,
$(w_{1},\dots,w_{\ell-1}):w_{\ell}$ is generated by variables. Consider now a
generator $x_{n}z:w_{\ell}$ with $z\in\textup{HS}_{k}(L)$ or
$z\in\textup{HS}_{k-1}(J)$. Then $x_{n}$ divides $x_{n}z:w_{\ell}$. On the
other hand $w_{\ell}/x_{t}\in\textup{HS}_{k-1}(J)$ for some $t$. But then
$x_{n}w_{\ell}/x_{t}:w_{\ell}=x_{n}\in Q$ divides our generator.
The three cases above show that (5) is an admissible order, as desired.
Since any tree can be constructed iteratively by adding a whisker to a tree on
a smaller vertex set at each step, the previous proposition implies
immediately
###### Corollary 3.3.
Let $G$ be a graph such that $G^{c}$ is a tree. Then $I(G)$ has homological
linear quotients.
The second operation we consider consists in joining disjoint graphs. Two
graphs $\Gamma_{1}$ and $\Gamma_{2}$ are called disjoint if $V(\Gamma_{1})\cap
V(\Gamma_{2})=\emptyset$. The join of $\Gamma_{1}$ and $\Gamma_{2}$ is the
graph $\Gamma$ with vertex set $V(\Gamma)=V(\Gamma_{1})\cup V(\Gamma_{2})$ and
edge set $E(\Gamma)=E(\Gamma_{1})\cup E(\Gamma_{2})$.
###### Proposition 3.4.
Let $\Gamma_{1}$ and $\Gamma_{2}$ be disjoint chordal graphs such that
$I(\Gamma_{1}^{c}),I(\Gamma_{2}^{c})$ have homological linear quotients. Let
$\Gamma$ be the join of $\Gamma_{1}$ and $\Gamma_{2}$ and set $G=\Gamma^{c}$.
Then $I(G)$ has homological linear quotients, too.
###### Proof.
Obviously $\Gamma$ is chordal, too. Let $G_{1}=\Gamma_{1}^{c}$,
$G_{2}=\Gamma_{2}^{c}$, $V(G_{1})=[n]$ and $V(G_{2})=\\{n+1,\dots,n+m\\}$. Set
$L=(x_{1},\dots,x_{n})(x_{n+1},\dots,x_{m})$. Then,
$I(G)=I(G_{1})+I(G_{2})+L$
Suppose $x_{1}>\dots>x_{n}$ and $x_{n+1}>\dots>x_{n+m}$ are perfect
elimination orders of $\Gamma_{1}$ and $\Gamma_{2}$. Then $G=\Gamma^{c}$ is
cochordal. Indeed, $x_{1}>\dots>x_{n}>x_{n+1}>\dots>x_{n+m}$ is a perfect
elimination order of $\Gamma$. Let $>_{\textup{lex}}$ be the lexicographic
order induced by such an ordering of the variables. Set, $I=I(G)$,
$I_{1}=I(G_{1})$ and $I_{2}=I(G_{2})$. Then, $I,I_{1},I_{2}$ and $J$ have
linear quotients with respect to $>_{\textup{lex}}$.
Let $k\geq 0$ and $u\in G(\textup{HS}_{k}(I))$ such that $x_{i}x_{j}$ divides
$u$ for some integers $i\in[n]$, $n+1\\!\leq\\!j\leq\\!n+m$. We claim that
$u\in G(\textup{HS}_{k}(L))$. Let $i_{0}=\max\\{i\in[n]:x_{i}\
\textup{divides}\ u\\}$ and $j_{0}=\max\\{j\in\\{n+1,\dots,n+m\\}:x_{j}\
\textup{divides}\ u\\}$. Let $u/(x_{i_{0}}x_{j_{0}})={\bf x}_{F}$. Then
$F\subseteq\\{1,\dots,i_{0}-1\\}\cup\\{n+1,\dots,j_{0}-1\\}=\textup{set}_{I}(x_{i_{0}}x_{j_{0}})$
and $x_{i_{0}}x_{j_{0}}\in L$. Thus, by equation (1),
$u=x_{i_{0}}x_{j_{0}}{\bf x}_{F}\in\textup{HS}_{k}(L)$, as desired. This
argument shows that any squarefree monomial $w\in K[x_{1},\dots,x_{n+m}]$ of
degree $k+2$, containing as a factor any monomial $x_{i}x_{j}$ with $i\in[n]$
and $n+1\leq j\leq n+m$, is a generator of $\textup{HS}_{k}(L)$.
From this remark, for all $k\geq 0$, it follows that
$\textup{HS}_{k}(I)=\textup{HS}_{k}(L)+\textup{HS}_{k}(I_{1})+\textup{HS}_{k}(I_{2}).$
Note that $L$ is the edge ideal of a complete bipartite graph. By Examples
2.9(a), $L$ has homological linear quotients. Let $u_{1},\dots,u_{m}$ be an
admissible order of $\textup{HS}_{k}(L)$. Moreover, let $v_{1},\dots,v_{r}$
and $w_{1},\dots,w_{s}$ be admissible orders of $\textup{HS}_{k}(I_{1})$ and
$\textup{HS}_{k}(I_{2})$, respectively. Note that the monomials
$u_{i},v_{j},w_{t}$ are all different, because all monomials $u_{i}$ contain a
factor $x_{i_{0}}x_{j_{0}}$ with $i_{0}\in[n]$ and
$j_{0}\in\\{n+1,\dots,n+m\\}$. Whereas, the $v_{j}$ are monomials in
$K[x_{1},\dots,x_{n}]$ and the $w_{t}$ are monomials in
$K[x_{n+1},\dots,x_{n+m}]$.
We claim that
(6) $u_{1},\dots,u_{m},\ v_{1},\dots,v_{r},\ w_{1},\dots,w_{s}$
is an admissible order of $\textup{HS}_{k}(I)$.
Let $\ell\in\\{1,\dots,m\\}$. Then $(u_{1},\dots,u_{\ell-1}):u_{\ell}$ is
generated by variables.
Let $\ell\in\\{1,\dots,r\\}$. We show that
$Q=(u_{1},\dots,u_{m},v_{1},\dots,v_{\ell-1}):v_{\ell}$
is generated by variables. Clearly $(v_{1},\dots,v_{\ell-1}):v_{\ell}$ is
generated by variables. Consider now $u_{q}:v_{\ell}$, $1\leq q\leq m$. Recall
that $v_{\ell}$ is a monomial in $K[x_{1},\dots,x_{n}]$. Thus $x_{j}$ divides
$u_{q}:v_{\ell}$ for some $j\in\\{n+1,\dots,n+m\\}$. Consider $v_{\ell}/x_{t}$
for some $t$. Then $u=x_{j}(v_{\ell}/x_{t})\in\textup{HS}_{k}(L)$ and
$u:v_{\ell}=x_{j}\in Q$, as desired.
Finally, let $\ell\in\\{1,\dots,s\\}$. We show that
$Q=(u_{1},\dots,u_{m},v_{1},\dots,v_{r},w_{1},\dots,w_{\ell-1}):w_{\ell}$
is generated by variables. Since $w_{1},\dots,w_{s}$ is an admissible order,
$(w_{1},\dots,w_{\ell-1}):w_{\ell}$ is generated by variables. Consider now a
generator $z:w_{\ell}$ with $z=u_{q}$ or $z=v_{q}$, for some $q$. Since
$w_{\ell}$ is a monomial in $K[x_{n+1},\dots,x_{n+m}]$, $z:w_{\ell}$ is
divided by a variable $x_{i}$, where $i\in[n]$. Consider $w_{\ell}/x_{t}$ for
some $t$. Then $u=x_{i}(w_{\ell}/x_{t})\in\textup{HS}_{k}(L)$ and
$u:w_{\ell}=x_{i}\in Q$, as desired.
The three cases above show that (6) is an admissible order, as desired.
###### Proof of Theorem 3.1.
Let $\Gamma=G^{c}$ be a forest and let $c$ be the number of connected
components of $\Gamma$. If $c=1$, then $\Gamma$ is a tree, and by Corollary
3.3, $I(G)$ has homological linear quotients. Suppose $c>1$ and write
$\Gamma=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_{1}$ and $\Gamma_{2}$ are
disjoint forests. The the numbers of connected components of $\Gamma_{1}$ and
$\Gamma_{2}$ are smaller than $c$. Thus, by induction $I(\Gamma_{1}^{c})$ and
$I(\Gamma_{2}^{c})$ have homological linear quotients. Applying Proposition
3.4, it follows that $I(G)$ has homological linear quotients, too.
Let $G$ be a complete multipartite graph, then $G^{c}$ is the disjoint union
of some complete graphs. Repeated applications of Proposition 3.4 yield
###### Corollary 3.5.
Let $G$ be a complete multipartite graph. Then $I(G)$ has homological linear
quotients.
## 4\. Polymatroidal ideals generated in degree two
A polymatroidal ideal $I\subset S=K[x_{1},\dots,x_{n}]$ is a monomial ideal
$I$ generated in a single degree verifying the following exchange property:
for all $u,v\in G(I)$ with $u\neq v$ and all $i$ such that
$\deg_{x_{i}}(u)>\deg_{x_{i}}(v)$, there exists $j$ such that
$\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$ and $x_{j}(u/x_{i})\in G(I)$.
The name polymatroidal ideal is justified by the fact that their minimal
generating set corresponds to the set of bases of a discrete polymatroid. A
squarefree polymatroidal ideal is called matroidal. Any polymatroidal ideal
also satisfy a dual version of the exchange property.
###### Lemma 4.1.
([11, Lemma 2.1]). Let $I\subset S$ be a polymatroidal ideal. Then, for all
$u,v\in G(I)$ and all $i$ such that $\deg_{x_{i}}(u)>\deg_{x_{i}}(v)$, there
exists $j$ such that $\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$ and $x_{i}(v/x_{j})\in
G(I)$.
There are many useful characterization of polymatroidal ideals. The following
one is due Bandari and Rahmati-Asghar.
###### Theorem 4.2.
([1, Theorem 2.4]). Let $I\subset S$ be a monomial ideal generated in a single
degree. Then, $I$ is polymatroidal if and only if $I$ has linear quotients
with respect to the lexicographic order induced by any ordering of the
variables.
It is expected by Bandari, Bayati and Herzog that the homological shift ideals
$\textup{HS}_{k}(I)$ of a polymatroidal ideal $I$ are all polymatroidal, see
[2, 13]. In this section, we provide an affirmative answer to this conjecture
for all polymatroidal ideals generated in degree two.
Firstly, we deal with the squarefree case.
###### Lemma 4.3.
Let $I\subset S$ be a matroidal ideal generated in degree two, and let $G$ be
the simple graph on $[n]$ such that $I=I(G)$. Then, any ordering of the
variables is a perfect elimination order of $G^{c}$.
###### Proof.
Up to relabeling, we can consider the ordering $x_{1}>x_{2}>\dots>x_{n}$. Let
$j,k\in N_{G^{c}}(i)$ with $j,k>i$. We must prove that $\\{j,k\\}\in
E(G^{c})$. By our assumption, $\\{i,j\\},\\{i,k\\}\notin E(G)$, that is
$x_{i}x_{j},x_{i}x_{k}\notin I(G)=I$. Suppose by contradiction that
$\\{j,k\\}\notin E(G^{c})$, then $\\{j,k\\}\in E(G)$, that is, $x_{j}x_{k}\in
I(G)$. Pick any monomial $x_{i}x_{s}\in I(G)$. Then
$\deg_{x_{i}}(x_{i}x_{s})>\deg_{x_{i}}(x_{j}x_{k})$. By Lemma 4.1, we can find
$\ell$ with $\deg_{x_{\ell}}(x_{i}x_{s})<\deg_{x_{\ell}}(x_{j}x_{k})$ and
$x_{i}(x_{j}x_{k})/x_{\ell}\in I(G)$. Thus, either $x_{i}x_{j}\in I(G)$ or
$x_{i}x_{k}\in I(G)$. This is a contradiction. Hence $\\{j,k\\}\in E(G^{c})$,
as desired.
###### Corollary 4.4.
Let $I\subset S$ be a matroidal ideal generated in degree two. Then
$\textup{HS}_{k}(I)$ is a matroidal ideal, for all $k\geq 0$.
###### Proof.
Let $G$ be the simple graph on $[n]$ such that $I=I(G)$. By Lemma 4.3 and
Theorem 2.2, $G^{c}$ is a reversible chordal graph and any ordering of the
variables is a reversible perfect elimination order of $G^{c}$. By Theorem
2.6, for all $k\geq 0$, $\textup{HS}_{k}(I)$ has linear quotients with respect
to the lexicographic order induced by any ordering of the variables. Thus, by
Theorem 4.2, $\textup{HS}_{k}(I)$ is matroidal, for all $k\geq 0$.
Now, we turn to the non-squarefree case.
###### Theorem 4.5.
Let $I\subset S$ be a polymatroidal ideal generated in degree two. Then,
$\textup{HS}_{k}(I)$ is a polymatroidal ideal, for all $k\geq 0$.
###### Proof.
If $I$ is squarefree, the thesis follows from Corollary 4.4. Suppose $I$ is
not squarefree. Up to a suitable relabeling, we can write
$I=(J,x_{1}^{2},x_{2}^{2},\dots,x_{t}^{2})$, where $J$ is the squarefree part
of $I$, _i.e._ , $G(J)=\\{u\in G(I):u\ \textup{is squarefree}\\}$ and $1\leq
t\leq n$. Then $J$ is a matroidal ideal. Let $G$ be the simple graph on $[n]$
with $J=I(G)$, then $G^{c}$ is cochordal. Let $u_{1},\dots,u_{m}$ be an
admissible order of $J$. We claim that
$u_{1},\dots,u_{m},x_{1}^{2},x_{2}^{2},\dots,x_{t}^{2}$
is an admissible order of $I$. We only need to prove that
$Q=(u_{1},\dots,u_{m},x_{1}^{2},\dots,x_{\ell-1}^{2}):x_{\ell}^{2}=(J,x_{1}^{2},\dots,x_{\ell-1}^{2}):x_{\ell}^{2}$
is generated by variables. Indeed, let $x_{i}x_{j}:x_{\ell}^{2}\in Q$ be a
generator with $i\leq j$. If $x_{i}x_{j}:x_{\ell}^{2}$ is a variable, there is
nothing to prove. Otherwise $x_{i}x_{j}:x_{\ell}^{2}=x_{i}x_{j}$, and
$\ell\neq i,j$. Since
$\deg_{x_{\ell}}(x_{\ell}^{2})>\deg_{x_{\ell}}(x_{i}x_{j})$, by the exchange
property, $w=x_{k}(x_{\ell}^{2})/x_{\ell}=x_{k}x_{\ell}\in I$, with $k=i$ or
$k=j$. Then $k\neq\ell$, $w=x_{k}x_{\ell}\in J$ and $w:x_{\ell}^{2}=x_{k}\in
Q$ is a variable that divides $x_{i}x_{j}:x_{\ell}^{2}$, as desired.
We claim that $\textup{set}(x_{\ell}^{2})=[n]\setminus\\{\ell\\}$, for all
$\ell=1,\dots,t$. Let $i\in[n]\setminus\\{\ell\\}$. Then $x_{i}x_{j}\in G(I)$
for some $j$. If $j=\ell$, then $x_{i}x_{\ell}\in I$. Suppose $j\neq\ell$,
then $\deg_{x_{j}}(x_{i}x_{j})>\deg_{x_{j}}(x_{\ell}^{2})$. By the exchange
property, $x_{i}x_{\ell}\in I$, as desired.
By equation (1), for all $k>0$,
$\displaystyle\textup{HS}_{k}(I)\ $ $\displaystyle=\
\textup{HS}_{k}(J)+\sum_{\ell=1}^{t}x_{\ell}^{2}\cdot\textup{HS}_{k-1}((x_{i}:i\in[n]\setminus\\{\ell\\})).$
We set $J_{\ell}=(x_{i}:i\in[n]\setminus\\{\ell\\})$, $\ell=1,\dots,t$. Since
$J$ is matroidal, $\textup{HS}_{k}(J)$ is matroidal by Corollary 4.4.
Moreover, each ideal $J_{\ell}$ is generated by variables, and so it is
matroidal. Hence all ideals $x_{\ell}^{2}\cdot\textup{HS}_{k-1}(J_{\ell})$ are
polymatroidal.
To verify that $\textup{HS}_{k}(I)$ is polymatroidal, we check the exchange
property. Let $u,v\in G(\textup{HS}_{k}(I))$ and $i$ such that
$\deg_{x_{i}}(u)>\deg_{x_{i}}(v)$.
To achieve our goal, we note the following fact. Let $w\in S$ be any
squarefree monomial of degree $k+1$ and let $\ell\in[t]$. Then
$x_{\ell}w\in\textup{HS}_{k}(I)$. Indeed, if $x_{\ell}$ divides $w$, then
$x_{\ell}w\in
x_{\ell}^{2}\cdot\textup{HS}_{k-1}(J_{\ell})\subset\textup{HS}_{k}(I)$.
Suppose $x_{\ell}$ does not divide $w$. For all $i$ such that $x_{i}$ divides
$w$, $x_{i}x_{\ell}\in J$ because $i\neq\ell$. Fix a lexicographic order
$\succ$ such that $x_{\ell}>x_{i}$ for all $i\in[n]\setminus\ell$. Up to
relabeling, we can assume $\ell=1$ and that $\succ$ is induced by
$x_{1}>x_{2}>\dots>x_{n}$. Writing $x_{\ell}w=x_{\ell}x_{j_{2}}\cdots
x_{j_{k+2}}$ with $\ell=1<j_{2}<\dots<j_{k+2}\leq n$, then
$x_{\ell}x_{j_{k+2}}\in J$, $x_{\ell}x_{j_{i}}\in J$ and
$x_{\ell}x_{j_{i}}\succ x_{\ell}x_{j_{k+2}}$, for $i=2,\dots,k+1$. Hence
$\\{j_{2},\dots,j_{k+1}\\}\subseteq\big{\\{}j\ |\ x_{j}\in(u\in G(J):u\succ
x_{\ell}x_{j_{k+2}}):x_{\ell}x_{j_{k+2}}\big{\\}}.$
This shows that $x_{\ell}w\in\textup{HS}_{k}(J)\subset\textup{HS}_{k}(I)$,
because by Theorem 4.2, $J$ has linear quotients with respect to $\succ$.
If $u,v\in\textup{HS}_{k}(J)$ or $u,v\in
x_{\ell}^{2}\cdot\textup{HS}_{k-1}(J_{\ell})$, we can find $j$ with
$\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$ such that
$x_{j}(u/x_{i})\in\textup{HS}_{k}(I)$, because both
$\textup{HS}_{k}(J),x_{\ell}^{2}\cdot\textup{HS}_{k-1}(J_{\ell})$ are
polymatroidal.
Suppose now $u\in\textup{HS}_{k}(J)$ and $v\in
x_{\ell}^{2}\cdot\textup{HS}_{k-1}(J_{\ell})$. Then
$\deg_{x_{\ell}}(u)<\deg_{x_{\ell}}(v)$ and
$x_{\ell}(u/x_{i})\in\textup{HS}_{k}(I)$, because $u/x_{i}$ is a squarefree
monomial of degree $k+1$.
Suppose $u\in x_{\ell}^{2}\cdot\textup{HS}_{k-1}(J_{\ell})$ and
$v\in\textup{HS}_{k}(J)$. Let $j$ such that $\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$.
Then $\deg_{x_{j}}(u)=0$. If $i=\ell$, then
$x_{j}(u/x_{\ell})\in\textup{HS}_{k}(I)$ because it is the product of
$x_{\ell}$ times a squarefree monomial of degree $k+1$. If $i\neq\ell$, then
$x_{j}(u/x_{i})$ can also be written as such a product. In any case
$x_{j}(u/x_{i})\in\textup{HS}_{k}(I)$.
Finally, suppose $u\in x_{\ell}\cdot\textup{HS}_{k-1}(J_{\ell})$ and $v\in
x_{h}^{2}\cdot\textup{HS}_{k-1}(J_{h})$ with $\ell\neq h$. Suppose $i=\ell$
and let $j$ such that $\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$. Then
$u^{\prime}=x_{j}(u/x_{i})$ is either $x_{\ell}$ times a squarefree monomial
of degree $k+1$, or is equal to $x_{h}$ times a squarefree monomial of degree
$k+1$. In both cases, $u^{\prime}\in\textup{HS}_{k}(I)$. Suppose now
$i\neq\ell$. If there exist more than one $j$ with
$\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$, we can choose $j\neq h$. Then
$\deg_{x_{j}}(v)=1$ and so $x_{j}$ does not divide $u$. Consequently
$x_{j}(u/x_{i})$ is equal to $x_{\ell}$ times a squarefree monomial of degree
$k+1$, and so $x_{j}(u/x_{i})\in\textup{HS}_{k}(I)$. If there is only one $j$
such that $\deg_{x_{j}}(u)<\deg_{x_{j}}(v)$, then $j=h$. We claim that $x_{h}$
does not divide $u$, then $x_{h}(u/x_{i})$ is equal to $x_{\ell}$ times a
squarefree monomial of degree $k+1$, and so
$x_{h}(u/x_{i})\in\textup{HS}_{k}(I)$, as wanted. Writing
$v=x_{h}^{2}x_{j_{1}}\cdots x_{j_{k}}$, with $j_{p}\in[n]\setminus\\{h\\}$,
$p=1,\dots,k$, then $\deg_{x_{j_{p}}}(v)=1\leq\deg_{x_{j_{p}}}(u)$, for all
$p=1,\dots,k$. Then $x_{j_{1}}\cdots x_{j_{k}}$ divides $u/(x_{i}x_{\ell})$
because $\deg_{x_{\ell}}(u)>1\geq\deg_{x_{\ell}}(v)$ and
$\deg_{x_{i}}(u)=1>\deg_{x_{i}}(v)$. This implies that $u=x_{i}x_{\ell}\cdot
x_{j_{1}}\cdots x_{j_{k}}$. From this presentation it follows that $x_{h}$
does non divide $u$, because $i,\ell\neq h$ and $j_{p}\neq h$ for
$p=1,\dots,k$, as well.
The cases above show that the exchange property holds for all monomials of
$G(\textup{HS}_{k}(I))$. Hence $\textup{HS}_{k}(I)$ is polymatroidal and the
proof is complete.
## References
* [1] S. Bandari, R. Rahmati-Asghar, On the Polymatroidal Property of Monomial Ideals with a View Towards Orderings of Minimal Generators, Bulletin of the Iranian Mathematical Society, 45 (2019), 657–666.
* [2] S. Bayati, Multigraded shifts of matroidal ideals, Arch. Math., (Basel) 111 (2018), no. 3, 239–246.
* [3] S. Bayati, I. Jahani, N. Taghipour, Linear quotients and multigraded shifts of Borel ideals, Bull. Aust. Math. Soc. 100 (2019), no. 1, 48–57.
* [4] M. Crupi, A. Ficarra, Very well–covered graphs by Betti splittings, preprint 2022.
* [5] G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 38 (1961), 71–76.
* [6] A. Ficarra. Homological shifts of polymatroidal ideals, available at arXiv preprint https://arxiv.org/abs/2205.04163[math.AC] (2022).
* [7] A. Ficarra. HomologicalShiftsIdeals, Macaulay2 package (2022).
* [8] C. A. Francisco, H. T. Ha, A. Van Tuyl, Splittings of monomial ideals, Proc. Amer. Math. Soc., 137 (10) (2009), 3271-3282.
* [9] R. Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988), 57–70, Banach Center Publ., 26, Part 2, PWN, Warsaw, 1990.
* [10] D. R. Grayson, M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
* [11] J. Herzog, T. Hibi, Cohen-Macaulay polymatroidal ideals, European J. Combin., 27 (2006), no. 4, 513–517.
* [12] J. Herzog, T. Hibi. _Monomial ideals_ , Graduate texts in Mathematics 260, Springer–Verlag, 2011.
* [13] J. Herzog, S. Moradi, M. Rahimbeigi, G. Zhu, Homological shift ideals. Collect. Math. 72 (2021), 157–174.
* [14] J. Herzog, S. Moradi, M. Rahimbeigi, G. Zhu, Some homological properties of borel type ideals, 2021, to appear in Comm. Algebra.
* [15] J. Herzog, Y. Takayama, Resolutions by mapping cones, in: The Roos Festschrift volume Nr.2(2), Homology, Homotopy and Applications 4, (2002), 277–294.
* [16] A.S. Jahan, X. Zheng, Ideals with linear quotients, J. Combin. Theory Ser. A 117 (2010), 104–110. MR2557882 (2011d:13030)
* [17] P.J. Looges and S. Olariu, Optimal greedy algorithms for indifference graphs, Comput. Math. Appl. 25, (1993), 15–25.
|
# Quantum reservoir computing in finite dimensions
Rodrigo Martínez-Peña<EMAIL_ADDRESS>Instituto de Física
Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Campus Universitat de
les Illes Balears, 07122 Palma de Mallorca, Spain Juan-Pablo Ortega juan-
<EMAIL_ADDRESS>Division of Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, Singapore 637371
###### Abstract
Most existing results in the analysis of quantum reservoir computing (QRC)
systems with classical inputs have been obtained using the density matrix
formalism. This paper shows that alternative representations can provide
better insights when dealing with design and assessment questions. More
explicitly, system isomorphisms are established that unify the density matrix
approach to QRC with the representation in the space of observables using
Bloch vectors associated with Gell-Mann bases. It is shown that these vector
representations yield state-affine systems (SAS) previously introduced in the
classical reservoir computing literature and for which numerous theoretical
results have been established. This connection is used to show that various
statements in relation to the fading memory (FMP) and the echo state (ESP)
properties are independent of the representation, and also to shed some light
on fundamental questions in QRC theory in finite dimensions. In particular, a
necessary and sufficient condition for the ESP and FMP to hold is formulated
using standard hypotheses, and contractive quantum channels that have
exclusively trivial semi-infinite solutions are characterized in terms of the
existence of input-independent fixed points.
Suggested keywords
## I Introduction
The development of noisy intermediate-scale quantum (NISQ) devices is
attracting a great deal of attention from the quantum community. Recent
advancements in fields such as quantum computation [1], quantum simulation
[2], and quantum communications [3] are just examples of the prosperous future
that awaits these technologies. Nevertheless, NISQ devices are already
demonstrating in the meantime that they can be very useful for diverse
research fields like physics, chemistry, and optimization [4], even providing
quantum advantage [1, 5, 6]. Machine learning (ML) is another example of
thriving synergies with NISQ technologies. In this context, quantum machine
learning (QML) aims to exploit the specific features of quantum mechanics to
obtain an advantage over its classical counterparts when dealing with machine
learning tasks, both with classical and quantum data [7]. There is already
positive evidence in this direction, both from a theoretical point of view in
the fault-tolerant picture [8] and in experiments [9].
The flexibility and range of action of ML and QML techniques is typically
specified by the so-called universality approximation theorems. A universal
approximation property takes place when a proposed restricted family of
functions can approximate any function in a much larger class with arbitrary
precision. There are many results of this type that are part of classical
analysis dealing with, for instance, polynomials and Fourier series, and
others that were added in the early days of ML like, for example, feed-forward
neural networks [10, 11, 12]. Further results of this type have been proved
for various ML paradigms like recurrent neural networks [13], support vector
machines [14], extreme learning machines [15, 16], or kernel methods [17, 18,
19]. Universality results have also been obtained in the QML context, such as
for one qubit algorithms [20, 21], and general quantum circuits [22, 23]. A
framework in which we are particularly interested is quantum reservoir
computing (QRC). As in classical reservoir computing (RC) [24, 25, 26], QRC
harnesses the rich dynamics of (quantum) dynamical systems to solve tasks
where memory and prediction capabilities are required. Examples of application
of these techniques are found in the prediction of chaotic time-series [27,
28, 29] and complex spatiotemporal dynamics [30, 31, 32, 33]. Since the first
work on QRC [34] many have followed (see [35, 36] for reviews). This includes
both experimental implementations [37, 38, 39, 40, 41] as well as theoretical
contributions on the universal approximation question [42, 37, 43]. The latter
was inspired by the works in the classical framework [44, 45, 46, 47], where
the discrete-time setting of RC theory fits well within the quantum dynamical
map description.
The universal approximation property in the RC context brings to the table
some conditions that dynamical systems should meet to ensure it. The most
prominent ones are the echo state property (ESP) [24] and the fading memory
property (FMP) [48]. A system has the FMP if inputs that are close in the
recent past produce outputs that are also close, independently of what
happened in the distant past. The ESP guarantees that a well-defined
input/output map can be associated with our system and amounts to an existence
and uniqueness property with respect to the input sequence that is fed into
the system. The ESP and FMP are standard requirements in many stochastic and
deterministic learning paradigms since they mathematically encode the
asymptotic decorrelation (and even independence) between physical states and
initial conditions that most physical systems exhibit as time goes by. There
are many physical mechanisms that lead to the declining relevance of a given
initial condition in a system state as the temporal distance between them
increases. For instance, two pervasive phenomena in applications in this
direction are chaos (high sensitivity to initial conditions) and dissipation.
Fading memory is an important modeling feature when using physical systems
because a reservoir system can only store a finite amount of information in
its trainable parameters [49]; this implies that information must be erased as
time goes by in order to store new input information. Under very mild
mathematical conditions, ESP and FMP are equivalent to the input-forgetting
property [50] which mathematically encodes the information removal process
that is needed to learn the newly fed one.
The ESP property has been studied in the context of QRC in different works.
Nurdin and Chen [42] provided sufficient conditions for the ESP (and FMP) to
hold in terms of the contractivity of the quantum map acting on a restricted
domain [42, 37, 51], while Tran and Nakajima also define the ESP using
contractivity of the quantum map but without considering restrictions. They
also numerically connect the ESP with the spectra of the quantum maps [32,
52]. Both approaches describe the ESP in the density matrix language, that is,
the dynamical equations are defined in the space of quantum states. However,
this description might be limited somehow, because, as we will see later on,
more information can be extracted if we choose a description in terms of
observables.
This paper aims to fill some gaps in the description of QRC with finite
dimensional systems and classical inputs. More precisely, we first unify the
quantum ESP defined in previous works [42, 37, 51, 32, 52] using the norm of
quantum maps, with the classical notion of ESP using observables, and show
that they are equivalent for finite-dimensional quantum systems. We extend in
passing these results to the FMP. More specifically, we establish explicit
system isomorphisms between the description of QRC systems using the space of
density matrices as state space and the one that uses the space of Bloch
vectors associated to Gell-Mann bases. That choice of basis, which is
customary in the study of quantum systems, happens to yield non-homogeneous
affine state dynamics (the corresponding systems are called state-affine
(SAS)) of the type introduced in [45]. The connection between QRC and SAS
systems has important consequences. Our results in that context are contained
in Proposition 3 and in Theorems 5 and 9. The proposition presents a necessary
and sufficient condition for the ESP and FMP to hold in different
representations with just a few requirements, such as the compactness of the
input space. We emphasize that this hypothesis is satisfied in most RC tasks
and when dealing with implementations of RC systems with dedicated hardware
since the experimental ranges of the physical systems involved are always
finite [26]. The theorems exhibit common situations that should be avoided in
the design of quantum channels so that fully operational QRC systems are
obtained. We shall work in an idealized framework in which observables are
obtained after an infinite number of measurements, with no statistical error.
Even in such an idealized setting, we expect that our results can contribute
to the general understanding of QRC experiments in finite dimensional systems,
as they have already been implemented in [37, 38, 40, 41].
The structure of the paper is as follows. Section II introduces the general
framework and the definitions that will be needed along the paper. Definitions
of the spaces of operators and quantum maps are included in Section II.1,
while all the RC ingredients will be presented in Section II.2, together with
some preliminary results. The main results are contained in Section III.
Section IV includes a brief discussion on some of the consequences of Theorems
5 and 9, and Section V concludes the paper.
## II Definitions
### II.1 Quantum definitions
We start by defining the space of quantum systems. See, for example, [53, 54]
for further details. Consider a complex Hilbert space $\mathcal{H}$. The set
of all bounded operators $\mathcal{B}(\mathcal{H})$ that act on $\mathcal{H}$
is a complex vector space under point-wise addition and scalar multiplication,
and it forms an algebra under composition. If we add the involution
$A\rightarrow A^{\dagger}$ given by the adjoint operation, then
$\mathcal{B}(\mathcal{H})$ is also a $C^{*}$-algebra with respect to the
operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{op}$
defined by
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|A\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{op}:=\sup_{||\psi||=1}\\{||A\psi||,\
\psi\in\mathcal{H}\\},$ (1)
where $A\in\mathcal{B}(\mathcal{H})$. Along this manuscript, we will denote
all the induced operator and matrix norms with the symbol
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$.
The predual of $\mathcal{B}(\mathcal{H})$ is the Banach space
$\mathcal{T}(\mathcal{H})$ of all trace-class operators on $\mathcal{H}$ that
have finite trace norm $||A||_{1}:=\text{tr}\sqrt{AA^{\dagger}}$. The trace
norm is a particular case (with $p=1$) of the Schatten norms defined by
$||A||_{p}:=\left(\text{tr}\left(\left(\sqrt{AA^{\dagger}}\right)^{p}\right)\right)^{1/p}$.
The space of quantum density matrices $\mathcal{S}(\mathcal{H})$ is a compact
convex subset (see Section II.2 and the argument above (15)) of the normed
vector space $\mathcal{T}(\mathcal{H})$ defined by
$\mathcal{S}(\mathcal{H})=\\{\rho\in\mathcal{T}(\mathcal{H})\ |\
\rho^{\dagger}=\rho,\ \rho\geq 0,\ \text{tr}(\rho)=1\\}.$ (2)
As $\mathcal{S}(\mathcal{H})$ is a closed subset of the Banach space
$\mathcal{T}(\mathcal{H})$, it is then a complete metric space when using the
distance induced by $\left\|\cdot\right\|_{1}$. All along this paper, we
restrict ourselves to finite-dimensional Hilbert spaces, for which the spaces
of bounded and trace class operators coincide, and we shall use the symbol
$\mathcal{B}(\mathcal{H})$ to refer to both of them. Moreover, in that case,
$\mathcal{S}(\mathcal{H})$ is a complete metric space with respect to any norm
since all the norms are equivalent. We shall reserve the symbol
$d\in{\mathbb{N}}$ for the dimension of ${\mathcal{H}}$.
We now introduce the notion of quantum channel. All definitions and further
properties of these maps can be found in [55, 56, 57, 58] and references
therein. A quantum channel is a linear map
$T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ that is
completely positive and trace preserving (CPTP). We recall here that a trace
preserving map $T$ is the one that satisfies that
$\text{tr}(T(A))=\text{tr}(A)$ for any $A\in\mathcal{B}({\mathcal{H}})$.
Moreover, we say that $T$ is positive when it maps positive semi-definite
operators to positive semi-definite operators. Finally, completely positive
maps $T$ are those positive maps which, when extended to a larger space using
the tensor map $T\otimes I_{k}$, with $I_{k}$ the identity map in dimension
$k$, they also yield a positive map for any $k\in\mathbb{N}$. A linear map is
CPTP if and only if it is possible to find a Kraus decomposition associated to
a set of operators $\\{K_{i}\\}_{i\in X}$ such that for all
$A\in\mathcal{B}(\mathcal{H})$:
$T(A)=\sum_{i\in X}K_{i}AK^{\dagger}_{i},$ (3)
where $\sum_{i\in X}K^{\dagger}_{i}K_{i}=I$ and $X$ is an index set of
cardinality at most $d^{2}$, with $d$ the dimension of ${\mathcal{H}}$. CPTP
maps obviously leave $\mathcal{S}(\mathcal{H})$ invariant and hence induce a
restricted map
$T:\mathcal{S}(\mathcal{H})\longrightarrow\mathcal{S}(\mathcal{H})$ that we
shall denote with the same symbol and use interchangeably. Note that, unlike
$\mathcal{B}({\mathcal{H}})$, the set $\mathcal{S}({\mathcal{H}})$ is not a
vector space, and hence it is only when use the map
$T:\mathcal{B}(\mathcal{H})\longrightarrow\mathcal{B}(\mathcal{H})$ that we
can talk about matrix expressions and eigenvalues for the operator $T$. It can
be shown that any CPTP map
$T:\mathcal{S}(\mathcal{H})\longrightarrow\mathcal{S}(\mathcal{H})$ is non-
expansive in the trace norm, which means that after applying $T$ to two input
states $\rho_{1},\rho_{2}\in\mathcal{S}(\mathcal{H})$, the distance between
these density matrices is either contracted or remains equal:
$||T(\rho_{1})-T(\rho_{2})||_{1}\leq||\rho_{1}-\rho_{2}||_{1}.$ (4)
We recall that the eigenvalues of a CPTP map
$T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ are the
complex numbers $\lambda$ that make the map $T-\lambda\,\text{id}$ non-
invertible. We will denote this set by $\text{spec}(T)$. The spectral radius
$\rho(T):=\max\\{|\lambda|\ \mid\lambda\in\text{spec}(T)\\}$ of any CPTP map
is one as a consequence of (4) and of the fact that $1$ is always an
eigenvalue. Additionally, the set $\text{spec}(T)$ is invariant under complex
conjugation (see [57]). On the other hand, since the finite-dimensional vector
space $\mathcal{B}(\mathcal{H})$ is isomorphic to $\mathbb{C}^{d^{2}}$, we can
represent $T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ as a
$d^{2}\times d^{2}$ matrix, which we write as $\widehat{T}$. This matrix
representation can be obtained by fixing an orthonormal basis in
$\mathcal{B}(\mathcal{H})$ for the Hilbert-Schmidt inner product, that is,
$\\{B_{i}\in\mathcal{B}(\mathcal{H})\\}$ with
$\text{tr}(B_{i}^{\dagger}B_{j})=\delta_{ij}$, and by setting
$\widehat{T}_{ij}=\text{tr}(B_{i}^{\dagger}T(B_{j}))$. The spectrum of the
matrix $\widehat{T}$ is given by the roots of the characteristic polynomial
$\det(\widehat{T}-\lambda I)=0$ and, since we are working with finite
dimensions, it coincides with $\text{spec}(T)$, as well as with the set of
complex values $\lambda$ that satisfy $T(X)=\lambda X$, for some non-trivial
eigenvector $X\in\mathcal{B}(\mathcal{H})$ [58].
We are particularly interested in the eigenvectors corresponding to the
eigenvalue $\lambda=1$ and that we call fixed points, that is, they are
elements $A\in\mathcal{B}(\mathcal{H})$ such that $T(A)=A$. The set of fixed
points of a CPTP map is always non-empty in the space of density matrices.
This is a consequence of Schauder’s Fixed-Point Theorem together with the
continuity of $T$ and the compactness of $\mathcal{S}(\mathcal{H})$. We will
work most of the time with quantum channels with single fixed points
$\rho^{*}\in\mathcal{S}(\mathcal{H})$ in the space of density matrices. Such
maps are called ergodic. Ergodicity of quantum channels is equivalent to the
eigenspace of $T$ associated with the eigenvalue $\lambda=1$ in
$\mathcal{B}(\mathcal{H})$ having dimension one, and to being made out of the
complex multiples of $\rho^{*}\in\mathcal{S}(\mathcal{H})$. In that case, we
obviously have (see Corollary 2 in [57]) that $T(A)=A$ with
$A\in\mathcal{B}(\mathcal{H})$ if and only if $A=\text{tr}(A)\rho^{*}$. It is
worth mentioning that the rest of the eigenvectors of an ergodic CPTP map must
be traceless. Indeed, given an eigenvalue $\lambda\neq 1$ of the CPTP map $T$
and $A$ a corresponding eigenvector, we find that
$\text{tr}(A)=\text{tr}(T(A))=\lambda\text{tr}(A)$, which implies that
$(\lambda-1)\text{tr}(A)=0$, and hence that $\text{tr}(A)=0$. Therefore, the
spectral set of a CPTP map can be decomposed as:
$\text{spec}(T)=\\{1\\}\cup\text{spec}(T|_{\mathcal{B}_{0}(\mathcal{H})}),$
(5)
where $\mathcal{B}_{0}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ is defined
as the vector subspace of traceless operators, where the restriction
$T|_{\mathcal{B}_{0}(\mathcal{H})}$ and its corresponding matrix
representation are obviously well defined.
Repeated applications of an ergodic CPTP map do not necessarily converge to a
fixed point (see [57] for an example). If convergence to a fixed point takes
place, we say that the CPTP map is mixing. More specifically, a CPTP map is
mixing if and only if its repeated applications converge in the trace norm,
that is,
$\lim_{n\rightarrow\infty}||T^{n}(\rho)-\rho^{*}||_{1}=0,\quad\forall\rho\in\mathcal{S}(\mathcal{H}).$
(6)
Equation (6) implies that the sequence $\\{T^{n}(\rho)\\}$ converges to
$\rho^{*}$ with respect to the trace norm, but in infinite-dimensional
situations this does not necessarily imply that convergence takes place with
respect to other norms (see Definition 5.4.1 in [59]). However, in the finite-
dimensional case mixing becomes a topological property where
$\lim_{n\rightarrow\infty}T^{n}(\rho)=\rho^{*}$ for any norm and for any
$\rho\in\mathcal{S}(\mathcal{H})$ [56].
Although all mixing maps are ergodic, the converse is not true in general.
However, for continuous-time Markovian evolution, both are equivalent and such
maps are called relaxing [57]. Another important consequence of mixing
condition is the following: a CPTP map is mixing if and only if the fixed
point $\rho^{*}$ is the only eigenvector with eigenvalue $|\lambda|=1$. Then,
it is straightforward to show that
$\max\\{|\lambda|\mid\lambda\in\text{spec}(T|_{\mathcal{B}_{0}(\mathcal{H})})\\}<1.$
(7)
We say that a CPTP map is called primitive when it is mixing and its unique
fixed point $\rho^{*}>0$ has full rank. The smallest natural number $n$ for
which $T^{n}$ sends positive semidefinite matrices to positive definite
matrices is called the index of primitivity of $T$ and is denoted by
$\omega(T)$. With that notation, we say that $T^{\omega(T)}$ is a strictly
positive map. Bounds for this number are given by the quantum version of the
Wietland inequality [60, 61].
Finally, we say that a quantum channel is strictly contractive when
$||T(\rho_{1})-T(\rho_{2})||_{1}\leq r||\rho_{1}-\rho_{2}||_{1}$ (8)
for all $\rho_{1},\rho_{2}\in\mathcal{S}(\mathcal{H})$, where $0\leq r<1$. As
it is customary in the quantum channels literature, we reserve the term
strictly contractive for the trace norm, unless a different norm is explicitly
specified. It can be shown that strictly contractive channels with a full-rank
fixed point must be primitive. Indeed, the contractivity condition, together
with Banach’s Fixed Point Theorem (using that the convex closed subset of
density matrices with the trace norm is a complete metric space) guarantees
that the channel is mixing. A mixing channel with a strictly positive fixed
point is a primitive channel. The converse holds when the primitive map
becomes strictly positive, that is, it sends positive semidefinite matrices to
positive definite ones: given a primitive channel $T$, $T^{\omega(T)}$ is
strictly contractive (see Theorem VI.3 in [61]). Contractive maps can be also
constructed by composing a strictly contractive channel with a general CPTP
map.
An important conclusion that can be drawn from all these considerations is
that strictly contractive channels, which will be relevant along this work,
can be constructed by either using a map like $T^{\omega(T)}$, where $T$ is a
primitive channel, or by composing a strictly contractive channel with any
other CPTP map, as it has been done in some examples in QRC [40, 39]. Examples
of mixing/relaxing CPTP maps with full-rank single fixed points can be found
in the Lindblad-Gorini-Kossakowski-Sudarshan equation (see [62] for a summary
of the necessary and sufficient conditions). As these maps belong to the
quantum Markov semigroup, iterative applications of the channels yield
$T_{\Delta\tau}^{\omega(T)}=T_{\omega(T)\Delta\tau}$, where $\Delta\tau$
represents the time of action of the map $T_{\Delta\tau}$. Therefore, taking
$\Delta\tau^{\prime}\geq\omega(T)\Delta\tau$ we obtain a strictly contractive
map.
### II.2 RC definitions
We now define quatum reservoir computing (QRC) systems in the density matrix
formalism. Classical RC definitions and mathematical details can be found in,
for instance, [44]. QRC maps are determined by two equations, namely, the
state-space and the readout or observation equations. The state equation is
given by a family of CPTP maps
$T:\mathcal{B}(\mathcal{H})\times\mathbb{R}^{n}\rightarrow\mathcal{B}(\mathcal{H})$,
with $n\in\mathbb{N}$ being the number of input features, which are taken to
be real values (classical inputs). The maps $T$ and $h$ will be, most of the
time, tacitly assumed to be continuous. The output is obtained from the
readout map $h:\mathcal{B}(\mathcal{H})\rightarrow\mathbb{R}^{m}$, with
$m\in\mathbb{N}$, which maps operators in $\mathcal{B}(\mathcal{H})$ to the
Euclidean space $\mathbb{R}^{m}$. Inputs are typically bi-infinite discrete-
time sequences of the form ${\bf z}=(\dots,{\bf z}_{-1},{\bf z}_{0},{\bf
z}_{1},\dots)\in(\mathbb{R}^{n})^{\mathbb{Z}}$, and outputs
$\textbf{y}\in(\mathbb{R}^{m})^{\mathbb{Z}}$ have the same structure. A QRC
system is hence determined by the state-space transformations:
$\begin{cases}&A_{t}=T(A_{t-1},{\bf z}_{t}),\\\
&\textbf{y}_{t}=h(A_{t}),\end{cases}$ (9)
where $t\in\mathbb{Z}$ denotes the time index. Analogously, one can define the
same setting for semi-infinite discrete-time sequences:
$(\mathbb{R}^{n})^{\mathbb{Z}_{-}}=\\{{\bf z}=(\dots,{\bf z}_{-1},{\bf
z}_{0})\ |\ {\bf z}_{i}\in\mathbb{R}^{n},i\in\mathbb{Z}_{-}\\}$ for left-
infinite sequences and $(\mathbb{R}^{n})^{\mathbb{Z}_{+}}=\\{{\bf z}=({\bf
z}_{0},{\bf z}_{1},\dots)\ |\ {\bf
z}_{i}\in\mathbb{R}^{n},i\in\mathbb{Z}_{+}\\}$ for right-infinite sequences.
Similar definitions apply to $(D_{n})^{\mathbb{Z}}$,
$(D_{n})^{\mathbb{Z}_{-}}$, and $(D_{n})^{\mathbb{Z}_{+}}$ with elements in
the subset $D_{n}\subset\mathbb{R}^{n}$. We can also construct sequence spaces
$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}}$ for the space of bounded (trace-
class) operators:
$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}}=\\{\textbf{A}=(\dots,A_{-1},A_{0},A_{1},\dots)\\\
\mid A_{i}\in\mathcal{B}(\mathcal{H}),\ i\in\mathbb{Z}\\}.$
Analogous definitions for
$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}_{-}}$,$(\mathcal{B}(\mathcal{H}))^{\mathbb{Z}_{+}}$,
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$,
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}}$, and
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{+}}$ follow immediately.
A natural way to construct CPTP state-space transformations is to insert the
input dependence using the Kraus decomposition that we introduced in (3), that
is,
$T(A,{\bf z})=\sum_{i\in X}K_{i}({\bf z})AK^{\dagger}_{i}({\bf z}),$ (10)
and for any input ${\bf z}\in\mathbb{R}^{n}$, the matrices $\\{K_{i}({\bf
z})\\}_{i\in X}$ satisfy that $\sum_{i\in X}K^{\dagger}_{i}({\bf z})K_{i}({\bf
z})=I$. The by-design CPTP character of the map
$T:\mathcal{B}(\mathcal{H})\times\mathbb{R}^{n}\rightarrow\mathcal{B}(\mathcal{H})$
in (10) implies that it naturally restricts to a state equation
$T:\mathcal{S}(\mathcal{H})\times\mathbb{R}^{n}\rightarrow\mathcal{S}(\mathcal{H})$
with density matrices as state space, that we shall use interchangeably in the
sequel and denote using the same symbol.
The Echo State Property (ESP). Consider the QRC system defined in (9) or its
analog for the subsets
$\mathcal{S}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ and
$D_{n}\subset\mathbb{R}^{n}$, that is, $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$. Given an input sequence ${\bf
z}\in(D_{n})^{\mathbb{Z}}$, we say that
$\bm{\rho}\in(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$ is a solution of (9) for
the input ${\bf z}$ if the components of the sequences ${\bf z}$ and
$\bm{\rho}$ satisfy the first relation in (9) for any $t\in\mathbb{Z}$. We say
that the QRC system has the echo state property (ESP) when it has a unique
solution for each input ${\bf z}\in(D_{n})^{\mathbb{Z}}$. More explicitly, for
each ${\bf z}\in(D_{n})^{\mathbb{Z}}$, there exists a unique sequence
$\bm{\rho}\in(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$ such that
$\rho_{t}=T(\rho_{t-1},{\bf z}_{t}),\ \text{for all}\ t\in\mathbb{Z}.$ (11)
Filters and functionals. Let
$\mathcal{S}(\mathcal{H})\subset\mathcal{B}(\mathcal{H})$ be the space of
density matrices and let $D_{n}\subset\mathbb{R}^{n}$ be a subset in the input
space. A map of the type
$U:(D_{n})^{\mathbb{Z}}\rightarrow(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}}$ is
called a filter associated to the QRC system (9) when it satisfies that
$U({\bf z})_{t}=T\left(U({\bf z})_{t-1},{\bf z}_{t}\right),\ \mbox{for all
${\bf z}\in(D_{n})^{\mathbb{Z}}$ and $t\in\mathbb{Z}$.}$
Filters induce what we call functionals
$H:(D_{n})^{\mathbb{Z}}\rightarrow\mathcal{S}(\mathcal{H})$ via the relation
$H({\bf z})=U({\bf z})_{0}$. It is clear that a uniquely determined filter can
be associated with a QRC system that satisfies the ESP. The filter maps, in
that case, any input sequence to the unique solution of the QRC system
associated with it. A filter is called causal if it only produces outputs that
depend on present and past inputs. More formally, causality means that for any
two inputs ${\bf z},\textbf{v}\in(D_{n})^{\mathbb{Z}}$ that satisfy ${\bf
z}_{\tau}=\textbf{v}_{\tau}$ for any $\tau\leq t$, for a given
$t\in\mathbb{Z}$, we have $U({\bf z})_{t}=U(\textbf{v})_{t}$. The filter $U$
is called time-invariant if there is no explicit time dependence on the system
that determines it, that is, it commutes with the time delay operator defined
as $\mathcal{T}_{\tau}({\bf z})_{t}:={\bf z}_{t-\tau}$. Filters associated to
QRC systems of the type (9) are always causal and time-invariant (Proposition
2.1 in [44]). As noted in previous works [48, 44, 45], there is a bijection
between causal and time-invariant filters and functionals on
$(D_{n})^{\mathbb{Z}_{-}}$. Then, we can restrict our work to causal and time-
invariant filters with target and domain in spaces of left-infinite sequences.
The Fading Memory Property (FMP). Infinite product spaces can be endowed with
Banach space structures associated to the supremum norm and weighted norms.
The supremum norm for input sequences is defined as $||{\bf
z}||_{\infty}:=\sup_{t\in\mathbb{Z}}\\{||{\bf z}_{t}||\\}$, and for operator
sequences as
$||\textbf{A}||_{\infty}:=\sup_{t\in\mathbb{Z_{-}}}\\{||A_{t}||\\}$, where
$||\cdot||$ represents a given vector and matrix norm, respectively. The
symbols $l^{\infty}(\mathbb{R}^{n})$, $l_{\pm}^{\infty}(\mathbb{R}^{n})$,
$l^{\infty}(\mathcal{B}(\mathcal{H}))$ and
$l_{\pm}^{\infty}(\mathcal{B}(\mathcal{H}))$ denote the Banach spaces formed
by the elements in the corresponding infinite product spaces with a finite
supremum norm.
We now define the weighted norm. Let $w:\mathbb{N}\rightarrow(0,1]$ be a
decreasing sequence with zero limit and $w_{0}=1$. The weighted norm
$||\cdot||_{w}$ on $(\mathbb{R})^{\mathbb{Z_{-}}}$ is defined as
$||{\bf z}||_{w}:=\sup_{t\in\mathbb{Z}_{-}}\\{w_{-t}||{\bf z}_{t}||\\},$ (12)
and the space
$l^{w}_{-}(\mathbb{R}^{n})=\\{{\bf z}\in(\mathbb{R}^{n})^{\mathbb{Z_{-}}}|\
||{\bf z}||_{w}<\infty\\},$ (13)
with weighted norm $||\cdot||_{w}$ forms a Banach space (see Appendix A.2 in
[44]). In the same vein, we can define
$\begin{split}&||\textbf{A}||_{w}:=\sup_{t\in\mathbb{Z}_{-}}\\{w_{-t}||A||\\},\\\
&l^{w}_{-}(\mathcal{B}(\mathcal{H}))=\\{\textbf{A}\in(\mathcal{B}(\mathcal{H}))^{\mathbb{Z_{-}}}|\
||\textbf{A}||_{w}<\infty\\}.\end{split}$ (14)
It can be shown that $l^{w}_{-}(\mathcal{B}(\mathcal{H}))$ is a Banach space
as well.
We now turn to the space of density matrices $\mathcal{S}(\mathcal{H})$ and
recall that since for positive semidefinite matrices the trace operator is
submultiplicative (see Exercise 7.2.P26 in [59]) we can conclude that
$\operatorname{tr}(\rho^{2})\leq 1$ for all $\rho\in\mathcal{S}(\mathcal{H})$.
This observation implies that the elements in $\mathcal{S}(\mathcal{H})$ have
Frobenius norms bounded by one. Since we are in finite dimensions, this
statement holds for any other matrix norm (eventually with a bounding constant
different from one) and allows us to conclude that
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}}\subset
l^{\infty}_{-}(\mathcal{B}(\mathcal{H}))\subset
l^{w}_{-}(\mathcal{B}(\mathcal{H})),$ (15)
for any weighting sequence $w$. The previous boundedness consideration,
together with the closedness of $\mathcal{S}(\mathcal{H})$ in
$\mathcal{B}(\mathcal{H})$ implies that $\mathcal{S}(\mathcal{H})$ is
necessary compact. An important consequence of this fact is that the relative
topology induced by the $l^{w}_{-}(\mathcal{B}(\mathcal{H}))$ on
$(\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}}$ coincides with the product
topology (see Corollary 2.7 in [44]).
Take now a subset $D_{n}\subset\mathbb{R}^{n}$ such that
$(D_{n})^{\mathbb{Z}_{-}}\subset l^{w}_{-}(\mathbb{R}^{n})$ and consider a QRC
system $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ that has the ESP. We say that $T$
has the fading memory property (FMP) when the corresponding functional
$H:(D_{n})^{\mathbb{Z}_{-}}\rightarrow\mathcal{S}(\mathcal{H})$ is a
continuous map between the metric spaces
$((D_{n})^{\mathbb{Z}_{-}},||\cdot||_{w})$ and
$((\mathcal{S}(\mathcal{H}))^{\mathbb{Z}_{-}},||\cdot||_{w})$, for some
weighting sequence $w$. If $D_{n}$ is compact, once $H$ is continuous for a
given weighting sequence $w$, then it is continuous for all weighting
sequences (see [44, Theorem 2.6]).
The compactness of $\mathcal{S}(\mathcal{H})$ implies that we can apply with
straightforward modifications Theorem 3.1 in [44] to prove the following
statement.
###### Proposition 1.
Let $D_{n}\subset\mathbb{R}^{n}$ be a compact subset of $\mathbb{R}^{n}$. Let
$T:\mathcal{S}(\mathcal{H})\times D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ be
a continuous QRC system such that the CPTP maps $T(\cdot,{\bf
z}):\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H})$ are strictly
contractive for all ${\bf z}\in D_{n}$ as in (8) with a common contraction
constant $0\leq r<1$ associated to some norm in $\mathcal{B}(\mathcal{H})$
(not necessarily $\left\|\cdot\right\|_{1}$). Then, the QRC system induced by
$T$ has the ESP and the FMP.
## III Results
Finding the expression for the QRC filter of a system that has the ESP is not
straightforward in the language of density matrices since the dependence on
the initial condition has to be addressed. More explicitly, given a general
CPTP map expressed using its Kraus decomposition, the relation (10) can be
iterated $n$-time steps into the past in order to obtain a quantum state
$\rho_{t}^{n}(\rho^{0}_{t-n})\in\mathcal{S}({\mathcal{H}})$ at time $t$ out of
an initial condition $\rho^{0}_{t-n}\in\mathcal{S}({\mathcal{H}})$ specified
at time $t-n$ via the formula:
$\footnotesize\begin{split}&\\!\\!\\!\\!\rho_{t}^{n}(\rho^{0}_{t-n})=\\\
&\sum_{i_{0},\ldots,i_{n-1}\in
X}\left(\overleftarrow{\prod}^{n-1}_{l=0}K_{i_{l}}({\bf
z}_{t-l})\right)\rho^{0}_{t-n}\left(\overrightarrow{\prod}^{n-1}_{l=0}K^{\dagger}_{i_{l}}({\bf
z}_{t-l})\right),\end{split}$ (16)
where $\overleftarrow{\prod}^{n-1}_{l=0}K_{i_{l}}({\bf
z}_{t-l})=K_{i_{0}}({\bf z}_{t})\cdots K_{i_{n-1}}({\bf z}_{t-n-1})$ and
$\overrightarrow{\prod}^{n-1}_{l=0}K^{\dagger}_{i_{l}}({\bf
z}_{t-l})=K^{\dagger}_{i_{n-1}}({\bf z}_{t-n-1})\cdots
K^{\dagger}_{i_{0}}({\bf z}_{t})$. If the QRC system has the ESP and hence a
filter
$U:(D_{n})^{\mathbb{Z}}\longrightarrow\left(\mathcal{S}({\mathcal{H}})\right)^{\mathbb{Z}}$
can be associated to it, it necessarily has to satisfy
$U({\bf z})_{t}=\lim_{n\rightarrow\infty}\rho_{t}^{n}(\rho^{0}_{t-n}),$ (17)
and this value has to be independent of the initial conditions
$\rho^{0}_{t-n}$. This fact has been explicitly shown in [42] (and in Chapter
2 of [51] in more detail) in the case of strictly contractive CPTP maps with
respect to the operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}$
associated to Schatten norms. We recall that given
$T:\mathcal{B}({\mathcal{H}})\rightarrow\mathcal{B}({\mathcal{H}})$, we define
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}:=\sup_{||A||_{p}=1}\\{||T(A)||_{p}\mid
A\in\mathcal{B}({\mathcal{H}})\\},$ (18)
for some $p\in[1,\infty)$ and $\left\|\cdot\right\|_{p}$ the $p$-Schatten
norm. Using this notation, the contractivity condition on the CPTP map $T$ is
stated by requiring that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}<1$,
for some $p\in[1,\infty)$. It is obvious that this condition ensures that the
hypotheses of Proposition 1 are satisfied, which in turn implies that $T$ has
the ESP and the FMP (notice that given
$\rho_{1},\rho_{2}\in\mathcal{S}({\mathcal{H}})$ and ${\bf z}\in D_{n}$
arbitrary, the difference $T(\rho_{1},{\bf z})-T(\rho_{2},{\bf
z})\in\mathcal{B}_{0}(\mathcal{H})$ and hence the hypothesis
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}<1$
implies that $\left\|T(\rho_{1},{\bf z})-T(\rho_{2},{\bf
z})\right\|_{p}<\left\|\rho_{1}-\rho_{2}\right\|_{p}$). Since we are in a
finite-dimensional context, it suffices to impose contractivity for just one
norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}$,
$p\in[1,\infty)$, in order to ensure that the limit in (17) exists and that
its value is independent of the initial condition.
The expression (16) shows that it is not possible to write down a closed-form
expression for the filter of a QRC system with the ESP when using the density
matrix representation, in which the initial dependence condition has been
eliminated. This feature is ultimately due to the linearity of the setup.
Already in the classical framework (see [45]), it has been shown that this can
be avoided by working with affine instead of purely linear systems. In the
quantum context, it can also be observed (see [37, 41]) that the non-
homogeneous state-space system given by
$\rho_{t}=(1-\epsilon)T(\rho_{t-1},{\bf z}_{t})+\epsilon\sigma$, where
$T(\rho_{t-1},{\bf z}_{t})$ is a CPTP map, $\sigma$ an arbitrary density
matrix, and $0<\epsilon<1$, defines a unique filter $U({\bf
z})_{t}=\epsilon\sigma+\epsilon\sum^{\infty}_{j=1}(1-\epsilon)^{j}T(\sigma,{\bf
z}_{t+1-j})$ in which the dependence on initial conditions has disappeared.
In the following subsections, we shall circumvent this problem by showing that
certain matrix representations of finite-dimensional QRC systems on density
matrices have a built-in non-homogeneous affine structure that makes them into
non-homogeneous state-affine systems (SAS) of the type introduced in [45].
More specifically, we shall be working with the Bloch vector representation of
quantum finite dimensional systems [63, 64] associated to a given Gell-Mann
basis. This idea is not new in QRC and it can already be seen in the seminal
work [34] or, more recently, in the Methods section of [40]. In the paragraphs
that follow, we shall explore in depth this representation, mostly in
connection with the available literature on SAS systems (Section III.2), which
will allow us later on in Section III.3 to identify various design constraints
on quantum channels.
### III.1 Matrix representation of quantum channels
We will start by introducing the notation necessary for the matrix
representation of quantum channels. In the next section, we shall focus on a
specific choice of basis adapted to density matrices. Let
$\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ be an orthonormal basis
for the vector space $\mathcal{B}(\mathcal{H})$, when endowed with the
Hilbert-Schmidt inner product, that is,
$\text{tr}(B^{\dagger}_{i}B_{j})=\delta_{ij}$. Using any such basis we can
represent any operator $A\in\mathcal{B}(\mathcal{H})$ as
$A=\sum_{i=1}^{d^{2}}a_{i}B_{i}$, with $a_{i}=\text{tr}(B_{i}^{\dagger}A)$.
Analogously, we can express any linear map
$T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$ as
$T(A)=\sum^{d^{2}}_{i,j=1}\widehat{T}_{ij}a_{j}B_{i},\ \mbox{where
$\widehat{T}_{ij}=\text{tr}(B^{\dagger}_{i}T(B_{j}))$.}$ (19)
This observation implies that QRC systems $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ admit an equivalent representation
as a system $\widehat{T}:V\times D_{n}\rightarrow V$, where
$V\subset\mathbb{R}^{d^{2}}$ is the subset of real Euclidean space that
contains the coordinate representations of the elements in
$\mathcal{S}(\mathcal{H})$ using the basis
$\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$.
This statement can be formalized using the language of system morphisms (see
[65, 66] for the standard definitions and elementary facts). Consider the
state-space systems determined by the triples $({\cal X}_{i},F_{i},h_{i})$,
$i\in\left\\{1,2\right\\}$, with $F_{i}:{\cal X}_{i}\times{\cal
Z}\longrightarrow{\cal X}_{i}$ and $h_{i}:{\cal X}_{i}\longrightarrow{\cal
Y}$. A map $f:{\cal X}_{1}\longrightarrow{\cal X}_{2}$ is a morphism between
the systems $({\cal X}_{1},F_{1},h_{1})$ and $({\cal X}_{2},F_{2},h_{2})$
whenever it satisfies the following two properties:
1. (i)
System equivariance: $f(F_{1}({\bf x}_{1},{\bf z}))=F_{2}(f({\bf x}_{1}),{\bf
z})$, for all ${\bf x}_{1}\in{\cal X}_{1}$ and ${\bf z}\in{\cal Z}$.
2. (ii)
Readout invariance: $h_{1}({\bf x}_{1})=h_{2}(f({\bf x}_{1}))$, for all ${\bf
x}_{1}\in{\cal X}_{1}$.
When the map $f$ has an inverse $f^{-1}$ and this inverse is also a morphism
between the systems determined by $({\cal X}_{2},F_{2},h_{2})$ and $({\cal
X}_{1},F_{1},h_{1})$, we say that $f$ is a system isomorphism and the systems
$({\cal X}_{1},F_{1},h_{1})$ and $({\cal X}_{2},F_{2},h_{2})$ are isomorphic.
We note that given a system $F_{1}:{\cal X}_{1}\times{\cal
Z}\longrightarrow{\cal X}_{1},h_{1}:{\cal X}_{1}\longrightarrow{\cal Y}$ and a
bijection $f:{\cal X}_{1}\longrightarrow{\cal X}_{2}$, the map $f$ is a system
isomorphism with respect to the system $F_{2}:{\cal X}_{2}\times{\cal
Z}\longrightarrow{\cal X}_{2},h_{2}:{\cal X}_{2}\longrightarrow{\cal Y}$
defined by
$\displaystyle F_{2}({\bf x}_{2},{\bf z})$ $\displaystyle:=f(F_{1}(f^{-1}({\bf
x}_{2}),{\bf z})),$ (20) $\displaystyle h_{2}({\bf x}_{2})$
$\displaystyle:=h_{1}(f^{-1}({\bf x}_{2})).$ (21)
for all ${\bf x}_{2}\in{\cal X}_{2}$, ${\bf z}\in{\cal Z}$.
Consider now the QRC system given by the quantum channel
$T:\mathcal{S}(\mathcal{H})\times D_{n}\rightarrow\mathcal{S}(\mathcal{H})$
and the readout map $h:\mathcal{B}(\mathcal{H})\rightarrow\mathbb{R}^{m}$,
$m\in\mathbb{N}$. Using the orthonormal basis
$\mathcal{B}=\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ and the
discussion above define the map
$\begin{array}[]{cccc}G_{\mathcal{B}}:&\mathbb{C}^{d^{2}}&\longrightarrow&\mathcal{B}({\mathcal{H}})\\\
&\mathbf{a}&\longmapsto&\sum_{i=1}^{d^{2}}a_{i}B_{i}.\end{array}$ (22)
This map is a linear homeomorphism. Define
$V=G_{\mathcal{B}}^{-1}(\mathcal{S}(\mathcal{H}))\subset\mathbb{R}^{d^{2}}$ as
well as the map (that we denote with the same symbol)
$G_{\mathcal{B}}:V\longrightarrow\mathcal{S}(\mathcal{H})$ that we obtain by
restriction of the domain and codomain in (22). This restricted map is also a
homeomorphism when $V$ and $\mathcal{S}(\mathcal{H})$ are endowed with their
relative topologies [67, Theorem 18.2]. With all these ingredients, it is
straightforward to verify that the QRC system $(\mathcal{S}(\mathcal{H}),T,h)$
is system isomorphic to $(V,\widehat{T},\widehat{h})$ with
$\widehat{T}:V\times D_{n}\longrightarrow V$ and
$\widehat{h}:V\longrightarrow\mathbb{R}^{m}$ given by
$\displaystyle\widehat{T}(\mathbf{a},{\bf z})$
$\displaystyle:=G_{\mathcal{B}}^{-1}\left(T\left(G_{\mathcal{B}}(\mathbf{a}),{\bf
z}\right)\right),$ (23) $\displaystyle\widehat{h}(\mathbf{a})$
$\displaystyle:=h(G_{\mathcal{B}}(\mathbf{a})),$ (24)
and that the isomorphism is given by the map
$G_{\mathcal{B}}:V\longrightarrow\mathcal{S}(\mathcal{H})$. The procedure that
we just spelled out can be reproduced for any other (orthonormal) basis
$\mathcal{B}^{\prime}$ of $\mathcal{B}({\mathcal{H}})$, in which case we would
obtain another system $(V^{\prime},\widehat{T}^{\prime},\widehat{h}^{\prime})$
which is obviously isomorphic to both $(V,\widehat{T},\widehat{h})$ and
$(\mathcal{S}(\mathcal{H}),T,h)$.
The system isomorphisms that we just defined and the compactness of the state
spaces where they are defined allow us to establish important connections
between the filters that they define. The following proposition describes
those connections in detail.
###### Proposition 2.
Let $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$ be a quantum channel,
$h:\mathcal{S}(\mathcal{H})\longrightarrow\mathbb{R}^{m}$ a readout, and let
$\mathcal{B}=\\{B_{i}\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ be an
orthonormal basis for $\mathcal{B}(\mathcal{H})$. Let $\widehat{T}:V\times
D_{n}\longrightarrow V$ be the isomorphic system defined in (22) and
$\widehat{h}:V\longrightarrow\mathbb{R}^{m}$ the corresponding readout defined
in (24). Then:
1. (i)
Given an input ${\bf z}\in({\mathbb{R}}^{n})^{\mathbb{Z}}$, a sequence
$\bm{\rho}\in\left(\mathcal{S}(\mathcal{H})\right)^{\mathbb{Z}}$ is a solution
for that input for the system determined by $T$ if and only if the sequence
${\cal
G}_{\mathcal{B}}^{-1}\left(\bm{\rho}\right)\in\left(V\right)^{\mathbb{Z}}$ is
a solution for the system associated to $\widehat{T}$. The symbol ${\cal
G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}:(V)^{\mathbb{Z}}\longrightarrow\left({\cal
S}({\mathcal{H}})\right)^{\mathbb{Z}}$ stands for the product map.
2. (ii)
$T$ has the ESP if and only if $\widehat{T}$ has the ESP. In that case, the
filters $U_{T}$ and $U_{\widehat{T}}$ (respectively, $U_{T}^{h}$ and
$U_{\widehat{T}}^{\widehat{h}}$) determined by $T$ and $\widehat{T}$
(respectively, by $(T,h)$ and $(\widehat{T},\widehat{h})$) satisfy that
$U_{T}={\cal G}_{\mathcal{B}}\circ U_{\widehat{T}}$ (respectively,
$U_{T}^{h}=U_{\widehat{T}}^{\widehat{h}}$).
3. (iii)
$U_{T}$ has the FMP if and only if $U_{\widehat{T}}$ has the FMP.
4. (iv)
Let $V^{\prime}$ be a set homeomorphic to $V$. The relations (20)-(21)
determine in that case an isomorphic system
$\widehat{T}^{\prime}:V^{\prime}\times D_{n}\longrightarrow V^{\prime}$,
$\widehat{h}^{\prime}:V^{\prime}\rightarrow\mathbb{R}^{m}$ that has the ESP
and the FMP if and only if $\widehat{T}$ and $\widehat{h}$ have that property.
###### Proof.
Parts (i), (ii), and (iv) are a straightforward consequence of Proposition 2.2
in [44]. As to part (iii), given that by (ii) $U_{T}={\cal
G}_{\mathcal{B}}\circ U_{\widehat{T}}$, it suffices to prove that the
bijection ${\cal
G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}:(V)^{\mathbb{Z}}\longrightarrow\left({\cal
S}({\mathcal{H}})\right)^{\mathbb{Z}}$ is a homeomorphism when the domain and
the target are endowed with a weighted norm. In order to prove that, recall
first that, using the observation right under (15), the weighted norms in
$(V)^{\mathbb{Z}}$ and $\left({\cal S}({\mathcal{H}})\right)^{\mathbb{Z}}$
induce the product topology due to the compactness of $V$ and
$\mathcal{S}(\mathcal{H})$. This immediately implies that (Theorem 19.6 in
[67]) ${\cal G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}$ is continuous
due to the continuity of $G_{\mathcal{B}}$. The same argument can be
immediately applied to the inverse map ${\cal G}_{\mathcal{B}}^{-1}$, which
proves the statement. ∎
### III.2 Non-homogenous state affine system representation
The goal of this section is to choose a specific basis in
$\mathcal{B}({\mathcal{H}})$ for the representation of the quantum channel
$T:\mathcal{B}(\mathcal{H})\times D_{n}\rightarrow\mathcal{B}(\mathcal{H})$
(equivalently, $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$) in which the associated system
isomorphic representation $\widehat{T}:\mathbb{C}^{d^{2}}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}}$ (equivalently, $\widehat{T}:V\times
D_{n}\longrightarrow V$) has a non-homogeneous state-affine form of the type
introduced in [45].
More specifically, we choose a generalized Gell-Mann basis [63, 64, 68]. This
is an orthonormal basis made of Hermitian operators in which, by convention,
its first element is the normalized identity, namely, $B_{1}=I/\sqrt{d}$. The
remaining $(d^{2}-1)$ traceless Hermitian operators are the generators
$\mathcal{B}_{0}=\left\\{\sigma_{i}\right\\}_{i\in\left\\{1,\ldots,d^{2}-1\right\\}}$
(25)
of the fundamental representation of the Lie algebra $\mathfrak{su}(d)$ of
SU($d$). The case $d=2$ corresponds to the case of one qubit, and the Gell-
Mann basis is made of the standard Pauli matrices. The orthonormality of the
Gell-Mann basis is guaranteed by the product property of the fundamental
representation of $\mathfrak{su}(d)$, namely,
$\sigma_{a}\sigma_{b}=\delta_{ab}I/(2d)+\sum^{d^{2}-1}_{c}f_{c}\sigma_{c}$,
where $\sigma_{a},\sigma_{b}$ are two elements of the Gell-Mann basis for
$\mathfrak{su}(d)$, and $f_{c}$ are complex coefficients. The resulting Gell-
Mann basis
$\mathcal{B}=\left\\{B_{i}\right\\}_{i\in\left\\{1,\ldots,d^{2}\right\\}}$ of
${\cal B}({\mathcal{H}})$ is hence given by $B_{1}=I/\sqrt{d}$ and
$B_{i}=\sigma_{i-1}$, $1<i\leq d^{2}$. Note that the subset
$\mathcal{B}_{0}=\left\\{B_{i}\right\\}_{i\in\left\\{2,\ldots,d^{2}\right\\}}$
is a basis for the vector subspace ${\cal B}_{0}({\mathcal{H}})\subset{\cal
B}({\mathcal{H}})$ of codimension $1$ made of traceless operators.
If our system is made of $N$ $d$-dimensional systems (qudits), we can extend
this basis to $\mathcal{B}\left({\mathcal{H}}^{\otimes^{N}}\right)$ by
tensorization. The orthonormality of the tensorized basis with respect to the
Hilbert-Schmidt inner product is preserved since $\text{tr}(A\otimes
B)=\text{tr}(A)\text{tr}(B)$ for any two $A,B\in\mathcal{B}({\mathcal{H}})$.
We now go back to the system constituted by one qudit and spell out the matrix
expression $\widehat{T}:\mathbb{C}^{d^{2}}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}}$ of a CPTP map
$T:\mathcal{B}({\mathcal{H}})\times
D_{n}\longrightarrow\mathcal{B}({\mathcal{H}})$ in the basis $\mathcal{B}$
that we just introduced, by using the prescription introduced in (19). We
first note that the choice $B_{1}=I/\sqrt{d}$ and the trace-preserving
character of $T(\cdot,{\bf z})$ for any ${\bf z}\in D_{n}$, imply that
$\widehat{T}(\cdot,{\bf z})_{11}=1$. Analogously, $\widehat{T}(\cdot,{\bf
z})_{1j}=\text{tr}(T(B_{j},{\bf z}))/\sqrt{d}=0$ for $1<j\leq d^{2}$, since
$T$ is trace-preserving. This implies that the matrix $\widehat{T}(\cdot,{\bf
z})$ can be written as
$\widehat{T}(\cdot,{\bf z})=\left(\begin{matrix}1&{\bf 0}_{d^{2}-1}\\\ q({\bf
z})&p({\bf z})\end{matrix}\right)$ (26)
where $p({\bf z})$ is the square matrix of dimension $d^{2}-1$ with complex
entries
$p({\bf z})_{ij}:=\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\mathcal{B}_{0}(\mathcal{H})\right)}\right)_{ij}\\\
=\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right)_{ij}=\text{tr}(B_{i}T(B_{j},{\bf
z})),$ (27)
$1<i,j\leq d^{2}$, and $q({\bf z})\in\mathbb{C}^{d^{2}-1}$ is given by $q({\bf
z})_{i}=\text{tr}(B_{i}T(I))/\sqrt{d}$, $1<i\leq d^{2}$. The symbol ${\bf
0}_{d^{2}-1}$ denotes a zero-row of length $d^{2}-1$.
Now, given that any element $\rho\in\mathcal{S}({\mathcal{H}})$ has
coordinates in the Gell-Mann basis of the form
$\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}\in\mathbb{C}^{d^{2}}$ with
$\mathbf{x}\in\mathbb{C}^{d^{2}-1}$ called the Bloch vector, the matrix form
(26) implies that
$\widehat{T}(\cdot,{\bf z})\left(\begin{matrix}1/\sqrt{d}\\\
\mathbf{x}\end{matrix}\right)=\left(\begin{matrix}1/\sqrt{d}\\\ p({\bf
z})\textbf{x}+q({\bf z})\end{matrix}\right).$ (28)
This expression implies that $\widehat{T}:V\times D_{n}\longrightarrow V$
admits a system-isomorphic representation $\widehat{T}_{0}:V_{0}\times
D_{n}\longrightarrow V_{0}$ on the set
$V_{0}=\left\\{\mathbf{x}\in\mathbb{C}^{d^{2}-1}\mid\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}\in
V\right\\}\subset\mathbb{C}^{d^{2}-1}$ given by
$\begin{array}[]{cccc}\widehat{T}_{0}:&V_{0}\times
D_{n}&\longrightarrow&V_{0}\\\ &(\mathbf{x},{\bf z})&\longmapsto&p({\bf
z})\textbf{x}+q({\bf z}),\end{array}$ (29)
with readout
$\widehat{h}_{0}(\mathbf{x})=\widehat{h}\left(\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}\right)$.
The system isomorphism is in this case, given by the map
$\begin{array}[]{cccc}i_{0}:&V_{0}&\longrightarrow&V\\\ &\mathbf{x}\in
V_{0}&\longmapsto&\left(1/\sqrt{d},\mathbf{x}^{\top}\right)^{\top}.\end{array}$
(30)
Part (iv) of Proposition 2 guarantees that the dynamical properties of the
system $(\widehat{T},\widehat{h},V)$ (and hence those of the QRC system
$(T,h,\mathcal{S}({\mathcal{H}}))$) are equivalent to those of
$(\widehat{T}_{0},\widehat{h}_{0},V_{0})$.
The importance of this observation is that it links
$(T,h,\mathcal{S}({\mathcal{H}}))$ to the non-homogeneous state-affine system
(SAS) introduced in [45] and for which various universality properties have
been additionally proved in [46, 69]. SAS are defined as state equations that
have the form spelled out in (29). Strictly speaking, the SAS systems studied
in the above-cited references impose polynomial or trigonometric dependences
of $q$ and $p$ on the inputs, while in our situation, the prescription
introduced in (10) is capable of accommodating more general forms.
It has been shown in [45] that when such a system has the ESP, the
corresponding filter
$U_{\widehat{T}_{0}}:\left(D_{n}\right)^{\mathbb{Z}}\rightarrow\left(V_{0}\right)^{\mathbb{Z}}$
can be written as
$U_{\widehat{T}_{0}}({\bf
z})_{t}=\sum^{\infty}_{j=0}\left(\prod^{j-1}_{k=0}p({\bf
z}_{t-k})\right)q({\bf z}_{t-j}),$ (31)
where $\prod_{k=0}^{j-1}p({\bf z}_{t-k}):=p({\bf z}_{t})\cdot p({\bf
z}_{t-1})\cdots p({\bf z}_{t-j+1})$. A first sufficient condition for ESP and
FMP has been formulated in [45] by imposing that $\sigma_{\text{max}}(p({\bf
z}))<1$ for all ${\bf z}\in D_{n}$, where $\sigma_{\text{max}}$ is the maximum
singular value of matrix $p({\bf z})$. Given a matrix $A$, the maximum
singular value is equal to the 2-Schatten induced norm:
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|A\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}=\sigma_{\text{max}}(A)$,
where the singular values of $A$ are the square-roots of the eigenvalues of
$AA^{\dagger}$. An improved sufficient condition could be potentially found
using other matrix norms as in [70].
The next proposition uses this hint and, moreover, spells out equivalent
necessary and sufficient conditions for the ESP and FMP to hold in the three
different equivalent representations for QRC systems that we have introduced
in this section. More explicitly, the statement addresses the ESP and the FMP
for the operator representation $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$, the matrix representation
$\widehat{T}:\mathbb{C}^{d^{2}}\times D_{n}\longrightarrow\mathbb{C}^{d^{2}}$
associated to the Gell-Mann basis $\mathcal{B}$, and the SAS representation
$\widehat{T}_{0}:\mathbb{C}^{d^{2}-1}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}-1}$ introduced in (29).
###### Proposition 3.
Let $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$ be a continuous QRC system. The
following three statements are equivalent:
1. (i)
There exists an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and $\epsilon>0$ such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon,\
\mbox{for all ${\bf z}\in D_{n}$}.$ (32)
2. (ii)
There exists a matrix norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
in the space of complex $d^{2}\times d^{2}$ matrices such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon,\
\mbox{for all ${\bf z}\in D_{n}$},$ (33)
with $\mathcal{B}_{0}$ the trace-zero elements in the basis $\mathcal{B}$
defined in (25) and $G_{\mathcal{B}}$ the isomorphism defined in (22) with
respect to the Gell-Mann basis.
3. (iii)
There is a matrix norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
in the space of complex $(d^{2}-1)\times(d^{2}-1)$ matrices such that the SAS
representation $\widehat{T}_{0}:\mathbb{C}^{d^{2}-1}\times
D_{n}\longrightarrow\mathbb{C}^{d^{2}-1}$ introduced in (29) satisfies that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p({\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon,\ \mbox{for
all ${\bf z}\in D_{n}$},$ (34)
If any of these three equivalent statements hold and $D_{n}$ is compact, then:
1. 1.
The isomorphic systems $T:\mathcal{S}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{S}(\mathcal{H})$, $\widehat{T}:V\times
D_{n}\rightarrow V$, and $\widehat{T}_{0}:V_{0}\times D_{n}\rightarrow V_{0}$,
have the ESP and the FMP, and hence continuous filters
$U_{T}:(D_{n})^{\mathbb{Z}}\longrightarrow\left(\mathcal{S}(\mathcal{H})\right)^{\mathbb{Z}}$,
$U_{\widehat{T}}:(D_{n})^{\mathbb{Z}}\longrightarrow V^{\mathbb{Z}}$, and
$U_{\widehat{T}_{0}}:(D_{n})^{\mathbb{Z}}\longrightarrow V_{0}^{\mathbb{Z}}$
can be associated to them.
2. 2.
In such case, the filter $U_{\widehat{T}_{0}}$ is then given by (31).
$U_{\widehat{T}}$ is determined by $U_{\widehat{T}}={\cal I}_{0}\circ
U_{\widehat{T}_{0}}$, with ${\cal I}_{0}=\prod_{\mathbb{Z}}i_{0}$ and $i_{0}$
as in (30). Finally, $U_{T}={\cal G}_{\mathcal{B}}\circ U_{\widehat{T}}$, with
${\cal G}_{\mathcal{B}}=\prod_{\mathbb{Z}}G_{\mathcal{B}}$ the map introduced
in Proposition 2.
3. 3.
The contraction conditions (32)-(34) are necessary for the ESP and the FMP to
hold.
###### Proof.
The equivalences between the contraction conditions (32)-(34) can be shown by
using norms in $\mathbb{C}^{d^{2}}$ and $\mathcal{B}({\mathcal{H}})$ that make
the map $G_{\mathcal{B}}$ in (22) into an isometry. More explicitly, let us
start with an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
for which the maps $T(\cdot,{\bf
z}):\mathcal{B}({\mathcal{H}})\longrightarrow\mathcal{B}({\mathcal{H}})$
satisfy the condition (32). Assume that this operator norm is associated with
a given norm $\left\|\cdot\right\|_{\mathcal{B}({\mathcal{H}})}$ in
$\mathcal{B}({\mathcal{H}})$, that is,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=\sup_{A\neq
0}\left\\{\left\|T(A,{\bf
z})\right\|_{\mathcal{B}({\mathcal{H}})}/\left\|A\right\|_{\mathcal{B}({\mathcal{H}})}\right\\}$.
Take now the norm $\left\|\cdot\right\|_{\mathbb{C}^{d^{2}}}$ in
$\mathbb{C}^{d^{2}}$ with respect to which $G_{\mathcal{B}}$ is an isometry,
that is, set
$\left\|\mathbf{a}\right\|_{\mathbb{C}^{d^{2}}}=\left\|G_{\mathcal{B}}(\mathbf{a})\right\|_{\mathcal{B}({\mathcal{H}})}$,
for all $\mathbf{a}\in\mathbb{C}^{d^{2}}$, and denote by
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{\prime}=\sup_{\mathbf{a}\neq
0}\left\\{\left\|\widehat{T}(\mathbf{a},{\bf
z})\right\|_{\mathbb{C}^{d^{2}}}/\left\|\mathbf{a}\right\|_{\mathbb{C}^{d^{2}}}\right\\}$.
Using just these definitions, it is easy to see that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{\prime}$ and,
moreover, that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{\prime},$
which can be easily used to prove the equivalence between (32) and (33). The
equivalence between (33) and (34) follows from the fact that, as it can be
seen in (26),
$\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}=p({\bf
z}).$
The claims $1$ and $2$ in the second part of the proposition are
straightforward consequences of Propositions 1 and 2. We now prove the last
claim in point $3$ using the SAS representation (29) for the continuous QRC
system $T$ that, this time, it is assumed to have the ESP and the FMP. If we
iterate $n$ times this state equation using an arbitrary vector
$\mathbf{x}_{0}\in\mathbb{C}^{d^{2}-1}$ as an initial condition, we obtain the
vector $U^{n,\mathbf{x}_{0}}_{\widehat{T}_{0}}({\bf
z})_{t}\in\mathbb{C}^{d^{2}-1}$ defined by:
$U^{n,\mathbf{x}_{0}}_{\widehat{T}_{0}}({\bf
z})_{t}=\sum^{n-1}_{j=0}\left(\prod^{j-1}_{k=0}p({\bf z}_{t-k})\right)q({\bf
z}_{t-j})+\prod^{n-1}_{j=0}p({\bf z}_{t-j})\mathbf{x}_{0}.$
If we now assume that the system has the ESP, we necessarily have that
$U_{\widehat{T}_{0}}({\bf
z})_{t}=\lim_{n\rightarrow\infty}U^{n,\mathbf{x}_{0}}_{\widehat{T}_{0}}({\bf
z})_{t}=\lim_{n\rightarrow\infty}U^{n,\mathbf{x}_{0}^{\prime}}_{\widehat{T}_{0}}({\bf
z})_{t},$ (35)
with $U_{\widehat{T}_{0}}$ the filter in (31) and
$\mathbf{x}_{0},\mathbf{x}_{0}^{\prime}\in\mathbb{C}^{d^{2}-1}$ arbitrary
vectors. The last equality in (35) and the arbitrary character of
$\mathbf{x}_{0},\mathbf{x}_{0}^{\prime}\in\mathbb{C}^{d^{2}-1}$ imply that
$\prod^{\infty}_{j=0}p({\bf
z}_{t-j})=\lim_{n\rightarrow\infty}\prod^{n-1}_{j=0}p({\bf z}_{t-j})=0,$
necessarily. We now notice that since $T$ is a continuous map, then so is the
dependence of $p({\bf z})$ on the inputs ${\bf z}$. Moreover, since, in this
case, $D_{n}$ is assumed to be a compact set, then so is the matrix set
$\left\\{p({\bf z})\mid{\bf z}\in D_{n}\right\\}$. Now, Corollary 6.4 in [71]
guarantees the existence of matrix norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
for which (34) is satisfied, as required. ∎
###### Remark 4.
Using the characterization of the mixing property in (7), it is clear that a
necessary condition for the QRC $T$ to satisfy the contractivity hypothesis in
the previous proposition and, in passing, satisfy the ESP and the FMP, is that
all the maps $T(\cdot,{\bf z})$ are mixing for all ${\bf z}\in D_{n}$. Indeed,
since the spectral radius is a lower bound for any matrix norm (see [59,
Theorem 5.6.9]) we have that $\lambda_{\text{max}}\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right)\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$.
Consequently, if
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon$
then $\lambda_{\text{max}}\left(\widehat{T}(\cdot,{\bf
z})|_{G_{\mathcal{B}}^{-1}\left(\operatorname{span}\left\\{\mathcal{B}_{0}\right\\}\right)}\right)\leq
1-\epsilon$, which is equivalent to $\lambda_{\text{max}}\left(T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right)\leq 1-\epsilon$ and then all maps
$T(\cdot,{\bf z})$ are mixing by (7). We emphasize that this mixing condition
is necessary but not sufficient since, as the composition of two mixing maps
is not necessarily mixing, the existence of the limit in (17) is not
guaranteed even if each of the factor operators is mixing.
### III.3 Some constrains on CPTP maps for QRC
The SAS representation allows us to easily characterize situations like the
one in Proposition 3 in which a QRC system has the ESP and the FMP and,
moreover, it allows us to write explicitly down the corresponding filter (31).
As we shall now see in this section, more interesting facts can be derived
from this representation having to do with design features that should be
avoided, as they produce systems with only trivial solutions. The first one
concerns unital quantum channels, that is, channels that satisfy
$T(I,{\bf z})=I\quad\mbox{for all ${\bf z}\in D_{n}$}.$
That situation is studied in the next theorem, which will be generalized in
Theorem 9 to the case of QRC systems that exhibit an input-independent fixed
point.
###### Theorem 5.
Let $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$ be a QRC system for which there
exists an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and $\epsilon>0$ such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon$,
for all ${\bf z}\in D_{n}$. Then, the corresponding filter $U_{T}$ is constant
with $U_{T}({\bf z})_{t}=I/d$ for all ${\bf z}\in(D_{n})^{\mathbb{Z}}$
(equivalently $U_{\widehat{T}_{0}}({\bf z})_{t}={\bf 0}$) if and only if $T$
is unital.
###### Proof.
We prove this statement by using the SAS representation associated to the
Gell-Mann basis $\mathcal{B}$. If $T$ is unital, that is, $T(I,{\bf z})=I$,
then the expressions (19) and (26) imply that $q({\bf
z})_{i}=\text{tr}(B_{i}T(I,{\bf z}))/\sqrt{d}=\text{tr}(B_{i})/\sqrt{d}=0$,
for all $i\in\left\\{2,\ldots,d^{2}\right\\}$. Therefore, the SAS state
equation becomes, in this case, $\textbf{x}_{t}=p({\bf
z}_{t})\textbf{x}_{t-1}$, which is a homogeneous equation whose only solution
under the contractivity hypotheses is the trivial one, that is,
$\textbf{x}_{t}={\bf 0}$ for all $t\in\mathbb{Z}$. Consequently,
$U_{\widehat{T}_{0}}={\bf 0}$. Proposition 3 implies then that $U_{T}={\cal
G}_{\mathcal{B}}\circ{\cal I}_{0}\circ U_{\widehat{T}_{0}}=I/d$.
Conversely, suppose that $U_{T}({\bf z})_{t}=I/d$ for all ${\bf
z}\in(D_{n})^{\mathbb{Z}}$. Since the filter $U_{T}$ is determined by the
recursions $U_{T}({\bf z})_{t}=T\left(U_{T}({\bf z})_{t-1},{\bf
z}_{t}\right)$, then, this relation implies that $T(I,{\bf z})=I$, for all
${\bf z}\in D_{n}$. ∎
###### Remark 6.
There is a close relation between unital quantum channels and contractivity.
Indeed, it has been shown that unital maps are contractive for all Schatten
$p$-norms, that is,
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{p}\leq 1$ (see
Theorem 2.4 in [72]) when $T(\cdot,{\bf z})$ is unital.
###### Example 7.
Consider the depolarizing channel
$\mathcal{E}:\mathcal{S}({\mathcal{H}})\longrightarrow\mathcal{S}({\mathcal{H}})$
defined by
$\mathcal{E}(\rho)=(1-\lambda)\rho+\lambda\frac{I}{d},$ (36)
where $0\leq\lambda\leq 1$ denotes the probability of finding the system at
the maximally mixed state $I/d$. If we arrange the previous equation as an
input-dependent channel, we can write the following state equation:
$\rho_{t}=\mathcal{E}(\rho_{t-1},{\bf
z}_{t})=(1-\lambda_{t})\rho_{t-1}+\lambda_{t}\frac{I}{d},$ (37)
where $\lambda_{t}:=\lambda({\bf z}_{t})$ is a function of the input at each
time step. This map is strictly contractive whenever
$r=\sup_{t\in\mathbb{Z}}(1-\lambda_{t})<1$ in which case $r$ is the
contraction rate. Let us compute the state after $n$ backwards iterations:
$\begin{split}\rho^{n}_{t}&=\left(\prod^{n-1}_{i=0}(1-\lambda_{t-i})\right)\rho_{t-n}\\\
&+\sum^{n-1}_{i=0}\left(\lambda_{t-i}\left(\prod^{i-1}_{j=0}(1-\lambda_{t-j})\right)\right)\frac{I}{d}.\end{split}$
(38)
Taking the limit $n\rightarrow\infty$, it is easy to see that the first
summand vanishes because $1-\lambda_{t-i}\leq r<1$ for all $t\in\mathbb{Z}$,
and hence all possible dependence on the initial condition disappears.
Regarding the second summand, since the map is strictly contractive, the limit
$\lim_{n\rightarrow\infty}\rho^{n}_{t}$ exists and must yield a density
matrix, which means that
$\sum^{\infty}_{i=0}\left(\lambda_{t-i}\left(\prod^{i-1}_{j=0}(1-\lambda_{t-j})\right)\right)=1$
because of the normalization. This limit hence defines the filter
$U_{\mathcal{E}}({\bf z})_{t}=I/d$, which is consistent with the conclusion of
Theorem 5 since (37) is a strictly contractive unital map.
We could have found the same result by directly using (29), for which we need
to define the extension of the depolarizing channel to the whole space of
bounded operators $\mathcal{B}(\mathcal{H})$. We define then the CPTP map
$\mathcal{E}^{\prime}:\mathcal{B}({\mathcal{H}})\longrightarrow\mathcal{B}({\mathcal{H}})$
$\mathcal{E}^{\prime}(A)=(1-\lambda)A+\lambda\text{tr}(A)\frac{I}{d}.$ (39)
The SAS representation associated with the input-dependent version of this map
is such that $q({\bf z}_{t})=0$ because the map is unital. It is also easy to
see that the system equation becomes $\textbf{x}_{t}=p({\bf
z}_{t})\textbf{x}_{t-1}=(1-\lambda_{t})\textbf{x}_{t-1}$. The filter
$U_{\widehat{\mathcal{E}}^{\prime}_{0}}$ of this QRC equation satisfies that
$U_{\widehat{\mathcal{E}}^{\prime}_{0}}({\bf z})_{t}={\bf 0}$, for all
$t\in\mathbb{Z}$.
###### Example 8.
The next case is an example of “poorly engineered” QRC system which can be
detected using Theorem 5. Indeed, we shall introduce a model of dissipation
using tuneable local losses, but Theorem 5 will discard its long-term
applicability because it is unital.
Let us define the Markovian master equation that governs the dynamics between
input injections:
$\dot{\rho}=-i[H({\bf z}_{t}),\rho]+\gamma L\rho
L^{\dagger}-\frac{\gamma}{2}\\{L^{\dagger}L,\rho\\},$ (40)
where $L$ is the jump operator and $\\{L^{\dagger}L,\rho\\}$ denotes the
anticommutator. We define the input-dependent Hamiltonian as $H({\bf
z}_{t})=h({\bf z}_{t})\sigma^{x}/2$, where $h({\bf z}_{t})$ will be an
arbitrary function of the input, and the jump operator as $L=\sigma^{z}$. This
is a single qubit under the influence of an external magnetic field in the $x$
direction of the real space (whose intensity varies between inputs) with a
local dephasing. Notice that the Hamiltonian is considered as time-independent
when integrating the dynamics since it is constant between input injections.
Going from the density matrix language to the real variable linear description
requires to find the CPTP map representation of (40). Since Markovian master
equations are CPTP linear transformations on their own, we just need to find
the Kraus decomposition that represents the dynamics of this master equation
as a map. We will follow the procedure as explained in [73] (see Section 2 in
the reference for the details of the algorithm). In particular, the Kraus
decomposition of a single qubit can be written in the following form:
$T(\rho)=\sum_{i,j}S^{(U)}_{ij}B_{i}\rho B^{\dagger}_{j},$ (41)
where $B_{i}$ are the basis elements of a single qubit in the operator space
($\\{I,\sigma^{x},\sigma^{y},\sigma^{z}\\}$) and $S^{(U)}_{ij}:=U^{\dagger}SU$
is a unitary transformation of the Choi matrix $S$. Applying (41) to the
definition of matrix $T$ in (19), we can find the map of the single qubit
observables:
$\left(\begin{matrix}1\\\ \braket{\sigma^{x}}_{t}\\\
\braket{\sigma^{y}}_{t}\\\
\braket{\sigma^{z}}_{t}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&\widehat{T}_{22}&0&0\\\ 0&0&\widehat{T}_{33}&\widehat{T}_{34}\\\
0&0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right)\left(\begin{matrix}1\\\
\braket{\sigma^{x}}_{t-1}\\\ \braket{\sigma^{y}}_{t-1}\\\
\braket{\sigma^{z}}_{t-1}\end{matrix}\right),$ (42)
where $\braket{\sigma^{a}}:=\text{tr}(\sigma^{a}\rho)$ is the expected value
of the spin projection in the $a$ direction of the real space. The expressions
for each matrix element are shown below:
$\begin{matrix}\widehat{T}_{22}=e^{-2\gamma\Delta\tau},\\\
\widehat{T}_{33}=e^{-\gamma\Delta\tau}\left(\cosh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)-\frac{\gamma}{\sqrt{\gamma^{2}-h_{t}^{2}}}\sinh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)\right),\\\
\widehat{T}_{44}=e^{-\gamma\Delta\tau}\left(\cosh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-h_{t}^{2}}}\sinh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)\right),\\\
\widehat{T}_{34}=\frac{h_{t}e^{-\gamma\Delta\tau}}{\sqrt{\gamma^{2}-h_{t}^{2}}}\sinh\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right),\\\
\widehat{T}_{43}=-\widehat{T}_{34},\end{matrix}$ (43)
where have shortened notation by setting $h_{t}=h({\bf z}_{t})$. The
eigenvalues of matrix $\widehat{T}$ can be computed analytically:
$\lambda_{1}=1$, $\lambda_{2}=e^{-2\gamma\Delta\tau}$,
$\lambda_{3}=e^{-(\gamma+\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}$ and
$\lambda_{4}=e^{-(\gamma-\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}$. The moduli
of the eigenvalues are $|\lambda_{1}|=1$,
$|\lambda_{2}|=e^{-2\gamma\Delta\tau}<1$ and
$|\lambda_{3}|=e^{-(\gamma+\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}<1$.
Eigenvalue $|\lambda_{4}|=e^{-(\gamma-\sqrt{\gamma^{2}-h^{2}_{t}})\Delta\tau}$
is smaller than one if and only if $h_{t}\neq 0$. Under that condition, the
map is a mixing channel with a single fixed point. It can be checked that the
map is unital so the fixed point is the maximally mixed state:
$\rho^{*}=\left(\begin{matrix}1/2&0\\\ 0&1/2\end{matrix}\right).$ (44)
Let us prove that there exists some norm where
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1$
for all inputs. A straightforward induced norm to evaluate is the 2-Schatten
induced norm. The singular values of the restriction
$p(\textbf{z})=\left(\begin{matrix}\widehat{T}_{22}&0&0\\\
0&\widehat{T}_{33}&\widehat{T}_{34}\\\
0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right)$ (45)
are $\sigma_{1}=e^{-2\gamma\Delta\tau}<1$,
$\sigma_{2}=e^{-\gamma\Delta\tau}\sqrt{f_{+}}$ and
$\sigma_{3}=e^{-\gamma\Delta\tau}\sqrt{f_{-}}$, where
$\begin{split}f_{\pm}&=\frac{1}{\gamma^{2}-h^{2}_{t}}\left(-h^{2}_{t}+\gamma^{2}\cosh\left(2\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)\right.\\\
&\left.\pm\gamma\sqrt{\sinh^{2}\left(\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)}\sqrt{-4h^{2}_{t}+2\gamma^{2}+2\gamma^{2}\cosh\left(2\Delta\tau\sqrt{\gamma^{2}-h_{t}^{2}}\right)}\right).\end{split}$
(46)
Figure 1: Density plot of the singular values (a) $\sigma_{2}$ and (b)
$\sigma_{3}$. The diagonal elements $h_{t}=\gamma$ are not determined because
of the denominator $1/(\gamma^{2}-h^{2}_{t})$.
Figure 1 numerically shows that for $\gamma\neq h_{t}$ and away from the axes
$h_{t}=0$ and $\gamma=0$, we find $\sigma_{2},\sigma_{3}<1$. Therefore, the
system has the ESP and the FMP. As we showed in Theorem 5, the corresponding
filter (17) is necessarily trivial and given by
$U_{T}({\bf z})_{t}=\rho^{*}=\left(\begin{matrix}1/2&0\\\
0&1/2\end{matrix}\right).$ (47)
Since the Bloch vector for this constant matrix is $(0,0,0)^{\top}$, this
shows that in the SAS representation $U_{\widehat{T}_{0}}({\bf
z})_{t}=(0,0,0)^{\top}$.
Unital quantum channels are very common in the quantum information literature
because of their practical advantages and well-known mathematical properties,
see for example [74, 75], and references therein. However, Theorem 5 discards
the possibility of relying on unital contractive channels for QRC for long
input sequences (see also Section IV for more details). The physical
explanation for this behavior stems from the fact that the fixed point of
these maps does not depend on the input. Then, after each application of the
channel, decoherence always leads to the same stationary state (the maximally
mixed state), which does not keep track of these inputs. This hinders any
possibility of storing the input information into the degrees of freedom of
the quantum system since all coherences fade out and the diagonal elements of
the density matrix become equal.
We believe that the observation that we just made is very relevant, namely
that quantum channels with input-independent fixed points become memoryless in
the long-term. This fact is proved in the next theorem that generalizes
Theorem 5.
###### Theorem 9.
Let $T:\mathcal{B}(\mathcal{H})\times
D_{n}\rightarrow\mathcal{B}(\mathcal{H})$ be a QRC system for which there
exists an operator norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and $\epsilon>0$ such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|T(\cdot,{\bf
z})|_{\mathcal{B}_{0}(\mathcal{H})}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1-\epsilon$,
for all ${\bf z}\in D_{n}$. Then, $T$ has an input-independent fixed point
$\rho^{\ast}\in\mathcal{S}({\mathcal{H}})$, that is, $T(\rho^{*},{\bf
z})=\rho^{*}$, for all ${\bf z}\in D_{n}$, if and only if the corresponding
filter $U_{T}$ is constant, that is, $U_{T}({\bf
z})_{t}=\rho^{\ast}\in\mathcal{S}({\mathcal{H}})$, for all ${\bf
z}\in(D_{n})^{\mathbb{Z}}$ (equivalently, $U_{\widehat{T}_{0}}({\bf
z})_{t}=i_{0}^{-1}\left(G_{\mathcal{B}}^{-1}(\rho^{\ast})\right)$) .
###### Proof.
We first note that the contractivity hypothesis implies by Proposition 3 that
the system associated to $T$ has the ESP and hence has a unique solution for
each input. The hypothesis $T(\rho^{*},{\bf z})=\rho^{*}$, for all ${\bf z}\in
D_{n}$, obviously implies that the constant sequence equal to $\rho^{*}$ is a
solution for any input ${\bf z}\in(D_{n})^{\mathbb{Z}}$ and hence $U_{T}({\bf
z})_{t}=\rho^{\ast}\in\mathcal{S}({\mathcal{H}})$, for all ${\bf
z}\in(D_{n})^{\mathbb{Z}}$. Conversely, since the filter $U_{T}$ is determined
by the recursions
$U_{T}({\bf z})_{t}=T\left(U_{T}({\bf z})_{t-1},{\bf z}_{t}\right),$ (48)
then, if we always have that $U_{T}({\bf z})_{t}=\rho^{\ast}$, the relation
(48) implies that $T(\rho^{*},{\bf z})=\rho^{*}$, for all ${\bf z}\in D_{n}$.
∎
###### Remark 10.
The differences between the hypotheses in Theorems 5 and 9 on fixed points are
apparent when the QRC system is expressed using the SAS representation in
terms of the functions $q$ and $p$. Indeed, as we saw in the proof of Theorems
5, $q({\bf z})=0$ for and ${\bf z}\in D_{n}$, in that case, and the input
dependence takes place only through $p$. This is the case in Example 7.
Theorem 9 allows for an input dependence through $q$ too.
###### Example 11.
We define a Markovian master equation for the dynamics between input
injections:
$\dot{\rho}=-i[H({\bf z}_{t}),\rho]+\gamma L\rho
L^{\dagger}-\frac{\gamma}{2}\\{L^{\dagger}L,\rho\\},$ (49)
where $H({\bf z}_{t})=h({\bf z}_{t})\sigma^{z}/2$ and $L=\sigma^{-}$. This is
a single qubit under the influence of an external magnetic field in the $z$
direction with local dissipation. The matrix expression $\widehat{T}$ for the
associated system is:
$\left(\begin{matrix}1\\\ \braket{\sigma^{x}}_{t}\\\
\braket{\sigma^{y}}_{t}\\\
\braket{\sigma^{z}}_{t}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&0\\\
0&-e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&0\\\
e^{-\gamma\Delta\tau}-1&0&0&e^{-\gamma\Delta\tau}\end{matrix}\right)\left(\begin{matrix}1\\\
\braket{\sigma^{x}}_{t-1}\\\ \braket{\sigma^{y}}_{t-1}\\\
\braket{\sigma^{z}}_{t-1}\end{matrix}\right).$ (50)
The eigenvalues of $\widehat{T}$ can be computed analytically:
$\lambda_{1}=1$, $\lambda_{2}=e^{-\gamma\Delta\tau}$,
$\lambda_{3}=e^{-\frac{\gamma\Delta\tau}{2}-ih_{t}\Delta\tau}$ and
$\lambda_{4}=e^{-\frac{\gamma\Delta\tau}{2}+ih_{t}\Delta\tau}$. The moduli of
the eigenvalues are $|\lambda_{1}|=1$, $|\lambda_{2}|=e^{-\gamma\Delta\tau}<1$
and $|\lambda_{3}|=|\lambda_{4}|=e^{-\frac{\gamma\Delta\tau}{2}}<1$, so $T$ is
a mixing channel. In this case, the single fixed point is a pure input-
independent state with density matrix
$\rho^{*}=\left(\begin{matrix}0&0\\\ 0&1\end{matrix}\right).$ (51)
Let us see if it is true that there exists some
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}<1$
for all inputs. The singular values of the restriction
$p(\textbf{z})=\left(\begin{matrix}e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&0\\\
-e^{-\frac{\gamma\Delta\tau}{2}}\sin(h_{t}\Delta\tau)&e^{-\frac{\gamma\Delta\tau}{2}}\cos(h_{t}\Delta\tau)&0\\\
0&0&e^{-\gamma\Delta\tau}\end{matrix}\right)$ (52)
are $\sigma_{1}=e^{-\gamma\Delta\tau}<1$ and
$\sigma_{2}=\sigma_{3}=e^{-\frac{\gamma\Delta\tau}{2}}<1$. Therefore, the
system has the ESP and FMP. As we showed in Theorem 9, the associated filter
is necessarily constant, and (17) necessarily yields the filter
$U_{T}({\bf z})_{t}=\rho^{*}=\left(\begin{matrix}0&0\\\
0&1\end{matrix}\right).$ (53)
Since the Bloch vector for this constant matrix is $(0,0,-1)^{\top}$, this
shows that in the SAS representation $U_{\widehat{T}_{0}}({\bf
z})_{t}=(0,0,-1)^{\top}$.
We can double-check the solution by explicitly computing the filter (31). Take
$p({\bf z}_{t})$ as in (52) and $q({\bf
z}_{t})^{\top}=(0,0,e^{-\gamma\Delta\tau}-1)$. Since
$\sigma_{\text{max}}(p({\bf z}_{t}))<1$, the filter in (31) exists. Using now
that $q({\bf z}_{t})$ is input-independent, we can write
$\begin{split}U_{\widehat{T}_{0}}({\bf
z})_{t}&=\sum^{\infty}_{j=0}\left(\prod^{j-1}_{k=0}p({\bf
z}_{t-k})\right)q({\bf z}_{t-j})\\\
&=\left(\sum^{\infty}_{j=0}\prod^{j-1}_{k=0}p({\bf
z}_{t-k})\right)q=(e^{-\gamma\Delta\tau}-1)M_{3},\end{split}$ (54)
where $M_{3}$ is the third column of the matrix
$M=\sum^{\infty}_{j=0}\prod^{j-1}_{k=0}p({\bf z}_{t-k})$. Notice that the
third column of the product $\prod^{j-1}_{k=0}p({\bf z}_{t-k})$ is
$(0,0,e^{-k\gamma\Delta\tau})^{\top}$. Then, the column $M_{3}$ equals
$M_{3}=\left(\begin{matrix}0\\\ 0\\\
\sum^{\infty}_{j=0}e^{-k\gamma\Delta\tau}\end{matrix}\right)=\left(\begin{matrix}0\\\
0\\\ \frac{1}{1-e^{-\gamma\Delta\tau}}\end{matrix}\right),$ (55)
and it hence follows that $U_{\widehat{T}_{0}}({\bf z})_{t}=(0,0,-1)^{\top}$.
To conclude, we show in Figure 2 a numerical example of input driving. We
choose $h(z_{t})=z_{t}\frac{h}{2}\sigma^{z}$ as the external magnetic field
function, where $h$ is a constant and $z_{t}$ is a unidimensional random
input. The input will be drawn from a random uniform distribution in the
interval $[0,1]$. As can be seen, the observables exhibit a transient time
after which they converge to the input-independent stationary state.
Figure 2: Dynamics of the spin projections $\braket{\sigma^{a}}_{t}$ for
$a=x,y,z$ when driven with a random input sequence with Hamiltonian
$H=z_{t}\frac{h}{2}\sigma^{z}$. The initial condition is the maximal coherent
state $\rho=1/2\sum^{1}_{i,j=0}\ket{i}\bra{j}$ and the system parameters are
$\Delta\tau=1$, $\gamma=1$ and $h=1$.
## IV Discussion
Theorems 5 and 9 bring up a connection between long-term computation and noisy
intermediate-scale quantum (NISQ) devices: they have a finite time of
operation due to decoherence. Consider a model of the type
$\rho_{t}=T(\rho_{t},{\bf z}_{t})=\mathcal{E}_{\text{deco}}(\mathcal{U}({\bf
z}_{t})\rho_{t-1}\mathcal{U}^{\dagger}({\bf z}_{t})),$ (56)
where $\mathcal{E}_{\text{deco}}$ represents the decoherence produced by the
contact of the system with an external environment. This model is present in
QRC experimental works like [40, 39], where the unitary dynamics
$\mathcal{U}({\bf z}_{t})$ is given by a quantum circuit. Theorem 5 explains
what happens in the extreme case in which the quantum noise of the device is
unital. If the decoherence channel $\mathcal{E}_{\text{deco}}$ is a unital
strictly contractive map then
$T\left(\frac{I}{d},{\bf
z}_{t}\right)=\mathcal{E}_{\text{deco}}\left(\mathcal{U}({\bf
z}_{t})\frac{I}{d}\mathcal{U}^{\dagger}({\bf
z}_{t})\right)=\frac{\mathcal{E}_{\text{deco}}(I)}{d}=\frac{I}{d},$
that is, $T(\cdot,{\bf z})$ becomes a unital strictly contractive map for all
${\bf z}\in D_{n}$. Theorem 5 shows, in this case, that the filter becomes
trivial after the injection of long input sequences. Instances of unital
decoherence can be found in depolarizing channels, like in Example 7, or in
dephasing channels. A dephasing channel damps the coherences of the density
matrix but does not affect the diagonal elements.
More generally, Theorem 9 can be interpreted as a “common sense” warning: it
explicitly states that the input codification must have a measurable influence
on the attractor of the natural dynamics of the CPTP map. Otherwise, there is
no possibility of storing input information in the long run. We emphasize that
this is independent of the type of dissipation that the quantum channel
produces. We can connect the implications of this theorem with the NISQ
discussion and (56). If the input is codified in some particular coherences of
the system, one must be careful that decoherence does not destroy those matrix
elements, because then input-dependence would vanish, and the resulting filter
would become trivial.
This discussion does not imply that models like (56) are useless. Indeed, the
opposite has been proven in previous works for short-term memory tasks [40,
39]. Theorems 5 and 9 only rigorously establish something that was already
known about NISQ devices, that is, that there is a coherence time in which
they can be exploited. Equivalently, models affected by these theorems have
the ESP, but the resulting input/output dynamics becomes trivial for long
input sequences.
The coherence time limitation affects to all quantum platforms to a greater or
lesser extent. Then, either QRC proposals are subject to operate on shorter
time scales than the natural noise time scale (as done in [40, 39]), or the
QRC system is carefully design to integrate it. For example, as we will see in
Section IV.1, dephasing can be integrated as part of a QRC system that is not
affected by Theorems 5 and 9.
We could further extend this analysis to QRC models with measurements. As an
example we take the model proposed in [76]. In this reference, the quantum
measurement is applied at each time step after the input dependent CPTP map
$T$. The measurement scheme is introduced by modeling an indirect measurement
with a continuous-variable ancilla [77, 78, 79, 80, 81], producing a quantum
reservoir with stochastic dynamics. For simplicity, we will restrict ourselves
to the case of a single qubit, and we will average the quantum states over the
limit of infinite measurements, yielding an unconditional state which is led
by a deterministic CPTP quantum channel [82]. Besides, we choose, without loss
of generality, to take measurements in the $z$ direction. Under these
conditions, the CPTP map is
$\rho_{t}=M\odot T(\rho_{t-1},\textbf{z}_{t}),$ (57)
where $\odot$ represents the Hadamard or element-wise matrix product and $M$
is defined as
$M=\begin{pmatrix}1&e^{-\frac{g^{2}}{2}}\\\
e^{-\frac{g^{2}}{2}}&1\end{pmatrix}.$ (58)
The measurement strength $g$ allows us to quantify the decoherence introduced
by sharp measurements ($g\gg 1$), while for $g\ll 1$ the state is weakly
perturbed. It is straightforward to see that this model is introducing
dephasing, such that we can rewrite (57) as
$\rho_{t}=\mathcal{E}_{\text{deph}}\left(T(\rho_{t-1},\textbf{z}_{t})\right),$
(59)
where the dephasing channel $\mathcal{E}_{\text{deph}}$ is defined as
$\mathcal{E}_{\text{deph}}(\rho)=e^{-\frac{g^{2}}{2}}\rho+(1-e^{-\frac{g^{2}}{2}})\sigma^{z}\rho\sigma^{z}.$
(60)
As we explained above, unitary dynamics for the map $T$ would lead to a
memoryless reservoir in the long-term. However, one could engineer a mixing
map $T$ such that there is a competition between the attractors of maps $T$
and $\mathcal{E}_{\text{deph}}$. The final fixed point of (59) would be
somewhere between the original fixed point of $T$ and a diagonal state (which
is the shape of the fixed points of $\mathcal{E}_{\text{deph}}$).
We conclude by presenting an example of a qubit with tunable local dissipation
that fulfills all the requirements to be a “properly engineered” QRC system.
Then, we extend it with the measurement model of [76] to show that it still
constitutes a proper QRC system.
### IV.1 A “properly engineered” QRC system
We start by introducing the model without measurements. The Markovian master
equation that governs the dynamics between input injections is:
$\dot{\rho}=-i[H({\bf z}_{t}),\rho]+\gamma L\rho
L^{\dagger}-\frac{\gamma}{2}\\{L^{\dagger}L,\rho\\},$ (61)
where $H({\bf z}_{t})=h({\bf z}_{t})\sigma^{x}/2$, and $L=\sigma^{-}$. The
corresponding matrix expression $\widehat{T}$ is
$\left(\begin{matrix}1\\\ \braket{\sigma^{x}}_{t}\\\
\braket{\sigma^{y}}_{t}\\\
\braket{\sigma^{z}}_{t}\end{matrix}\right)=\left(\begin{matrix}1&0&0&0\\\
0&\widehat{T}_{22}&0&0\\\
\widehat{T}_{31}&0&\widehat{T}_{33}&\widehat{T}_{34}\\\
\widehat{T}_{41}&0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right)\left(\begin{matrix}1\\\
\braket{\sigma^{x}}_{t-1}\\\ \braket{\sigma^{y}}_{t-1}\\\
\braket{\sigma^{z}}_{t-1}\end{matrix}\right),$ (62)
where the expressions for the matrix elements are shown below:
$\begin{matrix}\widehat{T}_{22}=e^{-\frac{\gamma\Delta\tau}{2}},\\\
\widehat{T}_{33}=e^{-\frac{3\gamma\Delta\tau}{4}}\left(\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right),\\\
\widehat{T}_{44}=e^{-\frac{3\gamma\Delta\tau}{4}}\left(\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)-\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right),\\\
\widehat{T}_{34}=\frac{4h_{t}e^{-\frac{3\gamma\Delta\tau}{4}}}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right),\\\
\widehat{T}_{43}=-\widehat{T}_{34},\\\ \widehat{T}_{31}=\frac{2\gamma
h_{t}}{\gamma^{2}+2h^{2}_{t}}\left\\{-1+e^{-\frac{3\gamma\Delta\tau}{4}}\left(\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{3\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)\right\\},\\\
\widehat{T}_{41}=\frac{\gamma}{\gamma^{2}+2h^{2}_{t}}\left\\{-\gamma+e^{-\frac{3\gamma\Delta\tau}{4}}\left(\gamma\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)-\frac{\gamma^{2}+8h^{2}_{t}}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)\right\\}.\end{matrix}$
(63)
Figure 3: Density plot of the singular values (a) $\sigma_{2}$ and (b)
$\sigma_{3}$. The elements $\gamma=4h_{t}$ are not determined because of the
denominator $1/(\gamma^{2}-16h^{2}_{t})$.
The eigenvalues of matrix $\hat{T}$ are: $\lambda_{1}=1$,
$\lambda_{2}=e^{-\frac{\gamma\Delta\tau}{2}}$,
$\lambda_{3}=e^{-(3\gamma^{2}+\sqrt{\gamma^{2}-16h^{2}_{t}})\frac{\Delta\tau}{4}}$
and
$\lambda_{4}=e^{-(3\gamma^{2}-\sqrt{\gamma^{2}-16h^{2}_{t}})\frac{\Delta\tau}{4}}$.
The three eigenvalues $\lambda_{2}$, $\lambda_{3}$ and $\lambda_{4}$ have
always modulus smaller than one when $\gamma\neq 0$. Then, the master equation
fulfills the conditions for having a single full-rank fixed point [62], whose
density matrix is
$\rho^{*}=\frac{1}{\gamma^{2}+2h^{2}_{t}}\left(\begin{matrix}h^{2}_{t}&i\gamma
h_{t}\\\ -i\gamma h_{t}&\gamma^{2}+h^{2}_{t}\end{matrix}\right).$ (64)
Finally, we compute the singular values of the restriction to the traceless
hyperplane. These values are $\sigma_{1}=e^{-\frac{\gamma\Delta\tau}{2}}<1$,
$\sigma_{2}=e^{-\frac{3\gamma\Delta\tau}{4}}\sqrt{f_{+}}$ and
$\sigma_{3}=e^{-\frac{3\gamma\Delta\tau}{4}}\sqrt{f_{-}}$, where
$\begin{split}f_{\pm}&=\frac{1}{\gamma^{2}-16h^{2}_{t}}\left(-16h^{2}_{t}+\gamma^{2}\cosh\left(\frac{\Delta\tau}{2}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right.\\\
&\left.\pm\gamma\sqrt{\sinh^{2}\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}\sqrt{-64h^{2}_{t}+2\gamma^{2}+2\gamma^{2}\cosh\left(\frac{\Delta\tau}{2}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}\right).\end{split}$
(65)
Figure 3 shows that for $\gamma\neq 4h_{t}$ and away from the axis $\gamma=0$,
we find $\sigma_{2},\sigma_{3}<1$, demonstrating the ESP and the FMP. Since
the fixed point $\rho^{*}$ is input dependent, this system exhibits the
necessary ingredients to be a competent QRC system.
As in Example 11, we show in Figure 4 a numerical example of input driving. We
choose again $h(z_{t})=z_{t}\frac{h}{2}\sigma^{x}$ as the external magnetic
field function, modifying its direction. Now, the observables
$\braket{\sigma^{z}}$ and $\braket{\sigma^{y}}$ exhibit an explicit response
to the driving, while $\braket{\sigma^{x}}$ converges to its input-independent
stationary value (which can be predicted from (62)).
Figure 4: Dynamics of the spin projections $\braket{\sigma^{a}}_{t}$ for
$a=x,y,z$ when driven with a random input sequence with Hamiltonian
$H=z_{t}\frac{h}{2}\sigma^{x}$. The initial condition is the maximal coherent
state $\rho=1/2\sum^{1}_{i,j=0}\ket{i}\bra{j}$ and the system parameters are
$\Delta\tau=1$, $\gamma=1$ and $h=1$.
Now we further extend the model to incorporate the measurement formalism
described in [76]. As the composition of CPTP maps can be described as the
product of their matrix representations [58], we just need to obtain the Kraus
operators of (60), which are $K_{0}=\sqrt{e^{-\frac{g^{2}}{2}}}I$,
$K_{1}=\sqrt{1-e^{-\frac{g^{2}}{2}}}\ket{0}\bra{0}$ and
$K_{2}=\sqrt{1-e^{-\frac{g^{2}}{2}}}\ket{1}\bra{1}$, where $\ket{0}$ and
$\ket{1}$ are the basis states in the $z$ axis. The matrix representation of
$\mathcal{E}_{\text{deph}}$ in the Pauli matrix basis is then
$\widehat{\mathcal{E}}_{\text{deph}}=\left(\begin{matrix}1&0&0&0\\\
0&e^{-\frac{g^{2}}{2}}&0&0\\\ 0&0&e^{-\frac{g^{2}}{2}}&0\\\
0&0&0&1\end{matrix}\right).$ (66)
It is straightforward to check that the maximum singular value of
$\widehat{\mathcal{E}}_{\text{deph}}$ restricted to the traceless hyperplane
is equal to one. The final matrix
$\widehat{T}^{\prime}=\widehat{\mathcal{E}}_{\text{deph}}\widehat{T}$ is
$\widehat{T}^{\prime}=\left(\begin{matrix}1&0&0&0\\\
0&e^{-\frac{g^{2}}{2}}\widehat{T}_{22}&0&0\\\
e^{-\frac{g^{2}}{2}}\widehat{T}_{31}&0&e^{-\frac{g^{2}}{2}}\widehat{T}_{33}&e^{-\frac{g^{2}}{2}}\widehat{T}_{34}\\\
\widehat{T}_{41}&0&\widehat{T}_{43}&\widehat{T}_{44}\end{matrix}\right).$ (67)
Given that the maximum singular value of $p(\textbf{z})$ is smaller than one
(for $\gamma\neq 4h_{t}$ and $\gamma\neq 0$), we find that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p^{\prime}(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p_{\text{deph}}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}\cdot{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|p(\textbf{z})\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{2}<1-\epsilon$
for some $\epsilon>0$ in the 2-Schatten norm, where $p(\textbf{z})$,
$p^{\prime}(\textbf{z})$ and $p_{\text{deph}}$ are the restrictions to the
traceless hyperplane of matrices $\widehat{T}$, $\widehat{T}^{\prime}$ and
$\widehat{\mathcal{E}}_{\text{deph}}$ respectively (given by (27)). Then, the
QRC system has the ESP and the FMP. The single fixed point of the map is given
by
$\rho^{*}=\frac{1}{\gamma^{2}+2h^{2}_{t}}\left(\begin{matrix}h^{2}_{t}-f_{1}&i\gamma
h_{t}(1-f_{2})\\\ -i\gamma
h_{t}(1-f_{2})&\gamma^{2}+h^{2}_{t}+f_{1}\end{matrix}\right),$ (68)
where
$\begin{split}&f_{1}=\frac{4\gamma
h^{2}_{t}\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}{\sqrt{\gamma^{2}-16h_{t}^{2}}\left(\cosh\left(\frac{g^{2}+3\gamma\Delta\tau}{4}\right)-\cosh\left(\frac{g^{2}}{4}\right)\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)+\gamma\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)},\\\
&f_{2}=\frac{\sinh\left(\frac{g^{2}}{4}\right)\left(e^{\frac{3\gamma\Delta\tau}{4}}-\cosh\left(\frac{g^{2}}{4}\right)\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)\right)}{\cosh\left(\frac{g^{2}+3\gamma\Delta\tau}{4}\right)-\cosh\left(\frac{g^{2}}{4}\right)\cosh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)+\frac{\gamma}{\sqrt{\gamma^{2}-16h_{t}^{2}}}\sinh\left(\frac{g^{2}}{4}\right)\sinh\left(\frac{\Delta\tau}{4}\sqrt{\gamma^{2}-16h_{t}^{2}}\right)}.\end{split}$
(69)
This fixed point is produced by the competition between the original fixed
point in (64) ($g\rightarrow 0$) and a diagonal state ($g\rightarrow\infty$).
With this we can conclude that this engineered model, even including the
measurement protocol, leads to an operational QRC system in the long-term run.
## V Conclusions
In this paper, we have unified the density matrix approach of previous works
in QRC with the Bloch vector representation. Moreover, we have shown that
these representations are linked by system isomorphisms and that various
results concerning the ESP and FMP are independent of the chosen
representation. We have also observed that the QRC dynamics in the Bloch
vectors representation amounts to that of a state-affine system (SAS) of the
type introduced in [45] and for which numerous theoretical results have been
established. We have capitalized on this connection to shed some light on
fundamental questions in QRC theory in finite dimensions. In particular, we
found a necessary and sufficient condition for the ESP and FMP in terms of the
existence of an induced norm that bounds the CPTP map for all inputs,
determining a guideline for its election. The necessity of this boundedness
hypothesis emerges out of the compactness of the input space, which is a
common requirement in the RC literature. If the input space is not compact,
sufficient conditions can still be found in terms of the weighting sequence
[50]. Besides, we described common situations in which QRC systems become
useless in long term runs which can be summarized by saying that quantum
channels that exhibit input-independent fixed points yield trivial
input/output dynamics.
Our work sets the grounds for further analysis and exploration of the QRC
theory. Future work can follow several paths, such as studying the connection
between spectral properties of QRC models and their performance in memory and
information processing tasks, studying infinite-dimensional quantum
reservoirs, or including generalized measurements (positive operator-valued
measures) and the effect of a finite number of measurements in the statistics
of expected values, given the effect that they imprint in the resources of QRC
algorithms [76].
## VI Acknowledgments
We thank A. Sannia for useful discussion and inspiration for this work, and D.
Burgarth and M. Rahaman for answering our questions regarding their work and
its relation to our study. We also thank the editor and two anonymous referees
for input that has significantly improved the paper. R.M.-P. and J.-P.O.
acknowledge partial financial support from the Swiss National Science
Foundation (Grant No. 200021 175801/1). R.M.-P. acknowledges the Spanish State
Research Agency for support through the Severo Ochoa and María de Maeztu
Program for Centers and Units of Excellence in R&D (Grant No. MDM-2017-0711)
and through the QUARESC project (Projects No. 2019-109094GB-C21 and
2019-109094GB-C22/AEI/10.13039/501100011033). Part of this work was funded by
MICINN/AEI/FEDER and the University of the Balearic Islands through a
predoctoral fellowship (Grant No. MDM-2017-0711-18-1) for R.M.-P. R.M.-P. also
is grateful for the hospitality of the Division of Mathematical Sciences of
the Nanyang Technological University, where most of these results were
obtained.
## References
* Arute _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, _et al._ , Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
* Choi _et al._ [2016] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
* Chen _et al._ [2021] Y.-A. Chen, Q. Zhang, T.-Y. Chen, W.-Q. Cai, S.-K. Liao, J. Zhang, K. Chen, J. Yin, J.-G. Ren, Z. Chen, _et al._ , An integrated space-to-ground quantum communication network over 4,600 kilometres, Nature 589, 214 (2021).
* Bharti _et al._ [2022] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, _et al._ , Noisy intermediate-scale quantum algorithms, Reviews of Modern Physics 94, 015004 (2022).
* Zhong _et al._ [2020] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, _et al._ , Quantum computational advantage using photons, Science 370, 1460 (2020).
* Madsen _et al._ [2022] L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, _et al._ , Quantum computational advantage with a programmable photonic processor, Nature 606, 75 (2022).
* Biamonte _et al._ [2017] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature 549, 195 (2017).
* Liu _et al._ [2021] Y. Liu, S. Arunachalam, and K. Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nature Physics 17, 1013 (2021).
* Huang _et al._ [2022] H.-Y. Huang, M. Broughton, J. Cotler, S. Chen, J. Li, M. Mohseni, H. Neven, R. Babbush, R. Kueng, J. Preskill, _et al._ , Quantum advantage in learning from experiments, Science 376, 1182 (2022).
* Cybenko [1989] G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of control, signals and systems 2, 303 (1989).
* Hornik _et al._ [1989] K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators, Neural networks 2, 359 (1989).
* Hornik [1991] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural networks 4, 251 (1991).
* Schäfer and Zimmermann [2006] A. M. Schäfer and H. G. Zimmermann, Recurrent neural networks are universal approximators, in _International Conference on Artificial Neural Networks_ (Springer, 2006) pp. 632–640.
* Hammer and Gersmann [2003] B. Hammer and K. Gersmann, A note on the universal approximation capability of support vector machines, neural processing letters 17, 43 (2003).
* Huang _et al._ [2006] G.-B. Huang, L. Chen, C. K. Siew, _et al._ , Universal approximation using incremental constructive feedforward networks with random hidden nodes, IEEE Trans. Neural Networks 17, 879 (2006).
* Gonon _et al._ [2022a] L. Gonon, L. Grigoryeva, and J.-P. Ortega, Approximation error estimates for random neural networks and reservoir systems, To appear in the Annals of Applied Probability (2022a).
* Steinwart [2001] I. Steinwart, On the influence of the kernel on the consistency of support vector machines, Journal of machine learning research 2, 67 (2001).
* Micchelli _et al._ [2006] C. A. Micchelli, Y. Xu, and H. Zhang, Universal Kernels, Journal of Machine Learning Research 7, 2651 (2006).
* Gonon _et al._ [2022b] L. Gonon, L. Grigoryeva, and J.-P. Ortega, Reservoir kernels and Volterra series, Preprint (2022b).
* Pérez-Salinas _et al._ [2020] A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Data re-uploading for a universal quantum classifier, Quantum 4, 226 (2020).
* Pérez-Salinas _et al._ [2021] A. Pérez-Salinas, D. López-Núñez, A. García-Sáez, P. Forn-Díaz, and J. I. Latorre, One qubit as a universal approximant, Physical Review A 104, 012405 (2021).
* Schuld _et al._ [2021] M. Schuld, R. Sweke, and J. J. Meyer, Effect of data encoding on the expressive power of variational quantum-machine-learning models, Physical Review A 103, 032430 (2021).
* Goto _et al._ [2021] T. Goto, Q. H. Tran, and K. Nakajima, Universal approximation property of quantum machine learning models in quantum-enhanced feature spaces, Physical Review Letters 127, 090506 (2021).
* Jaeger [2001] H. Jaeger, The “echo state” approach to analysing and training recurrent neural networks-with an erratum note, Bonn, Germany: German National Research Center for Information Technology GMD Technical Report 148, 13 (2001).
* Lukoševičius and Jaeger [2009] M. Lukoševičius and H. Jaeger, Reservoir computing approaches to recurrent neural network training, Computer Science Review 3, 127 (2009).
* Tanaka _et al._ [2019] G. Tanaka, T. Yamane, J. B. Héroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano, and A. Hirose, Recent advances in physical reservoir computing: A review, Neural Networks 115, 100 (2019).
* Jaeger and Haas [2004] H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, science 304, 78 (2004).
* Grigoryeva _et al._ [2014] L. Grigoryeva, J. Henriques, L. Larger, and J.-P. Ortega, Stochastic time series forecasting using time-delay reservoir computers: performance and universality, Neural Networks 55, 59 (2014).
* Shahi _et al._ [2022] S. Shahi, F. H. Fenton, and E. M. Cherry, Prediction of chaotic time series using recurrent neural networks and reservoir computing techniques: A comparative study, Machine learning with applications 8, 100300 (2022).
* Pathak _et al._ [2018] J. Pathak, A. Wikner, R. Fussell, S. Chandra, B. R. Hunt, M. Girvan, and E. Ott, Hybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model, Chaos 28, 10.1063/1.5028373 (2018), arXiv:1803.04779 .
* Vlachas _et al._ [2020] P. R. Vlachas, J. Pathak, B. R. Hunt, T. P. Sapsis, M. Girvan, E. Ott, and P. Koumoutsakos, Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics, Neural Networks 126, 191 (2020).
* Tran and Nakajima [2020] Q. H. Tran and K. Nakajima, Higher-order quantum reservoir computing, arXiv preprint arXiv:2006.08999 (2020).
* Wikner _et al._ [2021] A. Wikner, J. Pathak, B. R. Hunt, I. Szunyogh, M. Girvan, and E. Ott, Using data assimilation to train a hybrid forecast system that combines machine-learning and knowledge-based components, Chaos: An Interdisciplinary Journal of Nonlinear Science 31, 53114 (2021).
* Fujii and Nakajima [2017] K. Fujii and K. Nakajima, Harnessing disordered-ensemble quantum dynamics for machine learning, Physical Review Applied 8, 024030 (2017).
* Ghosh _et al._ [2021] S. Ghosh, K. Nakajima, T. Krisnanda, K. Fujii, and T. C. Liew, Quantum neuromorphic computing with reservoir computing networks, Advanced Quantum Technologies 4, 2100053 (2021).
* Mujal _et al._ [2021] P. Mujal, R. Martínez-Peña, J. Nokkala, J. García-Beni, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Opportunities in quantum reservoir computing and extreme learning machines, Advanced Quantum Technologies 4, 2100027 (2021).
* Chen _et al._ [2020] J. Chen, H. I. Nurdin, and N. Yamamoto, Temporal information processing on noisy quantum computers, Physical Review Applied 14, 024065 (2020).
* Dasgupta _et al._ [2020] S. Dasgupta, K. E. Hamilton, and A. Banerjee, Designing a nisq reservoir with maximal memory capacity for volatility forecasting, arXiv preprint arXiv:2004.08240 (2020).
* Suzuki _et al._ [2022] Y. Suzuki, Q. Gao, K. C. Pradel, K. Yasuoka, and N. Yamamoto, Natural quantum reservoir computing for temporal information processing, Scientific reports 12, 1 (2022).
* Kubota _et al._ [2022] T. Kubota, Y. Suzuki, S. Kobayashi, Q. H. Tran, N. Yamamoto, and K. Nakajima, Quantum noise-induced reservoir computing, arXiv preprint arXiv:2207.07924 (2022).
* Molteni _et al._ [2022] R. Molteni, C. Destri, and E. Prati, Optimization of the memory reset rate of a quantum echo-state network for time sequential tasks, arXiv preprint arXiv:2210.01052 (2022).
* Chen and Nurdin [2019] J. Chen and H. I. Nurdin, Learning nonlinear input–output maps with dissipative quantum systems, Quantum Information Processing 18, 1 (2019).
* Nokkala _et al._ [2021] J. Nokkala, R. Martínez-Peña, G. L. Giorgi, V. Parigi, M. C. Soriano, and R. Zambrini, Gaussian states of continuous-variable quantum systems provide universal and versatile reservoir computing, Communications Physics 4, 1 (2021).
* Grigoryeva and Ortega [2018a] L. Grigoryeva and J.-P. Ortega, Echo state networks are universal, Neural Networks 108, 495 (2018a).
* Grigoryeva and Ortega [2018b] L. Grigoryeva and J.-P. Ortega, Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems, Journal of Machine Learning Research 19, 1 (2018b).
* Gonon and Ortega [2019] L. Gonon and J.-P. Ortega, Reservoir computing universality with stochastic inputs, IEEE transactions on neural networks and learning systems 31, 100 (2019).
* Gonon and Ortega [2021] L. Gonon and J.-P. Ortega, Fading memory echo state networks are universal, Neural Networks 138, 10 (2021).
* Boyd and Chua [1985] S. Boyd and L. Chua, Fading memory and the problem of approximating nonlinear operators with volterra series, IEEE Transactions on circuits and systems 32, 1150 (1985).
* Wright and McMahon [2020] L. G. Wright and P. L. McMahon, The capacity of quantum neural networks, in _CLEO: Science and Innovations_ (Optica Publishing Group, 2020) pp. JM4G–5.
* Grigoryeva and Ortega [2019] L. Grigoryeva and J.-P. Ortega, Differentiable reservoir computing, Journal of Machine Learning Research 20, 1 (2019).
* Chen [2022] J. Chen, _Nonlinear Convergent Dynamics for Temporal Information Processing on Novel Quantum and Classical Devices_ , Ph.D. thesis, UNSW Sydney (2022).
* Tran and Nakajima [2021] Q. H. Tran and K. Nakajima, Learning temporal quantum tomography, Physical Review Letters 127, 260401 (2021).
* Rédei and Summers [2007] M. Rédei and S. J. Summers, Quantum probability theory, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38, 390 (2007).
* Bengtsson and Życzkowski [2017] I. Bengtsson and K. Życzkowski, _Geometry of quantum states: an introduction to quantum entanglement_ (Cambridge university press, 2017).
* Raginsky [2002] M. Raginsky, _Dynamical aspects of information storage in quantum-mechanical systems_ (Northwestern University, 2002).
* Burgarth and Giovannetti [2007] D. Burgarth and V. Giovannetti, The generalized lyapunov theorem and its application to quantum channels, New Journal of Physics 9, 150 (2007).
* Burgarth _et al._ [2013] D. Burgarth, G. Chiribella, V. Giovannetti, P. Perinotti, and K. Yuasa, Ergodic and mixing quantum channels in finite dimensions, New Journal of Physics 15, 073045 (2013).
* Wolf [2012] M. M. Wolf, Quantum channels & operations: Guided tour, Lecture notes available at http://www-m5. ma. tum. de/foswiki/pub M 5 (2012).
* Horn and Johnson [2013] R. A. Horn and C. R. Johnson, _Matrix Analysis_ , 2nd ed. (Cambridge University Press, 2013) p. 643.
* Sanz _et al._ [2010] M. Sanz, D. Perez-Garcia, M. M. Wolf, and J. I. Cirac, A quantum version of wielandt’s inequality, IEEE Transactions on Information Theory 56, 4668 (2010).
* Rahaman [2019] M. Rahaman, A new bound on quantum wielandt inequality, IEEE Transactions on Information Theory 66, 147 (2019).
* Nigro [2019] D. Nigro, On the uniqueness of the steady-state solution of the lindblad–gorini–kossakowski–sudarshan equation, Journal of Statistical Mechanics: Theory and Experiment 2019, 043202 (2019).
* Kimura [2003] G. Kimura, The bloch vector for n-level systems, Physics Letters A 314, 339 (2003).
* Byrd and Khaneja [2003] M. S. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation, Physical Review A 68, 062322 (2003).
* Gonon _et al._ [2020] L. Gonon, L. Grigoryeva, and J.-P. Ortega, Memory and forecasting capacities of nonlinear recurrent networks, Physica D 414, 1 (2020).
* Grigoryeva and Ortega [2021] L. Grigoryeva and J.-P. Ortega, Dimension reduction in recurrent networks by canonicalization, Journal of Geometric Mechanics 13, 647 (2021).
* Munkres [2000] J. R. Munkres, _Topology_ , Vol. 2 (Prentice Hall Upper Saddle River, 2000).
* Siewert [2022] J. Siewert, On orthogonal bases in the hilbert-schmidt space of matrices, Journal of Physics Communications 6, 055014 (2022).
* Cuchiero _et al._ [2021] C. Cuchiero, L. Gonon, L. Grigoryeva, J.-P. Ortega, and J. Teichmann, Discrete-time signatures and randomness in reservoir computing, IEEE Transactions on Neural Networks and Learning Systems 10.1109/TN, 1 (2021).
* Buehner and Young [2006] M. Buehner and P. Young, A tighter bound for the echo state property, IEEE transactions on neural networks 17, 820 (2006).
* Hartfiel [2002] D. J. Hartfiel, _Nonhomogeneous matrix products_ (World Scientific, 2002).
* Perez-Garcia _et al._ [2006] D. Perez-Garcia, M. M. Wolf, D. Petz, and M. B. Ruskai, Contractivity of positive and trace-preserving maps under $l^{p}$ norms, Journal of Mathematical Physics 47, 083506 (2006).
* Andersson _et al._ [2007] E. Andersson, J. D. Cresser, and M. J. Hall, Finding the kraus decomposition from a master equation and vice versa, Journal of Modern Optics 54, 1695 (2007).
* Mendl and Wolf [2009] C. B. Mendl and M. M. Wolf, Unital quantum channels–convex structure and revivals of birkhoff’s theorem, Communications in Mathematical Physics 289, 1057 (2009).
* Watrous [2018] J. Watrous, _The theory of quantum information_ (Cambridge university press, 2018).
* Mujal _et al._ [2022] P. Mujal, R. Martínez-Peña, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Time series quantum reservoir computing with weak and projective measurements, arXiv preprint arXiv:2205.06809 (2022).
* Naghiloo [2019] M. Naghiloo, Introduction to experimental quantum measurement with superconducting qubits, arXiv preprint arXiv:1904.09291 (2019).
* Murch _et al._ [2013] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, Observing single quantum trajectories of a superconducting quantum bit, Nature 502, 211 (2013).
* Weber _et al._ [2016] S. J. Weber, K. W. Murch, M. E. Kimchi-Schwartz, N. Roch, and I. Siddiqi, Quantum trajectories of superconducting qubits, Comptes Rendus Physique 17, 766 (2016), quantum microwaves / Micro-ondes quantiques.
* Lecocq _et al._ [2021] F. Lecocq, F. Quinlan, K. Cicak, J. Aumentado, S. A. Diddams, and J. D. Teufel, Control and readout of a superconducting qubit using a photonic link, Nature 591, 575 (2021).
* Hatridge _et al._ [2013] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Quantum back-action of an individual variable-strength measurement, Science 339, 178 (2013).
* Wiseman and Milburn [2009] H. M. Wiseman and G. J. Milburn, _Quantum measurement and control_ (Cambridge university press, 2009).
|
# Fusion systematics for weakly bound nuclei using neutron flow and collective
degrees of freedom
S<EMAIL_ADDRESS>V. V<EMAIL_ADDRESS>V<EMAIL_ADDRESS>S<EMAIL_ADDRESS>1Department of Nuclear Physics, Andhra University, Visakhapatnam - 530003,
India 2Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai -
400085, India 3Homi Bhabha National Institute, Anushaktinagar, Mumbai -
400094, India 4UM-DAE Centre for Excellence in Basic Sciences, Mumbai -
400098, India
###### Abstract
A systematic analysis of the fusion cross sections around the Coulomb barrier
energies with stable weakly bound (6Li,7Li,9Be) and strongly bound 12C
projectiles on various targets was performed by using neutron flow model and
coupled channels approach. The analysis show that both the models are
successful in explaining the near barrier fusion data. Further, it is also
observed that the collective degrees of freedom as well as the neutron flow
influence the near barrier fusion process involving weakly bound projectiles.
## I Introduction
Fusion of two nuclei at energies below the Coulomb barrier is a topic of great
interest due to its relation in understanding nucleosynthesis of elements and
probing underlying rich nuclear dynamics. It is well established that the
fusion cross-sections are significantly enhanced at sub-barrier energies
compared to the one dimensional barrier penetration model (1DBPM) Dasgupta et
al. (1998); Beckerman (1988); Balantekin and Takigawa (1998). The enhancement
of the fusion cross-sections is explained due to coupling of relative motion
with the various internal degrees of freedom of the interacting nuclei
Zagrebaev (2003); Stephanini et al. (1995); Bierman et al. (1996). In addition
to this, the role of neutron rearrangement process on the fusion cross-
sections has been also explored for various projectile-target systems
Zagrebaev (2003).
There are several theoretical prescriptions proposed to describe the
enhancement in the fusion cross-sections at near barrier energies and it is
observed that two different models are very much successful in explaining the
enhancement in the cross-sections Dasso et al. (1983); Broglia et al. (1983);
Stelson (1988); Stelson et al. (1990). The first model is the Coupled Channels
(CC) model, which is based on the effect of couplings of the incident
projectile and the target which leads to lowering the barrier height of the
interacting system. Because of reduction of barrier height, an enhancement of
the fusion cross-section compared to the 1DBPM Dasso et al. (1983); Broglia et
al. (1983) is observed. The second model is the Stelson model, which is based
on the neutron flow due to the exchange of neutrons between the interacting
projectile and target nuclei Stelson (1988); Stelson et al. (1990). It is
observed that both these models predict different barrier distributions for
various reactions studied. In the past, there were only a few studies carried
out in order to understand the enhancement of the sub barrier fusion cross-
sections by using both the models Kailas and Navin (1993); Vinodkumar et al.
(1996); Shapira and Stelson (1993); Vandenbosch (1992) and make a comparison
between these models. Systematic study of several reactions using Stelson
model indicates that there is an early onset of free neutron flow between the
interacting projectile and the target at relatively larger inter nuclear
distances. This leads to an enhancement of the fusion cross-sections at below
barrier energies and the correlation of barrier shifts for various systems can
also be successfully explained by this model Shapira and Stelson (1993);
Vinodkumar et al. (1996). On the other hand, the coupled channel approach is
well established and widely used method of calculating sub-barrier fusion,
which is very successful in explaining the experimental data for a variety of
nuclear systems Kailas and Navin (1993). There have been only a few attempts
in literature, in order to make comparative study of these two methods and
identify which of these mechanisms is more appropriate to explain the
experimental data on sub barrier fusion. In one of the studies, it was
reported that the coupled channels method is better correlated with the
experimental data, when compared to the neutron flow model Kailas and Navin
(1993). In the past, reactions involving strongly bound stable projectiles
were investigated by using both the models discussed above. These
investigations show that there is a very good correlation between the
experimental data and the theoretical calculations Kailas and Navin (1993);
Vinodkumar et al. (1996); Shapira and Stelson (1993); Vandenbosch (1992). It
is very interesting to make similar study for the reactions involving weakly
bound projectiles (WBP) and compare the results with the systematics of
strongly bound projectile (SBP) on various targets. It is well known that for
the reactions involving WBPs such as, 6Li,7Li and 9Be, due to their small
breakup thresholds, there is an enhancement in the fusion cross-sections at
sub barrier energies. There are several reports in literature showing that the
reaction cross-sections involving WBPs were very high around the barrier
energies Jha et al. (2020); Canto et al. (2006). In this context, it is very
much interesting to investigate the systematic of WBPs on various targets by
using both the formalisms discussed above Kailas and Navin (1993).
In this paper, we investigate the systematics of WBPs on various targets by
using both the coupled channel and Stelson model and compare the results with
SBP 12C on various targets. The paper is organized as follows: In section II,
the methodology for neutron flow based model and coupled channel models for
calculation of fusion cross-sections are described. In section III, the
results from WBP and SBP 12C projectile on various targets has been presented.
The summary and conclusions are given in section IV.
## II Methodology
In this paper, the complete fusion (CF) cross-sections for 6Li,7Li, 9Be and
12C projectiles on various targets was analyzed by using neutron flow model
and coupled channels model. For completeness of the paper, a brief
introduction about the two models was discussed here. The well known
expression for the fusion cross-section for reactions with projectile energies
greater than the barrier (B) can be written as
$\sigma_{fus}=\pi R_{b}^{2}(1-\frac{B}{E})$ (1)
where B, Rb, E are the Coulomb barrier, Coulomb radius and energy in the c.m.
frame respectively Frobrich and Lipperheide (1996). According to the Stelson
model, at near barrier energies the fusion barriers can be explained by a flat
distribution of barriers with a threshold energy cutoff (Texp) Stelson (1988);
Stelson et al. (1990). This barrier corresponds to the energy at which the
interacting nuclei come sufficiently close to each other for neutrons to flow
freely between target and projectile. So, the above expression transforms at
near barrier energies to
$\sigma_{fus}=\pi R_{b}^{2}\frac{(E-T_{exp})^{2}}{4E(B-T_{exp})}$ (2)
The maximum value of the merged neutron potential Vmax can be calculated by
assuming the neutron shell potential centered on each of the interacting
nuclei. In this configuration, the distance between the interacting nuclei is
given by Rt, the distance at which the threshold barrier (Tcal) is reached.
According to this model, if the merged neutron potential (Vmax) is equal or
lower than the binding energy of the valence neutron of the two interacting
nuclei then only the neutron flow takes place. Further the extent of the
barriers discussed above B - Texp are correlated by the difference between the
Coulomb barrier and the threshold barrier B - Tcal.
In the coupled channel formalism, it is well established that the coupling
between the incident projectile and target channels (vibrational, rotational,
transfer) can modify the barrier heights Dasso et al. (1983); Broglia et al.
(1983). By including the transmission probabilities and strength of the
couplings (F) through the modified barriers, the fusion cross-section can be
calculated. If only inelastic couplings are considered in the analyses, the
channel couplings (F) may lead to the decrease or increase in the barrier
height and it is expected that the B - Texp values will be related to F. The
coupling strength for inelastic excitations to collective states can be
calculated from the deformation parameter $\beta_{\lambda,k}$, where $\lambda$
is the multi-polarity of the transition and k is the excited state of the
nuclei (target or projectile) Kailas and Navin (1993).
By using the above discussed formalism, complete fusion data of 6Li,7Li, 9Be
and 12C projectiles on various targets was analyzed. The inelastic states of
the target were considered in the coupled channel calculations for all the WBP
(6Li,7Li, 9Be) and SBP (12C) systems, while the first projectile inelastic
state for 12C (4.4 MeV) and 7Li (0.48 MeV) were considered. In the case 6Li
and 9Be, inelastic excitation corresponding to 2.18 MeV and 2.43 MeV resonance
states were considered respectively in the coupled channel calculations.
Depending upon the correlation plots between B - Texp versus F and B - Texp
versus B - Tcal, one can estimate which method is more reliable to explain the
fusion cross-sections around the barrier energies. The values of B and $R_{b}$
are calculated by fitting the fusion data at above barrier energies by using
the Eq.1 and the value of Texp was calculated by fitting the data at near
barrier energies by using Eq.2. In order to calculate the effective value of
the strength of the couplings (F), CCFULL code Hagino et al. (1999) was used.
Fusion cross section was calculated with and without coupling at deep sub-
barrier energies and equated the ratio to exp(F.$\epsilon$) where $\epsilon$
is the barrier curvature. Knowing $\epsilon$, F can be determined. The
deformation, multipolarity and transition strengths of the excited states of
the projectile/ target used in the CCFULL code are taken from the literature
Raman et al. (1987); Kibedi and Spear (2002); Pritychenkov et al. (2016).
Previously, it was shown that at sub-barrier energies, this formalism is very
much valid in order to extract the effective value of F Kailas and Navin
(1993); Landowne and Pieper (1984).
B - Tcal values were calculated by the following procedure. The average
one/two neutron separation energies for various projectiles and targets were
taken from the mass table Wapstra et al. (1988). As the Sn values of WBPs
6,7Li and 9Be are smaller than those of the targets, the neutron flow is from
the projectile to the target. However, in the case of 12C as the Sn value is
nearly 19 MeV, the neutron flow is from the target to the projectile. By using
the neutron potential values given in Ref. Stelson et al. (1990) for both the
target and the projectile, the interacting distance Rt between the two has
been optimized such that the merged neutron potential Vmax at this distance is
equal to the Sn value of the target or projectile (whichever is lower). The
Tcal value has been computed using Rt as discussed in Ref. Kailas and Navin
(1993). The merged neutron potentials for the two reactions 12C + 208Pb and
9Be + 208Pb calculated are shown in Fig.1. One can observe that, the Vmax
values match with the Sn values of 208Pb and 9Be respectively. Here, we have
used S2n/2 for the targets to take care of the odd – even effects Kailas and
Navin (1993).
Figure 1: Merged neutron potentials at Rt, for the reactions 12C + 208Pb
(upper panel) and 9Be + 208Pb (lower panel).
## III Results and Discussion
By using the above discussed methodologies, the different parameters B, Rb,
Texp, Sn, Rt,Tcal and F were extracted for 12C, 6,7Li and 9Be projectiles on
various targets. The different parameters calculated by using both the
methodologies are given in Table 1.
Table 1: Summary of near barrier fusion analysis for 12C, 6,7Li and 9Be projectiles on various targets. Reaction | B | R0 | Texp (err.) | S2n/2 | Rt | Tcal | B - Tcal | B - Texp (err.) | F | Refs.
---|---|---|---|---|---|---|---|---|---|---
| (MeV) | (fm) | (MeV) | Sn (MeV) | (fm) | (MeV) | (MeV) | (MeV) | (MeV) |
12C + 46Ti | 21.09 (0.47) | 9.65 | 18.57 (0.10) | 11.36 | 10.07 | 18.87 | 2.22 | 2.52 (0.48) | 0.34 | Bozek et al. (1986)
12C + 48Ti | 20.32 (0.66) | 7.83 | 17.46 (0.22) | 10.25 | 10.29 | 18.46 | 1.86 | 2.86 (0.70) | 0.32 | Bozek et al. (1986)
12C + 50Ti | 19.32 (1.09) | 7.68 | 17.76 (0.16) | 9.54 | 10.46 | 18.16 | 1.16 | 1.56 (1.10) | 0.27 | Bozek et al. (1986)
12C + 92Zr | 31.90 (0.14) | 9.13 | 29.66 (0.02) | 7.91 | 11.65 | 29.66 | 2.28 | 2.24 (0.14) | 0.41 | Newton et al. (2001)
12C + 144Sm | 45.98 (0.46) | 10.49 | 44.23 (0.08) | 9.56 | 12.39 | 43.22 | 2.75 | 1.75 (0.47) | 0.47 | Abriola et al. (1992)
12C + 152Sm | 47.27 (0.92) | 11.30 | 42.0 (0.19) | 6.92 | 12.99 | 41.24 | 6.03 | 5.27 (0.94) | 1.54 | Broda et al. (1975)
12C + 181Ta | 52.97 (1.17) | 10.83 | 48.62 (0.1) | 7.11 | 13.35 | 47.26 | 5.17 | 4.35 (1.17) | 0.71 | Crippa et al. (1994)
12C + 194Pt | 54.99 (0.51) | 10.54 | 52.35 (0.04) | 7.30 | 13.47 | 50.03 | 4.96 | 2.64 (0.51) | 0.64 | Shrivastava et al. (2001)
12C + 198Pt | 54.86 (0.34) | 10.32 | 52.76 (0.02) | 6.70 | 13.64 | 49.39 | 5.47 | 2.1 (0.34) | 0.81 | Shrivastava et al. (2001)
12C + 204Pb | 54.15 (1.59) | 9.85 | 51.37(0.33) | 7.65 | 13.52 | 52.39 | 1.76 | 2.78 (1.62) | 0.64 | Sagaidak et al. (2003)
12C + 206Pb | 56.75 (2.29) | 11.20 | 53.19 (0.59) | 7.41 | 13.59 | 52.11 | 4.64 | 3.56 (2.37) | 0.65 | Sagaidak et al. (2003)
12C + 208Pb | 55.47 (0.69) | 9.04 | 54.17 (0.06) | 7.05 | 13.69 | 51.75 | 3.72 | 1.3 (0.50) | 0.64 | Mukherjee et al. (2007)
6Li + 28Si | 5.73 (0.7)) | 6.22 | 5.01 (0.12) | 5.66 | 9.84 | 5.27 | 0.46 | 0.72 (0.7) | 0.19 | Mandira Sinha et al. (2010)
6Li + 64Ni | 11.27 (0.9)) | 7.81 | 9.74 (0.06) | | 10.88 | 11.12 | 0.15 | 1.53 (0.9) | 0.29 | Md. Moin Shaikh et al. (2014)
6Li + 90Zr | 18.46 (0.3) | 8.24 | 15.14 (0.07) | | 11.63 | 14.85 | 3.61 | 3.32 (0.3) | 0.31 | Kumawat et al. (2012)
6Li + 124Sn | 20.84 (0.4)) | 8.67 | 17.08 (0.11) | | 12.26 | 17.62 | 3.22 | 3.76 (0.41) | 0.35 | Parkar et al. (2018a)
6Li + 144Sm | 26.03 (0.7)) | 7.43 | 22.71 (0.03) | | 12.58 | 21.29 | 4.73 | 3.32 (0.7) | 0.33 | Rath et al. (2009)
6Li + 152Sm | 25.71 (0.59) | 8.24 | 21.75 (0.04) | | 12.69 | 21.09 | 4.61 | 3.96 (0.6) | 0.33 | Rath et al. (2012)
6Li + 198Pt | 28.2 (1.01) | 8.71 | 24.3 (0.16) | | 13.30 | 25.32 | 2.87 | 3.9 (1.02) | 0.37 | Shrivastava et al. (2009)
6Li + 197Au | 28.42 (0.04) | 8.39 | 25.2 (0.09) | | 13.29 | 27.67 | 0.74 | 3.2 (0.1) | 0.34 | Palshetkar et al. (2014)
6Li + 208Pb | 27.72 (1.07) | 7.41 | 26.58 (0.09) | | 13.42 | 26.39 | 1.33 | 1.14 (1.1) | 0.35 | Wu et al. (2003)
6Li + 209Bi | 29.96 (0.83) | 8.78 | 26.57 (0.10) | | 13.44 | 28.69 | 1.27 | 3.39 (0.84) | 0.35 | Dasgupta et al. (2002)
7Li + 59Co | 11.61 (0.26) | 7.64 | 9.44 (0.1) | 7.25 | 10.66 | 10.94 | 0.68 | 2.17 (0.3) | 0.24 | Beck et al. (2003)
7Li + 124Sn | 21.46 (0.18) | 8.92 | 17.8 (0.03) | | 12.02 | 17.97 | 3.49 | 3.66 (0.18) | 0.29 | Parkar et al. (2018b)
7Li + 144Sm | 24.82 (0.15) | 8.71 | 21.7 (0.06) | | 12.28 | 21.81 | 3.01 | 3.12 (0.16) | 0.31 | Rath et al. (2013)
7Li + 152Sm | 24.18 (0.13) | 8.50 | 20.89 (0.04) | | 12.45 | 21.50 | 2.67 | 3.29 (0.13) | 0.40 | Rath et al. (2013)
7Li + 198Pt | 28.81 (0.45) | 9.71 | 26.09 (0.07) | | 13.06 | 25.79 | 3.02 | 2.71 (0.45) | 0.32 | Shrivastava et al. (2013)
7Li + 197Au | 28.45 (0.99) | 9.96 | 26.09 (0.34) | | 13.05 | 26.15 | 2.30 | 2.36 (1.05) | 0.31 | Palshetkar et al. (2014)
7Li + 209Bi | 29.52 (0.67) | 9.62 | 26.56 (0.10) | | 13.20 | 27.18 | 2.34 | 2.96 (0.68) | 0.33 | Dasgupta et al. (2002)
9Be + 89Y | 22.49 (0.46) | 7.78 | 20.77 (0.08) | 1.66 | 13.66 | 16.44 | 6.05 | 1.72 (0.47) | 0.38 | Palshetkar et al. (2010)
9Be + 124Sn | 26.97 (0.26) | 11.29 | 24.11 (0.04) | | 14.31 | 20.13 | 6.84 | 2.86 (0.26) | 0.29 | Parkar et al. (2010)
9Be + 144Sm | 32.13 (1.38) | 9.43 | 28.75 (0.17) | | 14.63 | 24.42 | 7.71 | 3.38 (1.39) | 0.41 | Gomes et al. (2005)
9Be + 208Pb | 40.19 (0.23) | 9.72 | 36.69 (0.05) | | 15.47 | 30.53 | 9.66 | 3.5 (0.24) | 0.51 | Dasgupta et al. (2004)
9Be + 209Bi | 37.84 (0.94) | 7.29 | 35.23 (0.06) | | 15.48 | 30.88 | 6.96 | 2.61 (0.94) | 0.49 | Dasgupta et al. (2004)
As a typical example, the fusion cross section data for the reaction 9Be +
208Pb Dasgupta et al. (2004) along with the Stelson model calculations are
shown in Fig. 2.
Figure 2: Fusion cross section data available for 9Be + 208Pb system Dasgupta
et al. (2004) along with the Stelson model calculations. Continuous and dashed
lines are Stelson model fits from the Equations 1 and 2, respectively. Figure
3: The extracted values of B - Texp as a function of F (left panel) and B -
Tcal (right panel) for the strongly bound 12C and weakly bound 6,7Li and 9Be
projectiles on various targets Bozek et al. (1986); Newton et al. (2001);
Abriola et al. (1992); Broda et al. (1975); Crippa et al. (1994); Shrivastava
et al. (2001); Sagaidak et al. (2003); Mukherjee et al. (2007); Mandira Sinha
et al. (2010); Md. Moin Shaikh et al. (2014); Kumawat et al. (2012); Parkar et
al. (2018a); Rath et al. (2009, 2012); Shrivastava et al. (2009); Palshetkar
et al. (2014); Wu et al. (2003); Beck et al. (2003); Parkar et al. (2018b);
Dasgupta et al. (2002); Rath et al. (2013); Shrivastava et al. (2013);
Palshetkar et al. (2010); Parkar et al. (2010); Gomes et al. (2005); Dasgupta
et al. (2004). The lines are the best linear fits to the data. Figure 4: The
extracted values of Texp values as a function of Tcal for the (a) strongly
bound 12C and weakly bound (b) 9Be, (c) 7Li, and (d) 6Li projectiles on
various targets. Lines are fit to the data (See text for details). Bozek et
al. (1986); Newton et al. (2001); Abriola et al. (1992); Broda et al. (1975);
Crippa et al. (1994); Shrivastava et al. (2001); Sagaidak et al. (2003);
Mukherjee et al. (2007); Mandira Sinha et al. (2010); Md. Moin Shaikh et al.
(2014); Kumawat et al. (2012); Parkar et al. (2018a); Rath et al. (2009,
2012); Shrivastava et al. (2009); Palshetkar et al. (2014); Wu et al. (2003);
Beck et al. (2003); Parkar et al. (2018b); Dasgupta et al. (2002); Rath et al.
(2013); Shrivastava et al. (2013); Palshetkar et al. (2010); Parkar et al.
(2010); Gomes et al. (2005); Dasgupta et al. (2004).
The values of the B - Texp as a function of F (left panel) and B - Tcal (right
panel) are plotted in Fig. 3. From the present analysis (Fig.3), it can be
observed that for the reactions induced by 12C on various targets, the B -
Texp values on the average increase with increasing F and the B - Texp values
also increase with increasing B - Tcal. The present results are very much
similar to the correlation results for reactions induced by 16O projectiles on
various targets Kailas and Navin (1993). It can also be observed that the
correlation between B - Texp vs F is very much similar when compared to the
correlation between B - Texp vs B -Tcal. Further, Fig.3 shows that there is a
strong correlation for the data analyzed by using the two models, since the
results are well reproduced by a simple linear fit. Followed by these results,
the experimental data for the reactions induced by WBPs on various targets was
analyzed in a similar way as discussed above for SBP 12C. From Fig.3, it is
very interesting to observe a similar trend for WBPs as observed for 12C
projectile on various targets. In case of WBPs also, one can observe that
there is a good correlation between B - Texp vs B - Tcal and F and all the
analyzed experimental data is very well reproduced by linear fits. The
$\chi^{2}$ fits for the two models are similar. While both the Coupled
Channels calculations and Stelson model describe the fusion data for a large
number of systems well, there is an important difference between the two
models. As per the Stelson model, for all isotopes of a given Z the Sn values
are always lowest for the heaviest isotope, giving lowest T values for it
irrespective of collectivity. In contrast, in the case of coupled channel
effects due to collectivity, the isotope which is more collective has larger F
value leading to lower T values. We have tried to look for this feature also
in our data. However, as the B - Texp has large error bars and therefore we
are not in a position to confirm the above expectation from our analysis.
To understand the neutron transfer mechanism in Stelson model, further the
Tcal values for 12C, 6,7Li and 9Be projectiles on various targets as a
function of Texp have been plotted in Fig. 4 which shows a very good
correlation between Texp and Tcal values. In general, the fusion cross section
at near and below barrier energies is enhanced (for both strongly and weakly
bound projectiles) when compared to the 1DBPM. Further, there is a small
reduction at well above barrier for strongly bound projectiles, while there is
a significant reduction 10 to 30 % in the case of weakly bound projectiles.
The behavior of fusion excitation function is understood in terms of coupling
to various degrees of freedom like collective, transfer and breakup, where the
latter two processes are more important for weakly bound nuclei. Diffuseness
parameter has been varied by 20 % to see its effect on the Rt and Tcal values.
The change in Tcal values with diffuseness parameter varying from 0.65 to 0.8
fm is around 1 MeV. Further, it is observed that a 20 % change in fusion cross
section changes the barrier radius by about 10 % and negligible change of the
corresponding fusion barrier. Therefore, B-T values are not affected
significantly.
In order to understand the correlation between Texp and Tcal, the experimental
data has been fitted by using linear function. From different kinds of linear
fits, one can conclude that for the reactions induced by 12C, 6,7Li
projectiles on various targets a simple linear fit function y = x is
sufficient to show a very good correlation between the calculated and
experimental results. But for the reactions induced by 9Be projectile on
various targets a simple linear fit is not sufficient to explain the analyzed
data. Good correlation between Texp and Tcal is obtained by using the function
y = x + c, where c is a constant. From this analysis, we can clearly conclude
that Tcal values are in good agreement with the values of Texp for the
reactions induced by 12C, 6,7Li projectile on various targets and for the
reactions induced by 9Be projectiles on various targets Texp values are higher
by few MeV when compared to the values of Tcal. We need to understand, why
this anomaly/different behavior exists for weakly bound 9Be projectile on
various targets, when compared to the other reactions. Further, from this
analysis it can be concluded that in the case of strongly bound stable
projectile 12C and weakly bound 6,7Li, 9Be, both the models (Stelson model and
coupled channels formalism) are very much successful in explaining the near
barrier fusion data. However from the neutron transfer model, one has to
understand why the Texp values are few MeV higher than the Tcal values for
weakly bound 9Be projectiles.
From this analysis, it can be observed that both these models (Stelson and
coupled channels) describe the trend of B - Texp variation successfully for
strongly bound and weakly bound projectiles, indicating the validity of these
models for sub barrier fusion data. In the Coupled Channels calculations, the
excited states of the targets have been included. In the case of 6Li and 9Be,
the resonance states have been included as discussed above. Stelson neutron
flow model considers only the transfer of neutrons between the interacting
nuclei at distances close to the grazing distance. However, the weak or strong
binding of the projectile is taken into account is through the neutron binding
energy of the interacting nuclei.
## IV Summary and Conclusions
In the present paper, the fusion data for weakly bound projectiles (6,7Li and
9Be) and strongly bound projectile 12C around the coulomb barrier energies has
been analyzed on various targets. The present results suggest that both the
neutron flow and coupled channels models are successful in explaining the near
barrier fusion data. Further from the Stelson model, it has been observed that
there is a good correlation between Texp and Tcal. For the reactions induced
by 9Be projectiles on various targets, the Texp values are a few MeV higher
than the corresponding Tcal values and this feature needs to be investigated.
The present results show that for reactions induced by WBPs on various
targets, the collective degrees of freedom as well as the neutron flow are
important in influencing the near barrier fusion phenomenon. In the case of Sm
isotopes considered in the present work, while the Sn (and hence the T values)
values decrease with increase of the mass number of the isotopes, the
collectivity increases with increase of mass number. Hence B-T ( Stelson
model) and F (collectivity) values increase with increase of mass number and
we can not choose as to which out of the above two models is more appropriate
to describe the data. However, if a series of isotopes were choosen, which
have both Sn and collectivity decrease with increase of mass number of the
target, then it will be possible to select between these two models. In this
case, if collectivity is the dominant mechanism, then B-T determined will
increase with decrease of mass number as lower mass number has higher
collectivity when compared to the heavier. If neutron flow is more important
mechanism, then B-T will increase with increase of mass number as the heavier
isotope has a lower value of Sn and hence T. It will be interesting to extend
these studies to other WBPs and radioactive ions for a range of isotopes of a
given target.
## V Acknowledgments
The authors SK acknowledges INSA, India for the senior scientist fellowship
and SA acknowledges the financial support from UGC, India.
## References
* Dasgupta et al. (1998) M. Dasgupta, D. J. Hinde, N. Rowley, and A. M. Stefanini, Annu. Rev. Nucl. Part. Sci. 48, 401 (1998).
* Beckerman (1988) M. Beckerman, Rep. Prog. Phys. 51, 1047 (1988).
* Balantekin and Takigawa (1998) A. B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70, 77 (1998).
* Zagrebaev (2003) V. I. Zagrebaev, Phys. Rev. C 67, 061601[R] (2003).
* Stephanini et al. (1995) A. M. Stephanini, D. Ackermann, L. Corradi, D. R. Napoli, C. Petrache, P. Spolaore, P. Bednarczyk, H. Q. Zhang, S. Beghini, G. Montagnoli, et al., Phys. Rev. Lett. 74, 864 (1995).
* Bierman et al. (1996) J. D. Bierman, P. Chan, J. F. Liang, M. P. Kelly, A. A. Sonzogni, and R. Vandenbosch, Phys. Rev. Lett. 76, 1587 (1996).
* Dasso et al. (1983) C. H. Dasso, S. Landowne, and A. Winther, Nucl. Phys. A 405, 381 (1983).
* Broglia et al. (1983) R. A. Broglia, C. H. Dasso, S. Landowne, and G. Pollarolo, Phys. Lett. B 133, 34 (1983).
* Stelson (1988) P. H. Stelson, Phys. Lett. B 205, 190 (1988).
* Stelson et al. (1990) P. H. Stelson, H. J. Kim, M. Beckerman, D. Shapira, and R. L. Robinson, Phys. Rev. C 41, 1584 (1990).
* Kailas and Navin (1993) S. Kailas and A. Navin, Pramana 41, 163 (1993).
* Vinodkumar et al. (1996) A. M. Vinodkumar, A. K. Sinha, N. V. S. V. Prasad, K. M. Varier, and P. Sugathan, Phys. Rev. C 54, 791 (1996).
* Shapira and Stelson (1993) D. Shapira and P. H. Stelson, Phys. Rev. C 47, 1666 (1993).
* Vandenbosch (1992) R. Vandenbosch, Annu. Rev. Nucl. Part. Sci. 42, 447 (1992).
* Jha et al. (2020) V. Jha, V. V. Parkar, and S. Kailas, Phys. Rep. 845, 1 (2020), and references therein.
* Canto et al. (2006) L. F. Canto, P. R. S. Gomes, R. Donangelo, and M. S. Hussein, Phys. Rep. 424, 1 (2006).
* Frobrich and Lipperheide (1996) P. Frobrich and R. Lipperheide, _Theory of Nuclear Reactions_ (Oxford University Press Inc., New York, 1996).
* Hagino et al. (1999) K. Hagino, N. Rowley, and A. T. Kruppa, Comput. Phys. Commun. 123, 143 (1999).
* Raman et al. (1987) S. Raman, C. H. Malarkey, W. T. Milner, C. W. Nestor, and P. H. Stelson, At. Data Nucl. Data Tables 36, 1 (1987).
* Kibedi and Spear (2002) T. Kibedi and R. H. Spear, At. Data Nucl. Data Tables 80, 35 (2002).
* Pritychenkov et al. (2016) B. Pritychenkov, M. Birch, B. Singh, and M. Horoi., At. Data Nucl. Data Tables 107, 1 (2016).
* Landowne and Pieper (1984) S. Landowne and S. C. Pieper, Phys. Rev. C 29, 1352 (1984).
* Wapstra et al. (1988) A. H. Wapstra, G. Audi, and H. Hoekstra, At. Data Nucl. Data Tables 39, 281 (1988).
* Bozek et al. (1986) E. Bozek, D. M. D. Castro-Rizzo, S. Cavallaro, B. Delaunay, J. Delaunay, H.Dumont, A. D’onofrio, M. G. Saint-Laurent, L. Sperduto, and F. Terrasi, Nucl. Phys. A 451, 171 (1986).
* Newton et al. (2001) J. O. Newton, C. R. Morton, M. Dasgupta, J. R. Leigh, J. C. Mein, D. J. Hinde, H. Timmers, and K. Hagino, Phys. Rev. C 64, 064608 (2001).
* Abriola et al. (1992) D. Abriola, A. A. Sonzogni, M. di Tada, A. Etchegoyen, M. C. Etchegoyen, J. O. Fernández Niello, S. Gil, A. O. Macchiavelli, A. J. Pacheco, R. Piegaia, et al., Phys. Rev. C 46, 244 (1992).
* Broda et al. (1975) R. Broda, M. Ishihara, B. Herskind, H. Oeschler, S. Ogaza, and H. Ryde, Nucl. Phys. A 248, 356 (1975).
* Crippa et al. (1994) M. Crippa, E. Gadioli, P. Vergani, G. Ciavola, M. C., and M. Bonardi, Z. Phys. A 350, 121 (1994).
* Shrivastava et al. (2001) A. Shrivastava, S. Kailas, A. Chatterjee, A. Navin, A. M. Samant, P. Singh, S. Santra, B. S. Mahata, K. Tomar, and G. Pollarolo, Phys. Rev. C 63, 054602 (2001).
* Sagaidak et al. (2003) R. N. Sagaidak, G. N. Kniajeva, I. M. Itkis, M. G. Itkis, N. A. Kondratiev, E. M. Kozulin, I. V. Pokrovsky, A. I. Svirikhin, V. M. Voskressensky, A. V. Yeremin, et al., Phys. Rev. C 68, 014603 (2003).
* Mukherjee et al. (2007) A. Mukherjee, D. J. Hinde, M. Dasgupta, K. Hagino, J. O. Newton, and R. D. Butt, Phys. Rev. C 75, 044608 (2007).
* Mandira Sinha et al. (2010) Mandira Sinha, H. Majumdar, P. Basu, Subinit Roy, M. Biswas, M. K. Pradhan, R. Bhattacharya, R. Palit, I. Mazumdar, and S. Kailas, Eur. Phys. J. A 44, 403 (2010).
* Md. Moin Shaikh et al. (2014) Md. Moin Shaikh, Subinit Roy, S. Rajbanshi, M. K. Pradhan, A. Mukherjee, P. Basu, S. Pal, V. Nanal, R. G. Pillay, and A. Shrivastava, Phys. Rev. C 90, 024615 (2014).
* Kumawat et al. (2012) H. Kumawat, V. Jha, V. V. Parkar, B. J. Roy, S. K. Pandit, R. Palit, P. K. Rath, C. S. Palshetkar, Sushil K. Sharma, Shital Thakur, et al., Phys. Rev. C 86, 024607 (2012).
* Parkar et al. (2018a) V. V. Parkar, S. K. Pandit, A. Shrivastava, R. Palit, K. Mahata, V. Jha, K. Ramachandran, Shilpi Gupta, S. Santra, Sushil K. Sharma, et al., Phys. Rev. C 98, 014601 (2018a).
* Rath et al. (2009) P. K. Rath, S. Santra, N. L. Singh, R. Tripathi, V. V. Parkar, B. K. Nayak, K. Mahata, R. Palit, Suresh Kumar, S. Mukherjee, et al., Phys. Rev. C 79, 051601 (2009).
* Rath et al. (2012) P. K. Rath, S. Santra, N. L. Singh, K. Mahata, R. Palit, B. K. Nayak, K. Ramachandran, V. V. Parkar, R. Tripathi, S. K. Pandit, et al., Nucl. Phys. A 874, 14 (2012).
* Shrivastava et al. (2009) A. Shrivastava, A. Navin, A. Lemasson, K. Ramachandran, V. Nanal, M. Rejmund, K. Hagino, T. Ichikawa, S. Bhattacharyya, A. Chatterjee, et al., Phys. Rev. Lett. 103, 232702 (2009).
* Palshetkar et al. (2014) C. S. Palshetkar, Shital Thakur, V. Nanal, A. Shrivastava, N. Dokania, V. Singh, V. V. Parkar, P. C. Rout, R. Palit, R. G. Pillay, et al., Phys. Rev. C 89, 024607 (2014).
* Wu et al. (2003) Y. W. Wu, Z. H. Liu, C. J. Lin, H. Q. Zhang, M. Ruan, F. Yang, Z. C. Li, M. Trotta, and K. Hagino, Phys. Rev. C 68, 044605 (2003).
* Dasgupta et al. (2002) M. Dasgupta, D. J. Hinde, K. Hagino, S. B. Moraes, P. R. S. Gomes, R. M. Anjos, R. D. Butt, A. C. Berriman, N. Carlin, C. R. Morton, et al., Phys. Rev. C 66, 041602 (2002).
* Beck et al. (2003) C. Beck, F. A. Souza, N. Rowley, S. J. Sanders, N. Aissaoui, E. E. Alonso, P. Bednarczyk, N. Carlin, S. Courtin, A. Diaz-Torres, et al., Phys. Rev. C 67, 054602 (2003).
* Parkar et al. (2018b) V. V. Parkar, Sushil K. Sharma, R. Palit, S. Upadhyaya, A. Shrivastava, S. K. Pandit, K. Mahata, V. Jha, S. Santra, K. Ramachandran, et al., Phys. Rev. C 97, 014607 (2018b).
* Rath et al. (2013) P. K. Rath, S. Santra, N. L. Singh, B. K. Nayak, K. Mahata, R. Palit, K. Ramachandran, S. K. Pandit, A. Parihari, A. Pal, et al., Phys. Rev. C 88, 044617 (2013).
* Shrivastava et al. (2013) A. Shrivastava, A. Navin, A. Diaz-Torres, V. Nanal, K. Ramachandran, M. Rejmund, S. Bhattacharyya, A. Chatterjee, S. Kailas, A. Lemasson, et al., Phys. Lett. B 718, 931 (2013).
* Palshetkar et al. (2010) C. S. Palshetkar, S. Santra, A. Chatterjee, K. Ramachandran, Shital Thakur, S. K. Pandit, K. Mahata, A. Shrivastava, V. V. Parkar, and V. Nanal, Phys. Rev. C 82, 044608 (2010).
* Parkar et al. (2010) V. V. Parkar, R. Palit, Sushil K. Sharma, B. S. Naidu, S. Santra, P. K. Joshi, P. K. Rath, K. Mahata, K. Ramachandran, T. Trivedi, et al., Phys. Rev. C 82, 054601 (2010).
* Gomes et al. (2005) P. R. S. Gomes, I. Padron, E. Crema, O. A. Capurro, J. O. Fernández Niello, A. Arazi, G. V. Martí, J. Lubian, M. Trotta, A. J. Pacheco, et al., Phys. Rev. C 73, 064606 (2005).
* Dasgupta et al. (2004) M. Dasgupta, P. R. S. Gomes, D. J. Hinde, S. B. Moraes, R. M. Anjos, A. C. Berriman, R. D. Butt, N. Carlin, J. Lubian, C. R. Morton, et al., Phys. Rev. C 70, 024606 (2004).
|
# Rethinking Two Consensuses of Transferability in Deep Learning
Yixiong Chen 1, Jingxian Li 2, Chris Ding 1, Li Liu 1,3 Corresponding author:
<EMAIL_ADDRESS>
###### Abstract
Deep transfer learning (DTL) has formed a long-term quest toward enabling deep
neural networks (DNNs) to reuse historical experiences as efficiently as
humans. This ability is named knowledge transferability. A commonly used
paradigm for DTL is firstly learning general knowledge (pre-training) and then
reusing (fine-tuning) them for a specific target task. There are two
consensuses of transferability of pre-trained DNNs: (1) a larger domain gap
between pre-training and downstream data brings lower transferability; (2) the
transferability gradually decreases from lower layers (near input) to higher
layers (near output). However, these consensuses were basically drawn from the
experiments based on natural images, which limits their scope of application.
This work aims to study and complement them from a broader perspective by
proposing a method to measure the transferability of pre-trained DNN
parameters. Our experiments on twelve diverse image classification datasets
get similar conclusions to the previous consensuses. More importantly, two new
findings are presented, i.e., (1) in addition to the domain gap, a larger data
amount and huge dataset diversity of downstream target task also prohibit the
transferability; (2) although the lower layers learn basic image features,
they are usually not the most transferable layers due to their domain
sensitivity.
## Introduction
Deep transfer learning (DTL) enables deep neural networks (DNNs) to learn
faster and gain better generalization with limited data. This successful
bionic design came from the knowledge transferability of human brains (Ausubel
et al. 1968), which are capable of learning new concepts based on the original
cognitive structure. DNNs transfer knowledge in a similar way. They learn
general recognition abilities from large pre-training datasets (e.g., ImageNet
(Deng et al. 2009)) and fine-tune the network parameters on specific
downstream datasets (Tajbakhsh et al. 2016).
Figure 1: The motivation of our transferability measurement. Pre-training on
task A makes the fine-tuning only needs shorter optimization distance to reach
the solution of task B than training from scratch.
Previous literature (Yosinski et al. 2014; Fayek, Cavedon, and Wu 2018;
Tajbakhsh et al. 2016) used semi-quantitative methods (i.e., usually by
comparing the network performance with/without transfer learning) to judge the
DNNs’ transferabilities. Two consensuses were drawn from these works. Firstly,
the transferability decreases as the distance between the source and target
dataset (domain gap) increases. We regard its root cause to be the poor
transferability of DNNs to the samples dissimilar to pre-training data, which
stimulates our interest in exploring other important factors leading to a
greater extent of dissimilarity (e.g., dataset diversity). Secondly, the
lowest layer (i.e., the layer that is close to the input) learns general
knowledge like colors and edges, while the highest layer (i.e., the layer that
is close to the output) learns semantic knowledge specific to the task,
indicating there is a decreasing trend of transferability from low to high
layers. Although this conclusion is generally correct, we conjecture that the
pre-trained features of lower layers might not be that applicable to the
downstream tasks. If the local pattern distributions of data change
drastically (e.g., from natural images to cartoons or even medical data), the
shallower local features might suffer more than the deeper semantic features.
In these cases, the decreasing trend of layer-wise transferability may need
further validations.
Instead of using semi-quantitative measuring methods, this work revisits these
consensuses with a quantitative method for more precise results. We mainly
study the following two questions:
* •
In addition to the domain gap, what other critical factors about datasets
affect the transferability?
* •
Do the transferabilities for sequential layers of a pre-trained DNN always
follow a downward trend?
To answer these two questions, we start with the nature of knowledge transfer,
a procedure of reusing and adjusting learned abilities for new tasks, to
design our measuring method. For DNNs, pre-trained parameters can be regarded
as learned knowledge, and it offers a better parameter initialization than
random initialization (Fig. 1). In this case, a few gradient steps of fine-
tuning can produce good results on a new task (Finn, Abbeel, and Levine 2017).
On this basis, we quantify transferability as how much the pre-training task A
pushes the parameters towards the optimal point of the downstream task B. If
the fine-tuning phase needs a shorter optimization distance than training from
scratch, pre-training helps to make the downstream adaptation easier. This
quantification method enables us to compare the transferabilities between a
pre-training dataset and different downstream tasks under the same standard.
It also provides a way to derive the transferabilities of different layers
precisely. With this measurement, the above questions can be answered with
solid justifications.
For the under-explored factors that affect transferability, downstream data
diversity (named domain width in this work) and data amount come to our mind
for simple reasons: (1) When the domain width of a target dataset goes larger,
there must exist more out-of-distribution samples that the DNN did not see in
the pre-training phase, which impairs the transferability; (2) For data amount
of the downstream task, it is shown by previous works (He, Girshick, and
Dollár 2019; Newell and Deng 2020) that the greater amount of data decreases
the benefit brought by pre-training. In the experiments, our results verify
that domain width and data amount negatively affect the transferability even
to a greater extent than the domain gap. For the layer-wise transferability,
this work finds that the lowest layers may be domain-sensitive, which makes
the trends downward when only excluding the first few layers.
To conclude, the main contributions of this work are as follows:
1. 1.
A novel method for quantifying the transferability of DNN parameters is
proposed.
2. 2.
By revisiting the previous consensuses through rich experimental validations
on twelve datasets, we find two important phenomena in DTL: (1) the target
domain width and data amount highly affect the transferability, even more than
the domain gap; (2) the transferabilities of the lowest layers are usually
prohibited by the domain shift. For all of our experiments with ResNet50, the
lowest layer is not the most transferable among all layers.
## Related Work
### Transferability of DNNs
Transferability of DNNs is the ability to gain transferable knowledge from the
source domain and reuse the knowledge to decrease the generalization error on
the target domain, under the distribution shift or the task discrepancy (Jiang
et al. 2022). A common method to assess the transferability of DNNs is
comparing the performance between fine-tuning and training from scratch
(Yosinski et al. 2014; Matsoukas et al. 2022). Given a downstream task, more
significant performance improvement means the DNN has better transferability.
There are several key factors affecting how transferable the parameters of a
DNN are. First, the performance of the DNN on the pre-training dataset is
positively correlated to its transferability to downstream tasks (Kornblith,
Shlens, and Le 2019) in most cases. Second, the more similar the source and
target task are, the more transferable the DNNs would be (Yosinski et al.
2014; Dwivedi and Roig 2019). Third, the transferabilities of parameters are
fragile to the large learning rate (Ro and Choi 2021). A drastic change of
highly transferable parameters may destroy their original knowledge, making
them less transferable. This work also tries to excavate the properties for
DNNs’ transferabilities, but from the perspective of the dataset.
### Pre-training
To improve DNNs’ transferabilities, many pre-training datasets and methods are
proposed. The most dominant pre-training dataset for vision tasks is ImageNet
(Deng et al. 2009) for its large-scale data with high-quality annotations.
Earlier literature mainly pre-train models in a supervised way (He, Girshick,
and Dollár 2019) with categorical labels. However, researchers found that
ImageNet cannot cover the need for all visual applications because the large
domain gap between natural and special images (e.g., medical images) leads to
poor transferability. Pre-training datasets and methods for specific domains
(Sermanet et al. 2018; Chen et al. 2021; Zhang et al. 2022) began to emerge
for this reason. This work tests the transferability for large domain gaps,
and tries to provide some practical guides when pre-trained models on specific
domains are unavailable. Other than training schemes, models themselves are
crucial for good transferabilities. Pre-training with big models like
convolutional neural networks (CNNs) with hundreds of layers (Huang et al.
2017) and Vision Transformers (ViTs) (Dosovitskiy et al. 2020) are proved to
bring more benefits to downstream tasks than smaller ones. The measurement
proposed in this work also validates that larger pre-trained models have
higher transferabilities (see the supplementary material).
Figure 2: The method for measuring parameter transferability. (a) The
transferability of the parameter $\theta_{A}$ is defined as
$D(\theta_{r},\theta_{B})/D(\theta_{A},\theta_{B})$, but there is an ambiguity
of $\theta_{B}$ between fine-tuning and training from scratch. We denote them
as $\theta_{FB}$ and $\theta_{SB}$ respectively and use them to calculate two
distance variants, $D(\theta_{r},\theta_{FB})$ and
$D(\theta_{r},\theta_{SB})$. (b) The transferability of the whole network is
calculated as the average transferability of each layer to avoid the parameter
scale difference.
### Fine-tuning
How to leverage the transferabilities of pre-trained DNNs for a new task has
become a research focus in recent years. The practices to transfer feature
extraction capability from an off-the-shelf network to a new network can be
roughly categorized into two classes: initialization-based and teacher-
student-based (Song et al. 2021). The former is the so-called fine-tuning
method and is the default setting to measure transferabilities in this work.
In addition to the common fine-tuning scheme that involves all layers, another
powerful paradigm is tuning the last few layers of a pre-trained DNN
(Tajbakhsh et al. 2016), so that the low-level general features inherent in
the shallow layers can be kept intact. Likewise, many advanced fine-tuning
schemes (Long et al. 2015; Li et al. 2019; Zhong et al. 2020; Ro and Choi
2021) assumed that the higher-level layers are more specific to the task,
therefore, are less transferable and need more updates. However, recent
research demonstrated some exceptions (Chen et al. 2022) when domain shift is
large, where lower layers also need to be adjusted to adapt to the different
visual feature distributions. This finding is consistent with our experimental
results, which reveal that the first few layers are not necessarily as
transferable as researchers previously considered.
## Methodology
To answer the above two questions, a general transferability measurement,
domain-agnostic, task-agnostic, and architecture-agnostic, is proposed to
compare parameter transferabilities considering different domains, tasks, and
layers.
#### Definition of the Transferability.
The aim for pre-training is to find a good initialization parameter
$\theta_{A}$ through task A so that the optimal solution $\theta_{B}$ on task
B can be obtained by fine-tuning as few steps as possible. We denote the
distance between $\theta_{A}$ and $\theta_{B}$ as $D(\theta_{A},\theta_{B})$,
which reflects the transferability of $\theta_{A}$ to task B. However, since
the optimization difficulties for different parameters in a DNN on different
tasks are different, $D(\theta_{A},\theta_{B})$ cannot be regarded as
transferability directly. To tackle this problem, the distance between random
initialized parameters $\theta_{r}$ and $\theta_{B}$,
$D(\theta_{r},\theta_{B})$, is utilized for normalization. This work defines
the transferability of random initialized parameters to be 1, and measures the
transferability of parameter $\theta_{A}$ to task B as follows:
$T_{B}(\theta_{A})=\frac{D(\theta_{r},\theta_{B})}{D(\theta_{A},\theta_{B})}.$
(1)
When transferability is greater than 1, pre-training shortens the optimization
distance, meaning that new knowledge is learned with the help of experiences.
In practice, the distance metric can be calculated with mean absolute
difference
$D(\theta_{A},\theta_{B})=\sum_{i=1}^{K}|\theta_{A}^{i}-\theta_{B}^{i}|/K$,
where $K$ is the number of parameters.
#### Two Different Optimal Points.
One crucial problem for this measurement is, that in most cases, the pre-
trained DNNs would not converge to the same (local) optima as the random
initialized ones do. Therefore, this work proposes two variants for measuring
$D(\theta_{r},\theta_{B})$ (illustrated in Fig. 2 (a)). In our experiments, we
use $\theta_{SB}$ to denote the converged parameters on B trained from
scratch, and $\theta_{FB}$ to denote the fine-tuned parameters.
Correspondingly, the transferabilities measured with $\theta_{SB}$ and
$\theta_{FB}$ are denoted as $T_{SB}(\theta_{A})$ and $T_{FB}(\theta_{A})$
respectively.
Table 1: Descriptions and statistics of the datasets used in this work. Dataset | Size (train/test) | Classes | Image description
---|---|---|---
ImageNet | 1,281,167/50,000 | 1000 | Photos of common objects
CIFAR-10 | 50,000/10,000 | 10 | Photos of common objects, image sizes $32\times 32$
CIFAR-100 | 50,000/10,000 | 100 | Photos of common objects, image sizes $32\times 32$
Caltech-101 | 3,060/6,084 | 102 | Photos/paintings/sketches of common objects
CUB-200 | 5,994/5,794 | 200 | Find-grained photos of birds
FGVC Aircraft | 6,667/3,333 | 100 | Find-grained photos of aircrafts
Flowers | 1,088/272 | 17 | Find-grained photos of flowers
UC Merced Land Use | 1,680/420 | 21 | Remote sensing images of different land uses
POCUS | 1,692/424 | 3 | Ultrasound images of COVID-19
DTD | 1,880/1,880 | 47 | Photos mainly contain textures of objects
DomainNet real | 120,906/52,041 | 345 | Photos of common objects
DomainNet painting | 50,416/21,850 | 345 | Oil Paintings, murals, drawings, tattoos
DomainNet clipart | 33,525/14,604 | 345 | Clip art images
#### Different Parameter Scales across Layers.
Random initialization for modern DNNs considers the dimension of features in
each layer an important factor. For the inputs following the standard Gaussian
distribution, reasonable random initialization methods (Glorot and Bengio
2010; He et al. 2015) make the corresponding outputs of the layer to be also
standard Gaussian. This work adapts Kaiming random initialization (He et al.
2015). It initializes the parameters with Gaussian distribution with mean
value $\mu=0$ and variance $\sigma^{2}=2/n_{in}$, where $n_{in}$ is the input
dimension. When measuring the transferability of a DNN, the scale difference
of parameters between layers causes an imbalance. The shallow layers of DNNs
usually have fewer channels, which leads to larger initialized parameters and
a large distance after training. Therefore, the variation of shallow layers’
parameters affects $D(\theta_{A},\theta_{B})$ more than the deeper layers.
This problem can be solved by calculating the average transferability of
multiple layers (see Fig. 2 (b)):
$T_{B}(\theta_{A})=\frac{1}{L}\sum_{l=1}^{L}T_{B}(\theta_{A}^{(l)}),$ (2)
where $\theta_{A}^{(l)}$ and $L$ denote the parameters of layer $l$ and the
total number of layers, respectively. It is worth noticing that, the
transferability of layer $l$, $T_{B}(\theta_{A}^{(l)})$, can also be used to
measure the trend of transferability between layers.
## Experiments and Results
This section describes our experimental results in detail. We begin with the
datasets and metrics to evaluate domain gaps and widths. Then we describe our
implementation of transfer learning. Finally, we demonstrate and discuss the
transferabilities of pre-trained parameters in different scenarios.
### Datasets and Metrics
Our experiments include transfer learning from ImageNet to twelve image
classification datasets (CIFAR-10/CIFAR-100 (Krizhevsky, Hinton et al. 2009),
Caltech-101 (Fei-Fei, Fergus, and Perona 2004), CUB-200 (Wah et al. 2011),
FGVC Aircraft (Maji et al. 2013), Flowers (Nilsback and Zisserman 2006), UC
Merced Land Use (Yang and Newsam 2010), POCUS (Born et al. 2021), DTD (Sharan,
Rosenholtz, and Adelson 2014), DomainNet (Peng et al. 2019)) from a wide
range. This is because we find the experiments conducted by previous works
(Yosinski et al. 2014; Azizpour et al. 2015) are primarily on natural images
(i.e., photos), missing the images from farther domains (e.g., medical images,
remote sensing images, paintings) or wider domains (e.g., DomainNet). But in
many transfer learning applications, the distribution of the target domain is
often significantly different from the source domain. The brief information of
the datasets used in this work is summarized in Tab.1
To quantify the domain gaps of datasets to ImageNet as well as the domain
widths, previous methods used trained DNNs to extract semantic features of
different datasets (Liu and Zhang 2022; Stacke et al. 2020). An obvious
drawback is that DNNs trained on specific tasks would have skewed performance
on other tasks. In this work, we need the domain metrics to be domain-agnostic
so that the gaps between different datasets can be more meaningful and
comparable. The basic image color and texture features are desirable for
evaluating the feature distributions of different datasets because they are
not related to the categorical semantics (Saito et al. 2019). First, the RGB
channels’ mean value and standard deviation are considered the color features.
Second, Gray-Level Co-occurrence Matrix features (Haralick, Shanmugam, and
Dinstein 1973) (Angular Second Moment, Entropy, Contrast, and Inverse
Differential Moment) of 4 directions are adopted as the texture features. The
total feature dimension is $3+3+4\times 4=22$. Given feature distributions,
the domain gap is measured as the Maximum Mean Discrepancy (Gretton et al.
2012) between two datasets. Finally, the domain width is calculated as the
maximum eigenvalue of the covariance matrix of features in a dataset. The
domain gaps and widths of datasets used in this work are illustrated in Fig.
3, which is highly consistent with the common expectation (see supplementary
material for the illustration of all datasets included in this work).
Figure 3: The domain gaps with ImageNet and the domain widths of downstream datasets. For good illustration, the domain gap between ImageNet and itself is set to 0.1. Table 2: Dataset information (i.e., the domain gap compared with ImageNet, domain width, and training data amount), training performance (random initialization vs. fine-tuning), and transferabilities on different downstream tasks. | CIFAR-10 | CIFAR-100 | Caltech-101 | CUB-200 | Aircraft | Flowers | Land Use | POCUS | DTD | DomainNet-r | DomainNet-p | DomainNet-c
---|---|---|---|---|---|---|---|---|---|---|---|---
Domain gap | 2.27 | 2.39 | 0.63 | 1.62 | 2.95 | 1.64 | 1.89 | 13.12 | 0.74 | 1.45 | 0.47 | 6.56
Domain width | 186.0 | 242.9 | 385.0 | 220.0 | 114.6 | 145.3 | 181.2 | 46.9 | 822.3 | 467.7 | 455.5 | 823.2
Data amount | 50k | 50k | 3.1k | 6.0k | 6.7k | 1.1k | 1.7k | 1.7k | 1.9k | 120k | 50k | 34k
Random init (%) | 84.9 | 54.6 | 64.6 | 32.4 | 41.5 | 72.8 | 83.6 | 85.9 | 29.3 | 71.4 | 48.5 | 53.5
Fine-tuning (%) | 90.0 | 66.3 | 94.6 | 76.4 | 81.6 | 98.9 | 98.8 | 94.6 | 73.1 | 81.7 | 69.0 | 74.9
Relative error rate (%) $\downarrow$ | 33.8 | 25.8 | 84.7 | 65.1 | 68.5 | 96.0 | 92.7 | 61.7 | 62.0 | 36.0 | 39.8 | 46.0
$T_{FB}(\theta_{A})$ | 7.79 | 5.59 | 23.92 | 21.96 | 12.45 | 106.30 | 39.09 | 53.58 | 24.15 | 6.32 | 7.12 | 8.91
$T_{SB}(\theta_{A})$ | 2.79 | 1.93 | 3.15 | 2.61 | 1.83 | 6.94 | 3.15 | 2.73 | 2.64 | 3.17 | 2.30 | 2.32
### Implementation Details
We utilize the ImageNet dataset to pre-train ResNet-50 (He et al. 2016)
backbone (experiments with other backbones are demonstrated in the
supplementary material, which shows similar conclusions). The hyperparameters
are set to be the same as the original paper. For fine-tuning, we use
$32\times 32$ CIFAR images and the input sizes of the other datasets are set
to $224\times 224$. Batch size of 128, and training time of 100 epochs are
used for all datasets. We use the initial learning rate (LR) of 0.01, SGD
optimizer with the momentum of 0.9, and weight decay of 0.0001. The SGD
optimizer is crucial for keeping learning rates for different layers
consistent (adaptive optimizers control LRs in an unexpected way). The LRs
decrease with a cosine LR scheduler. All images are augmented with random
cropping and horizontal flipping with the probability of 0.5. When training
DNNs from scratch, all hyperparameters are set to be identical as fine-tuning
for a fair comparison. For all experiments with random initialization, initial
parameters are also kept the same.
When measuring the transferabilities of the whole DNN, the last fully
connected layer will be excluded because it is meaningless to transfer
parameters that only work for specific tasks.
### Verification of the Transferability Measurement
In Tab. 2, the information of datasets, training performance from random
initialization and fine-tuning, and the transferabilities on different tasks
are demonstrated. The quantified transferabilities, $T_{FB}(\theta_{A})$ and
$T_{SB}(\theta_{A})$, are highly correlated with the error rate decreasing
ratio (p = 0.0001 and 0.0002 in F-test, respectively). This result indicates
the rationality of the proposed transferability measurement because, with a
higher transferability, a DNN can improve the downstream performance to a
larger extent. Compared with the downstream performance, such as accuracy, the
transferability defined in this work is less limited because it depends only
on parameter variation, which is model-agnostic and task-agnostic.
### Factors Affecting Transferability
In previous works, researchers usually regard DNNs to be less transferable
when domain gaps are larger (Yosinski et al. 2014; Azizpour et al. 2015;
Tajbakhsh et al. 2016). We find that this law does hold generally, and
transferability is even more negatively affected by the domain width and data
amount of downstream tasks.
In Tab. 2, datasets having larger domain gaps with ImageNet tend to have lower
transferabilities. For example, CIFAR-10 and CIFAR-100 have similar domain
widths and data amounts, but CIFAR-100 has a larger domain gap than CIFAR-10,
leading to lower transferability. The same phenomenon can also be observed
between the Flowers and Land Use datasets. However, we can also see that
CUB-200 and Flowers have similar domain gaps with ImageNet, but pre-trained
DNNs have different transferabilities. This fact motivates our interest in
exploring other crucial factors for transferability. Domain width and data
amount are chosen as possible factors, as discussed in the introduction.
Figure 4: Linear analysis of different factors. The correlation coefficient is
0.90.
In this work, we validate the three factors by linear analysis. By fitting
$log(T_{B}(\theta_{A}))$ to the domain gap $\mathcal{G}$, domain width
$\mathcal{W}$, and log data amount $log(\mathcal{N})$, we find a high
coefficient of determination for $log(T_{FB}(\theta_{A}))$ ($r^{2}$=0.90, see
Fig. 4). The fitted linear relationship is
$log(\hat{T}_{FB}(\theta_{A}))=-0.02\mathcal{G}-0.14\mathcal{W}-0.80log(\mathcal{N})+2.86.$
(3)
We can see that, despite the slightly negative effect of the domain gap, there
are stronger negative correlations between transferabilities and domain width
and data amount. The reason will be elaborated on later. For
$log(T_{SB}(\theta_{A}))$, linear regression performs poorly ($r^{2}=0.26$),
indicating that the three factors can hardly interpret $T_{SB}(\theta_{A})$.
This result may be due to the unstable convergence points of random
initialized DNNs. Some convergence points are too bad to be considered in the
“solution set” of the tasks (e.g., CUB and DTD). Therefore,
$D(\theta_{r},\theta_{SB})$ is not a desirable metric for measuring
transferability. $T_{FB}(\theta_{A})$ is mainly used for the following
analysis.
#### Domain Width.
We try to remove distractions from domain gap and data amount, and explore the
relationship between domain width and transferability $T_{FB}(\theta_{A})$.
DTD and DomainNet-c are selected as the target datasets as they have the
largest domain width. For DTD, we keep the training data amount to 1000 and
use 12, 20, and 47 classes to calculate the transferabilities. It is worth
noting that the domain gaps with ImageNet change slightly (from 6.56 to 6.44)
when the number of classes decreases. For DomainNet-c, the domain gap is also
stable when we randomly sample 2,000 images from 35, 100, or 345 classes. The
results are shown in Fig. 5, where the transferabilities decrease as the
domain widths increase, which validates Eqn. (3).
#### Downstream Data Amount.
This experiment verifies the negative correlation between transferabilities
and data amounts. We uniformly sample data from all classes of CIFAR-100 and
DomainNet-c with different amounts, and the transferability curves are shown
in Fig. 6. It is worth mentioning that the random sampling scheme makes the
domain gap and domain width stable in this experiment. With the increasing
amount of data, the transferabilities decrease monotonically. Moreover,
comparing Fig. 5 with Fig. 6, the effect of data amount seems greater than
that of domain width, which is consistent with the Eqn. (3).
Figure 5: The negative correlation between domain widths and
transferabilities.
Figure 6: The negative correlation between downstream data amounts and
transferabilities.
Figure 7: A conceptual visualization of source dataset and target dataset.
“OOD” and “ID” are the abbreviations for “out-of-distribution” and “in-
distribution”.
#### A General Explanation.
We try to explain the negative effects of domain gap, domain width, and data
amount with a general framework (Fig. 7). In the pre-training phase, a DNN
learns the basic knowledge (also called visual representations for visual
tasks) from the source dataset. Within these representations, there are some
in-distribution representations that can be used for the downstream task
directly. However, there are still new visual representations that the DNN
needs to learn from the downstream task for better performance. In the fine-
tuning phase, the pre-trained DNN updates its parameters to learn these out-
of-distribution representations. Such a larger parameter change corresponds to
lower transferability. When the domain gap becomes greater or the domain width
becomes larger, the more out-of-distribution representations the DNN needs to
learn, and the lower transferability the DNN has. Similarly, given the fixed
domain gap and domain width, higher downstream data amount makes the
representations that need to be learned denser. The wider range and denser new
representations decrease the DNN transferability by increasing knowledge that
cannot be transferred from experiences.
### Layer-wise Transferability
Figure 8: The layer-wise transferabilities on different datasets. The “Block”
denotes the first convolutional layer, the bottle-neck (a three-conv-layer
residual block), or the last fully connected layer of ResNet.
Previous literature usually regards the lower layers to be more transferable
(Yosinski et al. 2014), due to the generality of the low-level feature
extraction abilities. This work shows that this is not always the case. The
trend of transferabilities across layers is illustrated in Fig. 8. There are
two noteworthy laws. First, nearly all layer-wise transferabilities are
inverted U-shaped instead of downward trends, mainly caused by the first two
poorly transferable layers. Second, discarding the first two layers, nine
datasets have descending layer-wise transferabilities across layers consistent
with our previous consensus. However, three datasets (CIFAR-10, CIFAR-100, and
POCUS) still do not present any meaningful upward/downward trend. For these
three datasets, we find that the low-level features may not be transferable as
the low-level image pattern are not similar anymore. For CIFAR-10 and
CIFAR-100, the image sizes are only $32\times 32$. The pre-trained ResNets
have learned visual representations from $224\times 224$ ImageNet data, so
these representations may be poorly transferable to the seriously different
local pattern distributions. For the POCUS dataset, the ultrasound images also
have quite different image patterns compared with natural images, making the
pre-trained layers for extracting local features harder to transfer. All in
all, the different pattern distributions of pre-training and downstream
datasets make the layer-wise transferability no longer decreasing.
#### Domain-sensitivity of the Lowest Layers.
In Fig. 8, we can see a common phenomenon: the first layer has very low
transferability on all datasets, including those natural image datasets with
small domain gaps. This result may indicate that the first layer is more
sensitive to the domain shift from ImageNet than the subsequent layers. We
assume that when a pre-trained DNN “sees” a new sample with out-of-
distribution basic visual patterns, the first layer weights would get large
gradients and update drastically. These kinds of out-of-distribution basic
visual patterns may be common in downstream datasets. To verify this
assumption, we calculate the layer-wise gradients of pre-trained DNNs on two
natural image datasets, i.e., CUB-100 and Flowers (see Fig. 9), and we find
that both curves are U-shaped. It is reasonable for the last layers to have
large gradients, as they are replaced and randomly initialized for specific
tasks. But for the large gradients of the pre-trained lower layers, they
suggest that the parameters of these layers are somehow “unfit” for the
current tasks, which validates our assumption.
Figure 9: The layer-wise mean absolute gradients on CUB and Flower datasets.
Both curves are U-shaped.
#### Visualization of Pre-trained/Fine-tuned Features.
To further illustrate the weaker transferabilities of lower layers, we choose
CUB-200 to show how first-layer features of a pre-trained DNN change after
fine-tuning. In Fig. 10, the pre-trained first layer can successfully activate
meaningful edges. But some features are not good enough (in red boxes), with
unclear edges or activating the background gauze too much. After fine-tuning,
these imperfect feature maps become clearer and more helpful for recognizing
the bird. As reported in (Donahue et al. 2014), a unit’s activation is better
be maximized to extract more meaningful image features, meaning that the fine-
tuned parameters are more capable of finding key patterns on the images. The
benefit of fine-tuning the first layer is increasing the activations of the
target object in the image (mean value increases from 0.268 to 0.284). The
activation distributions of pre-trained/fine-tuned DNNs significantly differ
with the $t$ statistic of -16.99, $p=4.6\times 10^{-57}$, which shows
significant improvement brought by updating the first layer. This result
indicates that some previous methods of only tuning higher layers (Tajbakhsh
et al. 2016) can still be improved concerning the first-layer transferability.
Figure 10: Input image (left), first-layer features of pre-trained DNN (upper
right), and the corresponding features after fine-tuning (lower right). For
feature maps, lighter means more activated.
## Conclusion
In this work, we have proposed a novel method for quantifying the
transferabilities of DNNs. We conducted extensive experiments on twelve
datasets with this method and showed how transferability is more negatively
affected by domain width and downstream data amount than domain gap. This
phenomenon suggests that downstream tasks with narrower domains and fewer data
benefit more from pre-training. Transfer learning with a large domain gap
(e.g., medical data) may reach higher performance considering this result. In
addition, we observed that the layer-wise transferabilities of DNNs are not
monotonically decreasing as the traditional consensus regarded. The root cause
is that the first layers of DNNs are sensitive to domains. They have larger
gradients than the middle layers and are usually imperfect for downstream
tasks. Fine-tuning techniques with the least transferable layers from both
ends may achieve better performance than previous works with only the last
layers.
## References
* Ausubel et al. (1968) Ausubel, D. P.; Novak, J. D.; Hanesian, H.; et al. 1968. _Educational psychology: A cognitive view_ , volume 6. holt, rinehart and Winston New York.
* Azizpour et al. (2015) Azizpour, H.; Razavian, A. S.; Sullivan, J.; Maki, A.; and Carlsson, S. 2015. Factors of transferability for a generic convnet representation. _IEEE TPAMI_ , 38(9): 1790–1802.
* Born et al. (2021) Born, J.; Wiedemann, N.; Cossio, M.; Buhre, C.; Brändle, G.; Leidermann, K.; Aujayeb, A.; Moor, M.; Rieck, B.; and Borgwardt, K. 2021. Accelerating Detection of Lung Pathologies with Explainable Ultrasound Image Analysis. _Applied Sciences_ , 11(2): 672.
* Chen et al. (2022) Chen, Y.; Li, J.; Jiang, H.; Liu, L.; and Ding, C. 2022. MetaLR: Layer-wise Learning Rate based on Meta-Learning for Adaptively Fine-tuning Medical Pre-trained Models. _arXiv preprint arXiv:2206.01408_.
* Chen et al. (2021) Chen, Y.; Zhang, C.; Liu, L.; Feng, C.; Dong, C.; Luo, Y.; and Wan, X. 2021. Uscl: Pretraining deep ultrasound image diagnosis model through video contrastive representation learning. In _MICCAI_ , 627–637. Springer.
* Deng et al. (2009) Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-Fei, L. 2009. Imagenet: A large-scale hierarchical image database. In _CVPR_ , 248–255.
* Donahue et al. (2014) Donahue, J.; Jia, Y.; Vinyals, O.; Hoffman, J.; Zhang, N.; Tzeng, E.; and Darrell, T. 2014. Decaf: A deep convolutional activation feature for generic visual recognition. In _ICML_ , 647–655. PMLR.
* Dosovitskiy et al. (2020) Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S.; et al. 2020\. An image is worth 16x16 words: Transformers for image recognition at scale. _arXiv preprint arXiv:2010.11929_.
* Dwivedi and Roig (2019) Dwivedi, K.; and Roig, G. 2019. Representation similarity analysis for efficient task taxonomy & transfer learning. In _CVPR_ , 12387–12396.
* Fayek, Cavedon, and Wu (2018) Fayek, H. M.; Cavedon, L.; and Wu, H. R. 2018. On the transferability of representations in neural networks between datasets and tasks. _arXiv preprint arXiv:1811.12273_.
* Fei-Fei, Fergus, and Perona (2004) Fei-Fei, L.; Fergus, R.; and Perona, P. 2004. Learning generative visual models from few training examples: An incremental bayesian approach tested on 101 object categories. In _CVPR workshop_ , 178–178. IEEE.
* Finn, Abbeel, and Levine (2017) Finn, C.; Abbeel, P.; and Levine, S. 2017. Model-agnostic meta-learning for fast adaptation of deep networks. In _ICML_ , 1126–1135. PMLR.
* Glorot and Bengio (2010) Glorot, X.; and Bengio, Y. 2010. Understanding the difficulty of training deep feedforward neural networks. In _Proceedings of the thirteenth international conference on artificial intelligence and statistics_ , 249–256. JMLR Workshop and Conference Proceedings.
* Gretton et al. (2012) Gretton, A.; Borgwardt, K. M.; Rasch, M. J.; Schölkopf, B.; and Smola, A. 2012\. A kernel two-sample test. _The Journal of Machine Learning Research_ , 13(1): 723–773.
* Haralick, Shanmugam, and Dinstein (1973) Haralick, R. M.; Shanmugam, K.; and Dinstein, I. H. 1973. Textural features for image classification. _IEEE Transactions on systems, man, and cybernetics_ , (6): 610–621.
* He, Girshick, and Dollár (2019) He, K.; Girshick, R.; and Dollár, P. 2019. Rethinking imagenet pre-training. In _ICCV_ , 4918–4927.
* He et al. (2015) He, K.; Zhang, X.; Ren, S.; and Sun, J. 2015. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In _ICCV_ , 1026–1034.
* He et al. (2016) He, K.; Zhang, X.; Ren, S.; and Sun, J. 2016. Deep residual learning for image recognition. In _CVPR_ , 770–778.
* Huang et al. (2017) Huang, G.; Liu, Z.; Van Der Maaten, L.; and Weinberger, K. Q. 2017. Densely connected convolutional networks. In _CVPR_ , 4700–4708.
* Jiang et al. (2022) Jiang, J.; Shu, Y.; Wang, J.; and Long, M. 2022. Transferability in Deep Learning: A Survey. _arXiv preprint arXiv:2201.05867_.
* Kornblith, Shlens, and Le (2019) Kornblith, S.; Shlens, J.; and Le, Q. V. 2019. Do better imagenet models transfer better? In _CVPR_ , 2661–2671.
* Krizhevsky, Hinton et al. (2009) Krizhevsky, A.; Hinton, G.; et al. 2009. Learning multiple layers of features from tiny images.
* Li et al. (2019) Li, X.; Xiong, H.; Wang, H.; Rao, Y.; Liu, L.; Chen, Z.; and Huan, J. 2019. Delta: Deep learning transfer using feature map with attention for convolutional networks. _arXiv preprint arXiv:1901.09229_.
* Liu and Zhang (2022) Liu, X.; and Zhang, S. 2022. Who is closer: A computational method for domain gap evaluation. _Pattern Recognition_ , 122: 108293.
* Long et al. (2015) Long, M.; Cao, Y.; Wang, J.; and Jordan, M. 2015. Learning transferable features with deep adaptation networks. In _ICML_ , 97–105. PMLR.
* Maji et al. (2013) Maji, S.; Rahtu, E.; Kannala, J.; Blaschko, M.; and Vedaldi, A. 2013. Fine-grained visual classification of aircraft. _arXiv preprint arXiv:1306.5151_.
* Matsoukas et al. (2022) Matsoukas, C.; Haslum, J. F.; Sorkhei, M.; Söderberg, M.; and Smith, K. 2022\. What Makes Transfer Learning Work For Medical Images: Feature Reuse & Other Factors. In _CVPR_ , 9225–9234.
* Newell and Deng (2020) Newell, A.; and Deng, J. 2020. How useful is self-supervised pretraining for visual tasks? In _CVPR_ , 7345–7354.
* Nilsback and Zisserman (2006) Nilsback, M.-E.; and Zisserman, A. 2006. A visual vocabulary for flower classification. In _CVPR_ , volume 2, 1447–1454. IEEE.
* Peng et al. (2019) Peng, X.; Bai, Q.; Xia, X.; Huang, Z.; Saenko, K.; and Wang, B. 2019. Moment matching for multi-source domain adaptation. In _ICCV_ , 1406–1415.
* Ro and Choi (2021) Ro, Y.; and Choi, J. Y. 2021. AutoLR: Layer-wise pruning and auto-tuning of learning rates in fine-tuning of deep networks. In _AAAI_ , volume 35, 2486–2494.
* Saito et al. (2019) Saito, K.; Ushiku, Y.; Harada, T.; and Saenko, K. 2019. Strong-weak distribution alignment for adaptive object detection. In _CVPR_ , 6956–6965.
* Sermanet et al. (2018) Sermanet, P.; Lynch, C.; Chebotar, Y.; Hsu, J.; Jang, E.; Schaal, S.; Levine, S.; and Brain, G. 2018. Time-contrastive networks: Self-supervised learning from video. In _ICRA_ , 1134–1141. IEEE.
* Sharan, Rosenholtz, and Adelson (2014) Sharan, L.; Rosenholtz, R.; and Adelson, E. H. 2014. Accuracy and speed of material categorization in real-world images. _Journal of vision_ , 14(9): 12–12.
* Song et al. (2021) Song, L.; Wu, J.; Yang, M.; Zhang, Q.; Li, Y.; and Yuan, J. 2021. Robust Knowledge Transfer via Hybrid Forward on the Teacher-Student Model. In _AAAI_ , 2558–2566.
* Stacke et al. (2020) Stacke, K.; Eilertsen, G.; Unger, J.; and Lundström, C. 2020. Measuring domain shift for deep learning in histopathology. _IEEE journal of biomedical and health informatics_ , 25(2): 325–336.
* Tajbakhsh et al. (2016) Tajbakhsh, N.; Shin, J. Y.; Gurudu, S. R.; Hurst, R. T.; Kendall, C. B.; Gotway, M. B.; and Liang, J. 2016. Convolutional neural networks for medical image analysis: Full training or fine tuning? _IEEE TMI_ , 35(5): 1299–1312.
* Wah et al. (2011) Wah, C.; Branson, S.; Welinder, P.; Perona, P.; and Belongie, S. 2011. The caltech-ucsd birds-200-2011 dataset.
* Yang and Newsam (2010) Yang, Y.; and Newsam, S. 2010. Bag-of-visual-words and spatial extensions for land-use classification. In _Proceedings of the 18th SIGSPATIAL international conference on advances in geographic information systems_ , 270–279.
* Yosinski et al. (2014) Yosinski, J.; Clune, J.; Bengio, Y.; and Lipson, H. 2014. How transferable are features in deep neural networks? _NeurIPS_ , 27.
* Zhang et al. (2022) Zhang, C.; Chen, Y.; Liu, L.; Liu, Q.; and Zhou, X. 2022. HiCo: Hierarchical Contrastive Learning for Ultrasound Video Model Pretraining. In _Proceedings of the Asian Conference on Computer Vision_ , 229–246.
* Zhong et al. (2020) Zhong, J.; Wang, X.; Kou, Z.; Wang, J.; and Long, M. 2020. Bi-tuning of pre-trained representations. _arXiv preprint arXiv:2011.06182_.
|
# Free $\mathbb{Q}$-groups are residually torsion-free nilpotent
Andrei Jaikin-Zapirain Departamento de Matemáticas, Universidad Autónoma de
Madrid and Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM
<EMAIL_ADDRESS>
(Date: November 2022)
###### Abstract.
We develop a method to show that some (abstract) groups can be embedded into a
free pro-$p$ group. In particular, we show that every finitely generated
subgroup of a free $\mathbb{Q}$-group can be embedded into a free pro-$p$
group for almost all primes $p$. This solves an old problem raised by G.
Baumslag: free $\mathbb{Q}$-groups are residually torsion-free nilpotent.
###### Key words and phrases:
Free $\mathbb{Q}$-groups, free pro-$p$ groups, mod-$p$ $L^{2}$-Betti numbers,
Lück approximation, universal division ring of fractions
###### 2010 Mathematics Subject Classification:
Primary: 20E06, Secondary: 16K40, 20C07, 20E18, 20E26
## 1\. Introduction
A group $G$ is called a $\mathbb{Q}$-group if for any $n\in\mathbb{N}$ and
$g\in G$ there exists exactly one $h\in G$ satisfying $h^{n}=g$. These groups
were introduced by G. Baumslag in [4] under the name of
$\operatorname{\mathcal{D}}$-groups. He observed that $\mathbb{Q}$-groups may
be viewed as universal algebras, and as such they constitute a variety. Every
variety of algebras contains free algebras (in that variety). In the variety
of $\mathbb{Q}$-groups we call such free algebras free $\mathbb{Q}$-groups. G.
Baumslag dedicated several papers to the study of residual properties of free
$\mathbb{Q}$-groups [5, 7, 9]. For example, in [5] he showed that a free
$\mathbb{Q}$-group is residually periodic-by-soluble and locally residually
finite-by-soluble. He wrote in [5] “It is, of course, still possible that,
locally, free $\operatorname{\mathcal{D}}$-groups are, say, residually finite
$p$-groups” or in [7] “In particular it seems likely that free
$\operatorname{\mathcal{D}}$-groups are residually torsion-free nilpotent.
However the complicated nature of free $\operatorname{\mathcal{D}}$-groups
makes it difficult to substantiate such a remark.” This conjecture is part of
two main collections of problems in group theory ([10, Problem F12] and [38,
Problem 13.39 (a),(c)]), and in addition to mentioned works of Baumslag, it
was also studied in [15, 22]. In this paper we solve Baumslag’s conjecture.
###### Theorem 1.1.
A free $\mathbb{Q}$-group is residually torsion-free nilpotent.
The structure of a finitely generated subgroup of a free $\mathbb{Q}$-group
was studied already in [4] (see also [53, Section 8] and Proposition 5.3). It
was shown that it is the end result of repeatedly freely adjoining $n$th roots
to a finitely generated free group. The key point of our proof of Theorem 1.1
is to show that any finitely generated subgroup of a free $\mathbb{Q}$-group
can be embedded into a finitely generated free pro-$p$ group for some prime
$p$. We actually prove the following more precise result.
###### Theorem 1.2.
Let $p$ be a prime. Let $H_{0}$ be a finitely generated free group and let
$H_{0}\hookrightarrow\mathbf{F}$ be the canonical embedding of $H_{0}$ into
its pro-$p$ completion $\mathbf{F}$. Let $(H_{i})_{i\geq 0}$ be a sequence of
subgroups of $\mathbf{F}$ such that for $i\geq 0$,
1. (1)
$H_{i+1}=\langle H_{i},B_{i}\rangle$, where $B_{i}$ is a finitely generated
abelian subgroup of $\mathbf{F}$ and
2. (2)
$A_{i}=H_{i}\cap B_{i}$ is a maximal abelian subgroup of $H_{i}$.
Then for every $i\geq 0$, the canonical map
$H_{i}*_{A_{i}}B_{i}\to H_{i+1}$
is an isomorphism,
Theorem 1.2 is actually an application of the slightly more technical Theorem
5.1.
Let us make a few remarks about the groups $A_{i}$ and $B_{i}$. It is
relatively easy to describe abelian subgroups of amalgamated products. In
particular, the conclusion of the theorem implies that all abelian subgroups
of $H_{i}$ are finitely generated. Thus, an implicit hypothesis, which appears
in the theorem, that maximal abelian subgroups $A_{i}$ of $H_{i}$ are finitely
generated, is automatically fulfilled.
A maximal abelian subgroup of $\mathbf{F}$ is isomorphic to the additive group
of the ring of $p$-adic numbers $(\mathbb{Z}_{p},+)$. Therefore, for any
finitely generated (abstract) abelian subgroup $A$ of $\mathbf{F}$ and any
finitely generated torsion-free abelian group $B$ which contains $A$ and such
that $B/A$ has no $p$-torsion, it is posible to extend the embedding
$A\hookrightarrow\mathbf{F}$ to an embedding $B\hookrightarrow\mathbf{F}$.
This extension is unique if and only if $B/A$ is finite.
Given a commutative ring $A$, we will introduce in Section 5 the notions of
$A$-group and free $A$-group $F^{A}(X)$. For example, a free pro-$p$ group is
an example of a $\mathbb{Z}_{p}$-group. We have the following consequence of
Theorem 1.2.
###### Corollary 1.3.
Let $F(X)$ be the free group on a finite free generating set $X$, let
$\mathbf{F}$ be its pro-$p$ completion. Then the canonical homomorphism
$\phi:F^{\mathbb{Z}_{p}}(X)\to\mathbf{F}$
is injective.
Let $H$ be a group and $A$ the centralizer of a non-trivial element. Then the
group $G=H*_{A}(A\times\mathbb{Z}^{k})$ is said to be obtained from $H$ by
extension of a centralizer. A group is called an ICE group if it can be
obtained from a free group using iterated centralizer extensions. A group $G$
is a limit group if and only if it is a finitely generated subgroup of an ICE
group (see [37, 14]). All centralizers of non-trivial elements of an ICE group
are abelian. Thus, Theorem 1.1 provides explicit realizations of ICE groups
(and so limit groups) as subgroups of a non-abelian free pro-$p$-groups (for
this application we only need the case where all $B_{i}/A_{i}$ are torsion-
free). Non-explicit realizations of limit groups as subgroups of a non-abelian
free pro-$p$ group (in fact, as subgroups of every compact group containing a
non-abelian free group) was obtained in [2] (see also [12]).
In Section 5 we recall the definition of the $\mathbb{Q}$-completion of a
group $G$. For example, a free $\mathbb{Q}$-group is the
$\mathbb{Q}$-completion of a free group. Theorem 1.2 allows also to show that
the $\mathbb{Q}$-completion of a limit group is residually torsion-free
nilpotent.
###### Theorem 1.4.
The $\mathbb{Q}$-completion of a limit group is residually torsion-free
nilpotent.
A group $G$ is called parafree if it is residually nilpotent and for some free
group $F$, we have that for all $i$, $G/\gamma_{i}(G)\cong F/\gamma_{i}(F)$
where $\gamma_{i}(G)$ denotes the terms of the lower central series of $G$.
Baumslag introduced this family of groups and produced many examples of them
[6]. In [35] we apply the method of the proof of Theorem 1.2 in order to
construct new examples of finitely generated parafree groups.
Our proof of Theorem 1.2 is by induction on $i$. In the inductive step
argument we start with the following situation. We have a finitely generated
subgroup $H$ of $\mathbf{F}$, a maximal abelian subgroup $A$ of $H$ and an
abelian subgroup $B$ of $\mathbf{F}$ containing $A$. We want to show that the
canonical homomorphism $H*_{A}B\to\langle H,B\rangle$ is an isomorphism.
Unfortunately, we do not know how to show this statement in such a generality,
but we prove it in Theorem 5.1 under an additional assumption that the
embedding $H\hookrightarrow\mathbf{F}$ is strong (see Definition 3.8). Theorem
5.1 is the main result of the paper. Its proof uses in an essential way the
results of [32], where we proved a particular case of the Lück approximation
in positive characteristic.
The paper is organized as follows. In Section 2 we give basic preliminaries.
The proof of Theorem 5.1 uses the theory of mod-$p$ $L^{2}$-Betti numbers. In
Section 3 we explain how to define them for subgroups $G$ of a free pro-$p$
group. In Section 4 we introduce a technical notion of $\mathcal{D}$-torsion-
free modules and show that some relevant $\mathbb{F}_{p}[G]$-modules are
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free (see Proposition
4.10). In Section 5 we prove Theorem 5.1 and obtain all the results mentioned
in the introduction. In Section 6 we discuss the following two well-known
problems concerning linearity of free pro-$p$ groups and free
$\mathbb{Q}$-groups:
###### Question 1.5.
1. (1)
(I. Kapovich) Is a free $\mathbb{Q}$-group linear?
2. (2)
(A. Lubotzky) Is a free pro-$p$ group linear?
## Acknowledgments
I am very grateful to Pavel Zalesski who explained to me how a surface group
can be embedded into a free pro-$p$ group. This was a starting point of
Theorem 1.2. I would like to thank Warren Dicks for providing the references
needed in the proof of Proposition 2.13, Yago Antolin, Ashot Minasyan and a
referee for explaining to me the proof of Theorem 6.1, Emmanuel Breulliard for
pointing out the paper [2] and Ismael Morales and anonymous referees for
useful suggestions and comments.
This paper is partially supported by the grants MTM2017-82690-P and
PID2020-114032GB-I00 of the Ministry of Science and Innovation of Spain and by
the ICMAT Severo Ochoa project CEX2019-000904-S4.
## 2\. Preliminaries
### 2.1. $R$-rings
All rings in this paper are associative and have the identity element. All
ring homomorphisms send the identity to the identity. We denote the invertible
elements of a ring $R$ by $R^{*}$. An $R$-module means a left $R$-module. By
an $R$-ring we understand a ring homomorphism $\varphi:R\to S$. We will often
refer to $S$ as $R$-ring and omit the homomorphism $\varphi$ if $\varphi$ is
clear from the context. Two $R$-rings $\varphi_{1}:R\to S_{1}$ and
$\varphi_{2}:R\to S_{2}$ are said to be isomorphic if there exists a ring
isomorphism $\alpha:S_{1}\to S_{2}$ such that
$\alpha\circ\varphi_{1}=\varphi_{2}$.
If $Y=\\{y_{i}\colon i\in I\\}$, we denote by $A\langle\\!\langle
Y\rangle\\!\rangle$ the ring of of formal power series with coefficients in
$A$ on the non-commuting indeterminates $Y$.
### 2.2. Left ideals in group algebras
Let $G$ be a group and $k$ a commutative ring. We denote by $I_{G}$ the
augmentation ideal of $k[G]$. If $H$ is a subgroup of $G$ we denote by
$I_{H}^{G}$ the left ideal of $k[G]$ generated by $I_{H}$. The following lemma
gives an alternative description of the $k[G]$-module $I_{H}^{G}$.
###### Lemma 2.1.
Let $H\leq T$ be subgroups of $G$. Then the following holds.
1. (a)
The canonical map
$k[G]\otimes_{k[H]}I_{H}\to I_{H}^{G}$
sending $a\otimes b$ to $ab$, is an isomorphism of $k[G]$-modules.
2. (b)
The canonical map $k[G]\otimes_{k[T]}(I_{T}/I_{H}^{T})\to
I_{T}^{G}/I_{H}^{G}$, sending $a\otimes(b+I_{H}^{T})$ to $ab+I_{H}^{G}$, is an
isomorphism of $k[G]$-modules.
###### Proof.
(a) Consider an exact sequence
$0\to I_{H}\to k[H]\to k\to 0.$
The freeness of $k[G]$ as $k[H]$-module implies that the sequence
$0\to
k[G]\otimes_{k[H]}I_{H}\xrightarrow{\alpha}k[G]\xrightarrow{\beta}k[G]\otimes_{k[H]}k\to
0$
is also exact. Here $\alpha$ sends $a\otimes b$ to $ab$ and $\beta$ sends $a$
to $a\otimes 1$. Thus, $\alpha$ establishes an isomorphisms of $k[G]$-modules
between $k[G]\otimes_{k[H]}I_{H}$ and $\ker\beta=I_{H}^{G}$. This proves the
first claim of the lemma.
(b) Consider now the exact sequence
$0\to I_{H}^{T}\to I_{T}\to I_{T}/I_{H}^{T}\to 0.$
Applying $k[G]\otimes_{k[T]}$ and taking again into account that $k[G]$ is a
free $k[T]$-module, we obtain the exact sequence
$0\to I_{H}^{G}\to I_{T}^{G}\to k[G]\otimes_{k[T]}(I_{T}/I_{H}^{T})\to 0.$
This proves the second claim. ∎
### 2.3. Profinite modules over pro-$p$ groups
In this paper the letters $\mathbf{F}$, $\mathbf{G}$, $\mathbf{H}$, etc. will
denote pro-$p$ groups. When we speak about pro-$p$ groups, the finite
generation, the finite presentation, the freeness etc. will always be
considered in the category of pro-$p$ groups. For example, $d(\mathbf{G})$
denotes the minimal number of topological generators of $\mathbf{G}$.
Almost all pro-$p$ groups that we consider are free pro-$p$ groups. Recall
that a closed subgroup of a free pro-$p$ group is also free pro-$p$ ([56,
Corollary 7.7.5]). As a consequence we obtain the following result which we
will use often in this paper.
###### Lemma 2.2.
Every maximal abelian subgroup of a non-trivial free pro-$p$ group is
isomorphic to $(\mathbb{Z}_{p},+)$.
Let $\mathbf{G}$ be a pro-$p$ group. We denote by
$\mathbb{F}_{p}[[\mathbf{G}]]$ the inverse limit of
$\mathbb{F}_{p}[\mathbf{G}/\mathbf{U}]$, where the limit is taken over all
open normal subgroups $\mathbf{U}$ of $\mathbf{G}$.
$\mathbb{F}_{p}[[\mathbf{G}]]$ is called the completed group algebra of
$\mathbf{G}$ over $\mathbb{F}_{p}$. In the case where $\mathbf{G}$ is a free
pro-$p$ group we have the following useful description of
$\mathbb{F}_{p}[[\mathbf{G}]]$.
###### Proposition 2.3.
[39, Section II, Proposition 3.1.4] Let $\mathbf{F}$ be a finitely generated
free pro-$p$ group freely generated by $x_{1},\ldots,x_{d}$. Then the
continuous $\mathbb{F}_{p}$-algebra homomorphism
$\mathbb{F}_{p}\langle\\!\langle
y_{1},\ldots,y_{d}\rangle\\!\rangle\to\mathbb{F}_{p}[[\mathbf{F}]]$
that sends $y_{i}$ to $x_{1}-1$ (for $1\leq i\leq d$) is an isomorphism.
A discrete $\mathbb{F}_{p}[[\mathbf{G}]]$-module is an
$\mathbb{F}_{p}[[\mathbf{G}]]$-module $M$ such that for any $m\in M$,
$\operatorname{Ann}_{\mathbb{F}_{p}[[\mathbf{G}]]}(m):=\\{a\in\mathbb{F}_{p}[[\mathbf{G}]]\colon
am=0\\}$
is open in $\mathbb{F}_{p}[[\mathbf{G}]]$.
A profinite $\mathbb{F}_{p}[[\mathbf{G}]]$-module is an inverse limit of
finite discrete $\mathbb{F}_{p}[[\mathbf{G}]]$-modules.
###### Lemma 2.4.
Let $\mathbf{G}$ be a pro-$p$ group. Let $\alpha:M\to N$ be a homomorphism of
profinite $\mathbb{F}_{p}[[\mathbf{G}]]$-modules. If $M$ is finitely generated
as an $\mathbb{F}_{p}[[\mathbf{G}]]$-module, then $\alpha$ is continuous.
###### Remark 2.5.
This lemma resembles a well-known result of Nikolov and Segal [55] that says
that every homomorphism from a finitely generated profinite group to a
profinite group is always continuous. This is equivalent to that every
subgroup of finite index in a finitely generated profinite group is open. In
the case of $\mathbb{F}_{p}[[\mathbf{G}]]$-modules, the situation is
different: it is not always true that every left ideal of
$\mathbb{F}_{p}[[\mathbf{G}]]$ of finite index is open.
###### Proof.
Without loss of generality we may assume that $N$ is a finite discrete
$\mathbb{F}_{p}[[\mathbf{G}]]$-module and we have to show that $\ker\alpha$ is
open.
We put
$I=\operatorname{Ann}_{\mathbb{F}_{p}[[G]]}=\\{a\in\mathbb{F}_{p}[[\mathbf{G}]]\colon
aN=\\{0\\}\\}$. Then $I$ is an open, and so also closed, ideal of
$\mathbb{F}_{p}[[\mathbf{G}]]$. Let $IM$ be the submodule of $M$ generated by
$\\{a\cdot m\colon a\in I,m\in M\\}$. Since $M$ is finitely generated we can
write
$M=\sum_{i=1}^{s}\mathbb{F}_{p}[[\mathbf{G}]]m_{i}.$
Therefore, $IM=\sum_{i=1}^{s}Im_{i}$.
From the definition of a profinite $\mathbb{F}_{p}[[\mathbf{G}]]$-module it
follows that the multiplicative map $\mathbb{F}_{p}[[\bf G]]\times M\to M$ is
continuous. Hence, since $I$ is closed in $\mathbb{F}_{p}[[\mathbf{G}]]$,
$IM=\sum_{i=1}^{s}Im_{i}$ is a closed submodule of $M$. But $IM$ is also of
finite index. Hence, it is open in $M$. Since $IM\leq\ker\alpha$, $\ker\alpha$
is also open. ∎
If $M=\varprojlim_{i\in I}M_{i}$ and $N=\varprojlim_{j\in J}N_{i}$ are right
and left, respectively, profinite $\mathbb{F}_{p}[[\mathbf{G}]]$-modules
($M_{i}$ and $N_{j}$ are finite discrete
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules), then the profinite tensor product is
denoted by $\widehat{\otimes}$ and it is defined as the inverse limit of
$M_{i}\otimes_{\mathbb{F}_{p}[[\mathbf{G}]]}N_{j}$.
###### Lemma 2.6.
Let $\mathbf{H}$ be a closed subgroup of $\mathbf{G}$ and let $M$ be a
finitely presented $\mathbb{F}_{p}[[\mathbf{H}]]$-module. Then $M$ is a
profinite module and the canonical map
$\gamma:\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M\to\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}M$
is an isomorphism of $\mathbb{F}_{p}[[\mathbf{G}]]$-modules.
###### Remark 2.7.
It is claimed in [56, Proposition 5.5.3 (d)] that it is enough to assume that
$M$ is a finitely generated profinite $\mathbb{F}_{p}[[\mathbf{H}]]$-module.
However, we want to warn the reader that the proof is incorrect.
###### Proof.
Since $M$ is finitely presented, there exists an exact sequence of
$\mathbb{F}_{p}[[\mathbf{H}]]$-modules
(1)
$\mathbb{F}_{p}[[\mathbf{H}]]^{r}\xrightarrow{\alpha}\mathbb{F}_{p}[[\mathbf{H}]]^{d}\to
M\to 0.$
Thus, we can write $M\cong\mathbb{F}_{p}[[\mathbf{H}]]^{d}/I$, where
$I=\operatorname{Im}\alpha$. By Lemma 2.4, $\alpha$ is continuous. Since
$\mathbb{F}_{p}[[\mathbf{H}]]^{r}$ is compact and Hausdorff, $I$ is closed in
$\mathbb{F}_{p}[[\mathbf{H}]]^{d}$. Hence
$\mathbb{F}_{p}[[\mathbf{H}]]^{d}/I$, and so $M$, are profinite.
It is clear that
$\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}\mathbb{F}_{p}[[\mathbf{H}]]\cong\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\mathbb{F}_{p}[[\mathbf{H}]]\cong\mathbb{F}_{p}[[\mathbf{G}]].$
Thus, after applying
$\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}$ and
$\mathbb{F}_{p}[[\mathbf{G}]]{\widehat{\otimes}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$
to the sequence (1) we obtain a commutative diagram of
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules:
$\begin{array}[]{cccccc}\mathbb{F}_{p}[[\mathbf{G}]]^{r}&\to&\mathbb{F}_{p}[[\mathbf{G}]]^{d}&\to&\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M&\to
0\\\ ||&&||&&\downarrow^{\gamma}&\\\
\mathbb{F}_{p}[[\mathbf{G}]]^{r}&\to&\mathbb{F}_{p}[[\mathbf{G}]]^{d}&\to&\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}M&\to
0.\end{array}.$
Since $\otimes$ and $\widehat{\otimes}$ are right exact (see [57, Theorem
2.6.3] and [56, Proposition 5.5.3(a)]) the horizontal sequences are exact.
This clearly implies that $\gamma$ is an isomorphism. ∎
Recall that if a pro-$p$ group $\mathbf{G}$ is finitely generated, then the
augmentation ideal $I_{\mathbf{G}}$ is finitely generated as an
$\mathbb{F}_{p}[[\mathbf{G}]]$-module. In particular, the trivial
$\mathbb{F}_{p}[[\mathbf{G}]]$-module $\mathbb{F}_{p}$ is finitely presented.
We can extend this to all finite discrete
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules.
###### Lemma 2.8.
Let $\mathbf{G}$ be a finitely generated pro-$p$ group and let $M$ be a finite
discrete $\mathbb{F}_{p}[[\mathbf{G}]]$-module. Then $M$ is finitely presented
as an $\mathbb{F}_{p}[[G]]$-module.
###### Proof.
We have to show that every open submodule of a finitely generated profinite
$\mathbb{F}_{p}[[\mathbf{G}]]$-module $M$ is finitely generated. Arguing by
induction on the index of the open submodule, we easily obtain this statement
from the fact that $I_{\mathbf{G}}$ is finitely generated. ∎
### 2.4. Left ideals in completed group algebras
Now we apply the results of the previous subsection in a particular situation
that interests us.
Let $\mathbf{G}$ be a pro-$p$ group. The definition of free profinite
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules can be found in [56, Section 5.2]. We
will need the the following fact.
###### Lemma 2.9.
Let $\mathbf{G}$ be a pro-$p$ group and $\mathbf{H}$ a closed subgroup of
$\mathbf{G}$. Then $\mathbb{F}_{p}[[\mathbf{G}]]$ is a free profinite
$\mathbb{F}_{p}[[\mathbf{H}]]$-module. In particular the functor
$\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}-$
is exact.
###### Proof.
The first part follows from [56, Corollary 5.7.2] and the second one from [56,
Proposition 5.5.3 (e)]. ∎
We denote by $I_{\mathbf{G}}$ the augmentation ideal of
$\mathbb{F}_{p}[[\mathbf{G}]]$. The following lemma is well-known. We provide
a proof for the convenience of the reader.
###### Lemma 2.10.
Let $\mathbf{H}$ be a finitely presented pro-$p$ group. Then $I_{\mathbf{H}}$
is finitely presented as a $\mathbb{F}_{p}[[\mathbf{H}]]$-module.
###### Proof.
Since $\mathbb{F}_{p}[[\mathbf{H}]]$ is a local ring, a profinite
$\mathbb{F}_{p}[[\mathbf{H}]]$-module $M$ is finitely presented if and only if
$H_{0}(\mathbf{H};M)$ and $H_{1}(\mathbf{H},M)$ are finite. Since $\mathbf{H}$
is finitely presented as a pro-$p$ group, [56, Theorem 7.8.1 and Theorem
7.8.3] imply that $H_{1}(\mathbf{H};\mathbb{F}_{p})$ and
$H_{2}(\mathbf{H};\mathbb{F}_{p})$ are finite. However, we have that for
$i\geq 0$,
$H_{i}(\mathbf{H},I_{\mathbf{H}})=H_{i+1}(\mathbf{H};\mathbb{F}_{p})$. Thus,
$I_{\mathbf{H}}$ is finitely presented as a
$\mathbb{F}_{p}[[\mathbf{H}]]$-module. ∎
If $\mathbf{H}$ is a closed subgroup of $\mathbf{G}$, then
$I_{\mathbf{H}}^{\mathbf{G}}$ denotes the closed left ideal of
$\mathbb{F}_{p}[[\mathbf{G}]]$ generated by $I_{\mathbf{H}}$.
###### Lemma 2.11.
Let $\mathbf{G}$ be a pro-$p$ group and let $\mathbf{H}\leq\mathbf{T}$ be
closed subgroups of $\mathbf{G}$. Then the following holds.
1. (a)
The continuous map
$\displaystyle\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}\to
I_{\mathbf{H}}^{\mathbf{G}}$
that sends $a\otimes b$ to $ab$, is an isomorphism of
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules.
2. (b)
If $\mathbf{H}$ is finitely presented, the map
$\displaystyle\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}\to
I_{\mathbf{H}}^{\mathbf{G}}$
that sends $a\otimes b$ to $ab$, is an isomorphism of
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules.
3. (c)
If $\mathbf{T}$ is finitely presented and $\mathbf{H}$ is finitely generated,
the map
$\displaystyle\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{T}]]}(I_{\mathbf{T}}/I_{\mathbf{H}}^{\mathbf{T}})\to
I_{\mathbf{T}}^{\mathbf{G}}/I_{\mathbf{H}}^{\mathbf{G}}$
sending $a\otimes(b+I_{\mathbf{H}}^{\mathbf{T}})$ to
$ab+I_{\mathbf{H}}^{\mathbf{G}}$, is an isomorphism of
$\mathbb{F}_{p}[[\mathbf{G}]]$-modules.
###### Proof.
(a) Consider an exact sequence
$0\to I_{\mathbf{H}}\to\mathbb{F}_{p}[[\mathbf{H}]]\to\mathbb{F}_{p}\to 0.$
By Lemma 2.9, the sequence
$0\to\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}\xrightarrow{\alpha}\mathbb{F}_{p}[[\mathbf{G}]]\xrightarrow{\beta}\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}\mathbb{F}_{p}\to
0$
is also exact. Thus,
$\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}$
is isomorphic to $\ker\beta=I_{\mathbf{H}}^{\mathbf{G}}$.
(b) Since $\mathbf{H}$ is finitely presented, by Lemma 2.10, $I_{\mathbf{H}}$
is finitely presented as $\mathbb{F}_{p}[[\mathbf{H}]]$-module, and so, by
Lemma 2.6,
$\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}\cong\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}.$
Thus we conclude that the natural map from
$\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}I_{\mathbf{H}}$
to $I_{\mathbf{H}}^{\mathbf{G}}$ is also an isomorphism by (a).
(c) Consider now the exact sequence
$0\to I_{\mathbf{H}}^{\mathbf{T}}\to I_{\mathbf{T}}\to
I_{\mathbf{T}}/I_{\mathbf{H}}^{\mathbf{T}}\to 0.$
Applying
$\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{T}]]}$
and taking again into account that $\mathbb{F}_{p}[[\mathbf{G}]]$ is a free
profinite $\mathbb{F}_{p}[[\mathbf{T}]]$-module (Lemma 2.10), we obtain the
exact sequence
$0\to I_{\mathbf{H}}^{\mathbf{G}}\to
I_{\mathbf{T}}^{\mathbf{G}}\to\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{T}]]}(I_{\mathbf{T}}/I_{\mathbf{H}}^{\mathbf{T}})\to
0.$
Since $\mathbf{T}$ is finitely presented and $\mathbf{H}$ is finitely
generated, by Lemma 2.10, $I_{\mathbf{T}}/I_{\mathbf{H}}^{\mathbf{T}}$ is
finitely presented as $\mathbb{F}_{p}[[\mathbf{T}]]$-module. Thus, by Lemma
2.6, the canonical map
$\mathbb{F}_{p}[[\mathbf{G}]]\otimes_{\mathbb{F}_{p}[[\mathbf{T}]]}(I_{\mathbf{T}}/I_{\mathbf{H}}^{\mathbf{T}})\to\mathbb{F}_{p}[[\mathbf{G}]]\widehat{\otimes}_{\mathbb{F}_{p}[[\mathbf{T}]]}(I_{\mathbf{T}}/I_{\mathbf{H}}^{\mathbf{T}})$
is an isomorphism. This proves the last claim. ∎
###### Corollary 2.12.
Let $\mathbf{H}$ be a finitely generated free subgroup of a pro-$p$ group
$\mathbf{G}$. Then $I_{\mathbf{H}}^{\mathbf{G}}$ is a free
$\mathbb{F}_{p}[[\mathbf{G}]]$-module of rank $d(\mathbf{H})$.
###### Proof.
By Proposition 2.3, $I_{\mathbf{H}}$ is a free
$\mathbb{F}_{p}[[\mathbf{H}]]$-module of rank $d(\mathbf{H})$. Therefore, by
Lemma 2.11(b), $I_{\mathbf{H}}^{\mathbf{G}}$ is a free
$\mathbb{F}_{p}[[\mathbf{G}]]$-module of rank $d(\mathbf{H})$. ∎
### 2.5. On amalgamated products of groups
Let $G$ be a group and $H_{1}$ and $H_{2}$ two subgroups that generate $G$ and
have intersection $A=H_{1}\cap H_{2}$. The following result gives an algebraic
condition for $G$ to be isomorphic to the amalgamated product of $H_{1}$ and
$H_{2}$ over $A$.
###### Proposition 2.13.
[40] Let $k$ be a non-trivial commutative ring. Then the canonical map
$H_{1}*_{A}H_{2}\to G$ is an isomorphism if and only if $I_{H_{1}}^{G}\cap
I_{H_{2}}^{G}=I_{A}^{G}$ in $k[G]$.
###### Proof.
This is [40, Theorem 1], which is proved for $k=\mathbb{Z}$ but the proof
works over an arbitrary nonzero commutative ring $k$. ∎
### 2.6. On convergence of Sylvester rank functions
Let $R$ be a ring. A Sylvester matrix rank function $\operatorname{rk}$ on $R$
is a function that assigns a non-negative real number to each matrix over $R$
and satisfies the following conditions.
1. (SMat1)
$\operatorname{rk}(M)=0$ if $M$ is any zero matrix and
$\operatorname{rk}(1)=1$ (where 1 denotes the identity matrix of size one);
2. (SMat2)
$\operatorname{rk}(M_{1}M_{2})\leq\min\\{\operatorname{rk}(M_{1}),\operatorname{rk}(M_{2})\\}$
for any matrices $M_{1}$ and $M_{2}$ which can be multiplied;
3. (SMat3)
$\operatorname{rk}\left(\begin{array}[]{cc}M_{1}&0\\\
0&M_{2}\end{array}\right)=\operatorname{rk}(M_{1})+\operatorname{rk}(M_{2})$
for any matrices $M_{1}$ and $M_{2}$;
4. (SMat4)
$\operatorname{rk}\left(\begin{array}[]{cc}M_{1}&M_{3}\\\
0&M_{2}\end{array}\right)\geq\operatorname{rk}(M_{1})+\operatorname{rk}(M_{2})$
for any matrices $M_{1}$, $M_{2}$ and $M_{3}$ of appropriate sizes.
We denote by $\mathbb{P}(R)$ the set of Sylvester matrix rank functions on
$R$, which is a compact convex subset of the space of functions on matrices
over $R$ considered with pointwise convergence.
Many problems can be reinterpreted in terms of convergence in $\mathbb{P}(R)$.
For example, if $G$ is group and $G\geq G_{1}\geq G_{2}\geq\ldots$ is a chain
of normal subgroups of $G$ of finite index with trivial intersection, then the
fact of the existence of the Lück approximation over $\mathbb{Q}$ and
$\mathbb{C}$ can be viewed as the convergence of $\operatorname{rk}_{G/G_{i}}$
to $\operatorname{rk}_{G}$ in $\mathbb{P}(\mathbb{Q}[G])$ and
$\mathbb{P}(\mathbb{C}[G])$ respectively (see [45, 30] for details).
An alternative way to introduce Sylvester rank functions is via Sylvester
module rank functions. A Sylvester module rank function $\dim$ on $R$ is a
function that assigns a non-negative real number to each finitely presented
$R$-module and satisfies the following conditions.
1. (SMod1)
$\dim\\{0\\}=0$, $\dim R=1$;
2. (SMod2)
$\dim(M_{1}\oplus M_{2})=\dim M_{1}+\dim M_{2}$;
3. (SMod3)
if $M_{1}\to M_{2}\to M_{3}\to 0$ is exact then
$\dim M_{1}+\dim M_{3}\geq\dim M_{2}\geq\dim M_{3}.$
By [50, Theorem 4] (see also [43, Proposition 1.2.8]), there exists a natural
bijection between Sylvester matrix and module rank functions over a ring.
Given a Sylvester module rank function $\dim$ on $R$ and a finitely presented
$R$-module $M\cong R^{n}/R^{m}A$ ($A$ is a matrix over $R$), we define the
corresponding Sylvester matrix rank function $\operatorname{rk}$ by means of
$\operatorname{rk}(A)=n-\dim M$.
By a recent result of Li [41], any Sylvester module rank function $\dim$ on
$R$ can be extended to a unique function (satisfying some natural conditions)
on arbitrary modules over $R$. We will call this extension, the extended
Sylvester module rank function and denote it also by $\dim$. In this paper we
will mostly use this extension for finitely generated modules $M$. In this
case $\dim M$ is defined as
(2) $\dim M=\inf\\{\dim\widetilde{M}:\ \widetilde{M}\textrm{\ is finitely
presented and\ }M\textrm{\ is a quotient of\ }\widetilde{M}\\}.$
In the case of an arbitrary $R$-module $M$ the formula for $\dim M$ is more
complex. In this case we put
$\dim M=\sup_{M_{1}}\inf_{M_{2}}(\dim M_{2}-\dim(M_{2}/M_{1})),$
where $M_{1}\leq M_{2}$ are finitely generated $R$-submodules of $M$. Observe
that we allow $+\infty$ to be a value of $\dim M$.
###### Remark 2.14.
If $\operatorname{rk}$, $\operatorname{rk}_{i}\in\mathbb{P}(R)$
$(i\in\mathbb{N})$ are Sylvester matrix rank functions corresponding to
Sylvester module rank functions $\dim$, $\dim_{i}$, respectively, then
$\operatorname{rk}=\displaystyle\lim_{i\to\infty}\operatorname{rk}_{i}$ in the
space $\mathbb{P}(R)$ if and only if for any finitely presented $R$-module
$M$, $\dim M=\displaystyle\lim_{i\to\infty}\dim_{i}M$.
However, the existence of the limit
$\operatorname{rk}=\displaystyle\lim_{i\to\infty}\operatorname{rk}_{i}$ does
not imply that $\dim M=\displaystyle\lim_{i\to\infty}\dim_{i}M$ for any
finitely generated $R$-module $M$. This phenomenon is well-known. For example,
it explains why the Lück approximation of the first $L^{2}$-Betti numbers is
valid for finitely presented groups but not always valid for finitely
generated groups (see [46]).
If $M$ is a finitely generated $R$-module, we only always have that
(3) $\dim M\geq\limsup_{i\to\infty}\dim_{i}M.$
Indeed, let $\mathcal{F}$ be the set of all finitely presented $R$-modules
$\tilde{M}$ such that $M$ is a quotient of $\tilde{M}$. Then
$\dim
M=\inf_{\tilde{M}\in\mathcal{F}}\dim\tilde{M}=\inf_{\tilde{M}\in\mathcal{F}}\lim_{i\to\infty}\dim_{i}\tilde{M}\geq\\\
\limsup_{i\to\infty}\inf_{\tilde{M}\in\mathcal{F}}\dim_{i}\tilde{M}=\limsup_{i\to\infty}\dim_{i}M.$
In this subsection we will explain how to overcome this problem in some
situations. For two Sylvester rank functions $\operatorname{rk}_{1}$ and
$\operatorname{rk}_{2}\in\mathbb{P}(R)$ we write
$\operatorname{rk}_{1}\geq\operatorname{rk}_{2}$ if
$\operatorname{rk}_{1}(A)\geq\operatorname{rk}_{2}(A)$ for any matrix $A$ over
$R$. If $\dim_{1}$ and $\dim_{2}$ are the Sylvester module rank functions on
$R$ corresponding to $\operatorname{rk}_{1}$ and $\operatorname{rk}_{2}$, then
the condition $\operatorname{rk}_{1}\geq\operatorname{rk}_{2}$ is equivalent
to the condition $\dim_{1}\leq\dim_{2}$, meaning that $\dim_{1}M\leq\dim_{2}M$
for any finitely presented $R$-module $M$.
###### Proposition 2.15.
Let $R$ be a ring and let $\operatorname{rk}$,
$\operatorname{rk}_{i}\in\mathbb{P}(R)$ $(i\in\mathbb{N})$ be Sylvester matrix
rank functions on $R$ corresponding to (extended) Sylvester module rank
functions $\dim$, $\dim_{i}$ respectively. Assume that
$\operatorname{rk}=\displaystyle\lim_{i\to\infty}\operatorname{rk}_{i}$ and
for all $i$, $\operatorname{rk}\geq\operatorname{rk}_{i}$. Then, for any
finitely generated $R$-module $M$,
$\dim M=\displaystyle\lim_{i\to\infty}\dim_{i}M.$
###### Proof.
Fix $\varepsilon>0$. Let $k$ be such that
$\liminf_{i\to\infty}\dim_{i}M\geq\dim_{k}M-\varepsilon.$
There exists a finitely presented $R$-module $\tilde{M}$ such that $M$ is a
quotient of $\tilde{M}$ and $\dim_{k}M\geq\dim_{k}\tilde{M}-\varepsilon$.
Since $\operatorname{rk}\geq\operatorname{rk}_{k}$, we have that
$\dim_{k}\tilde{M}\geq\dim\tilde{M}$. Thus, we obtain
$\liminf_{i\to\infty}\dim_{i}M\geq\dim_{k}M-\varepsilon\geq\dim_{k}\tilde{M}-2\varepsilon\geq\dim\tilde{M}-2\varepsilon\geq\dim
M-2\varepsilon.$
Since $\varepsilon$ is arbitrary, we conclude that
$\displaystyle\liminf_{i\to\infty}\dim_{i}M\geq\dim M$. In view of (3), we are
done. ∎
### 2.7. Epic division $R$-rings
Let $R$ be a ring. An epic division $R$-ring is a $R$-ring
$\phi:R\to\operatorname{\mathcal{D}}$ where $\operatorname{\mathcal{D}}$ is a
division ring generated by $\phi(R)$. Moreover, we say that
$\operatorname{\mathcal{D}}$ is a division $R$-ring of fractions if $\phi$ is
injective. In this case we will normally omit $\phi$ and see $R$ as a subring
of $\operatorname{\mathcal{D}}$.
Each epic division $R$-ring $\operatorname{\mathcal{D}}$ induces a Sylvester
module rank function $\dim_{\operatorname{\mathcal{D}}}$ on $R$: for every a
finitely presented $R$-module $M$ we define
$\dim_{\operatorname{\mathcal{D}}}M$ to be equal to the dimension of
$\operatorname{\mathcal{D}}\otimes_{R}M$ as a
$\operatorname{\mathcal{D}}$-module.
The extended Sylvester rank function $\dim_{\operatorname{\mathcal{D}}}$ is
calculated in the same way: for an $R$-module $M$,
$\dim_{\operatorname{\mathcal{D}}}M$ is equal to the dimension of
$\operatorname{\mathcal{D}}\otimes_{R}M$ as a
$\operatorname{\mathcal{D}}$-module.
We will use $\dim_{\operatorname{\mathcal{D}}}$ for the (extended) Sylvester
module rank function on $R$ and for the $\operatorname{\mathcal{D}}$-dimension
of $\operatorname{\mathcal{D}}$-spaces. This is a coherent notation because,
since $\operatorname{\mathcal{D}}$ is epic,
$\operatorname{\mathcal{D}}\otimes_{R}\operatorname{\mathcal{D}}$ is
isomorphic to $\operatorname{\mathcal{D}}$ as $R$-bimodule (see [31, Section
4]).
The following result of P. M. Cohn will be used several times in the paper.
###### Proposition 2.16.
[18, Theorem 4.4.1] Two epic division $R$-rings
$\operatorname{\mathcal{D}}_{1}$ and $\operatorname{\mathcal{D}}_{2}$ are
isomorphic as $R$-rings if and only if
$\dim_{\operatorname{\mathcal{D}}_{1}}M=\dim_{\operatorname{\mathcal{D}}_{2}}M$
for every finitely presented $R$-module $M$.
### 2.8. Natural extensions of Sylvester rank functions
Let $G$ be a group with trivial element $e$. We say that a ring $R$ is
$G$-graded if $R$ is equal to the direct sum $\oplus_{g\in G}R_{g}$ and
$R_{g}R_{h}\subseteq R_{gh}$ for all $g$ and $h$ in $G$. If for each $g\in G$,
$R_{g}$ contains an invertible element $u_{g}$, then we say that $R$ is a
crossed product of $R_{e}$ and $G$ and we will write $R=S*G$ if $R_{e}=S$.
Let $R=S*G$ be a crossed product. Let $\operatorname{rk}$ be a Sylvester
matrix rank function on $S$ and $\dim$ its associated Sylvester module rank
function. We say that $\operatorname{rk}$ (and $\dim$) are $R$-compatible if
for any $g\in G$ and any matrix $A$ over $S$,
$\operatorname{rk}(A)=\operatorname{rk}(u_{g}Au_{g}^{-1})$. If $G$ is finite
and $M$ is a finitely presented $R$-module, then $M$ is also a finitely
presented $S$-module. Thus, if $\dim$ is $R$-compatible, we can define
(4) $\widetilde{\dim}\ M=\frac{\dim M}{|G|},$
where $M$ is a finitely presented $R$-module. This defines a Sylvester module
rank function on $R$, called the natural extension of $\dim$. This notion was
studied, for example, in [36]. We notice that the same formula (4) holds also
for extended Sylvester module rank functions (that is, when $M$ is an
arbitrary $R$-module). In this subsection we prove the following result.
###### Proposition 2.17.
Let $R=S*G$ be a crossed product with $G$ finite and let
$R\hookrightarrow\operatorname{\mathcal{D}}$ be a division $R$-ring of
fractions. Denote by $\operatorname{\mathcal{D}}_{e}$ the division closure of
$S$ in $\operatorname{\mathcal{D}}$. Then the following are equivalent.
1. (a)
$\widetilde{\dim_{\operatorname{\mathcal{D}}_{e}}}=\dim_{\operatorname{\mathcal{D}}}$
as Sylvester functions on $R$.
2. (b)
$\dim_{\operatorname{\mathcal{D}}_{e}}\operatorname{\mathcal{D}}=|G|$.
3. (c)
$\operatorname{\mathcal{D}}$ is isomorphic to a crossed product
$\operatorname{\mathcal{D}}_{e}*G$.
4. (d)
$\operatorname{\mathcal{D}}$ is isomorphic to
$\operatorname{\mathcal{D}}_{e}\otimes_{S}R$ as
$(\operatorname{\mathcal{D}}_{e},R)$-bimodule.
5. (e)
$\operatorname{\mathcal{D}}$ is isomorphic to
$R\otimes_{S}\operatorname{\mathcal{D}}_{e}$ as
$(R,\operatorname{\mathcal{D}}_{e})$-bimodule.
###### Proof.
For any $h,g\in G$, define $\alpha(g,h)=u_{g}u_{h}(u_{gh})^{-1}\in R$. Since
the conjugation by $u_{g}$ fixes $S$, it also fixes
$\operatorname{\mathcal{D}}_{e}$. Therefore, we can define a ring structure on
$T=\oplus_{g\in G}\operatorname{\mathcal{D}}_{e}v_{g}$ defining the
multiplication on homogenous elements by means of
$(d_{1}v_{g})\cdot(d_{2}v_{h})=(d_{1}u_{g}d_{2}(u_{g})^{-1}\alpha(g,h))v_{gh},\
d_{1},d_{2}\in\operatorname{\mathcal{D}}_{e},\ g,h\in G.$
It is clear that $T=\operatorname{\mathcal{D}}_{e}*G$ is a crossed product, it
contains $R$ as a subring, and $T$ is isomorphic to
$\operatorname{\mathcal{D}}_{e}\otimes_{S}R$ as
$(\operatorname{\mathcal{D}}_{e},R)$-bimodule.
There exists a natural map $\gamma:T\to\operatorname{\mathcal{D}}$, sending
$\sum_{g\in G}d_{g}v_{g}$ ($d_{g}\in D_{e}$) to $\sum_{g\in G}d_{g}u_{g}$.
Observe that $\gamma(T)$ is a domain and of finite dimension over
$\operatorname{\mathcal{D}}_{e}$. Thus, $\gamma(T)$ is a division subring of
$\operatorname{\mathcal{D}}$. Since $T$ contains $R$,
$\operatorname{\mathcal{D}}=\gamma(T)$. This implies that (c) and (d) are
equivalent.
Since $\dim_{D_{e}}T=|G|$, (b) implies that $\gamma$ is an isomorphism, and
so, (b) implies (c).
Now, let us assume (d). Let $M$ be a finitely presented $R$-module. We have
the following.
$\widetilde{\dim_{\operatorname{\mathcal{D}}_{e}}}\
M=\frac{\dim_{\operatorname{\mathcal{D}}_{e}}(\operatorname{\mathcal{D}}_{e}\otimes_{S}M)}{|G|}=\frac{\dim_{\operatorname{\mathcal{D}}_{e}}(\operatorname{\mathcal{D}}_{e}\otimes_{S}(R\otimes_{R}M))}{|G|}=\\\
\frac{\dim_{\operatorname{\mathcal{D}}_{e}}((\operatorname{\mathcal{D}}_{e}\otimes_{S}R)\otimes_{R}M))}{|G|}\stackrel{{\scriptstyle\text{(d)}}}{{=}}\frac{\dim_{\operatorname{\mathcal{D}}_{e}}(\operatorname{\mathcal{D}}\otimes_{R}M)}{|G|}=\\\
\dim_{\operatorname{\mathcal{D}}}(\operatorname{\mathcal{D}}\otimes_{R}M)=\dim_{\operatorname{\mathcal{D}}}M.$
This proves (a).
Now, we assume that (a) holds. Since $\operatorname{\mathcal{D}}_{e}$ es an
epic $S$-ring
$\operatorname{\mathcal{D}}_{e}\otimes_{S}\operatorname{\mathcal{D}}_{e}$ is
isomorphic to $\operatorname{\mathcal{D}}_{e}$ as
$\operatorname{\mathcal{D}}_{e}$-bimodule and by the same reason,
$\operatorname{\mathcal{D}}\otimes_{R}\operatorname{\mathcal{D}}$ is
isomorphic to $\operatorname{\mathcal{D}}$ as
$\operatorname{\mathcal{D}}$-bimodule. Consider $M=\operatorname{\mathcal{D}}$
as an $R$-module and $N=\operatorname{\mathcal{D}}_{e}$ as a $S$-module. Then
$1=\dim_{\operatorname{\mathcal{D}}}(\operatorname{\mathcal{D}}\otimes_{R}M)=\dim_{\operatorname{\mathcal{D}}}M\stackrel{{\scriptstyle\text{(a)}}}{{=}}\widetilde{\dim_{\operatorname{\mathcal{D}}_{e}}}M=\\\
\frac{\dim_{\operatorname{\mathcal{D}}_{e}}(\operatorname{\mathcal{D}}_{e}\otimes_{S}M)}{|G|}=\frac{\dim_{\operatorname{\mathcal{D}}_{e}}(\operatorname{\mathcal{D}}_{e}\otimes_{S}(N^{\dim_{\operatorname{\mathcal{D}}_{e}}\operatorname{\mathcal{D}}}))}{|G|}=\frac{\dim_{\operatorname{\mathcal{D}}_{e}}\operatorname{\mathcal{D}}}{|G|}$
This implies (b).
Let $R^{op}$ denotes the opposite ring, that is the ring with the same
elements and addition operation, but with the multiplication performed in the
reverse order. Then $R^{op}\cong S^{op}*G$ and the condition (c) is equivalent
to
1. (c’)
$\operatorname{\mathcal{D}}^{op}$ is isomorphic to a crossed product
$(\operatorname{\mathcal{D}}_{e})^{op}*G$.
Our previous proof gives that (c’) is equivalent to
1. (d’)
$\operatorname{\mathcal{D}}^{op}$ is isomorphic to
$(\operatorname{\mathcal{D}}_{e})^{op}\otimes_{S^{op}}R^{op}$ as
$((\operatorname{\mathcal{D}}_{e})^{op},R^{op})$-bimodule.
Now, observe that (d’) is equivalent to (e). ∎
## 3\. On mod-$p$ $L^{2}$-Betti numbers of subgroups of a free pro-$p$ groups
### 3.1. Universal division ring of fractions
Given two epic division $R$-rings $\operatorname{\mathcal{D}}_{1}$ and
$\operatorname{\mathcal{D}}_{2}$ the condition
$\dim_{\operatorname{\mathcal{D}}_{1}}\leq\dim_{\operatorname{\mathcal{D}}_{2}}$
is equivalent to the existence of a specialization from
$\operatorname{\mathcal{D}}_{1}$ to $\operatorname{\mathcal{D}}_{2}$ in the
sense of P. Cohn ([18, Subsection 4.1]). We say that an epic division $R$-ring
$\operatorname{\mathcal{D}}$ is universal if for every division $R$-ring
$\mathcal{E}$, $\dim_{\operatorname{\mathcal{D}}}\leq\dim_{\mathcal{E}}$. If a
universal epic division $R$-ring exists, it is unique up to $R$-isomorphism.
We will denote it by $\operatorname{\mathcal{D}}_{R}$ and instead of
$\dim_{\operatorname{\mathcal{D}}_{R}}$ we will simply write $\dim_{R}$.
We say that a ring $R$ is a semifir if every finitely generated left ideal of
$R$ is free and free modules of distinct finite rank are non-isomorphic. For
example, if $K$ is a field, the ring $K\langle\\!\langle X\rangle\\!\rangle$
of non-commutative power series is a semifir ([17, Theorem 2.9.4]). By a
theorem of P. M. Cohn [16] a semifir $R$ has a universal division $R$-ring. P.
M. Cohn proved that in this case $\operatorname{\mathcal{D}}_{R}$ can be
obtained from $R$ by formally inverting all full matrices over $R$. In
particular, this implies the following result.
###### Proposition 3.1.
Let $R$ be a semifir. Then $\dim_{R}$ is the smallest Sylvester module rank
function among all the Sylvester module rank functions on $R$.
We will need the following result.
###### Proposition 3.2.
[32, Proposition 2.2] Let $R$ be a semifir. Then
$\operatorname{Tor}^{R}_{1}(\operatorname{\mathcal{D}}_{R},M)=0$
for any $R$-submodule of a $\operatorname{\mathcal{D}}_{R}$-module.
Let $G$ be a residually torsion-free nilpotent group (for example, $G$ is a
subgroup of a free pro-$p$ group). Let $K$ be a field. Then the universal
division ring of fractions $\operatorname{\mathcal{D}}_{K[G]}$ exists (see
[33]). It can be constructed in the following way. Since $G$ is residually
torsion free nilpotent, $G$ is bi-orderable. Fix a bi-invariant order
$\preceq$ on $G$. A. Malcev [49] and B. Neumann [54] (following an idea of H.
Hahn [24]) showed independently that the set $K((G,\preceq))$ of formal power
series over $G$ with coefficients in $K$ having well-ordered support has a
natural structure of a ring and, moreover, it is a division ring.
$\operatorname{\mathcal{D}}_{K[G]}$ can be defined as the division closure of
$K[G]$ in $K((G,\preceq))$. The universality of this division ring is shown in
[33, Theorem 1.1].
If $A$ is a torsion-free abelian group, then
$\operatorname{\mathcal{D}}_{K[A]}$ coincides with the classical ring of
fractions $\mathcal{Q}(K[A])$ of $K[A]$.
### 3.2. The division ring
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$
If $\mathbf{F}$ is a free pro-$p$ group freely generated by
$f_{1},\ldots,f_{d}$, then, by Proposition 2.3, the continuous homomorphism
$\mathbb{F}_{p}\langle\\!\langle
x_{1},\ldots,x_{d}\rangle\\!\rangle\to\mathbb{F}_{p}[[\mathbf{F}]]$ that sends
$x_{i}$ to $f_{1}-1$, is an isomorphism. Thus, there exists a universal
division ring of fraction
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$. Using results of
[32] we establish the following formula for
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}$ which is one of main ingredients of the
proof of Theorem 1.2.
###### Proposition 3.3.
Let $\mathbf{F}=\mathbf{N}_{1}>\mathbf{N}_{2}>\mathbf{N}_{3}>\ldots$ be a
chain of open normal subgroups of a finitely generated free pro-$p$ group
$\mathbf{F}$ with trivial intersection. Let $M$ be a finitely generated
$\mathbb{F}_{p}[[\mathbf{F}]]$-module. Then
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}M=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M)}{|\mathbf{F}:\mathbf{N}_{i}|}.$
###### Proof.
Let $\mathbf{N}$ be a normal open subgroup of $\mathbf{F}$ and let
$\dim_{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]}$ be a Sylvester module rank
function on $\mathbb{F}_{p}[[\mathbf{F}]]$ defined by
$\dim_{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]}L=\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}L)}{|\mathbf{F}:\mathbf{N}|},$
where $L$ is a finitely presented $\mathbb{F}_{p}[[\mathbf{F}]]$-module.
Let $M=\mathbb{F}_{p}[[\mathbf{F}]]^{n}/I$ be a finitely generated
$\mathbb{F}_{p}[[\mathbf{F}]]$-module. Since
${\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M\cong\mathbb{F}[[\mathbf{G}]]^{n}/(I+(I_{\mathbf{N}}^{\mathbf{G}})^{n})$
is finite, it is finitely presented by Lemma 2.10. Therefore, there exists a
finitely generated $\mathbb{F}_{p}[[\mathbf{G}]]$-submodule $J$ of $I$ such
that $I+(I_{N}^{G})^{n}=J+(I_{N}^{G})^{n}$. Put
$\widetilde{M}=\mathbb{F}_{p}[[\mathbf{F}]]^{n}/J$. Then $\widetilde{M}$ is a
finitely presented $\mathbb{F}_{p}[[\mathbf{F}]]$-module $\widetilde{M}$
satisfying the following conditions:
1. (1)
$M$ is a quotient of $\widetilde{M}$ and
2. (2)
${\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}\widetilde{M}\cong{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M.$
Therefore, from (2) we obtain that
$\dim_{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]}M=\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}\widetilde{M})}{|\mathbf{F}:\mathbf{N}|}=\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M)}{|\mathbf{F}:\mathbf{N}|}.$
In the case where $M$ is finitely presented, [32, Theorem 1.4] and Remark 2.14
impliy that
(5)
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}M=\lim_{i\to\infty}\dim_{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]}M.$
By Proposition 3.1, we also have that
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}M\leq\dim_{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]}M$
for all $i$. Therefore, Proposition 2.15 implies that (5) holds also when $M$
is finitely generated. Hence,
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}M=\lim_{i\to\infty}\dim_{\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]}M=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M)}{|\mathbf{F}:\mathbf{N}_{i}|}.$
∎
In the following proposition we collect some basic properties of
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$.
###### Proposition 3.4.
Let $\mathbf{H}$ be a finitely generated closed subgroup of $\mathbf{F}$. The
following holds.
1. (a)
Let $\operatorname{\mathcal{D}}_{\mathbf{H}}$ be the division closure of
$\mathbb{F}_{p}[[\mathbf{H}]]$ in
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$. Then
$\operatorname{\mathcal{D}}_{\mathbf{H}}$ is isomorphic to
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$ as an
$\mathbb{F}_{p}[[\mathbf{H}]]$-ring.
2. (b)
If $M$ is a finitely generated $\mathbb{F}_{p}[[\mathbf{H}]]$-module, then
$\dim_{\mathbb{F}_{p}[[\mathbf{H}]]}(M)=\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M).$
3. (c)
If $\mathbf{H}$ is open then,
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\cong\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$
as $(\mathbb{F}_{p}[[\mathbf{F}]],\mathbb{F}_{p}[[\mathbf{H}]])$-bimodules.
###### Proof.
(a) Fix a normal chain
$\mathbf{F}=\mathbf{N}_{1}>\mathbf{N}_{2}>\mathbf{N}_{3}>\ldots$ of open
normal subgroups of $\mathbf{F}$, and let
$\mathbf{H}_{i}=\mathbf{N}_{i}\cap\mathbf{H}$. Let $M$ be a finitely generated
$\mathbb{F}_{p}[[\mathbf{H}]]$-module. Observe first, that by Proposition 3.3,
(6)
$\dim_{\mathbb{F}_{p}[[\mathbf{H}]]}M=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{H}/\mathbf{H}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)}{|\mathbf{H}:\mathbf{H}_{i}|}.$
Considering $\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]$ as a right
$\mathbb{F}_{p}[[\mathbf{H}]]$-module, we obtain that
$\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\cong\mathbb{F}_{p}[\mathbf{H}/\mathbf{H}_{i}]^{|\mathbf{F}:\mathbf{N}_{i}\mathbf{H}|}$
as right $\mathbb{F}_{p}[[\mathbf{H}]]$-modules. Thus,
(7)
$\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{H}/\mathbf{H}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)=\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)}{|\mathbf{F}:\mathbf{N}_{i}\mathbf{H}|}.$
Therefore, from (6), (7) and Proposition 3.3, we conclude that
(8)
$\dim_{\mathbb{F}_{p}[[\mathbf{H}]]}M=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)}{|\mathbf{F}:\mathbf{N}_{i}|}=\\\
\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M).$
On the other hand, we have
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)=\\\
\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M))=\\\
\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M))=\dim_{\operatorname{\mathcal{D}}_{\mathbf{H}}}(\operatorname{\mathcal{D}}_{\mathbf{H}}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M).$
Thus, we conclude that
$\dim_{\mathbb{F}_{p}[[\mathbf{H}]]}M=\dim_{\operatorname{\mathcal{D}}_{\mathbf{H}}}(\operatorname{\mathcal{D}}_{\mathbf{H}}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M),$
and so, by Proposition 2.16, $\operatorname{\mathcal{D}}_{\mathbf{H}}$ is
isomorphic to $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$ as
an $\mathbb{F}_{p}[[\mathbf{H}]]$-ring.
(b) This is the equality (8).
(c) First assume that $\mathbf{H}$ is normal in $\mathbf{F}$. Observe that for
large $i$, $\mathbf{N}_{i}\leq\mathbf{H}$. Let $M$ be a finitely presented
$\mathbb{F}_{p}[[F]]$-module. Then we obtain that
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}M=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[[\mathbf{N}_{i}]]}M)}{|\mathbf{F}:\mathbf{N}_{i}|}=\\\
\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[[\mathbf{N}_{i}]]}M)}{|\mathbf{F}:\mathbf{H}||\mathbf{H}:\mathbf{N}_{i}|}=\frac{\dim_{\mathbb{F}_{p}[[\mathbf{H}]]}M}{|\mathbf{F}:\mathbf{H}|}.$
Therefore,
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}=\widetilde{\dim_{\mathbb{F}_{p}[[\mathbf{H}]]}}$.
Now, the result follows from Proposition 2.17.
Now we assume that $\mathbf{H}$ is arbitrary, We argue by induction on
$|\mathbf{F}:\mathbf{H}|$. If $\mathbf{H}$ has index $p$ in $\mathbf{F}$, then
it is normal. If $|\mathbf{F}:\mathbf{H}|>p$, we find $\mathbf{H}_{1}$ of
index $p$ in $\mathbf{F}$ containing $\mathbf{H}$. Then by induction,
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\cong\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}_{1}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}_{1}]]}\cong\\\
\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}_{1}]]}(\mathbb{F}_{p}[[\mathbf{H}_{1}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]})\cong\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}.$
∎
In view of the previous proposition, we will identify
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$ and the division
closure of $\mathbb{F}_{p}[[\mathbf{H}]]$ in
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$, and see
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$ as a subring of
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$.
### 3.3. The division rings
$\operatorname{\mathcal{D}}(\mathbb{F}_{p}[G],\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]})$
Let $G$ be a subgroup of $\mathbf{F}$. As we have explained in Subsection 3.1
there exists the universal division $\mathbb{F}_{p}[G]$-ring of fractions
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$. Let
$\operatorname{\mathcal{D}}_{G}=\operatorname{\mathcal{D}}(\mathbb{F}_{p}[G],\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]})$
be the division closure of $\mathbb{F}_{p}[G]$ in
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$. In this
subsection we will show that $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$
and $\operatorname{\mathcal{D}}_{G}$ are isomorphic as
$\mathbb{F}_{p}[G]$-rings. In the case $G=F$ is a finitely generated free
group and $\mathbf{F}$ is the pro-$p$ completion of $F$, this result follows
from [17, Corollary 2.9.16].
###### Proposition 3.5.
Let $\mathbf{F}$ be a finitely generated free pro-$p$ group and let $G$ be a
finitely generated subgroup of $\mathbf{F}$. Let
$\mathbf{F}=\mathbf{N}_{1}>\mathbf{N}_{2}>\mathbf{N}_{3}>\ldots$ be a chain of
open normal subgroups of $\mathbf{F}$ with trivial intersection. We put
$G_{j}=G\cap\mathbf{N}_{j}$. Let
$\operatorname{\mathcal{D}}_{G}=\operatorname{\mathcal{D}}(\mathbb{F}_{p}[G],\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]})$
be the division closure of $\mathbb{F}_{p}[G]$ in
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$. Then for every
finitely generated $\mathbb{F}_{p}[G]$-module $M$,
$\dim_{\operatorname{\mathcal{D}}_{G}}(\operatorname{\mathcal{D}}_{G}\otimes_{\mathbb{F}_{p}[G]}M)=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[G_{i}]}M)}{|G:G_{i}|}=\\\
\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}\otimes_{\mathbb{F}_{p}[G]}M).$
In particular, the divison rings $\operatorname{\mathcal{D}}_{G}$ and
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$ are isomorphic as
$\mathbb{F}_{p}[G]$-rings.
###### Proof.
Without loss of generality we assume that $G$ is dense in $\mathbf{F}$. First
observe that
(9)
$\dim_{\operatorname{\mathcal{D}}_{G}}(\operatorname{\mathcal{D}}_{G}\otimes_{\mathbb{F}_{p}[G]}M)=\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[G]}M)=\\\
\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[G]}M))=\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[G]}M).$
Observe also that, since $G$ is dense in $\mathbf{F}$,
$|\mathbf{F}:\mathbf{N}_{i}|=|G:G_{i}|$ and
$\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[G_{i}]}M\cong\mathbb{F}_{p}[G/G_{i}]\otimes_{\mathbb{F}_{p}[G]}M\cong\\\
\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\otimes_{\mathbb{F}_{p}[G]}M\cong\mathbb{F}_{p}[\mathbf{F}/\mathbf{N}_{i}]\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[G]}M)).$
Thus, Proposition 3.3 implies that
(10)
$\dim_{\operatorname{\mathcal{D}}_{G}}(\operatorname{\mathcal{D}}_{G}\otimes_{\mathbb{F}_{p}[G]}M)=\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[G_{i}]}M)}{|G:G_{i}|}.$
Let $\mathbf{F}_{i}=\gamma_{i}(\mathbf{F})$ and we put
$H_{i}=G\cap\mathbf{F}_{i}$. The ring $\mathbb{F}_{p}[G/H_{i}]$ is a
Noetherian domain and its classical field of fractions
$\mathcal{Q}(\mathbb{F}_{p}[G/H_{i}])$ is universal. Moreover, by [33, Theorem
1.2], we have that
(11)
$\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}\otimes_{\mathbb{F}_{p}[G]}M)=\lim_{i\to\infty}\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G/H_{i}]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G/H_{i}]}\otimes_{\mathbb{F}_{p}[G]}M).$
Observe that $\mathbb{F}_{p}[[\mathbf{F}/\mathbf{F}_{i}]]$ is also a
Noetherian domain, and so the division closure of $\mathbb{F}_{p}[G/H_{i}]$ in
$\mathcal{Q}(\mathbb{F}_{p}[[\mathbf{F}/\mathbf{F}_{i}]])$ is isomorphic to
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G/H_{i}]}$ (as a
$\mathbb{F}_{p}[G]$-ring). Therefore,
(12)
$\dim_{\mathcal{Q}(\mathbb{F}_{p}[[\mathbf{F}/\mathbf{F}_{i}]])}(\mathcal{Q}(\mathbb{F}_{p}[[\mathbf{F}/\mathbf{F}_{i}]])\otimes_{\mathbb{F}_{p}[G]}M)=\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G/H_{i}]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G/H_{i}]}\otimes_{\mathbb{F}_{p}[G]}M).$
Using Proposition 3.3 and arguing as in the proof of [29, Theorem 2.3], we
obtain that
(13)
$\dim_{\mathbb{F}_{p}[[\mathbf{F}]]}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[G]}M)=\lim_{i\to\infty}\dim_{\mathcal{Q}(\mathbb{F}_{p}[[\mathbf{F}/F_{i}]])}(\mathcal{Q}(\mathbb{F}_{p}[[\mathbf{F}/\mathbf{F}_{i}]])\otimes_{\mathbb{F}_{p}[G]}M).$
Therefore, putting together (11), (12), (13), (9) and (10), we obtain that
$\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}\otimes_{\mathbb{F}_{p}[G]}M)=\dim_{\operatorname{\mathcal{D}}_{G}}(\operatorname{\mathcal{D}}_{G}\otimes_{\mathbb{F}_{p}[G]}M)=\\\
\lim_{i\to\infty}\frac{\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[G_{i}]}M)}{|G:G_{i}|}.$
Applying Proposition 2.16, we obtain that the divison rings
$\operatorname{\mathcal{D}}_{G}$ and
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$ are isomorphic as
$\mathbb{F}_{p}[G]$-rings. ∎
An alternative approach of proving that $\operatorname{\mathcal{D}}_{G}$ is
isomorphic to $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$ as
$\mathbb{F}_{p}[G]$-ring is taken in [51, Lemma 7.5.5], where the result is
proved by using a variation from [58, Theorem 6.3] of the uniqueness of
Hughes-free division rings [27] (see also [19]).
### 3.4. Mod-$p$ $L^{2}$-Betti numbers
$L^{2}$-Betti numbers play an important role in the solution of many problems
in group theory. In the last years there was an attempt to develop a theory of
mod-$p$ $L^{2}$-Betti numbers for different families of groups (see [31]). If
$G$ is torsion-free and satisfies the Atiyah conjecture, P. Linnell [42]
showed that $L^{2}$-Betti numbers of $G$ can be defined as
$b_{i}^{(2)}(G)=\dim_{\operatorname{\mathcal{D}}(G)}H_{i}(G;\operatorname{\mathcal{D}}(G)),$
where $\operatorname{\mathcal{D}}(G)$ is the division ring obtained as the
division closure of $\mathbb{Q}[G]$ in the ring of affilated operators
$\mathcal{U}(G)$. It turns out that if $G$ is residually torsion-free
nilpotent, $\operatorname{\mathcal{D}}(G)$ is isomorphic to the universal
division ring of fractions $\operatorname{\mathcal{D}}_{\mathbb{Q}[G]}$ (see,
for example, [34]). Therefore, we have
$b_{i}^{(2)}(G)=\dim_{\operatorname{\mathcal{D}}_{\mathbb{Q}[G]}}H_{i}(G;\operatorname{\mathcal{D}}_{\mathbb{Q}[G]}).$
Thus, by analogy, if $G$ is a residually torsion-free nilpotent group, we
define the $i$th mod-$p$ $L^{2}$-Betti number of $G$ as
$\beta^{\operatorname{mod}p}_{i}(G)=\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}}H_{i}(G;\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}).$
In the case, where $G$ is a subgroup of a free pro-$p$ group, we also obtain
the following formula, which can be seen as a mod-$p$ analogue of the Lück
approximation theorem [45].
###### Proposition 3.6.
Let $\mathbf{F}$ be a finitely generated free pro-$p$ group and let $G$ be a
subgroup of $\mathbf{F}$ of type $FP_{k}$ for some $k\geq 1$. Let
$\mathbf{F}=\mathbf{N}_{1}>\mathbf{N}_{2}>\mathbf{N}_{3}>\ldots$ be a chain of
open normal subgroups of $\mathbf{F}$ with trivial intersection. We put
$G_{j}=G\cap\mathbf{N}_{j}$. Then
$\beta_{k}^{\operatorname{mod}p}(G)=\displaystyle\lim_{j\to\infty}\frac{\dim_{\mathbb{F}_{p}}H_{k}(G_{j};\mathbb{F}_{p})}{|G:G_{j}|}.$
###### Proof.
There exists a resolution of the $\mathbb{F}_{p}[G]$-module $\mathbb{F}_{p}$
$0\to
R_{k}\to\mathbb{F}_{p}[G]^{n_{k}}\xrightarrow{\phi_{k}}\ldots\to\mathbb{F}_{p}[G]^{n_{1}}\xrightarrow{\phi_{1}}\mathbb{F}_{p}\xrightarrow{\phi_{0}}0$
with $R_{k}$ finitely generated. The relevant part of the sequence for
calculation of $H_{k}(G;*)$ is the following exact sequence
$0\to R_{k}\to\mathbb{F}_{p}[G]^{n_{k}}\to R_{k-1}\to 0,$
($R_{k-1}=\operatorname{Im}\phi_{k}=\ker\phi_{k-1}$), because for any right
$\mathbb{F}_{p}[G]$-module $M$ we have
$0\to H_{k}(G;M)\to M\otimes_{\mathbb{F}_{p}[G]}R_{k}\to M^{n_{k}}\to
M\otimes_{\mathbb{F}_{p}[G]}R_{k-1}\to 0$
Then we obtain that
$\beta_{k}^{\operatorname{mod}p}(G)=\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}}H_{k}(G;\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]})=\dim_{\mathbb{F}_{p}[G]}R_{k}-{n_{k}}+\dim_{\mathbb{F}_{p}[G]}R_{k-1}\textrm{\
and\ }$
$\dim_{\mathbb{F}_{p}}H_{k}(G_{j};\mathbb{F}_{p})=\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[G_{j}]}R_{k})-n_{k}|G:G_{j}|+\dim_{\mathbb{F}_{p}}(\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[G_{j}]}R_{k-1}).$
Thus, Proposition 3.5 implies the proposition.∎
In this paper we will work only with $\beta_{1}^{\operatorname{mod}p}(G)$.
Observe that in this case, if $G$ is infinite, we have the formula
$\beta_{1}^{\operatorname{mod}p}(G)=\dim_{\mathbb{F}_{p}[G]}I_{G}-1.$
Also observe that if $A$ is a non-trivial torsion-free abelian group, then
since
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[A]}=\mathcal{Q}(\mathbb{F}_{p}[A])$
is the field of fractions of $\mathbb{F}_{p}[A]$,
(14)
$\beta_{1}^{\operatorname{mod}p}(A)=\dim_{\mathbb{F}_{p}[A]}I_{A}-1=\dim_{\mathcal{Q}(\mathbb{F}_{p}[A])}(\mathcal{Q}(\mathbb{F}_{p}[A])\otimes_{\mathbb{F}_{p}[A]}I_{A})-1=0$
### 3.5. Strong embeddings into free pro-$p$ groups
Assume that a finitely generated group $G$ is a subgroup of a free pro-$p$
group $\mathbf{F}$. Since a closed subgroup of a free pro-$p$ group is free
(see [56, Corollary 7.7.5]), changing $\mathbf{F}$ by the closure of $G$ in
$\mathbf{F}$, we may assume that $G$ is dense in $\mathbf{F}$. Let
$\mathbf{F}=\mathbf{N}_{1}>\mathbf{N}_{2}>\mathbf{N}_{3}>\ldots$ be a chain of
open normal subgroups of $\mathbf{F}$ with trivial intersection and put
$G_{j}=G\cap\mathbf{N}_{j}$. Observe that the closure of $G_{j}$ in
$\mathbf{F}$ is equal to $\mathbf{N}_{j}$, and so
(15)
$|G_{j}:G_{j}^{p}[G_{j},G_{j}]|\geq|\mathbf{N}_{j}:\mathbf{N}_{j}^{p}[\mathbf{N}_{j},\mathbf{N}_{j}]|.$
Denote by $d(\mathbf{F})$ the number of profinite generators of $\mathbf{F}$.
By [56, Theorem 7.8.1],
$d(\mathbf{F})=\log_{p}|\mathbf{F}:\mathbf{F}^{p}[\mathbf{F},\mathbf{F}]|$
and by the index Schreier formula
$d(\mathbf{N}_{j})=(d(\mathbf{F})-1)|\mathbf{F}:\mathbf{N}_{j}|+1.$
Therefore, we obtain
$\dim_{\mathbb{F}_{p}}H_{1}(G_{j};\mathbb{F}_{p})=\log_{p}|G_{j}:G_{j}^{p}[G_{j},G_{j}]|\geq\log_{p}|\mathbf{N}_{j}:\mathbf{N}_{j}^{p}[\mathbf{N}_{j},\mathbf{N}_{j}]|=\\\
d(\mathbf{N}_{j})=(d(\mathbf{F})-1)|\mathbf{F}:\mathbf{N}_{j}|+1=(d(\mathbf{F})-1)|G:G_{j}|+1.$
Thus, Proposition 3.6 implies the following corollary.
###### Corollary 3.7.
Let $G$ be a finitely generated dense subgroup of a free pro-$p$ group
$\mathbf{F}$. Then
(16) $\dim_{\mathbb{F}_{p}G}I_{G}=\beta_{1}^{\operatorname{mod}p}(G)+1\geq
d(\mathbf{F}).$
This result suggests the following definition.
###### Definition 3.8.
We say that an embedding $G\hookrightarrow\mathbf{F}$ of a finitely generated
group $G$ into a free pro-$p$ group $\mathbf{F}$ is strong if $G$ is dense in
$\mathbf{F}$ and
$\dim_{\mathbb{F}_{p}G}I_{G}=\beta_{1}^{\operatorname{mod}p}(G)+1=d(\mathbf{F})$.
A finitely generated group $G$ is called strongly embeddable in a free pro-$p$
group (SE($p$) for simplicity) if there are a free pro-$p$ group $\mathbf{F}$
and a strong embedding $G\hookrightarrow\mathbf{F}$.
Let $G$ be a parafree group. Observe that $G$ is residually-$p$ for every
prime $p$. Thus, if $G$ is finitely generated, its pro-$p$ completion
$G_{\widehat{p}}$ is a finitely generated free pro-$p$ group and $G$ is a
subgroup of $G_{\widehat{p}}$. In this case the inequality (15) is an
equality, and so in the same way as we obtained the inequality (16), we obtain
that $\beta_{1}^{\operatorname{mod}p}(G)=d(G_{\widehat{p}})-1$. Therefore, the
embedding $G\hookrightarrow G_{\widehat{p}}$ is strong. Thus, all finitely
generated parafree groups are SE($p$). On the other hand, not every subgroup
of a parafree group is SE($p$). For example, the fundamental group of an
oriented surface of genus greater than 1 is not SE($p$). However the
fundamental group of an oriented surface of genus greater than 2 can be
embedded in a parafree group (see [8, Section 4.1]).
By [13, Proposition 7.5], if $G$ is a finitely generated dense subgroup of a
finitely generated free pro-$p$ group $\mathbf{F}$, then $b_{1}^{(2)}(G)\geq
d(\mathbf{F})-1$. On the other hand, by [21, Theorem 1.6] and Proposition 3.6
, $b_{1}^{(2)}(G)\leq\beta_{1}^{\operatorname{mod}p}(G)$. Thus, if
$G\hookrightarrow\mathbf{F}$ is a strong embedding, we have
$b_{1}^{(2)}(G)=\beta_{1}^{\operatorname{mod}p}(G)=d(\mathbf{F})-1$.
## 4\. On $\mathcal{D}$-torsion-free modules.
### 4.1. General results
Let $R$ be a ring and let $R\hookrightarrow\mathcal{D}$ be an embedding of $R$
into a division ring $\mathcal{D}$. Let $M$ be a left $R$-module. We say that
$M$ is $\mathcal{D}$-torsion-free if the canonical map
$M\to\mathcal{D}\otimes_{R}M,\ m\mapsto 1\otimes m,$
is injective. The following lemma describes several equivalent definitions of
$\mathcal{D}$-torsion-free modules.
###### Lemma 4.1.
The following statements for a left $R$-module $M$ are equivalent.
1. (a)
$M$ is $\mathcal{D}$-torsion-free.
2. (b)
There are a $\operatorname{\mathcal{D}}$-module $N$ and an injective
homomorphism $\varphi:M\to N$ of $R$-modules.
3. (c)
For any $0\neq m\in M$, there exists a homomorphism of $R$-modules
$\varphi:M\to\operatorname{\mathcal{D}}$, such that $\varphi(m)\neq 0$.
###### Proof.
The proof is straightforward and we leave it to the reader.∎
Let $M$ be a left $R$-module. Recall that we use
$\dim_{\operatorname{\mathcal{D}}}M$ to denote the dimension of
$\operatorname{\mathcal{D}}\otimes_{R}M$ as a
$\operatorname{\mathcal{D}}$-module. Observe that if
$\dim_{\operatorname{\mathcal{D}}}M$ is finite, it is also equal to the
dimension of $\operatorname{Hom}_{R}(M,\operatorname{\mathcal{D}})$ as a right
$\mathcal{D}$-module.
###### Lemma 4.2.
Let $0\to M_{1}\to M_{2}\to M_{3}\to 0$ be an exact sequence of $R$-modules.
Assume that
1. (1)
$M_{1}$ and $M_{3}$ are $\operatorname{\mathcal{D}}$-torsion-free,
2. (2)
$\dim_{\operatorname{\mathcal{D}}}M_{1}$ and
$\dim_{\operatorname{\mathcal{D}}}M_{3}$ are finite and
3. (3)
$\dim_{\operatorname{\mathcal{D}}}M_{2}=\dim_{\operatorname{\mathcal{D}}}M_{1}+\dim_{\operatorname{\mathcal{D}}}M_{3}$.
Then $M_{2}$ is also $\operatorname{\mathcal{D}}$-torsion-free.
###### Proof.
We are going to use the third characterization of
$\operatorname{\mathcal{D}}$-torsion-free modules from Lemma 4.1. Consider the
following exact sequence of right $\mathcal{D}$-modules.
$0\to\operatorname{Hom}_{R}(M_{3},\operatorname{\mathcal{D}})\to\operatorname{Hom}_{R}(M_{2},\operatorname{\mathcal{D}})\to\operatorname{Hom}_{R}(M_{1},\operatorname{\mathcal{D}}).$
Since
$\dim_{\operatorname{\mathcal{D}}}M_{2}=\dim_{\operatorname{\mathcal{D}}}M_{1}+\dim_{\operatorname{\mathcal{D}}}M_{3}$,
the last map is surjective.
Let $m\in M_{2}$. If $m\not\in M_{1}$, then since $M_{3}$ is
$\mathcal{D}$-torsion-free, there exists
$\varphi\in\operatorname{Hom}_{R}(M_{2}/M_{1},\operatorname{\mathcal{D}})$
such that $\varphi(m+M_{1})\neq 0$. Hence there exists
$\widetilde{\varphi}\in\operatorname{Hom}_{R}(M_{2},\operatorname{\mathcal{D}})$,
such that $\widetilde{\varphi}(m)\neq 0$.
If $m\in M_{1}$, then since $M_{1}$ is $\mathcal{D}$-torsion-free, there
exists $\varphi\in\operatorname{Hom}_{R}(M_{1},\operatorname{\mathcal{D}})$
such that $\varphi(m)\neq 0$. Using that the restriction map
$\operatorname{Hom}_{R}(M_{2},\operatorname{\mathcal{D}})\to\operatorname{Hom}_{R}(M_{1},\operatorname{\mathcal{D}})$
is surjective, we obtain again that there exists
$\widetilde{\varphi}\in\operatorname{Hom}_{R}(M_{2},\operatorname{\mathcal{D}})$,
such that $\widetilde{\varphi}(m)\neq 0$.∎
In the calculations of $\dim_{\operatorname{\mathcal{D}}}$ the following
elementary lemma will be useful.
###### Lemma 4.3.
Let $\operatorname{\mathcal{D}}$ be a division $R$-ring and $M$ be a
$\operatorname{\mathcal{D}}$-torsion-free $R$-module of finite
$\operatorname{\mathcal{D}}$-dimension. Let $L$ be a non-trivial $R$-submodule
of $M$. Then
$\dim_{\operatorname{\mathcal{D}}}(M/L)<\dim_{\operatorname{\mathcal{D}}}M$.
Moreover, if $\dim_{\operatorname{\mathcal{D}}}L=1$, then
$\dim_{\operatorname{\mathcal{D}}}(M/L)=\dim_{\operatorname{\mathcal{D}}}M-1$.
###### Proof.
Since $M$ is $\operatorname{\mathcal{D}}$-torsion-free,
$\operatorname{\mathcal{D}}\otimes_{R}(M/L)$ is a proper quotient of
$\operatorname{\mathcal{D}}\otimes_{R}M$. Hence
$\dim_{\operatorname{\mathcal{D}}}M/L<\dim_{\operatorname{\mathcal{D}}}M$.
Now assume that $\dim_{\operatorname{\mathcal{D}}}L=1$. In this case
$\dim_{\operatorname{\mathcal{D}}}(M/L)\geq\dim_{\operatorname{\mathcal{D}}}M-1$.
Therefore,
$\dim_{\operatorname{\mathcal{D}}}(M/L)=\dim_{\operatorname{\mathcal{D}}}M-1$.∎
### 4.2. $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-
free modules
Let $\mathbf{F}$ be a finitely generated free pro-$p$ group. The main purpose
of this subsection is to prove the following result.
###### Proposition 4.4.
Assume that $1\neq z\in\mathbf{F}$ is not a proper $p$-power of an element of
$\mathbf{F}$. Denote by $\mathbf{Z}$ the closed subgroup of $\mathbf{F}$
generated by $z$. Then the module $I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$
is $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free.
Before proving the proposition we have to establish several preliminary
results.
###### Lemma 4.5.
Let $\mathbf{H}$ be an open subgroup of $\mathbf{F}$ Let $M$ be a
$\mathbb{F}_{p}[[\mathbf{H}]]$-module. Then
$\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free if
and only if $M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$-torsion-free.
###### Proof.
Assume that $M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$-torsion-free. We
have that the map
$M\to\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M$
is injective. Then, since $\mathbb{F}_{p}[[\mathbf{F}]]$ is a free right
$\mathbb{F}_{p}[[\mathbf{H}]]$-module, the map
$\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M\xrightarrow{\alpha}\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)$
is also injective.
Consider the canonical isomorphism between tensor products
$\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M)\xrightarrow{\beta}(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]})\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M.$
By Propositopn 3.4(c),
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\cong\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$
as $(\mathbb{F}_{p}[[\mathbf{F}]],\mathbb{F}_{p}[[\mathbf{H}]])$-bimodules.
Thus, there exists an isomorphism of $\mathbb{F}_{p}[[\mathbf{F}]]$-modules
$(\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H})]]})\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M\xrightarrow{\gamma}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M.$
We put $\varphi=\gamma\circ\beta\circ\alpha$ and apply Lemma 4.1. Since
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M$
is a $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-module and
$\varphi$ is an injective $\mathbb{F}_{p}[[\mathbf{F}]]$-homomorphism,
$\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free.
Another direction of the proposition is clear because $M$ is a
$\mathbb{F}_{p}[[\mathbf{H}]]$-submodule of
$\mathbb{F}_{p}[[\mathbf{F}]]\otimes_{\mathbb{F}_{p}[[\mathbf{H}]]}M$.∎
###### Lemma 4.6.
Let $\mathbf{H}$ be an open normal subgroup of $\mathbf{F}$ and assume that
$1\neq z\in\mathbf{H}$. Let $\mathbf{Z}$ be the closed subgroup of
$\mathbf{H}$ generated by $z$. Then the following are equivalent.
1. (a)
The $\mathbb{F}_{p}[[\mathbf{F}]]$-module
$I_{\mathbf{H}}^{\mathbf{F}}/I^{\mathbf{F}}_{\mathbf{Z}}$ is not
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free.
2. (b)
The $\mathbb{F}_{p}[[\mathbf{H}]]$-module
$I_{\mathbf{H}}/I^{\mathbf{H}}_{\mathbf{Z}}$ is not
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$-torsion-free.
3. (c)
There are $a\in I_{\mathbf{H}}^{\mathbf{F}}$ and $b\in I_{\mathbf{F}}$ such
that $ba=z-1$.
4. (d)
There are $a\in I_{\mathbf{H}}^{\mathbf{F}}$ and $b\in I_{\mathbf{F}}$ such
that $ab=z-1$.
###### Proof.
(a)$\Longleftrightarrow$(b): This follows from Lemma 4.5 and Lemma 2.11.
(c)$\Longrightarrow$(a): Put
$N=\mathbb{F}_{p}[[\mathbf{F}]]a/I_{\mathbf{Z}}^{\mathbf{F}}$, Then
$N=\mathbb{F}_{p}[[\mathbf{F}]]a/\mathbb{F}_{p}[[\mathbf{F}]](z-1)=\mathbb{F}_{p}[[\mathbf{F}]]a/\mathbb{F}_{p}[[\mathbf{F}]]ba.$
Since $b$ is not invertible in $\mathbb{F}_{p}[[\mathbf{F}]]$, $N$ is a non-
trivial submodule of $I_{\mathbf{H}}^{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$
and since $\mathbb{F}_{p}[[\mathbf{F}]]$ does not have non-trivial zero-
divisors
$\mathbb{F}_{p}[[\mathbf{F}]]a/\mathbb{F}_{p}[[\mathbf{F}]]ba\cong\mathbb{F}_{p}[[\mathbf{F}]]/\mathbb{F}_{p}[[\mathbf{F}]]b.$
Clearly
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}N=0$,
and so (a) holds.
(a)$\Longrightarrow$(c): We put
$M=I_{\mathbf{H}}^{\mathbf{F}}/I^{\mathbf{F}}_{\mathbf{Z}}$ and let
$\phi:M\to\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M$
be the canonical map. Let $\overline{L}=L/I_{\mathbf{Z}}^{\mathbf{F}}$ be the
kernel of $\phi$ and $\overline{M}$ the image of $\phi$. Hence we have the
exact sequence
$0\to L\to I_{\mathbf{H}}^{\mathbf{F}}\to\overline{M}\to 0.$
After applying
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}$
we obtain the exact sequence
(17)
$0\to\operatorname{Tor}_{1}^{\mathbb{F}_{p}[[\mathbf{F}]]}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]},\overline{M})\to\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}L\to\\\
(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]})^{d(\mathbf{H})}\to\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}\overline{M}\to
0.$
By Corrollary 2.12, the $\mathbb{F}_{p}[[\mathbf{F}]]$-module
$I_{\mathbf{H}}^{\mathbf{F}}$ is free of rank $d(\mathbf{H})$ and the
$\mathbb{F}_{p}[[\mathbf{F}]]$-module $I_{\mathbf{Z}}^{\mathbf{F}}$ is cyclic.
Thus, by Lemma 4.3,
$\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M)=d(\mathbf{H})-1.$
Therefore,
$\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}\overline{M})=\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}M)=d(\mathbf{H})-1.$
Observe also that by Proposition 3.2,
$\operatorname{Tor}_{1}^{\mathbb{F}_{p}[[\mathbf{F}]]}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]},\overline{M})=0$.
Therefore, (17) implies that
$\dim_{\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}}(\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}\otimes_{\mathbb{F}_{p}[[\mathbf{F}]]}L)=1.$
By [29, Lemma 3.1] a profinite submodule of a free profinite
$\mathbb{F}_{p}[[\mathbf{F}]]$-module is again free. Hence $L$ is a free
profinite $\mathbb{F}_{p}[[\mathbf{F}]]$-module, and so $L$ should be a cyclic
$\mathbb{F}_{p}[[\mathbf{F}]]$-module. We write
$L=\mathbb{F}_{p}[[\mathbf{F}]]a$ for some $a\in I_{\mathbf{H}}^{\mathbf{F}}$.
Then there exists $b\in\mathbb{F}_{p}[[\mathbf{F}]]$ such that $ba=z-1$. By
our assumption $L\neq I_{\mathbf{Z}}^{\mathbf{F}}$. Thus, $b$ is not
invertible, and so $b\in I_{\mathbf{F}}$.
(c)$\Longrightarrow$(d): The map $g\mapsto g^{-1}$ on $\mathbf{F}$ can be
extended to a continuous anti-isomorphism
$\alpha:\mathbb{F}_{p}[[\mathbf{F}]]\to\mathbb{F}_{p}[[\mathbf{F}]]$. If
$z-1=ba$, then $z^{-1}-1=\alpha(z-1)=\alpha(a)\alpha(b)$ and so
$z-1=(-z\alpha(a))\alpha(b)$. Now note that $-z\alpha(a)\in
I_{\mathbf{H}}\mathbb{F}_{p}[[\mathbf{F}]]$ and $\alpha(b)\in I_{\mathbf{F}}$.
Since $\mathbf{H}$ is normal in $\mathbf{F}$,
$I_{\mathbf{H}}\mathbb{F}_{p}[[\mathbf{F}]]=I_{\mathbf{H}}^{\mathbf{F}}$, and
we obtain (d).
(d)$\Longrightarrow$(c): It is proved in the same way as
(c)$\Longrightarrow$(d).∎
Now we are ready to prove Proposition 4.4.
###### Proof of Proposition 4.4.
If $\mathbf{F}$ is cyclic, then $I_{\mathbf{F}}=I_{\mathbf{Z}}^{F}$ and so
$I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free.
Now we assume that $\mathbf{F}$ is not cyclic. There exists a normal open
subgroup $\mathbf{N}$ of $\mathbf{F}$ such that $z\mathbf{N}$ is not a
$p$-power in $\mathbf{F}/\mathbf{N}$. We will prove the proposition by
induction on $|\mathbf{F}/\mathbf{N}|$.
If $\mathbf{F}/\mathbf{N}$ is cyclic, then $z\not\in\Phi(\mathbf{F})$ and so
$z$ is a member of a free generating system of $\mathbf{F}$. If
$\\{z,x_{1},\ldots.x_{k}\\}$ is a free generating set of $F$, then
$I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}=(\mathbb{F}_{p}[[\mathbf{F}]](z-1)\oplus(\oplus_{i=1}^{k}\mathbb{F}_{p}[[\mathbf{F}]](x_{i}-1)))/\mathbb{F}_{p}[[\mathbf{F}]](z-1)\cong\mathbb{F}_{p}[[\mathbf{F}]]^{k}$
is a free $\mathbb{F}_{p}[[\mathbf{F}]]$-module and we are done.
Assume now that $\mathbf{F}/\mathbf{N}$ is not cyclic. Since
$\Phi(\mathbf{F}/\mathbf{N})=\mathbf{N}\Phi(\mathbf{F})/\mathbf{N}$, the
pro-$p$ group $\mathbf{F}/\mathbf{N}\Phi(\mathbf{F})$ is not cyclic as well.
Let $\mathbf{M}$ be the closed subgroup of $\mathbf{F}$ containing the
commutator subgroup $[\mathbf{F},\mathbf{F}]$ and the element $z$ and such
that $\mathbf{M}/([\mathbf{F},\mathbf{F}]\mathbf{Z})$ is the torsion part of
$\mathbf{F}/([\mathbf{F},\mathbf{F}]\mathbf{Z})$. Since $\mathbf{Z}$ is cyclic
and $\mathbf{F}/([\mathbf{F},\mathbf{F}])$ is torsion-free and abelian,
$\mathbf{M}\Phi(\mathbf{F})/\Phi(\mathbf{F})$ is non-trivial cyclic (if
$z\not\in\Phi(\mathbf{F})$) or trivial (if $z\in\Phi(\mathbf{F})$). Therefore,
since $\mathbf{F}/\mathbf{N}\Phi(\mathbf{F})$ is not cyclic,
$\mathbf{M}\mathbf{N}$ is a proper subgroup of $\mathbf{F}$.
By the construction of $\mathbf{M}$,
$\mathbf{F}/\mathbf{M}\cong\mathbb{Z}_{p}^{k}$ for some $k\geq 1$. Since
$\mathbf{M}\mathbf{N}$ is a proper subgroup of $\mathbf{F}$,
$\mathbf{M}\mathbf{N}/\mathbf{M}$ is a proper subgroup of
$\mathbf{F}/\mathbf{M}$. Therefore, there exists a surjective map
$\sigma:\mathbf{F}\to\mathbb{Z}_{p}$ such that $\mathbf{M}\leq\ker\sigma$ and
$\mathbf{N}\ker\sigma\neq\mathbf{F}$. We put $\mathbf{H}=\mathbf{N}\ker\sigma$
and extend $\sigma$ to the map
$\widetilde{\sigma}:\mathbb{F}_{p}[[\mathbf{F}]]\to\mathbb{F}_{p}[[\mathbb{Z}_{p}]]$.
Observe that $\ker\widetilde{\sigma}=I_{\ker\sigma}^{\mathbf{F}}$.
By way of contradiction, assume that
$I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$ is not
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free. Then
by Lemma 4.6, there are $a,b\in I_{\mathbf{F}}$ such that $ab=z-1$. Applying
$\widetilde{\sigma}$ we obtain that
$\widetilde{\sigma}(a)\widetilde{\sigma}(b)=0$. Since
$\mathbb{F}_{p}[[\mathbb{Z}_{p}]]$ is a domain, either $a$ or $b$ lie in
$\ker\widetilde{\sigma}=I_{\ker\sigma}^{\mathbf{F}}\subset
I_{\mathbf{H}}^{\mathbf{F}}$. Applying again Lemma 4.6, we conclude that
$I_{\mathbf{H}}/I_{\mathbf{Z}}^{\mathbf{H}}$ is not
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$-torsion-free.
However, observe that $\mathbf{N}$ is also a normal subgroup of $\mathbf{H}$,
$z\mathbf{N}$ is not a $p$-power in $\mathbf{H}/\mathbf{N}$ and
$|\mathbf{H}/\mathbf{N}|<|\mathbf{F}/\mathbf{N}|$. Thus, we can apply the
inductive assumption and conclude that
$I_{\mathbf{H}}/I_{\mathbf{Z}}^{\mathbf{H}}$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{H}]]}$-torsion-free. We
have arrived to a contradiction. Thus,
$I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-torsion-free. ∎
### 4.3. $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free
modules
In this subsection we assume that $\mathbf{F}$ is a finitely generated free
pro-$p$ group and $G$ is an (abstract) dense finitely generated subgroup of
$\mathbf{F}$. First we prove the following analogue of Lemma 4.5.
###### Lemma 4.7.
Let $H$ be a subgroup of $G$ and let $M$ be a
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[H]}$-torsion-free left
$\mathbb{F}_{p}[H]$-module. Then
$\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free.
###### Proof.
Let $\operatorname{\mathcal{D}}_{H}$ be the division closure of
$\mathbb{F}_{p}[H]$ in $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$.
Observe that $\operatorname{\mathcal{D}}_{H}$ and
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[H]}$ are isomorphic as
$\mathbb{F}_{p}[H]$-rings (it follows, for example, from Proposition 3.5).
We have that the map
$M\to\operatorname{\mathcal{D}}_{H}\otimes_{\mathbb{F}_{p}[H]}M$ is injective.
Then, since $\mathbb{F}_{p}[G]$ is a free right $\mathbb{F}_{p}[H]$-module,
the map
$\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}M\xrightarrow{\alpha}\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}(\operatorname{\mathcal{D}}_{H}\otimes_{\mathbb{F}_{p}[H]}M)$
is also injective.
Consider the canonical isomorphism between tensor products
$\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}(\operatorname{\mathcal{D}}_{H}\otimes_{\mathbb{F}_{p}[H]}M)\xrightarrow{\beta}(\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}\operatorname{\mathcal{D}}_{H})\otimes_{\mathbb{F}_{p}[H]}M.$
By [23], $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$ is strongly Hughes-
free. This means that the canonical map of
$(\mathbb{F}_{p}[G],\mathbb{F}_{p}[H])$-bimodules
$\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}\operatorname{\mathcal{D}}_{H}\to\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$
is injective. Moreover, the image of
$\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}\operatorname{\mathcal{D}}_{H}$
is a direct summand of $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$ as a
right $\operatorname{\mathcal{D}}_{H}$-submodule (and so, it is also a direct
summand as a right $\mathbb{F}_{p}[H]$-submodule). Thus, the following
canonical map of $\mathbb{F}_{p}[G]$-modules
$(\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}\operatorname{\mathcal{D}}_{H})\otimes_{\mathbb{F}_{p}[H]}M\xhookrightarrow{\gamma}\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}\otimes_{\mathbb{F}_{p}[H]}M$
is injective. We put $\varphi=\gamma\circ\beta\circ\alpha$ and apply Lemma
4.1. Since
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}\otimes_{\mathbb{F}_{p}[H]}M$
is a $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-module and $\varphi$ is
an injective $\mathbb{F}_{p}[G]$-homomorphism,
$\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free.
∎
Now we can present our first example of a
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free
$\mathbb{F}_{p}[G]$-module.
###### Proposition 4.8.
Let $H$ be a non-trivial subgroup of $G$ and $A$ a maximal abelian subgroup of
$H$. Then the $\mathbb{F}_{p}[G]$-module $I^{G}_{H}/I^{G}_{A}$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free.
###### Proof.
From Lemma 2.1 we know that
$I^{G}_{H}/I^{G}_{A}\cong\mathbb{F}_{p}[G]\otimes_{\mathbb{F}_{p}[H]}(I_{H}/I^{H}_{A}).$
Thus, in view of Lemma 4.7, it is enough to show that $I_{H}/I^{H}_{A}$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[H]}$-torsion-free.
Let $\mathbf{Z}=C_{\mathbf{F}}(A)$. Since $\mathbf{F}$ is a free pro-$p$
group, a centralizator of a non-trivial element is maximal cyclic pro-$p$
subgroup. Therefore, since $A$ is abelian and non-trivial, $\mathbf{Z}$ is a
maximal cyclic pro-$p$ subgroup of $\mathbf{F}$.
###### Claim 4.9.
The canonical map
$\mathbb{F}_{p}[H/A]\to\mathbb{F}_{p}[[\mathbf{F}/\mathbf{Z}]]$ is injective.
###### Proof.
Since $A$ is maximal abelian in $H$, we have that $A=\mathbf{Z}\cap H$. Hence
the obvious map $\mathbb{F}_{p}[H/A]\to\mathbb{F}_{p}[\mathbf{F}/\mathbf{Z}]$
in injective. The map
$\mathbb{F}_{p}[\mathbf{F}/\mathbf{Z}]\to\mathbb{F}_{p}[[\mathbf{F}/\mathbf{Z}]]$
is also injective, because $\mathbf{Z}$ is closed in $\mathbf{F}$. This
finishes the proof of the claim.∎
Observe that $\mathbb{F}_{p}[H/A]\cong\mathbb{F}_{p}[H]/I_{A}^{H}$ and
$\mathbb{F}_{p}[[\mathbf{F}]]/I_{\mathbf{Z}}^{\mathbf{F}}\cong\mathbb{F}_{p}[[\mathbf{F}/\mathbf{Z}]]$.
Therefore, by Claim 4.9, $I_{H}/I^{H}_{A}$ is a $\mathbb{F}_{p}[H]$-submodule
of $I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$. By Proposition 4.4, we can
embed $I_{\mathbf{F}}/I_{\mathbf{Z}}^{\mathbf{F}}$ in a
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-module. By
Proposition 3.5, every
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[[\mathbf{F}]]}$-module is also a
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[H]}$-module. Therefore, by Lemma
4.1, $I_{H}/I^{H}_{A}$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[H]}$-torsion-free. ∎
The following proposition shows another example of a
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free
$\mathbb{F}_{p}[G]$-module. This is the main result of this section.
###### Proposition 4.10.
Let $\mathbf{F}$ be a finitely generated free pro-$p$ group and let $G$ be an
(abstract) dense finitely generated subgroup of $\mathbf{F}$. Let $H$ be a
non-trivial subgroup of $G$ and $A$ a maximal abelian subgroup of $H$. Let $B$
be an abelian subgroup of $G$ containing $A$. We put
$J=\\{(x,-x)\in I^{G}_{H}\oplus I^{G}_{B}:\ x\in I^{G}_{A}\\}.$
Then $M=(I^{G}_{H}\oplus I^{G}_{B})/J$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free and
$\dim_{\mathbb{F}_{p}[G]}M=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}$.
###### Proof.
Let $L=(I^{G}_{A}\oplus I^{G}_{B})/J\leq M$. Then $L\cong I^{G}_{B}$ is a
submodule of $\mathbb{F}_{p}[G]$, and so it is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free. The
$\mathbb{F}_{p}[G]$-module $M/L$ is isomorphic to $I_{H}^{G}/I^{G}_{A}$, and
so it is $\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free by
Proposition 4.8.
By (14), $\dim_{\mathbb{F}_{p}[A]}I_{A}=1$. Therefore, by Proposition 3.4(b),
$\dim_{\mathbb{F}_{p}[G]}I^{G}_{A}=\dim_{\mathbb{F}_{p}[A]}I_{A}=1.$
In the same way we obtain that $\dim_{\mathbb{F}_{p}[G]}I^{G}_{B}=1$.
Since $\dim_{\mathbb{F}_{p}[G]}(I^{G}_{H}\oplus
I^{G}_{B})=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}+1,$ by Lemma 4.3,
$\dim_{\mathbb{F}_{p}[G]}M=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}+1-1=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}\textrm{\
and}\\\
\dim_{\mathbb{F}_{p}[G]}(I_{H}^{G}/I^{G}_{A})=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}-1=\dim_{\mathbb{F}_{p}[G]}M-1.$
Therefore,
$\dim_{\mathbb{F}_{p}[G]}(M/L)=\dim_{\mathbb{F}_{p}[G]}(I_{H}^{G}/I^{G}_{A})=\dim_{\mathbb{F}_{p}[G]}M-1.$
Thus, we have obtained that $M/L$ and $L$ are
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free and
$\dim_{\mathbb{F}_{p}[G]}M=\dim_{\mathbb{F}_{p}[G]}(M/L)+\dim_{\mathbb{F}_{p}[G]}L.$
Applying Lemma 4.2, we conclude that $M$ is also
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free. ∎
## 5\. Proof of main results
### 5.1. The inductive step in the proof of Theorem 1.2
The following theorem is the main result of the paper. Theorem 1.2 follows
from it directly.
###### Theorem 5.1.
Let $\mathbf{F}$ be a finitely generated free pro-$p$ group and let
$H\hookrightarrow\mathbf{F}$ be a strong embedding of a finitely generated
group $H$. Let $A$ be a maximal abelian subgroup of $H$ and let $B$ be an
abelian finitely generated subgroup of $\mathbf{F}$ containing $A$. Put
$G=\langle H,B\rangle$. Then the canonical homomorphism $H*_{A}B\to G$ is an
isomorphism, and the embedding $G\hookrightarrow\mathbf{F}$ is strong.
###### Proof.
In view of Proposition 2.13 we have to show that $I^{G}_{H}\cap
I_{B}^{G}=I_{A}^{G}$ in $\mathbb{F}_{p}[G]$. Let
$J=\\{(x,-x)\in I^{G}_{H}\oplus I^{G}_{B}:\ x\in I^{G}_{A}\\}\textrm{\ and\
}M=(I^{G}_{H}\oplus I^{G}_{B})/J.$
Then by Proposition 4.10,
$\dim_{\mathbb{F}_{p}[G]}M=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}$. Therefore,
$\dim_{\mathbb{F}_{p}[G]}M=\dim_{\mathbb{F}_{p}[G]}I^{G}_{H}=\dim_{\mathbb{F}_{p}[H]}I_{H}=\beta_{1}^{\operatorname{mod}p}(H)+1=d(\mathbf{F}).$
Since $I_{G}=I_{H}^{G}+I_{B}^{G}$, we have that the natural map $\alpha:M\to
I_{G}$ is surjective. In particular
$\beta^{\operatorname{mod}p}_{1}(G)=\dim_{\mathbb{F}_{p}[G]}I_{G}-1\leq\dim_{\mathbb{F}_{p}[G]}M-1=d(\mathbf{F})-1.$
Thus, using (16) we obtain that
$\beta^{\operatorname{mod}p}_{1}(G)=d(\mathbf{F})-1$ and
$\dim_{\mathbb{F}_{p}[G]}I_{G}=\dim_{\mathbb{F}_{p}[G]}M$. This shows that the
embedding $G\hookrightarrow\mathbf{F}$ is strong.
By Proposition 4.10, $M$ is
$\operatorname{\mathcal{D}}_{\mathbb{F}_{p}[G]}$-torsion-free. Therefore, by
Lemma 4.3, for any proper quotient $\overline{M}$ of $M$,
$\dim_{\mathbb{F}_{p}[G]}\overline{M}<\dim_{\mathbb{F}_{p}[G]}M$. This implies
that $\alpha$ is an isomorphism, and so $I^{G}_{H}\cap I_{B}^{G}=I_{A}^{G}$.
Hence, Proposition 2.13 implies that $G\cong H*_{A}B$.∎
Another direct consequence of Theorem 5.1 is the following corollary.
###### Corollary 5.2.
Let $\mathbf{F}$ be a finitely generated free pro-$p$ group and let
$H\hookrightarrow\mathbf{F}$ be a strong embedding of finitely generated group
$H$. Let $A$ be a maximal abelian subgroup of $H$. Assume that $A$ is finitely
generated. Let $B$ be a finitely generated torsion-free abelian group
containing $A$ and such that $B/A$ has no $p$-torsion. Then there exists an
embedding of $H*_{A}B$ into $\mathbf{F}$ that extends
$H\hookrightarrow\mathbf{F}$. In particular, $H*_{A}B$ is SE($p$) (see
Definition 3.8).
###### Proof.
Let $H\hookrightarrow\mathbf{F}$ be a strong embedding. Since $B/A$ has no
$p$-torsion and $B$ is torsion-free abelian, the embedding of $A$ into
$\mathbf{F}$ can be extended to an embedding of $B$ into $\mathbf{F}$. Now, we
can apply Theorem 5.1.∎
### 5.2. $A$-groups
Let $A$ be a commutative ring. An $A$-group is a group $G$ along with a map
$G\times A\to G$, called “action”, satisfying the following axioms:
1. (a)
$g^{1_{A}}=g$, $g^{0_{A}}=1$, $1^{\alpha}=1$;
2. (b)
$g^{\alpha}g^{\beta}=g^{\alpha+\beta}$,
$(g^{\alpha})^{\beta}=g^{\alpha\beta}$;
3. (c)
$(h^{-1}gh)^{\alpha}=h^{-1}g^{\alpha}h$;
4. (d)
If $gh=hg$, $(gh)^{\alpha}=g^{\alpha}h^{\alpha}$;
for all $g,h\in G$ and $\alpha,\beta\in A$. This definition generalizes the
definition of $\mathbb{Q}$-group which appears in the introduction.
Given a group $G$ and a commutative ring $A$, an $A$-completion of $G$ is an
$A$-group $G^{A}$ with a group homomorphism $\lambda:G\to G^{A}$ such that
$G^{A}$ is generated by $\lambda(G)$ as an $A$-group and for any $A$-group $H$
and any homomorphism $\phi:G\to H$ there exists a unique $A$-homomorphism
$\psi:G^{A}\to H$ such that $\phi=\psi\circ\lambda$. It is shown in [52,
Theorems 1 and 2] that an $A$-completion of $G$ exists and it is unique up to
an $A$-isomorphism.
An $A$-group $F^{A}(X)$ with the set of $A$-generators $X$ is said to be a
free $A$-group with base $X$, if for every $A$-group $G$ an arbitrary mapping
$\phi_{0}:X\to G$ can be extended to an $A$-homomorphism $\phi:F^{A}(X)\to G$.
Thus, $F^{A}(X)$ is the $A$-completion of the ordinary free group $F(X)$ with
free generating set $X$.
A CSA-group is a group in which the centralizer of every nontrivial element is
abelian and malnormal. In [53, Theorem 5] it is shown that every torsion-free
extension of centralizer of a torsion-free CSA group is again CSA. This is
used in [53, Theorem 8] to describe the group $F^{A}(X)$. In the following
proposition we extract the information that we will need later.
###### Proposition 5.3.
Let $X$ be a finite set and $A$ a commutative ring with a torsion-free
additive group. Then there are subgroups $\\{W_{n}\\}_{n\geq 0}$ of $F^{A}(X)$
such that
1. (a)
$W_{0}=F(X)$;
2. (b)
$W_{n+1}=\langle W_{n},z_{n}^{A}\rangle$, where $z_{n}\in W_{n}$ generates a
maximal abelian subgroup in $W_{n}$ and the canonical map $\displaystyle
W_{n}*_{z_{n}=1_{A}}A\to W_{n+1}$ is an isomorphism;
3. (c)
$F(X)^{A}=\bigcup_{i=0}^{\infty}W_{n}$.
Now we are ready to prove Corollary 1.3.
###### Proof of Corollary 1.3.
If $\phi$ is not injective, then there exists a finitely generated subgroup
$G$ of $F^{\mathbb{Z}_{p}}(X)$ such that $G\cap\ker\phi$ is not trivial. By
Proposition 5.3, there exists a non-negative integer $k$ and a sequence
$G_{0}\leq G_{1}\leq\ldots\leq G_{k}$ of subgroups of $F^{A}(X)$, where
1. (a)
$G_{0}=F(X_{0})$ is the group generated by a finite subset $X_{0}$ of $X$;
2. (b)
for $0\leq i\leq k-1$ there exists an element $z_{i}\in G_{i}$, generating a
maximal abelian subgroup in $G_{i}$, and a finitely generated subgroup $T_{i}$
of $(A,+)$ containing $1_{A}$, such that $z_{i}^{T_{i}}\leq G_{i+1}$ and the
canonical map
$G_{i}*_{z_{i}=1_{A}}T_{i}\to G_{i+1}$
is an isomorphism;
3. (c)
$G$ is a subgroup of $G_{k}$.
Put $H_{i}=\phi(G_{i})$ $A_{i}=\langle\phi(z_{i})\rangle$ and
$B_{i}=\phi(z_{i}^{T_{i}})$.
Let us show by induction on $i$ that $\phi_{|G_{i}}:G_{i}\to H_{i}$ is an
isomorphism. It is clear for $i=0$. Assume we have proved it for $i<k$.
Observe that, since $\phi(z_{i})\neq 1$,
$B_{i}=\phi(z_{i}^{T_{i}})=\phi(z_{i})^{T_{i}}\cong T_{i}\textrm{\ and \
}A_{i}=\phi(\langle z_{i}\rangle)=\phi(G_{i}\cap z_{i}^{T_{i}})=B_{i}\cap
H_{i}.$
Then by Theorem 1.2, $\phi_{|G_{i+1}}:G_{i+1}\to H_{i+1}$ is also an
isomorphism.
Thus, $\ker\phi\cap G_{k}=\\{1\\}$. This is a contradiction. ∎
We will also need the following consequence of Proposition 5.3.
###### Corollary 5.4.
[53, Corollary 5] Let $B$ be a ring with a torsion-free additive group and $A$
a subring of $B$. Then the canonical map $F^{A}(X)\to F^{B}(X)$ is injective.
### 5.3. Proof of Theorem 1.1
In this subsection we will prove that $F^{A}(X)$ is residually nilpotent for
every commutative ring $A$ with a torsion-free additive group. The proof is
based on the following general result communicated to us by a referee.
###### Proposition 5.5.
Let $G$ be a group. Assume that for each $n\geq 0$ there are subgroups $G_{n}$
and $B_{n}$ of $G$ such that
1. (a)
$G_{n+1}=\langle G_{n},B_{n}\rangle$;
2. (b)
$B_{n}$ is abelian and $G_{n}$ is residually torsion-free nilpotent;
3. (c)
$A_{n}=G_{n}\cap B_{n}$ and the canonical map $G_{n}*_{A_{n}}B_{n}\to G_{n+1}$
is an isomorphism;
4. (d)
$G=\bigcup_{i=0}^{\infty}G_{n}$.
Then $G$ is residually torsion-free nilpotent.
###### Proof.
Let $x\in G$. We want to show that $G$ has a torsion-free nilpotent quotient
such that the image of $x$ in this quotient is not trivial.
Let $x\in G_{m}$ for some $m\geq 0$. Since $G_{m}$ is residually torsion-free
nilpotent, there exists a torsion-free nilpotent group $N$ and the map
$\phi_{m}:G_{m}\to N$ such that $\phi_{m}(x)\neq 1$. We can embed $N$ in its
Malcev $\mathbb{Q}$-completion. Thus, without loss of generality, we may
assume that $N$ is also a $\mathbb{Q}$-group.
We will show that for each $n\geq m$ there exist $\phi_{n}:G_{n}\to N$ such
that the restriction of $\phi_{n}$ on $G_{n-1}$ is $\phi_{n-1}$. We construct
$\phi_{n}$ by induction. Assume that we constructed $\phi_{n}$ for $m\leq
n\leq k$. Since $N$ is a $\mathbb{Q}$-group, we can extend $\phi_{k}$ from
$A_{k}$ to $A_{k}^{\mathbb{Q}}$ and so to $B_{k}^{\mathbb{Q}}$ and $B_{k}$.
Thus we have a homomorphism $\tilde{\phi}_{k}:B_{k}\to N$ that extends
$(\phi_{k})_{|A_{k}}:A_{k}\to N$. Now, by the universal property of the
amalgamated product, there exists a homomorphism $\phi_{k+1}:G_{k+1}\to N$
whose restriction on $G_{k}$ is $\phi_{k}$ and on $B_{k}$ is
$\tilde{\phi}_{k}$.
Let $\phi:G\to N$ be the homomorphism satisfying $\phi(g)=\phi_{n}(g)$ if
$g\in G_{n}$. Then $\phi$ is well-defined and $\phi(x)\neq\\{1\\}$. ∎
###### Corollary 5.6.
Let $X$ be a set and $A$ a countable commutative ring with a torsion-free
additive group. Then $F^{A}(X)$ is residually torsion-free nilpotent.
###### Proof.
First assume that $X$ is finite. Since $A$ is countable, by Proposition 5.3
there are subgroups $\\{G_{n}\\}_{n\geq 0}$ and $\\{B_{n}\\}_{n\geq 0}$ of
$F^{A}(X)$ such that
1. (a)
$G_{n+1}=\langle G_{n},B_{n}\rangle$;
2. (b)
$B_{n}$ is abelian and finitely generated;
3. (c)
$A_{n}=G_{n}\cap B_{n}$ is a maximal abelian subgroup of $G_{n}$ and the
canonical map $G_{n}*_{A_{n}}B_{n}\to G_{n+1}$ is an isomorphism;
4. (d)
$F^{A}(X)=\bigcup_{i=0}^{\infty}G_{n}$.
By Theorem 1.2, $G_{n}$ are residually torsion-free nilpotent. Thus,
Proposition 5.5 implies that $F^{A}(X)$ is residually torsion-free nilpotent.
Now assume that $X$ is an arbitrary set. Given an element $1\neq g$ of
$F^{A}(X)$, it belongs to $F^{A}(X_{0})$ for some finite subset $X_{0}$ of
$X$, and $F^{A}(X_{0})$ is a retract of $F^{A}(X)$. As we have already proved,
there exists a homomorphism $\phi:F^{A}(X_{0})\to N$, where $N$ is torsion-
free nilpotent such that $\phi(g)\neq 1$. This finishes the proof. ∎
###### Remark 5.7.
The proof of the previous corollary can also be adapted to the case where $A$
is not countable. We prove it, using a different method, in Corollary 5.9.
Let $X=\\{x_{i}\colon i\in I\\}$ and $Y=\\{y_{i}\colon i\in I\\}$ be two sets
indexed by the elements of a set $I$. Let $A$ be a commutative ring. We say
that $A$ is a binomial domain if $A$ is a domain and $a\choose n$ belongs to
$A$ for every $a\in A$. For example $\mathbb{Z}_{p}$ and $\mathbb{Q}$-algebras
are binomial domains.
Assume that $A$ is a binomial domain. If $\Delta_{A}$ denotes the ideal of
$A\langle\\!\langle Y\rangle\\!\rangle$ generated by $Y$ then $1+\Delta_{A}$
is a subgroup of the group of units of $A\langle\\!\langle
Y\rangle\\!\rangle$. We can define an action of $A$ on $1+\Delta_{A}$ in the
following way:
$(1+f)^{a}=1+\sum_{n=1}^{\infty}{a\choose n}f^{n}\ (a\in A,f\in\Delta_{A}).$
Then, by [60], $1+\Delta_{A}$ provided with this action is an $A$-group.
Therefore, the map $x_{i}\mapsto 1+y_{i}$ can be uniquely extended to an
$A$-homomorphism
$\phi_{A}:F^{A}(X)\to 1+\Delta_{A}$
called the Magnus representation of $F^{A}(X)$. Magnus (see, for example,
[47]) proved that $\phi_{\mathbb{Z}}$ is injective. Now we prove the main
result of this subsection.
###### Theorem 5.8.
Let $X$ be an arbitrary set and $A$ be a binomial domain.Then the map
$\phi_{A}:F^{A}(X)\to 1+\Delta_{A}$ is injective.
###### Proof.
By Corollary 5.4, we can assume that $A$ is a $\mathbb{Q}$-algebra. If
$\ker\phi_{A}$ is not trivial, then there exists a finite subset $X_{0}$ of
$X$ and countable $\mathbb{Q}$-subalgebra of $A$ such that the kernel of the
map $F^{A_{0}}(X_{0})\to 1+\Delta_{A_{0}}$ is not trivial. Hence, we can also
assume that $X$ is finite and $A$ is countable.
Let $F_{n}=F(X)/\gamma_{n}(F(X))$. Denote by $K_{n}$ the kernel of the
canonical map $F^{A}(X)\to F_{n}^{A}$. Using [52, Property 2], we obtain that
$F_{n}^{A}$ is nilpotent of nilpotency class $n-1$. By Proposition 5.6,
$F^{A}(X)$ is residually torsion-free nilpotent. Hence
$\cap_{n}K_{n}=\\{1\\}$.
The map $F_{n}\to 1+\Delta_{A}/1+(\Delta_{A})^{n}$ induces the map
$\phi_{n}:F_{n}^{A}\to 1+\Delta_{A}/1+(\Delta_{A})^{n}$. It is known that
$\phi_{n}$ is injective (see, for example, [22, Theorem 23]). Thus $K_{n}$ is
the kernel of the canonical map $F^{A}(X)\to 1+\Delta_{A}/1+(\Delta_{A})^{n}$.
This implies that $\phi_{A}$ is injective. ∎
The following definition has been suggested to us by a referee. We say that an
$A$-group $G$ is $A$-torsion-free if for every $1\neq g\in G$, the map $A\to
G$ that sends $a$ to $g^{a}$ is injective.
###### Corollary 5.9.
Let $A$ be a commutative ring with a torsion-free additive group. Then the
group $F^{A}(X)$ is residually-($A$-torsion-free nilpotent).
###### Proof.
By Corollary 5.4, we can assume that $A$ is a $\mathbb{Q}$-algebra. Hence the
corollary follows from Theorem 5.8 because the $A$-groups
$1+\Delta_{A}/1+(\Delta_{A})^{n}$ are $A$-torsion-free and nilpotent. ∎
### 5.4. Proof of Theorem 1.4
The proof of Theorem 1.4 uses a particular case of [53, Theorem 8] that we
describe now.
###### Proposition 5.10.
Let $G$ be a countable torsion-free CSA-group. Then the
$\mathbb{Q}$-completion $G^{\mathbb{Q}}$ of $G$ is a direct union of subgroups
$W_{0}\leq W_{1}\leq\ldots$ such that
1. (a)
$W_{0}=G$;
2. (b)
for $i\geq 0$ there exists a maximal abelian subgroup $A_{i}$ of $W_{i}$ such
that $W_{i+1}$ is the image of the canonical map
$W_{i}*_{A_{i}}A_{i}^{\mathbb{Q}}\to G^{\mathbb{Q}}$.
Moreover, if $H$ is a subgroup $G^{\mathbb{Q}}$ and $A$ is a maximal abelian
subgroup of $H$, then the canonical map $H*_{A}A^{\mathbb{Q}}\to
G{{}^{\mathbb{Q}}}$ is injective.
###### Proof of Theorem 1.4.
Our definition of a limit group from the introduction and Proposition 5.3 show
that the limit groups are exactly finitely generated subgroups of
$F^{\mathbb{Z}[t]}(X)$. Since $F^{\mathbb{Z}[t]}(X)$ is a CSA-group, its
$\mathbb{Q}$-completion can be calculated using Proposition 5.10. In
particular, $G^{\mathbb{Q}}$ is a subgroup of the $\mathbb{Q}$-completion of
$F^{\mathbb{Z}[t]}(X)$ and so it is a subgroup of $F^{\mathbb{Q}[t]}(X)$.
Hence by Corollary 5.9, it is residually torsion-free nilpotent. ∎
## 6\. Linearity of free $\mathbb{Q}$-groups and free pro-$p$ groups
We finish this paper with a discussion on another two well-known problems
concerning linearity of free $\mathbb{Q}$-groups and free pro-$p$ groups.
The problem of whether a free $\mathbb{Q}$-group $F^{\mathbb{Q}}(X)$ is linear
appears in [10, Problem F13] and it is attributed to I. Kapovich (see also
[38, Problem 13.39(b)]). The problem of whether a free pro-$p$ group
$\mathbf{F}$ is linear is usually attributed to A. Lubotzky (for example, we
discussed this question in Jerusalem in November, 2001).
In the context of profinite groups, one can consider two kinds of linearity
(see, for example, [28]). On one hand, we say that a profinite group $G$ is
linear if it is linear as an abstract group, that is it has a faithful
representation by matrices of fixed degree over a field. On the other hand,
the concept of $t$-linear profinite group takes into account the topology of
$G$ and means that $G$ can be faithfully represented as a closed subgroup of
the group of invertible matrices of fixed degree over a profinite commutative
ring.
It is commonly believed that a non-abelian free pro-$p$ group is not
$t$-linear (see, [44, Conjecture 3.8], the discussion after [3, Theorem 1.1]
and [59, Section 5.3]). An equivalent reformulation of this statement is that
a $p$-adic analytic pro-$p$ group satisfies a non-trvial pro-$p$ identity. A.
Zubkov [64] proved that if $p>2$, then a non-abelian free pro-$p$ group cannot
be represented by 2-by-2 matrices over a profinite commutative ring. E.
Zelmanov announced that given a fixed $n$, a non-abelian free pro-$p$ group
cannot be represented by $n$-by-$n$ matrices over a profinite commutative ring
for every large enough prime $p>>n$ (see [62, 63]). Recently, D. El-Chai Ben-
Ezra, E. Zelmanov showed that a free pro-2 group cannot be represented by
2-by-2 matrices over a profinite commutative ring of characteristic 2 [20].
Recall that by a result of A. Malcev [48, Theorem IV], a group can be
represented by matrices of degree $n$ over a field if and only if every one of
its finitely generated subgroup has this property. Thus, in order to decide
whether $F^{\mathbb{Q}}(X)$ or $\mathbf{F}$ are linear, we have to analyze the
structure of their finitely generated (abstract) subgroups. In order to apply
the Malcev criterion one should find a uniform $n$ which does not depend on a
finitely generated subgroup. We may ask a weaker question of whether
$F^{\mathbb{Q}}(X)$ and $\mathbf{F}$ are locally linear. Using recent advances
in geometric group theory one can answer this positively in the case of
$F^{\mathbb{Q}}(X)$.
###### Theorem 6.1.
The groups $F^{\mathbb{Q}}(X)$ are locally linear over $\mathbb{Z}$.
###### Proof.
Let $G$ be a finitely generated subgroup of $F^{\mathbb{Q}}(X)$. Then by
Proposition 5.3, there exists $k\geq 0$ and a sequence $G_{0}\leq
G_{1}\leq\ldots\leq G_{k}$ of subgroups of $F^{\mathbb{Q}}(X)$, such that
$G_{0}$ is free, $G_{i+1}$ is obtained from $G_{i}$ by adjoining a root and
$G\leq G_{k}$. By [53, Theorem 5], the groups $G_{i}$ are CSA. Thus, from a
corollary on the page 100 of [11] we obtain that $G$ is hyperbolic. Therefore,
[26, Corollary C] ensures, by induction on $i$, that each $G_{i}$ acts
properly and cocompactly on a CAT(0) cube complex. Hence, by [1, Theorem 1.1],
$G_{k}$ has a finite index subgroup acting faithfully and specially on a
CAT(0) cube complex. Finally from [25, Theorem 1.1] it follows that $G_{k}$,
and so $G$, are linear over $\mathbb{Z}$.∎
I wouldn’t be surprised if the groups $F^{\mathbb{Q}[t]}(X)$ are also locally
linear. However, to decide whether $F^{\mathbb{Q}}(X)$ and
$F^{\mathbb{Q}[t]}(X)$ are linear will require a completely new approach.
## References
* [1] I. Agol, The virtual Haken conjecture. With an appendix by Agol, Groves, and Manning. Doc. Math. 18 (2013), 1045–1087.
* [2] J. Barlev, T. Gelander, Compactifications and algebraic completions of limit groups. J. Anal. Math. 112 (2010), 261–287.
* [3] Y. Barnea, M. Larsen, A non-abelian free pro-$p$ group is not linear over a local field. J. Algebra 214 (1999), 338–341.
* [4] G. Baumslag, Some aspects of groups with unique roots. Acta Math. 104 (1960), 217–303.
* [5] G. Baumslag, On free $\mathcal{D}$-groups. Comm. Pure Appl. Math. 18 (1965), 25-30.
* [6] G. Baumslag, Groups with the same lower central sequence as a relatively free group. I. The groups. Trans. Amer. Math. Soc. 129 (1967), 308–321.
* [7] G. Baumslag, On the residual nilpotence of certain one-relator groups. Comm. Pure Appl. Math. 21 (1968), 491–506.
* [8] G. Baumslag, Groups with the same lower central sequence as a relatively free group. II. Properties. Trans. Amer. Math. Soc. 142 (1969), 507–538.
* [9] G. Baumslag, Some reflections on proving groups residually torsion-free nilpotent. I. Illinois J. Math. 54 (2010), 315–325.
* [10] G. Baumslag, A. Myasnikov, V. Shpilrain, Open problems in combinatorial group theory. Second edition. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 1–38, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002.
* [11] M. Bestvina, M. Feighn, A combination theorem for negatively curved groups. J. Differential Geom. 35 (1992), 85–101.
* [12] E. Breuillard, T. Gelander, J. Souto, P. Storm, Dense embeddings of surface groups. Geom. Topol. 10 (2006), 1373–1389.
* [13] M. Bridson, A. Reid, Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag. Int. Math. Res. Not. IMRN 2015, 2111–2140.
* [14] C. Champetier, V. Guirardel, Limit groups as limits of free groups. Israel J. Math., 146 (2005), 1–75.
* [15] B. Chandler, The representation of a generalized free product in an associative ring. Comm. Pure Appl. Math. 21 (1968), 271–288.
* [16] P. M. Cohn, Localization in semifirs. Bull. London Math. Soc. 6 (1974), 13–20.
* [17] P. M. Cohn, Free rings and their relations. Second edition. London Mathematical Society Monographs, 19. Academic Press, Inc., London, 1985.
* [18] P. M. Cohn, Skew fields. Theory of general division rings. Encyclopedia of Mathematics and its Applications, 57. Cambridge University Press, Cambridge, 1995.
* [19] W. Dicks, D. Herbera, J. Sánchez, On a theorem of Ian Hughes about division rings of fractions. Comm. Algebra 32 (2004), 1127–1149.
* [20] D. El-Chai Ben-Ezra, E. Zelmanov, On pro-2 identities of $2\times 2$ linear groups, Trans. Amer. Math. Soc. 374 (2021), 4093–4128
* [21] M. Ershov, W. Lück, The first $L^{2}$-Betti number and approximation in arbitrary characteristic, Doc. Math. 19 (2014), 313–332.
* [22] A. Gaglione, A. Myasnikov, V. Remeslennikov, D. Spellman, Formal power series representations of free exponential groups. Comm. Algebra 25 (1997), 631–648.
* [23] J. Gräter, Free Division Rings of Fractions of Crossed Products of Groups With Conradian Left-Orders, Forum Math. 32 (2020), 739–772. .
* [24] H. Hahn, Über die nichtarchimedischen Grössensysteme, Sitzungsberichte der K. Akademie der Wissenschaften, Vienna 116 (1907), 601–655.
* [25] F. Haglund, D. Wise, Special cube complexes. Geom. Funct. Anal. 17 (2008), 1551–1620.
* [26] T. Hsu, D. Wise, Cubulating malnormal amalgams. Invent. Math. 199 (2015), 293–331.
* [27] I. Hughes, Division rings of fractions for group rings. Comm. Pure Appl. Math., 23 (1970), 181–188.
* [28] A. Jaikin-Zapirain, On linear just infinite pro-$p$ groups. J. Algebra 255 (2002), 392–404.
* [29] A. Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Math. J. 166 (2017), 1955–1987.
* [30] A. Jaikin-Zapirain, The base change in the Atiyah and the Lück approximation conjectures. Geom. Funct. Anal. 29 (2019), 464–538.
* [31] A. Jaikin-Zapirain, $L^{2}$-Betti numbers and their analogues in positive characteristic, Groups St Andrews 2017 in Birmingham, 346–406, London Math. Soc. Lecture Note Ser., 455, Cambridge Univ. Press, Cambridge, 2019.
* [32] A. Jaikin-Zapirain, An explicit construction of the universal division ring of fractions of $E\langle\\!\langle x_{1},\ldots,x_{d}\rangle\\!\rangle$, J. Comb. Algebra 4 (2020), 369–395.
* [33] A. Jaikin-Zapirain, The universality of Hughes-free division rings, Sel. Math. New Ser. 27 (2021), 74.
* [34] A. Jaikin-Zapirain, D. López-Álvarez, The strong Atiyah and Lück approximation conjectures for one-relator groups, Math. Ann. 376 (2020), 1741–1793.
* [35] A. Jaikin-Zapirain, I. Morales, Parafree fundamental groups of graph of groups with cyclic edge subgroups, preprint (2021), arXiv:2110.11655.
* [36] B. Jiang, H. Li, Sylvester rank functions for amenable normal extensions, Journal of Functional Analysis 280 (2021), 108913, 47 pp.
* [37] O. Kharlampovich, A. Miasnikov. Elementary theory of free nonabelian groups, J. Algebra, 302 (2006), 451–552.
* [38] The Kourovka notebook. Unsolved problems in group theory. Nineteenth edition. March 2019 update. Edited by E. I. Khukhro and V. D. Mazurov. Sobolev Institute of Mathematics. Russian Academy of Sciences. Siberian Branch, Novosibirsk, 2018.
* [39] M. Lazard, Groupes analytiques p-adiques, Inst. Hautes Études Sci. Publ. Math. No. 26 (1965), 389–603.
* [40] J. Lewin, On the intersection of augmentation ideals. J. Algebra 16 (1970), 519–522.
* [41] H. Li, Bivariant and extended Sylvester rank functions, J. London Math. Soc. (2021), 222–249.
* [42] P. Linnell, Division rings and group von Neumann algebras. Forum Math. 5 (1993), 561–576.
* [43] D. López-Álvarez, Sylvester rank functions, epic division rings, and the strong Atiyah conjecture for locally indicable groups, Ph.D. Thesis 2021, Universidad Autónoma de Madrid, https://www.icmat.es/Thesis/2021/Tesis_Diego_Lopez_Alvarez.pdf.
* [44] A. Lubotzky, A. Shalev, On some $\Lambda$-analytic pro-$p$ groups. Israel J. Math. 85 (1994), 307–337.
* [45] W. Lück, Approximating $L^{2}$-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4 (1994), 455–481.
* [46] W. Lück, D. Osin, Approximating the first $L^{2}$-Betti number of residually finite groups. J. Topol. Anal. 3 (2011), 153–160.
* [47] W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations. Dover Publications, Inc., Mineola, NY, 2004.
* [48] A. Malcev, On isomorphic matrix representations of infinite groups. (Russian) Rec. Math. [Mat. Sbornik] N.S. 8 (50), (1940), 405–422.
* [49] A. I. Malcev, On the embedding of group algebras in division algebras. (Russian) Doklady Akad. Nauk SSSR (N.S.) 60 (1948). 1499–1501.
* [50] P. Malcolmson, Determining homomorphisms to skew fields. J. Algebra 64 (1980), 399–413.
* [51] I. Morales, Embeddings into pro-$p$ groups and the construction of parafree groups, Master Thesis, Universidad Autónoma de Madrid (2021), arXiv:2109.12341.
* [52] A. Myasnikov, V. Remeslennikov, Exponential groups I: foundations of the theory and tensor completion. Siberian Math. J., 5 (1994), 986–996.
* [53] A. Myasnikov, V. Remeslennikov, Exponential groups. II. Extensions of centralizers and tensor completion of CSA-groups. Internat. J. Algebra Comput. 6 (1996), 687–711.
* [54] B. H. Neumann, On ordered division rings. Trans. Amer. Math. Soc. 66, (1949).
* [55] N. Nikolov, D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds. Ann. of Math. (2) 165 (2007), no. 1, 171–238.
* [56] L Ribes, P. Zalesskii, Profinite groups. Second edition. A Series of Modern Surveys in Mathematics, 40. Springer-Verlag, Berlin, 2010.
* [57] J. J. Rotman, An introduction to homological algebra. Second edition. Universitext. Springer, New York, 2009.
* [58] J. Sánchez, On division rings and tilting modules, Ph. D. Thesis, Universitat Autònoma de Barcelona (2008), www.tdx.cat/bitstream/handle/10803/3107/jss1de1.pdf.
* [59] A. Shalev, Lie methods in the theory of pro-$p$ groups. New horizons in pro-$p$ groups, 1–54, Progr. Math., 184, Birkhäuser Boston, Boston, MA, 2000.
* [60] R. B. Warfield, Nilpotent groups. Lecture Notes in Mathematics, Vol. 513. Springer-Verlag, Berlin-New York, 1976.
* [61] D. Wise, Research announcement: the structure of groups with a quasiconvex hierarchy. Electron. Res. Announc. Math. Sci. 16 (2009), 44–55.
* [62] E. Zelmanov, Infinite algebras and pro-$p$ groups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math. 248, Birkhäuser, Basel, 2005, 403–413.
* [63] E. Zelmanov, Groups with identities, Note. Mat. 36, 2016, 101–113.
* [64] A. N. Zubkov, Nonrepresentability of a free nonabelian pro-$p$-group by second-order matrices. (Russian) Sibirsk. Mat. Zh. 28 (1987), 64–69, English translation: Siberian Math. J. 28 (1987), 742–746.
|
# Fast convolution kernels on pascal GPU with high memory efficiency
Qiong Chang
Grad. of Systems and Information Engineering
University of Tsukuba
Tsukuba, Japan, 305-8577
<EMAIL_ADDRESS>
&Masaki Onishi
National Institute of Advanced Industrial Science and Technology (AIST)
Tsukuba, Japan, 305-8560
<EMAIL_ADDRESS>
&Tsutomu Maruyama
Grad. of Systems and Information Engineering
University of Tsukuba
Tsukuba, Japan, 305-8577
<EMAIL_ADDRESS>
###### Abstract
The convolution computation is widely used in many fields, especially in CNNs.
Because of the rapid growth of the training data in CNNs, GPUs have been used
for the acceleration, and memory-efficient algorithms are focused because of
thier high performance. In this paper, we propose two convolution kernels for
single-channel convolution and multi-channel convolution respectively. Our two
methods achieve high performance by hiding the access delay of the global
memory efficiently, and achieving high ratio of floating point Fused Multiply-
Add operations per fetched data from the global memory. In comparison to the
latest Cudnn library developed by Nvidia aimed to accelerate the deep-learning
computation, the average performance improvement by our research is 2.6X for
the single-channel, and 1.4X for the multi-channel.
## 1 Introduction
Convolution is widely used as a fundamental operation in many applications
such as computer vision, natural language processing, signal processing.
Especially, the Convolution Neural Network (CNN), a popular model used for
deep-learning, is widely used in many applications such as image recognition
[2][6], video analysis[3], natural language processing[7], and has yielded
remarkable results.
Recently, many CNN models have been proposed, such as AlexNet [15], GoogleNet
[11], VGGNet [6], ResNet [9], etc. They are used in many areas, and are
improved steadily. The sizes of their networks have grown larger, and this
leads to the increase of their processing time. The CNNs have several layers,
but a large portion of the total processing time is used for the convolution
layers. Because of the high inherent parallelism of the convolution
algorithms, many researchers are exploring to use GPUs to accelerate them.
They can be divided into four categories: 1) Direct-based method, 2) FFT-based
method, 3) Winograd-based method and 4) General matrix multiplication (GEMM)
based method.
Recently, many of the algorithms and their modified versions have been
aggregated into a public library called Cudnn by Nvidia, which aims to
accelerate the deep-learning platforms like Chainer[4], Caffe[14], etc. As the
volume of the processing data used in deep-learning increases, the memory-
efficient algorithms play an increasingly more important role, resulting in
the constant proposal of many improved versions. Among them, Implicit-GEMM[12]
has been included in the Cudnn because of its high efficiency. It divides the
feature map and the filter data into many sub-blocks and converts them to many
sub-matrices by using only on-chip memory of GPU, not using global memory.
This method is very memory-efficient, and achieved a high ratio of floating
point Fused Multiply-Add (FMA) operations per data transferred from the global
memory. In [1], two memory-efficient methods were proposed. Both of them are
faster than the Implicit-GEMM, however their performances are negatively
affected when the feature map size is smaller than 32, because it fixes the
amount of the data assigned to each $SM$, which sometimes is not suitable to
the small feature map. In CNN models such as [15][11][6][9], more than half of
the convolution layers are used for the calculation of the images smaller than
32 (such as 28, 14, 7). This means that [1] cannot handle the modern CNN
models efficiently.
In this paper, we propose two methods to accelerate the convolution kernels
for single-channel and multi-channel respectively. In these methods, the
memory access is optimized to achieve higher performance. For the single-
channel kernel, our approach follows the computation method proposed in [1],
however, the data are divided and assigned to each $SM$ carefully to hide the
access delay of the global memory considering the input data size and the
hardware features of our target GPUs (Pascal GPUs). For the multi-channel
kernel, we propose a stride-fixed block method. This method aims to maximize
the number of FMA operations per loaded data because the total amount of data
that have to be loaded to each $SM$ is much larger than the single-channel
convolution, and the access delay can be hidden by data prefetching.
## 2 Optimization Methods
In this section, we first introduce the CNN models, and then discuss what kind
of optimization method is applicable on Pascal GPUs.
### 2.1 The Convolution Models
The multi-channel convolution can be defined as follows:
$\begin{split}O^{m}(x,y)=\sum_{ch=1}^{C}\sum_{i=0}^{K-1}\sum_{j=0}^{K-1}I^{ch}(x+i,y+j)\cdot
F^{ch,m}(i,j),\\\ where\hskip
14.22636ptx\in[0,W_{x}-K+1),y\in[0,W_{y}-K+1),m\in[1,M].\end{split}$ (1)
Here, $I$ is the input feature map set and $F$ is the input filter set. $O$ is
the output feature map set which is generated from $I$ and $F$. $x$ and $y$
are the coordinates of the pixels of the feature maps. $W_{x}$ is the width
and $W_{y}$ is the height of the input feature map. $i$ and $j$ are the
offsets, and are added to the coordinates. Their upper bound $K$ decides the
size of filter. $ch$ represents the channel of the input in the range of
$[1,C]$ ($C$ is the number of channels), and all of the convolution results
are added along the dimension $ch$. $m$ represents the filter number, and each
filter has $C$ channels. $M$ is the number of filters, and it is defined in
each convolution layer. When $C=1$, it is called single-channel convolution,
and its definition is given the following equation.
$\begin{split}O^{m}(x,y)=\sum_{i=0}^{K-1}\sum_{j=0}^{K-1}I(x+i,y+j)\cdot
F^{m}(i,j)\\\ \end{split}$ (2)
### 2.2 Acceleration models on GPU
Here, we discuss the acceleration methods of the convolution calculation.
However, this discussion is not restricted to the convolution, and can be
applied to other applications.
In GPUs, the on-chip memory including registers and shared memory, and the
off-chip memory, mainly the global memory are supported. The register is the
fastest, and the global memory is the slowest and largest. To fully utilize
this hierarchy, many studies such as [1] have been proposed. However,
throughout all computation, data loading time from the global memory to the
on-chip memory is most critical, and hiding the latency of the global memory
is the most important point for the acceleration.
To hide the latency of the global memory, two methods can be considered:
1. 1.
keep the operation units busy (mainly Fused Multiply-Add (FMA) operation units
in convolution) by executing more than $N_{FMA}$ operations (the lowest value
to make the units busy) in each $SM$ for the current data set until the next
data set arrive from the global memory by data prefetching, and
2. 2.
transfer a large volume of data ($V_{s}$) from the global memory continuously.
In most cases, the first approach is preferable, because the data loading
overhead from the global memory can be relatively reduced more by executing
more number of operations per loaded data. In the multi-channel convolution,
the data size is large enough, and it is possible to find the division of the
feature maps and filters that makes it possible to execute more than $N_{FMA}$
operations in each $SM$. However, in the single-channel convolution, when the
size of feature maps is small, the number of executable operations becomes
less than $N_{FMA}$ even with the data prefetching, and the second approach is
required. Thus, it is necessary to make it clear under what conditions which
method shows better performance.
Table 1 shows several parameters of GTX 1080Ti and its performance for
accessing single precision data. As shown in Table 1, in GTX 1080Ti, 2 FMA
operations can be executed in one clock cycle in each core, namely 256 FMA
operations in each $SM$ (each $SM$ has $N_{cores}$ = 128 cores). According to
the method proposed in [5], the global memory latency of the GTX 1080Ti is 258
clock cycles. In order to hide this 258 clock cycles, $N_{FMA}=66,048$ FMA
operations ($66,048=258\times N_{cores}\times 2$) are required in each $SM$
for the current data set (the set of divided feature maps and filters).
The volume size $V_{s}$ can be calculated as follows. The Geforce GTX 1080Ti
has a base clock of 1480 MHz and the bandwidth of 484 GB/s, which means the
transfer rate is roughly 327 bytes per clock cycle. Therefore, the volume size
which is needed to hide the latency (258 clock cycles) becomes 84,366 = 327
$\times$ 258 bytes. To realize the data transfer of this size, 21,120 ($=$
84,366 / 4) threads are required because each thread fetches a 4 byte data in
single precision. Thus, in each of 28 $SMs$ ($N_{sm}=28$ is the total number
of $SMs$ in the GTX 1080Ti), 768 threads are required to fetch one 4-byte word
respectively (in total, it becomes 768 $\times 4\times 28=$ 86,016 > 84,366).
This means that the minimum volume size to make the global memory busy is
$V_{s}=$ 86,016 bytes. For dividing the feature maps and filters, and
assigning them to each $SM$, the following procedure should be taken:
1. 1.
Divide the feature maps and filters so that the total size of data that are
assigned to each $SM$ is smaller than the size of the shared memory
$S_{shared}$ (96KB in GTX 1080Ti).
2. 2.
Evaluate the number of FMA operations that can be executed for the data in
each $SM$.
3. 3.
If it is larger than $N_{FMA}$, use the the first method which is based on the
data prefetching.
4. 4.
If not, redivide the feature maps and filters so that the total size of data
that are transferred to all $SMs$ becomes larger than $V_{s}$, and use the
second approach.
Additionally, for accessing global memory, it is necessary to confirm that the
starting address and the size of the sequential accessing segment is a
multiple of 32-byte. In Pascal GPU, a multiple of 128-byte shows better
performance than that of 32-byte and 64-byte, but the performance for 32-byte
and 64-byte is acceptable.
Table 1: Parameters to access single precision data | GTX 1080Ti
---|---
Architecture | Pascal
Global Memory Latency (clock cycles) | 258
Bandwidth (Gb/s) | 484
Base clock cycle (MHz) | 1480
SM | 28
Transmission Rate (Byte/clock cycle) | 327
Data Requirement (bytes) | 84,366
Thread Requirement/SM | 768
Warp Requirement/SM | 24
Data Requirement/SM (bytes) | 3072
Flops/clock cycle/core | 2
### 2.3 Data Mapping
As shown in Fig.1(a), in the single-channel convolution ($C=1$), the size of
each filter is $K\times K\times{4\mbox{-}byte}$, and they are stored in the
global memory continuously. With this data mapping, the filters can be divided
only along the dimension $m$, and the filters can be loaded from the global
memory efficiently because they are stored continuously. Three approaches for
dividing the feature maps and filters can be considered.
1. 1.
Only the filters are divided. They are assigned to each $SM$, and in each
$SM$, the assigned filters are applied to the whole feature maps (the feature
map is processed sequentially against each filter).
2. 2.
Only the feature maps are divided. They are assigned to each $SM$, and in each
$SM$, the assigned feature maps are processed by all filters (the filters are
applied sequentially).
3. 3.
Both feature maps and filters are divided, and they are assigned to each $SM$
(the combination of the first and the second approach).
By using different approach, the amount of data that has to be loaded to the
shared memory memory from the global memory, and the number of FMA operations
that can be executed in parallel become different. Therefore, finding a good
balance between the size of divided feature maps and filters becomes a key
point.
(a) Single-Channel (b) Multi-Channel
---
Figure 1: Memory storage form of the Filter (a) (b)
---
(c) (d)
(e)
Figure 2: Assignment of the input data
In the multi-channel convolution ($C>1$), which is the typical case in the
convolution layers of the CNN except for the first one, the data size becomes
much larger than the single-channel convolution. Fig.1(b) shows how the
filters are stored in the global memory. They are stored along the dimension
$ch$ first, and then along the dimension $m$. In this case, the dividing
method of the filters along the dimension $m$ as used for the single-channel
convolution can not be applied as it is, because the data size of each filter
is normally not a multiple of 32-byte. Especially when $K=1$ (the filter size
is only 4 bytes), the filters are accessed as 4-byte segments, and it causes
serious performance reduction because of non-coalescing memory access.To solve
this problem, several approaches can be considered. Fig.2(a) shows the whole
data structure before the data division.
1. 1.
In Fig.2(b), both the filters and the feature maps are divided along the
dimension $ch$, and the data for $C^{\prime}=C/N_{sm}$ channels are assigned
to each $SM$ ($N_{sm}$ is the total number of $SMs$). With this division, the
data calculated in each thread have to be summed along dimension $ch$. This
means that add operations among $SMs$ are required, and
$W_{x}\times{W_{y}\times{C^{\prime}}\times{4\mbox{-}byte}}$ byte in the global
memory are used for this summation. The global memory accesses to this area
and the synchronous operations required for this summation considerably reduce
the overall performance.
2. 2.
In Fig.2(c), only the filters are divided along the dimension $m$.
$M^{\prime}\times C=M/N_{sm}\times C$ filters are assigned to each $SM$, and
the whole feature map are loaded to $SMs$ from the global memory. In this
approach, if the total size of the filters is less than the total size of all
shared memory ($K\times K\times C\times M\times 4\mbox{-}byte<N_{sm}\times
S_{shared}$), the divided filters can be cached in each $SM$, and no
additional access to the global memory is required.
3. 3.
On the other hand, in Fig.2(d), only the feature maps are divided along
dimension $y$. The divided feature maps are assigned to each $SM$, and the
whole filters are loaded to $SMs$ from the global memory. In this case, if the
total size of the feature maps is smaller than the total size of the shared
memory, the divided feature maps can be cached in each $SM$, and no additional
access to the global memory is required.
4. 4.
However, the total size of the filters and feature maps are larger than the
total size of the shared memory in general. Thus, as shown in Fig.2(e), both
the filters and feature maps have to be divided respectively, and each divided
segments is cached in each $SM$ or loaded from the global memory. In this
case, there exist many alternatives for how to divide the filters and features
maps.
According to our preliminary evaluation, the performance with the data
dividing method along the dimension $ch$ (Fig.2(b)) is obviously slower than
other dividing methods because of the additional access to the global memory
for the addition. For achieving higher performance, it is necessary to choose
other dividing methods considering the data size and the hardware features of
the target GPUs so that in each $SM$ the number of FMA operations that can be
executed per loaded data from the global memory is maximized.
## 3 GPU Implementation
According to the discussion in Section 2, in both the single-channel and
multi-channel convolution, it is important to make the number of FMA
operations per pre-fetched data higher than $N_{FMA}$ in order to achieve
higher performance. However, in some cases of the single-channel convolution,
for example when the size of feature maps is small, the number of FMA
operations cannot be kept high enough by data prefetching. This means that, in
case of single channel convolution, according to the size of input data, we
need to choose one of the two methods described in Section 2.2: data
prefetching or data transfer larger than $V_{s}$.
In the multi-channel convolution, the size of input data is large enough, and
the number of FMA operations can be kept high enough by data prefetching.
However, the performance can be improved more by achieving higher FMA
operation ratio for the fetched data, because to fetch the data from the
global memory, each thread has to issue the instruction to read data, and the
clock cycles are spent for issuing these read instructions. Therefore, in the
multi-channel convolution, to find the data dividing method that maximizes the
number of FMA operations for each divided data is the key to achieve higher
performance.
### 3.1 Single-Channel Convolution
Here, we describe how to divide the input data to achieve higher performance
in the single channel convolution. As for the convolution calculation in each
$SM$, we follow the method proposed in [1].
As shown in equation (4), the total amount of the input data is given by:
$\begin{split}D_{input}=D_{filter}+D_{map}=(K\times{K}\times{M}+W_{x}\times{W_{y}})\times{4}\,Bytes.\end{split}$
(3)
Let $N_{sm}$ be the number of $SMs$. There are two ways to divide the input
data and assign them to each $SM$. In the first method, the input data is
divided along the dimension $m$ of the filter. $D_{1}$, the size of input data
assigned to each $SM$, becomes
$\begin{split}D_{1}=\frac{D_{filter}}{N_{sm}}+D_{map}=(K\times{K}\times{\lceil\frac{M}{N_{sm}}\rceil}+W_{x}\times{W_{y}})\times{4}\,Bytes\end{split}$
(4)
In general, $D_{map}$ is too large to be stored in the on-chip memory of each
$SM$. Thus, $D_{map}$ is divided into $P$ pieces along the dimension $y$. The
size of each piece becomes $D_{Inc1}=D_{map}/{P}$. Here, for each line of
feature map, since the convolution requires additional $K-1$ lines, the amount
of data that have to be held in the on-chip memory becomes
$\begin{split}D_{1}=\frac{D_{filter}}{N_{sm}}+D_{Inc1}+(K-1)*W_{x}=(K\times{K}\times{\lceil\frac{M}{N_{sm}}\rceil}+(\lceil\frac{W_{y}}{P}\rceil+K-1)\times{W_{x}})\times{4}\,Bytes\end{split}$
(5)
and the number of FMA operation that can be executed for these data in each
$SM$ is given by
$\begin{split}Th_{1}=\frac{D_{filter}}{N_{sm}}\times{D_{Inc1}}=K\times{K}\times{\lceil\frac{M}{N_{sm}}\rceil}\times{\lceil\frac{W_{y}}{P}\rceil\times{W_{x}}}.\end{split}$
(6)
In the second method, the input data is divided along the dimension $y$ of the
feature map. In this case, $D_{2}$, the amount of the input data assigned to
each $SM$, becomes
$\begin{split}D_{2}=D_{filter}+\frac{D_{map}}{N_{sm}}=(K\times{K}\times{M}+({\lceil\frac{W_{y}}{N_{sm}}\rceil+K-1})\times{W_{x}})\times{4}\,Bytes.\end{split}$
(7)
$D_{filter}$ is too large to be stored in the on-chip memory in general, and
it is divided into $Q$ pieces. The size of each piece becomes
$D_{Inc2}=D_{filter}/{Q}$. Then, $D_{2}$ becomes
$\begin{split}D_{2}=D_{Inc2}+\frac{D_{map}}{N_{sm}}=\frac{D_{filter}}{Q}+\frac{D_{map}}{N_{sm}}=(K\times{K}\times{\lceil\frac{M}{Q}\rceil}+(\lceil\frac{W_{y}}{N_{sm}}\rceil+K-1)\times{W_{x}})\times{4}\,Bytes\end{split}$
(8)
and the number of FMA operation that can be executed for these data in each
$SM$ is given by
$\begin{split}Th_{2}=D_{Inc2}\times\frac{D_{map}}{N_{sm}}=K\times{K}\times{\lceil\frac{M}{Q}\rceil}\times({\lceil\frac{W_{y}}{N_{sm}}\rceil})\times{W_{x}}.\end{split}$
(9)
The values of $P$ and $Q$ are decided considering if $D_{1}$ or $D_{2}$ is
smaller than $S_{shared}$, and if $Th_{1}$ or $Th_{2}$ is larger than
$N_{FMA}$. If $P=1$ or $Q=1$, the feature maps or the filters are not divided,
and they are transferred to the on-chip memory at a time. If $P>1$ or $Q>1$,
the feature maps or the filters are divided into several pieces, and the
pieces are transferred to each $SM$ by using the data prefetching. With
smaller $P$ and $Q$, $D_{1}$, $D_{2}$ and $Th_{1}$, $Th_{2}$ become larger.
The lower bound of $P$ and $Q$ is given by the requirement that $D_{1}$ and
$D_{2}$ have to be smaller than $S_{shared}$, and the upper bound is given by
the requirement that $Th_{1}$ and $Th_{2}$ should be larger than $N_{FMA}$.
$P$ and $Q$ should be chosen so that these requirements can be satisfied. In
our implementation, $P$ and $Q$ are decided as follows.
1. 1.
$Th_{1}$ or $Th_{2}$ should be larger than the number of FMA operation
$N_{FMA}$.
$Th_{1}\geq N_{FMA}$ and $Th_{2}\geq N_{FMA}$
Thus, the upper bound of $P$ and $Q$ (they must be smaller than $W_{y}$ and
$M$ respectively) is given as follows
$P\leq\frac{K\times{K}\times{\lceil\frac{M}{N_{sm}}\rceil}\times{W_{y}}\times{W_{x}}}{N_{FMA}}\,and\,P\leq{W_{y}}$,
$Q\leq\frac{K\times{K}\times{M}\times{\lceil\frac{W_{y}}{N_{sm}}\rceil}\times{W_{x}}}{N_{FMA}}\,and\,Q\leq{M}$
2. 2.
$D_{1}$ and $D_{2}$ must be smaller than the size of on-chip memory. The lower
bound of $P$ and $Q$ is given as follows.
$P\geq\frac{4\times{W_{y}}\times{W_{x}}}{S_{shared}-4\times{K\times{K\times{\lceil\frac{M}{N_{SM}}}\rceil}}+(1-K)\times{4}\times{W_{x}}}$,
$Q\geq\frac{4\times{M}\times{K}\times{K}}{S_{shared}-4\times{W_{x}}\times{(\lceil\frac{W_{y}}{N_{SM}}+K-1)}}$
Actually, there exist one more requirement to decide this lower bound. The
number of required registers for the computation must be smaller than that
supported in each $SM$. Its detail is not shown here, but considering this
requirement, the lower bound is calculated.
3. 3.
If there exist $P$ and $Q$ ($P$ and $Q$ must be an integer) in the range
specified by (2) and (3), Any of them can be used. In our current
implementation, the minimum ones are chosen as $P$ and $Q$, because the
smaller values means less number of division, and make the processing sequence
simpler. If no value exists, $P$ and $Q$ are set to 1.
4. 4.
Using the obtained $P$ and $Q$, $D_{1}$ and $D_{2}$ are calculated and
compared. If $D_{1}$ is smaller than $D_{2}$, $Q$ is reset to 1 to use the
first dividing method described above, and otherwise, $P$ is reset to 1 to use
the second one. Both methods can be used because they both satisfy the
requirements, but for the safety (for leaving more memory space on the on-chip
memory), the smaller one is chosen.
Following this procedure, the input data are divided and allocated to each
$SM$ in the best balance.
### 3.2 Multi-Channel Convolution
As described above, in the multi-channel convolution, both feature maps and
filters are divided, and prefetching is used to transfer them to each $SM$
from the global memory. Recently, the block-based methods show high
performance in convolution due to their continuous and simple memory access
sequence. Fig.3 shows the data mapping of the filters and feature map, and how
they are divided and calculated in each $SM$. In the block-based method, as
shown in Fig.3(a), the following data are loaded to the on-chip memory in each
$SM$.
1. 1.
$S$ bytes of each filter along the dimension $ch$ (called segment in the
following discussion) of $M^{\prime}$ filters ($S\times{M^{\prime}}$ bytes in
total), and
2. 2.
a part of feature map, $W^{\prime}_{x}\times{W^{\prime}_{y}}\times{4\,bytes}$
in the same channel ($W^{\prime}_{x}$ is an arbitrary value that is decided by
the size of on-chip memory, but $W^{\prime}_{y}$ is specified as
$\lceil\frac{S}{K\times{4bytes}}\rceil$, because when $S$ bytes are fetched
along the dimension $ch$, $\lceil\frac{S}{K\times{4bytes}}\rceil$ lines in the
feature map are required to apply the filter).
Then, the convolution is calculated for these data, and the next data (next
$S\times M^{\prime}$ bytes of filters and $W^{\prime}_{y}\times
W^{\prime}_{x}$ bytes of feature maps are loaded by data prefetching. In [1],
the filter size is chosen as $S$ ($S=K\times{K}\times{4\,bytes}$), and only
the filters of the target channel and a part of feature map of the same
$channel$ are loaded to the on-chip memory. However, the filter size $K\times
K$ is usually odd and often small, and the performance is seriously degraded
because of non-coalescing memory access. [16] tried to solve this problem by
extending $S$ to 128-bytes. By fetching continuous 128 bytes on the global
memory, the highest memory throughput can be achieved in GPUs. In this method,
the filters of several channels (and a part of the next channel) are fetched
at the same time, and they are kept in the on-chip memory. First, only the
filters of the first channel are used for the computation, and then, the
filters of the next channel are used. With this larger $S$, $M^{\prime}$ has
to be kept small because of the limited size of on-chip memory, and smaller
$M^{\prime}$ means less parallelism ($M^{\prime}$ filters are applied in
parallel to the feature map of the same channel). In [1], higher parallelism
comes first, while in [16], lower access delay has a higher priority.
Figure 3: Multi-Channel Convolution Kernel
Here, we propose a stride-fixed block method not only to maintain the
efficient global memory access, but also to achieve high parallelism in each
$SM$.
1. 1.
$S$ is set to a multiple of 32-bytes. Actually, 32 or 64 is used. Small $S$
allows larger $M^{\prime}$, namely higher parallelism, under the limited size
of on-chip memory. When $S$ is 32 or 64 bytes, the memory throughput from the
global memory becomes a bit worse than $S=128$ bytes (the highest throughput),
but it is acceptable. $S=32$ is the minimum value to maintain efficient global
memory access.
2. 2.
Next, we fix $W^{\prime}_{x}$. $W^{\prime}_{x}$ pixels in the feature map are
fetched along the dimension $x$ from the global memory. Thus, $W^{\prime}_{x}$
should be a multiple of 128-bytes to achieve the highest memory throughput.
Larger $W^{\prime}_{x}$ is preferable because it increases the Instruction
Level Parallelism (ILP), which can improve the performance of the convolution.
3. 3.
After deciding the values of $S$ and $W^{\prime}_{x}$, the most suitable
$M^{\prime}$ can be found by the requirements of the number of FMA operations.
$M^{\prime}\geq\frac{N_{FMA}\times{4\mbox{-}bytes}}{S\times{W^{\prime}_{x}}}$.
4. 4.
Because the data prefetching is used to fetch the next data set while the
current data set is being used for the current calculation, the size of data
set cannot exceeds the half of the size of shared memory. Thus,
$(S\times{M^{\prime}}+\lceil\frac{S}{K\times{4bytes}}\rceil\times{W^{\prime}_{x}})\leq\frac{S_{shared}}{2}$
Here, $\lceil\frac{S}{K\times{4bytes}}\rceil=W^{\prime}_{y}$ is the number of
feature maps required for the calculation.
With this approach, for given $S$, $W^{\prime}_{x}$ and $M^{\prime}$ to
achieve high performance based on block method can be obtained.
From here, we describe how the convolution calculation is executed in each
$SM$. As shown in Fig.3(a)(b), first, each $SM$ loads $S\,bytes$ of
$M^{\prime}$ filters to the shared memory. At the same time, $W^{\prime}_{x}$
pixels on $\lceil\frac{S}{K\times{4bytes}}\rceil=W^{\prime}_{y}$ lines of the
feature maps are also loaded. After the first round loading of these data, the
same size of data for the next round are pre-fetched: the next $S\times
M^{\prime}$ bytes along the dimension $ch$, and the next $W^{\prime}_{x}$
pixels of the $W^{\prime}_{y}$ lines. During the second round loading, the
convolutions for the first round data set are calculated on the chip as shown
in Fig.3(b)(c). On the chip, each thread corresponds to one target pixel of
the feature map as shown in Fig.3(b). Because the accessing speed of registers
is faster than that of shared memory, it is required to transfer each data in
the shared memory to the registers in order to achieve high performance. In
the convolution computation, the target pixel and its neighbors in the feature
map are sent to the corresponding registers by each thread. Here, one
important point is that only $S/{4bytes}$ pixels in the feature map have to be
loaded onto the registers. The rest pixels, the red ones in Fig.3(b), are just
held in the shared memory for the next round. The filter data are also
transferred to the registers by the corresponding thread, but in this case,
all data are transferred to the registers because all of them are used. After
that, each pixel data is multiplied by the corresponding filter data, and
their products are added. When all computations for the data stored in on-chip
memory has been finished, data prefetching for the third round is started.
During this loading, the convolution for the second round data is calculated.
By using this method, the size of $S$ can be kept small, and the number of
filters $M^{\prime}$ can be increased. This ensures that more filters can be
applied in parallel to the same feature map. This does not increase the number
of data loading of the feature maps, and hides the latency caused by global
memory access.
## 4 Experimental analysis
We implemented our two convolution kernels on Pascal series GPU Geforce GTX
1080Ti by using the CUDA 8.0. Their performances were evaluated using many
convolutions which are commonly used in popular CNN models[15][9][6][11], and
compared with the latest public library Cudnn v7.1[12].
In the single-channel convolution, we changed the sample size of the feature
maps from 28 to 1K and the size of the corresponding channels from 512 to 32\.
The filter size is 1, 3 or 5, which is usually used in many CNN models. In
CUDA programming, by assigning more number of blocks to each $SM$, the $SMs$
can be kept busy. In our current implementation,
$N_{block}=2\times{N_{SM}}=2\times{28}$ blocks are used. Two blocks are
assigned to each $SM$, and 512 threads are assigned to each block. Thus, the
maximum number of registers for each thread is constrained to 128. For each
tested case, $P$ and $Q$ are decided following the method described in Section
3.1. Fig.5 shows the results of the single-channel convolution. Our method is
faster than Cudnn v7.1 in all tested cases. The performance gain is 1.5X to
5.6X, and its average is 2.6X. In the multi-channel convolution, we changed
the sample size of the feature maps from 7 to 512, and the size of the
corresponding channels from 64 to 512. The filter size is also 1, 3, or 5. As
discussed in Section 3.2, larger $M^{\prime}$ is preferable for making data
prefetching more effective. Therefore, we fixed the segment size $S$ as 32 or
64 bytes, and then $M^{\prime}$ and $W^{\prime}_{x}$ are decided following the
method described in Section 3.2. According to our preliminary evaluation, when
$M^{\prime}=64$ and $W^{\prime}_{x}=128$, the performance becomes best, and we
used these values for this comparison. As shown in Fig.5, our method is faster
than Cudnn in all tested cases, and the throughput has been increased by 1.05X
to 2X, with an average increase of 1.39X. In [1], a different GPU is used, and
a direct comparision is not possible. However, when $K=3$, our performance is
4X faster than [1] on GPU the peak performance of which is 2.4X faster than
that used in [1].
We also implemented our two kernels on Maxwell series GPU GTX Titan X, and it
also showed that our performance is faster than Cudnn on the same GPU by 1.3X
to 3.7X in the single-channel convolution and 1.08X to 1.8X in the multi-
channel convolution.
Filter Size = 1 Filter Size = 3 Filter Size = 5
---
Figure 4: Performance of the Single-Channel Convolution Kernel
Filter Size = 1 Filter Size = 3 Filter Size = 5
---
Figure 5: Performance of the Multi-Channel Convolution Kernel
## 5 CONCLUSIONS
In this paper, we proposed two convolution kernels on Pascal series GPUs for
single-channel and multi-channel respectively. For single-channel convolution,
we introduced an effective method of data mapping, which can hide the access
delay of the global memory efficiently. For multi-channel convolution, we
introduced a method that not only guarantees the memory access efficiency, but
also achieves high FMA operation ratio per loaded data. Performance comparison
with the public library Cudnn shows that our approaches are faster in all
tested cases: 1.5X to 5.5X in the single-channel convolution and 1.05X to 2X
in the multi-channel convolution. Our approaches was designed assuming Pascal
architecture, but the performance is also faster than Cudnn on Maxwell
architecture. This practice shows that our approaches can be applied to the
wide range of CNN models on various GPUs. In our current implementation, the
throughput is still lower than the theoretical maximum. It means that the
convolution kernel still has the room for improvement, and this is our main
future work.
## References
* [1] Xiaoming Chen, Jianxu Chen, Danny Z. Chen, Xiaobo Sharon Hu. “Optimizing Memory Efficiency for Convolution Kernels on Kepler GPUs,”In DAC, 2017.
* [2] A. Poznanski, L. Wolf, “Cnn-n-gram for handwriting word recognition,”in: CVPR, pp. 2305-2314, 2016.
* [3] D. Yu, W. Xiong, J. Droppo, A. Stolcke, G. Ye, J. Li, and G. Zweig, “Deep convolutional neural networks with layer-wise context expansion and attention,”in Proc.Interspeech, 2016.
* [4] Eddie Bell. “A implementation of squeezenet in chainer.”https://github.com/ejlb/squeezenet-chainer, 2016.
* [5] X. Mei, X. Chu, “Dissecting GPU memory hierarchy through microbenchmarking,”IEEE Trans. Parallel Distrib. Syst, 2016.
* [6] K. Simonyan and A. Zisserman. “Very deep convolutional networks for large-scale image recognition,”In ICLR, 2015.
* [7] Zhang, Y. and Wallace, B. “A sensitivity analysis of (and practitioners guide to) convolutional neural networks for sentence classification.”arXiv:1510.03820, 2015.
* [8] A. Lavin. “Fast algorithms for convolutional neural networks.”arXiv:1509.09308, 2015.
* [9] K. He, X. Zhang, et al. “Deep residual learning for image recognition,”arXiv:1512.03385, 2015.
* [10] S. Li, Y. Zhang, C. Xiang, and L. Shi. “Fast Convolution Operations on Many-Core Architectures.”In HPCC, pages 316-323, 2015.
* [11] Szegedy, C. et al. “Going deeper with convolutions.”Preprint at http://arxiv.org/abs/1409.4842, 2014.
* [12] Sharan Chetlur, Cliff Woolley, et al. “cuDNN: Efficient primitives for deep learning.”arXiv:1410.0759, 2014.
* [13] M. Mathieu, M. Henaff, et al. “Fast training of convolutional networks through ffts.”In CoRR, 2013.
* [14] Jia, Y. “Caffe: An open source convolutional architecture for fast feature embedding.” http://caffe.berkeleyvision.org/, 2013.
* [15] Krizhevsky, A., Sutskever, I., and Hinton, G. E. “ImageNet classification with deep convolutional neural networks,”In NIPS, pp. 1106-1114, 2012.
* [16] Guangming Tan, Linchuan Li, et al. “Fast implementation of DGEMM on Fermi GPU.”In Supercomputing 2011, pages 35:15:11, New York, NY, USA, 2011. ACM.
* [17] R. Nath, S. Tomov, and J. Dongarra. “An improved magma gemm for fermi gpus.”Technical Report 227, LAPACK Working Note, 2010.
|
# Quasidisorder Induced Topology
M. F. Madeira Departamento de Física and CeFEMA, Instituto Superior Técnico,
Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal P. D.
Sacramento Departamento de Física and CeFEMA, Instituto Superior Técnico,
Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(August 28, 2024)
###### Abstract
We study the effects of quasidisorder and Anderson disorder on a two
dimensional topological superconductor with an applied external magnetic
field. The cases of a $p$-wave superconductor and a noncentrosymmetric
superconductor with mixed $p$ and $s$-wave pairings and Rashba spin-orbit
coupling are studied. We show that, for a perpendicular magnetic field, the
introduction of quasidisorder leads to the appearance of topological phases in
new regions, characterised by an integer value of the Chern number. For a
parallel magnetic field, we identify regimes with the appearance of new
Majorana flat bands and also new unidirectional Majorana edge states, as
quasidisorder is introduced. We show that the Majorana flat bands have a
quantized Berry phase of $\pi$ and identify it as a topological invariant. Two
topological transitions are identified and the values of the critical
exponents $z$ and $\nu$ are obtained. The fractal nature of the eigenstates is
discussed both for Anderson disorder and Aubry-André disorder.
††preprint: APS/123-QED
## I Introduction
The search and study of topological properties of matter has proved fruitful
in recent years in research in materials science and condensed matter physics.
Superconductors have long been a focus of interest due to their promising
applications. Superconductors with intrinsic topological properties, in
particular, have recently attracted theoretical and experimental interest due
to phenomena associated with surface or edge Majorana modes, which appear from
an interplay between topology and bulk-boundary correspondence [1, 2, 3].
These Majorana zero modes emerge with non-Abelian exchange statistics and are
sought after due to their promising expected applications in quantum
computing, being candidates for the building blocks of a quantum qubit [4, 5].
It has been theoretically predicted that Majorana states appear as flat
dispersion bands in gapless superconducting phases, such as in the
$d_{xy}$+$p$-wave pairing noncentrosymmetric superconductor in two dimensions
with preserved time-reversal symmetry [6, 7], or for a $p$-wave topological
superconductor in two dimensions, with broken time reversal symmetry by an
applied magnetic field parallel to the two dimensional plane of the system
[8]. Flat bands also emerge on the surface of three dimensional
noncentrosymmetric superconductors, with spin-orbit coupling and which
preserve time-reversal symmetry [9, 10]. It is predicted that flat bands can
increase the critical temperature for superconductivity, and even give rise to
room-temperature superconductivity [11, 12]. Similar behavior has been found
when one has finite-size systems (with increased fluctuations of the density
of states) [13, 14], non-homogeneous order parameters [15, 16, 17, 18, 19,
20], or fractal (critical) states [21, 22, 23, 24, 25, 26, 27, 28, 29] with
corresponding spatial fluctuations of the amplitude of the wave functions. The
difference between an isolated flat band and a flat band with band touchings
has also been recently discussed [30]. It was shown that isolated flat bands
are not needed to achieve a higher superconducting temperature, and that band
touchings can actually increase it. Flat electronic bands can also be found in
some Kagome-type superconductors [31]. A growing interest has been seen in
these types of materials, AV3Sb5 (with A$=$K, Rb, Cs), which can host exotic
quantum properties, displaying topological phases, an unconventional charge
density wave, and evidence of time-reversal symmetry breaking [32, 33, 34,
35].
The study of perturbations in condensed matter systems, namely through the
introduction of disorder, is a central issue. On one hand, introducing
disorder can destroy some phases and their properties, preventing their
experimental observation. In this sense, the study of their robustness becomes
crucial. On the other hand, disorder can by itself lead to new phenomena or
stabilize previously existing phases. One type of disorder that has been
attracting interest in the research field is quasiperiodic disorder. These
systems are somewhat in between periodic and truly random systems, and exhibit
interesting phenomena, in transport [36, 37], topological properties [38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], and critical behaviour [36,
52, 53, 54]. It is possible to realize these types of systems in experimental
setups of ultracold atoms [55, 56], in optical lattices [57, 58] or in
photonics systems [59]. In addition to systems subject to quasiperiodic
potentials, as in the Aubry-André model [60], there has been growing interest
in Moiré systems in which two incommensurate lattices are connected, or in
which layers of lattices are put in contact and rotated, such as the 2d
twisted bilayer graphene [61, 62, 63, 64, 65]. In such systems, a superlattice
potential is created from proximity coupling between the two lattices, which,
depending on the angle of rotation between the two, may exhibit
quasiperiodicity.
An example of the study of coexisting quasidisorder and superconductivity,
which is significant in the context of this work, is the one dimensional
Kitaev chain with Aubry-André modulation [66, 67, 68, 69]. Without
superconductivity the model has a topological nature revealed by its mapping
to a $2d$ quantum Hall system [70], maintaining a topological nature as we add
superconducting pairing. In general, the mappings involve a corresponding
model in a higher dimension in the form of some parent Hofstadter generalized
Hamiltonian. Topology in quasicrystals may be understood considering mappings
to higher dimensions, typically of the types $1d$ to $2d$ and $2d$ to $4d$. In
$1d$ with no superconductivity the model is self-dual (position and momentum
space) and there is a single transition from an extended state phase to a
phase where all the states are localized. At the transition point the system
has critical states. Generalized models show the existence of mobility edges,
such that there is a separation as a function of energy between extended and
localized states [71, 72, 73, 74, 75, 76, 77, 78] and the existence of hidden
dualities leads to a rich class of systems, where such edges appear [79]. The
introduction of $p$-wave pairing in the Aubry-André model leads to the
appearance of a finite extent region of critical (fractal) states, between the
regions of extended and localized states. Remarkably, the transitions between
localised and critical regimes have been studied and were found to deviate
from the known Aubry-André universality class [68, 69].
In this work we study a model of a two-dimensional superconductor with spin
triplet $p$-wave pairing, or mixed $p$ and $s$-wave pairings with Rashba spin-
orbit coupling, in the presence of a time reversal symmetry breaking magnetic
field. Some materials which are candidates for realizing triplet pairing
superconductivity include Sr2RuO4 [80], UPt3 [81] and CuxBi2Se3 [82]. In the
presence of $s$-wave pairing and Rashba spin orbit coupling, the model
describes a noncentrosymmetric superconductor, of which are examples
$\text{CePt}_{3}\text{Si}$ [83], $\text{CeIrSi}_{3}$ [84] and
$\text{CeRhSi}_{3}$ [85]. In the noncentrosymmetric regime the breaking of
inversion symmetry allows for the mixture of spin-triplet and spin-singlet
pairings. This mixing is expected to lead to novel phenomena such as higher
than usual values of the upper critical field [86, 87].
The clean model has been studied, in both the centrosymmetric and the
noncentrosymmetric regimes, and is known to possess diverse topological
properties. If time-reversal symmetry is preserved, the model displays gapless
Majorana edge states and is characterised by a $\mathbf{Z}_{2}$ invariant. The
observed properties when time-reversal symmetry is broken by an external
magnetic field are found to be very dependent on its direction in relation to
the two-dimensional superconducting plane. If the magnetic field is such that
it is perpendicular to the plane of the superconductor, the model has a rich
phase diagram indexed by the Chern number [88]. When the magnetic field is
parallel to the plane of the system, interesting phenomena, such as Majorana
flat bands or Majorana unidirectional states, appear on phases with a gapless
bulk [8, 89].
The effect of disorder may be considered in different ways. One possibility is
to consider a non-homogeneous magnetic field, achieved by inserting magnetic
impurities in the clean superconductor [90] which may give rise to or change
topological properties in the system. Examples include the addition of chains
of magnetic adatoms [91, 92], islands of magnetic impurities [93] or fully
random distributions of impurities [94]. Another possibility is to consider
potential scattering impurities on the superconductor in the presence of a
constant magnetic field, either perpendicular or parallel to the system. We
are interested in studying the effects of quasidisorder in these regimes.
Besides Aubry-André disorder, we will also consider Anderson disorder as a
comparison to the effects of quasi-periodicity.
Anderson localization does not require full randomness. If differences in
potential between sites are large enough compared to hoppings, one may expect
a transition to localized states. In addition to full randomness, a
quasidisordered potential leads to localization if the disorder amplitude is
large enough [95]. One expects that the Aubry-André quasiperiodic potential
should affect the long-range nature of states, and in particular topological
states that are by themselves of long-range nature. Aubry-André is expected to
be naturally of a multifractal nature. Anderson and Aubry-André are different
and, in particular, critical states due to Anderson appear at the transition
to localization while in the Aubry-André added to the Kitaev $1d$ model one
finds phases with this behavior (or in $2d$ a mixture of critical states in
the crossover to localization). Multifractality probes long distances and
therefore one expects that it may enhance superconductivity due to Chalker
scaling [96, 97], as expected and observed with other inhomogeneities.
Multifractal wave functions have larger spatial overlap and stronger state to
state correlations for states with similar energies.
As stated previously, quasiperiodicity may also lead to topological properties
[98, 99]. A $2d$ topological insulator plus quasiperiodic potential shows a
transition from a trivial insulator to a topological insulator. Flat
topological bands and eigenstate criticality have also been shown as a result
of a quasiperiodic perturbation in the context of the Bernevig-Hughes-Zhang
model plus $2d$ quasiperiodic potential [100].
The presence of gapless states in a system may also be associated with long-
distance behavior. For instance, nodal points of Weyl semimetals may lead to
interesting behavior in the presence of disorder. It has been shown that they
survive the presence of moderate disorder [101]. On the other hand, in the
case of gapless states of the form of nodal loops, any amount of disorder
mixes states. Disorder-driven multifractality has been shown in Weyl nodal
loops [102]. In the case of magic angle semimetals quasiperiodicity
generically leads to flat bands in nodal, semi-metallic structrures. A
transition from a Weyl semimetal to metal driven by quasiperiodic potential
has been found in $3d$ [52, 103].
While the influence of disorder, either Anderson or quasidisorder has been
extensively considered in the case of one-dimensional systems, including in
the presence of superconductivity, it is interesting to consider their effects
on a two-dimensional $p$-wave superconductor, and in particular in the
presence of a magnetic field. In the clean system the topology is influenced
by the orientation of the magnetic field and, in particular, the gapped or
gapless nature of the states may lead to different responses to disorder. As
stated before, the difference of symmetry classes plus disorder gives rise to
new universality classes. Also, topology may be induced by quasiperiodicity,
which leads to the expectation of new universality classes (beyond the usual
classification), as found in the one-dimensional case. In particular, one may
expect interesting effects with the interplay of quasiperiodicity due to the
presence of critical bulk states, and the existence of Majorana flat bands.
The long-range nature of the quasiperiodic potential and the intrinsic long-
range nature of the gapless states may lead to an interesting competition. A
distinction between Anderson disorder (with moderate intensity) and
quasidisorder is therefore interesting to consider, as shown in non-
superconducting systems, where for instance nodal points and nodal loops are
affected differently by Anderson disorder, or on a semimetal where imposing a
quasiperiodic potential leads to flat bands.
The rest of the paper is organized as follows. Section II introduces the model
of the Hamiltonian and the topological properties of the clean system are
discussed, first under a perpendicular and second under a parallel magnetic
field, respectively in subsections II.1 and II.2. In subsection II.2 we derive
the regions where the model is topological, and show that the topological
regions are characterized by a Berry phase of $\pi$. In section III we present
the results for the disordered model under a perpendicular magnetic field. We
show that the introduction of Aubry-André disorder leads to the appearance of
topological phases in new regions. In IV we present the results for the
disordered model under a parallel magnetic field. First we discuss the
localization properties of the system in real space under different types of
disorder, using the inverse participation ratio (IPR). We then turn to a mixed
space description and discuss the evolution of the system as Anderson or
Aubry-André disorder are introduced. We show that the introduction of Aubry-
André disorder leads to the appearance of new regimes: for the $p$-wave
superconductor, new gapless regimes with Majorana flat bands appear, and for
the noncentrosymmetric superconductor, new regimes with unidirectional edge
states appear. We then obtain the Berry phase using twisted boundary
conditions and show it is quantized to a value of $\pi$ for the quasidisorder
induced flat bands. Identifying it as a topological invariant, we study two
topological transitions and obtain the critical exponents $z$ and $\nu$, which
we find to deviate from the known universality classes. Finally, using the IPR
we study the nature of the eigenfunctions distinguishing between localized,
single-fractal and multifractal regimes in the thermodynamic limit for both
Anderson and Aubry-André disorder. We conclude in section V. Three appendices
discuss some further results on the disorder driven transitions under a
perpendicular magnetic field in Appendix A, the influence of the
dimensionality of the quasidisorder potential in Appendix B and the energy
spectra and density of states for the disordered noncentrosymmetric
superconductor in Appendix C.
## II Model Hamiltonian
In momentum space, the Bogoliubov-de Gennes (BdG) Hamiltonian matrix of the
two dimensional model is written as
$\mathcal{H}(\mathbf{k})=\left(\begin{array}[]{cc}\xi(\mathbf{k})+\mathbf{B}\cdot\bm{\sigma}&\Delta(\mathbf{k})\\\
\Delta^{\dagger}(\mathbf{k})&-\xi^{T}(-\mathbf{k})-\mathbf{B}\cdot\bm{\sigma}^{*}\end{array}\right)$
(1)
in a basis
$(\bm{c}_{\mathbf{k}}^{\dagger},\bm{c}_{-\mathbf{k}})=(c_{\mathbf{k}\uparrow}^{\dagger},c_{\mathbf{k}\downarrow}^{\dagger},c_{-\mathbf{k}\uparrow},c_{-\mathbf{k}\downarrow})$
with $c_{\mathbf{k}\sigma}^{\dagger}$ ($c_{\mathbf{k}\sigma}$) the creation
(annihilation) operator for an electron with momentum $\bm{k}=(k_{x},k_{y})$
and spin projection $\sigma$. In the BdG Hamiltonian,
$\xi(\bm{k})=\epsilon_{\bm{k}}\sigma_{0}+\mathbf{s}\cdot\bm{\sigma}$, where
$\epsilon_{\mathbf{k}}=\left[-2t\left(\cos k_{x}+\cos
k_{y}\right)-\mu\right]\sigma_{0}$ is the kinetic term, with $t$ the nearest-
neighbour hopping integral and $\mu$ the chemical potential,
$\mathbf{s}\cdot\bm{\sigma}=-\alpha(-\sin k_{y},\sin
k_{x},0)\cdot\bm{\sigma}=-\alpha\left[-\sin k_{y}\sigma_{x}+\sin
k_{x}\sigma_{y}\right]$ is the Rashba spin-orbit term with $\mathbf{s}$ the
spin-orbit vector. The term $\mathbf{B}\cdot\bm{\sigma}$ describes the Zeeman
coupling of the electrons with an external magnetic field $\mathbf{B}$ and
$\hat{\Delta}(\mathbf{k})=\left[\Delta_{s}+\mathbf{d}(\mathbf{k})\cdot\bm{\sigma}\right]\left(i\sigma_{y}\right)$
is the superconducting gap function. The pairing vector is chosen as
$\mathbf{d}=d(-\sin k_{y},\sin k_{x},0)$, so that $d$ is the $p$-wave pairing
amplitude and $\Delta_{s}$ is the $s$-wave pairing amplitude. The simultaneous
existence of $s$ and $p$-wave terms is possible with a nonzero spin-orbit
term, which breaks the parity symmetry.
The case of study is that of a system with periodic boundary conditions along
the $x$ direction and open boundary conditions in the $y$ direction, such as
in a cylinder geometry. Thus we can also write the Hamiltonian in a mixed
space, $(k_{x},y)$, where a Fourier transform to the reciprocal space is only
done in the $x$ direction. In this case, for each value of $k_{x}$ the
Hamiltonian matrix has a dimension $(4\times N_{y})\times(4\times N_{y})$,
where $N_{y}$ is the number of sites in $y$. It is also of interest to write
the Hamiltonian in real space. In this case the Hamiltonian matrix has
dimension $(4\times N)\times(4\times N)$ with $N=N_{x}\times N_{y}$ the total
number of sites and $N_{x}$, $N_{y}$ the number of sites in the $x$ and $y$
directions, respectively.
When $\mathbf{B}=0$, the system respects the time-reversal symmetry (TRS)
$\mathcal{T}=(\sigma_{0}\otimes\mathrm{i}\sigma_{y})$ and the particle-hole
symmetry (PHS) $\mathcal{P}=(\sigma_{x}\otimes\sigma_{o})$ such that
$\begin{split}&\mathcal{P}\mathcal{H}(\mathbf{k})\mathcal{P^{\dagger}}=-\mathcal{H}^{*}(-\mathbf{k}),\\\
&\mathcal{T}\mathcal{H}(\mathbf{k})\mathcal{T}^{\dagger}=\mathcal{H}^{*}(-\mathbf{k}),\end{split}$
(2)
and $\mathcal{T}^{2}=-1$, $\mathcal{P}^{2}=1$. Therefore the Hamiltonian
belongs to the DIII symmetry class, and if $|d|>|\Delta_{s}|$ the system has a
nontrivial $\mathbf{Z}_{2}$ number, displaying gapless counterpropagating
Majorana edge states [88, 8].
For $\mathbf{B}\neq 0$ the time-reversal symmetry is broken. The system
exhibits different topological properties whether the applied magnetic field
is perpendicular or parallel to the plane of the system, as will be now
discussed.
### II.1 Perpendicular Magnetic Field
Let us first consider the case in which the external magnetic field is
perpendicular to the plane of the system, $\mathbf{B}=(0,0,B_{z})$. We have a
gap closing point if one of the equations is satisfied [88]:
$\begin{split}(-4t-\mu)^{2}+\Delta^{2}_{s}=B^{2}_{z},\\\
\mu^{2}+\Delta^{2}_{s}=B^{2}_{z},\\\
(4t-\mu)^{2}+\Delta^{2}_{s}=B^{2}_{z}.\end{split}$ (3)
Eqs. 3 define the boundaries between regions in which the system has different
topological properties. At the gap closing points the $D$ class system with
broken time reversal symmetry undergoes topological transitions between gapped
phases with different Chern numbers. The phase diagram of the system (indexed
by the Chern number) is presented in Fig. 1(a) for $t=1$, $\Delta_{s}=0$, and
$d>0$.
Figure 1: Phase diagram for a) Chern number and b) winding number
$I(k_{y}=0,\pi)$ as a function of $\mu$ and $B_{z}$, for $t=1$,
$\Delta_{s}=0$, $d>0$.
The regimes with a Chern number of zero and $B_{z}<2$, $0<|\mu|<4t$ exhibit
edge states, besides having $C=0$. This can be explained by one additional
topological invariant. It can be defined noting that the Hamiltonian obeys a
particle hole symmetry $\mathcal{P}=(\sigma_{x}\otimes\sigma_{0})$ with
$\mathcal{P}\mathcal{H}(\mathbf{k})\mathcal{P}^{\dagger}=-\mathcal{H}^{*}(-\mathbf{k}).$
(4)
For the values $k_{y}=0$ and $k_{y}=\pi$, the Hamiltonian obeys
$\mathcal{H}^{*}(-\mathbf{k})=\mathcal{H}(\mathbf{k})$ and thus anticommutes
with $\mathcal{P}$, $\\{\mathcal{H}(\mathbf{k}),\mathcal{P}\\}=0$. Therefore
the basis which diagonalizes $\mathcal{P}$ anti-diagonalizes the Hamiltonian.
A winding number $I(k_{y})$ can then be defined as [88]
$\begin{split}I\left(k_{y}\right)=\frac{1}{4\pi
i}\int_{-\pi}^{\pi}dk_{x}\operatorname{tr}\big{[}q^{-1}(k_{x})\partial_{k_{x}}q(k_{x})-\\\
q^{\dagger-1}(k_{x})\partial_{k_{x}}q^{\dagger}(k_{x})\big{]},\quad
k_{y}=0,\pi,\end{split}$ (5)
with
$q(k_{x})=\left(\begin{array}[]{cc}-\epsilon_{\mathbf{k}}-B_{z}+id\sin
k_{x}&\Delta_{s}-i\alpha\sin k_{x}\\\ -\Delta_{s}+i\alpha\sin
k_{x}&-\epsilon_{\mathbf{k}}+B_{z}+id\sin k_{x}\end{array}\right)$ (6)
the anti-diagonal block of the Hamiltonian matrix. The values of $I(0)$ and
$I(\pi)$ inside each phase are represented in Fig. 1(b).
The invariant $I(k_{y})$ loses its meaning if a finite magnetic field in the
$y$ direction, $B_{y}$, is applied. However, we found that this is not true
for the Chern number. Fig. 2 shows phase diagrams indexed by the Chern number
as a function of $B_{z}$ and $B_{y}$ for three different values of $\mu$. In
this case the Chern number depends only on the value of
$\sqrt{B^{2}_{y}+B^{2}_{z}}$. Also note that the diagrams only concern values
of $B_{z}>0$, excluding the points where $B_{z}=0$ and $B_{y}\neq 0$. In Fig.
3 we present the phase diagram of the system as a function of $\mu$ and
$B_{z}$ for constant values of $B_{y}$.
Figure 2: Phase diagrams for ($B_{z}>0,B_{y}$), indexed by the Chern number,
obtained numerically for a) $\mu=0$, b) $\mu=1$ and c) $\mu=-3.5$ for
$\Delta_{s}=0$. Figure 3: Phase diagram indexed by the Chern number as a
function of $\mu$ and $B_{z}$ (with $B_{z}>0$), for $t=1$, $\Delta_{s}=0$,
$d>0$, and a) $B_{y}=2$ and b) $B_{y}=4.5$.
### II.2 Parallel Magnetic Field
Now let us consider the case in which the applied magnetic field is parallel
to the system, $\mathbf{B}=(B_{x},B_{y},0)$. This could be realized, for
instance, by threading a wire through the center of the superconductor in a
cylindrical geometry.
Taking first the $s$-wave term $\Delta_{s}$ and the spin-orbit term $\alpha$
to be zero, the eigenvalues of the Hamiltonian are given by
$E(\mathbf{k})=\pm\sqrt{z_{1}\pm 2\sqrt{z_{2}}},$ (7)
with
$\begin{split}z_{1}=\mathbf{d}\cdot\mathbf{d}+\epsilon^{2}_{\mathbf{k}}+\mathbf{B}\cdot\mathbf{B},\\\
z_{2}=\epsilon^{2}_{\mathbf{k}}(\mathbf{B}\cdot\mathbf{B})+(\mathbf{B}\cdot\mathbf{d})^{2}.\end{split}$
(8)
The gap closing points are solutions of the equation $z_{1}=2\sqrt{z_{2}}$,
which is equivalent to the two equations being simultaneously satisfied:
$\begin{split}&\mathbf{d}\cdot\mathbf{d}+\epsilon^{2}_{\mathbf{k}}=\mathbf{B}\cdot\mathbf{B},\\\
&(\mathbf{B}\cdot\mathbf{B})(\mathbf{d}\cdot\mathbf{d})=(\mathbf{B}\cdot\mathbf{d})^{2}.\end{split}$
(9)
Eqs. 9 simplify if we consider the magnetic field aligned with one of the
axes. Let us then take the magnetic field aligned with the $y$ direction,
$\mathbf{B}=(0,B_{y},0)$. In this case, the second equation simplifies to
$\sin{k_{y}}=0$ which implies the bulk gap will close at
$k_{y,0}=n\pi,n\in\mathbb{Z}$, provided there are values of $k_{x}$ that
satisfy the equations
$d^{2}\sin^{2}{k_{x}}+(-2t(\cos{k_{x}}\pm 1)-\mu)^{2}=B^{2}_{y}.$ (10)
When the $p$-wave superconductor is in a gapless phase, and for a certain
range of magnetic field, Majorana flat bands (MFBs) will appear in the system.
This will be discussed next.
When finite spin-orbit $\alpha$ and $s$-wave pairing $\Delta_{s}$ terms are
also considered, the flat bands will (for certain values of the magnetic
field) acquire a slope, giving origin to unidirectional Majorana edge states
(MESs). The appearance of such states is only possible with a gapless bulk,
where a counter-propagating bulk current is created to cancel the edge current
[8].
#### II.2.1 Flat bands: winding number and Berry phase quantization
When the system is subject to an applied magnetic field, it no longer respects
time-reversal symmetry. If the applied field has a generic form
$\mathbf{B}=(B_{x},B_{y},0)$ we can, however, take $k_{x}$ as a fixed
parameter of the Hamiltonian and find a set of symmetries that are only
satisfied in the $y$ direction. It is found that the Hamiltonian respects the
symmetries:
$\begin{split}\mathcal{T}_{k_{y}}^{-1}\mathcal{H}(k_{x},k_{y})\mathcal{T}_{k_{y}}=\mathcal{H}(k_{x},-k_{y}),\\\
\mathcal{P}_{k_{y}}^{-1}\mathcal{H}(k_{x},k_{y})\mathcal{P}_{k_{y}}=-\mathcal{H}(k_{x},-k_{y}),\\\
\end{split}$ (11)
where $\mathcal{T}_{k_{y}}=(\sigma_{z}\otimes\sigma_{z})K$ and
$\mathcal{P}_{k_{y}}=(\sigma_{y}\otimes\sigma_{y})K$ are, respectively,
defined as a ”time-reversal-like” symmetry and a ”particle-hole-like” symmetry
[8] with $\mathcal{T}_{k_{y}}^{2}=\mathcal{P}_{k_{y}}^{2}=1$ ($K$ is the
complex conjugate operator). From these we can define a third chiral-like
symmetry $\mathcal{S}_{k_{y}}=\mathcal{T}_{k_{y}}\mathcal{P}_{k_{y}}$:
$\begin{split}\mathcal{S}_{k_{y}}^{-1}\mathcal{H}(k_{x},k_{y})\mathcal{S}_{k_{y}}=-\mathcal{H}(k_{x},k_{y}).\\\
\end{split}$ (12)
Since we have that $\mathcal{T}_{k_{y}}^{2}=\mathcal{P}_{k_{y}}^{2}=1$, the
Hamiltonian belongs to the BDI symmetry class and, since the problem is
effectively reduced to one dimension, the system can be characterized by an
integer topological invariant. We can then write the Hamiltonian in the basis
where $\mathcal{S}_{k_{y}}$ is diagonal, in which the Hamiltonian takes an
anti-diagonal form. From here it is possible to obtain a winding number
$\mathcal{W}$ at each value of $k_{x}$. It can be shown [8] that the winding
number is calculated as
$\mathcal{W}(k_{x})=\frac{i}{\pi}\left[\log{\left(\frac{sgn(\mathcal{M}(k_{y}=0))}{sgn(\mathcal{M}(k_{y}=\pi))}\right)}\right]$
(13)
with
$\begin{split}&\mathcal{M}\left(k_{x},k_{y}\right)=\\\ &\left[\mu+2t\left(\cos
k_{x}+\cos
k_{y}\right)\right]^{2}+d^{2}\sin^{2}k_{x}-B_{y}^{2}+B_{x}^{2}.\end{split}$
(14)
In the regimes with $|\mathcal{W}|=1$ the system has a topological nature and
Majorana flat bands appear, as is shown in Fig. 4. These are protected by the
chiral symmetry $\mathcal{S}_{k_{y}}$ as defined in Eq. 12.
Figure 4: Energy spectrum, absolute value of the winding number $\mathcal{W}$
and Berry phase $\gamma$ normalized by $2\pi$, as a function of $k_{x}/\pi$.
The values of the parameters are $t=1$, $d=1/6$, $\mu=-3.5$ and a) $B_{y}=d$,
b) $B_{y}=3.5d$.
The existence of topological flat bands may also be identified by a non-
trivial Berry phase. In general, the Berry phase can take any real value. In
the presence of certain symmetry constraints, the Berry phase can become
quantized to 0 or $\pi$ and carry topological information (at the value of
$\pi$). This quantization can happen in the presence of inversion or chiral
symmetries, also leading to the quantization of polarization [104]. As the
problem is reduced to one dimension, we can obtain a Berry phase $\gamma_{B}$
at each value of $k_{x}$, given by:
$\gamma_{B}(k_{x})=i\int_{0}^{2\pi}dk_{y}\langle\Psi(k_{x},k_{y})\mid\frac{\partial}{\partial
k_{y}}\Psi(k_{x},k_{y})\rangle$ (15)
with $\Psi$ the ground-state wavefunction. The calculation is done numerically
by discretizing the Brillouin zone [104, 105, 106, 107] in the $y$ direction.
As is shown in Fig. 4, we have found that in the regimes with
$|\mathcal{W}|=1$, the Berry phase is also quantized to a value of $\pi$.
#### II.2.2 Domain of flat band existence: topological and gapless regions
From Eq. 13 it is found that $|\mathcal{W}|=1$ in the regimes where
$\mathcal{M}\left(k_{x},k_{y}=0\right)$ and
$\mathcal{M}\left(k_{x},k_{y}=\pi\right)$ have opposite signs. This is only
possible if $|B_{y}|>|B_{x}|$, thus this is a necessary condition for the
appearance of MFBs. The flat band regions can be summarized in (with
$\tilde{B}^{2}=B_{y}^{2}-B_{x}^{2}$):
* •
$(1)$ $\mu\geq 2t$
$\mathcal{D}_{+}>\tilde{B}^{2}>\mathcal{D}_{-}$ (16)
* •
$(2)$ $\mu\leq-2t$
$\mathcal{D}_{-}>\tilde{B}^{2}>\mathcal{D}_{+}$ (17)
* •
$(3)$ $-2t<\mu<2t$
$(\mathcal{D}_{+}>\tilde{B}^{2}>\mathcal{D}_{-})\vee(\mathcal{D}_{-}>\tilde{B}^{2}>\mathcal{D}_{+})$
(18)
where
$\mathcal{D}_{\pm}=\left[\mu+2t\left(\cos k_{x}\pm
1\right)\right]^{2}+d^{2}\sin^{2}k_{x}.$ (19)
Eqs. 16, 17 and 18 define the regions where the superconductor is in a
nontrivial regime with $|\mathcal{W}|=1$, for a certain value of $k_{x}$.
Furthermore, since MFBs can only appear in a gapless phase, the equations also
define the regions where the bulk is gapless, as a function of the in-plane
magnetic field.
Figure 5: a) Domain of existence of Majorana flat bands (shaded region) for
$B_{y}$ vs. $k_{x}$ for the parameters $t=1$, $d=1/6$, $\mu=-3.5$. b) Closeup
of a) in the region $B_{y}\in$ [$-1,1$] and $k_{x}\in$ [$-1.5,1.5$].
Note that the chiral-like symmetry that protects the flat bands is broken by
either a non-zero $s$-wave pairing term $\Delta_{s}$ or a non-zero spin-orbit
term $\alpha$. A finite perpendicular magnetic field $B_{z}$ is also found to
break the chiral-like symmetry, leading to the absence of flat bands. If the
flat band includes the point $k_{x}=0$, the addition of a finite $B_{z}$ will
lead to the appearance of bands with a finite slope that cross at zero energy
at $k_{x}=0$. Otherwise, the bands will be lifted to finite energy.
Figure 6: Phase diagrams indexed by the Chern number $C$ for a system with
20x20 sites, for several values of disorder strength $\lambda$ and
perpendicular magnetic field $B_{z}$, obtained for an average over 10 disorder
configurations. For Aubry-André disorder, each random disorder configuration
is obtained by selecting a random value of $\phi$. The first row with panels
a)-c) concerns the case of Anderson disorder (2d), the second row with panels
d)-f) concerns the case of Anderson disorder (1d along $y$, uniform along
$x$), the third row with panels g)-i) concerns Aubry-André disorder (1d along
$y$, uniform along $x$), and the fourth row with panels j)-l) concerns Aubry-
André disorder (2d). The values of the parameters are $t=1$, $d=0.6$ and
$\mu=0$ (left), $d=0.6$ and $\mu=1$ (middle), $d=1/6$ and $\mu=3d-4t=-3.5$
(right).
## III Disordered model under a perpendicular magnetic field
We first want to investigate the effects of quasidisorder and disorder on the
system subject to an applied magnetic field in the perpendicular direction,
$\mathbf{B}=(0,0,B_{z})$. Here we limit ourselves to the study of the system
in real space and, to classify the topological nature of the system, the Chern
number is obtained numerically [108]. We consider four different types of
disorder potentials:
1. 1.
Anderson disorder (2d), where the disorder term is random at each site and
varies with uniform probability within an interval:
$\Lambda(x,y)\in[-\lambda,\lambda].$ (20)
2. 2.
Anderson disorder (1d along $y$, uniform along $x$), where the potential is of
the same type as described above but varies only along the $y$ direction,
being uniform along the $x$ direction:
$\Lambda(x,y)=\Lambda(y)\in[-\lambda,\lambda].$ (21)
3. 3.
Aubry-André disorder (1d along $y$, uniform along $x$), where the disorder
term is a quasiperiodic potential of the form:
$\Lambda(x,y)=\Lambda(y)=\lambda\cos(2\pi\beta f(x,y)+\phi)$ (22)
with $f(x,y)$ a function of the lattice sites, $\beta=\frac{\sqrt{5}-1}{2}$
the inverse golden ratio, and $\phi$ a phase between $0$ and $2\pi$. Here we
take $f(x,y)=y$, so that the considered quasiperiodic potential is uniform in
the $x$ direction.
4. 4.
Aubry-André disorder (2d), where the disorder term is a sum of two
quasiperiodic potentials of the form:
$\Lambda(x,y)=\lambda\cos(2\pi\beta x+\phi)+\lambda\cos(2\pi\beta y+\phi)$
(23)
so that disorder potentials are introduced in both the $x$ and $y$ directions.
In Fig. 6 we show the phase diagrams indexed by the Chern number, for three
different values of $\mu$ and $d$ (with $t=1$ in all cases) and for a system
with size $20\times 20$.
When Anderson disorder is introduced in the system (first row), the
topological regimes are destroyed as the disorder strength is increased. There
is, however, some difference in robustness as a function of the magnetic
field. This is noticeable in Figs. 6(a) and (b), where we see that the
robustness of the topological phases increases with the increase of $B_{z}$.
In Fig. 6(a) and for a small region of magnetic field (for $B_{z}>4$) we
observe reentrant topology as disorder is increased, as in Fig. 6(b), for
lower values of magnetic field ($B_{z}<1$).
The second row of the figure is obtained when disorder is considered with
uniformity in the $x$ direction. Unexpectedly, the topological regions are to
be less robust if compared with the previous case where Anderson disorder was
considered with no modulation. Small traces of induced topology are observed
for $B_{z}<1$ in Fig. 6(e) and $B_{z}<0.5$ in Fig. 6(f).
For Aubry-André disorder uniform in the $x$ direction (third row) we obtain
phase diagrams with well defined boundaries, and with induced topological
regions. Here, the topological phases show an interesting and unexpected
response to the increase of quasidisorder. There is a clear difference in
robustness for different values of $B_{z}$, which originates the seemingly
effect of “peaks” and “valleys” in the phase diagram, respectively at more
robust and more vulnerable values of $B_{z}$. Induced topology is visible in
panels g)-i), with topological transitions to finite values of $C$ happening
at low and high values of the magnetic field with the increase of disorder.
The last row of Fig. 6 concerns the case of two-dimensional Aubry-André
disorder. The introduction of disorder leads to the appearance of new
topological regions, where several are characterized by values of $C$ that are
not seen in the clean system, in the range of $[-4,4]$. Also, some regions
appear where the Chern number oscillates within an interval between two
integer values, without tending clearly to one of them.
We may argue that by adding disorder, local fluctuations of $\mu$ may lead to
changes of the Chern number. This is particularly seen in the presence of
quasidisorder. This suggests that the long-range quasiperiodicity resonates
more with the calculation of the Chern number, that reflects the global
structure of the states. Further details on the effect of disorder are shown
in Appendix A.
Figure 7: Average IPR of the whole system as a function of disorder strength,
for a) Anderson disorder (2d), b) Anderson disorder (1d along $y$, uniform
along $x$), c) Aubry-André disorder (1d along $y$, uniform along $x$), d)
Aubry-André disorder (2d). The IPR is averaged over all eigenstates of the
system in a given disorder configuration, and averaged over 10 disorder
configurations. In a) fits are done to functions of the form
$IPR=C_{1}\exp{C_{2}\lambda}$ in the range $\lambda\in[1.5,3]$, giving the
values $(C_{1},C_{2})=(5.6\times 10^{-4},1.13)$ for $B_{y}=0.5d$ and
$B_{y}=3.5d$, and $(C_{1},C_{2})=(3.6\times 10^{-4},1.22)$ for $B_{y}=4d$,
$\alpha=0.2d$, $\Delta_{s}=0.5d$. In b) fits are done to functions of the form
$IPR=C_{1}\lambda+C_{2}$ in the range $\lambda\in[0.5,3]$, giving the values
$(C_{1},C_{2})=(4.4\times 10^{-3},5\times 10^{-5})$ for $B_{y}=0.5d$ and
$B_{y}=3.5d$, and $(C_{1},C_{2})=(2.8\times 10^{-3},1.4\times 10^{-4})$ for
$B_{y}=4d$, $\alpha=0.2d$, $\Delta_{s}=0.5d$.
## IV Disordered model under a parallel magnetic field
We now introduce disorder on the system with an applied parallel magnetic
field in the $y$ direction, $\mathbf{B}=(0,B_{y},0)$. In section A we consider
the system in real space, where the disorder term takes the same form as in
the previous section (Eqs. 20, 21, 22,23).
From section B onwards we study the system in a mixed $(k_{x},y)$ space. In
this case the disorder term is either of the form of Eq. 21 or Eq. 22, with
the potential varying in the $y$ direction and being the same for all $k_{x}$,
as:
1. 1.
Anderson disorder:
$\Lambda(y)\in[-\lambda,\lambda],$ (24)
2. 2.
Aubry-André disorder:
$\Lambda(y)=\lambda\cos(2\pi\beta y+\phi),$ (25)
where, as before, $\beta=\frac{\sqrt{5}-1}{2}$ is the inverse golden ratio and
$\phi$ is a phase between $0$ and $2\pi$.
### IV.1 Localization properties in real space
Here we briefly consider the system in real space and study its localization
properties in three different regimes. We fix the parameter values as $t=1$,
$d=1/6$, $\mu=-3.5$ and consider three different cases: the case of a $p$-wave
superconductor for which a magnetic field $B_{y}=0.5d$ is added, such that the
system is in a phase with a gapped bulk but gapless edge states; a $p$-wave
superconductor with an added magnetic field of $B_{y}=3.5d$, where the system
has a gapless bulk and is in the MFB regime; and a case of the
noncentrosymmetric superconductor, with $B_{y}=4d$ and added $s$-wave pairing
and spin-orbit terms, $\Delta_{s}=0.3d$ and $\alpha=0.2d$, where the system
has a gapless bulk and unidirectional MESs.
To quantify the effects of disorder on the system’s localization we use the
inverse participation ratio, IPR. For a given eigenstate labeled by $m$, the
IPR is defined as:
$\text{IPR}_{m}=\sum_{i}\left|\psi_{i}^{m}\right|^{4},$ (26)
with $\psi_{i}^{m}$ the wavefunction of the eigenstate $m$ at a site $i$. For
perfectly localized states we have that $\text{IPR}_{m}\sim 1$ and for
delocalized states $\text{IPR}_{m}\sim 1/N$. In Fig. 7 we present results for
the average IPR as a function of disorder for a system of size $N=N_{x}\times
N_{y}=41\times 41$ [109] and for the same types of disorder as before: Fig.
7(a) Anderson disorder (2d), Fig. 7(b) Anderson disorder (1d along $y$,
uniform along $x$), Fig. 7(c) Aubry-André disorder (1d along $y$, uniform
along $x$), and Fig. 7(d) Aubry-André disorder (2d).
From observation of Figs. 7(a)-(d) we find four qualitatively different
behaviours. In Fig. 7(a) (Anderson disorder) we see that the IPR shows an
exponential-like behaviour for $\lambda>1.5$. A fit of the form
$\text{IPR}=C_{1}\exp{C_{2}\lambda}$ is done in the range $\lambda\in[1.5,3]$,
and is presented in Fig. 7(a) in dashed lines. We find that for $\lambda>1.5$
the IPR follows an exponential behaviour closely, while for $\lambda<1.5$
there is a deviation from it. As disorder is increased, the low energy states
become increasingly localized inside the bulk. From inspection of the
wavefunctions we observed that the edge states quickly lose their structure
for low values of disorder, although they do not become as quickly localized
as the remaining bulk states. Accordingly, the IPR of these low energy states
shows a slower increase than what is observed in Fig. 7(a).
Figure 8: Energy spectra evolution with a) Anderson disorder and b) Aubry-
André disorder, for $B_{y}=0.5d$.
For Anderson disorder along $y$ and with $x$ uniformity, we find a different
localization behaviour. The IPR grows linearly with the increase of disorder,
although with some fluctuations and a deviation for $\lambda<0.5$. A fit of
the form $\text{IPR}=C_{1}\lambda+C_{2}$ is done to the range
$\lambda\in[0.5,3]$ and presented in red dashed lines. The change in behaviour
in relation to 7(a) is a result of imposing periodicity in the $x$ direction
on the disorder term, which unexpectedly causes the behaviour of the IPR to
become linear. The IPR of the low energy states follows a similar behaviour to
what is seen for the average IPR. By increasing disorder the low energy edge
states are removed from the edges and localize inside the bulk, while
remaining periodic in the $x$ direction.
In Fig. 7(c) (Aubry-André disorder with $x$ uniformity) we see a threshold
behaviour where a transition happens around $\lambda=2$. For $\lambda<2$
(approximately) there is a slow increase of the IPR, while for $\lambda>2$ the
IPR greatly increases. This resembles some known results: in the one-
dimensional Aubry-André model, where the system undergoes an extended-
localized transition at $\lambda=2t$, after which the average IPR shows a
marked increase; for a one dimensional $p$-wave superconductor with an Aubry-
André potential this transition point changes to $\lambda$ = 2(t+d) with $d$
the $p$-wave pairing amplitude (when the chemical potential is taken as zero)
[41, 42]. For values of $\lambda$ before the transition, we observed that some
bulk states acquire a critical like behaviour in the $y$ direction, while
remaining periodic in the $x$ direction. The low energy states are more robust
to disorder if compared with the Anderson disorder cases, and are only removed
from the system at the transition: after the threshold value of $\lambda$
there are no edge states in the system.
Fig. 7(d) concerns the case of two-dimensional Aubry-André disorder. The IPR
shows again a threshold behaviour, and as in c) a transition is seen slightly
below $\lambda=2$. However, the transition between two different regimes is
abrupt in the IPR, and more closely resembles that of the one dimensional
Aubry-André chain. Also, unlike cases a)-c), the IPR follows the same
behaviour for the three regimes considered. By a closer inspection of the IPR
we see that this is only true for values of disorder over $\lambda=0.7$, as
for $\lambda<0.7$ the noncentrosymmetric regime shows a consistently lower
IPR, as before. As disorder is increased for $\lambda>2$, states localize
along both the $x$ and $y$ directions.
The comparison between $1d$ Aubry-André disorder and $2d$ Aubry-André
disorder, as well as a comparison between a perpendicular and a parallel
magnetic field, is detailed in Appendix B, with particular emphasis on the
existence of critical states and the apparent absence of a transition between
extended and critical states, in contrast to what is found in the one-
dimensional case.
Although results are not explicitly shown, the effect of edge disorder was
also briefly studied, extending previous results obtained for a time-reversal
invariant system [110]. We considered both Anderson and Aubry-André disorder
potentials which were introduced locally at the edges at $y=0$ and $y=N_{y}$,
varying along the $x$ direction (along the edge) also for a system of size
$N=41\times 41$. We found that the bulk states and the system as a whole are
almost unaffected by edge disorder, and the average IPR of the system remains
nearly constant. However, the edge states are affected, and their behaviour
depends on the type of disorder introduced. For Anderson disorder, the states
localize continuously along the edges, while for Aubry-André disorder there is
also a threshold behaviour, similarly to what is presented in Fig. 7(c).
Figure 9: Energy spectra evolution with a) Anderson disorder and b) Aubry-
André disorder, for $B_{y}=d$.
### IV.2 Energy spectra evolution and density of states
We now consider the system in a mixed $(k_{x},y)$ space, with finite width
along $y$ and OBC. We fix the parameter values as $t=1$, $d=1/6$, $\mu=-3.5$,
and $B_{y}=0.5d$ or $B_{y}=d$ (such that the system describes a $p$-wave
superconductor) and obtain the evolution of the energy spectra for several
values of disorder strength $\lambda$ for both Anderson and Aubry-André
disorder. Since the values of $t$, $d$ and $\mu$ will be kept constant we will
now omit them.
In Fig. 8 we show the energy spectra for $B_{y}=0.5d$ with a) Anderson
disorder and b) Aubry-André disorder. The clean system has gapless edge states
and the bulk gap is not closed by $B_{y}$. As Anderson disorder is increased,
the edge states lose their structure and the bulk gap is closed. Accordingly,
there is an increase in the density of states at $E=0$ and around zero energy
as it can be seen in Fig. 10(a).
Introducing quasidisorder, as seen in panel b), leads to a closing of the bulk
gap with the appearance of new Majorana flat bands. As disorder is increased,
the flat band then splits in two and disappears as a gap opens in the system
for around $\lambda=1.8$. The appearance of MFBs leads to an increase of the
density of states at zero energy, as can be seen in Fig. 10(b) for the value
of $\lambda=1.4$. At higher values of disorder, the system is gapped and the
DOS at $E=0$ goes to zero. The reopening of the gap contrasts with what was
found for Anderson disorder, where the bulk remains gapless as disorder is
increased. We observed that the edge states inside the quasidisorder induced
flat bands appear localized at both edges simultaneously. While the edge
states of the clean system are localized symmetrically on both edges, the flat
band states lose this symmetry and localize more near one of the edges if
quasidisorder is present. Near the edge on which a given state appears less
localized, there is also a deviation from the edge, and the state mostly
localizes on the subsequent sites in $y$.
Figure 10: Density of states evolution with a) Anderson disorder and b)
Aubry-André (AA) disorder, for $B_{y}=0.5d$ and with c) Anderson disorder and
d) Aubry-André disorder for $B_{y}=d$.
In Fig. 9 the clean system with $B_{y}=d$ is in a gapless phase with both edge
states and a range of $k_{x}$ supporting Majorana flat bands. As Anderson
disorder is increased, the bulk remains gapless and there is a sharp increase
in the density of states at zero energy, as the bulk states come from finite
energies to lower energies. The sharp peak in the DOS observed at $E=0$ is
reminiscent of the characteristic behaviour of a two-dimensional disordered
superconductor with broken time-reversal invariance in the thermal metal
regime [111] in which the density of states displays a logarithmic divergence
at zero energy.
In Fig. 9(b) when Aubry-André disorder is introduced, the edge states appear
to be robust up until around $\lambda\approx 0.8$. However, the MFBs which are
present at $\lambda=0$ are more robust if compared with the edge states, with
the band staying at zero energy but the initial range of $k_{x}$ hosting flat
bands decreasing as $\lambda$ increases. Simultaneously, flat bands appear for
new values of $k_{x}$, as is can be seen in the figure for $\lambda=1.4$, and
accordingly, the density of states at zero energy increases. At higher values
of disorder there is a collapse of states to lower energies and the density of
states exhibits a peak at $E=0$ which is reminiscent of the behaviour found
for Anderson disorder for the same parameter values. Contrary to what is
observed in for a lower magnetic field, there is no opening of the bulk gap
for larger values of $\lambda$. When quasiperiodic disorder is introduced, a
gap will only open for larger values of $\lambda$ if the bulk was gapped prior
to introducing disorder, as in Fig. 8, otherwise the bulk will remain gapless.
Let us now consider the addition of finite values of $\alpha$ and
$\Delta_{s}$. The addition of finite values of spin-orbit coupling and
$s$-wave pairing potential breaks the chiral-like symmetry
$\mathcal{S}_{k_{y}}$ (defined in Eq. 12) that protects the flat bands. As a
result, the latter are lifted to a finite energy and the spectrum acquires a
tilt. For certain regimes of $B_{y}$, the noncentrosymmetric superconductor in
the clean system shows unidirectional edge states. In such regimes, the
addition of Aubry-André disorder leads to the appearance of ”flipped”
unidirectional states in the system. This can be seen in Fig. 11 for
$\lambda=1.4$.
Figure 11: Energy spectra evolution with Aubry-André disorder for $B_{y}=4d$,
$\alpha=0.2d$, $\Delta_{s}=0.3d$ and a) $\lambda=0$, b) $\lambda=1.4$.
### IV.3 Topological nature of quasidisorder induced flat bands
We want to investigate if the Majorana flat bands that arise in the presence
of a quasiperiodic potential have a topological nature, such as is the case of
the flat bands in the ordered system. Since the Berry phase was found to be
quantized to a value of $\pi$ in the clean system in the region of flat bands,
we calculate it here for the disordered case. The Berry phase $\gamma_{B}$ is
obtained in real space using twisted boundary conditions. Considering a
twisted boundary phase $\theta_{y}$ we have:
$\gamma_{B}(k_{x})=i\int_{0}^{2\pi}d\theta_{y}\langle\Psi(k_{x},\theta_{y})\mid\frac{\partial}{\partial\theta_{y}}\Psi(k_{x},\theta_{y})\rangle$
(27)
where $\Psi$ denotes the ground-state many body wavefunction, which is given
by the Slater determinant of the single particle wavefunctions. We can
represent the ground state wavefunction by an $M\times N$ matrix
$\mathbf{\Psi}^{\theta_{y}}$ where $N$ is the number of sites in $y$ and $M$
is the number of occupied states (negative energy states). Numerically, the
twist variable is discretized into $L$ points between $0$ and $2\pi$, such
that $\theta_{y}$ is constrained to take the values
$\theta_{y,n}=\frac{2\pi}{L}n$, with $n$ an integer that goes from $0$ to
$L-1$. A link variable can then be defined as
$U(\theta_{y,n})=\text{det}\left[\mathbf{\Psi^{\dagger}}_{\theta_{y,n}}\mathbf{\Psi}_{\theta_{y,n+1}}\right]$,
and the Berry phase is obtained as
$\gamma_{B}=-\mathrm{i}\sum_{n=1}^{L}\log{U(\theta_{y,n})}.$ (28)
Figure 12: Energy spectrum and Berry phase $\gamma$ normalized by $2\pi$, as a
function of $k_{x}$. The values of the parameters are $t=1$, $d=t/6$,
$\mu=3d-4t$ and a) $B_{y}=0.5d$, $\lambda=1.4$ b) $B_{y}=d$, $\lambda=1.6$,
with $\lambda$ the strength of the quasiperiodic Aubry-André potential.
We find that at the values of $k_{x}$ where Majorana flat bands appear the
Berry phase is quantized to $\pi$, as shown in Fig. 12. We have also found
that, for the considered system sizes, the values of $k_{x}$ at which the
Berry phase is quantized to $\pi$ are independent of the phase $\phi$ in the
Aubry-André potential.
To quantify the induced bands at zero energy and study the transition to a
$\pi$-quantized Berry phase, we use the concept of Majorana pair density,
defined as [112]:
$\rho_{\gamma}=\frac{N_{\gamma}}{N_{k}},$ (29)
where $N_{k}$ is the number of discrete points of $k_{x}$ taken inside the
interval $\left[-\pi,\pi\right]$, and $N_{\gamma}$ is the number of such
points which support MFBs at the edges. Numerically it is more convenient to
consider the number of $k_{x}$ points for which the Berry phase is quantized
to $\pi$, $N_{\pi}$, since it was found that $N_{\pi}=N_{\gamma}$. A
transition from $\rho_{\gamma}=0$ to $\rho_{\gamma}\neq 0$ then signals a
transition from a trivial to a topological regime ($\pi$-quantized Berry
phase). Fig. 13(a) shows the evolution of $\rho_{\gamma}$ as a function of
quas-idisorder strength for the case $t=1$, $d=t/6$, $\mu=-3.5$ and
$B_{y}=0.5d$, and for the range $\lambda\in[1,2]$. A transition
$\rho_{\gamma}=0\rightarrow\rho_{\gamma}\neq 0$ occurs between $\lambda=1.22$
and $\lambda=1.23$ at a certain critical value $\lambda_{C,1}$. The value of
$\rho_{\gamma}$ grows until $1.49\pm 0.01$ when the flat band splits in two
and the behaviour of $\rho_{\gamma}$ changes, with an abrupt change in the
sign of the second derivative. A second transition occurs between $1.79$ and
$1.8$, at a critical value $\lambda_{C,2}$, where $\rho_{\gamma}$ becomes
zero.
Figure 13: a) Values of $\rho_{\gamma}$ for the case $t=1$, $d=t/6$,
$\mu=-3.5$ and $B_{y}=0.5d$ vs. quasidisorder strength $\lambda$. Obtained for
a system with $76$ sites in $y$. b) Value of the DOS at $E=0$ for the same
parameter values as in a), vs. quasidisorder strength $\lambda$, and the
contribution for $\rho(E=0)$ which comes from the Majorana flat bands in the
corresponding regime.
In Fig. 13(b) the density of states at zero energy $\rho(E=0)$ (normalized by
the system size) is shown, for the same parameters as in Fig. 13(a) and for
$N_{y}=76$, along with the corresponding contribution for the zero energy
density of states which comes from the MFB, $\rho(E=0)_{\gamma}$. Inside the
topological phase, which is highlighted, we can see that the finite value of
$\rho(E=0)$ observed in the system with OBC comes almost entirely from the
presence of flat bands.
$N_{y}$ | $\lambda_{C,1}$ | $\lambda_{C,2}$
---|---|---
76 | $1.225\pm 0.005$ | $1.800\pm 0.005$
100 | $1.215\pm 0.005$ | $1.775\pm 0.005$
175 | $1.230\pm 0.005$ | $1.805\pm 0.005$
200 | $1.220\pm 0.005$ | $1.800\pm 0.005$
400 | $1.230\pm 0.005$ | $1.805\pm 0.005$
800 | $1.225\pm 0.005$ | $1.805\pm 0.005$
Table 1: Values of the critical points $\lambda_{C,1}$ and $\lambda_{C,2}$,
for the parameter values $t=1$, $d=t/6$, $\mu=-3.5$ and $B_{y}=0.5d$ and for
the system sizes $\\{76,100,175,200,400,800\\}$.
Table 1 shows the values of $\lambda_{C,1}$ and $\lambda_{C,2}$ for several
system sizes, obtained for random values of the phase $\phi$ in the Aubry-
André potential, where the uncertainty is taken as the minimum interval
considered between values of $\lambda$. Is is found that the values of the
critical points show little variation with the system size, and we also found
that the critical points are independent of $\phi$ for the system sizes
considered.
### IV.4 Scaling of the density of states: critical exponents
#### IV.4.1 A detour to the clean system
Let us first briefly consider the clean system, without disorder. For the
clean case, it is possible to obtain the values of the dynamical exponent $z$
and of the critical exponent $\nu$ analytically, for the transition that
occurs as $B_{y}$ is increased, corresponding to a transition from a winding
number of 0 to 1 or a Berry phase of 0 to $\pi$. Here we consider the case of
$\mu<-2t$ (such that the topological phase is within the region described by
Eq. 17). At the topological transition to a gapless phase, the gap closing
points in $k_{x}$, $k_{x,0}$, are given by
$k_{x,0}=\pm\arccos{\left[-\frac{2(t\mu+2t^{2})}{-d^{2}+4t^{2}}\right]}+2n\pi,n\in\mathbb{Z}.$
(30)
The values of $k_{y}$ for which the gap closes are given by $k_{y,0}=n\pi$,
$n\in\mathbb{Z}$ (general solution). In this case the transition happens at
$k_{y,0}=2n\pi$, $n\in\mathbb{Z}$. The gap closes at a critical value of the
magnetic field, $B_{y_{C}}$, which, fixing $k_{y}=k_{y,0}$, is defined from
the value of $k_{x,0}$ as
$B^{2}_{y_{C}}=\left[\mu+2t\left(\cos
k_{x,0}+1\right)\right]^{2}+d^{2}\sin^{2}k_{x,0}.$ (31)
We can now first expand the expressions for the bulk energy around $k_{x,0}$
to find the dependence of the energy on $k_{x}$. We only need to consider the
first positive energy band, $E_{+}(k_{x})$. Taking $k_{y}=k_{y,0}$ and
expanding around $k_{x}=k_{x,0}$ we find
$\begin{split}E_{+}(k_{x})\propto(k_{x}-k_{x,0}),\end{split}$ (32)
implying a value of the dynamical exponent $z=1$ for the transition. We can
now take $k_{x}=k_{x,0}$ and see how the gap closes as a function of $B_{y}$.
We find
$\begin{split}E_{+}(k_{x}=k_{x,0})=\left||B_{y_{C}}|-|B_{y}|\right|.\end{split}$
(33)
Near a quantum phase transition as a critical point $\lambda_{C}$ is
approached, the gap behaves as $\Delta\sim|\lambda-\lambda_{C}|^{z\nu}$,
therefore at $k_{x}=k_{x,0}$ the gap vanishes linearly, with an exponent
$z\nu=1$. Since $z=1$, this implies $\nu=1$, and
$z=1,\quad\nu=1.$ (34)
#### IV.4.2 Quasidisorder: numerical calculation of the critical exponents
Around a critical point, the density of states $\rho(E)$ follows [113]
$\rho(E)=\delta^{(D-z)\nu}f(|E|\delta^{-z\nu}),$ (35)
with $D$ the dimension of the system (here $D=2$),
$\delta=\frac{|\lambda-\lambda_{C}|}{\lambda_{C}}$ the normalized distance to
the critical point $\lambda_{C}$, and $f$ a scaling function. Right at the
critical point, when $\delta=0$, the DOS behaves as
$\rho(E)\sim|E|^{\frac{D}{z}-1}.$ (36)
From the behaviour of the density of states near the phase transition and
using Eqs. 35 and 36 it is possible to obtain the values of the critical
exponents numerically.
Here we study a system with $N_{y}=800$ sites in $y$ and consider the obtained
critical values $\lambda_{C,1}=1.225$ and $\lambda_{C,2}=1.805$ (as shown in
table 1 for this system size). A fit of the form of Eq. 36 for the density of
states at the critical points, done in the interval E $\in[0.005,0.025]$,
gives the values of the critical exponents $z=1.27\pm 0.04$ for the first
transition and $z=1.23\pm 0.03$ for the second transition. To determine the
value of $\nu$ we take values of $\lambda$ inside the topological (gapless)
phase, $\lambda>1.225$ and $\lambda<1.805$, and obtain the density of states
close to zero energy. For small values of $\delta$ and close to zero energy a
collapse of the scaled values of the density of states according to Eq. 35 is
expected.
Figure 14: Density of states for $E\in[0.005,0.025]$ and several values of
$\lambda$ close to the critical values, for a) $\lambda_{C,1}=1.225$ and b)
$\lambda_{C,2}=1.805$, scaled according to Eq. 35 for a) $z=1.27$ and
$\nu=0.95$ and b) $z=1.23$ and $\nu=1.00$.
In Fig. 14 we show the results for the scaled density of states for: a) values
close to the first transition at $\lambda_{C,1}=1.225$, and b) values close to
the second transition at $\lambda_{C,2}=1.805$. The density of states shows a
collapse for a) $z=1.27$ and $\nu=0.95$ and b) $z=1.23$ and $\nu=1.00$.
The quantum phase transitions in the disordered regime are therefore in a
different universality class than that of the clean case, which was found to
behave with $z=\nu=1$. The obtained values also differ significantly from the
known results for the Anderson or the Aubry-André transitions in one
dimension, the first belonging to an universality class with with $\nu=2$ and
$z=2/3$, and the second case with critical exponents $\nu=1$ and $z=2.375$
[114]. Recent results show that for a one dimensional system with $p$-wave
superconductivity subject to an Aubry-André potential the quasidisorder driven
transitions also deviate from the normal Aubry-André class. For the localized-
critical transition line and when the $p$-wave pairing term is finite, the
correlation length exponent has been obtained as $\nu=0.997$ and the dynamical
exponent as $z=1.373$ in [68], and as $\nu=1.000$, $z=1.388$ in [69]. Note,
however, that the referred results are for $D=1$ while we are studying a two
dimensional system, and concern systems with no applied magnetic field.
Nevertheless, one could say that the aforementioned results make it so that
deviations from the known universality classes are also expected for
transitions in the system at study. Up to numerical errors, the values of
$\nu$ obtained for the disordered driven transitions coincide with that of the
Aubry-André transition; nevertheless the value of $z$ deviates from that of
the known classes, which suggests these transitions belong to new universality
classes. The identified transitions, where Marojana flat bands appear as a
result of a quasidisorder induced gap closing, and the subsequent opening of
the bulk gap, are found to happen for other values of the imposed parameters.
Considering the values of the parameters $\mu$, $t$ and $d$ are such that the
topological regions of the superconductor are described by Eq. 17, then as
long as $B_{y}<B_{y,C}$ (when the bulk is gapless) with $B_{y,C}$ defined as
in Eq. 31) the same type of transitions will take place with the increase of
$\lambda$.
$\lambda_{C}$ | $z$ | $\nu$
---|---|---
1.225 | $1.27\pm 0.04$ | $0.95\pm 0.05$
1.805 | $1.23\pm 0.03$ | $1.00\pm 0.05$
Table 2: Values of $z$ and $\nu$ obtained numerically for the topological
transitions for $N_{y}=800$.
### IV.5 Fractal Analysis
Figure 15: Results of $\tau$ vs. $q$, for several values of disorder strength,
$\lambda$, for $k_{x}=0.02\pi$ and $k_{x}=0.2\pi$, for a) Anderson disorder
and b) Aubry-André disorder. In all cases, the IPR is averaged for the states
within the energy range $E\in[0.05,1]$.
One of the effects of Anderson transitions is the emergence of
multifractality, which is characterized by fluctuations of eigenstates. These
fluctuactions are manifested in the generalized inverse participation ratio.
For a given eigenstate labeled by $m$, the generalized IPR is defined as:
$\text{IPR}(q)_{m}=\sum_{i}\left|\psi_{i}^{m}\right|^{2q},$ (37)
where, as before, $\psi_{i}^{m}$ is the wavefunction of the eigenstate $m$ at
a site $i$. At criticality, the generalized IPR behaves as [115]
$\text{IPR}(q)\sim L^{\tau(q)}$ (38)
where $L$ is the system size and the exponent $\tau(q)$ is defined in terms of
a generalized dimension $D(q)$ as $\tau(q)=D(q)(q-1)$. In a metallic phase,
$D(q)=d$ and for an insulating phase $D(q)=0$. Wavefunction multifractality is
characterized by a $q$ dependent value of $D(q)$, whereas the cases of a
constant $D(q)$ are single fractals [116].
Here we want to make a simple fractal analysis of the system both for disorder
and quasidisorder. We take $k_{x}$ at fixed values, such that system is
reduced to an effective one dimension. The IPR as a function of $q$ is
calculated and averaged within the energy range $E\in[0.05,1]$. We fix the
parameters $t=1$, $d=t/6$, $\mu=3d-4t$ and $B_{y}=0.5d$ and consider both the
cases of Aubry-André disorder and Anderson disorder. The following
subintervals of $L$ are considered, to which a fit of an equation of the form
of Eq. 38 is done:
$\begin{split}&L_{1}=\\\
&\\{75,100,150,175,200,255,275,400,475,600,675,800\\},\\\
&L_{2}=\\{150,175,200,255,275,400,475,600,675,800\\},\\\
&L_{3}=\\{200,255,275,400,475,600,675,800\\},\\\
&L_{4}=\\{275,400,475,600,675,800\\}.\end{split}$ (39)
The obtained results are presented in Figs. 15 and 16.
#### IV.5.1 Anderson disorder
Fig. 15(a) shows the values of $\tau(q)$ for $k_{x}=0.02\pi$ and
$k_{x}=0.2\pi$, for several values of $\lambda$ and considering the system
size interval $L_{1}$. One thing that can be immediately noticed is that for
the clean system, $\lambda=0$, the values of $\tau(q)$ closely follow the line
$\tau(q)=(q-1)$, indicating that $D(q)$ is $q-$independent and equal to $1$.
This is the expected behaviour of the clean system (taking a fixed $k_{x}$
where the system is reduced to one dimension) and reveals that the bulk states
are extended in the $y$ direction. For higher values of disorder, $\tau(q)$
approaches the line $\tau(q)=0$, where $D(q)=0$, suggesting the states are
localized. For other values of disorder strength, starting at $\lambda=0.1$,
$\tau(q)$ does not follow a behaviour characteristic either of $D(q)=1$ or
$D(q)=0$. In order to take a conclusion, it is necessary to evaluate $\tau(q)$
as the system size tends to infinity. To do this, the subintervals of $L$ in
Eq. 39 are considered, to which a fit of equation of the form of Eq. 38 is
done. The results are presented in Fig. 16(a). We find that for the values
$\lambda=0.1$ and above, as larger values of $L$ are considered, the curves
$\tau(q)$ approach $\tau(q)=0$. This confirms a localization of the bulk
states in the thermodynamic limit for small values of disorder.
Figure 16: a) Values of $\tau$ at different values of $q$ and Anderson
disorder strength $\lambda$, for $k_{x}=0.02\pi$ and $k_{x}=0.2\pi$. b) Values
of $\tau$ at different values of $q$ and quasidisorder strength $\lambda$, for
$k_{x}=0.02\pi$ and $k_{x}=0.2\pi$. In all cases, the IPR is averaged for the
states within the energy range $E\in[0.05,1]$.
#### IV.5.2 Aubry-André disorder
Fig. 16(b) shows the values of $\tau(q)$ for $k_{x}=0.02\pi$ and
$k_{x}=0.2\pi$, for several values of quasidisorder strength $\lambda$ for the
size interval $L_{1}$. Unlike the previous case with Anderson disorder, we see
that the results differ for each $k_{x}$, and that for some values of disorder
strength $\tau(q)$ follows the line $q-1$ closely until some value of $q$
where the behaviour suddenly changes. In Fig. 16(b) we show, as before, values
of $\tau(q)$ fitted for the considered size intervals $L_{1}$, $L_{2}$,
$L_{3}$ and $L_{4}$. or lower values of $q$, $\tau$ remains at the values
defined by the Eq. $\tau(q)=D(q)(q-1)$ with $D(q)=1$. However, at higher
values of $q$, this behaviour changes. Contrary to the case with Anderson
disorder, there is no clear tendency of $\tau(q)$ at increased system sizes,
and the behaviour also depends on the value of $q$. This deviation from the
$D(q)=1$ line is verified as soon as disorder is introduced, and suggests the
system is in a multifractal regime. Accordingly, we see the appearance of
critical bulk states in the system. From inspection of Fig. 15(b) and of the
corresponding values of $\tau(q)$ at larger system sizes, we identify a
transition to a localized phase around $\lambda\in[2.0,2,1]$.
## V Conclusions
In this work we studied a two-dimensional topological superconductor in the
presence of a magnetic field. We introduced disorder and quasidisorder in the
system with the aim of studying the effects on topological and localization
properties. Considering previous results on other systems such as insulators,
semimetals and one-dimensional superconductors, and the results we found on
the effect of quasidisorder in two-dimensional superconductors, we may expect
that the results may be general considering gapless systems or topological
systems (gapped or gapless), in which regions displaying similar topological
and localization properties may be found.
The system was first studied under a perpendicular magnetic field $B_{z}$.
Four types of disorder were considered: Anderson disorder (two-dimensional),
Anderson disorder (one-dimensional along $y$, uniform along $x$), Aubry-André
disorder (one-dimensional along $y$, uniform along $x$) and Aubry-André
disorder (two-dimensional). We observed that the response of the topological
phases of the system differs depending on the type of disorder, and that
quasidisorder induces topological phases in new regions of $B_{z}$,
characterized by integer values of the Chern number $C$. The critical points
at these phase boundaries were shown to become sharper as the system size
increases, allowing us to conclude that the obtained phase diagrams apply to
bigger system sizes.
The real space system was also briefly studied when a parallel magnetic field
is applied in the $y$ direction. We studied the cases of bulk disorder, with
the same four different spatial configurations. For two-dimensional Anderson
disorder, we found that the average IPR of the system increases with an
exponential behaviour as a function of $\lambda$ for $\lambda>1.5$. When
uniformity in the $x$ direction is imposed in the Anderson disorder potential,
we found that the IPR shows a linear increase as a function of $\lambda$, for
$\lambda>0.5$. For Aubry-André disorder, the behaviour of the average IPR of
the system reveals the existence of two different regimes. In the first, the
average IPR shows a slow increase with $\lambda$, and in the second the IPR
greatly increases. The transition between the two regimes is located around
$\lambda=2$.
We then studied the system in a mixed ($k_{x},y$) space with an applied
parallel magnetic field. The clean superconducting system is known to possess
flat bands in the gapless regime. At the corresponding values of $k_{x}$ these
have a winding number $\mathcal{W}$ of 1, which is defined from reducing the
two dimensional system to an effective one dimension. We showed that these are
also characterized by a $\pi-$quantized Berry phase at the same values of
$k_{x}$.
We showed that the introduction of quasidisorder induces new gapless phases in
parameter regimes where they were absent in the clean case. For the $p$-wave
system subject to a parallel magnetic field this leads to new regimes with
Majorana flat bands. This is not only true for phases with a gapless bulk but
also for gapped phases, where quasidisorder closes the bulk gap and Majorana
flat bands appear. We then obtained the Berry phase with twisted boundary
conditions and concluded the quasidisorder induced flat bands also have a
quantized Berry phase of $\pi$. For the noncentrosymmetric superconductor with
added $s$-wave superconducting pairing and Rashba spin orbit coupling, we
showed that new regimes with unidirectional Majorana edge states appear. In
particular, we showed that for a phase where right-moving unidirectional edge
states were present in the system, the introduction of quasidisorder leads to
the appearance of edge modes in the opposite moving direction, and for a
certain quasidisorder range these modes coexist in the system.
Figure 17: Phase diagrams of a system with 20x20 sites indexed by the Chern
number $C$, for several values of Aubry-André disorder strength $\lambda$ and
perpendicular magnetic field $B_{z}$, obtained for the average over 20
disorder configurations. The values of the parameters are $t=1$, $d=1/6$ and
$\mu=4.5$ . Figure 18: Values of the Chern number $C$ vs. one-dimensional
Aubry-André disorder strength $\lambda$ for the system sizes $20\times 20$,
$30\times 30$ and $41\times 41$ and for a) $t=1$, $\mu=0$, $d=0.6$,
$B_{z}=0.4$, b) $\mu=1$, $d=0.6$, $B_{z}=1.1$, c) $\mu=-3.5$, $d=1/6$,
$B_{z}=0.3$. The results were averaged over 20 disorder configurations. Figure
19: Values of the Chern number $C$ vs. two-dimensional Aubry-André disorder
strength $\lambda$ for the system sizes $20\times 20$, $30\times 30$ and
$41\times 41$ and for a) $t=1$, $\mu=0$, $d=0.6$, $B_{z}=0.4$, b) $\mu=1$,
$d=0.6$, $B_{z}=1.1$, c) $\mu=-3.5$, $d=1/6$, $B_{z}=0.3$. The results were
averaged over 10 disorder configurations.
The identification of the quasidisorder induced topologically non-trivial flat
bands with a quantized Berry phase of $\pi$ allowed us to study in detail two
topological transitions, for the $p$-wave superconductor with a parallel
applied magnetic field $B_{y}$. The two critical points were identified and
studied by obtaining the density of induced Majorana bound states in relation
to $k_{x}$ points. We found that the values of the critical points show almost
no variation with the system size for systems bigger than $76$ sites along
$y$. The values of the dynamical critical exponents and correlation length
critical exponents were obtained as $z=1.27\pm 0.04$ and $\nu=0.95\pm 0.05$
for the first critical point and $z=1.23\pm 0.03$, $\nu=1.00\pm 0.05$ for the
second critical point, which puts these transitions in novel universality
classes. We then investigated the multifractal nature of the wavefunctions by
calculating the values of $\tau(q)$ from the IPR values at several values of
disorder, at the same parameter values as those in which the topological
transition was studied. From the behaviour as the thermodynamic limit is
approached, we concluded that the introduction of quasidisorder induces
multifractality in the system. A transition to a localized regime was
identified for $\lambda_{AA}\in[2.0,2.1]$. The same analysis was made for the
system with Anderson disorder. The behaviour of $\tau(q)$ as the system size
tends to infinity suggests that the introduction of Anderson disorder will
drive the system to a localized phase (in the thermodynamic limit).
We have also shown that the average inverse participation ratio is not very
sensitive to the magnitude of $B_{z}$ or $B_{y}$. Although $B_{y}$ leads to
gapless behavior and $B_{z}$ in general leads to gapped behavior, and although
each magnetic field direction leads to different topological properties and
symmetries in the clean system, the localization properties are similar and
the existence of critical states is also similar. It seems that a magnetic
field in the $y$ direction, $B_{y}$, leads to a more localized behavior in the
presence of a quasidisordered potential. For both magnetic field directions we
found critical states and no mobility edges were found. We found a crossover
as a function of $\lambda$ with a mixture of extended and critical states that
grow in number as quasidisorder increases. In this context, Aubry-André along
$1d$ in the two dimensional system or along $2d$ does not lead to
qualitatively different results (in the sense that the crossover in
localization is seen for both cases), besides the differences in induced
topology.
###### Acknowledgements.
We acknowledge partial support from FCT through the Grant UID/CTM/04540/2019.
M.F.M. acknowledges partial support through the grant (1801P.01102.1.01)
QMSP2021 - CEFEMA - IST-ID.
## Appendix A Additional details on the disordered model under a
perpendicular magnetic field
In Figs. 6(h) and 6(i), we observe that for low magnetic fields the increase
of quasidisorder induces topological phases in regions for which the Chern
number was zero. In the clean system, however, these regions correspond to a
phase that is topological and characterized by a finite value of $I(k_{y})$.
In Fig. 17 we show a phase diagram for the parameters $d=1/6$ and $\mu=4.5$.
In this case, the topological invariant $I(k_{y}=0,\pi)$ is trivial for low
values of magnetic field ($B_{z}<0.5$) when $C=0$. Contrarily to what is
observed in Fig. 6(i), there is no induced topological region with $C=-1$.
This thus suggests that the reentrant regions observed in Fig. 6(h) and Fig.
6(i) can possibly be related with the topological nature of the phases
characterized by $I(k_{y})$ and with $C=0$.
To see how the different critical values for quasiperiodic disorder scale with
the system size, three transitions for the phase diagrams obtained with Aubry-
André disorder along $y$ with uniformity along $x$ (third row in Fig. 6) and
Aubry-André in two dimensions (fourth row in Fig. 6) were considered, at fixed
values of $B_{z}$ and $\mu$. The results are presented in Figs. 18 and 19 for
the system sizes $20\times 20$, $30\times 30$ and $41\times 41$. We found that
within the considered system size range, the transitions become sharper as the
size increases, thus suggesting that the obtained phase diagrams in Fig. 6
should apply to larger systems.
Figure 20: Average participation ratio, APR, for a $p$-wave superconductor in
the presence of Aubry-André quasidisorder for a) one-dimensional system and b)
two-dimensional system, as a function of the superconducting pairing term $d$
and disorder strength $\lambda$. Figure 21: Average participation ratio,
APR, for a $2d$ system with Anderson or Aubry-André (AA) quasidisorder, for
$d=t/6,\mu=-3.5$, in a a) perpendicular or b) parallel magnetic field.
## Appendix B Participation ratio of $2d$ Aubry-André quasidisorder
In Fig. 7 we considered the average inverse participation ratio regarding both
disorder or quasidisorder along one spatial direction (the $y$ axis) and
disorder or quasidisorder in the plane, comparing a set of values of parallel
magnetic field. In this Appendix we carry out a more extensive analysis. We
want to focus our attention on the regime of increasing disorder strength,
from small values to larger values as localization takes place, in particular
on the possible separation between extended, critical and localized regimes.
The inverse participation ratio is particularly useful to study the transition
to the localized regime, but is not as revealing in the extended-critical
regimes. In this Appendix we will consider instead the participation ratio,
which is given by the inverse of Eq. 26. It is of the order of one for
extended states, and becomes of the order of the inverse of the system size in
the localized regime.
Figure 22: Average participation ratio as a function of $\lambda$ for
perpendicular magnetic field, $B_{z}$, and a) $1d$ and b) $2d$ quasidisorder
and parallel magnetic field, $B_{y}$, and c) $1d$ and $2d$ quasidisorder.
Figure 23: Participation ratios of the eigenstates for
$\lambda=1,d=t/6,\mu=-3.5$, as a function of energy for a) $B_{z}=2$ and b)
$B_{y}=2$.
In the case of a one-dimensional $p$-wave superconductor (Kitaev model) in the
presence of an Aubry-André potential and with no magnetic field, it is known
that the three regimes of extended, critical and localized states are present,
as one changes the amplitude of the pairing, $d$, and the quasidisorder
strength, $\lambda$ [42]. The average participation ratio (APR), considering a
single disorder configuration, is shown in Fig. 20(a), where clear transitions
are shown separating the various regimes (here we only consider a disorder
configuration, but the results are characteristic of a larger set of disorder
configurations). The average participation ratio has different plateaus as the
parameters change. Considering a two-dimensional $p$-wave superconductor with
two-dimensional quasidisorder, there is no clear transition between the
regimes and one finds crossovers as $\lambda$ increases. The system sizes
considered in Fig. 20(b) are small ($21\times 21$), but a smooth transition
showing the decreasing of the average participation ratio seems to indicate no
clear separation of a critical regime before the localized phase takes place.
In Fig. 21 we consider the average participation ratio for the two-dimensional
case in the presence of a perpendicular or parallel magnetic field, for the
cases of Anderson disorder and quasidisorder. These results highlight the
extended/critical regimes at lower values of $\lambda$. Anderson disorder
behaves similarly for the two magnetic field directions, and quasidisorder
leads to higher values of the average participation ratio as disorder
increases, with a sharper transition to the localized regime, as shown in Fig.
7. In the case of the parallel magnetic field quasidisorder has a stronger
localization effect. A difference with respect to the perpendicular magnetic
field is the existence of gapless states, more sensitive to the long-range
disorder associated with the Aubry-André potential. In the case of
perpendicular magnetic field, the system remains gapped (except at the
transitions between the various topological regimes) and therefore is expected
to be less sensitive to the quasidisorder potential.
The crossover behavior is clearly seen in Fig. 22, independently of the
magnetic field direction ($B_{z}$ or $B_{y}$). Also, the consideration of
quasidisorder along the $y$ direction and periodic along $x$ or quasidisorder
that is fully two-dimensional leads to similar results. Some differences are
visible for small magnetic fields or small values of $\lambda$. Except for
these regions, the average participation ratio is quite independent of the
amplitude of the magnetic field, but the effect of a parallel magnetic field
is more significant, as discussed.
Figure 24: Wave function amplitudes for the parameters of Fig. 23, as a
function of space location in a $21\times 21$ system, for states with a)
$B_{z}=2$ and participation ratio $PR=0.781$, b) $B_{z}=2$ and $PR=0.186$, c)
$B_{y}=2$ and $PR=0.643$, d) $B_{y}=2$ and $PR=0.169$.
In order to have a better understanding of the possible existence of critical
states in the regime prior to the transition to localization, we show a
typical case in Fig. 23 of the participation ratios of the various
eigenstates, as a function of their energies, for perpendicular and parallel
magnetic fields. The parameters are chosen so that we are in an intermediate
regime, where the average participation ratio is in the crossover between
fully extended and localized states. There are significant fluctuations
between states with high participation ratios (characteristic of extended
states) and low participation ratios (characteristic of intermediate,
critical, states) but still larger than values that correspond to the
localized regime. The results do not show a mobility edge, and the states mix
throughout the energy range. Also, as $\lambda$ increases, we have found that
the percentage of critical-like states increases, explaining the crossover
behavior. The extended versus critical character nature of the states is shown
in Fig. 24, where a few wave functions are shown (for the system with periodic
boundary conditions in both directions), characteristic of extended and
critical states coexisting in the same energy spectrum.
## Appendix C Energy spectra evolution for the noncentrosymmetric
superconductor with Aubry-André and Anderson disorder in $(k_{x},y)$ space
Figure 25: Energy spectra evolution with a) Anderson disorder and b) Aubry-
André disorder, for $B_{y}=d$, $\alpha=0.2d$ and $\Delta_{s}=0.5d$.
Here we present results for the evolution of the energy spectra for the
noncentrosymmetric superconductor, with spin orbit coupling $\alpha$ and mixed
$p$ and $s$-wave pairings, subject to Anderson and Aubry-André disorder in the
$(k_{x},y)$ space. We take the same parameter values for $t$, $d$ and $\mu$ as
in section IV.2, and consider two different cases: $B_{y}=d$, $\alpha=0.2d$,
$\Delta_{s}=0.5d$ and $B_{y}=4d$, $\alpha=0.2d$ and $\Delta_{s}=0.3d$.
In Fig. 25 we present the evolution of the energy spectrum for a) Anderson
disorder and b) Aubry-André disorder, for $B_{y}=d$, $\alpha=0.2d$,
$\Delta_{s}=0.5d$. As Anderson disorder is introduced in the system, the edge
states are destroyed and the considered energy range gets filled with bulk
states, but the tilt of the spectrum is preserved. As a result, the flat bands
which were previously lifted due to the introduction of finite values of
$\alpha$ and $\Delta_{s}$ do not collapse to zero energy. For high values of
$\lambda$ the density of states exhibits two peaks which result from the
inclination of the bulk energy spectrum (Fig. 27 a) ).
Figure 26: Energy spectra evolution with a) Anderson disorder and b) Aubry-
André disorder, for $B_{y}=4d$, $\alpha=0.2d$, and $\Delta_{s}=0.3d$.
In Fig. 25(b), as Aubry-André disorder is introduced, we see an evolution that
is similar to Fig. 9(b), but instead of new flat band regimes, new
unidirectional edge states appear. Unlike what happens for Anderson disorder,
at high values of $\lambda$ a gap opens for values of $k_{x}$ around $k_{x}=0$
(although the bulk as a whole remains gapless). This is reflected in the
density of states, that drops around $E=0$ for higher disorder values.
Similarly to what was observed for Anderson disorder, the tilt of the energy
spectrum is preserved as disorder increases.
Figure 27: Density of states evolution with a) Anderson disorder and b)
Aubry-André (AA) disorder, for $B_{y}=d$, $\alpha=0.2d$ and $\Delta_{s}=0.5d$
and with c) Anderson disorder and d) Aubry-André disorder for $B_{y}=4d$,
$\alpha=0.2d$, and $\Delta_{s}=0.3d$.
In Fig. 26, the clean system is in the regime where unidirectional MESs
appear. The values of the $p$-wave pairing and spin orbit term are kept
constant in relation to the case of Fig. 25, but the $s$-wave pairing term is
decreased from $\Delta_{s}=0.5d$ to $\Delta_{s}=0.3d$ and the magnetic field
is increased from $B_{y}=d$ to $B_{y}=4d$. The spectrum acquires a tilt in the
opposite direction if compared to the clean system in Fig. 25, as a result of
the increased magnetic field. With Anderson disorder, Fig. 26(a), The
unidirectional Majorana edge states are robust to small values of disorder
strength but as disorder increases the structure of the band is lost, as bulk
states fill the lower energy values. This differs from Fig. 25(a) where the
tilt of the spectrum is preserved even at higher values of disorder. As
disorder is increased, there is at first an increase in the value of the DOS
at zero energy, which then decreases for higher values of disorder. For
$\lambda>1.8$ the density of states becomes nearly constant in the considered
range of $E\in[-0.4,0.4]$.
In Fig. 26(b), when a certain value of Aubry-André disorder is reached,
”flipped” unidirectional states appear in the system. This is seen clearly in
Fig. 26(b) for the values of $\lambda=1.2$ and $\lambda=1.4$, as a band with
negative slope appears for values of $k_{x}$ around $k_{x}=0$. At
$\lambda=1.2$ there is a coexistence of unidirectional ”flipped” left-moving
edge modes (with negative slope) around $k_{x}=0$ and right-moving edge modes
(with positive slope) for higher (absolute) values of $k_{x}$. A backflow
current that balances the current on the edges is created on the bulk: extra
right or left moving modes will appear depending on the net current on the
edges.
## References
* [1] C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013).
* [2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
* [3] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
* [4] J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).
* [5] M. Sato and Y. Ando, Rep. Prog. Phys. 80, 076501 (2017).
* [6] Y. Tanaka, Y. Mizuno, T. Yokoyama, K. Yada, and M. Sato, Phys. Rev. Lett. 105, 097002 (2010).
* [7] M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Phys. Rev. B 83, 224511 (2011).
* [8] C. L. M. Wong, J. Liu, K. T. Law, and P. A. Lee, Phys. Rev. B 88, 060504 (2013).
* [9] A. P. Schnyder and S. Ryu, Phys. Rev. B 84, 060504 (2011).
* [10] A. P. Schnyder, P. M. R. Brydon, and C. Timm, Phys. Rev. B 85, 024522 (2012).
* [11] G. E. Volovik, Jetp Lett. 107, 516 (2018).
* [12] S. Peotta and P. Törmaä, Nature Communications 6, 8944 (2015).
* [13] S. Bose, A. M. García-García, M. M. Ugeda, J. D. Urbina, C. H. Michaelis, I. Bruhega, and K. Kern, LABEL:https://www.nature.com/articles/nmat2768Nat. Materials 9, 550 (2010).
* [14] M. A. N. Araújo, A. M. García-García, and P. D. Sacramento, Phys. Rev. B 84, 172502 (2011).
* [15] I. S. Burmistrov, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. Lett. 108, 017002 (2012).
* [16] J. Mayoh and A. M. García-García, Phys. Rev. B 90, 134513 (2014).
* [17] B. Fan and A. M. García-García, Phys. Rev. B 102, 184507 (2020).
* [18] C. Carbillet, V. Cherkez, M. A. Skvortsov, M. V. Feigelḿan, F. Debontridder, L. B. Ioffe, V. S. Stolyarov, K. Ilin, M. Siegel, D. Roditchev, T. Cren, and C. Brun, Phys. Rev. B 102, 024504 (2020).
* [19] A. Sĺebarski, M. Fijalkowski, P. Zajdel, M.M. Maśka, J. Deniszczyk, M. Zubko, O. Pavlosiuk, K. Sasmal, and M. B. Maple, Phys. Rev. B 102, 054514 (2020).
* [20] A. Sĺebarski, M. Fijalkowski, M. M. Maśka, J. Deniszczyk, P. Zajdel, B. Trump, and A. Yakovenko, Phys. Rev. B 103, 155133 (2021).
* [21] M. V. Feigelḿan, L. B. Ioffe, V. E. Kravtsov, and E. A. Yuzbashyan, Phys. Rev. Lett. 98, 027001 (2007).
* [22] M. V. Feigelḿan, L. B. Ioffe, V. E. Kravtsov, and E. Cuevas, Ann. Phys. 325, 1390 (2010).
* [23] J. Mayoh, and A. M. García-García, Phys. Rev. B 92, 174526 (2015).
* [24] K. Zhao, H. Lin, X. Xiao, W. Huang, W. Yao, M. Yan, Y. Xing, Q. Zhang, Z.-X. Li, S. Hoshino, J. Wang, S. Zhou, L. Gu, M. S. Bahramy, H. Yao, N. Nagaosa, Q.-K. Xue, K. T. Law, X. Chen, and S.-H. Ji, Nat. Phys. 15, 904 (2019).
* [25] C. Rubio-Verdú, A. M. García-García, H. Ryu, D.-J. Choi, J. Zaldívar, S. Tang, B. Fan, Z.-X. Shen, S.-K. Mo, J. I. Pascual, and M. M. Ugeda, Nano Letters 20, 5111 (2020).
* [26] B. Fan and A. M. García-García, Phys. Rev. B 101, 104509 (2020).
* [27] Z. Fan, G.-W. Chern, and S.-Z. Lin, Phys. Rev. Research 3, 023195 (2021).
* [28] M. Stosiek, F. Evers, and I. S. Burmistrov, Phys. Rev. Research 3, L042016 (2021).
* [29] X. Zhang and M. Foster, Phys. Rev. B 106, L180503 (2022).
* [30] K.-E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevig, and P. Törmä, Phys. Rev. B 106, 014518 (2022).
* [31] Y. Hu, S. M. L. Teicher, B. R. Ortiz, Y. Luo, S. Peng, L. Huai, J. Ma, N. C. Plumb, S. D. Wilson, J. He, and M. Shi, Science Bulletin 67, 495 (2022).
* [32] K. Jiang, T. Wu, J.-X. Yin, Z. Wang, M. Z. Hasan, S. D. Wilson, X. Chen, and J. Hu, National Science Review nwac199 (2022).
* [33] X. Wu, T. Schwemmer, T. Müller, A. Consiglio, G. Sangiovanni, D. Di Sante, Y. Iqbal, W. Hanke, A. P. Schnyder, M. M. Denner, M. H. Fischer, T. Neupert, and R. Thomale, Phys. Rev. Lett. 127, 177001 (2021).
* [34] B. R. Ortiz, S. M. L. Teicher, Y. Hu, J. L. Zuo, P. M. Sarte, E. C. Schueller, A. M. M. Abeykoon, M. J. Krogstad, S. Rosenkranz, R. Osborn, R. Seshadri, L. Balents, J. He, and S. D. Wilson, Phys. Rev. Lett. 125, 247002 (2020).
* [35] C. Mielke, D. Das, J.-X. Yin, H. Liu, R. Gupta, Y.-X. Jiang, M. Medarde, X. Wu, H. C. Lei, J. Chang, P. Dai, Q. Si, H. Miao, R. Thomale, T. Neupert, Y. Shi, R. Khasanov, M. Z. Hasan, H. Luetkens, and Z. Guguchia, Nature 602, 245 (2022).
* [36] J. Sutradhar, S. Mukerjee, R. Pandit, and S. Banerjee, Phys. Rev. B 99, 224204 (2019).
* [37] M. Saha, B. P. Venkatesh, and B. K. Agarwalla, Phys. Rev. B 105, 224204 (2022).
* [38] L.-J. Lang, X. Cai, and S. Chen, Phys. Rev. Lett. 108, 220401 (2012).
* [39] M. Tezuka and N. Kawakami, Phys. Rev. B 85, 140508 (2012).
* [40] W. DeGottardi, D. Sen, and S. Vishveshwara, Phys. Rev. Lett. 110, 146404 (2013).
* [41] X. Cai, L.-J. Lang, S. Chen, and Y. Wang, Phys. Rev. Lett. 110, 176403 (2013).
* [42] J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phys. Rev. B 93, 104504 (2016).
* [43] X. Cai, J. Phys.: Condens. Matter 26, 155701 (2014).
* [44] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012).
* [45] S. Nakajima, N. Takei, K. Sakuma, Y. Kuno, P. Marra, and Y. Takahashi, Nat. Phys. 17, 844 (2021).
* [46] S. Longhi, Opt. Lett. 45, 4036 (2020).
* [47] O. Zilberberg, Opt. Mater. Express 11, 1143 (2021).
* [48] T. Liu and H. Guo, Phys. Letters A 382, 3287 (2018).
* [49] M. I. N. Rosa, Y. Guo, and M. Ruzzene, Appl. Phys. Lett. 118, 131901 (2021)
* [50] M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Phys. Rev. Lett. 110, 076403 (2013).
* [51] Y. E. Kraus and O. Zilberberg, Phys. Rev. Lett. 109, 116404 (2012).
* [52] J. H. Pixley, J. H. Wilson, D. A. Huse, and S. Gopalakrishnan, Phys. Rev. Lett. 120, 207604 (2018).
* [53] V. Goblot, A. Štrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaître, L. Le Gratiet, A. Harouri, I. Sagnes, S. Ravets, A. Amo, J. Bloch, and O. Zilberberg, Nat. Phys. 16, 832 (2020).
* [54] T. Xiao, D. Xie, Z. Dong, T. Chen, W. Yi, and B. Yan, Science Bulletin 66, 2175 (2021).
* [55] F. A. An, E. J. Meier, and B. Gadway, Phys. Rev. X 8, 031045 (2018).
* [56] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
* [57] L. Guidoni, C. Triché, P. Verkerk, and G. Grynberg, Phys. Rev. Lett. 79, 3363 (1997).
* [58] H. P. Lüschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bordia, X. Li, S. Das Sarma, and I. Bloch, Phys. Rev. Lett. 120, 160404 (2018).
* [59] Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Phys. Rev. Lett. 103, 013901 (2009).
* [60] S. Aubry and G. André, Ann. Israel Phys. Soc 3, 18 (1980).
* [61] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007).
* [62] R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. U.S.A. 108, 12233 (2011).
* [63] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. B 86, 155449 (2012).
* [64] D. Mao and T. Senthil, Phys. Rev. B 103, 115110 (2021).
* [65] M. Gonçalves, H. Z. Olyaei, B. Amorim, R. Mondaini, P. Ribeiro, and E. V. Castro, 2D Mater. 9, 011001 (2022).
* [66] J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phys. Rev. B 93, 104504 (2016).
* [67] J. Fraxanet, U. Bhattacharya, T. Grass, D. Rakshit, M. Lewenstein, and A. Dauphin, Phys. Rev. Research 3, 013148 (2021).
* [68] X. Tong, Y.-M. Meng, X. Jiang, C. Lee, G. D. de M. Neto, and G. Xianlong, Phys. Rev. B 103, 104202 (2021).
* [69] T. Lv, T.-C. Yi, L. Li, G. Sun, and W.-L. You, Phys. Rev. A 105, 013315 (2022).
* [70] D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
* [71] S. Das Sarma, Song He, and X. C. Xie Phys. Rev. Lett. 61, 2144 (1988).
* [72] S. Das Sarma, Song He, and X. C. Xie, Phys. Rev. B 41, 5544 (1990).
* [73] J. Biddle and S. Das Sarma, Phys. Rev. Lett. 104, 070601 (2010).
* [74] S. Ganesham, J. H. Pixley and S. Das Sarma, Phys. Rev. Lett. 114, 146601 (2015).
* [75] S. Gopalakrishnan, Phys. Rev. B 96, 054202 (2017).
* [76] A. Duthie, S. Roy, and D. E. Logan, Phys. Rev. B 103, L060201 (2021).
* [77] T. Liu, X. Xia, S. Longhi, L. S.-Palencia, SciPost Phys. 12, 027 (2022).
* [78] S. Ganeshan, K. Sun, and S. Das Sarma, Phys. Rev. Lett. 110, 180403 (2013).
* [79] M. Gonçalves, B. Amorim, E. V. Castro and P. Ribeiro, SciPost Phys. 13, 046 (2022).
* [80] A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003).
* [81] R. Joynt and L. Taillefer, Rev. Mod. Phys. 74, 235 (2002).
* [82] K. Matano, M. Kriener, K. Segawa, Y. Ando, and G. Zheng, Nature Phys 12, 852 (2016).
* [83] E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A. Gribanov, Yu. Seropegin, H. Noël, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004).
* [84] I. Sugitani, Y. Okuda, H. Shishido, T. Yamada, A. Thamizhavel, E. Yamamoto, T. D. Matsuda, Y. Haga, T. Takeuchi, R. Settai, and Y. Ōnuki, J. Phys. Soc. Jpn. 75, 043703 (2006).
* [85] N. Kimura, K. Ito, K. Saitoh, Y. Umeda, H. Aoki, and T. Terashima, Phys. Rev. Lett. 95, 247004 (2005).
* [86] S. Fujimoto, J. Phys. Soc. Jpn. 76, 051008 (2007).
* [87] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004).
* [88] M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).
* [89] A. Daido and Y. Yanase, Phys. Rev. B 95, 134507 (2017).
* [90] A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. 78, 373 (2006).
* [91] I. Reis, D. J. J. Marchand, and M. Franz, Phys. Rev. B 90, 085124 (2014).
* [92] P. D. Sacramento, J. Phys.: Condens. Matter 27, 445702 (2015).
* [93] K. Pöyhönen, I. Sahlberg, A. Westström, and T. Ojanen, Nat Commun 9, 2103 (2018).
* [94] T. C̆adez̆, R. Mondaini, E. V. Castro, and P. D. Sacramento, Acta Phys. Pol. A 135, 1180 (2019).
* [95] T. Devakul and D. A. Huse, Phys. Rev. B 96, 214201 (2017).
* [96] J. T. Chalker and G. J. Daniell, Phys. Rev. Lett. 61, 593 (1988).
* [97] J. T. Chalker, Physica A: Stat. Mech. Appl. 167, 253 (1990).
* [98] B. Huang and W. V. Liu, Phys. Rev. B 100, 144202 (2019).
* [99] J. H. Pixley, P. Goswami, and S. Das Sarma, Phys. Rev. Lett. 115, 076601 (2015).
* [100] Y. Fu, J. H. Wilson and J. H. Pixley, Phys. Rev. B 104, L041106 (2021).
* [101] M. Buchhold, S. Diehl, and A. Altland, Phys. Rev. B 98, 205134 (2018).
* [102] M. Gonçalves, P. Ribeiro, E. V. Castro and M. A. N. Araújo, Phys. Rev. Lett. 124, 136405 (2020).
* [103] Y. Fu, E. J. König, J. H. Wilson, Y.-Z. Chou, and J. H. Pixley, npj Quantum Materials (2020)5:71.
* [104] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B 96, 245115 (2017).
* [105] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
* [106] R. Resta, J. Phys.: Condens. Matter 22, 123201 (2010).
* [107] T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005).
* [108] Y.-F. Zhang, Y.-Y. Yang, Y. Ju, L. Sheng, R. Shen, D.-N. Sheng, and D.-Y. Xing, Chinese Phys. B 22, 117312 (2013).
* [109] Here we limit ourselves to the analysis of a system of size $41\times 41$, although draft results for a system of size $20\times 20$ followed the same behaviour as the one discussed.
* [110] R. Queiroz and A. P. Schnyder, Phys. Rev. B 89, 054501 (2014).
* [111] A. Mildenberger, F. Evers, A. D. Mirlin, and J. T. Chalker, Phys. Rev. B 75, 245321 (2007).
* [112] N. Sedlmayr, J. M. Aguiar-Hualde, and C. Bena, Phys. Rev. B 91, 115415 (2015).
* [113] K. Kobayashi, T. Ohtsuki, K.-I. Imura, and I. F. Herbut, Phys. Rev. Lett. 112, 016402 (2014).
* [114] J. C. C. Cestari, A. Foerster, M. A. Gusmão, and M. Continentino, Phys. Rev. A 84, 055601 (2011).
* [115] F. Wegner, Z Physik B 36, 209 (1980).
* [116] F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
|
§ INTRODUCTION
The Neil Gehrels Swift Observatory (Swift hereafter; [Gehrels et al. , 2004]) was launched on 2004 November 20, with the primary goal of discovering and localising Gamma-Ray Bursts (GRBs) to arcsec accuracy within a few hundred seconds, and then following the X-ray and optical afterglows as they fade. While GRBs are still an important part of Swift's work, over the years the satellite has become the go-to mission for observations of all kinds of transients as they vary in the optical, ultraviolet (UV), X-ray and $\gamma$-ray regimes. In this paper, we focus on novae observed in the UV by the UV/Optical Telescope (UVOT; [Roming et al. , 2005]).
Novae are thermonuclear explosions formed in accreting white dwarf (WD) systems. Material is transferred from the secondary, or companion, star onto the surface of the WD primary, until the pressure and temperature are sufficient to trigger a thermonuclear runaway (see [Bode & Evans , 2008, Woudt & Ribeiro , 2014] for review articles), flinging material outwards, and shrouding the WD surface from view. At this point, a new optical source – the nova – is formed, with the peak optical brightness occurring when the photosphere reaches maximum expansion[While most novae are first discovered at optical wavelengths, V959 Mon was initially detected in $\gamma$-rays by the Large Area Telescope onboard Fermi, when too close to the Sun for ground-based telescopes to observe [Cheung et al. , 2012, Cheung et al. , 2012]. Models of nova eruptions also predict there should be a brief, soft X-ray flash after hydrogen ignition, before the optical emission is detectable. This was finally observed for the first time in 2020 by eROSITA (extended Roentgen Survey with an Imaging Telescope Array) for the nova YZ Ret [König et al. , 2022].].
As the ejecta expand, they become optically thin, typically allowing the surface nuclear burning (and hence soft X-rays) to become visible; this is the so-called super-soft source (SSS) phase[It is, however, possible for the nuclear burning to cease before the ejecta fully clear, in which case the soft X-rays may fade away before they can be detected. In the case of V745 Sco [Page et al. , 2015], it was suggested that only the cooling, tail-end of the SSS emission was seen, with the actual hydrogen burning having ended very quickly, placing it close to this `unobservable region'.]. Once the hydrogen shell has been burnt, the system fades back to quiescence.
There are two main groups of novae, termed Classical Novae (CNe) and Recurrent Novae (RNe), respectively.
CNe – which make up the majority – have only been seen in eruption once. These systems have orbital periods of a few hours, and the secondary star is typically a late-type main sequence (MS) object <cit.>.
RNe, on the other hand, have had multiple detected eruptions, with recurrence timescales of up to around 100 yr (though this is, of course, something of a selection effect, dependent on historical records; even the so-called CNe are expected to erupt more than once, with duty cycles of typically thousands of years or longer <cit.>). The orbital periods of RNe cover a wide range, from a few hours to a year or more [Schaefer , 2010, Anupama , 2008], with the longer periods corresponding to systems where the secondary star is an evolved giant; the short period systems are more similar to CNe, with an MS or subgiant companion star.
A binary system with a higher-mass WD and higher accretion rate is expected to have a shorter recurrence time [Starrfield , 1989] (though T Pyx is unusual, in that it is thought to have a comparatively low mass WD; <cit.>). Given the shorter timescales involved, RNe accrete, and then eject, less material during each nova cycle <cit.>, which means the SSS phase tends to become visible more rapidly in these systems. The optical/UV light-curves in some RNe have been noted to show a flat, plateau phase, which is speculated to arise from the re-radiation of the SSS emission from an accretion disc [Schaefer , 2010, Hachisu et al. , 2006, Hachisu et al. , 2008].
There are ten confirmed Galactic RNe [Schaefer , 2010], with more known in other galaxies, such as M31 [Shafter et al. , 2015, Darnley , 2021].
Our sample contains some novae (classical and recurrent) which occurred in symbiotic binaries.
These are systems where the secondary star is a late-type giant, with outflowing wind encompassing the WD; they are sometimes known as embedded novae, to differentiate them from the so-called `symbiotic novae', which are a different population of slowly-evolving eruptive variable stars [Mürset & Nussbaumer , 1994, Mikołajewska , 2007, Mikołajewska , 2008], and are not considered here.
The first nova which Swift monitored in detail was the 2006 eruption of the RN RS Oph [Bode et al. , 2006, Osborne et al. , 2011]. This impressive dataset showed that Swift was well suited for monitoring such sources, and this subsequently led to many more novae being observed in detail with the observatory. Examples within our own Galaxy include V458 Vul [Ness et al. , 2009, Schwarz et al. , 2011], V2491 Cyg [Ibarra et al. , 2009, Page et al. , 2010], HV Cet [Beardmore et al. , 2012], V5668 Sgr [Gehrz et al. , 2018], V745 Sco [Page et al. , 2015], V407 Lup [Aydi et al. , 2018] and V3890 Sgr [Page et al. , 2020].
The best monitored Swift light-curves up until the end of 2017 were collated by [Page et al. , 2020], where both X-ray and UV results were presented. In this paper, we focus in more detail on the evolution of the UV emission in a sample of 12 novae. These were chosen to be the sources best monitored by Swift-UVOT over an extended period of time, in at least one UV filter, where the light-curves follow a series of power-law decays.
While this paper considers only novae, this approach to fitting UV light-curves could be extended to other transients, such as supernovae or active galactic nuclei.
§ OBSERVATIONS
The Swift-UVOT is one of two narrow field instruments onboard Swift; the X-ray Telescope (XRT; [Burrows et al. , 2005]) is the other. The UVOT has, among other observing capabilities, six broadband filters: three optical and three UV. Given that there are a large number of ground-based observers who frequently follow novae at optical wavelengths, Swift typically utilises one or more of the UV filters when monitoring novae: $uvw1$ (central wavelength, $\lambda_c$ = 2600 Å; Full Width at Half Maximum, FWMH = 693 Å), $uvm2$ ($\lambda_c$ = 2246 Å; FWHM = 498 Å) and $uvw2$ ($\lambda_c$ = 1928 Å; FWHM = 657 Å). More details on the instrument can be found in [Poole et al. , 2008, Breeveld et al. , 2011].
Novae can peak at very bright magnitudes, some even reaching naked eye visibility – RS Oph, for example, peaks around magnitude 4.5 in the visual band. While this is good news for the typical stargazer, sources brighter than around magnitude 10–11 lead to significant coincidence loss in the UVOT, which is not recovered when using the standard Swift photometric tools. A technique was developed by [Page et al. , 2013] whereby the read-out streak formed by bright sources can be used to obtain measurements up to 2.4 mag brighter than the previous limit, thus extending the useful dynamic range of the instrument. We have utilised this method where relevant in the following analysis.
In 2020 September, a more accurate UVOT calibration file was released, to account for the loss of sensitivity over time. The light-curves presented here have been corrected for this degradation[See details at <https://www.swift.ac.uk/analysis/uvot/>.], leading to small differences between these results and those previously published for the older novae (up to $\sim$ 0.02 mag for the sources in this sample).
Not only do novae show intrinsic variability as the source itself evolves, novae can be very different from each other, showing a variety of light-curve shapes <cit.> across all wavebands. Some show periodic modulations (e.g., HV Cet; [Beardmore et al. , 2012]), while others brighten and fade apparently randomly, sometimes with the UV in antiphase with the X-ray emission (e.g., V458 Vul; [Ness et al. , 2009, Schwarz et al. , 2011]); see Fig. <ref>. Many, however, do just (mainly) fade in the UV after peak brightness, and, in this paper, we concentrate on these novae. Specifically, we present a sample where detailed UVOT monitoring, starting promptly (within a few days) after the eruption[V959 Mon is the exception, with observations starting some time after the actual nova eruption, because of its location with respect to the Sun at outburst (see footnote 1).], was performed in at least one UV filter, over an extended period of time. Table <ref> lists the names of the novae, together with their eruption dates, orbital periods, interstellar reddening E(B$-$V), and (approximate) distances[Over the years, distance estimates to novae have changed. We use the most recent derivations by [Schaefer , 2022], using parallax data from the third Gaia data release (DR3) among other methods, for all our sample bar V407 Cyg, which was not included in this catalogue.].
Those novae which occurred in symbiotic systems are noted in the sixth column.
Examples of light-curves where the UV emission is not simply a series of power-law decays. Left: HV Cet. This nova shows a strong periodic oscillation in both the UV and X-ray bands, as well as an underlying slower modulation. The inset shows a zoom-in of the 1.77-d period, plotted in linear time. Right: V458 Vul. This nova shows aperiodic UV modulation, which is approximately anti-correlated with the corresponding X-ray emission. In both cases the XRT count rate is over 0.3–10 keV.
Names, eruption details, orbital periods, interstellar reddening E(B$-$V), and distances of the novae in this paper, ordered by eruption date; symbiotic systems are marked in the sixth column.
The novae marked in bold are recurrent; the eruption date given is that for the data included in this article.
$^a$ Date of $\gamma$-ray discovery by Fermi-LAT.
$^b$ While [Joshi et al. , 2015] refer to V1534 Sco as a symbiotic system, [Munari et al. , 2017] note that there are some inconsistencies with the evidence for a cool giant. We will therefore not treat V1534 Sco as a symbiotic in this work.
$^c$ There has been significant uncertainty in the distance to RS Oph over the years, with 1.6 kpc being assumed for a long time, although it now appears this larger estimate is preferred; see discussions in [Schaefer , 2022, Orio et al. , 2022] for example.
$^d$ https://www.aavso.org/aavso-alert-notice-752
Nova Alternative name Eruption date UT P$_{\rm orb}$ E(B$-$V) Symbiotic? distance (kpc) References
V2491 Cyg Nova Cyg 2008 No. 2 2008-04-10.73 2.3 h 0.23 4.8 [Nakano et al. , 2008, Baklanov et al. , 2008, Helton et al. , 2008, Rudy et al. , 2008, Schaefer , 2022]
U Sco – 2010-01-28.44 1.23 d 0.20 6.3 [Schaefer , 2010, Schaefer et al. , 2010, Pagnotta et al. , 2015, Schaefer , 2022]
V407 Cyg – 2010-03-10.80 43 y 0.5 Y 3.9 [Nishiyama et al. , 2010, Munari et al. , 1990, Hachisu & Kato , 2019, Shore et al. , 2011]
T Pyx – 2011-04-14.24 1.83 h 0.25 3.6 [Schaefer , 2010, Waagan et al. , 2011, Shore et al. , 2011, Schaefer , 2022]
V959 Mon Nova Mon 2012 2012-06-22$^{a}$ 7.1 h 0.38 2.9 [Fujikawa et al. , 2012, Cheung et al. , 2012, Cheung et al. , 2012, Page et al. , 2013, Shore et al. , 2013, Schaefer , 2022]
V339 Del Nova Del 2012 2013-08-14.58 0.163 d 0.18 1.6 [Nakano , 2013, Shore et al. , 2016, Schaefer , 2022]
V745 Sco – 2014-02-06.69 2440 d 1.00 Y 8.0 [Schaefer , 2010, Waagen & Pearce , 2014, Banerjee et al. , 2014, Schaefer , 2022]
V1534 Sco Nova Sco 2014 2014-03-26.85 520 d 1.11 ?$^b$ 8.2 [Nishiyama & Kabashima , 2014, Joshi et al. , 2015, Munari et al. , 2017, Hachisu & Kato , 2019, Schaefer , 2022, Schaefer , 2022]
V1535 Sco Nova Sco 2015 2015-02-11.84 50 d 0.80 Y 7.8 [Nakano , 2015, Walter , 2015, Nelson et al. , 2015, Linford et al. , 2017, Hachisu & Kato , 2019, Schaefer , 2022, Schaefer , 2022]
V3890 Sgr – 2019-08-27.87 747.6 d 0.59 Y 8.5 [Schaefer , 2010, Pereira , 2019, Munari & Walter , 2019, Evans et al. , 2022, Schaefer , 2022, Mikołajewska et al. , 2021]
V1674 Her Nova Her 2021 2021-06-12.19 0.153 d 0.50 3.2 [Patterson et al. , 2021, Drake et al. , 2021, Woodward et al. , 2021, Schaefer , 2022]
RS Oph – 2021-08-09.54 453.6 d 0.73 Y 2.7$^c$ AAVSO$^d$; [Brandi et al. , 2009, Schaefer , 2010, Page et al. , 2022, Schaefer , 2022, Snijders , 1987]
§ ANALYSIS AND RESULTS
Following [Page et al. , 2013, Page et al. , 2015, Page et al. , 2020], we parameterise the UV decay as a series of power-laws. That is, we consider the magnitudes to be proportional to log(time), which is equivalent to flux proportional to time: f $\propto$ (t/1 day)$^{-\alpha}$; T$_0$ is defined as the start of the nova eruption.
While optical/UV light-curves are traditionally plotted in terms of magnitude versus linear time, a log scale allows us to infer useful information, with trends easily visible to the eye.
A series of power-laws was thus fitted to each UVOT light-curve, using a $\chi^2$ minimisation routine for both slopes and break times; breaks were included if they were significant at the 3$\sigma$ level.
The power-law slope $\alpha$ was then calculated as
\begin{equation}
\alpha = \frac{1}{2.5} . \frac{m_2 - m_1}{log\frac{t_2}{t_1}}
\label{eqn}
\end{equation}
where m$_1$ and m$_2$ are the magnitudes from the fitted model at the break times of t$_1$ and t$_2$.
For each nova, the different UV filter data were fitted independently, and all breaks are assumed to be instantaneous, with no smoothing applied.
Figs. <ref>–<ref> show the X-ray and UV light-curves for each of the novae in the sample, plotted as days since eruption. The solid grey lines show the broken power-laws fitted to the UVOT data; the corresponding power-law indices for each segment are listed in Table <ref>. Where the value of $\alpha$ is negative, this indicates that the source has rebrightened. The error bars on the data points are mostly smaller than the marker size, and are usually $\sim$ 0.02–0.05 mag, occasionally up to $\sim$ 0.3 mag for some of the shorter (and fainter) observations.
The XRT and UVOT light-curve of V2491 Cyg, with the best-fit broken power-law decline shown as the grey solid line. The UV light-curve shows a rebrightening `cusp' after day 12, so only the data after day 19 (marked with the vertical dashed line) were fitted. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of U Sco, with the best-fit broken power-law declines shown as the grey solid lines. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V407 Cyg, with the best-fit broken power-law decline shown as the grey solid line. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of T Pyx, with the best-fit broken power-law decline shown as the grey solid line. Only data after the UV peak on day 70 (marked with the vertical dashed line) were considered. The observations of T Pyx are plotted as days since discovery; the optical peak occurred around 28 days later. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V959 Mon, with the best-fit broken power-law declines shown as the grey solid lines. Late-time data beyond day 300 were excluded from the fits, because there are not enough bins to constrain the break times accurately. The observations of V959 Mon are plotted as days since Fermi discovery. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V339 Del, with the best-fit broken power-law declines shown as the grey solid lines. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V745 Sco, with the best-fit broken power-law declines shown as the grey solid lines. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V1534 Sco, with the best-fit broken power-law declines shown as the grey solid lines. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V1535 Sco, with the best-fit broken power-law decline shown as the grey solid line. Data were only collected with the $uvw1$ filter intermittently for the first 19 days, so have not been included. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V3890 Sgr, with the best-fit broken power-law decline shown as the grey solid line. $uvw2$ data were only collected until day 25, so have not been included. Late-time data after day 100 have been excluded from the fits. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of V1674 Her, with the best-fit broken power-law declines shown as the grey solid lines. The earliest bin in the $uvm2$ light-curve has been excluded from the fit, since the break time cannot be constrained. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
The XRT and UVOT light-curves of RS Oph, with the best-fit broken power-law declines shown as the grey solid lines. Only the UVOT data before the first solar conjunction have been fitted. The X-ray data (top) are in count s$^{-1}$ over 0.3–10 keV.
Parameterising the UV light-curves as a series of power-law decays. The uncertainties on the fitted break times are given at the 90% level, and the power-law slopes are calculated using Equation <ref>. The errors on the slopes have been estimated using the uncertainties of the magnitudes from the fits.
Nova filter $\alpha_1$ T$_{\rm break,1}$ $\alpha_2$ T$_{\rm break,2}$ $\alpha_3$ T$_{\rm break,3}$ $\alpha_4$ T$_{\rm break,4}$ $\alpha_5$ T$_{\rm break,5}$ $\alpha_6$
(day) (day) (day) (day) (day)
V2491 $uvw2$ 1.93$^{+0.02}_{-0.02}$ 40.2$^{+0.1}_{-0.1}$ 4.60$^{+0.01}_{-0.01}$ 55.1$^{+0.1}_{-0.1}$ 2.91$^{+0.09}_{-0.09}$ 124$^{+2}_{-3}$ 1.5$^{+0.1}_{-0.1}$
U Sco $uvw1$ 0.54$^{+0.05}_{-0.05}$ 4.4$^{+0.1}_{-0.3}$ 3.0$^{+0.1}_{-0.1}$ 12.52$^{+0.01}_{-0.01}$ $-$0.28$^{+0.01}_{-0.01}$ 24.19$^{+0.06}_{-0.06}$ 4.99$^{+0.01}_{-0.01}$ 44.51$^{+0.06}_{-0.14}$ $-$2.85$^{+0.06}_{-0.06}$ 52.28$^{+0.07}_{-0.03}$ 8.34$^{+0.02}_{-0.02}$
$uvw2$ 0.44$^{+0.02}_{-0.02}$ 3.9$^{+0.1}_{-0.1}$ 3.20$^{+0.02}_{-0.02}$ 12.53$^{+0.01}_{-0.09}$ 0.02$^{+0.01}_{-0.01}$ 25.15$^{+0.01}_{-0.06}$ 5.61$^{+0.02}_{-0.02}$ 41.8$^{+0.1}_{-0.2}$ $-$0.33$^{+0.04}_{-0.04}$ 52.34$^{+0.02}_{-0.07}$ 7.56$^{+0.02}_{-0.02}$
V407 $uvm2$ 0.35$^{+0.01}_{-0.01}$ 41.4$^{+0.3}_{-0.4}$ 4.39$^{+0.07}_{-0.07}$ 63$^{+1}_{-1}$ 2.23$^{+0.06}_{-0.06}$
T Pyx $uvm2$ 6.54$^{+0.01}_{-0.01}$ 113.90$^{+0.01}_{-0.01}$ 1.27$^{+0.01}_{-0.01}$ 142.59$^{+0.01}_{-0.01}$ 7.03$^{+0.05}_{-0.05}$ 150.2$^{+0.2}_{-0.3}$ 4.05$^{+0.01}_{-0.01}$ 268.7$^{+0.9}_{-0.8}$ 1.38$^{+0.02}_{-0.02}$
V959 $uvw1$ 2.02$^{+0.05}_{-0.05}$ 212$^{+1}_{-1}$ 5.33$^{+0.08}_{-0.08}$
Mon $uvm2$ 1.83$^{+0.01}_{-0.01}$ 205$^{+1}_{-1}$ 4.91$^{+0.07}_{-0.07}$
$uvw2$ 2.07$^{+0.04}_{-0.04}$ 205$^{+1}_{-1}$ 5.29$^{+0.06}_{-0.06}$
V339 $uvw1$ – – 1.1$^{+0.2}_{-0.2}$ 138$^{+16}_{-14}$ 2.4$^{+0.2}_{-0.2}$ 258$^{+12}_{-4}$ 1.5$^{+0.5}_{-0.5}$
Del $uvm2$ 3.0$^{+0.5}_{-0.5}$ 81$^{+15}_{-7}$ 0.8$^{+0.5}_{-0.2}$ 129$^{+2}_{-1}$ 2.17$^{+0.07}_{-0.07}$ 266$^{+11}_{-9}$ 1.7$^{+0.2}_{-0.2}$
$uvw2$ – – 0.94$^{+0.08}_{-0.08}$ 133$^{+5}_{-4}$ 2.96$^{+0.02}_{-0.02}$ 264$^{+2}_{-6}$ 2.1$^{+0.1}_{-0.1}$
V745 $uvw1$ 0.68$^{+0.01}_{-0.01}$ 6.3$^{+0.1}_{-0.1}$ 3.01$^{+0.05}_{-0.05}$ 15.7$^{+0.5}_{-0.4}$ 1.45$^{+0.03}_{-0.03}$
Sco $uvm2$ 0.53$^{+0.01}_{-0.01}$ 5.5$^{+0.1}_{-0.1}$ 2.4$^{+0.1}_{-0.1}$ 18$^{+1}_{-1}$ 1.40$^{+0.03}_{-0.03}$
$uvw2$ 0.73$^{+0.01}_{-0.01}$ 7.8$^{+0.1}_{-0.1}$ 3.62$^{+0.06}_{-0.06}$ 15.5$^{+0.4}_{-0.4}$ 1.46$^{+0.04}_{-0.04}$
V1534 $uvw1$ 0.75$^{+0.02}_{-0.02}$ 9.5$^{+0.4}_{-0.4}$ 3.65$^{+0.06}_{-0.06}$ 22$^{+1}_{-1}$ 1.1$^{+0.1}_{-0.1}$
Sco $uvm2$ 0.5$^{+0.1}_{-0.1}$ 4.9$^{+1.6}_{-0.6}$ 1.7$^{+0.2}_{-0.2}$ – –
$uvw2$ 0.43$^{+0.09}_{-0.09}$ 9.4$^{+1.1}_{-0.7}$ 3.7$^{+0.3}_{-0.3}$ – –
V1535 $uvm2$ 0.71$^{+0.01}_{-0.01}$ 16.2$^{+0.1}_{-0.1}$ 5.20$^{+0.01}_{-0.01}$ 19.0$^{+0.2}_{-0.1}$ 2.0$^{+0.1}_{-0.1}$ 31.4$^{+0.3}_{-0.8}$ 3.7 $^{+0.2}_{-0.2}$ 44.4$^{+0.6}_{-0.6}$ $-$1.32$^{+0.02}_{-0.02}$ 67$^{+1}_{-1}$ 5.00$^{+0.06}_{-0.06}$
Sco $uvw2$ 0.85$^{+0.05}_{-0.05}$ 16.0$^{+0.2}_{-0.2}$ 6.4$^{+0.4}_{-0.4}$ 18.2$^{+0.6}_{-0.4}$ 2.2$^{+0.1}_{-0.1}$ – – – – – –
V3890 $uvm2$ 0.88$^{+0.06}_{-0.06}$ 13$^{+1}_{-1}$ 1.4$^{+0.1}_{-0.1}$ 21.6$^{+0.2}_{-0.2}$ 4.7$^{+0.1}_{-0.1}$ 30.2$^{+0.6}_{-0.6}$ 2.9$^{+0.3}_{-0.3}$ 41$^{+3}_{-3}$ 2.1$^{+0.1}_{-0.1}$
V1674 $uvw1$ – – 2.2$^{+0.2}_{-0.2}$ 30$^{+2}_{-4}$ 2.1$^{+0.1}_{-0.1}$ 77$^{+8}_{-3}$ 1.2$^{+0.2}_{-0.2}$
Her $uvm2$ 2.1$^{+0.2}_{-0.2}$ 12.3$^{+1.2}_{-0.3}$ 1.96$^{+0.01}_{-0.01}$ 32$^{+8}_{-5}$ 2.02$^{+0.01}_{-0.01}$ 86$^{+6}_{-5}$ 0.80$^{+0.06}_{-0.06}$
$uvw2$ 1.6$^{+0.2}_{-0.2}$ 15$^{+1}_{-1}$ 2.6$^{+0.1}_{-0.1}$ 30$^{+2}_{-3}$ 2.2 $^{+0.2}_{-0.2}$ – –
RS uvm2 0.99$^{+0.08}_{-0.08}$ 27$^{+1}_{-1}$ 1.70$^{+0.03}_{-0.03}$ 76$^{+2}_{-3}$ 3.6$^{+0.6}_{-0.6}$
Oph uvw2 1.1$^{+0.2}_{-0.2}$ 40$^{+3}_{-2}$ 1.8$^{+0.6}_{-0.6}$ 68$^{+17}_{-9}$ 2.6$^{+0.10}_{-0.10}$
§.§ Light-curve descriptions
Below we briefly summarise the fits for each nova. We note that the models applied may not be a unique description of the data. For example, in some cases consecutive slope changes are in the same direction – i.e., we see a gradual flattening across two break times, rather than a flattening followed by a steepening – which may suggest a smoother, slower change is occurring, rather than the instantaneous breaks we fit. The series of sharply-broken power-laws is, however, a simple and convenient characterisation of the decline in UV brightness.
§.§.§ V2491 Cyg
Data were collected in the $uvw2$ filter between days 7 and 236; the earliest magnitudes were estimated using the read-out streak method [Page et al. , 2013] (Fig. <ref>). The UV light-curve shows a rebrightening starting around day 12 (a `cusp'; [Strope, Schaefer & Henden , 2010]), so only the data after day 19 were fitted. The data before the start of the cusp align with an extrapolation of the earliest fitted power-law, however. During the post-cusp time interval, the rate of the UV decline steepened around day 40, before starting to flatten off at day 55 and flattening further after day 133.
§.§.§ U Sco
Following the 2010 eruption of the RN U Sco, the majority of the UVOT data were collected using the $uvw1$ and $uvw2$ filters, between days 0.6 and 63 post-eruption; the earliest magnitudes were estimated using the read-out streak method [Page et al. , 2013] (Fig. <ref>). U Sco is an eclipsing system <cit.>, with the 1.23 d period clearly visible in the UVOT data (Fig. <ref>). However, the underlying long-term evolution (including all the data within and outside the eclipse times) can be approximated by a series of power-law segments, showing both fading and flattening/brightening intervals. The fitted break times in the light-curves agree between the filters to within
a few days.
Zoom-in of part of the U Sco $uvw1$ light-curve, showing a series of eclipses.
§.§.§ V407 Cyg
Between days 5 and 141, observations were performed using the $uvm2$ filter (Fig. <ref>). During this time, the decay steepened on day 41, before flattening off again after day 63.
§.§.§ T Pyx
T Pyx was observed while the UV emission was still rising; these early data, starting on day 7, have been excluded, with the decay fitting performed after day 70, once the emission had peaked, and running through till day 369. The $uvm2$ light-curve (Fig. <ref>) showed a complex decay, with at least five separate segments, including a short, week-long interval between days 142 and 149, which showed a significant steepening.
§.§.§ V959 Mon
V959 Mon was initially monitored only using the $uvm2$ filter between days 58 and 150; after this time, the other two UV filters were also brought into play (Fig. <ref>). There was no notable change in decay rate at the earlier times, and all three filters show breaks from a flatter to steeper decay around day 205-212.
Only data until day 256 are considered; after this time there were observations on days 423 (all three filters), 687 ($uvw1$) and 1002 ($uvm2$), which do not provide enough data points to constrain additional fitting. The measurements do suggest a further flattening of the decay between days 256 and 423, however.
§.§.§ V339 Del
The very earliest observations of V339 Del with Swift were performed with the UVOT blocked, because of the extreme brightness (it peaked at $V$ $\sim$ 4.3; [Munari et al. , 2015]); the UV data collection commenced on day 57 in the $uvm2$ filter, and day 60 in $uvw1$ and $uvw2$, continuing until day 291 in $uvw1$, and day 409 in both $uvm2$ and $uvw2$ (Fig. <ref>). Again, the read-out streak method [Page et al. , 2013] was required for data brighter than around magnitude 10–11. Since the observations were performed mainly with the high time resolution event mode, data were subsequently averaged over bins of $\sim$ 1 d in length, to improve the statistics. The $uvm2$ filter was the one used most frequently at the start, and this higher density of early data points is likely why an additional early-time break, from steep to flat can be constrained in this light-curve. All three UV light-curves subsequently steepen around day 130–140, before flattening between day 258–266.
§.§.§ V745 Sco
V745 Sco was observed with all three filters from 0.5 day after eruption until day 66. There was also an earlier $uvm2$ observation at day 0.16, and further observations out till day 173 in $uvw1$ (Fig. <ref>). All three light-curves steepen around day 5–8, and flatten off again between $\sim$ day 16–19.
§.§.§ V1534 Sco
While V1534 Sco was initially observed with all UV filters, beyond three weeks after eruption only the $uvw1$ observations were continued (Fig. <ref>). Considering the longest ($uvw1$) dataset, which runs between days 0.3 and 83, the decay broke twice, first steepening after day 9.5, then flattening again after day 22. The $uvm2$ and $uvw2$ light-curves only cover the time of the first break, with data collected between days 0.9–15.2 and 0.9–19.8, respectively. While the break time fitted to the $uvw2$ data, and the post-break slope, are in close agreement with $uvw1$, the initial slope is flatter. The break in $uvm2$ appears to occur days earlier.
§.§.§ V1535 Sco
V1535 Sco was observed between days 3.8 and 80.3 with the $uvm2$ filter, and 6.0 to 42.2 with $uvw2$ (Fig. <ref>). The $uvw1$ filter was only used for seven observations, so these data have not been included. Because of the longer duration of the $uvm2$ dataset, it is better fitted with a further three breaks, in addition to the two which are required for the $uvw2$ light-curve, around 16 and 18–19 days. After day 44, the $uvm2$ data show a brightening trend for around 23 days, before starting to fade again.
§.§.§ V3890 Sgr
The vast majority of the UVOT observations of V3890 Sgr were obtained using the $uvm2$ filter, so these are the data considered here (Fig. <ref>). Observations in this filter began on day 3, continuing until the nova entered the solar observing constraint for Swift on day 75. Further observations were taken after the source re-emerged from behind the Sun, until day 250, but these have not been included in the overall fit (though are shown in Fig. <ref>). The light-curve decline steepens twice and then flattens off again, in two stages. The UV brightness appears to have stayed constant during the observing constraint (or the source faded and rebrightened by about the same amount), before starting to fade again, with $\alpha$ $\sim$ 2.8. We note that this flat interval is much later than the X-ray SSS phase, so will not correspond to the plateau phase of [Hachisu et al. , 2008].
§.§.§ V1674 Her
An early observation was obtained on day 1.3 using the $uvm2$ filter, with following observations on days 6.5, 7.6 and 10.3 in $uvw2$, $uvm2$ and $uvw1$ respectively, before subsequent observations used all three filters until day 86, after which filters were again alternated (Fig. <ref>). The last observations in $uvw1$, $uvm2$ and $uvw2$ were on days 126.5, 341.4 and 86.5, respectively.
There is clearly a change in slope between the first two observations obtained with the $uvm2$ filter on days 1.3 and 7.6, but a break time cannot be sensibly estimated with no measurements in between. Therefore the initial $uvm2$ data point was not included in the analysis.
While the $uvm2$ data show evidence for three breaks in the power-law decay, the $uvw1$ data do not require the earliest break, most likely since this filter dataset starts latest. Similarly, the $uvw2$ light-curve does not require the final change in slope because of the lack of data beyond day 86. Although $\alpha_2$ and $\alpha_3$ are very similar for the $uvm2$ data, the slight change is an improvement (the break is significant at the 3$\sigma$ level), and, given the requirement for a break around day 30 in the other two filters, it seems likely to be real.
§.§.§ RS Oph
The very earliest Swift observations of RS Oph in 2021 had the UVOT blocked, because of the optical brightness of the source. The first photometric observation was obtained 15.6 days after the optical peak, and ran through till the solar observing constraint began on day 87; observations began again on day 197.5 and continued until the following solar constraint, on day 448. The read-out streak method [Page et al. , 2013] was used for the first two months. The majority of observations were obtained with either the $uvm2$ or $uvw2$ filter (sometimes with grism exposures as well; [Azzollini et al. , 2022]), and these are shown in Fig. <ref>. Both filters show two steepening breaks, the first around day 30–40, the second, day 70–75. Following solar conjunction, the $uvm2$ light-curve shows signs of slow rebrightening; these data have not been included in the fits.
§.§ Normalised break times
In order to compare the different novae more directly, in Table <ref> the break times have been normalised to the end of the X-ray SSS phase as T$_{\rm break}$/T$_{\rm SSSend}$. Each SSS end point has been estimated from the X-ray light-curves (Figs. <ref>-<ref>; see also [Page et al. , 2020, Page et al. , 2020, Drake et al. , 2021]), and is defined to be the time at which the X-ray count rate starts to decrease steadily. There can obviously be some uncertainty in this time, given that there are gaps in the light-curve coverage, as well as short-term variability, but this is typically only a couple of days or less[The SSS phase of V339 Del ended during the solar observing constraint, when no observations could be performed. However, an extrapolation backwards of the decline once observations restarted suggests the X-ray fading began around the same time as the constraint.], as can be seen from the top panels of Figs. <ref>–<ref>.
The nth break time normalised to the end time of the X-ray SSS: T$_{\rm break,n}$ (norm.) = T$_{\rm break,n}$/T$_{\rm SSSend}$. The power-law indices are the same as in Table <ref>, so have not been listed.
$^{a}$ The SSS switched off during the Sun constraint, which ran from days 144–202; given the shape of the decay, it seems likely the SSS ended around the time V339 Del entered the observing constraint.
$^b$ In V1534 Sco, the early brightening of the X-ray emission appears to be caused by the absorbing column declining, rather than a new soft component appearing. The X-ray count rate peaked on day 7, and this is the value used for the break time normalisation.
Nova T$_{\rm SSSstart}$–T$_{\rm SSSend}$ filter T$_{\rm break,1}$ T$_{\rm break,2}$ T$_{\rm break,3}$ T$_{\rm break,4}$ T$_{\rm break,5}$
(day) (norm.) (norm.) (norm.) (norm.) (norm.)
V2491 33–43 $uvw2$ 0.9 1.3 2.9
U Sco 12–33 $uvw1$ 0.1 0.4 0.7 1.4 1.6
$uvw2$ 0.1 0.4 0.8 1.3 1.6
V407 12–40 $uvm2$ 1.0 1.6
T Pyx 123–180 $uvm2$ 0.6 0.79 0.83 1.5
V959 150–200 $uvw1$ 1.1
Mon $uvm2$ 1.0
$uvw2$ 1.0
V339 60–144$^{a}$ $uvw1$ – 0.9 1.8
Del $uvm2$ 0.6 0.9 1.8
$uvw2$ – 0.9 1.8
V745 3–6 $uvw1$ 1.1 2.6
Sco $uvm2$ 0.9 3.2
$uvw2$ 1.3 2.6
V1534 7$^b$ $uvw1$ 1.4 3.2
Sco $uvm2$ 0.7 –
$uvw2$ 1.2 –
V1535 12–25 $uvm2$ 0.6 0.8 1.3 1.8 2.7
Sco $uvw2$ 0.6 0.7 – – –
V3890 8–20 $uvm2$ 0.7 1.1 1.5 2.1
V1674 19–45 $uvw1$ – 0.7 1.7
Her $uvm2$ 0.3 0.7 1.9
$uvw2$ 0.3 0.7 –
RS 21–62 uvm2 0.4 1.2
Oph uvw2 0.6 1.1
Each light-curve shows a break close in time to the end of the SSS phase (i.e., there is a T$_{\rm break}$ (norm) close to 1 – in the range 0.7–1.3 – in Table <ref>), with the possible exception of V1534 Sco, which does not break until a little later in the $uvw1$ band.
However, as mentioned in the caption of Table <ref>, V1534 Sco did not show an obvious SSS phase, but rather became softer and brighter as the absorbing column decreased, so this discrepancy (and the mis-match between the break times in the different filters) can be disregarded.
Fig. <ref> shows the normalised break times (using the best-sampled light-curve for each nova which was observed using more than one UVOT filter) plotted as a histogram. The inset shows only the break which is closest in time to the end of the SSS phase, since some of the light curves have multiple breaks around this interval. There is a clear peak in the number of breaks around T$_{\rm SSSend}$. The fact that some breaks occur before the apparent end of the SSS emission in the X-ray band will be returned to in Section <ref>.
A histogram showing the normalised break times fitted in the UVOT light-curves for all novae in the sample; the best-sampled light-curve has been used in each case where observations were performed in more than one filter. The inset plots only the break for each nova light-curve which is closest in time to T$_{\rm break}$/T$_{\rm SSSend}$ = 1.
§ DISCUSSION
The UV light-curves presented here were chosen to be well sampled, and it is obvious that a higher cadence of observations is more likely to highlight changes in slope, especially over shorter time intervals[U Sco was observed at a particularly high cadence when trying to map the 1.23-day eclipses.]. This should be borne in mind when comparing the fits to different novae.
§.§ Individual novae
Two thirds of the novae in this sample have multi-filter light-curves. While there are distinct similarities between the different filters, the best fits are not always identical.
Considering the break times normalised with respect to the end of the SSS in Table <ref>, there is a slight difference in the third and fourth break times for U Sco (with T$_{\rm break,4}$ being around 2.7 d earlier in $uvw2$ from Table <ref>), causing different slopes in the plateau, fading and late re-brightening.
V959 Mon shows a later break (by about a week) in $uvw1$, although the $uvw1$ and $uvw2$ slopes are very similar; $uvm2$ shows flatter declines both pre- and post-break.
In V745 Sco, the first break happens in the order $uvm2$, $uvw1$, $uvw2$; however, the second break time is consistent for $uvw1$ and $uvw2$, with $uvm2$ apparently breaking a couple of days later.
We note that the $uvw1$ and $uvw2$ filter transmission curves have extended red tails – the so-called `red leak'[<https://swift.gsfc.nasa.gov/analysis/uvot_digest/redleak.html>]. This might in general affect the break times in these two filters compared with $uvm2$ (which does not suffer from this red leak) in systems with a red companion.
In the case of V1534 Sco, only the first break was covered by all three filters, and the break in $uvm2$ is noticeable earlier (around 4.5 d), leading to a flatter post-break decay slope. The longest dataset, $uvw1$, fades more steeply than the other two before this early-time break.
V1535 Sco has very similar break times for both filters used for the early monitoring. The slope between the breaks is steeper in the $uvw2$ data, however.
V1674 Her shows an earlier first break in $uvm2$ than $uvw2$ (no break required in $uvw1$ due to lack of early data). Around this break, the $uvm2$ light-curve flattens, whereas the $uvw2$ data steepen. The break around day 30 is consistent in all three filters, and the third break agrees between the two filters used at that time, though the error bars on the time are relatively large.
For RS Oph, the first break occurs slightly earlier in $uvm2$ than $uvw2$, but the second break occurs a little later – although the second break time is consistent within the uncertainties. The initial decay slopes are similar for both filters, though the final decay is steeper in $uvm2$.
In some of these situations, the differences in observation times (and sometimes cadences) between the UVOT filters may account for some of the discrepancies found. There do seem to be some intrinsic differences between the results at different wavelengths, but there are no obvious trends which are the same in each case: we don't always see that the bluer $uvw2$ data fade in a different manner from the redder $uvw1$, for example. As noted above, $uvw1$ and $uvw2$ are affected by a `red leak', which could be a more significant issue where the secondary star is redder. UV spectra of novae often show strong emission lines, which can vary over time. Such emission will affect the overall magnitudes in different filters by differing amounts, depending on the wavelengths of the lines, which may lead to changes in the apparent break times between filters.
We recall that [Page et al. , 2015] extended the fitting to the optical and infrared (IR) bands for V745 Sco, finding that the optical curves changed more gradually than the UV, while the IR was more strongly affected by light from the red giant (RG) secondary.
§.§ Comparing the sample as a whole
If the pre-day-70 data for T Pyx, and the single early $uvm2$ bin for V1674 Her, are included, then, with the exception of the $uvm2$ data for V339 Del, all the light-curves in the sample show a first break from a flatter to a steeper decay. This initial break occurs before or at the time of the SSS switch-off in the X-ray band[V1534 Sco is a possible exception, as previously noted.].
Whether it is the earliest fitted break or not, all the novae (with obvious supersoft emission) show a change in slope around the end of the SSS phase (given the uncertainties in determining the exact SSS end time), as has been previously noted by, e.g., [Page et al. , 2010, Page et al. , 2013, Page et al. , 2015]; see also Fig. <ref>.
The end of the SSS phase corresponds to the point at which nuclear burning ceases, thus leading to a drop in the ionisation of the RG wind in the symbiotic-like systems, and a general decline in any UV emission coming directly from the hot WD photosphere.
All the breaks are from flatter to steeper at this time, with the exception of V1674 Her, for which $uvw1$ and $uvw2$ both show a flattening; the $uvm2$ curve shows a (slight) steepening, however. T Pyx shows two breaks in quick succession (only a week apart) close to 0.8 T$_{\rm SSSend}$, with the first showing a flat to steep transition. The V1535 Sco $uvm2$ light-curve has two breaks relatively close in time to the SSS end point (0.76$\times$ and 1.26$\times$ T$_{\rm SSSend}$), with the second of these times corresponding to a steepening, which may suggest that is the break related to the end of the SSS phase (and is therefore the normalised break time included in the inset of Fig. <ref>).
Fig. <ref> shows that, for some novae, the break closest in time to the end of the SSS phase actually occurs a little before T$_{\rm SSSend}$ measured from the X-ray light-curves. While it is not immediately obvious why the UV emission would start to fade before the nuclear burning switches off, it could be that fitting the light-curves with sharp breaks affects the results, given that some breaks, at least, might in reality be a smoother transition between power-law slopes. In addition, changes in the continuum spectral shape, or the emergence (or variation) of emission lines could affect the rate at which the overall UV brightness changes.
All the novae then show further breaks in their UV decline after the end of the SSS phase. There are limited later-time measurements for V959 Mon, but it is clear that the decay flattened again; there just isn't enough information to constrain the fit. Following the 100+ day gap caused by the solar conjunction (which started around 10–15 days after the SSS phase started to fade), the data for RS Oph show a brightening trend. These later changes in slope do not occur at fixed times with respect to the SSS turn-off, neither do different novae show similar numbers of breaks over given time intervals. The earliest fitted break occurs at only 10% of the SSS end time in U Sco, corresponding to around day 4 post-eruption (followed by a further early break at 40% of the SSS time – day 12.5 – corresponding to the start of the plateau interval, which then persists for around 12 days). The UV decay in V745 Sco also breaks early on, around day 6, though that already equates to the end of the SSS phase in this rapidly-evolving nova. We note that T Pyx shows a short `wiggle' at the end of its month-long plateau (days 114–143), fading steeply with $\alpha$ $\sim$ 7 for about a week, before continuing to decay with $\alpha$ $\sim$ 4 for more than 100 days.
The different novae were followed for differing lengths of time overall, with respect to the end of the SSS phase (or, indeed, in general: usually out till at least 100 days post-eruption, though). The duration of the monitoring was generally set by the brightness of the fading nova (either X-ray or UV), though sometimes will also have been affected by Swift observing constraints. This means that we cannot sensibly compare the numbers of late-time (i.e., after the end of the SSS emission) breaks.
There is a wide range of decay slopes seen, from the flat plateau phase in some RNe, to steep declines of $\alpha$ $>$ 5. Two of the novae (U Sco and V1535 Sco) show temporary rebrightenings at later times; in fact, both of these happen around day 40 post-eruption, though these equate to different normalised times with respect to the SSS end point. The UV emission of RS Oph also rebrightens at late times; this rebrightening has not yet ended with the observations collected to date.
Besides the break in decay caused by the end of nuclear burning, the other observed changes in slope are likely to be related to physical processes in the expanding ejecta, such as declining density, or (before the end of the SSS phase) decreasing incident flux due to greater distance from the WD. In some novae, the accretion process may remain unsettled for quite some time after the eruption, rather than promptly relaxing back into a typical accretion disc [Mason et al. , 2021], and this could affect the rate at which the UV emission fades.
§.§ Populations
In Fig. <ref>, we show the power-law fits to the light-curves in this sample (one per nova, choosing the filter which covers the longest interval of time if more than one was used), transformed into absolute magnitude using the distances in Table <ref>, and plotted against the time normalised to the SSS end. Corrections for reddening have also been applied at the wavelength of the UV filter in use each time, based on values of E(B$-$V) taken from the literature (mainly from [Schaefer , 2022]; see Table <ref>), and assuming that A$_{\rm V}$ = 3.1 $\times$ E(B$-$V) for the Milky Way [Draine , 2003].
The early times for T Pyx and V2491 Cyg which were excluded from the power-law fitting are not shown, likewise the V959 Mon and RS Oph late time data.
The sub-samples of CNe, RNe with short periods, RNe with long periods, and symbiotic-like systems (which have not been seen to recur) are plotted in black, red, magenta and blue, respectively. Different line styles highlight which filter is plotted. The symbiotic RNe with long periods are at the brighter absolute magnitude end of the sample, with the CNe mainly at a much lower luminosity (V1534 Sco being the exception, though there is a possibility that this source is actually a symbiotic system); there do not appear to be any other obvious differences between the populations.
Power-law fits to the UVOT light-curves, converted to absolute magnitude, and plotted against time normalised to the end of the SSS. In the case where a nova was observed in multiple filters, the best sampled curve is shown. Classical novae are plotted in black, short-period recurrent nova in red, long-period recurrent novae with RG companions in magenta, and non-recurrent symbiotic-like systems (those with giant companions) in blue. Solid lines mark the light-curves collected using the $uvw1$ filter, dashed – $uvm2$ and dotted – $uvw2$.
§.§.§ Classical novae
The classical (non-symbiotic) novae in this current sample are V2491 Cyg, V959 Mon, V339 Del, V1674 Her and (probably) V1534 Sco. The UV light-curves of V959 Mon started later after eruption because of the proximity to the Sun delayed observations, so early-time breaks in the decay may have been missed. Despite this, these four objects show similar power-law decays and breaks.
The spread of unreddened, absolute magnitudes is not large, either.
§.§.§ Recurrent novae
The recurrent novae in this paper consist of both short-period (U Sco and T Pyx), and long-period (with a giant companion) systems (V745 Sco, V3890 Sgr and RS Oph). In this sample of 12 novae, the long-period, symbiotic RNe are typically the overall brightest in terms of absolute magnitude (although we did not measure the peak for CN V339 Del, since the UVOT was operated in blocked mode for instrument safety). These measurements do, of course, rely on having accurate distance and extinction measurements. For example, if RS Oph were at 1.6 kpc (the previously favoured distance) instead of 2.7 kpc, it would be about one magnitude fainter in absolute terms.
U Sco and T Pyx, the two short-period systems, both show early time breaks and the (oft-seen) plateau corresponding to the X-ray SSS interval. None of the long period, symbiotic systems in this sample shows a plateau in their UV decline. The earliest break in the V745 Sco light-curve corresponds to the end of the SSS phase, which occurred very early, running only between days 3–6 post-eruption; there might not have been time for an optical/UV plateau to form during this brief interval. V3890 Sgr showed a longer SSS duration of $\sim$ 12 d, though this is still shorter than in U Sco ($\sim$ 21 d) and T Pyx ($\sim$ 57 d) so, again, this might account for the lack of plateau. While RS Oph does not show an obvious plateau in these UV data, its optical light-curves do, as shown by [Strope, Schaefer & Henden , 2010, Schwarz et al. , 2011].
§.§.§ Symbiotic-like novae
In this sample there are five systems which have cool RG companions: V407 Cyg, V1535 Sco, V745 Sco, V3890 Sgr and RS Oph. The first two are CNe, having only been seen in eruption once, while the other three are known RNe. V1535 Sco has an orbital period of 50 d, while the other four have even longer P$_{\rm orb}$ $>$ 1 yr. Following a nova explosion in these embedded systems, there will be shock interactions between the expanding ejecta and the RG wind. As well as producing hard X-rays <cit.>, these shocks also lead to UV line emission [Azzollini , 2021, Azzollini et al. , 2022].
The presence or otherwise of a giant companion and accompanying wind does not directly have an effect on the end time of the X-ray SSS emission. In a summary paper of Swift-XRT observations of SSS novae, [Schwarz et al. , 2011] also showed that there is no correlation between orbital period (typically longer for the symbiotic-like novae) and the SSS turn-off time. The SSS interval is generally accepted to be related to the WD mass, with an early onset and short duration being signs of a massive WD <cit.> – as expected for RNe. We do indeed see that the RNe U Sco, V745 Sco and V3890 Sgr show early and short SSS phases; T Pyx does not, but this system is known to be unusual, evolving more slowly than most other recurrents [Schaefer , 2010]. The start of the SSS phase in RS Oph is not quite as early as some, but has a well-defined beginning and end.
For the symbiotic-like sample here, V745 Sco has a very short SSS phase (ending on day 6), while the other four switched off after 20–60 d. The SSS phases of the non-symbiotic systems end between days 33–200[As noted in Table <ref>, V1534 Sco does not show an obvious SSS phase, with day 7 actually corresponding to the time when the X-ray emission peaked due to the decrease of the absorption. This nova has therefore be omitted from this range.].
In the case of the symbiotic-like novae, there will be a RG wind `bubble' surrounding the system. The outer extent of this region depends on the recurrence time of the nova: the longer the time between eruptions (or if only one nova event has so far occurred), the further the RG wind can flow before disruption. Once the nova ejecta reach this limit (where the wind density becomes too low to be significant), the shock `breaks out' of the medium, which will likely lead to a drop in the UV flux as the shock-related emission decreases, and so a steepening in the light-curve decay. As an example, in the 2006 eruption of RS Oph (not included in this sample, since the majority of the UVOT observations were taken using the grism; [Azzollini , 2021, Azzollini et al. , 2022]), the shock breakout time was found to occur around day 80 post-outburst [Anupama , 2008].
There is no obvious difference between the UV light-curve evolution for the symbiotic-like systems and the others in this sample (Fig. <ref>) – although it is likely that one of the later breaks (possibly later than Swift has followed, and therefore not included in the plots above) in the UV decline in the symbiotic/embedded systems is related to the drop-off of the RG wind.
§.§ Comparison with previous work
Strope et al. [Strope, Schaefer & Henden , 2010] published almost 100 detailed nova optical light-curves, including some of the sources in our sample, categorising the results depending on the shapes of the light-curves. Those which follow a series of power-law declines in magnitude-log(time) space, as in the sample considered here, are classified as `S' for smooth (or stereotypical), and constitute almost 40% of their sample. As mentioned above, RNe often show a plateau in their optical/UV decay, coincident with the SSS phase seen in X-rays; these are included in the `P'-class by [Strope, Schaefer & Henden , 2010], meaning an `S'-type curve, but with a long, almost-flat interval a few magnitudes below peak. Considering the RNe in this sample, U Sco and T Pyx are indeed both marked as `P' by [Strope, Schaefer & Henden , 2010], and a flat interval is also seen in our fits (see Table <ref>, and Figs. <ref> and <ref>). RS Oph is also classified as as `P', and does indeed show a plateau in the optical light-curve [Strope, Schaefer & Henden , 2010], but no such flattening is obvious in the UV data (Fig. <ref>).
RN V3890 Sgr does not show a plateau, and is marked as `S' in [Strope, Schaefer & Henden , 2010]; see also [Page et al. , 2020]. RN V745 Sco is not classified by [Strope, Schaefer & Henden , 2010], but does not show a plateau in the UVOT data presented here.
V2491 Cyg is one of only a few novae which have been found to show a cusp-shaped secondary maximum (`C' in the Strope classification scheme: “power-law decline plus secondary maximum with steepening rise then steep decline”), which is clearly visible in the UVOT light-curve in Fig. <ref>.
We do note that the specific RN eruptions considered by [Strope, Schaefer & Henden , 2010] are not the exact same events as those presented here, with our data corresponding to more recent outbursts. Optical/UV light-curves from recurrent eruptions are frequently very similar – possibly identical – for both Galactic and extragalactic novae, however <cit.>.
Hachisu & Kato [Hachisu & Kato , 2006] proposed a `universal decline law' for (classical) novae, considering optical and IR light-curves, which shows a similar shape to the `S'-class. Modelling the eruption as a radiatively-driven wind, their template has a decay slope steepening from $\alpha$ $\sim$ 1.75 to $\sim$ 3.5, around 6 mag below peak; this break is taken to be caused by an abrupt decrease in the wind mass-loss rate. As the wind ceases, the decline flattens to $\alpha$ $\sim$ 3. However, [Strope, Schaefer & Henden , 2010] find a much larger scatter in their measurements than can be physically explained by this universal model by [Hachisu & Kato , 2006], and suggest this means there is an additional mechanism at work.
Indeed, [Shen & Quataert , 2022] show how the initial mass loss can be driven by binary interaction rather than radiation pressure.
§ SUMMARY
Nova UV light-curves show a variety of shapes, evolutions and variabilities following the initial eruption. In some cases, the UV emission varies in phase (or anti-phase) with the X-rays, in others the wavebands seem completely unrelated (at least during the super-soft X-ray phase). In this paper, we concentrated on this latter population, where the UV decline can typically be modelled by a series of (usually) declining power-law slopes, while the X-rays first brighten through the SSS phase, and then fade. We have presented the first uniform analysis of nova UV light-curves – including classical and recurrent, short-period and long – which were well monitored by the Neil Gehrels Swift Observatory. Within this sample we find that all sources show a break in the UV decay slope around the point the SSS phase ends in the X-ray band, while additional breaks also occur over a range of times, both earlier and later. While the light-curves using different filters for a given nova sometimes show very similar break times and slopes, at other times they disagree; there are no definitive trends between the different filter wavelengths in all cases.
Overall, there are no strong population trends between the evolution of the UV light-curves of the classical, symbiotic-like, short-period recurrent and long-period recurrent novae, although it is likely that the symbiotic-like, embedded systems will show a late-time break when the ejecta shock reaches the outer edge of the wind bubble formed by the giant companion. There is also no large difference in the range of unreddened, absolute magnitudes for the four populations considered, although the three symbiotic, long-period RNe are at the brighter end of the sample, while the majority of the CNe are significantly less luminous.
Conceptualization, K.L.P., N.P.M.K., J.P.O.; methodology, K.L.P., N.P.M.K., J.P.O.; validation, K.L.P., N.P.M.K., J.P.O.; formal analysis, K.L.P., J.P.O.; visualization, K.L.P.; writing — original draft preparation, K.L.P.; writing — review and editing, K.L.P., N.P.M.K., J.P.O.
All authors have read and agreed to the published version of the manuscript.
The Swift project in the UK is funded by the UK Space Agency.
The data underlying this paper are available in the Swift
archive at
<https://www.swift.ac.uk/swift_live/> and the HEASARC
Browse archive at
The authors declare no conflict of interest.
[Gehrels et al. , 2004] Gehrels, N.; Chincarini, G.; Giommi, P.; Mason, K.O.; Nousek, J.A.; Wells, A.A.; White, N.E.; Barthelmy, S.D.; Burrows, D.N.; Cominsky, L.R.; et al. The Swift Gamma-Ray Burst Mission. Astrophys. J. 2004, 611, 1005–1020.
[Roming et al. , 2005] Roming, P.W.A.; Kennedy, T.E.; Mason, K.O.; Nousek, J.A.; Ahr, L.; Bingham, R.E., Broos, P.S.; Carter, M.J.; Hancock, B.K.; Huckle, H.E.; et al. The Swift Ultra-Violet/Optical Telescope. Space Sci. Rev. 2005, 120, 95–142.
[Bode & Evans , 2008] Bode M.F.; Evans A. Classical Novae, 2nd ed.; Cambridge Astrophysics Series 43, Cambridge Univ. Press, UK, 2008
[Woudt & Ribeiro , 2014]Woudt P.A.; Ribeiro V.A.R.M. Stella Novae: Past and Future Decades; Astronomical Society of the Pacific Conference Series, Vol. 490, San Francisco: Astronomical Society of the Pacific, 2014
[Cheung et al. , 2012] Cheung, C.C.; Hays, E.; Venters, T.; Donato, D.; Corbet, R. H. D. Fermi LAT Detection of a New Gamma-ray Transient in the Galactic Plane: Fermi J0639+0548. Astron. Telegram 2012, 4224, 1–1.
[Cheung et al. , 2012] Cheung, C.C.; Shore, S.N.; De Gennaro Aquino, I.; Charbonnel, S.; Edlin, J.; Hays, E.; Corbet, R.H.D.; Wood, D.L. Possible Association of the Gamma-ray Transient Fermi J0639+0548 with Nova Mon 2012. Astron. Telegram 2012b, 4310, 1–1.
[König et al. , 2022] König, O.; Wilms, J.; Arcodia, R.; Dauser, T.; Dennerl, K.; Doroshenko, V.; Haberl, F.; Hämmerich, S.; Kirsch, C.; Kreykenbohm, I.; et al. X-ray detection of a nova in the fireball phase. Nature 2022, 605, 248–250.
[Page et al. , 2015] Page, K.L.; Osborne, J.P.; Kuin, N.P.M.; Henze, M.; Walter, F.M.; Beardmore, A.P.; Bode, M.F.; Darnley, M.J.; Delgado, L.; Drake, J.J.; Hernanz, M.; Mukai, K.; Nelson, T.; Ness, J.-U.; Schwarz, G.J.; Shore, S.N.; Starrfield, S.; Woodward, C.E. Swift detection of the super-swift switch-on of the super-soft phase in nova V745 Sco (2014). Mon. Not. R. Astron. Soc. 2015, 454, 3108–3120.
[Darnley et al. , 2012] Darnley, M.J.; Ribeiro, V.A.R.M.; Bode, M.F.; Hounsell, R.A.; Williams, R.P. On the Progenitors of Galactic Novae. em Astrophys. J. 2012, 746, 61:1–61:10.
[Truran & Livio , 1986]Truran, J.W.; Livio, M. On the Frequency of Occurrence of Oxygen-Neon-Magnesium White Dwarfs in Classical Nova Systems. Astrophys. J. 1986, 308, 721–727.
[Schaefer , 2010]Schaefer B.E. Comprehensive Photometric Histories of All Known Galactic Recurrent Novae. Astrophys. J. Supp. 2010a, 187, 275–373.
[Anupama , 2008]Anupama, G.C. One Year of Monitoring of the 2006 Outburst in Optical and Low Radio Frequencies in RS Ophiuchi (2006) and the Recurrent Nova Phenomenon, ed. A. Evans et al., ASP Conf. Ser. 401, (San Francisco, CA: ASP), 2008, 251–255.
[Starrfield , 1989]Starrfield S., Thermonuclear processes and the classical nova outburst in Classical Novae, eds. Bode M., Evans A., Wiley, NY, USA, 1989, 39–60.
[Tofflemire et al. , 2013] Tofflemire, B.M.; Orio, M.; Page, K.L.; Osborne, J.P.; Ciroi, S.; Cracco, V.; Di Mille, F.; Maxwell, M. X-Ray Grating Observations of Recurrent Nova T Pyxidis during the 2011 Outburst. Astrophys. J. 2013, 779, 22:1–22:13.
[Nelson et al. , 2014] Nelson, T.; Chomiuk, L.; Roy, N.; Sokoloski, J.L.; Mukai, K.; Krauss, M.I.; Mioduszewski, A.J.; Rupen, M.P.; Weston, J. The 2011 Outburst of Recurrent Nova T Pyx: Radio Observations Reveal the Ejecta Mass and Hint at Complex Mass Loss. Astrophys. J. 2014, 785, 78:1–78:13.
[Chomiuk et al. , 2014] Chomiuk, L.; Nelson, T.; Mukai, K.; Sokoloski, J.L.; Rupen, M.P.; Page, K.L.; Osborne, J.P.; Kuulkers, E.; Mioduszewski, A.J.; Roy, N.; Weston, J.; Krauss, M.I. The 2011 Outburst of Recurrent Nova T Pyx: X-ray Observations Expose the White Dwarf Mass and Ejection Dynamics. Astrophys. J. 2014, 788, 130:1–130:13.
[Wolf et al. , 2013]Wolf, W.M.; Bildsten, L.; Brooks, J.; Paxton, B. Hydrogen Burning on Accreting White Dwarfs: Stability, Recurrent Novae, and the Post-nova Supersoft Phase. Astrophys. J. 2013, 777, 136:1–136:15.
[Hachisu et al. , 2006] Hachisu, I.; Kato, M.; Kiyota, S.; Kubotera, K.; Maehara, H.; Nakajima, K.; Ishii, Y.; Kamada, M.; Mizoguchi, S.; Nishiyama, S.; et al. The Hydrogen-Burning Turnoff of RS Ophiuchi (2006). Astrophys. J. Lett. 2006, 651, L141–144.
[Hachisu et al. , 2008]Hachisu, I.; Kato, M.; Kiyota, S.; Kubotera, K.; Maehara, H.; Nakajima, K.; Ishii, Y.; Kamada, M.; Mizoguchi, S.; Nishiyama, S.; Sumitomo, N.; Tanaka, K.; Yamanaka, M.; Sadakane, K. Optical Light Curves of RS Oph (2006) and Hydrogen Burning Turnoff in RS Ophiuchi (2006) and the Recurrent Nova Phenomenon, ed. A. Evans et al., ASP Conf. Ser. 401, (San Francisco, CA: ASP), 2008, 206–209.
[Shafter et al. , 2015] Shafter, A.W.; Henze, M.; Rector, T.A.; Schweizer, F.; Hornoch, K.; Orio, M.; Pietsch, W.; Darnley, M.J.; Williams, S.C.; Bode, M.F.; Bryan, J. Recurrent Novae in M31. Astrophys. J. Supp. 2015, 216, 34:1–34:35.
[Darnley , 2021]Darnley, M.J. Accrete, Accrete, Accrete... Bang! (and repeat): The Remarkable Recurrent Novae. PoS 2021, 368, 44:0–44:18
[Mürset & Nussbaumer , 1994] Mürset, U.; Nussbaumer, H. Temperatures and luminosities of symbiotic novae. em Astron. Astrophs. 1994, 282, 586–604.
[Mikołajewska , 2007]Mikołajewska, J. Symbiotic Stars: Continually Embarrassing Binaries. BaltA bf 2007, 16, 1–9.
[Mikołajewska , 2008]Mikołajewska, J. The Place of Recurrent Novae Among the Symbiotic Stars, in RS Ophiuci (2006) and the Recurrent Nova Phenomenon, ed. A. Evans et al., San Francisco, CA, ASP Conf. Ser. 401, 2008, 42-51.
[Bode et al. , 2006] Bode, M.F.; O'Brien, T.J.; Osborne, J.P.; Page, K.L.; Senziani, F.; Skinner, G.K.; Starrfield, S.; Ness, J.-U.; Drake, J.J.; Schwarz, G.; et al. Swift Observations of the 2006 Outburst of the Recurrent Nova RS Ophiuchi. I. Early X-Ray Emission from the Shocked Ejecta and Red Giant Wind. Astrophys. J. 2006, 652, 629–635.
[Osborne et al. , 2011] Osborne, J.P.; Page, K.L.; Beardmore, A.P.; Bode, M.F.; Goad, M.R.; O'Brien, T.J.; Starrfield, S.; Rauch, T.; Ness, J.-U.; Krautter, J.; et al. The Supersoft X-ray Phase of Nova RS Ophiuchi 2006. Astrophys. J. 2011, 727, 124:1–124:10.
[Ness et al. , 2009]Ness, J.-U.; Drake, J.J.; Beardmore, A.P.; Boyd, D.; Bode, M.F.; Brady, S.; Evans, P.A.; Gaensicke, B.T.; Kitamoto, S.; Knigge, C.; Miller, I.; Osborne, J.P.; Page, K.L.; Rodriguez-Gil, P.; Schwarz, G.; Staels, B.; Steeghs, D.; Takei, D.; Tsujimoto, M.; Wesson, R. ; Zijlstra, A. Swift X-Ray and Ultraviolet Monitoring of the Classical Nova V458 Vul (Nova Vul 2007). AJ 2009, 137, 4160–4168.
[Schwarz et al. , 2011] Schwarz, G.J.; Ness, J.U.; Osborne, J.P.; Page, K.L.; Evans, P.A.; Beardmore, A.P.; Walter, F.M.; Helton, L.A.; Woodward, C.E.; Bode, M.; Starrfield, S.; Drake, J.J. Swift X-Ray Observations of Classical Novae. II. The Super Soft Source Sample. Astrophys. J. Supp. 2011, 197, 31:1–31:26.
[Ibarra et al. , 2009]Ibarra, A.; Kuulkers, E.; Osborne, J.P.; Page, K.; Ness, J.U.; Saxton, R.D.; Baumgartner, W.; Beckmann, V.; Bode, M.F.; Hernanz, M.; Mukai, K.; Orio, M.; Sala, G.; Starrfield, S.; Wynn, G.A. Pre-nova X-ray observations of V2491 Cygni (Nova Cyg 2008b). Astron. Astrophys. 2009, 497, L5–L8.
[Page et al. , 2010] Page, K.L.; Osborne, J.P.; Evans, P.A.; Wynn, G.A.; Beardmore, A.P.; Starling, R.L.C.; Bode, M.F.; Ibarra, A.; Kuulkers, E.; Ness, J.-U.; Schwarz, G.J. Swift observations of the X-ray and UV evolution of V2491 Cyg (Nova Cyg 2008 No. 2). Mon. Not. R. Astron. Soc. 2010, 401, 121–130.
[Beardmore et al. , 2012] Beardmore, A.P.; Osborne, J.P.; Page, K.L.; Hakala, P.J.; Schwarz, G.J.; Rauch, T.; Balman; S.; Evans, P.A.; Goad, M.R.; Ness, J.-U; Starrfield, S.; Wagner, R.M. The outburst of Nova CSS 081007:030559+054715 (HV Ceti). Astron. Astrophys. 2012, 545, A116:1–A116:10.
[Gehrz et al. , 2018] Gehrz, R.D.; Evans, A.; Woodward, C.E.; Helton, L.A.; Banerjee, D.P.K.; Srivastava, M.K.; Ashok, N.M.; Joshi, V.; Eyres, S.P.S.; Krautter, J.; Kuin, N.P.M.; Page, K.L.; Osborne, J.P.; Schwarz, G.J.; Shenoy, D.P.; Shore, S.N.; Starrfield, S.G.; Wagner, R. M. The Temporal Development of Dust Formation and Destruction in Nova Sagittarii 2015#2 (V5668 SGR): A Panchromatic Study. Astrophys. J. 2018, 858, 78:1–78:17.
[Aydi et al. , 2018]Aydi, E.; Orio, M.; Beardmore, A.P.; Ness, J.-U.; Page, K.L.; Kuin, N.P.M.; Walter, F.M.; Buckley, D.A.H.; Mohamed, S.; Whitelock, P.; Osborne, J.P.; Strader, J.; Chomiuk, L.; Darnley, M.J.; Dobrotka, A.; Kniazev, A.; Miszalski, B.; Myers, G.; Ospina, N.; Henze, M.; Starrfield, S.; Woodward, C.E. Multiwavelength observations of V407 Lupi (ASASSN-16kt) - a very fast nova erupting in an intermediate polar. Mon. Not. R. Astron. Soc. 2018, 480, 572–609.
[Page et al. , 2020] Page, K.L.; Kuin, N.P.M.; Beardmore, A.P.; Walter, F.M.; Osborne, J.P.; Markwardt, C.B.; Ness, J.-U.; Orio, M.; Sokolovsky, K.V. The 2019 eruption of recurrent nova V3890 Sgr: observations by Swift, NICER, and SMARTS. Mon. Not. R. Astron. Soc. 2020a, 499, 4814-4831.
[Page et al. , 2020] Page, K.L.; Beardmore, A.P.; Osborne, J.P. Neil Gehrels Swift Observatory studies of supersoft novae. Ad. Space Res. 2020b, 66, 1169-1192.
[Burrows et al. , 2005] Burrows, D.N.; Hill, J.E.; Nousek, J.A.; Kennea, J.A.; Wells, A.; Osborne, J.P.; Abbey, A.F.; Beardmore, A.; Mukerjee, K.; Short, A.D.T.; et al. The Swift X-Ray Telescope. Space Sci. Rev. 2005, 120, 165–195.
[Poole et al. , 2008] Poole, T.S.; Breeveld, A.A.; Page, M.J.; Landsman, W.; Holland, S.T.; Roming, P.; Kuin, N.P.M.; Brown, P.J.; Gronwall, C.; Hunsberger, S.; et al. Photometric calibration of the Swift ultraviolet/optical telescope. Mon. Not. R. Astron. Soc. 2008, 383, 627–645.
[Breeveld et al. , 2011] Breeveld, A.A.; Landsman, W.; Holland, S.T.; Roming, P.; Kuin, N.P.M.; Page, M.J. An Updated Ultraviolet Calibration for the Swift/UVOT in Gamma-Ray Bursts 2010), AIP Conf. Proc. 1358, 2011, 373–376.
[Page et al. , 2013]Page, M.J.; Kuin, N.P.M.; Breeveld, A.A.; Hancock, B.; Holland, S.T.; Marshall, F.E.; Oates, S.; Roming, P.W.A.; Siegel, M.H.; Smith, P.J.; Carter, M.; De Pasquale, M.; Symeonidis, M.; Yershov, V.; Beardmore, A.P. The use and calibration of read-out streaks to increase the dynamic range of the Swift Ultraviolet/Optical Telescope. Mon. Not. R. Astron. Soc. 2013, 436, 1684-1693.
[Schaefer , 2022]Scahefer, B.E. Comprehensive Catalogue of the Overall Best Distances and Properties of 402 Galactic Novae. Mon. Not. R. Astron. Soc. 2022a, 517, 6150–6169.
[Nakano et al. , 2008] Nakano, S.; Beize, J.; Jin, Z.-W.; Gao, X.; Yamaoka, H.; Haseda, K.; Guido, E.; Sostero, G.; Klingenberg, G.; Kadota, K. V2491 Cygni. IAU Circ. 2008, 8934, 1–1.
[Baklanov et al. , 2008]Baklanov, A.; Pavlenko, E.; Berezina, E. Discovery of the optical variability of Nova V2491 Cyg. Astron. Telegram 2008, 1514, 1–1.
[Helton et al. , 2008] Helton, L.A.; Woodward, C.E.; Vanlandingham, K.; Schwarz, G.J. V2491 Cygni. IAU Circ. 2008, 1379, 1–1.
[Rudy et al. , 2008]Rudy, R.J.; Lynch, D.K.; Russell, R.W.; Woodward, C.E.; Covey, K. V2491 Cygni. IAU Circ. 2008, 8938, 2–2.
[Schaefer et al. , 2010]Schaefer, B.E.; Harris, B.G.; Dvorak, S.; Templeton, M.; Linnolt, M. U Scorpii. IAU Circ. 2010, 9111, 1–1.
[Pagnotta et al. , 2015] Pagnotta, A.; Schaefer, B.E.; Clem, J.L.; Landolt, A.U.; Handler, G.; Page, K.L.; Osborne, J.P.; Schlegel, E.M.; Hoffman, D.I.; Kiyota, S.; Maehara, H. The 2010 Eruption of the Recurrent Nova U Scorpii: The Multi-wavelength Light Curve Astrophys. J. 2015, 811, 32:1–32:42.
[Nishiyama et al. , 2010] Nishiyama, K.; Kabashima, F.; Kojima, T.; Sakaniwa, K.; Tago, A.; Schmeer, P.; Munari, U. V407 Cygni. IAU Circ. 2010, 9130, 1–1.
[Munari et al. , 1990] Munari, U.; Margoni, R.; Stagni, R. The extreme, possible symbiotic Mira V407 CYG and its relevance to the OH/IR sources. Mon. Not. R. Astron. Soc. 1990, 242, 563–569.
[Hachisu & Kato , 2019]Hachisu, I., Kato, M. A Light-curve Analysis of 32 Recent Galactic Novae: Distances and White Dwarf Masses. Astrophys. J. Supp. 2019, 242, 1–18.
[Shore et al. , 2011] Shore, S.N.; Wahlgren, G.M.; Augusteijn, T.; Liimets, T.; Page, K.L.; Osborne, J.P.; Beardmore, A.P.; Koubsky, P.; Šlechta, M.; Votruba, V. The spectroscopic evolution of the symbiotic-like recurrent nova V407 Cygni during its 2010 outburst. I. The shock and its evolution. Astron. Astrophys. 2011a, 527, A98:1–A98:15.
[Waagan et al. , 2011]Waagan, E.; Linnolt, M.; Bolzoni, S.; Amorim, A.; Plummer, A.; Williams, P. F.; Matsuyama, H.; Kerr, S.; Pearce, A. T Pyxidis. CBET 2011, 2700, 1–1.
[Shore et al. , 2011] Shore, S.N.; Augusteijn, T.; Ederoclite, A.; Uthas, H. The spectroscopic evolution of the recurrent nova T Pyxidis during its 2011 outburst. I. The optically thick phase and the origin of moving lines in novae . Astron. Astrophys. 2011b, 533, L8:1–L8:5
[Fujikawa et al. , 2012] Fujikawa, S.; Yamaoka, H.; Nakano, S. Nova Monocerotis 2012 = PNV J06393874+0553520. CBET 2012, 3202, 1–1.
[Page et al. , 2013] Page, K.L.; Osborne, J.P.; Wagner, R.M.; Beardmore, A.P.; Shore, S.N.; Starrfield, S.; Woodward, C.E. The 7.1 Hr X-Ray-Ultraviolet-Near-infrared Period of the $\gamma$-Ray Classical Nova Monocerotis 2012. Astrophys. J. 2013, 768, L26:1–L26:6.
[Shore et al. , 2013] Shore, S.N.; De Gennaro Aquino, I.; Schwarz, G.J.; Augusteijn, T.; Cheung, C.C.; Walter, F.M.; Starrfield, S. The spectroscopic evolution of the $\gamma$-ray emitting classical nova Nova Mon 2012. I. Implications for the ONe subclass of classical novae. Astron. Astrophys. 2013, 553, A123:1–A123:17.
[Nakano , 2013]Nakano, S. V339 Delphini = Nova Delphini 2013 = PNV J20233073+2046041. IAUC 9258, 1–1.
[Shore et al. , 2016] Shore, S.N.; Mason, E.; Schwarz, G.J.; Teyssier, F.M.; Buil, C.; De Gennaro Aquino, I.; Page, K.L.; Osborne, J.P.; Scaringi, S.; Starrfield, S.; van Winckel, H.; Williams, R.E.; Woodward, C.E. The panchromatic spectroscopic evolution of the classical CO nova V339 Delphini (Nova Del 2013) until X-ray turnoff. Astron. Astrophys. 2016, 590, A123:1–A123:16.
[Waagen & Pearce , 2014] Waagen, E.O.; Pearce, A. V745 Scorpii. CBET 2014, 3803, 1–1.
[Banerjee et al. , 2014] Banerjee, D.P.K.; Joshi, V.; Venkataraman, V.; Ashok, N.M.; Marion, G.H.; Hsiao, E.Y.; Raj, A. Near-IR Studies of Recurrent Nova V745 Scorpii during its 2014 Outburst. Astrophys. J. 2014, 785 L11:1–L11:5.
[Nishiyama & Kabashima , 2014] Nishiyama, K.; V1534 Scorpii = Nova Scorpii 2014
= TCP J17154683-3128303. IAU Circ. 2014, 9271, 3–3.
[Joshi et al. , 2015] Joshi, Vishal; Banerjee, D.P.K.; Ashok, N.M.; Venkataraman, V.; Walter, F.M. Infrared studies of Nova Scorpii 2014: an outburst in a symbiotic system sans an accompanying blast wave. Mon. Not. R. Astron. Soc. 2015, 452, 3696-3703.
[Munari et al. , 2017] Munari, U.; Hambsch, F.-J.; Frigo, A. Photometric evolution of seven recent novae and the double-component characterizing the light curve of those emitting in gamma rays. Mon. Not. R. Astron. Soc. 2017, 469, 4341-4358.
[Schaefer , 2022] Schaefer, B.E. Comprehensive listing of 156 reliable orbital periods for novae, including 49 new periods. Mon. Not. R. Astron. Soc. 2022b, 517, 3640-3659.
[Nakano , 2015]Nakano, S. Nova Scorpii 2015 = PNV J17032620-3504140. IAU Circ. 2015, 9274, 1–1.
[Walter , 2015]Walter, F. Nova Sco 2015 = PNV J17032620-3504140. Astron. Telegram 2015, 7060, 1–1.
[Nelson et al. , 2015] Nelson, T.; Linford, J.; Chomiuk, L.; Sokoloski, J.; Mukai, K.; Finzell, T.; Weston, J.; Rupen, M.; Mioduszewski, A. Early X-ray and radio observations of Nova Sco 2015 implicate strong shocks against a red giant wind. Astron. Telegram 2015, 7085, 1–1.
[Linford et al. , 2017] Linford, J.D.; Chomiuk, L.; Nelson, T.; Finzell, T.; Walter, F.M.; Sokoloski, J.L.; Mukai, K.; Mioduszewski, A.J.; van der Horst, A.J.; Weston, J.H.S.; Rupen, M.P. The Peculiar Multiwavelength Evolution Of V1535 Sco. Astrophys. J. 2017, 842, 1–16.
[Pereira , 2019]Pereira, A. V3890 Sgr outburst. vsnet-alert 2019 23505.
[Munari & Walter , 2019]Munari, U., Walter, F.M. The reddening and distance to recurrent nova V3890 Sgr from interstellar features. Astron. Telegram 2019, 13069, 1–1.
[Evans et al. , 2022] Evans, A.; Geballe, T.R.; Woodward, C.E.; Banerjee, D.P.K.; Gehrz, R.D.; Starrfield, S.; Shahbandeh, M. Infrared spectroscopy of the 2019 eruption of the recurrent nova V3890 Sgr: separation into equatorial and polar winds revealed. Mon. Not. R. Astron. Soc. 2022, 517, 6077–6090.
[Mikołajewska et al. , 2021] Mikołajewska, J.; Iłkiewicz, K.; Gałan, C.; Monard, B.; Otulakowska-Hypka, M.; Shara, M.M.; Udalski, A. The symbiotic recurrent nova V3890 Sgr: binary parameters and pre-outburst activity. Mon. Not. R. Astron. Soc. 2021, 504, 2122-2132.
[Patterson et al. , 2021]Patterson, J.; Epstein-Martin, M.; Vanmunster, T.; Kemp, J. The Orbital and Pulse Periods of V1674 Herculis. Astron. Telegram 2021, 14856, 1–1.
[Drake et al. , 2021] Drake, J.J.; Ness, J.-U.; Page, K.L.; Luna, G.J.M.; Beardmore, A.P.; Orio, M.; Osborne, J.P.; Mròz, P.; Starrfield, S.; Banerjee, D.P.K.; et al. The Remarkable Spin-down and Ultrafast Outflows of the Highly Pulsed Supersoft Source of Nova Herculis 2021. Astrophys. J. 2021, 922, L42:1–L42:9.
[Woodward et al. , 2021] Woodward, C.E.; Banerjee, D.P.K.; Geballe, T.R.; Page, K.L.; Starrfield, S.; Wagner, R. M. Near-infrared Studies of Nova V1674 Herculis: A Shocking Record Breaker. Astrophys. J. 2021, 922, L10:1–L10:7.
[Brandi et al. , 2009] Brandi, E.; Quiroga, C.; Mikołajewska, J.; Ferrer, O.E.; García, L.G. Spectroscopic orbits and variations of RS Ophiuchi. Astron. Astrophys. 2009, 497, 815–825.
[Page et al. , 2022] Page, K.L.; Beardmore, A.P.; Osborne, J.P.; Munari, U.; Ness, J.-U.; Evans, P.A.; Bode, M.F.; Darnley, M.J.; Drake, J.J.; Kuin, N.P.M.; O'Brien, T.J.; Orio, M.; Shore, S.N.; Starrfield, S.; Woodward, C.E. The 2021 outburst of the recurrent nova RS Ophiuchi observed in X-rays by the Neil Gehrels Swift Observatory: a comparative study. Mon. Not. R. Astron. Soc. 2022, 514, 1557–1574.
[Snijders , 1987]Snijders, M.A.J. Multi-frequency observations of the 1985 outburst of RS Ophiuchi. Astrophys. Space Science 1987, 130, 243–254.
[Orio et al. , 2022] Orio, M.; Behar, E.; Luna, G.J.M.; Drake, J.J.; Gallagher, J.; Nichols, J.S.; Ness, J.U.; Dobrotka, A.; Mikolajewska, J.; Della Valle, M.; Ignace, R.; Rahin, R. Shocks in the Outflow of the RS Oph 2021 Eruption Observed with X-Ray Gratings. Astrophys. J 2022, 938, 34:1–34:14.
[Strope, Schaefer & Henden , 2010]Strope, R.J.; Schaefer, B.E.; Henden, A.A. Catalog of 93 Nova Light Curves: Classification and Properties. Astron. J. 2010, 140, 34–62.
[Schaefer et al. , 2010]Schaefer, B.E.; Pagnotta, A.; Xiao, L.; Darnley, M.J.; Bode, M.F.; Harris, B.G.; Dvorak, S.; Menke, J.; Linnolt, M.; Templeton, M.; et al. Discovery of the Predicted 2010 Eruption and the Pre-eruption Light Curve for Recurrent Nova U Scorpii. Astron. J 2010c, 140, 925-932.
[Ness et al. , 2012] Ness, J.-U.; Schaefer, B.E.; Dobrotka, A.; Sadowski, A.; Drake, J.J.; Barnard, R.; Talavera, A.; Gonzalez-Riestra, R.; Page, K.L.; Hernanz, M.; Sala, G.; Starrfield, S. From X-Ray Dips to Eclipse: Witnessing Disk Reformation in the Recurrent Nova U Sco. Astrophys. J. 2012, 745, 43:1–43:16
[Munari et al. , 2015] Munari, U.; Maitan, A.; Moretti, S.; Tomaselli, S. 500 days of Stromgren b, y and narrow-band [OIII], H$\alpha$ photometric evolution of gamma-ray Nova Del 2013 (=V339 Del) . New Astronomy 2015, 40, 28–40.
[Mason et al. , 2021] Mason, E.; Shore, S.N.; Drake, J.; Howell, S.B.; Kuin, P.; Magaudda, E. The aftermath of nova Centauri 2013 (V1369 Centauri). Astron. Astrophys. 2021, 649, A28:1–A28:15.
[Draine , 2003]Draine, B.T. Interstellar Dust Grains. Ann. Rev. Astron. Astrophys. 2003, 41, 241-289.
[Azzollini , 2021]Azzollini, A. Multi-wavelength Spectroscopic Study of Shock Phenomena Driven by Explosive Outbursts in Symbiotic-like Recurrent Novae. Master thesis, Univ. Pisa, Pisa, Italy, 2021.
[Azzollini et al. , 2022]Azzollini, A.; Shore, S.N.; Kuin, N.P.M.; Page, K.L. Multiwavelength spectroscopic study of shock driven phenomena in explosive outbursts in symbiotic-like recurrent novae with emphasis on RS Ophiuchi. Astron. Astrophys. 2022, submitted.
[Kato , 1997]Kato, M. Optically Thick Winds from Degenerate Dwarfs. I. Classical Novae of Populations I and II. Astrophys. J. Supp. 1997, 113, 121–129.
[Yaron et al. , 2005] Yaron, O.; Prialnik, D.; Shara, M.M.; Kovetz, A. An Extended Grid of Nova Models. II. The Parameter Space of Nova Outbursts. Astrophys. J. 2005, 623, 398–410.
[Henze et al. , 2011] Henze, M.; Pietsch, W.; Haberl, F.; Hernanz, M.; Sala, G.; Hatzidimitriou, D.; Della Valle, M.; Rau, A.; Hartmann, D.H.; Burwitz, V. X-ray monitoring of classical novae in the central region of M 31. II. Autumn and winter 2007/2008 and 2008/2009. Astron. Astrophys. 2011, 533, A52:1–A52:30.
[Henze et al. , 2018] Henze, M.; Darnley, M.J.; Williams, S.C.; Kato, M.; Hachisu, I.; Anupama, G.C.; Arai, A.; Boyd, D.; Burke, D.; Ciardullo, R.; et al. Breaking the Habit: The Peculiar 2016 Eruption of the Unique Recurrent Nova M31N 2008-12a. Astrophys. J. 2018, 857, 68:1–68:29.
[Hachisu & Kato , 2006] Hachisu, I.; Kato, M. A Universal Decline Law of Classical Novae. Astrophys. J. Supp. 2006, 167, 59–80.
[Shen & Quataert , 2022]Shen, K.J.; Quataert, E. Binary Interaction Dominates Mass Ejection in Classical Novae. Astrophys. J. 2022, 938, 31:1–31:13.
|
email<EMAIL_ADDRESS>email<EMAIL_ADDRESS>
# Assessing quantum thermalization in physical and configuration spaces
via many-body weak values
Carlos F. Destefani1 [ Xavier Oriols1 [ 1 Department of Electronic
Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona,
Spain
###### Abstract
We explore the origin of the arrow of time in an isolated quantum system
described by the Schrödinger equation. We provide an explanation from weak
values in the configuration space, which are understood as operational
properties obtained in the laboratory following a well-defined protocol. We
show that quantum systems satisfying the eigenstate thermalization hypothesis
can simultaneously provide thermalized ensemble expectation values and
nonthermalized weak values of the momentum, both from the same operational
probability distribution. The reason why weak values of the momentum may
escape from the eigenstate thermalization hypothesis is because they are
linked only to off-diagonal elements of the density matrix in the energy
representation. For indistinguishable particles, however, operational
properties can not be defined in the configuration space. Therefore, we state
that the origin of the arrow of time in isolated quantum systems described by
the Schrödinger equation comes from dealing with properties obtained by
averaging (tracing out) some degrees of freedom of the configuration space. We
then argue that thermalization does not occur in the properties defined in the
configuration space, and our argument is compatible with defending that
thermalization is a real phenomenon in the properties defined in the physical
space. All of these conclusions are testable in the laboratory through many-
body weak values.
## I Introduction
The arrow of time has always been a topic of lively debate [1, 2, 3, 4, 5, 6,
7, 8]. It appears in many disciplines as, for example, a cosmological arrow of
time pointing in the direction of the Universe expansion [2]. A related arrow
of time appears in the time evolution of time-irreversible macroscopic systems
governed by the second law of thermodynamics, where entropy always increases
with time [1, 7]. The puzzle implicit in such irreversibility is that most
fundamental microscopic laws have no arrow of time. They are time-reversible
laws, in the sense that time appears as a variable, just like position,
without any privileged direction. But, if macroscopic laws emerge from
microscopic laws, what can make them so different? A possible explanation is
that such fundamental laws are in fact not time-reversal. For example, it has
been argued that the time-reversible Schrödinger equation is not the true law
at a fundamental level, and that it should be substituted by laws from
spontaneous collapse theories which, by construction, are time-irreversible
[3]. Another explanation argues that real systems are never perfectly
isolated, so that time-reversible fundamental laws, despite being the true
laws, are not directly applicable [4]. And yet another argumentation claims
that the evolution of a real system depends, apart from the true time-
reversible microscopic laws, on the initial conditions which provide an
irreversible time evolution [6].
In this paper we explore a different path, via weak values in the
configuration space, to understand the physical origins of the arrow of time
in perfectly isolated nonrelativistic systems described by the Schrödinger
equation, assumed as a true reversible law where initial conditions are not
relevant to explain the observed time-irreversibility in our results. Our goal
is to link the evolution of microscopic (or macroscopic) properties of a model
system with the configuration (or physical) space where such properties are
defined. The wave function solution of the Schrödinger equation is defined in
a $3N$-dimensional configuration space, while typical properties where time-
irreversibility is observed, are defined in smaller spaces where some degrees
of freedom of the configuration space are integrated. Thus, the question that
motivates us is whether the presence or absence of an arrow of time in the
time evolution of the properties of a quantum system is a consequence of
defining them in the full configuration space or in a smaller space.
For our goal, the renewed interest in statistical mechanics of closed quantum
systems [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
provides the perfect scenario. Such interest has been generated by the
successful experimental ability to isolate and manipulate bosonic [25, 26, 27,
28, 29] and fermionic [30, 31, 32, 33, 34] many-body systems built on ultra-
cold atomic gases subjected to optical lattices. Such quantum systems can be
described by the many-particle Schrödinger equation; an arrow of time appears
because, despite these systems being expected to present unitary evolution,
some of their initial nonequilibrium nonthermalized properties may later
thermalize.
A preliminary consideration is that it is not at all obvious whether the
configuration space is more or less fundamental than the ordinary physical
space. The nonrelativistic Schrödinger equation can be seen as a sort of
approximation to the relativistic quantum field theory [35]. For
indistinguishable particles, quantum field theory does not require knowledge
of the exact position of each particle in the configuration space, but only of
how many particles are present in a position of the physical space and, as
such, configuration space seems less fundamental than physical space.
Another consideration is that thermalization is typically reported in
properties mensurable in a laboratory, so that one needs to discuss, within
similar empirical protocols, nonthermalized properties also mensurable in a
laboratory. But making conclusions testable in a laboratory opens new
difficulties since, strictly speaking, a measured closed system is no longer a
closed system because of its interaction with the measuring apparatus [36].
Such measurement produces a collapse of the quantum state of the isolated
system, which is at the origin of quantum randomness and backaction. Within
the orthodox theory, such a collapse requires a time-asymmetric law, different
from the time-symmetric Schrödinger equation, so that an “orthodox quantum-
mechanical arrow of time” seems to enter into play [37]. We will see how the
weak values protocol provides operational properties of the quantum system
without backaction or quantum randomness (avoiding the role of the collapse in
our discussion).
It is important to differentiate between operational (empirical) and
ontological (real) properties of a system. Operational properties are those
whose definition comes exclusively from operations done on the laboratory over
the system (with or without ontological meaning for that property), which are
defined independently on any quantum theory. Ontological properties, on the
other hand, are those that a specific quantum theory postulates to exist.
Therefore, it is possible that a given operational property coincides with an
ontological property in one theory, but not in another. A typical example is
the velocity of a particle, known to be an operational property computed from
weak values, independently on any quantum theory; such operational property
coincides with an ontological property in the Bohmian theory (the velocity
itself), but it is not an ontological property in the Orthodox theory. It is
far from our scope to discuss whether or not thermalization is an ontological
phenomenon or not occurring in the configuration space or in the physical
space, since this depends on which quantum theory is invoked. Our less
controversial focus here is to approach thermalization in closed systems from
operational properties testable in laboratory.
The structure of the paper is the following. Sect. II presents the many-body
generalization of the single-particle weak values, stating them as operational
properties in configuration space without explicit dependence on quantum
randomness and backaction, for both distinguishable and indistinguishable
particles. Sect. III discusses thermalization and equilibration concepts as
found in the literature for closed quantum systems, and address the eigenstate
thermalization hypothesis [38, 39]. Sect. IV defines our model system and its
nonequilibrium dynamics, and summarizes our results for both expectation
values and weak values from the Schrödinger equation dynamics. In Sect. V we
conclude.
## II Operational properties in the configuration space
To simplify notation, along the paper we use natural units and consider a
$1$-dimensional physical space with degree of freedom $x$, so that
$\mathbf{x}=\\{x_{1},...,x_{N}\\}$ is the position in the $N$-dimensional
configuration space; the extension to a $3N$-dimensional space should be
straightforward. As already mentioned, a measured closed system is no longer a
closed system, and a strong measurement, for example, of the momentum operator
$\hat{p}$ yields the eigenvalue $p$, and produces the initial state to be
converted into the momentum eigenstate. On the other hand, expectation values
and weak values yield operational information of the system without backaction
and quantum randomness.
### II.1 Expectation values
We define the expectation value of the momentum $\langle p(t)\rangle$ as an
operational property of the system in the sense that it is linked to a well
defined protocol in the laboratory,
$\langle p(t)\rangle=\int dp\;p\;\mathbb{P}(p,t),$ (1)
where the probability distribution $\mathbb{P}(p,t)$ can be obtained as
follows: i) many identical initial states $|\Psi(t)\rangle$ are prepared at
time $t$; ii) for each initial state, a (weak or strong) measurement of
momentum is done, yielding the value $p$ at time $t$; iii) $\mathbb{P}(p,t)$
is constructed by counting how many $p$ occurs when repeating ii) on ensemble
i).
When then applying Born law to predict the value of $\mathbb{P}(p,t)$ one can
easily identifies
$\langle p(t)\rangle=\int
dp\;p\;\mathbb{P}(p,t)=\langle\Psi(t)|\hat{p}|\Psi(t)\rangle.$ (2)
Notice that the right hand side of (2) depends on the state of the system
$|\Psi(t)\rangle$ before a measurement is done, without neither randomness nor
backaction. In fact, $\langle p(t)\rangle$ is a typical property used to
analyze when an isolated quantum system thermalizes. We are here interested in
discussing thermalization from operational properties of the isolated quantum
system requiring the measurement of both momentum and position simultaneously.
Let us then start by discussing weak values in physical space.
### II.2 Weak values in the physical space
It has recently been shown that weak values [40] are able to yield dynamic
information on two noncommuting operators at a single time avoiding backaction
[41, 42] and quantum randomness. Weak values have attracted a lot of
theoretical [41, 42, 43, 44, 45, 46] and experimental [47, 48, 49] interests
in many research fields.
At the laboratory the single-particle weak value of momentum $p_{W}(x,t)$ is
given by
$p_{W}(x,t)=\frac{\int dp\;p\;\mathbb{P}(p,x,t)}{\int dp\;\mathbb{P}(p,x,t)},$
(3)
computed from the probability distribution $\mathbb{P}(p,x,t)$ via the
following procedure: i) many identical initial states $|\Psi(t)\rangle$ are
prepared at time $t$; ii) for each initial state, a weak measurement of the
momentum is done, yielding the value $p$ at time $t$; iii) subsequently, a
strong measurement of the position is done, yielding the value $x$ at time
$t$; iv) $\mathbb{P}(p,x,t)$ is constructed by counting how many $p$ and $x$
occurs when repeating ii) and iii) on ensemble i).
Since $x$ is post-selected in (3), but not integrated out, one gets
information on how the expectation value of the momentum is distributed in the
physical space. Again, when applying Born law to predict the value of
$\mathbb{P}(p,x,t)$ one can easily identifies
$p_{W}(x,t)=\frac{\int dp\;p\;\mathbb{P}(p,x,t)}{\int
dp\;\mathbb{P}(p,x,t)}=\text{Real}\left(\frac{\langle
x|\hat{p}|\Psi(t)\rangle}{\langle x|\Psi(t)\rangle}\right).$ (4)
Similarly to (2), the ensemble-over-identical-experiments in the right hand
side in (4) eliminates the undesired backaction and quantum randomness induced
by the first measuring apparatus [40, 42, 46, 50, 51], so that $p_{W}(x,t)$ in
(4) depends only on the initial state before the measurements took place.
Expression (4) allows us to give the weak value a very simple interpretation.
In a single-particle system, from $\mathbb{I}=\int dx|x\rangle\langle x|$, one
can rewrite the expectation value of the momentum in (2) (see Appendix A) as
$\langle p\rangle(t)=\int dx\langle\Psi(t)|x\rangle\langle
x|\hat{p}|\Psi(t)\rangle=\int dx|\Psi(x,t)|^{2}p_{W}(x,t),$ (5)
so that the same probability distribution $\mathbb{P}(p,x,t)$ used to compute
$p_{W}(x,t)$ in (4) can be employed to obtain $\mathbb{P}(p,t)$ used to
compute $\langle p(t)\rangle$ in (2), since
$\mathbb{P}(p,t)=\int dx\;\mathbb{P}(p,x,t).$ (6)
Using the mathematics (and not necessarily the ontology) of Bohmian mechanics,
one can also re-interpret $p_{W}(x,t)$ as the (operational) velocity of the
particle at position $x$ and time $t$ [43, 44, 45, 46, 52, 53, 54, 55, 56,
57],
$p_{W}(x,t)=\text{Imag}\left(\frac{1}{\Psi(x,t)}\frac{\partial\Psi(x,t)}{\partial
x}\right)=\frac{J(x,t)}{|\Psi(x,t)|^{2}},$ (7)
with $J(x,t)=\text{Imag}(\Psi^{*}(x,t)\partial\Psi(x,t)/\partial x)$ the
current density (see Appendix B).
### II.3 Weak values in the configuration space for distinguishable particles
Notice that $p_{W}(x,t)$ is an operational property in the ordinary physical
space, but we now need to define an operational property in the configuration
space for dealing with an isolated quantum system with $N$ particles.
Therefore, in this paper, we extend the original single-particle weak values
in (4) to $N$-particle scenarios for both distinguishable and
indistinguishable cases. For the former case, the weak values for the
$j$-particle is
$p_{W}^{j}(\mathbf{x},t)=\frac{\int
dp_{j}\;p_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)}{\int
dp_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)},$ (8)
where now the probability distribution $\mathbb{P}(p_{j},\mathbf{x},t)$ is
obtained as follows: i) many identical initial states $|\Psi(t)\rangle$ are
prepared at time $t$; ii) for each initial state, a weak measurement of the
momentum of the $j$-particle is done, yielding the value $p_{j}$ at time $t$;
iii) subsequently, a strong measurement of the positions of particles
$1$,$2$,…,$N$ yielding respectively the values $x_{1}$,$x_{2}$,…,$x_{N}$ is
done; iv) $\mathbb{P}(p_{j},\mathbf{x},t)$ is constructed by counting how many
$p_{j}$,$x_{1}$,$x_{2}$,…,$x_{N}$ occurs when repeating ii) and iii) on
ensemble i).
Our definition of distinguishable particles above is operational in the sense
that the measuring apparatus is somehow able to distinguish particles, for
example, by measuring their masses but, to avoid unnecessary notation, we have
not indicated this extra measurement in the above protocol. Again Born law
allows us to rewrite (8) as
$p_{W}^{j}(\mathbf{x},t)=\frac{\int
dp_{j}\;p_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)}{\int
dp_{j}\;\mathbb{P}(p_{j},\mathbf{x},t)}=\text{Real}\left(\frac{\langle\mathbf{x}|\hat{p}_{j}|\Psi(t)\rangle}{\langle\mathbf{x}|\Psi(t)\rangle}\right),$
(9)
where once more from the mathematics (and not necessarily from the ontology)
of Bohmian mechanics, one can also re-interpret $p_{W}^{j}(\mathbf{x},t)$ as
the (operational) velocity of the $j$-particle at the position $\mathbf{x}$ in
the configuration space and time $t$ [43, 44, 45, 46, 52, 53, 54, 55, 56, 57],
$p_{W}^{j}(\mathbf{x},t)=\frac{J^{j}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}},$
(10)
with
$J^{j}(\mathbf{x},t)=\text{Imag}(\Psi^{*}(\mathbf{x},t)\partial\Psi(\mathbf{x},t)/\partial
x_{j})$ the current density in the $x_{j}$ direction.
### II.4 Weak values for indistinguishable particles
The most common situation in the laboratory however relates to identical
particles, for which a proper many-body wave function should implicitly
include the exchange symmetry among the particles. Equation (9) then becomes
inaccessible in a laboratory because it is no longer possible to know,
operationally, which position belongs to each particle. To deal with
indistinguishable particles, one needs to construct a many-body weak value
defined in physical space coordinate $x$ by averaging (integrating) all
degrees of freedom (see Appendix A). By doing so one obtains
$\tilde{p}_{W}(x,t)=\frac{\int dp\;p\;\mathbb{\tilde{P}}(p,x,t)}{\int
dp\;\mathbb{\tilde{P}}(p,x,t)},$ (11)
where the probability distribution $\mathbb{\tilde{P}}(p,x,t)$ is now obtained
as follows: i) many identical initial states $|\Psi(t)\rangle$ are prepared at
time $t$; ii) for each initial state, a weak measurement of the momentum of
one nonidentified particle is done, yielding the value $p$ at time $t$; iii)
subsequently, a strong measurement of the position of the same or another
nonidentified particle is done, yielding the value $x$ at time $t$; iv)
$\mathbb{\tilde{P}}(p,x,t)$ is constructed by counting how many $p$ and $x$
occurs when repeating ii) and iii) on ensemble i).
Born law again allows us to rewrite (11) as (see Appendix A)
$\tilde{p}_{W}(x,t)=\frac{\int dp\;p\;\mathbb{\tilde{P}}(p,x,t)}{\int
dp\;\mathbb{\tilde{P}}(p,x,t)}=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t),$
(12)
with
$p_{W}^{j,k}(x,t)=\frac{\int dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;p_{W}^{j}(..,x_{k-1},x,x_{k+1},..,t)|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}{\int
dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}.$ (13)
Notice that $\tilde{p}_{W}(x,t)$ is, in fact, the local velocity as used in
quantum hydrodynamic models [52, 53, 54, 55], being empirically accessible in
both distinguishable and indistinguishable scenarios. The operational protocol
for computing $\mathbb{\tilde{P}}(p,x,t)$ for indistinguishable particles is
related to $\mathbb{P}(p_{k},\mathbf{x},t)$ for distinguishable particles as
$\mathbb{\tilde{P}}(p,x,t)=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}\int
dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;\mathbb{P}(p_{j},..,x_{k-1},x,x_{k+1},..,t).$ (14)
## III Thermalization in isolated systems from expectation values
Our main contribution in the study of quantum thermalization, as detailed in
next section, is the inclusion of many-body weak values in the configuration
space as operational properties. However, such a study has usually been done
in the literature in terms of expectation values as in (2), and as such we
summarize in this section the role of expectation values to characterize
thermalization.
For an initial nonequilibrium pure state $|\Psi(0)\rangle$, the Schrödinger
equation provides its unitary evolution as
$|\Psi(t)\rangle=\sum_{n}c_{n}e^{-iE_{n}t}|n\rangle$, where $|n\rangle$ is an
energy eigenstate with eigenvalue $E_{n}$, and $c_{n}=\langle
n|\Psi(0)\rangle$ is defined by the initial conditions. The expectation value
of some observable $\hat{A}$ is given by
$\langle A\rangle(t)=\sum_{n}\rho_{n,n}A_{n,n}+\sum_{n,m\neq
n}\rho_{n,m}(t)A_{m,n},$ (15)
with the time-dependent off-diagonal elements of the density matrix in the
energy representation defined as
$\rho_{n,m}(t)=c_{m}^{*}c_{n}e^{i(E_{m}-E_{n})t},$ (16)
and the time-independent diagonal elements as
$\rho_{n,n}(t)=c_{n}^{*}c_{n}=\rho_{n,n}(0),$ (17)
with the operator $\hat{A}$ in the energy representation being
$A_{m,n}=\langle m|\hat{A}|n\rangle.$ (18)
A system is said to equilibrate if, after some time $t_{eq}$ enough for full
dephasing between different energy eigenstates, the off-diagonal terms
(coherences) cancel out so that (15) can be computed solely from the time-
independent diagonal terms (populations), that is, $\langle
A\rangle(t)\approx\sum_{n}\rho_{n,n}A_{n,n}$ for most times $t>t_{eq}$ (except
for some recurrence times). The properties of the system after equilibration
are fully determined by the initial conditions $\rho_{n,n}(0)=|c_{n}|^{2}$,
since the density matrix populations are time-independent. The closed system
is then said to thermalize when $\langle A\rangle(t)$ becomes roughly equal to
the expectation value as computed from the classical density matrix in the
microcanonical ensemble, $\rho_{cl}$, in which equal probabilities are
attached to each microstate within an energy window defined by the initial
conditions; that is, a subset $N_{act}$ of the energy eigenstates are
initially activated at $t=0$, and they will remain as the only states
dictating the system dynamics at any $t$. It is important to notice that
$N_{act}$ refers to a given number of relevant elements in a basis set, but it
has no relation with the number of particles $N$ of the system. In other
words, one can also expect thermalization even in few-particle systems, as
detailed in next section; in fact, thermalization has also been studied in
laboratories in small systems with as little as 6 [27], 5 [14], or 2-4 bosons
[58, 59], 3 qubits [60], and even single-particle systems [61, 62, 63].
The eigenstate thermalization hypothesis (ETH) [38, 39] has become the
standard theory dealing with quantum thermalization in closed systems. It
states that the dephasing above mentioned is typical to nondegenerate and
chaotic many-body nonintegrable systems, where the off-diagonals $A_{m,n}$ in
(18) become exponentially smaller than $A_{n,n}$. In recent years a large
amount of numerical experiments has successfully tested such a hypothesis by
directly diagonalizing some sort of short range many-body lattice Hamiltonian,
like Fermi- or Bose-Hubbard [26, 31, 32, 64, 65, 66], and XXZ- or XYZ-
Heisenberg [14, 23, 27, 67, 68, 69, 70], in the search of chaotic signatures
in the statistics of their spectra, as in general induced by local impurities,
without the need to explicitly evolve $|\Psi(0)\rangle$. The ETH states that
nonintegrable systems (where total energy may be the only conserved quantity),
after a quench (which may create a nonequilibrium initial state by activating
a subset $N_{act}$ of excited eigenstates), can present a ‘chaotic’ spectrum
ruled by the Wigner-Dyson statistics (which contains level repulsion), and
that the long time average of the expectation value of some observable roughly
equals its thermal equilibrium value in an (microcanonical) ensemble. Such a
hypothesis claims that thermalization is indeed hidden in the chaotic initial
nature of the Hamiltonian eigenstates themselves. Lattice models typically
handle $\approx 24$ sites with $\approx 1/3$ filling, and the above local
impurities are added to break their otherwise integrable character, as the ETH
overall claims that integrable systems are not expected to thermalize.
On the other hand, our time evolution dwells in true configuration space with
an antisymmetrized wave function and full long range electron-electron
interaction; due to the tensorial nature of our problem, and since we need to
employ a grid with $M\approx 10^{3}$ points per degree of freedom for decent
position and momentum resolutions, we can realistically only deal with
$N\lesssim 4$ particles ($N=3$ already implies $M\approx 10^{9}$ grid points
at each time step).
## IV Numerical results for expectation values and weak values
We now apply the many-body weak values machinery for the analysis of quantum
thermalization in a model with $N$ spinless electrons [20, 71, 72, 73, 74, 75,
76, 77], typical of condensates in harmonic oscillator traps under a speckle
field. Such a field translates to a ‘chaotic’ random disorder potential, which
can yield a chaotic energy spectrum as requested by the ETH and so induce
thermalization even in systems with small $N$. Our model can attach a disorder
to each grid point, and an initial velocity at $t=0$ is given to the electrons
as to simulate the initial quench of the confining potential, for each of the
considered $N=1,2,3$ systems. In fact, most of the thermalization literature
dealing with identical particles employs some lattice-based model, since such
models avoid the need of explicit knowledge of the exact position of each
particle in a point of the configuration space; instead, they only need to
know how many particles are present in each site of the physical space. From a
computational point of view lattice models have unquestionable advantages but
are inappropriate for our goal of discussing whether or not thermalization
occurs in configuration space, a goal that forces us to directly solve the
time-evolution of the few-body Schrödinger equation in configuration space,
and to analyze thermalization by monitoring the time-evolution of both
expectation values and weak values.
### IV.1 Initial state
The pure initial $N$-electron nonequilibrium antisymmetric state is
$\langle\mathbf{x}|\Psi(0)\rangle=\frac{1}{\mathcal{C}}\sum_{n=1}^{N!}\text{sign}(\vec{p}(n))\prod_{j=1}^{N}\psi_{j}(x_{p(n)_{j}},0),$
(19)
with $\mathcal{C}$ a normalization constant and $\text{sign}(\vec{p}(n))$ the
sign of the permutation $\vec{p}(n)=\\{p(n)_{1},..,P(n)_{N}\\}$. Each initial
Gaussian state in (19) is
$\psi_{j}(x)=\exp\left[-\frac{(x-x_{0j})^{2}}{2\sigma_{j}^{2}}\right]\exp\left[ip_{0j}(x-x_{0j})\right],$
(20)
with spatial dispersion $\sigma_{j}$, central position $x_{0j}$, and central
velocity $p_{0j}$. The dynamical evolution of $|\Psi(t)\rangle$ is determined
by the Schrödinger equation, $i\partial|\Psi(t)\rangle/\partial
t=\hat{H}|\Psi(t)\rangle$, where the Hamiltonian $\hat{H}$ is described in
next section.
### IV.2 Full Hamiltonian
The $N$-electron Hamiltonian of our model system is
$\hat{H}=\hat{H}_{0}+\hat{D}$, with
$\hat{H}_{0}=\sum_{j=1}^{N}\left[\hat{k}_{j}+\hat{v}_{j}+\sum_{k<j}^{N}\hat{e}_{k,j}\right],\;\;\;\;\;\;\hat{D}=\sum_{j=1}^{N}\hat{d}_{j}.$
(21)
In $\hat{H}_{0}$, $\langle
x_{j}|\hat{e}_{k,j}|x_{k}\rangle=1/\sqrt{(x_{j}-x_{k})^{2}+\alpha^{2}}$ takes
care of the Coulomb repulsion with a smooth parameter $\alpha$, $\langle
x_{j}|\hat{k}_{j}|x_{j}\rangle=-\partial^{2}/(2\partial x_{j}^{2})$ stands for
the kinetic energy, and $\langle
x_{j}|\hat{v}_{j}|x_{j}\rangle=\omega^{2}x_{j}^{2}/2$ is the harmonic trap
potential. On the other hand, $\hat{D}$ introduces random disorder at every
grid point, where $\langle
x_{j}|\hat{d}_{j}|x_{j}\rangle=\gamma_{D}\sum_{k=1}^{M}a_{k}\exp[-4(x_{j}-g_{k})^{2}/\sigma_{D}^{2}]$,
with $\gamma_{D}$ its strength and $\sigma_{D}$ its spatial dispersion, while
$g_{k}$ runs through $M$ grid points; the set of random numbers $a_{k}$
satisfies $\langle a_{k}\rangle=0$ and $\langle a_{k}^{2}\rangle=1$, and the
disorder potential is normalized via $\int\langle
x_{j}|\hat{d}_{j}|x_{j}\rangle^{2}dx_{j}=\gamma_{D}^{2}$. Such random disorder
can be mapped onto speckle field potentials typical in some optical lattice
experiments. All simulation parameters are found in [78].
Figure 1: (a): Zoom of a disordered harmonic potential for $N=1$, $\langle
x_{1}|\hat{v}_{1}+\hat{d}_{1}|x_{1}\rangle$, from a larger simulation box.
(b): Successive energy splittings $\Delta E=E_{n}-E_{n-1}$ from a grid
diagonalization of the respective Hamiltonian $\hat{H}$. (d): zoom around the
peak of the corresponding density matrix modulus $|\rho_{n,m}(t=0)|$, which
remains the same at any time $t$ in an unitary evolution. (c): initial
positions $x_{0j}$ for $N=1,2,3$, and initial velocity $p_{0j}$ which is the
same in every case.
Figure 1(a) exemplifies, for $N=1$, a typical shape of the disordered harmonic
potential, $\langle x_{1}|\hat{v}_{1}+\hat{d}_{1}|x_{1}\rangle$, while figure
1(b) shows the corresponding successive energy splittings $\Delta
E=E_{n}-E_{n-1}$, as obtained from a direct diagonalization of $\hat{H}$,
which oscillate around the pure harmonic oscillator value of $1$. Figure 1(d)
shows the related shape of the density matrix modulus $|\rho_{n,m}(t=0)|$,
from where one identifies $N_{act}\approx 70$ (counting every level above
$10\%$ of peak value), while figure 1(c) shows initial positions $x_{0j}$,
$j=1...N$, and the initial velocity $p_{0}$, the same for any $j$ and $N$, for
the $N=1,2,3$ systems. Notice that $p_{0}$ not only takes the role of
simulating the initial quench of the trap and so to initiate the
nonequilibrium dynamics, but it is also responsible for determining the size
of the energy window and so the value of $N_{act}$. The initial energy $E_{0}$
of the wave packet defines, in the language of the ETH, the center of the
microcanonical energy window; since $E_{0}\approx\omega+p_{0}^{2}/2=201$, the
peak is at $n=m=201$. As mentioned in (15), the diagonal terms are time-
independent and, although real and imaginary parts of the off-diagonals terms
oscillate in time, their modulus remain constant so that, in an unitary
evolution, the modulus seen in figure 1(d) remains the same at any $t$ (see
Appendix B).
Both initial wave packet and Hamiltonian in (19)-(21) have many adjustable
parameters in [78] upon which the value of $t_{eq}$ depends on: (i) Coulomb
correlation by varying $\omega$ or $\alpha$; (ii) initial Gaussian by varying
$\sigma_{j}$, $x_{0j}$, or $p_{0j}$; (iii) disorder potential by varying
$\gamma_{D}$ or $\sigma_{D}$. All of them play a role in determining the
$\Delta E$ splittings of the involved $N_{act}$ activated eigenstates, which
is what drives the nonequilibrium dynamics of both expectation values and weak
values. It is not our goal to fully characterize the equilibration process as
a function of all those parameters. Neither to address the topic of many-body
localization [79, 80, 81], which should work against thermalization, nor to go
deeper in the issue of quantum-to-classical transition at $t\gg t_{eq}$; these
two latter issues are beyond the scope of our work and are focus of extensive
studies elsewhere. The main goal of our paper is to present a distinct
perspective in the understanding of quantum thermalization by looking at the
many-body weak values of the momentum in the configuration space.
Each numerical experiment corresponds to a static realization of a disorder
pattern; $t_{eq}$ hardly changes among different runs, given that all other
model parameters remain unchanged. The disorder amplitude should not be too
strong, to avoid localization due to all small wells created on the top of the
trap, neither too weak, to avoid too long simulation times until reaching
$t_{eq}$. We emphasize that even in the absence of electron-electron
collision, disorder collision is able to create correlation among distinct
degrees of freedom in configuration space when $N>1$.
### IV.3 Time evolution
Figure 2: Wave packet dynamics. Upper panels for $N=3$: initial
$|\Psi(x_{1},x_{2},[x_{3}],t=0)|^{2}$ (a) and final
$|\Psi(x_{1},x_{2},[x_{3}],t=150)|^{2}$ (b) shapes. Middle panels for $N=2$:
initial $|\Psi(x_{1},x_{2},t=0)|^{2}$ (c) and final
$|\Psi(x_{1},x_{2},t=150)|^{2}$ (d) shapes. Lower panels: (e) for $N=1$:
initial (dotted) $|\Psi(x_{1},t=0)|^{2}$ and final (solid)
$|\Psi(x_{1},t=150)|^{2}$ shapes; (f): $1$D-view for the three systems:
$|\Psi(x_{1},[x_{2}],[x_{3}],t)|^{2}$ for $N=3$ (red),
$|\Psi(x_{1},[x_{2}],t)|^{2}$ for $N=2$ (blue), and $|\Psi(x_{1},t)|^{2}$ for
$N=1$ (green), with solid (dotted) lines for $t=150$ ($t=0$); also shown the
classical microcanonical harmonic oscillator distribution $\rho_{cl}(x_{1})$
(black dashed line). All plots are in log-scale, and the horizontal axis in
(a),(c) ((b),(d)) is the same as in (e) ((f)).
Figure 2 plots in configuration space the time evolution of the $N$-electron
wave function for the three $N=1,2,3$ systems from the initial nonequilibrium
state. We show $|\Psi(x_{1},x_{2},[x_{3}],t)|^{2}$ for $N=3$ and
$|\Psi(x_{1},x_{2},t)|^{2}$ for $N=2$ at initial ($t=0$, panels (a),(c)) and
final ($t=150$, panels (b),(d)) simulation times, while panel (e) shows
$|\Psi(x_{1},t)|^{2}$ for $N=1$ for both initial ($t=0$, dotted) and final
($t=150$, solid) simulation times; the notation $[x_{j}]$ means that the
degree $j$ is integrated out, with results independent on chosen $j$ due to
the antisymmetry of the problem. The initially localized wave packets fully
spread out due to random scattering generated by both disorder potential and
Coulomb repulsion, with such spreading becoming more homogeneous as $N$
increases. In panel (f) we show the respective $1$D plots of
$|\Psi(x_{1},[x_{2}],[x_{3}],t)|^{2}$ ($N=3$, red),
$|\Psi(x_{1},[x_{2}],t)|^{2}$ ($N=2$, blue), and $|\Psi(x_{1},t)|^{2}$ ($N=1$,
green), with initial (final) results at dotted (solid) lines; the
probabilities at large $t$, as one expects to have crossed $t_{eq}$, approach
the microcanonical distribution of a classical harmonic oscillator (dashed
black line), $\rho_{cl}(x)=[\pi l_{0}\sqrt{p_{0}^{2}-x^{2}/l_{0}^{2}}]^{-1}$,
so that the dynamics develops in between the classical turning points at
$x_{TP}=\pm 20$ ($=p_{TP}$ since $\omega=1/l_{0}^{2}=1$); the deviation from
$\rho_{cl}(x)$ decreases as $N$ increases. In Appendix C we analyze the same
nonequilibrium dynamics in momentum representation.
### IV.4 Thermalized expectation values
Figures 3 and 4 show time evolution and thermalization of some typical
expectation values $\langle
A\rangle(t)=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle$, for $N=1,2,3$ respectively
in panels (a), (b), (c). On one hand, figure 3 focus on energy terms
normalized by $N$: kinetic $\langle K\rangle(t)$, with
$\hat{K}=\sum_{j}\hat{k}_{j}$, potential $\langle V\rangle(t)=\langle
V_{HO}\rangle(t)+\langle V_{D}\rangle(t)+\langle V_{Cou}\rangle(t)$, with
$\hat{V}_{HO}=\sum_{j}\hat{v}_{j}$, $\hat{V}_{D}=\sum_{j}\hat{d}_{j}$, and
$\hat{V}_{Cou}=\sum_{j,k<j}\hat{e}_{k,j}$, and half of total energy $\langle
E\rangle(t)=\langle K\rangle(t)+\langle V\rangle(t)$. On other hand, figure 4
focus on position $\langle x_{j}\rangle(t)$ and momentum $\langle
p_{j}\rangle(t)$ for the $j$-electron, and on their RMS values
$z_{j,RMS}(t)=\sqrt{\langle z^{2}_{j}\rangle(t)-\langle z_{j}\rangle^{2}(t)}$,
with $z=x,p$, and results independing on chosen $j$. _Without_ random
disorder, such expectation values would only exhibit harmonic oscillations
with period $2\pi/\omega$: while $\langle x_{j}\rangle(t)$ and $\langle
p_{j}\rangle(t)$ would respectively oscillate within position $\pm x_{TP}=\pm
20$ and momentum $\pm p_{TP}=\pm 20$ turning points, $\langle V\rangle(t)/N$
and $\langle K\rangle(t)/N$ would respectively oscillate within $0$ and
$x_{TP}^{2}/2=200$ and within 0 and $p_{TP}^{2}/2=200$; these latter values
increase a little upon $N$ due to Coulomb repulsion, whose isolated
contribution is shown ($100\times$-magnified) in figure 3.
Figure 3: Energy expectation values from the dynamics in figure 2. Panels (a),
(b), (c) respectively for $N=1,2,3$. Kinetic $\langle K\rangle(t)$, potential
$\langle V\rangle(t)$, half of total energy $\langle E\rangle(t)=\langle
K\rangle(t)+\langle V\rangle(t)$, and isolated contribution of $\langle
V_{Cou}\rangle(t)$ ($100\times$-magnified) are all normalized by $N$. Inset
for $N=2$ shows a longer propagation time, $t=[150-300]$. Legend in (a) and
horizontal axis in (c) apply to all panels.
The _presence_ of random disorder, even though $\langle V_{D}\rangle(t)\approx
0$ at any $t$, brings the initial nonequilibrium state into a final
_equilibrium_ state after a relaxation time $t_{eq}$. We know from the
discussions in (15) and in figure 1(d) that thermalization of an observable
$\hat{A}$ is determined by the diagonal populations of the density matrix,
while its off-diagonal coherences should dephase and only yield small
fluctuations around the relaxed value (see Appendix B). So one may estimate
the value of $t_{eq}$ either from figure 3, when the virial theorem $\langle
K\rangle\approx\langle V\rangle\approx\langle E\rangle/2$ is roughly satisfied
(since $\langle V_{Cou}\rangle\ll\langle V_{HO}\rangle$) [74], or from figure
4, when $\langle p_{j}\rangle\approx\langle x_{j}\rangle\approx 0$ seemingly
indicating a frozen dynamics after thermalization. From this latter result the
RMS values become $z_{j,RMS}(t)\approx\sqrt{\langle z^{2}_{j}\rangle(t)}$,
becoming also constant in figure 4 at $t>t_{eq}$ (from $\approx 14.2$ for
$N=1$ to $\approx 14.4$ for $N=3$), so that the values of
$p^{2}_{j,RMS}/2=\langle p^{2}_{j}\rangle/2$ and $x^{2}_{j,RMS}/2=\langle
x^{2}_{j}\rangle/2$ roughly yield the respective values of $\langle
K\rangle/N$ and $\langle V\rangle/N$ at $t>t_{eq}$ in figure 3. As expected
for an unitary evolution, $\langle E\rangle(t)/N$ remains conserved at any
$t$, from $\approx 201$ for $N=1$ to $\approx 207$ for $N=3$.
Figure 4: Position $\langle x_{j}\rangle(t)$ and momentum $\langle
p_{j}\rangle(t)$ expectation values from the dynamics in figure 2. Panels (a),
(b), (c) respectively for $N=1,2,3$. Their respective RMS values, as defined
in the text, are also shown, where results do not depend on chosen $j$. Inset
for $N=2$ shows a longer propagation time, $t=[150-300]$. Legend in (a) and
horizontal axis in (c) apply to all panels.
The value of $t_{eq}$ depends on all parameters [78] in (19)-(21), e.g., the
smaller is $p_{0j}$ or the higher is $\gamma_{D}$ the smaller is $t_{eq}$. By
increasing the influence of $\langle V_{Cou}\rangle$ in comparison to $\langle
K\rangle$ (by decreasing $\omega$ or $\alpha$), $t_{eq}$ increases since the
oscillation period and so $x_{TP}$ increases. The plots of $\langle
V_{Cou}\rangle(t)$ in figure 3 show that the Coulomb repulsion is more
effective at the turning points for $t\ll t_{eq}$, where electrons spend more
time reversing their movements, while the disorder potential overall acts
through a whole oscillation, although one may take it as more effective at the
origin. Coulomb correlation has a striking influence on the thermalization
process: in configuration space the only scattering mechanism for $N=1$ is due
to disorder, while for $N>1$ Coulomb scattering also makes more difficult for
the system to relax. This is seen by the slightly increasing values of
$t_{eq}$ as one moves in figure 3 from (a) ($t_{eq}\approx 70$) to (b)
($t_{eq}\approx 80$) to (c) ($t_{eq}\approx 90$). The vanishing of $\langle
p\rangle(t)$ in figure 4 seems more effective as one moves from (a) to (b) to
(c) but, however, it does not necessarily imply that electrons have achieved a
stationary-state null velocity at $t\gg t_{eq}$, as our following analysis on
weak values of the momentum will clarify (see also ‘phase-space’ in Appendix
C).
### IV.5 Nonthermalized weak values
We can at last elaborate on how the many-body weak values of the momentum may
improve our understanding on thermalization. In table 1 we summarize the five
types of operational properties accessible for the three different types of
quantum systems considered in this section: single-particle, distinguishable
particles, indistinguishable particles.
$N=1$ | $\langle p(t)\rangle$ (2) | $p_{W}(x,t)$ (4) | | |
---|---|---|---|---|---
$N>1$ (Dis) | $\langle p(t)\rangle$ (2) | | $p_{W}^{j}(\mathbf{x},t)$ (9) | $\tilde{p}_{W}(x,t)$ (12) | $p_{W}^{j,k}(x,t)$ (13)
$N>1$ (Ind) | $\langle p(t)\rangle$ (2) | | | $\tilde{p}_{W}(x,t)$ (12) |
Table 1: Five operational properties accessible in the laboratory for each of
the three quantum systems considered in our work: (i) with $N=1$ particles,
(ii) with $N>1$ distinguishable particles, and (iii) with $N>1$
indistinguishable particles.
The plot in figure 5(a) corresponds to the quantum system $N=1$ in table 1.
Although the expectation value $\langle p_{j}\rangle(t)$ seems to indicate
that the quantum behavior at $t\gg t_{eq}$ roughly equals the behavior of a
diagonal density matrix in (17), the weak values $p_{W}^{1,1}(0,t)$ (which
obviously corresponds to $p_{W}(0,t)$ in (4)) certifies that the off-diagonal
terms in (16) do not vanish after thermalization. For $N=1$, the fact that
expectation values thermalize is just a result that positive and negative off-
diagonals elements can roughly compensate each other, but they certainly do
not disappear as indicated by $p_{W}^{1,1}(x,t)$. We remind that $\langle
p\rangle(t)=\int dx\int dp\;p\;\mathbb{P}(p,x,t)$ and $p_{W}(x,t)$ can, both,
be computed from the same empirical probability $\mathbb{P}(p,x,t)$. In other
words, $\mathbb{P}(p,x,t)$ provides simultaneous thermalized and
nonthermalized results depending on how it is treated.
Figure 5: Local-in-position many-body weak values of the momentum from the
dynamics in figure 2 for $N=1$ in (a), and for distinguishable $N=2$ particles
in (b),(c) where Coulomb/exchange terms are disconsidered. Panels (a), (b)
show $p_{W}^{j,j}(x,t)$ from (13), which does not depend on $j$. Panel (c)
shows both $p_{W}^{j,k}(x,t)$ from (13) and $\tilde{p}_{W}(x,t)$ from (12).
Values of $x$ are the respective initial $x_{0j}$ values. The expectation
value of the momentum $\langle p_{j}\rangle(t)$ is also shown. Horizontal axis
in (a),(b) the same as in (c).
The plots in figures 5(b) and 5(c) correspond to the quantum system
$N>1(\text{Dis})$ in table 1, because neither exchange nor Coulomb interaction
among the particles are included; that is, the many-body wave function here
could be written as $\Psi(\mathbf{x},t)=\psi_{1}(x_{1},t)\psi_{2}(x_{2},t)$.
The weak values $p_{W}^{1,1}(0,t)$ and $p_{W}^{1,1}(-4,t)$ in panel (b)
confirm the nonthermalized operational properties even at $t\gg t_{eq}$. In
panel (c), one notices that $p_{W}^{1,2}(0,t)$, which corresponds to the weak
measurement of the momentum of particle 1 and strong measurement of the
position of particle 2, shows a thermalized behavior as it overlaps $\langle
p_{j}\rangle(t)$; but this is because here, as the particles have no
correlations among them, one gets $p_{W}^{j,k}(x,t)=\langle p_{j}\rangle(t)$
when $j\neq k$, as discussed in (32). Also in panel (c) $\tilde{p}_{W}(0,t)$,
from (12), which here is just a particle-average of the oscillating term
$p_{W}^{1,1}(0,t)$ and the non-oscillating term $p_{W}^{1,2}(0,t)$, presents
less oscillations but still clearly showing a non-thermalized behavior of
these operational properties.
Figure 6: Local-in-position many-body weak values of the momentum from the
dynamics in figure 2 for indistinguishable particles with $N=2$ in (a),(b) and
$N=3$ in (c). Panels (a), (c) show $p_{W}^{j,j}(x,t)$ from (13), which does
not depend on $j$, for the respective initial $x_{0j}$ values. Panel (b) shows
both $p_{W}^{j,k}(x,t)$ from (13) and $\tilde{p}_{W}(x,t)$ from (12). The
expectation value of the momentum $\langle p_{j}\rangle(t)$ is also shown.
Inset in (a) shows a longer propagation time, $t=[150-300]$. Horizontal axis
in (a),(b) the same as in (c).
Figure 6 corresponds to the (most common) quantum system $N>1(\text{Ind})$ in
table 1, in which one only has operational access in the laboratory to the
expectation value $\langle p\rangle(t)$ in (2) and to the weak value
$\tilde{p}_{W}(x,t)$ in (12). Panels (a) and (b) for $N=2$; panel (c) for
$N=3$. Since we also have mathematical (not operational) access in our
simulations (directly from configuration space) to the weak value
$p_{W}^{j,k}(x,t)$, we have also plotted it to help us to understand the
behavior of the operational weak value $\tilde{p}_{W}(x,t)$, which is a
particle average over different $p_{W}^{j,k}(x,t)$. We see in Fig. 6(a) that
$p_{W}^{1,1}(x,t)$ has smaller oscillations than in Fig. 5(b), but it still
does not thermalize, while $\langle p_{j}\rangle(t)$ effectivelly thermalize
(this later result is independent on $j$); the larger time window in the inset
so confirms. In figure 6(b) one notices that both $p_{W}^{1,2}(0,t)$ and
$p_{W}^{1,1}(0,t)$ have a similar non-thermalizing behavior. From these two
latter values we obtain
$\tilde{p}_{W}(0,t)=(p_{W}^{1,1}(0,t)+p_{W}^{1,2}(0,t))/2$ (thanks to the
properties in (31)), whose plot in panel (b) shows that this operational
parameter for identical particles does not fully thermalize. The $N=3$ case in
panel (c) starts to show the trend that, at higher $N$, $p_{W}^{1,1}(x,t)$
will approach the expectation value $\langle p_{1}\rangle(t)$ and so will also
thermalize, which is seem for all used $x_{0j}$ values. This is understood
from the fact that, for identical particles, $p_{W}^{j,k}(x,t)$ (and
$p_{W}^{j,j}(x,t))$ contains $N-1$ spatial integrals, while $\langle
p_{j}\rangle(t)$ contains $N$ and so, as one increases $N$, one gets
$p_{W}^{j,k}(x,t)\approx\langle p_{j}\rangle(t)$ because $N-1\approx N$.
The fact that the weak value of the momentum does not thermalize in the
configuration space can be understood when $p_{W}^{j}(\mathbf{x},t)$ in (9) is
mathematically interpreted as a Bohmian velocity, satisfying all of its
mathematical properties without any ontologic implications. Without external
perturbation, the initial state $|\Psi(0)\rangle$ of a closed system cannot
change with time its condition of being or not an energy eigenstate [36]. In
other words, the only energy eigenstates are the ones that are so at all
times. We know that for a closed system with $\Psi(\mathbf{x},t)$ vanishing at
the boundaries, the only Bohmian velocities in (9) that are zero are those
linked to an energy eigenstate [56]. From (10) we see that the weak value of
the momentum depends on the current density, and a time-dependent
$J^{j}(\mathbf{x},t)$ (strictly different from zero) is required for a time-
dependent probability presence in such a space, due to the continuity equation
in the configuration space implicit in the Schrödinger equation. Thus, since
our initial nonequilibrium state is not an energy eigenstate, from all
previous arguments, we conclude that $p_{W}^{j}(\mathbf{x},t)$ will never
vanish in the configuration space no matter the value of $N$, independently on
the thermalization or not of the expectation value of the momentum $\langle
p_{j}\rangle(t)$; that is, $p_{W}^{j}(\mathbf{x},t)$ will never thermalize
when evaluated at a point $\mathbf{x}$ of the configuration space (the same
conclusions apply to the presence probability in such space). Most of the
developments done in this paper come from the fact that, in most cases, the
weak value $p_{W}^{j}(\mathbf{x},t)$ in the configuration space is not
empirically accessible in the laboratory, and as such it is not an operational
property for indistinguishable particles; in such cases, though, the weak
value $\tilde{p}_{W}(x,t)$ in the physical space tends to thermalize as $N$
increases.
## V Conclusions
We have seen how, from the empirical knowledge of a unique distribution
probability $\mathbb{P}(p,\textbf{x},t)$, it is possible to get,
simultaneously, both thermalized (expectation values) and nonthermalized (weak
values) operational properties of a closed quantum system described by the
Schrödinger equation. A possible origin of the arrow of time in such systems
comes from dealing with operational properties obtained by averaging (tracing
out) some of the degrees of freedom defined in configuration space. As such
there is no contradiction in that there are properties defined in the
configuration space that do not thermalize, while some properties defined in
the physical space (by averaging or tracing out some degrees of freedom) have
their own time irreversible equations of motion. Such conclusion can be tested
in the laboratory through the many-body weak values of the momentum for
distinguishable particles where, at least conceptually, it is possible to get
information of the position of all particles in the laboratory after their
strong measurement.
However, for indistinguishable particles, a position measurement at $x$ cannot
be linked to a specific particle since there is no experimental protocol to
tag identical particles and, as such, instead of $\mathbb{P}(p,\textbf{x},t)$
in configuration space, one has operational access to $\mathbb{P}(p,x,t)$ in
physical space. But we have shown that $\mathbb{P}(p,x,t)$ can be understood
as an averaging of $\mathbb{P}(p,\textbf{x},t)$ over degrees of freedom in the
configuration space. For the simpler single-particle case, where obviously the
distinction between distinguishable/indistinguishable particles and between
configuration/physical spaces makes no sense, we have also seen that
expectation values can thermalize while weak values may not. The simple
explanation is that the expectation value has one integration over the single
valid coordinate while the weak value has not. Even for a system with $N=2$
identical particles, we see different behaviors for the expectation value
$\langle p\rangle(t)$ and the weak value $\tilde{p}_{W}(x,t)$ in the physical
space. In general, for identical particles, $\tilde{p}_{W}(x,t)$ contains
$N-1$ spatial integrals, while $\langle p_{j}\rangle(t)$ contains $N$ and so,
as one increases $N$, one gets $p_{W}^{j,k}(x,t)\approx\langle
p_{j}\rangle(t)$ because $N-1\approx N$; that is, the former thermalizes when
the latter also does. But such thermalization is seen in the physical space,
not in the configuration space.
So does thermalization occur in configuration space? No thermalization occurs
in the most common operational properties of a quantum system when evaluated
in a point $\mathbf{x}$ of the configuration space, even when other properties
defined in simpler spaces are thermalized. Notice that we have avoided along
the paper to discuss ontologic properties. Instead, our operational approach
has the advantage that the conclusions do not depend on which quantum theory
is invoked, but it also forbids the discussion whether or not thermalization
is a real (ontologic) phenomenon occurring in physical space. This latter
discussion depends on the ontology of each quantum theory; there are quantum
theories where the configuration space is not the fundamental space.
In summary, as mentioned in our Intro [1, 2, 3, 4, 5, 6, 7, 8], there are many
arrows of time which could require different explanations. We here only
discuss the origin of the arrow of time in the non-relativistic many-body
Schrödinger equation in closed quantum systems: a non-thermalized property in
the configuration space, when some of its degrees of freedom are averaged,
leads to a thermalized property in the physical space. We have also developed
the operational many-body weak values of the momentum to make such a
conclusion testable in the laboratory.
###### Acknowledgements.
This research was funded by Spain’s Ministerio de Ciencia, Innovación y
Universidades under Grant No. RTI2018-097876-B-C21 (MCIU/AEI/FEDER, UE), Grant
PID2021-127840NB-I00 (MICINN/AEI/FEDER, UE), the “Generalitat de Catalunya”
and FEDER for the project 001-P-001644 (QUANTUMCAT), the European Union’s
Horizon 2020 research and innovation programme under Grant No. 881603
GrapheneCore3 and under the Marie Skłodowska-Curie Grant No. 765426 TeraApps.
## Appendix A Weak values equations in many-body systems
In this appendix we develop the main weak values equations in the paper.
### A.1 Development of equations (4) and (9) in the paper
For a single-particle system described by the wave function $\psi(x,t)=\langle
x|\psi(t)\rangle$, the expression for the weak values of the momentum,
$p_{W}(x,t)$, can be obtained from the position distribution of the mean
momentum $\langle p\rangle(t)$ of the single-particle operator,
$\hat{p}=p|p\rangle\langle p|$, as
$\langle p\rangle(t)=\langle\psi(t)|\left(\int dx|x\rangle\langle
x\right)|\hat{p}|\psi(t)\rangle=\int dx|\psi(x,t)|^{2}\frac{\langle
x|\hat{p}|\psi(t)\rangle}{\langle x|\psi(t)\rangle}=\int
dx|\psi(x,t)|^{2}p_{W}(x,t),$ (22)
where we have defined the weak values as
$p_{W}(x,t)=\text{Real}\left(\frac{\langle x|\hat{p}|\psi(t)\rangle}{\langle
x|\psi(t)\rangle}\right)$. Since $\langle p\rangle(t)$ is real, as $\hat{p}$
is an hermitian operator, we have $\int
dx|\psi(x,t)|^{2}\text{Imag}\left(\frac{\langle
x|\hat{p}|\psi(t)\rangle}{\langle x|\psi(t)\rangle}\right)=0$. Thus, only the
real part is considered for defining the weak values and so we reproduce
equation (4) in the paper.
From a similar development we can find the weak values for an $N$-particle
system, with $\Psi(\mathbf{x},t)=\langle\mathbf{x}|\psi(t)\rangle$. The mean
momentum $\langle p_{j}\rangle(t)$ of degree of freedom $j$ belonging to
operator
$\hat{P}_{j}\equiv\hat{1}\otimes...\otimes\hat{p}_{j}\otimes...\otimes\hat{1}$
with $\hat{p}_{j}=p_{j}|p_{j}\rangle\langle p_{j}|$ is
$\displaystyle\langle p_{j}\rangle(t)$ $\displaystyle=$
$\displaystyle\langle\Psi(t)|\hat{P}_{j}|\Psi(t)\rangle=\int
d\mathbf{x}\langle\Psi(t)|\mathbf{x}\rangle\langle\mathbf{x}|\hat{P}_{j}|\Psi(t)\rangle=\int
d\mathbf{x}\langle\Psi(t)|\mathbf{x}\rangle\langle
x_{1}|\otimes...\otimes\left(\langle
x_{j}|\hat{p}_{j}\right)\otimes...\otimes\langle x_{N}|\Psi(t)\rangle$ (23)
$\displaystyle=$ $\displaystyle\int
d\mathbf{x}\Psi^{*}(\mathbf{x},t)(-i)\frac{\partial\Psi(\mathbf{x},t)}{\partial
x_{j}}=\int
d\mathbf{x}|\Psi(\mathbf{x},t)|^{2}\text{Imag}\left(\frac{\frac{\partial\Psi(\mathbf{x},t)}{\partial
x_{j}}}{\Psi(\mathbf{x},t)}\right)=\int
d\mathbf{x}|\Psi(\mathbf{x},t)|^{2}p_{W}^{j}(\mathbf{x},t),$
where we have used $\int d\mathbf{x}=\int dx_{1}...\int dx_{N}$,
$|\mathbf{x}\rangle=|x_{1}\rangle\otimes...\otimes|x_{N}\rangle$, and
$\left(\langle x_{j}|\hat{p}_{j}\right)=\int dx_{j}^{\prime}\langle
x_{j}|\hat{p}_{j}|x_{j}^{\prime}\rangle\langle x_{j}^{\prime}|=\langle
x_{j}|(-i)\frac{\partial}{\partial x_{j}}$. This result shows that $\langle
p_{j}\rangle(t)$ can be decomposed into different components along the
positions $\mathbf{x}$ on the configuration space. Each component
$p_{W}^{j}(\mathbf{x},t)$ is the many-body weak values of the momentum of the
$j$-th particle,
$p_{W}^{j}(\mathbf{x},t)=\text{Real}\left(\frac{\langle\mathbf{x}|\hat{P}_{j}|\Psi(t)\rangle}{\langle\mathbf{x}|\Psi(t)\rangle}\right)=\text{Imag}\left(\frac{\frac{\partial\Psi(\mathbf{x},t)}{\partial
x_{j}}}{\Psi(\mathbf{x},t)}\right)=\frac{J^{j}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}},$
(24)
with $|\Psi(\mathbf{x},t)|^{2}$ the probability of finding a particle at the
given configuration position $\mathbf{x}$. Expression (24) so reproduces
equation (9) in the paper. The last identity in (24) shows that the weak
values of the momentum is just the Bohmian velocity of the $j$-th particle in
the configuration position $\mathbf{x}$. If one deals with few particles well-
separated in the physical space, then the measurement of the many-body weak
values of each particle in the laboratory is unproblematic.
### A.2 Development of equation (13) in the paper
The problem with the weak values in (24) appears in the laboratory when we
consider $N$ particles in the same region of the physical space, since
$p_{W}^{j}(\mathbf{x},t)$ depends on all positions of the $N$ particles. Then,
it seems impossible for practical purposes to develop a measurement protocol
identifying the $N$ positions of the particles simultaneously. Thus, we want
to rewrite (23) in a way that it only depends on one position of one of the
$N$ particles. We are interested in an expression for computing $\langle
p_{j}\rangle(t)$ as a product of a probability in the physical space,
$\mathbb{P}^{k}(x,t)$, by a weak values which is also local in the physical
space, $p_{W}^{j,k}(x,t)$, which goes like
$\langle p_{j}\rangle(t)=\int dx\mathbb{P}^{k}(x,t)\;p_{W}^{j,k}(x,t),$ (25)
where
$\mathbb{P}^{k}(x,t)=\int dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;|\Psi(\mathbf{x},t)|^{2}.$ (26)
From (23), (24), and (26), one exactly gets equation (13) in the paper,
$p_{W}^{j,k}(x,t)=\frac{\int dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}\;p_{W}^{j}(..,x_{k-1},x,x_{k+1},..,t)|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}{\int
dx_{1}...\int dx_{k-1}\int dx_{k+1}...\int
dx_{N}|\Psi(..,x_{k-1},x,x_{k+1},..,t)|^{2}}.$ (27)
It is straightforward to show that $p_{W}^{j,k}(x,t)$ in (27) pondered by
$\mathbb{P}^{k}(x,t)$ in (26) exactly gives $\langle p_{j}\rangle(t)$.
### A.3 Development of equation (12) in the paper
The problem with the weak values in (27) is that it seems very difficult to
identify on which $j$-th particle the momentum is (weakly) measured, and on
which $k$-particle the position is (strongly) measured. Even worst, it seems
not possible to repeat the experiment and get the position and momentum of the
same two particles as in the previous measurement. In fact, the evaluation of
$\langle p_{j}\rangle(t)$ is independent on which $k$-th particle one does the
position measurement. If there are $N$ particles in the system we can write
$\langle p_{j}\rangle(t)=\frac{1}{N}\sum_{k=1}^{N}\int
dx\;\mathbb{P}^{k}(x,t)p_{W}^{j,k}(x,t)=\int
dx\;\mathbb{P}(x,t)\frac{1}{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t),$ (28)
since $\mathbb{P}^{k}(x,t)=\mathbb{P}(x,t)$ for identical particles. Finally,
if we assume that we will not identify the $j$-th particle for the momentum
measurement, then instead of trying to get $\langle p_{j}\rangle(t)$ we will
get an average over the $N$ particles,
$\displaystyle\langle p\rangle(t)\equiv\frac{1}{N}\sum_{j=1}^{N}\langle
p_{j}\rangle(t)=\frac{1}{N}\frac{1}{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\int
dx\;\mathbb{P}(x,t)p_{W}^{j,k}(x,t)=\int
dx\;\mathbb{P}(x,t)\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t).$
(29)
As such, we arrive to the weak values of the momentum as in equation (12) in
the paper,
$\tilde{p}_{W}(x,t)=\frac{1}{N^{2}}\sum_{j=1}^{N}\sum_{k=1}^{N}p_{W}^{j,k}(x,t),$
(30)
which satisfies $\langle p\rangle(t)=\int
dx\;\mathbb{P}(x,t)\;\tilde{p}_{W}(x,t)$.
For identical particles, either fermions or bosons, we have
$\mathbb{P}^{k}(x,t)=\mathbb{P}^{j}(x,t)\equiv\mathbb{P}(x,t)$ for all $j,k$
since $|\Psi(..,x_{k},..,x_{j},..,t)|^{2}=|\Psi(..,x_{j},..,x_{k},..,t)|^{2}$.
We also have
$p_{W}^{j}(..,x_{l},..,x_{j},..,t)=p_{W}^{l}(..,x_{j},..,x_{l},..,t)$ when
$j,l\neq k$ because
$J^{j}(..,x_{l},..,x_{j},..,t)=J^{l}(..,x_{j},..,x_{l},..,t)$ when $j,l\neq
k$. As such, we obtain
$\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$ $\displaystyle
p_{W}^{l,k}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j,l\neq k,$ $\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$
$\displaystyle
p_{W}^{j,l}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }k,l\neq j,$ $\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$
$\displaystyle
p_{W}^{k,j}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j\neq k,$ $\displaystyle p_{W}^{j,j}(x,t)$ $\displaystyle=$
$\displaystyle
p_{W}^{k,k}(x,t)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j,k.$ (31)
For a separable wave function,
$\Psi(\mathbf{x},t)=\psi_{1}(x_{1},t)...\psi_{N}(x_{N},t)$, we obtain
$\displaystyle p_{W}^{j,k}(x,t)$ $\displaystyle=$ $\displaystyle\frac{\int
dx\;p_{W}^{j}(x,t)|\psi_{j}(x,t)|^{2}}{\int dx|\psi_{j}(x,t)|^{2}}=\int
dx\;J^{j}(x,t)=\langle p_{j}\rangle(t)\;\;\;\;\;\;\;\text{for all }j\neq k,$
(32) $\displaystyle p_{W}^{j,j}(x,t)$ $\displaystyle=$ $\displaystyle
p_{W}^{j}(x,t)=\frac{J^{j}(x,t)}{|\psi_{j}(x,t)|^{2}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{for
all }j,$ (33)
where $J^{j}(x,t)$ and $p_{W}^{j}(x,t)$ are the current density and the weak
values, respectively, linked to the single-particle $\psi_{j}(x_{j},t)$. So,
for separable systems, the terms $p_{W}^{j,k}(x,t)$ are spatially uniform,
while $p^{j}_{W}(x,t)$ depend strongly on the position. For nonseparable
systems such differences are not true. Thus, $p_{W}^{j,k}(x,t)$ also provides
a procedure to quantify the interaction between distinct particles.
### A.4 Time-averaging of weak values
Figure 7: Time-averaged many-body weak values of the momentum,
$\bar{f}_{W}(\\{x_{0j}\\},t,T)=\bar{p}_{W}^{j,k}(\\{x_{0j}\\},t,T)$, from the
weak values presented in figure 6 in the paper. Only $j=k=1$ is plotted, which
is the same as $j=k=2$ for $N=2$ and $j=k=2,3$ for $N=3$. The values of
$\\{x_{0j}\\}$ are taken as the set of initial values of the respective wave
packets for $N=1$ in (a),(b), $N=2$ in (c),(d), and $N=3$ in (e),(f). Left and
right panels consider respectively a time $t=20<t_{eq}$ and $t=120>t_{eq}$,
while the integration period $T/2$ in all panels runs from $0$ to $20$.
We have seen in figure 6 in the paper that, contrary to the momentum
expectation value in figure 4 in the paper, the weak values of the momentum
only approach $0$ at higher $N$. For $N=2$ they remain within a finite range
of values after $t_{eq}$, while for $N=1$ such range is much larger. We have
also discussed how the weak values (as well as the density matrix coherences)
have a random nature for $t>t_{eq}$. Thus we compute the time-average of such
weak values on a period of time $T$,
$\bar{f}_{W}(x,t,T)=\frac{1}{T}\int_{t-T/2}^{t+T/2}dt^{\prime}\;f_{W}(x,t^{\prime}),$
(34)
where $f_{W}(x,t)$ can be any of the $4$ different types of weak values
expressed in the paper, that is, $p_{W}(x,t)$, $p_{W}^{j}(x,t)$,
$p_{W}^{j,k}(x,t)$, and $\tilde{p}_{W}(x,t)$. We consider in figure 7 the case
$f_{W}(x,t)=p_{W}^{j,k}(x,t)$, and then plot
$\bar{f}_{W}(x,t,T)=\bar{p}_{W}^{j,k}(x,t,T)$ from the weak values presented
in figure 6 in the paper, as a function of $T$. Upper, middle, lower panels
respectively for $N=1$, $N=2$, $N=3$; panels in left and right column
respectively related to a time before ($t=20<t_{eq}$) and after
($t=120>t_{eq}$) thermalization. For $N>1$, the time-average of the weak
values always vanish at $t>t_{eq}$, even at small $T$ ((d),(f)); for
$t<t_{eq}$ this may only happen at much higher $T$ ((c),(e)). The $N=1$ case
is, once more, much less smooth due to the many nodes of the wave function.
## Appendix B Diagonal and off-diagonal elements of the density matrix in the
energy representation and its connection with weak values of the momentum
The dynamics presented in our paper is based on an initial nonequilibrium
_pure_ state evolving in a closed system, while being affected by some
‘chaotic’ disorder potential. Without disorder, such a pure state simply
evolves periodically in the underneath harmonic oscillator potential. With
disorder, we have seen its role in bringing the nonequilibrium state into an
equilibrium regime characterized e.g. by the behaviour of the expectation
values shown in figures 3 and 4 in the paper. But what is the role of disorder
on the density matrix in the energy representation?
### B.1 Density matrix populations and coherences
We consider a pure state as a superposition of (single-particle or many-
particle) energy eigenstates,
$\langle x_{1}|\otimes...\otimes\langle
x_{N}|\Psi(t)\rangle=\langle\mathbf{x}|\Psi(t)\rangle=\sum_{n}c_{n}\langle\mathbf{x}|n\rangle
e^{-iE_{n}t}=\sum_{n}\;c_{n}R_{n}(\mathbf{x})e^{-iE_{n}t},$ (35)
being $|n\rangle$ an energy eigenstate with eigenvalue $E_{n}$, $c_{n}=\langle
n|\Psi(0)\rangle$, and $\langle\mathbf{x}|n\rangle=R_{n}(\mathbf{x})$ a real
function. The $n$-sum runs within a set $N_{act}$ of activated states, which
is defined by some quench responsible for creating the nonequilibrium initial
state; in our model, the quench is a sudden shift of the harmonic trap
translated to an initial velocity for the electrons at $t=0$. The pure state
density matrix, $\hat{\rho}(t)=|\Psi(t)\rangle\langle\Psi(t)|$, in the energy
basis becomes
$\rho_{n,m}(t)=\langle m|\hat{\rho}(t)|n\rangle=\langle
m|\Psi(t)\rangle\langle\Psi(t)|n\rangle=c_{n}\;c_{m}^{*}\;e^{i(E_{m}-E_{n})t},$
(36)
where the diagonal elements, $\rho_{n,n}$ (_populations_), are clearly time-
independent, while the off-diagonal terms (_coherences_) are not. When time-
averaged, within a time interval $T\to\infty$, the off-diagonal elements
vanish and the density matrix becomes diagonal,
$\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}\;\rho_{n,m}(t)\;dt=c_{n}\;c_{m}^{*}\;\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}e^{i(E_{m}-E_{n})t}dt=c_{n}\;c_{m}^{*}2\pi\delta_{n,m},$
(37)
with $\delta_{n,m}$ the Kronecker delta. This does not imply that the density
matrix in (36) (without time averaging) becomes diagonal as $t\to\infty$, but
only that the off-diagonal elements oscillate around zero.
In figure 8 we show the evolution of real and imaginary parts of a few off-
diagonal elements $\rho_{n,m}(t)$ ($N=1$), which can be written from (36) by
employing $c_{l}=|c_{l}|e^{i\theta_{l}}$ as
$\rho_{n,m}(t)=|c_{n}|\;|c_{m}|\text{cos}(\theta_{n}-\theta_{m}+(E_{m}-E_{n})t)+i|c_{n}|\;|c_{m}|\text{sin}(\theta_{n}-\theta_{m}+(E_{m}-E_{n})t),$
(38)
such that the modulus $|\rho_{n,m}(t)|=|c_{n}||c_{m}|$ is also time-
independent, as seen in panel (a). The oscillation period depends inversely on
their level splittings, $\Delta E=E_{m}-E_{n}$, as shown in panels (b)-(d),
which relate to the same level $n_{p}=201$ at the peak of $\rho_{n,m}(t)$: as
one increases the splitting, by considering a matrix element with $1$ (b), $3$
(c), and $6$ (d) levels apart, the oscillation period decreases. Also, if
considering both $n,m$ values away from the peak at $n_{p}=201$, the
respective matrix elements have exponentially smaller amplitudes. In a pure
harmonic oscillator all level splittings are simply proportional to the
difference in number of levels, but the disorder potential creates _random_
splittings. The full density matrix is given in figure 1(d) in the paper,
which shows $N_{act}\approx 70$ activated states at $t=0$, which will remain
as the main states determining the system unitary evolution at any $t$. From
these results we conclude that the thermalized system keeps memory of its
initial state through the propagation of its off-diagonal coherences (in
particular by keeping the information $c_{l}=|c_{l}|e^{i\theta_{l}}$), which
never disappear. Most importantly for the current operator as discussed in the
paper, since its diagonal populations are zero by construction.
Figure 8: Density matrix coherences for $N=1$. From the grid-diagonalization
as depicted in the inset (b) of figure 1 in the paper, the time-evolution of a
few of the off-diagonal coherences $\rho_{n,m}(t)$ is shown with respect to
the peak at $n_{p}=201$, with real (imaginary) part in blue (red). As the
energy splitting, $\Delta E=E_{m}-E_{n}$, increases from (b) to (c) to (d),
the respective oscillation period decreases. In (a) one verifies that the
respective moduli $|\rho_{n,m}(t)|^{2}$ remain constant.
### B.2 Connection between density matrix coherences and weak values
First, we need the presence probability density $|\Psi(\mathbf{x},t)|^{2}$,
given by
$\displaystyle|\Psi(\mathbf{x},t)|^{2}$ $\displaystyle=$
$\displaystyle\text{trace}\left[\hat{\rho}(|x_{1}\rangle\otimes...\otimes|x_{N}\rangle\langle
x_{1}|\otimes...\otimes\langle x_{N}|)\right]$ (39) $\displaystyle=$
$\displaystyle\sum_{n}\sum_{m}c_{n}c_{m}^{*}\;e^{i(E_{m}-E_{n})\;t}R_{n}(\mathbf{x})R^{*}_{m}(\mathbf{x}),$
and that can be decomposed into time-independent populations and time-
dependent coherences as
$|\Psi(\mathbf{x},t)|^{2}=|\Psi_{dia}(\mathbf{x},t)|^{2}+|\Psi_{off}(\mathbf{x},t)|^{2},$
(40)
with
$\displaystyle|\Psi_{dia}(\mathbf{x},t)|^{2}$ $\displaystyle=$
$\displaystyle\sum_{n}\rho_{n,n}|R_{n}(\mathbf{x})|^{2},$
$\displaystyle|\Psi_{off}(\mathbf{x},t)|^{2}$ $\displaystyle=$
$\displaystyle\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)R_{n}(\mathbf{x})R^{*}_{m}(\mathbf{x}).$ (41)
We also need the current density $J^{j}(\mathbf{x},t)$ due to the $j$-th
particle at one given position in the configuration space,
$\displaystyle J^{j}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\text{trace}\left[\hat{\rho}\left(\hat{p}_{j}|x_{1}\rangle\otimes...\otimes|x_{N}\rangle\langle
x_{1}|\otimes...\otimes\langle
x_{N}|+|x_{1}\rangle\otimes...\otimes|x_{N}\rangle\langle
x_{1}|\otimes...\otimes\langle x_{N}|\hat{p}_{j}\right)\right]$ (42)
$\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{n}\sum_{m}c_{n}c_{m}^{*}\;e^{i(E_{m}-E_{n})\;t}\left[R_{n}(\mathbf{x},t)\;\frac{\partial
R_{m}(\mathbf{x},t)}{\partial x_{j}}-R_{m}(\mathbf{x},t)\;\frac{\partial
R_{n}(\mathbf{x},t)}{\partial x_{j}}\right],$
that can also be decomposed in terms of diagonal and off-diagonal components
as
$J^{j}(\mathbf{x},t)=J^{j}_{dia}(\mathbf{x},t)+J^{j}_{off}(\mathbf{x},t),$
(43)
with
$\displaystyle J^{j}_{dia}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{n}\rho_{n,n}\left[R_{n}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial x_{j}}-R_{n}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial
x_{j}}\right]=\sum_{n}\rho_{n,n}(t)\;J^{j}_{n,n}(\mathbf{x}),$ $\displaystyle
J^{j}_{off}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)\;\left[R_{n}(\mathbf{x})\;\frac{\partial
R_{m}(\mathbf{x})}{\partial x_{j}}-R_{m}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial x_{j}}\right]=\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)\;J^{j}_{m,n}(\mathbf{x}),$ (44)
where
$J^{j}_{m,n}(\mathbf{x})=\frac{i}{2}\left[R_{n}(\mathbf{x})\;\frac{\partial
R_{m}(\mathbf{x})}{\partial x_{j}}-R_{m}(\mathbf{x})\;\frac{\partial
R_{n}(\mathbf{x})}{\partial x_{j}}\right]$. The relevant point is that only
off-diagonal elements $\rho_{n,m}(t)\;J^{j}_{m,n}(\mathbf{x})$ for $n\neq m$
provide current densities, since the contribution of diagonal elements vanish,
$J^{j}_{dia}(\mathbf{x},t)=0$, in closed systems with wave function vanishing
at the boundaries. The diagonal terms $\rho_{n,n}(t)\;J^{j}_{n,n}(\mathbf{x})$
do not contribute to the total current because energy eigenstates are pure
real (or pure imaginary), so that their current $J^{j}_{n,n}(\mathbf{x})=0$ in
first equation in (44).
Both results from (40) and (43) are in agreement with the well-known
continuity equation,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\partial|\Psi(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}(\mathbf{x},t)}{\partial x_{j}}$ (45)
$\displaystyle=$
$\displaystyle\frac{\partial|\Psi_{dia}(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}_{dia}(\mathbf{x},t)}{\partial
x_{j}}+\frac{\partial|\Psi_{off}(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}_{off}(\mathbf{x},t)}{\partial x_{j}}$
$\displaystyle=$
$\displaystyle\frac{\partial|\Psi_{off}(\mathbf{x},t)|^{2}}{\partial
t}+\sum_{j=1}^{N}\frac{\partial J^{j}_{off}(\mathbf{x},t)}{\partial x_{j}},$
where we have used the trivial results
$\frac{\partial|\Psi_{dia}(\mathbf{x},t)|^{2}}{\partial t}=0$ (because
$|\Psi_{dia}(\mathbf{x},t)|$ is time-independent) and
$\sum_{j=1}^{N}\frac{\partial J^{j}_{dia}(\mathbf{x},t)}{\partial x_{j}}=0$
(because $J^{j}_{dia}(\mathbf{x},t)=0$).
Since we know from (36) and (38) that the coherences never vanish
($\rho_{n,m}(t)\neq 0$ for $n\neq m$) and always oscillate, we conclude that
$\frac{\partial|\Psi_{off}(\mathbf{x},t)|^{2}}{\partial t}\neq 0$, so that
(45) means that the off-diagonal probability presence
$|\Psi_{off}(\mathbf{x},t)|^{2}$ and the off-diagonal current density
$J^{j}_{off}(\mathbf{x},t)$ are dynamically changing during the whole
simulation, before and after $t_{eq}$. We notice now that
$|\Psi(\mathbf{x},t)|^{2}$ and $J^{j}_{off}(\mathbf{x},t)$ are the elements
that define the weak values in equation (3) in the paper,
$p_{W}^{j}(\mathbf{x},t)=\frac{J^{j}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}}=\frac{J^{j}_{off}(\mathbf{x},t)}{|\Psi(\mathbf{x},t)|^{2}}=\frac{1}{|\Psi(\mathbf{x},t)|^{2}}\left(\sum_{n}\sum_{m\neq
n}\rho_{n,m}(t)\;J^{j}_{m,n}(\mathbf{x})\right).$ (46)
This provides the required connection between weak values and coherences.
## Appendix C Momentum representation and ‘phase-space’
In this appendix we provide the dynamics evolution in momentum representation
and construct a pseudo phase-space of the system.
Figure 9: Wave packet dynamics in momentum representation. This figure is
similar to figure 2 in the paper, which employed position representation, but
instead it considers momentum representation. Upper panels: wave packet
$|\Psi(p_{1},p_{2},[p_{3}],t)|^{2}$ for $N=3$ at $t=0$ in (a) and $t=150$ in
(b). Middle panels: $|\Psi(p_{1},p_{2},t)|^{2}$ for $N=2$ at $t=0$ in (c) and
$t=150$ in (d). Panel (e): wave packet $|\Psi(p_{1},t)|^{2}$ for $N=1$ at
$t=0$ (dotted) and at $t=150$ (solid). Panel (f): $1$D-view of
$|\Psi(p_{1},[p_{2}],[p_{3}],t)|^{2}$ for $N=3$ (red),
$|\Psi(p_{1},[p_{2}],t)|^{2}$ for $N=2$ (blue), and $|\Psi(p_{1},t)|^{2}$ for
$N=1$ (green), with solid (dotted) lines for the wave packet at $t=150$
($t=0$), where all $t=0$ plots overlap at $p_{0j}=20$. All plots are in log-
scale. Horizontal axis in (a),(c) ((b),(d)) is the same as in (e) ((f)).
Figure 10: ‘Phase-space’ analysis. From position $\langle x_{1}\rangle$(t) and
momentum $\langle p_{1}\rangle(t)$ expectation values in figure 4 in the
paper, we compile a respective ‘phase-space’ for $N=1$ in (a), $N=2$ in (c),
and $N=3$ in (e). Panels (b),(d),(f) are a zoom at the origin for the
respective $N$. The initial value of $\langle x_{1}\rangle(t=0)$ for each $N$
is the average of the set of respective initial positions $\\{x_{0j}\\}$,
while $\langle p_{1}\rangle(t=0)=20$ for any $N$. As one approaches the final
simulation time, $t=150>t_{eq}$, the ‘phase-space’ looks noisier for $N=1,2$
as a consequence of the recurrences as seen in the respective expectation
values in the paper. Horizontal axis in (a),(c) ((b),(d)) is the same as in
(e) ((f)).
### C.1 Dynamics in momentum representation
Figure 2 in the paper summarizes the dynamics for the systems with $N=1,2,3$
by showing the initial and final snapshots of the respective wave functions in
_position_ representation. Since our algorithm for propagating the Schrödinger
equation uses a split-operator method where the kinetic energy is handled in
momentum representation, where it is diagonal, and then Fourier-transformed
back to position representation (where the potential terms are diagonal), for
consistency, we show in figure 9 exactly the same plots as in figure 2 in the
paper, but in _momentum_ representation. That is, we plot in the upper panels
$|\Psi(p_{1},p_{2},[p_{3}],t)|^{2}$, with $[p_{3}]$ integrated out, for $N=3$
at $t=0$ in (a) and $t=150$ in (b) (results would be the same if integrating
on $[p_{1}]$ or $[p_{2}]$). The middle panels show $|\Psi(p_{1},p_{2},t)|^{2}$
for $N=2$ at $t=0$ in (c) and $t=150$ in (d). Panel (e) shows
$|\Psi(p_{1},t)|^{2}$ for $N=1$ at $t=0$ ($t=150$) in dotted (solid) lines,
while panel (f) compiles all respective $1$D plots of
$|\Psi(p_{1},[p_{2}],[p_{3}],t)|^{2}$ for $N=3$, $|\Psi(p_{1},[p_{2}],t)|^{2}$
for $N=2$, and $|\Psi(p_{1},t)|^{2}$ for $N=1$, with solid (dotted) lines for
$t=150$ ($t=0$). Since the initial velocity is the same and centered at
$p_{0j}=20$ for each degree of freedom and for every $N$, all $1$D plots at
$t=0$ overlap. One also notices in momentum representation the same full
spread of the wave function after thermalization, at $t\gg t_{eq}$, which then
remains in between the ’turning points’ $p_{TP}=\pm 20$ (since $\omega=1$). As
in position representation, the higher the $N$ the more homogeneous is the
wave packet spread.
### C.2 Pseudo phase-space
We have shown in the paper that thermalization provides time-independent
expectation values, so that another useful plot is a pseudo ‘phase-space’ as
if that could be simply compiled from the expectation values of $\langle
p_{j}\rangle(t)$ and $\langle x_{j}\rangle(t)$ in figure 4 in the paper.
Figure 10 shows the ‘phase space’ for $N=1,2,3$ respectively in upper, middle,
lower panels; left column the full range, right column a zoom at the origin
$(0,0)$. The figures look the same no matter which $j$-electron is considered
at each $N$. For any $N$ the curves start at $\langle p_{1}\rangle(0)=20$ and
$\langle x_{1}\rangle(0)=0$, only exception being $N=2$ which starts at
$\langle x_{1}\rangle(0)=-2$. The curves after thermalization (right column)
are about the same at any $N$, with $\langle p_{1}\rangle(t)\approx\langle
x_{1}\rangle(t)\approx 0$; they look visually different since some recurrences
may happen causing a noisier ‘phase space’ as time keeps rolling after
$t_{eq}$; however, once more, the behaviour is more smooth as $N$ increases.
## References
* [1] R. Penrose, The Emperor’s New Mind: Concerning computers, Brains and the Laws of Physics (Oxford Univ. Press, Oxford 1989).
* [2] S. W. Hawking, Phys. Rev. D 32, 2489 (1985).
* [3] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986).
* [4] L. Sklar, Physics and Change: Philosophical Issues in the Foundations of Statistical Mechanics (Cambridge Univ. Press, Cambridge 1993).
* [5] D. Z. Albert, Time and Change (MA: Harvard Univ. Press, Cambridge 2000).
* [6] J. Bricmont, D. Durr, M. Galavotti, F. Petruccione, and N. Zanghi, Change in Physics: Foundations and Perspectives (Springer, Berlin 2001).
* [7] R. Penrose, J. Stat. Phys. 77, 217 (1994).
* [8] H. D. Zeh, The Physical Basis of the Direction of Time (5th ed., Springer, Berlin 2007).
* [9] M. Ueda, Nat. Rev. Phys. 2, 669 (2020).
* [10] C. Gogolin and J. Eisert, Rep. Prog. Phys. 79, 056001 (2016).
* [11] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015).
* [12] V. I. Yukalov, Laser Phys. Lett. 8, 485 (2011).
* [13] P. Reimann, New J. Phys. 21, 053014 (2019).
* [14] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452, 854 (2008).
* [15] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
* [16] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016).
* [17] J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018).
* [18] T. Langen, R. Geiger, and J. Schmiedmayer, Annu. Rev. Condens. Matter Phys. 6, 201 (2015).
* [19] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), and U. Sen, Adv. Phys. 56, 243 (2007).
* [20] L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, and A. Aspect, New J. Phys. 10, 045019 (2008).
* [21] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
* [22] P. Reimann, New J. Phys. 17, 055025 (2015).
* [23] T. R. Oliveira, C. Charalambous, D. Jonathan, M. Lewenstein, and A. Riera, New J. Phys. 20, 033032 (2018).
* [24] H. Kim, T. N. Ikeda, and D. A. Huse, Phys. Rev. E 90, 052105 (2014).
* [25] T. Kinoshita, T. Wenger, and D. Weiss, Nature 440, 900 (2006).
* [26] S. Trotzky, Y-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Nat. Phys. 8, 325 (2012).
* [27] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
* [28] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P. McCulloch, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Phys. Rev. Lett. 110, 205301 (2013).
* [29] D. Dries, S. E. Pollack, J. M. Hitchcock, and R. G. Hulet, Phys. Rev. A 82, 033603 (2010).
* [30] S. Will, D. Iyer, and M. Rigol, Nat. Commun. 6, 6009 (2015).
* [31] J. Kajala, F. Massel, and P. Torma, Phys. Rev. Lett. 106, 206401 (2011).
* [32] U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch, and A. Rosch, Science 322, 1520 (2008).
* [33] B. Nagler, K. Jagering, A. Sheikhan, S. Barbosa, J. Koch, S. Eggert, I. Schneider, and A. Widera, Phys. Rev. A 101, 053633 (2020).
* [34] U. Schneider, L. Hackermuller, J. P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch, and A. Rosch, Nat. Phys. 8, 213 (2012).
* [35] W. C. Myrvold, Synthese 192, 3247 (2015).
* [36] H-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford Univ. Press, Oxford, 2002).
* [37] In other quantum theories, like Bohmian mechanics or many-worlds, the collapse is just an ‘effective’ phenomenon whose consequences can still be treated from a bigger unitary Schrödinger equation that includes the measuring apparatus.
* [38] M. Srednicki, Phys. Rev. E 50, 888 (1994).
* [39] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
* [40] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
* [41] B. E. Y. Svensson, Quanta 2, 18 (2013).
* [42] D. Pandey, R. Sampaio, T. Ala-Nissila, G. Albareda, and X. Oriols, Phys. Rev. A 103, 052219 (2021).
* [43] H. M. Wiseman, New J. Physics 9, 165 (2007).
* [44] D. Dürr, S. Goldstein, and N. Zanghì, J. Stat. Phys. 134, 1023 (2009).
* [45] D. Marian, N. Zanghi, and X. Oriols, Phys. Rev. Lett. 116, 110404 (2016).
* [46] F. L. Traversa, G. Albareda, M. Di Ventra, and X. Oriols, Phys. Rev. A 87, 052124 (2013).
* [47] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
* [48] A. Hariri, D. Curic, L. Giner, and J. S. Lundeen, Phys. Rev. A 100, 032119 (2019).
* [49] R. Ramos, D. Spierings, I. Racicot, and A. M. Steinberg, Nature 583, 529 (2020).
* [50] A.G. Kofman, S. Ashhab, and F. Nori, Physics Reports 520, 43 (2012).
* [51] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys. 86, 307 (2014).
* [52] K. Renziehausen and I. Barth, Found. Phys. 50, 772 (2020).
* [53] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Phys. Rev. X 6, 041065 (2016).
* [54] V. Alba, B. Bertini, M. Fagotti, L. Piroli, and P. Ruggiero, J. Stat. Mech., 114004 (2021).
* [55] I. Bouchole and J. Dubail, J. Stat. Mech., 014003 (2022).
* [56] X. Oriols and J. Mompart, Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology (Pan Stanford, Singapore, 2012).
* [57] A. O. T. Pang, H. Ferretti, N. Lupu-Gladstein, W.-K. Tham, A. Brodutch, K. Bonsma-Fisher, J. E. Sipe, and A. M. Steinberg, Quantum 4, 365 (2020).
* [58] T. Fogarty, M. Á. García-March, L. F. Santos, and N. L. Harshman, Quantum 5, 486 (2021).
* [59] G. Zisling, L. F. Santos, and Y. B. Lev, SciPost. Phys. 10, 088 (2021).
* [60] C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodrubetz, Z. Chen, A. Megrant, R. Barends, B. Campbell, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, J. Mutus, P. J. J. O’Malley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. Polkovnikov, and J. M. Martinis, Nat. Phys. 12, 1037 (2016).
* [61] Md. M. Ali, W.-M. Huang, and W.-M. Zhang, Sci. Rep. 10, 13500 (2020).
* [62] J.-Q. Liao, H. Dong, and C. P. Sun, Phys. Rev. A 81, 052121 (2010).
* [63] P. Lydzba, Y. Zhang, M. Rigol, and L. Vidmar, Phys. Rev. B 104, 214203 (2021).
* [64] C. Nation and D. Porras, Phys. Rev. E 102, 042115 (2020).
* [65] M. Rigol, Phys. Rev. A 80, 053607 (2009).
* [66] L. F. Santos and M. Rigol, Phys. Rev. E 81, 036206 (2010).
* [67] Y. Tang, W. Kao, K.-Y. Li, S. Seo, K. Mallayya, M. Rigol, S. Gopalakrishnan, and B. L. Lev, Phys. Rev. X 8, 021030 (2018).
* [68] C. Gogolin, M. P. Muller, and J. Eisert, Phys. Rev. Lett. 106, 040401 (2011).
* [69] M. Brenes, T. LeBlond, J. Goold, and M. Rigol, Phys. Rev. Lett. 125, 070605 (2020).
* [70] M. Brenes, J. Goold, and M. Rigol, Phys. Rev. B 102, 075127 (2020).
* [71] Y.-W. Hsueh, C.-H. Hsueh, and W.-C. Wu, Entropy 22, 855 (2020).
* [72] L. Pezze, B. Hambrecht, and L. Sanchez-Palencia, EPL 88, 30009 (2009).
* [73] L. Pezze, and L. Sanchez-Palencia, Phys. Rev. Lett. 106, 040601 (2011).
* [74] C.-H. Hsueh, R. Ong, J.-F. Tseng, M. Tsubota, and W.-C. Wu, Phys. Rev. A 98, 063613 (2018).
* [75] R. Ong, C.-H. Hsueh, and W.-C. Wu, Phys. Rev. A 100, 053619 (2019).
* [76] D. Clément, A. F. Varón, J. A. Retter, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, New J. Phys. 8, 165 (2006).
* [77] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and Alain Aspect, Nature 453, 891 (2008).
* [78] All figures in the paper consider $\alpha=0.1$, $\omega=1.0$, $\sigma_{j}=l_{0}=1/\sqrt{\omega}=1.0$; initial velocities are $p_{0j}=20$ for any $j$ and $N$; initial positions are $x_{01}=0$ for $N=1$, $(x_{01},x_{02})=(-4,0)$ for $N=2$, $(x_{01},x_{02},x_{03})=(-4,0,4)$ for $N=3$, having in mind that the initial wave packet is antisymmetrized such that the label $j$ in $x_{0j}$ is redundant; for the disorder, $\gamma_{D}=25$ and $\sigma_{D}\approx\Delta x$, with $\Delta x$ being the position resolution of the respective grid, defined in a way that its momentum resolution is $\Delta p\approx\Delta x$, since our propagator uses both position and momentum representations. Initial time is $t_{0}=0$, final time is $t_{f}=150$ (for $N=2$ $t_{f}=300$), with time step $\Delta t=0.001$. Simulation boxes are roughly twice the extension of the turning points, and $\approx 10^{3}$ points per dimension are used; so e.g for $N=3$ our configuration space has $\approx 10^{9}$ points.
* [79] A. Maksymov, P. Sierant, and J. Zakrzewski, Phys. Rev. B 102, 134205 (2020).
* [80] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nat. Phys. 12, 907 (2016).
* [81] L. F. Santos and M. Rigol, Phys. Rev. E 82, 031130 (2010).
|
# Randomized Milstein algorithm for approximation of solutions of jump-
diffusion SDEs
Paweł Przybyłowicz Verena Schwarz Michaela Szölgyenyi
###### Abstract
We investigate the error of the randomized Milstein algorithm for solving
scalar jump-diffusion stochastic differential equations. We provide a complete
error analysis under substantially weaker assumptions than known in the
literature. In case the jump-commutativity condition is satisfied, we prove
optimality of the randomized Milstein algorithm by proving a matching lower
bound. Moreover, we give some insight into the multidimensional case by
investigating the optimal convergence rate for the approximation of jump-
diffusion type Lévys’ areas. Finally, we report numerical experiments that
support our theoretical findings.
Keywords: jump-diffusion SDEs, randomized Milstein algorithm, Lévy’s area,
$n$-th minimal error, optimality of algorithms, information-based complexity
MSC (2020): 68Q25, 65C30, 60H10
## 1 Introduction
Consider the following jump-diffusion stochastic differential equation (SDE)
$\displaystyle\mathop{}\\!\mathrm{d}X(t)=\mu(t,X(t))\mathop{}\\!\mathrm{d}t+\sigma(t,X(t))\mathop{}\\!\mathrm{d}W(t)+\rho(t,X(t-))\mathop{}\\!\mathrm{d}N(t),\quad
t\in[0,T],\quad X(0)=X_{0},$ (1)
where $\mu,\sigma,\rho:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ are (at least)
measurable functions, $T\in(0,\infty)$, $W=(W(t))_{t\in[0,T]}$ is a standard
Wiener process, and $N=(N(t))_{t\in[0,T]}$ is a homogeneous Poisson process
with intensity $\lambda>0$ on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},{\mathbb{P}})$ with a
filtration $(\mathcal{F}_{t})_{t\geq 0}$ that satisfies the usual conditions.
Furthermore, we assume $p\in[2,\infty)$ and $X_{0}$ to be an
$\mathcal{F}_{0}$-measurable random variable with
${\mathbb{E}}[|X_{0}|^{2p}]<\infty$.
Due to their numerous applications in mathematical finance, control theory,
and the modelling of energy markets, cf. [15, 20, 21, 23], jump-diffusion SDEs
continue to gain scientific interest. Only in very special cases exact
solutions are available. It is therefore important to develop efficient (or
even in some sense optimal) numerical algorithms.
In this paper, we define the randomized Milstein algorithm, which is a
Milstein-type scheme that uses randomization of the drift coefficient in time.
We prove $L^{p}$-error and optimality of the scheme applied to SDE (1). We
provide appropriate upper and lower error estimates in the multidimensional
case, extending the findings from [2]. In the case that only a finite number
of evaluations of $W$ and $N$ are allowed, this enables us to address the
problem of the optimal approximation of jump-diffusion type Lévys’ areas. As
these Lévys’ areas are naturally present in approximation schemes for jump-
diffusion SDEs, our result implies lower error bounds. Analysis of the lower
bounds and optimality is provided in the Information-Based Complexity (IBC)
framework, see [24]. This setting is widely used for investigating optimal
algorithms for approximation of solutions of SDEs, see, for example, [5, 6,
11], [12, 13, 17, 18, 19, 22, 10].
The approximate the solutions of SDEs using randomized algorithms is, for
example, studied in [11, 12, 17, 18, 22], where the authors consider the
randomized Euler–Maruyama scheme for SDEs in the jump-free case, and provide
error bounds and optimality results. The articles [4] and [5] discuss the
properties of randomized quadrature rules used for approximating stochastic
Itô integrals. Error bounds and optimality results of the randomized Milstein
scheme for SDEs without jumps were investigated in [13] and [8]. The latter
constructed a two-stage version of the randomized Milstein scheme and examined
its error. In this paper we extend the results from [13] and [2] to provide
results for jump-diffusion SDEs.
In the scalar case, we consider SDEs (1) with coefficients that are Hölder
continuous in time and Lipschitz continuous and differentiable with Lipschitz
continuous derivative in space. Under these assumptions we provide upper
bounds for the error of the randomized Milstein algorithm. Our assumptions are
significantly weaker than any other in the literature, where it is usually
assumed that the coefficients are at least twice continuously differentiable
in space, cf. [13, 15]. In addition to that, in case jump-commutativity
condition (JCC) is satisfied, we prove optimality of the randomized Milstein
algorithm among those randomized algorithms that use finitely many evaluations
of the driving processes. It turns out that randomization of the drift
coefficient in time improves the convergence rate, see Remark 4.2 and Theorem
4.1. In the multidimensional case we establish upper and lower bounds for the
approximation of jump-diffusion Lévys’ areas using the trapezoidal rule. These
error bounds imply lower error bounds for any class of coefficients of
multidimensional SDEs for which the two-dimensional SDE generating the Levy’s
area is a subproblem. In particular, it implies optimality of the
multidimensional Euler–Maruyama algorithm in the class of algorithms that use
only finitly many evaluations of $W$ and $N$. Therefore, the scalar and
multidimensional case can differ a lot from a point of view of optimality of
algorithms under certain admissible information about $W$, $N$, and under
certain regularity assumptions. Our numerical experiments match the
theoretical results on the convergence of the randomized Milstein algorithm.
Most interestingly our experiments suggest that for the simulation of jump-
diffusion SDEs the $L^{p}$-convergence rate is indeed dependent on $p$.
The main contributions of the paper are:
* •
We perform rigorous error analysis for the randomized Milstein algorithm for
scalar jump-diffusion SDEs (1) under relatively mild assumptions on the
coefficients (Theorem 3.2).
* •
We investigate lower error bounds in the worst-case setting in the scalar
(Theorem 4.1) and multidimensional case (Theorem 4.3). This essentially allows
us to establish optimality of the randomized Milstein algorithm in the scalar
case with $p=2$.
* •
We show that numerical experiments match our theoretical results.
The paper is organized as follows. Section 2 states the assumptions under
which we perform error analysis for the randomized Milstein algorithm. Section
3 is devoted to error analysis of the randomized Milstein process. Lower
bounds and optimality analysis in the IBC framework are given in Section 4. In
Section 5 we show the results of the numerical experiments. Finally, some
auxiliary results used in the proofs can be found in the Appendix.
## 2 Preliminaries
For a random variable $X\colon\Omega\to\mathbb{R}$ we denote by
$\|X\|_{L^{p}(\Omega)}=(\mathbb{E}[|X|]^{p})^{1/p}$, where $p\in[2,\infty)$.
We take $\displaystyle{\mathcal{F}_{\infty}=\sigma\Big{(}\bigcup_{t\geq
0}\mathcal{F}_{t}\Big{)}}$. Moreover, for $Z\in\\{W,N\\}$ we define
$\displaystyle{\mathcal{F}_{\infty}^{Z}=\sigma\Big{(}\bigcup_{t\geq
0}\mathcal{F}_{t}^{Z}\Big{)}}$ where
$\displaystyle{\mathcal{F}_{t}^{Z}=\sigma\Big{(}\bigcup_{0\leq s\leq
t}\sigma\big{(}Z(s)\big{)}\Big{)}}$. The processes $W$ and $N$ are
independent, i.e. $\mathcal{F}_{\infty}^{N}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathcal{F}_{\infty}^{W}$, cf. [23, p. 64,
Theorem 97]. We denote for all functions $f\in
C^{0,1}([0,T]\times{\mathbb{R}};{\mathbb{R}})$ the partial derivative of $f$
with respect to $y$ by $f^{\prime}_{y}$. Further we define for all functions
$f\in C^{0,1}([0,T]\times{\mathbb{R}};{\mathbb{R}})$,
$L_{1}f(t,y)=\sigma(t,y)f^{\prime}_{y}(t,y)$ and
$L_{-1}f(t,y)=f(t,y+\rho(t,y))-f(t,y)$ for all $t\in[0,T]$,
$y\in{\mathbb{R}}$.
We impose the following assumptions on the coefficient functions.
###### Assumption 2.1.
For the functions
$\mu,\sigma,\rho\colon[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ and for
$p\in[2,\infty)$ we assume that there exist constants
$\varrho_{1},\varrho_{2},\varrho_{3}\in(0,1]$ such that:
* (i)
For all $f\in\\{\mu,\sigma,\rho\\}$, $f\in
C^{0,1}([0,T]\times{\mathbb{R}};{\mathbb{R}})$.
* (ii)
There exists a constant $K_{1}>0$ such that for all $t,s\in[0,T]$,
$y,z\in{\mathbb{R}},$ and all $f\in\\{\mu,\sigma,\rho\\}$ it holds that
$\displaystyle|f(t,y)-f(t,z)|\leq K_{1}|y-z|,$ (2)
$|f^{\prime}_{y}(t,y)-f^{\prime}_{y}(t,z)|\leq K_{1}|y-z|,$ (3)
$|f(t,y)-f(s,y)|\leq K_{1}(1+|y|)|t-s|^{\varrho_{f}},$ (4)
where
$(\varrho_{f},f)\in\\{(\varrho_{1},\mu),(\varrho_{2},\sigma),(\varrho_{3},\rho)\\}$.
* (iii)
There exists a constant $K_{2}>0$ such that for all $y\in{\mathbb{R}}$,
$t,s\in[0,T]$,
$|\mu^{\prime}_{y}(t,y)-\mu^{\prime}_{y}(s,y)|\leq
K_{2}(1+|y|)|t-s|^{\varrho_{1}}.$ (5)
* (iv)
There exists a constant $K_{3}>0$ such that for all $t\in[0,T]$,
$y,z\in{\mathbb{R}}$, and all $f\in\\{\sigma,\rho\\}$ it holds that
$\displaystyle|L_{1}f(t,y)-L_{1}f(t,z)|\leq K_{3}|y-z|.$ (6)
* (v)
For the initial value $X_{0}$ we assume that it is an
$\mathcal{F}_{0}$-measurable random variable and that
$\|X_{0}\|_{L^{2p}(\Omega)}<\infty.$ (7)
By the Lipschitz assumption (2) we obtain that for all
$(t,y)\in[0,T]\times{\mathbb{R}}$ and $f\in\\{\mu,\sigma,\rho\\}$ we have
$\displaystyle|f(t,y)|\leq K_{4}(1+|y|),$ (8)
with
$K_{4}=\max\limits_{i=1,2,3}\\{\max\\{|f(0,0)|,K_{1}\\}+K_{1}T^{\varrho_{i}}\\}$,
and by (i) and (2),
$\displaystyle|f^{\prime}_{y}(t,y)|\leq K_{1}.$ (9)
Furthermore, we know that for $f\in\\{\mu,\sigma,\rho\\}$ it holds that for
all $t\in[0,T]$ the first partial derivative $f^{\prime}_{y}(t,\cdot)$ is
absolutely continuous, since it is Lipschitz continuous. Hence, for all
$t\in[0,T]$ the second partial derivative $f^{\prime\prime}_{yy}(t,\cdot)$
exists almost everywhere on ${\mathbb{R}}$. For all $t\in[0,T]$ denote by
$S_{f}(t)$ the set of Lebesgue measure $0$ for which the second partial
derivative $f^{\prime\prime}_{yy}(t,\cdot)$ does not exist. Then for
$f\in\\{\mu,\sigma,\rho\\}$, $t\in[0,T]$, and all $y\in{\mathbb{R}}\setminus
S_{f}(t)$ it holds that
$\displaystyle|f^{\prime\prime}_{yy}(t,y)|\leq K_{1}.$ (10)
On $S_{f}(t)$ we define $f^{\prime\prime}_{yy}(t,\cdot)\equiv 0$. At this
point, we like to emphasise that the choice of the values of
$f^{\prime\prime}_{yy}(t,\cdot)$ on $S_{f}(t)$ does not influence the proof of
the main result, since by using the local time theory we see that the suitable
bounds we compute are not dependent on these values. Morever, for
$f\in\\{\sigma,\rho\\}$ we have that there exists a constant
$K_{5}\in(0,\infty)$ such that for all $(t,y)\in[0,T]\times{\mathbb{R}}$,
$\max\\{|L_{1}f(t,y)|,|L_{-1}f(t,y)|\\}\leq K_{5}(1+|y|).$ (11)
Under Assumption 2.1 the existence and uniqueness of a strong solution to the
SDE (1) is well-known, see, for example [16, p. 255, Theorem 6]. Since
$\mathbb{E}[|X_{0}|^{2p}]<\infty$, there exists $K_{6}\in(0,\infty)$ such that
$\displaystyle{\mathbb{E}}\Big{[}\sup_{0\leq t\leq T}|X(t)|^{2p}\Big{]}\leq
K_{6},$ (12)
and for all $s,t\in[0,T]$,
$\displaystyle{\mathbb{E}}\Big{[}|X(t)-X(s)|^{p}\Big{]}\leq K_{6}|t-s|,$ (13)
see [22, Lemma 1]. The estimate (13) can be improved if $\rho\equiv 0$.
Moreover, under the Assumption 2.1 (i), (ii), for $f\in\\{\mu,\sigma,\rho\\}$
and fixed $t\in[v_{1},v_{2}]\subset[0,T]$, it is possible to apply the Meyer-
Itô formula [16, p. 221, Theorem 71] to the function ${\mathbb{R}}\ni y\mapsto
f(t,y)\in\mathbb{R}$ and to the solution process $(X(s))_{s\in[v_{1},v_{2}]}$.
This gives the following parametric version of the Meyer-Itô formula: For all
$s,t\in[v_{1},v_{2}]$ it holds that
$f(t,X(s))=f(t,X(v_{1}))+\int\limits_{v_{1}}^{s}\alpha(f,t,u)\mathop{}\\!\mathrm{d}u+\int\limits_{v_{1}}^{s}\beta(f,t,u)\mathop{}\\!\mathrm{d}W(u)+\int\limits_{v_{1}}^{s}\gamma(f,t,u)\mathop{}\\!\mathrm{d}N(u),$
(14)
where
$\displaystyle\alpha(f,t,u)$
$\displaystyle=\alpha_{1}(f,t,u)+\alpha_{2}(f,t,u),$ (15)
$\displaystyle\alpha_{1}(f,t,u)$
$\displaystyle=f^{\prime}_{y}(t,X(u))\mu(u,X(u)),$
$\displaystyle\alpha_{2}(f,t,u)$
$\displaystyle=\frac{1}{2}f^{\prime\prime}_{yy}(t,X(u))\sigma^{2}(u,X(u)),$
$\displaystyle\beta(f,t,u)$
$\displaystyle=f^{\prime}_{y}(t,X(u))\sigma(u,X(u)),$
$\displaystyle\gamma(f,t,u)$
$\displaystyle=f(t,X(u-)+\rho(u,X(u-)))-f(t,X(u-)).$
Lemma A.1 states basic estimates for the functions above. For approximating
the solution of SDE (1) we use the randomized Milstein algorithm. For
$n\in{\mathbb{N}}$ we set $\delta=T/n$ and let $t_{i}=i\delta$ for
$i\in\\{0,\ldots,n\\}$. Moreover, we use the notation $\Delta
Y_{i}=Y(t_{i+1})-Y(t_{i})$ for $i\in\\{0,1,\ldots,n-1\\}$, and
$\displaystyle{I_{s,t}(Y,Z)=\int\limits_{s}^{t}\int\limits_{s}^{u-}\mathop{}\\!\mathrm{d}Y(v)\mathop{}\\!\mathrm{d}Z(u)}$
for $Y,Z\in\\{W,N\\}$ and $s,t\in[0,T]$. Note that
$I_{s,t}(N,W)+I_{s,t}(W,N)=(W(t)-W(s))(N(t)-N(s)),$ (16)
and the $\sigma$-fields $\sigma(I_{s,t}(Y,Z))$ and $\mathcal{F}_{s}$ are
independent, cf. [7, Fact B.28 (ii)]. Let $\\{\xi_{i}\\}_{i=0}^{n-1}$ be
independent random variables on the probability space
$(\Omega,\mathcal{F},\mathbb{P})$, such that the $\sigma$-fileds
$\sigma(\xi_{0},\xi_{1},\ldots,\xi_{n-1})$ and $\mathcal{F}_{\infty}$ are
independent, with $\xi_{i}$ being uniformly distributed on $[t_{i},t_{i+1}]$.
Then the randomized Milstein algorithm $X^{(\delta)}$ is defined recursively
through
$\displaystyle X^{(\delta)}(t_{0})$ $\displaystyle=X_{0},$ (17) $\displaystyle
X^{(\delta)}(t_{i+1})$
$\displaystyle=X^{(\delta)}(t_{i})+\mu(\xi_{i},X^{(\delta)}(t_{i}))\delta+\sigma(t_{i},X^{(\delta)}(t_{i}))\Delta
W_{i}+\rho(t_{i},X^{(\delta)}(t_{i}))\Delta N_{i}$
$\displaystyle\quad+L_{1}\sigma(t_{i},X^{(\delta)}(t_{i}))I_{t_{i},t_{i+1}}(W,W)+L_{-1}\rho(t_{i},X^{(\delta)}(t_{i}))I_{t_{i},t_{i+1}}(N,N)$
$\displaystyle\quad+L_{-1}\sigma(t_{i},X^{(\delta)}(t_{i}))I_{t_{i},t_{i+1}}(N,W)+L_{1}\rho(t_{i},X^{(\delta)}(t_{i}))I_{t_{i},t_{i+1}}(W,N),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
i\in\\{0,\ldots,n-1\\}.$
In order to analyse the error of the randomized Milstein algorithm we use the
so-called time-continuous Milstein approximation
$(X^{(\delta)}_{c}(t))_{t\in[0,T]}$, called the randomized Milstein process.
It is defined as follows
$\displaystyle X^{(\delta)}_{c}(t_{0})$ $\displaystyle=X_{0},$ (18)
$\displaystyle X^{(\delta)}_{c}(t)$
$\displaystyle=X^{(\delta)}_{c}(t_{i})+\mu(\xi_{i},X^{(\delta)}_{c}(t_{i}))(t-t_{i})+\sigma(t_{i},X^{(\delta)}_{c}(t_{i}))(W(t)-W(t_{i}))$
$\displaystyle\quad+\rho(t_{i},X^{(\delta)}_{c}(t_{i}))(N(t)-N(t_{i}))$
$\displaystyle\quad+L_{1}\sigma(t_{i},X^{(\delta)}_{c}(t_{i}))I_{t_{i},t}(W,W)+L_{-1}\rho(t_{i},X^{(\delta)}_{c}(t_{i}))I_{t_{i},t}(N,N)$
$\displaystyle\quad+L_{-1}\sigma(t_{i},X^{(\delta)}_{c}(t_{i}))I_{t_{i},t}(N,W)+L_{1}\rho(t_{i},X^{(\delta)}_{c}(t_{i}))I_{t_{i},t}(W,N),$
for $t\in(t_{i},t_{i+1}]$, $i\in\\{0,\ldots,n-1\\}$.
This implies that for all $i\in\\{0,\ldots,n\\}$,
$X^{(\delta)}(t_{i})=X^{(\delta)}_{c}(t_{i})$. Now, analog to [13], we extend
the filtration $(\mathcal{F}_{t})_{t\geq 0}$ in the following way: we take
$\mathcal{\bar{F}}^{n}_{t}=\sigma(\mathcal{F}_{t}\cup\mathcal{G}^{n})$, where
$\mathcal{G}^{n}=\sigma(\xi_{0},\ldots,\xi_{n-1})$. Since $\mathcal{G}^{n}$
and $\mathcal{F}_{\infty}$ are independent, $W$ and $N$ are still Wiener and
Poisson processes with respect to $(\mathcal{\bar{F}}^{n}_{t})_{t\geq 0}$,
respectively. Since in the paper we are integrating
* •
$(\mathcal{\bar{F}}^{n}_{t})_{t\geq 0}$-progressively measurable processes
with respect to the continuous $(\mathcal{\bar{F}}^{n}_{t})_{t\geq 0}$-semi-
martingales $(t)_{t\in[0,T]}$, $(W(t))_{t\in[0,T]}$,
* •
$(\mathcal{\bar{F}}^{n}_{t})_{t\geq 0}$-adapted càglàd processes with respect
to the càdlàg $(\mathcal{\bar{F}}^{n}_{t})_{t\geq
0}$-semimartingale$(N(t))_{t\in[0,T]}$,
the (stochastic) integrals are well-defined, see, for example, [16]. Moreover,
the randomized Milstein process is $(\mathcal{\bar{F}}^{n}_{t})_{t\geq
0}$-progressively measurable, since it is càdlàg and adapted.
Note that the randomised Milstein process is not an implementable algorithm
since it uses all values of $W$ and $N$ and these are not accessible. However
we will use it as an auxiliary scheme for our proof that the randomized
Milstein algorithm has convergence orders
$\delta^{\min\\{\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{3}\\}}$.
## 3 Error analysis for the randomized Milstein process
Let for all $i\in\\{1,\ldots,n\\}$,
$U_{i}=(t_{i},X^{(\delta)}(t_{i})),\quad V_{i}=(\xi_{i},X^{(\delta)}(t_{i})).$
(19)
The processes $X$ and $X^{(\delta)}_{c}$ can be written for all $t\in[0,T]$ as
$\displaystyle X(t)=X(0)+A(t)+B(t)+C(t),$ (20) $\displaystyle
X^{(\delta)}_{c}(t)=X(0)+A^{(\delta)}(t)+B^{(\delta)}(t)+C^{(\delta)}(t),$
(21)
where
$\displaystyle A(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\mu(s,X(s))\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
(22) $\displaystyle B(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\sigma(s,X(s))\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s),$
$\displaystyle C(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\rho(s,X(s-))\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s),$
$\displaystyle A^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\mu(V_{i})\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
(23) $\displaystyle B^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Big{(}\sigma(U_{i})+\int\limits_{t_{i}}^{s}L_{1}\sigma(U_{i})\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s}L_{-1}\sigma(U_{i})\mathop{}\\!\mathrm{d}N(u)\Big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s),$
$\displaystyle C^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Big{(}\rho(U_{i})+\int\limits_{t_{i}}^{s}L_{1}\rho(U_{i})\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s-}L_{-1}\rho(U_{i})\mathop{}\\!\mathrm{d}N(u)\Big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s).$
###### Lemma 3.1.
Under the Assumption 2.1 it holds that there exists a constant
$K_{8}\in(0,\infty)$ such that for all $n\in{\mathbb{N}}$ it holds that
$\displaystyle\sup_{0\leq t\leq
T}{\mathbb{E}}\big{[}|X^{(\delta)}_{c}(t)|^{p}\big{]}\leq K_{8}.$ (24)
###### Proof.
By induction and the fact that $\mathbb{E}[|X_{0}|^{p}]<\infty$ we get that
$\max\limits_{0\leq i\leq
n}{\mathbb{E}}\big{[}|X^{(\delta)}(t_{i})|^{p}\big{]}<\infty.$ (25)
Moreover, by (25) and (18) we obtain that for all $n\in{\mathbb{N}}$ there
exists a constant $c_{1}\in(0,\infty)$ such that
$\sup\limits_{0\leq t\leq
T}\mathbb{E}\big{[}|X^{\delta}_{c}(t)|^{p}\big{]}\leq
c_{1}(1+\max\limits_{0\leq i\leq
n-1}\mathbb{E}\big{[}|X^{(\delta)}(t_{i})|^{p}\big{]})<\infty.$ (26)
Now, we denote for all $t\in[0,T]$,
$X_{c}^{(\delta)}(t)=X(0)+\int\limits_{0}^{t}\Psi_{1,n}(s)\mathop{}\\!\mathrm{d}s+\int\limits_{0}^{t}\Psi_{2,n}(s)\mathop{}\\!\mathrm{d}W(s)+\int\limits_{0}^{t}\Psi_{3,n}(s)\mathop{}\\!\mathrm{d}N(s),$
(27)
where
$\Psi_{1,n}(s)=\sum_{i=0}^{n-1}\mu(V_{i})\mathds{1}_{(t_{i},t_{i+1}]}(s),$
(28)
$\Psi_{2,n}(s)=\sum_{i=0}^{n-1}\Bigg{(}\sigma(U_{i})+\int\limits_{t_{i}}^{s}L_{1}\sigma(U_{i})\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s}L_{-1}\sigma(U_{i})\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s),$
(29)
$\Psi_{3,n}(s)=\sum_{i=0}^{n-1}\Bigg{(}\rho(U_{i})+\int\limits_{t_{i}}^{s}L_{1}\rho(U_{i})\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s-}L_{-1}\rho(U_{i})\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s).$
(30)
By Lemma A.6 we have for all $(k,Z)\in\\{(1,s),(2,W),(3,N)\\}$ that
$\mathbb{E}\Bigg{[}\Biggl{|}\int\limits_{0}^{t}\Psi_{k,n}(s)dZ(s)\Biggl{|}^{p}\Bigg{]}\leq\hat{c}\int\limits_{0}^{t}\mathbb{E}\big{[}|\Psi_{k,n}(s)|^{p}\big{]}ds.$
(31)
From (8) we get that there exist constants $c_{2},c_{3}\in(0,\infty)$ such
that
$\mathbb{E}\big{[}|\Psi_{1,n}(s)|^{p}\big{]}\leq
K_{4}^{p}\sum\limits_{i=0}^{n-1}\mathbb{E}\big{[}(1+|X^{(\delta)}(t_{i})|)^{p}\big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\leq
c_{2}+c_{3}\sum\limits_{i=0}^{n-1}\mathbb{E}\big{[}|X^{(\delta)}(t_{i})|^{p}\big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s).$
(32)
By (8), (11), and Lemma A.6 we obtain that there exist constants
$c_{4},c_{5}\in(0,\infty)$ such that for all
$(k,f)\in\\{(2,\sigma),(3,\rho)\\}$ it holds that
$\displaystyle\int\limits_{0}^{t}\mathbb{E}\big{[}|\Psi_{k,n}(s)|^{p}\big{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\hat{c}\,{\mathbb{E}}\Bigg{[}\sum_{i=0}^{n-1}\int\limits_{0}^{t}|f(U_{i})|^{p}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{]}$
(33)
$\displaystyle\quad+\hat{c}\,{\mathbb{E}}\Bigg{[}\sum_{i=0}^{n-1}\int\limits_{0}^{t}\Bigg{|}\int\limits_{t_{i}}^{s}L_{1}f(U_{i})\mathop{}\\!\mathrm{d}W(u)\Bigg{|}^{p}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{]}$
$\displaystyle\quad+\hat{c}\,{\mathbb{E}}\Bigg{[}\sum_{i=0}^{n-1}\int\limits_{0}^{t}\Bigg{|}\int\limits_{t_{i}}^{s}L_{-1}f(U_{i})\mathop{}\\!\mathrm{d}N(u)\Bigg{|}^{p}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{]}$
$\displaystyle\leq
c_{4}+c_{5}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\big{[}|X^{(\delta)}(t_{i})|^{p}\big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
Here we used that
$\displaystyle{\int\limits_{t_{i}}^{s}L_{-1}f(U_{i})\mathop{}\\!\mathrm{d}N(u)}$
and
$\displaystyle{\int\limits_{t_{i}}^{s-}L_{-1}f(U_{i})\mathop{}\\!\mathrm{d}N(u)}$
differ only at finitely many points. Combining (27), (32), and (33) we obtain
that there exist constants $c_{6},c_{7},c_{8}\in(0,\infty)$ such that
$\displaystyle{\mathbb{E}}\Big{[}\big{|}X_{c}^{(\delta)}(t)\big{|}^{p}\Big{]}\leq
c_{6}\Bigg{(}{\mathbb{E}}\Big{[}|X(0)|^{p}\Big{]}+\sum_{k=1}^{3}\int\limits_{0}^{t}{\mathbb{E}}\Big{[}|\Psi_{k,n}(s)|^{p}\Big{]}\mathop{}\\!\mathrm{d}s\Bigg{)}$
(34) $\displaystyle\leq
c_{7}\Big{(}{\mathbb{E}}\Big{[}|X(0)|^{p}\Big{]}+1\Big{)}+c_{8}\int\limits_{0}^{t}\sup_{0\leq
u\leq
s}{\mathbb{E}}\Big{[}\big{|}X_{c}^{(\delta)}(u)\big{|}^{p}\Big{]}\mathop{}\\!\mathrm{d}s.$
Hence,
$\displaystyle\sup_{0\leq s\leq
t}{\mathbb{E}}\Big{[}\big{|}X_{c}^{(\delta)}(s)\big{|}^{p}\Big{]}\leq
c_{7}\Big{(}{\mathbb{E}}\Big{[}|X(0)|^{p}\Big{]}+1\Big{)}+c_{8}\int\limits_{0}^{t}\sup_{0\leq
u\leq
s}{\mathbb{E}}\Big{[}\big{|}X_{c}^{(\delta)}(u)\big{|}^{p}\Big{]}\mathop{}\\!\mathrm{d}s.$
(35)
The mapping $t\mapsto\sup_{0\leq s\leq
t}{\mathbb{E}}\Big{[}\big{|}X_{c}^{(\delta)}(s)\big{|}^{p}\Big{]}$ is monotone
and hence Borel measurable. Moreover, by (26) it is bounded. Hence, applying
Gronwall’s lemma proves the claim. ∎
Next we prove the convergence rate of the randomized Milstein algorithm.
###### Theorem 3.2.
Let Assumption 2.1 hold. Then there exists $C\in(0,\infty)$ such that for all
$n\in{\mathbb{N}}$ it holds that
$\displaystyle\sup_{0\leq t\leq
T}\|X(t)-X^{(\delta)}_{c}(t)\|_{L^{p}(\Omega)}\leq
C\delta^{\min\\{\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{3}\\}}.$
(36)
###### Proof.
For all $t\in[0,T]$ it holds that
$\displaystyle
X(t)-X^{(\delta)}_{c}(t)=\big{(}A(t)-A^{(\delta)}(t)\big{)}+\big{(}B(t)-B^{(\delta)}(t)\big{)}+\big{(}C(t)-C^{(\delta)}(t)\big{)}.$
(37)
We first rewrite each summand of the right hand side of equation (37). We
obtain
$\displaystyle A(t)-A^{(\delta)}(t)$
$\displaystyle=\tilde{A}^{(\delta)}_{1}(t)+\tilde{A}^{(\delta)}_{2}(t)+\tilde{A}^{(\delta)}_{3}(t),$
(38)
where
$\displaystyle\tilde{A}^{(\delta)}_{1}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\mu(s,X(s))-\mu(s,X(t_{i}))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
(39) $\displaystyle\tilde{A}^{(\delta)}_{2}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\mu(s,X(t_{i}))-\mu(\xi_{i},X(t_{i}))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
$\displaystyle\tilde{A}^{(\delta)}_{3}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\mu(\xi_{i},X(t_{i}))-\mu(\xi_{i},X^{(\delta)}_{c}(t_{i}))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
For $\tilde{A}^{(\delta)}_{1}(t)$ we apply the parametric version of the
Meyer-Itô formula (14), that is
$\mu(s,X(s))-\mu(s,X(t_{i}))=\int\limits_{t_{i}}^{s}\alpha(\mu,s,u)\mathop{}\\!\mathrm{d}u+\int\limits_{t_{i}}^{s}\beta(\mu,s,u)\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s}\gamma(\mu,s,u)\mathop{}\\!\mathrm{d}N(u).$
(40)
Hence,
$\tilde{A}_{1}^{(\delta)}(t)=\sum\limits_{j=1}^{3}\tilde{M}_{j}^{(\delta)}(t),$
(41)
where
$\displaystyle\tilde{M}_{1}^{(\delta)}(t)=\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\alpha(\mu,s,u)\mathop{}\\!\mathrm{d}u\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
(42)
$\displaystyle\tilde{M}_{2}^{(\delta)}(t)=\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\beta(\mu,s,u)\mathop{}\\!\mathrm{d}W(u)\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
$\displaystyle\tilde{M}_{3}^{(\delta)}(t)=\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s}\gamma(\mu,s,u)\mathop{}\\!\mathrm{d}N(u)\Biggr{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
Next we obtain for the second summand of (37),
$\displaystyle B(t)-B^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\sigma(s,X(s))-\sigma(t_{i},X(s))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s)$
(43)
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Bigg{(}\sigma(t_{i},X(s))-\sigma(t_{i},X(t_{i}))$
$\displaystyle\quad\quad\quad\quad\quad\quad-\int\limits_{t_{i}}^{s}L_{1}\sigma(U_{i})\mathop{}\\!\mathrm{d}W(u)-\int\limits_{t_{i}}^{s}L_{-1}\sigma(U_{i})\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\sigma(t_{i},X(t_{i}))-\sigma(U_{i})\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s).$
Again we apply the parametric version of the Meyer-Itô formula (14) and obtain
$\displaystyle\sigma(t_{i},X(s))-\sigma(t_{i},X(t_{i}))-\int\limits_{t_{i}}^{s}L_{1}\sigma(U_{i})\mathop{}\\!\mathrm{d}W(u)-\int\limits_{t_{i}}^{s}L_{-1}\sigma(U_{i})\mathop{}\\!\mathrm{d}N(u)$
(44)
$\displaystyle=\int\limits_{t_{i}}^{s}\alpha(\sigma,t_{i},u)\mathop{}\\!\mathrm{d}u+\int\limits_{t_{i}}^{s}\Bigl{(}\beta(\sigma,t_{i},u)-L_{1}\sigma(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s}\Bigl{(}\gamma(\sigma,t_{i},u)-L_{-1}\sigma(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u).$
Hence,
$\displaystyle B(t)-B^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Bigg{(}\int\limits\limits_{t_{i}}^{s}\alpha(\sigma,t_{i},u)\mathop{}\\!\mathrm{d}u\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s)$
(45)
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Biggl{(}\int\limits\limits_{t_{i}}^{s}\Bigl{(}\beta(\sigma,t_{i},u)-L_{1}\sigma(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s}\Bigl{(}\gamma(\sigma,t_{i},u)-L_{-1}\sigma(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\sigma(s,X(s))-\sigma(t_{i},X(s))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\sigma(t_{i},X(t_{i}))-\sigma(U_{i})\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}W(s).$
For the third summand of (37) we get
$\displaystyle C(t)-C^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\rho(s,X(s-))-\rho(t_{i},X(s-))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s)$
(46)
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Bigg{(}\rho(t_{i},X(s-))-\rho(t_{i},X(t_{i}))$
$\displaystyle\quad\quad\quad\quad\quad\quad-\int\limits_{t_{i}}^{s}L_{1}\rho(U_{i})\mathop{}\\!\mathrm{d}W(u)-\int\limits_{t_{i}}^{s-}L_{-1}\sigma(U_{i})\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\rho(t_{i},X(t_{i}))-\rho(U_{i})\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s).$
Again we apply the parametric version of the Meyer-Itô formula (14) to obtain
$\displaystyle\rho(t_{i},X(s-))-\rho(t_{i},X(t_{i}))-\int\limits_{t_{i}}^{s}L_{1}\rho(U_{i})\mathop{}\\!\mathrm{d}W(u)-\int\limits_{t_{i}}^{s-}L_{-1}\rho(U_{i})\mathop{}\\!\mathrm{d}N(u)$
(47)
$\displaystyle=\int\limits_{t_{i}}^{s}\alpha(\rho,t_{i},u)\mathop{}\\!\mathrm{d}u+\int\limits_{t_{i}}^{s}\Bigl{(}\beta(\rho,t_{i},u)-L_{1}\rho(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)+\int\limits_{t_{i}}^{s-}\Bigl{(}\gamma(\rho,t_{i},u)-L_{-1}\rho(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u);$
due to the continuity of the processes
$\int\limits_{t_{i}}^{s}\alpha(\rho,t_{i},u)\mathop{}\\!\mathrm{d}u=\int\limits_{t_{i}}^{s-}\alpha(\rho,t_{i},u)\mathop{}\\!\mathrm{d}u,\quad\int\limits_{t_{i}}^{s}\beta(\rho,t_{i},u)\mathop{}\\!\mathrm{d}W(u)=\int\limits_{t_{i}}^{s-}\beta(\rho,t_{i},u)\mathop{}\\!\mathrm{d}W(u).$
(48)
Therefore, for all $t\in[0,T]$,
$\displaystyle C(t)-C^{(\delta)}(t)$
$\displaystyle=\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\alpha(\rho,t_{i},u)\mathop{}\\!\mathrm{d}u\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s)$
(49)
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s}\Bigl{(}\beta(\rho,t_{i},u)-L_{1}\rho(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s-}\Bigl{(}\gamma(\rho,t_{i},u)-L_{-1}\rho(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\rho(s,X(s-))-\rho(t_{i},X(s-))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s)$
$\displaystyle\quad+\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}\rho(t_{i},X(t_{i}))-\rho(U_{i})\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}N(s).$
Now we estimate all terms in (38), (41), (45), and (49). We apply Lemma A.6
and Assumption 2.1 (i) for
$(f,v,Z)\in\\{(\mu,\xi_{i},\operatorname{Id}),(\sigma,t_{i},W),(\rho,t_{i},N)\\}$
. This shows that there exists a constant $c_{1}\in(0,\infty)$ such that
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{(}f(v,X(t_{i}))-f(v,X^{(\delta)}_{c}(t_{i}))\big{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(50)
$\displaystyle\leq\hat{c}\,{\mathbb{E}}\Bigg{[}\int\limits_{0}^{t}\sum_{i=0}^{n-1}\big{|}f(v,X(t_{i}))-f(v,X^{(\delta)}_{c}(t_{i}))\big{|}^{p}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{]}$
$\displaystyle\leq
c_{1}\int\limits_{0}^{t}\sum_{i=0}^{n-1}{\mathbb{E}}\Big{[}\big{|}X(t_{i})-X^{(\delta)}_{c}(t_{i})\big{|}^{p}\Big{]}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
Moreover, we obtain that there exists $c_{2}\in(0,\infty)$ such that for
$(f,Z)\in\\{(\sigma,W),(\rho,N)\\}$ and for all $t\in[0,T]$ it holds that
$\displaystyle\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\Bigl{(}f(s,X(s-))-f(t_{i},X(s-))\Bigr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(51)
$\displaystyle\leq\hat{c}\sum\limits_{i=0}^{n-1}\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{t_{i+1}}|f(s,X(s-))-f(t_{i},X(s-))|^{p}\mathop{}\\!\mathrm{d}s\Bigg{]}$
$\displaystyle\leq\hat{c}K_{1}^{p}\sum\limits_{i=0}^{n-1}\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{t_{i+1}}(1+|X(s-)|)^{p}\cdot(s-t_{i})^{p\varrho_{f}}\mathop{}\\!\mathrm{d}s\Bigg{]}$
$\displaystyle\leq\hat{c}K_{1}^{p}\delta^{p\varrho_{f}}\sum\limits_{i=0}^{n-1}\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{t_{i+1}}(1+|X(s)|)^{p}\mathop{}\\!\mathrm{d}s\Bigg{]}\leq
2^{p-1}\hat{c}K_{1}^{p}\delta^{p\varrho_{f}}\Bigl{(}1+\mathbb{E}\Big{[}\sup\limits_{0\leq
t\leq T}|X(t)|^{p}\Big{]}\Bigr{)}\leq c_{2}\delta^{p\varrho_{f}}.$
By Lemma A.6 we get that there exist constants $c_{3},c_{4}\in(0,\infty)$ such
that for
$(f,v,Z)\in\\{(\mu,s,\operatorname{Id}),(\sigma,t_{i},W),(\rho,t_{i},N)\\}$
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\alpha(f,v,u)\mathop{}\\!\mathrm{d}u\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(52) $\displaystyle\leq
c_{3}\,\sum_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}|\alpha_{1}(f,v,u)|\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}\mathop{}\\!\mathrm{d}s+c_{4}\,\sum_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}|\alpha_{2}(f,v,u)|\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}\mathop{}\\!\mathrm{d}s.$
Next we estimate the expectations in equation (LABEL:Meq11) separately. For
the first term we use (8), (9), and (12) to obtain that there exists a
constant $c_{5}\in(0,\infty)$ such that for all $s\in[t_{i},t_{i+1}]$,
$v\in\\{s,t_{i}\\}$,
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}|\alpha_{1}(f,v,u)|\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}={\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}\big{|}f^{\prime}_{y}(v,X(u))\big{|}\cdot\big{|}\mu(u,X(u))\big{|}\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}$
(53)
$\displaystyle\leq(K_{1}K_{4})^{p}\,{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}\big{(}1+|X(u)|\big{)}\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}\leq(K_{1}K_{4})^{p}\delta^{p}\,{\mathbb{E}}\Big{[}\big{(}1+\sup_{0\leq
t\leq T}|X(t)|\big{)}^{p}\Big{]}\leq c_{5}\,\delta^{p}.$
For the second term we obtain that there exist constants
$c_{6},c_{7},c_{8}\in(0,\infty)$ such that for all $s\in[t_{i},t_{i+1}]$,
$v\in\\{s,t_{i}\\}$,
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}|\alpha_{2}(f,v,u)|\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}={\mathbb{E}}\Bigg{[}\Bigg{(}\frac{1}{2}\int\limits_{t_{i}}^{s}\big{|}f^{\prime\prime}_{yy}(v,X(u))\big{|}\cdot\big{|}\sigma^{2}(u,X(u))\big{|}\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}$
(54)
$\displaystyle\leq\Big{(}\frac{K_{1}}{2}\Big{)}^{p}{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{s}\sigma^{2}(u,X(u))\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}\leq
c_{6}\,{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{i}}^{t_{i+1}}\big{(}1+|X(u)|^{2}\big{)}\mathop{}\\!\mathrm{d}u\Bigg{)}^{p}\Bigg{]}$
$\displaystyle\leq c_{7}\,\delta^{p}\,{\mathbb{E}}\Big{[}\big{(}1+\sup_{0\leq
t\leq T}|X(t)|^{2}\big{)}^{p}\Big{]}\leq c_{8}\,\delta^{p},$
since $|f_{yy}^{\prime\prime}(t,y)|\leq K_{1}$ for all $t\in[0,T]$ and
$y\in{\mathbb{R}}$. Hence, combining equations (LABEL:Meq11), (LABEL:Meq12),
and (LABEL:Meq13) we obtain that there exists a constant $c_{9}\in(0,\infty)$
such that
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\alpha(f,v,u)\mathop{}\\!\mathrm{d}u\Bigg{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}\leq
c_{9}\,\delta^{p}.$ (55)
For $(f,Z)\in\\{(\sigma,W),(\rho,N)\\}$ and $t\in[0,T]$ we get
$\displaystyle\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s}\Bigl{(}\beta(f,t_{i},u)-L_{1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(56)
$\displaystyle\leq\hat{c}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{t_{i}}^{s}\Bigl{(}\beta(f,t_{i},u)-L_{1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)\Bigg{|}^{p}\Bigg{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
Further, there exists a constant $c_{10}\in(0,\infty)$ such that for all
$s\in[t_{i},t_{i+1}]$,
$\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{t_{i}}^{s}\Bigl{(}\beta(f,t_{i},u)-L_{1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)\Bigg{|}^{p}\Bigg{]}\leq
c_{10}(s-t_{i})^{\frac{p}{2}-1}\cdot\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{s}\big{|}\beta(f,t_{i},u)-L_{1}f(U_{i})\big{|}^{p}\mathop{}\\!\mathrm{d}s\Bigg{]}.$
(57)
Moreover, for $u\in[t_{i},t_{i+1}]$,
$\displaystyle|\beta(f,t_{i},u)-L_{1}f(U_{i})|\leq|\beta(f,t_{i},u)-L_{1}f(t_{i},X(u))|+|L_{1}f(t_{i},X(u))-L_{1}f(t_{i},X^{(\delta)}(t_{i}))|$
(58) $\displaystyle\leq
K_{3}|X(u)-X(t_{i})|+K_{3}|X(t_{i})-X^{(\delta)}(t_{i})|+K_{1}^{2}(1+|X(u)|)\cdot|u-t_{i}|^{\varrho_{2}},$
and by (12), (13) we have that there exist constants
$c_{11},c_{12},c_{13}\in(0,\infty)$ such that
$\mathbb{E}\big{[}|\beta(f,t_{i},u)-L_{1}f(U_{i})|^{p}\Big{]}\leq
c_{11}(u-t_{i})+c_{12}(u-t_{i})^{p\varrho_{2}}+c_{13}\mathbb{E}\big{[}|X(t_{i})-X^{(\delta)}(t_{i})|^{p}\big{]}.$
(59)
Hence, there exist constants $c_{14},c_{15},c_{16}\in(0,\infty)$ such that
$\displaystyle\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s}\Bigl{(}\beta(f,t_{i},u)-L_{1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}W(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(60) $\displaystyle\leq
c_{14}\delta^{\frac{p}{2}+1}+c_{15}\delta^{p(\varrho_{2}+\frac{1}{2})}+c_{16}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Big{[}\big{|}X(t_{i})-X^{(\delta)}(t_{i})\big{|}^{p}\Big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
Moreover, for $(f,Z)\in\\{(\sigma,W),(\rho,N)\\}$ and $t\in[0,T]$ we obtain
$\displaystyle\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s-}\Bigl{(}\gamma(f,t_{i},u)-L_{-1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(61)
$\displaystyle\leq\hat{c}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{t_{i}}^{s}\Bigl{(}\gamma(f,t_{i},u)-L_{-1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u)\Bigg{|}^{p}\Bigg{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
where
$\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{t_{i}}^{s}\Bigl{(}\gamma(f,t_{i},u)-L_{-1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u)\Bigg{|}^{p}\Bigg{]}\leq\hat{c}\,\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{s}|\gamma(f,t_{i},u)-L_{-1}f(U_{i})|^{p}\mathop{}\\!\mathrm{d}s\Bigg{]}.$
(62)
Further, there exist constants $c_{17},c_{18},c_{19}\in(0,\infty)$ such that
for all $s\in[t_{i},t_{i+1}]$,
$\displaystyle\int\limits_{t_{i}}^{s}|\gamma(f,t_{i},u)-L_{-1}f(U_{i})|^{p}\mathop{}\\!\mathrm{d}u$
(63) $\displaystyle\leq
c_{17}\int\limits_{t_{i}}^{s}|X(u)-X(t_{i})|^{p}\mathop{}\\!\mathrm{d}u+c_{18}\delta|X(t_{i})-X^{(\delta)}(t_{i})|^{p}+c_{19}\int\limits_{t_{i}}^{s}(1+|X(u)|)^{p}\cdot|u-t_{i}|^{p\varrho_{3}}\mathop{}\\!\mathrm{d}u,$
hence, by (12) and (13) there exist constants
$c_{20},c_{21},c_{22}\in(0,\infty)$ such that
$\displaystyle\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t}\sum_{i=0}^{n-1}\Biggl{(}\int\limits_{t_{i}}^{s-}\Bigl{(}\gamma(f,t_{i},u)-L_{-1}f(U_{i})\Bigr{)}\mathop{}\\!\mathrm{d}N(u)\Biggr{)}\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}Z(s)\Bigg{|}^{p}\Bigg{]}$
(64) $\displaystyle\leq
c_{20}\delta^{2}+c_{21}\delta^{p(\varrho_{3}+\frac{1}{p})}+c_{22}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Big{[}|X(t_{i})-X^{(\delta)}(t_{i})|^{p}\Big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
With this preliminary calculations we can estimate the error of the randomized
Milstein algorithm as follows. It holds that there exists a constant
$c_{23}\in(0,\infty)$ such that
$\displaystyle{\mathbb{E}}\Big{[}|X(t)-X^{(\delta)}_{c}(t)|^{p}\Big{]}$ (65)
$\displaystyle\leq
c_{23}\Big{(}{\mathbb{E}}\Big{[}\big{|}A(t)-A^{(\delta)}(t)\big{|}^{p}\Big{]}+{\mathbb{E}}\Big{[}\big{|}B(t)-B^{(\delta)}(t)\big{|}^{p}\Big{]}+{\mathbb{E}}\Big{[}\big{|}C(t)-C^{(\delta)}(t)\big{|}^{p}\Big{]}\Big{)}.$
Combining (45) resp. (49) with (LABEL:Meq10), (LABEL:Meq10a), (LABEL:Meq13a),
(LABEL:Meq13b), and (LABEL:Meq13c), we obtain that there exist constants
$c_{24},c_{25},c_{26},c_{27}\in(0,\infty)$ such that for all $t\in[0,T]$,
$\mathbb{E}\Big{[}\big{|}B(t)-B^{(\delta)}(t)\big{|}^{p}\Big{]}\leq
c_{24}\delta^{p\min\\{\frac{2}{p},\varrho_{2},\varrho_{3}+\frac{1}{p}\\}}+c_{25}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Big{[}\big{|}X(t_{i})-X^{(\delta)}(t_{i})\big{|}^{p}\Big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s,$
(66) $\mathbb{E}\Big{[}\big{|}C(t)-C^{(\delta)}(t)\big{|}^{p}\Big{]}\leq
c_{26}\delta^{p\min\\{\frac{2}{p},\varrho_{3},\varrho_{2}+\frac{1}{2}\\}}+c_{27}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Big{[}\big{|}X(t_{i})-X^{(\delta)}(t_{i})\big{|}^{p}\Big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
(67)
Next we estimate the remaining terms in (38) and (41). The estimation of
$\mathbb{E}[|\tilde{M}_{1}^{(\delta)}(t)|^{p}]$ is already included in
(LABEL:Meq13a). Analog to the steps in [13, pages 8–10] and by applying Lemma
A.1 we obtain that there exists a constant $c_{28}\in(0,\infty)$ such that for
all $t\in[0,T]$,
$\mathbb{E}\Big{[}\big{|}\tilde{M}^{(\delta)}_{2}(t)\big{|}^{p}\Big{]}\leq
c_{28}\delta^{p\min\\{\frac{1}{2}+\varrho_{1},1\\}}.$ (68)
We now show the upper bound for
$\mathbb{E}[|\tilde{M}^{(\delta)}_{3}(t)|^{p}]$. There exists a constant
$c_{29}\in(0,\infty)$ such that for all $t\in[0,T]$ there exists
$\ell\in\\{0,1,\ldots,n-1\\}$ with $t\in[t_{\ell},t_{\ell+1}]$ and
$\displaystyle\mathbb{E}\Big{[}\big{|}\tilde{M}^{(\delta)}_{3}(t)\big{|}^{p}\Big{]}\leq
c_{29}\Bigg{(}\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t_{\ell}}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}(\gamma(\mu,s,u)-\gamma(\mu,t_{i},u))\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}$
(69)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t_{\ell}}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\gamma(\mu,t_{i},u)\mathop{}\\!\mathrm{d}\tilde{N}(u)\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\lambda^{p}\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t_{\ell}}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\gamma(\mu,t_{i},u)\mathop{}\\!\mathrm{d}u\bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad+\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{t_{\ell}}^{t}\Bigg{(}\int\limits_{t_{\ell}}^{s}\gamma(\mu,s,u)\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}\Big{)}.$
By Lemmas A.6 and A.1 we obtain that there exists a constant
$c_{30}\in(0,\infty)$ such that
$\displaystyle\mathbb{E}\Bigg{[}\Bigg{|}\int\limits_{0}^{t_{\ell}}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}(\gamma(\mu,s,u)-\gamma(\mu,t_{i},u))\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}$
(70)
$\displaystyle\leq\hat{c}\sum\limits_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{s}|\gamma(\mu,s,u)-\gamma(\mu,t_{i},u)|^{p}\mathop{}\\!\mathrm{d}u\Bigg{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\hat{c}K_{7}\sum\limits_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}\mathbb{E}\Bigg{[}\int\limits_{t_{i}}^{s}(1+|X(u-)|)^{p}\cdot(s-t_{i})^{p\varrho_{1}}\mathop{}\\!\mathrm{d}u\Bigg{]}\mathop{}\\!\mathrm{d}s\leq
c_{30}\delta^{p(\varrho_{1}+\frac{1}{p})}.$
Similar as above, by the Hölder inequality, (12), and Lemma A.1 we obtain that
there exists a constant $c_{31}\in(0,\infty)$ such that for all
$t\in[t_{\ell},t_{\ell+1}]$,
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{|}\int\limits_{t_{\ell}}^{t}\Bigg{(}\int\limits_{t_{\ell}}^{s}\gamma(\mu,s,u)\mathop{}\\!\mathrm{d}N(u)\Bigg{)}\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}\leq{\mathbb{E}}\Bigg{[}\Bigg{(}\int\limits_{t_{\ell}}^{t}\Bigg{|}\int\limits_{t_{\ell}}^{s}\gamma(\mu,s,u)\mathop{}\\!\mathrm{d}N(u)\Bigg{|}\mathop{}\\!\mathrm{d}s\Bigg{)}^{p}\Bigg{]}$
(71)
$\displaystyle\leq\delta^{p-1}\,\int\limits_{t_{\ell}}^{t_{\ell+1}}{\mathbb{E}}\Bigg{[}\Bigg{|}\int\limits_{t_{\ell}}^{s}\gamma(\mu,s,u)\mathop{}\\!\mathrm{d}N(u)\Bigg{|}^{p}\Bigg{]}\mathop{}\\!\mathrm{d}s\leq\hat{c}\delta^{p-1}\,\int\limits_{t_{\ell}}^{t_{\ell+1}}{\mathbb{E}}\Bigg{[}\int\limits_{t_{\ell}}^{s}|\gamma(\mu,s,u)|^{p}\mathop{}\\!\mathrm{d}u\Bigg{]}\mathop{}\\!\mathrm{d}s\leq
c_{31}\delta^{p+1}.$
Further we obtain that there exists a constant $c_{32}\in(0,\infty)$ such that
$\displaystyle{\mathbb{E}}\Bigg{[}\Bigg{|}\int\limits_{0}^{t_{\ell}}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\gamma(\mu,t_{i},u)\mathop{}\\!\mathrm{d}u\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}$
(72) $\displaystyle\leq
T^{p-1}\sum\limits_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\Biggl{[}\Biggl{|}\int\limits_{t_{i}}^{s}\gamma(\mu,t_{i},u)\mathop{}\\!\mathrm{d}u\Biggl{|}^{p}\Biggr{]}\mathop{}\\!\mathrm{d}s\leq
T^{p-1}\delta^{p-1}\sum\limits_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\Biggl{[}\int\limits_{t_{i}}^{s}|\gamma(\mu,t_{i},u)|^{p}\mathop{}\\!\mathrm{d}u\Biggr{]}\mathop{}\\!\mathrm{d}s\leq
c_{32}\delta^{p}.$
Moreover, we have
$\mathbb{E}\Bigg{[}\Biggl{|}\int\limits_{0}^{t_{\ell}}\sum\limits_{i=0}^{n-1}\Bigg{(}\int\limits_{t_{i}}^{s}\gamma(\mu,t_{i},u)\mathop{}\\!\mathrm{d}\tilde{N}(u)\Bigg{)}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s\Bigg{|}^{p}\Bigg{]}={\mathbb{E}}\Big{[}\big{|}\tilde{Z}_{\ell-1}\big{|}^{p}\Big{]},$
(73)
where
$\tilde{Z}_{k}=\sum\limits_{i=0}^{k}\tilde{Y}_{i},\quad
k\in\\{0,1,\ldots,n-1\\},$ (74)
with $Z_{-1}=0$, and
$\tilde{Y}_{i}=\int\limits_{t_{i}}^{t_{i+1}}\Bigl{(}\int\limits_{t_{i}}^{s}\gamma(\mu,t_{i},u)\mathop{}\\!\mathrm{d}\widetilde{N}(u)\Bigr{)}\mathop{}\\!\mathrm{d}s.$
(75)
Let $\mathcal{G}_{k}:=\mathcal{F}_{t_{k+1}}$. Then it holds that
$\\{\tilde{Z}_{k},\mathcal{G}_{k}\\}_{k=0,1,\ldots,n-1}$ is a discrete-time
martingale, since $\widetilde{Z}_{k}$ is adapted to $\mathcal{G}_{k}$ for
$k\in\\{0,\ldots,n-1\\}$ and from Fubini’s theorem for conditional
expectations (see [1]) we have
$\displaystyle{\mathbb{E}}\big{[}\widetilde{Z}_{k+1}-\widetilde{Z}_{k}\big{|}\mathcal{G}_{k}\big{]}$
$\displaystyle={\mathbb{E}}\Bigg{[}\int\limits_{t_{k+1}}^{t_{k+2}}\Bigg{(}\int\limits_{t_{k+1}}^{s}\gamma(\mu,t_{k+1},u)\mathop{}\\!\mathrm{d}\widetilde{N}(u)\Bigg{)}\mathop{}\\!\mathrm{d}s\Bigg{|}\mathcal{F}_{t_{k+1}}\Bigg{]}$
(76)
$\displaystyle=\int\limits_{t_{k+1}}^{t_{k+2}}{\mathbb{E}}\Bigg{[}\int\limits_{t_{k+1}}^{s}\gamma(\mu,t_{k+1},u)\mathop{}\\!\mathrm{d}\widetilde{N}(u)\Bigg{|}\mathcal{F}_{t_{k+1}}\Bigg{]}\mathop{}\\!\mathrm{d}s=0.$
Hence, by the discrete version of the Burkholder-Davis-Gundy inequality and
Jensen’s inequality we obtain that there exist constants
$c_{33},c_{34}\in(0,\infty)$ such that
${\mathbb{E}}\Big{[}\big{|}\tilde{Z}_{k}\big{|}^{p}\Big{]}\leq
c_{33}{\mathbb{E}}\Bigl{[}\Bigl{(}\sum\limits_{i=0}^{k}|\tilde{Y}_{i}|^{2}\Bigr{)}^{p/2}\Bigr{]}\leq
c_{33}n^{\frac{p}{2}-1}\sum\limits_{i=0}^{n-1}{\mathbb{E}}\Big{[}\big{|}\tilde{Y}_{i}\big{|}^{p}\Big{]}\leq
c_{34}\delta^{\frac{p}{2}+1},$ (77)
for $k\in\\{0,1,\ldots,n-1\\}$. Combining (LABEL:mainM3p1), (LABEL:mainM3p2),
(LABEL:mainM3p3), (LABEL:mainM3p4), (73), and (77) we get that there exists a
constant $c_{35}\in(0,\infty)$ such that for all $t\in[0,T]$,
${\mathbb{E}}\Big{[}\big{|}\tilde{M}_{3}(t)\big{|}^{p}\Big{]}\leq
c_{35}\delta^{p\min\\{\varrho_{1}+\frac{1}{p},\frac{1}{2}+\frac{1}{p},1\\}}.$
(78)
By (LABEL:Meq13a), (68), and (78) we get that there exists a constant
$c_{36}\in(0,\infty)$ such that
${\mathbb{E}}\Big{[}\big{|}\tilde{A}_{1}^{(\delta)}(t)\big{|}^{p}\Big{]}\leq
c_{36}\delta^{p\min\\{\varrho_{1}+\frac{1}{p},\frac{1}{2}+\frac{1}{p},1\\}}.$
(79)
Analog to the proof of [13, equation 33] we get that there exists a constant
$c_{37}\in(0,\infty)$ such that for all $t\in[0,T]$ it holds that
$\mathbb{E}\Big{[}\big{|}\tilde{A}^{(\delta)}_{2}(t)\big{|}^{p}\Big{]}\leq
c_{37}\delta^{p(\varrho_{1}+\frac{1}{2})}.$ (80)
Moreover, we can estimate $\mathbb{E}[|\tilde{A}_{3}^{(\delta)}(t)|^{p}]$ by
using (LABEL:Meq10). Therefore, by (LABEL:Meq10), (79), and (80) we have that
there exist constants $c_{38},c_{39}\in(0,\infty)$ such that for all
$t\in[0,T]$ it holds that
${\mathbb{E}}\Big{[}\big{|}A(t)-A^{(\delta)}(t)\big{|}^{p}\Big{]}\leq
c_{38}\delta^{p\min\\{\varrho_{1}+\frac{1}{p},\frac{1}{2}+\frac{1}{p},1\\}}+c_{39}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Big{[}\big{|}X(t_{i})-X^{(\delta)}(t_{i})\big{|}^{p}\Big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s.$
(81)
Using (LABEL:part_est_XXd), (66), (67), and (81) we have that there exist
constants $c_{40},c_{41}\in(0,\infty)$ such that for all $t\in[0,T]$ it holds
that
$\displaystyle\mathbb{E}\Big{[}\big{|}X(t)-X^{(\delta)}_{c}(t)\big{|}^{p}\Big{]}\leq
c_{40}\delta^{p\min\\{\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{3}\\}}+c_{41}\int\limits_{0}^{t}\sum\limits_{i=0}^{n-1}\mathbb{E}\Big{[}\big{|}X(t_{i})-X^{(\delta)}(t_{i})\big{|}^{p}\Big{]}\cdot\mathds{1}_{(t_{i},t_{i+1}]}(s)\mathop{}\\!\mathrm{d}s$
(82) $\displaystyle\leq
c_{40}\delta^{p\min\\{\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{3}\\}}+c_{41}\int\limits_{0}^{t}\sup\limits_{0\leq
u\leq
s}\mathbb{E}\Big{[}\big{|}X(u)-X^{(\delta)}_{c}(u)\big{|}^{p}\Big{]}\mathop{}\\!\mathrm{d}s,$
and hence,
$\sup\limits_{0\leq u\leq
t}\mathbb{E}\Big{[}\big{|}X(u)-X^{(\delta)}_{c}(u)\big{|}^{p}\Big{]}\leq
c_{40}\delta^{p\min\\{\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{3}\\}}+c_{41}\int\limits_{0}^{t}\sup\limits_{0\leq
u\leq
s}\mathbb{E}\Big{[}\big{|}X(u)-X^{(\delta)}_{c}(u)\big{|}^{p}\Big{]}\mathop{}\\!\mathrm{d}s.$
(83)
The function $\displaystyle{[0,T]\ni t\mapsto\sup\limits_{0\leq u\leq
t}{\mathbb{E}}\Big{[}\big{|}X(u)-X^{(\delta)}_{c}(u)\big{|}^{p}\Big{]}\in[0,\infty)}$
is Borel measurable, since it is non-decreasing and bounded due to (12) and
(24). Therefore, by applying Grownall’s lemma to (83) we obtain that there
exists a constant $C\in(0,\infty)$ such that for all $t\in[0,T]$,
$\displaystyle\sup_{0\leq u\leq
t}{\mathbb{E}}\Big{[}\big{|}X(u)-X^{(\delta)}_{c}(u)\big{|}^{p}\Big{]}\leq
C\delta^{p\min\\{\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{3}\\}}.$
(84)
∎
###### Remark 3.3.
Note that for the classical Milstein scheme $\bar{X}^{(\delta)}$, which can be
seen as a randomized Milstein process with $\xi_{i}$ fixed to $t_{i}$ for all
$i\in\\{0,1,\ldots,n-1\\}$, we have that there exists a constant
$K_{9}\in(0,\infty)$ such that for all $n\in{\mathbb{N}}$,
$\displaystyle\sup_{0\leq t\leq
T}\|X(t)-\bar{X}^{(\delta)}_{c}(t)\|_{L^{p}(\Omega)}\leq
K_{9}\delta^{\min\\{\frac{2}{p},\varrho_{1},\varrho_{2},\varrho_{3}\\}}.$ (85)
This follows from a straightforward modification of the proof of Theorem 3.2.
###### Remark 3.4.
In the jump-free case ($\rho=0$) we get from the proof of Theorem 3.2 that
there exists $K_{10}\in(0,\infty)$ such that for all $n\in{\mathbb{N}}$,
$\displaystyle\sup_{0\leq t\leq
T}\|X(t)-X^{(\delta)}_{c}(t)\|_{L^{p}(\Omega)}\leq
K_{10}\delta^{\min\\{\varrho_{1}+\frac{1}{2},\varrho_{2}\\}}.$ (86)
Hence our result implies the same upper error bound for the randomized
Milstein process as in [13, Proposition 1] but under slightly weaker
assumptions on $\mu$ and $\sigma$. Moreover, for
$\varrho_{2}=\min\\{\frac{1}{2}+\varrho_{1},1\\}$ we recover the upper error
bound from [8], which therein is obtained for a two-stage randomized Milstein
scheme.
## 4 Lower bounds and optimality
In this section we first consider the scalar case under the JCC and then we
study the multidimensional case. In both cases we set $p=2$ and assume
availability only of standard information, given by values of $W$ and $N$ at a
finite number of points. We investigate lower bounds and optimality in the IBC
framework, cf. [24].
### 4.1 Scalar case and optimality of the randomized Milstein algorithm
We investigate lower bound and optimality of the randomized Milstein algorithm
in the case where $p=2$ and when the JCC is satisfied, that is
$L_{-1}\sigma(t,y)=L_{1}\rho(t,y),\ (t,y)\in[0,T]\times\mathbb{R},$ (87)
see [15]. It follows that under condition (87) the randomized Milstein
algorithm uses only standard discrete information about $W,N$, i.e., the
values $W(t_{1}),\ldots,W(t_{n}),N(t_{1}),\ldots,N(t_{n})$, since, by (16) and
(17), in that case it has the form
$\displaystyle X^{(\delta)}(t_{0})=X_{0},$ (88) $\displaystyle
X^{(\delta)}(t_{i+1})=X^{(\delta)}(t_{i})+\mu(\xi_{i},X^{(\delta)}(t_{i}))\delta+\sigma(t_{i},X^{(\delta)}(t_{i}))\Delta
W_{i}+\rho(t_{i},X^{(\delta)}(t_{i}))\Delta N_{i}$
$\displaystyle\quad\quad\quad\quad\quad\quad+L_{1}\sigma(t_{i},X^{(\delta)}(t_{i}))I_{t_{i},t_{i+1}}(W,W)+L_{-1}\rho(t_{i},X^{(\delta)}(t_{i}))I_{t_{i},t_{i+1}}(N,N)$
$\displaystyle\quad\quad\quad\quad\quad\quad+L_{-1}\sigma(t_{i},X^{(\delta)}(t_{i}))\Delta
W_{i}\Delta N_{i},\quad\quad\hbox{for }i\in\\{0,\ldots,n-1\\}.$
Hence, if the JCC holds, the scheme is implementable.
In order to provide worst-case error and optimality analysis we define the
following function classes. For $K\in(0,\infty)$, and $\gamma\in(0,1]$, a
function $f:[0,T]\times\mathbb{R}\to\mathbb{R}$ belongs to the function class
$F_{K}^{\gamma}$ if and only if for all $t,s\in[0,T]$ and all
$y,z\in\mathbb{R}$ it satisfies
* (i)
$f\in C^{0,1}\left([0,T]\times\mathbb{R}\right)$,
* (ii)
$|f(0,0)|\leq K$,
* (iii)
$|f(t,y)-f(t,z)|\leq K|y-z|$,
* (iv)
$|f(t,y)-f(s,y)|\leq K(1+|y|)|t-s|^{\gamma}$,
* (v)
$\left|\frac{\partial f}{\partial y}(t,y)-\frac{\partial f}{\partial
y}(t,z)\right|\leq K|y-z|$.
In this paper we consider drift coefficients $\mu$ from the class
$\mathcal{M}^{\varrho_{1}}_{K}=\Biggl{\\{}\mu\in
F^{\varrho_{1}}_{K}\colon\left|\frac{\partial\mu}{\partial
y}(t,y)-\frac{\partial\mu}{\partial y}(s,y)\right|\leq
K(1+|y|)|t-s|^{\varrho_{1}}\ \hbox{for all}\
t,s\in[0,T],y\in\mathbb{R}\Biggr{\\}},$
while we assume that the diffusion and jump coefficients $(\sigma,\rho)$ are
from the class
$\displaystyle\mathcal{B}^{\varrho_{2},\varrho_{3}}_{K}=\Bigl{\\{}(\sigma,\rho)\in
F^{\varrho_{2}}_{K}\times
F^{\varrho_{3}}_{K}\colon|L_{1}\sigma(t,y)-L_{1}\sigma(t,z)|\leq K|y-z|,$ (89)
$\displaystyle|L_{1}\rho(t,y)-L_{1}\rho(t,z)|\leq K|y-z|,\
L_{-1}\sigma(t,y)=L_{1}\rho(t,y),\ \ \hbox{for all}\
t\in[0,T],y,z\in\mathbb{R}\Bigr{\\}},$
where we recall that $L_{1}f(t,y)=\sigma(t,y)f^{\prime}_{y}(t,y)$ and
$L_{-1}f(t,y)=f(t,y+\rho(t,y))-f(t,y)$. Moreover, for all $p\in[2,\infty)$,
$\mathcal{J}^{p}_{K}=\\{X_{0}\colon\Omega\to\mathbb{R}\colon X_{0}\ \hbox{is}\
\mathcal{F}_{0}-\hbox{measurable},\mathbb{E}[|X_{0}|^{2p}]\leq K\\}.$
The class of input data $(\mu,\sigma,\rho,X_{0})$ is defined by
$\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},p,K)=\mathcal{M}^{\varrho_{1}}_{K}\times\mathcal{B}^{\varrho_{2},\varrho_{3}}_{K}\times\mathcal{J}^{p}_{K}.$
(90)
We call $\varrho_{1},\varrho_{2},\varrho_{3},p,K,T$ the parameters of the
class $\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},p,K)$.
We now describe the model of computation. An information vector has the form
$\displaystyle\mathcal{N}(\mu,\sigma,\rho,X_{0},W,N)=$
$\displaystyle[\mu(\xi_{0},y_{0}),\ldots,\mu(\xi_{k_{1}-1},y_{k_{1}-1}),\sigma(t_{0},y_{0}),\ldots,\sigma(t_{k_{1}-1},y_{k_{1}-1}),$
(91) $\displaystyle\ \
\rho(t_{0},y_{0}),\ldots,\rho(t_{k_{1}-1},y_{k_{1}-1}),\frac{\partial\sigma}{\partial
y}(t_{0},y_{0}),\ldots,\frac{\partial\sigma}{\partial
y}(t_{k_{1}-1},y_{k_{1}-1}),$ $\displaystyle\ \
\sigma(t_{0},z_{0}),\ldots,\sigma(t_{k_{1}-1},z_{k_{1}-1}),\rho(t_{0},v_{0}),\ldots,\rho(t_{k_{1}-1},v_{k_{1}-1}),$
$\displaystyle\ \
W(s_{0}),\ldots,W(s_{k_{2}-1}),N(q_{0}),\ldots,N(q_{k_{3}-1}),X_{0}],$
where $k_{1}$, $k_{2}$, $k_{3}\in{\mathbb{N}}$ and
$[\xi_{0},\xi_{1},\ldots,\xi_{k_{1}-1}]$ is a random vector on
$(\Omega,\mathcal{F},\mathbb{P})$ with values in $[0,T]^{k_{1}}$. We assume
that the $\sigma$-fields $\sigma(\xi_{0},\xi_{1},\ldots,\xi_{k_{1}-1})$ and
$\mathcal{F}_{\infty}$ are independent. Moreover,
$t_{0},t_{1},\ldots,t_{k_{1}-1}\in[0,T]$,
$s_{0},s_{1},\ldots,s_{k_{2}-1}\in[0,T]$, and
$q_{0},q_{1},\ldots,q_{k_{3}-1}\in[0,T]$ are given time points. We assume that
$s_{i}\neq s_{j}$, $q_{i}\neq q_{j}$ for all $i\neq j$. The evaluation points
$y_{j},z_{j},v_{j}$ for the spatial variables of $\mu,\sigma$,
$\partial\sigma/\partial y$, and $\rho$ can be given in an adaptive way with
respect to $(\mu,\sigma,\rho,X_{0})$ and $[W,N]$. This means that for some
measurable mappings $\psi_{j}$, $j\in\\{0,1,\ldots,k_{1}-1\\}$, it holds that
$(y_{0},z_{0},v_{0})=\psi_{0}(W(s_{0}),\ldots,W(s_{k_{2}-1}),N(q_{0}),\ldots,N(q_{k_{3}-1}),X_{0})$
(92)
and
$\displaystyle(y_{j},z_{j},v_{j})=\psi_{j}(\mu(\xi_{0},y_{0}),\ldots,\mu(\xi_{j-1},y_{j-1}),\sigma(t_{0},y_{0}),\ldots,\sigma(t_{j-1},y_{j-1}),$
(93)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\rho(t_{0},y_{0}),\ldots,\rho(t_{j-1},y_{j-1}),\frac{\partial\sigma}{\partial
y}(t_{0},y_{0}),\ldots,\frac{\partial\sigma}{\partial y}(t_{j-1},y_{j-1}),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\sigma(t_{0},z_{0}),\ldots,\sigma(t_{j-1},z_{j-1}),\rho(t_{0},v_{0}),\ldots,\rho(t_{j-1},v_{j-1}),$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad
W(s_{0}),\ldots,W(s_{k_{2}-1}),N(q_{0}),\ldots,N(q_{k_{3}-1}),X_{0}).$
The total number of evaluations of $\mu,\sigma,\rho$, $W$, and $N$ is
$l=6k_{1}+k_{2}+k_{3}$.
Any algorithm $\mathcal{A}$ that uses the information
$\mathcal{N}(\mu,\sigma,\rho,X_{0},W,N)$ and computes the approximation to
$X(T)$, is of the form
$\mathcal{A}(\mu,\sigma,\rho,X_{0},W,N)=\varphi(\mathcal{N}(\mu,\sigma,\rho,X_{0},W,N)),$
(94)
where $\varphi:\mathbb{R}^{3k_{1}+k_{2}+k_{3}+1}\to\mathbb{R}$ is a Borel
measurable function. For a fixed $n\in\mathbb{N}$ we denote by $\Phi_{n}$ the
class of all algorithms (94) with total number of evaluations $l\leq n$.
For
$(\mu,\sigma,\rho,X_{0})\in\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},p,K)$
we define the error of $\mathcal{A}\in\Phi_{n}$ as
$e^{(p)}(\mathcal{A},\mu,\sigma,\rho,X_{0},W,N)=\|\mathcal{A}(\mu,\sigma,\rho,X_{0},W,N)-X(\mu,\sigma,\rho,X_{0})(T)\|_{p}.$
(95)
The worst-case error of $\mathcal{A}$ in a subclass $\mathcal{G}$ of
$\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},p,K)$ is defined by
$e^{(p)}(\mathcal{A},\mathcal{G},W,N)=\sup\limits_{(\mu,\sigma,\rho,X_{0})\in\mathcal{G}}e^{(p)}(\mathcal{A},\mu,\sigma,\rho,X_{0},W,N),$
(96)
while the $n$-th minimal error in $\mathcal{G}$ is
$e^{(p)}_{n}(\mathcal{G},W,N)=\inf\limits_{\mathcal{A}\in\Phi_{n}}e^{(p)}(\mathcal{A},\mathcal{G},W,N).$
(97)
The aim is to find sharp bounds for
$e^{(p)}_{n}(\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},p,K),W,N)$, i.e.
lower and upper error bounds which match up to constants.
The randomized Milstein algorithm can be written as
$\mathcal{A}^{RM}_{n}(\mu,\sigma,\rho,X_{0},W,N)=X^{(\delta)}(T),$ (98)
where $X^{(\delta)}(T)$ is defined in (17), and we have that
$\mathcal{A}^{RM}_{n}\in\Phi_{8n}$.
###### Theorem 4.1.
It holds that
$e^{(2)}_{n}(\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)=\Theta(n^{-\min\\{\varrho_{1}+\frac{1}{2},\varrho_{2},\varrho_{3}\\}})$
(99)
as $n\to+\infty$.
###### Proof.
The upper bound
$O(n^{-\min\\{\varrho_{1}+\frac{1}{2},\varrho_{2},\varrho_{3}\\}})$ on
$e^{(2)}_{n}(\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)$
follows from Theorem 3.2 and the fact that
$e^{(2)}_{n}(\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)\leq
e^{(2)}(\mathcal{A}^{RM}_{n},\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)$.
We now turn to the lower bound. Let $\mathcal{A}$ be any algorithm from
$\Phi_{n}$ that uses at most $n$ evaluations of $(\mu,\sigma,\rho)$, $W$, and
$N$. We consider the following subclasses of
$\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K)$:
$\mathcal{G}_{1}(\varrho_{1},1,1,2,K)=\mathcal{\bar{M}}_{K}^{\varrho_{1}}\times\\{(0,0)\\}\times\\{0\\},$
where
$\mathcal{\bar{M}}_{K}^{\varrho_{1}}=\\{\mu\in\mathcal{M}_{K}^{\varrho_{1}}\
|\ \mu(t,x)=\mu(t,0)\text{ for all }t\in[0,T],x\in\mathbb{R}\\},$
and
$\mathcal{G}_{2}(1,\varrho_{2},1,2,K)=\\{0\\}\times\mathcal{\bar{B}}^{\varrho_{2},1}_{K}\times\\{0\\},$
where
$\mathcal{\bar{B}}^{\varrho_{2},1}_{K}=\Bigl{\\{}(\sigma,0)\in\mathcal{B}^{\varrho_{2},1}_{K}\,\Bigl{|}\,\sigma(t,y)=\sigma(t,0)\
\hbox{for all}\ t\in[0,T],y\in\mathbb{R}\Bigr{\\}},$
and
$\mathcal{G}_{3}(1,1,\varrho_{3},2,K)=\\{0\\}\times\mathcal{\bar{B}}^{1,\varrho_{3}}_{K}\times\\{0\\},$
where
$\mathcal{\bar{B}}^{1,\varrho_{3}}_{K}=\Bigl{\\{}(0,\rho)\in\mathcal{B}^{1,\varrho_{3}}_{K}\,\Bigl{|}\,\rho(t,y)=\rho(t,0)\
\hbox{for all}\ t\in[0,T],y\in\mathbb{R}\Bigr{\\}}.$
For $(\mu,\sigma,\rho,X_{0})\in\mathcal{G}_{1}(\varrho_{1},1,1,2,K)$ we have
$\displaystyle{X(\mu,\sigma,\rho,X_{0})(T)=\int_{0}^{T}\mu(t,0)\mathop{}\\!\mathrm{d}t}$.
Since $k_{1}=O(n)$ by [14, Section 2.2.9, Proposition 2] we obtain that
$e(\mathcal{A},\mathcal{G}_{1}(\varrho_{1},1,1,2,K))=\Omega(n^{-(\varrho_{1}+\frac{1}{2})}).$
(100)
Next, for $(\mu,\sigma,\rho,X_{0})\in\mathcal{G}_{2}(1,\varrho_{2},1,2,K)$ we
have
$\displaystyle{X(\mu,\sigma,\rho,X_{0})(T)=\int_{0}^{T}\sigma(t,0)\mathop{}\\!\mathrm{d}W(t)}$.
Since $k_{2}=O(n)$, [11, Proposition 5.1(i)] gives
$e(\mathcal{A},\mathcal{G}_{2}(1,\varrho_{2},1,2,K))=\Omega(n^{-\varrho_{2}}).$
(101)
Finally, for $(\mu,\sigma,\rho,X_{0})\in\mathcal{G}_{3}(1,1,\varrho_{3},2,K)$
we have
$\displaystyle{X(\mu,\sigma,\rho,X_{0})(T)=\int_{0}^{T}\rho(t,0)\mathop{}\\!\mathrm{d}N(t)}$.
Since $k_{3}=O(n)$, [22, Lemma 6] yields
$e(\mathcal{A},\mathcal{G}_{3}(1,1,\varrho_{3},2,K))=\Omega(n^{-\varrho_{3}}).$
(102)
Due to the fact that
$\mathcal{G}_{1}(\varrho_{1},1,1,2,K)\cup\mathcal{G}_{2}(1,\varrho_{2},1,2,K)\cup\mathcal{G}_{3}(1,1,\varrho_{3},2,K)\subset\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K)$,
it holds that
$\displaystyle
e(\mathcal{A},\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K))$ (103)
$\displaystyle\geq\max\\{e(\mathcal{A},\mathcal{G}_{1}(\varrho_{1},1,1,2,K)),e(\mathcal{A},\mathcal{G}_{2}(1,\varrho_{2},1,2,K)),e(\mathcal{A},\mathcal{G}_{3}(1,1,\varrho_{3},2,K))\\}$
$\displaystyle=\Omega(n^{-\min\\{\varrho_{1}+\frac{1}{2},\varrho_{2},\varrho_{3}\\}}).$
This, together with the upper bound proves the claim. ∎
###### Remark 4.2.
For $\varrho_{2}=\varrho_{3}=1$ and $\varrho_{1}\in(1/2,1)$ we compare the
worst case errors for the classical Euler–Maruyama algorithm
$\mathcal{A}^{E}_{n}$, randomized Euler–Maruyama algorithm
$\mathcal{A}^{RE}_{n}$, classical Milstein algorithm $\mathcal{A}^{M}_{n}$,
and randomized Milstein algorithm $\mathcal{A}^{RE}_{n}$ in the class
$\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K)$:
$\displaystyle
e^{(2)}(\mathcal{A}^{E}_{n},\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)=O(n^{-1/2}),\
e^{(2)}(\mathcal{A}^{RE}_{n},\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)=O(n^{-1/2}),$
(104) $\displaystyle
e^{(2)}(\mathcal{A}^{M}_{n},\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)=O(n^{-\varrho_{1}}),\
e^{(2)}(\mathcal{A}^{RM}_{n},\mathcal{F}(\varrho_{1},\varrho_{2},\varrho_{3},2,K),W,N)=O(n^{-1}).$
We observe that in the considered case the randomized Milstein algorithm
outperforms the other (classical) algorithms.
### 4.2 Multidimensional case and optimality of the Euler–Maruyama algorithm
In this section we show results that give insight into lower error bounds for
the problem of approximation of solutions of systems of jump-diffusion SDEs in
the case when only standard information about $W$ and $N$ is available. For
this we extend results from [2] and consider the following jump-diffusion
Lévy’s area
$\displaystyle
J(N,W)=I_{0,T}(N,W)=\int\limits_{0}^{T}\int\limits_{0}^{t-}\mathop{}\\!\mathrm{d}N(s)\mathop{}\\!\mathrm{d}W(t)=\int\limits_{0}^{T}N(t-)\mathop{}\\!\mathrm{d}W(t)=\int\limits_{0}^{T}N(t)\mathop{}\\!\mathrm{d}W(t).$
(105)
The last equality holds since $W$ is continuous while $N(t)$ and $N(t-)$
differ in an at most finite number of time points almost surely. Note that
$J(N,W)=X(T)$ where $X$ is the solution of the two-dimensional SDE
$\displaystyle dY(t)=\mathop{}\\!\mathrm{d}N(t),$ (106) $\displaystyle
dX(t)=Y(t)\mathop{}\\!\mathrm{d}W(t),\ t\in[0,T].$
In order to approximate (LABEL:LA1) we consider an arbitrary algorithm of the
form
$\displaystyle\mathcal{A}_{n}(N,W)=\varphi_{n}(\mathcal{N}_{n}(N,W))$ (107)
for some Borel-measurable function
$\varphi_{n}\colon{\mathbb{R}}^{2n}\to{\mathbb{R}}$, where
$\displaystyle\mathcal{N}_{n}(N,W)=[N(t_{1}),\ldots,N(t_{n}),W(t_{1}),\ldots,W(t_{n})]$
(108)
and
$0=t_{0}<t_{1}<\ldots<t_{n}=T$ (109)
is a fixed discretization of $[0,T]$. In particular, we consider the
trapezoidal method $\mathcal{A}_{n}^{T}(N,W)$ based on the mesh (109), which
is defined as
$\mathcal{A}_{n}^{T}(N,W)=\sum_{i=0}^{n-1}\frac{1}{2}\big{(}W(t_{i+1})-W(t_{i})\big{)}\big{(}N(t_{i+1})+N(t_{i})\big{)}.$
(110)
###### Theorem 4.3.
For the trapezoidal method (110) based on the equidistant mesh $t_{i}=iT/n$,
$i\in\\{0,1,\ldots,n\\}$, it holds that
$\lim_{n\to\infty}n^{1/2}\cdot\|J(N,W)-\mathcal{A}_{n}^{T}(N,W)\|_{2}=\lim\limits_{n\to\infty}n^{1/2}\cdot\inf\limits_{\mathcal{A}_{n}}\|J(N,W)-\mathcal{A}_{n}(N,W)\|_{2}=\frac{\lambda^{1/2}T}{2},$
(111)
and therefore it is the optimal method among all methods of the form (107).
###### Proof.
From the projection property for the conditional expectation we get for any
algorithm (107) that
$\displaystyle{\mathbb{E}}\Big{[}\big{|}J(N,W)-\mathcal{A}_{n}(N,W)\big{|}^{2}\Big{]}\geq{\mathbb{E}}\Big{[}\big{|}J(N,W)-{\mathbb{E}}\big{[}J(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}\big{|}^{2}\Big{]},$
(112)
since $\mathcal{A}_{n}(N,W)$ is $\sigma(\mathcal{N}_{n}(N,W))$-measurable.
Therefore, we also have
$\inf_{\mathcal{A}_{n}}{\mathbb{E}}\Big{[}\big{|}J(N,W)-\mathcal{A}_{n}(N,W)\big{|}^{2}\Big{]}\geq\inf\limits_{0=t_{0}<t_{1}\ldots<t_{n}=T}{\mathbb{E}}\Big{[}\big{|}J(N,W)-{\mathbb{E}}\big{[}J(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}\big{|}^{2}\Big{]}.$
(113)
Hence, we need to compute
$\displaystyle{\mathbb{E}}\big{[}J(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}=\sum_{i=0}^{n-1}{\mathbb{E}}\Bigg{[}\int\limits_{t_{i}}^{t_{i+1}}N(t)\mathop{}\\!\mathrm{d}W(t)\Big{|}\mathcal{N}_{n}(N,W)\Bigg{]}.$
(114)
For all $i\in\\{0,\ldots,n-1\\}$ we define
$J_{t_{i},t_{i+1}}(N,W)=\int\limits_{t_{i}}^{t_{i+1}}N(t)\mathop{}\\!\mathrm{d}W(t).$
(115)
From the definition of the Itô integral we get for all
$i\in\\{0,\ldots,n-1\\}$ that
$\displaystyle J_{t_{i},t_{i+1}}(N,W)=\lim_{m\to\infty}J_{m}^{i}(N,W)\text{ in
}L^{2}({\mathbb{R}}),$ (116)
where
$\displaystyle
J_{m}^{i}(N,W)=\sum_{j=0}^{m-1}N(s_{j}^{i})(W(s_{j+1}^{i})-W(s_{j}^{i})),$
(117)
with $s_{j}^{i}=t_{i}+j(t_{i+1}-t_{i})/m$ for all $j\in\\{0,\ldots,m\\}$.
Further, we denote $\Delta W_{j}^{i}=W(s_{j+1}^{i})-W(s_{j}^{i})$ for all
$i\in\\{1,\ldots,n-1\\}$ and $j\in\\{1,\ldots,m-1\\}$. Then
$\displaystyle{\mathbb{E}}\big{[}J_{m}^{i}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}=\sum_{j=0}^{m-1}{\mathbb{E}}\big{[}N(s_{j}^{i})\Delta
W_{j}^{i}\big{|}\mathcal{N}_{n}(N,W)\big{]}.$ (118)
Since by Proposition A.5 the processes $N$ and $W$ are conditionally
independent given the $\sigma$-algebra $\sigma(\mathcal{N}_{n}(N,W))$, we have
that
$\displaystyle{\mathbb{E}}\big{[}N(s_{j}^{i})\Delta
W_{j}^{i}\big{|}\mathcal{N}_{n}(N,W)\big{]}={\mathbb{E}}\big{[}N(s_{j}^{i})\big{|}\mathcal{N}_{n}(N)\big{]}\cdot{\mathbb{E}}\big{[}\Delta
W_{j}^{i}\big{|}\mathcal{N}_{n}(W)\big{]}.$ (119)
Moreover, by [6, Lemma 8] and [19, Lemma 3.1], we have for all
$s\in[t_{i},t_{i+1}]$
$\displaystyle{\mathbb{E}}\big{[}N(s)\big{|}\mathcal{N}_{n}(N)\big{]}=\frac{N(t_{i+1})(s-t_{i})+N(t_{i})(t_{i+1}-s)}{t_{i+1}-t_{i}}$
(120)
and
${\mathbb{E}}\big{[}\Delta
W_{j}^{i}\big{|}\mathcal{N}_{n}(W)\big{]}=\frac{\big{(}W(t_{i+1})-W(t_{i})\big{)}(s_{j+1}^{i}-s_{j}^{i})}{t_{i+1}-t_{i}}.$
(121)
Plugging (119),(LABEL:LA14), and (121) into (118), we obtain
$\displaystyle{\mathbb{E}}\big{[}J_{m}^{i}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}=\sum_{j=0}^{m-1}{\mathbb{E}}\big{[}N(s_{j}^{i})\big{|}\mathcal{N}_{n}(N)\big{]}\cdot{\mathbb{E}}\big{[}\Delta
W_{j}^{i}\big{|}\mathcal{N}_{n}(W)\big{]}$ (122)
$\displaystyle=\frac{W(t_{i+1})-W(t_{i})}{t_{i+1}-t_{i}}\sum_{j=0}^{m-1}{\mathbb{E}}\big{[}N(s_{j}^{i})\big{|}\mathcal{N}_{n}(N)\big{]}\cdot(s_{j+1}^{i}-s_{j}^{i}).$
Since
$\displaystyle{\sum_{j=0}^{m-1}{\mathbb{E}}\big{[}N(s_{j}^{i})\big{|}\mathcal{N}_{n}(N)\big{]}\cdot(s_{j+1}^{i}-s_{j}^{i})}$
is a (pathwise) Riemann sum for the stochastic process
$\big{(}{\mathbb{E}}\big{[}N(s)\big{|}\mathcal{N}_{n}(N)\big{]}\big{)}_{s\in[t_{i},t_{i+1}]}$
with continuous sample paths, it holds for all $i\in\\{0,\ldots,m-1\\}$ almost
surely that
$\displaystyle\lim_{m\to\infty}{\mathbb{E}}\big{[}J_{m}^{i}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}=\frac{W(t_{i+1})-W(t_{i})}{t_{i+1}-t_{i}}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\big{[}N(t)\big{|}\mathcal{N}_{n}(N)\big{]}\mathop{}\\!\mathrm{d}t.$
(123)
Moreover, from (117) and by Jensen’s inequality for the conditional
expectation we obtain
$\displaystyle{\mathbb{E}}\Big{[}\big{|}{\mathbb{E}}[J_{t_{i},t_{i+1}}(N,W)|\mathcal{N}_{n}(N,W)]-{\mathbb{E}}[J_{m}^{i}(N,W)|\mathcal{N}_{n}(N,W)]\big{|}^{2}\Big{]}$
(124)
$\displaystyle\leq{\mathbb{E}}\Big{[}\big{|}J_{t_{i},t_{i+1}}(N,W)-J_{m}^{i}(N,W)\big{|}^{2}\Big{]}\to
0\text{ as }m\to\infty.$
Hence, by (LABEL:LA18)
$\displaystyle{\mathbb{E}}\big{[}J_{m}^{i}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}\to{\mathbb{E}}\big{[}J_{t_{i},t_{i+1}}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}\text{
as }m\to\infty\text{ in }L^{2}(\Omega)$ (125)
and by (123)
$\displaystyle{\mathbb{E}}\big{[}J_{m}^{i}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}\to\frac{W(t_{i+1})-W(t_{i})}{t_{i+1}-t_{i}}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\big{[}N(t)\big{|}\mathcal{N}_{n}(N)\big{]}\mathop{}\\!\mathrm{d}t\text{
as }m\to\infty\text{ a.s.}$ (126)
Convergence in $L^{2}(\Omega)$ as well as almost sure convergence imply
convergence in probability. Moreover by the uniqueness of the limit in
probability we get that for all $i\in\\{0,\ldots,m-1\\}$,
$\displaystyle{\mathbb{E}}\big{[}J_{t_{i},t_{i+1}}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}=\frac{W(t_{i+1})-W(t_{i})}{t_{i+1}-t_{i}}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\big{[}N(t)\big{|}\mathcal{N}_{n}(N)\big{]}\mathop{}\\!\mathrm{d}t\text{
a.s.}$ (127)
Moreover, we have
$\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\big{[}N(t)\big{|}\mathcal{N}_{n}(N)\big{]}\mathop{}\\!\mathrm{d}t=\frac{1}{2}\big{(}N(t_{i+1})+N(t_{i})\big{)}(t_{i+1}-t_{i}).$
(128)
Hence by plugging (128) into (LABEL:LA21) we get for all
$i\in\\{0,\ldots,m-1\\}$ that
$\displaystyle{\mathbb{E}}\big{[}J_{t_{i},t_{i+1}}(N,W)\big{|}\mathcal{N}_{n}(N,W)\big{]}=\frac{1}{2}(W(t_{i+1})-W(t_{i}))(N(t_{i+1})+N(t_{i})).$
(129)
Combining (129) and (114) we get that
${\mathbb{E}}[J(N,W)|\mathcal{N}_{n}(N,W)]=\mathcal{A}^{T}_{n}(N,W)$
corresponds to the trapezoidal method.
Now we investigate its error in order to get the minimal possible error among
all methods of the form (107). To do so, let us consider the step process
given for all $t\in[0,T]$ by
$\displaystyle\hat{N}_{n}(t):=\sum_{i=0}^{n-1}\mathds{1}_{(t_{i},t_{i+1}]}(t)\frac{N(t_{i})+N(t_{i+1})}{2}.$
(130)
The process $(\hat{N}_{n}(t))_{t\in[0,T]}$ is not adapted to the initial
filtration $(\mathcal{F}_{t})_{t\geq 0}$. However, it is adapted to
$\mathcal{\widetilde{F}}_{t}:=\sigma(\mathcal{F}_{t}^{W}\cup\mathcal{F}_{\infty}^{N})$,
$t\geq 0$. Moreover, $(W(t))_{t\geq 0}$ is still a one-dimensional Wiener
process with respect to the filtration $(\mathcal{\widetilde{F}}_{t})_{t\geq
0}$, since $\mathcal{F}_{\infty}^{N}$ and $\mathcal{F}_{\infty}^{W}$ are
independent. Hence
$J(\hat{N}_{n},W)=\int\limits_{0}^{T}\hat{N}_{n}(t)\mathop{}\\!\mathrm{d}W(t)$
(131)
is a well-defined Itô integral of the $(\mathcal{\widetilde{F}}_{t})_{t\geq
0}$-simple process $(\hat{N}_{n}(t))_{t\in[0,T]}$ and therefore
$\displaystyle
J(\hat{N}_{n},W)=\sum_{i=0}^{n-1}\frac{N(t_{i})+N(t_{i+1})}{2}(W(t_{i+1})-W(t_{i}))=\mathcal{A}_{n}^{T}(N,W).$
(132)
Since $(N(t)-\hat{N}_{n}(t))_{t\in[0,T]}$ is a
$(\mathcal{\widetilde{F}}_{t})_{t\geq 0}$-progressively measurable process, by
the Itô isometry and Jensen’s inequality we get
$\displaystyle{\mathbb{E}}\Big{[}\big{|}J(N,W)-\mathcal{A}_{n}^{T}(N,W)\big{|}^{2}\Big{]}=\sum_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}{\mathbb{E}}\Big{[}\big{|}N(t)-\hat{N}_{n}(t)\big{|}^{2}\Big{]}\mathop{}\\!\mathrm{d}t$
(133)
$\displaystyle=\frac{1}{4}\sum_{i=0}^{n-1}\int\limits_{t_{i}}^{t_{i+1}}\Big{(}{\mathbb{E}}\Big{[}\big{(}N(t)-N(t_{i})\big{)}^{2}\Big{]}-2{\mathbb{E}}\Big{[}N(t)-N(t_{i})\Big{]}\cdot{\mathbb{E}}\Big{[}N(t_{i+1})-N(t)\Big{]}$
$\displaystyle\quad\quad\quad\quad+{\mathbb{E}}\Big{[}\big{(}N(t_{i+1})-N(t)\big{)}^{2}\Big{]}\Big{)}\mathop{}\\!\mathrm{d}t$
$\displaystyle=\frac{\lambda}{4}\sum_{i=0}^{n-1}(t_{i+1}-t_{i})^{2}+\frac{\lambda^{2}}{12}\sum_{i=0}^{n-1}(t_{i+1}-t_{i})^{3}\geq\frac{\lambda}{4n}\Big{(}\sum_{i=0}^{n-1}(t_{i+1}-t_{i})\Big{)}^{2}+\frac{1}{n^{2}}\,\frac{\lambda^{2}}{12}\Big{(}\sum_{i=0}^{n-1}(t_{i+1}-t_{i})\Big{)}^{3}$
$\displaystyle\geq\frac{\lambda T^{2}}{4n}+\frac{\lambda^{2}T^{3}}{12n^{2}}.$
Therefore,
$\inf\limits_{0=t_{0}<t_{1}<\ldots
t_{n}=T}{\mathbb{E}}\Big{[}\big{|}J(N,W)-\mathcal{A}_{n}^{T}(N,W)\big{|}^{2}\Big{]}\geq\frac{\lambda
T^{2}}{4n}+\frac{\lambda^{2}T^{3}}{12n^{2}}.$ (134)
Hence, by (113) we conclude
$\displaystyle
n^{1/2}\cdot\inf_{\mathcal{A}_{n}}\|J(N,W)-\mathcal{A}_{n}(N,W)\|_{L^{2}(\Omega)}\geq\sqrt{\frac{\lambda
T^{2}}{4}+\frac{\lambda^{2}T^{3}}{12n}}.$ (135)
Moreover, for the trapezoidal method $\mathcal{A}^{T}_{n}(N,W)$ based on the
equidistant mesh $t_{i}=iT/n$, $i\in\\{0,1,\ldots,n\\}$, we get by
(LABEL:LA29) that
${\mathbb{E}}\Big{[}\big{|}J(N,W)-\mathcal{A}_{n}^{T}(N,W)\big{|}^{2}\Big{]}=\frac{\lambda
T^{2}}{4n}+\frac{\lambda^{2}T^{3}}{12n^{2}},$ (136)
and hence
$n^{1/2}\cdot\inf\limits_{\mathcal{A}_{n}}\|J(N,W)-\mathcal{A}_{n}(N,W)\|_{2}\leq
n^{1/2}\cdot\|J(N,W)-\mathcal{A}^{T}_{n}(N,W)\|_{2}=\sqrt{\frac{\lambda
T^{2}}{4}+\frac{\lambda^{2}T^{3}}{12n}}.$ (137)
From (135) and (137) we arrive at
$\displaystyle
n^{1/2}\cdot\inf\limits_{\mathcal{A}_{n}}\|J(N,W)-\mathcal{A}_{n}(N,W)\|_{2}=\sqrt{\frac{\lambda
T^{2}}{4}+\frac{\lambda^{2}T^{3}}{12n}},$ (138)
which together with (136) imply the thesis. ∎
###### Remark 4.4.
Consider any class of coefficients of multidimensional SDEs for which
(LABEL:2dim_sde) is a subproblem. Then by Theorem 4.3, in the worst case
setting with respect to the coefficients, the error cannot be smaller than
$\Omega(n^{-1/2})$. Therefore, no matter if the JCC (87) is satisfied or not,
we can apply the Euler–Maruyama (or randomized Euler–Maruyama) scheme in order
to achieve the optimal error bound $O(n^{-1/2})$ if the error is measured in
the $L^{2}(\Omega)$-norm, see, for example, [11], [12].
###### Remark 4.5.
In Theorems 4.1 and 4.3 we have considered only the $L^{2}$-error. Matching
upper and lower bounds that depend on $p$ remain an open problem. Our
numerical experiments in Section 5 suggest that for jump-diffusion SDEs the
error indeed depends on $p$.
## 5 Numerical experiments
We implement111The program code is available as ancillary file from the arXiv
page of this paper. the randomized Milstein algorithm for the SDE
$\displaystyle\left\\{\begin{array}[]{lll}\mathop{}\\!\mathrm{d}X(t)=\sin(M\cdot
X(t)(1+t)^{\varrho_{1}})\mathop{}\\!\mathrm{d}t+\cos(M\cdot
X(t)\cdot(1+t)^{\varrho_{2}})\mathop{}\\!\mathrm{d}W(t)&\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\Big{(}-X(t)+\frac{\pi}{2M\cdot(1+t)^{\varrho_{2}}}\Big{)}\mathop{}\\!\mathrm{d}N(t),&t\in[0,1],\\\
X(0)=1.&\\\ \end{array}\right.$ (139)
that has already been considered in the jump-free case in [13]. The jump
coefficient is chosen so that the JCC is satisfied. The verification of
Assumption 2.1 is straight forward, for the JCC (87), see Remark 5.2. In our
simulations we set $\lambda=100$, $M=100$, $\varrho_{1}=0.1$, and
$\varrho_{2}=0.6$.
We estimate the $L^{p}$-error similar as in [20, p. 14] by
$\displaystyle\operatorname{error}(k)=\operatorname{mean}\big{(}\big{|}X^{(k)}(T)-X^{(k-1)}(T)\big{|}^{p}\big{)}^{\frac{1}{p}}.$
(140)
Here $X^{(k)}(T)$ is the approximation of $X(T)$ with step size
$\delta^{(k)}$, where $\delta^{(k)}=2^{-k}$ for $k\in{\mathbb{N}}$. The
$\operatorname{mean}$ is taken over $2^{16}$ sample paths.
###### Remark 5.1.
In the implementation the interesting part is how the values $\xi_{i}$ are
computed. For the randomization we first simulate independent uniformly
distributed random variables $\xi_{i}$ on the corresponding intervals for the
finest discretization grid. Iteratively we compute the values for the
discretization grid with doubled step size as follows: One time interval in
the larger grid consists of two time intervals of equal length in the finer
grid. For those two intervals we have simulated two values $\xi_{i}$. Now we
simulate an independent Bernoulli$(0.5)$ random variable that determines which
of the values $\xi_{i}$ we take. This choice is then uniformly distributed on
the interval of the large grid and hence consistent with the randomized
Milstein algorithm.
For $p\in[2,\infty)$ we obtain by Theorem 3.2 the theoretical convergence rate
$\displaystyle{\min\Big{\\{}\frac{2}{p},\varrho_{1}+\frac{1}{p},\varrho_{2},\varrho_{2}\Big{\\}}=\min\Big{\\{}\frac{2}{p},0.1+\frac{1}{p},0.6\Big{\\}}}.$
For $p=1$ we take as theoretical convergence rate the same rate as for $p=2$,
because the $L^{1}$-error can be estimated by the $L^{2}$-error using the
Cauchy-Schwarz inequality. In Figure 1 we plot the
$\log_{2}(\operatorname{error}(k))$ over $\log_{2}(\delta^{(k)}))$ for
$p\in\\{1,2,3,4\\}$ and the corresponding theoretical convergence orders.
Figure 1: Error estimates and theoretical convergence order for
$p\in\\{1,2,3,4\\}$
We see that the observed convergence order is decreasing with increasing $p$.
Further we notice that for $p=1$ the convergence of the simulation is higher
than the theoretical convergence rate. This is reasonable because we took the
rate of the $L^{2}$-error. For $p=2$ we observe that the simulation confirms
the theoretical results; the slope of the simulation matches the convergence
rate, which we proved to be optimal. Also for $p=3$ and $p=4$ the simulations
confirm the theoretical results, since the simulation converges at least as
fast as the theoretically obtained upper bound; we have not proven any lower
bound.
Next, we regress the slope of the simulated
$\log_{2}(\operatorname{error}(k))$ in dependence of the corresponding
$\log_{2}(\delta^{(k)}))$ for all $p\in\\{1,\ldots,8\\}$ and compare it to the
theoretical upper bounds on the convergence rates we have proven, see Figure
2. We observe that for the simulations the convergence order is dependent on
$p$, which confirms also this theoretical finding.
Figure 2: Slopes of the simulation (estimated by linear regression) in
comparison to theoretical convergence rates
###### Remark 5.2.
Let us assume that the diffusion coefficient is of the form
$\sigma(t,y)=F(\alpha(t)y+\beta(t))$ while the jump coefficient
$\rho(t,y)=-y+\gamma(t)$ for some functions $F:\mathbb{R}\to\mathbb{R}$ and
$\alpha,\beta,\gamma:[0,T]\to\mathbb{R}$. Moreover, let us assume that there
exists $x_{0}\in\mathbb{R}$ such that
* •
$F(x_{0})=0$,
* •
$\alpha(t)\cdot\gamma(t)+\beta(t)=x_{0}$ for all $t\in[0,T]$.
Then the JCC (87) is satisfied for the pair $(\sigma,\rho)$. This provides a
new class of functions $(\sigma,\rho)$ satisfying the JCC which may, in
contrast to the class considered in [15], be nonlinear.
## Appendix A Appendix
The proof of the following lemma is straightforward and will be omitted.
###### Lemma A.1.
Under Assumption 2.1 there exists a constant $K_{7}\in(0,\infty)$ such that
for $f\in\\{\mu,\sigma,\rho\\}$ and for all $t_{1},t_{2},t,u\in[0,T]$,
$\displaystyle|\alpha_{1}(f,t,u)|$ $\displaystyle\leq K_{7}(1+|X(u)|),$ (141)
$\displaystyle|\beta(f,t,u)|$ $\displaystyle\leq K_{7}(1+|X(u)|),$
$\displaystyle|\beta(\mu,t_{1},u)-\beta(\mu,t_{2},u)|$ $\displaystyle\leq
K_{7}(1+|X(u)|^{2})\cdot|t_{1}-t_{2}|^{\varrho_{1}},$
$\displaystyle|\gamma(f,t,u)|$ $\displaystyle\leq K_{7}(1+|X(u-)|),$
$\displaystyle|\gamma(\mu,t_{1},u)-\gamma(\mu,t_{2},u)|$ $\displaystyle\leq
K_{7}(1+|X(u-)|)|t_{1}-t_{2}|^{\varrho_{1}}.$
Following [3] and [7] we recall the notion of conditional independence and
some of its useful consequences.
###### Definition A.2 ([3, p. 36-II, Definition 43]).
Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space and let
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, and $\mathcal{F}_{3}$ be three
sub-$\sigma$-fields of $\mathcal{F}$. $\mathcal{F}_{1}$ and $\mathcal{F}_{3}$
are called conditionally independent given $\mathcal{F}_{2}$, if for all
positive random variables $Y_{1}$ and $Y_{3}$, which are measurable with
respect to $\mathcal{F}_{1}$ respectively $\mathcal{F}_{3}$ it holds that
$\displaystyle{\mathbb{E}}[Y_{1}Y_{3}|\mathcal{F}_{2}]={\mathbb{E}}[Y_{1}|\mathcal{F}_{2}]\cdot{\mathbb{E}}[Y_{3}|\mathcal{F}_{2}]\text{
a.s.}$ (142)
###### Theorem A.3 ([3, p. 36-II, Theorem 45]).
Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space and let
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, and $\mathcal{F}_{3}$ be three
sub-$\sigma$-fields of $\mathcal{F}$. Further let $\mathcal{F}_{12}$ be the
$\sigma$-field generated by $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$. Then
$\mathcal{F}_{1}$ and $\mathcal{F}_{3}$ are called conditionally independent
given $\mathcal{F}_{2}$, if and only if for all $\mathcal{F}_{3}$-measurable
and integrable random variables $Y_{3}$ it holds that
$\displaystyle{\mathbb{E}}[Y_{3}|\mathcal{F}_{12}]={\mathbb{E}}[Y_{3}|\mathcal{F}_{2}]\text{
a.s.}$ (143)
###### Proposition A.4 ([7, Proposition A.23]).
Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space and let
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, and $\mathcal{F}_{3}$ be three
sub-$\sigma$-fields of $\mathcal{F}$. Further let
$Y_{1},Y_{3}\colon\Omega\to{\mathbb{R}}$ be integrable random variables such
that ${\mathbb{E}}[|Y_{1}Y_{3}|]<\infty$. Assume that
$\sigma(Y_{1})\subset\mathcal{F}_{1}$ and
$\sigma(Y_{3})\subset\mathcal{F}_{3}$. Further assume that $\mathcal{F}_{1}$
and $\mathcal{F}_{3}$ are conditionally independent given $\mathcal{F}_{2}$.
Then it holds that
$\displaystyle{\mathbb{E}}[Y_{1}Y_{3}|\mathcal{F}_{2}]={\mathbb{E}}[Y_{1}|\mathcal{F}_{2}]\cdot{\mathbb{E}}[Y_{3}|\mathcal{F}_{2}]\text{
a.s.}$ (144)
###### Proposition A.5 ([7, Lemma B.18]).
Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space and let
$X,Y\colon\Omega\times[0,\infty)\to{\mathbb{R}}$ be stochastic processes,
which are both
$\mathcal{F}\otimes\mathcal{B}([0,\infty))|\mathcal{B}({\mathbb{R}})$-measurable
and independent, i.e. $\mathcal{F}^{X}_{\infty}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathcal{F}^{Y}_{\infty}$, where
$\mathcal{F}^{Z}_{\infty}=\sigma(\bigcup_{t\geq 0}\sigma(Z(t)))$ for both
$Z\in\\{X,Y\\}$. Further, assume that ${\mathbb{E}}[|X(t)|]<\infty\text{ and
}{\mathbb{E}}[|Y(t)|]<\infty\text{ for all }t\geq 0$. Additionally, let
$m,n\in{\mathbb{N}}$, $t_{i}^{X},t_{j}^{Y}\in[0,\infty)$ for all
$i\in\\{1,\ldots,n\\}$ and $j\in\\{1,\ldots,m\\}$ be such that $0\leq
t_{1}^{X}<t_{2}^{X}<\ldots<t_{n}^{X}$, $0\leq
t_{1}^{Y}<t_{2}^{Y}<\ldots<t_{m}^{Y}$. Then $\mathcal{F}_{\infty}^{X}$ and
$\mathcal{F}_{\infty}^{Y}$ are conditionally independent given
$\sigma(X(t_{1}^{X}),\ldots,X(t_{n}^{X}),Y(t_{1}^{Y}),\ldots,Y(t_{m}^{Y}))$.
The following estimate is a direct consequence of the Hölder, the Burkholder-
Davis-Gundy, and the Kunita inequalitiy, see [9].
###### Lemma A.6.
Let $q\in[2,\infty)$, $a,b\in[0,T]$ with $a<b$,
$Z\in\\{\operatorname{Id},W,N\\}$, $Y=(Y(t))_{t\in[a,b]}$ is a predictable
stochastic process such that
${\mathbb{E}}\Big{[}\int\limits_{a}^{b}|Y(t)|^{q}\mathop{}\\!\mathrm{d}t\Big{]}<\infty$
(145)
Then there exists a constant $\hat{c}\in(0,\infty)$ such that for all
$t\in[a,b]$ it holds that
$\displaystyle{\mathbb{E}}\Biggl{[}\sup_{s\in[a,t]}\Biggl{|}\int\limits_{a}^{s}Y(u)\mathop{}\\!\mathrm{d}Z(u)\Biggl{|}^{q}\Biggr{]}\leq\hat{c}\int\limits_{a}^{t}{\mathbb{E}}[|Y(u)|^{q}]\mathop{}\\!\mathrm{d}u.$
(146)
## Acknowledgements
V. Schwarz and M. Szölgyenyi are supported by the Austrian Science Fund (FWF):
DOC 78.
## References
* Brooks [1972] R. A. Brooks. Conditional expectations associated with stochastic processes. _Pacific Journal of Mathematics_ , 41:33–42, 1972.
* Clark and Cameron [1980] J. M. C. Clark and R. J. Cameron. The maximum rate of convergence of discrete approximations for stochastic differential equations, in: Stochastic differential systems filtering and control. _Stochastic Differential Systems Filtering and Control_ , pages 162–171, 1980.
* Dellacherie and Meyer [1978] C. Dellacherie and P. Meyer. _Probabilities and Potential_. North-Holland mathematic studies. Elsevier North-Holland, inc., 1978.
* Eisenmann and Kruse [2018] M. Eisenmann and R. Kruse. Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity. _Communications in Mathematical Sciences_ , 16:2125–2146, 2018.
* Heinrich [2019] S. Heinrich. Complexity of stochastic integration in Sobolev classes. _Journal of Mathematical Analysis and Applications_ , 476:177–195, 2019.
* Hertling [2001] P. Hertling. Nonlinear Lebesgue and Itô integration problems of high complexity. _Journal of Complexity_ , 17:366–387, 2001.
* Kałuża [2020] A. Kałuża. _Optimal algorithms for solving stochastic initial-value problems with jumps_. PhD thesis, AGH University of Science and Technology, Kraków. https://winntbg.bg.agh.edu.pl/rozprawy2/11743/full11743.pdf, 2020.
* Kruse and Wu [2019] R. Kruse and Y. Wu. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. _Discrete and Continuous Dynamical Systems - Series B_ , 24:3475–3502, 2019.
* Kunita [2004] H. Kunita. Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In _Real and stochastic analysis_ , pages 305–373. Springer, 2004\.
* Müller-Gronbach and Yaroslavtseva [2020] T. Müller-Gronbach and L. Yaroslavtseva. Sharp lower error bounds for strong approximation of sdes with discontinuous drift coefficient by coupling if noise. 2020\. arXiv:2010.00915.
* Morkisz and Przybyłowicz [2014] P. Morkisz and P. Przybyłowicz. Strong approximation of solutions of stochastic differential equations with time-irregular coefficients via randomized Euler algorithm. _Applied Numerical Mathematics_ , 78:80–94, 2014.
* Morkisz and Przybyłowicz [2017] P. M. Morkisz and P. Przybyłowicz. Optimal pointwise approximation of SDE’s from inexact information. _Journal of Computational and Applied Mathematics_ , 324:85–100, 2017.
* Morkisz and Przybyłowicz [2021] P. M. Morkisz and P. Przybyłowicz. Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information. _Journal of Computational and Applied Mathematics_ , 383, 2021. 113112\.
* Novak [1988] E. Novak. _Deterministic and Stochastic Error Bounds in Numerical Analysis_. Lecture Notes in Mathematics, vol. 1349. Springer, 1988.
* Platen and Bruti-Liberati [2010] E. Platen and N. Bruti-Liberati. _Numerical Solution of Stochastic Differential Equations with Jumps in Finance_. Springer Verlag, Berlin, Heidelberg, 2010.
* Protter [2005] P. Protter. _Stochastic Integration and Differential Equations_. Stochastic Modelling and Applied Probability. Springer, Berlin-Heidelberg, 2005.
* Przybyłowicz [2015a] P. Przybyłowicz. Minimal asymptotic error for one-point approximation of sdes with time-irregular coefficients. _Journal of Computational and Applied Mathematics_ , 282:98–110, 2015a.
* Przybyłowicz [2015b] P. Przybyłowicz. Optimal global approximation of SDEs with time-irregular coefficients in asymptotic setting. _Applied Mathematics and Computation_ , 270:441–457, 2015b.
* Przybyłowicz [2016] P. Przybyłowicz. Optimal global approximation of stochastic differential equations with additive poisson noise. _Numerical Algorithms_ , 73:323–348, 2016.
* Przybyłowicz and Szölgyenyi [2021] P. Przybyłowicz and M. Szölgyenyi. Existence, uniqueness, and approximation of solutions of jump-diffusion SDEs with discontinuous drift. _Applied Mathematics and Computation_ , 403:126191, 2021\.
* Przybyłowicz et al. [2021] P. Przybyłowicz, M. Szölgyenyi, and F. Xu. Existence and uniqueness of solutions of SDEs with discontinuous drift and finite activity jumps. _Statistics and Probability Letters_ , 174:109072, 2021\.
* Przybyłowicz et al. [2022] P. Przybyłowicz, M. Sobieraj, and Ł. Stȩpień. Efficient approximtion of SDEs driven by countably dimensional Wiener process and Poisson random measure. _SIAM Journal of Numerical Analysis_ , 60:824–855, 2022\.
* Situ [2005] R. Situ. _Theory of Stochastic Differential Equations with Jumps and Applications_. Mathematical and Analytical Techniques with Applications to Engineering. Springer, 2005.
* Traub et al. [1988] J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski. _Information-Based Complexity_. Academic Press, New York, 1988.
Paweł Przybyłowicz
Faculty of Applied Mathematics, AGH University of Science and Technology, Al.
Mickiewicza 30, 30-059 Krakow, Poland
<EMAIL_ADDRESS>
Michaela Szölgyenyi
Department of Statistics, University of Klagenfurt, Universitätsstraße 65-67,
9020 Klagenfurt, Austria
<EMAIL_ADDRESS>
Verena Schwarz 🖂
Department of Statistics, University of Klagenfurt, Universitätsstraße 65-67,
9020 Klagenfurt, Austria
<EMAIL_ADDRESS>
|
# The interior Backus problem:
local resolution in Hölder spaces
Toru Kan Department of Mathematics, Osaka Metropolitan University, 1-1
Gakuen-cho, Naka-ku, Sakai, 599-8531, Japan<EMAIL_ADDRESS>, Rolando
Magnanini Dipartimento di Matematica ed Informatica “U. Dini”, Università di
Firenze, viale Morgagni 67/A, 50134 Firenze, Italy<EMAIL_ADDRESS>http://web.math.unifi.it/users/magnanin and Michiaki Onodera Department of
Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo
152-8551, Japan<EMAIL_ADDRESS>
###### Abstract.
We prove an existence result for the interior Backus problem in the Euclidean
ball. The problem consists in determining a harmonic function in the ball from
the knowledge of the modulus of its gradient on the boundary. The problem is
severely nonlinear. From a physical point of view, the problem can be
interpreted as the determination of the velocity potential of an
incompressible and irrotational fluid inside the ball from measurements of the
velocity field’s modulus on the boundary. The linearized problem is an
irregular oblique derivative problem, for which a phenomenon of loss of
derivatives occurs. As a consequence, a solution by linearization of the
Backus problem becomes problematic. Here, we linearize the problem around the
vertical height solution and show that the loss of derivatives does not occur
for solutions which are either (vertically) axially symmetric or oddly
symmetric in the vertical direction. A standard fixed point argument is then
feasible, based on ad hoc weighted estimates in Hölder spaces.
###### Key words and phrases:
Backus problem, fully nonlinear boundary conditions, irregular oblique
derivative problem
###### 2010 Mathematics Subject Classification:
35J65, 35C15, 35B07
## 1\. Introduction
Let $\Omega$ be a bounded domain in the Euclidean space $\mathbb{R}^{N}$,
$N\geq 2$, with boundary $\Gamma$. Let $g$ be a positive continuous function
on $\Gamma$. The interior Backus problem consists in determining a function
$u\in C^{1}(\overline{\Omega})\cap C^{2}(\Omega)$ such that
(1.1) $\Delta u=0\ \mbox{ in }\ \Omega,\quad|\nabla u|=g\ \mbox{ on }\
\Gamma.$
This problem was first considered and completely solved in [2] (see also
[12]), for $N=2$. There, Backus was motivated by a problem in geophysics,
which entails the reconstruction of the gravitational or geomagnetic
terrestrial field from measurements of its intensity on the Earth’s surface
${\mathcal{S}}$. In fact, if one models the Earth as the unit ball $B$, then
the relevant geophysical problem amounts to determine solutions $u$ of (1.1),
with $\Omega=\mathbb{R}^{N}\setminus\overline{B}$ and $\Gamma={\mathcal{S}}$,
such that
(1.2) $u(x)\to 0\ \mbox{ as }\ |x|\to\infty.$
This is what we call the exterior Backus problem.
When $N=2$, by the Riemann mapping theorem, we know that any simply connected
(seen as a domain on the Riemann sphere) proper subdomain of the complex plane
$\mathbb{C}$ is conformally equivalent to the unit disk. Moreover, the
harmonicity of functions is preserved by conformal mappings, while the modulus
of their gradients changes just by a positive factor (which depends on the
derivative of the conformal map). So, that is one reason why problem (1.1) has
some interest.
Another physical motivation, which genuinely pertains the interior problem
setting, has to do with the study of incompressible and irrotational fluid
flows. Let $\vec{V}$ be the velocity field of a fluid and let $\rho$ be its
density. Any fluid flow obeys the continuity equation:
$\mathop{\mathrm{div}}(\rho\,\vec{V})+\rho_{t}=0.$
If a fluid is incompressible, its density is constant. In particular, we have
that $\rho_{t}=0$, and hence the continuity equation reduces to
$\mathop{\mathrm{div}}(\vec{V})=0$. If the fluid is irrotational, then
$\mathop{\mathrm{curl}}\,(\vec{V})=0$, and hence there exists a harmonic
velocity potential $u$ such that $\nabla u=\vec{V}$. Thus, solving (1.1) can
be interpreted as the determination of the velocity of the fluid inside the
domain $\Omega$ from measurements of its modulus on the boundary.
It is worthwhile to clarify from this point of view the results obtained in
[2] for planar domains. As already mentioned, in this case we can always
assume that $\Omega$ is the unit disk. We shall describe the situation in the
simplest case in which $g$ is assumed to be constant, say $g\equiv 1$. As
shown in [2], or by simply invoking Weierstrass factorization theorem (see
[16]), in the complex variable $z=x+i\,y$, when $g\equiv 1$, the complex
gradient $u_{x}-i\,u_{y}$ of $u$ is uniquely determined by the Blaschke
product
$u_{x}-i\,u_{y}=e^{i\,\alpha}\prod_{j=1}^{n}\frac{z-z_{j}}{1-\overline{z}_{j}\,z},$
where $\alpha\in\mathbb{R}$ and $z_{1},\dots,z_{n}\in\Omega$ are given
parameters. In fact, each factor in the product has unitary modulus on the
unit circle. Thus, up to a rotation of an angle $\alpha$, the velocity field
of the fluid can be uniquely determined from its modulus on the boundary if we
know the position and the nature (the multiplicity, so as to speak) of its
stagnation points $z_{1},\dots,z_{n}$.
In order to conclude our motivations, it may be of interest to mention [6], in
which it can be found a possible application to encephalography by magnetic
means.
When $N\geq 3$, we can still use the Kelvin transformation to map the exterior
of the ball to its interior (or an exterior domain to a bounded one) by
preserving the harmonicity of functions. However, the boundary condition in
(1.1) changes quite considerably. In fact, if
${\mathcal{K}}w(y)=|y|^{2-N}\,w\left(\frac{y}{|y|^{2}}\right),\ \ y\neq 0,$
is the standard Kelvin transformation of a function
$w:\mathbb{R}^{N}\setminus\\{0\\}\to\mathbb{R}$, we have that
$\nabla{\mathcal{K}}w(y)=|x|^{N}\left\\{\nabla w(x)-\Bigl{[}2\,x\cdot\nabla
w(x)+(N-2)\,w(x)\Bigr{]}\frac{x}{|x|^{2}}\right\\},$
with $y=x/|x|^{2}$. In particular, if we apply the transformation to the
exterior of unit ball $B$ centered at the origin, we obtain that the condition
$|\nabla u|=g$ on ${\mathcal{S}}$ changes into
$|\nabla U+(N-2)\,U\,\nu|=g\ \mbox{ on }\ {\mathcal{S}},$
with $U={\mathcal{K}}u$. Here, $\nu$ denotes the exterior unit normal vector
field on ${\mathcal{S}}$.
Another feature for which the interior and exterior problems differ from one
another is that the latter admits solutions $u$ whose normal derivative
$u_{\nu}$ does not change sign on ${\mathcal{S}}$ (e.g. the fundamental
solution or, more in general, the capacity potential of a bounded domain),
while in the former the divergence theorem tells us that the mean value on
${\mathcal{S}}$ of $u_{\nu}$ must be equal to zero. The positivity of the
normal derivative of the solution has been useful in [8] to obtain the local
resolution of the exterior gravitational Backus problem for the Earth near the
so-called monopole $\Phi(x)=1/|x|$, based on a fixed-point argument. In other
words, when $N=3$ and $\Omega=\mathbb{R}^{3}\setminus\overline{B}$, the
existence and uniqueness of a solution $u$ of (1.1)-(1.2) is obtained as the
perturbation
$u=\Phi+v,$
where $v$ is harmonic in $\Omega$ and decays to zero at infinity. This result
holds if the data $g$ is sufficiently close to $1$ in a Hölder norm. (Notice
that $1$ is the modulus of the gradient of the monopole on ${\mathcal{S}}$.)
We also mention that the positivity of $u_{\nu}$ is also used in [4] to
construct a comparison principle for suitably defined viscosity solutions for
the exterior Backus problem and hence develop a nonlinear approach to the
problem.
When the positivity property is not available, the only existence result up to
date is given in [10]. There, we consider, in physical dimesion $N=3$, the
local resolution of the exterior Backus problem (1.1)-(1.2) near the so-called
dipole:
$d(x)=\frac{x_{3}}{|x|^{3}}.$
The gradient $\nabla d$ models the terrestrial geomagnetic field.
The problem of finding solutions of (1.1)-(1.2) of the type
$u=d+v,$
with $v$ harmonic in $\Omega$, which decays to zero at infinity, has another
level of difficulty, though. We can see this if we linearize (1.1)-(1.2) near
$d$. Indeed, we obtain the boundary value problem:
(1.3) $\Delta v=0\ \mbox{ in }\ \Omega,\quad\nabla d\cdot\nabla v=\varphi\
\mbox{ on }\ {\mathcal{S}},\quad v(x)\to 0\ \mbox{ as }\ |x|\to\infty.$
This can be classified as an irregular oblique derivative problem. In fact, in
contrast with the monopole case, in which the vector field $\nabla\Phi$
governing the linearized problem is nothing else than the normal field on
${\mathcal{S}}$, in (1.3), instead, the field $\nabla d$ points outward to the
Earth’s surface on the southern hemisphere, becomes tangential on the equator
$\mathcal{E}=\\{x\in{\mathcal{S}}:x_{N}=0\\}$, and points inward on the
northern hemisphere. For this reason, (1.3) suffers of two drawbacks. The
first one is a severe lack of uniqueness, since its solutions can be uniquely
determined only up to prescribing Dirichlet boundary values on $\mathcal{E}$.
The second one is a loss of regularity: the expected solution $v$ does not
gain the desired regularity. In other words, the regularity of $v$ does not
improve that of the data $\varphi$ by one order — it can be seen that it falls
short of $1/2$. This inconvenience makes the perturbation approach more
complicated, because the iterative scheme on which a fixed-point argument is
based upon loses derivatives at each step.
In presence of a loss of derivatives, the Nash-Moser implicit function theorem
has worked in other contexts (see the pioneering works [15, 13, 14, 7], for
instance). Unfortunately, in attacking the Backus problem, this plan is so far
out of reach. In fact, one would need sufficiently precise estimates for the
relevant oblique derivative problems. Namely, it is necessary to have an
accurate control not only for the solution of (1.3), but also for those of a
class of oblique derivative problems obtained by perturbing $\nabla d$.
Nevertheless, in [10] we showed that (1.1)-(1.2) is solvable near $d$ for
solutions which are axially symmetric around the Earth’s axis
$\mathcal{A}=\\{\lambda\,(0,0,1):\lambda\in\mathbb{R}\\}$. In fact, we show
that the solutions of the linearized problem (1.3) with this symmetry no
longer lose derivatives in an appropriate scale of fractional Sobolev spaces
on ${\mathcal{S}}$. This result is made possible by the use of spherical
harmonics on ${\mathcal{S}}$. As a consequence, the relevant fixed-point
scheme can be mended and the existence of a solution of (1.1)-(1.2) is
obtained if $g$ is sufficiently close to $|\nabla d|$ in some fractional
Sobolev norm.
In the present paper, for $N\geq 3$, we turn our attention to the local
resolution of the interior Backus problem (1.1) in the framework of Hölder
spaces. This framework is that used in [8] for the gravitational case. The
simplest instance in this case is to consider solutions of (1.1) in the form:
$u(x)=x_{N}+v(x),$
where $v$ is harmonic in $\Omega$. If we place the $x_{N}$-axis horizontally,
in the fluidmechanical framework mentioned above, we want to determine the
velocity of the fluid inside a domain from measurements of its modulus on the
boundary as a perturbation of that of an horizontal laminar flow with
potential
$f(x)=x_{N}.$
In this case, the associated linearized problem is simply:
(1.4) $\Delta v=0\ \mbox{ in }\ \Omega,\quad\partial_{x_{N}}v=\varphi\ \mbox{
on }\ {\mathcal{S}},$
where $\varphi$ is a given function. It is clear that the vector field
governing (1.4) is $e_{N}=(0,\dots,0,1)$, that shows similar qualitative
features to those of $\nabla d$, in the sense that $-e_{N}$ points inward on
the northern hemisphere, is tangential on $\mathcal{E}$, and points outward on
the southern hemisphere. Also in this case, though, a loss of derivatives
occurs for problem (1.4). Nevertheless, we shall see that the somewhat easier
oblique derivative condition in (1.4) allows a treatment of (1.1) in the
framework of Hölder spaces, in the cases where solutions are oddly symmetric
with respect to the hyperplane $x_{N}=0$ or axially symmetric around
$x_{N}$-axis.
In order to describe the main result of this paper, we need to introduce some
notation. For $k=0,1,2,\dots$ and $\alpha\in(0,1)$, we define the function
spaces:
$\displaystyle C_{\rm even}^{k+\alpha}(\overline{B})=\\{\varphi\in
C^{k+\alpha}(\overline{B}):\varphi(x^{\prime},x_{N})=\varphi(x^{\prime},-x_{N})\\},$
$\displaystyle C_{\rm odd}^{k+\alpha}(\overline{B})=\\{\varphi\in
C^{k+\alpha}(\overline{B}):\varphi(x^{\prime},x_{N})=-\varphi(x^{\prime},-x_{N})\\},$
$\displaystyle C_{\rm ax}^{k+\alpha}(\overline{B})=\\{\varphi\in
C^{k+\alpha}(\overline{B}):\varphi(x^{\prime},x_{N})=\varphi(|x^{\prime}|e_{1}^{\prime},x_{N})\\}.$
Here $e_{1}^{\prime}=(1,0,\ldots,0)\in\mathbb{R}^{N-1}$. We note that these
are closed subspaces of the space $C^{k+\alpha}(\overline{B})$ of
$k$-differentiable functions whose derivatives up to the order $k$ are
$\alpha$-Hölder continuous. The usual Hölder seminorm and norm on
$C^{k+\alpha}(\overline{B})$ are denoted by $[\cdot]_{\alpha,\overline{B}}$
and $|\cdot|_{k+\alpha,\overline{B}}$, respectively, and defined as:
$\displaystyle[u]_{\alpha,\overline{B}}=\sup_{x,y\in\overline{B},\,x\neq
y}\frac{|u(x)-u(y)|}{|x-y|^{\alpha}},$
$\displaystyle|u|_{k+\alpha,\overline{B}}=\sum_{|\gamma|\leq
k}|D^{\gamma}u|_{0,\overline{B}}+\sum_{|\gamma|=k}[D^{\gamma}u]_{\alpha,\overline{B}}.$
Here, $|\cdot|_{0,\overline{B}}$ stands for the standard (uniform) maximum
norm.
With these premises, we can state the main result of this paper as follows.
###### Theorem 1.1.
Let $\alpha\in(0,1)$ and set $\Omega=B$. Then there exist positive constants
$\delta_{0}$ and $C$ with the following properties.
* (i)
If $g\in C^{1+\alpha}_{\rm even}(\overline{B})$ and
$|g-1|_{1+\alpha,\overline{B}}\leq\delta_{0},$
then problem (1.1) has a solution $u\in C^{2}(B)\cap C^{1+\alpha}_{\rm
odd}(\overline{B})$ satisfying
$|u-f|_{1+\alpha,\overline{B}}\leq C\,|g-1|_{1+\alpha,\overline{B}}.$
* (ii)
If $g\in C^{1+\alpha}_{\rm ax}(\overline{B})$, $h\in\mathbb{R}$ and
$|g-1|_{1+\alpha,\overline{B}}\leq\delta_{0},$
then problem (1.1) has a solution $u\in C^{2}(B)\cap C^{1+\alpha}_{\rm
ax}(\overline{B})$ satisfying
$u=h\ \mbox{ on }\ \mathcal{E},$
and
$|u-h-f|_{1+\alpha,\overline{B}}\leq
C\,\left(|g-1|_{1+\alpha,\overline{B}}\right).$
Note that the solution $u$ obtained in (i) of Theorem 1.1 automatically
satisfies the condition $u=0$ on $\mathcal{E}$, since it is odd in the
variable $x_{N}$; while in (ii) we need to impose the additional boundary
condition on $\mathcal{E}$, due to the fact that (1.1) is invariant under the
addition of constants.
It is worthwhile noticing at this point that, even if the height $f$ and the
dipole $d$ are the Kelvin trasformations of one another, ${\mathcal{K}}$ maps
the linearized exterior problem (1.3) into the interior oblique derivative
problem:
$\Delta v=0\ \mbox{ in }\
B,\quad\partial_{x_{N}}v+(N-2)\,f\,\partial_{\nu}v+(N-1)(N-2)\,f\,v=\varphi\
\mbox{ on }\ {\mathcal{S}}.$
This problem is more difficult to treat, even if the relevant vector field
governing it has the same qualitative properties of $e_{N}$. A study of this
problem will be the theme of future work.
The proof of Theorem 1.1 hinges on some a priori estimates for the solution of
the linearized problem (1.4) subject to the Dirichlet-type condition
(1.5) $v=\psi\ \mbox{ on }\ \mathcal{E}.$
Thus, in Section 2, we will first derive an explicit representation formula
for the solutions of (1.4)-(1.5). We will also adapt a couple of lemmas in [1]
to our purposes, by making explicit the dependence of the relevant norms of
$v$ on those of the data $\varphi$. Then, in Section 3, we shall derive
crucial a priori estimates for the linearized problem (1.4)-(1.5) (see Theorem
3.1). Finally, in Section 4, based on these estimates, we will carry out the
proof of Theorem 1.1.
## 2\. An explicit integral representation formula
for the oblique derivative problem
In this section, we carry out explicit computations which lead to the
construction of an integral representation formula for the linearized problem
(1.4)-(1.5). We follow the scheme introduced in [1].
### 2.1. Uniqueness for problem (1.4)-(1.5)
Let a function space $C^{1}_{N}(\overline{B})$ be defined by
$C^{1}_{N}(\overline{B})=\\{v\in C(\overline{B}):\partial_{x_{N}}v\mbox{
exists and }\partial_{x_{N}}v\in C(\overline{B})\\}.$
Then $C^{1}_{N}(\overline{B})\cap C^{2}(B)$ is one of the natural spaces for
solutions of (1.4)-(1.5). We show that a uniqueness result holds in this
space. The argument follows the lines of one used in [9] and is based on
Hopf’s boundary lemma.
###### Proposition 2.1 (Uniqueness).
For any given $\varphi\in C({\mathcal{S}})$ and $\psi\in C(\mathcal{E})$, the
problem (1.4)-(1.5) has at most one solution of class
$C^{1}_{N}(\overline{B})\cap C^{2}(B)$.
###### Proof.
Let $v_{1}$ and $v_{2}$ be two solutions of class $C^{1}_{N}(\overline{B})\cap
C^{2}(B)$ of (1.4)-(1.5), and set $v=v_{1}-v_{2}$. Then $v$ solves (1.4)-(1.5)
with $\varphi\equiv 0$ and $\psi\equiv 0$. If $v(x)$ is a (positive) maximum
for $v$ on $\overline{B}$, then we have that $x\in{\mathcal{S}}$, by the
maximum principle, and $x\notin\mathcal{E}$, being as $v(x)>0$.
Now, if $x$ were in the upper hemisphere of ${\mathcal{S}}$, then we would
have that $\partial_{x_{N}}v(x)>0$, by Hopf’s boundary lemma, since $e_{N}$ is
an outward direction on that hemisphere. This is a contradiction. By a similar
argument, we infer that $x$ cannot belong to the lower hemisphere of
${\mathcal{S}}$. Thus, we conclude that $v\leq 0$ on $\overline{B}$. By
considering the minimum of $v$, we can infer that $v\geq 0$ on $\overline{B}$,
and hence $v\equiv 0$ on $\overline{B}$. ∎
We can easily use this proposition to infer that the solution $v$ of
(1.4)-(1.5) inherits possible symmetries of the data $\varphi$ and $\psi$. For
instance, if $\varphi$ and $\psi$ are axially symmetric around $x_{N}$-axis,
then $v$ is so.
### 2.2. Estimates for the Dirichlet problem for the Laplace equation
In the next result, we derive estimates for solutions of the Dirichlet problem
in $B$:
(2.1) $\Delta w=0\ \mbox{ in }\ B,\quad w=\varphi\ \mbox{ on }\
{\mathcal{S}}.$
It is well-known that if $\varphi\in C({\mathcal{S}})$, this problem has the
unique solution $w\in C(\overline{B})\cap C^{2}(B)$ given by the Poisson
integral formula:
(2.2) $w(x)=\int_{\mathcal{S}}P_{B}(x;y)\,\varphi(y)\,dS_{y}.$
Here, $P_{\Omega}$ stands for the Poisson kernel for a bounded domain
$\Omega$. In particular, $P_{B}$ is explicitly given by
$P_{B}(x;y)=\frac{1}{N\omega_{N}}\frac{1-|x|^{2}}{|x-y|^{N}}\ \mbox{ for }\
x\in B,\ y\in{\mathcal{S}},$
where $\omega_{N}$ is the volume of $B$. For our aims, we need the following
refinement of [1, Lemma 2.2].
###### Proposition 2.2.
Suppose that $\varphi\in C^{k+\alpha}(\overline{B})$ for some non-negative
integer $k$ and $\alpha\in[0,1)$. Let $\beta$ be a multi-index with
$|\beta|>k+\alpha$.
If $w$ is the solution of (2.1), then there exists a positive constant
$C=C(N,\beta,k,\alpha)$ such that
(2.3) $|D^{\beta}w(x)|\leq
C\,|\varphi|_{k+\alpha,\overline{B}}\,(1-|x|^{2})^{-|\beta|+k+\alpha}$
for all $x\in B$.
To prove Proposition 2.2, we first recall the following a priori estimate for
the Laplace equation. For the proofs, see for instance [5, Theorem 6.6,
Problem 6.2].
###### Lemma 2.3.
Suppose that $\varphi$ is of class $C^{k+\alpha}(\overline{B})$ for some
integer $k\geq 2$ and $\alpha\in(0,1)$. Then the solution $w$ of (2.1)
satisfies the inequality
$|w|_{k+\alpha,\overline{B}}\leq C|\varphi|_{k+\alpha,\overline{B}}$
for some positive constant $C=C(N,k,\alpha)$.
Next, we derive a pointwise estimate for the Poisson kernel.
###### Lemma 2.4.
For any multi-index $\beta$, it holds that
$D^{\beta}_{x}P_{B}(x;y)=\frac{a_{\beta}(x,y)}{|x-y|^{|\beta|+N-1}}\ \mbox{
for }\ x\in B,\ y\in{\mathcal{S}},$
where $|a_{\beta}(x,y)|\leq C_{*}$ for some positive constant $C_{*}$ which
only depends on $N$ and $\beta$.
###### Proof.
We have that
$P_{B}(x;y)=\frac{1}{N\omega_{N}}\frac{1-|x|^{2}}{|x-y|^{N}}=\frac{1}{N\omega_{N}}\frac{|y|^{2}-|x|^{2}}{|x-y|^{N}}=\frac{1}{N\omega_{N}}\frac{(x-y)\cdot(x+y)}{|x-y|^{N}}.$
Hence, if we set $z=x-y$, we obtain that
$P_{B}(z+y;y)=\frac{1}{N\omega_{N}}\frac{1}{|z|^{N-2}}+\frac{2}{N\omega_{N}}\,\frac{y\cdot
z}{|z|^{N}}.$
This function of $z$ is the sum of a $(2-N)$-homogeneous and a
$(1-N)$-homogeneous function. Thus, we infer that
$D^{\beta}_{z}P_{B}(z+y;y)=A(z;y)+B(z;y),$
where $A(z;y)$ and $B(z;y)$ are homogeneous of degree $2-N-|\beta|$ and
$1-N-|\beta|$ in $z$. As a consequence, we get:
$A(z;y)+B(z;y)=|z|^{1-N-|\beta|}\bigl{[}|z|\,A(z/|z|;y)+B(z/|z|;y)\bigr{]}.$
The function in the brackets is bounded since both $z/|z|$ and $y$ have a
unitary norm and $z\in 2B$. Therefore, we conclude by setting
$a_{\beta}(x,y)=|x-y|\,A\left(\frac{x-y}{|x-y|};y\right)+B\left(\frac{x-y}{|x-y|};y\right),$
for $x\in B$ and $y\in{\mathcal{S}}$. ∎
We also need the following bound.
###### Lemma 2.5.
Let a multi-index $\beta$ and a nonnegative number $\kappa$ satisfy
$|\beta|>\kappa$. Then, there exists a positive constant $C=C(N,\beta,\kappa)$
such that
$\int_{{\mathcal{S}}}|D^{\beta}_{x}P_{B}(x;y)|\,|y-x_{0}|^{\kappa}\,dS_{y}\leq
C(1-|x|)^{-|\beta|+\kappa},$
for all $x\in B\setminus\\{0\\}$, where $x_{0}=x/|x|\in{\mathcal{S}}$. When
$x=0$, we can choose $x_{0}$ to be any point in ${\mathcal{S}}$.
###### Proof.
When $x=0$, the bound easily follows from Lemma 2.4. Let $x\in
B\setminus\\{0\\}$, $x_{0}=x/|x|$, and set $r=1-|x|$. Then, for every
$y\in{\mathcal{S}}$, we have that
$|y-x|=\frac{2}{3}|y-x|+\frac{1}{3}|(y-x_{0})-(x-x_{0})|\geq\\\
\frac{2}{3}(|y|-|x|)+\frac{1}{3}(|y-x_{0}|-|x-x_{0}|)=\frac{1}{3}r+\frac{1}{3}|y-x_{0}|.$
This with Lemma 2.4 shows that
$\int_{{\mathcal{S}}}|D^{\beta}_{x}P_{B}(x;y)|\,|y-x_{0}|^{\kappa}\,dS_{y}=\int_{{\mathcal{S}}}\frac{|a_{\beta}(x,y)||y-x_{0}|^{\kappa}}{|y-x|^{|\beta|+N-1}}\,dS_{y}\leq\\\
3^{1-N-|\beta|}C_{*}\int_{\mathcal{S}}\frac{|y-x_{0}|^{\kappa}}{(r+|y-x_{0}|)^{|\beta|+N-1}}\,dS_{y}=\\\
3^{1-N-|\beta|}C_{*}r^{-|\beta|+\kappa}\int_{|rz+e_{N}|=1}\frac{|z|^{\kappa}}{(1+|z|)^{|\beta|+N-1}}\,dS_{z},$
where we have used the change of variables $y=x_{0}+r{\mathcal{R}}z$ with the
orthogonal matrix ${\mathcal{R}}$ satisfying
${\mathcal{R}}^{-1}x_{0}=e_{N}=(0,\ldots,0,1)$. The last integral on the
right-hand side of the above inequality is bounded with respect to $r$,
because as $r\to 0$ it converges to the integral
$\int_{\mathbb{R}^{N-1}}\frac{|z|^{\kappa}}{(1+|z|)^{|\beta|+N-1}}\,dS_{z},$
which is finite, being as $|\beta|>\kappa$. We thus obtain the desired
inequality. ∎
###### Proof of Proposition 2.2.
As usual, in this proof $C$ will denote a generic constant possibly depending
on $N,\beta,k$, and $\alpha$.
Pick any point $x_{0}\in{\mathcal{S}}$. Since $\varphi\in
C^{k+\alpha}(\overline{B})$, we can write the following standard Taylor
expansion for $\varphi$:
$\varphi(y)=\sum_{j=0}^{k}\sum_{|\gamma|=j}\frac{D^{\gamma}\varphi(x_{0})}{\gamma!}\,(y-x_{0})^{\gamma}+\sum_{|\gamma|=k}\frac{D^{\gamma}\varphi(x_{0}+\theta\,(y-x_{0}))-D^{\gamma}\varphi(x_{0})}{\gamma!}\,(y-x_{0})^{\gamma}.$
Here, we use the standard conventions on the multi-index notation. Thus,
integrating $\varphi(y)$ for $y\in{\mathcal{S}}$ against
$D^{\beta}_{x}P_{B}(x;y)$ gives that
(2.4) $D^{\beta}w(x)=\sum_{|\gamma|\leq
k}\frac{D^{\gamma}\varphi(x_{0})}{\gamma!}\,D^{\beta}h_{\gamma}(x)+R_{k}(x),$
where $h_{\gamma}$ is the solution of (2.1) with $\varphi=(\cdot-
x_{0})^{\gamma}$ and
$R_{k}(x)=\sum_{|\gamma|=k}\int_{\mathcal{S}}D^{\beta}_{x}P_{B}(x;y)\,\frac{D^{\gamma}\varphi(x_{0}+\theta\,(y-x_{0}))-D^{\gamma}\varphi(x_{0})}{\gamma!}\,(y-x_{0})^{\gamma}\,dS_{y}.$
Now, let $x_{*}$ be any point in $B\setminus\\{0\\}$ such that
$x_{*}=|x_{*}|x_{0}$. Then, Lemma 2.3 gives that
(2.5)
$|D^{\beta}h_{\gamma}(x_{*})|\leq|D^{\beta}h_{\gamma}|_{0,\overline{B}}\leq
C.$
Moreover, if $\alpha\in(0,1)$, Lemma 2.5 shows that
(2.6)
$|R_{k}(x_{*})|\leq\sum_{|\gamma|=k}\frac{[D^{\gamma}\varphi]_{\alpha,\overline{B}}}{\gamma!}\int_{\mathcal{S}}|D^{\beta}_{x}P_{B}(x_{*};y)|\,|y-x_{0}|^{k+\alpha}\,dS_{y}\leq\\\
C\sum_{|\gamma|=k}[D^{\gamma}\varphi]_{\alpha,\overline{B}}\,(1-|x_{*}|)^{-|\beta|+k+\alpha}.$
This inequality is also valid for $\alpha=0$, if
$[D^{\gamma}\varphi]_{\alpha,\overline{B}}$ is replaced by
$|D^{\gamma}\varphi|_{0,\overline{B}}$. Plugging (2.5) and (2.6) into (2.4),
then gives that
$|D^{\beta}w(x_{*})|\leq
C\,|\varphi|_{k+\alpha,\overline{B}}\,(1-|x_{*}|)^{-|\beta|+k+\alpha}.$
which yields (2.3), after an update of the constant $C$. ∎
### 2.3. Representation formulas for problem (1.4)-(1.5)
In order to obtain a representation formula, we consider the Dirichlet problem
(2.7) $-\Delta_{x^{\prime}}Z(x^{\prime})=\partial_{x_{N}}w(x^{\prime},0)\
\mbox{ in }\ D,\quad Z=\psi\ \mbox{ on }\ \partial D,$
where $w$ is the solution of (2.1) and
$D=\\{x^{\prime}\in\mathbb{R}^{N-1}:|x^{\prime}|<1\\}$. We identify $D$ and
$\partial D$ with the equatorial ball $\\{x=(x^{\prime},x_{N})\in
B:x_{N}=0\\}$ and the equator $\mathcal{E}$, respectively. From Lemma 2.5, we
see that $w$ satisfies
$|\partial_{x_{N}}w(x^{\prime},0)|\leq|\varphi|_{0,{\mathcal{S}}}\int_{{\mathcal{S}}}|\partial_{x_{N}}P_{B}(x^{\prime},0;y)|\,dS_{y}\leq
C\,|\varphi|_{0,{\mathcal{S}}}\,(1-|x^{\prime}|)^{-1}$
for some constant $C$. Therefore, for any $\varphi\in C({\mathcal{S}})$ and
$\psi\in C(\mathcal{E})$, the existence and uniqueness of solutions of (2.7)
in $C(\overline{D})\cap C^{2}(D)$ are guaranteed by [5, Theorem 4.9].
###### Proposition 2.6 (Existence and representation formula).
Suppose that $\varphi\in C({\mathcal{S}})$ and $\psi\in C(\mathcal{E})$. Let
$w$ and $Z$ be the solutions of (2.1) and (2.7), respectively, and set
(2.8) $W(x)=\int_{0}^{x_{N}}w(x^{\prime},t)\,dt,\quad
x=(x^{\prime},x_{N})\in\overline{B},$
where $w$ is defined in (2.2). Then the unique solution $v$ of class
$C^{1}_{N}(\overline{B})\cap C^{2}(B)$ of (1.4)-(1.5) is given by
(2.9) $v(x)=W(x)+Z(x^{\prime}),\quad x=(x^{\prime},x_{N})\in\overline{B}.$
###### Proof.
Let $v$ be defined by (2.9). Then $v=W+Z\in C(\overline{B})\cap C^{2}(B)$,
since we know that $w\in C(\overline{B})\cap C^{2}(B)$ and $Z\in
C(\overline{D})\cap C^{2}(D)$. Moreover, we have that $\partial_{x_{N}}v=w\in
C(\overline{B})$, and therefore $v\in C^{1}_{N}(\overline{B})\cap C^{2}(B)$.
Since $W(x^{\prime},0)=0$, we see that
$\partial_{x_{N}}v=w=\varphi\ \mbox{ on }\ {\mathcal{S}},\quad v=Z=\psi\
\mbox{ on }\ \mathcal{E}.$
Hence the assertion follows if we show that $v$ is harmonic in $B$. By a
direct calculation, we have that
$\Delta
W(x)=\int_{0}^{x_{N}}\Delta_{x^{\prime}}w(x^{\prime},t)\,dt+\partial_{x_{N}}w(x^{\prime},x_{N})=\\\
-\int_{0}^{x_{N}}\partial_{x_{N}x_{N}}^{2}w(x^{\prime},t)\,dt+\partial_{x_{N}}w(x^{\prime},x_{N})=\partial_{x_{N}}w(x^{\prime},0).$
We thus infer that
$\Delta v(x)=\Delta
W(x)+\Delta_{x^{\prime}}Z(x^{\prime})=\partial_{x_{N}}w(x^{\prime},0)-\partial_{x_{N}}w(x^{\prime},0)=0,$
as desired. ∎
Even if it is not needed in the proof of Theorem 1.1, we also derive for
future reference an explicit integral representation formula. The formula may
be helpful for numerical approximations. To derive the formula, we recall that
the fundamental solution $\Gamma_{d}$ of the Laplace equation in a
$d$-dimensional Euclidean space ($d\geq 2$) is given by
$\Gamma_{2}(x)=\frac{1}{2\pi}\,\log\frac{1}{|x|},\qquad\Gamma_{d}(x)=\frac{1}{d(d-2)\,\omega_{d}}|x|^{2-d}\
\mbox{ if }\ d\geq 3.$
Then, the Green’s function for $D$ is written as
$G_{D}(x^{\prime};y^{\prime})=\Gamma_{N-1}(x^{\prime}-y^{\prime})-\Gamma_{N-1}\left(|x^{\prime}|({\mathcal{I}}(x^{\prime})-y^{\prime})\right),$
where ${\mathcal{I}}$ denotes the inversion
${\mathcal{I}}(x^{\prime})=x^{\prime}/|x^{\prime}|^{2}$ for $x^{\prime}\neq
0$.
If we now define the kernel
$K(x;y)=\int_{0}^{x_{N}}P_{B}(x^{\prime},t;y)\,dt+\int_{D}G_{D}(x^{\prime},z^{\prime})\,\partial_{x_{N}}P_{B}(z^{\prime},0;y)\,dz^{\prime},$
for $x=(x^{\prime},x_{N})\in B$ and $y\in{\mathcal{S}}$, the representation
formula is given as follows.
###### Proposition 2.7 (Integral representation formula).
Suppose that $\varphi\in C({\mathcal{S}})$ and $\psi\in C(\mathcal{E})$. Then,
the function defined by
(2.10)
$v(x)=\int_{\mathcal{S}}K(x;y)\,\varphi(y)\,dS_{y}+\int_{\mathcal{E}}P_{D}(x^{\prime};y^{\prime})\,\psi(y^{\prime})\,dS_{y^{\prime}}\
\mbox{ for }\ x\in B$
coincides with the unique solution of class $C^{1}_{N}(\overline{B})\cap
C^{2}(B)$ of the problem (1.4)-(1.5).
###### Proof.
By the well-known representation formula for the Dirichlet problem for the
Poisson equation, we know that
$Z(x^{\prime})=\int_{D}G_{D}(x^{\prime};z^{\prime})\,\partial_{x_{N}}w(z^{\prime},0)\,dz^{\prime}+\int_{\mathcal{E}}P_{D}(x^{\prime};z^{\prime})\,\psi(z^{\prime})\,dS_{z^{\prime}}.$
With the definition of $w$ in mind, by the Fubini theorem we then infer that
$\int_{D}G_{D}(x^{\prime};z^{\prime})\,\partial_{x_{N}}w(z^{\prime},0)\,dz^{\prime}=\int_{S}\left[\int_{D}G_{D}(x^{\prime};z^{\prime})\,\partial_{x_{N}}P_{B}(z^{\prime},0;y)\,dz^{\prime}\right]\varphi(y)\,dS_{y}.$
Being as $x^{\prime}\in D$, the Fubini theorem is applicable in this formula,
because the function $F$ defined a.e. on $D\times{\mathcal{S}}$ by
$F(z^{\prime},y)=\varphi(y)\,\partial_{x_{N}}P_{B}(z^{\prime},0;y)\,G_{D}(x^{\prime};z^{\prime})$
is in $L^{1}(D\times{\mathcal{S}})$. In fact, we have that
$\int_{D\times{\mathcal{S}}}|F(z^{\prime},y)|(dz^{\prime}\times
dS_{y})=\int_{D}G_{D}(x^{\prime};z^{\prime})\left[\int_{\mathcal{S}}|\partial_{x_{N}}P_{B}(z^{\prime},0;y)||\varphi(y)|\,dS_{y}\right]dz^{\prime}$
(see [11, Theorem 1.12]). The right-hand side is finite thanks to the
properties of $G_{D}$ and Lemma 2.5 with $\kappa=0$ and $|\beta|=1$.
Finally, that
$W(x)=\int_{\mathcal{S}}\left[\int_{0}^{x_{N}}P_{B}(x^{\prime},t;y)\,dt\right]\varphi(y)\,dS_{y}$
follows from (2.8), again by a straightforward application of the Fubini
theorem. We have thus proved that the right-hand side of (2.10) coincides with
$W+Z$. ∎
## 3\. A priori estimates for the linearized problem
A crucial a priori bound we will use to prove Theorem 1.1 is contained in the
following theorem.
###### Theorem 3.1.
Let $\alpha\in(0,1)$ and suppose that $\varphi\in C^{1+\alpha}(\overline{B})$
and $\psi\in C^{3/2+\alpha}(\overline{D})$. Then a solution $v$ of (1.4)-(1.5)
has the properties
$v\in C^{1+\alpha}(\overline{B}),\quad\partial_{x_{N}}v\in
C^{1+\alpha}(\overline{B}),\quad x_{N}D^{2}_{x^{\prime}}v\in
C^{\alpha}(\overline{B}).$
Moreover, the following inequality holds for some positive constant $C$
independent of $\varphi$ and $\psi$:
(3.1)
$|v|_{1+\alpha,\overline{B}}+|\partial_{x_{N}}v|_{1+\alpha,\overline{B}}+|x_{N}D^{2}_{x^{\prime}}v|_{\alpha,\overline{B}}\leq
C\left(|\varphi|_{1+\alpha,\overline{B}}+|\psi|_{3/2+\alpha,\overline{D}}\right).$
We will use the following simple estimate, which is a refinement of [1, Lemma
3.2].
###### Lemma 3.2.
Let $\kappa>0$. Then there exists a constant $C>0$ such that
$|x_{N}|\int_{0}^{|x_{N}|}\frac{dt}{(1-|x^{\prime}|^{2}-t^{2})^{1+\kappa}}\leq\frac{C}{(1-|x|^{2})^{\kappa}}$
for all $x\in B$.
###### Proof.
Set $|x_{N}|=\sigma\,\sqrt{1-|x^{\prime}|^{2}}$; it holds that $0\leq\sigma<1$
for $x=(x^{\prime},x_{N})\in B$. By the change of variable
$t=s\,\sqrt{1-|x^{\prime}|^{2}}$, we have that
$\int_{0}^{|x_{N}|}\frac{dt}{(1-|x^{\prime}|^{2}-t^{2})^{1+\kappa}}=\frac{1}{(1-|x^{\prime}|^{2})^{1/2+\kappa}}\int_{0}^{\sigma}\frac{ds}{(1-s^{2})^{1+\kappa}},$
and hence
$|x_{N}|(1-|x|^{2})^{\kappa}\int_{0}^{|x_{N}|}\frac{dt}{(1-|x^{\prime}|^{2}-t^{2})^{1+\kappa}}=\sigma(1-\sigma^{2})^{\kappa}\int_{0}^{\sigma}\frac{ds}{(1-s^{2})^{1+\kappa}}.$
The right-hand side is bounded by some constant $C$, since L’Hôpital’s rule
shows that its limit as $\sigma\to 1^{-}$ is equal to $1/(2\kappa)$. Thus the
lemma follows. ∎
The following lemma is essentially shown in [1, Lemma 2.5]. For the sake of
completeness, we give a proof.
###### Lemma 3.3.
Let $v\in C^{1}(B)\cap C(\overline{B})$. Suppose that there exist a positive
constant $M$ and an exponent $\alpha\in(0,1)$ such that
$|\nabla v(x)|\leq M(1-|x|^{2})^{-1+\alpha}\ \mbox{ for all }\ x\in B.$
Then $v\in C^{\alpha}(\overline{B})$ and it holds that
(3.2) $[v]_{\alpha,\overline{B}}\leq CM,$
for some positive constant $C$ only depending on $\alpha$.
###### Proof.
The assumption on $v$ gives that
(3.3) $|\nabla v(x)|\leq
M\,(1-|x|)^{-1+\alpha}=M\,|x-\overline{x}|^{-1+\alpha}\ \mbox{ with }\
\overline{x}=x/|x|,$
since $\alpha<1$.
(i) Let $\vartheta=|x-\overline{x}|$ and
$\ell=(x-\overline{x})/\vartheta=-\overline{x}$. Then, we have that
$|v(x)-v(\overline{x})|=\left|\int_{0}^{\vartheta}\nabla
v(\overline{x}+t\,\ell)\cdot\ell\,dt\right|\leq
M\,\int_{0}^{\vartheta}t^{-1+\alpha}dt=\frac{M}{\alpha}\vartheta^{\alpha}.$
(ii) Let $\overline{x}$ and $\overline{y}$ be arbitrary points on
${\mathcal{S}}$ with $\vartheta=|\overline{x}-\overline{y}|<1$. Take
$x=(1-\vartheta)\,\overline{x}$ and $y=(1-\vartheta)\,\overline{y}$. Then, for
an intermediate point $\xi$ between $x$ and $y$, we have that
$|v(\overline{x})-v(\overline{y})|\leq|v(x)-v(\overline{x})|+|v(x)-v(y)|+|v(y)-v(\overline{y})|\leq\\\
\frac{2\,M}{\alpha}\vartheta^{\alpha}+|\nabla
v(\xi)|\,|x-y|\leq\frac{2\,M}{\alpha}\vartheta^{\alpha}+M\,\vartheta^{-1+\alpha}|x-y|\leq\\\
\frac{2\,M}{\alpha}\vartheta^{\alpha}+M\,\vartheta^{\alpha}.$
Here, we have used (i), (3.3) and the fact that
$|\xi-\overline{\xi}|\geq|x-\overline{x}|=\vartheta$.
If $\vartheta=|\overline{x}-\overline{y}|\geq 1$, then one can choose a finite
number of points on ${\mathcal{S}}$, say
$\overline{x}=\overline{x}_{0},\overline{x}_{1},\overline{x}_{2},\overline{x}_{3},\overline{x}_{4}=\overline{y}$
with $|\overline{x}_{i}-\overline{x}_{i+1}|<1$ so that the previous estimate
applies, and the combination of the estimates yields the desired inequality.
Therefore, $|v(\overline{x})-v(\overline{y})|\leq
CM\,|\overline{x}-\overline{y}|^{\alpha}$, for some constant $C$.
(iii) Now, take $x,y\in B$, set $\vartheta=|x-y|$, and let $\overline{x}$ and
$\overline{y}$ be the usual projections of $x$ and $y$ on ${\mathcal{S}}$. We
can always assume that $|y-\overline{y}|\geq|x-\overline{x}|$. If
$|x-\overline{x}|\geq\vartheta$, then, for an intermediate point $\xi$ between
$x$ and $y$,
$|v(x)-v(y)|\leq|\nabla v(\xi)|\,|x-y|\leq
M\,\vartheta^{-1+\alpha}|x-y|=M\,\vartheta^{\alpha},$
thanks to the inequalities
$|\xi-\overline{\xi}|\geq|x-\overline{x}|\geq\vartheta$. If
$|x-\overline{x}|<\vartheta$ instead, we first infer that
$|y-\overline{y}|\leq|y-\overline{x}|\leq|y-x|+|x-\overline{x}|<2\,\vartheta.$
Thus, (i) gives that $|v(x)-v(\overline{x})|\leq M\,\vartheta^{\alpha}/\alpha$
and $|v(y)-v(\overline{y})|\leq 2^{\alpha}\,M\,\vartheta^{\alpha}/\alpha$,
while (ii) yields:
$|v(\overline{x})-v(\overline{y})|\leq
CM\,|\overline{x}-\overline{y}|^{\alpha}\leq
4^{\alpha}\,CM\,\vartheta^{\alpha}.$
We then conclude thanks to the triangle inequality. The bound (3.2) then
follows at once. ∎
###### Proof of Theorem 3.1.
Throughout the proof $C$ will denote a generic positive constant only
depending on $N$ and $\alpha$.
From Proposition 2.6, the unique solution $v\in C^{1}_{N}(\overline{B})\cap
C^{2}(B)$ of (1.4)-(1.5) is given by
$v(x)=W(x)+Z(x^{\prime})=\int_{0}^{x_{N}}w(x^{\prime},t)\,dt+Z(x^{\prime}),\quad
x=(x^{\prime},x_{N})\in B,$
$w$ and $Z$ being the solutions of (2.1) and (2.7), respectively. We note that
$Z$ is expressed as $Z=Z_{1}+Z_{2}$, where $Z_{1}$ is the solution of (2.7)
with $\partial_{x_{N}}w(x^{\prime},0)$ replaced by $0$ and $Z_{2}$ is the
solution of (2.7) with $\psi=0$. Hence, it will be enough to prove the three
estimates:
(3.4)
$\displaystyle|W|_{1+\alpha,\overline{B}}+|\partial_{x_{N}}W|_{1+\alpha,\overline{B}}+|x_{N}D^{2}_{x^{\prime}}W|_{\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}},$ (3.5)
$\displaystyle|Z_{1}|_{1+\alpha,\overline{D}}+|x_{N}D^{2}_{x^{\prime}}Z_{1}|_{\alpha,\overline{B}}\leq
C\,|\psi|_{3/2+\alpha,\overline{D}}.$ (3.6)
$\displaystyle|Z_{2}|_{1+\alpha,\overline{D}}+|x_{N}D^{2}_{x^{\prime}}Z_{2}|_{\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}.$
We first derive these inequalities under the additional assumptions
$\varphi\in C^{2+\alpha}(\overline{B})$ and $\psi\in
C^{2+\alpha}(\overline{D})$. We then have
(3.7) $w\in C^{2+\alpha}(\overline{B}),\quad W\in
C^{2+\alpha}(\overline{B}),\quad Z_{1}\in C^{2+\alpha}(\overline{D}).$
We note that the following inequality holds:
(3.8) $|w|_{1+\alpha,\overline{B}}\leq C\,|\varphi|_{1+\alpha,\overline{B}}.$
Indeed, this is shown as follows. Since
$\int_{{\mathcal{S}}}P_{B}(x;y)\,dS_{y}=1,\quad\int_{{\mathcal{S}}}P_{B}(x;y)\,y\,dS_{y}=x,$
we have
$\nabla
w(x)=\nabla\varphi(x)+\int_{{\mathcal{S}}}\nabla_{x}P_{B}(x;y)\,[\varphi(y)-\varphi(x)-\nabla\varphi(x)\cdot(y-x)]\,dS_{y}.$
This with Lemma 2.4 shows that
$|\nabla
w(x)|\leq|\nabla\varphi(x)|+C[\nabla\varphi]_{\alpha,\overline{B}}\int_{{\mathcal{S}}}|x-y|^{-N+1+\alpha}dS_{y}.$
We easily find that the integral on the right is finite and is bounded by some
constant independent of $x$, and hence $|\nabla w|_{0,\overline{B}}\leq
C|\nabla\varphi|_{\alpha,\overline{B}}$. Since Proposition 2.2 gives the
inequality $|D^{2}w(x)|\leq
C|\varphi|_{1+\alpha,\overline{B}}(1-|x|^{2})^{-1+\alpha}$, we have that
$[\nabla w]_{\alpha,\overline{B}}\leq C|\varphi|_{1+\alpha,\overline{B}}$ by
Lemma 3.3. Therefore (3.8) holds.
First, we observe that (3.6) easily follows from the Schauder estimates for
the Poisson equation and (3.8). In fact, we have that
$|Z_{2}|_{2+\alpha,\overline{D}}\leq
C\,|\partial_{x_{N}}w(\cdot,0)|_{\alpha,\overline{D}}\leq
C\,|w|_{1+\alpha,\overline{B}}\leq C\,|\varphi|_{1+\alpha,\overline{B}}.$
Next, we prove (3.4). It is clear that
$|W|_{1+\alpha,\overline{B}}\leq C|w|_{1+\alpha,\overline{B}}.$
This together with (3.8) and the fact that $\partial_{x_{N}}W=w$ then yields:
$|W|_{1+\alpha,\overline{B}}+|\partial_{x_{N}}W|_{1+\alpha,\overline{\Omega}}\leq
C\,|w|_{1+\alpha,\overline{B}}\leq C\,|\varphi|_{1+\alpha,\overline{B}}.$
Therefore, we only need to verify that
(3.9) $|x_{N}D^{2}_{x^{\prime}}W|_{\alpha,\overline{B}}\leq
C|\varphi|_{1+\alpha,\overline{B}}.$
To prove this inequality, we examine pointwise estimates of
$D^{2}_{x^{\prime}}W$. Proposition 2.2 yields that
$|D^{2}_{x^{\prime}}W(x)|=\left|\int_{0}^{x_{N}}D^{2}_{x^{\prime}}w(x^{\prime},t)dt\right|\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\int_{0}^{|x_{N}|}\frac{dt}{(1-|x^{\prime}|^{2}-t^{2})^{1-\alpha}}.$
We estimate the last integral in two ways. First, by monotonicity in $t$ and
$|x_{N}|$, we see that the integral can be bounded by
$(1-|x|^{2})^{-1+\alpha}$. From this, we infer that
(3.10) $|D^{2}_{x^{\prime}}W(x)|\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\,(1-|x|^{2})^{-1+\alpha}.$
Secondly, by the inequality
(3.11) $x_{N}^{2}\leq 1-|x^{\prime}|^{2}\ \mbox{ for }\ (x^{\prime},x_{N})\in
B,$
we get that
$\int_{0}^{|x_{N}|}\frac{dt}{(1-|x^{\prime}|^{2}-t^{2})^{1-\alpha}}\leq\int_{0}^{|x_{N}|}\frac{dt}{(x_{N}^{2}-t^{2})^{1-\alpha}}=|x_{N}|^{-1+2\alpha}\int_{0}^{1}\frac{ds}{(1-s^{2})^{1-\alpha}},$
after the change of variable $t=|x_{N}|\,s$. This gives the bound:
(3.12) $|x_{N}D^{2}_{x^{\prime}}W|_{0,\overline{B}}\leq
C|\varphi|_{1+\alpha,\overline{B}}.$
In order to estimate the Hölder seminorm
$[x_{N}D^{2}_{x^{\prime}}W]_{\alpha,\overline{B}}$, we consider the
derivatives of $D^{2}_{x^{\prime}}W$. We use Proposition 2.2 to obtain
$|D^{3}_{x^{\prime}}W(x)|=\left|\int_{0}^{x_{N}}D^{3}_{x^{\prime}}w(x^{\prime},t)dt\right|\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\int_{0}^{|x_{N}|}\frac{dt}{(1-|x^{\prime}|^{2}-t^{2})^{2-\alpha}}.$
Applying Lemma 3.2 to the right-hand side, we infer that
(3.13) $|x_{N}D^{3}_{x^{\prime}}W(x)|\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\,(1-|x|^{2})^{-1+\alpha}.$
Furthermore, we see from Proposition 2.2 that
(3.14) $|\partial_{x_{N}}D^{2}_{x^{\prime}}W(x)|=|D^{2}_{x^{\prime}}w(x)|\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\,(1-|x|^{2})^{-1+\alpha}.$
Combining (3.10), (3.13) and (3.14), we deduce that
$|x_{N}D^{3}_{x^{\prime}}W(x)|+|\partial_{x_{N}}(x_{N}D^{2}_{x^{\prime}}W(x))|\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\,(1-|x|^{2})^{-1+\alpha}.$
Thus, by (3.7) and Lemma 3.3, we obtain that
$[x_{N}D^{2}_{x^{\prime}}W]_{\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}$. This together with (3.12) shows that
(3.9) holds.
Finally, we verify (3.5). Note that the same estimates as in Proposition 2.2
and (3.8) hold if $w$, $\varphi$ and $B$ are replaced by $Z_{1}$, $\psi$ and
$D$, respectively. By (3.8), we see that
(3.15) $|Z_{1}|_{1+\alpha,\overline{D}}\leq
C\,|\psi|_{1+\alpha,\overline{D}}\leq C\,|\psi|_{3/2+\alpha,\overline{D}}.$
Proposition 2.2 and (3.11) show that
$\displaystyle|x_{N}D_{x^{\prime}}^{2}Z_{1}(x^{\prime})|\leq
C\,|\psi|_{3/2,\overline{D}}\,|x_{N}|\,(1-|x^{\prime}|)^{-\frac{1}{2}}\leq
C\,|\psi|_{3/2+\alpha,\overline{D}},$
$\displaystyle|x_{N}D_{x^{\prime}}^{3}Z_{1}(x^{\prime})|\leq
C\,|\psi|_{3/2+\alpha,\overline{D}}\,|x_{N}|\,(1-|x^{\prime}|)^{-\frac{3}{2}+\alpha}\leq
C\,|\psi|_{3/2+\alpha,\overline{D}}\,(1-|x|)^{-1+\alpha},$
and
$|\partial_{x_{N}}(x_{N}D_{x^{\prime}}^{2}Z_{1}(x^{\prime}))|=|D_{x^{\prime}}^{2}Z_{1}(x^{\prime})|\leq\\\
C\,|\psi|_{1+\alpha,\overline{D}}\,(1-|x^{\prime}|)^{-1+\alpha}\leq
C\,|\psi|_{3/2+\alpha,\overline{D}}\,(1-|x|)^{-1+\alpha}.$
Hence it follows from (3.7) and Lemma 3.3 that
$|x_{N}D_{x^{\prime}}^{2}Z_{1}|_{\alpha,\overline{B}}\leq
C\,|\psi|_{3/2+\alpha,\overline{D}}.$
This and (3.15) gives (3.5).
Now, if $\varphi\in C^{1+\alpha}(\overline{B})$ and $\psi\in
C^{3/2+\alpha}(\overline{D})$, we take sequences $\\{\varphi_{j}\\}\subset
C^{2+\alpha}(\overline{B})$ and $\\{\psi_{j}\\}\subset
C^{2+\alpha}(\overline{D})$ such that
(3.16) $\displaystyle\varphi_{j}\to\varphi\ \mbox{ in }\
C(\overline{B}),\quad\psi_{j}\to\psi\ \mbox{ in }\ C(\overline{D})\ \mbox{ as
}\ j\to\infty,$ (3.17) $\displaystyle|\varphi_{j}|_{1+\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}},\quad|\psi_{j}|_{3/2+\alpha,\overline{D}}\leq
C\,|\psi|_{3/2+\alpha,\overline{D}},\quad j=1,2,\ldots.$
Let $w_{j}$ be a solution of (2.1) with $\varphi=\varphi_{j}$. Then, (3.16)
and the Schauder interior estimates for Poisson’s equation give that $w_{j}\to
w$ in $C^{2}_{\rm loc}(B)$ as $j\to\infty$. Hence
$W_{j}(x)=\int_{0}^{x_{N}}w_{j}(x^{\prime},t)\,dt$ converges to $W$ in
$C^{2}_{\rm loc}(B)$. Since the inequality
$|x_{N}D^{2}_{x^{\prime}}W_{j}|_{\alpha,\overline{B}}\leq
C|\varphi_{j}|_{1+\alpha,\overline{B}}$ is valid, we obtain (3.9) by using
(3.17) and letting $j\to\infty$. In a similar way, approximating $\psi$ by
$\psi_{j}$ gives that (3.5) is still valid for $\psi\in
C^{3/2+\alpha}(\overline{D})$. Thus the proof is complete. ∎
## 4\. Local existence for the
interior Backus problem with symmetric data
This section is devoted to the proof of Theorem 1.1.
### 4.1. The nonlinear operator ${\mathcal{T}}$
We define an operator ${\mathcal{T}}$ by setting
${\mathcal{T}}[\varphi]=|\nabla v|^{2},$
where $v$ satisfies (1.4)-(1.5) with $\psi=0$. The next proposition shows that
${\mathcal{T}}$ is locally bounded and locally Lipschitz continuous on bounded
subsets of $C_{\rm even}^{1+\alpha}(\overline{B})$.
###### Proposition 4.1.
We have that ${\mathcal{T}}$ is a mapping from $C_{\rm
even}^{1+\alpha}(\overline{B})$ into itself. Furthermore, there are positive
constants $C_{1}$ and $C_{2}$ such that
(4.1)
$\displaystyle\bigl{|}{\mathcal{T}}[\varphi]\bigr{|}_{1+\alpha,\overline{B}}\leq
C_{1}\,|\varphi|_{1+\alpha,\overline{B}}^{2},$ (4.2)
$\displaystyle\bigl{|}{\mathcal{T}}[\varphi_{1}]-{\mathcal{T}}[\varphi_{2}]\bigr{|}_{1+\alpha,\overline{B}}\leq
C_{2}\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}$
for all $\varphi,\varphi_{1},\varphi_{2}\in C_{\rm
even}^{1+\alpha}(\overline{B})$.
To prove this proposition, we need the following simple lemma.
###### Lemma 4.2.
Let $k$ be a non-negative integer and take $\alpha\in(0,1)$. Suppose that a
function $v$ defined on $\overline{B}$ is such that $\partial_{x_{N}}v$ in
$C^{k+\alpha}(\overline{B})$ and is zero on $\overline{D}\times\\{0\\}$. Then
the function defined by
(4.3) $\omega(x)=\begin{cases}\displaystyle\frac{v(x)}{x_{N}}&\mbox{ for }\
x_{N}\neq 0,\\\ \partial_{x_{N}}v(x^{\prime},0)&\mbox{ for }\
x_{N}=0,\end{cases}$
belongs to $C^{k+\alpha}(\overline{B})$ and satisfies the inequality:
(4.4)
$|\omega|_{k+\alpha,\overline{B}}\leq|\partial_{x_{N}}v|_{k+\alpha,\overline{B}}.$
###### Proof.
By the fundamental theorem of calculus, we have that
$v(x^{\prime},x_{N})=x_{N}\int_{0}^{1}\partial_{x_{N}}v(x^{\prime},x_{N}t)\,dt,$
and hence $\omega$ can be written as
$\omega(x)=\int_{0}^{1}\partial_{x_{N}}v(x^{\prime},x_{N}t)\,dt.$
Thus, we see that the partial derivative $D_{x}^{\beta}\omega$ exists for any
multi-index $\beta=(\beta_{1},\ldots,\beta_{N})$ with $|\beta|\leq k$ and is
given by
$D^{\beta}_{x}\omega(x)=\int_{0}^{1}t^{\beta_{N}}D^{\beta}_{x}\partial_{x_{N}}v(x^{\prime},x_{N}t)dt.$
The assertion then easily follows from this formula. ∎
###### Proof of Proposition 4.1.
Throughout the proof, $i$ is any index in $\\{1,2,\ldots,N-1\\}$ and $C$ is a
generic positive constant only depending on $N$ and $\alpha$.
(i) Let $\varphi\in C_{\rm even}^{1+\alpha}(\overline{B})$ and let $v$ denote
the solution of the problem (1.4)–(1.5) with $\psi=0$. Since $\varphi$ is even
in $x_{N}$, we see that the function $-v(x^{\prime},-x_{N})$ is also a
solution of (1.4)–(1.5) with $\psi=0$. By uniqueness, we infer that $v$ is odd
in $x_{N}$. In particular, $v(x^{\prime},0)=0$, and hence
(4.5)
$\partial_{x_{j}}(\partial_{x_{i}}v)^{2}=2\partial_{x_{i}}v\,\partial^{2}_{x_{i}x_{j}}v=2\left(\partial_{x_{i}}\omega\right)\left(x_{N}\partial^{2}_{x_{i}x_{j}}v\right)\
\mbox{ for }\ j=1,\ldots,N,$
where $\omega$ is a function given by (4.3). From Theorem 3.1 and Lemma 4.2,
we see that the right-hand side of this equality is in
$C^{\alpha}(\overline{B})$. Therefore, we have that
$(\partial_{x_{i}}v)^{2}\in C^{1+\alpha}(\overline{B})$. This together with
the fact that $(\partial_{x_{N}}v)^{2}\in C^{1+\alpha}(\overline{B})$, which
follows from Theorem 3.1, gives:
${\mathcal{T}}[\varphi]=|\nabla v|^{2}\in C^{1+\alpha}(\overline{B}).$
Since the fact that $v$ is odd in $x_{N}$ yields that $|\nabla
v(x^{\prime},x_{N})|=|\nabla v(x^{\prime},-x_{N})|$, we conclude that
${\mathcal{T}}$ is a mapping from $C_{\rm even}^{1+\alpha}(\overline{B})$ to
itself.
(ii) Let us derive (4.1). By (4.5), we have:
$\left|(\partial_{x_{i}}v)^{2}\right|_{1+\alpha,\overline{B}}=\left|(\partial_{x_{i}}v)^{2}\right|_{0,\overline{B}}+\sum_{j=1}^{N}\left|\partial_{x_{j}}(\partial_{x_{i}}v)^{2}\right|_{\alpha,\overline{B}}=\\\
\left|(\partial_{x_{i}}v)^{2}\right|_{0,\overline{B}}+2\sum_{j=1}^{N-1}\left|\left(\partial_{x_{i}}\omega\right)\left(x_{N}\partial^{2}_{x_{i}x_{j}}v\right)\right|_{\alpha,\overline{B}}+2\left|\partial_{x_{i}}v\,\partial^{2}_{x_{i}x_{N}}v)\right|_{\alpha,\overline{B}}.$
From (3.1), the first and third terms of the rightest-hand side are handled as
$\displaystyle\left|(\partial_{x_{i}}v)^{2}\right|_{0,\overline{B}}\leq\left|\partial_{x_{i}}v\right|_{0,\overline{B}}^{2}\leq|v|_{1+\alpha,\overline{B}}^{2}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}^{2},$
$\displaystyle\left|\partial_{x_{i}}v\,\partial^{2}_{x_{i}x_{N}}v\right|_{\alpha,\overline{B}}\leq\left|\partial_{x_{i}}v\right|_{\alpha,\overline{B}}\left|\partial^{2}_{x_{i}x_{N}}v\right|_{\alpha,\overline{B}}\leq|v|_{1+\alpha,\overline{B}}\left|\partial_{x_{N}}v\right|_{1+\alpha,\overline{B}}\leq
C|\varphi|_{1+\alpha,\overline{B}}^{2}.$
Furthermore, (3.1) and (4.4) show that
$\left|\left(\partial_{x_{i}}\omega\right)\left(x_{N}\partial^{2}_{x_{i}x_{j}}v\right)\right|_{\alpha,\overline{B}}\leq\left|\partial_{x_{i}}\omega\right|_{\alpha,\overline{B}}\left|x_{N}\partial^{2}_{x_{i}x_{j}}v\right|_{\alpha,\overline{B}}\leq\\\
\left|\omega\right|_{1+\alpha,\overline{B}}\left|x_{N}\partial^{2}_{x_{i}x_{j}}v\right|_{\alpha,\overline{B}}\leq\left|\partial_{x_{N}}v\right|_{1+\alpha,\overline{B}}\left|x_{N}\partial^{2}_{x_{i}x_{j}}v\right|_{\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}^{2},$
for $j=1,\ldots,N-1$. From these estimates it follows that
(4.6) $\left|(\partial_{x_{i}}v)^{2}\right|_{1+\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}^{2}.$
In order to estimate $(\partial_{x_{N}}v)^{2}$, we use (3.1) to find that
(4.7) $\left|(\partial_{x_{N}}v)^{2}\right|_{1+\alpha,\overline{B}}\leq
C\,\left|\partial_{x_{N}}v\right|_{1+\alpha,\overline{B}}^{2}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}^{2}.$
The combination of (4.6) and (4.7) then gives (4.1).
(iii) It remains to prove (4.2). For $m=1,2$, let $v_{m}$ be the solution of
the problem (1.4)-(1.5) with $\varphi=\varphi_{m}\in C_{\rm
even}^{1+\alpha}(\overline{B})$ and $\psi=0$. Also, let $\omega_{m}$ be
defined by (4.3) with $v=v_{m}$.
It is clear that
$\left|(\partial_{x_{i}}v_{1})^{2}-(\partial_{x_{i}}v_{2})^{2}\right|_{0,\overline{B}}\leq\\\
\left(|v_{1}|_{0,\overline{B}}+|v_{2}|_{0,\overline{B}}\right)|v_{1}-v_{2}|_{0,\overline{B}}\leq\left(|v_{1}|_{1+\alpha,\overline{B}}+|v_{2}|_{1+\alpha,\overline{B}}\right)|v_{1}-v_{2}|_{1+\alpha,\overline{B}}.$
Hence, (3.1) easily gives that
(4.8)
$\left|(\partial_{x_{i}}v_{1})^{2}-(\partial_{x_{i}}v_{2})^{2}\right|_{0,\overline{B}}\leq
C\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}.$
Next, we write the differential identity:
$\sum_{j=1}^{N}\partial_{x_{j}}\left[(\partial_{x_{i}}v_{1})^{2}-(\partial_{x_{i}}v_{2})^{2}\right]=2\,\sum_{j=1}^{N}\bigl{[}\partial_{x_{i}}v_{1}\,\partial^{2}_{x_{i}x_{j}}v_{1}-\partial_{x_{i}}v_{2}\,\partial^{2}_{x_{i}x_{j}}v_{2}\bigr{]}=\\\
2\,\sum_{j=1}^{N-1}\bigl{[}(\partial_{x_{i}}\omega_{1})(x_{N}\partial^{2}_{x_{i}x_{j}}v_{1}-x_{N}\partial^{2}_{x_{i}x_{j}}v_{2})+x_{N}\,(\partial^{2}_{x_{i}x_{j}}v_{2})(\partial_{x_{i}}\omega_{1}-\partial_{x_{i}}\omega_{2})\bigr{]}+\\\
2\,\bigl{[}(\partial_{x_{i}}v_{1})(\partial^{2}_{x_{i}x_{N}}v_{1}-\partial^{2}_{x_{i}x_{N}}v_{2})+(\partial^{2}_{x_{i}x_{N}}v_{2})(\partial_{x_{i}}v_{1}-\partial_{x_{i}}v_{2})\bigr{]}.$
We then take care of the last summand:
$\left|(\partial_{x_{i}}v_{1})(\partial^{2}_{x_{i}x_{N}}v_{1}-\partial^{2}_{x_{i}x_{N}}v_{2})+(\partial^{2}_{x_{i}x_{N}}v_{2})(\partial_{x_{i}}v_{1}-\partial_{x_{i}}v_{2})\right|_{\alpha,\overline{B}}\leq\\\
\left|\partial_{x_{i}}v_{1}\right|_{\alpha,\overline{B}}\left|\partial^{2}_{x_{i}x_{N}}v_{1}-\partial^{2}_{x_{i}x_{N}}v_{2}\right|_{\alpha,\overline{B}}+\left|\partial^{2}_{x_{i}x_{N}}v_{2}\right|_{\alpha,\overline{B}}\left|\partial_{x_{i}}v_{1}-\partial_{x_{i}}v_{2}\right|_{\alpha,\overline{B}}\leq\\\
\left|v_{1}\right|_{1+\alpha,\overline{B}}\left|\partial_{x_{N}}v_{1}-\partial_{x_{N}}v_{2}\right|_{1+\alpha,\overline{B}}+\left|\partial_{x_{N}}v_{2}\right|_{1+\alpha,\overline{B}}\left|v_{1}-v_{2}\right|_{1+\alpha,\overline{B}}.$
Thus, we have that
$\left|(\partial_{x_{i}}v_{1})(\partial^{2}_{x_{i}x_{N}}v_{1}-\partial^{2}_{x_{i}x_{N}}v_{2})+(\partial^{2}_{x_{i}x_{N}}v_{2})(\partial_{x_{i}}v_{1}-\partial_{x_{i}}v_{2})\right|_{\alpha,\overline{B}}\leq\\\
C\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}.$
Moreover, for $j=1,\ldots,N-1$, we have:
$\left|(\partial_{x_{i}}\omega_{1})(x_{N}\partial^{2}_{x_{i}x_{j}}v_{1}-x_{N}\partial^{2}_{x_{i}x_{j}}v_{2})+x_{N}\,(\partial^{2}_{x_{i}x_{j}}v_{2})(\partial_{x_{i}}\omega_{1}-\partial_{x_{i}}\omega_{2})\right|_{\alpha,\overline{B}}\leq\\\
\left|\omega_{1}\right|_{1+\alpha,\overline{B}}\left|x_{N}\partial^{2}_{x_{i}x_{j}}v_{1}-x_{N}\partial^{2}_{x_{i}x_{j}}v_{2}\right|_{\alpha,\overline{B}}+\left|x_{N}\partial^{2}_{x_{i}x_{j}}v_{2}\right|_{\alpha,\overline{B}}\left|\omega_{1}-\omega_{2}\right|_{1+\alpha,\overline{B}}\leq\\\
\left|\partial_{x_{N}}v_{1}\right|_{1+\alpha,\overline{B}}\left|x_{N}\partial^{2}_{x_{i}x_{j}}v_{1}-x_{N}\partial^{2}_{x_{i}x_{j}}v_{2}\right|_{\alpha,\overline{B}}+\left|x_{N}\partial^{2}_{x_{i}x_{j}}v_{2}\right|_{\alpha,\overline{B}}\left|\partial_{x_{N}}v_{1}-\partial_{x_{N}}v_{2}\right|_{1+\alpha,\overline{B}}.$
where we have used (4.4). Hence, (3.1) gives:
(4.9)
$\left|(\partial_{x_{i}}\omega_{1})(x_{N}\partial^{2}_{x_{i}x_{j}}v_{1}-x_{N}\partial^{2}_{x_{i}x_{j}}v_{2})+x_{N}\,(\partial^{2}_{x_{i}x_{j}}v_{2})(\partial_{x_{i}}\omega_{1}-\partial_{x_{i}}\omega_{2})\right|_{\alpha,\overline{B}}\leq\\\
C\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}.$
All in all, by (4.8)–(4.9) we deduce that
$\left|(\partial_{x_{i}}v_{1})^{2}-(\partial_{x_{i}}v_{2})^{2}\right|_{1+\alpha,\overline{B}}\leq
C\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}},$
for $i=1,\dots,N-1$.
On the other hand, we can also use (3.1) to infer that
$\left|(\partial_{x_{N}}v_{1})^{2}-(\partial_{x_{N}}v_{2})^{2}\right|_{1+\alpha,\overline{B}}\leq\\\
C\,\left(|\partial_{x_{N}}v_{1}|_{1+\alpha,\overline{B}}+|\partial_{x_{N}}v_{2}|_{1+\alpha,\overline{B}}\right)|\partial_{x_{N}}v_{1}-\partial_{x_{N}}v_{2}|_{1+\alpha,\overline{B}}\leq\\\
C\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}.$
In conclusion, we obtain (4.2), and the lemma follows. ∎
### 4.2. The nonlinear operator $\tilde{\mathcal{T}}_{\psi}$
We fix a cut-off function $\eta\in C^{\infty}(\mathbb{R})$ satisfying
$\eta(t)=1\ \mbox{ if }\ |t|\leq\frac{1}{3},\qquad\eta(t)=0\ \mbox{ if }\
|t|\geq\frac{2}{3}.$
For a function $\phi$ defined on $\overline{B}$, we set
${{\mathcal{J}}}[\phi](x)=\eta(x_{N})\phi\left(\sqrt{1-x_{N}^{2}}e_{1}^{\prime},x_{N}\right)+(1-\eta(x_{N}))\phi(x),\quad
x=(x^{\prime},x_{N})\in\overline{B}.$
Here $e_{1}^{\prime}=(1,0,\ldots,0)\in\mathbb{R}^{N-1}$. Then, for fixed
$\psi\in C^{3/2+\alpha}(\overline{D})$, we define an operator
$\tilde{\mathcal{T}}_{\psi}$ by
$\tilde{\mathcal{T}}_{\psi}[\varphi]={\mathcal{J}}\left[|\nabla
v|^{2}\right],$
where $v$ is the solution of (1.4)-(1.5). Our goal here is to obtain the
properties of $\tilde{\mathcal{T}}_{\psi}$, which enables us to solve problem
(1.1) in $C^{1+\alpha}_{\rm ax}(\overline{B})$.
###### Proposition 4.3.
Let $\psi\in C^{3/2+\alpha}(\overline{D})$. Then the following hold.
* (i)
The operator $\tilde{\mathcal{T}}_{\psi}$ is a mapping from
$C^{1+\alpha}(\overline{B})$ into itself. Moreover, there exist positive
constants $C_{3}$ and $C_{4}$ such that
(4.10)
$\bigl{|}\tilde{\mathcal{T}}_{\psi}[\varphi]\bigr{|}_{1+\alpha,\overline{B}}\leq
C_{3}\,\left(|\varphi|_{1+\alpha,\overline{B}}^{2}+|\psi|_{3/2+\alpha,\overline{D}}^{2}\right),$
(4.11)
$\bigl{|}\tilde{\mathcal{T}}_{\psi}[\varphi_{1}]-\tilde{\mathcal{T}}_{\psi}[\varphi_{2}]\bigr{|}_{1+\alpha,\overline{B}}\leq\\\
C_{4}\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}+|\psi|_{3/2+\alpha,\overline{D}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}$
for all $\varphi,\varphi_{1},\varphi_{2}\in C^{1+\alpha}(\overline{B})$.
* (ii)
If $\varphi\in C^{1+\alpha}_{\rm ax}(\overline{B})$ and $\psi$ is constant,
then
(4.12) $\tilde{\mathcal{T}}_{\psi}[\varphi]\in C^{1+\alpha}_{\rm
ax}(\overline{B})\ \mbox{ and }\ \tilde{\mathcal{T}}_{\psi}[\varphi]=|\nabla
v|^{2}\ \mbox{ on }\ {\mathcal{S}}.$
Here $v$ is the solution of (1.4)-(1.5).
Before proving the proposition, we examine properties of ${\mathcal{J}}$.
###### Lemma 4.4.
If a function $\phi$ defined on $\overline{B}$ satisfies
$\phi(x^{\prime},x_{N})=\phi(|x^{\prime}|e_{1}^{\prime},x_{N})$, then
${{\mathcal{J}}}[\phi]=\phi\ \mbox{ on }\ {\mathcal{S}}.$
###### Proof.
Let $x=(x^{\prime},x_{N})\in{\mathcal{S}}$. Then, since
$|x^{\prime}|^{2}+x_{N}^{2}=1$, we have that
$\phi\left(\sqrt{1-x_{N}^{2}}e_{1}^{\prime},x_{N}\right)=\phi(|x^{\prime}|e_{1}^{\prime},x_{N})=\phi(x).$
This gives that
${{\mathcal{J}}}[\phi](x)=\eta(x_{N})\phi(x)+(1-\eta(x_{N}))\phi(x)=\phi(x)$,
as desired. ∎
###### Lemma 4.5.
Suppose that $\phi\in C^{\alpha}(\overline{B})\cap C^{1}(B)$ satisfies
$\partial_{x_{N}}\phi\in C^{\alpha}(\overline{B})$ and
$x_{N}\nabla_{x^{\prime}}\phi\in C^{\alpha}(\overline{B})$. Then
${{\mathcal{J}}}[\phi]\in C^{1+\alpha}(\overline{B})$ and
$|{{\mathcal{J}}}[\phi]|_{1+\alpha,\overline{B}}\leq
C\left(|\phi|_{\alpha,\overline{B}}+|\partial_{x_{N}}\phi|_{\alpha,\overline{B}}+|x_{N}\nabla_{x^{\prime}}\phi|_{\alpha,\overline{B}}\right),$
where $C$ is a positive constant independent of $\phi$.
###### Proof.
Let
$B_{1}=\left\\{(x^{\prime},x_{N})\in B:|x_{N}|<\frac{2}{3}\right\\},\qquad
B_{2}=\left\\{(x^{\prime},x_{N})\in B:|x_{N}|>\frac{1}{3}\right\\},$
and define the mapping
$\xi:\overline{B}\ni(x^{\prime},x_{N})\mapsto\xi(x)=\left(\sqrt{1-x_{N}^{2}}e_{1}^{\prime},x_{N}\right)\in\overline{B}.$
We first show that
(4.13) $\displaystyle\phi\circ\xi\in
C^{1+\alpha}(\overline{B_{1}}),\qquad\phi\in C^{1+\alpha}(\overline{B_{2}}),$
(4.14)
$\displaystyle|\phi\circ\xi|_{1+\alpha,\overline{B_{1}}}+|\phi|_{1+\alpha,\overline{B_{2}}}\leq
C\left(|\phi|_{\alpha,\overline{B}}+|\partial_{x_{N}}\phi|_{\alpha,\overline{B}}+|x_{N}\nabla_{x^{\prime}}\phi|_{\alpha,\overline{B}}\right).$
Here and subsequently, $C$ denotes a positive constant independent of $\phi$.
To prove $\phi\circ\xi\in C^{1+\alpha}(\overline{B_{1}})$, we observe that
$\phi\circ\xi$ depends only on $x_{N}$ and
$\partial_{x_{N}}(\phi\circ\xi)=(\partial_{x_{N}}\phi)\circ\xi-\frac{1}{\sqrt{1-x_{N}^{2}}}(x_{N}\partial_{x_{1}}\phi)\circ\xi.$
By assumption and the fact that $\xi$ and $1/\sqrt{1-x_{N}^{2}}$ are smooth on
$\overline{B_{1}}$, we see that the right-hand side of this equality is in
$C^{\alpha}(\overline{B_{1}})$. Moreover,
$|\partial_{x_{N}}(\phi\circ\xi)|_{\alpha,\overline{B_{1}}}\leq\left|(\partial_{x_{N}}\phi)\circ\xi\right|_{\alpha,\overline{B_{1}}}+\left|\frac{1}{\sqrt{1-x_{N}^{2}}}(x_{N}\partial_{x_{1}}\phi)\circ\xi\right|_{\alpha,\overline{B_{1}}}\leq\\\
C|\partial_{x_{N}}\phi|_{\alpha,\overline{B}}+C|(x_{N}\partial_{x_{1}}\phi)\circ\xi|_{\alpha,\overline{B_{1}}}\leq
C|\partial_{x_{N}}\phi|_{\alpha,\overline{B}}+C|x_{N}\partial_{x_{1}}\phi|_{\alpha,\overline{B}}.$
Therefore, we have that $\phi\circ\xi\in C^{1+\alpha}(\overline{B_{1}})$ and
(4.15)
$|\phi\circ\xi|_{1+\alpha,\overline{B_{1}}}=|\phi\circ\xi|_{0,\overline{B_{1}}}+|\partial_{x_{N}}(\phi\circ\xi)|_{\alpha,\overline{B}}\leq\\\
|\phi|_{0,\overline{B}}+C|\partial_{x_{N}}\phi|_{\alpha,\overline{B}}+C|x_{N}\partial_{x_{1}}\phi|_{\alpha,\overline{B}}.$
Since $x_{N}^{-1}$ is smooth on $\overline{B_{2}}$, we deduce that
$\phi=x_{N}^{-1}\cdot(x_{N}\phi)\in C^{1+\alpha}(\overline{B_{2}})$ and
(4.16)
$|\phi|_{1+\alpha,\overline{B_{2}}}=|\phi|_{0,\overline{B_{2}}}+|\partial_{x_{N}}\phi|_{\alpha,\overline{B_{2}}}+|x_{N}^{-1}\cdot
x_{N}\nabla_{x^{\prime}}\phi|_{\alpha,\overline{B_{2}}}\leq\\\
|\phi|_{0,\overline{B}}+|\partial_{x_{N}}\phi|_{\alpha,\overline{B}}+C|x_{N}\nabla_{x^{\prime}}\phi|_{\alpha,\overline{B}}.$
By (4.15) and (4.16), we obtain (4.14).
We note that $\eta$ and $1-\eta$ vanish on $\overline{B}\setminus B_{1}$ and
$\overline{B}\setminus B_{2}$, respectively. This with (4.13) shows that
${\mathcal{J}}[\phi]=\eta(\phi\circ\xi)+(1-\eta)\phi\in
C^{1+\alpha}(\overline{B})$. Furthermore,
$|{{\mathcal{J}}}[\phi]|_{1+\alpha,\overline{B}}\leq
C|\eta(\phi\circ\xi)|_{1+\alpha,\overline{B_{1}}}+C|(1-\eta)\phi|_{1+\alpha,\overline{B_{2}}}\leq
C|\phi\circ\xi|_{1+\alpha,\overline{B_{1}}}+C|\phi|_{1+\alpha,\overline{B_{2}}}.$
Combining this and (4.14) proves the lemma. ∎
###### Proof of Proposition 4.3.
Let $\varphi,\varphi_{1},\varphi_{2}\in C^{1+\alpha}(\overline{B})$ and
$\psi\in C^{3/2+\alpha}(\overline{D})$. In the proof, $C$ stands for a generic
positive constant only independent of these functions.
Let $v$ stand for the solution of (1.4)-(1.5). Then Theorem 3.1 shows that the
function $\phi=|\nabla v|^{2}$ satisfies $\phi\in C^{\alpha}(\overline{B})\cap
C^{1}(B)$, $\partial_{x_{N}}\phi=2\nabla v\cdot\nabla\partial_{x_{N}}v\in
C^{\alpha}(\overline{B})$ and $x_{N}\partial_{x_{j}}\phi=2x_{N}\nabla
v\cdot\nabla\partial_{x_{j}}v\in C^{\alpha}(\overline{B})$ for
$j=1,\ldots,N-1$. Hence, using Lemma 4.5, we see that
$\tilde{\mathcal{T}}_{\psi}[\varphi]={{\mathcal{J}}}[\phi]\in
C^{1+\alpha}(\overline{B})$ and
$\bigl{|}\tilde{\mathcal{T}}_{\psi}[\varphi]\bigr{|}_{1+\alpha,\overline{B}}\leq
C\left(\bigl{|}|\nabla v|^{2}\bigl{|}_{\alpha,\overline{B}}+\bigl{|}\nabla
v\cdot\nabla\partial_{x_{N}}v\bigl{|}_{\alpha,\overline{B}}+\sum_{j=1}^{N-1}\bigl{|}x_{N}\nabla
v\cdot\nabla\partial_{x_{j}}v\bigl{|}_{\alpha,\overline{B}}\right).$
Each term of the right-hand side is estimated as
$\bigl{|}|\nabla
v|^{2}\bigl{|}_{\alpha,\overline{B}}\leq|v|_{1+\alpha,\overline{B}}^{2},\qquad\bigl{|}\nabla
v\cdot\nabla\partial_{x_{N}}v\bigl{|}_{\alpha,\overline{B}}\leq
C|v|_{1+\alpha,\overline{B}}|\partial_{x_{N}}v|_{1+\alpha,\overline{B}},$
and
$\sum_{j=1}^{N-1}\bigl{|}x_{N}\nabla
v\cdot\nabla\partial_{x_{j}}v\bigl{|}_{\alpha,\overline{B}}\leq
C\sum_{j=1}^{N-1}|v|_{1+\alpha,\overline{B}}|x_{N}\nabla\partial_{x_{j}}v|_{\alpha,\overline{B}}\leq\\\
C|v|_{1+\alpha,\overline{B}}\left(|x_{N}D_{x^{\prime}}^{2}v|_{\alpha,\overline{B}}+|\partial_{x_{N}}v|_{1+\alpha,\overline{B}}\right).$
Therefore, by (3.1), we conclude that $\tilde{\mathcal{T}}_{\psi}$ is a
mapping from $C^{1+\alpha}(\overline{B})$ into itself and that (4.10) holds.
Inequality (4.11) is shown as follows. For $i=1,2$, we denote by $v_{i}$ the
solution of (1.4)-(1.5) with $\varphi=\varphi_{i}$ and $\psi=\psi_{i}$. We see
from Lemma 4.5 that
(4.17)
$\bigl{|}\tilde{\mathcal{T}}_{\psi}[\varphi_{1}]-\tilde{\mathcal{T}}_{\psi}[\varphi_{2}]\bigr{|}_{1+\alpha,\overline{B}}=\bigl{|}{\mathcal{J}}\bigl{[}|\nabla
v_{1}|^{2}-|\nabla v_{2}|^{2}\bigr{]}\bigr{|}_{1+\alpha,\overline{B}}\leq\\\
C\left(\bigl{|}|\nabla v_{1}|^{2}-|\nabla
v_{2}|^{2}\bigr{|}_{0,\overline{B}}+\bigl{|}\partial_{x_{N}}\left(|\nabla
v_{1}|^{2}-|\nabla v_{2}|^{2}\right)\bigr{|}_{\alpha,\overline{B}}+\right.\\\
\left.\sum_{j=1}^{N-1}\bigl{|}x_{N}\partial_{x_{j}}\left(|\nabla
v_{1}|^{2}-|\nabla v_{2}|^{2}\right)\bigr{|}_{\alpha,\overline{B}}\right).$
For abbreviation, we write $w_{1}=v_{1}+v_{2}$ and $w_{2}=v_{1}-v_{2}$. Then
the first and second terms on the right of the above inequality can be handled
as
(4.18) $\bigl{|}|\nabla v_{1}|^{2}-|\nabla
v_{2}|^{2}\bigr{|}_{0,\overline{B}}=\bigl{|}\nabla w_{1}\cdot\nabla
w_{2}\bigr{|}_{0,\overline{B}}\leq
C|w_{1}|_{1+\alpha,\overline{B}}|w_{2}|_{1+\alpha,\overline{B}},$
and
$\bigl{|}\partial_{x_{N}}\left(|\nabla v_{1}|^{2}-|\nabla
v_{2}|^{2}\right)\bigr{|}_{\alpha,\overline{B}}=\bigl{|}\nabla\partial_{x_{N}}w_{1}\cdot\nabla
w_{2}+\nabla
w_{1}\cdot\nabla\partial_{x_{N}}w_{2}\bigr{|}_{\alpha,\overline{B}}\leq\\\
C|\partial_{x_{N}}w_{1}|_{1+\alpha,\overline{B}}|w_{2}|_{1+\alpha,\overline{B}}+C|w_{1}|_{1+\alpha,\overline{B}}|\partial_{x_{N}}w_{2}|_{1+\alpha,\overline{B}}.$
The other terms are estimated as
$\bigl{|}x_{N}\partial_{x_{j}}\left(|\nabla v_{1}|^{2}-|\nabla
v_{2}|^{2}\right)\bigr{|}_{\alpha,\overline{B}}=\bigl{|}x_{N}\nabla\partial_{x_{j}}w_{1}\cdot\nabla
w_{2}+x_{N}\nabla
w_{1}\cdot\nabla\partial_{x_{j}}w_{2}\bigr{|}_{\alpha,\overline{B}}\leq\\\
|x_{N}\nabla\partial_{x_{j}}w_{1}|_{\alpha,\overline{B}}|w_{2}|_{1+\alpha,\overline{B}}+|w_{1}|_{1+\alpha,\overline{B}}|x_{N}\nabla\partial_{x_{j}}w_{2}|_{\alpha,\overline{B}},$
and hence
$\sum_{j=1}^{N-1}\bigl{|}x_{N}\partial_{x_{j}}\left(|\nabla v_{1}|^{2}-|\nabla
v_{2}|^{2}\right)\bigr{|}_{\alpha,\overline{B}}\leq\\\
\left(|x_{N}D_{x^{\prime}}^{2}w_{1}|_{\alpha,\overline{B}}+|\partial_{x_{N}}w_{1}|_{1+\alpha,\overline{B}}\right)|w_{2}|_{1+\alpha,\overline{B}}+\\\
|w_{1}|_{1+\alpha,\overline{B}}\left(|x_{N}D_{x^{\prime}}^{2}w_{2}|_{\alpha,\overline{B}}+|\partial_{x_{N}}w_{2}|_{1+\alpha,\overline{B}}\right).$
We note that $w_{1}$ satisfies (1.4)-(1.5) with $\varphi$ and $\psi$ replaced
by $\varphi_{1}+\varphi_{2}$ and $2\psi$, respectively. Hence it follows from
Theorem 3.1 that
$|w_{1}|_{1+\alpha,\overline{B}}+|\partial_{x_{N}}w_{1}|_{1+\alpha,\overline{B}}+|x_{N}D_{x^{\prime}}^{2}w_{1}|_{\alpha,\overline{B}}\leq\\\
C\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}+|\psi|_{3/2+\alpha,\overline{D}}\right).$
Moreover, since $w_{2}$ solves (1.4)-(1.5) with
$\varphi=\varphi_{1}-\varphi_{2}$ and $\psi=0$, Theorem 3.1 shows that
(4.19)
$|w_{2}|_{1+\alpha,\overline{B}}+|\partial_{x_{N}}w_{2}|_{1+\alpha,\overline{B}}+|x_{N}D_{x^{\prime}}^{2}w_{2}|_{\alpha,\overline{B}}\leq
C|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}.$
We thus obtain (4.11) by plugging (4.18)-(4.19) into (4.17).
It remains to prove (ii). To this end, we assume that $\varphi\in
C^{1+\alpha}_{\rm ax}(\overline{B})$ and that $\psi$ is constant. Then, we can
directly check that for any $(N-1)\times(N-1)$ orthogonal matrix
${{\mathcal{R}}}$, the function $v({{\mathcal{R}}}x^{\prime},x_{N})$ also
satisfies (1.4)-(1.5). Hence Proposition 2.1 gives that
$v(x^{\prime},x_{N})=v({{\mathcal{R}}}x^{\prime},x_{N})$. In particular, we
have that $|\nabla v(x^{\prime},x_{N})|^{2}=|\nabla
v(|x^{\prime}|e_{1}^{\prime},x_{N})|^{2}$. Therefore (i) of Proposition 4.3
and Lemma 4.4 show that (4.12) holds, and the proof is complete. ∎
### 4.3. Contraction mappings and the proof of Theorem 1.1
For $g\in C^{1+\alpha}(\overline{B})$ and $h\in\mathbb{R}$, we define
operators $\Psi_{g}$ and $\tilde{\Psi}_{g,h}$ by
$\Psi_{g}[\varphi]=\frac{1}{2}\left(g^{2}-1-{\mathcal{T}}[\varphi]\right),\qquad\tilde{\Psi}_{g,h}[\varphi]=\frac{1}{2}\left(g^{2}-1-\tilde{\mathcal{T}}_{h}[\varphi]\right).$
Note that, by Propositions 4.1 and 4.3, $\Psi_{g}$ is a mapping from $C_{\rm
even}^{1+\alpha}(\overline{B})$ into itself and $\tilde{\Psi}_{g,h}$ is a
mapping from $C_{\rm ax}^{1+\alpha}(\overline{B})$ into itself. We also define
closed sets of $C^{1+\alpha}(\overline{B})$ by
$\displaystyle X_{g}=\\{\varphi\in C_{\rm
even}^{1+\alpha}(\overline{B}):|\varphi|_{1+\alpha,\overline{B}}\leq|g^{2}-1|_{1+\alpha,\overline{B}}\\},$
$\displaystyle\tilde{X}_{g,h}=\\{\varphi\in C_{\rm
ax}^{1+\alpha}(\overline{B}):|\varphi|_{1+\alpha,\overline{B}}\leq|g^{2}-1|_{1+\alpha,\overline{B}}+|h|\\},$
and positive constants $\delta_{1}$ and $\delta_{2}$ by
$\delta_{1}=\min\left\\{\frac{1}{C_{1}},\frac{\lambda}{C_{2}}\right\\},\qquad\delta_{2}=\min\left\\{\frac{1}{2C_{3}},\frac{2\lambda}{3C_{4}}\right\\}.$
Here, $\lambda\in(0,1)$ is an arbitrary fixed constant, and $C_{1}$, $C_{2}$,
$C_{3}$, $C_{4}$ are the constants given in Propositions 4.1 and 4.3.
###### Lemma 4.6.
The following hold.
* (i)
If $g\in C_{\rm even}^{1+\alpha}(\overline{B})$ satisfies
$|g^{2}-1|_{1+\alpha,\overline{B}}\leq\delta_{1}$, then $\Psi_{g}$ has a
unique fixed point in $X_{g}$.
* (ii)
If $g\in C_{\rm ax}^{1+\alpha}(\overline{B})$ and $h\in\mathbb{R}$ satisfy
$|g^{2}-1|_{1+\alpha,\overline{B}}+|h|\leq\delta_{2}$, then
$\tilde{\Psi}_{g.h}$ has a unique fixed point in $\tilde{X}_{g.h}$.
###### Proof.
We show (i). Set $\delta=|g^{2}-1|_{1+\alpha,\overline{B}}$, so that
$\delta\leq\delta_{1}$. For $\varphi\in X_{g}$, we see from Proposition 4.1
that $\Psi_{g}[\varphi]\in C_{\rm even}^{1+\alpha}(\overline{B})$ and
$\bigl{|}\Psi_{g}[\varphi]\bigr{|}_{1+\alpha,\overline{B}}\leq\frac{1}{2}\left(\delta+\bigl{|}{\mathcal{T}}[\varphi]\bigr{|}_{1+\alpha,\overline{B}}\right)\leq\frac{1}{2}\left(\delta+C_{1}|\varphi|_{1+\alpha,\overline{B}}^{2}\right)\leq\frac{1}{2}\left(1+C_{1}\delta_{1}\right)\delta\leq\delta.$
Hence we have that $\Psi_{g}(X_{g})\subseteq X_{g}$.
Furthermore, inequality (4.2) shows that for $\varphi_{1},\varphi_{2}\in
X_{g}$,
$\bigl{|}\Psi_{g}[\varphi_{1}]-\Psi_{g}[\varphi_{2}]\bigr{|}_{1+\alpha,\overline{B}}=\frac{1}{2}\,\bigl{|}{\mathcal{T}}[\varphi_{1}]-{\mathcal{T}}[\varphi_{2}]\bigr{|}_{1+\alpha,\overline{B}}\leq\\\
\frac{1}{2}\,C_{2}\,\left(|\varphi_{1}|_{1+\alpha,\overline{B}}+|\varphi_{2}|_{1+\alpha,\overline{B}}\right)|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}\leq
C_{2}\,\delta_{1}|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}}.$
Thus, we infer that
$\bigl{|}\Psi_{g}[\varphi_{1}]-\Psi_{g}[\varphi_{2}]\bigr{|}_{1+\alpha,\overline{B}}\leq\lambda\,|\varphi_{1}-\varphi_{2}|_{1+\alpha,\overline{B}},$
i.e. we have proved that $\Psi_{g}$ is a contraction mapping in $X_{g}$. The
Banach fixed point theorem then gives the desired conclusion.
The assertion (ii) can be proved in the same way, by applying Proposition 4.3
instead of Proposition 4.1. ∎
We are now in a position to prove Theorem 1.1.
###### Proof of Theorem 1.1.
We first prove (i). Note that
(4.20) $|g^{2}-1|_{1+\alpha,\overline{\Omega}}\leq
C_{0}|g+1|_{1+\alpha,\overline{\Omega}}|g-1|_{1+\alpha,\overline{\Omega}}\leq\\\
C_{0}\left(|g-1|_{1+\alpha,\overline{\Omega}}+2\right)|g-1|_{1+\alpha,\overline{\Omega}}$
for some constant $C_{0}>0$. Hence, if
$|g-1|_{1+\alpha,\overline{\Omega}}\leq\sqrt{1+\frac{\delta_{1}}{C_{0}}}-1,$
we have that $|g^{2}-1|_{1+\alpha,\overline{\Omega}}\leq\delta_{1}$.
Suppose that the above condition is satisfied. Then, $\Psi_{g}$ has a unique
fixed point $\varphi\in X_{g}$, by Lemma 4.6. Let $v$ be the solution of
(1.4)-(1.5) with $\psi=0$. Then $u=f+v$ is harmonic in $B$ and
(4.21) $|\nabla u|^{2}=|\nabla f|^{2}+2\nabla f\cdot\nabla v+|\nabla
v|^{2}=1+2\partial_{x_{N}}v+{\mathcal{T}}[\varphi]=\\\
1+2\varphi+(g^{2}-1-2\Psi_{g}[\varphi])=g^{2}\ \mbox{ on }\ {\mathcal{S}}.$
This shows that $u$ is a solution of (1.1). Moreover, Theorem 3.1 and the fact
that $\varphi\in X_{g}$ give that $u$ is in $C_{\rm
odd}^{1+\alpha}(\overline{B})$ and
$|u-f|_{1+\alpha,\overline{B}}=|v|_{1+\alpha,\overline{B}}\leq
C\,|\varphi|_{1+\alpha,\overline{B}}\leq
C\,|g^{2}-1|_{1+\alpha,\overline{B}}\leq C\,|g-1|_{1+\alpha,\overline{B}},$
where $C>0$ is a constant. The assertion (i) thus holds if we take
$\delta_{0}$ such that $\delta_{0}\leq\sqrt{1+\delta_{1}/C_{0}}-1$.
For (ii), we may assume $h=0$, by considering $\tilde{u}=u-h$ instead of $u$.
Then, (ii) can be shown in a similar way. We see from (4.20) that
$|g^{2}-1|_{1+\alpha,\overline{\Omega}}\leq\delta_{2}$ if
$|g-1|_{1+\alpha,\overline{\Omega}}\leq\sqrt{1+\frac{\delta_{2}}{C_{0}}}-1.$
Under this condition, Lemma 4.6 shows that $\tilde{\Psi}_{g,0}$ has a fixed
point $\tilde{\varphi}\in\tilde{X}_{g,0}$. We put $\tilde{u}=f+\tilde{v}$ for
the solution $\tilde{v}$ of (1.4)-(1.5) with $\varphi=\tilde{\varphi}$ and
$\psi=0$. Then $\tilde{u}$ solves (1.1), since the same computation as in
(4.21) is valid thanks to (4.12). The fact that $\tilde{u}\in C_{\rm
ax}^{1+\alpha}(\overline{B})$ and the inequality
$|\tilde{u}-f|_{1+\alpha,\overline{B}}\leq C\,|g-1|_{1+\alpha,\overline{B}}$
follow from Theorem 3.1. We have thus shown (ii), and the proof is complete. ∎
## Acknowledgements
The authors would like to thank the anonymous referee for useful suggestions
for improving the readability of the presentation. The first author was
partially supported by the Grant-in-Aid for Early-Career Scientists 19K14574,
Japan Society for the Promotion of Science. The second author was partially
supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e
le loro Applicazioni (GNAMPA) dell’Istituto Nazionale di Alta Matematica
(INdAM). The third author was supported in part by the Grant-in-Aid for
Scientific Research (C) 20K03673, Japan Society for the Promotion of Science.
This research started while the second author was visiting the Department of
Mathematics of Tokyo Institute of Technology. He wants to thank their kind
hospitality.
## References
* [1] Sh. A. Alimov, _On a problem with an oblique derivative_ , (Russian) Differentsial’nye Uravneniya 17 (1981), no. 10, 1738–1751, 1915. English translation: Differential Equations 17 (1981), no. 10, 1073–1083 (1982).
* [2] G. E. Backus, _Application of a non-linear boundary value problem for the Laplace’s equation to gravity and geomagnetic surveys_ , Quart. J. Mech. Appl. Math. XXI (1968), 195–221.
* [3] D. Colton, R. Kress, Integral equation methods in scattering theory. John Wiley & Sons, Inc., New York, 1983.
* [4] G. Díaz, J. J. Díaz, J. Otero, _Construction of the maximal solution of Backus’ problem in geodesy and geomagnetism_ , Stud. Geophys. Geod. 55 (2011), 415-440.
* [5] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order. Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.
* [6] M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, O. V. Lounasmaa, _Magnetoencephalography – theory, instrumentation, and applications to noninvasinve studies of the working human brain_ , Review of Modern Physics, 65 (1993), 413–497.
* [7] R. S. Hamilton, _The inverse function theorem of Nash and Moser_ , Bull. Amer. Math. Soc. 7 (1982), 65–222.
* [8] M. C. Jorge, _Local existence of the solution to a nonlinear inverse problem in gravitation_ , Quart. Appl. Math. 45 (1987), 287–292.
* [9] M. C. Jorge, R. Magnanini, _Explicit calculation of the solution to Backus problem with condition for uniqueness_ , J. Math. Anal. Appl. 173 (1993), 515–522.
* [10] T. Kan, R. Magnanini, M. Onodera, _Backus problem in geophysics: a resolution near the dipole in fractional Sobolev spaces_ , NoDEA Nonlinear Differential Equations Appl. 29 (2022), no. 3, Paper No. 21, 29 pp.
* [11] E. H. Lieb, M. Loss, Analysis. American Mathematical Society, Providence, RI, 1997.
* [12] R. Magnanini, _A fully nonlinear boundary value problem for the Laplace equation in dimension two_ , Appl. Anal. 39 (2-3), 185–192.
* [13] J. Moser, _A new technique for the construction of solutions of nonlinear differential equations_ , Proc. N. A. S. 47 (1961), 1824–1831.
* [14] J. Moser, _A rapidly convergent iteration method and non-linear partial differential equations – I_ , Ann. Sc. Norm. Sup. Pisa (3) 20 (1966), 499–435.
* [15] J. Nash, _$C^{1}$ isometric imbeddings_, Ann. of Math. 60 (1954), 383–396.
* [16] W. Rudin, Real and Complex Analysis. Second edition. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.
|
# Conservative algebras of $2$-dimensional algebras, IV
Amir Fernández Ouaridi A. Fernández Ouaridi: CMUC, Universidade de Coimbra,
Coimbra, Portugal; Universidad de Cádiz, Cádiz, Spain.
<EMAIL_ADDRESS>, Ivan Kaygorodov I. Kaygorodov: CMA-UBI,
Universidade da Beira Interior, Covilhã, Portugal; Moscow Center for
Fundamental and Applied Mathematics, Moscow, Russia; Saint Petersburg State
University, Russia<EMAIL_ADDRESS>and Cándido Martín González
C. Martín: Departamento de Álgebra Geometría y Topología, Facultad de
Ciencias, Universidad de Málaga, Campus de Teatinos s/n. 29071 Málaga. Spain.
<EMAIL_ADDRESS>
###### Abstract.
The notion of conservative algebras appeared in a paper of Kantor in 1972.
Later, he defined the conservative algebra $W(n)$ of all algebras (i.e.
bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra
$W(n)$ does not belong to any well-known class of algebras (such as
associative, Lie, Jordan, or Leibniz algebras). It looks like that $W(n)$ in
the theory of conservative algebras plays a similar role with the role of
$\mathfrak{gl}_{n}$ in the theory of Lie algebras. Namely, an arbitrary
conservative algebra can be obtained from a universal algebra $W(n)$ for some
$n\in\mathbb{N}.$ The present paper is a part of a series of papers, which
dedicated to the study of the algebra $W(2)$ and its principal subalgebras.
Keywords: bilinear maps, conservative algebra, contraction, identities.
MSC2020: 17A30111Corresponding author: Ivan Kaygorodov
<EMAIL_ADDRESS>
## Introduction
A multiplication on a vector space $W$ is a bilinear mapping $W\times W\to W$.
We denote by $(W,P)$ the algebra with underlining space $W$ and multiplication
$P$. Given a vector space $W$, a linear mapping ${\bf A}:W\rightarrow W$, and
a bilinear mapping ${\bf B}:W\times W\to W$, we can define a multiplication
$[{\bf A},{\bf B}]:W\times W\to W$ by the formula
$[{\bf A},{\bf B}](x,y)={\bf A}({\bf B}(x,y))-{\bf B}({\bf A}(x),y)-{\bf
B}(x,{\bf A}(y))$
for $x,y\in W$. For an algebra ${\bf A}$ with a multiplication $P$ and
$x\in{\bf A}$ we denote by $L_{x}^{P}$ the operator of left multiplication by
$x$. If the multiplication $P$ is fixed, we write $L_{x}$ instead of
$L_{x}^{P}$.
In 1990 Kantor [14] defined the multiplication $\cdot$ on the set of all
algebras (i.e. all multiplications) on the $n$-dimensional vector space
$V_{n}$ as follows:
${\bf A}\cdot{\bf B}=[L_{e}^{\bf A},{\bf B}],$
where ${\bf A}$ and ${\bf B}$ are multiplications and $e\in V_{n}$ is some
fixed vector. Let $W(n)$ denote the algebra of all algebra structures on
$V_{n}$ with multiplication defined above. If $n>1$, then the algebra $W(n)$
does not belong to any well-known class of algebras (such as associative, Lie,
Jordan, or Leibniz algebras). The algebra $W(n)$ turns out to be a
conservative algebra (see below).
In 1972 Kantor [11] introduced conservative algebras as a generalization of
Jordan algebras (also, see a good written survey about the study of
conservative algebras and superalgebras [25]). Namely, an algebra ${\bf
A}=(W,P)$ is called a conservative algebra if there is a new multiplication
$F:W\times W\rightarrow W$ such that
(1) $\displaystyle[L_{b}^{P},[L_{a}^{P},P]]=-[L_{F(a,b)}^{P},P]$
for all $a,b\in W$. In other words, the following identity holds for all
$a,b,x,y\in W$:
(2) $b(a(xy)-(ax)y-x(ay))-a((bx)y)+(a(bx))y+(bx)(ay)\\\
-a(x(by))+(ax)(by)+x(a(by))=-F(a,b)(xy)+(F(a,b)x)y+x(F(a,b)y).$
The algebra $(W,F)$ is called an algebra associated to ${\bf A}$. The main
subclass of conservative algebras is the variety of terminal algebras, which
defined by the identity (2) with $F(a,b)=\frac{1}{3}(2ab+ba).$ It includes the
varieties of Leibniz and Jordan algebras as subvarieties.
Let us recall some well-known results about conservative algebras. In [11]
Kantor classified all simple conservative algebras and triple systems of
second-order and defined the class of terminal algebras as algebras satisfying
some certain identity. He proved that every terminal algebra is a conservative
algebra and classified all simple finite-dimensional terminal algebras with
left quasi-unit over an algebraically closed field of characteristic zero
[12]. Terminal trilinear operations were studied in [13]. After that,
Cantarini and Kac classified simple finite-dimensional (and linearly compact)
super-commutative and super-anticommutative conservative superalgebras and
some generalization of these algebras (also known as “rigid” or quasi-
conservative superalgebras) over an algebraically closed field of
characteristic zero [3]. The classification of all $2$-dimensional
conservative and rigid (in sense of Kac-Cantarini) algebras is given in [2];
and also, the algebraic and geometric classification of nilpotent low
dimensional terminal algebras is given in [16, 17].
The algebra $W(n)$ plays a similar role in the theory of conservative algebras
as the Lie algebra of all $n\times n$ matrices $\mathfrak{gl}_{n}$ plays in
the theory of Lie algebras. Namely, in [14] Kantor considered the category
$\mathscr{S}_{n}$ whose objects are conservative algebras of non-Jacobi
dimension $n$. It was proven that the algebra $W(n)$ is the universal
attracting object in this category, i.e., for every $M\in\mathscr{S}_{n}$
there exists a canonical homomorphism from $M$ into the algebra $W(n)$. In
particular, all Jordan algebras of dimension $n$ with unity are contained in
the algebra $W(n)$. The same statement also holds for all noncommutative
Jordan algebras of dimension $n$ with unity. Some properties of the product in
the algebra $W(n)$ were studied in [5, 15]. The universal conservative
superalgebra was constructed in [20]. The study of low dimensional
conservative algebras was started in [18]. The study of properties of
$2$-dimensional algebras is also one of popular topic in non-associative
algebras (see, for example, [6, 22, 23, 4, 24]) and as we can see the study of
properties of the algebra $W(2)$ could give some applications on the theory of
$2$-dimensional algebras. So, from the description of idempotents of the
algebra $W(2)$ it was received an algebraic classification of all
$2$-dimensional algebras with left quasi-unit [21]. Derivations and
subalgebras of codimension 1 of the algebra $W(2)$ and of its principal
subalgebras $W_{2}$ and $S_{2}$ were described [18]. Later, the automorphisms,
one-sided ideals, idempotents, local (and $2$-local) derivations and
automorphisms of $W(2)$ and its principal subalgebras were described in [1,
21]. Note that $W_{2}$ and $S_{2}$ are simple terminal algebras with left
quasi-unit from the classification of Kantor [12]. The present paper is
devoted to continuing the study of properties of $W(2)$ and its principal
subalgebras. Throughout this paper, unless stated otherwise, $\mathbb{F}$
denotes a field of characteristic zero. All algebras are defined over
$\mathbb{F}$.
The multiplication table of $W(2)$ is given by the following table.
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{7}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{8}$ | $0$ | $e_{2}$ | $-e_{3}$ | $-2e_{4}$ | $0$ | $-e_{6}$ | $-e_{7}$ | $0$
## 1\. Inönü-Wigner contractions of $W(2)$ and its subalgebras
The class of conservative algebras includes the variety of terminal algebras,
which includes all Leibniz and Jordan algebras. On the other hand, the variety
of terminal algebras is ”dual” to the variety of commutative algebras (in the
sense of generalized TTK-functor). The algebra $W(2)$ is not terminal, but its
principal subalgebras $W_{2}$ and $S_{2}$ are terminal. Our main aim is to try
to understand how the algebra $W(2)$ ”far” from terminal algebras. For a
particular answer for our question, we will consider contractions to some
certain subalgebras of $W(2)$ and study its relations with the variety of
terminal algebras. The standard Inönü-Wigner contraction was introduced in
[8]. We will call it IW contraction for short.
###### Definition 1.
Let $\mu,\chi$ represent algebras ${\bf A}$ and ${\bf B}$ respectively defined
on a vector space $V$. Suppose that there are some elements $E_{i}^{t}\in V$
$(1\leq i\leq n$, $t\in{\mathbb{F}}^{*})$ such that
$E^{t}=(E_{1}^{t},\dots,E_{n}^{t})$ is a basis of $V$ for any
$t\in{\mathbb{F}}^{*}$ and the structure constants of $\mu$ in this basis are
$\mu_{i,j}^{k}(t)$ for some polynomials $\mu_{i,j}^{k}(t)\in{\mathbb{F}}[t]$.
If $\mu_{i,j}^{k}(0)=\chi_{i,j}^{k}$ for all $1\leq i,j,k\leq n$, then ${\bf
A}\to{\bf B}$. To emphasize that the parametrized basis
$E^{t}=(E_{1}^{t},\dots,E_{n}^{t})$ $(t\in{\mathbb{F}}^{*})$ gives a
degeneration between the algebras represented by the structures $\mu$ and
$\chi$, we will write $\mu\xrightarrow{E^{t}}\chi$. Suppose that ${\bf A}_{0}$
is an $(n-m)$-dimensional subalgebra of the $n$-dimensional algebra ${\bf A}$
and $\mu$ is a structure representing ${\bf A}$ such that ${\bf A}_{0}$
corresponds to the subspace $\langle e_{m+1},\dots,e_{n}\rangle$ of $V$. Then
$\mu\xrightarrow{(te_{1},\dots,te_{m},e_{m+1},\dots,e_{n})}\chi$ for some
$\chi$ and the algebra ${\bf B}$ represented by $\chi$ is called the IW
contraction of ${\bf A}$ with respect to ${\bf A}_{0}$.
### 1.1. IW contraction of $W(2)$
#### 1.1.1. The algebra $\overline{W(2)}$
The description of all subalgebras of codimension $1$ for the algebra $W(2)$
is given in [18]. Namely, $W(2)$ has only one $7$-dimensional subalgebra. It
is generated by elements $e_{1},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8},$ and it
is terminal. Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},te_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8})}\overline{W(2)}.$
It is easy to see, that the multiplication table of $\overline{W(2)}$ is given
by the following table.
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{3}$ | $-2e_{3}$ | $0$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $0$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{7}$ | $2e_{3}$ | $0$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
$e_{8}$ | $0$ | $e_{2}$ | $-e_{3}$ | $-2e_{4}$ | $0$ | $-e_{6}$ | $-e_{7}$ | $0$
After a carefully checking of the dimension of the algebra of derivation of
$\overline{W(2)},$ we have $\dim\mathfrak{Der}(\overline{W(2)})=3.$ Since
$\dim\mathfrak{Der}(W(2))=2,$ it follows that the degeneration
$W(2)\to\overline{W(2)}$ is primary, that is, there is no algebra ${\bf A}$
such that ${W(2)}\to{\bf A}$ and ${\bf A}\to\overline{W(2)}$, where ${\bf A}$
is neither isomorphic to $W(2)$ or $\overline{W(2)}$ (see [9]).
###### Lemma 2.
The algebra $\overline{W(2)}$ is a non-terminal conservative non-simple
algebra.
###### Proof.
The subspace $\langle e_{2},e_{3},e_{4},e_{6},e_{7},e_{8}\rangle$ gives a
$6$-dimensional ideal, it gives that $\overline{W(2)}$ is non-simple. The non-
terminal property is following from the direct verification of the terminal
identity (for example, using a modification of the Wolfram code presented in
[10]). The conservative property is following from the direct verification of
the conservative identity with the additional multiplication $*:$
$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{1}*e_{5}=-2e_{1}$ | $e_{2}*e_{1}=e_{2}$ | $e_{2}*e_{5}=-e_{2}$
---|---|---|---|---
$e_{2}*e_{8}=-e_{2}$ | $e_{5}*e_{1}=-2e_{1}$ | $e_{5}*e_{2}=-2e_{2}$ | $e_{5}*e_{5}=-4e_{1}.$ |
∎
###### Lemma 3.
Let $\mathscr{S}$ be a subalgebra of $\overline{W(2)}$ of codimension $1,$ the
$\mathscr{S}$ is one of the following conservative subalgebras
${\mathscr{S}}_{1}=\langle e_{2},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\rangle$ | ${\mathscr{S}}_{2}=\langle e_{1},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\rangle$
---|---
${\mathscr{S}}_{5}=\langle e_{1},e_{2},e_{3},e_{4},e_{6},e_{7},e_{8}\rangle$ | ${\mathscr{S}}_{\alpha,\beta}=\langle e_{1}+\alpha e_{8},e_{2},e_{3},e_{4},e_{5}+\beta e_{8},e_{6},e_{7}\rangle_{\alpha,\beta\in{\mathbb{F}}},$
where only ${\mathscr{S}}_{1},$ ${\mathscr{S}}_{2}$, ${\mathscr{S}}_{0,0}$ and
${\mathscr{S}}_{-1,1}$ are terminal.
###### Proof.
Let $\mathscr{S}$ be generated by the following set
$\\{e_{1},\ldots,\widehat{e_{i}},\ldots,e_{8}\\}.$ By some easy verification
of $8$ possibilities, we have that there are only $4$ subalgebras of this
type: for $i=1,2,5,8.$
Let us consider the situation when $\mathscr{S}$ is generated by seven vectors
of the following type:
$\\{\sum\alpha_{i1}e_{i},\ldots,\sum\alpha_{i7}e_{i}\\}.$ By some linear
combinations, we can reduce this basis to a basis considered above, or a basis
of the following type:
$\\{e_{1}+\alpha_{1}e_{8},\ldots,e_{7}+\alpha_{1}e_{8}\\}.$ It is easy to see,
that
$(e_{2}+\alpha_{2}e_{8})^{2}=\alpha_{2}e_{2}\in{\mathscr{S}}$ | $(e_{3}+\alpha_{3}e_{8})^{2}=-3e_{4}-\alpha_{3}(e_{3}+e_{7})\in{\mathscr{S}}$
---|---
$(e_{4}+\alpha_{4}e_{8})^{2}=-2\alpha_{4}e_{4}\in{\mathscr{S}}$ | $(e_{6}+\alpha_{6}e_{8})^{2}=-\alpha_{6}(e_{6}-e_{7})\in{\mathscr{S}},$
which gives that $e_{2},e_{4}\in{\mathscr{S}}$ and there are four cases:
I. $e_{3},e_{6}\in{\mathscr{S}}$ | II. $e_{3},e_{6}-e_{7}\in{\mathscr{S}}$
---|---
III. $e_{3}+e_{7},e_{6}\in{\mathscr{S}}$ | IV. $e_{3}+e_{7},e_{6}-e_{7}\in{\mathscr{S}}$
Analysing all these cases, we have that ${\mathscr{S}}$ is a subalgebra
considered above, or it has the following basis $\langle e_{1}+\alpha
e_{8},e_{2},e_{3},e_{4},e_{5}+\beta
e_{8},e_{6},e_{7}\rangle_{\alpha,\beta\in{\mathbb{F}}}.$
The conservative property of the subalgebra ${\mathscr{S}}_{5}$ is following
from the direct verification of the conservative identity with the additional
multiplication $*:$
$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{2}*e_{1}=e_{2}$ | $e_{2}*e_{8}=-e_{2}$
---|---|---|---
Let us give the multiplication table of $\overline{W(2)}$ in more useful way
(here the subalgebra $\langle e_{1},\ldots,e_{7}\rangle$ gives
${\mathscr{S}}_{\alpha,\beta}$):
$e_{1}e_{1}=-e_{1}$ | $e_{1}e_{2}=(-3+\alpha)e_{2}$ | $e_{1}e_{3}=(1-\alpha)e_{3}$ | $e_{1}e_{4}=(3-2\alpha)e_{4}$
---|---|---|---
$e_{1}e_{5}=-e_{5}$ | $e_{1}e_{6}=(1-\alpha)e_{6}$ | $e_{1}e_{7}=(1-\alpha)e_{7}$ | $e_{1}e_{8}=-e_{8}$
$e_{2}e_{1}=3e_{2}$ | $e_{3}e_{1}=-2e_{3}-\alpha e_{7}$ | $e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{5}=e_{6}-\beta e_{7}$
$e_{3}e_{8}=-e_{7}$ | $e_{5}e_{1}=-2e_{1}$ | $e_{5}e_{2}=(-3+\beta)e_{2}$ | $e_{5}e_{3}=(-1-\beta)e_{3}$
$e_{5}e_{4}=-2\beta e_{4}$ | $e_{5}e_{5}=-2e_{5}$ | $e_{5}e_{6}=(-1-\beta)e_{6}$ | $e_{5}e_{7}=(-1-\beta)e_{7}$
$e_{5}e_{8}=-2e_{8}$ | $e_{6}e_{1}=2e_{3}+\alpha e_{7}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{5}=-e_{6}+\beta e_{7}$
$e_{6}e_{8}=e_{7}$ | $e_{7}e_{1}=2e_{3}+\alpha e_{7}$ | $e_{7}e_{3}=3e_{4}$ | $e_{7}e_{5}=-e_{6}+\beta e_{7}$
$e_{7}e_{8}=e_{7}$ | $e_{8}e_{2}=e_{2}$ | $e_{8}e_{3}=-e_{3}$ | $e_{8}e_{4}=-2e_{4}$
| $e_{8}e_{6}=-e_{6}$ | $e_{8}e_{7}=-e_{7}$ |
The conservative property of the subalgebra ${\mathscr{S}}_{\alpha,\beta}$ is
following from the direct verification of the conservative identity with the
additional multiplication $*:$
$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{1}*e_{5}=-2e_{1}$ | $e_{2}*e_{1}=(1-\alpha)e_{2}$
---|---|---|---
$\par e_{2}*e_{5}=(-1-\beta)e_{2}$ | $e_{5}*e_{1}=-2e_{1}$ | $e_{5}*e_{2}=-2e_{2}$ | $e_{5}*e_{5}=-4e_{1}$
∎
#### 1.1.2. Algebras $\overline{{\mathscr{S}}}$
In the present subsection, we have to talk about contractions of the algebra
$\overline{W(2)}$ to its subalgebra of codimension $1.$
$\bullet$
$\overline{W(2)}\xrightarrow{(te_{1},{\mathscr{S}}_{1})}\overline{{\mathscr{S}}_{1}}.$
It is easy to see that the multiplication of $\overline{{\mathscr{S}}_{1}}$ is
given by the following table.
$e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{5}=e_{6}$ | $e_{3}e_{8}=-e_{7}$ | $e_{5}e_{1}=-2e_{1}$ | $e_{5}e_{2}=-3e_{2}$ | $e_{5}e_{3}=-e_{3}$ | $e_{5}e_{5}=-2e_{5}$
---|---|---|---|---|---|---
$e_{5}e_{6}=-e_{6}$ | $e_{5}e_{7}=-e_{7}$ | $e_{5}e_{8}=-2e_{8}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{5}=-e_{6}$ | $e_{6}e_{8}=e_{7}$ | $e_{7}e_{3}=3e_{4}$
$e_{7}e_{5}=-e_{6}$ | $e_{7}e_{8}=e_{7}$ | $e_{8}e_{2}=e_{2}$ | $e_{8}e_{3}=-e_{3}$ | $e_{8}e_{4}=-2e_{4}$ | $e_{8}e_{6}=-e_{6}$ | $e_{8}e_{7}=-e_{7}$
###### Lemma 4.
The algebra $\overline{{\mathscr{S}}_{1}}$ is terminal.
$\bullet$
$\overline{W(2)}\xrightarrow{(te_{5},{\mathscr{S}}_{5})}\overline{{\mathscr{S}}_{5}}.$
It is easy to see that the multiplication of $\overline{{\mathscr{S}}_{5}}$ is
given by the following table.
$e_{1}e_{1}=-e_{1}$ | $e_{1}e_{2}=-3e_{2}$ | $e_{1}e_{3}=e_{3}$ | $e_{1}e_{4}=3e_{4}$ | $e_{1}e_{5}=-e_{5}$ | $e_{1}e_{6}=e_{6}$
---|---|---|---|---|---
$e_{1}e_{7}=e_{7}$ | $e_{1}e_{8}=-e_{8}$ | $e_{2}e_{1}=3e_{2}$ | $e_{3}e_{1}=-2e_{3}$ | $e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{8}=-e_{7}$
$\par e_{6}e_{1}=2e_{3}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{8}=e_{7}$ | $e_{7}e_{1}=2e_{3}$ | $e_{7}e_{3}=3e_{4}$ | $e_{7}e_{8}=e_{7}$
$\par e_{8}e_{2}=e_{2}$ | $e_{8}e_{3}=-e_{3}$ | $e_{8}e_{4}=-2e_{4}$ | $e_{8}e_{6}=-e_{6}$ | $e_{8}e_{7}=-e_{7}$ |
###### Lemma 5.
The algebra $\overline{{\mathscr{S}}_{5}}$ is a non-terminal conservative
algebra.
###### Proof.
The conservative property of the algebra $\overline{{\mathscr{S}}_{5}}$ is
following from the direct verification of the conservative identity with the
additional multiplication $*:$
$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{2}*e_{1}=e_{2}$ | $e_{2}*e_{8}=-e_{2}$
---|---|---|---
∎
$\bullet$
$\overline{W(2)}\xrightarrow{(te_{8},{\mathscr{S}}_{\alpha,\beta})}\overline{{\mathscr{S}}_{\alpha,\beta}}.$
It is easy to see that the multiplication of
$\overline{{\mathscr{S}}_{\alpha,\beta}}$ is given by the following table.
$e_{1}e_{1}=-e_{1}$ | $e_{1}e_{2}=(-3+\alpha)e_{2}$ | $e_{1}e_{3}=(1-\alpha)e_{3}$ | $e_{1}e_{4}=(3-2\alpha)e_{4}$
---|---|---|---
$\par e_{1}e_{5}=-e_{5}$ | $e_{1}e_{6}=(1-\alpha)e_{6}$ | $e_{1}e_{7}=(1-\alpha)e_{7}$ | $e_{1}e_{8}=-e_{8}$
$\par e_{2}e_{1}=3e_{2}$ | $e_{3}e_{1}=-2e_{3}-\alpha e_{7}$ | $e_{3}e_{3}=-3e_{4}$ | $e_{3}e_{5}=e_{6}-\beta e_{7}$
$\par e_{5}e_{1}=-2e_{1}$ | $e_{5}e_{2}=(-3+\beta)e_{2}$ | $e_{5}e_{3}=(-1-\beta)e_{3}$ | $e_{5}e_{4}=-2\beta e_{4}$
$\par e_{5}e_{5}=-2e_{5}$ | $e_{5}e_{6}=(-1-\beta)e_{6}$ | $e_{5}e_{7}=(-1-\beta)e_{7}$ | $e_{5}e_{8}=-2e_{8}$
$\par e_{6}e_{1}=2e_{3}+\alpha e_{7}$ | $e_{6}e_{3}=3e_{4}$ | $e_{6}e_{5}=-e_{6}+\beta e_{7}$ | $e_{7}e_{1}=2e_{3}+\alpha e_{7}$
| $\par e_{7}e_{3}=3e_{4}$ | $e_{7}e_{5}=-e_{6}+\beta e_{7}$ |
###### Lemma 6.
The algebra $\overline{{\mathscr{S}}_{\alpha,\beta}}$ is a conservative
algebra; and it is a terminal algebra if and only if $(\alpha,\beta)=(-1,1)$
or $(\alpha,\beta)=(0,0)$.
###### Proof.
The conservative property of the algebra
$\overline{{\mathscr{S}}_{\alpha,\beta}}$ is following from the direct
verification of the conservative identity with the additional multiplication
$*:$
$e_{1}*e_{1}=-e_{1}$ | $e_{1}*e_{2}=-e_{2}$ | $e_{1}*e_{5}=-2e_{1}$ | $e_{2}*e_{1}=(1-\alpha)e_{2}$
---|---|---|---
$\par e_{2}*e_{5}=(-1-\beta)e_{2}$ | $e_{5}*e_{1}=-2e_{1}$ | $e_{5}*e_{2}=-2e_{2}$ | $e_{5}*e_{5}=-4e_{1}$
∎
#### 1.1.3. The algebra $\widehat{W(2)}$
The second interesting ”big” subalgebra of $W(2)$ is $W_{2},$ which is
generated by $e_{1},\ldots,e_{6}.$ Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},te_{7},te_{8})}{\widehat{W(2)}}.$
It is easy to see, that the multiplication table (for nonzero products) of
${\widehat{W(2)}}$ is given by the following table.
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
$e_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{5}$ | $-2e_{1}$ | $-3e_{2}$ | $-e_{3}$ | $0$ | $-2e_{5}$ | $-e_{6}$ | $-e_{7}$ | $-2e_{8}$
$e_{6}$ | $2e_{3}$ | $e_{1}$ | $3e_{4}$ | $0$ | $-e_{6}$ | $0$ | $0$ | $e_{7}$
###### Lemma 7.
The algebra ${\widehat{W(2)}}$ is terminal.
#### 1.1.4. The algebra $\widehat{\widehat{W(2)}}$
The next interesting subalgebra of $W(2)$ is $S_{2},$ which is generated by
$e_{1},\ldots,e_{4}.$ Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},e_{2},e_{3},e_{4},te_{5},te_{6},te_{7},te_{8})}{\widehat{\widehat{W(2)}}}.$
It is easy to see, that the multiplication table (for nonzero products) of
${\widehat{\widehat{W(2)}}}$ is given by the following table.
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $2e_{1}$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $-e_{1}$ | $-3e_{4}$ | $0$ | $e_{6}$ | $0$ | $0$ | $-e_{7}$
###### Corollary 8.
The algebra ${\widehat{\widehat{W(2)}}}$ is terminal.
#### 1.1.5. The algebra $\widetilde{W(2)}$
The next interesting subalgebra of $W(2)$ is generated by $e_{1}$ and $e_{2}.$
Let us consider the IW contraction
$W(2)\xrightarrow{(e_{1},e_{2},te_{3},te_{4},te_{5},te_{6},te_{7},te_{8})}{\widetilde{W(2)}}.$
It is easy to see, that the multiplication table (for nonzero products) of
${\widetilde{W(2)}}$ is given by the following table.
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $0$ | $e_{3}$ | $0$ | $-e_{5}$ | $e_{8}$ | $0$
$e_{3}$ | $-2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{6}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{7}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
###### Lemma 9.
The algebra ${\widetilde{W(2)}}$ is a non-Leibniz, non-Jordan terminal
algebra.
#### 1.1.6. The algebra $\widetilde{\widetilde{W(2)}}$
The last interesting subalgebra of $W(2)$ is generated by $e_{1}.$ Let us
consider the IW contraction
$W(2)\xrightarrow{(e_{1},te_{2},te_{3},te_{4},te_{5},te_{6},te_{7},te_{8})}\widetilde{\widetilde{W(2)}}.$
It is easy to see, that the multiplication table (for nonzero products) of
$\widetilde{\widetilde{W(2)}}$ is given by the following table.
| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{6}$ | $e_{7}$ | $e_{8}$
---|---|---|---|---|---|---|---|---
$e_{1}$ | $-e_{1}$ | $-3e_{2}$ | $e_{3}$ | $3e_{4}$ | $-e_{5}$ | $e_{6}$ | $e_{7}$ | $-e_{8}$
$e_{2}$ | $3e_{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{3}$ | $-2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{6}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$e_{7}$ | $2e_{3}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
###### Lemma 10.
The algebra $\widetilde{\widetilde{W(2)}}$ is a non-Leibniz, non-Jordan
terminal algebra.
### 1.2. IW contraction of $S_{2}$ and $W_{2}$
Thanks to [18], algebras $S_{2}$ and $W_{2}$ have also only one subalgebra of
codimension $1,$ which are $\langle e_{1},e_{3},e_{4}\rangle$ and $\langle
e_{1},e_{3},e_{4},e_{5},e_{6}\rangle.$ Other important subalgebras of $S_{2}$
and $W_{2}$ are $\langle e_{1}\rangle,$ $\langle e_{1},e_{2}\rangle$ and
$\langle e_{1},e_{2},e_{3},e_{4}\rangle.$ All contractions of $S_{2}$ (and
$W_{2}$) with respected to all cited subalgebras can be obtained as
$4$-dimensional subalgebras $\langle e_{1},e_{2},e_{3},e_{4}\rangle$
($6$-dimensional subalgebras $\langle e_{1},\ldots,e_{6}\rangle$) of the
following algebras $\overline{W(2)},$ ${\widehat{\widehat{W(2)}}},$
${\widetilde{W(2)}}$ and $\widetilde{\widetilde{W(2)}}.$ All these subalgebras
are non-Leibniz, non-Jordan terminal algebras.
## 2\. Varieties related to $W(2)$ and its subalgebras
### 2.1. Identities
Let ${\mathbb{F}}$ be a field of characteristic zero and ${\mathbb{F}}\langle
x_{1},x_{2},\ldots,x_{n}\rangle$ the free nonassociative
${\mathbb{F}}$-algebra in $n$ indeterminates. Let ${\bf A}$ be any algebra and
${\mathfrak{S}}^{n}_{\bf A}$ the subspace of ${\mathbb{F}}\langle
x_{1},x_{2},\ldots,x_{n}\rangle$ of all $n$-linear
$w(x_{1},x_{2},\ldots,x_{n})$ which vanish on ${\bf A}$ (so
$w(x_{1},x_{2},\ldots,x_{n})$ is of degree one in each variable). We have
studied by direct verification the subspace ${\mathfrak{S}}^{n}_{\bf A}$ for
$n=3,4$ of $W(2)$ and of its IW contractions mentioned in this paper.
###### Proposition 11.
In the following table, we summarize the dimension of the subspaces
${\mathfrak{S}}^{n}_{\bf A}$ for ${\bf
A}\in\\{W(2),\overline{W(2)},\overline{\mathscr{S}_{1}},\overline{\mathscr{S}_{5}},\overline{\mathscr{S}_{-1,1}},\overline{\mathscr{S}_{0,0}},\widehat{W(2)},\widehat{\widehat{W(2)}},{\widetilde{W(2)}}\\}$
and $n=3,4$.
Algebra (${\bf A}$) $\textrm{dim}({\mathfrak{S}}^{3}_{\bf A})$
$\textrm{dim}({\mathfrak{S}}^{4}_{\bf A})$ Comments $W(2)$ 0 0 non-terminal
conservative $\overline{W(2)}$ 0 20 non-terminal, conservative
$\overline{\mathscr{S}_{1}}$ 0 64 terminal $\overline{\mathscr{S}_{5}}$ 0 40
non-terminal, conservative $\overline{\mathscr{S}_{-1,1}}$ 0 64 terminal
$\overline{\mathscr{S}_{0,0}}$ 0 44 terminal $\widehat{W(2)}$ 0 24 terminal
$\widehat{\widehat{W(2)}}$ 0 47 terminal ${\widetilde{W(2)}}$ 0 82 terminal
${\widetilde{\widetilde{W(2)}}}$ 2 101 terminal
Moreover, if ${\bf A}=\overline{\mathscr{S}_{\alpha,\beta}}$ then
$\dim({\mathfrak{S}}^{3}_{\bf A})=0$ for $(\alpha,\beta)\neq(2,1)$ and
$(\alpha,\beta)\neq(0,-3)$.
###### Proof.
We have determined the spaces ${\mathfrak{S}}^{n}_{\bf A}$ for $n=3,4$ by
constructing an arbitrary $n$-linear map $w(x_{1},\ldots,x_{n})$ and solving
$w(x_{1},\ldots,x_{n})=0$ in ${\bf A}$ using Wolfram.
∎
###### Proposition 12.
If ${\bf A}=\overline{\mathscr{S}_{2,1}}$ then $\dim({\mathfrak{S}}^{3}_{\bf
A})=3$ and a basis of the ${\mathbb{F}}$-vector space ${\mathfrak{S}}^{3}_{\bf
A}$ is the set of identities:
1. (1)
$x_{1}(x_{2}x_{3})-x_{2}(x_{1}x_{3}),$
2. (2)
$x_{2}(x_{3}x_{1})-x_{3}(x_{2}x_{1}),$
3. (3)
$x_{3}(x_{1}x_{2})-x_{1}(x_{3}x_{2}).$
Now, consider the following identities:
$\mathfrak{st}^{n}_{1}=\sum\limits_{\sigma\in\mathbb{S}_{n}}(-1)^{\sigma}(\ldots(x_{\sigma(1)}x_{\sigma(2)})x_{\sigma(3)}\ldots)x_{\sigma(n)}\mbox{
and
}\mathfrak{st}^{n}_{2}=\sum\limits_{\sigma\in\mathbb{S}_{n}}(-1)^{\sigma}x_{\sigma(n)}(\ldots
x_{\sigma(3)}(x_{\sigma(2)}x_{\sigma(1)})\ldots).$
It is clear that if an algebra satisfies $\mathfrak{st}^{n}_{1}$ (resp.
$\mathfrak{st}^{n}_{2}$) then it also satisfies $\mathfrak{st}^{n+1}_{1}$
(resp. $\mathfrak{st}^{n+1}_{2}$).
###### Proposition 13.
If ${\bf A}=\overline{\mathscr{S}_{0,-3}}$ then $\dim({\mathfrak{S}}^{3}_{\bf
A})=1$. This ${\mathbb{F}}$-vector space is generated by
$2\mathfrak{st}^{3}_{1}-3\mathfrak{st}^{3}_{2}.$
###### Proposition 14.
If ${\bf A}=\widetilde{\widetilde{W(2)}}$ then $\dim({\mathfrak{S}}^{3}_{\bf
A})=2$ and a basis of the ${\mathbb{F}}$-vector space ${\mathfrak{S}}^{3}_{\bf
A}$ is the set of identities $\mathfrak{st}^{3}_{1},\mathfrak{st}^{3}_{2}$.
We have studied the space ${\mathfrak{S}}^{n}_{\bf A}$ for the subalgebras of
$W(2)$ mentioned in this paper. We have also studied the identities
$\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{2}$ for these subalgebras for
$n=3,4,5$.
###### Proposition 15.
In the following table, we summarize the dimension of the subspaces
${\mathfrak{S}}^{n}_{\bf A}$ for ${\bf A}$ a subalgebra of $W(2)$ and $n=3,4$.
Subalgebra $({\bf A})$ $\textrm{dim}({\mathfrak{S}}^{3}_{\bf A})$
$\textrm{dim}({\mathfrak{S}}^{4}_{\bf A})$ Comments $B_{2}:=\langle
e_{1},e_{3},e_{4},e_{5},e_{6},e_{7},e_{8}\rangle$ 0 64 no
$\mathfrak{st}^{4}_{1}$, no $\mathfrak{st}^{4}_{2}$,
$\mathfrak{st}^{5}_{1},\mathfrak{st}^{5}_{2}$ $W_{2}=\langle
e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\rangle$ 0 24 no $\mathfrak{st}^{5}_{1}$,
no $\mathfrak{st}^{4}_{2},$ $\mathfrak{st}^{5}_{2}$ $C_{2}:=\langle
e_{1},e_{3},e_{4},e_{5},e_{6}\rangle$ 0 64 no $\mathfrak{st}^{4}_{1}$, no
$\mathfrak{st}^{4}_{2}$, $\mathfrak{st}^{5}_{1},$ $\mathfrak{st}^{5}_{2}$
$S_{2}=\langle e_{1},e_{2},e_{3},e_{4}\rangle$ 3 86 no
$\mathfrak{st}^{4}_{1}$, no $\mathfrak{st}^{4}_{2}$, $\mathfrak{st}^{5}_{1},$
$\mathfrak{st}^{5}_{2}$ $D_{2}:=\langle e_{1},e_{3},e_{4}\rangle$ 6 110
$\mathfrak{st}^{3}_{1},\mathfrak{st}^{3}_{2}$ $E_{2}:=\langle
e_{1},e_{2}\rangle$ 8 115 $\mathfrak{st}^{3}_{1},\mathfrak{st}^{3}_{2}$
Moreover, ${\mathfrak{S}}^{4}_{B_{2}}={\mathfrak{S}}^{4}_{C_{2}};$ all present
algebras are terminal, non-Leibniz and non-Jordan.
###### Proposition 16.
If ${\bf A}=S_{2}$, the subalgebra of $W(2)$ generated by
$e_{1},\ldots,e_{4}$, then a basis of the ${\mathbb{F}}$-vector space
${\mathfrak{S}}^{3}_{\bf A}$ is the set of identities:
1. (1)
$30x_{1}(x_{2}x_{3})-42x_{1}(x_{3}x_{2})-25x_{2}(x_{1}x_{3})+7x_{2}(x_{3}x_{1})+39x_{3}(x_{1}x_{2})-9x_{3}(x_{2}x_{1})+10(x_{1}x_{3})x_{2}+15(x_{2}x_{1})x_{3}-19(x_{3}x_{1})x_{2}-6(x_{3}x_{2})x_{1}$,
2. (2)
$-5x_{1}(x_{2}x_{3})+11x_{1}(x_{3}x_{2})+5x_{2}(x_{1}x_{3})-11x_{2}(x_{3}x_{1})-12x_{3}(x_{1}x_{2})+12x_{3}(x_{2}x_{1})-5(x_{1}x_{3})x_{2}+5(x_{2}x_{3})x_{1}+2(x_{3}x_{1})x_{2}-2(x_{3}x_{2})x_{1}$,
3. (3)
$-3x_{1}(x_{2}x_{3})-3x_{1}(x_{3}x_{2})+4x_{2}(x_{1}x_{3})-4x_{2}(x_{3}x_{1})+3x_{3}(x_{1}x_{2})+3x_{3}(x_{2}x_{1})+3(x_{1}x_{2})x_{3}+2(x_{1}x_{3})x_{2}-2(x_{3}x_{1})x_{2}-3(x_{3}x_{2})x_{1}$.
Finally, we have studied the family of identities $\mathfrak{st}^{n}_{1}$ and
$\mathfrak{st}^{n}_{2}$ for $W(2)$ and its contractions.
###### Proposition 17.
In the following table, we summarize which identities from the families
$\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{1}$ are satisfies for every
contraction of $W(2)$, for $n=3,4,5$.
Algebra $\mathfrak{st}^{3}_{1}$ $\mathfrak{st}^{4}_{1}$
$\mathfrak{st}^{5}_{1}$ $\mathfrak{st}^{3}_{2}$ $\mathfrak{st}^{4}_{2}$
$\mathfrak{st}^{5}_{2}$ $W(2)$ ✗ ✗ ✗ ✗ ✗ ✓ $\overline{W(2)}$ ✗ ✗ ✓ ✗ ✗ ✓
$\overline{\mathscr{S}_{1}}$ ✗ ✗ ✓ ✗ ✗ ✓ $\overline{\mathscr{S}_{5}}$ ✗ ✗ ✓ ✗
✗ ✓ $\overline{\mathscr{S}_{\alpha,\beta}}$ ✗ $\text{\char
51}_{\alpha=\frac{3+\beta}{2}}$ ✓ $\text{\char 51}_{(\alpha,\beta)=(2,1)}$
$\text{\char 51}_{\alpha=\frac{3+\beta}{2}}$ ✓ $\widehat{W(2)}$ ✗ ✗ ✗ ✗ ✗ ✓
$\widehat{\widehat{W(2)}}$ ✗ ✗ ✓ ✗ ✗ ✓ ${\widetilde{W(2)}}$ ✗ ✓ ✓ ✗ ✓ ✓
###### Corollary 18.
${\mathfrak{S}}^{4}_{W(n)}=0$ and ${\mathfrak{S}}^{5}_{W(2)}\neq 0$.
The present corollary gives the following question.
###### Open question.
Find minimal $k,$ such that ${\mathfrak{S}}^{k}_{W(n)}\neq 0$. In this case,
is $W(n)$ satisfying $\mathfrak{st}^{k}_{2}$?
### 2.2. Other degree five identities for $W(2)$
In this subsection we are interested in finding other degree five identities
for $W(2)$. Consider the set of free monomials
$w(x_{1},x_{2},x_{3},x_{4},x_{5})$ of degree five up to permutations of the
variables. There are exactly fourteen monomials:
$w_{1}(x_{1},x_{2},x_{3},x_{4},x_{5})=(((x_{1}x_{2})x_{3})x_{4})x_{5}$ | $w_{2}(x_{1},x_{2},x_{3},x_{4},x_{5})=((x_{1}x_{2})x_{3})(x_{4}x_{5})$
---|---
$w_{3}(x_{1},x_{2},x_{3},x_{4},x_{5})=((x_{1}x_{2})(x_{3}x_{4}))x_{5}$ | $w_{4}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}x_{2})((x_{3}x_{4})x_{5})$
$w_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}x_{2})(x_{3}(x_{4}x_{5}))$ | $w_{6}(x_{1},x_{2},x_{3},x_{4},x_{5})=((x_{1}(x_{2}x_{3}))x_{4})x_{5}$
$w_{7}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}(x_{2}x_{3}))(x_{4}x_{5})$ | $w_{8}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}((x_{2}x_{3})x_{4}))x_{5}$
$w_{9}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}(((x_{2}x_{3})x_{4})x_{5})$ | $w_{10}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}((x_{2}x_{3})(x_{4}x_{5}))$
$w_{11}(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1}(x_{2}(x_{3}x_{4})))x_{5}$ | $w_{12}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}((x_{2}(x_{3}x_{4}))x_{5})$
$w_{13}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}(x_{2}((x_{3}x_{4})x_{5}))$ | $w_{14}(x_{1},x_{2},x_{3},x_{4},x_{5})=x_{1}(x_{2}(x_{3}(x_{4}x_{5})))$
Now, consider the $\mathbb{F}$-vector spaces ${\mathfrak{Z}}^{i}$ generated by
the set:
$\left\\{w_{i}(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},x_{\sigma(4)},x_{\sigma(5)}):\sigma\in\mathbb{S}_{5}\right\\}.$
Denote by ${\mathfrak{Z}}^{i}_{\bf A}$ the subspace of ${\mathfrak{Z}}^{i}$ of
all $5$-linear polynomials vanishing on $\bf A$. Then we have the following
result regarding the dimension of these subspaces:
###### Proposition 19.
If ${\bf A}=W(2)$, then $\textrm{dim}({\mathfrak{Z}}^{i}_{\bf A})=0$ for
$1\leq i\leq 13$ and $\textrm{dim}({\mathfrak{Z}}^{14}_{\bf A})=5$. A basis of
the ${\mathbb{F}}$-vector space ${\mathfrak{Z}}^{14}_{\bf A}$ is the set of
identities:
1. (1)
$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(2)}(x_{\sigma(3)}(x_{\sigma(4)}(x_{\sigma(5)}x_{1}))).$
2. (2)
$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(3)}(x_{\sigma(4)}(x_{\sigma(5)}x_{2}))).$
3. (3)
$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(2)}(x_{\sigma(4)}(x_{\sigma(5)}x_{3}))).$
4. (4)
$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(2)}(x_{\sigma(3)}(x_{\sigma(5)}x_{4}))).$
5. (5)
$\sum\limits_{\sigma\in\mathbb{S}_{4}}(-1)^{\sigma}x_{\sigma(1)}(x_{\sigma(2)}(x_{\sigma(3)}(x_{\sigma(4)}x_{5}))).$
Moreover, the linear combination with parameters $(1,-1,1,-1,1)$ is
$\mathfrak{st}^{5}_{2}$.
### 2.3. Central extensions
The notion of central extensions appeared in the study of Lie algebras, but it
can be considered in an arbitrary variety of algebras (see, for example,
[19]). The calculation of central extensions of an algebra ${\bf A}$ of
dimension $n$ from a certain variety of algebras gives the classification of
all algebras with $(k-n)$-dimensional annihilator, such that its factor
algebra by the annihilator is isomorphic to ${\bf A}$ (see, for example, [7]).
These calculations are carried out by studying the cohomology, with respect to
a polynomial identity, of the algebra ${\bf A}$. In this section, we are
interested in the central extensions of the contractions and subalgebras of
$W(2)$ considered in this paper. Some of these contractions and subalgebras
have turned out to be terminal, i.e., they satisfy the terminal identity
(degree four). The following result is about these particular algebras.
###### Proposition 20.
There are no terminal central extensions of the terminal contractions of
$W(2)$: $\overline{\mathscr{S}_{1}}$, $\overline{\mathscr{S}_{-1,1}}$,
$\overline{\mathscr{S}_{0,0}}$, $\widehat{W(2)}$, $\widehat{\widehat{W(2)}}$,
$\widetilde{W(2)}$ and $\widetilde{\widetilde{W(2)}}.$
###### Proof.
Recall that if $Z^{2}_{P}\left({\bf A},{\mathbb{F}}\right)$ denotes the space
of cocycles with respect to the polynomial identity $P$ of the algebra ${\bf
A}$, $B^{2}\left({\bf A},{\mathbb{F}}\right)$ denotes the space of coborders
of the algebra ${\bf A}$ and $H^{2}_{P}\left({\bf
A},{\mathbb{F}}\right):=Z^{2}_{P}\left({\bf
A},{\mathbb{F}}\right)/B^{2}\left({\bf A},{\mathbb{F}}\right)$ denotes the
cohomology space with respect to the polynomial identity $P$ of the algebra
${\bf A}$, then if $H^{2}_{P}\left({\bf A},{\mathbb{F}}\right)$ is trivial, we
have that ${\bf A}$ has no central extensions for the identity $P$. Now, fix
$P=T$ the terminal identity. Thus, the result is proven by direct calculation
of the cohomology space, obtaining that $H^{2}_{T}\left({\bf
A},{\mathbb{F}}\right)$ is trivial for any of the terminal contractions ${\bf
A}$ considered. ∎
###### Proposition 21.
There are no terminal central extensions of the terminal subalgebras of
$W(2)$: $B_{2},W_{2},C_{2},S_{2},D_{2},E_{2}$.
###### Proof.
The result follows by the direct calculation of the cohomology space,
obtaining that $H^{2}_{T}\left({\bf A},{\mathbb{F}}\right)$ is trivial for any
of the terminal subalgebras ${\bf A}$ considered. ∎
Similarly, we can determine if there are central extensions for the rest of
identities mentioned in the previous section.
###### Proposition 22.
By calculating the correspoding cohomology space, we conclude the following.
1. (1)
There are no central extensions of $\overline{\mathscr{S}_{2,1}}$ in the
variety defined by one identity from the proposition 12.
2. (2)
$\textrm{dim}\,Z^{2}_{P}\left(\overline{\mathscr{S}_{0,-3}},{\mathbb{F}}\right)=31$
and
$\textrm{dim}\,H^{2}_{P}\left(\overline{\mathscr{S}_{0,-3}},{\mathbb{F}}\right)=23$,
where $P$ is the identity $2\mathfrak{st}^{3}_{1}-3\mathfrak{st}^{3}_{2}$ from
proposition 13.
3. (3)
There are no central extensions of $S_{2}$ in the variety defined by one
identity from the proposition 16.
Regarding the central extensions with respect to the identities
$\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{2}$ for $n=3,4,5$ (see
Proposition 15 and Proposition 17), we have the following result.
###### Proposition 23.
The dimensions of the spaces of cocycles and coborders of the subalgebras of
$W(2)$ are given.
Algebra $\textrm{dim}\,B^{2}$ $\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{2}}$ $B_{2}$ 7 - - 44 - - 44 $W_{2}$
6 - - - - - 30 $C_{2}$ 5 - - 24 - - 24 $S_{2}$ 4 - - 16 - - 16 $D_{2}$ 3 8 9 9
8 9 9 $E_{2}$ 2 4 4 4 4 4 4
###### Proposition 24.
The dimensions of the spaces of cocycles and coborders of the subalgebras of
$W(2)$ are given.
Algebra $\textrm{dim}\,B^{2}$ $\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{1}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{3}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{4}_{2}}$
$\textrm{dim}\,Z^{2}_{\mathfrak{st}^{5}_{2}}$ $W(2)$ 8 - - - - - 38
$\overline{W(2)}$ 8 - - 54 - - 54 $\overline{\mathscr{S}_{1}}$ 8 - - 60 - - 60
$\overline{\mathscr{S}_{5}}$ 8 - - 60 - - 60 $\overline{\mathscr{S}_{-1,1}}$ 8
- - 60 - - 60 $\overline{\mathscr{S}_{0,0}}$ 8 - - 60 - - 60 $\widehat{W(2)}$
8 - - - - - 40 $\widehat{\widehat{W(2)}}$ 8 - - 50 - - 54 ${\widetilde{W(2)}}$
8 - 52 64 - 52 64 $\widetilde{\widetilde{W(2)}}$ 8 43 64 64 43 64 64
By Proposition 23 and Proposition 24, we can conclude that for any of the
subalgebras and contractions considered there are central extensions with
respect to the identities $\mathfrak{st}^{n}_{1}$ and $\mathfrak{st}^{n}_{2}$
for $n=3,4,5$.
## Acknowledgements
Funding The first part of this work is supported by the Spanish Government
through the Ministry of Universities grant ‘Margarita Salas’, funded by the
European Union - NextGenerationEU; by the Junta de Andalucía through projects
UMA18-FEDERJA-119 and FQM-336 and by the Spanish Ministerio de Ciencia e
Innovación through project PID 2019-104236GB-I00, all of them with FEDER
funds; FCT UIDB/00212/2020 and UIDP/00212/2020. The second part of this work
is supported by the Russian Science Foundation under grant 22-11-00081.
## References
* [1] F. Arzikulov, N. Umrzaqov, Conservative algebras of 2-dimensional algebras. III, Communications in Mathematics, 29 (2021), 2, 255–267.
* [2] A. Calderón, A. Fernández Ouaridi, I. Kaygorodov, The classification of $2$-dimensional rigid algebras, Linear and Multilinear Algebra, 68 (2020), 4, 828–844.
* [3] N. Cantarini, V. Kac, Classification of linearly compact simple rigid superalgebras, International Mathematics Research Notices, 17 (2010), 3341–3393.
* [4] M. Cardoso Gonçalves, D. Gonçalves, D. Martín Barquero, C. Martín González, M. Siles Molina, Squares and associative representations of two-dimensional evolution algebras, Journal of Algebra and its Applications 20 (2021), 6, 2150090.
* [5] R. Fehlberg Júnior, I. Kaygorodov, On the Kantor product, II, Carpatian Mathematical Publications, 2022, to appear, arXiv:2201.00174
* [6] A. Giambruno, S. Mishchenko, M. Zaicev, Codimension growth of two-dimensional non-associative algebras, Proceedings of the American Mathematical Society, 135 (2007), 11, 3405–3415.
* [7] A. Hegazi, H. Abdelwahab, A. J. Calderon Martin, The classification of n-dimensional non-Lie Malcev algebras with $(n-4)$-dimensional annihilator, Linear Algebra and its Applications, 505 (2016), 32–56.
* [8] E. Inönü, E.P. Wigner, On the contraction of groups and their representations, Proceedings of the National Academy of Sciences USA, 39 (1953), 510–524.
* [9] A. Fernández Ouaridi, I. Kaygorodov, M. Khrypchenko, Yu. Volkov, Degenerations of nilpotent algebras, Journal of Pure and Applied Algebra, 226 (2022), 3, 106850, 21 pp.
* [10] Sh. Kadyrov, F. Mashurov, Unified computational approach to nilpotent algebra classification problems, Communications in Mathematics, 29 (2021), 2, 215–226.
* [11] I. Kantor, Certain generalizations of Jordan algebras (Russian), Trudy Sem. Vektor. Tenzor. Anal., 16 (1972), 407–499.
* [12] I. Kantor, On an extension of a class of Jordan algebras, Algebra and Logic, 28 (1989), 2, 117–121.
* [13] I. Kantor, Terminal trilinear operations, Algebra and Logic, 28 (1989), 1, 25–40.
* [14] I. Kantor, An universal conservative algebra, Siberian Mathematical Journal, 31 (1990), 3, 388–395.
* [15] I. Kaygorodov, On the Kantor Product, Journal of Algebra and Its Applications, 16 (2017), 9, 1750167 (17 pages).
* [16] I. Kaygorodov, M. Khrypchenko, Yu. Popov, The algebraic and geometric classification of nilpotent terminal algebras, Journal of Pure and Applied Algebra, 225 (2021), 6, 106625.
* [17] I. Kaygorodov, A. Khudoyberdiyev, A. Sattarov, One-generated nilpotent terminal algebras, Communications in Algebra, 48 (2020), 10, 4355–4390.
* [18] I. Kaygorodov, A. Lopatin, Yu. Popov, Conservative algebras of 2-dimensional algebras, Linear Algebra and its Applications, 486 (2015), 255–274.
* [19] I. Kaygorodov, S. Lopes, P. Páez-Guillán, Non-associative central extensions of null-filiform associative algebras, Journal of Algebra, 560 (2020), 1190–1210.
* [20] I. Kaygorodov, Yu. Popov, A. Pozhidaev, The universal conservative superalgebra, Communications in Algebra, 47 (2019), 10, 4064–4074.
* [21] I. Kaygorodov, Yu. Volkov, Conservative algebras of 2-dimensional algebras. II, Communications in Algebra, 45 (2017), 8, 3413-3421.
* [22] I. Kaygorodov, Yu. Volkov, The variety of $2$-dimensional algebras over algebraically closed fields, Canadian Journal of Mathematics, 71 (2019), 4, 819–842.
* [23] H. Petersson, The classification of two-dimensional nonassociative algebras, Results in Mathematics, 37 (2000), 1-2, 120–154.
* [24] H. Petersson, M. Scherer, The number of nonisomorphic two-dimensional algebras over a finite field, Results in Mathematics, 45 (2004), no. 1-2, 137–152.
* [25] Yu. Popov, Conservative algebras and superalgebras: a survey, Communications in Mathematics, 28 (2020), 2, 231–251.
|
# Microscopic Optical Potentials: recent achievements and future perspectives
Paolo Finelli Dipartimento di Fisica ed Astronomia, Università degli Studi di
Bologna and INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy
<EMAIL_ADDRESS>Matteo Vorabbi Department of Physics, University of
Surrey, Guildford, Surrey, GU2 7XH, UK<EMAIL_ADDRESS>Carlotta Giusti
Dipartimento di Fisica, Università degli Studi di Pavia and INFN, Sezione di
Pavia, I-27100 Pavia, Italy<EMAIL_ADDRESS>
###### Abstract
Few years ago we started the investigation of microscopic Optical Potentials
(OP) in the framework of chiral effective field theories [1, 2] and published
our results in a series of manuscripts. Starting from the very first work [3],
where a microscopic OP was introduced following the multiple scattering
procedure of Watson [4], and then followed by Refs. [5, 6], where the
agreement with experimental data and phenomenological approaches was
successfully tested, we finally arrived at a description of elastic scattering
processes off non-zero spin nuclei [7]. Among our achievements, it is worth
mentioning the partial inclusion of three-nucleon forces [8], and the
extension of our OP to antiproton-nucleus elastic scattering [9]. Despite the
overall good agreement with empirical data obtained so far, we do believe that
several improvements and upgrades of the present approach are still to be
achieved.
In this short essay we would like to address some of the most relevant
achievements and discuss an interesting development that, in our opinion, is
needed to further improve microscopic OPs in order to reach in a near future
the same level of accuracy of the phenomenological ones.
## 1 Introduction
It is well known that a suitable and successful framework to describe nucleon-
nucleus processes is provided by the concept of the nuclear optical potential.
Within this approach it is possible to compute the scattering observables
across wide regions of the nuclear landscape and to extend calculations to
inelastic channels or to a wide variety of nuclear reactions, e.g., nucleon
transfer, knockout, capture, or breakup. Even if it is true that a
phenomenological approach is generally preferred to achieve a more accurate
description of the available experimental data, nowadays, with the upcoming
facilities for exotic nuclei (FRIB at MSU just to mention one of the most
important projects [10]), we strongly believe that a microscopic approach will
be the preferred way to make robust predictions and to assess the unavoidable
theoretical uncertainties already in a near future [11].
## 2 Rooting optical potentials within ab initio approaches
The calculation of a microscopic optical potential requires, in principle, the
solution of the full many-body nuclear problem for the incident nucleon and
the A nucleons of the target. In practice, with suitable approximations,
microscopic optical potentials are usually calculated from two basic
quantities: the nucleon-nucleon ($NN$) $t$ matrix and the matter distribution
of the nucleus in the coordinate $\rho({\bm{r}},{\bm{r}^{\prime}})$, or in the
momentum $\rho({\bm{k}},{\bm{k}^{\prime}})$ representation space. Formally we
have to start from the the full $(A+1)$-body Lippmann-Schwinger equation that
allows to determine the two-body transition matrix $T$ defined as
$V|\Psi\rangle=T|\phi\rangle\;,$ (1)
where $|\Psi\rangle$ is the scattered state and $|\Phi\rangle$ is a free wave
(asymptotic) solution, in terms of an external two-body interaction $V$
$T=V+VG_{0}(E)T\,.$ (2)
The previous relation can be manipulated in order to obtain a set of two
equations in terms of an auxiliary quantity, i.e. the optical potential $U$.
The first approximation we need is given by the spectator expansion [12], in
which the scattering relation is expanded in a finite series of terms where
the target nucleons interact directly with the incident proton. In particular,
the first term of this series only involves the interaction of the projectile
with a single target nucleon, the second term involves the interaction of the
projectile with two target nucleons, and so on to the subsequent orders.
As a consequence we can split Eq. (2) into a set of two equations. The first
one is an integral equation for $T$
$T=U+UG_{0}(E)PT\,,$ (3)
where $U$ is the optical potential operator, and the second one is an integral
equation for $U$
$U=V+VG_{0}(E)QU\,.$ (4)
$G_{0}(E)$ is the free propagator for the $(A+1)$-nucleon system and the
operators $P$ and $Q$ in Eqs. (3) and (4) are projection operators,
$P+Q=\hbox{$1\hskip-1.2pt\vrule depth=0.0pt,height=6.88889pt,width=0.7pt\vrule
depth=0.0pt,height=0.3pt,width=1.19995pt$}\,.$ (5)
For elastic scattering processes $P$ selects only the ground state
$|\Phi_{0}\rangle$, i.e.
$P=|\Phi_{0}\rangle\langle\Phi_{0}|\;.$ (6)
Eq. (4) can be numerically solved if we assume the validity of another
approximation, namely the impulse approximation where, since the projectile
energy scales involved are large compared to nuclear binding forces of the
interacting target nucleon, we can safely use the free $NN$ $t$ matrix in the
Eq. (3). Most of the theoretical derivations can be found in Refs. [3, 12].
Figure 1: In this infographic the main building blocks of the optical
potential are schematically shown along the main features of each components.
For practical reasons the auxiliary optical potential $U$ is expressed as a
function of the center-of-mass momentum ${\bm{K}}\equiv
1/2({\bm{k}}+{\bm{k}}^{\prime})$, and the momentum transfer
${\bm{q}}\equiv({\bm{k}}-{\bm{k}}^{\prime})$. It can be factorized into three
pieces [13, 14, 15, 16]: the Möller correction
$\eta({\bm{q}},{\bm{K}},{\bm{P}})$ that ensures the conservation of the
transmitted flux after the transformation in the center-of-mass reference
frame, the projectile-target potential $t_{NN}({\bm{q}},{\bm{K}};\omega)$ that
depends also on the energy of the projectile $\omega$, and the structure
component $\rho_{N}({\bm{q}},{\bm{P}})$, i.e. the Fourier transform of the
non-local matter density in the momentum space. For both inputs we show into
the coloured insertions the main features. For the potential part, at the
moment, we are able to include the full $NN$ interaction and averaged $NNN$
contributions in terms of a medium density-dependent $NN$ interaction [17].
This solution requires to fix a density $\bar{\rho}$ at which the calculations
are performed. So far [7, 8] we decided to explore the impact of $NNN$ terms
performing calculations for several values of $\bar{\rho}$ (usually in the
range $0.08$ fm${}^{-3}\leq\bar{\rho}\leq 0.13$ fm-3, see Figs 1-6 of Ref.
[7]). Investigations in this direction are currently underway [18]. For the
structure part we used the NCSM [21] for light nuclei that allows us to fully
use both $NN$ and $NNN$ terms (see Sect II and III. of Ref. [7]). For medium
nuclei other approaches should be explored, like Self-Consistent Green
Functions (SCGF) [22, 23].
Then, from a practical point of view, the OP can be computed in momentum space
as follows (after some mathematical manipulations that can be found in Refs.
[13, 14, 15, 16])
$\begin{split}U({\bm{q}},{\bm{K}};E)=\sum_{N=p,n}&\int
d{\bm{P}}\;\eta({\bm{q}},{\bm{K}},{\bm{P}})\,t_{NN}\left[{\bm{q}},\frac{1}{2}\left(\frac{A+1}{A}{\bm{K}}+\sqrt{\frac{A-1}{A}}{\bm{P}}\right);E\right]\\\
&\times\rho_{N}\left({\bm{P}}+\sqrt{\frac{A-1}{A}}\frac{{\bm{q}}}{2},{\bm{P}}-\sqrt{\frac{A-1}{A}}\frac{{\bm{q}}}{2}\right)\,,\end{split}$
(7)
where ${\bm{q}}$ and ${\bm{K}}$ represent the momentum transfer and the
average momentum, respectively. Here ${\bm{P}}$ is an integration variable,
$t_{NN}$ is the $NN$ $t$ matrix and $\rho_{N}$ is the one-body nuclear density
matrix. The parameter $\eta$ is the Möller factor, that imposes the Lorentz
invariance of the flux when we pass from the $NA$ to the $NN$ frame in which
the $t$ matrices are evaluated. Finally, $E$ is the energy at which the $t$
matrices are evaluated and it is fixed at one half the kinetic energy of the
incident nucleon in the laboratory frame.
As shown in Fig. 1, the calculation of Eq. (7) requires two external inputs:
the $NN$ $t$ matrix and the one-body density $\rho_{N}$.
The calculation of the density matrix could be performed using different ab
initio approaches. The method followed in Refs. [19, 20], where one-body
translationally invariant densities were computed within the ab initio No-Core
Shell Model (NCSM) [21] approach proved to be very successful in our own works
[7, 9]. Alternatively other methods could be used, i.e. Self-Consistent Green
Functions (SCGF) [22, 23], that allow to overcome the main limitation of the
NCSM method, namely the severe limits on the atomic mass of the target nucleus
due to the computational overload. Work in this direction is currently
underway and a thorough investigation of nickel and calcium isotopes will soon
appear [24]. For heavier nuclei, the only available method is Density
Functional Theory (DFT). Even if most of the parametrizations have a
phenomenological derivation, in recent years some authors [25] started to
employ such approach from an ab initio point of view.
The main advantages of using ab initio methods in our framework is twofold. On
one hand we can preserve the consistency of our theoretical framework if the
same realistic interactions are used both for the structure part and the
projectile-target interaction (with some caveats explained in Refs. [3, 5, 6,
8, 7]) and, on the other hand, we are able to provide more theoretically
founded predictions for exotic nuclei since we do not have to rely on any
fitting procedure on any selections of stable nuclei.
To calculate the projectile-target potential we rely on modern $NN$ potentials
derived in the framework of Chiral Perturbation Theory (ChPT). Among the many
available choices of realistic potentials, i.e. those that reproduce phase-
shifts and binding energies of light nuclei with $\chi^{2}|_{\rm data}\simeq
1$, the choice of chiral potentials over phenomenological ones responds to
some important requirements that will be listed shortly below.
Generally speaking, ChPT is a perturbative technique for the description of
hadron scattering amplitudes based on expansions in powers of a parameter that
can be generally defined as $(p,m_{\pi})/\Lambda_{b}$, where $p$ is the
magnitude of three-momenta of the external particles, $m_{\pi}$ is the pion
mass, and $\Lambda_{b}$ is the chiral symmetry breaking scale [26]. ChPT
respects the low-energy symmetries of Quantum ChromoDynamics (QCD) and, up to
a certain extent, is model independent and systematically improvable by an
order-by-order expansion, with controlled uncertainties from neglected higher-
order terms [27]. Both the fact that ChPT is directly connected to QCD and
that the $NN$ components (and also $NNN$) are organized in terms of a
systematic theoretical expansion is a huge improvement respect to conventional
$NN$ potentials, like CD-Bonn [28] or Nijmegen [29]. The hierarchy of terms
could also be exploited in order to provide an estimate for the theoretical
uncertainties.
From a practical point of view, for our own calculations [3, 5, 6, 7, 8, 9] we
mainly employed the $NN$ chiral interactions developed by Entem et al. [30,
31] up to the fifth order (N4LO) or, alternatively, the ones developed by
Epelbaum et al. [32, 33] following a different strategy. The main difference
between the two theoretical approaches of the chiral potentials concerns the
renormalization procedures. The strategy followed in Refs. [32, 33] consists
in a coordinate space regularization for the long-range contributions, and a
conventional momentum space regularization for the contact (short-range)
terms. On the other hand, for Refs. [30, 31], a slightly more conventional
approach was pursued. A spectral function regularization was employed to
regularize the loop contributions and a conventional regulator function to
deal with divergences in the Lippman-Schwinger equation.
The three-nucleon forces deserve a separate treatment since so far we can
employ genuine $3N$ forces only to compute the one-body densities of the
target nuclei (up to the third order N2LO with a local and nonlocal
regularization [34, 35]). At N2LO order we include the $2\pi$ exchange diagram
between three nucleons, a one-$\pi$-exchange plus a $NN$ contact term and a
$3N$ contact term. For more details and an explicit derivation of the relevant
formulae, we refer the reader to Refs. [34]. For the projectile-target part we
follow the prescriptions by Holt et al. [17] to derive a suitable density-
dependent NN interaction.
### 2.1 A selection of results
To show the robustness of our proposed approach we will show a couple of non
trivial calculations perfomed in the last years. The first case is the elastic
proton scattering on 7Li, in which the target nucleus has spin and parity
quantum numbers $J^{\pi}=3/2^{-}$. Theoretical predictions along with the
experimental data are presented in Fig. 2.
Figure 2: Differential cross section (upper panel) and analyzing power (lower
panel), as functions of the center-of-mass scattering angle, for 200 MeV
protons elastically scattered from 7Li (Jπ = 3/2-). The results were obtained
using the $NN$ $t$ matrix computed with the $NN$ chiral interaction at N3LO
order supplemented by a density dependent NN interaction (where the baryon
density is varied in the range between 0.08 fm-3 and 0.13 fm-3) and the one-
body nonlocal density matrices computed with the NCSM method using realistic
nucleon-nucleon and three-nucleon local-nonlocal chiral interactions.
The differential cross section and analyzing power were measured at the
Indiana University Cyclotron Facility using a polarized proton beam at a
laboratory bombarding energy of 200.4 MeV [36]. The agreement between the
theoretical prediction and the empirical data is good for the differential
cross section, over all the angular distribution shown in the figure, and
satisfactory for the analyzing power for values of the scattering angle up to
$\sim 45^{\rm o}$.
The second test case is an extension much needed for future applications:
inverse kinematics. In fact, in recent years, experimental efforts have
multiplied to develop the technologies necessary to study the elastic
scattering of protons (and ions) by using inverse kinematics. This
configuration is necessary to study exotic nuclei that have very short average
lifetimes. A very interesting experiment performed at the GSI was the study of
the scattering of 56,58Ni nuclei on hydrogen targets at energies near 400
MeV/nucleon in inverse kinematics for the determination of the distribution of
nuclear matter [37]. Since the only theoretical approach used so far to
analyze the experimental data is the Glauber model which contains
phenomenological inputs, it is mandatory to establish a framework in which
microscopic calculations could help understading experimental results.
In Fig. 3 we show the results only for $t\leq 0.1$ (GeV/c)2 in the laboratory
system because, due to the challenges of inverse kinematics, the regime of the
elastic scattering needs to be focused at small center-of-mass angles. The
agreement with experimental data is impressive despite the different
kinematics and the high-energy regime at which the measurements are performed.
Figure 3: Differential cross section as a function of the Mandelstam variable
$t$ in the laboratory frame for 56Ni(p, p)56Ni elastic scattering at $E=400$
MeV/u. Calculations are performed with different microscopic OPs derived from
the $NN$ chiral potentials of Epelbaum et al. (EKM) [32, 33] and Entem et al.
(EMN) [30, 31] at N4LO in comparison with the well known CD-Bonn [28]. With
empty black squares and circles are shown the empirical data along with the
corresponding error bars [37].
## 3 Necessary extensions
Among the different extensions under development, we would like to mention
probably the most relevant one: the inclusion in our framework of inelastic
reactions.
In the previous sections we mentioned a class of experiments that usually need
the subtraction of contributions from the inelastic channel to perform a
correct data analysis. In this perspective, if we wish to establish a
consistent microscopic approach for inelastic NA scattering, which is our
long-term goal, it is mandatory to develop a two-step theory program: first a
reliable description of OPs for states with spin-parity quantum numbers
J${}^{\pi}\neq 0^{+}$ (already under study with good results [7]) and then an
extension of our formalism that, so far, is restricted to the elastic channel.
Let’s suppose to extend the definition of the projection operator $P$
$P\equiv
P_{0}+P_{1}=\ket{\Psi_{0}}\bra{\Psi_{0}}+\ket{\Psi_{1}}\bra{\Psi_{1}}\,,$ (8)
with still $P+Q=\hbox{$1\hskip-1.2pt\vrule
depth=0.0pt,height=6.88889pt,width=0.7pt\vrule
depth=0.0pt,height=0.3pt,width=1.19995pt$}$. Inserting this relation into the
many-body Lippmann-Schwinger equation we could follow the same prescriptions
adopted for elastic case to define the following quantities
$T_{00}\equiv P_{0}TP_{0}\,,\quad T_{01}\equiv P_{0}TP_{1}\,,\dots\quad
G_{00}\equiv P_{0}G_{0}(E)P_{0}\,,\ldots$ (9)
that represent the new scattering amplitudes that can be accomodated in a
matrix form as follows (see Fig. 4)
${\bm{T}}={\bm{U}}+{\bm{U}}{\bm{G}}{\bm{T}}\,,$ (10)
with the previous terms defined as
${\bm{T}}\equiv\begin{pmatrix}T_{00}&T_{01}\\\
T_{10}&T_{11}\end{pmatrix}\,,\qquad{\bm{U}}\equiv\begin{pmatrix}U_{00}&U_{01}\\\
U_{10}&U_{11}\end{pmatrix}\,,\qquad{\bm{G}}\equiv\begin{pmatrix}G_{00}&0\\\
0&G_{11}\end{pmatrix}\,.$ (11)
When projected on the momentum basis, the equation for the $00$ component of
the scattering matrix becomes
$T_{00}({\bm{k}}^{\prime},{\bm{k}})=U_{00}({\bm{k}}^{\prime},{\bm{k}})+\int
d{\bm{p}}\frac{U_{00}({\bm{k}}^{\prime},{\bm{p}})T_{00}({\bm{p}},{\bm{k}})}{E-E(p)+i\epsilon}+\int
d{\bm{p}}\frac{U_{01}({\bm{k}}^{\prime},{\bm{p}})T_{10}({\bm{p}},{\bm{k}})}{E-E(p)-e_{1}+i\epsilon}\,,$
(12)
and similarly for the other equations. Here $e_{1}$ is the excitation energy
of the first excited state. Of course the previous equation has to be expanded
in partial waves along with the other 3 equations. Work along this direction
is in progress [38].
Figure 4: This infographic shows how the structure of Eq. 2 must change in
order to include the description of inelastic processes. In order to
accommodate the inclusion of the relevant excited state, it is necessary to
calculate four optical potentials that describe all the available scattering
channels. According to this picture we would need both stationary and
transition densities.
## References
## References
* [1] Epelbaum E, Hammer H W and Meissner U G 2009 Rev. Mod. Phys. 81 1773
* [2] Machleidt R and Entem D R 2011 Phys. Rep. 503 1
* [3] Vorabbi M, Finelli P and Giusti C 2016 Phys. Rev. C 93 034619
* [4] Riesenfeld W B and Watson K M 1956 Phys. Rev. 102 1157
* [5] Vorabbi M, Finelli P and Giusti C 2017 Phys. Rev. C 96 044001
* [6] Vorabbi M, Finelli P and Giusti C 2018 Phys. Rev. C 98 064602
* [7] Vorabbi M, Gennari M, Finelli P, Giusti C, Navrátil P and Machleidt R 2022 Phys. Rev. C 105 014621
* [8] Vorabbi M, Gennari M, Finelli P, Giusti C, Navrátil P and Machleidt R 2021 Phys. Rev. C 103 024604
* [9] Vorabbi M, Gennari M, Finelli P, Giusti C and Navrátil P 2020 Phys. Rev. Lett. 124 162501
* [10] Castelvecchi D 2022 Nature 605 201
* [11] Hebborn C et al., arXiv:2210.07293 [nucl-th]
* [12] Chinn C R, Elster Ch, Thaler R M and Weppner S P 1992 Phys. Rev. C 52 1992
* [13] Elster Ch and Tandy P C 1989 Phys. Rev. C 40 81
* [14] Elster Ch, Cheon T, Redish E F and Tandy P C 1990 Phys. Rev. C 41 814
* [15] Elster Ch, Weppner S P and Chinn C R 1997 Phys. Rev. C 56 2080
* [16] Weppner S P, Elster Ch and Huber D 1998 Phys. Rev. C 57 1378
* [17] Holt J W, Kaiser N and Weise W 2010 Phys. Rev. C 81 024002
* [18] Vorabbi M, Finelli P and Giusti C, in preparation
* [19] Burrows M, Elster Ch, Popa G, Launey K, Nogga A and Maris P, 2017 Phys. Rev. C 97 024325
* [20] Gennari M, Vorabbi M, Calci A and Navrátil P 2017 Phys. Rev. C 97 034619
* [21] Barrett B R, Navrátil P and Vary J P 2013 Prog. Part. Nucl. Phys. 69 131
* [22] Dickhoff W H and Barbieri C 2004 Prog. in Part. and Nuc. Phys. 52 377
* [23] Carbone A, Cipollone A, Barbieri C, Rios A and Polls A 2013 Phys. Rev. C 88 054326
* [24] Vorabbi M, Barbieri C, Somà V, Finelli P and Giusti C, in preparation
* [25] Marino F, Barbieri C, Carbone A, Colò G, Lovato A, Pederiva F, Roca-Maza X and Vigezzi E, 2021 Phys. Rev. C 104 024315
* [26] Scherer S 2003 Adv. Nucl. Phys. 27 277
* [27] Georgi G 1993 Ann. Rev. Nucl. Part. Sci. 43 209
* [28] Machleidt R 2001 Phys. Rev. C 63 024001
* [29] de Swart J J, Klomp R A M M, Rentmeester M C M and Rijken T A, 1995 Few Body Syst. Suppl. 8 438
* [30] Entem D R, Kaiser N, Machleidt R and Nosyk Y 2015 Phys. Rev. C 91 014002
* [31] Entem D R, Machleidt R and Nosyk Y 2017 Phys. Rev. C 96 024004
* [32] Epelbaum E, Krebs H and Meißner U - G 2015 Phys. Rev. Lett. 115 122301
* [33] Epelbaum E, Krebs H and Meißner U - G 2015 Eur. Phys. J. A 51 53
* [34] Navràtil P 2007 Few-Body Syst. 41 117
* [35] Gysbers P, Hagen G, Holt J D, Jansen G R , Morris T D, Navrátil P, Papenbrock T, Quaglioni S, Schwenk A, Stroberg S R and Wendt K A 2019 Nat. Phys. 15 428
* [36] Glover C W, Foster C C, Schwandt P, Comfort J R, Rapaport J, Taddeucci T N, Wang D, Wagner G J, Seubert J, Carpenter A W, Carr J A, Petrovich F, Philpott R J and Threapleton M J 1991 Phys. Rev. C 43 1664
* [37] Von Schmid M 2015 Nuclear matter distribution of 56Ni measured with EXL, https://d-nb.info/1112044388/34, Ph. D. Thesis
* [38] Vorabbi M, Gennari M, Finelli P, Giusti C and Navrátil P, in preparation
|
# What Physical Layer Security Can Do for 6G Security
Miroslav Mitev1, Arsenia Chorti1,2, H. Vincent Poor3, Gerhard Fettweis1,4 1
Barkhausen Institut, Dresden, Germany; 2 ETIS, UMR 8051 CY Cergy Paris
Université, ENSEA, CNRS, Cergy, France; 3 School of Engineering and Applied
Science, Princeton University, Princeton, NJ, USA; 4 Vodafone Chair for
Mobile Communications Systems, Technische Universität Dresden, Dresden,
Germany; Corresponding author<EMAIL_ADDRESS>
###### Abstract
While existing security protocols were designed with a focus on the core
network, the enhancement of the security of the B5G access network becomes of
critical importance. Despite the strengthening of 5G security protocols with
respect to LTE, there are still open issues that have not been fully
addressed. This work is articulated around the premise that rethinking the
security design bottom up, starting at the physical layer, is not only viable
in 6G but importantly, arises as an efficient way to overcome security hurdles
in novel use cases, notably massive machine type communications (mMTC), ultra
reliable low latency communications (URLLC) and autonomous cyberphysical
systems. Unlike existing review papers that treat physical layer security
orthogonally to cryptography, we will try to provide a few insights of
underlying connections. Discussing many practical issues, we will present a
comprehensive review of the state-of the-art in i) secret key generation from
shared randomness, ii) the wiretap channels and fundamental limits, iii)
authentication of devices using physical unclonable functions (PUFs),
localization and multi-factor authentication, and, iv) jamming attacks at the
physical layer. We finally conclude with the proposers’ aspirations for the 6G
security landscape, in the hyper-connectivity and semantic communications era.
## I Introduction
The rollout of fifth-generation (5G) mobile networks and the forthcoming
sixth-generation (6G) will bring about fundamental changes in the way we
communicate, access services and entertainment. In the context of security,
inarguably, 5G security enhancements present a big improvement with respect to
LTE. However, as the complexity of the application scenarios increases with
the introduction of novel use cases, notably ultra-reliable low latency
(URLLC), massive machine type communications (mMTC) and autonomous
cyberphysical systems (drones, autonomous cars, robots, etc.), novel security
challenges arise that might be difficult to address using the standard
paradigm of complexity based classical cryptographic solutions.
Specific use cases with open security issues are described in detail in a
number of 3GPP technical reports, e.g., on the false base station attack
scenario [1] and on the security issues in URLLC [2]. Indeed, for beyond 5G
(B5G) systems, there exist security aspects that can be further enhanced by
exploiting different approaches, as classical mechanisms either fall short in
guaranteeing all the security and privacy relevant aspects, or, can be
strengthened with mechanisms that could provide a second layer of protection.
In the past years, physical layer security (PLS) [3] has been studied and
indicated as a possible way to emancipate networks from classical, complexity
based, security approaches. Multiple white papers on the vision for 6G
incorporate physical layer security, e.g., [4, 5, 6], as well as in the IEEE
International Network Generations Roadmap (INGR) 1st and 2nd Editions [7].
Motivated by the above, a key point of this paper is to showcase how PLS and
in general security controls at the PHY level can be exploited towards
securing future networks.
One of the most promising and mature PLS technologies concern the distillation
of symmetric keys from shared randomness, typically in the form of wireless
fading coefficients. Within the channel’s coherence time, small scale fading
is reciprocal, time-varying and random in nature and therefore, offers a
valid, inherently secure source for key agreement (KA) protocols between two
communicating parties. This is pertinent to many forthcoming B5G applications
that will require strong, but nevertheless, lightweight KA mechanisms, notably
in the realm of Internet of things (IoT).
With respect to authentication, there are multiple PLS possibilities,
including physical unclonable functions (PUFs), wireless fingerprinting and
high precision localization. Combined with more classical approaches, these
techniques could enhance authentication in demanding scenarios, including (but
not limited to) device to device (D2D) and Industry 4.0. Note that according
to the 6G vision, as a network of (sub)networks, authentication might be
required independently for access to the local (sub)network and to the core
network, making the adoption of RF and device fingerprints a viable
alternative for fast authentication of local wireless connections.
In parallel, mmWave and subTHz bands require the use of a huge number of
antennas and pencil sharp beamforming. Consequently, a viable scenario for the
wiretap channel can be substantiated, without any assumptions regarding the
hardware (number of antennas, noise figure, etc.) or the position of a
potential eavesdropper. Similarly, visible light communications (VLC) systems
offer respective use cases. It is therefore pertinent to discuss advancements
in wiretap secrecy encoders. The interplay between secrecy and privacy in
finite blocklengths is another aspect that emerged from recent fundamental
results in finite blocklength secrecy coding and should be highlighted.
Furthermore, new types of attacks have to be accounted for. In particular,
there is mounting concern for potential jamming attacks and pilot
contamination attacks during beam allocation and entry phases of nodes into
the network [8]. Clearly, such attacks cannot be addressed with standard
cryptographic tools and the required solutions can only emerge at the PHY,
potentially in the form of jamming-resilient waveform and code design.
Finally, a less considered aspect relates to anomaly / intrusion detection by
monitoring hardware metrics. This can be either used for distributed anomaly
detection in low-end IoT networks, i.e., by monitoring memory usage, Tx and Rx
time, debug interface of devices, or, for more generalized anomaly detection
of devices of untrusted manufacturers, etc. Such approaches could help lessen
the monitoring overhead of centralized approaches and could provide new
approaches towards the identification of the source of the anomaly [9].
Looking at the bigger picture, future security controls will be adaptive and
context-aware [10]. In this framework, rethinking the security design bottom
up can provide low-cost alternatives. In particular,
1. 1
PLS can provide information-theoretic security guarantees with lightweight
mechanisms (e.g., using LDPC, Polar codes, etc.);
2. 2
Hybrid crypto-PLS protocols can provide fast, low-footprint and low-complexity
solutions for issues such as in [1] and [2];
3. 3
PLS can act as an extra security layer, complementing other approaches,
enhancing the trustworthiness of the radio access network (RAN);
4. 4
PLS is inherently adaptive and can leverage the context and the semantics of
the data exchanged.
In the following we will provide a comprehensive review of fundamental,
cutting edge results in PLS and showcase how PLS can be employed to achieve
many of the standard security goals, notably confidentiality, authentication,
integrity. To this end, and, in order to provide a platform for a fair
comparison to standard crypto schemes and a discussion on the potential
advantages of hybrid PLS-crypto systems, we will first review fundamental
cryptographic concepts and goals in Section II. Next, Section III gives a
brief motivation on why PLS should be considered for the 6G. In Section IV the
wiretap channel theory will be presented (focusing on information theoretic
characterizations for the finite blocklength) along with some recent results
for privacy in sensing systems. Subsequently Section V discusses the topic of
secret key generation (SKG) from shared randomness and highlights two subtle
points concerning the pre-processing of the observation channel coefficients
and coding methods in the short blocklength, furthermore, jamming attacks and
countermeasures are discussed [11, 12]. In Section VI hardware based and
statistical methods used in authentication will be visited, focusing on
localization based authentication [13, 14] and physical unclonable functions.
Finally, future directions and the authors’ aspirations for security controls
at all layers in 6G will be presented in Section VII.
## II Background Concepts in Cryptography and Network Security
Starting with some fundamental concepts in cryptography, we will address
questions that arise in the systematic study of any system. In particular, we
will provide answers to the following questions: ”what do we want to
achieve?”; ”what is the system model?”; ”what are the underlying assumptions,
and what are the desirable properties?”
With respect to what we aim to achieve, typically any security system aims at
reaching one or multiple of four fundamental goals. The first goal is to be
able to provide data confidentiality, i.e., security against eavesdropping
(passive attackers). The corresponding threat model involves two legitimate
parties communicating in the presence of an eavesdropper. Typically, with the
aid of encryption, confidentiality is ensured against passive attackers. The
second major goal is that of data integrity, i.e., providing guarantees that
as the data traverses through the network, any modification or alteration of a
message will be perceptible at the destination. The corresponding threat model
involves an active attacker that in addition to intercepting messages also
performs modifications. The third major security goal is authentication (user
or device), while access control is a closely related topic. The threat model
involves again an active attacker that potentially attempts to gain
unauthorized access. Finally, the fourth goal is that of availability, i.e.,
users should not be denied services. The network should be resilient to active
attacks that fall in the general category of “denial of service”.
With respect to the system model, as noted above, the basic system setting
includes three nodes. Two legitimate parties, that are referred here Alice and
Bob and an adversarial node that is typically referred to as Eve (passive
eavesdropper) or Mallory (active attacker, i.e., man-in-the-middle). To
securely transmit a message (plaintext) to Bob, Alice uses a secret key to
first encrypt it to a ciphertext. The ciphertext is then propagated through
the transmission medium and received at Bob. Bob can decrypt the ciphertext by
using the same or a different type of key, depending on the underlying
algorithm.
### II-A Confidentiality
To perform the operations above, i.e., encryption / decryption, Alice and Bob
rely on the use of ciphers. A key feature of modern block ciphers is to
exploit highly non-linear operations to induce confusion, i.e., to render
statistical inference attacks impossible. A textbook example of a linear
cipher that is badly broken is the substitution cipher in which each letter of
the alphabet is moved $k$ positions to the right (or to the left), with $k$
changing per letter. Considering the English alphabet, this results in $25!$
possible key combinations, making a brute force attack impractical. However,
due to the linearity of the operations (permutations), a frequency analysis of
a (long enough) ciphertext suffices to guess the plaintext.
A revolutionizing result in security was presented by Shannon in $1949$ [15],
when he demonstrated that perfect secrecy can be achieved if and only if (iff)
the entropy of the secret key is greater or equal to the entropy of the
plaintext. The corresponding scheme, known as one-time-pad, is implemented by
xor-ing the plaintext with the key. Unfortunately, to perform the above, the
key size must be at least equal to that of the data which raises the problem
of key distribution.
While one-time pad is impractical, it provided insight into how secrecy can be
achieved. In particular, it inspired the family of stream ciphers that rely on
the idea of inflating short key sequences to psedorandom sequences of the same
size as the plaintext and xor-ing them. This is achieved through the use of
pseudorandom number generators (PRNGs). Although they cannot provide perfect
secrecy (entropy cannot increase by data processing as a consequence of the
data processing inequality), their usage led to the introduction of a more
practical concept, i.e., semantic security.
The definition of semantic security for PRGNs relies on the
indistinguishability between their output and the output of a truly random
source. More generally, semantic security ensures that a non-negligible
statistical advantage cannot be accumulated by an adversary in polynomial
time. For all practical purposes, if a statistical advantage happens with
probability higher that $2^{-30}$, e.g., one bit is leaked in one gigabyte of
data, the system is considered broken (not semantically secure).
A canonical example of modern block ciphers is the advanced encryption
standard (AES). AES is a semantically secure symmetric block cipher which
takes a $n$-bit plaintext ($n=128$) and a $k$-bit key ($k$ chosen from $128$,
$192$, or $256$ bits, with AES-$256$ considered to be quantum resistant) as
input and outputs a $n$-bit ciphertext. AES relies on a set of substitution
and permutation operations including the use of substitution (S) boxes. A well
structured S-box removes the relation and dependency between bits, making a
(linear or differential) cryptanalysis attack impossible. To allow the re-use
of a single key for multiple blocks, nonces can be used. Nonces are
deterministic (e.g., a counter) or random (initialization vectors), chosen
such that a pair (key, nonce) never repeats. The important message here is
that, today’s cryptographic mechanisms allow the use of a short key sequence
(e.g., 96 Bytes of key material in TLS v1.3) for the encryption of very long
data sequences (in the order of GBs), allowing to overcome the key issue with
one-time pad.
### II-B Data integrity
Data integrity is achieved with message authentication codes (MACs). The
principle of MACs is to append a small label (tag) to each message, which
validates its integrity. A MAC consists of two algorithms: signing and
verification. Similarly to confidentiality schemes, there are historical
examples of broken integrity algorithms in which linear functions (e.g.,
cyclic redundancy checks) have been used to generate MACs. Modern signing
algorithms (tag generation) leverage the use of secret keys and symmetric
block ciphers to generate a $t$-bit tag for a $n$-bit message, with $t<<n$.
Upon reception, the verification algorithm uses the key, the received message
and the tag and outputs a binary decision, i.e., the integrity check is either
successful or not.
Building on the above, a naturally arising concept is the one of authenticated
encryption (AE) which combines both confidentiality and integrity. Various
options exist on how to perform the two operations. One approach, that is
always correct and provably secure, is the so called encrypt-then-sign, i.e.,
after a plaintext is encrypted a tag is generated over the ciphertext. The
receiver would first check the integrity and iff successful would continue
with decryption.
### II-C Authentication
The process of authentication relies on digital signatures, which in turn, are
used to produce digital certificates. Digital certificate is data signed by a
trusted third party (certificate authority (CA)) that ensures the authenticity
of the its owner. A certificate contains information about the CA, the owner
of the certificate, the validity of the certificate, etc. As an example, when
a user accesses a public server, the server proves its authenticity by
presenting a certificate signed from a CA. To achieve mutual authentication
the user must enter a password information, provide biometric data, etc.
## III Motivation for Considering Physical Layer Security
Given the fact that all schemes discussed in the previous section are widely
deployed and trusted, one question remains: What is the motivation in
considering PLS?
PLS technologies can offer multiple security techniques: i) secrecy encoders
for wiretap channels, ii) privacy preserving transmission, iii) secret key
generation from shared randomness iv) physical unclonable functions for device
authentication, and v) localization or RF fingerprinting based authentication.
While crypto solutions can provide these functionalities for current
standards, they face number of challenges when considering new and emerging
technologies. First, latency requirements are getting more stringent than
ever, bringing the need for faster authentication and integrity checks.
Second, large scale IoT deployment requires flexible and easily scalable
security solutions that could simultaneously satisfy different security
levels. A third element comes from the rise of quantum computing which opens
the need for quantum secure algorithms. Finally, a fourth motivation comes
from the new PHY infrastructures where the number of operations performed at
the edge are expected to rise dramatically. Therefore, it is of utmost
importance to separate the security of the core network from the one at the
edge and introduce new faster and lightweight security algorithms. The
statements above are complemented with the following list:
1. 1.
Regarding latency, 3GPP has recently noted that delays should be minimized in
two directions, delays incurred by the communication and delays incurred due
to computational overhead. A particular case where computational overhead of
current standards do not comply with the requirements is security. As an
example, it has been shown that the verification of a digital signature, in a
vehicular networking scenario using a $400$ MHz processor, exceeds the
tolerated delays and requires approximately $20$ ms [16]. Such results hint
that a revolutionizing actions are needed in that direction.
2. 2.
Next, deploying billions of IoT devices is not inconceivable anymore. In 2016,
it has been demonstrated that a Mirai sized attack (e.g., $6\times 10^{5}$
bots) is plausible. The attack has been demonstrated over simple machines,
e.g. water heater, however, controlling $6\times 10^{5}$ can instantly change
the demand in the smart grid by $3$ GW, which is comparable to having an
access to a nuclear plant. Examples like this raise a lot of questions on the
security of the IoT.
3. 3.
In 2017, the NIST started the investigation on the topic of quantum resistance
and post-quantum cryptography. However, as it stands now, the state of the art
is based on using longer keys and increased complexity. This makes the
mechanisms heavier which contradicts with the need for low latency and low
footprint. Hence, post-quantum innovations at the moment are not well aligned
to the expectations towards 6G networks.
4. 4.
Finally, new PHY and networking structures are being developed for the next
generation of communication technologies. The central idea is to enhance the
role of AI edge intelligence. This is a key component, that can enable the use
of PLS in 6G. More details regarding this point will be discussed in Sec. VII.
In the following sections it will be discussed how PLS technologies can be
employed and some fundamental results in the area will be showed.
## IV Confidentiality and Privacy Using PLS
### IV-A Confidential transmission
In this section two aspects of physical layer security will be discussed,
i.e., data confidentiality and data privacy. In detail, the information
theoretic formulations of these problems will be investigated.
As noted in Section I secure data transmission tends to be a higher layer
issue, e.g., enabled by encryption. However, confidential data transmission
becomes difficult when considering massive numbers of low cost and low
complexity devices. This is where physical layer security can play an
important role. The idea is, instead of having reliability encoding, i.e.,
error control coding separated from the encryption, we can use joint encoding
schemes that provide both reliability and security.
This approach, known as wiretap coding, was proposed approximately half a
century ago by A. Wyner [17]. Wyner looked at a three terminal wireless
channel, i.e., two legitimate users Alice and Bob, and an eavesdropper, Eve.
He recognized that the channels between the terminals are not perfect, i.e.,
their transmission will be impacted by noise. Therefore, when Alice transmits,
Bob and Eve will not see exactly what has been transmitted. Moreover, Bob and
Eve will have different received signals as they have different noisy
channels. Wyner was interested in whether Alice could send a message reliably
to Bob, while keeping it secret from Eve. To answer, he looked at the reliable
rate to Bob, versus the equivocation at Eve (conditional entropy of the
message at Eve’s receiver). Note that, perfect secrecy can be achieved if the
reliable rate at which data is being transmitted to Bob equals to the
equivocation of Eve. To measure these quantities Wyner introduced a new
metric, named secrecy capacity, which is the maximum reliable rate that equals
the equivocation. He further showed that, achieving positive secrecy capacity
is possible, hence, confidential transmission can be performed without the use
of secret keys. However, achieving positive secrecy capacity is possible iff,
the measurements at Eve are degraded with respect to those at Bob. A plausible
example is when the signal to noise ratio (SNR) at Bob is higher than the SNR
at Eve.
Now, thinking about the physical layer, it is clear that the properties of
radio propagation, i.e., diffusion and superposition, provide opportunities to
achieve positive secrecy capacity. For example, by using the natural
degradeness over time (e.g., fading), by introducing an artificial degradeness
to the eavesdropper (e.g., interference and jamming), or, by leveraging
spatial diversity (e.g., multiple antenna systems and relays can create
secrecy degrees of freedom).
Based on the above, over the last fifteen years the idea of wiretap coding has
been further examined considering several fundamental channel models:
broadcast channel (one transmitter, multiple receivers), multiple access
channel (multiple transmitters, one receiver), interference channels (multiple
transmitters, multiple receivers); see e.g. [18]. To illustrate the main
results in the area, this work focuses on the broadcast channel [19]. First,
consider a Gaussian broadcast channel with Alice being a transmitter and Bob
and Eve receivers. Assume two messages are transmitted: $M_{1}$ intended for
both receivers and $M_{2}$ a secret message that is intended only for Bob. To
define the capacity region we consider a degraded channel at Eve. In
particular, it is assumed that the SNR level at Bob equals $10$ dB, and the
SNR at Eve is $5$ dB. This is illustrated in Figure 1 where the horizontal
axis gives the range of possible rates for the common message $M_{1}$, and the
vertical axis gives the range of possible rates for the secret message
$M_{2}$. The capacity region without secrecy constraints is shown with red
solid curve and the secrecy capacity is indicated by the dashed blue curve. It
can be observed that, if secrecy is required, part of the available capacity
must be sacrificed in order to confuse the eavesdropper for that message. It
is important to note that the amount to be sacrificed depends upon choosing a
codeword that randomizes the message w.r.t. Eve, but allows Bob to
successfully verify it.
Figure 1: Achievable rates for the Gaussian broadcast channel.
Next, Figure 2 shows the impact when the SNR at Eve varies. Similarly, the
horizontal axis gives the common rate and the vertical axis gives the secrecy
rate. The arrow shows that, if the SNR at Eve decreases, the range for the
common rate shrinks and the range of secrecy rates increases. On the other
hand, if the SNR at Eve reaches $10$ dB, the same level as Bob’s SNR, the
secrecy region collapses. That is, if the second receiver is not degraded,
secrecy rate becomes zero.
Figure 2: Achievable rates for the Gaussian broadcast channel considering
variable SNR at Eve.
Interestingly, things change when looking at a fading Gaussian broadcast
channel. To illustrate this scenario we consider the same model, i.e., one
transmitter, two receivers, one common message, and one secret message, but we
assume that both the receivers have the same level of Gaussian noise, i.e.,
Bob and Eve have $5$ dB SNR. This is given in Figure 3. The difference between
Bob and Eve is the fading parameter, i.e., Bob’s experiences Rayleigh fading
with a unit parameter, and Eve has Rayleigh fading with parameter
$\sigma_{2}$. Note, a smaller $\sigma_{2}$, results in more intense fading. As
before, when Eve’s channel gets worse, i.e., $\sigma_{2}$ decreases, it can be
seen that the range of common rates on the horizontal axis shrinks and the
range of secret rates on the vertical axis increases. However, a distinction
here is that if the two receivers observe the statistically identical channels
(this is the case when $\sigma_{2}=1$), the secrecy capacity does not collapse
as in the case of the Gaussian channel. This result holds under the assumption
of perfect channel knowledge and follows from the fact that fading provides
additional degrees of freedom leading to advantage during the time when other
receivers experience deeper fade.
Figure 3: Achievable rates for the Rayleigh fading broadcast channel
considering variable $\sigma^{2}$ at Eve. (From [19].).
A major issue concerning the results above comes from an information theoretic
perspective. In particular, they are based on the assumption of infinite
coding blocklength. Hence, it concerns the following scenario. Assume that a
message $W$, that is encoded into a length-$n$ codeword, is transmitted into
the channel. After passing through the wireless medium noisy instances of the
codeword are obtained by Bob and Eve. These codewords are then fed into Bob’s
and Eve’s decoders. The desired property for this scenario is that for Bob to
be able to reconstruct the codeword perfectly while at the same time, the
leakage of the codeword to Eve is bounded by the quantity $\delta$. In the
original formulation by Wyner, the considered blocklength is infinity, i.e,
$n$, the number of channel uses, is infinity. When $n\rightarrow\infty$, the
probability of error at Bob, i.e., probability that he decodes to a $\hat{W}$
which is different compared to $W$ goes to zero. Additionally, the information
leakage $\delta$ also goes to zero. The secrecy capacity for this case has
been formulated as the difference between the mutual information between
Alice, $X_{A}$, and Bob, $X_{B}$, and the mutual information between Alice and
Eve, $X_{E}$, when considering the maximum from the channel input distribution
$P_{X}$, i.e.,:
$C_{S}=\underset{P_{X}}{\max}\\{I(X_{A};X_{B})-I(X_{A};X_{E})\\}.$ (1)
This is an intuitive result, i.e., achieving positive secrecy capacity relies
on the degradation of Eve’s channel. The limitation of this theory is that it
gives only asymptotic results that are not suitable for low latency
applications, such as in an IoT scenario. This opens the question: What is
achievable in the non-asymptotic case?, and the answer depends on the finite
blocklength information theory. Assume we have a source $W$, which can take
$1,2,\dots,M$ possible values, i.e., it has $\log_{2}M$ bits. The source is
mapped using an encoder to a sequence, $X^{n}$, which is then passed through a
channel. Due to noise, the receiver will observe a corrupted version of the
transmission, i.e., $Y^{n}$, which is then decoded to $\hat{W}$. If the errors
between $\hat{W}$ and $W$ are less than a particular value, $\epsilon$, the
decoder could reconstruct the original source. In systems like this, the
design of $nM\epsilon$ codes is of particular interest: $M$ the number of
source symbols, $n$ the number of channel uses, and $\epsilon$ the upper bound
on the reconstruction fidelity of the source at the output of the decoder. The
fundamental limit for such a system is defined by the maximum $M$, i.e., the
largest possible number of source symbols that can be transmitted through the
channel in $n$ channel uses and be reconstructed at the decoder with error
probability $\leq\epsilon$. Note that,
$\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\log_{2}(M)$ gives the
Shannon’s capacity where $\epsilon\rightarrow 0$. However, in an actual system
$n$ and $\epsilon$ are finite values. Considering this, an approximation for
$M^{*}$ was derived in [20], and it is given as
$\log M^{*}(n,\epsilon)=nC-\sqrt{n}VQ^{-1}(\epsilon)+\mathcal{O}(\log n),$ (2)
where $C$ gives the Shannon’s capacity, $Q^{-1}(\epsilon)$ defines the tail of
a standard Gaussian distribution evaluated at $\epsilon$, and $V$ is the
channel dispersion, which is the variance of the information density (note
that Shannon’s capacity is the mean of the information density).
The result from Equation (2) is illustrated in Figure 4, where an AWGN channel
is assumed with SNR equal to $0$ dB, $\epsilon=10^{-3}$ and $C=1/2$. The
figure shows the upper bound and lower bound for the capacity for finite block
lengths, denoted here by “Converse”, and “Best achievability”, respectively.
Hence, the actual capacity, which remains to be found, lies between those two
curves. While the gap between the curves is small for high values of $n$, it
can be observed that for small values of $n$ the gap remains large, hence,
further work in the area is required to obtain a more precise solution.
Figure 4: Upper and lower bounds on the capacity regions for short block
length communication. SNR is equal to $0$ dB and $\epsilon=10^{-3}$. (From
[20])
Following the result for channel capacity, it has been just recently shown
that the secrecy capacity in the finite blocklength scenario can also be
approximated [21]. Fixing the error probability at Bob, $\epsilon$, the
leakage at Eve, $\delta$, and the block length $n$, an approximation for the
secrecy capacity is given as
$R^{*}(n,\epsilon,\delta)=C_{S}-\sqrt{\frac{V}{n}}Q^{-1}\left(\frac{\delta}{1-\epsilon}\right)+\mathcal{O}\left(\frac{\log
n}{n}\right),$ (3)
where $V$ is defined similarly to the channel dispersion of (2). The result
from Equation (3) is illustrated in Figure 5. The figure considers a binary
symmetric wiretap channel with crossover probability $p=0.11$,
$\delta=\epsilon=10^{-3}$ and $C_{S}=1/2$. A similar trend is observed as in
the previous figure, the gap between upper bound (Converse) and lower bound
(Best achievability) shrinks and widens as $n$ gets larger or smaller,
respectively.
Figure 5: Upper and lower bounds on the secrecy capacity for short block
length communication in a binary symmetric channel with crossover probability
$p=0.11$ and $\delta=\epsilon=10^{-3}$. (From [21].)
This has also been evaluated for a Gaussian wiretap channel and the result is
illustrated in Figure 6. The SNR at Bob here equals $3$ dB, and the SNR at Eve
equals $-3$ dB. It can be observed that the gap between achievability and
converse is even larger for this scenario. However, what is important to
mention here is that the upper bound, when considering finite block lengths,
is far from the asymptotic secrecy capacity, $C_{S}$. This shows that research
on emerging IoT technologies should not rely on asymptotic results and should
focus on the investigation of short block length communications.
Figure 6: Upper and lower bounds on the secrecy capacity for short block
length communication in a Gaussian wiretap channel. SNR at Bob is equal to $3$
dB and SNR at Eve equals to $-3$ dB. (From [21].)
### IV-B Privacy in sensing systems
Differently from secrecy, where the concern is about restricting a malicious
party from getting access to the transmission, in the case of privacy, the
goal is to keep part of the information secret from other parties, including
the legitimate receiver (Bob). A simple way to ensure there is no privacy
leakage is to deny access to Bob, however, without having a recipient the data
source becomes useless. Therefore, it is important to study, which part of the
data can be shared, such that the message is successfully and securely
transmitted, while the privacy leakage is minimized.
This section focuses on the problem of privacy leakage with particular focus
on sensing systems. Such systems include smart meters, cameras, motion
sensors, i.e., devices that generate useful data for companies who provide
users with particular service (alarm, power supply, etc.). While companies can
use the data to improve their services, the full access to it endangers the
privacy of users.
The above hints towards that, there is a fundamental trade-off between privacy
and usefulness of data (distortion). This is illustrated in Figure 7. If the
data is completely private, i.e., its equivocation at Bob is high, the data
becomes useless and it is fully distorted. Contrarily, if the data is fully
accessible, i.e., it has low distortion at Bob, then its equivocation goes to
zero and the data is not private.
Figure 7: Trade-off between privacy and usefulness of data.
Now, when considering a specific application, i.e., smart meters, the trade-
off can be specified as follows: a smart meter measures the electricity usage
in almost real time, hence, having the utility of providing users with
information on their usage, but in the same time it leaks this information to
the power supply company who can use it to trace in-home activities [22]. One
way to model this problem is through a hidden Gauss-Markov model. This is
given in Figure 8 where the hidden state is the intermittent state, e.g.,
turning your toaster on, your kettle on, etc. The figure captures a smart
meter trace, and shows that the privacy-utility trade-off for this model can
be characterized by a reverse water-filling [23]. The trade-off here is
defined by the water level $\phi$, such that all signals with power lower than
$\phi$ are being suppressed by the meter, while all signals above are being be
transmitted (and leaked) by the meter. Therefore, the value of $\phi$ defines
the amount of privacy that the user is willing to sacrifice to increase his
utility.
Figure 8: Privacy-utility trade-off characterized by a reverse water-filling.
Another way to approach the same problem is through using control, i.e.,
actively controlling what the meter sees based on storage and energy
harvesting [24]. This is illustrated in Figure 9, where the utility-privacy
trade-off for this model is captured by measuring wasted energy versus
information leakage. Presenting this control approach as a Markov model allows
to numerically determine the efficient frontier. This is given in Figure 10,
where the red curve gives the optimal trade-off of wasted power versus
information leakage.
Figure 9: Privacy-utility trade-off characterized by a measuring wasted energy
versus information leakage. Figure 10: Wasted power versus information leakage
when considering a control approach.
Another example is when considering the case of competitive privacy. In
competitive privacy, there are multiple agents (Bobs) each having own privacy
utility trade-offs. On one hand, there are multiple interacting agents who are
competing with one another, but, on the other hand, the agents have coupled
measurements. In detail, each agent wants to estimate its own parameters and
can help other agents by sharing data but does not want to compromise his own
privacy.
This competitive scenario can be represented as a linear measurement model
[25]. Utility can be measured in terms of mean squared error on the state
estimation and privacy can be measured in terms of information leakage. In
fact, it has been shown that this reduces to a classical problem, known as the
Wyner-Ziv problem or the distributed source coding problem. Particularly, it
has not been discussed what is the optimal amount of information that must be
exchanged, but it has been shown that the optimal way to exchange information
is by using Wyner-Ziv coding. Next, depending on the scenario a simple way to
find the optimal amount of information is through the use of game theory.
Finally, an important conclusion for this section is that information theory
can help us understand the fundamental limits of security and privacy. While
mainly theoretical constructs have been discussed, it is clear that there is a
need to connect the theoretical analyses to real networks. Building on the
above, some emerging research directions include finite blocklength analysis
(short packet low latency communication), scaling laws for large networks
(channel models that consider massive networks) and practical coding schemes.
## V Secret Key Generation Using PLS
This section focuses on several aspects concerning SKG. First, it provides an
overview on how to extract symmetric keys from shared randomness, then it
shows how SKG can be incorporated in actual crypto systems, and finally, it
discusses how the SKG process can be made resilient to active attacks.
### V-A Secret key generation
Generally, the SKG protocol consists of three steps: advantage distillation,
information reconciliation, and, privacy amplification. Assuming two
legitimate parties, e.g., Alice and Bob, the steps can be summarized as
follows. In the first step, Alice and Bob exchange pilot signals during the
coherence time of the channel, and obtain correlated observations $Z_{A}$ and
$Z_{B}$, respectively. In the second step, their observations are first
quantized and then passed through a distributed source code type of decoder.
During this step Alice (or Bob) shares side information, which is used by Bob
to correct errors at the output of his decoder. Hence, at the end of this step
both parties obtain a common binary sequence. Finally, to produce a maximum
entropy key and suppress the leaked information, privacy amplification is
performed. In this last step, Alice and Bob apply an irreversible compression
function (e.g., hash function) over the reconciled bit sequence. This produces
a uniform key that is unobservable by adversaries.
There are few important points that need to be taken into account for the
success of the SKG process. First, channel measurements represent a mixture of
large scale and small scale fading components. In multiple studies, it has
been demonstrated that the large scale component is strongly dependent on the
location and the distance between users, which makes it predictable for
eavesdroppers. Therefore, to distill a secret key, Alice and Bob should either
remove this part from their measurements and generate the key using the
unpredictable small scale components or should compress more at the privacy
amplification. This point is further discussed in Section VII. Second, the SKG
protocol should follow all the steps described above, and no steps should be
skipped. As an example, skipping the privacy amplification would give Alice
and Bob longer key sequence, however, the key sequence is vulnerable to
different attacks [26]. Third, it is important that, Alice and Bob do not
transmit information related to their observations, as this could be exposed
to eavesdroppers in the vicinity. Forth, Alice and Bob should respect the
coherence time and coherence bandwidth of the channel, such that their
subsequent measurements are decorrelated in time and frequency. This allows
them to generate random and unpredictable bit sequences. Finally, as mentioned
in the previous section, further testing of short blocklength encoders is
necessary in order to identify the optimal solution for SKG.
Regarding the last point, Figures 11 and 12 show a comparison between an upper
bound, evaluated in [27], versus information reconciliation rates achieved
using of LDPC, polar codes and BCH codes [28]. Both figures $n=128$ and
$n=512$ show that polar codes with CRC and BCH codes with list decoding
outperform the other approaches, making them good candidates for
reconciliation decoding. Note that such type of encoders are already used in
5G for different purposes.
Figure 11: FER performance of reconciliation codes compared to the lower
bound from [27] for $n=128$. (From [28].) Figure 12: FER performance of
reconciliation codes compared to the lower bound from [27] for $n=512$. (From
[28].)
### V-B Secret key generation in hybrid crypto systems
Building on the above, we continue with a particular example on how SKG can be
incorporated in hybrid security cryptographic schemes. In detail, it will be
discussed how to build a SKG-based authenticated encryption. Three ingredients
are needed to formulate this problem:
* •
A SKG scheme $\verb"G":\mathbb{C}\rightarrow\mathcal{K}\times\mathcal{S}$,
that takes channel measurements as input and generates a key $\mathbf{k}$ and
side information $\mathbf{s}$.
* •
A symmetric encryption algorithm, i.e., a pair of functions
$\verb"Es":\mathcal{K}\times\mathcal{M}\rightarrow\mathcal{C_{T}}$ and
$\verb"Ds":\mathcal{K}\times\mathcal{C_{T}}\rightarrow\mathcal{M}$, for
encryption and decryption, respectively, where $\mathcal{C_{T}}$ defines the
ciphertext space and $\mathcal{M}$ the message space.
* •
A message authentication code (MAC) algorithm, given as
$\verb"Sign":\mathcal{K}\times\mathcal{M}\rightarrow\mathcal{T}$, for signing
and
$\verb"Ver":\mathcal{K}\times\mathcal{M}\times\mathcal{T}\rightarrow(yes,no)$,
for verification, where $\mathcal{T}$ defines the tag space.
Now, the components can be combined as follows:
1. 1.
SKG is performed between Alice and Bob as:
$\verb"G"(\mathbf{h})=\left(\mathbf{k},\mathbf{s_{A}}\right),$ (4)
where $\mathbf{h}$ represents the channel measurements, $\mathbf{k}$ the
generated key after privacy amplification and $\mathbf{s_{A}}$ is Alice’s side
information that has to be transmitted to Bob to finalize the process.
2. 2.
Before transmitting $\mathbf{s_{A}}$ to Bob, Alice breaks her key into two
parts $\mathbf{k}=\\{\mathbf{k}_{e},\mathbf{k}_{i}\\}$, generates a ciphertext
as $\mathbf{c}=\verb"Es"(\mathbf{k}_{e},\mathbf{m})$ and signs it as
$\mathbf{t}=\verb"Sign"(\mathbf{k}_{i},\mathbf{c})$. Afterwards she transmits
to Bob the concatenation of $[\mathbf{s}_{A}||\mathbf{c}||\mathbf{t}]$, i.e.,
in a single message she can transmit the side information and her message.
3. 3.
Upon receiving the above, Bob uses the side information $\mathbf{s}_{A}$, to
finish the SKG process, i.e., to obtains the key $\mathbf{k}$. Then, he checks
the integrity of the received ciphertext as
$\verb"Ver"(\mathbf{k}_{i},\mathbf{c},\mathbf{t})$ and if successful he
decrypts and obtain the message $\mathbf{m}$.
Differently from the standard SKG scheme, where SKG is performed in parallel
at both nodes and data exchange happens only after the key generation is
finalized, in the scheme above Alice completes the SKG locally and then
transmits in a single go the ciphertext, the tag, and, the side information
(e.g., syndrome). Then Bob uses the syndrome to complete the SKG and performs
the authenticated decryption. This small change in the standard procedure
shows how PLS can be easily combined with standard crypto schemes.
Such approaches bring new opportunities. For example, the scheme above opens
the problem of transmission optimization. Consider a scenario with multiple
subcarriers used for transmission. The subcarriers can then be split into two
subsets, a subset $\mathcal{D}$ used for transmitting encrypted data and a
subset $\bar{\mathcal{D}}$ used for transmitting side information (syndromes).
This transmission scheme can be optimized considering several constraints. The
first constraint comes from the world of cryptography, i.e., based on the
choice of cryptographic cipher we can define the amount of data to be
encrypted with a single key. This can be captured by the following constraint:
$C_{SKG}\geq\beta C_{D},\quad 0<\beta\leq 1,$ (5)
where $C_{SKG}$ defines the key generation rate, $C_{D}$ defines the data rate
and $\beta$ is a quantity that relates the key size to the data size that will
be encrypted, e.g., $\beta=1$ corresponds to a one-time pad cipher. The second
constraint comes from the world of information theory. It relates the
necessary (side information) syndrome rate $C_{R}$ and the SKG rate as
follows:
$C_{R}\geq\kappa C_{SKG},$ (6)
where $\kappa$ defines minimum number of reconciliation bits with respect to
the key bits. It is a parameter defined by the type of the encoder/decoder
used for SKG, e.g., for a $\frac{k}{n}$ block encoder $\kappa=\frac{n-k}{k}$.
Further constraints that can be incorporated are power constraint:
$\sum_{j=1}^{N}p_{j}\leq NP,\;\;p_{j}\geq 0,\;\forall j\in\\{1,\dots,N\\},$
(7)
and a channel capacity constraint, i.e.,
$C_{D}+C_{R}\leq C,$ (8)
where $N$ gives the number of subcarriers, $P$ is the power limit per
subcarrier and $C$ is the total capacity of the channel. The objective of the
problem can then be defined as:
$\max_{p_{j},j\in\mathcal{D}}C_{D}\quad\text{s.t. \eqref{eq:beta_constraint},
\eqref{eq:kappa}, \eqref{eq:power}, and \eqref{eq:C}}$ (9)
The problem can be turned into a combinatorial optimization problem which can
be solved optimally using dynamic programming techniques or sub-optimally
using heuristic approaches. Overall, this problem shows how physical layer
aspects can be related to cryptographic schemes, in the form of a hybrid
security scheme, and provide new opportunities for cross layer optimization.
The problem was solved in [29] and the main result is depicted in Figure 13.
The figure shows the long term efficiency (expected sum data rate normalized
to the capacity of the channel) of the proposed parallel approach, i.e., the
transmission of side information and encrypted data are done simultaneously on
$\mathcal{\bar{D}}$ and $\mathcal{D}$, respectively, versus a standard
sequential transmission approach. It can be seen that, for most values of
$\beta$, the parallel approach outperforms the sequential one. Another
observations is that as $\beta$ increases, the efficiency decreases. This is
expected result as higher $\beta$ will required more frequent key generation,
hence, less data transmission. Finally, an important result that can be
observed on the graph is that the authors proposed a simple heuristic approach
for the parallel scheme that gives an equivalent efficiency to the optimal
solution solved using dynamic programming approach (i.e., as a Knapsack
problem). Further interesting aspects that can be included in this analysis
are factors such as handover or other aspects that may cause frequent key
generation.
Figure 13: Efficiency comparison for $N=64$, SNR$=10$ dB and $\kappa=2$. (From
[29].)
This problem has been further investigated in [30], where a general quality of
service (QoS) delay constraint was introduced. The work is based on leveraging
the theory of the effective capacity and identifies the maximum supported
transmission rate when considering a delay constraints, i.e., instead of
maximizing the data rate $C_{D}$ the problem focuses on maximizing the
effective data rate $E_{C}(\alpha)$, given as
$E_{C}(\alpha)=-\frac{1}{\alpha}\log_{2}\left(\mathbb{E}\left[e^{-\alpha
C_{D}}\right]\right),$ (10)
where $\alpha=\frac{\theta T_{f}B}{\ln(2)}$ with $\theta$ being a MAC sub-
layer parameter that captures the packet arrival rate and introduces a delay
requirement into the problem, $T_{f}$ is the frame duration and $B$ denotes
the bandwidth. Considering that, [30] identified the optimal power allocation
policy that maximizes $E_{C}(\alpha)$ as
$p_{i}^{*}=\frac{1}{g_{0}^{\frac{N}{\alpha+N}}\hat{g}_{i}^{\frac{\alpha}{\alpha+N}}}-\frac{1}{\hat{g}_{i}},$
(11)
where $g_{0}$ is a cut-off value that can be found from the power constraint
and $\hat{g}_{i}$ $i=1,\dots,N$ denote the imperfectly estimated channel
gains. If the system can tolerate looser delay requirements, i.e.,
$\theta\rightarrow 0$ the result above converges to the well-known water-
filling algorithm and if stringent delay constraints are implied, i.e.,
$\theta\rightarrow\infty$ the optimal power allocation converges to total
channel inversion. Similarly to the previous case, it has been demonstrated
that the parallel approach outperforms the sequential approach, in terms of
efficiency, regardless of the values of $\theta$ and $\beta$ [30].
### V-C Secret key generation under active attacks
The previous section discussed how SKG can be used to build authenticated
encryption protocols. However, the above scheme could only be secure under the
assumption that the advantage distillation phase is robust against active
attacks. Therefore, this section focuses on active attacks during SKG, in
particular the injection attack is investigated. The idea of this attack is
illustrated in Figure 14.
AliceMalloryBob$XH$$W=\mathbf{H_{A}}^{T}\mathbf{P}\
X_{J}$$W=\mathbf{H_{B}}^{T}\mathbf{P}\ X_{J}$ Figure 14: Alice and Bob have
single transmit and receive antennas and exchange pilot signals $X$ over a
Rayleigh fading channel $H$. A MiM, Mallory, with multiple transmit antennas
injects a pre-coded signal $\mathbf{P}X_{J}$, such that the received signals
at Alice and Bob are equal
$W=\mathbf{H_{A}}^{T}\mathbf{P}=\mathbf{H_{B}}^{T}\mathbf{P}$.
Differently from previous sections, instead of an eavesdropper, an active man-
in-the-middle (MiM) attacker is considered, referred to as Mallory. The system
model assumes two legitimate users, Alice and Bob, each having a single
antenna and Mallory, who has two antennas. The goal of the attacker is to
inject an equivalent signal $W$ at both, Alice and Bob, such that their
channel observations $Z_{A}$ and $Z_{B}$, respectively, will also include the
injected signal:
$\displaystyle Z_{A}=XH+W+N_{A}$ (12) $\displaystyle Z_{B}=XH+W+N_{B},$ (13)
where the channel realization between Alice-Bob is denoted by
$H\sim\mathcal{CN}(0,\sigma^{2})$, the exchanged signal over this channel is
given as $X$, $\mathbb{E}[|X|^{2}]\leq P$, the noise observations at Alice and
Bob are given as $N_{A},N_{B}\sim\mathcal{CN}\left(0,1\right)$ and the
injected signals over the link Eve-Alice (given as $\mathbf{H}_{A}$) and Eve-
Bob (given as $\mathbf{H}_{B}$) are given as
$W=\mathbf{H_{A}}^{T}\mathbf{P}X_{J}=\mathbf{H_{B}}^{T}\mathbf{P}X_{J}$. The
received signals are equal, thanks to the precoding matrix $\mathbf{P}$. A
simple mathematical operation can reveal that, as long as Mallory has one
extra antenna, as compared to Alice and Bob, the design of the pre-coding
matrix is straight forward, i.e.,
$\displaystyle\mathbf{H_{A}}^{T}\mathbf{P}X_{J}$
$\displaystyle=\mathbf{H_{B}}^{T}\mathbf{P}X_{J}\Rightarrow$ $\displaystyle
P_{1}$ $\displaystyle=\frac{H_{B2}-H_{A2}}{H_{A1}-H_{B1}}P_{2}.$ (14)
Overall, this is a simple attack to mount its consequences are crucial. As it
can be seen in Equations (12) and (13), by injecting the signals, Mallory adds
additional term to the shared randomness between Alice and Bob, turning it
into $XH+W$. Hence, this allows Mallory to obtain partial information with
respect to the generated key.
Fortunately, a simplistic countermeasure has been proposed in [11]. The idea
is instead of using deterministic pilot signals $X$, as described above, Alice
and Bob can transmit independent and randomized probe signals $X$ and $Y$,
respectively. This turns their observations into
$\displaystyle Z_{A}=YH+W+N_{A},$ (15) $\displaystyle Z_{B}=XH+W+N_{B},$ (16)
which allows them to simply post-multiply by their own transmission resulting
into the following:
$\displaystyle\tilde{Z}_{A}$ $\displaystyle=$ $\displaystyle
XZ_{A}=XYH+XW+XN_{A},$ (17) $\displaystyle\tilde{Z}_{B}$ $\displaystyle=$
$\displaystyle YZ_{B}=XYH+YW+YN_{B},$ (18)
where, as it can be seen, $W$ is not anymore part of the shared randomness.
Therefore, as long as $X$ and $Y$ are uncorrelated this simple approach can
successfully reduce an injection attack to a less harmful uncorrelated jamming
attack. In detail, the jamming attack has impact on the achievable key rate
but does not reveal anything about the key to Mallory.
Now, when Mallory’s attack is reduced to jamming, a smart thing she can do, is
to act as a reactive jammer. A reactive jammer would first sense the spectrum
and jam only subcarriers where she detects a transmission. Considering a
multicarrier system, Mallory can choose a sensing threshold and jam only
subcarriers where she detects signals with power greater than the chosen
threshold. A thorough analysis considering this scenario has be performed in
[11], where this problem has been investigated using game theory. In fact, the
scenario can be formulated as a non-cooperative zero-sum game with two
players, i.e., player $L$, (legitimate users act as a single player), and
player $J$, (the jammer). Based on the fact that player $J$ jams only after
observing the action from player $L$, this is formed as a hierarchical game
with $L$ being the leader of the game and $J$ being the follower. Note that in
hierarchical games, the optimal action is the Stackelberg equilibrium (SE).
What was shown in this study is that the SE is based on two things: i) the
sensitivity of the receiver at player $J$, and more specifically how well the
sensing threshold is chosen, and ii) the available power at the legitimate
users. The SE is defined as:
* •
If the jammer has badly chosen threshold, depending on the available power at
the legitimate users they would optimally:
1. 1.
equally distribute their power below the sensing threshold and do not comprise
their communication.
2. 2.
transmit with full power on all subcarriers, hence being sensed and jammed.
* •
If the jammer has chosen a low threshold that allows to detect all ongoing
transmissions, Alice and Bob have no choice but to transmit at full power.
Overall, SKG is a promising PLS technology and could help solving the key
distribution issue for emerging 6G applications, e.g., addressing scalability
for massive IoT [31].
## VI Authentication Using PLS
One of the main motivations to look at PLS authentication schemes is the
increasing complexity of standard crypto schemes. In fact, it has been shown
in multiple studies that there exists a trade-off between delay and key sizes
used in the cryptographic schemes.
A particular example that focuses on addressing such issues is the zero-round-
trip-time (0-RTT) protocol introduced in the TLS version $1.3$ for session
resumption. The idea is based on using resumption keys to quickly resume a
session, in a 0-RTT, as opposed to re-authenticating users every subsequent
session. Unfortunately, it has been shown that this scheme is vulnerable a set
of attacks (e.g., replay attack), however, the community answer was “But too
big a win not to do” [32].
This section gives a hint on what PLS can do in terms of authentication for 6G
systems. In particular, it first gives a brief background on physical
unclonable functions (PUFs), then discusses how localization can be used as an
authentication factor, and finally, it introduces a secure 0-RTT
authentication protocol that leverages multiple PLS technologies.
### VI-A Physical unclonable functions
PUFs can be referred to as device fingerprints. The idea is that, the
manufacturing of a circuit is a process with unique characteristics (e.g., due
to change in the temperature, vibrations), which makes each device unique on
its own. While devices operate in a similar manner, they always have small
variations in terms of delays, power-on-state, jitter, etc. This gives an
opportunity to leverage these uniqueness, and use it for authentication.
Given that, a standard PUF based authentication protocol follows two phases.
An enrolment phase which takes place offline, and an authentication phase
which is performed online. During the enrolment phase, a set of challenges are
run on a device’s PUF. A set of challenge could refer to measuring propagation
delays over different propagation paths. Due to the presence of noise, these
measurements are passed through a suitable encoder to generate helper data.
Following that, a verifier (e.g., a server) creates a database where
challenge-response pairs (CRPs) are stored along with the corresponding helper
data. Next, during the online authentication phase, the verifier sends a
random challenge to the device, and the device replies with a new PUF
measurement. The authentication is successful if the verifier can regenerate
the response saved during enrolment by using the new response and the helper
data in its database. Note that, to avoid replay type of attacks a CRP should
not be re-used. A major advantage of the scheme above is that the device does
not need to store any key information and relies only on PUF measurements.
Hence, if the device is compromised (e.g., “captured by an enemy”), no useful
information can be extracted.
### VI-B Location-based authentication
Localization precision is continuously increasing and the goal of 6G
technologies is to achieve centimeter level accuracy. Popular approaches for
fingerprinting rely on measuring received signal strength (RSS), carrier
frequency offsets, I-Q imbalances, CSI measurements and more. This section
presents a lightweight example for location based authentication, through a
low-complexity proximity estimation.
Consider a mobile low-end device with a single antenna and low computational
power. Assume that the device has a map of a premise and knows the location of
the access points within this premise. A simple strategy to perform reverse
authentication (i.e., the device authenticates an access point) is to move in
an unpredictable manner and measure the RSS from multiple positions. As the
RSS is strongly related to the distance between devices, this simple approach
allows to confirm the location of the access point. Typically, localization
would require either the deployment of multiple nodes that measure the RSS
simultaneously or advanced hardware/computational capabilities when
considering a single device. The approach above does not have such
requirements and can still be used as an authentication factor. In fact, the
proximity detection described above can provide resilience to impersonation
type of attacks, e.g., in the presence of a malicious access point.
Now, we summarize some open research issues in the direction of using
fingerprint based authentication. A concern that naturally arises is about the
resilience of such schemes to jamming and man-in-the-middle type of attacks.
In particular, how to cope with interference transmissions, or pilot
contamination type of attacks, both of which can alter the precision of the
localization information. Another issue concerns the trustworthiness of the
localization information, i.e., depending whether we operate at short or long
distance, the variability of measurements can change, hence, bringing
uncertainty into the system. Finally, another aspect concerns the type of
application where such approach could be useful. A good example comes from the
idea presented above, e.g., reverse authentication. Reverse authentication can
help in mitigating attacks that fall into the general category of false base
station attacks (which are open issues in 5G). However, we note that before
deploying location-based authentication technologies all concerns must be
addressed.
### VI-C Multi-factor PLS authentication
A recent publication [14], has shown how three PLS credentials (PUFs, SKG and
location fingerprints) can be combined into a multi-factor PLS based
authentication protocol. The proposed scheme uses PUFs as a mutual
authentication factor between a mobile node (Alice) and a static server (Bob).
The protocol is realized following a typical PUF approach, i.e., following two
steps, enrolment and authentication. The use of PUFs provides several security
guaranties, including protection against physical and cloning attacks. Next,
Alice uses proximity estimation as a second authentication factor. This simple
technique re-assures her for the legitimacy of Bob and provides resistance to
impersonation attacks (e.g., false base station attacks). To provide anonymity
for Alice, the scheme introduces one-time alias IDs. After a successful
authentication, both parties exchange resumption secrets, following a standard
TLS 1.3 procedure. The resumption secrets are used for a fast 0-RTT re-
authentication between Alice and Bob, i.e., session resumption (as opposed to
performing a full authentication procedure). While the standard approach for
session resumption is not forward secure and is vulnerable to replay attacks,
the scheme in [14] uses SKG keys to randomize the resumption secrets. It is
shown that adding SKG ensures both perfect forward security and resistance
against replay attacks.
In general, using the physical layer for authentication is a well investigated
topic. Schemes like the one above, show that there are already multiple PHY
schemes which can contribute for the system security. Some of the research
problems in the area include design of high-entropy PUFs and accurate and
privacy-preserving location-based authentication.
## VII Conclusions and Future Directions
This paper highlights the role that PLS could play in 6G, in view of the
evolution in terms of security, with the concepts of trust, context awareness,
and quality of security.
6G is expected to introduce new features to communication standards including
sensing, subTHz communication, massive MIMO, extreme beamforming, learning and
actuating, ultra reliable low latency computing and more. While it is still
not clear how the transition from 5G to 6G will look like, there is growing
interest on the use of semantics, semantic communications, semantic
compression, and context awareness in 6G.
Another perspective was introduced with quality of security (QoSec), i.e.,
different slices of the network have different security and privacy
requirements. This brings the need of adaptive security levels. A series of
questions arise based on the above: How to define other security levels? How
to perform adaptive identity management? How to make an intelligent risk
assessment?
PLS emerges as a contestant for the next generation of security systems in 6G.
One key advantage of PLS is that it is inherently adaptive. This is due to the
fact that in physical technologies, the secrecy outage probability can be
directly tuned through adjusting the transmission rate.
In particular, wireless channels can be treated as a source of two things, a
source of uniqueness, and a source of entropy. For example, in a slow flat
fading scenario (e.g. LoS) then the channel could be treated as a good source
of uniqueness. As discussed in Section VI, uniqueness can be easily used for
authentication purposes. On the other hand, if the channel changes very fast,
due to small scale fading, it could be treated as a good source of entropy.
The variability of the channel can then be directly used to either distill
keys, or perform keyless transmission. An important observation is that if one
is not available, e.g., uniqueness, then the other will be, e.g., entropy.
Following the above, an open research question is, how to characterize the
channel properties and particularly, which part of the channel should be
considered as predictable and which as unpredictable. It is not an easy
question to answer as it would require the characterization of the channel
correlations in time, frequency and space domains; but it is an important one
as it would allow the alignment of PLS metrics to semantic security metrics.
Finally, we believe it is now time to start defining the security levels based
on the usage of multiple elements. Here, we list several elements:
1. 1.
Criticality of information - how important the information is from user or the
network perspective;
2. 2.
Value of information for the attacker - this captures, who is the attacker and
how much effort is expected to put into compromising the system;
3. 3.
System resilience - this includes the stability and repair time after an
attack;
4. 4.
Threat level - the usage of context to recognize “abnormal” events (could
include location, behavior and communication information);
5. 5.
QoS constraint - systems are expected to comply with particular QoS index.
Today, the deployment of PLS in systems is still lacking traction. However,
there is a growing interest by industry and academia. This paper shows the
potential of PLS for upcoming wireless system designs. It gives concrete
examples of use cases for PLS, reaching far beyond addressing encryption. By
doing so, greatly improving the security of 6G networks. For PLS it is
instrumental to characterize and exploit the wireless channel from a security
point of view. A key advantage is seen for developing light-weight security
solutions for low-latency and massive IoT use cases.
## Acknowledgement
This work is financed by the Saxon State government out of the State budget
approved by the Saxon State Parliament.
## References
* [1] 3GPP TR33.809, “Study on 5G security enhancements against false base stations (Rel 16),” 2018.
* [2] “3GPP TR 33.825 V0.3.0, Study on the Security for 5G URLLC (Release 16),” 3rd Generation Partnership Project; Technical Specification Group Services and System Aspects.
* [3] M. Bloch and J. Barros, _Physical-Layer Security: From Information Theory to Security Engineering_. Cambridge University Press, 2011.
* [4] M. Latva-aho and K. Leppänen, “Key drivers and research challenges for 6G ubiquitous wireless intelligence,” October 2019, published online by the University of Oulu.
* [5] V. Ziegler, P. Schneider, H. Viswanathan, M. Montag, S. Kanugovi, and A. Rezaki, “Security and trust in the 6G era,” _IEEE Access_ , vol. 9, pp. 142 314–142 327, 2021.
* [6] “Deliverable D1.3 Targets and requirements for 6G - initial E2E architecture, 2022,” A flagship for B5G/6G vision and intelligent fabric of technology enablers connecting human, physical, and digital worlds, Hexa-X.
* [7] “International Network Generations Roadmap (INGR),” IEEE , available online https://futurenetworks.ieee.org/roadmap.
* [8] L. Sanguinetti, E. Björnson, and J. Hoydis, “Toward massive MIMO 2.0: Understanding spatial correlation, interference suppression, and pilot contamination,” _IEEE Transactions on Communications_ , vol. 68, no. 1, pp. 232–257, 2020.
* [9] G. A. N. Segura, A. Chorti, and C. B. Margi, “Centralized and distributed intrusion detection for resource-constrained wireless SDN networks,” _IEEE Internet of Things Journal_ , vol. 9, no. 10, pp. 7746–7758, 2022.
* [10] A. Chorti, A. N. Barreto, S. Köpsell, M. Zoli, M. Chafii, P. Sehier, G. Fettweis, and H. V. Poor, “Context-aware security for 6G wireless: The role of physical layer security,” _IEEE Communications Standards Magazine_ , vol. 6, no. 1, pp. 102–108, 2022.
* [11] M. Mitev, A. Chorti, E. V. Belmega, and M. Reed, “Man-in-the-middle and denial of service attacks in wireless secret key generation,” in _IEEE Global Communication Conference (GLOBECOM)_ , Dec 2019, pp. 1–6.
* [12] E. Belmega and A. Chorti, “Protecting secret key generation systems against jamming: Energy harvesting and channel hopping approaches,” _IEEE Transactions on Information Forensics and Security_ , vol. 12, no. 11, pp. 2611–2626, 2017.
* [13] L. Xiao, Q. Yan, W. Lou, G. Chen, and Y. T. Hou, “Proximity-based security techniques for mobile users in wireless networks,” _IEEE Transactions on Information Forensics and Security_ , vol. 8, no. 12, pp. 2089–2100, 2013.
* [14] M. Mitev, M. Shakiba-Herfeh, A. Chorti, M. Reed, and S. Baghaee, “A physical layer, zero-round-trip-time, multifactor authentication protocol,” _IEEE Access_ , vol. 10, pp. 74 555–74 571, 2022.
* [15] C. E. Shannon, “Communication theory of secrecy systems,” _The Bell System Technical Journal_ , vol. 28, no. 4, pp. 656–715, 1949.
* [16] A. Teniou and B. A. Bensaber, “Efficient and dynamic elliptic curve qu‐vanstone implicit certificates distribution scheme for vehicular cloud networks,” _Security and Privacy_ , Jan. 2018.
* [17] A. D. Wyner, “The wire-tap channel,” _The Bell System Technical Journal_ , vol. 54, no. 8, pp. 1355–1387, 1975.
* [18] H. V. Poor and R. F. Schaefer, “Wireless physical layer security,” _Proceedings of the National Academy of Sciences_ , vol. 114, no. 1, pp. 19–26, 2017.
* [19] Y. Liang, H. V. Poor, and S. Shamai, “Secure communication over fading channels,” _IEEE Transactions on Information Theory_ , vol. 54, no. 6, pp. 2470–2492, 2008.
* [20] Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” _IEEE Transactions on Information Theory_ , vol. 56, no. 5, pp. 2307–2359, 2010.
* [21] W. Yang, R. F. Schaefer, and H. V. Poor, “Wiretap channels: Nonasymptotic fundamental limits,” _IEEE Transactions on Information Theory_ , vol. 65, no. 7, pp. 4069–4093, 2019.
* [22] G. Giaconi, D. Gunduz, and H. V. Poor, “Privacy-aware smart metering: Progress and challenges,” _IEEE Signal Processing Magazine_ , vol. 35, no. 6, pp. 59–78, 2018.
* [23] L. Sankar, S. R. Rajagopalan, S. Mohajer, and H. Poor, “Smart meter privacy: A theoretical framework,” _IEEE Transactions on Smart Grid_ , vol. 4, no. 2, pp. 837–846, 2013.
* [24] G. Giaconi, D. Gündüz, and H. V. Poor, “Privacy-cost trade-offs in smart electricity metering systems,” _IET Smart Grid_ , vol. 3, no. 5, pp. 596–604, 2020.
* [25] E. V. Belmega, L. Sankar, and H. V. Poor, “Enabling data exchange in two-agent interactive systems under privacy constraints,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 9, no. 7, pp. 1285–1297, 2015.
* [26] C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, “Generalized privacy amplification,” _IEEE Transactions on Information Theory_ , vol. 41, no. 6, 1995.
* [27] H. Tyagi and S. Watanabe, “Converses for secret key agreement and secure computing,” _IEEE Transactions on Information Theory_ , vol. 61, no. 9, 2015\.
* [28] M. Shakiba-Herfeh and A. Chorti, “Comparison of short blocklength Slepian-Wolf coding for key reconciliation,” in _2021 IEEE Statistical Signal Processing Workshop (SSP)_ , 2021, pp. 111–115.
* [29] M. Mitev, A. Chorti, and M. Reed, “Subcarrier scheduling for joint data transfer and key generation schemes in multicarrier systems,” in _IEEE Global Communication Conference (GLOBECOM)_ , Dec 2019, pp. 1–6.
* [30] M. Mitev, A. Chorti, M. Reed, L. Musavian, “Authenticated secret key generation in delay-constrained wireless systems,” _EURASIP Journal on Wireless Communications and Networking_ , no. 122, June 2020.
* [31] M. Mitev, T. M. Pham, A. Chorti, A. N. Barreto, and G. Fettweis, “Physical Layer Security - from Theory to Practice,” _TechRxiv preprint_ , 11 2022\.
* [32] E. Rescorla, “TLS 1.3,” Stanford University, November 2015.
|
# Locally Frobenius algebras and Hopf algebras
Andrew Baker School of Mathematics & Statistics, University of Glasgow,
Glasgow G12 8QQ, Scotland<EMAIL_ADDRESS>http://www.maths.gla.ac.uk/$\sim$ajb
(Date: this version 26/12/2022 – version 3 arXiv:2212.00437 )
###### Abstract.
We develop a theory of _locally Frobenius algebras_ which are colimits of
certain directed systems of Frobenius algebras. A major goal is to obtain
analogues of the work of Moore & Peterson and Margolis on _nearly Frobenius
algebras_ and _$P$ -algebras_ which was applied to graded Hopf algebras such
as the Steenrod algebra for a prime.
Such locally Frobenius algebras are coherent and in studying their modules we
are naturally led to focus on coherent and finite dimensional modules. Indeed,
the category of coherent modules over locally Frobenius algebra $A$ is abelian
with enough projectives and injectives since $A$ is injective relative to the
coherent modules; however it only has finite limits and colimits. The finite
dimensional modules also form an abelian category but finite dimensional
modules are never coherent. The minimal ideals of a locally Frobenius algebra
are precisely the ones which are isomorphic to coherent simple modules; in
particular it does not contain a copy of any finite dimensional simple module
so it is not a Kasch algebra. We discuss possible versions of stable module
categories for such algebras. We also discuss possible monoidal structures on
module categories of a locally Frobenius Hopf algebra: for example tensor
products of coherent modules turn out to be pseudo-coherent.
Examples of locally Frobenius Hopf algebras include group algebras of locally
finite groups, already intensively studied in the literature.
###### Key words and phrases:
Frobenius algebra, Hopf algebra, stable module category
###### 2020 Mathematics Subject Classification:
Primary 16T05; Secondary 16S99, 57T05
I would like to thank the following: The Max-Planck-Institut für Mathematik in
Bonn for supporting my visit during April and May 2022; Scott Balchin, Tobias
Barthel, Ken Brown, Bob Bruner, John Rognes and Chuck Weibel for sharing their
mathematical knowledge and insights. This project really took off during the
first Covid-19 lock-down in the Spring of 2020, the social isolation in that
strange period at least proved conducive to mathematical research
###### Contents
1. 1 Recollections on rings and modules
2. 2 Frobenius algebras and Frobenius extensions
3. 3 Locally Frobenius algebras
4. 4 Modules over a locally Frobenius algebra
5. 5 Stable module categories for locally Frobenius algebras
6. 6 Locally Frobenius Hopf algebras
7. 7 Some examples of locally Frobenius Hopf algebras
1. 7.1 Group algebras of locally finite groups
2. 7.2 Dual profinite group algebras
## Introduction
The aim of the paper is to develop a non-graded version of the theories of
nearly _Frobenius algebras_ and _$P$ -algebras_ introduced half a century ago
by Moore & Peterson, and Margolis [JCM&FPP:NearlyFrobAlg, HRM:Book,
AB:Palgebras]; these were motivated by topological applications involving the
Steenrod algebra for a prime number.
In these graded connected versions, the properties obtained for the algebras
and their modules are reminiscent of properties of Poincaré duality algebras
(the graded equivalent of Frobenius algebras) and we are able to prove similar
results. However there are some difficulties which are overcome in the graded
context by concentrating on bounded below modules, and it is not clear how to
obtain analogous results in our setting.
Our most complete results involve coherent modules over _locally Frobenius
algebras_ which are themselves coherent rings, although some results on finite
dimensional modules are also obtained.
One motivation for setting up this theory is to introduce stable module
categories for such algebras and we discuss options for doing this, exploiting
the fact that a locally Frobenius algebra is self injective at least relative
to coherent modules and in some cases also relative to all finite dimensional
modules.
Instead of indexing a family of subalgebras on natural numbers as in the
graded theory, we use a family of augmented Frobenius subalgebras indexed on a
directed set; this allows us to include examples such as group algebras of
locally finite groups in our theory. When the indexing is countable this has
implications for vanishing of derived functors of limits taken over the
indexing set but otherwise we do not make use of its cardinality.
We give some general examples of locally Frobenius algebras, but we leave
detailed exploration of examples to future work. The special case of group
algebras of locally finite groups and its literature was drawn to my attention
by my Glasgow colleague Ken Brown.
Notation & conventions: Throughout, $\Bbbk$ will denote a field of
characteristic $p\geqslant 0$. All rings, algebras, modules and their
homomorphisms will be unital.
A _directed set_ $(\Lambda,\preccurlyeq)$ will mean a filtered partially
ordered set, i.e., every pair (or finite set) of elements of $\Lambda$ has an
upper bound. A system of objects and morphisms in a category indexed on such a
$(\Lambda,\preccurlyeq)$ will be referred to as a _$(\Lambda,\preccurlyeq)$
-filtered system_ or _$\Lambda$ -filtered system_.
## 1\. Recollections on rings and modules
We will require results on projective, injective and flat modules over
Noetherian and coherent rings which are thoroughly covered in Lam
[TYL:LectModules&Rings].
The following definitions are standard except for projectively finitely
related.
###### Definition 1.1.
Let $R$ be a ring and $M$ a left/right $R$-module.
* •
$M$ is _finitely generated_ (f.g.) if there is an exact sequence
$F\to M\to 0$
with $F$ finitely generated and free (or equivalently f.g. projective).
* •
$M$ is _finitely related_ (f.r.) if there is a short exact sequence
$0\to K\to F\to M\to 0$
with $K$ finitely generated and $F$ free.
* •
$M$ is _projectively finitely related_ (p.f.r.) if there is a short exact
sequence
$0\to K\to P\to M\to 0$
with $K$ finitely generated and $P$ projective.
* •
$M$ is _finitely presented_ (f.p.) if there is a short exact sequence
$0\to K\to F\to M\to 0$
with $K$ f.g. and $F$ f.g. and free (or equivalently f.g. projective).
The notion of p.f.r. is of course redundant for local rings since every
projective module is free by a celebrated result of Kaplansky [IK:ProjMod].
The next result is a basic tool used in arguments about such conditions, see
for example Lam [TYL:LectModules&Rings]*lemma (5.1).
###### Lemma 1.2 (Schanuel’s Lemma).
Let $M$ be an $R$-module. Suppose that there are short exact sequences
$0\to K\to P\to M\to 0,\quad 0\to L\to Q\to M\to 0$
where $P$ is projective. Then there is a short exact sequence of the form
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\oplus
P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
and moreover, if $Q$ is also projective,
$P\oplus L\cong K\oplus Q.$
###### Corollary 1.3.
Suppose that $M$ is f.p. and that there is a short exact sequence
$0\to L\to Q\to M\to 0$
where $Q$ is f.g., then so is $L$.
Using Schanuel’s Lemma, it is easy to see that $\text{(f.g. \&
f.r.)}\Longleftrightarrow\text{f.p.}$, so we use these descriptions
interchangeably as well as using f.g. projective modules in place of f.g. free
modules.
The first two parts of the following result are taken from
[TYL:LectModules&Rings]*chapter 2§4.
###### Theorem 1.4.
Let $R$ be a ring and $M$ a left/right $R$-module.
_(a)_ $M$ is f.r. if and only if $M\cong M_{0}\oplus F$ where $M_{0}$ is f.p.
and $F$ is free.
_(b)_ If $M$ is f.r. then it is flat if and only if it is projective.
_(c)_ $M$ is p.f.r. if and only if $M\oplus F^{\prime}\cong M_{0}\oplus
F^{\prime\prime}$ where $M_{0}$ is f.p. and $F^{\prime},F^{\prime\prime}$ are
free.
###### Proof.
(a) See [TYL:LectModules&Rings]*theorem 2.4.26(c).
(b) See [TYL:LectModules&Rings]*theorem 2.4.30.
(c) If $M$ is p.f.r. there is a short exact sequence
$0\to K\to P\to M\to 0$
with $K$ finitely generated and $P$ projective. Choose a projective module so
that $F=Q\oplus P$ is free and write $\displaystyle
F^{\infty}=\bigoplus_{i\in\mathbb{N}}F$. By the Eilenberg swindle,
$P\oplus F^{\infty}\cong F^{\infty}$
so there is an exact sequence
$0\to K\to F^{\infty}\to M\oplus F^{\infty}\to 0.$
Therefore $M\oplus F^{\infty}$ is f.r. and the result follows from (a). ∎
Part (c) can be rephrased as saying that saying that being p.f.r. is
equivalent to being _stably f.p._ , or _stably coherent_ when the ring itself
is coherent.
We also recall the characterisation of flat modules provided by the theorem of
Lazard and Govorov [TYL:LectModules&Rings]*theorem 4.34.
###### Theorem 1.5 (Lazard-Govorov theorem).
Let $R$ be a ring and $M$ an $R$-module. Then $M$ is flat if and only if it is
a filtered colimit of f.g. free modules.
### Coherence for rings and modules
We recall the notion of (pseudo-)coherence since we will make heavy use of it.
For basic properties of (pseudo-)coherent modules see Bourbaki
[Bourbaki:HomAlg]*X.§3, ex. 10.
###### Definition 1.6.
Let $R$ be a ring.
* •
An $R$-module $M$ is _pseudo-coherent_ if every f.g. submodule is f.p..
* •
A f.g. pseudo-coherent module is called _coherent_.
* •
$R$ is _left/right coherent_ if it is coherent as a left/right $R$-module.
* •
$R$ is _coherent_ if it is coherent as a left and right $R$-module.
###### Remark 1.7.
Over a coherent ring, a module is f.p. if and only if it is coherent, and
every p.f.r. module is pseudo-coherent. Moreover, its coherent modules form an
abelian category (see Theorem 4.1).
The notion of coherence has an obvious meaning for algebras over a field, and
we will use it without further comment.
Of course every Noetherian ring is coherent, so Frobenius algebras are
coherent. Here is a well known result that can be used to produce many more
examples of coherent rings, this version appears in Bourbaki
[Bourbaki:HomAlg]*X.§3, ex. 11e, and it can be used to show that an infinitely
generated polynomial ring over a coherent commutative ring is coherent.
###### Proposition 1.8.
Let $(\Lambda,\preccurlyeq)$ be a directed set and let $R(\lambda)$
$(\lambda\in\Lambda)$ be a $\Lambda$-directed system of rings and
homomorphisms $\varphi_{\alpha}^{\beta}\colon R(\alpha)\to R(\beta)$ for
$\alpha\preccurlyeq\beta$. Assume that
* •
whenever $\alpha\preccurlyeq\beta$, $R(\beta)$ is flat as a right/left
$R(\alpha)$-module;
* •
each $R(\lambda)$ is coherent.
Then the ring $\displaystyle
R=\operatorname*{colim}_{(\Lambda,\preccurlyeq)}R(\lambda)$ is left/right
coherent and for every $\lambda\in\Lambda$, $R$ is a flat right/left
$R(\lambda)$-module.
Noetherianness and coherence of rings are characterised by homological
properties.
###### Theorem 1.9.
Let $R$ be a ring.
_(a)_ $R$ is left/right Noetherian if and only if all coproducts and directed
colimits of left/right injectives are injective.
_(b)_ $R$ is left/right coherent if and only if all products and directed
limits of flat left/right modules are flat.
###### Proof.
(a) See [TYL:LectModules&Rings]*theorem 1.3.46.
(b) This is a result of Chase, see [TYL:LectModules&Rings]*theorem 1.4.47. ∎
We will frequently make use of faithful flatness, so for the convenience of
the reader we state some of the main properties.
###### Proposition 1.10.
Let $R$ be a ring and $P$ a flat right $R$-module.
_(a)_ $P$ is faithfully flat if it satisfies any and hence all of the
following equivalent conditions.
* •
A sequence of left $R$-modules
$0\to L\to M\to N\to 0$
is short exact if and only if the induced sequence
$0\to P\otimes_{R}L\to P\otimes_{R}M\to P\otimes_{R}N\to 0$
is short exact.
* •
For a left $R$-module $M$, $P\otimes_{R}M=0$ if and only if $M=0$.
* •
A homomorphism of left $R$-modules $\varphi\colon M\to N$ is zero if and only
if the induced homomorphism $\operatorname{Id}_{P}\otimes\varphi\colon
P\otimes_{R}M\to P\otimes_{R}N$ is zero.
_(b)_ If $P$ is faithfully flat then $P\otimes_{R}(-)$ reflects monomorphisms,
epimorphisms and isomorphisms.
_(c)_ Let $R\to S$ be a ring homomorphism so that $S$ is a faithfully flat
right $R$-module, and let $M$ be a left $R$-module. If $S\otimes_{R}M$ is a
simple $S$-module, then $M$ is simple.
###### Proof.
(a) See [TYL:LectModules&Rings]*theorem 4.70.
(b) This is immediate from (a).
(c) By flatness, a short exact sequence of $R$-modules
$0\to L\to M\to N\to 0$
on tensoring with $S$ yields a short exact sequence of $S$-modules
$0\to S\otimes_{R}L\to S\otimes_{R}M\to S\otimes_{R}N\to 0.$
If $S\otimes_{R}M$ is simple then one of $S\otimes_{R}L$ or $S\otimes_{R}N$
must be trivial, so by faithful flatness of $S$, one of $L$ or $N$ must be
trivial. Hence $M$ is simple. ∎
## 2\. Frobenius algebras and Frobenius extensions
Recall that a Frobenius $\Bbbk$-algebra $R$ is self-injective, i.e., injective
as a left/right $R$-module. The following is a fundamental consequence.
Recall that there are induction and coinduction functors
$\operatorname{ind}_{\Bbbk}^{R}\colon\mathbf{Mod}_{\Bbbk}\to\mathbf{Mod}_{R},\quad\operatorname{coind}_{\Bbbk}^{R}:\mathbf{Mod}_{\Bbbk}\to\mathbf{Mod}_{R}$
where
$\operatorname{ind}_{\Bbbk}^{R}(-)=R\otimes_{\Bbbk}(-),\quad\operatorname{coind}_{\Bbbk}^{R}(-)=\operatorname{Hom}_{\Bbbk}(R,-).$
Notice that for a $\Bbbk$-vector space $W$, the $R$-module
$\operatorname{ind}_{\Bbbk}^{R}W$ is projective (and in fact free) while
$\operatorname{coind}_{\Bbbk}^{R}W$ is injective.
For future use we recall that since we are working over a field $\Bbbk$, every
$R$-module $M$ admits an embedding into an injective,
(2.1) $M\to\operatorname{coind}_{\Bbbk}^{R}M;\quad m\longmapsto(r\mapsto rm).$
The next result is a standard reformulation of what it means to be a Frobenius
algebra.
###### Theorem 2.1.
If $R$ is a Frobenius $\Bbbk$-algebra, then the functors
$\operatorname{ind}_{\Bbbk}^{R}$ and $\operatorname{coind}_{\Bbbk}^{R}$ are
naturally isomorphic. Hence for a $\Bbbk$-vector space $W$,
$\operatorname{ind}_{\Bbbk}^{R}W\cong\operatorname{coind}_{\Bbbk}^{R}W$ is
both projective and injective.
###### Corollary 2.2.
An $R$-module is injective if and only if it is projective.
###### Proof.
Let $J$ be an injective $R$-module. As a special case of (2.1) there is a
monomorphism of $R$-modules $J\to\operatorname{coind}_{\Bbbk}^{R}J$ and by
injectivity there is a commutative diagram with exact row
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}_{J}}$$\textstyle{\operatorname{coind}_{\Bbbk}^{R}J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J}$
so $J$ is a retract of a projective module, hence projective. A similar
argument shows that for a projective $P$, there is a commutative diagram
$\textstyle{\operatorname{ind}_{\Bbbk}^{R}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}_{P}}$
which shows that $P$ is injective. ∎
A useful consequence of this result is that every monomorphism of $R$-modules
$R\to M$ splits.
The next result summarises the fundamental properties of modules over a
Frobenius algebra which motivate much of our work on modules over locally
Frobenius algebras.
###### Proposition 2.3.
Let $R$ be a Frobenius $\Bbbk$-algebra and $M$ an $R$-module. Then the
following are equivalent:
* •
$M$ is injective;
* •
$M$ is projective;
* •
$M$ is flat.
###### Proof.
The first two are equivalent by Corollary 2.2, while projectivity implies
flatness.
Since $R$ is self-injective and Noetherian, by using the Lazard-Govorov
Theorem 1.5, Corollary 2.2 and Theorem 1.9(a) we find that flatness implies
injectivity. ∎
Now recall that if $S$ is a simple $R$-module then Schur’s Lemma implies that
then $\operatorname{End}_{R}(S)$ is a division algebra central over $\Bbbk$,
so $S$ can be viewed as a vector space over $\operatorname{End}_{R}(S)$ and
$\dim_{\Bbbk}S=\dim_{\Bbbk}\operatorname{End}_{R}(S)\,\dim_{\operatorname{End}_{R}(S)}S.$
###### Proposition 2.4.
Let $R$ be a Frobenius $\Bbbk$-algebra.
_(a)_ Every $R$-module embeds into a free module. In particular, every f.g.
$R$-module embeds into a f.g. free module.
_(b)_ Let $S$ be a simple left $R$-module. Then $S$ embeds as a submodule of
$R$ with multiplicity equal to $\dim_{\operatorname{End}_{R}(S)}S$.
###### Proof.
We focus on left modules but the case of right modules is similar.
(a) For an $R$-module $M$, there is an injective homomorphism as in (2.1),
$M\to\operatorname{coind}_{\Bbbk}^{R}M\cong\operatorname{ind}_{\Bbbk}^{R}M$
and the induced module is a free $R$-module. When $M$ is f.g. $R$-module it is
a f.d. $\Bbbk$-vector space so the induced module is a f.g. free $R$-module.
(b) Every non-trivial homomorphism $S\to R$ is injective. By Schur’s Lemma its
endomorphism algebra is a division algebra which is central over $\Bbbk$. The
multiplicity of $S$ in $R$ is equal to
$\dim_{\operatorname{End}_{R}(S)^{\mathrm{op}}}\operatorname{Hom}_{R}(S,R)$,
where $\operatorname{End}_{R}(S)$ acts on the right of
$\operatorname{Hom}_{R}(S,R)$ by precomposition.
The coinduced module
$\operatorname{coind}_{\Bbbk}^{R}\Bbbk=\operatorname{Hom}_{\Bbbk}(R,\Bbbk)$ is
a left $R$-module through the right action on the domain, and then there are
isomorphisms of right $\operatorname{End}_{R}(S)$-vector spaces
$\displaystyle\operatorname{Hom}_{R}(S,R)$
$\displaystyle\cong\operatorname{Hom}_{R}(S,\operatorname{coind}_{\Bbbk}^{R}\Bbbk)$
$\displaystyle\cong\operatorname{Hom}_{\Bbbk}(R\otimes_{R}S,\Bbbk)$
$\displaystyle\cong\operatorname{Hom}_{\Bbbk}(S,\Bbbk),$
hence $\operatorname{Hom}_{R}(S,R)$ is non-trivial and the multiplicity of $S$
is
$\dim_{\operatorname{End}_{R}(S)^{\mathrm{op}}}\operatorname{Hom}_{R}(S,R)=\dim_{\operatorname{End}_{R}(S)}S.\qed$
An important property of finite dimensional Hopf algebras is that they have
non-zero _integrals_. Although in general Frobenius algebras do not, Frobenius
algebras augmented over the ground field do. If $\varepsilon\colon R\to\Bbbk$
is the augmentation, its subspaces of left and right _integrals with respect
to $\varepsilon$_ are defined by
${\smallint}^{\mathrm{l}}_{R}=\\{h\in R:\forall y\in
R,\;yh=\varepsilon(y)h\\},\quad{\smallint}^{\mathrm{r}}_{R}=\\{h\in R:\forall
y\in R,\;hy=\varepsilon(y)h\\}.$
Then there is an isomorphism
${\smallint}^{\mathrm{l}}_{R}\cong\operatorname{Hom}_{R}(\Bbbk,R)$ and a
similar identification of ${\smallint}^{\mathrm{r}}_{R}$ with right module
homomorphisms.
###### Proposition 2.5.
Let $R$ be a Frobenius $\Bbbk$-algebra which is augmented over $\Bbbk$. Then
viewing $R$ and $\Bbbk$ as left or right $R$-modules we have
$\dim_{\Bbbk}\operatorname{Hom}_{R}(\Bbbk,R)=1$ and therefore
$\dim_{\Bbbk}{\smallint}^{\mathrm{l}}_{R}=1=\dim_{\Bbbk}{\smallint}^{\mathrm{r}}_{R}.$
###### Proof.
This follows from Proposition 2.4(b). ∎
If ${\smallint}^{\mathrm{r}}_{R}={\smallint}^{\mathrm{l}}_{R}$ then the
augmented Frobenius algebra is _unimodular_ ; see Farnsteiner
[RF:FrobExtnHopfAlgs] for more on this.
We will be interested in Frobenius algebras which are _symmetric_ , i.e., they
have a Frobenius form $\lambda\colon A\to\Bbbk$ which induces a symmetric
bilinear form. It follows that the associated Nakayama automorphism is the
identity and so by Farnsteiner [RF:FrobExtnHopfAlgs]*lemma 1.1 it is
unimodular.
For a Hopf algebra the counit is the natural choice of augmentation and we
will always choose it. If the Hopf algebra is commutative or cocommutative
then its antipode is a self-inverse automorphism making it _involutive_ , and
since the identity function is inner, by [RF:FrobExtnHopfAlgs]*proposition 2.3
it is a symmetric Frobenius algebra if and only if it is unimodular.
For finite dimensional Hopf algebras there is an analogue of Maschke’s
Theorem.
###### Theorem 2.6.
Let $H$ be a finite dimensional Hopf algebra and $\varepsilon$ its counit.
Then $H$ is semisimple if and only if
$\varepsilon{\smallint}^{\mathrm{l}}_{H}\neq\\{0\\}\neq\varepsilon{\smallint}^{\mathrm{r}}_{H}.$
###### Proof.
See [ML:TourRepThy]*page 553 for example. The proof of semisimplicity uses a
non-zero idempotent $e\in{\smallint}^{\mathrm{l}}_{H}$ or
$e\in{\smallint}^{\mathrm{r}}_{H}$ and the coproduct applied to it. ∎
###### Remark 2.7.
This result does not apply to all augmented Frobenius algebras. As an example,
consider $R=\Bbbk\times R_{0}$ with the augmentation being projection onto the
first factor and $R_{0}$ a local Frobenius algebra that is not semisimple. Let
$\lambda_{0}$ be a Frobenius form on $R_{0}$; then the form on $R$ given by
$\lambda(x,y)=x+\lambda_{0}(y)$ is Frobenius and
$(1,0)\in{\smallint}^{\mathrm{l}}_{R}$ but $R$ is not semisimple.
Now we turn to _Frobenius extensions_ , introduced by Kasch
[FK:ProjFrobExtns]. For a concise account which highlights aspects relevant to
our work, see Lorenz [ML:TourRepThy], especially the exercises for sections
2.2 and 12.4. Other useful sources are Farnsteiner [RF:FrobExtnHopfAlgs] and
Fischman et al [DF&SM&H-JS:FrobExtns]. We adopt the notation $A:B$ rather than
$A/B$ to indicate an extension of algebras.
Suppose given a homomorphism $B\to A$ of $\Bbbk$-algebras so we can view $A$
as a $B$-bimodule and consider the extension of $\Bbbk$-algebras $A:B$.
###### Definition 2.8.
The extension $A:B$ is a _Frobenius extension_ if there is a $B$-bimodule
homomorphism $\Lambda\colon A\to B$ and elements $x_{i},y_{i}\in A$
$(1\leqslant i\leqslant n)$ such that for all $a\in A$,
$\sum_{1\leqslant i\leqslant n}y_{i}\Lambda(x_{i}a)=a=\sum_{1\leqslant
i\leqslant n}\Lambda(ay_{i})x_{i}.$
###### Proposition 2.9.
Suppose that $A:B$ is a Frobenius extension with associated $B$-bimodule
homomorphism $\Lambda\colon A\to B$ and elements $x_{i},y_{i}\in A$
$(1\leqslant i\leqslant n)$. Then $A$ is projective as a left and right
$B$-module. Furthermore, there is an isomorphism of functors
$\operatorname{coind}^{A}_{B}\xrightarrow{\;\cong\;}\operatorname{ind}^{A}_{B};$
which is defined on each $B$-module $M$ by
$\operatorname{coind}^{A}_{B}M=\operatorname{Hom}_{B}(A,M)\mapsto\operatorname{coind}^{A}_{B}M;\quad
f\mapsto\sum_{1\leqslant i\leqslant n}y_{i}f(x_{i}).$
Conversely, if $A$ is projective as a left and right $B$-module and
$\operatorname{coind}^{A}_{B}\cong\operatorname{ind}^{A}_{B}$, then $A:B$ is a
Frobenius extension.
In practice we will be considering the situation in following result.
###### Proposition 2.10.
Let $\varepsilon\colon A\to\Bbbk$ be an augmented symmetric Frobenius algebra
and let $B\subseteq A$ be a subalgebra augmented by the restriction of
$\varepsilon$ and which is also a symmetric Frobenius algebra. Then $A:B$ is a
Frobenius extension.
###### Proof.
Since the Nakayama automorphisms of $A$ and $B$ are the identity functions,
the result is an immediate consequence of [RF:FrobExtnHopfAlgs]*theorem 1.3. ∎
## 3\. Locally Frobenius algebras
Let $(\Lambda,\preccurlyeq)$ be an infinite partially ordered set which has
the following properties, in particular it is a _directed set_ :
* •
it is filtered: for any $\lambda^{\prime},\lambda^{\prime\prime}\in\Lambda$
there is a common upper bound $\lambda\in\Lambda$, so
$\lambda^{\prime}\preccurlyeq\lambda$ and
$\lambda^{\prime\prime}\preccurlyeq\lambda$;
* •
there is a unique initial element $\lambda_{0}$;
* •
for every $\lambda\in\Lambda$ there is a
$\lambda^{\prime\prime\prime}\in\Lambda$ with
$\lambda\precneqq\lambda^{\prime\prime\prime}$.
We will denote such a directed set by $(\Lambda,\preccurlyeq,\lambda_{0})$ but
often just refer to it as $\Lambda$.
If $\Lambda^{\prime}\subseteq\Lambda$ is an infinite filtered subset
containing $\lambda_{0}$, then will refer to
$(\Lambda^{\prime},\preccurlyeq,\lambda_{0})$ as a _subdirected set_ and write
$(\Lambda^{\prime},\preccurlyeq,\lambda_{0})\subseteq(\Lambda,\preccurlyeq,\lambda_{0})$.
###### Definition 3.1.
A (necessarily infinite dimensional) $\Bbbk$-algebra $A$ is _locally Frobenius
of shape $\Lambda$_ if it satisfies the following conditions.
(a) There is a $\Lambda$-directed system of symmetric Frobenius algebras
$A(\lambda)$ ($\lambda\in\Lambda$) augmented over $\Bbbk$ and proper inclusion
homomorphisms $\iota_{\lambda^{\prime}}^{\lambda^{\prime\prime}}\colon
A(\lambda^{\prime})\hookrightarrow A(\lambda^{\prime\prime})$ with
$A(\lambda_{0})=\Bbbk$ and $\displaystyle
A=\bigcup_{\lambda\in\Lambda}A(\lambda)$.
(b) Each inclusion $\iota_{\lambda^{\prime}}^{\lambda^{\prime\prime}}$ makes
$A(\lambda^{\prime\prime}):A(\lambda^{\prime})$ a free Frobenius extension
(i.e., $A(\lambda^{\prime\prime})$ is free as a left and right
$A(\lambda^{\prime})$-module).
Of course the Frobenius condition in (b) is a consequence of Proposition 2.10,
while the freeness is an additional assumption.
###### Definition 3.2.
Let $A$ be a locally Frobenius algebra of shape $\Lambda$ and let
$(\Lambda^{\prime},\preccurlyeq,\lambda_{0})\subseteq(\Lambda,\preccurlyeq,\lambda_{0})$
be a subdirected set. A subalgebra $B\subseteq A$ is a _locally Frobenius
subalgebra of shape $\Lambda^{\prime}$_ if $\displaystyle
B=\bigcup_{\lambda\in\Lambda^{\prime}}B(\lambda)$ is locally Frobenius algebra
of shape $\Lambda^{\prime}$ and for every $\lambda\in\Lambda^{\prime}$,
$B(\lambda)\subseteq A(\lambda)$.
###### Notation.
We will denote the kernel of the augmentation $A(\lambda)\to\Bbbk$ by
$A(\lambda)^{+}$; clearly $A$ is also augmented so we similarly denote its
augmentation ideal by $A^{+}$. All of these are completely prime maximal
ideals (recall that an ideal $P$ in a ring is _completely prime_ if $xy\in P$
implies $x\in P$ or $y\in P$); in general this is stronger the notion of prime
(an ideal $Q$ in a ring $R$ is _prime_ if $xRy\subseteq P$ implies $x\in P$ or
$y\in P$). All nilpotent elements of $A$ are contained in $A^{+}$, and if
$e\in A$ is an idempotent then exactly one of $e$ or $1-e$ is in $A^{+}$.
The augmentation condition in (a) may seem unnecessary but we do require it
for some technical results and it is automatically satisfied in the Hopf
algebra case because the counit is an augmentation. The Frobenius extension
condition in (b) might be weakened, and even in the Hopf algebra case it is
not otherwise guaranteed except in special circumstances such as when the
$A(\lambda)$ are all local.
We make a trivial observation.
###### Lemma 3.3.
Let $A$ be a locally Frobenius algebra. Then for every finite subset
$Z\subseteq A$ there is a $\lambda\in\Lambda$ such that $Z\subseteq
A(\lambda)$. Hence every finite dimensional subspace $V\subseteq A$ is
contained in some $A(\lambda)$.
###### Proof.
Each element $z\in Z$ is contained in some $A(\lambda_{z})$ and by the
filtering condition there is an upper bound $\lambda$ of the $\lambda_{z}$, so
that $Z\subseteq A(\lambda)$. ∎
Of course a f.d. subalgebra of $A$ is f.d. subspace so it is a subalgebra of
some $A(\lambda)$.
The next result provides an important class of examples of locally Frobenius
algebras which we will refer to as _locally Frobenius Hopf algebras_.
###### Proposition 3.4.
Let $H$ be a Hopf algebra over $\Bbbk$. Then $H$ is a locally Frobenius
algebra of shape $\Lambda$ if the following conditions are satisfied.
_(a)_ There is a $\Lambda$-directed system of f.d. subHopf algebras
$H(\lambda)\subseteq H$ $(\lambda\in\Lambda)$ and inclusion homomorphisms
$\iota_{\lambda^{\prime}}^{\lambda^{\prime\prime}}\colon
H(\lambda^{\prime})\to H(\lambda^{\prime\prime})$ with $H(\lambda_{0})=\Bbbk$
and $\displaystyle H=\bigcup_{\lambda\in\Lambda}H(\lambda)$.
_(b)_ Each $H(\lambda)$ is involutive and unimodular.
###### Proof.
The Larson-Sweedler theorem [LarsonSweedlerThm] implies that each $H(\lambda)$
is Frobenius, while [RF:FrobExtnHopfAlgs]*proposition 2.3 implies that it is
symmetric since the square of the antipode is the identity which is also a
Nakayama automorphism. Finally, the Nichols-Zoeller theorem
[WDN&MBZ:HAfreeness]. implies that each pair
$H(\lambda^{\prime\prime}):H(\lambda^{\prime})$ is free. ∎
Of course if each Hopf algebra $H(\lambda)$ is commutative or cocommutative
then it is involutive.
It is not necessarily true that an infinite dimensional Hopf algebra $H$ is
free as a left/right module over a finite dimensional subHopf algebra $K$,
although some results on when this holds are known; for the case where $K$ is
semisimple see [WDN&MBR:HAfreenessinfdim], while for the case where $K$ is a
normal subalgebra see [H-JS:RemQGps]. However, for our purposes this is not
required.
Later we will discuss some examples of locally Frobenius Hopf algebras such as
group algebras of locally finite groups.
The next definition builds on Definition 3.2.
###### Definition 3.5.
Let $H$ be a locally Frobenius Hopf algebra of shape $\Lambda$ and let
$(\Lambda^{\prime},\preccurlyeq,\lambda_{0})\subseteq(\Lambda,\preccurlyeq,\lambda_{0})$
be a subdirected set. A subHopf algebra $K\subseteq H$ is a _locally Frobenius
subHopf algebra of shape $\Lambda^{\prime}$_ if $\displaystyle
K=\bigcup_{\lambda\in\Lambda^{\prime}}K(\lambda)$ is locally Frobenius Hopf
algebra of shape $\Lambda^{\prime}$ and for every
$\lambda\in\Lambda^{\prime}$, $K(\lambda)\subseteq H(\lambda)$.
We end this section with a result suggested by Richardson
[JSR:GpRngsnonzerosocle]*lemma 2.2. This shows that the augmentation ideal of
a locally Frobenius subalgebra generates a ‘large’ submodule.
###### Proposition 3.6.
Let $A$ be a locally Frobenius algebra of shape $\Lambda$ and $B$ a locally
Frobenius subalgebra of shape $\Lambda^{\prime}\subseteq\Lambda$. Then the
submodule $AB^{+}\subseteq A$ is essential.
###### Proof.
This proof is a straightfoward adaption of that in Richardson
[JSR:GpRngsnonzerosocle].
Suppose that $Az\cap AB^{+}=0$ for some non-zero $z\in A$. Choose
$\alpha\in\Lambda$ such that $z\in A(\alpha)$. As $\Lambda^{\prime}$ is
infinite we can choose a $\beta\in\Lambda^{\prime}$ for which
$\dim_{\Bbbk}A(\beta)>\frac{\dim_{\Bbbk}A(\beta)}{\dim_{\Bbbk}A(\beta)z}.$
Now choose $\gamma\in\Lambda^{\prime}$ so that
$A(\alpha)\subseteq A(\gamma)\supseteq A(\beta).$
and then
$A(\gamma)z\cap A(\gamma)A(\beta)^{+}\subseteq Az\cap A(\gamma)B^{+}=0.$
Now recall that $A(\gamma)$ is a free module over each of $A(\alpha)$ and
$A(\beta)$, so
$A(\gamma)z\cong A(\gamma)\otimes_{A(\alpha)}A(\alpha)z,\quad
A(\gamma)A(\beta)^{+}\cong A(\gamma)\otimes_{A(\beta)}A(\beta)^{+},$
giving
$\displaystyle\dim_{\Bbbk}A(\gamma)z$
$\displaystyle=\frac{\dim_{\Bbbk}A(\gamma)}{\dim_{\Bbbk}A(\alpha)}\dim_{\Bbbk}A(\alpha)z,$
$\displaystyle\dim_{\Bbbk}A(\gamma)A(\beta)^{+}$
$\displaystyle=\frac{\dim_{\Bbbk}A(\gamma)}{\dim_{\Bbbk}A(\beta)}\bigl{(}\dim_{\Bbbk}A(\beta)-1\bigr{)}.$
Using these we obtain
$\displaystyle\dim_{\Bbbk}A(\gamma)$
$\displaystyle\geqslant\dim_{\Bbbk}\bigl{(}A(\gamma)z\oplus
A(\gamma)A(\beta)^{+}\bigr{)}$
$\displaystyle=\dim_{\Bbbk}A(\gamma)z+\dim_{\Bbbk}A(\gamma)A(\beta)^{+}$
$\displaystyle=\frac{\dim_{\Bbbk}A(\gamma)\dim_{\Bbbk}A(\alpha)z}{\dim_{\Bbbk}A(\alpha)}+\frac{\dim_{\Bbbk}A(\gamma)\bigl{(}\dim_{\Bbbk}A(\beta)-1\bigr{)}}{\dim_{\Bbbk}A(\beta)}$
$\displaystyle>\frac{\dim_{\Bbbk}A(\gamma)}{\dim_{\Bbbk}A(\beta)}+\frac{\dim_{\Bbbk}A(\gamma)\bigl{(}\dim_{\Bbbk}A(\beta)-1\bigr{)}}{\dim_{\Bbbk}A(\beta)}$
$\displaystyle=\dim_{\Bbbk}A(\gamma),$
which is impossible. Therefore $AB^{+}\subseteq A$ must be an essential
submodule. ∎
## 4\. Modules over a locally Frobenius algebra
Now we will describe some basic properties of modules over locally Frobenius
algebras. Throughout this section we will suppose that $A$ is a locally
Frobenius of shape $\Lambda$.
First we state a result for coherent $A$-modules; since $A$ is coherent these
are exactly the f.p. $A$-modules.
###### Theorem 4.1.
The category $\mathbf{Mod}^{\mathrm{coh}}_{A}$ of coherent $A$-modules and
their homomorphisms is an abelian category with all finite limits and colimits
as well as enough projectives and injectives.
###### Proof.
Clearly this category has finite products and coproducts, so it is sufficient
to check that it has kernels, images and cokernels; this is an exercise in
Bourbaki [Bourbaki:HomAlg]*ex. §3.10(b), see also Cohen [JMC:Coherent]*section
1. For projectives and injectives see Lemma 4.17 below. ∎
###### Proposition 4.2.
_(a)_ For each $\lambda\in\Lambda$, $A$ is injective, projective and flat as a
left or right $A(\lambda)$-module.
_(b)_ The $\Bbbk$-algebra $A$ is coherent.
_(c)_ Suppose that $M$ is a coherent $A$-module. Then for some $\lambda$ there
is an $A(\lambda)$-module $M^{\prime}$ with a finite presentation
$0\longleftarrow M^{\prime}\longleftarrow A(\lambda)^{k}\longleftarrow
A(\lambda)^{\ell}$
inducing a finite presentation
$0\longleftarrow A\otimes_{A(\lambda)}M^{\prime}\longleftarrow
A^{k}\longleftarrow A^{\ell}$
where $A\otimes_{A(\lambda)}M^{\prime}\cong M$.
_(d)_ Let $\varphi\colon M\to N$ be a homomorphisms of coherent $A$-modules.
Then there is a $\lambda\in\Lambda$, and a homomorphism of finitely generated
$A(\lambda)$-modules $\varphi^{\prime\prime}\colon M^{\prime\prime}\to
N^{\prime\prime}$ fitting into a commutative diagram of $A$-module
homomorphisms.
$\textstyle{A\otimes_{A(\lambda)}M^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}\otimes\varphi^{\prime\prime}}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(\lambda)}N^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{N}$
###### Proof.
We will give brief indications of the proofs.
(a) By Proposition 1.8 $A$ is a flat $A(\lambda)$-module, hence by Proposition
2.3 it is also injective and projective.
(b) Since each $A(\lambda)$ is Noetherian over $\Bbbk$, Proposition 1.8
implies that $A$ is left and right coherent.
(c) If $M$ is an f.p. $A$-module, there is an exact sequence of $A$-modules
$0\longleftarrow M\xleftarrow{\rho_{0}}A^{m}\xleftarrow{\rho_{1}}A^{n}$
for some $m,n$. The image of $\rho_{1}$ must be contained in
$A(\lambda)^{m}\subseteq A^{m}$ for some $\lambda\in\Lambda$, so we obtain an
exact sequence of $A(\lambda)$-modules
$0\longleftarrow
M^{\prime}\xleftarrow{\rho^{\prime}_{0}}A(\lambda)^{m}\xleftarrow{\rho^{\prime}_{1}}A(\lambda)^{n}$
and on tensoring with $A$, by (a) this yields an exact sequence of $A$-modules
$0\longleftarrow
A\otimes_{A(\lambda)}M^{\prime}\xleftarrow{\operatorname{Id}\otimes\rho^{\prime}_{0}}A\otimes_{A(\lambda)}A(\lambda)^{m}\xleftarrow{\operatorname{Id}\otimes\rho^{\prime}_{1}}A\otimes_{A(\lambda)}A(\lambda)^{n}$
which is equivalent to the original one.
(d) Using (c) we can represent $M$ and $N$ as induced up from f.g.
$A(\lambda_{1})$-modules $M^{\prime},N^{\prime}$ for some $\lambda_{1}$. The
image of the restriction of $\varphi$ to $M^{\prime}$ in
$A\otimes_{A(\lambda_{1})}N^{\prime}$ lies in some
$A(\lambda_{2})\otimes_{A(\lambda_{1})}N^{\prime}$ where
$\lambda_{1}\preccurlyeq\lambda_{2}$. Base changing gives a homomorphism
$\varphi^{\prime\prime}\colon
M^{\prime\prime}=A(\lambda_{2})\otimes_{A(\lambda_{1})}M^{\prime}\to
A(\lambda_{2})\otimes_{A(\lambda_{1})}N^{\prime}=N^{\prime\prime}$
with the required properties. ∎
The faithful flatness condition of Proposition 4.9 gives another useful
property.
###### Corollary 4.3.
Every short exact sequence of coherent $A$-modules
(4.1) $0\to L\xrightarrow{\varphi}M\xrightarrow{\theta}N\to 0$
is induced up from a short exact sequence of f.g. $A(\alpha)$-modules
$0\to
L^{\prime}\xrightarrow{\varphi^{\prime}}M^{\prime}\xrightarrow{\theta^{\prime}}N^{\prime}\to
0$
for some $\alpha\in\Lambda$, i.e., there is a commutative diagram of
$A$-modules of the following form.
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\cong}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\cong}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{A(\alpha)}L^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}\otimes\varphi^{\prime}}$$\textstyle{A\otimes_{A(\alpha)}M^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}\otimes\theta^{\prime}}$$\textstyle{A\otimes_{A(\alpha)}N^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
In particular, an isomorphism $\varphi\colon L\to M$ is induced up from an
isomorphism of f.g. $A(\alpha)$-modules for some $\alpha\in\Lambda$.
###### Proof.
Choose $\alpha\in\Lambda$ so that there are homomorphisms of f.g.
$A(\alpha)$-modules
$L^{\prime}\xrightarrow{\varphi^{\prime}}M^{\prime}\xrightarrow{\theta^{\prime}}N^{\prime}$
such that
$0\to
A\otimes_{A(\alpha)}L^{\prime}\xrightarrow{\operatorname{Id}\otimes\varphi^{\prime}}A\otimes_{A(\alpha)}M^{\prime}\xrightarrow{\operatorname{Id}\otimes\theta^{\prime}}A\otimes_{A(\alpha)}N^{\prime}\to
0$
corresponds to the short exact sequence (4.1). Then by Proposition 1.10(a),
$0\to
L^{\prime}\xrightarrow{\varphi^{\prime}}M^{\prime}\xrightarrow{\theta^{\prime}}N^{\prime}\to
0$
is a short exact sequence of $A(\alpha)$-modules.
The statement about isomorphisms also follows using faithful flatness of the
$A(\alpha)$-module $A$ and Proposition 1.10(b). ∎
Now we give some results on the dimension of coherent $A$-modules.
###### Lemma 4.4.
Let $\lambda\in\Lambda$ and let $L\subseteq A(\lambda)$ be a left ideal. Then
as left $A$-modules,
$A\otimes_{A(\lambda)}A(\lambda)/L\cong A/AL.$
and the $\Bbbk$-vector space $A/AL$ is infinite dimensional.
###### Proof.
By a standard criterion for flatness of [TYL:LectModules&Rings]*(4.12),
multiplication induces an isomorphism
$A\otimes_{A(\lambda)}L\xrightarrow{\cong}AL$
where $AL\subseteq A$ is the left ideal generated by $L$. Therefore there is a
commutative diagram of left $A$-modules
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{A(\lambda)}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(\lambda)}A(\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(\lambda)}A(\lambda)/L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{AL\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/AL\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
with exact rows. It follows that the right hand vertical arrow is an
isomorphism.
Whenever $\alpha\preccurlyeq\beta$, $A(\beta)$ is a finite rank free
$A(\alpha)$-module, therefore
$\dim_{\Bbbk}A(\beta)=\dim_{\Bbbk}A(\alpha)\operatorname{rank}_{A(\alpha)}A(\beta)$
and
$\operatorname{rank}_{A(\alpha)}A(\beta)\leqslant\operatorname{rank}_{A(\alpha)}A$
where is strictly increasing as a function of $\beta$. ∎
###### Proposition 4.5.
Let $M$ be a coherent left/right $A$-module. Then $M$ is an infinite
dimensional $\Bbbk$-vector space.
###### Proof.
We will assume that $M$ is a left module, the proof when it is a right module
is similar.
When $M$ is cyclic, $M\cong A/L$ for some f.g. left ideal $L$. A finite set of
generators of $L$ must lie in some $A(\lambda)$ so the ideal
$L(\lambda)=A(\lambda)\cap L$ satisfies $AL(\lambda)=L$ and by Lemma 4.4,
$M\cong AL(\lambda)$ is infinite dimensional.
Now for a general coherent module, suppose that $M$ has $m$ generators
$x_{1},\dots,x_{m}$ where $m$ is minimal. Then for the proper submodule
$M^{\prime}\subseteq M$ generated by $x_{1},\dots,x_{m-1}$, $M/M^{\prime}$ is
a non-trivial cyclic coherent module which is infinite dimensional. Since
there is an epimorphism $M\to M/M^{\prime}$, $M$ must be infinite dimensional.
∎
This gives an important fact about f.d. modules.
###### Proposition 4.6.
A non-trivial f.d. $A$-module is not coherent.
For simple modules we have the following.
###### Corollary 4.7.
Let $S=A/L$ be a f.d. simple left/right $A$-module, where $L$ is a maximal
left/right ideal of $A$. Then $S$ is not coherent and $L$ is not finitely
generated. In particular, the trivial $A$-module $\Bbbk$ is not coherent and
the augmentation ideal $A^{+}$ is not f.g. as a left/right module.
Of course this shows that a locally Frobenius algebra is never Noetherian. As
we will see later, it also implies that the trivial module is not isomorphic
to a submodule of $A$.
###### Corollary 4.8.
Let $z\in A$ be nilpotent. Then $A/Az$ and $A/zA$ are both infinite
dimensional.
###### Proof.
Every nilpotent element of $A$ is contained in the completely prime ideal
$A^{+}$. ∎
We end this section with another important observation.
###### Proposition 4.9.
For $\alpha,\beta\in\Lambda$,
* •
if $\alpha\preccurlyeq\beta$ then $A(\beta)$ is a faithfully flat
$A(\alpha)$-module;
* •
$A$ is a faithfully flat $A(\alpha)$-module.
###### Proof.
We know that $A(\beta)$ and $A$ are flat as $A(\alpha)$-modules; we also know
that the inclusions $A(\alpha)\hookrightarrow A(\beta)$ and
$A(\alpha)\hookrightarrow A$ split as $A(\alpha)$-module homomorphisms since
$A(\alpha)$ is self-injective.
For an $A(\alpha)$-module $M$, the unit induces a split homomorphism of
$A(\alpha)$-modules $M\to A(\beta)\otimes_{A(\alpha)}M$, showing that
$A(\beta)$ is a faithful $A(\alpha)$-module.
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{A(\alpha)\otimes_{A(\alpha)}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A(\beta)\otimes_{A(\alpha)}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A(\alpha)\otimes_{A(\alpha)}M}$
A similar argument with $A$ in place of $A(\beta)$ shows that $A$ is also
faithful. ∎
Combining Lemma 1.10(c) with Proposition 4.9 we obtain
###### Corollary 4.10.
Let $\alpha,\beta\in\Lambda$ and let $M$ be a left $A(\alpha)$-module. Then
* •
for $\alpha\preccurlyeq\beta$, if $A(\beta)\otimes_{A(\alpha)}M$ is a simple
$A(\beta)$-module then $M$ is simple;
* •
if $A\otimes_{A(\alpha)}M$ is a simple $A$-module then $M$ is simple.
Here are some useful consequences of faithful flatness of $A$.
###### Corollary 4.11.
Let $M$ be a simple coherent $A$-module. Then there is a $\lambda\in\Lambda$
for which $M$ has the form $M\cong A\otimes_{A(\lambda)}M^{\prime}$ for a
simple $A(\lambda)$-module $M^{\prime}$.
###### Proof.
By Proposition 4.2 there is a $\lambda\in\Lambda$ such that $M$ is induced up
from an $A(\lambda)$-module $M^{\prime}$. By Corollary 4.10, $M^{\prime}$ is
simple. ∎
###### Corollary 4.12.
Let $\lambda\in\Lambda$, and let $\varphi\colon M_{\lambda}\to N_{\lambda}$ be
a homomorphism of f.g. left $A(\lambda)$-modules. If
$\operatorname{Id}\otimes\varphi\colon A\otimes_{A(\lambda)}M_{\lambda}\to
A\otimes_{A(\lambda)}N_{\lambda}$ is a monomorphism/an epimorphism/an
isomorphism, then so is $\varphi$.
###### Proof.
See Proposition 1.10(b). ∎
The Frobenius extension condition for each inclusion $A(\lambda)\subseteq
A(\lambda^{\prime})$ ensures that the induction and coinduction functors
$\operatorname{ind}_{A(\lambda)}^{A(\lambda^{\prime})}\colon\mathbf{Mod}_{A(\lambda)}\to\mathbf{Mod}_{A(\lambda^{\prime})}$
and
$\operatorname{coind}_{A(\lambda)}^{A(\lambda^{\prime})}:\mathbf{Mod}_{A(\lambda)}\to\mathbf{Mod}_{A(\lambda^{\prime})}$
are naturally isomorphic, where for a left $A(\lambda)$-module $M$,
$\operatorname{ind}_{A(\lambda)}^{A(\lambda^{\prime})}M=A(\lambda^{\prime})\otimes_{A(\lambda)}M,\quad\operatorname{coind}_{A(\lambda)}^{A(\lambda^{\prime})}M=\operatorname{Hom}_{A(\lambda)}(A(\lambda^{\prime}),M).$
Here we use the right multiplication of $A(\lambda^{\prime})$ on itself to
define the left $A(\lambda^{\prime})$-module structure on
$\operatorname{Hom}_{A(\lambda)}(A(\lambda^{\prime}),M)$. Of course we can
specialise to the case $\lambda=\lambda_{0}$ and $A(\lambda_{0})=\Bbbk$.
For an $A(\lambda)$-module $M$, there is injective composition
$M\xrightarrow{\cong}\operatorname{Hom}_{\Bbbk}(\Bbbk,M)\to\operatorname{Hom}_{\Bbbk}(A(\lambda),M);\quad
x\mapsto(a\mapsto ax)$
which is an $A(\lambda)$-module homomorphism with
$\operatorname{Hom}_{\Bbbk}(A(\lambda),M)\cong\operatorname{coind}_{\Bbbk}^{A(\lambda)}M\cong\operatorname{ind}_{\Bbbk}^{A(\lambda)}M$
being both an injective $A(\lambda)$-module and a free module
$A(\lambda)$-module. Of course if $M$ is f.g. then so is
$\operatorname{ind}_{\Bbbk}^{A(\lambda)}M$.
###### Proposition 4.13.
Let $M$ be a coherent $A$-module. Then there is an embedding of $M$ into a
f.g. free $A$-module.
###### Proof.
We know that $M\cong\operatorname{ind}_{A(\lambda)}^{A}M^{\prime}$ for some
f.g. $A(\lambda)$-module with $\lambda\in\Lambda$. There is also an embedding
of $M^{\prime}$ into a f.g. free $A(\lambda)$-module $F^{\prime}$ say.
Inducing up and using flatness of $A$ over $A(\lambda)$, we obtain an
injection
$M\xrightarrow{\cong}\operatorname{ind}_{A(\lambda)}^{A}M^{\prime}\to\operatorname{ind}_{A(\lambda)}^{A}F^{\prime}=F$
where $F$ is a f.g. free $A$-module. ∎
###### Lemma 4.14.
If $M$ is a coherent $A$-module, then for $s>0$,
$\operatorname{Ext}_{A}^{s}(M,A)=0$. Hence $A$ is injective in the category of
coherent $A$-modules. More generally, this holds if $M$ is a coproduct of
coherent $A$-modules.
###### Proof.
By Proposition 4.2(c), for some $\lambda\in\Lambda$ there is an
$A(\lambda)$-module $M^{\prime}$ such that
$A\otimes_{A(\lambda)}M^{\prime}\cong M$. Then
$\operatorname{Ext}^{*}_{A}(M,A)\cong\operatorname{Ext}^{*}_{A}(A\otimes_{A(\lambda)}M^{\prime},A)\cong\operatorname{Ext}^{*}_{A(\lambda)}(M^{\prime},A).$
Now recall that by Proposition 4.2(a), for $\lambda\in\Lambda$, $A$ is an
injective $A(\lambda)$-module, hence for $s>0$,
$\operatorname{Ext}^{s}_{A(\lambda)}(M^{\prime},A)=0$.
An alternative argument uses Proposition 4.9. A sequence of coherent
$A$-modules
$0\to U\to V$
is induced up from a sequence of $A(\lambda)$-modules for some
$\Lambda\in\Lambda$,
$0\to A\otimes_{A(\lambda)}U^{\prime}\to A\otimes_{A(\lambda)}V^{\prime}$
which is exact if and only if the sequence
$0\to U^{\prime}\to V^{\prime}$
is exact. Now given the solid diagram of $A$-modules
$\textstyle{A\otimes_{A(\lambda)}U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{A\otimes_{A(\lambda)}V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A}$
with exact row, by using the adjunction there is a solid diagram of
$A(\lambda)$-modules
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A}$
with exact row. Since $A$ is a flat $A(\lambda)$-module, it is injective and
it follows that we can complete this diagram with a dashed arrow. Again using
the adjunction, we can complete the original diagram, showing that $A$ is
relatively injective.
For a coproduct of coherent modules, it is standard that
$\operatorname{Ext}_{A}^{s}(-,N)$ sends coproducts to products. ∎
###### Corollary 4.15.
If $M$ is a f.r. $A$-module, then for $s>0$,
$\operatorname{Ext}_{A}^{s}(M,A)=0$.
###### Proof.
By Theorem 1.4(a), such a module has the form $M\cong M_{0}\oplus F$, with
$M_{0}$ f.p. and $F$ free. Then for $s>0$,
$\operatorname{Ext}_{A}^{s}(M,A)\cong\operatorname{Ext}_{A}^{s}(M_{0},A)=0.\qed$
###### Proposition 4.16.
Let $A$ be a locally Frobenius $\Bbbk$-algebra and $M$ an $A$-module.
_(a)_ If $M$ is f.p. then there is an embedding of $M$ into a finite rank free
module.
_(b)_ If $M$ is f.r. then there is an embedding of $M$ into a free module.
###### Proof.
(a) As in the proof of Lemma 4.14, a finite presentation
$A^{m}\to A^{n}\to M\to 0$
is induced up from an exact sequence of $A(\lambda)$-modules
$A(\lambda)^{m}\to A(\lambda)^{n}\to M^{\prime}\to 0$
for some $\lambda\in\Lambda$, where $A\otimes_{A(\lambda)}M^{\prime}\cong M$.
By Proposition 2.4(a), there is a monomorphism $M^{\prime}\to F^{\prime}$
where $F^{\prime}$ is a f.g. free $A(\lambda)$-module. By flatness of $A$ over
$A(\lambda)$, this gives a monomorphism
$M\xrightarrow{\cong}A\otimes_{A(\lambda)}M^{\prime}\to
A\otimes_{A(\lambda)}F^{\prime}=F,$
where $F$ is a f.g. free $A$-module.
(b) By Theorem 1.4(a), $M\cong F\oplus M_{0}$ with $F$ free and $M_{0}$ f.p.,
so the result follows using (a). ∎
We now have a result which completes the proof of Theorem 4.1.
###### Lemma 4.17.
The abelian category $\mathbf{Mod}^{\mathrm{coh}}_{A}$ has enough projectives
and injectives which are the summands of f.g. free modules.
###### Proof.
The existence of projectives is obvious. Lemma 4.14 implies the existence of
injectives in $\mathbf{Mod}^{\mathrm{coh}}_{A}$, and by Proposition 4.16(a),
every coherent module embeds in a f.g. free module which is also injective. ∎
###### Remark 4.18.
Of course we should not expect $A$ to be an injective in $\mathbf{Mod}_{A}$ so
it is only relatively injective with respect to the full subcategory
$\mathbf{Mod}^{\mathrm{coh}}_{A}$; the basic notions of relative homological
algebra can be found in Eilenberg & Moore [SE&JCM:RelHomAlg]. We can use
finitely generated free modules to build projective resolutions in
$\mathbf{Mod}^{\mathrm{coh}}_{A}$ for computing $\operatorname{Ext}_{A}$ since
for a coherent module $M$, $\operatorname{Ext}_{A}^{*}(-,M)$ are isomorphic to
derived functors on $\mathbf{Mod}^{\mathrm{coh}}_{A}$. On the category of all
$A$-modules the left exact functor $\operatorname{Hom}_{A}(M,-)$ has the right
derived functors $\operatorname{Ext}_{A}^{*}(M,-)$. By Lemma 4.14, for each
finitely generated free module $F$, $\operatorname{Ext}_{A}^{s}(M,F)=0$ if
$s>0$. This means that $F$ is $\operatorname{Hom}_{A}(M,-)$-acyclic, and it is
well-known that these right derived functors can be computed using resolutions
by such modules which always exist here; see [CAW:HomAlg] on $F$-acyclic
objects and dimension shifting.
### Injective, projective and flat modules
Lemma 4.14 shows that $A$ is relatively injective with respect to the category
of coherent $A$-modules; it follows that f.g. free and projective modules are
also relatively injective. We can relate flatness to relatively injectivity.
###### Lemma 4.19.
Let $P$ be a flat $A$-module. Then $P$ is relatively injective with respect to
the category of coherent $A$-modules.
###### Proof.
By [TYL:LectModules&Rings]*theorem 4.32, every homomorphism $M\to P$ from a
finitely presented $A$-module factors as
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$
where $F$ is a finitely generated free module. Then given a diagram of solid
arrows with exact row where $U,V$ are coherent,
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$
we can extend it with the dashed arrows using the factorisation above, and
since $F$ is relatively injective there is a dotted arrow making the whole
diagram commute.
This can also be proved using the Lazard-Govorov Theorem 1.5. ∎
Since products of (relative) injectives are (relative) injectives, this is
consistent with Chase’s Theorem 1.9(b) which says that products of flat
modules over a coherent ring are flat.
###### Lemma 4.20.
Let $J$ be a relatively injective $A$-module with respect to the category of
coherent $A$-modules. Then $J$ is flat. In particular, every injective
$A$-module is flat.
###### Proof.
Let $M$ be a coherent $A$-module. By Proposition 4.13 there is a monomorphism
$i\colon M\to F$ where $F$ is a f.g. free module. Now for any homomorphism
$f\colon M\to J$, the diagram of solid arrows with exact row
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J}$
can be extended by a dashed arrow by injectivity. By Lam
[TYL:LectModules&Rings]*theorem 4.32, $J$ is flat. ∎
Here is a summary of what we have established.
###### Proposition 4.21.
Let $M$ be an $A$-module. Consider the following conditions:
* (A)
$M$ is relatively injective for the category of coherent $A$-modules;
* (B)
$M$ is flat;
* (C)
$M$ is projective;
* (D)
$M$ is a product of flat modules.
Then we have the implications shown.
(A)(B)(C)(D)
Furthermore, if $M$ is coherent or more generally p.f.r., then
$\text{\rm(B)}\Longrightarrow\text{\rm(C)}$.
###### Proof.
The equivalence of (A) and (B) follows from Lemmas 4.19 and 4.20. The
implication $\text{(D)}\Longrightarrow\text{(B)}$ follows from Chase’s theorem
[TYL:LectModules&Rings]*theorem 4.47; in particular it applies to the case of
a product of free modules. Actually we don’t really need to use Chase’s
Theorem since a product of (relative) injectives is a (relative) injective.
The last statement follows by Theorem 1.4(a) and Lam
[TYL:LectModules&Rings]*theorem (4.30). ∎
We also mention another result on flat modules whose proof requires Corollary
4.27 which will be given later.
###### Proposition 4.22.
Let $M$ be a f.d. $A$-module and $P$ a flat $A$-module. Then
$\operatorname{Hom}_{A}(M,P)=0.$
###### Proof.
By the Lazard-Govorov Theorem 1.5, $P$ is a filtered colimit of f.g. free
modules $F_{\alpha}$,
$P=\operatorname*{colim}_{\alpha}F_{\alpha}.$
Since $M$ is f.g. and by Corollary 4.27,
$\operatorname{Hom}_{A}(M,P)=\operatorname*{colim}_{\alpha}\operatorname{Hom}_{A}(M,F_{\alpha})=0.\qed$
### Jacobson Radicals
Recall for $\alpha\preccurlyeq\beta$, $A(\alpha)$ is left/right self-injective
and so each inclusion $A(\alpha)\hookrightarrow A(\beta)$ is split as a
left/right $A(\alpha)$-module homomorphism. So by Lam
[TYL:NonCommRings]*proposition 5.6,
(4.2)
$A(\alpha)\cap\operatorname{rad}A(\beta)\subseteq\operatorname{rad}A(\alpha).$
Similarly, the inclusion $A(\alpha)\hookrightarrow A$ is split since $A$ is an
injective $A(\alpha)$-module and
(4.3) $A(\alpha)\cap\operatorname{rad}A\subseteq\operatorname{rad}A(\alpha).$
Since $A(\alpha)$ is Artinian, $\operatorname{rad}A(\alpha)$ is actually
nilpotent and so nil, therefore $\operatorname{rad}A$ is also a nil ideal (in
fact the largest one); see also Lam [TYL:NonCommRings]*proposition 4.19.
If all of the $A(\alpha)$ are semisimple, then for $z\in\operatorname{rad}A$
there must be some $\gamma$ such that $z\in A(\gamma)$, hence
$z\in
A(\gamma)\cap\operatorname{rad}A\subseteq\operatorname{rad}A(\gamma)=\\{0\\}.$
Therefore $\operatorname{rad}A=\\{0\\}$ and $A$ is semiprimitive (or Jacobson
semisimple in the terminology of Lam [TYL:NonCommRings]). In fact since each
$A(\alpha)$ is von Neumann regular, it easily follows that $A$ is too; see Lam
[TYL:NonCommRings]*corollary 4.24. Since $A$ is not Noetherian, it cannot be
semisimple by Lam [TYL:NonCommRings]*corollary 4.25.
For primitive ideals we have the following. If $\operatorname{ann}_{A}(S)$ is
the annihilator of a simple $A$-module $S$ which restricts to a simple
$A(\alpha)$-module for some $\alpha$, then for any $\beta\succcurlyeq\alpha$,
$A(\beta)\cap\operatorname{ann}_{A}(S)=\operatorname{ann}_{A(\beta)}(S).$
###### Proposition 4.23.
Suppose that each $A(\alpha)$ is local with maximal ideal $A(\alpha)^{+}$, so
that each homomorphism $A(\alpha)\to A(\beta)$ is local. Then $A$ is also
local.
###### Proof.
The kernel of the augmentation $\varepsilon\colon A\to\Bbbk$ is the maximal
ideal $A^{+}\lhd A$ for which
$A(\alpha)\cap A^{+}\subseteq A(\alpha)^{+}=\operatorname{rad}A(\alpha).$
Now given $z\in A^{+}$ and $a\in A$ we can assume that $z,a\in A(\gamma)$ for
some $\gamma$ and so $1+az$ has a left inverse in $A(\gamma)\subseteq A$; a
similar argument shows that $1+za$ has a right unit, therefore
$A^{+}=\operatorname{rad}A$. ∎
### Annihilators
We will discuss left annihilators and modules, but similar considerations
apply to right annihilators and modules.
Let $a\in A$. Then the left $A$-module homomorphism
$A\to Aa;\quad x\mapsto xa$
fits into a short exact sequence
$0\to\operatorname{ann}^{\mathrm{l}}_{A}(a)\to A\to Aa\to 0$
where $Aa\subseteq A$ is a f.g. submodule and so f.p., therefore
$\operatorname{ann}^{\mathrm{l}}_{A}(a)$ is also a f.p. module. More generally
we have
###### Proposition 4.24.
Let $M$ be a coherent $A$-module and $m\in M$. Then the left ideal
$\operatorname{ann}_{A}(m)\subseteq A$ is a f.p. submodule, hence it is a
coherent $A$-module.
More generally, if $W\subseteq M$ is a f.d. $\Bbbk$-subspace then its
annihilator $\operatorname{ann}_{A}(W)$ is f.g. as a left ideal, hence
$\operatorname{ann}_{R}(W)$ and $M/\operatorname{ann}_{R}(W)$ are coherent
modules.
###### Proof.
For the first part, see [TYL:LectModules&Rings]*theorem 2.4.58.
If $n_{1},\ldots,n_{k}$ span the $\Bbbk$-vector space $W$, then
$\operatorname{ann}_{R}(W)=\operatorname{ann}_{R}(n_{1})\cap\cdots\cap\operatorname{ann}_{R}(n_{k}).$
By a well-known argument, the intersection of two f.g. submodules of a
coherent module is f.g. and so coherent. Hence, $\operatorname{ann}_{R}(W)$
and $M/\operatorname{ann}_{R}(W)$ are coherent modules. ∎
### Finite and simple modules
Although $A$ will have simple modules, in general they will not all be
coherent. Frobenius algebras are Kasch algebras (i.e., every left/right simple
module is isomorphic to a left/right ideal). The situation for a locally
Frobenius algebra $A$ is less straightforward. For example, the trivial module
$\Bbbk$ can never be isomorphic to a submodule because of the next result
characterising the simple modules which occur as minimal left/right ideals of
$A$.
###### Proposition 4.25.
Let $S$ be a simple $A$-module. Then $S$ is isomorphic to submodule of $A$ if
and only if it is coherent.
###### Proof.
Assume that $S\subseteq A$ and $0\neq s\in S$. Since $As=S$, Proposition 4.24
implies that there is a short exact sequence
$0\to\operatorname{ann}^{\mathrm{l}}_{A}(s)\to A\to S\to 0$
with $\operatorname{ann}^{\mathrm{l}}_{A}(s)$ f.p., so $S$ is coherent.
For the converse, suppose that $S$ is a coherent simple $A$-module. By
Proposition 4.13, there is an embedding $j\colon S\hookrightarrow F$ into a
f.g. free module. At least one of the compositions of $j$ with the projection
onto a copy of $A$ must be non-zero and so a monomorphism by simplicity, hence
$S$ is isomorphic to a submodule of $A$. ∎
We already know that f.d. simple modules are not coherent by Corollary 4.7, so
combining this with Proposition 4.25 we obtain an important consequence.
###### Corollary 4.26.
A f.d. simple module $S$ is not isomorphic to a submodule of $A$, or
equivalently $\operatorname{Hom}_{A}(S,A)=0$. More generally, for a coherent
$A$-module $N$, $\operatorname{Hom}_{A}(S,N)=0$.
In particular this applies to the trivial module $\Bbbk$.
###### Proof.
For the second statement, recall that there is an embedding $N\hookrightarrow
F$ into a finitely generated free module. The short exact sequence
$0\to N\to F\to F/N\to 0$
induces a long exact sequence beginning with
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{A}(S,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{A}(S,F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{A}(S,F/N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{0}$
so $\operatorname{Hom}_{A}(S,M)=0$. ∎
We can extend this result to arbitrary f.d. modules.
###### Corollary 4.27.
Suppose that $M$ is a f.d. $A$-module. Then for any coherent module $N$,
$\operatorname{Hom}_{A}(M,N)=0.$
###### Proof.
This can be proved by induction on the length of a composition series of $M$.
The inductive step uses the fact that for a f.d. simple module $S$, we have
$\operatorname{Hom}_{A}(S,N)=0$ by Propositions 4.25 and 4.13. ∎
We also have the following.
###### Proposition 4.28.
Suppose that $L\subseteq A$ is a minimal left ideal and $0\neq s\in L$. Then
at least one of $L\subseteq A^{+}$ or
$\operatorname{ann}^{\mathrm{l}}_{A}(s)\subseteq A^{+}$ must be true.
Furthermore, if $\operatorname{ann}^{\mathrm{l}}_{A}(s)\subseteq A^{+}$ then
$\dim_{\Bbbk}L=\infty$.
Similar results are true for minimal right ideals.
###### Proof.
By Proposition 4.25, $\operatorname{ann}^{\mathrm{l}}_{A}(s)$ is f.g., and
since $\operatorname{ann}^{\mathrm{l}}_{A}(s)s=0$ we have $s\in A^{+}$ or
$\operatorname{ann}_{A}(s)\subseteq A^{+}$ because $A^{+}$ is completely
prime. By Proposition 4.5, if $\operatorname{ann}_{A}(s)\subseteq A^{+}$ then
$L\cong A/\operatorname{ann}_{A}(s)$ is infinite dimensional. ∎
###### Proposition 4.29.
Suppose that $S$ is a finite dimensional simple $A$-module. Then there is a
$\lambda\in\Lambda$ for which the restriction of $S$ to $A(\lambda)$ is
simple. Hence for $\lambda^{\prime}\in\Lambda$ with
$\lambda\preccurlyeq\lambda^{\prime}$, the restriction of $S$ to
$A(\lambda^{\prime})$ is also simple.
###### Proof.
I would like to thank Ken Brown for the following proof.
The quotient algebra $A/\operatorname{ann}_{A}(S)$ is a finite dimensional
simple $\Bbbk$-algebra with dimension
$\dim_{\Bbbk}A/\operatorname{ann}_{A}(S)=m$ say. Lift a $\Bbbk$-basis of
$A/\operatorname{ann}_{A}(S)$ to a linearly independent set
$E=\\{e_{1},\ldots,e_{m}\\}\subseteq A$ which is contained in some
$A(\lambda)$. We also have the ideal
$J=A(\lambda)\cap\operatorname{ann}_{A}(S)\lhd A(\lambda)$
and the composition
$A(\lambda)\hookrightarrow A\to A/\operatorname{ann}_{A}(S)$
factors through an injective homomorphism $A(\lambda)/J\to
A/\operatorname{ann}_{A}(S)$. Since the images of the elements of $E$ in
$A(\lambda)/J$ are still linearly independent, we have
$\dim_{\Bbbk}(A(\lambda)/J)\geqslant m$; but as
$\dim_{\Bbbk}A(\lambda)/J\leqslant m$ we obtain
$\dim_{\Bbbk}A(\lambda)/J=\dim_{\Bbbk}A/\operatorname{ann}_{A}(S)$. Therefore
we have $A(\lambda)/J\cong A/\operatorname{ann}_{A}(S)$, making a
$A(\lambda)/J$ simple Artinian ring. But we know that $S$ is the (unique)
simple $A/\operatorname{ann}_{A}(S)$-module so it is also simple as an
$A(\lambda)/J$-module and as an $A(\lambda)$-module.
The other statement is clear. ∎
###### Corollary 4.30.
Suppose that $S$ is a f.d. non-coherent simple $A$-module. Then there is an
$\alpha\in\Lambda$ such that
* •
$S$ is a simple $A(\alpha)$-module;
* •
there is an embedding of $A(\alpha)$-modules $S\hookrightarrow A(\alpha)$;
* •
there is a $\beta\succcurlyeq\alpha$ such that the composition
$S\hookrightarrow A(\alpha)\hookrightarrow A(\beta)$ is not a homomorphism of
$A(\beta)$-modules. Therefore there is an $a\in A(\beta)$ such that
$aS\nsubseteq S$ and in particular $aS\setminus S\neq\varnothing$.
###### Proof.
The first statement follows from Proposition 4.29 and the second is a
consequence of a Frobenius algebras being a Kasch algebra. If for every
$\beta\succcurlyeq\alpha$ this composition were a $A(\beta)$-module
homomorphism then we would obtain an $A$-module embedding $S\hookrightarrow
A$, contradicting Proposition 4.25. ∎
Of course there may be infinite dimensional minimal left/right ideals which
satisfy one of the conditions $L^{2}=0$ or $L^{2}=L$. If $L=Az$ and $L^{2}=0$
then $z\in A^{+}$ since this ideal is completely prime.
The next results were suggested by analogous results of Richardson
[JSR:GpRngsnonzerosocle].
###### Lemma 4.31.
Let $L$ be a minimal left/right ideal in $A$. Then there is a
$\lambda\in\Lambda$ for which $L(\lambda)=A(\lambda)\cap L$ is a non-trivial
minimal left/right ideal in $A(\lambda)$ and
$L=AL(\lambda)\cong A\otimes_{A(\lambda)}L(\lambda).$
###### Proof.
We outline the proof for left ideals.
Write $L=Az$ and choose $\lambda\in\Lambda$ so that $z\in A(\lambda)$. Then
$L(\lambda)=A(\lambda)\cap L$ is a non-trivial left ideal in $A(\lambda)$ and
$AL(\lambda)=L$. By the well-known criterion for flatness of
[TYL:LectModules&Rings]*(4.12), multiplication gives an isomorphism
$A\otimes_{A(\lambda)}L(\lambda)\xrightarrow{\cong}AL(\lambda)=L$
and Corollary 4.11 implies that $L(\lambda)$ is simple. ∎
###### Corollary 4.32.
Let $L$ be a minimal left ideal in $A$. Then either $L^{2}=0$ or there is an
idempotent $e\in A$ such that $L=Ae$.
Similarly for a minimal right ideal in $A$, either $L^{2}=0$ or $L$ is
generated by an idempotent.
###### Proof.
If $L^{2}\neq 0$ then suppose $L(\lambda)=A(\lambda)\cap L\neq 0$. In the
Artinan ring $A(\lambda)$, the minimal ideal $L(\lambda)=L(\lambda)^{2}$ is
generated by an idempotent $e$ and $L=Ae$. ∎
Our next result and its proof are direct generalisations of Richardson
[JSR:GpRngsnonzerosocle]*proposition 2.8.
###### Proposition 4.33.
Let $0\neq w\in A$. Then $Aw$ is a minimal left ideal of $A$ if and only if
$wA$ is a minimal right ideal. Hence the left and right socles of $A$ agree.
###### Proof.
If $Aw$ is a minimal left ideal then by Lemma 4.31 there is a
$\lambda\in\Lambda$ such that $w\in A(\lambda)$ and $A(\lambda)\cap Aw$ is a
minimal left ideal in $A(\lambda)$. Since $wA$ is the union of the
$wA(\lambda)$ over all such $\lambda$, it suffices to show that $wA(\lambda)$
is a minimal right ideal. There is an isomorphism of left $A(\lambda)$-modules
$A(\lambda)w\xrightarrow{\cong}A(\lambda)/\operatorname{ann}^{\mathrm{l}}_{A(\lambda)}(w)$
so
$\operatorname{ann}^{\mathrm{l}}_{A(\lambda)}(w)=\operatorname{ann}_{A(\lambda)}(wA(\lambda))$
which is a maximal left ideal. As $A(\lambda)$ is Frobenius its left and right
submodule lattices correspond under an inclusion reversing bijection,
therefore $wA(\lambda)$ is the annihilator ideal of $A(\lambda)w$ which is a
minimal right ideal. ∎
Another result of Richardson [JSR:GpRngsnonzerosocle]*lemma 2.7 (see also
Hartley & Richardson [BH&JSR:SocleGpRngs]) can be extended to a locally
Frobenius algebra. We know that a minimal left ideal of $A$ is a coherent
$A$-module and so infinite dimensional; nevertheless this result gives some
sort of finiteness albeit over a division algebra which is infinite
dimensional over $\Bbbk$.
###### Proposition 4.34.
Let $L=Aw$ be a minimal left ideal of $A$.
_(a)_ The ideal $L$ is finite dimensional over the division algebra
$\operatorname{End}_{A}(L)$.
_(b)_ Every finite subset of $\operatorname{End}_{A}(L)$ lies in a finite
dimensional separable $\Bbbk$-subalgebra.
_(c)_ If $\operatorname{char}\Bbbk>0$, then $\operatorname{End}_{A}(L)$ is a
field.
###### Proof.
This involves a routine reworking of that for locally finite group algebras
and is left to the reader, the main point to notice is that group algebras of
finite subgroups need to be replaced by Frobenius algebras $A(\lambda)$. ∎
For locally Frobenius Hopf algebras our results on f.d. modules should be
compared with the following.
###### Proposition 4.35 (See [ML:TourRepThy]*proposition 10.6).
Suppose that $H$ is a Hopf algebra over $\Bbbk$. If $H$ is infinite
dimensional then it has no non-trivial f.d. left/right ideals.
Combining this with Proposition 4.25 we obtain a result which also follows
from Proposition 4.5.
###### Corollary 4.36.
If $H$ is a locally Frobenius Hopf algebra then no f.d. simple module is
coherent.
### Coherent projective covers
In general f.g. modules over a coherent ring may not have projective covers in
the usual sense. But we can define something similar for coherent modules over
a locally Frobenius algebra $A$.
Consider the coherent $A$-module $A\otimes_{A(\alpha)}M_{\alpha}$ where
$M_{\alpha}$ is a f.g. (hence f.d.) $A(\alpha)$-module. If
$\alpha\preccurlyeq\beta$ then
$M_{\beta}=A(\beta)\otimes_{A(\alpha)}M_{\alpha}$
has dimension
$\dim_{\Bbbk}M_{\beta}=\frac{\dim_{\Bbbk}A(\beta)\dim_{\Bbbk}M_{\alpha}}{\dim_{\Bbbk}A(\alpha)}.$
so
$\frac{\dim_{\Bbbk}M_{\beta}}{\dim_{\Bbbk}A(\beta)}=\frac{\dim_{\Bbbk}M_{\alpha}}{\dim_{\Bbbk}A(\alpha)}.$
By Corollary 4.3, if there is an isomorphism of $A$-modules
$A\otimes_{A(\alpha)}M_{\alpha}\cong A\otimes_{A(\beta)}N_{\beta}$ then for
some large enough $\gamma$ there is an isomorphism of $A(\gamma)$-modules
$M_{\gamma}=A(\gamma)\otimes_{A(\alpha)}M_{\alpha}\cong
A(\gamma)\otimes_{A(\beta)}N_{\beta}=N_{\gamma},$
therefore
$\frac{\dim_{\Bbbk}M_{\gamma}}{\dim_{\Bbbk}A(\gamma)}=\frac{\dim_{\Bbbk}N_{\gamma}}{\dim_{\Bbbk}A(\gamma)}.$
This shows that to any coherent $A$-module we can assign an isomorphism
invariant rational number by taking its _coherent dimension_ to be
$\operatorname{coh-
dim}(A\otimes_{A(\alpha)}M_{\alpha})=\frac{\dim_{\Bbbk}M_{\alpha}}{\dim_{\Bbbk}A(\alpha)}.$
Now let $M$ be a coherent $A$-module. Then there exist epimorphisms of
$A$-modules $P\to M$ where $P$ is a coherent projective module. By Corollary
4.12, such an epimorphism can be taken to be induced up from an epimorphism of
$A(\alpha)$-modules $P_{\alpha}\to M_{\alpha}$ for some $\alpha\in\Lambda$,
where $P_{\alpha}$ is projective; we will refer to such epimorphisms as
_coherent epimorphisms_. Notice also that
$\dim_{\Bbbk}P_{\alpha}\geqslant\dim_{\Bbbk}M_{\alpha}$ so
$\operatorname{coh-
dim}(A\otimes_{A(\alpha)}P_{\alpha})\geqslant\operatorname{coh-
dim}(A\otimes_{A(\alpha)}M_{\alpha}).$
In fact we have the following useful observation. Let $\varphi\colon P\to N$
be a homomorphism of coherent $A$-modules with $P$ coherent projective, and
let $\theta\colon M\to N$ be a coherent epimorphism. Then we can factor
$\varphi$ through $M$.
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\widetilde{\varphi}}$$\textstyle{N}$
By choosing a large enough $\alpha\in\Lambda$ we can find underlying
homomorphisms of f.g. $A(\alpha)$-modules
$\textstyle{M_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\alpha}}$$\textstyle{P_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\alpha}}$$\scriptstyle{\widetilde{\varphi}_{\alpha}}$$\textstyle{N_{\alpha}}$
which on applying $A\otimes_{A(\alpha)}(-)$ induce the diagram of $A$-modules.
We need to check that this second diagram commutes. Since
$\theta_{\alpha}\circ\widetilde{\varphi}_{\alpha}$ and $\varphi_{\alpha}$
induce the same homomorphism $P\to M$, so by Proposition 1.10(a)
$\theta_{\alpha}\circ\widetilde{\varphi}_{\alpha}-\varphi_{\alpha}$ must be
trivial and this diagram also commutes.
Now suppose that we have two coherent epimorphisms
$P\xrightarrow{\varphi}M\xleftarrow{\theta}Q$ where $P$ and $Q$ are coherent
projectives. The preceding discussion shows that we can assume that for some
$\lambda\in\Lambda$ there are there are projective $A(\lambda)$-modules
$P_{\lambda}$ and $Q_{\lambda}$ for which $P\cong
A\otimes_{A(\lambda)}P_{\lambda}$ and $Q\cong
A\otimes_{A(\lambda)}Q_{\lambda}$ together with a f.g. $A(\lambda)$-module
$M_{\lambda}$ for which $M\cong A\otimes_{A(\lambda)}M_{\lambda}$; furthermore
there is a commutative diagram of $A(\lambda)$-modules
$\textstyle{P_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\lambda}}$$\scriptstyle{\rho}$$\textstyle{Q_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\lambda}}$$\scriptstyle{\sigma}$$\textstyle{P_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\lambda}}$$\textstyle{M_{\lambda}}$
where $\varphi_{\lambda}$ and $\theta_{\lambda}$ induce $\varphi$ and
$\theta$. On tensoring with $A$ and using faithful flatness we see that
$\sigma\circ\rho=\operatorname{Id}-{P_{\lambda}}$, so $P_{\lambda}$ is a
summand of $Q_{\lambda}$ and
$\dim_{\Bbbk}P_{\lambda}\geqslant\dim_{\Bbbk}Q_{\lambda}$.
We can now define the _coherent rank of $M$_ to be
$\operatorname{coh-rank}M=\min\\{\operatorname{coh-dim}P:\text{there is a
coherent epi. $P\to M$}\\}\geqslant\operatorname{coh-dim}(M).$
If we realise this minimum with a coherent epimorphism $P\to M$, then for any
other such coherent epimorphism $Q\to M$ the above discussion shows that
$\operatorname{coh-dim}P=\operatorname{coh-dim}Q$ and so $P\cong Q$. Thus we
can regard $P\to M$ as a _coherent projective cover_.
### Pseudo-coherent modules
In Bourbaki [Bourbaki:HomAlg]*ex. §3.10, as well as coherent modules, pseudo-
coherent modules are considered, where a module is _pseudo-coherent_ if every
finitely generated submodule is finitely presented (so over a coherent ring it
is coherent). The reader is warned that pseudo-coherent is sometimes used in a
different sense; for example, over a coherent ring the definition of Weibel
[CAW:Ktheory]*example II.7.1.4 corresponds to our coherent.
Examples of pseudo-coherent modules over a coherent ring $A$ include
coproducts of coherent modules such as the f.r. modules of Definition 1.1.
However pseudo-coherent modules do not form a full abelian subcategory of
$\mathbf{Mod}_{A}$ since for example cokernels of homomorphisms between
pseudo-coherent modules need not be pseudo-coherent. Nevertheless, pseudo-
coherent modules do occur quite commonly when working with coherent modules
over locally Frobenius algebras.
###### Lemma 4.37.
Let $A$ be a ring and $M$ a pseudo-coherent $A$-module. Then $M$ is the union
of its coherent submodules and therefore their colimit.
###### Proof.
Every element $m\in M$ generates a cyclic submodule which is coherent. ∎
###### Lemma 4.38.
Let $A$ be a coherent ring and $B$ an Artinian subring where $A$ is flat as a
right $B$-module. Then every extended $A$-module $A\otimes_{B}N$ is pseudo-
coherent.
###### Proof.
Let $U\subseteq A\otimes_{B}N$ be a finitely generated submodule. Taking a
finite generating set and expressing each element as a sum of basic tensors we
find that $U\subseteq A\otimes_{B}U^{\prime}$ where $U^{\prime}\subseteq N$ is
a finitely generated submodule. As $B$ is Artinian and so Noetherian,
$U^{\prime}$ is finitely presented, so by flatness of $A$,
$A\otimes_{B}U^{\prime}$ is a finitely presented $A$-module. Since $A$ is
coherent, $U$ is also finitely presented. ∎
An important special case of this occurs when $A$ is a coherent algebra over a
field $\Bbbk$ and $B$ is a finite dimensional subalgebra. If $N$ is a
$B$-module then it is locally finite, i.e., every element is contained in a
finite dimensional submodule.
We will discuss pseudo-coherence for modules over a locally Frobenius Hopf
algebra in Section 6.
The following stronger notions of pseudo-coherence are perhaps more likely to
be important for example for computational purposes.
###### Definition 4.39.
* •
A module over a coherent ring is _strongly pseudo-coherent_ if it is a
coproduct of coherent modules.
* •
A module $M$ over a locally Frobenius algebra $A$ indexed on $\Lambda$ is
_$\lambda$ -strongly pseudo-coherent_ for $\lambda\in\Lambda$ if
$M\cong A\otimes_{A(\lambda)}M^{\prime}$
where the $A(\lambda)$-module $M^{\prime}$ is a coproduct of finitely
generated $A(\lambda)$-modules.
Over a coherent ring every free module is strongly pseudo-coherent as is every
finitely related module since it is the sum of a coherent module and a free
module. Clearly a $\lambda$-strongly pseudo-coherent module over a locally
Frobenius algebra $A$ is strongly pseudo-coherent.
### Cohomology for finite dimensional $A$-modules
To end this section we discuss $\operatorname{Ext}$ groups for finite
dimensional modules. We will work with left modules, but a similar discussion
applies to right modules.
If $W$ is an $A(\lambda)$-module for some $\lambda\in\Lambda$ and $N$ is an
$A$-module (hence an $A(\lambda)$-module via restriction), then
$A\otimes_{A(\lambda)}W$ is an $A$-module and
$\operatorname{Hom}_{A(\lambda)}(W,N)\cong\operatorname{Hom}_{A(\lambda)}(A\otimes_{A(\lambda)}W,N).$
More generally, since $A$ is $A(\lambda)$-flat,
$\operatorname{Ext}^{s}_{A(\lambda)}(W,N)\cong\operatorname{Ext}^{s}_{A}(A\otimes_{A(\lambda)}W,N).$
If $M,N$ are $A$-modules viewed as $A(\lambda)$-modules through restriction,
then
$\operatorname*{colim}_{(\Lambda,\preccurlyeq)}A\otimes_{A(\lambda)}M\cong M$
as $A$-modules and
(4.4)
$\operatorname{Hom}_{A}(M,N)\cong\operatorname{Hom}_{A}(\operatorname*{colim}_{(\Lambda,\preccurlyeq)}A\otimes_{A(\lambda)}M,N)\cong\lim_{(\Lambda,\preccurlyeq)}\operatorname{Hom}_{A(\lambda)}(M,N).$
There is a spectral sequence of Jensen [LNM254]*théorème 4.2 of the form
$\mathrm{E}_{2}^{s,t}={\lim_{(\Lambda,\preccurlyeq)}}^{\\!s}\operatorname{Ext}^{t}_{A}(A\otimes_{A(\lambda)}M,N)\Longrightarrow\operatorname{Ext}^{s+t}_{A}(M,N),$
where $\displaystyle{\lim_{(\Lambda,\preccurlyeq)}}^{\\!s}$ is the $s$-th
right derived functor of $\displaystyle\lim_{(\Lambda,\preccurlyeq)}$. The
$\mathrm{E}_{2}$-term can be rewritten to give
(4.5)
$\mathrm{E}_{2}^{s,t}={\lim_{(\Lambda,\preccurlyeq)}}^{\\!s}\operatorname{Ext}^{t}_{A(\lambda)}(M,N)\Longrightarrow\operatorname{Ext}^{s+t}_{A}(M,N).$
When $\Lambda$ is countable, by a result of Jensen [LNM254], for $s>1$,
$\displaystyle{\lim_{\lambda\in\Lambda}}^{\\!s}$ is trivial, so for each
$n\geqslant 1$ there is an exact sequence
(4.6)
$0\to{\lim_{(\Lambda,\preccurlyeq)}}^{\\!1}\operatorname{Ext}^{n-1}_{A(\lambda)}(M,N)\to\operatorname{Ext}^{n}_{A}(M,N)\to\lim_{(\Lambda,\preccurlyeq)}\operatorname{Ext}^{n}_{A(\lambda)}(M,N).\to
0$
Suppose that $J$ is an injective $A$-module and
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V}$$\textstyle{J}$
is a diagram of $A(\lambda)$-modules with an exact row. By flatness of $A$ as
an $A(\lambda)$-module,
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{A(\lambda)}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{A(\lambda)}V}$
is an exact sequence of $A$-modules and on applying
$\operatorname{Hom}_{A(\lambda)}(-,J)$ we obtain a commutative diagram
$\textstyle{\operatorname{Hom}_{A(\lambda)}(U,J)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{Hom}_{A(\lambda)}(V,J)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{Hom}_{A}(A\otimes_{A(\lambda)}U,J)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{Hom}_{A}(A\otimes_{A(\lambda)}V,J)}$
where the lower row is exact by injectivity of the $A$-module $J$. It follows
that $J$ is an injective $A(\lambda)$-module.
Now taking $N=J$ and using (4.6) with $n=1$ we obtain
${\lim_{(\Lambda,\preccurlyeq)}}^{\\!1}\operatorname{Hom}_{A(\lambda)}(M,J)=0,$
and by using (4.4),
$\operatorname{Hom}_{A}(M,J)\cong\lim_{(\Lambda,\preccurlyeq)}\operatorname{Hom}_{A(\lambda)}(M,J).$
We know that $A$ is relatively injective with respect to coherent $A$-modules.
For a finite dimensional non-coherent $A$-module $M$, the spectral sequence
(4.5) gives
$\mathrm{E}_{2}^{s,t}={\lim_{(\Lambda,\preccurlyeq)}}^{\\!s}\operatorname{Ext}^{t}_{A(\lambda)}(M,A)\Longrightarrow\operatorname{Ext}^{s+t}_{A}(M,A)$
and as $A$ is $A(\lambda)$-injective for every $\lambda$, while if $t=0$,
$\mathrm{E}_{2}^{s,t}=\begin{dcases*}{\lim_{(\Lambda,\preccurlyeq)}}^{\\!s}\operatorname{Hom}_{A(\lambda)}(M,A)&if
$t=0$,\\\ \quad\quad\quad\quad\quad 0&if $t>0$.\end{dcases*}$
So we have
$\operatorname{Ext}^{n}_{A}(M,A)\cong{\lim_{(\Lambda,\preccurlyeq)}}^{\\!n}\operatorname{Hom}_{A(\lambda)}(M,A).$
In particular, if $\Lambda$ is countable, $\operatorname{Ext}^{n}_{A}(M,A)=0$
when $n>1$.
###### Definition 4.40.
If $S$ is a finite dimensional simple $A$-module then $A$ is _$S$
-Margolisian_ if $\operatorname{Ext}^{n}_{A}(S,A)=0$ for all $n>0$.
If $A$ is $S$-Margolisian for all finite dimensional simple $A$-modules then
$A$ is _Margolisian_.
So $A$ is Margolisian if for every finite dimensional non-coherent simple
$A$-module $S$ and every $s>0$,
${\lim_{(\Lambda,\preccurlyeq)}}^{\\!s}\operatorname{Hom}_{A(\lambda)}(M,A)=0.$
We can summarise this in a result which is obtained by combining these ideas
with what we already know for coherent finite dimensional $A$-modules.
###### Proposition 4.41.
$A$ is Margolisian if and only if $A$ is relatively injective with respect to
the abelian category $\mathbf{Mod}_{A}^{\mathrm{f.d.}}$ of all finite
dimensional $A$-modules.
Margolis [HRM:Book] shows that every $P$-algebra is $\Bbbk$-Margolisian in our
sense. In this case $\Bbbk$ is the only simple module. His proof makes
essential use of the fact that its modules are graded and we have neither been
able to find an argument that works in our situation nor a counter example, so
it seems possible that every locally Frobenius algebra is Margolisian, or at
least $\Bbbk$-Margolisian.
###### Remark 4.42.
As in Remark 4.18, for a f.d. $A$-module $M$, the left exact functor
$\operatorname{Hom}_{A}(M,-)$ on the category of all $A$-modules has the right
derived functors $\operatorname{Ext}_{A}^{*}(M,-)$. If $A$ is Margolisian then
for each finitely generated free module $F$,
$\operatorname{Ext}_{A}^{*}(M,F)=0$ and $F$ is
$\operatorname{Hom}_{A}(M,-)$-acyclic, so these right derived functors can be
computed using resolutions by such modules which are the injectives in
$\mathbf{Mod}_{A}^{\mathrm{coh}}$. So they are also right derived functors on
$\mathbf{Mod}_{A}^{\mathrm{coh}}$. But then
$\operatorname{Ext}_{A}^{*}(M,N)=0$ for every coherent module $N$.
We now discuss the finite dual of a locally Frobenius $\Bbbk$-algebra $A$ and
some related ideas. As we could not find a convenient reference we give
details which are probably well known.
Recall that the _finite dual_ of $A$ is
$A^{\circ}=\\{f\in\operatorname{Hom}_{\Bbbk}(A,\Bbbk):\text{$\exists I\lhd A$
s.t. $I$ is cofinite and $I\subseteq\ker
f$}\\}\subseteq\operatorname{Hom}_{\Bbbk}(A,\Bbbk).$
Here $I$ is _cofinite_ if $\dim_{\Bbbk}A/I<\infty$. It is clear that
$A^{\circ}\subseteq\operatorname{Hom}_{\Bbbk}(A,\Bbbk)$ is a $\Bbbk$-subspace.
There are two $A$-module structures on $\operatorname{Hom}_{\Bbbk}(A,\Bbbk)$
namely a left one induced by right multiplication on the domain and a right
one induced by left multiplication on the domain.
###### Lemma 4.43.
The left and right $A$-module structures on $A^{\circ}$ each restrict to make
$A^{\circ}$ a submodule of $\operatorname{Hom}_{\Bbbk}(A,\Bbbk)$.
Furthermore, if $M$ is a f.d. $A$-module the image of every $A$-module
homomorphism $M\to\operatorname{Hom}_{\Bbbk}(A,\Bbbk)$ is contained in
$A^{\circ}$.
###### Proof.
We give the argument for the left module structure.
Let $f\in A^{0}$ be trivial on a cofinite ideal $I\lhd A$. If $a\in A$ and
$x\in I$ then
$(af)(x)=f(xa)=0$
since $xa\in I$.
Since $M$ is f.d., the associated algebra homomorphism
$A\to\operatorname{End}_{\Bbbk}(M)$ has a cofinite kernel $J\lhd A$ say. Now
for an $A$-module homomorphism $M\to\operatorname{Hom}_{\Bbbk}(A,\Bbbk)$, the
image of each $m\in M$ must vanish on $J$. ∎
If $W$ is a f.d. vector space then we can similarly define
$\operatorname{Hom}_{\Bbbk}^{\mathrm{fin}}(A,W)=\\{f\in\operatorname{Hom}_{\Bbbk}(A,W):\text{$\exists
I\lhd A$ s.t. $I$ is cofinite and $I\subseteq\ker
f$}\\}\subseteq\operatorname{Hom}_{\Bbbk}(A,W).$
Then the obvious left and right $A$-module structures on
$\operatorname{Hom}_{\Bbbk}(A,W)$ also restrict to make
$\operatorname{Hom}_{\Bbbk}^{\mathrm{fin}}(A,W)$ a left or right submodule. If
we choose a basis for $W$ we obtain isomorphisms of left and right $A$-modules
$\operatorname{Hom}_{\Bbbk}^{\mathrm{fin}}(A,W)\cong(A^{\circ})^{\dim_{\Bbbk}W}.$
An analogue of this is also true when $W$ is infinite dimensional.
It is standard that injectives in the module category $\mathbf{Mod}_{A}$ are
summands of modules of the form $\operatorname{Hom}_{\Bbbk}(A,W)$. In
particular every $A$-module $M$ admits an embedding
$M\hookrightarrow\operatorname{Hom}_{\Bbbk}(A,M)$ where the copy of $M$ in the
codomain is viewed just as a vector space.
###### Lemma 4.44.
Suppose that $W$ is a f.d. vector space. Then
$\operatorname{Hom}_{\Bbbk}^{\mathrm{fin}}(A,W)$ is a relative injective with
respect to the category of f.d. $A$-modules. Furthermore every f.d. $A$-module
admits a resolution by such relative injectives.
###### Proof.
It suffices to show this for $A^{\circ}$ itself.
Suppose we have the solid diagram of $A$-modules
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{\Bbbk}(A,\Bbbk)}$
where $U$ and $V$ are f.d. and the row is exact. Because
$\operatorname{Hom}_{\Bbbk}(A,\Bbbk)$ is a genuine injective $A$-module there
is a dotted arrow making the diagram commute. By the second part of Lemma 4.43
its image is contained in $A^{\circ}$ so we obtain a dashed arrow making the
diagram commute. ∎
This means that to calculate $\operatorname{Ext}_{A}^{*}(M,N)$ where $M$ and
$N$ are f.d., we can use a resolution of $N$ by $A$-modules of the form
$\operatorname{Hom}_{\Bbbk}^{\mathrm{fin}}(A,W)$. Of course this is not a
resolution in $\mathbf{Mod}_{A}^{\mathrm{f.d.}}$, but it is in the category of
_locally finite_ $A$-modules $\mathbf{Mod}_{A}^{\mathrm{l.f.}}$ which is a
full abelian subcategory of $\mathbf{Mod}_{A}$.
## 5\. Stable module categories for locally Frobenius algebras
For an arbitrary ring there are two distinct ways to define a stable module
category starting with the category of modules, namely by treating as trivial
those homomorphisms which factor through either projectives or injectives; for
a Frobenius algebra these coincide. A locally Frobenius algebra $A$ is not
self-injective, but it is injective relative to certain types of $A$-modules,
in particular coherent modules and f.r. modules. So starting with the abelian
category of coherent $A$-modules we obtain a stable module category using
either approach. However, there are grounds for thinking that f.r. or pseudo-
coherent modules should also be included although neither form an abelian
category in an obvious way.
We adopt some standard notions for stable module categories for the stable
module category of coherent $A$-modules which we will denote by
$\mathbf{Stmod}^{\mathrm{coh}}_{A}$. We will also use the notation
$\\{M,N\\}=\mathbf{Stmod}^{\mathrm{coh}}_{A}(M,N)=\mathbf{Mod}^{\mathrm{coh}}_{A}(M,N)/\approx$
for the morphisms, where for two homomorphisms $f,g\colon M\to N$, $f\approx
g$ if and only if there is a factorisation
$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f-g}$$\textstyle{N}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
where $F$ is a f.g. free module. For a coherent module $M$, the objects
$\Omega M$ and $\mho M$ are defined by taking exact sequences
$P\xrightarrow{p}M\to 0,\quad 0\to M\xrightarrow{j}J$
where $P$ and $J$ are f.g. free and hence injective modules. Using Schanuel’s
Lemma, $\ker p$ and $\operatorname{coker}j$ are well-defined up to isomorphism
in $\mathbf{Stmod}^{\mathrm{coh}}_{A}$ so we may use these as representatives
of $\Omega M$ and $\mho M$.
###### Lemma 5.1.
In $\mathbf{Stmod}^{\mathrm{coh}}_{A}$, every object $M$ is isomorphic to
objects of the form $\Omega M^{\prime}$ and $\mho M^{\prime\prime}$ for some
coherent modules $M^{\prime}$ and $M^{\prime\prime}$. Then $\Omega$ and $\mho$
define mutually inverse endofunctors of $\mathbf{Stmod}^{\mathrm{coh}}_{A}$.
###### Proof.
By Proposition 4.16(a) we know that $M$ is a submodule of a f.g. free module
$F$ so there is an exact sequence
$0\to M\to F\to M^{\prime}\to 0$
showing that $M=\Omega M^{\prime}$. On the other hand there is also an
epimorphism $F^{\prime}\to M$ for some f.g. free module $F^{\prime}$ and by
Lemma 4.14 this is injective, hence there is an exact sequence
$0\to M^{\prime\prime}\to F^{\prime}\to M\to 0$
and therefore $M=\mho M^{\prime\prime}$. This also shows that
$M^{\prime\prime}\cong\Omega M$ and so $M\cong\mho\Omega M$. A similar
argument shows that $M\cong\Omega\mho M$. ∎
###### Proposition 5.2.
The functors
$\Omega,\mho\colon\mathbf{Stmod}_{A}^{\mathrm{coh}}\to\mathbf{Stmod}_{A}^{\mathrm{coh}}$
satisfy the following for all coherent $A$-modules $M$ and $N$ and these
isomorphisms are natural:
$\\{\Omega M,\Omega N\\}\cong\\{M,N\\}\cong\\{\mho M,\mho N\\}.$
We extend $\\{-,-\\}$ to a graded $\mathbb{Z}$-bifunctor by setting
$\\{M,N\\}^{n}=\\{M,N\\}_{-n}=\begin{dcases*}\\{M,\mho^{n}N\\}&if $n\geqslant
0$,\\\ \\{\Omega^{-n}M,N\\}&if $n<0$.\end{dcases*}$
This leads to a triangulated structure on $\mathbf{Stmod}_{A}^{\mathrm{coh}}$.
Of course the point of this definition is that when $n>0$,
$\operatorname{Ext}_{A}^{n}(M,N)\cong\\{M,N\\}^{n}.$
A major defect of this stable module category in the case when $A$ is a
locally Frobenius Hopf algebra is its lack of an obvious monoidal structure:
the tensor product $M\otimes_{\Bbbk}N$ of two coherent modules with the
diagonal action need not be coherent as an $A$-module. This problem disappears
if one of the factors is instead a f.d. module. We will discuss this further
in Section 6.
## 6\. Locally Frobenius Hopf algebras
In this section we consider results on locally Frobenius Hopf algebras where
the Hopf structure plays a rôle. We are particularly motivated by the goal of
extending results on locally finite group algebras. A convenient source for
background material on Hopf algebras is provided by Montgomery
[SM:HopfAlgActions].
###### Assumption 6.1.
We will assume that $H$ is a Hopf algebra over the field $\Bbbk$ with
coproduct $\psi\colon H\to H\otimes H$ and antipode $\chi\colon H\to H$, where
will use the notation for these:
$\psi(h)=\sum_{i}h^{\prime}_{i}\otimes
h_{i}^{\prime\prime},\quad\overline{h}=\chi(h).$
We will also assume that $K\subseteq H$ is a subHopf algebra with $H$ being
flat as a left and right $K$-module.
Now given two left $H$-modules $L$ and $M$, their tensor product $L\otimes M$
is a left $H$-module with the diagonal action given by
$h\cdot(\ell\otimes m)=\sum_{i}h^{\prime}_{i}\ell\otimes
h^{\prime\prime}_{i}m,$
where the coproduct on $h$ is
$\psi h=\sum_{i}h^{\prime}_{i}\otimes h^{\prime\prime}_{i}.$
In particular, given a left $H$-module $M$ and a left $K$-module $N$, the
tensor product of $M$ and $H\otimes_{K}N$ is a left $H$-module
$M\otimes(H\otimes_{K}N)$. There is an isomorphism of $H$-modules
(6.1) $M\otimes(H\otimes_{K}N)\xrightarrow[\cong]{\Theta}H\otimes_{K}(M\otimes
N);\quad m\otimes(h\otimes
n)\mapsto\sum_{i}h^{\prime}_{i}\otimes(\overline{h^{\prime\prime}_{i}}m\otimes
n)$
where $\overline{x}=\chi(x)$ and $M\otimes N$ is a left $K$-module with the
diagonal action.
A particular instance of this is
(6.2) $(H/\\!/K)\otimes(H/\\!/K)\cong H\otimes_{K}(H/\\!/KB)$
where
$H/\\!/K=H/HK^{+}\cong H\otimes_{K}\Bbbk.$
###### Lemma 6.2.
Suppose that $H$ is coherent and $K$ is finite dimensional. Let $M$ be a left
$H$-module and $N$ a left $K$-module. Then the $H$-module
$M\otimes(H\otimes_{K}N)$ is pseudo-coherent.
###### Proof.
Using (6.1),
$M\otimes(H\otimes_{K}N)\cong H\otimes_{K}(M\otimes N)$
which is pseudo-coherent by Lemma 4.38. ∎
In general, the tensor product of two coherent modules over a locally
Frobenius Hopf algebra is not a coherent module. However, it turns out that it
is pseudo-coherent.
###### Proposition 6.3.
Suppose that $H$ is a locally Frobenius Hopf algebra and that $L$ and $M$ are
two coherent left $H$-modules. Then the $H$-module $L\otimes M$ is pseudo-
coherent.
###### Proof.
Every coherent $H$-module is induced from a finitely generated
$H(\lambda)$-module for some $\lambda$. By choosing a large enough $\lambda$
we can assume that $L\cong H\otimes_{H(\lambda)}L^{\prime}$ for some finitely
generated $H(\lambda)$-module $L^{\prime}$. Then as $H$-modules,
$L\otimes M\cong H\otimes_{H(\lambda)}(L^{\prime}\otimes M),$
which is a pseudo-coherent $H$-module. ∎
If $H$ is a locally Frobenius Hopf algebra, then for $\lambda$, the cyclic
$H$-module
$H/\\!/H(\lambda)=H/HH(\lambda)^{+}\cong H\otimes_{H(\lambda)}\Bbbk$
can be viewed as an $H(\lambda)$-module, and by (6.2) there is an isomorphism
of $H$-modules
$H/\\!/H(\lambda)\otimes H/\\!/H(\lambda)\cong
H\otimes_{H(\lambda)}H/\\!/H(\lambda).$
So if $H/\\!/H(\lambda)$ is a coproduct of f.g./f.d. $H(\lambda)$-modules then
$H/\\!/H(\lambda)\otimes H/\\!/H(\lambda)$ is $\lambda$-strongly pseudo-
coherent $H$-module.
Now we consider _normality_ for subHopf algebras of locally Frobenius Hopf
algebras.
###### Definition 6.4.
The left and right _adjoint actions_ of $h\in H$ on $x\in H$ are given by
$h\cdot x=\sum_{i}h^{\prime}_{i}x\overline{h_{i}^{\prime\prime}},\quad x\cdot
h=\sum_{i}\overline{h^{\prime}_{i}}xh_{i}^{\prime\prime}.$
When $H=\Bbbk G$ is a group algebra then if $g\in G$, the adjoint action
agrees with conjugation by $g$.
The adjoint action of $H$ on itself is compatible with the product
$\varphi\colon H\otimes H\to H$ and coproduct $\psi\colon H\to H\otimes H$
provided $H\otimes H$ is given the diagonal coactions for which
$h\cdot(x\otimes y)=\sum_{i}h^{\prime}_{i}\cdot x\otimes
h^{\prime\prime}_{i}\cdot y,\quad(x\otimes y)\cdot h=\sum_{i}x\cdot
h^{\prime}_{i}\otimes y\cdot h^{\prime\prime}_{i}.$
In the following definition, the notion of _strongly normalised_ is motivated
by the case of the group algebra of a locally finite group $G$. If a finite
subgroup $H\leqslant G$ normalises a subgroup $K$, then for any finite
subgroup $K^{\prime}\leqslant K$, the finite set
$\bigcup_{h\in H}hK^{\prime}h^{-1}\subseteq K$
is contained in some finite subgroup $K^{\prime}\leqslant K$. This has
implications for the image of the adjoint action of $\Bbbk H$ on $\Bbbk
K^{\prime}$ which is contained in $\Bbbk K^{\prime\prime}$. For a locally
Frobenius Hopf algebra our definition specialises to this case.
###### Definition 6.5.
A sub(Hopf) algebra $K\subseteq H$ is _normal_ if for all $h\in H$, $h\cdot
K\subseteq K$ and $K\cdot h\subseteq K$.
If $H^{\prime}\subseteq H$ is a subHopf algebra, then $K$ is _normalised by
$H^{\prime}$_ if for every $h\in H^{\prime}$, $h\cdot K\subseteq K$ and
$K\cdot h\subseteq K$.
If $K\subseteq H$ is a locally Frobenius subHopf algebra of shape
$\Lambda^{\prime}\subseteq\Lambda$ then $K$ is _strongly normalised_ by a
subHopf algebra $H^{\prime}\subseteq H$ if for all $h\in H^{\prime}$ and
$\lambda\in\Lambda^{\prime}$, there is a $\lambda_{h}\in\Lambda^{\prime}$ such
that
$h\cdot K(\lambda)\subseteq K(\lambda_{h}),\quad K(\lambda)\cdot h\subseteq
K(\lambda_{h}).$
Notice that in the strongly normalised case, if $H^{\prime}$ is f.d., then
$\sum_{h\in H^{\prime}}h\cdot K(\lambda)$
is a f.d. subspace so is contained in some $K(\widetilde{\lambda}_{h})$.
Here is an omnibus proposition combining results found in Montgomery
[SM:HopfAlgActions]*section 3.4
###### Proposition 6.6.
Let $K\subseteq H$ be a subHopf algebra.
_(a)_ Let $K$ be normal. Then $HK^{+}=K^{+}H$ and this is a Hopf ideal in $H$;
furthermore, the quotient homomorphism $H\to H/HK^{+}$ is a morphism of Hopf
algebras.
_(b)_ If $H$ is faithfully flat as a left/right $K$-module and
$HK^{+}=K^{+}H$, then $K$ is normal.
_(c)_ If $H$ is finite dimensional then $K$ is normal if and only if
$HK^{+}=K^{+}H$.
The next result is probably standard but we do not know a convenient
reference.
###### Proposition 6.7.
Suppose that $H$ is a Hopf algebra over a field and that $K,L$ are subHopf
algebras where $K$ is normalised by $L$. Then $KL=LK$ and this is a subHopf
algebra of $H$.
###### Proof.
This is similar to the proof of Proposition 6.6(a). The key point is that for
$k\in K$ and $\ell\in L$, using the definition of the counit and
coassociativity we have,
$\displaystyle\ell k$
$\displaystyle=\sum_{i}\ell^{\prime}_{i}k\varepsilon(\ell^{\prime\prime}_{i})$
$\displaystyle=\sum_{i}\ell^{\prime}_{i}\cdot k\ell^{\prime\prime}_{i}\in KL$
so $LK\subseteq KL$. Similarly, $KL\subseteq LK$. ∎
###### Proposition 6.8.
Let $H$ be a locally Frobenius Hopf algebra of shape $\Lambda$ and let
$K\subseteq H$ be a locally Frobenius subHopf algebra if shape
$\Lambda^{\prime}\subseteq\Lambda$. Suppose that $L\subseteq H$ is a finite
dimensional subHopf algebra and $K$ is normalised by $L$. Then for any finite
dimensional subHopf algebra $K^{\prime}\subseteq K$ there is a
$\lambda\in\Lambda$ such that $H(\lambda)$ contains $K^{\prime}$ and $L$;
furthermore $K\cap H(\lambda)$ is normalised by $L$ and $L(K\cap
H(\lambda))=(K\cap H(\lambda))L\subseteq H(\lambda)$ is a finite dimensional
subHopf algebra containing $K^{\prime}$ and $L$.
###### Proof.
This is a straightforward consequence of Lemma 3.3 and Proposition 6.7. ∎
### Locally finite $H$-modules and $H^{\circ}$-comodules
For a locally Frobenius Hopf algebra $H$ the finite dual $H^{\circ}$ is also a
Hopf algebra. It is standard that the categories of locally finite $H$-modules
$\mathbf{Mod}_{H}^{\mathrm{l.f.}}$ and left $H^{\circ}$-comodules
$\mathbf{Comod}_{H^{\circ}}$ are equivalent (in fact isomorphic). The latter
category has as injectives summands of extended comodules $H^{\circ}\otimes
W$, so when viewed as locally finite $H$-modules these are the injectives of
$\mathbf{Mod}_{H}^{\mathrm{l.f.}}$. It is also well known that
$\mathbf{Comod}_{H^{\circ}}$ and $\mathbf{Mod}_{H}^{\mathrm{l.f.}}$ lack
projectives so only the right derived functors of
$\operatorname{Hom}_{H}(M,-)\cong\operatorname{Cohom}_{H^{\circ}}(M,-)=\mathbf{Comod}_{H^{\circ}}(M,-)$
are defined and $\operatorname{Coext}_{H^{\circ}}(-,-)$ is not a balanced
functor. We study the graded analogue of this for $P$-algebras in
[AB:Palgebras].
Of course the existence of injectives makes it possible to define a stable
module category using these. The tensor product of two locally finite
$H$-modules is also locally finite so $\mathbf{Mod}_{H}^{\mathrm{l.f.}}$ has a
monoidal structure and this passes to the stable module category since for any
$H^{\circ}$-comodule $M$, $H^{\circ}\otimes M$ is isomorphic to an extended
comodule.
## 7\. Some examples of locally Frobenius Hopf algebras
In this section we decribe some examples which occur in the literature whose
understanding might be aided by viewing them as locally Frobenius algebras but
we leave detailed investigation for future work.
### 7.1. Group algebras of locally finite groups
Recall that a countable group $G$ is _locally finite_ if every finite subset
$S\subseteq G$ is contained in a finite subgroup. The group algebras of such
groups have been studied in the literature, for example in the work of
Hartley, Richardson and Musson [BH&JSR:SocleGpRngs, JSR:GpRngsnonzerosocle,
IMM:InjModGpAlgLOcFinGps].
###### Proposition 7.1.
Let $\Bbbk$ be a field and $G$ a locally finite group. Then $\Bbbk G$ is a
locally Frobenius Hopf algebra.
###### Proof.
We take the indexing set $\lambda$ to be the set of finite subgroups ordered
by inclusion and for each $H\in\lambda$ take the finite group algebras $\Bbbk
H$ which is a subHopf algebra of $\Bbbk G$.
A finite dimensional subspace $V\subseteq\Bbbk G$ has a basis whose elements
are expressible as linear combinations of finitely many elements of $G$,
therefore it must be contained in some finite subgroup $H\leqslant G$, so
$V\subseteq\Bbbk H$.
For two finite subgroups $H\leqslant K\leqslant G$, there are left and right
integrals
$\sum_{h\in H}h\in{\smallint}_{\Bbbk H},\quad\sum_{k\in
K}h\in{\smallint}_{\Bbbk K}.$
Then taking a complete set of right coset representatives
$k_{1},\ldots,k_{|K:H|}$ for $K/H$ we have
$\biggl{(}\Bbbk_{1}+\cdots+k_{|K:H|}\biggr{)}\sum_{h\in H}h=\sum_{k\in K}k,$
so ${\smallint}_{\Bbbk K}\subseteq\Bbbk K{\smallint}_{\Bbbk H}$. The condition
of Lorenz [ML:TourRepThy]* 12.4.1(a) shows that $\Bbbk K:\Bbbk H$ is a
Frobenius extension. ∎
### 7.2. Dual profinite group algebras
Let $G$ be a profinite group,
$G=\lim_{\begin{subarray}{c}N\lhd G\\\ |G:N|<\infty\end{subarray}}G/N.$
For a field $\Bbbk$ the pro-group algebra
$\Bbbk G=\lim_{\begin{subarray}{c}N\lhd G\\\ |G:N|<\infty\end{subarray}}\Bbbk
G/N$
is a complete topological Hopf algebra. However, there is also dual object
$\Bbbk(G)=\operatorname*{colim}_{\begin{subarray}{c}N\lhd G\\\
|G:N|<\infty\end{subarray}}\Bbbk(G/N)$
where
$\Bbbk(G/N)=\operatorname{Map}(G/N,\Bbbk)\cong\operatorname{Hom}_{\Bbbk}(\Bbbk
G/N,\Bbbk)$
is the dual group ring. Alternatively, $\Bbbk(G)$ agrees with the algebra of
locally constant (continuous) functions $G\to\Bbbk$ with respect to the
profinite topology. Since each $\Bbbk(G/N)$ is a commutative Hopf algebra so
is $\Bbbk(G)$. In fact, $\Bbbk(G)$ also agrees with the finite dual of $\Bbbk
G$, commonly denoted by $(\Bbbk G)^{\mathrm{o}}$.
###### Proposition 7.2.
For a profinite group $G$, $\Bbbk(G)$ is a commutative locally Frobenius Hopf
algebra.
###### Proof.
We need to check that for cofinite normal subgroups $M\lhd G$ and $N\lhd G$
with $M\lhd N$, $\Bbbk(G/M):\Bbbk(G/N)$ is a Frobenius extension.
The functions $\delta_{1_{G/M}}\in\Bbbk(G/M)$ and
$\delta_{1_{G/N}}\in\Bbbk(G/N)$ with
$\displaystyle\delta_{1_{G/M}}(gM)$ $\displaystyle=\begin{dcases*}1&if $g\in
M$,\\\ 0&otherwise,\end{dcases*}$ $\displaystyle\delta_{1_{G/N}}(gN)$
$\displaystyle=\begin{dcases*}1&if $g\in N$,\\\ 0&otherwise,\end{dcases*}$
are integrals for $\Bbbk(G/M)$ and $\Bbbk(G/N)$. The quotient homomorphism
$\pi\colon G/M\to G/N$ satisfies
$\pi^{*}\delta_{1_{G/N}}=\sum_{gM\in\ker\pi}\delta_{gM}$
and
$\delta_{1_{G/M}}\pi^{*}\delta_{1_{G/N}}=\delta_{1_{G/M}},$
so the condition of Lorenz [ML:TourRepThy]* 12.4.1(a) shows that
$\Bbbk(G/M):\Bbbk(G/N)$ is a Frobenius extension. ∎
## References
|
# Metric approximation of set-valued functions of bounded variation by
integral operators
Elena E. Berdysheva 111 University of Cape Town, South Africa, Nira Dyn
222Tel-Aviv University, School of Mathematical Sciences, Elza Farkhi
22footnotemark: 2 Alona Mokhov 444Afeka, Tel-Aviv Academic College of
Engineering
Abstract. We introduce an adaptation of integral approximation operators to
set-valued functions (SVFs, multifunctions), mapping a compact interval
$[a,b]$ into the space of compact non-empty subsets of ${\mathbb{R}}^{d}$. All
operators are adapted by replacing the Riemann integral for real-valued
functions by the weighted metric integral for SVFs of bounded variation with
compact graphs. For such a set-valued function $F$, we obtain pointwise error
estimates for sequences of integral operators at points of continuity, leading
to convergence at such points to $F$. At points of discontinuity of $F$, we
derive estimates, which yield the convergence to a set, first described in our
previous work on the metric Fourier operator. Our analysis uses recently
defined one-sided local quasi-moduli at points of discontinuity and several
notions of local Lipschitz property at points of continuity.
We also provide a global approach for error bounds. A multifunction $F$ is
represented by the set of all its metric selections, while its approximation
(its image under the operator) is represented by the set of images of these
metric selections under the operator. A bound on the Hausdorff distance
between these two sets of single-valued functions in $L^{1}$ provides our
global estimates.
The theory is illustrated by presenting the examples of two concrete
operators: the Bernstein-Durrmeyer operator and the Kantorovich operator.
Key words: Set-valued functions, functions of bounded variation, metric
integral, metric approximation, integral operators, positive linear operators,
rate of convergence
Mathematics Subject Classification 2020: 26E25, 28B20, 41A35, 41A36, 41A25,
26A45
## 1 Introduction
We study set-valued functions (SVFs, multifunctions) that map a compact
interval $[a,b]\subset{\mathbb{R}}$ into the space of compact non-empty
subsets of ${\mathbb{R}}^{d}$. These functions appear in different fields such
as dynamical systems, control theory, optimization, game theory, differential
inclusions, economy, geometric modeling. See the book [2] for foundations of
set-valued analysis.
Approximation methods for SVFs has been developing in the last decades. Older
works deal mostly with convex-valued multifunctions and their approximation
based on Minkowski linear combinations, e.g., [24, 13, 15, 23, 5, 9]. In [28],
such an adapation of the classical Bernstein polynomial operator is proved to
converge to SVFs with convex compact images (values). Yet, it is shown that
this adaptation fails to approximate general SVFs (with general compact not
necessarily convex images). In general, approximation methods developed for
multifunctions with convex images usually are not suitable for general SVFs.
A first successful attempt to approximate general SVFs from their samples is
accomplished by Z. Artstein in [1], where piecewise linear approximants are
constructed. This is done by replacing binary Minkowski average of two sets
with the metric average, which is further extended in [16] to the metric
linear combination of several sets. Based on the metric linear combination, N.
Dyn, E. Farkhi and A. Mokhov developed in a series of works [14, 20, 16, 17,
18] adaptation of classical sample-based approximation operators to continuous
general SVFs. For these adapted operators, termed metric operators, error
estimates are obtained, which for most operators are similar to those obtained
in the real-valued case. Special attention is given to Bernstein polynomial
operators, Schoenberg spline operators and polynomial interpolation operators.
Later in [6], the above metric approach is extended to SVFs of bounded
variation.
The metric approach is applied in [19] to introduce and study the metric
integral for general SVFs of bounded variation. The metric integral is not
necessarily convex in contrast to the Aumann integral, which is always convex,
even if the integrand is not convex-valued [3]. In [7] the metric integral is
extended to the weighted metric integral, which is used, with the Dirichlet
kernels as weight functions, to define metric Fourier partial sums for SVFs of
bounded variation. The convergence of these partial sums is analyzed at points
of continuity of a multifunction as well as at points of discontinuity. An
important tool in the analysis at points of discontinuity is the notion of
one-sided local quasi-moduli of a function of bounded variation.
In this paper we adapt integral approximation operators for real-valued
functions to general SVFs of bounded variation. Previous adaptations of
integral operators to SVFs are limited to convex-valued multifunctions and are
based on the Aumann integral (see e.g. [4, 10]). Our adaptation is based on
the weighted metric integral, and its analysis applies and extends the
techniques developed in [7].
The outline of the paper is as follows. Section 2 gives a short overview of
notions we use in the paper, and also discusses different regularity
properties of functions with values in a metric space. In Section 3 we refine
known results concerning approximation of real-valued functions by sequences
of integral approximation operators, which are necessary for the adaptation of
these operators to SVFs.
The core part of the paper is Section 4 where we construct an adaptation of
integral approximation operators to general SVFs. For set-valued functions of
bounded variation with compact graphs, we study pointwise convergence, in the
Hausdorff metric, of sequences of such operators at points of continuity of
the function as well as at points of discontinuity, and derive estimates for
the rate of convergence. In Section 5 we illustrate our theory by considering
examples of two particular integral approximation operators, the Bernstein-
Durrmeyer operator and the Kantorovich operator.
In the final Section 6 we provide global error bounds. The multifunction $F$
is represented by the set of all its metric selections (see [18] for more
information on representations of SVFs), while its approximation (its image
under the operator) is represented by the set of images of these metric
selections under the operator. A bound of the Hausdorff distance between these
two sets of single-valued functions in $L^{1}$ is obtained using results from
[8].
## 2 Preliminaries
In this section we introduce some notation and basic notions related to sets
and set-valued functions. We discuss notions of regularity of functions in
metric spaces. We review the notions of metric selections and the metric
integral of set-valued functions.
### 2.1 On sets
All sets considered from now on are sets in ${{\mathbb{R}}}^{d}$. We denote by
$\mathrm{K}({{\mathbb{R}}}^{d})$ the collection of all compact non-empty
subsets of ${{\mathbb{R}}}^{d}$. The metric in ${{\mathbb{R}}}^{d}$ is of the
form $\rho(u,v)=|u-v|$, where $|\cdot|$ is any fixed norm on
${{\mathbb{R}}}^{d}$. Recall that ${{\mathbb{R}}}^{d}$ endowed with this
metric is a complete metric space and that all norms on ${{\mathbb{R}}}^{d}$
are equivalent.
To measure the distance between two non-empty sets
${A,B\in\mathrm{K}({{\mathbb{R}}}^{d})}$, we use the Hausdorff metric based on
$\rho$
$\mathrm{haus}(A,B)_{\rho}=\max\left\\{\sup_{a\in
A}\mathrm{dist}(a,B)_{\rho},\;\sup_{b\in
B}\mathrm{dist}(b,A)_{\rho}\right\\},$
where the distance from a point $c$ to a set $D$ is
$\mathrm{dist}(c,D)_{\rho}=\inf_{d\in D}\rho(c,d)$. It is well known that
$\mathrm{K}({{\mathbb{R}}}^{d})$ endowed with the Hausdorff metric is a
complete metric space [25, 27].
In the following, we keep the metric in ${{\mathbb{R}}}^{d}$ fixed, and omit
the notation $\rho$ as a subscript.
We denote by $|A|=\mathrm{haus}(A,\\{0\\})$ the “norm” of the set
$A\in\mathrm{K}({{\mathbb{R}}}^{d})$. The set of projections of
$a\in{{\mathbb{R}}}^{d}$ on a set $B\in\mathrm{K}({{\mathbb{R}}}^{d})$ is
$\Pi_{B}{(a)}=\\{b\in B\ :\ |a-b|=\mathrm{dist}(a,B)\\},$ and the set of
metric pairs of two sets $A,B\in\mathrm{K}({{\mathbb{R}}}^{d})$ is
$\Pi\big{(}{A},{B}\big{)}=\\{(a,b)\in A\times B\ :\
a\in\Pi_{A}{(b)}\;\,\mbox{or}\;\,b\in\Pi_{B}{(a)}\\}.$
Using metric pairs, we can rewrite
$\mathrm{haus}(A,B)=\max\\{|a-b|\ :\ (a,b)\in\Pi\big{(}{A},{B}\big{)}\\}.$
We recall the notions of a metric chain and of a metric linear combination
[19].
###### Definition 2.1.
[19] Given a finite sequence of sets
$A_{0},\ldots,A_{n}\in\mathrm{K}({{\mathbb{R}}}^{d})$, $n\geq 1$, a metric
chain of $A_{0},\ldots,A_{n}$ is an $(n+1)$-tuple $(a_{0},\ldots,a_{n})$ such
that $(a_{i},a_{i+1})\in\Pi\big{(}{A_{i}},{A_{i+1}}\big{)}$,
$i=0,1,\ldots,n-1$. The collection of all metric chains of
$A_{0},\ldots,A_{n}$ is denoted by
${\mathrm{CH}}(A_{0},\ldots,A_{n})=\left\\{(a_{0},\ldots,a_{n})\ :\
(a_{i},a_{i+1})\in\Pi\big{(}{A_{i}},{A_{i+1}}\big{)},\
i=0,1,\ldots,n-1\right\\}.$
The metric linear combination of the sets
$A_{0},\ldots,A_{n}\in\mathrm{K}({{\mathbb{R}}}^{d})$, $n\geq 1$, is
$\bigoplus_{i=0}^{n}\lambda_{i}A_{i}=\left\\{\sum_{i=0}^{n}\lambda_{i}a_{i}\
:\
(a_{0},\ldots,a_{n})\in{\mathrm{CH}}(A_{0},\ldots,A_{n})\right\\},\quad\lambda_{0},\ldots,\lambda_{n}\in{\mathbb{R}}.$
###### Remark 2.2.
For any $j\in{\mathbb{N}}$, $0\leq j\leq n$ and for any $a\in A_{j}$ there
exists a metric chain
$(a_{0},\ldots,a_{n})\in{\mathrm{CH}}(A_{0},\ldots,A_{n})$ such that
$a_{j}=a$. For a possible construction see [16], Figure 3.2.
Note that the metric linear combination depends on the order of the sets, in
contrast to the Minkowski linear combination of sets which is defined by
$\sum_{i=0}^{n}\lambda_{i}A_{i}=\left\\{\sum_{i=0}^{n}\lambda_{i}a_{i}\ :\
a_{i}\in A_{i}\right\\},\quad n\geq 1.$
The upper Kuratowski limit of a sequence of sets $\\{A_{n}\\}_{n=1}^{\infty}$
is the set of all limit points of converging subsequences
$\\{a_{n_{k}}\\}_{k=1}^{\infty}$, where ${a_{n_{k}}\in A_{n_{k}}}$,
$k\in{\mathbb{N}}$, namely
$\limsup_{n\to\infty}A_{n}=\left\\{a\ :\
\exists\,\\{n_{k}\\}_{k=1}^{\infty},\,n_{k+1}>n_{k},\,k\in{\mathbb{N}},\
\exists\,a_{n_{k}}\in A_{n_{k}}\text{ such that
}\lim_{k\to\infty}a_{n_{k}}=a\right\\}.$ (1)
### 2.2 Notions of regularity of functions with values in a metric space
In this paper we consider functions defined on a fixed compact interval
$[a,b]\subset{\mathbb{R}}$ with values in a complete metric space $(X,\rho)$,
where $X$ is either ${{\mathbb{R}}}^{d}$ or $\mathrm{K}({{\mathbb{R}}}^{d})$.
We recall the notion of the variation of ${f:[a,b]\rightarrow X}$. Let
${\,\chi=\\{x_{0},\ldots,x_{n}\\}}$, $a=x_{0}<x_{1}<\cdots<x_{n}=b$, be a
partition of the interval $[a,b]$ with the norm
${\displaystyle|\chi|=\max_{0\leq i\leq n-1}(x_{i+1}-x_{i})}.$
The variation of $f$ on the partition $\chi$ is defined as
$V(f,\chi)=\sum_{i=1}^{n}\rho(f(x_{i}),f(x_{i-1}))\,.$ The total variation of
$f$ on $[a,b]$ is
$V_{a}^{b}(f)=\sup_{\chi}V(f,\chi),$
where the supremum is taken over all partitions of $[a,b]$.
A function $f$ is said to be of bounded variation on $[a,b]$ if
${V_{a}^{b}(f)<\infty}$. We call functions of bounded variation BV functions
and write $f\in\mathrm{BV}[a,b]$. If $f$ is also continuous, we write
$f\in\mathrm{CBV}[a,b]$.
For $f\in\mathrm{BV}[a,b]$ the function $v_{f}:[a,b]\rightarrow{\mathbb{R}}$,
$v_{f}(x)=V_{a}^{x}(f)$ is called the variation function of $f$. Note that
$V_{z}^{x}(f)=v_{f}(x)-v_{f}(z)\quad\mbox{for}\quad a\leq z<x\leq b,$
and that $v_{f}$ is monotone non-decreasing.
For a BV function ${f:{\mathbb{R}}\rightarrow X}$ the following property holds
(see e.g. Lemma 2.4 in [6]),
$\int_{a}^{b}V_{x-\delta}^{x+\delta}(f)dx\leq 2\delta V_{a}^{b}(f).$ (2)
We recall the notion of the local modulus of continuity [26], which is central
to the approximation of functions at continuity points.
For $f:[a,b]\to X$ the local modulus of continuity at $x^{*}\in[a,b]$ is
$\omega\big{(}{f},{x^{*}},{\delta}\big{)}=\sup\left\\{\,\rho(f(x_{1}),f(x_{2})):\;x_{1},x_{2}\in\left[x^{*}-\delta/2,x^{*}+\delta/2\right]\cap[a,b]\,\right\\},\quad\delta>0.$
It follows from the definition of the variation that
###### Result 2.3.
For a function $f:[a,b]\to X$, $f\in\mathrm{BV}[a,b]$,
$\omega(f,x^{*},\delta)\leq\omega(v_{f},x^{*},\delta),\quad
x^{*}\in[a,b],\quad\delta>0.$
Moreover, $f$ is continuous at $x^{*}\in[a,b]$ if and only if $v_{f}$ is
continuous at $x^{*}$.
###### Result 2.4.
A function $f:[a,b]\to X$, $f\in\mathrm{BV}[a,b]$ is left continuous at
$x^{*}\in(a,b]$ if and only if $v_{f}$ is left continuous at $x^{*}$. The
function $f$ is right continuous at $x^{*}\in[a,b)$ if and only if $v_{f}$ is
right continuous at $x^{*}$.
A function $f:[a,b]\to X$ of bounded variation with values in a complete
metric space $(X,\rho)$ is not necessarily continuous, but has right and left
limits at any point $x$ [11]. We denote the one-sided limits by
$f(x+)=\lim_{t\to x+0}f(t),\quad f(x-)=\lim_{t\to x-0}f(t).$
In [7], we introduced the notion of the left and right local quasi-moduli. For
a function $f:[a,b]\to X$ of bounded variation, the left local quasi-modulus
at point $x^{*}$ is
$\varpi^{-}\big{(}{f},{x^{*}},{\delta}\big{)}=\sup{\big{\\{}\rho(f(x^{*}-),f(x))\
:\ x\in[x^{*}-\delta,x^{*})\cap[a,b]\big{\\}}},\quad\delta>0\
,\;x^{*}\in(a,b].$ (3)
Similarly, the right local quasi-modulus is
$\varpi^{+}\big{(}{f},{x^{*}},{\delta}\big{)}=\sup{\\{\rho(f(x^{*}+),f(x))\ :\
x\in(x^{*},x^{*}+\delta]\cap[a,b]\\}},\quad\delta>0\ ,\;x^{*}\in[a,b).$ (4)
Clearly, for $f\in\mathrm{BV}[a,b]$ the local quasi-moduli satisfy
$\lim_{\delta\to 0^{+}}\varpi^{-}\big{(}{f},{x^{*}},{\delta}\big{)}=0,\quad
x^{*}\in(a,b],\quad\text{and}\quad\lim_{\delta\to
0^{+}}\varpi^{+}\big{(}{f},{x^{*}},{\delta}\big{)}=0,\quad x^{*}\in[a,b).$
Defining
$\varpi\big{(}{f},{x^{*}},{\delta}\big{)}=\max\\{\varpi^{-}\big{(}{f},{x^{*}},{\delta}\big{)},\varpi^{+}\big{(}{f},{x^{*}},{\delta}\big{)}\\},$
we obtain
$\lim_{\delta\to 0^{+}}\varpi\big{(}{f},{x^{*}},{\delta}\big{)}=0.$ (5)
###### Lemma 2.5.
Let $f:[a,b]\to X$, $f\in\mathrm{BV}[a,b]$, then for any $x^{*}\in(a,b]$ or
$[a,b)$, respectively, and $\delta>0$ we have
$\varpi^{-}\big{(}{f},{x^{*}},{\delta}\big{)}\leq\varpi^{-}\big{(}{v_{f}},{x^{*}},{\delta}\big{)},\quad\varpi^{+}\big{(}{f},{x^{*}},{\delta}\big{)}\leq\varpi^{+}\big{(}{v_{f}},{x^{*}},{\delta}\big{)}.$
###### Proof.
The first inequality follows from the fact that for $x^{*}\in(a,b]$ and
$\max\\{x^{*}-\delta,a\\}\leq x<x^{*}$ we have
$\rho(f(x^{*}-),f(x))\leq\lim_{t\to x^{*}-0}V_{x}^{t}(f)=\lim_{t\to
x^{*}-0}v_{f}(t)-v_{f}(x)=\rho(v_{f}(x^{*}-),v_{f}(x)).$
Similarly one can show the second inequality.
∎
Below we discuss several notions of Lipschitz regularity. A functions
$f:[a,b]\to(X,\rho)$ is Lipschitz continuous with a constant $\mathcal{L}>0$,
if
$\rho(f(x),f(y))\leq{\mathcal{L}}|x-y|,\quad\forall\,x,y\in[a,b].$
###### Definition 2.6.
Let $f:[a,b]\to(X,\rho)$.
* (a)
We say that $f$ is locally Lipschitz around a point $x$ with the Lipschitz
constant $\mathcal{L}>0$ if there exists $\delta>0$ such that
$\rho(f(x_{1}),f(x_{2}))\leq{\mathcal{L}}|x_{1}-x_{2}|,\quad\forall\,x_{1},x_{2}\in\left(x-\delta/2,x+\delta/2\right)\cap[a,b].$
(6)
* (b)
A function $f$ is locally Lipschitz at a point $x$ with the Lipschitz constant
$\mathcal{L}>0$ if there exists $\delta>0$ such that
$\rho(f(z),f(x))\leq{\mathcal{L}}|z-x|,\quad\forall\,z\in\left(x-\delta/2,x+\delta/2\right)\cap[a,b].$
(7)
We denote by $\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$ the collection of all
functions $f$ satisfying (7).
* (c)
A function $f$ is globally Lipschitz at a point $x$ if there exists
$\mathcal{L}>0$ such that
$\rho(f(z),f(x))\leq{\mathcal{L}}|z-x|,\quad\forall\,z\in[a,b].$ (8)
###### Remark 2.7.
Note that if $f$ is locally Lipschitz around a point $x$ with the Lipschitz
constant $\mathcal{L}$, then $f\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$, but
the inverse implication does not hold. For example, the function
$f(x)=x\sin(1/x)$ for $x\neq 0$ and $f(0)=0$ is not locally Lipschitz around
$x=0$, while $f\in\mathrm{Lip}\\{{0},{1}\\}$.
We say that $f:[a,b]\to X$ is bounded on $[a,b]$ if there exists $y^{*}\in X$
such that
$M(f,y^{*})=\sup_{x\in[a,b]}\rho(f(x),y^{*})<\infty.$
The following lemmas deal with relations between the above notions.
###### Lemma 2.8.
If $f\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$ and $f$ is bounded on $[a,b]$,
then $f$ is also globally Lipschitz at $x$.
###### Proof.
By Definition 2.6 (b), there is $\delta>0$ such that for
$|z-x|<\frac{\delta}{2}$ we have $\rho(f(z),f(x))\leq\mathcal{L}|z-x|$. Also,
since $f$ is bounded there is $y^{*}$ such that
$\rho(f(z),f(x))\leq\rho(f(z),y^{*})+\rho(y^{*},f(x))\leq 2M(f,y^{*}).$
If $|z-x|\geq\frac{\delta}{2}$ we obtain from the above inequality the
estimate $\rho(f(z),f(x))\leq\frac{4M(f,y^{*})}{\delta}|z-x|$.
Altogether (8) holds with
$\widetilde{\mathcal{L}}=\max\left(\mathcal{L},\frac{4M(f,y^{*})}{\delta}\right)$.
∎
###### Lemma 2.9.
If $f\in\mathrm{BV}[a,b]$ is locally Lipschitz around a point $x$ with a
constant $\mathcal{L}>0$, then
1. (i)
$v_{f}$ is locally Lipschitz around the point $x$ with the same constant
$\mathcal{L}$,
2. (ii)
$v_{f}\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$.
###### Proof.
Assume that $f$ satisfies (6) with some $\delta>0$. To prove (i), let
$x-\frac{\delta}{2}<y<z<x+\frac{\delta}{2}$, then
$v_{f}(z)-v_{f}(y)=V_{y}^{z}(f)=\sup_{\chi}\sum_{j=0}^{n-1}\rho(f(x_{j}),f(x_{j+1}))\leq\sup_{\chi}\sum_{j=0}^{n-1}\mathcal{L}|x_{j+1}-x_{j}|=\mathcal{L}|y-z|,$
where the supremum is taken over all partitions
${\chi=\\{y=x_{0}<\cdots<x_{n}=z\\}}$ of $[y,z]$.
By (i) and Remark 2.7 we obtain (ii). ∎
###### Lemma 2.10.
Let $f\in\mathrm{BV}[a,b]$.
1. (i)
If $v_{f}$ is locally Lipschitz around a point $x$ with some $\mathcal{L}>0$
and $\delta>0$, then $f$ is locally Lipschitz around $x$ with the same
$\mathcal{L}$ and $\delta$.
2. (ii)
If $v_{f}\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$, then
$f\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$.
###### Proof.
(i) For all $z,y$ such that ${x-\frac{\delta}{2}<y<z<x+\frac{\delta}{2}}$ we
have
$\rho(f(z),f(y))\leq V_{y}^{z}(f)=v_{f}(z)-v_{f}(y)\leq\mathcal{L}|z-y|\,.$
(9)
To prove (ii) we replace either $y$ or $z$ in (9) by $x$.
∎
###### Remark 2.11.
The inverse implication of Lemma 2.10 (ii) does not hold. Consider, for
example, the function $f(x)=x\sin(1/x)$ for $x\neq 0$ and $f(0)=0$. Clearly,
$f\in\mathrm{Lip}\\{{0},{\mathcal{L}}\\}$ with $\mathcal{L}=1$, but
$v_{f}\notin\mathrm{Lip}\\{{0},{\mathcal{L}}\\}$ for any $\mathcal{L}>0$.
In the following we consider ${f:[a,b]\rightarrow{{\mathbb{R}}}^{d}}$,
$f=\begin{pmatrix}f_{1}\\\ \vdots\\\ f_{d}\end{pmatrix}$. We recall that
$|\cdot|$ is a fixed norm on ${{\mathbb{R}}}^{d}$.
###### Lemma 2.12.
Let ${f:[a,b]\rightarrow{{\mathbb{R}}}^{d}}$ be Riemann integrable. Then
$\left|\int_{a}^{b}f(x)dx\right|\leq\int_{a}^{b}|f(x)|dx.$
###### Proof.
Consider a sequence of partitions
$\chi_{n}=\\{a=x_{0}^{n}<\cdots<x_{n}^{n}=b\\}$ with
$\displaystyle\lim_{n\to\infty}|\chi_{n}|=0$. Take some
$c_{i}^{n}\in[x_{i-1}^{n},x_{i}^{n}]$, $i=1,\ldots,n$. Using the triangle
inequality in the estimate for the Riemann sums we obtain
$\left|\int_{a}^{b}f(x)dx\right|=\left|\lim_{n\to\infty}\sum_{i=1}^{n}f(c_{i}^{n})(x_{i}^{n}-x_{i-1}^{n})\right|\leq\lim_{n\to\infty}\sum_{i=1}^{n}|f(c_{i}^{n})|(x_{i}^{n}-x_{i-1}^{n})=\int_{a}^{b}|f(x)|dx.$
∎
We recall the notion of integral moduli of continuity for functions with
values in ${{\mathbb{R}}}^{d}$. We extend a function
${f:[a,b]\rightarrow{{\mathbb{R}}}^{d}}$ outside $[a,b]$ in a simple way
preserving its variation on $[a,b]$
$f(x)=\left\\{\begin{array}[]{ll}f(a),&x<a,\\\ f(x),&a\leq x\leq b,\\\
f(b),&x>b.\end{array}\right.$
For ${f:[a,b]\rightarrow{{\mathbb{R}}}^{d}}$, $f=\begin{pmatrix}f_{1}\\\
\vdots\\\ f_{d}\end{pmatrix}$, we write $f\in L^{1}[a,b]$ if all its
components $f_{i}$, $i=1,\ldots,d$, are in $L^{1}[a,b]$.
The distance between two functions $f,g\in L^{1}[a,b]$ is given by
$\|f-g\|_{L^{1}}=\int_{a}^{b}|f(x)-g(x)|dx.$
We denote $\displaystyle\|f\|_{\infty}=\sup_{x\in[a,b]}|f(x)|$.
The first order integral modulus of continuity of
${f:[a,b]\rightarrow{{\mathbb{R}}}^{d}}$, $f\in L^{1}[a,b]$ is
$\vartheta(f,\delta)=\vartheta_{1}(f,\delta)=\sup_{0<h\leq\delta}\int_{a}^{b}|f(x+h)-f(x)|dx,\quad\delta>0.$
The second order integral modulus of continuity is
$\vartheta_{2}(f,\delta)=\sup_{0<h\leq\delta}\int_{a}^{b}|f(x+h)-2f(x)+f(x-h)|dx,\quad\delta>0.$
It is easy to see that
$\vartheta_{2}(f,\delta)\leq 2\vartheta(f,\delta).$ (10)
Using (2) one can easily obtain that for $f\in\mathrm{BV}[a,b]$
$\vartheta\big{(}{f},{\delta}\big{)}\leq\delta V_{a}^{b}(f).$ (11)
### 2.3 Metric selections and the weighted metric integral of multifunctions
We consider set-valued functions (SVFs, multifunctions) mapping a compact
interval $[a,b]\subset{\mathbb{R}}$ to $\mathrm{K}({{\mathbb{R}}}^{d})$. The
graph of a multifunction $F$ is the set of points in ${\mathbb{R}}^{d+1}$
${\mathrm{Graph}}(F)=\left\\{(x,y)\ :\ y\in F(x),\;x\in[a,b]\right\\}.$
It is easy to see that if $F\in\mathrm{BV}[a,b]$ then ${\mathrm{Graph}}(F)$ is
a bounded set and
$\|F\|_{\infty}=\sup_{x\in[a,b]}|F(x)|<\infty.$
We denote the class of SVFs of bounded variation with compact graphs by
$\mathcal{F}[a,b]$.
For a set-valued function $F:[a,b]\to\mathrm{K}({{\mathbb{R}}}^{d})$, a
single-valued function ${s:[a,b]\to{{\mathbb{R}}}^{d}}$ such that $s(x)\in
F(x)$ for all $x\in[a,b]$ is called a selection of $F$.
The notions of the metric selections and of the weighted metric integral are
central in our work. We recall their definitions.
Given a multifunction $F:[a,b]\to\mathrm{K}({{\mathbb{R}}}^{d})$, a partition
$\chi=\\{x_{0},\ldots,x_{n}\\}\subset[a,b]$, $a=x_{0}<\cdots<x_{n}=b$, and a
corresponding metric chain
$\phi=(y_{0},\ldots,y_{n})\in{\mathrm{CH}}\left(F(x_{0}),\ldots,F(x_{n})\right)$
(see Definition 2.1), the chain function based on $\chi$ and $\phi$ is
$c_{\chi,\phi}(x)=\left\\{\begin{array}[]{ll}y_{i},&x\in[x_{i},x_{i+1}),\quad
i=0,\ldots,n-1,\\\ y_{n},&x=x_{n}.\end{array}\right.$ (12)
A selection $s$ of $F$ is called a metric selection, if there is a sequence of
chain functions $\\{c_{\chi_{k},\phi_{k}}\\}_{k\in{\mathbb{N}}}$ of $F$ with
${\lim_{k\to\infty}|\chi_{k}|=0}$ such that
$s(x)=\lim_{k\to\infty}c_{\chi_{k},\phi_{k}}(x)\quad\mbox{pointwisely on}\
[a,b].$
We denote the set of all metric selections of $F$ by $\mathcal{S}(F)$.
Note that the definitions of chain functions and metric selections imply that
a metric selection $s$ of a multifunction $F$ is constant in any open interval
where the graph of $s$ stays in the interior of ${\mathrm{Graph}}(F)$.
Below we quote some results from [19] and [7] which are used in this paper.
###### Result 2.13.
[19, Theorem 3.6]
Let $s$ be a metric selection of $F\in\mathcal{F}[a,b]$. Then
$V_{a}^{b}(s)\leq V_{a}^{b}(F)$ and $\|s\|_{\infty}\leq\|F\|_{\infty}$.
###### Result 2.14.
[19, Corollary 3.7]
Let $F\in\mathcal{F}[a,b]$. Through any point $\alpha\in{\mathrm{Graph}}(F)$
there exists a metric selection which we denote by ${\,s_{\alpha}}$. Moreover,
$F$ has a representation by metric selections, namely
$F(x)=\\{s_{\alpha}(x)\ :\ \alpha\in{\mathrm{Graph}}(F)\\}.$
###### Result 2.15.
[7, Theorem 4.9]
Let $F\in\mathcal{F}[a,b]$, $s$ be a metric selection of $F$ and
$x^{*}\in[a,b]$. Then
$\omega\big{(}{s},{x^{*}},{\delta}\big{)}\leq\omega\big{(}{v_{F}},{x^{*}},{2\delta}\big{)},\quad\delta>0.$
In particular, if $F$ is continuous at $x^{*}$, then $s$ is continuous at
$x^{*}$.
###### Result 2.16.
[7, Theorem 4.13]
For $F\in\mathcal{F}[a,b]$, the pointwise limit (if exists) of a sequence of
metric selections of $F$ is a metric selection of $F$.
###### Result 2.17.
[7, Lemmas 6.7, 6.8]
Let $F\in\mathcal{F}[a,b]$ and $\delta>0$. Then any metric selection
$s\in\mathcal{S}(F)$ satisfies
$\varpi^{-}\big{(}{v_{s}},{x^{*}},{\delta}\big{)}\leq\varpi^{-}\big{(}{v_{F}},{x^{*}},{2\delta}\big{)},\;x^{*}\in(a,b],\qquad\varpi^{+}\big{(}{v_{s}},{x^{*}},{\delta}\big{)}\leq\varpi^{+}\big{(}{v_{F}},{x^{*}},{\delta}\big{)},\;x^{*}\in[a,b).$
A direct consequence of Lemma 2.5 and Result 2.17 is
###### Corollary 2.18.
Let $F\in\mathcal{F}[a,b]$, $s\in\mathcal{S}(F)$ and $\delta>0$. Then
$\varpi^{-}\big{(}{s},{x^{*}},{\delta}\big{)}\leq\varpi^{-}\big{(}{v_{F}},{x^{*}},{2\delta}\big{)},\;x^{*}\in(a,b],\qquad\varpi^{+}\big{(}{s},{x^{*}},{\delta}\big{)}\leq\varpi^{+}\big{(}{v_{F}},{x^{*}},{\delta}\big{)},\;x^{*}\in[a,b).$
Now we show that the metric selections of $F$ inherit the Lipschitz regularity
property at $x$ from $v_{F}$.
###### Lemma 2.19.
Let $F\in\mathcal{F}[a,b]$. If $v_{F}\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$
then $s\in\mathrm{Lip}\\{{x},{4\mathcal{L}}\\}$ for all metric selections
$s\in\mathcal{S}(F)$.
###### Proof.
Since $v_{F}\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$, there is $\delta>0$ such
that ${|v_{F}(x)-v_{F}(z)|\leq{\mathcal{L}}|x-z|}$, ${\forall
z\in\left(x-\frac{\delta}{2},x+\frac{\delta}{2}\right)\cap[a,b]}$, implying
that $\omega(v_{F},x,\eta)\leq\mathcal{L}\eta$ for $\eta\leq\delta$. Thus by
Result 2.15 we have for all
$z\in\left(x-\frac{\delta}{4},x+\frac{\delta}{4}\right)\cap[a,b]$
$|s(x)-s(z)|\leq\omega(s,x,2|x-z|)\leq\omega(v_{F},x,4|x-z|)\leq
4\mathcal{L}|x-z|.$
∎
The metric integral of SVFs is introduced in [19] and extended to the weighted
metric integral in [7]. We recall its definition.
For a multifunction $F:[a,b]\rightarrow\mathrm{K}({{\mathbb{R}}}^{d})$, a
weight function $\kappa:[a,b]\to{\mathbb{R}}$ and for a partition
${\chi=\\{x_{0},\ldots,x_{n}\\}}$, ${a=x_{0}<x_{1}<\cdots<x_{n}=b}$, we define
the weighted metric Riemann sum of $F$ by
$\displaystyle{\scriptstyle(\mathcal{M}_{\kappa})}S_{\chi}F$
$\displaystyle=\left\\{\sum_{i=0}^{n-1}(x_{i+1}-x_{i})\kappa(x_{i})y_{i}:\
(y_{0},\ldots,y_{n-1})\in{\mathrm{CH}}(F(x_{0}),\ldots,F(x_{n-1}))\right\\}$
$\displaystyle=\bigoplus_{i=0}^{n-1}(x_{i+1}-x_{i})\kappa(x_{i})F(x_{i}).$
The weighted metric integral of $F$ with the weight function $\kappa$ is
defined as the upper Kuratowski limit of weighted metric Riemann sums
${\scriptstyle(\mathcal{M}_{\kappa})}\int_{a}^{b}\kappa(x)F(x)dx=\limsup_{|\chi|\to
0}{\scriptstyle(\mathcal{M}_{\kappa})}S_{\chi}F.$
The set ${\scriptstyle(\mathcal{M}_{\kappa})}\int_{a}^{b}\kappa(x)F(x)dx$ is
non-empty whenever the set-valued function $\kappa F$ has a bounded range.
###### Result 2.20.
[7] Let $F\in\mathcal{F}[a,b]$ and $\kappa\in\mathrm{BV}[a,b]$. Then the set
${\scriptstyle(\mathcal{M}_{\kappa})}\int_{a}^{b}\kappa(x)F(x)dx$ is compact
and
${\scriptstyle(\mathcal{M}_{\kappa})}\int_{a}^{b}\kappa(x)F(x)dx=\left\\{\int_{a}^{b}\kappa(x)s(x)dx\
:\ s\in\mathcal{S}(F)\right\\}.$
Here and in such occasions below we understand that the integral is applied to
each component of $s=\begin{pmatrix}s_{1}\\\ \vdots\\\ s_{d}\end{pmatrix}$.
## 3 Rate of pointwise convergence of integral operators for real-valued
functions
Let $\\{\mathcal{K}_{n}(x,t)\\}_{n\in{\mathbb{N}}}$,
${\mathcal{K}}_{n}:[a,b]\times[a,b]\to{\mathbb{R}}$, be a sequence of
functions that are integrable with respect to $t$ for each $x$. We term the
functions $\mathcal{K}_{n}$ kernels. With the help of the sequence of the
kernels we define a sequence of linear integral operators
$\\{T_{n}\\}_{n\in{\mathbb{N}}}$ on real-valued functions in $L^{\infty}[a,b]$
by
$T_{n}f(x)=\int_{a}^{b}\mathcal{K}_{n}(x,t)f(t)dt,\quad x\in[a,b],\quad
n\in{\mathbb{N}}.$ (13)
###### Remark 3.1.
If $f:[a,b]\to{{\mathbb{R}}}^{d}$, then $T_{n}f$ is obtained by the operation
of $T_{n}$ on each component of $f$.
Next we introduce the following notation. For $x\in[a,b]$ let
$\alpha_{n}(x)=\left|\int_{a}^{b}\mathcal{K}_{n}(x,t)dt-1\right|$
and
$\beta_{n}(x,\delta)=\int_{|x-t|\geq\delta}|\mathcal{K}_{n}(x,t)|dt,\quad\delta>0.$
Furthermore, for each $x\in[a,b]$ let $M(x)\in[0,\infty]$ be such that
$\int_{a}^{b}|\mathcal{K}_{n}(x,t)|dt\leq M(x),\quad n\in{\mathbb{N}}.$ (14)
### 3.1 The case of continuity points
The theorem below can be considered as a refinement of Theorem 2.1 in [12,
Chapter 1]. For that we need the following result [22, page 12]
###### Lemma 3.2.
If the function $f$ is defined on $[a,b]$ and $I\subseteq[a,b]$ is a closed
interval such that $f$ is continuous at all the points of $I$, then for any
$\epsilon>0$ there exists $\delta>0$ such that
$|f(y)-f(x)|<\epsilon$
for all $x\in I$, $y\in[a,b]$ and $|y-x|<\delta$.
###### Theorem 3.3.
Let $f:[a,b]\rightarrow{\mathbb{R}}$ be bounded and measurable on $[a,b]$,
$x\in[a,b]$ and let $M(x)$ be as in (14). Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$|T_{n}f(x)-f(x)|\leq\omega(f,x,2\delta)M(x)+\|f\|_{\infty}2\beta_{n}(x,\delta)+|f(x)|\alpha_{n}(x).$
(15)
(ii) If $\displaystyle\lim_{n\to\infty}\alpha_{n}(x)=0$,
$\displaystyle\lim_{n\to\infty}\beta_{n}(x,\delta)=0$ for any sufficiently
small $\delta>0$, $M(x)<\infty$, and if $x$ is a point of continuity of $f$,
then
$\lim_{n\to\infty}{|T_{n}f(x)-f(x)|}=0.$ (16)
(iii) If $f$ is continuous at all points of a closed interval
$I\subseteq[a,b]$, if $\displaystyle\lim_{n\to\infty}\alpha_{n}(x)=0$ and
${\displaystyle\lim_{n\to\infty}\beta_{n}(x,\delta)=0}$ uniformly in $x\in I$
for any sufficiently small $\delta>0$, and if $M(x)$ is bounded on $I$, then
the convergence is uniform in $I$.
###### Proof.
(i) We have
$\displaystyle|T_{n}f(x)-f(x)|$
$\displaystyle=\left|\int_{a}^{b}\mathcal{K}_{n}(x,t)f(t)dt-\int_{a}^{b}\mathcal{K}_{n}(x,t)f(x)dt+f(x)\int_{a}^{b}\mathcal{K}_{n}(x,t)dt-f(x)\right|$
$\displaystyle\leq\int_{a}^{b}|\mathcal{K}_{n}(x,t)||f(t)-f(x)|dt+|f(x)|\left|\int_{a}^{b}\mathcal{K}_{n}(x,t)dt-1\right|$
$\displaystyle\leq\int_{|x-t|\leq\delta}|\mathcal{K}_{n}(x,t)||f(t)-f(x)|dt+\int_{|x-t|>\delta}|\mathcal{K}_{n}(x,t)||f(t)-f(x)|dt+|f(x)|\alpha_{n}(x).$
Using the fact that $|f(t)-f(x)|\leq\omega(f,x,2\delta)$ in the first integral
and that $|f(t)-f(x)|\leq 2\|f\|_{\infty}$, we obtain
$\displaystyle|T_{n}f(x)-f(x)|$
$\displaystyle\leq\omega(f,x,2\delta)\int_{|x-t|\leq\delta}|\mathcal{K}_{n}(x,t)|dt+2\|f\|_{\infty}\int_{|x-t|>\delta}|\mathcal{K}_{n}(x,t)|dt+|f(x)|\alpha_{n}(x)$
$\displaystyle\leq\omega(f,x,2\delta)M(x)+2\|f\|_{\infty}\beta_{n}(x,\delta)+|f(x)|\alpha_{n}(x)$
which gives (15).
(ii) Fix $\delta>0$ and let $n\to\infty$ in (15). With the assumption of (ii),
it follows from (15) that
$\limsup_{n\to\infty}{|T_{n}f(x)-f(x)|}\leq\omega(f,x,2\delta)M(x).$
Since this is valid for each $\delta>0$ and since
$\displaystyle\lim_{\delta\to 0+}\omega(f,x,2\delta)=0$ when $x$ is a point of
continuity, we obtain (16).
(iii) By Lemma 3.2, $\omega(f,x,2\delta)$ tends to zero uniformly in $x\in I$
when $\delta\to 0+$ and it follows from (15) that the convergence is uniform
in $I$. ∎
In view of Lemma 2.12, estimates in the proof of Theorem 3.3 can be repeated
verbatim for $f:[a,b]\to{{\mathbb{R}}}^{d}$, $f\in\mathrm{BV}[a,b]$, since
such $f$ is Riemann integrable. We obtain
###### Corollary 3.4.
If $f:[a,b]\to{{\mathbb{R}}}^{d}$, $f\in\mathrm{BV}[a,b]$ and
$\mathcal{K}_{n}(x,\cdot)\in\mathrm{BV}[a,b]$ for each $x\in[a,b]$, then the
estimates in Theorem 3.3 hold.
###### Remark 3.5.
It is easy to see that if $f:[a,b]\to{{\mathbb{R}}}^{d}$ is bounded and
measurable, then (15) becomes
$|T_{n}f(x)-f(x)|\leq
C{\omega}(f,x,2\delta)M(x)+C\|f\|_{\infty}2\beta_{n}(x,\delta)+C|f(x)|\alpha_{n}(x),\quad
x\in[a,b],\;n\in{\mathbb{N}},\;\delta>0,$
where $C$ depends only on the underlying norm $|\cdot|$ in
${{\mathbb{R}}}^{d}$.
### 3.2 The case of discontinuity points
In this section we follow and refine the analysis in [21] for real-valued
functions.
Let $f:[a,b]\to{\mathbb{R}}$, $f\in\mathrm{BV}[a,b]$. Fix $x\in(a,b)$,
following [21] we introduce the function $g_{x}:[a,b]\to{\mathbb{R}}$,
$g_{x}(t)=\begin{cases}f(t)-f(x-),&t\in[a,x),\\\ f(t)-f(x)=0,&t=x,\\\
f(t)-f(x+),&t\in(x,b].\end{cases}$
Note that $f\in\mathrm{BV}[a,b]$ implies that $g_{x}$ is well defined and is
continuous at $x$. It is easy to check that
$f(t)=g_{x}(t)+f(x-)\chi_{[a,x)}(t)+f(x+)\chi_{(x,b]}(t)+f(x)\chi_{\\{x\\}}(t),\quad
t\in[a,b],$
where $\chi_{A}$ denotes the characteristic function of a set $A$
$\chi_{A}(t)=\begin{cases}1,&t\in A,\\\ 0,&t\not\in A.\end{cases}$
A simple computation shows that for $t\in[a,b]$
$\displaystyle f(t)$
$\displaystyle-\frac{1}{2}\big{[}f(x+)+f(x-)\big{]}=g_{x}(t)+\frac{1}{2}\big{[}f(x+)-f(x-)\big{]}\big{(}\chi_{(x,b]}(t)-\chi_{[a,x)}(t)\big{)}$
$\displaystyle+\left(f(x)-\frac{1}{2}\big{[}f(x+)+f(x-)\big{]}\right)\chi_{\\{x\\}}(t)$
$\displaystyle=g_{x}(t)+\frac{1}{2}\big{[}f(x+)-f(x-)\big{]}{\mathrm{sign}}(t-x)+\left(f(x)-\frac{1}{2}\big{[}f(x+)+f(x-)\big{]}\right)\chi_{\\{x\\}}(t),$
where
${\mathrm{sign}}(t)=\begin{cases}1,&t>0,\\\ 0,&t=0,\\\ -1,&t<0.\end{cases}$
Inserting this representation into the operator (13), we obtain
$T_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\int_{a}^{b}\mathcal{K}_{n}(x,t)dt=T_{n}g_{x}(x)+\frac{1}{2}[f(x+)-f(x-)]T_{n}({\mathrm{sign}}(\cdot-x))(x).$
(17)
Taking into account that $g_{x}(x)=0$, the estimate (15) for $T_{n}g_{x}(x)$
takes the form
$|T_{n}g_{x}(x)|=|T_{n}g_{x}(x)-g_{x}(x)|\leq\omega(g_{x},x,2\delta)M(x)+2\|g_{x}\|_{\infty}\beta_{n}(x,\delta).$
(18)
By the definition of the function $g_{x}$ we easily see that
$\|g_{x}\|_{\infty}\leq 2\|f\|_{\infty}$. By the definition of the local
modulus of continuity and the local quasi-moduli (see Section 2.2), we get
$\omega(g_{x},x,2\delta)\leq
2\max{\\{\varpi^{-}\big{(}{f},{x},{\delta}\big{)},\varpi^{+}\big{(}{f},{x},{\delta}\big{)}\\}}=2\varpi\big{(}{f},{x},{\delta}\big{)}.$
(19)
Clearly, $\frac{1}{2}\big{|}f(x+)-f(x-)\big{|}\leq\|f\|_{\infty}$, since
$f\in\mathrm{BV}[a,b]$. Combining this with (17), (18), (19) we arrive at
$\begin{split}&\left|T_{n}f(x)-\frac{1}{2}\big{[}f(x+)+f(x-)\big{]}\int_{a}^{b}\mathcal{K}_{n}(x,t)dt\right|\\\
&\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}M(x)+\|f\|_{\infty}\left(4\beta_{n}(x,\delta)+\big{|}T_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\right).\end{split}$
(20)
Thus we obtain the following result,
###### Theorem 3.6.
Let $f:[a,b]\to{\mathbb{R}}$, $f\in\mathrm{BV}[a,b]$ and $x\in(a,b)$.
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$ we have
$\displaystyle\left|T_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|$
$\displaystyle\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}M(x)+\|f\|_{\infty}\left(4\beta_{n}(x,\delta)+\alpha_{n}(x)+\big{|}T_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\right).$
(ii) If $\displaystyle\lim_{n\to\infty}\alpha_{n}(x)=0$,
$\displaystyle\lim_{n\to\infty}\beta_{n}(x,\delta)=0$ for any sufficiently
small $\delta>0$,
${\displaystyle\lim_{n\to\infty}T_{n}({\mathrm{sign}}(\cdot-x))(x)=0}$, and
$M(x)<\infty$, then
$\lim_{n\to\infty}{\left|T_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|}=0.$
###### Proof.
By the triangle inequality and (4.2) we have
$\displaystyle\left|T_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|$
$\displaystyle\leq\left|T_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\int_{a}^{b}\mathcal{K}_{n}(x,t)dt\right|+\frac{1}{2}|f(x+)+f(x-)|\left|\int_{a}^{b}\mathcal{K}_{n}(x,t)dt-1\right|$
$\displaystyle\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}M(x)+\|f\|_{\infty}\left(4\beta_{n}(x,\delta)+\big{|}T_{n}({\mathrm{sign}}(\cdot-x))(x)\,\big{|}\,\right)\,+\|f\|_{\infty}\alpha_{n}(x),$
which leads to the first claim. The second claim follows directly from it by
applying (5). ∎
Similarly to Corollary 3.4 for vector-valued functions we obtain
###### Corollary 3.7.
Let $f:[a,b]\to{{\mathbb{R}}}^{d}$, $f\in\mathrm{BV}[a,b]$,
$\mathcal{K}_{n}(z,\cdot)\in\mathrm{BV}[a,b]$ for each $z\in[a,b]$, then for
$x\in(a,b)$
$\displaystyle\left|T_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|$
$\displaystyle\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}M(x)+\|f\|_{\infty}\big{(}4\beta_{n}(x,\delta)+\alpha_{n}(x)+\big{|}T_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\,\big{)}.$
## 4 Rate of pointwise convergence of integral operators for set-valued
functions
Let $F\in\mathcal{F}[a,b]$ and $\mathcal{K}_{n}(x,\cdot)\in\mathrm{BV}[a,b]$
for any $x\in[a,b]$. Denoting $\kappa_{n,x}(t)=\mathcal{K}_{n}(x,t)$ and using
the concept of the weighted metric integral (see Section 2.3), we define
$T_{n}F(x)={\scriptstyle(\mathcal{M}_{\kappa_{n,x}})}\int_{a}^{b}\kappa_{n,x}(t)F(t)dt,\quad
x\in[a,b],\quad n\in{\mathbb{N}}.$ (21)
By Result 2.20 and since $\kappa_{n,x}(t)=\mathcal{K}_{n}(x,t)$ we have for
$F\in\mathcal{F}[a,b]$
$T_{n}F(x)=\left\\{\int_{a}^{b}\mathcal{K}_{n}(x,t)s(t)dt\ :\
s\in\mathcal{S}(F)\right\\}=\left\\{T_{n}s(x)\ :\
s\in\mathcal{S}(F)\right\\},\quad x\in[a,b],$ (22)
where $\mathcal{S}(F)$ is the set of metric selections of $F$. By Result 2.14
we have as well
$F(x)=\left\\{s(x)\ :\ s\in\mathcal{S}(F)\right\\},\quad x\in[a,b].$ (23)
It is easy to obtain from (22) and (23) that
$\mathrm{haus}\left(T_{n}F(x),F(x)\right)\leq\sup{\left\\{\left|T_{n}s(x)-s(x)\right|\
:\ s\in\mathcal{S}(F)\right\\}}.$ (24)
The arguments leading to (24) are similar to those in the proof of Lemma 6.3
below.
###### Remark 4.1.
Any bounded linear operator defined for single-valued functions can be
extended to set-valued functions from the class $\mathcal{F}[a,b]$ by
$TF(x)=\\{Ts(x)\ :\ s\in\mathcal{S}(F)\\},\quad x\in[a,b],\quad
F\in\mathcal{F}[a,b].$
This approach was used for operators $T$ defined on continuous real-valued
functions in [18, Section 8.2].
### 4.1 The case of continuity points
Recall that by Results 2.13 and 2.15 for each selection $s\in\mathcal{S}(F)$
we have
$\|s\|_{\infty}\leq\|F\|_{\infty}\quad\mbox{and}\quad\omega(s,x,\delta)\leq\omega(v_{F},x,2\delta)\,,\;\,x\in[a,b]\,,\;\delta>0.$
Thus for each $s\in\mathcal{S}(F)$ the estimate in Corollary 3.4 turns into
$|T_{n}s(x)-s(x)|\leq\omega(v_{F},x,4\delta)M(x)+\|F\|_{\infty}(2\beta_{n}(x,\delta)+\alpha_{n}(x)).$
(25)
We arrive at
###### Theorem 4.2.
Let $F\in\mathcal{F}[a,b]$, $x\in[a,b]$,
$\\{\mathcal{K}_{n}(x,t)\\}_{n\in{\mathbb{N}}}$ be a sequence of kernels on
$[a,b]\times[a,b]$ such that $\mathcal{K}_{n}(x,\cdot)\in\mathrm{BV}[a,b]$ for
any $x\in[a,b]$ and let $\alpha_{n}(x)$, $\beta_{n}(x,\delta)$ and $M(x)$ be
as in Section 3. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$\mathrm{haus}(T_{n}F(x),F(x))\leq\omega(v_{F},x,4\delta)M(x)+\|F\|_{\infty}(2\beta_{n}(x,\delta)+\alpha_{n}(x))\,,\quad
x\in[a,b].$
(ii) If $\displaystyle\lim_{n\to\infty}\alpha_{n}(x)=0$,
$\displaystyle\lim_{n\to\infty}\beta_{n}(x,\delta)=0$ for any sufficiently
small $\delta>0$, $M(x)<\infty$ and if $x$ is a point of continuity of $F$,
then
$\lim_{n\to\infty}{\mathrm{haus}\left(T_{n}F(x),F(x)\right)}=0.$
(iii) If $F$ is continuous at all points of a closed interval
$I\subseteq[a,b]$, if ${\displaystyle\lim_{n\to\infty}\alpha_{n}(x)=0}$ and
${\displaystyle\lim_{n\to\infty}\beta_{n}(x,\delta)=0}$ uniformly in $x\in I$
for any sufficiently small $\delta>0$, and if $M(x)$ is bounded on $I$, then
the convergence is uniform in $I$.
###### Proof.
The statements (i) and (ii) follow from the first two claims of Theorem 3.3
combined with (24) and (25). The proof of the statements (iii) is based on
Result 2.3, which implies that $v_{F}$ is uniformly continuous on compact
intervals, and therefore $\omega(v_{F},x,4\delta)\to 0$ uniformly for $x\in
I$. ∎
### 4.2 The case of discontinuity points
For $x\in(a,b)$ we define the set (see also [7, Section 6.3])
$A_{F}(x)=\left\\{\frac{1}{2}\left(s(x+)+s(x-)\right)\ :\
s\in\mathcal{S}(F)\right\\}.$
Similarly to (24) we have
$\mathrm{haus}\left(T_{n}F(x),A_{F}(x)\right)\leq\sup{\left\\{\left|T_{n}s(x)-\frac{1}{2}(s(x+)+s(x-))\right|\
:\ s\in\mathcal{S}(F)\right\\}}.$ (26)
Recall that by Result 2.13, $\|s\|_{\infty}\leq\|F\|_{\infty}$ and
$\frac{1}{2}|s(x+)+s(x-)|\leq\|F\|_{\infty}$. In view of Corollary 2.18 we
have for each $s\in\mathcal{S}(F)$
$\varpi\big{(}{s},{x},{\delta}\big{)}\leq\varpi\big{(}{v_{F}},{x},{2\delta}\big{)}.$
By Corollary 3.7 and the above inequality we have for any $s\in\mathcal{S}(F)$
$\displaystyle\left|T_{n}s(x)-\frac{1}{2}[s(x+)+s(x-)]\right|$
$\displaystyle\leq
2\varpi\big{(}{v_{F}},{x},{2\delta}\big{)}M(x)+\|F\|_{\infty}\big{(}4\beta_{n}(x,\delta)+\alpha_{n}(x)+\big{|}T_{n}({\mathrm{sign}}(\cdot-x))(x)\,\big{|}\,\big{)}.$
This together with (26), by arguments as in the proof of (ii) of Theorem 3.3,
leads to the following result
###### Theorem 4.3.
Let $F\in\mathcal{F}[a,b]$ and let
$\\{\mathcal{K}_{n}(x,t)\\}_{n\in{\mathbb{N}}}$ be a sequence of kernels on
$[a,b]\times[a,b]$ such that $\mathcal{K}_{n}(x,\cdot)\in\mathrm{BV}[a,b]$ for
any $x\in[a,b]$. Then for any $x\in(a,b)$ the following statements hold.
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$\displaystyle\mathrm{haus}(T_{n}F(x),A_{F}(x))\leq
2\varpi\big{(}{v_{F}},{x},{2\delta}\big{)}M(x)+\|F\|_{\infty}\big{(}4\beta_{n}(x,\delta)+\alpha_{n}(x)+\big{|}T_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\big{)}.$
(ii) If $\displaystyle\lim_{n\to\infty}\alpha_{n}(x)=0$,
$\displaystyle\lim_{n\to\infty}\beta_{n}(x,\delta)=0$ for any sufficiently
small $\delta>0$,
${\displaystyle\lim_{n\to\infty}T_{n}({\mathrm{sign}}(\cdot-x))(x)=0}$, and
$M(x)<\infty$, then
$\lim_{n\to\infty}{\mathrm{haus}\left(T_{n}F(x),A_{F}(x)\right)}=0.$
## 5 Specific operators
Here we apply the results of the previous sections to two specific integral
operators. In particular, at points of discontinuity we obtain error estimates
that combine ideas from [21, 29] with the local quasi-moduli of continuity
(3), (4).
### 5.1 Bernstein-Durrmeyer operators
For $x\in[0,1]$, $n\in{\mathbb{N}}$ the Bernstein basis polynomials are
defined as
$p_{n,k}(x)={n\choose k}x^{k}(1-x)^{n-k}\,,\quad k=0,1,\ldots,n.$ (27)
Note that $\displaystyle\sum_{k=0}^{n}p_{n,k}(x)=1$ and
$\displaystyle\int_{0}^{1}p_{n,k}(x)dx=\frac{1}{n+1}$.
The Bernstein-Durrmeyer operator is defined for $f\in L^{1}[0,1]$ by
$M_{n}f(x)=(n+1)\sum_{k=0}^{n}p_{n,k}(x)\int_{0}^{1}p_{n,k}(t)f(t)dt=\int_{0}^{1}\mathcal{K}_{n}(x,t)f(t)dt\,,\quad
x\in[0,1]\,,\;n\in{\mathbb{N}},$
where
$\mathcal{K}_{n}(x,t)=(n+1)\sum_{k=0}^{n}p_{n,k}(x)p_{n,k}(t).$ (28)
The Bernstein-Durrmeyer operator for a set-valued function
$F\in\mathcal{F}[0,1]$ is
$M_{n}F(x)={\scriptstyle(\mathcal{M}_{\kappa_{n,x}})}\int_{0}^{1}\kappa_{n,x}(t)F(t)dt,\quad
x\in[0,1],\quad n\in{\mathbb{N}}\,,$
with $\kappa_{n,x}(t)=\mathcal{K}_{n}(x,t)$, where $\mathcal{K}_{n}(x,t)$ is
given in (28).
The properties of the Bernstein basis polynomials $p_{n,k}$ yield
$\mathcal{K}_{n}(x,t)\geq 0,\quad\int_{0}^{1}\mathcal{K}_{n}(x,t)dt=1,$
so that $\alpha_{n}(x)=0$ and $M(x)=1$, $x\in[0,1]$.
Denote $e_{i}(t)=t^{i}$, $i=0,1,2$. A direct calculation shows that (see e.g.
[21])
$M_{n}e_{0}(x)=1\,,\qquad M_{n}e_{1}(x)=\frac{nx+1}{n+2}\,,\qquad
M_{n}e_{2}(x)=\frac{n(n-1)x^{2}+4nx+2}{(n+2)(n+3)}.$
This implies
$M_{n}((\cdot-x)^{2})(x)=\frac{2[(n-3)x(1-x)+1]}{(n+2)(n+3)}.$ (29)
Following S. Guo (see Lemma 6 in [21]), we estimate $\beta_{n}(x,\delta)$ as
follows:
$\displaystyle\beta_{n}(x,\delta)$
$\displaystyle=\int_{|x-t|\geq\delta}\mathcal{K}_{n}(x,t)dt\leq\int_{|x-t|\geq\delta}\left(\frac{x-t}{\delta}\right)^{2}\mathcal{K}_{n}(x,t)dt$
$\displaystyle\leq\frac{1}{\delta^{2}}\int_{0}^{1}(x-t)^{2}\mathcal{K}_{n}(x,t)dt=\frac{1}{\delta^{2}}M_{n}((\cdot-x)^{2})(x)=\frac{1}{\delta^{2}}\frac{2[(n-3)x(1-x)+1]}{(n+2)(n+3)}.$
Using the inequality $\displaystyle x(1-x)\leq\frac{1}{4}$, $x\in[0,1]$, we
arrive at the estimate
$\displaystyle\beta_{n}(x,\delta)\leq\frac{n+1}{2\delta^{2}(n+2)(n+3)}\leq\frac{1}{2n\delta^{2}}.$
Thus, for the Bernstein-Durrmeyer operator we have
$M(x)=1\;,\;\alpha_{n}(x)=0\;,\;\beta_{n}(x,\delta)\leq\frac{1}{2n\delta^{2}}\,,\quad
x\in[0,1].$
The case of continuity points
The assumptions of Theorem 3.3 are fulfilled for the Bernstein-Durrmeyer
operator. Thus we obtain the following result, where part (i), which provides
rate of convergence, is new in this form, while parts (ii) and (iii) are
already known.
###### Corollary 5.1.
Let $f\in L^{1}[0,1]$ be bounded on $[0,1]$, $x\in[0,1]$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$|M_{n}f(x)-f(x)|\leq\omega(f,x,2\delta)+\|f\|_{\infty}\frac{1}{n\delta^{2}}.$
(ii) If $x$ is a point of continuity of $f$, then
$\lim_{n\to\infty}{|M_{n}f(x)-f(x)|}=0$.
(iii) If $f$ is continuous at all points of a closed interval
$I\subseteq[0,1]$, then the convergence is uniform in $I$.
From Theorem 4.2 for the Bernstein-Durrmeyer operator we get
###### Corollary 5.2.
Let $F\in\mathcal{F}[0,1]$, $x\in[0,1]$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$\mathrm{haus}(M_{n}F(x),F(x))\leq\omega(v_{F},x,4\delta)+\|F\|_{\infty}\frac{1}{n\delta^{2}}.$
(ii) If $x$ is a point of continuity of $F$, then
$\lim_{n\to\infty}{\mathrm{haus}\left(M_{n}F(x),F(x)\right)}=0$.
(iii) If $F$ is continuous at all points of a closed interval
$I\subseteq[0,1]$, then the convergence is uniform in $I$.
The case of discontinuity points
Theorem 3.6 for the Bernstein-Durrmeyer operator leads to
$\displaystyle\left|M_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}+\|f\|_{\infty}\left(\frac{2}{n\delta^{2}}+\big{|}M_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\right)$
for $x\in(0,1)$. S. Guo proved in [21] (see Proof of the Theorem) that
$|M_{n}({\mathrm{sign}}(\cdot-x))(x)|\leq\frac{13}{2\sqrt{nx(1-x)}}$
for $x\in(0,1)$ and $n$ large enough. Thus we get
###### Corollary 5.3.
Let $f:[0,1]\to{\mathbb{R}}$, $f\in\mathrm{BV}[a,b]$ and $x\in(0,1)$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$ large enough
$\left|M_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}+\|f\|_{\infty}\left(\frac{2}{n\delta^{2}}+\frac{13}{2\sqrt{nx(1-x)}}\right),$
(ii)
$\displaystyle\lim_{n\to\infty}{\left|M_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|}=0$.
Note that (ii) and an estimate for the order of convergence are obtained by S.
Guo in [21].
Theorem 4.3 for the Bernstein-Durrmeyer operator results in
###### Corollary 5.4.
Let $F\in\mathcal{F}[0,1]$, $x\in(0,1)$. Then
(i) For all $\delta>0$ and large enough $n\in{\mathbb{N}}$
$\mathrm{haus}(M_{n}F(x),A_{F}(x))\leq
2\varpi\big{(}{v_{F}},{x},{2\delta}\big{)}+\|F\|_{\infty}\left(\frac{2}{n\delta^{2}}+\frac{13}{2\sqrt{nx(1-x)}}\right).$
(ii)
$\displaystyle\lim_{n\to\infty}{\mathrm{haus}\left(M_{n}F(x),A_{F}(x)\right)}=0$.
### 5.2 Kantorovich operators
For $f\in L^{1}[0,1]$ the Kantorovich operator is defined by
$K_{n}f(x)=(n+1)\sum_{k=0}^{n}p_{n,k}(x)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}f(t)dt=\int_{0}^{1}\mathcal{K}_{n}(x,t)f(t)dt,\quad
x\in[0,1],\quad n\in{\mathbb{N}},$
where
$\mathcal{K}_{n}(x,t)=(n+1)\sum_{k=0}^{n}p_{n,k}(x)\chi_{\left[\frac{k}{n+1},\frac{k+1}{n+1}\right]}(t)$
and $p_{n,k}(x)$ are defined in (27).
By formula (21) we can extend this operator to set-valued functions
$F\in\mathcal{F}[0,1]$. Using the properties of $p_{n,k}(x)$, we obtain, as in
the case of the Bernstein-Durrmeyer operator, that $\mathcal{K}_{n}(x,t)\geq
0$, ${\int_{0}^{1}\mathcal{K}_{n}(x,t)dt=1}$, leading to $\alpha_{n}(x)=0$ and
$M(x)=1$, $x\in[0,1]$.
Calculating $K_{n}e_{i}(x)$, $i=0,1,2$ one can get (see e.g. [29])
$K_{n}e_{0}(x)=1,\qquad K_{n}e_{1}(x)=\frac{2nx+1}{2(n+1)},\qquad
K_{n}e_{2}(x)=\frac{3n(n-1)x^{2}+6nx+1}{3(n+1)^{2}},$
and
$K_{n}((\cdot-x)^{2})(x)=\frac{3(n-1)x(1-x)+1}{3(n+1)^{2}}.$ (30)
Thus, estimating $\beta_{n}(x,\delta)$ as in the case of the Bernstein-
Durrmeyer operator, we obtain
$\beta_{n}(x,\delta)\leq\frac{1}{\delta^{2}}K_{n}((\cdot-x)^{2})(x)=\frac{1}{\delta^{2}}\frac{3(n-1)x(1-x)+1}{3(n+1)^{2}}\leq\frac{1}{4n\delta^{2}}.$
The case of continuity points
Theorem 3.3 takes for the Kantorovich operator the following form, where part
(i) is new in this form (providing rate of convergence), while parts (ii) and
(iii) are well-known.
###### Corollary 5.5.
Let $f\in L^{1}[0,1]$ be bounded on $[0,1]$, $x\in[0,1]$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$|K_{n}f(x)-f(x)|\leq\omega(f,x,2\delta)+\|f\|_{\infty}\frac{1}{2n\delta^{2}}.$
(ii) If $x$ is a point of continuity of $f$, then
$\lim_{n\to\infty}{|K_{n}f(x)-f(x)|}=0$.
(iii) If $f$ is continuous at all points of a closed interval
$I\subseteq[0,1]$, then the convergence is uniform in $I$.
Applying Theorem 4.2, we get
###### Corollary 5.6.
Let $F\in\mathcal{F}[0,1]$, $x\in[0,1]$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$\mathrm{haus}(K_{n}F(x),F(x))\leq\omega(v_{F},x,4\delta)+\|F\|_{\infty}\frac{1}{2n\delta^{2}}.$
(ii) If $x$ is a point of continuity of $F$, then
$\lim_{n\to\infty}{\mathrm{haus}\left(K_{n}F(x),F(x)\right)}=0$.
(iii) If $F$ is continuous at all points of a closed interval
$I\subseteq[0,1]$, then the convergence is uniform in $I$.
The case of discontinuity points
Theorem 3.6 for the Kantorovich operator gives the estimate
$\displaystyle\left|K_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}+\|f\|_{\infty}\left(\frac{1}{n\delta^{2}}+\big{|}K_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\right)$
for $x\in(0,1)$. An estimate for
$\big{|}K_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}$ is given in [29]. Closely
following the consideration of Zeng and Piriou in [29], we derive here a very
similar estimate in a slightly different form. We also replace the estimate
$p_{n,k}(x)\leq\frac{1}{\sqrt{2e}}\frac{1}{\sqrt{nx(1-x)}}$ used in [29] and
quoted there from an unpublished paper, by the estimate
$p_{n,k}(x)\leq\frac{5}{2\sqrt{nx(1-x)}}$ (31)
of Guo, which is published in [21] with a full proof.
Let $x\in\left[\frac{\ell}{n+1},\frac{\ell+1}{n+1}\right)$ with a certain
$\ell\in\\{0,1,\ldots,n\\}$, then
$\displaystyle K_{n}({\mathrm{sign}}(\cdot-x))(x)$
$\displaystyle=(n+1)\sum_{k=0}^{n}p_{n,k}(x)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}{\mathrm{sign}}(t-x)dt$
$\displaystyle=-\sum_{k=0}^{\ell-1}p_{n,k}(x)+\sum_{k=\ell+1}^{n}p_{n,k}(x)+p_{n,\ell}(x)(n+1)\left(\frac{\ell+1}{n+1}-x-\left(x-\frac{\ell}{n+1}\right)\right)$
$\displaystyle=2\sum_{k=\ell+1}^{n}p_{n,k}(x)-1+2p_{n,\ell}(x)[(\ell+1)-(n+1)x],$
and, since $\ell\leq(n+1)x<\ell+1$,
$\big{|}K_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\leq
2\left|\sum_{(n+1)x<k\leq n}p_{n,k}(x)-\frac{1}{2}\right|+2p_{n,\ell}(x).$
It was proved in [29] (see proof of Lemma 2) that
$\left|\sum_{nx<k\leq
n}p_{n,k}(x)-\frac{1}{2}\right|\leq\frac{0.8(2x^{2}-2x+1)}{\sqrt{nx(1-x)}}<\frac{1}{\sqrt{nx(1-x)}},$
where the last inequality is easy to check. Combining this with (31), we
obtain
$\left|\sum_{(n+1)x<k\leq
n}p_{n,k}(x)-\frac{1}{2}\right|\leq\frac{7}{2\sqrt{nx(1-x)}}$
and finally
$\big{|}K_{n}({\mathrm{sign}}(\cdot-x))(x)\big{|}\leq\frac{12}{\sqrt{nx(1-x)}}$
for $x\in(0,1)$. Thus we obtain
###### Corollary 5.7.
Let $f:[0,1]\to{\mathbb{R}}$, $f\in\mathrm{BV}[a,b]$ and $x\in(0,1)$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$\left|K_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|\leq
2\varpi\big{(}{f},{x},{\delta}\big{)}+\|f\|_{\infty}\left(\frac{1}{n\delta^{2}}+\frac{12}{\sqrt{nx(1-x)}}\right),$
(ii)
$\displaystyle\lim_{n\to\infty}{\left|K_{n}f(x)-\frac{1}{2}[f(x+)+f(x-)]\right|}=0$.
Note that (ii) is known (see [29] and the bibliography therein), this is also
the case for the order of convergence $1/\sqrt{n}$.
Theorem 4.3 for the Kantorovich operator takes the following form.
###### Corollary 5.8.
Let $F\in\mathcal{F}[0,1]$, $x\in(0,1)$. Then
(i) For all $\delta>0$ and $n\in{\mathbb{N}}$
$\mathrm{haus}(K_{n}F(x),A_{F}(x))\leq
2\varpi\big{(}{v_{F}},{x},{2\delta}\big{)}+\|F\|_{\infty}\left(\frac{1}{n\delta^{2}}+\frac{12}{\sqrt{nx(1-x)}}\right).$
(ii)
$\displaystyle\lim_{n\to\infty}{\mathrm{haus}\left(K_{n}F(x),A_{F}(x)\right)}=0$.
### 5.3 Rate of convergence of the Bernstein-Durrmeyer operators and the
Kantorovich operators for locally Lipschitz functions
In this section the sequence $\\{T_{n}\\}$ is the sequence of Bernstein-
Durrmeyer operators $\\{M_{n}\\}$ or the sequence of Kantorovich operators
$\\{K_{n}\\}$.
Using the notion of locally Lipschitz functions we can state the following
result.
###### Theorem 5.9.
Let $F\in\mathcal{F}[0,1]$ be locally Lipschitz around a point $x\in[0,1]$.
Then
$\mathrm{haus}\left(T_{n}F(x),F(x)\right)=O\left(\frac{1}{\sqrt{n}}\right),\quad
n\to\infty.$
###### Proof.
By (24)
$\mathrm{haus}\left(T_{n}F(x),F(x)\right)\leq\sup{\left\\{\left|T_{n}s(x)-s(x)\right|\
:\ s\in\mathcal{S}(F)\right\\}}.$
Since $F$ is Lipschitz around the point $x$ with some $\mathcal{L}>0$, by (ii)
of Lemma 2.9, $v_{F}\in\mathrm{Lip}\\{{x},{\mathcal{L}}\\}$ and by Lemma 2.19
each metric selection $s\in\mathcal{S}(F)$ satisfies
$s\in\mathrm{Lip}\\{{x},{4\mathcal{L}}\\}$. By Lemma 2.8 there exists
$\widetilde{\mathcal{L}}>0$ such that
$|s(z)-s(x)|\leq\widetilde{\mathcal{L}}|z-x|,\quad\forall z\in[0,1].$ (32)
Using Lemma 2.12, (32), the Cauchy-Schwarz inequality and (29) for
$\\{M_{n}\\}$ or (30) for $\\{K_{n}\\}$, we get
$\displaystyle\left|T_{n}s(x)-s(x)\right|$
$\displaystyle\leq\int_{0}^{1}|s(t)-s(x)|\mathcal{K}_{n}(x,t)dt\leq\widetilde{\mathcal{L}}\int_{0}^{1}|t-x|\mathcal{K}_{n}(x,t)dt$
$\displaystyle\leq\widetilde{\mathcal{L}}\left(\int_{0}^{1}|t-x|^{2}\mathcal{K}_{n}(x,t)dt\right)^{\frac{1}{2}}\left(\int_{0}^{1}\mathcal{K}_{n}(x,t)dt\right)^{\frac{1}{2}}=O\left(\frac{1}{\sqrt{n}}\right),\quad
n\to\infty,$
and the statement follows. ∎
## 6 Approximation of the set of metric selections in $\mathbb{L^{1}[a,b]}$
In this section we consider the sequence of operators $\\{T_{n}\\}$ defined in
(13) and study two sets of functions, $\mathcal{S}(F)$ and the set
${T_{n}\mathcal{S}(F)}=\\{T_{n}s:\,s\in\mathcal{S}(F)\\}.$
Note that for $F\in\mathcal{F}[a,b]$, the set $\mathcal{S}(F)\subset
L^{1}[a,b]$, since it consists of functions of bounded variation. We assume
that $\mathcal{K}_{n}(\cdot,\cdot)\in C([a,b]^{2})$. This condition guaranties
that $T_{n}:\mathcal{S}(F)\to C[a,b]\subset L^{1}[a,b]$.
Motivated by (22), we regard the set of functions $T_{n}\mathcal{S}(F)$ as an
approximant to $F$, represented, in view of Result 2.14, by the set of
functions $\mathcal{S}(F)$. We show below that $\mathcal{S}(F)$ and
$T_{n}\mathcal{S}(F)$ are elements of the metric space $\widetilde{H}$ of
compact non-empty subsets of $L^{1}[a,b]$, endowed with the Hausdorff metric.
We use this metric to measure the approximation error of $\mathcal{S}(F)$ by
$T_{n}\mathcal{S}(F)$.
### 6.1 Two compact sets of functions
In the the next two lemmas we prove that the sets $\mathcal{S}(F)$,
$T_{n}\mathcal{S}(F)$ are compact in $L^{1}{[a,b]}$, thus they are elements of
$\widetilde{H}$. It is enough to show that they are sequentially compact, i.e.
that every sequence in $\mathcal{S}(F)$ (in $T_{n}\mathcal{S}(F)$,
respectively) has a convergent subsequence with a limit in $\mathcal{S}(F)$
(in $T_{n}\mathcal{S}(F)$, respectively).
###### Lemma 6.1.
For $F\in\mathcal{F}[a,b]$ the set $\mathcal{S}(F)$ is compact in
$L^{1}{[a,b]}$.
###### Proof.
By Result 2.13 any metric selection $s\in\mathcal{S}(F)$ satisfies
$\|s\|_{\infty}\leq\|F\|_{\infty}$ and $V_{a}^{b}(s)\leq V_{a}^{b}(F)$.
Applying Helly’s selection principle, we conclude that for any sequence of
metric selections there is a subsequence converging pointwisely at all points
$x\in[a,b]$. By Result 2.16 the pointwise limit function of such a subsequence
is a metric selection. Thus by Lebesgue Dominated Convergence Theorem this
subsequence converges in the $L^{1}$-norm to the same limit metric selection.
∎
###### Lemma 6.2.
Let $F\in\mathcal{F}[a,b]$ and assume that $\mathcal{K}_{n}(\cdot,\cdot)\in
C([a,b]^{2})$. Then the set $T_{n}\mathcal{S}(F)$ is compact in $L^{1}[a,b]$.
###### Proof.
We have to show that any sequence $\\{\sigma_{k}\\}_{k=1}^{\infty}\subset
T_{n}\mathcal{S}(F)$ has a subsequence converging in the $L^{1}$-norm and that
its $L^{1}$-limit is in $T_{n}\mathcal{S}(F)$.
By definition
$\sigma_{k}(x)=T_{n}s_{k}(x)=\int_{a}^{b}\mathcal{K}_{n}(x,t)s_{k}(t)dt,$
for some $s_{k}\in\mathcal{S}(F)$.
By Helly’s selection principle there exists a subsequence
$\\{s_{k_{\ell}}\\}_{\ell=1}^{\infty}$ such that
$\displaystyle\lim_{\ell\to\infty}s_{k_{\ell}}(t)=s^{\infty}(t)$ pointwisely
for all $t\in[a,b]$. By Result 2.16, $s^{\infty}\in\mathcal{S}(F)$. Since
$\mathcal{K}_{n}$ is bounded, we have
$\displaystyle\lim_{\ell\to\infty}\mathcal{K}_{n}(x,t)s_{k_{\ell}}(t)=\mathcal{K}_{n}(x,t)s^{\infty}(t)$,
for all $x,t\in[a,b]$. Clearly, the functions
$\mathcal{K}_{n}(x,t)s_{k_{\ell}}(t)$, $\ell\in{\mathbb{N}}$ are dominated by
$\|\mathcal{K}_{n}\|_{\infty}\|F\|_{\infty}$. Applying Lebesgue Dominated
Convergence Theorem we get
$\lim_{\ell\to\infty}\sigma_{k_{\ell}}(x)=\lim_{\ell\to\infty}\int_{a}^{b}\mathcal{K}_{n}(x,t)s_{k_{\ell}}(t)dt=\int_{a}^{b}\mathcal{K}_{n}(x,t)s^{\infty}(t)dt=\sigma^{\infty}(x),\qquad
x\in[a,b].$ (33)
Thus, we have that $\\{\sigma_{k_{\ell}}(x)\\}_{\ell=1}^{\infty}$ converges
pointwisely to $\sigma^{\infty}(x)$ for all $x\in[a,b]$. Also the sequence
$\\{\sigma_{k_{\ell}}\\}_{\ell=1}^{\infty}$ is dominated by
$\|F\|_{\infty}\|\mathcal{K}_{n}\|_{\infty}(b-a)$, indeed
$|\sigma_{k_{\ell}}(x)|\leq\int_{a}^{b}|\mathcal{K}_{n}(x,t)|\,|s_{k_{\ell}}(t)|dt\leq\|F\|_{\infty}\int_{a}^{b}|\mathcal{K}_{n}(x,t)|dt\leq\|F\|_{\infty}\|\mathcal{K}_{n}\|_{\infty}(b-a),\quad
x\in[a,b].$
It follows that the sequence of functions $\\{\sigma_{k_{\ell}}(x)\\}$,
${\ell\in{\mathbb{N}}}$, $x\in[a,b]$ satisfies the assumptions of the Lebesgue
Dominated Convergence Theorem and we get
$\lim_{\ell\to\infty}\int_{a}^{b}|\sigma_{k_{\ell}}(x)-\sigma^{\infty}(x)|dx=0.$
To finish the proof, note that by (33) $\sigma^{\infty}\in
T_{n}\mathcal{S}(F)$, since $s^{\infty}\in\mathcal{S}(F)$. ∎
The Hausdorff distance between the sets $\mathcal{S}(F)$ and
$T_{n}\mathcal{S}(F)$ is
$\mathrm{haus}(\mathcal{S}(F),T_{n}\mathcal{S}(F))=\max\left\\{\sup_{\sigma\in
T_{n}\mathcal{S}(F)}\,\inf_{s\in\mathcal{S}(F)}\|s-\sigma\|_{L^{1}}\,,\,\sup_{s\in\mathcal{S}(F)}\inf_{\sigma\in
T_{n}\mathcal{S}(F)}\|s-\sigma\|_{L^{1}}\right\\}.$
###### Lemma 6.3.
Let $F\in\mathcal{F}[a,b]$. For the sets $\mathcal{S}(F)$ and
$T_{n}\mathcal{S}(F)$
$\mathrm{haus}(\mathcal{S}(F),T_{n}\mathcal{S}(F))\leq\sup_{s\in\mathcal{S}(F)}\|s-T_{n}s\|_{L^{1}}.$
###### Proof.
For any fixed $s^{*}\in\mathcal{S}(F)$ we have
$\displaystyle\inf_{\sigma\in
T_{n}\mathcal{S}(F)}\|s^{*}-\sigma\|_{L^{1}}\leq\|s^{*}-T_{n}s^{*}\|_{L^{1}}\leq\sup_{s\in\mathcal{S}(F)}\|s-T_{n}s\|_{L^{1}}.$
Similarly, for any fixed $\sigma^{*}\in T_{n}\mathcal{S}(F)$ there is
$s^{*}\in\mathcal{S}(F)$ with $T_{n}s^{*}=\sigma^{*}$ and we get
$\displaystyle\inf_{s\in\mathcal{S}(F)}\|s-\sigma^{*}\|_{L^{1}}\leq\|s^{*}-T_{n}s^{*}\|_{L^{1}}\leq\sup_{s\in\mathcal{S}(F)}\|s-T_{n}s\|_{L^{1}}.$
Now the statement follows easily. ∎
### 6.2 Error estimates
For $e_{i}(x)=x^{i}$, denote
${\displaystyle\lambda_{n}=(\max_{i=0,1}\|T_{n}e_{i}-e_{i}\|_{L^{1}})^{1/2}}$.
Here we use the integral moduli $\vartheta$ and $\vartheta_{2}$ defined in
Section 2.2. In light of the Theorem in [8] we get for real-valued functions
(as a special case of the statement in [8] for $p=1$)
###### Result 6.4.
Let $T_{n}$ be a positive linear operator satisfying
$\|T_{n}g\|_{L^{1}}\leq\|g\|_{L^{1}}$ for any $g\in L^{1}[a,b]$. Then
$\|T_{n}f-f\|_{L^{1}}\leq
C_{1}\left(\lambda_{n}^{2}\|f\|_{L^{1}}+\vartheta_{2}(f,\lambda_{n})\right),\quad
f\in L^{1}[a,b].$ (34)
If, in addition $T_{n}e_{0}\equiv e_{0}$, then
$\|T_{n}f-f\|_{L^{1}}\leq
C_{2}\left(\lambda_{n}^{2}\,\vartheta\big{(}{f},{b-a}\big{)}+\vartheta_{2}\big{(}f,\lambda_{n}\big{)}\right),\quad
f\in L^{1}[a,b].$ (35)
Here the constants $C_{i}$, $i=0,1$ depend only on the interval $[a,b]$.
Applying (34) and (35) on a metric selection $s\in\mathcal{S}(F)$ and using
(10) and (11) we get
$\|T_{n}s-s\|_{L^{1}}\leq\widetilde{C}_{1}\left(\lambda_{n}^{2}\,\|s\|_{L^{1}}+2\vartheta\big{(}{s},{\lambda_{n}}\big{)}\right)\leq\widetilde{C}_{1}\left(\lambda_{n}^{2}\,\|s\|_{L^{1}}+2\lambda_{n}V_{a}^{b}(s)\right),$
and in the case when $T_{n}e_{0}\equiv e_{0}$
$\|T_{n}s-s\|_{L^{1}}\leq\widetilde{C}_{2}\left(\lambda_{n}^{2}\,\vartheta\big{(}{s},{b-a}\big{)}+2\vartheta\big{(}{s},{\lambda_{n}}\big{)}\right)\leq\widetilde{C}_{2}\left(\lambda_{n}^{2}\,(b-a)V_{a}^{b}(s)+2\lambda_{n}V_{a}^{b}(s)\right),$
where $\widetilde{C}_{i}$, $i=0,1$ depend only on $[a,b]$ and on the
underlying norm in ${{\mathbb{R}}}^{d}$.
Since by Result 2.13 $V_{a}^{b}(s)\leq V_{a}^{b}(F)$, we obtain for any
$s\in\mathcal{S}(F)$
$\sup_{s\in\mathcal{S}(F)}\|T_{n}s-s\|_{L^{1}}\leq\widetilde{C}_{1}\left(\lambda_{n}^{2}\,\sup_{s\in\mathcal{S}(F)}\|s\|_{L^{1}}+2\lambda_{n}V_{a}^{b}(F)\right),$
(36)
and if $T_{n}e_{0}\equiv e_{0}$
$\sup_{s\in\mathcal{S}(F)}\|T_{n}s-s\|_{L^{1}}\leq\widetilde{C}_{2}\left(\lambda_{n}^{2}\,(b-a)+2\lambda_{n}\right)V_{a}^{b}(F).$
(37)
Thus in view of Lemma 6.3, (36) and (37) we arrive at
###### Theorem 6.5.
Let $T_{n}$ be a positive linear operator satisfying
$\|T_{n}g\|_{L^{1}}\leq\|g\|_{L^{1}}$ for any $g\in L^{1}[a,b]$ and let
$F\in\mathcal{F}[a,b]$. Then
$\mathrm{haus}(\mathcal{S}(F),T_{n}\mathcal{S}(F))\leq\widetilde{C}_{1}\left(\lambda_{n}^{2}\,\sup_{s\in\mathcal{S}(F)}\|s\|_{L^{1}}+2\lambda_{n}V_{a}^{b}(F)\right)\leq\widetilde{C}_{1}\left(\lambda_{n}^{2}\,\|F\|_{\infty}(b-a)+2\lambda_{n}V_{a}^{b}(F)\right).$
If, in addition, $T_{n}e_{0}\equiv e_{0}$, then
$\mathrm{haus}(\mathcal{S}(F),T_{n}\mathcal{S}(F))\leq\widetilde{C}_{2}\left(\lambda_{n}^{2}\,(b-a)+2\lambda_{n}\right)V_{a}^{b}(F).$
Here $\widetilde{C}_{i}$, $i=0,1$ depend only on $[a,b]$ and on the underlying
norm in ${{\mathbb{R}}}^{d}$.
### 6.3 Examples
We apply the results of Section 6.2 to the Bernstein-Durrmeyer operator
$M_{n}$ and the Kantorovich operator $K_{n}$. As was shown in Section 5, these
operators satisfy $M_{n}e_{0}\equiv e_{0}$ , $K_{n}e_{0}\equiv e_{0}$.
Moreover,
$M_{n}e_{1}(x)-e_{1}(x)=\frac{nx+1}{n+2}-x=\frac{1-2x}{n+2},\quad
K_{n}e_{1}(x)-e_{1}(x)=\frac{2nx+1}{2(n+1)}-x=\frac{1-2x}{2(n+1)}\,.$
Combining this with the second claim of Theorem 6.5 we end up with the
following result.
###### Theorem 6.6.
Let $F\in\mathcal{F}[0,1]$ and let $T_{n}$ be the Bernstein-Durrmeyer operator
or the Kantorovich operator. Then
$\mathrm{haus}(\mathcal{S}(F),T_{n}\mathcal{S}(F))\leq
CV_{a}^{b}(F)\left(\lambda_{n}^{2}\,(b-a)+2\lambda_{n}\right)=O\left(\frac{1}{\sqrt{n}}\right),\quad
n\to\infty,$
with $C$ depending only on the underlying norm in ${{\mathbb{R}}}^{d}$.
Note that the rate $O\left(\frac{1}{\sqrt{n}}\right)$ here is obtained for any
$F\in\mathcal{F}[0,1]$, while in the pointwise estimates of Section 5.3 we
obtain this rate under the assumption of local Lipschitz continuity of
$F\in\mathcal{F}[0,1]$.
## References
* [1] Z. Artstein, Piecewise linear approximations of set-valued maps, J. Approx. Theory 56 (1989), 41–47.
* [2] J.-P. Aubin, H. Frankowska, Set-valued analysis, Birkhäuser Boston Inc., Boston (1990).
* [3] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12.
* [4] V. F. Babenko, V. V. Babenko, M. V. Polishchuk, Approximation of some classes of set-valued periodic functions by generalized trigonometric polynomials, Ukrainian Math. J. 68(4) (2016), 502–514.
* [5] R. Baier, G. Perria, Set-valued Hermite interpolation, J. Approx. Theory 163 (2011), 1349–1372.
* [6] E. E. Berdysheva, N. Dyn, E. Farkhi, A. Mokhov, Metric approximation of set-valued functions of bounded variation, J. Comput. Appl. Math. 349 (2019), 251–264.
* [7] E. E. Berdysheva, N. Dyn, E. Farkhi, A. Mokhov, Metric Fourier approximation of set-valued functions of bounded variation, J. Fourier Anal. Appl. (2021) 27:17.
* [8] H. Berens, R. A. DeVore, Quantitative Korovkin theorems for $L_{p}$-spaces, In: Approximation Theory II, Proc. Internat. Sympos., Univ. Texas, Austin, Tex., Academic Press, New York (1976), 289–298.
* [9] M. Campiti, Korovkin-type approximation in spaces of vector-valued and set-valued functions, Appl. Anal. 98 (2019), 2486–2496.
* [10] M. Campiti, Korovkin approximation of set-valued integrable functions, manuscript (2022).
* [11] V. V. Chistyakov, On mappings of bounded variation, J. Dyn. Control Syst. 3 (1997), 261–289.
* [12] R. A. De Vore, G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg (1993).
* [13] N. Dyn, E. Farkhi, Spline subdivision schemes for convex compact sets, J. Comput. Appl. Math. 119(1-2) (2000), 133–144.
* [14] N. Dyn, E. Farkhi, Spline subdivision schemes for compact sets with metric averages, In: Trends in approximation theory (Nashville, TN, 2000), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2001, 93–102.
* [15] N. Dyn, E. Farkhi, Set-valued approximations with Minkowski averages — convergence and convexification rates, Numer. Funct. Anal. Optim. 25 (2004), 363–377.
* [16] N. Dyn, E. Farkhi, A. Mokhov, Approximations of set-valued functions by metric linear operators, Constr. Approx. 25 (2007), 193–209.
* [17] N. Dyn, E. Farkhi, A. Mokhov, Approximation of univariate set-valued functions—an overview. Serdica Math. J. 33 (2007), no. 4, 495–514.
* [18] N. Dyn, E. Farkhi, A. Mokhov, Approximations of Set-Valued Functions: Adaptation of Classical Approximation Operators, Imperial College Press, London (2014).
* [19] N. Dyn, E. Farkhi, A. Mokhov, The metric integral of set-valued functions, Set-Valued Var. Anal. 26 (2018), 867–885.
* [20] N. Dyn, A. Mokhov, Approximations of set-valued functions based on the metric average, Rend. Mat. Appl., VII. Ser. 26(3-4) (2006), 249–266.
* [21] S. Guo, On the rate of convergence of the Durrmeyer operator for functions of bounded variation, J. Approx. Theory 51 (1987), 183–192.
* [22] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Corp., Delhi (1960).
* [23] M. Mureşan, Set-valued approximation of multifunctions, Stud. Univ. Babeş-Bolyai, Math. 55 (1) (2010), 107–148.
* [24] M. S. Nikolskiĭ, Approximation of continuous convex-valued multivalued mappings (Russian), Optimization 21(2) (1990), 209–214.
* [25] R. T. Rockafellar, R. Wets, Variational Analysis, Springer-Verlag, Berlin (1998).
* [26] B. Sendov, V. Popov, The Averaged Moduli of Smoothness: Applications in Numerical Methods and Approximation, Wiley (1988).
* [27] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge University Press, Cambridge (1993).
* [28] R. A. Vitale, Approximations of convex set-valued functions, J. Approx. Theory 26 (1979), 301–316.
* [29] X. Zeng, A. Piriou, On the rate of convergence of two Bernstein-Bézier type operators for bounded variation functions J. Approx. Theory 95 (1998), 369–387.
|
# Enveloping norms on the spaces of regularly $\cal{P}$-operators in Banach
lattices
###### Abstract
We introduce and study the enveloping norms of regularly $\cal{P}$-operators
between Banach lattices $E$ and $F$, where $\cal{P}$ is a subspace of the
space $\text{\rm L}(E,F)$ of continuous operators from $E$ to $F$. We prove
that if $\cal{P}$ is closed in $\text{\rm L}(E,F)$ in the operator norm then
the regularly $\cal{P}$-operators forms a Banach space under the enveloping
norm. Conditions providing that regularly $\cal{P}$-operators forms a Banach
lattice under the enveloping norm are given.
Safak Alpay1, Eduard Emelyanov2, Svetlana Gorokhova 3
$1$ Department of Mathematics, Middle East Technical University, 06800 Ankara,
Turkey<EMAIL_ADDRESS>
$2$ Sobolev Institute of Mathematics, Acad. Koptyug avenue, 4, 630090,
Novosibirsk, Russia<EMAIL_ADDRESS>
$3$ Uznyj matematiceskij institut VNC RAN, Vatutin str., 53, 362025,
Vladikavkaz, Russia<EMAIL_ADDRESS>
Keywords: Banach lattice, regularly $\cal{P}$-operator, enveloping norm
MSC2020: 46B25, 46B42, 46B50, 47B60
Throughout the paper, all vector spaces are real, all operators are linear,
letters $E$ and $F$ denote Banach latices, $B_{E}$ denotes the closed unit
ball of $X$, $\text{\rm L}(E,F)$ (K(E,F); W(E,F)) denotes the space of all
continuous (resp. compact; weakly compact) operators from $E$ to $F$, and
$E_{+}$ the positive cone of $E$. Let ${\cal P}\subseteq\text{\rm L}(E,F)$. We
call elements of ${\cal P}$ by ${\cal P}$-operators and say that ${\cal
P}$-operators satisfy the domination property if $0\leq S\leq T\in{\cal P}$
implies $S\in{\cal P}$. An operator $T\in\text{\rm L}(E,F)$ is called ${\cal
P}$-dominated if $\pm T\leq U$ for some $U\in{\cal P}$. Under the assumption
${\cal P}\pm{\cal P}\subseteq{\cal P}\neq\emptyset$, ${\cal P}$-operators
satisfy the domination property iff each ${\cal P}$-dominated operator lies in
${\cal P}$. We refer to [1, 10, 12] for unexplained terminology and notations.
## 0.1
In the present paper we continue the investigation of regularly ${\cal
P}$-operators following [9, 2] and introduce enveloping norms on spaces of
regularly ${\cal P}$-operators. An operator $T:E\to F$ is called regular if
$T=T_{1}-T_{2}$ for some $T_{1},T_{2}\in\text{\rm L}_{+}(E,F)$. We denote by
$\text{\rm L}_{r}(E,F)$ (resp. $\text{\rm L}_{ob}(E,F)$, $\text{\rm
L}_{oc}(E,F)$) the ordered space of all regular (resp. order bounded,
o-continuous) operators in $\text{\rm L}(E,F)$. The space $\text{\rm
L}_{r}(E,F)$ is a Banach space under the regular norm [12, Prop.1.3.6]
$\|T\|_{r}:=\inf\\{\|S\|:\pm T\leq S\in\text{\rm L}(E,F)\\}.$ (1)
Furthermore, for every $T\in\text{\rm L}_{r}(E,F)$,
$\|T\|_{r}=\inf\\{\|S\|:S\in\text{\rm L}(E,F),|Tx|\leq S|x|\ \forall x\in
E\\}\geq\|T\|,$ (2)
and if $F$ is Dedekind complete then $(\text{\rm L}_{r}(E,F),\|\cdot\|_{r})$
is a Banach lattice with $\|T\|_{r}=\|~{}|T|~{}\|$ for each $T\in\text{\rm
L}_{r}(E,F)$ [12, Prop.1.3.6]. The following notion was introduced in [9,
Def.2] (cf. also [2, Def.1.5.1]).
###### Definition 1.
Let ${\cal P}\subseteq\text{\rm L}(E,F)$. An operator $T:E\to F$ is called a
regularly ${\cal P}$-operator (shortly, an r-${\cal P}$-operator) if
$T=T_{1}-T_{2}$ with $T_{1},T_{2}\in{\cal P}\cap\text{\rm L}_{+}(E,F)$. We
denote by:
1. ${\cal P}(E,F):={\cal P}$ the set of all ${\cal P}$-operators in $\text{\rm L}(E,F)$;
2. ${\cal P}_{r}(E,F)$ the set of all regular operators in ${\cal P}(E,F)$;
3. $\text{\rm r-}{\cal P}(E,F)$ the set of all r-${\cal P}$-operators in $\text{\rm L}(E,F)$.
###### Assertion 2.
([2, Prop.1.5.2]) Let ${\cal P}\subseteq\text{\rm L}(E,F)$, ${\cal P}\pm{\cal
P}\subseteq{\cal P}\neq\emptyset$, and $T\in\text{\rm L}(E,F)$. Then the
following holds.
1. (i)
$T$ is an r-${\cal P}$-operator iff $T$ is a ${\cal P}$-dominated ${\cal
P}$-operator.
2. (ii)
Suppose ${\cal P}$-operators satisfy the domination property and the modulus
$|T|$ exists in $\text{\rm L}(E,F)$. Then $T$ is an r-${\cal P}$-operator iff
$|T|\in{\cal P}$.
The next fact was established in [2, Prop.1.5.3].
###### Assertion 3.
$\text{\rm r-}{\cal P}(E,F)$ is a Dedekind complete vector lattice if $F$ is
Dedekind complete and ${\cal P}$ is a subspace of $\text{\rm L}(E,F)$
satisfying the domination property.
## 0.2
In the definition (1) of the regular norm, we replace $\text{\rm L}(E,F)$ by
an arbitrary subspace ${\cal P}$ of $\text{\rm L}(E,F)$ as follows:
$\|T\|_{\text{\rm r-}{\cal P}}:=\inf\\{\|S\|:\pm T\leq S\in{\cal P}\\}\ \ \ \
(T\in\text{\rm r-}{\cal P}(E,F)).$ (3)
###### Lemma 4.
For a subspace ${\cal P}$ of $\text{\rm L}(E,F)$, the formula
$(\ref{enveloping norm})$ defines a norm on $\text{\rm r-}{\cal P}(E,F)$,
called the enveloping norm. Moreover,
$\|T\|_{\text{\rm r-}{\cal P}}=\inf\\{\|S\|:S\in{\cal P}\ \&\ (\forall x\in
E)\ |Tx|\leq S|x|\\}\ \ \ (T\in\text{\rm r-}{\cal P}(E,F)).$ (4)
If ${\cal P}_{1}$ is a subspace of ${\cal P}$ then
$\|T\|_{\text{\rm r-}{\cal P}_{1}}\geq\|T\|_{\text{\rm r-}{\cal
P}}\geq\|T\|_{r}\geq\|T\|\ \ \ \ \ (\forall\ T\in\text{\rm r-}{\cal
P}_{1}(E,F)).$ (5)
###### Proof.
Only the triangle inequality and the formula (4) require explanations.
(A) Let $T_{1},T_{2}\in\text{\rm r-}{\cal P}(E,F)$ and $\varepsilon>0$. Pick
$S_{1},S_{2}\in{\cal P}$ with $\pm T_{1}\leq S_{1}$, $\pm T_{2}\leq S_{2}$,
$\|S_{1}\|\leq\|T_{1}\|_{\text{\rm r-}{\cal P}}+\varepsilon$, and
$\|S_{2}\|\leq\|T_{2}\|_{\text{\rm r-}{\cal P}}+\varepsilon$. Then
$\pm(T_{1}+T_{2})\leq S_{1}+S_{2}\in{\cal P}$, and $\|T_{1}+T_{2}\|_{\text{\rm
r-}{\cal P}}\leq\|S_{1}+S_{2}\|\leq\|S_{1}\|+\|S_{2}\|\leq\|T_{1}\|_{\text{\rm
r-}{\cal P}}+\|T_{2}\|_{\text{\rm r-}{\cal P}}+2\varepsilon.$ Since
$\varepsilon>0$ is arbitrary, $\|T_{1}+T_{2}\|_{\text{\rm r-}{\cal
P}}\leq\|T_{1}\|_{\text{\rm r-}{\cal P}}+\|T_{2}\|_{\text{\rm r-}{\cal P}}$.
(B) Denote the right side of (4) by $R(T)$. If $\pm T\leq S\in{\cal P}$ then
$\pm Tx=\pm(T(x_{+})-T(x_{-}))=\pm T(x_{+})\mp T(x_{-})\leq
S(x_{+})+S(x_{-})=S|x|$
for all $x\in E$. Then $|Tx|\leq S|x|$ for all $x\in E$, and hence
$\|T\|_{\text{\rm r-}{\cal P}}\geq R(T)$.
If $S\in{\cal P}$ satisfies $|Tx|\leq S|x|$ for all $x\in E$ then, for all
$y\in E_{+}$, $|Ty|\leq Sy$ and consequently $\pm Ty\leq Sy$. Therefore $\pm
T\leq S$, and hence $R(T)\geq\|T\|_{\text{\rm r-}{\cal P}}$. ∎
Let ${\cal P}$ be a subspace of $\text{\rm L}(E,F)$. What are the conditions
for ${\cal P}_{r}(E,F)=\text{\rm r-}{\cal P}(E,F)$? Trivially it happens for
${\cal P}=\text{\rm L}(E,F)$. The following proposition shows that if
$\text{\rm L}(E,F)$ is non-trival then we can always find a closed subspace
${\cal P}$ such that $\text{\rm r-}{\cal P}(E,F)$ is a proper subspace of
${\cal P}_{r}(E,F)$. More interesting case of the question is included in
Example 20 below.
###### Proposition 5.
Let $\dim(\text{\rm L}(E,F))>1$. Then there exists a one-dimensional subspace
${\cal P}$ of $\text{\rm L}(E,F)$ such that $\text{\rm r-}{\cal
P}(E,F)\subsetneqq{\cal P}_{r}(E,F)$.
###### Proof.
If $\dim(E)>1$ then $\dim(E^{\prime})>1$ and hence there are two linearly
independent functionals $g_{1},g_{2}\in E^{\prime}_{+}$. We take a nonzero
element $y\in F_{+}$ and set ${\cal P}:=\text{\rm span}((g_{1}-g_{2})\otimes
y)$.
If $\dim(F)>1$, there are two linearly independent elements $y_{1},y_{2}\in
F_{+}$. We take a nonzero functional $g\in E^{\prime}_{+}$ and set ${\cal
P}:=\text{\rm span}(g\otimes(y_{1}-y_{2}))$.
In the both cases ${\cal P}$ is one-dimensional subspace of $\text{\rm
L}(E,F)$. Moreover, $\text{\rm r-}{\cal P}(E,F)=\\{0\\}\neq{\cal P}={\cal
P}_{r}(E,F)$. ∎
## 0.3
The following result is an extension of [12, Prop.1.3.6], [9, Lm.1] (see also
[6, Prop.2.2] and [7, Thm.2.3] for particular cases). Its proof is a
modification of the proof of [12, Prop.1.3.6].
###### Theorem 6.
Let ${\cal P}$ be a subspace of $\text{\rm L}(E,F)$ closed in the operator
norm. Then $\text{\rm r-}{\cal P}(E,F)$ is a Banach space under the enveloping
norm.
###### Proof.
Let $(T_{n})$ be a $\|\cdot\|_{\text{\rm r-}{\cal P}}$-Cauchy sequence in
$\text{\rm r-}{\cal P}(E,F)$, say $T_{n}=P_{n}-R_{n}$ for $P_{n},R_{n}\in{\cal
P}\cap{\cal L}_{+}(E,F)$. Without lost of generality, we can assume
$\|T_{n+1}-T_{n}\|_{\text{\rm r-}{\cal P}}<2^{-n}\ \ \ (\forall
n\in\mathbb{N}).$
Since $\|\cdot\|_{\text{\rm r-}{\cal P}}\geq\|\cdot\|$, there exists a
$T\in\text{\rm L}(E,F)$ with $\|T-T_{n}\|\to 0$. We obtain $T\in{\cal P}$
because $T_{n}\in{\cal P}$ and ${\cal P}$ is closed in the operator norm. Pick
$S_{n}\in{\cal P}$ with $\|S_{n}\|<2^{-n}$ and $\pm(T_{n+1}-T_{n})\leq S_{n}$.
By (4),
$|(T_{n+1}-T_{n})x|\leq S_{n}|x|\ \ \ \ \ (\forall x\in E)(\forall
n\in\mathbb{N}).$
As ${\cal P}$ is closed,
$Q_{n}:=\|\cdot\|\text{\rm-}\sum\limits_{k=n}^{\infty}S_{k}\in{\cal P}$ for
each $n\in\mathbb{N}$. Since
$|(T-T_{n})x|=\lim\limits_{k\to\infty}|(T_{k}-T_{n})x|\leq\sum\limits_{k=n}^{\infty}|(T_{k+1}-T_{n})x|\leq
Q_{n}|x|\ \ \ \ \ (\forall x\in E),$
then $\pm(T-T_{n})\leq Q_{n}$ for all $n\in\mathbb{N}$. Thus
$-Q_{n}\leq(T-T_{n})\leq Q_{n}$ and hence $0\leq(T-T_{n})+Q_{n}$ for all
$n\in\mathbb{N}$. In particular,
$T=[(T-T_{n})+Q_{n}]+[T_{n}-Q_{n}]=[(T-T_{n})+Q_{n}+P_{n}]-[R_{n}+Q_{n}]\in\text{\rm
r-}{\cal P}(E,F),$
and hence $(T-T_{n})\in\text{\rm r-}{\cal P}(E,F)$ for all $n\in\mathbb{N}$.
Now,
$\|T-T_{n}\|_{\text{\rm r-}{\cal P}}\leq\|Q_{n}\|<2^{1-n}$
implies $(T_{n})\stackrel{{\scriptstyle\|\cdot\|_{\text{\rm r-}{\cal
P}}}}{{\to}}T$. ∎
###### Theorem 7.
Let $F$ be Dedekind complete and let ${\cal P}$ be a closed in the operator
norm subspace of $\text{\rm L}(E,F)$ satisfying the domination property. Then
$\text{\rm r-}{\cal P}(E,F)$ is a Dedekind complete Banach lattice under the
enveloping norm.
###### Proof.
By Assertion 3, $\text{\rm r-}{\cal P}(E,F)$ is a Dedekind complete vector
lattice. Theorem 6 implies that $\text{\rm r-}{\cal P}(E,F)$ is
$\|\cdot\|_{\text{\rm r-}{\cal P}}$-complete. Let $|S|\leq|T|$ for some
$S,T\in\text{\rm r-}{\cal P}(E,F)$. By Assertion 2 (ii), $|S|,|T|\in{\cal P}$,
and hence
$\|S\|_{\text{\rm r-}{\cal P}}=\|~{}|S|~{}\|=\sup\limits_{x\in E_{+}\cap
B_{E}}\||S|x\|\leq\sup\limits_{x\in E_{+}\cap
B_{E}}\||T|x\|=\|~{}|T|~{}\|=\|T\|_{\text{\rm r-}{\cal P}}.$
Therefore, $(\text{\rm r-}{\cal P}(E,F),\|\cdot\|_{\text{\rm r-}{\cal P}})$ is
a Banach lattice. ∎
## 0.4
In general, $\text{\rm r-}{\cal P}(E,F)\subsetneqq\text{\rm r-}{\overline{\cal
P}}(E,F)$, where ${\overline{\cal P}}$ is the norm-closure of ${\cal P}$ in
$\text{\rm L}(E,F)$. From the other hand, for ${\cal P}:=\text{\rm
r-L}_{ob}(E,F)$ (which is almost never closed in $\text{\rm L}(E,F)$ in the
operator norm), $\text{\rm r-}{\cal P}(E,F)=\text{\rm r-}{\overline{\cal
P}}(E,F)=\text{\rm L}_{r}(E,F)$. The enveloping norms on these three spaces
agree with the regular norm. The next result coupled with Example 9 shows that
the enveloping norm on $\text{\rm r-}{\cal P}(E,F)$ can be complete even if
${\cal P}(E,F)$ is not complete in the operator norm.
###### Proposition 8.
Let the norm in $F$ be o-continuous. Then $\text{\rm r-L}_{oc}(E,F)$ is a
Banach space under the enveloping norm.
###### Proof.
Let $(T_{n})$ be a Cauchy sequence in $\text{\rm r-L}_{oc}(E,F)$ in the
enveloping norm. We can assume that $\|T_{n+1}-T_{n}\|_{\text{\rm
r-L}_{oc}(E,F)}<2^{-n}$ for all $n\in\mathbb{N}$. Let $T\in\text{\rm L}(E,F)$
satisfy $\|T-T_{n}\|\to 0$. Pick $S_{n}\in\text{\rm L}_{oc}(E,F)$ with
$\|S_{n}\|<2^{-n}$ and $\pm(T_{n+1}-T_{n})\leq S_{n}$. First, we claim
$Q_{n}:=\|\cdot\|\text{\rm-}\sum\limits_{k=n}^{\infty}S_{k}\in\text{\rm
L}_{oc}(E,F)$ for all $n\in\mathbb{N}$. To prove the claim, it suffices to
show that $Q_{1}\in\text{\rm L}_{oc}(E,F)$. So, let $x_{\alpha}\downarrow 0$
in $E$. Passing to a tail we can assume that $\|x_{\alpha}\|\leq
M\in\mathbb{R}$ for all $\alpha$. Since $Q_{1}\geq 0$ then
$Q_{1}x_{\alpha}\downarrow\geq 0$ and hence in order to show that
$Q_{1}x_{\alpha}\downarrow 0$ it is enough to prove that
$\|Q_{1}x_{\alpha}\|\to 0$. Let $\varepsilon>0$. Fix an $m\in\mathbb{N}$ with
$\|Q_{m+1}\|\leq\varepsilon$. Since the positive operators $S_{1},...,S_{m}$
are all o-continuous, and since the norm in $F$ is o-continuous, there exists
an $\alpha_{1}$ such that
$\sum\limits_{k=1}^{m}\|S_{k}x_{\alpha}\|\leq\varepsilon$ for all
$\alpha\geq\alpha_{1}$. Since $\varepsilon>0$ is arbitrary, it follows from
$\|Q_{1}x_{\alpha}\|\leq\|\sum\limits_{k=1}^{m}S_{k}x_{\alpha}\|+\|Q_{m+1}x_{\alpha}\|\leq\varepsilon+M\|Q_{m+1}\|\leq
2\varepsilon\ \ \ \ (\forall\alpha\geq\alpha_{1})$
that $\|Q_{1}x_{\alpha}\|\to 0$, which proves our claim that
$Q_{n}\in\text{\rm L}_{oc}(E,F)$ for all $n\in\mathbb{N}$.
Since $\pm(T_{n+1}-T_{n})\leq S_{n}$ then by formula (4),
$|(T-T_{n})x|=\lim\limits_{k\to\infty}|(T_{k}-T_{n})x|\leq\sum\limits_{k=n}^{\infty}|(T_{k+1}-T_{n})x|\leq\sum\limits_{k=n}^{\infty}S_{n}|x|=Q_{n}|x|$
for all $x\in E$. In particular, $|T-T_{1}|\leq Q_{1}\in\text{\rm
L}_{oc}(E,F)$ and, since $\text{\rm L}_{oc}(E,F)$ is an order ideal in
$\text{\rm L}_{r}(E,F)$, then $(T-T_{1})\in\text{\rm L}_{oc}(E,F)$. Since
$T_{1}\in\text{\rm L}_{oc}(E,F)$, it follows $T\in\text{\rm L}_{oc}(E,F)$.
Now, $\|T-T_{n}\|_{\text{\rm r-L}_{oc}(E,F)}\leq\|Q_{n}\|<2^{1-n}$ implies
$(T_{n})\stackrel{{\scriptstyle\|\cdot\|_{\text{\rm
r-L}_{oc}(E,F)}}}{{\longrightarrow}}T$. ∎
###### Example 9.
Consider the following modification of Krengel’s example [11] (cf. [1,
Ex.5.6]$)$. Define a sequence $(S_{n})$ in $\text{\rm
L}_{oc}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0})$ by
$S_{n}x:=(2^{-\frac{1}{3}}T_{1}x_{1},2^{-\frac{2}{3}}T_{2}x_{2},\dots
2^{-\frac{n}{3}}T_{n}x_{n},0,0,\dots),$
where $T_{n}:\ell^{2}_{2^{n}}\to\ell^{2}_{2^{n}}$ is an isometry as in [1,
Ex.5.6]. Then $\|S_{n}-S\|\to 0$ for an operator $S\in\text{\rm L}(E,F)$,
defined by
$Sx:=(2^{-\frac{1}{3}}T_{1}x_{1},2^{-\frac{2}{3}}T_{2}x_{2},\dots
2^{-\frac{n}{3}}T_{n}x_{n},\dots).$
Notice that $(\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0}$ has o-continuous
norm, and hence is Dedekind complete. However $|S|$ does not exist. Thus, $S$
is not order bounded and hence $S\not\in\text{\rm
L}_{oc}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0})$.
## 0.5
We include several applications of Theorem 6 and Assertion 2. We begin with
recalling the following definition (see [5, 8]).
###### Definition 10.
A continuous operator $T:E\to F$ is called limited (resp. almost limited) if
$T^{\prime}$ takes $\text{\rm w}^{\ast}$-null (resp. disjoint $\text{\rm
w}^{\ast}$-null) sequences of $F^{\prime}$ to norm null sequences of
$E^{\prime}$. By $\text{\rm Lm}(E,F)$ (by $\text{\rm a-Lm}(E,F)$) we denote
the space of limited (resp. almost limited) operators from $E$ to $F$.
###### Lemma 11.
The spaces $\text{\rm a-Lm}(E,F)$ and $\text{\rm Lm}(E,F)$ are both closed in
$\text{\rm L}(E,F)$ under the operator norm.
###### Proof.
Suppose $\text{\rm a-Lm}(E,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,F)$. Let
$(f_{n})$ be disjoint $\text{\rm w}^{\ast}$-null in $F^{\prime}$. We need to
show that $(T^{\prime}f_{n})$ is norm null $X^{\prime}$. Let $\varepsilon>0$.
Choose any $k$ with $\|T^{\prime}_{k}-T^{\prime}\|\leq\varepsilon$. Since
$T_{k}\in\text{\rm a-Lm}(X,F)$, there exists $n_{0}$ such that
$\|T_{k}^{\prime}f_{n}\|\leq\varepsilon$ whenever $n\geq n_{0}$. As $(f_{n})$
is $\text{\rm w}^{\ast}$-null, there exists $M\in\mathbb{R}$ such that
$\|f_{n}\|\leq M$ for all $n\in\mathbb{N}$. Then
$\|T^{\prime}f_{n}\|\leq\|T^{\prime}_{k}f_{n}\|+\|T^{\prime}_{k}f_{n}-T^{\prime}f_{n}\|\leq\varepsilon+\|T^{\prime}_{k}-T^{\prime}\|\|f_{n}\|\leq\varepsilon+M\varepsilon$
for $n\geq n_{0}$. Since $\varepsilon>0$ is arbitrary then $(T^{\prime}f_{n})$
is norm null, as desired.
Similarly, the space $\text{\rm Lm}(E,F)$ is closed. ∎
The next result follows from Theorem 6 and Lemma 11.
###### Corollary 12.
For arbitrary Banach lattices $E$ and $F$, $\text{\rm r-Lm}(E,F)$ and
$\text{\rm r-a-Lm}(E,F)$ are both Banach spaces (each under its enveloping
norm).
###### Definition 13.
(see [10]) A continuous operator $T:E\to F$ is called Grothendieck (resp.
almost Grothendieck) if $T^{\prime}$ takes $\text{\rm w}^{\ast}$-null (resp.
disjoint $\text{\rm w}^{\ast}$-null) sequences of $F^{\prime}$ to w-null
sequences of $E^{\prime}$. By $\text{\rm G}(E,F)$ (by $\text{\rm a-G}(E,F)$)
we denote the space of Grothendieck (resp. almost Grothendieck) operators from
$E$ to $F$.
By Definitions 10 and 13,
$\text{\rm K}(E,F)\subseteq\text{\rm Lm}(E,F)\subseteq\text{\rm
a-Lm}(E,F)\subseteq\text{\rm a-G}(E,F),$ (6)
and
$\text{\rm Lm}(E,F)\subseteq\text{\rm G}(E,F)\subseteq\text{\rm a-G}(E,F).$
(7)
###### Lemma 14.
The spaces $\text{\rm a-G}(E,F)$ and $\text{\rm G}(E,F)$ are both closed in
$\text{\rm L}(E,F)$ under the operator norm.
###### Proof.
Suppose $\text{\rm a-G}(E,F)\ni
T_{k}\stackrel{{\scriptstyle\|\cdot\|}}{{\to}}T\in\text{\rm L}(E,F)$. Let
$(f_{n})$ be disjoint $\text{\rm w}^{\ast}$-null in $F^{\prime}$. We need to
show that $(T^{\prime}f_{n})$ is w-null in $E^{\prime}$. So, take any $g\in
F^{\prime\prime}$. Let $\varepsilon>0$. Pick a $k$ with
$\|T^{\prime}_{k}-T^{\prime}\|\leq\varepsilon$. Since $T_{k}\in\text{\rm
a-G}(E,F)$, there exists $n_{0}$ such that
$|g(T_{k}^{\prime}f_{n})|\leq\varepsilon$ whenever $n\geq n_{0}$. Note that
$\|f_{n}\|\leq M$ for some $M\in\mathbb{R}$ and for all $n\in\mathbb{N}$.
Since $\varepsilon>0$ is arbitrary, it follows from
$|g(T^{\prime}f_{n})|\leq|g(T_{k}^{\prime}f_{n}-T^{\prime}f_{n})|+|g(T^{\prime}f_{n})|\leq\|g\|\|T_{k}^{\prime}-T^{\prime}\|\|f_{n}\|+\varepsilon\leq(\|g\|M+1)\varepsilon$
for $n\geq n_{0}$, that $g(T^{\prime}f_{n})\to 0$. Since $g\in
F^{\prime\prime}$ is arbitrary, $T\in\text{\rm a-G}(E,F)$.
Similarly, the space $\text{\rm G}(E,F)$ is closed. ∎
The next result follows from Theorem 6 and Lemma 14.
###### Corollary 15.
For arbitrary Banach lattices $E$ and $F$, $\text{\rm r-G}(E,F)$ and
$\text{\rm r-a-G}(E,F)$ are both Banach spaces (each under its enveloping
norm).
We continue with the following definition.
###### Definition 16.
A continuous operator $T:E\to F$ is called:
1. a)
Dunford–Pettis (shortly, $T\in\text{\rm DP}(E,F)$) if $T$ takes w-null
sequences to norm null ones (cf. [1, p.340]).
2. b)
weak Dunford–Pettis (shortly, $T\in\text{\rm wDP}(E,F)$) if $f_{n}(Tx_{n})\to
0$ whenever $(f_{n})$ is w-null in $F^{\prime}$ and $(x_{n})$ is w-null in $E$
[1, p.349].
3. c)
almost Dunford–Pettis (shortly, $T\in\text{\rm a-DP}(E,F)$) if $T$ takes
disjoint w-null sequences to norm null ones [13];
4. d)
almost weak Dunford–Pettis (shortly, $T\in\text{\rm a-wDP}(E,F)$) if
$f_{n}(Tx_{n})\to 0$ whenever $(f_{n})$ is w-null in $F^{\prime}$ and
$(x_{n})$ is disjoint w-null in $E$.
Clearly,
$\text{\rm K}(E,F)\subseteq\text{\rm DP}(E,F)\subseteq\text{\rm
wDP}(E,F)\bigcap\text{\rm a-DP}(E,F),$ (8)
and
$\text{\rm wDP}(E,F)\subseteq\text{\rm a-wDP}(E,F).$ (9)
The operator $L^{1}[0,1]\stackrel{{\scriptstyle T}}{{\to}}\ell^{\infty}$,
$Tf=\left(\int_{0}^{1}f(t)r^{+}_{k}(t)\,dt\right)_{k=1}^{\infty}$ is wDP yet
not DP, as $r_{n}\stackrel{{\scriptstyle\text{\rm w}}}{{\to}}0$ and
$\|Tr_{n}\|\geq\int_{0}^{1}r_{n}(t)r^{+}_{n}(t)\,dt\equiv\frac{1}{2}$. By [3,
Thm.4.1], $\text{\rm a-DP}(E,F)=\text{\rm DP}(E,F)$ for all $F$ iff the
lattice operations in $E$ are sequentially w-continuous. The identity
operator:
1. $I:L^{1}[0,1]\to L^{1}[0,1]$ is a-DP yet not DP;
2. $I:c\to c$ is a-wDP yet neither wDP nor a-DP.
The domination property for a-DP-operators was established in [4, Cor.2.3]. We
omit the proof of the next proposition as it is a modification of the proof of
the Kalton–Saab domination theorem (cf. [1, Thm.5.101].
###### Proposition 17.
Any positive operator dominated by an a-wDP-operator is likewise an a-wDP-
operator.
We also omit the straightforward proof of the following fact.
###### Lemma 18.
The spaces $\text{\rm a-DP}(E,F)$ and $\text{\rm a-wDP}(E,F)$ are both closed
in $\text{\rm L}(E,F)$ under the operator norm.
By [4, Cor.2.3], Proposition 17, and Lemma 18, the next result follows from
Theorem 7.
###### Corollary 19.
If $F$ is Dedekind complete then $\text{\rm r-a-DP}(E,F)$ and
$\text{\rm r-a-wDP}(E,F)$ are both Banach lattices under their enveloping
norms.
## 0.6
Finally, we discuss once more the Krengel example (cf. [1, Ex.5.6].
###### Example 20.
Let $A_{n}$ be a $2^{n}\times 2^{n}$ matrix constructed inductively:
$A_{1}=\left[\begin{matrix}1&1\\\ 1&-1\end{matrix}\right]\quad\text{\rm
and}\quad A_{n+1}=\left[\begin{matrix}A_{n}&A_{n}\\\
A_{n}&-A_{n}\end{matrix}\right].$
For each $n$, let $T_{n}:\ell^{2}_{2^{n}}\to\ell^{2}_{2^{n}}$ be an isometry
defined by the orthogonal matrix $2^{-\frac{n}{2}}A_{n}$. Then
$|T_{n}|\in\text{\rm L}(\ell^{2}_{2^{n}})$ is the operator, whose $2^{n}\times
2^{n}$ coordinate matrix has all entries equal to $2^{-\frac{n}{2}}$, in
particular $\||T_{n}|\|=2^{\frac{n}{2}}$ for all $n$. Consider the
$c_{0}$-direct sum $E:=(\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0}$. Note that
$E$ has o-continuous norm and hence is Dedekind complete. Define $T:E\to E$,
by
$Tx:=(2^{-\frac{n}{2}}T_{n}x_{n})\ \ \ \ (x=(x_{1},x_{2},\dots,x_{n},\dots)\in
E).$ (10)
It is straightforward to see that $T\in\text{\rm K}(E)$, and then
$T\in\text{\rm a-G}(E)$ by (6), and $T\in\text{\rm a-DP}(E)$ by (8). The
modulus $|T|$ exists and is given by
$|T|x=(2^{-\frac{n}{2}}|T_{n}|x_{n})\ \ \ \ (x\in E).$ (11)
However, $|T|\not\in\text{\rm a-G}(E)$. Indeed, consider a disjoint $\text{\rm
w}^{\ast}$-null sequence $(f_{n})$ in
$E^{\prime}=(\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{1}$, where all terms in
$f_{n}$ are zero, except the $n$-th term which is equal to
$(2^{-\frac{n}{2}},2^{-\frac{n}{2}},\dots,2^{-\frac{n}{2}})\in\ell^{2}_{2^{n}}$.
Since $|T|^{\prime}\in\text{\rm
L}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{1})$ satisfies
$|T|^{\prime}y=(2^{-\frac{n}{2}}|T_{n}|^{\prime}y_{n})$ for all $y\in
E^{\prime}$, where the coordinate matrix of $|T_{n}|^{\prime}$ has all entries
equal to $2^{-\frac{n}{2}}$, then all terms of $|T|^{\prime}f_{n}$ are zero,
except the $n$-th term which is equal to
$(2^{-\frac{n}{2}},2^{-\frac{n}{2}},\dots,2^{-\frac{n}{2}})\in\ell^{2}_{2^{n}}$.
Take $g\in E^{\prime\prime}=(\oplus_{k=1}^{\infty}\ell^{2}_{2^{k}})_{\infty}$
by letting each $k$-th term of $g$ equals to
$(2^{-\frac{k}{2}},2^{-\frac{k}{2}},\dots,2^{-\frac{k}{2}})\in\ell^{2}_{2^{k}}$,
then
$g(|T|^{\prime}f_{n})=\sum\limits_{i=1}^{2^{n}}2^{-\frac{n}{2}}\cdot
2^{-\frac{n}{2}}=\sum\limits_{i=1}^{2^{n}}2^{-n}=1\not\to 0,$
and hence $T\not\in\text{\rm a-G}(E)$. In particular, $T\not\in\text{\rm
a-Lm}(E)$ by (6). Now, take a disjoint w-null sequence $(x_{n})$ in
$E=(\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0}$, whose terms are zero, except
the $n$-th one which is equals to
$(2^{-\frac{n}{2}},2^{-\frac{n}{2}},\dots,2^{-\frac{n}{2}})\in\ell^{2}_{2^{n}}$.
Then $|T|x_{n}$ has all the terms zero, except the $n$-th term which is equal
to $(2^{-\frac{n}{2}},2^{-\frac{n}{2}},\dots,2^{-\frac{n}{2}})$. Thus
$\||T|x_{n}\|=1$ for all $n$, and hence $|T|\not\in\text{\rm a-DP}(E)$.
Since a-G\- and a-DP-operators satisfy the domination property, it follows
from Assertion 2 (ii) that $T\in\text{\rm
a-G}_{r}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0})\setminus\text{\rm
r-a-G}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0})$ and
$T\in\text{\rm
a-DP}_{r}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0})\setminus\text{\rm r-a-
DP}((\oplus_{n=1}^{\infty}\ell^{2}_{2^{n}})_{0})$.
## References
* [1] C. D. Aliprantis, O. Burkinshaw, Positive Operators. Springer, Dordrecht, 2006.
* [2] S. Alpay, E. Emelyanov, S. Gorokhova, Generalizations of L- and M-weakly compact operators. https://arxiv.org/abs/2206.02718v4
* [3] B. Aqzzouz, K. Bouras, (L) sets and almost (L) sets in Banach lattices. Quaest. Math. 36, no.1 (2013), 107–118.
* [4] B. Aqzzouz, A. Elbour, Some characterizations of almost Dunford–Pettis operators and applications. Positivity 15 (2011), 369–380.
* [5] J. Bourgain, J. Diestel, Limited operators and strict cosingularity. Math. Nachr. 119 (1984), 55–58.
* [6] Z. L. Chen, A. W. Wickstead, Incompleteness of the linear span of the positive compact operators. Proc. Amer. Math. Soc. 125 (1997), 3381–3389.
* [7] N. Cheng, Dedekind $\sigma$-complete vector lattice of b-AM-compact operators. Quaest. Math. 40 (2017), no. 3, 313–318.
* [8] A. Elbour, Some characterizations of almost limited operators. Positivity 21 (2017), 865–874.
* [9] E. Emelyanov, Algebras of Lebesgue and KB regular operators on Banach lattices. https://arxiv.org/abs/2203.08326v3
* [10] P. Galindo, V. C. C. Miranda, Grothendieck-type subsets of Banach lattices. J. Math. Anal. Appl. 506 (2022)
* [11] U. Krengel, Remark on the modulus of compact operators. Bull. Amer. Math. Soc. 72 (1966), 132–133.
* [12] P. Meyer-Nieberg, Banach Lattices. Universitext, Springer-Verlag, Berlin 1991.
* [13] J. A. Sanchez, Operators on Banach lattices. Ph. D. Thesis, Complutense University, Madrid, 1985.
|
# Nontrivial lower bounds for the $p$-adic valuations of some type of rational
numbers
Bakir FARHI
Laboratoire de Mathématiques appliquées
Faculté des Sciences Exactes
Université de Bejaia, 06000 Bejaia, Algeria
<EMAIL_ADDRESS>
http://farhi.bakir.free.fr/
###### Abstract
In this paper, we will show that the $p$-adic valuation (where $p$ is a given
prime number) of some type of rational numbers is unusually large. This
generalizes the very recent results by the author and by A. Dubickas, which
are both related to the special case $p=2$. The crucial point for obtaining
our main result is the fact that the $p$-adic valuation of the rational
numbers in question is unbounded from above. We will confirm this fact by
three different methods; the first two are elementary while the third one
leans on the $p$-adic analysis.
MSC 2010: Primary 11B83; Secondary 11A41, 05A10, 05A19, 11B65.
Keywords: $p$-adic valuations, binomial coefficients, combinatorial
identities, $p$-adic analysis.
## 1 Introduction and Notation
Throughout this paper, we let ${\mathbb{N}}$ denote the set of positive
integers and ${\mathbb{N}}_{0}:={\mathbb{N}}\cup\\{0\\}$ denote the set of
non-negative integers. For $x\in{\mathbb{R}}$, we let $\lfloor x\rfloor$
denote the integer part of $x$. For a given prime number $p$ and a given non-
zero rational number $r$, we let $\vartheta_{p}(r)$ denote the usual $p$-adic
valuation of $r$; if in addition $r$ is positive then we let $\log_{p}(r)$
denote its logarithm to the base $p$ (i.e.,
$\log_{p}(r):=\frac{\log{r}}{\log{p}}$). Next, the least common multiple of
given positive integers $u_{1},u_{2},\dots,u_{n}$ ($n\in{\mathbb{N}}$) is
denoted by $\mathrm{lcm}(u_{1},u_{2},\dots,u_{n})$ or by
$\mathrm{lcm}\\{u_{1},u_{2},\dots,u_{n}\\}$ if this is more convenient. In
several places of this paper, we will use the immediate estimate
$\vartheta_{p}(n)\leq\log_{p}(n)$ (for any prime $p$ and any
$n\in{\mathbb{N}}$). We also often use the immediate formula
$\vartheta_{p}\left(\mathrm{lcm}(1,2,\dots,n)\right)=\left\lfloor\log_{p}(n)\right\rfloor$
(for any prime $p$ and any $n\in{\mathbb{N}}$). At the end of the paper, we
need to use the $p$-adic logarithm function which we denote by $L_{p}$ (to
differentiate from the notation $\log_{p}$, which is reserved to denote the
logarithm to the base $p$). With the usual notation ${\mathbb{Q}}_{p}$ for the
field of $p$-adic numbers, ${\mathbb{C}}_{p}$ for the field of the $p$-adic
complex numbers, and ${|\cdot|}_{p}$ for the usual $p$-adic absolute value on
${\mathbb{C}}_{p}$, recall that $L_{p}$ can be defined by:
$-L_{p}(1-x):=\sum_{k=1}^{+\infty}\frac{x^{k}}{k}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
x\in{\mathbb{C}}_{p},{|x|}_{p}<1).$
(See [5]). The fundamental property of $L_{p}$ is that it satisfies the
functional equation:
$L_{p}(uv)=L_{p}(u)+L_{p}(v)$
(for all $u,v\in{\mathbb{C}}_{p}$, with ${|u-1|}_{p}<1$ and ${|v-1|}_{p}<1$).
In [2, 3], the author have obtained nontrivial lower bounds for the $2$-adic
valuation of the rational numbers of the form $\sum_{k=1}^{n}\frac{2^{k}}{k}$
($n\in{\mathbb{N}}$). The stronger one is
$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq
n-\left\lfloor\log_{2}(n)\right\rfloor~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}).$ (1.1)
In [3], the author also posed the problem of generalizing (1.1) to other prime
numbers $p$ other than $p=2$. In [1], Dubickas has found arguments to
improving and optimizing (1.1) by leaning only on the fact that the sequence
$\left\\{\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\right\\}_{n\geq
1}$ is unbounded from above. However, he does not established any way to prove
this fact without using (1.1). The main result in [1] states that we have for
any $n\in{\mathbb{N}}$:
$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq(n+1)-\log_{2}(n+1),$
(1.2)
with equality if and only if $n$ has the form $n=2^{\alpha}-1$
($\alpha\in{\mathbb{N}}$).
The goal of this paper is twofold. On the one hand, we expand and improve the
arguments in [1] to establish a general result providing to us nontrivial
lower bounds for the $p$-adic valuation of a sum of rational numbers under
some conditions (see Theorem 2.2). On the other hand, we solve the problem
posed in [3] by generalizing (1.1) and (1.2) to other prime numbers.
Precisely, we show (in different ways) that for any prime number $p$ and any
non-multiple integer $a$ of $p$, the sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\right\\}_{n\geq
1}$ (1.3)
is unbounded from above. Then, by using our general theorem 2.2, we derive an
optimal lower bound for the sequence in (1.3). It must be noted that the
crucial point of the non-boundness from above of the sequence in (1.3) is
established by three methods. The first two are elementary and effective while
the third one leans on the $p$-adic analysis and it is ineffective; precisely,
it uses the function $L_{p}$ described above. Personally, we consider that the
deep reason why the sequence in (1.3) is unbounded from above is rather given
by the third method.
## 2 The results and the proofs
Our main result is the following:
###### Theorem 2.1.
Let $p$ be a prime number and $a$ be an integer not multiple of $p$. Then we
have for all positive integer $n$:
$\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\geq(n+1)-\log_{p}\left(\frac{n+1}{2}\right).$
(2.1)
In addition, this inequality becomes an equality if and only if $n$ has the
form $n=2p^{\alpha}-1$ ($\alpha\in{\mathbb{N}}_{0}$).
Note that Theorem 2.1 generalizes the recent results of the author [2, 3] and
Dubickas [1], which are both related to the particular case $p=2$. Especially,
if we take $p=2$ and $a=1$ in Theorem 2.1, we exactly obtain (after some
obvious simplifications) the main result of [1], stating that:
$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq(n+1)-\log_{2}(n+1)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}),$
with equality if and only if $n$ has the form $(2^{\alpha}-1)$
($\alpha\in{\mathbb{N}}$).
The proof of Theorem 2.1 is based in part on the following result which can
serve us well in other situations for bounding from below the $p$-adic
valuation of a sum of rational numbers when it is unbounded from above. It
must be also noted that the result below is obtained by generalizing the
arguments in [1].
###### Theorem 2.2.
Let $p$ be a fixed prime number and ${(r_{n})}_{n\geq 1}$ be a sequence of
rational numbers such that the sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)\right\\}_{n\geq 1}$ is
unbounded from above. Let also ${(\ell_{k})}_{k\geq 2}$ be an increasing real
sequence satisfying the property:
$\ell_{k}\leq\vartheta_{p}(r_{k})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall k\geq
2).$ (2.2)
Then we have for any positive integer $n$:
$\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)\geq\min_{k\geq
n+1}\vartheta_{p}(r_{k})\geq\ell_{n+1}.$ (2.3)
In addition, the inequality
$\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)\geq\ell_{n+1}$ becomes an
equality if and only if we have
$\min_{k\geq n+1}\vartheta_{p}(r_{k})=\ell_{n+1}.$ (2.4)
Our main result (i.e., Theorem 2.1) is proved in two steeps: in the first one,
we suppose (in the situation of Theorem 2.1) that the sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\right\\}_{n\geq
1}$ is unbounded from above and we apply for it Theorem 2.2 to prove the lower
bound (2.1) and to characterize the $n$’s for which it is attained. In the
second one, we return to prove the non-boundness from above of the considered
sequence. We do this by three different methods: the first one is based on two
identities, one is combinatorial and the other is arithmetic. The second one
uses a certain functional equation and the Taylor polynomials. The third one
uses the $p$-adic analysis; precisely the $p$-adic logarithm function.
Let us begin by proving Theorem 2.2.
###### Proof of Theorem 2.2.
Let $n$ be a fixed positive integer. Let us show the first inequality of
(2.3). Since, by hypothesis, the sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{N}r_{k}\right)\right\\}_{N\geq 1}$ is
unbounded from above then there exists $m\in{\mathbb{N}}$, with $m>n$, such
that:
$\vartheta_{p}\left(\sum_{k=1}^{m}r_{k}\right)>\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right).$
Then, by using the elementary properties of the $p$-adic valuation, we have on
the one hand:
$\vartheta_{p}\left(\sum_{k=n+1}^{m}r_{k}\right)=\vartheta_{p}\left(\sum_{k=1}^{m}r_{k}-\sum_{k=1}^{n}r_{k}\right)=\min\left(\vartheta_{p}\left(\sum_{k=1}^{m}r_{k}\right),\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)\right)=\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)$
and on the other hand:
$\vartheta_{p}\left(\sum_{k=n+1}^{m}r_{k}\right)\geq\min_{n+1\leq k\leq
m}\vartheta_{p}(r_{k})\geq\min_{k\geq n+1}\vartheta_{p}(r_{k}).$
By comparing these two results, we deduce that:
$\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)\geq\min_{k\geq
n+1}\vartheta_{p}(r_{k}),$
which is nothing else the first inequality of (2.3). The second inequality of
(2.3) is immediately derived from its first one together with the properties
of the sequence ${(\ell_{k})}_{k\geq 2}$. Indeed, we have
$\displaystyle\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)$
$\displaystyle\geq\min_{k\geq
n+1}\vartheta_{p}(r_{k})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{by the first
inequality of \eqref{eq2}})$ $\displaystyle\geq\min_{k\geq
n+1}\ell_{k}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{by using
\eqref{eq1}})$
$\displaystyle=\ell_{n+1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{since
}{(\ell_{k})}_{k}\text{ is increasing by hypothesis}),$
confirming the second inequality of (2.3).
Now, let us prove the second part of Theorem 2.2. If
$\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)=\ell_{n+1}$ then we have
(according to (2.3), proved above): $\min_{k\geq
n+1}\vartheta_{p}(r_{k})=\ell_{n+1}$, as required. Conversely, suppose that
$\min_{k\geq n+1}\vartheta_{p}(r_{k})=\ell_{n+1}$ and let us show that
$\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)=\ell_{n+1}$. To do so, we first
show that:
$\vartheta_{p}(r_{n+1})<\vartheta_{p}(r_{k})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k>n+1).$ (2.5)
To prove (2.5), let us argue by contradiction. So, suppose that there is an
integer $k_{0}>n+1$ which satisfies
$\vartheta_{p}(r_{n+1})\geq\vartheta_{p}(k_{0})$. So we have
$\min_{k\geq n+1}\vartheta_{p}(r_{k})=\min_{k\geq
n+2}\vartheta_{p}(r_{k})\geq\min_{k\geq n+2}\ell_{k}=\ell_{n+2}>\ell_{n+1}$
(according to (2.2) and the increase of ${(\ell_{k})}_{k}$), contradicting the
supposition $\min_{k\geq n+1}\vartheta_{p}(r_{k})=\ell_{n+1}$. This
contradiction confirms (2.5). Now, we shall use (2.5) to prove the desired
equality $\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)=\ell_{n+1}$. On the
one hand, we have (according to the first part of this proof):
$\vartheta_{p}\left(\sum_{k=n+1}^{m}r_{k}\right)=\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right).$
But on the other hand, we have (according to (2.5) and the elementary
properties of the $p$-adic valuation):
$\vartheta_{p}\left(\sum_{k=n+1}^{m}r_{k}\right)=\min_{n+1\leq k\leq
m}\vartheta_{p}(r_{k})=\vartheta_{p}(r_{n+1})=\min_{k\geq
n+1}\vartheta_{p}(r_{k})=\ell_{n+1}.$
Comparing the two results, we derive the required equality:
$\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)=\ell_{n+1}$. This confirms the
second part of Theorem 2.2 and completes this proof. ∎
Next, we have the following fundamental result:
###### Theorem 2.3.
Let $p$ be a prime number and $a$ be an integer not multiple of $p$. Then the
sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\right\\}_{n\geq
1}$
is unbounded from above.
Admitting for the moment Theorem 2.3, our main result is obtained as an
application of Theorem 2.2.
###### Proof of Theorem 2.1 by admitting Theorem 2.3.
Let us put ourselves in the situation of Theorem 2.1. We apply Theorem 2.2
with $r_{k}:=\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}$
($\forall k\in{\mathbb{N}}$) and
$\ell_{k}:=k-\log_{p}\left(\frac{k}{2}\right)$ ($\forall k\geq 2$). The non-
boundness from above of the sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{n}r_{k}\right)\right\\}_{n\geq 1}$ is
guaranteed by Theorem 2.3 (admitted for the moment). Next, the increase of the
sequence ${(\ell_{k})}_{k\geq 2}$ can be derived from the increase of the
function $x\mapsto x-\log_{p}\left(\frac{x}{2}\right)$ in the interval
$[2,+\infty)$. Finally, we have for any integer $k\geq 2$:
$\displaystyle\vartheta_{p}(r_{k})$
$\displaystyle=\vartheta_{p}\left(\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)$
$\displaystyle=\vartheta_{p}\left(\frac{a^{k}+(p-a)^{k}}{\left(a(p-a)\right)^{k}}\cdot\frac{p^{k}}{k}\right)$
$\displaystyle=\vartheta_{p}\left(a^{k}+(p-a)^{k}\right)+k-\vartheta_{p}(k)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{since
}a\text{ is coprime with }p).$
If $k$ is even, we use
$\vartheta_{p}\left(a^{k}+(p-a)^{k}\right)\geq\vartheta_{p}(2)$ (for $p>2$,
this is obvious and for $p=2$, observe that $a^{k}+(p-a)^{k}$ is even). So, we
obtain
$\vartheta_{p}(r_{k})\geq k-\vartheta_{p}\left(\frac{k}{2}\right)\geq
k-\log_{p}\left(\frac{k}{2}\right)=\ell_{k}$
(because $k/2$ is a positive integer if $k$ is even). However, if $k$ is odd,
we use $\vartheta_{p}\left(a^{k}+(p-a)^{k}\right)\geq 1$ (since
$a^{k}+(p-a)^{k}\equiv a^{k}+(-a)^{k}\ (\mathrm{mod}\ p)\equiv 0\
(\mathrm{mod}\ p)$). So, we obtain again:
$\vartheta_{p}(r_{k})\geq
1+k-\vartheta_{p}(k)\geq\log_{p}(2)+k-\log_{p}(k)=k-\log_{p}\left(\frac{k}{2}\right)=\ell_{k}.$
Consequently, we have for any integer $k\geq 2$:
$\vartheta_{p}(r_{k})\geq\ell_{k}$. So, all the hypothesis of Theorem 2.2 are
satisfied; thus we can apply it for our situation. Applying the first part of
Theorem 2.2, we get for any positive integer $n$:
$\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\geq\min_{k\geq
n+1}\vartheta_{p}\left(\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\geq(n+1)-\log_{p}\left(\frac{n+1}{2}\right),$
confirming Inequality (2.1) of Theorem 2.1. Next, for a given positive integer
$n$, the second part of Theorem 2.2 tells us that (2.1) becomes an equality if
and only if we have
$\min_{k\geq
n+1}\vartheta_{p}\left(\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)=(n+1)-\log_{p}\left(\frac{n+1}{2}\right),$
that is
$\min_{k\geq
n+1}\vartheta_{p}\left(\left(a^{k}+(p-a)^{k}\right)\frac{p^{k}}{k}\right)=(n+1)-\log_{p}\left(\frac{n+1}{2}\right).$
(2.6)
So, it remains to prove that (2.6) holds if and only if $n$ has the form
$n=2p^{\alpha}-1$ ($\alpha\in{\mathbb{N}}_{0}$). Let us prove this last fact.
• Suppose that (2.6) holds. Then, we have
$\log_{p}\left(\frac{n+1}{2}\right)=(n+1)-\min_{k\geq
n+1}\vartheta_{p}\left(\left(a^{k}+(p-a)^{k}\right)\frac{p^{k}}{k}\right)\in{\mathbb{Z}}.$
But since $\log_{p}\left(\frac{n+1}{2}\right)\geq 0$, we have even
$\log_{p}\left(\frac{n+1}{2}\right)\in{\mathbb{N}}_{0}$. By setting
$\alpha:=\log_{p}\left(\frac{n+1}{2}\right)\in{\mathbb{N}}_{0}$, we get
$n=2p^{\alpha}-1$, as required.
• Conversely, suppose that $n=2p^{\alpha}-1$ for some
$\alpha\in{\mathbb{N}}_{0}$. We first claim that we have
$\vartheta_{p}\left(a^{n+1}+(p-a)^{n+1}\right)=\vartheta_{p}(2).$ (2.7)
To confirm (2.7), we distinguish two cases:
— 1st case: (If $p=2$). In this case, because $a$ is coprime with $p$ then $a$
and $(p-a)$ are both odd, implying that $a^{2}\equiv 1\ (\mathrm{mod}\ 4)$ and
$(p-a)^{2}\equiv 1\ (\mathrm{mod}\ 4)$. Then, because $n+1=2p^{\alpha}$ is
even, we have also $a^{n+1}\equiv 1\ (\mathrm{mod}\ 4)$ and $(p-a)^{n+1}\equiv
1\ (\mathrm{mod}\ 4)$; thus $a^{n+1}+(p-a)^{n+1}\equiv 2\ (\mathrm{mod}\ 4)$,
implying that
$\vartheta_{p}\left(a^{n+1}+(p-a)^{n+1}\right)=1=\vartheta_{p}(2)$.
— 2nd case: (If $p$ is odd). In this case, because $n+1=2p^{\alpha}$ is even,
we have $a^{n+1}+(p-a)^{n+1}\equiv a^{n+1}+(-a)^{n+1}\ (\mathrm{mod}\ p)\equiv
2a^{n+1}\ (\mathrm{mod}\ p)\not\equiv 0\ (\mathrm{mod}\ p)$ (since $p$ is
assumed odd and $a$ is coprime with $p$). Thus
$\vartheta_{p}\left(a^{n+1}+(p-a)^{n+1}\right)=0=\vartheta_{p}(2)$. Our claim
(2.7) is proved. Now, using (2.7), we have
$\vartheta_{p}\left(\left(a^{n+1}+(p-a)^{n+1}\right)\frac{p^{n+1}}{n+1}\right)=\vartheta_{p}(2)+(n+1)-\vartheta_{p}(n+1)=(n+1)-\vartheta_{p}\left(\frac{n+1}{2}\right)\\\
=n+1-\vartheta_{p}\left(p^{\alpha}\right)=n+1-\alpha=n+1-\log_{p}\left(\frac{n+1}{2}\right).$
This shows that (2.6) is equivalent to:
$\vartheta_{p}\left(\left(a^{k}+(p-a)^{k}\right)\frac{p^{k}}{k}\right)\geq(n+1)-\log_{p}\left(\frac{n+1}{2}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k\geq n+2),$
which is weaker than:
$k-(n+1)\geq\vartheta_{p}(k)-\log_{p}\left(\frac{n+1}{2}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k\geq n+2).$ (2.8)
Let us prove (2.8) for a given integer $k\geq n+2$. To do so, we distinguish
two cases according to whether $\vartheta_{p}(k)\leq\alpha+1$ or not.
— 1st case: (If $\vartheta_{p}(k)\leq\alpha+1$). In this case, we have
$k-(n+1)\geq
1~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\vartheta_{p}(k)-\log_{p}\left(\frac{n+1}{2}\right)=\vartheta_{p}(k)-\alpha\leq
1.$
Thus (2.8) is true.
— 2nd case: (If $\vartheta_{p}(k)\geq\alpha+2$). In this case, we have $k\geq
p^{\alpha+2}\geq 2p^{\alpha+1}$. Hence
$k-(n+1)=k-2p^{\alpha}\geq\frac{p-1}{p}k\geq\frac{k}{p}\geq
p^{\vartheta_{p}(k)-1}\geq
2^{\vartheta_{p}(k)-1}\geq\vartheta_{p}(k)\geq\vartheta_{p}(k)-\log_{p}\left(\frac{n+1}{2}\right),$
confirming (2.8) for this case also.
Consequently, (2.8) is valid for any integer $k\geq n+2$. This completes the
proof of the second part of Theorem 2.1 and achieves this proof. ∎
The rest of the paper is now devoted to prove Theorem 2.3. We achieve this by
three different methods:
### The first method
We lean on two identities. The first one (due to Mansour [6]) is combinatorial
and states that:
$\sum_{k=0}^{n}\frac{x^{k}y^{n-k}}{\binom{n}{k}}=\frac{n+1}{(x+y)\left(\frac{1}{x}+\frac{1}{y}\right)^{n+1}}\sum_{k=1}^{n+1}\frac{\left(x^{k}+y^{k}\right)\left(\frac{1}{x}+\frac{1}{y}\right)^{k}}{k}$
(2.9)
(for any $x,y\in{\mathbb{R}}^{*}$, with $x+y\neq 0$, and any
$n\in{\mathbb{N}}_{0}$). While the second one (due to the author [4]) is
arithmetic and states that:
$\mathrm{lcm}\left\\{\binom{n}{0},\binom{n}{1},\dots,\binom{n}{n}\right\\}=\frac{\mathrm{lcm}\left(1,2,\dots,n,n+1\right)}{n+1}$
(2.10)
(for any $n\in{\mathbb{N}}_{0}$).
Using (2.9) and (2.10), we are now ready to prove Theorem 2.3. Let $p$ be a
prime number and $a$ be an integer non-multiple of $p$. By applying (2.9) for
$x=a$ and $y=p-a$ and replacing $n$ by $(n-1)$ (where $n\in{\mathbb{N}}$), we
get (after simplifying and rearranging)
$\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}=\frac{p^{n+1}}{n\left(a(p-a)\right)^{n}}\sum_{k=0}^{n-1}\frac{a^{k}(p-a)^{n-1-k}}{\binom{n-1}{k}}.$
(2.11)
On the other hand, for any $n\in{\mathbb{N}}$, we have (according to (2.10)):
$1=\frac{n}{\mathrm{lcm}(1,2,\dots,n)}\mathrm{lcm}\left\\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{n-1}\right\\}.$
(2.12)
Then, for a given $n\in{\mathbb{N}}$, by multiplying side to side (2.11) and
(2.12), we obtain
$\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}=\frac{p^{n+1}}{\left(a(p-a)\right)^{n}\mathrm{lcm}(1,2,\dots,n)}\\\
\times\mathrm{lcm}\left\\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{n-1}\right\\}\sum_{k=0}^{n-1}\frac{a^{k}(p-a)^{n-1-k}}{\binom{n-1}{k}}.$
But since the rational number
$\mathrm{lcm}\left\\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{n-1}\right\\}\sum_{k=0}^{n-1}\frac{a^{k}(p-a)^{n-1-k}}{\binom{n-1}{k}}$
is obviously an integer, we derive from the last identity that:
$\displaystyle\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)$
$\displaystyle\geq\vartheta_{p}\left(\frac{p^{n+1}}{\left(a(p-a)\right)^{n}\mathrm{lcm}(1,2,\dots,n)}\right)$
$\displaystyle=n+1-\vartheta_{p}\left(\mathrm{lcm}(1,2,\dots,n)\right)~{}~{}~{}~{}~{}(\text{since
}a\text{ is not a multiple of }p)$
$\displaystyle=n+1-\left\lfloor\log_{p}(n)\right\rfloor,$
confirming the non-boundness from above of the sequence
$\left\\{\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\right\\}_{n\geq
1}$ (since $n+1-\left\lfloor\log_{p}(n)\right\rfloor\rightarrow+\infty$ as
$n\rightarrow+\infty$). $\square$
### The second method
Let $p$ be a prime number and $a$ be an integer non-multiple of $p$. For a
given $n\in{\mathbb{N}}$, consider the rational function $R_{n}$ defined by:
$R_{n}(X):=\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(X-a)^{k}}\right)\frac{X^{k}}{k}=\sum_{k=1}^{n}\dfrac{\left(\frac{X}{a}\right)^{k}}{k}+\sum_{k=1}^{n}\dfrac{\left(\frac{X}{X-a}\right)^{k}}{k}.$
Consider also the real function $f$ defined at the neighborhood of $0$ by:
$f(X):=-\log\left(1-X\right),$
which satisfies the functional equation:
$f\left(\frac{X}{a}\right)+f\left(\frac{X}{X-a}\right)=0$ (2.13)
and whose the $n$th degree Taylor polynomial at $0$ is
$\sum_{k=1}^{n}\frac{X^{k}}{k}$.
On the one hand, according to the well-known properties of Taylor polynomials,
the $n$th degree Taylor polynomial of the function $X\stackrel{{\scriptstyle
g}}{{\mapsto}}f\left(\frac{X}{a}\right)+f\left(\frac{X}{X-a}\right)$ at $0$ is
the same with the $n$th degree Taylor polynomial of
$\sum_{k=1}^{n}\dfrac{\left(\frac{X}{a}\right)^{k}}{k}+\sum_{k=1}^{n}\dfrac{\left(\frac{X}{X-a}\right)^{k}}{k}=R_{n}(X).$
But on the other hand, in view of (2.13), this $n$th degree Taylor polynomial
of $g$ at $0$ is zero. Comparing these two results, we deduce that the
multiplicity of $0$ in $R_{n}$ is at least $(n+1)$. Consequently, $R_{n}(X)$
can be written as:
$R_{n}(X)=X^{n+1}\cdot\frac{U_{n}(X)}{a^{n}(X-a)^{n}\mathrm{lcm}(1,2,\dots,n)},$
where $U_{n}\in{\mathbb{Z}}[X]$. In particular, we have
$R_{n}(p)=p^{n+1}\cdot\frac{U_{n}(p)}{a^{n}(p-a)^{n}\mathrm{lcm}(1,2,\dots,n)}.$
Next, because $U_{n}(p)\in{\mathbb{Z}}$ (since $U_{n}\in{\mathbb{Z}}[X]$) and
$a$ is not a multiple of $p$, then by taking the $p$-adic valuations in the
two sides of the last identity, we derive that:
$\vartheta_{p}\left(R_{n}(p)\right)\geq
n+1-\vartheta_{p}\left(\mathrm{lcm}(1,2,\dots,n)\right)=n+1-\left\lfloor\log_{p}(n)\right\rfloor,$
implying that the sequence
$\left\\{\vartheta_{p}\left(R_{n}(p)\right)\right\\}_{n\geq 1}$ is unbounded
from above, as required by Theorem 2.3. $\square$
###### Remark 2.4.
Curiously, the two previous methods give the same upper bound
$\vartheta_{p}\left(\sum_{k=1}^{n}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}\right)\geq
n+1-\left\lfloor\log_{p}(n)\right\rfloor.$
Furthermore, this last estimate is remarkably very close to the optimal one of
Theorem 2.1.
In the third method below, we will show the non-boundness of the sequence in
Theorem 2.3 without providing any estimate!
### The third method
Let $p$ be a prime number and $a$ be an integer non-multiple of $p$. For all
$n\in{\mathbb{N}}$, set
$r_{n}:=\left(\frac{1}{a^{n}}+\frac{1}{(p-a)^{n}}\right)\frac{p^{n}}{n}~{}~{}\text{and}~{}~{}s_{n}:=\sum_{k=1}^{n}r_{k}.$
The property we have to show is that the sequence
${\left(\vartheta_{p}(s_{n})\right)}_{n\geq 1}$ is unbounded from above; in
other words, we have that
$\limsup_{n\rightarrow+\infty}\vartheta_{p}(s_{n})=+\infty$. So, if we show
the stronger property
$\lim_{n\rightarrow+\infty}\vartheta_{p}(s_{n})=+\infty$, then we are done. To
do so, observe that:
$\displaystyle\lim_{n\rightarrow+\infty}\vartheta_{p}(s_{n})=+\infty$
$\displaystyle\Longleftrightarrow\lim_{n\rightarrow+\infty}{\left|s_{n}\right|}_{p}=0$
$\displaystyle\Longleftrightarrow\lim_{n\rightarrow+\infty}s_{n}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{in
the }p\text{-adic sense})$
$\displaystyle\Longleftrightarrow\sum_{k=1}^{+\infty}r_{k}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{in
the }p\text{-adic sense}).$
Consequently, it suffices to show that:
$\sum_{k=1}^{+\infty}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}=0$
(2.14)
(in the $p$-adic sense). Let us show (2.14). By using the $p$-adic logarithm
function (recalled in §1), we have
$\displaystyle\sum_{k=1}^{+\infty}\left(\frac{1}{a^{k}}+\frac{1}{(p-a)^{k}}\right)\frac{p^{k}}{k}$
$\displaystyle=\sum_{k=1}^{+\infty}\frac{\left(\frac{p}{a}\right)^{k}}{k}+\sum_{k=1}^{+\infty}\frac{\left(\frac{p}{p-a}\right)^{k}}{k}$
$\displaystyle=-L_{p}\left(1-\frac{p}{a}\right)-L_{p}\left(1-\frac{p}{p-a}\right)$
$\displaystyle=-\left[L_{p}\left(\frac{a-p}{a}\right)+L_{p}\left(\frac{-a}{p-a}\right)\right]$
$\displaystyle=-L_{p}\left(\frac{a-p}{a}\cdot\frac{-a}{p-a}\right)$
$\displaystyle=-L_{p}(1)=0,$
as required. The non-boundness from above of the sequence
${\left(\vartheta_{p}(s_{n})\right)}_{n\geq 1}$ follows. $\square$
## References
* [1] A. Dubickas. On a sequence of integers with unusual divisibility by a power of $2$, to appear in Miskolc Math. Notes.
* [2] B. Farhi. On a curious integer sequence, preprint 2022, available at https://arxiv.org/abs/2204.10136.
* [3] B. Farhi. The integrality of the Genocchi numbers obtained through a new identity and other results, Notes Number Theory Discrete Math., 28, n°4 (2022), p. 749-757.
* [4] B. Farhi. An identity involving the least common multiple of binomial coefficients and its application, Amer. Math. Monthly, 116 (2009), p. 836-839.
* [5] N. Koblitz. $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd edition, Springer-Verlag, New York, 1984.
* [6] T. Mansour. Combinatorial identities and inverse binomial coefficients, Adv. Appl. Math, 28 (2002), p. 196-202.
|
# Density Functional Theory for two-dimensional homogeneous materials with
magnetic fields
David Gontier CEREMADE, University of Paris-Dauphine, PSL University, 75016
Paris, France & ENS/PSL University, Département de Mathématiques et
Applications, F-75005, Paris, France<EMAIL_ADDRESS>, Salma
Lahbabi EMAMI, LRI, ENSEM, UHIIC, 7 Route d’El Jadida, B.P. 8118 Oasis,
Casablanca; MSDA, Mohammed VI Polytechnic University, Lot 660, Hay Moulay
Rachid Ben Guerir, 43150, Morocco<EMAIL_ADDRESS>and Abdallah
Maichine Department of Mathematics, Faculty of Sciences, Mohammed V
University in Rabat, 4 Avenue Ibn Battouta P.B. 1014 RP, Rabat, Morocco &
Ecole Centrale Casablanca, Bouskoura, Ville Verte, P.B. 27182, Morocco
<EMAIL_ADDRESS>
###### Abstract.
This paper studies DFT models for homogeneous 2D materials in 3D space, under
a constant perpendicular magnetic field. We show how to reduce the
three–dimensional energy functional to a one–dimensional one, similarly as in
our previous work. This is done by minimizing over states invariant under
magnetic translations and that commute with the Landau operator. In the
reduced model, the Pauli principle no longer appears. It is replaced by a
penalization term in the energy.
###### Contents
1. 1 Introduction
2. 2 States commuting with magnetic translations
1. 2.1 Two dimensional Landau operator
2. 2.2 Magnetic translations
3. 2.3 Diagonalisation of states commuting with magnetic translations
3. 3 Reduction of the kinetic energy, and applications
1. 3.1 Reduction of the kinetic energy
2. 3.2 Some properties of the function F
3. 3.3 Reduced DFT models
4. 3.4 The reduced Hartree–Fock case
4. 4 Models with spin
## 1\. Introduction
The analysis of quantum properties of two dimensional materials is an active
research area in physics and material science. Some 2D materials such as
graphene or phosphorene exhibits many interesting physical properties [16, 3,
10, 11] which has many applications such as High Electron Mobility Transistors
[12]. Some of these properties are not yet fully understood. This is the main
motivation to revisit Density Functional Theory (DFT) when applied to quantum
two dimensional systems (see [1, 9] for previous works).
As in our previous work [9], we study homogeneous two–dimensional slabs, when
embedded in three dimensional space, but this time, we include a constant
perpendicular magnetic field. We consider a charge distribution $\mu$ which is
equidistributed in the first two dimensions:
$\mu(x_{1},x_{2},x_{3})=\mu(x_{3})$, and with a constant perpendicular
magnetic field ${\mathbb{B}}=b{\mathbb{e}}_{3}$.
One key result of our previous work was an inequality for the kinetic energy
per unit surface for translationally invariant states. Let us quickly
summarize the result. Let
${\mathcal{P}}:=\left\\{\gamma\in\underline{\mathfrak{S}}^{1}(L^{2}(\mathbb{R}^{3})),\quad
0\leq\gamma\leq
1,\quad\forall{\mathbb{R}}\in\mathbb{R}^{2},\quad\tau_{\mathbb{R}}\gamma=\gamma\tau_{\mathbb{R}}\right\\}$
denote the set of one-body density matrices which commute with all
$\mathbb{R}^{2}$ translations. Here,
$\underline{\mathfrak{S}}^{1}(L^{2}(\mathbb{R}^{3}))$ stands for locally trace
class self–adjoint operators with finite trace per unit surface
$\underline{\rm Tr}(\gamma)<{\infty}$ (see Section 2.3). For
${\mathbb{R}}\in\mathbb{R}^{2}\subset\mathbb{R}^{3}$, we have denoted by
$\tau_{\mathbb{R}}f(x_{1},x_{2},x_{3}):=f(x_{1}-R_{1},x_{2}-R_{2},x_{3})$ the
usual translation along the first two dimensions. Let us also introduce the
set of reduced states
${\mathcal{G}}:=\left\\{G\in\mathfrak{S}^{1}(L^{2}(\mathbb{R})),\quad G\geq
0\right\\},$
where $\mathfrak{S}^{1}(L^{2}(\mathbb{R}))$ refers to the space of trace class
self-adjoint operators on $L^{2}(\mathbb{R})$. We have proved in [9] that, for
any (representable) density $\rho=\rho(x_{3})$ depending only on the third
variable, we have
(1.1)
$\inf_{\gamma\in{\mathcal{P}}\atop\rho_{\gamma}=\rho}\left\\{\frac{1}{2}\underline{\rm
Tr}(-\Delta_{3}\gamma)\right\\}=\inf_{G\in{\mathcal{G}}\atop\rho_{G}=\rho}\left\\{\frac{1}{2}{\rm
Tr}(-\Delta_{1}G)+\pi{\rm Tr}(G^{2})\right\\},$
where $\Delta_{d}$ denotes the Laplacian operator in $d$–space dimension. The
energy appearing in the right hand side leads to one–dimensional reduced
models for homogeneous semi-infinite slabs in the context of DFT. One
typically obtains a minimization problem of the form
(1.2) $\inf\left\\{\frac{1}{2}{\rm Tr}(-\Delta_{1}G)+\pi{\rm
Tr}(G^{2})+\frac{1}{2}{\mathcal{D}}_{1}(\rho_{G}-\mu)+E^{\rm
xc}(\rho_{G}),\quad G\in{\mathcal{G}}\right\\},$
where ${\mathcal{D}}_{1}(\cdot)$ is the one–dimensional Coulomb interaction
energy (see [9, Section 3.1] for a discussion about this term), and $E^{\rm
xc}$ some exchange-correlation energy per unit surface. Note that there is no
Pauli principle for the operator $G$; it has been replaced by the penalization
term $\pi{\rm Tr}(G^{2})$ in the energy, which prevents $G$ from having large
eigenvalues. We refer to [9] for details, where we also studied the reduced
Hartree–Fock model, which corresponds to the case $E^{\rm xc}=0$.
The scope of this paper is to apply a similar reduction when taking into
account magnetic effects. Without considering the spin (we refer to Section 4
for the case with spin), the kinetic energy per unit surface is of the form
$\frac{1}{2}\underline{\rm
Tr}\left((-{\mathrm{i}}\nabla_{3}+{\mathbb{A}})^{2}\gamma\right),$
where ${\mathbb{A}}=b(0,x_{1},0)$ is a vector potential so that ${\bf
curl}\,{\mathbb{A}}={\mathbb{B}}=b{\mathbb{e}}_{3}$. We chose a gauge which is
not symmetric, but which will simplify some computations. The Laplacian
operator $-\Delta_{3}$ has been replaced by the Landau operator
${\mathbb{L}}^{\mathbb{A}}_{3}:=(-{\mathrm{i}}\nabla_{3}+{\mathbb{A}})^{2}={\mathbb{L}}^{\mathbb{A}}_{2}-\partial_{x_{3}x_{3}}^{2},\quad\text{with}\quad{\mathbb{L}}^{\mathbb{A}}_{2}=-\partial_{x_{1}x_{1}}^{2}+(-{\mathrm{i}}\partial_{x_{2}}+bx_{1})^{2}.$
In analogy with [9], we only consider states commuting with the so–called
magnetic translations ${\mathfrak{m}}_{\mathbb{R}}$ and with the Landau
operator. We refer to Section 2 for the definition of these operators, and for
the justification of this choice. We denote the set of such states by
(1.3)
${\mathcal{P}}^{\mathbb{A}}:=\left\\{\gamma\in\underline{\mathfrak{S}}^{1}(L^{2}(\mathbb{R}^{3})),\
0\leq\gamma\leq 1,\ \forall{\mathbb{R}}\in\mathbb{R}^{2},\
{\mathfrak{m}}_{\mathbb{R}}\gamma=\gamma{\mathfrak{m}}_{\mathbb{R}},\;{\mathbb{L}}^{\mathbb{A}}_{2}\gamma=\gamma{\mathbb{L}}^{\mathbb{A}}_{2}\right\\}.$
In Theorem 2.7, we prove that any $\gamma\in{\mathcal{P}}^{\mathbb{A}}$ has a
simple decomposition, in terms of the different projectors on the Landau
levels. Using this decomposition, we prove in Theorem 3.1 that, in the
magnetic case, we have an equality similar to (1.1), which reads
(1.4)
$\boxed{\inf_{\gamma\in{\mathcal{P}}^{\mathbb{A}}\atop\rho_{\gamma}=\rho}\left\\{\frac{1}{2}\underline{\rm
Tr}({\mathbb{L}}^{\mathbb{A}}_{3}\gamma)\right\\}=\inf_{G\in{\mathcal{G}}\atop\rho_{G}=\rho}\left\\{\frac{1}{2}{\rm
Tr}(-\Delta_{1}G)+{\rm Tr}\left(F(b,G)\right)\right\\}}$
with the penalization term $F$ defined by
$F(b,g):=\pi g^{2}+\frac{b^{2}}{4\pi}\left\\{\frac{2\pi
g}{b}\right\\}\left(1-\left\\{\frac{2\pi g}{b}\right\\}\right),$
where $\\{x\\}$ denotes the fractional part of $x$. The function $F$ is
studied in Proposition 3.2. It is a piece-wise linear function, reflecting the
contribution of the different Landau levels. The function $F$ is not new, and
already appears in the context of the two-dimensional Thomas Fermi (TF) theory
under constant magnetic fields (see [14] and related references). The
(spinless) TF kinetic energy takes the form
$E_{\rm kin}^{\rm
TF}(b,\rho):=\int_{\mathbb{R}^{2}}F(b,\rho({\mathbf{x}})){\mathrm{d}}{\mathbf{x}}.$
For $b=0$, we recover the usual two–dimensional TF kinetic energy
$\int\pi\rho^{2}$. It is different from the three–dimensional TF kinetic
energy of a gas under a constant magnetic field, which has been derived and
studied in [6, 19], and is obtained by assuming that the electron density is
constant, hence also invariant under the third–direction translation.
Equation (1.4) allows the reduction of DFT models for two-dimensional
homogeneous slabs under constant magnetic field. In fact, one obtains a one-
dimensional problem of the form (compare with (1.2))
$\inf\left\\{\frac{1}{2}{\rm Tr}(-\Delta_{1}G)+{\rm
Tr}\left(F(b,G)\right)+\frac{1}{2}{\mathcal{D}}_{1}(\rho_{G}-\mu)+E^{\rm
xc}(b,\rho_{G}),\quad G\in{\mathcal{G}}\right\\}.$
Note that the exchange-correlation function may depend on the external
magnetic field $b$ (see [6, Eqn. (4.1)-(4.2)] for the expression of the
exchange energy for the Landau gas). In Section 3.3, we study the
corresponding reduced Hartree-Fock model, where $E^{\rm xc}=0$.
This article is structured as follows. In Section 2, we start by introducing
the Landau operator and studying its spectral decomposition, then we define
magnetic translations
$\\{{\mathfrak{m}}_{\mathbb{R}}\\}_{{\mathbb{R}}\in\mathbb{R}^{2}}$ with some
of their properties, and we characterize the states in
${\mathcal{P}}^{\mathbb{A}}$. Using this characterization, we explain in
section 3 how to reduce the kinetic energy, we give some properties of the
penalization term $F$ and we study the corresponding reduced Hartree-Fock
model. Finally, we show in Section 4 how to extend our results to systems with
spin.
### Acknowledgments
The research leading to these results has received funding from OCP grant AS70
“Towards phosphorene based materials and devices”.
## 2\. States commuting with magnetic translations
In this section, we prove that states $\gamma\in{\mathcal{P}}^{\mathbb{A}}$
have a particular structure.
### 2.1. Two dimensional Landau operator
We start by recalling some classical facts about the (two–dimensional) Landau
operator
${\mathbb{L}}^{\mathbb{A}}_{2}=-\partial_{x_{1}x_{1}}^{2}+(-{\mathrm{i}}\partial_{x_{2}}+bx_{1})^{2}$.
In what follows, we assume $b\neq 0$. For
$n\in\mathbb{N}_{0}=\\{0,1,2,\cdots\\}$, we introduce the function
$\varphi_{n}:\mathbb{R}\to\mathbb{R}$ defined by
(2.1)
$\varphi_{n}(x):=a_{n}|b|^{1/4}{\mathcal{H}}_{n}(\sqrt{|b|}x){\mathrm{e}}^{-\tfrac{1}{2}|b|x^{2}},$
where
${\mathcal{H}}_{n}(x)=(-1)^{n}{\mathrm{e}}^{x^{2}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}}x^{n}}{\mathrm{e}}^{-x^{2}}$
refers to the $n$-th Hermite polynomial and $a_{n}=(2^{n}n!)^{-1/2}/\pi^{1/4}$
is a normalization constant so that $\|\varphi_{n}\|_{L^{2}(\mathbb{R})}=1$.
###### Proposition 2.1.
The operator ${\mathbb{L}}^{A}_{2}$ has purely discrete spectrum
(2.2) $\sigma({\mathbb{L}}^{A}_{2})=b(2\mathbb{N}_{0}+1).$
The eigenvalue $\varepsilon_{n}:=b(2n+1)$ is of infinite multiplicity, with
eigenspace
(2.3)
$E_{n}:=\ker({\mathbb{L}}^{\mathbb{A}}_{2}-\varepsilon_{n})=\\{{\mathcal{W}}(\varphi_{n},g),\;g\in
L^{2}(\mathbb{R})\\},$
where ${\mathcal{W}}$ is a Wigner type transform defined on
$L^{2}(\mathbb{R})\times L^{2}(\mathbb{R})$ by
(2.4)
${\mathcal{W}}(\varphi,g)({\mathbf{x}}):=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}{\mathrm{e}}^{-{\mathrm{i}}kx_{2}}\varphi\left(x_{1}-\frac{k}{b}\right)g(k){\mathrm{d}}k.$
In particular, the spectral projector $\mathbf{P}_{n}$ onto
$\ker({\mathbb{L}}^{\mathbb{A}}_{2}-\varepsilon_{n})$ has kernel
(2.5)
$\mathbf{P}_{n}({\mathbf{x}};{\mathbf{y}})=\frac{1}{2\pi}\int_{\mathbb{R}}{\mathrm{e}}^{-{\mathrm{i}}k(x_{2}-y_{2})}\varphi_{n}\left(x_{1}-\frac{k}{b}\right)\varphi_{n}\left(y_{1}-\frac{k}{b}\right){\mathrm{d}}k.$
Its density
$\rho_{\mathbf{P}_{n}}({\mathbf{x}}):=\mathbf{P}_{n}({\mathbf{x}};{\mathbf{x}})=\frac{b}{2\pi}$
is constant and independent of $n$.
###### Remark 2.2.
The definition (2.4) is slightly different from the classical Wigner transform
(see for example [18, Chapter 2]) which is rather adapted to study Landau
operator with the gauge
$\tilde{{\mathbb{A}}}=\frac{b}{2}\begin{pmatrix}-x_{2}\\\ x_{1}\end{pmatrix}$,
for $b=1$. A gauge transformation links the two transforms.
###### Proof.
First, we remark that ${\mathbb{L}}^{\mathbb{A}}_{2}$ commutes with all
translations in the $x_{2}$–direction. We introduce the Fourier transform with
respect to the $x_{2}$–variable
(2.6)
${\mathcal{F}}[f](x_{1},k):=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x_{1},x_{2}){\mathrm{e}}^{{\mathrm{i}}kx_{2}}{\mathrm{d}}x_{2}$
and its inverse
${\mathcal{F}}^{-1}[\phi](x_{1},x_{2}):=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\phi(x_{1},k){\mathrm{e}}^{-{\mathrm{i}}kx_{2}}{\mathrm{d}}k.$
We have
(2.7)
${\mathcal{F}}{\mathbb{L}}^{A}_{2}{\mathcal{F}}^{-1}=\int_{\mathbb{R}}^{\oplus}{\mathfrak{h}}_{b,k}{\mathrm{d}}k,\quad\text{with}\quad{\mathfrak{h}}_{b,k}:=-\partial_{xx}^{2}+(bx-k)^{2}.$
The operator ${\mathfrak{h}}_{b,k}$ is a translation of the harmonic
oscillator ${\mathfrak{h}}:=-\partial_{xx}^{2}+b^{2}x^{2}$, whose spectral
decomposition is
${\mathfrak{h}}=\sum_{n=0}^{\infty}\varepsilon_{n}|\varphi_{n}\rangle\langle\varphi_{n}|,$
with $\varepsilon_{n}=b(2n+1)$ and $\varphi_{n}$ as defined in (2.1). We
deduce that the spectral decomposition of ${\mathfrak{h}}_{b,k}$ is
${\mathfrak{h}}_{b,k}:=\sum_{n=0}^{\infty}\varepsilon_{n}|\varphi_{n}(\cdot-\tfrac{k}{b})\rangle\langle\varphi_{n}(\cdot-\tfrac{k}{b})|,$
which proves (2.2). Using (2.7), we see that $W$ is an eigenfunction of
${\mathbb{L}}^{{\mathbb{A}}}_{2}$, corresponding to eigenvalue
$\varepsilon_{n}$, if and only if it is of the form
$W={\mathcal{F}}^{-1}\left[(x_{1},k)\mapsto
g(k)\varphi_{n}\left(x_{1}-\frac{k}{b}\right)\right]={\mathcal{W}}(\varphi_{n},g),$
where $g\in L^{2}(\mathbb{R})$. Thus
$E_{n}:=\ker({\mathbb{L}}^{\mathbb{A}}_{2}-\varepsilon_{n})=\left\\{{\mathcal{W}}(\varphi_{n},g),\;g\in
L^{2}(\mathbb{R})\right\\}.$
To compute the kernel $\mathbf{P}_{n}({\mathbf{x}},{\mathbf{y}})$ of the
projector on $E_{n}$, we use the Moyal identity, that we recall here.
###### Proposition 2.3.
Let $f_{1},g_{1},f_{2},g_{2}$ in $L^{2}(\mathbb{R})$. Then,
${\mathcal{W}}(f_{1},g_{1}),{\mathcal{W}}(f_{2},g_{2})\in
L^{2}(\mathbb{R}^{2})$ and (Moyal identity)
(2.8)
$\langle{\mathcal{W}}(f_{1},g_{1}),{\mathcal{W}}(f_{2},g_{2})\rangle_{L^{2}(\mathbb{R}^{2})}=\langle
f_{1},f_{2}\rangle_{L^{2}(\mathbb{R})}\,\langle
g_{1},g_{2}\rangle_{L^{2}(\mathbb{R})}.$
###### Proof.
We first prove the result for $f_{1},g_{1},f_{2},g_{2}\in
C^{\infty}_{0}(\mathbb{R})$ and conclude by density. Parseval identity gives
$\displaystyle\langle{\mathcal{W}}\left(f_{1},g_{1}\right),{\mathcal{W}}\left(f_{2},g_{2}\right)\rangle_{L^{2}(\mathbb{R}^{2})}$
$\displaystyle=\iint_{\mathbb{R}^{2}}(\overline{f_{1}}f_{2})\left(x_{1}-\frac{k}{b}\right)(\overline{g_{1}}g_{2})(k){\mathrm{d}}x_{1}{\mathrm{d}}k$
$\displaystyle=\langle f_{1},f_{2}\rangle_{L^{2}(\mathbb{R})}\,\langle
g_{1},g_{2}\rangle_{L^{2}(\mathbb{R})}.$
∎
In particular, if $(\psi_{m})_{m\in\mathbb{N}}$ is any basis of
$L^{2}(\mathbb{R})$, then
$\\{{\mathcal{W}}(\varphi_{n},\psi_{m})\\}_{m\in\mathbb{N}}$ is a basis of
$E_{n}$. Notice that, for any fixed ${\mathbf{x}}\in\mathbb{R}^{2}$, we have
${\mathcal{W}}(\varphi,g)({\mathbf{x}})=\langle\varphi_{{\mathbf{x}}},g\rangle_{L^{2}(\mathbb{R})},\quad\text{with}\quad\varphi_{\mathbf{x}}(k):=\frac{1}{\sqrt{2\pi}}{\mathrm{e}}^{{\mathrm{i}}kx_{2}}\overline{\varphi}\left(x_{1}-\frac{k}{b}\right).$
Hence
$\displaystyle\mathbf{P}_{n}({\mathbf{x}};{\mathbf{y}})$
$\displaystyle=\sum_{m\in\mathbb{N}}{\mathcal{W}}(\varphi_{n},\psi_{m})({\mathbf{x}})\overline{{\mathcal{W}}(\varphi_{n},\psi_{m})}({\mathbf{y}})=\sum_{m\in\mathbb{N}}\langle\varphi_{n,{\mathbf{x}}},\psi_{m}\rangle\langle\psi_{m},\varphi_{n,{\mathbf{y}}}\rangle.$
Using that $\sum_{m}|\psi_{m}\rangle\langle\psi_{m}|=\mathbb{I}_{L^{2}}$, we
obtain
$\mathbf{P}_{n}({\mathbf{x}};{\mathbf{y}})=\langle\varphi_{n,{\mathbf{x}}},\varphi_{n,{\mathbf{y}}}\rangle$,
which, given that $\varphi_{n}$ is real-valued, gives (2.5). The density of
$\mathbf{P}_{n}$ is thus
$\rho_{\mathbf{P}_{n}}({\mathbf{x}})=\mathbf{P}_{n}({\mathbf{x}};{\mathbf{x}})=\frac{1}{2\pi}\int_{\mathbb{R}}\left|\varphi_{n}\left(x_{1}-\frac{k}{b}\right)\right|^{2}{\mathrm{d}}k=\frac{b}{2\pi}.$
∎
### 2.2. Magnetic translations
The Landau operator does not commute with the usual translations, however it
commutes with the magnetic translations, that we define now. We write
${\mathbb{L}}_{2}^{\mathbb{A}}:=p_{{\mathbb{A}},1}^{2}+p_{{\mathbb{A}},2}^{2},\quad\text{with}\quad
p_{{\mathbb{A}},1}:=-{\mathrm{i}}\partial_{x_{1}},\quad
p_{{\mathbb{A}},2}:=-{\mathrm{i}}\partial_{x_{2}}+bx_{1}.$
The operators $p_{{\mathbb{A}},1}$ and $p_{{\mathbb{A}},2}$ do not commute,
and do not commute with $L_{2}^{\mathbb{A}}$. Actually, we have
$\left[p_{{\mathbb{A}},1},p_{{\mathbb{A}},2}\right]=-{\mathrm{i}}b,\quad\left[p_{{\mathbb{A}},1},L_{2}^{\mathbb{A}}\right]=-2{\mathrm{i}}bp_{{\mathbb{A}},2},\quad\left[p_{{\mathbb{A}},2},L_{2}^{\mathbb{A}}\right]=2{\mathrm{i}}bp_{{\mathbb{A}},1}.$
However, introducing the dual momentum operators
$\widetilde{p}_{{\mathbb{A}},1}:=-{\mathrm{i}}\partial_{x_{1}}+bx_{2},\quad\widetilde{p}_{{\mathbb{A}},2}:=-{\mathrm{i}}\partial_{x_{2}},$
we can check that
$\left[\widetilde{p}_{{\mathbb{A}},1},L_{2}^{\mathbb{A}}\right]=\left[\widetilde{p}_{{\mathbb{A}},1},L_{2}^{\mathbb{A}}\right]=0$.
The magnetic translation ${\mathfrak{m}}_{\mathbb{R}}$,
${\mathbb{R}}\in\mathbb{R}^{2}$, is the unitary operator
$\displaystyle{\mathfrak{m}}_{\mathbb{R}}$
$\displaystyle=\exp(-\tfrac{{\mathrm{i}}}{2}bR_{1}R_{2})\exp\left(-{\mathrm{i}}\widetilde{{\mathbb{p}}}_{\mathbb{A}}\cdot{\mathbb{R}}\right)$
$\displaystyle=\exp(-\tfrac{{\mathrm{i}}}{2}bR_{1}R_{2})\exp\left(-{\mathrm{i}}\left(\widetilde{p}_{{\mathbb{A}},1}R_{1}+\widetilde{p}_{{\mathbb{A}},2}R_{2}\right)\right).$
Note that we have added a phase factor in order to match the usual convention.
Using the Baker–Campbell–Haussdorf formula and the fact that
$\left[\widetilde{p}_{{\mathbb{A}},1},\widetilde{p}_{{\mathbb{A}},2}\right]={\mathrm{i}}b$
commutes with all operators, we obtain the explicit expression
${\mathfrak{m}}_{\mathbb{R}}=\exp(-{\mathrm{i}}bR_{1}x_{2})\tau_{\mathbb{R}},\quad\text{that
is}\quad\left({\mathfrak{m}}_{\mathbb{R}}f\right)({\mathbf{x}})=\exp(-{\mathrm{i}}bR_{1}x_{2})f({\mathbf{x}}-{\mathbb{R}}),$
where $\tau_{\mathbb{R}}f({\mathbf{x}}):=f({\mathbf{x}}-{\mathbb{R}})$ is the
usual translation operator. By construction, the magnetic translations commute
with $L^{\mathbb{A}}_{2}$ and $\mathbf{P}_{n}$, but they do not commute among
them. Actually, we have
${\mathfrak{m}}_{\mathbb{R}}{\mathfrak{m}}_{\tilde{{\mathbb{R}}}}={\mathrm{e}}^{{\mathrm{i}}bR_{2}\tilde{R}_{1}}{\mathfrak{m}}_{{\mathbb{R}}+\tilde{{\mathbb{R}}}}\quad\text{and}\quad{\mathfrak{m}}_{\mathbb{R}}^{*}={\mathfrak{m}}_{\mathbb{R}}^{-1}={\mathrm{e}}^{{\mathrm{i}}bR_{1}R_{2}}{\mathfrak{m}}_{-{\mathbb{R}}}.$
An important feature of magnetic translations is that they form an irreducible
family on each eigenspace $E_{n}$, in the sense of [2, Definition 2.3.7].
###### Proposition 2.4.
The set of magnetic translation operators
$({\mathfrak{m}}_{\mathbb{R}})_{\mathbb{R}}$ is an irreducible family of
operators on each $E_{n}$, in the sense that
(2.9) $\forall\Psi\in E_{n}\setminus\left\\{0\right\\},\quad
E_{n}={\mathrm{span}}\\{{\mathfrak{m}}_{\mathbb{R}}\Psi:{\mathbb{R}}\in\mathbb{R}^{2}\\}.$
###### Proof.
Assume otherwise, and let $\Psi$ so that
$\widetilde{E}_{n}(\Psi):={\mathrm{span}}\\{{\mathfrak{m}}_{\mathbb{R}}\Psi:{\mathbb{R}}\in\mathbb{R}^{2}\\}\subsetneq
E_{n}.$
Then there is $\Phi\in E_{n}\setminus\\{0\\}$ so that
$\Phi\perp\widetilde{E}_{n}(\Psi)$. Let $f,g\in L^{2}(\mathbb{R})$ so that
$\Psi={\mathcal{W}}(\varphi_{n},g)$ and $\Phi={\mathcal{W}}(\varphi_{n},f)$.
Using the Moyal identity, and the fact that
${\mathfrak{m}}_{\mathbb{R}}{\mathcal{W}}(\varphi_{n},g)={\mathcal{W}}(\varphi_{n},{\mathfrak{t}}_{\mathbb{R}}g),\quad\text{with}\quad{\mathfrak{t}}_{\mathbb{R}}g:k\mapsto{\mathrm{e}}^{-{\mathrm{i}}bR_{1}R_{2}}{\mathrm{e}}^{{\mathrm{i}}kR_{2}}g(k-bR_{1}),$
the condition $\langle\Phi,{\mathbb{m}}_{\mathbb{R}}\Psi\rangle=0$ for all
${\mathbb{R}}\in\mathbb{R}^{2}$ reads
$\forall{\mathbb{R}}\in\mathbb{R}^{2},\quad\langle
f,{\mathfrak{t}}_{\mathbb{R}}g\rangle=0,\quad\text{hence}\quad\int_{\mathbb{R}}\overline{f}(k){\mathrm{e}}^{{\mathrm{i}}kR_{2}}g(k-bR_{1}){\mathrm{d}}k=0.$
Applying the inverse Fourier transform to
$k\mapsto\overline{f}(k)g(k-R_{1})=0$ shows that $\overline{f}(k)g(k-R_{1})=0$
a.e. for all $R_{1}\in\mathbb{R}$. Squaring and integrating in $R_{1}$ gives
$f=0$, a contradiction. ∎
### 2.3. Diagonalisation of states commuting with magnetic translations
In what follows, we are interested in one-body density matrices which commute
with all magnetic translations. In the case without magnetic field, if a state
commutes with all usual translations $\tau_{\mathbb{R}}$, then it commutes
with the Laplacian operator. In the magnetic case, there are operators which
commute with all magnetic translations
$({\mathfrak{m}}_{\mathbb{R}})_{{\mathbb{R}}\in\mathbb{R}^{2}}$, but which do
not commute with the Landau operator (we give an example of such an operator
in Remark 2.6 below). So, we rather consider one-body density matrices which
commute with ${\mathfrak{m}}_{\mathbb{R}}$, and with the Landau operator. It
turns out that such operators have an explicit and simple characterization.
We first enunciate our result in dimension two before turning to the three
dimensional case.
###### Proposition 2.5.
Let $\eta\in{\mathcal{S}}(L^{2}(\mathbb{R}^{2}))$ be such that
$\eta{\mathfrak{m}}_{\mathbb{R}}={\mathfrak{m}}_{\mathbb{R}}\eta$ for all
${\mathbb{R}}\in\mathbb{R}^{2}$ and
$\eta{\mathbb{L}}_{\mathbb{A}}={\mathbb{L}}_{\mathbb{A}}\eta$. Then, there is
a family of real numbers $(\lambda_{n})_{n\in\mathbb{N}_{0}}$ so that
(2.10) $\eta=\sum_{n\in\mathbb{N}_{0}}\lambda_{n}\mathbf{P}_{n}.$
If $\eta$ is a locally trace class operator, then its density is constant
$\rho_{\eta}=\frac{b}{2\pi}\sum_{n\in\mathbb{N}_{0}}\lambda_{n}.$
###### Proof.
Since $\eta$ commutes with ${\mathbb{L}}^{\mathbb{A}}_{2}$, it commutes with
any spectral projector $\mathbf{P}_{n}$, hence leaves invariant $E_{n}={\rm
Ran}(\mathbf{P}_{n})$, for all $n\in\mathbb{N}$. The operator
$\eta_{n}:=\mathbf{P}_{n}\eta\mathbf{P}_{n}\in{\mathcal{S}}(E_{n})$ commutes
with all magnetic translations. Since the family
$\\{{\mathbb{m}}_{\mathbb{R}}\\}_{{\mathbb{R}}}$ is irreducible, it implies
that $\eta_{n}$ is proportional to the identity on $E_{n}$, hence is of the
form $\eta_{n}=\lambda_{n}\mathbb{I}_{E_{n}}$. This is a kind of Schur’s
Lemma, see [2, Proposition 2.3.8]. ∎
###### Remark 2.6.
The hypothesis
$\eta{\mathbb{L}}^{\mathbb{A}}_{2}={\mathbb{L}}^{\mathbb{A}}_{2}\eta$ is not a
consequence of the commutation with the operators
$({\mathfrak{m}}_{\mathbb{R}})_{{\mathbb{R}}}$. Indeed, consider for a
normalized $\zeta\in L^{2}(\mathbb{R})$, the projector $P_{\zeta}$ onto the
vectorial space
$E_{\zeta}:=\left\\{{\mathcal{W}}(\zeta,f),\ f\in L^{2}(\mathbb{R})\right\\}.$
Since
${\mathfrak{m}}_{\mathbb{R}}{\mathcal{W}}(\zeta,f)={\mathcal{W}}(\zeta,{\mathfrak{t}}_{\mathbb{R}}f)\in
E_{\zeta}$, the set $E_{\zeta}$ is invariant by ${\mathfrak{m}}_{\mathbb{R}}$,
hence $P_{\zeta}$ commutes with all magnetic translations. However, we have,
using the decomposition (2.7) that
$\displaystyle{\mathbb{L}}^{\mathbb{A}}_{2}\left[{\mathcal{W}}(\zeta,f)\right]$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}{\mathrm{e}}^{-{\mathrm{i}}kx_{2}}\left[(-\partial_{x_{1}x_{1}}^{2}+\left(bx_{1}-k\right)^{2})\zeta\left(x_{1}-\frac{k}{b}\right)\right]f(k){\mathrm{d}}k$
$\displaystyle={\mathcal{W}}(\widetilde{\zeta},f)$
with $\widetilde{\zeta}:=\left(-\partial_{xx}^{2}+b^{2}x^{2}\right)\zeta$. So
${\mathbb{L}}^{{\mathbb{A}}}_{2}E_{\zeta}=E_{\widetilde{\zeta}}$, and
${\mathbb{L}}^{\mathbb{A}}_{2}$ leaves $E_{\zeta}$ invariant iff
$\widetilde{\zeta}$ is collinear to $\zeta$. This happens if and only if
$\zeta$ is an eigenstate of the harmonic oscillator, that is
$\zeta=\varphi_{n}$ for some $n\in\mathbb{N}_{0}$.
The three–dimensional analogue of the previous Proposition reads as follows.
###### Theorem 2.7.
Let $\gamma$ be a bounded operator on $L^{2}(\mathbb{R}^{3})$ commuting with
all magnetic translations ${\mathfrak{m}}_{\mathbb{R}}$ and with
${\mathbb{L}}^{\mathbb{A}}_{2}\otimes\mathbb{I}$. Then, there exists a family
$(\gamma_{n})_{n\in\mathbb{N}}$ of bounded operators on $L^{2}(\mathbb{R})$,
with $\|\gamma_{n}\|\leq\|\gamma\|$, and so that
(2.11) $\gamma=\sum_{n=0}^{\infty}\mathbf{P}_{n}\otimes\gamma_{n}.$
If $\gamma$ is a locally trace class operator, then its density depends only
on $x_{3}$
$\rho_{\gamma}({\mathbf{x}})=\rho_{\gamma}(x_{3})=\frac{b}{2\pi}\sum_{n=0}^{\infty}\rho_{\gamma_{n}}(x_{3}).$
###### Proof.
Let us consider two fixed test functions $\phi,\psi\in L^{2}(\mathbb{R})$, and
define the operator $\eta_{\phi,\psi}$ on $L^{2}(\mathbb{R}^{2})$ by
$\forall f,g\in L^{2}(\mathbb{R}^{2}),\quad\langle
f,\eta_{\phi,\psi},g\rangle_{L^{2}(\mathbb{R}^{2})}:=\langle
f\otimes\phi,\gamma(g\otimes\psi)\rangle_{L^{2}(\mathbb{R}^{3})}.$
The conditions on $\gamma$ imply that $\eta_{\phi,\psi}$ is a bounded
self–adjoint operator that commutes with all ${\mathfrak{m}}_{\mathbb{R}}$ and
with ${\mathbb{L}}_{\mathbb{A}}$. Thus, using Proposition 2.5,
$\eta_{\phi,\psi}$ can be decomposed as
$\eta_{\phi,\psi}=\sum_{n\in\mathbb{N}_{0}}\lambda_{n}(\phi,\psi)\mathbf{P}_{n}.$
Since $\eta$ is a bounded operator, we have for any normalized $\Phi_{n}\in
L^{2}(\mathbb{R}^{2})$ in the range of $\mathbf{P}_{n}$,
$\left|\lambda_{n}(\phi,\psi)\right|=\left|\langle\Phi_{n}\otimes\phi,\gamma(\Phi_{n}\otimes\psi)\rangle_{L^{2}(\mathbb{R}^{3})}\right|\leq\|\gamma\|_{\rm
op}\|\psi\|_{L^{2}(\mathbb{R})}\left\|\phi\right\|_{L^{2}(\mathbb{R})}.$
We deduce that the map $(\phi,\psi)\mapsto\lambda_{n}(\phi,\psi)$ is
sesquilinear and bounded, with bound smaller than $\|\eta\|_{\rm op}$. The
result then follows by taking $\gamma_{n}$ the bounded self-adjoint operator
on $L^{2}(\mathbb{R})$ defined by
$\langle\phi,\gamma_{n}\psi\rangle:=\lambda_{n}(\phi,\psi).$
Finally, for $\eta$ of the form (2.11), we obtain, using
$\rho_{\mathbf{P}_{n}}=\frac{b}{2\pi}$,
$\rho_{\eta}({\mathbf{x}})=\eta({\mathbf{x}},{\mathbf{x}})=\sum_{n\in\mathbb{N}}\rho_{\mathbf{P}_{n}}(x_{1},x_{2})\rho_{\gamma_{n}}(x_{3})=\frac{b}{2\pi}\sum_{n\in\mathbb{N}}\rho_{\gamma_{n}}(x_{3}).$
∎
For an operator $\eta$ of the form (2.11), the trace per unit-surface, defined
as the limit
$\underline{\rm Tr}(\eta)=\lim_{L\to\infty}\frac{1}{L^{2}}{\rm
Tr}\left({\mathds{1}}_{\Gamma_{L}}\eta{\mathds{1}}_{\Gamma_{L}}\right),\quad\Gamma_{L}=[-\tfrac{L}{2},\tfrac{L}{2}]^{2}\times\mathbb{R}$
takes the simpler form
$\underline{\rm Tr}(\eta)=\frac{b}{2\pi}\sum_{n=0}^{\infty}{\rm
Tr}_{1}(\eta_{n}),$
where we have used that the density of any Landau level is
$\rho_{\mathbf{P}_{n}}=\frac{b}{2\pi}$.
## 3\. Reduction of the kinetic energy, and applications
We now exploit the particular structure of states
$\gamma\in{\mathcal{P}}^{\mathbb{A}}$ to deduce their kinetic energy.
### 3.1. Reduction of the kinetic energy
Recall that the set ${\mathcal{P}}^{\mathbb{A}}$ has been defined in (1.3) as
the set of one-body density matrices $\gamma$, satisfying the Pauli principle
$0\leq\gamma\leq 1$, which commute with all magnetic translations, and with
the Landau operator ${\mathbb{L}}^{\mathbb{A}}_{2}\otimes\mathbb{I}$. We also
recall that the set of reduced states ${\mathcal{G}}$ is defined by
${\mathcal{G}}:=\left\\{G\in{\mathcal{S}}(L^{2}(\mathbb{R})),\quad G\geq
0\right\\}.$
The main result of this section is the following.
###### Theorem 3.1.
For any $\gamma\in{\mathcal{P}}^{\mathbb{A}}$, there is an operator
$G\in{\mathcal{G}}$ satisfying $\rho_{G}=\rho_{\gamma}$ and
(3.1) $\frac{1}{2}\underline{\rm
Tr}({\mathbb{L}}^{\mathbb{A}}_{3}\gamma)\geq\frac{1}{2}{\rm
Tr}(-\Delta_{1}G)+{\rm Tr}(F(b,G)),$
where
(3.2) $F(b,g):=\pi g^{2}+\frac{b^{2}}{4\pi}\left\\{\frac{2\pi
g}{b}\right\\}\left(1-\left\\{\frac{2\pi g}{b}\right\\}\right).$
Conversely, for any $G\in{\mathcal{G}}$, there is
$\gamma\in{\mathcal{P}}^{\mathbb{A}}$ so that $\rho_{\gamma}=\rho_{G}$, and
for which there is equality in (3.1). In particular, for any (representable)
density $\rho$,
(3.3)
$\inf_{\gamma\in{\mathcal{P}}^{\mathbb{A}}\atop\rho_{\gamma}=\rho}\left\\{\frac{1}{2}\underline{\rm
Tr}({\mathbb{L}}^{\mathbb{A}}_{3}\gamma)\right\\}=\inf_{G\in{\mathcal{G}}\atop\rho_{G}=\rho}\left\\{\frac{1}{2}{\rm
Tr}(-\Delta_{1}G)+{\rm Tr}(F(b,G))\right\\}.$
###### Proof.
According to Theorem (2.7), any $\gamma\in{\mathcal{P}}^{\mathbb{A}}$ can be
decomposed as
(3.4)
$\gamma=\sum_{n=0}^{\infty}\mathbf{P}_{n}\otimes\gamma_{n}\quad\text{with}\quad\gamma_{n}\in{\mathcal{S}}(L^{2}(\mathbb{R})),\quad
0\leq\gamma_{n}\leq 1.$
For a state of the form (3.4), we define the operator
$G_{\gamma}\in{\mathcal{S}}(L^{2}(\mathbb{R}))$ by
$G_{\gamma}:=\frac{b}{2\pi}\sum_{n=0}^{\infty}\gamma_{n}.$
Since $\gamma_{n}\geq 0$, we have $G_{\gamma}\geq 0$ as well, so
$G_{\gamma}\in{\mathcal{G}}$. Also, since
$\rho_{\mathbf{P}_{n}}({\mathbf{x}})=\frac{b}{2\pi}$, we deduce that
$\rho_{G}=\rho_{\gamma}$.
Recalling that
${\mathbb{L}}^{\mathbb{A}}_{3}:={\mathbb{L}}^{\mathbb{A}}_{2}\otimes\mathbb{I}_{L^{2}(\mathbb{R})}+\mathbb{I}_{L^{2}(\mathbb{R}^{2})}\otimes(-\Delta_{1})$,
and using Proposition 2.1, we obtain that
$\displaystyle\frac{1}{2}\underline{\rm
Tr}\left({\mathbb{L}}^{\mathbb{A}}_{3}\gamma\right)$
$\displaystyle=\frac{1}{2}\sum_{n=0}^{\infty}\underline{\rm
Tr}\left(\varepsilon_{n}\mathbf{P}_{n}\otimes\gamma_{n}\right)+\frac{1}{2}\sum_{n=0}^{\infty}\underline{\rm
Tr}\left(\mathbf{P}_{n}\otimes(-\Delta_{1}\gamma_{n})\right)$
$\displaystyle=\frac{b}{4\pi}\sum_{n=0}^{\infty}\varepsilon_{n}{\rm
Tr}(\gamma_{n})+\frac{b}{4\pi}\sum_{n=0}^{\infty}{\rm
Tr}\left(-\Delta_{1}\gamma_{n}\right)$
$\displaystyle=\frac{b}{4\pi}\sum_{n=0}^{\infty}\varepsilon_{n}{\rm
Tr}(\gamma_{n})+\frac{1}{2}{\rm Tr}\left(-\Delta_{1}G\right).$
The first term cannot be expressed directly as a function of $G$, but we have
an inequality for this term. Since $G$ is a positive operator with finite
trace, it is compact, and admits a spectral decomposition of the form
$G=\sum_{j}g_{j}|\psi_{j}\rangle\langle\psi_{j}|$ with $g_{j}>0$ and
$\sum_{j}g_{j}<{\infty}$. Evaluating the trace of $\gamma_{n}$ in the
$\\{\psi_{j}\\}$ basis, and changing the order of the sums (all terms are
positive), we obtain
$\sum_{n=0}^{\infty}\varepsilon_{n}{\rm
Tr}(\gamma_{n})=\sum_{j=1}^{\infty}\left(\sum_{n=0}^{\infty}\varepsilon_{n}\left\langle\psi_{j},\gamma_{n}\psi_{j}\right\rangle\right).$
Since $0\leq\gamma_{n}\leq 1$, the quantity
$m_{j}(n):=\left\langle\psi_{j},\gamma_{n}\psi_{j}\right\rangle$ satisfies
$0\leq m_{j}(n)\leq 1$. In addition, we have
$\frac{b}{2\pi}\sum_{n}m_{j}(n)=\frac{b}{2\pi}\langle\psi_{j},\sum_{n}\gamma_{n}\psi_{j}\rangle=\langle\psi_{j},G\psi_{j}\rangle=g_{j}$.
So we have the inequality
(3.5) $\sum_{n=0}^{\infty}\varepsilon_{n}{\rm
Tr}(\gamma_{n})\geq\sum_{j=1}^{\infty}\inf_{m}\left\\{\sum_{n=0}^{\infty}\varepsilon_{n}m(n),\
0\leq m(n)\leq 1,\ \sum_{n=0}^{\infty}m(n)=\frac{2\pi g_{j}}{b}\right\\}.$
Since the $\varepsilon_{n}$ are ranked in increasing order, we can apply the
bathtub principle [13, Theorem 1.4]. The optimal $m$ for the above
minimization is given by
(3.6) $m_{j}^{*}(n)=\begin{cases}1&\text{for all}\quad 0\leq
n\leq\lfloor\frac{2\pi g_{j}}{b}\rfloor-1\\\ \\{\frac{2\pi
g_{j}}{b}\\}&\text{for}\quad n=\lfloor\frac{2\pi g_{j}}{b}\rfloor\\\
0&\text{otherwise}.\end{cases}.$
We now calculate the infimum in the RHS of (3.5) using the explicit formula of
the optimal function $m^{*}$. Recalling that $\varepsilon_{n}=b(2n+1)$ and
denoting by $x:=\frac{2\pi g_{j}}{b}$, we obtain
$\displaystyle\sum_{n=0}^{\infty}\varepsilon_{n}m_{j}^{*}(n)$
$\displaystyle=b\sum_{n=0}^{\lfloor x\rfloor-1}(2n+1)+b\left(2\left\lfloor
x\right\rfloor+1\right)\left\\{x\right\\}$
$\displaystyle=b\left(x^{2}+\\{x\\}(1-\\{x\\})\right)=\frac{4\pi}{b}F(b,g_{j}),$
with the function $F$ defined in (3.2). Summing in $j$ and gathering the terms
gives the inequality
$\frac{1}{2}\underline{\rm
Tr}\left({\mathbb{L}}^{\mathbb{A}}_{3}\gamma\right)\geq{\rm
Tr}\left(F(b,G)\right)+\frac{1}{2}{\rm Tr}\left(-\Delta_{1}G\right),$
which proves the first part of the Theorem. Conversely, given
$G=\sum_{j}g_{j}|\psi_{j}\rangle\langle\psi_{j}|\in{\mathcal{G}}$, we consider
the state
$\gamma^{*}:=\sum_{n=0}^{\infty}\mathbf{P}_{n}\otimes\gamma_{n}^{*},\quad\text{with}\quad\gamma_{n}^{*}:=\sum_{j=1}^{\infty}m_{j}^{*}(n)|\psi_{j}\rangle\langle\psi_{j}|$
and $m_{j}^{*}$ defined as in (3.6). The operator $\gamma^{*}$ belongs to
${\mathcal{P}}^{\mathbb{A}}$, satisfies $G_{\gamma^{*}}=G$, and gives an
equality in (3.1). ∎
### 3.2. Some properties of the function F
Let us collect some useful properties of the function $F$. A plot of $F$ is
displayed in Figure 1 below.
###### Proposition 3.2.
The function $F$ in (3.2) is continuous and satisfies
(3.7) $\pi g^{2}\leq F(b,g)\leq\pi g^{2}+\frac{b^{2}}{16\pi},$
with equality in the left for $\frac{2\pi g}{b}\in\mathbb{N}$, and equality in
the right for $\frac{2\pi g}{b}\in\mathbb{N}+\frac{1}{2}$, and $F(b,g)\to\pi
g^{2}$ as $b\to 0$. For any $b\geq 0$, the map $g\mapsto F(b,g)-\pi g^{2}$ is
$\frac{b}{2\pi}$ periodic and the map $g\in\mathbb{R}_{+}\mapsto F(b,g)$ is
piece-wise linear, increasing and convex. Finally, for all $0\leq
g<\frac{b}{2\pi}$, we have $F(b,g)=\frac{1}{2}bg$
###### Proof.
The first part is straightforward from the definition (3.2). To see that it is
convex, piece-wise linear and increasing, we use the alternative form
(3.8) $F(b,g)=\frac{b^{2}}{4\pi}\left(x(1+2\lfloor x\rfloor)-\lfloor
x\rfloor-\lfloor x\rfloor^{2}\right),$
where we have denoted by $x:=\frac{2\pi g}{b}$. When $0\leq g<\frac{b}{2\pi}$,
which corresponds to $0\leq x<1$, $F(b,g)=\frac{1}{2}bg$. ∎
###### Remark 3.3.
The left inequality of (3.7) implies ${\rm Tr}(F(b,G))\geq\pi{\rm Tr}(G^{2})$,
hence, together with (1.1), that
$\inf_{\gamma\in{\mathcal{P}}^{\mathbb{A}}\atop\rho_{\gamma}=\rho}\left\\{\frac{1}{2}\underline{\rm
Tr}\left({\mathbb{L}}^{\mathbb{A}}_{3}\gamma\right)\right\\}\geq\inf_{\gamma\in{\mathcal{P}}\atop\rho_{\gamma}=\rho}\left\\{\frac{1}{2}\underline{\rm
Tr}\left(-\Delta_{3}\gamma\right)\right\\}.$
In particular, the kinetic energy is higher with the magnetic field. This is a
kind of diamagnetic inequality for 2D materials.
###### Remark 3.4.
The fact that $F(b,g)\to\pi g^{2}$ as $b\to 0$, for all $g\in\mathbb{R}^{+}$,
means that, our reduction approach in this manuscript is coherent with the one
without magnetic field already treated in [9].
###### Remark 3.5.
Splitting $F$ into $F(b,G)=\pi g^{2}+\tilde{F}(b,g)$, we see that the effect
of adding a magnetic field ${\mathbb{B}}=(0,0,b)$ is a periodic perturbation
of the energy with no magnetic field.
### 3.3. Reduced DFT models
In the context of DFT, the previous result suggests modelling the electronic
state in a homogeneous slab of charge distribution
$\mu({\mathbf{x}})=\mu(x_{3})$ under a constant magnetic field
${\mathbb{B}}=b(0,0,x_{3})$ by a reduced state $G\in{\mathcal{G}}$ whose
energy per unit surface is given by
(3.9) ${\mathcal{E}}(G)=\frac{1}{2}{\rm Tr}(-\Delta_{1}G)+{\rm
Tr}\left(F_{b}(G)\right)+\frac{1}{2}{\mathcal{D}}_{1}(\rho_{G}-\mu)+E^{\rm
xc}(\rho_{G}).$
Here, $E^{\rm xc}$ models is an exchange-correlation energy per unit surface,
and ${\mathcal{D}}_{1}$ is the one–dimensional Hartree term. This last term
has been extensively studied in our previous work [9, Section 3.1], and is
defined as follows. For $f\in{\mathcal{C}}:=\left\\{f\in L^{1}(\mathbb{R}),\
W_{f}\in L^{2}(\mathbb{R})\right\\}$, where $W_{f}(x):=\int_{-\infty}^{x}f$ is
a primitive of $f$, we have
${\mathcal{D}}_{1}(f):=4\pi\int_{\mathbb{R}}|W_{f}|^{2}(x){\mathrm{d}}x.$
We have proved in [9, Proposition 3.3] that the elements $f\in{\mathcal{C}}$
have null integral $\int f=0$, that the map ${\mathcal{C}}\ni
f\mapsto{\mathcal{D}}_{1}(f)$ is convex, and that
${\mathcal{D}}_{1}(f)=4\pi\iint_{(\mathbb{R}_{+})^{2}\times(\mathbb{R}_{-})^{2}}\min\\{|x|,|y|\\}f(x)f(y){\mathrm{d}}x{\mathrm{d}}y=\int_{\mathbb{R}}\Phi_{f}(x)f(x){\mathrm{d}}x,$
with the mean-field potential
$\Phi_{f}(x):=4\pi\int_{\mathbb{R}^{\pm}}\min\\{|x|,|y|\\}f(y){\mathrm{d}}y,\quad
x\in\mathbb{R}^{\pm}.$
The function $\Phi_{f}$ is continuous, and is the (unique) solution to
$-\Phi_{f}^{\prime\prime}(x)=4\pi
f,\quad\Phi_{f}^{\prime}(x)\xrightarrow[x\to\pm\infty]{}0,\quad\Phi_{f}(0)=0.$
In practice, we restrict the minimization problem to the $G$ so that
$\rho_{G}-\mu\in{\mathcal{C}}$. This implies in particular the neutrality
condition ${\rm Tr}(G)=\int\rho_{G}=\int\mu=\nu$. On the other hand, if $G$ is
a trace class operator, then $\rho_{G}\in L^{1}$ (we assume that $\mu\in
L^{1}$ as well), and if $\rho_{G}-\mu$ has null integral, then
${\mathcal{D}}_{1}(\rho_{G}-\mu)<\infty$ iff $\rho_{G}-\mu\in{\mathcal{C}}$.
Note that there is no Pauli principle on $G$ for admissible states
${\mathcal{G}}$. It has been replaced by a penalization term $+F(b,G)$ in the
energy.
###### Remark 3.6.
The energy (3.9) is obtained when minimizing a 3-dimensional DFT model over
transitionally invariant states. In particular, this model does not include
possible spatial symmetry breaking along the first $2$ variables. Such
phenomena are known to exist in two-dimensional electron gas under magnetic
field due to the de Haas–van Alphen effect [5]. In some real-life systems e.g.
Br2, magnetic domains form, sometimes called Condon domains [4, 15]. Our
simple model is unable to capture these effects.
### 3.4. The reduced Hartree–Fock case
Let us illustrate the previous discussion in the particular case of the
reduced Hartree-Fock (rHF) model, in which $E^{\rm xc}=0$ in (3.9). We let
$0\leq\mu\in L^{1}(\mathbb{R})$ be a nuclear density describing a homogeneous
2D material and denote by $\nu=\int_{\mathbb{R}}\mu$ the total charge per unit
surface.
We denote by
(3.10) ${\mathcal{E}}_{b}^{\rm rHF}(G):=\frac{1}{2}{\rm Tr}(-\Delta_{1}G)+{\rm
Tr}(F(b,G))+\frac{1}{2}{\mathcal{D}}_{1}(\rho_{G}-\mu)$
the corresponding rHF energy per unit surface, and study the minimization
problem
${\mathcal{I}}_{b}^{\rm rHF}:=\inf\left\\{{\mathcal{E}}_{b}^{\rm
rHF}(G),\;G\in{\mathcal{G}}^{\nu}\right\\},\quad\text{with}\quad{\mathcal{G}}^{\nu}:=\left\\{G\in{\mathcal{G}},\
{\rm Tr}(G)=\nu\right\\}.$
Following the exact same lines as [9, Theorem 2.7], one has the following.
###### Theorem 3.7.
The problem ${\mathcal{I}}_{b}^{\rm rHF}$ admits a minimizer, and all
minimizers share the same density.
We skip the proof for brevity. The uniqueness of the density comes from the
fact that the problem is strictly convex in $\rho_{G}$. However, unlike the
case without magnetic field, $F(b,\cdot)$ is not strictly convex for $b>0$. It
is unclear to us whether the minimizer of ${\mathcal{I}}_{b}^{\rm rHF}$ is
unique.
We would like to write the Euler-Lagrange equations for a minimizer $G_{*}$.
Recall that $g\mapsto F(b,g)$ is continuous and convex (but is not smooth). We
denote by $f_{b}:=\partial_{g}F(b,\cdot)$ its subdifferential, a set-valued
function defined by
$f_{b}(g):=\left\\{a\in\mathbb{R},\quad\forall g^{\prime}\in\mathbb{R},\quad
F(b,g^{\prime})-F(b,g)\geq a(g^{\prime}-g)\right\\}.$
In our case, since the function $F_{b}$ is piece-wise linear, $f_{b}$ is
explicit. From (3.8), we obtain (in the following lines, $\\{a\\}$ denotes the
singleton $a$)
$f_{b}(g)=\begin{cases}\left\\{\frac{b^{2}}{4\pi}(2n+1)\right\\}\quad\text{if
}\exists n\in\mathbb{N}_{0},\;\quad n<\frac{2\pi g}{b}<n+1\\\
\left[\frac{b^{2}}{4\pi}(2n-1),\frac{b^{2}}{4\pi}(2n+1)\right]\quad\text{if}\quad\frac{2\pi
g}{b}=n\in\mathbb{N}_{0}.\end{cases}$
Its inverse map, noted $h_{b}$, is the set-valued function so that $y\in
f_{b}(x)$ iff $x\in h_{b}(y)$. One finds, for $y>0$,
$h_{b}(y)=\begin{cases}\left[n\frac{b}{2\pi},(n+1)\frac{b}{2\pi}\right]\quad\text{if}\quad
n:=\frac{1}{2}\left(\frac{4\pi}{b^{2}}y-1\right)\in\mathbb{N}_{0},\\\
\left\\{n\frac{b}{2\pi}\right\\}\quad\text{if there is $n\in\mathbb{N}_{0}$ so
that}\quad n-1<\frac{1}{2}\left(\frac{4\pi}{b^{2}}y-1\right)<n.\end{cases}$
We extend the definition of $h_{b}$ by setting $h_{b}(y)=0$ for $y<0$.
In order to work with functions, it is useful to introduce the maps
(3.11) $f_{b}^{\pm}(g):=\lim_{t\to
0^{\pm}}\frac{1}{t}\left(F(b,g+t)-F(b,g)\right)$
so that $f_{b}(g)=[f_{b}^{-}(g),f_{b}^{+}(g)]$ for all $g\in\mathbb{R}^{+}$.
Of course, if $g\notin\frac{b}{2\pi}\mathbb{N}_{0}$ is a regular point, then
$f_{b}(y)=f_{b}^{+}(y)=f_{b}^{-}(y)$. We define the maps $h_{b}^{\pm}$
similarly, so that $h_{b}(y)=[h_{b}^{-}(y),h_{b}^{+}(y)]$ for all
$y\in\mathbb{R}$.
The Euler–Lagrange equations for $G_{*}$ takes the following form (see end of
the section for the proof).
###### Proposition 3.8 (Euler-Lagrange equations).
Let $G_{*}$ be a minimizer of ${\mathcal{I}}^{{\rm rHF}}_{b}$. Then there is
$\lambda\in\mathbb{R}$ so that
(3.12) $\begin{cases}h_{b}^{-}(\lambda-H_{*})\leq G_{*}\leq h_{b}^{+}(\lambda-
H_{*})\\\ H_{*}:=-\frac{1}{2}\Delta_{1}+\Phi_{*}\\\
-\Phi_{*}^{\prime\prime}=4\pi(\rho_{*}-\mu),\quad\Phi_{*}^{\prime}(x)\xrightarrow[x\to\pm\infty]{}0,\quad\Phi_{*}(0)=0,\end{cases}$
where $\rho_{*}=\rho_{G_{*}}$ is the associated density of $G_{*}$, and
$\Phi_{*}$ is the mean-field potential, defined as the unique solution of the
last equation.
The first equation can also be written as
$G_{*}\in h_{b}(\lambda-H_{*}),$
and means that if $G_{*}=\sum_{j}g_{j}|\psi_{j}\rangle\langle\psi_{j}|$ is the
spectral decomposition of the optimizer, then $\psi_{j}$ is also an
eigenfunction of $H_{*}$ for an eigenvalue $\varepsilon_{j}$ so that
$g_{j}\in h_{b}(\lambda-\varepsilon_{j}),\quad\text{or
equivalently}\quad\varepsilon_{j}\in\lambda-f_{g}(g_{j}).$
Conversely, if $\varepsilon<\lambda$ is an eigenvalue of $H_{*}$, then
$\varepsilon=\varepsilon_{j}$ for some $j$.
In practice, for numerical purpose, one rather considers an approximation
$F^{\delta}_{b}$ of $F$, which is smooth, strictly convex, and so that
$\|F^{\delta}_{n}-F\|_{\infty}<\delta$. In this case, one can repeat the
arguments in [9, Theorem 2.7], and the first line of (3.12) becomes
$\left(F_{b}^{\delta}\right)^{\prime}(G_{*})=\lambda-H_{*},\quad\text{or,
equivalently}\quad
G_{*}=\left[\left(F_{b}^{\delta}\right)^{\prime}\right]^{-1}(\lambda-H_{*}).$
###### Remark 3.9 (Strong magnetic fields).
In the case where $b>2\pi\nu$, any $G\in{\mathcal{G}}^{\nu}$are positive and
satisfies ${\rm Tr}(G)=\nu$, hence all eigenvalues of $G$ are smaller than
$\nu$. In particular,
$F_{b}(G)=\frac{1}{2}bG,\quad\text{hence}\quad{\rm
Tr}\left(F_{b}(G)\right)=\frac{1}{2}b\nu$
is a constant, independent of $G\in{\mathcal{G}}^{\nu}$. In this case, $G_{*}$
is also the minimizer of
$\inf\left\\{\frac{1}{2}{\rm
Tr}(-\Delta_{1}G)+\frac{1}{2}{\mathcal{D}}_{1}(\rho_{G}-\mu),\quad
G\in{\mathcal{G}}^{\nu}\right\\}.$
This minimizer is therefore independent of $b>2\pi\nu$, reflecting the fact
that all electrons lie in the lowest Landau level. Following the previous
lines, we deduce that $G_{*}$ is a rank-$1$ operator, of the form
$G_{*}=\nu|\psi_{*}\rangle\langle\psi_{*}|$, with $\psi_{*}$ minimizing
$\inf\left\\{\frac{\nu}{2}\int_{\mathbb{R}}|\nabla\psi_{*}|^{2}+\frac{1}{2}{\mathcal{D}}_{1}\left(\nu|\psi_{*}|^{2}-\mu\right),\quad\psi_{*}\in
L^{2}(\mathbb{R}),\ \|\psi_{*}\|=1\right\\}.$
###### Proof of Proposition 3.8.
Let $G_{*}$ be a minimizer of ${\mathcal{I}}^{\rm rHF}_{b}$, and let
$\rho_{*}$ and $\Phi_{*}$ be the corresponding density and mean-field
potential, and set $H_{*}:=-\frac{1}{2}\Delta_{1}+\Phi_{*}$. Recall that
$\rho_{*}$ (hence $\Phi_{*}$ and $H_{*}$) is uniquely defined.
First, we claim that $G_{*}$ commutes with $H_{*}$. This is a standard result
in the case where the map $F$ is smooth (say of class $C^{1}$), using that
${\rm Tr}(F(G_{*}+H))={\rm Tr}(F(G_{*}))+{\rm Tr}(F^{\prime}(G_{*})H)+o(H).$
In our case however, the map $F$ is only piece-wise smooth, and we need a
direct proof. Let $A$ be a finite-rank symmetric operator on
$L^{2}(\mathbb{R})$, and set
$G_{t,A}:={\mathrm{e}}^{-{\mathrm{i}}tA}G_{*}{\mathrm{e}}^{{\mathrm{i}}tA}$.
Since $G_{t,A}$ is a unitary transformation of $G_{*}$, we have ${\rm
Tr}(G_{t,A})={\rm Tr}(G_{*})=\nu$, that is $G\in{\mathcal{G}}^{\nu}$ and ${\rm
Tr}(F(b,G_{t,A}))={\rm Tr}(F(b,G_{*}))$. In particular,
$\displaystyle{\mathcal{E}}^{\rm rHF}_{b}(G_{t,A})$
$\displaystyle={\mathcal{E}}^{\rm rHF}_{b}(G_{*})$
$\displaystyle+\frac{1}{2}\left({\rm
Tr}(-\Delta_{1}[G_{t,A}-G_{*}])+{\mathcal{D}}_{1}\left(\rho_{G_{t,A}}-\mu\right)-{\mathcal{D}}_{1}\left(\rho_{*}-\mu\right)\right).$
Together with the fact that
$G_{t,A}=G_{*}+{\mathrm{i}}t\left[G_{*},A\right]+o(t),$
and the definition of $H_{*}$, we deduce that
${\mathcal{E}}^{\rm rHF}_{b}(G_{t,A})={\mathcal{E}}^{\rm
rHF}_{b}(G_{*})+{\mathrm{i}}t{\rm Tr}\left(H_{*}[G_{*},A]\right)+o(t).$
Since the minimum of ${\mathcal{E}}^{\rm rHF}_{b}$ is obtained for $t=0$, the
linear term in $t$ must vanish, that is:
$\displaystyle 0$ $\displaystyle={\rm Tr}\left(H_{*}[G_{*},A]\right)={\rm
Tr}\left(H_{*}G_{*}A-H_{*}AG_{*}\right)$ $\displaystyle={\rm
Tr}\left(H_{*}G_{*}A-G_{*}H_{*}A\right)={\rm Tr}\left([H_{*},G_{*}]A\right),$
where we have used cyclicity of the trace, and the fact that $A$ is finite
rank (so all operators are trace-class). Since this is true for all finite
rank symmetric operators $A$, we deduce as wanted that $[H_{*},G_{*}]=0$.
Recall that $G_{*}$ is positive compact (even trace class). So there is
$M:={\rm rank}(G_{*})\in\mathbb{N}\cup\\{\infty\\}$ and an orthonormal family
$\\{\psi_{j}\\}_{j\in[1,M]}$ so that
$G_{*}=\sum_{j=1}^{M}g_{j}|\psi_{j}\rangle\langle\psi_{j}|,\quad\text{with}\quad
H\psi_{j}=\varepsilon_{j}\psi_{j},$
and with $g_{1}\geq g_{2}\geq\cdots\geq g_{M}>0$. The orthonormal family
$\\{\psi_{j}\\}_{1\leq j\leq M}$ spans ${\rm Ran}(G_{*})$. For two indices
$(i,j)\in[1,M]$, we consider the operator
$G_{t}^{(i,j)}:=G_{*}+t\left(|\psi_{i}\rangle\langle\psi_{i}|-|\psi_{j}\rangle\langle\psi_{j}|\right).$
For $t$ small enough ($|t|<\min\\{g_{i},g_{j}\\}$), the operator
$G_{t}^{(i,j)}$ is positive, and with ${\rm Tr}(G_{t}^{(i,j)})={\rm
Tr}(G_{*})=\nu$, hence $G_{t}^{(i,j)}\in{\mathcal{G}}^{\nu}$. In addition, we
have
$\displaystyle{\mathcal{E}}^{\rm rHF}_{b}(G_{t}^{(i,j)})-{\mathcal{E}}^{\rm
rHF}_{b}(G_{*})=$
$\displaystyle\left(F(b,g_{i}+t)-F(b,g_{i})\right)+\left(F(b,g_{j}-t)-F(b,g_{j})\right)$
$\displaystyle+t\underbrace{{\rm
Tr}(H_{*}(|\psi_{i}\rangle\langle\psi_{i}|-|\psi_{j}\rangle\langle\psi_{j}|))}_{=\varepsilon_{i}-\varepsilon_{j}}+o(t).$
By optimality of $G_{*}$, this quantity is always positive. Taking the limit
$t\to 0^{+}$, and recalling the definition of $f_{b}^{\pm}$ in (3.11), we
deduce that
$f_{b}^{+}(g_{i})+\varepsilon_{i}\geq f_{b}^{-}(g_{j})+\varepsilon_{j}.$
This inequality is valid for all $1\leq i,j\leq M$, so
$\inf_{1\leq i\leq
M}\left(f_{b}^{+}(g_{i})+\varepsilon_{i}\right)\geq\sup_{1\leq j\leq
M}\left(f_{b}^{-}(g_{j})+\varepsilon_{j}\right)=:\lambda$
For all $1\leq j\leq M$, we have
$f_{b}^{-}(g_{j})+\varepsilon_{j}\leq\lambda\leq
f_{b}^{+}(g_{j})+\varepsilon_{j},\quad\text{hence}\quad\lambda-\varepsilon_{j}\in[f_{b}^{-}(g_{j}),f_{b}^{+}(g_{j})].$
This is also $\lambda-\varepsilon_{j}\in f_{b}(g_{j})$, or equivalently
$g_{j}\in h_{b}(\lambda-\varepsilon_{j})$. Note that if there is an eigenvalue
$g_{j}\notin\frac{b}{2\pi}\mathbb{Z}$, then
$f_{b}^{-}(g_{j})=f_{b}^{+}(g_{j})$ for this eigenvalue, and there is
equality. In other words, we have proved that
$G_{*}\in h_{b}(\lambda-H_{*})\quad\text{on}\quad{\rm Ran}(G_{*}).$
It remains to prove the result on ${\rm Ker}(G_{*})$. Let $\psi\in{\rm
Ker}(G_{*})$. This time, we consider the perturbed state
$G_{t}^{(j)}:=G_{*}+t\left(|\psi\rangle\langle\psi|-|\psi_{j}\rangle\langle\psi_{j}|\right),$
which is in ${\mathcal{G}}^{\nu}$ for all $0\leq t\leq g_{j}$. Taking the
limit $t\to 0^{+}$, and reasoning as before, we get
$f_{b}^{+}(0)+\langle\psi,H_{*}\psi\rangle\geq
f_{b}^{-}(g_{j})+\varepsilon_{j},\quad\text{hence}\quad
f_{b}^{+}(0)+\langle\psi,H_{*}\psi\rangle\geq\lambda,$
where we took the supremum in $1\leq j\leq M$ in the last inequality. We
deduce that, for all $\psi\in{\rm Ker}(G_{*})$,
$f_{b}^{+}(\langle\psi,G_{*}\psi\rangle)=f_{b}^{+}(0)\geq\lambda-\langle\psi,H_{*}\psi\rangle,$
so
$f_{b}^{+}(G_{*})\geq\lambda-H_{*},\quad\text{on}\quad{\rm Ker}(G_{*}).$
Together with the fact that $f_{b}^{-}(G_{*})=0$ on ${\rm Ker}(G_{*})$, we
obtain $G_{*}\in h_{b}(\lambda-H_{*})$ on ${\rm Ker}(G_{*})$ as well. ∎
## 4\. Models with spin
In this section, we explain how to extend our results to the case where the
spin is taken into account. In this case, the density matrix is an operator
$\gamma\in{\mathcal{S}}(L^{2}(\mathbb{R}^{3},\mathbb{C}^{2}))$ satisfying the
Pauli-principle $0\leq\gamma\leq 1$. Such an operator can be decomposed as a
$2\times 2$ matrix of the form
${\gamma}=\begin{pmatrix}\gamma^{\uparrow\uparrow}&\gamma^{\uparrow\downarrow}\\\
\gamma^{\downarrow\uparrow}&\gamma^{\downarrow\downarrow}\end{pmatrix}.$
The spin–density $2\times 2$ matrix of $\gamma$ is
$R_{\gamma}({\mathbf{x}})=\gamma({\mathbf{x}},{\mathbf{x}})$ (see [7] for
details), and the total density is
$\rho_{\gamma}({\mathbf{x}})=R_{\gamma}^{\uparrow\uparrow}({\mathbf{x}})+R_{\gamma}^{\downarrow\downarrow}({\mathbf{x}})=\gamma^{\uparrow\uparrow}({\mathbf{x}},{\mathbf{x}})+\gamma^{\downarrow\downarrow}({\mathbf{x}},{\mathbf{x}})$.
The kinetic energy operator is now the Pauli operator
$\mathbf{P}^{\mathbb{A}}_{3}:=\left[\sigma\cdot\left(-{\mathrm{i}}\nabla+{\mathbb{A}}\right)\right]^{2},$
where $\sigma$ contains the Pauli matrices. In the case of constant magnetic
field ${\mathbb{B}}=(0,0,b)$ with the gauge ${\mathbb{A}}=b(0,x_{1},0)$, this
operator becomes
$\mathbf{P}^{\mathbb{A}}_{3}={\mathbb{L}}^{\mathbb{A}}_{3}\mathbb{I}_{2}+\begin{pmatrix}b&0\\\
0&-b\end{pmatrix}={\mathbb{L}}^{\mathbb{A}}_{2}\mathbb{I}_{2}+\begin{pmatrix}b&0\\\
0&-b\end{pmatrix}-\partial_{x_{3}x_{3}}^{2}\mathbb{I}_{2}.$
In what follows, we denote by ${\mathcal{B}}:=\begin{pmatrix}b&0\\\
0&-b\end{pmatrix}$. This term corresponds to the Zeeman term. There are
several ways to read this operator. Indeed, we have
$\displaystyle L^{2}(\mathbb{R}^{3},\mathbb{C}^{2})$ $\displaystyle\simeq
L^{2}(\mathbb{R}^{3})\otimes\mathbb{C}^{2}\simeq L^{2}(\mathbb{R}^{2})\otimes
L^{2}(\mathbb{R})\otimes\mathbb{C}^{2}$ (4.1) $\displaystyle\simeq
L^{2}(\mathbb{R}^{2})\otimes L^{2}(\mathbb{R},\mathbb{C}^{2})$ (4.2)
$\displaystyle\simeq L^{2}(\mathbb{R}^{2},\mathbb{C}^{2})\otimes
L^{2}(\mathbb{R}).$
In the decomposition (4.1) (resp. (4.2)), we split the Pauli operator as
$\mathbf{P}^{\mathbb{A}}_{3}={\mathbb{L}}^{\mathbb{A}}_{2}\otimes\mathbb{I}+\mathbb{I}\otimes({\mathcal{B}}-\partial_{x_{3}x_{3}}^{2}),\quad(\text{resp.
}\quad\mathbf{P}^{\mathbb{A}}_{3}=({\mathbb{L}}^{\mathbb{A}}_{2}+{\mathcal{B}})\otimes\mathbb{I}+\mathbb{I}\otimes(-\partial_{x_{3}x_{3}}^{2}).)$
First splitting: keeping the spin structure. This splitting is useful if one
wants to keep track of the spin–density structure of $R_{\gamma}$ as a
function of $x_{3}$. This happens for instance when using
exchange–correlations functionals which depend explicitly on the spin, as in
the LSDA model [17, 8]. In this case, we consider one-body density matrices
$\gamma$ of the form
(4.3)
$\gamma=\sum_{n=0}^{\infty}\mathbf{P}_{n}\otimes\gamma_{n},\quad\gamma_{n}\in{\mathcal{S}}(L^{2}(\mathbb{R},\mathbb{C}^{2})),\
0\leq\gamma_{n}\leq 1.$
This is similar to what we have studied in the present article, but the
$\gamma_{n}$ operators now act on $L^{2}(\mathbb{R},\mathbb{C}^{2})$.
Introducing the operator $G:=\frac{b}{2\pi}\sum\gamma_{n}$ and the set
${\mathcal{G}}^{\rm spin}:=\left\\{G\in L^{2}(\mathbb{R},\mathbb{C}^{2}),\
G\geq 0\right\\},$
we obtain as before that, for all representable spin–density $2\times 2$
matrix $R$, we have
$\displaystyle\inf_{\gamma\ \text{of the form\leavevmode\nobreak\
\eqref{eq:gamma-spin1}}\atop R_{\gamma}=R}\left\\{\frac{1}{2}\underline{\rm
Tr}(\mathbf{P}^{\mathbb{A}}_{3}\gamma)\right\\}$
$\displaystyle\qquad=\inf_{G\in{\mathcal{G}}^{\rm spin}\atop
R_{G}=R}\left\\{\frac{1}{2}{\rm Tr}(-\Delta_{1}G)+{\rm
Tr}(F(b,G))\right\\}+b\int_{\mathbb{R}}(R^{\uparrow\uparrow}-R^{\downarrow\downarrow}).$
The last term is the Zeeman term, where we have used that ${\rm
Tr}({\mathcal{B}}G)=b{\rm
Tr}(G^{\uparrow\uparrow}-G^{\downarrow\downarrow})=b\int(R^{\uparrow\uparrow}-R^{\downarrow\downarrow})$.
Second splitting: loosing the spin structure. If one is only interested in
keeping the total density $\rho_{\gamma}$ instead of the full spin–density
$2\times 2$ matrix $R_{\gamma}$, one should instead use the spectral
decomposition of the operator
$\mathbf{P}^{\mathbb{A}}_{2}:={\mathbb{L}}^{\mathbb{A}}_{2}+{\mathcal{B}}$.
This one is easily deduced from the one of ${\mathbb{L}}^{\mathbb{A}}_{2}$ in
Proposition 2.1. Using that $\varepsilon_{n}=b(2n+1)$, we obtain
$\mathbf{P}^{\mathbb{A}}_{2}=\sum_{n=0}^{\infty}\widetilde{\varepsilon}_{n}\widetilde{\mathbf{P}}_{n}$
with
$\widetilde{\varepsilon}_{n}=2nb,\quad\widetilde{\mathbf{P}}_{0}=\begin{pmatrix}0&0\\\
0&\mathbf{P}_{0}\end{pmatrix},\quad\widetilde{\mathbf{P}}_{n}=\begin{pmatrix}\mathbf{P}_{n-1}&0\\\
0&\mathbf{P}_{n}.\end{pmatrix}\text{ for }n\geq 1.$
The lowest Landau level has now energy $\widetilde{\varepsilon}_{0}=0$, and is
only occupied by spin-down electrons. The corresponding eigenspace has density
$\rho_{\widetilde{\mathbf{P}_{0}}}=\frac{b}{2\pi}$. The other Landau levels
are <<doubly>> occupied, with density
$\rho_{\widetilde{\mathbf{P}_{n}}}=\frac{b}{\pi}$.
In this case, we consider density matrices $\gamma$ of the form
(4.4)
$\gamma=\sum_{n=0}^{\infty}\widetilde{\mathbf{P}_{n}}\otimes\widetilde{\gamma}_{n},\quad\widetilde{\gamma}_{n}\in{\mathcal{S}}(L^{2}(\mathbb{R},\mathbb{C})),\
0\leq\widetilde{\gamma}_{n}\leq 1.$
###### Remark 4.1.
One can prove that the set of such states is smaller than the set of states
$\gamma$ commuting with $\mathbf{P}^{\mathbb{A}}_{2}$ and the magnetic
translations ${\mathfrak{m}}_{\mathbb{R}}$. This comes from the fact that the
family $({\mathfrak{m}}_{\mathbb{R}})$ is not irreducible on
$\widetilde{E}_{n}:={\rm Ran}\widetilde{P_{n}}$.
The conclusion of Theorem 3.1 still holds in the spin setting, and the proof
is similar with only minor modifications: we now consider the $G$ matrix
defined by
$G_{\gamma}=\frac{b}{2\pi}\left(\widetilde{\gamma}_{0}+2\sum_{n=1}^{\infty}\widetilde{\gamma}_{n}\right)\qquad\in{\mathcal{G}},$
and the optimal $m_{j}^{*}$ functions are given by $m_{j}^{*}(0)=\\{\frac{2\pi
g_{j}}{b}\\}$ and $m_{j}^{*}(n)=0$ if $\frac{2\pi g_{j}}{b}<1$, and, if
$\frac{2\pi g_{j}}{b}\geq 1$,
$m_{j}^{*}(n)=\begin{cases}1&\quad\text{for}\quad
n\leq\lfloor\frac{\pi}{b}g_{j}+\frac{1}{2}\rfloor-1\\\
\left\\{\frac{\pi}{b}g_{j}+\frac{1}{2}\right\\}&\quad\text{for}\quad
n=\lfloor\frac{\pi}{b}g_{j}+\frac{1}{2}\rfloor\\\
0&\quad\text{otherwise}.\end{cases}$
Denoting by $y=\frac{\pi}{b}g_{j}+\frac{1}{2}$, this gives the energy
$\displaystyle\frac{b}{2\pi}\sum_{n=0}^{\infty}\widetilde{\varepsilon}_{n}m_{j}^{*}(n)$
$\displaystyle=\frac{b^{2}}{2\pi}\sum_{n=1}^{\lfloor
y\rfloor-1}2n+\frac{b^{2}}{\pi}\lfloor
y\rfloor\left\\{y\right\\}=\frac{b^{2}}{2\pi}\left(y^{2}-y+\\{y\\}(1-\\{y\\})\right).$
We obtain that for any representable density $\rho$.
$\inf_{\gamma\ \text{of the form\leavevmode\nobreak\ \eqref{eq:gamma-
spin2}}\atop\rho_{\gamma}=\rho}\left\\{\frac{1}{2}\underline{\rm
Tr}(\mathbf{P}^{\mathbb{A}}_{3}\gamma)\right\\}=\inf_{G\in{\mathcal{G}}\atop\rho_{G}=\rho}\left\\{\frac{1}{2}{\rm
Tr}(-\Delta_{1}G)+{\rm Tr}(F^{\rm spin}(b,G))\right\\},$
with the new functional
$F^{\rm
spin}(b,g)=\frac{}{}\frac{\pi}{2}g^{2}-\frac{b^{2}}{8\pi}+\frac{b^{2}}{2\pi}\left\\{\frac{\pi
g}{b}+\frac{1}{2}\right\\}\left(1-\left\\{\frac{\pi
g}{b}+\frac{1}{2}\right\\}\right).$
A plot of $F^{\rm spin}(b=1,g)$ is displayed in Figure 1. The constant
$\frac{\pi}{2}$ in the $\frac{\pi}{2}g^{2}$ term is the two-dimensional
Thomas-Fermi constant, when the spin of the electron is included. Again, we
see that magnetic effects gives a correction to the Thomas-Fermi
approximation, but this time, including the Zeeman term, the magnetic energy
is always lower than the no-magnetic one, and we have
(4.5) $\frac{\pi}{2}g^{2}-\frac{b^{2}}{8\pi}\leq F^{\rm
spin}(b,g)\leq\frac{\pi}{2}g^{2}.$
Figure 1. (Left). The map $g\mapsto F(b=1,g)$ (red) together with its
lower/upper bounds in (3.7). (Right) The map $g\mapsto F^{\rm spin}(b=1,g)$
(red) and its bounds (4.5).
## References
* [1] X. Blanc and C. Le Bris. Thomas-Fermi type theories for polymers and thin films. Advances in Differential Equations, 5(7-9):977 – 1032, 2000.
* [2] O. Bratteli and D.W. Robinson. Operator Algebras and Quantum Statistical Mechanics: C*- and W*-algebras, symmetry groups, decomposition of states. Texts and monographs in physics. Springer-Verlag, 1987.
* [3] Vivek Chaudhary, Petr Neugebauer, Omar Mounkachi, Salma Lahbabi, and A. EL Fatimy. Phosphorene - an emerging two-dimensional material: recent advances in synthesis, functionalization, and applications. 2D Materials, 9, 05 2022.
* [4] J.H. Condon. Nonlinear de Haas-van Alphen effect and magnetic domains in Beryllium. Phys. Rev., 145(2):526, 1966.
* [5] W.J. De Haas and P.M. Van Alphen. The dependence of the susceptibility of diamagnetic metals upon the field. Proc. Netherlands Roy. Acad. Sci, 33(1106):170, 1930.
* [6] I. Fushiki, E.H. Gudmundsson, C.J. Pethick, and J. Yngvason. Matter in a magnetic field in the Thomas-Fermi and related theories. Annals of Physics, 216(1):29–72, 1992.
* [7] D. Gontier. N-representability in noncollinear spin-polarized density-functional theory. Phys. Rev. Lett., 111:153001, 2013.
* [8] D. Gontier. Existence of minimizers for Kohn–Sham within the local spin density approximation. Nonlinearity, 28(1):57, 2014.
* [9] D. Gontier, S. Lahbabi, and Maichine A. Density functional theory for homogeneous two dimensional materials. Comm. Math. Phys., 388:1475–1505, 2021.
* [10] D. Gupta, V. Chauhan, and R. Kumar. A comprehensive review on synthesis and applications of molybdenum disulfide (mos2) material: Past and recent developments. Inorganic Chemistry Communications, 121:108200, 2020.
* [11] Y. Kaddar, G. Tiouitchi, A. El Kenz, A . Benyoussef, P. Neugebauer, S. Lahbabi, A. El Fatimy, and O. Mounkachi. Black phosphorus synthesized by the chemical vapor transport: Magnetic properties induced by oxygen atoms. Submitted for publication, 2022.
* [12] L.L. Lev, I.O. Maiboroda, M.A. Husanu, and et al. k-space imaging of anisotropic 2D electron gas in GaN/GaAlN high-electron-mobility transistor heterostructures. Nature Communication, 9(2653), 2018.
* [13] E.H. Lieb and M. Loss. Analysis, volume 14. American Mathematical Soc., 2001.
* [14] E.H. Lieb, J.P. Solovej, and J. Yngvason. Ground states of large quantum dots in magnetic fields. Phys Rev B Condens Matter, 51(16):10646–10665, 1995.
* [15] R.S. Markiewicz, M. Meskoob, and C. Zahopoulos. Giant magnetic interaction (Condon domains) in two dimensions. Phys. Rev. Lett., 54(13):1436, 1985.
* [16] J. Saha and A. Dutta. A review of graphene: Material synthesis from biomass sources. Waste and Biomass Valorization, 13, 03 2022.
* [17] U. Von Barth and L. Hedin. A local exchange-correlation potential for the spin polarized case. I. Journal of Physics C: Solid State Physics, 5(13):1629, 1972.
* [18] M. W. Wong. Wigner Transforms. Universitext. Springer, New York, 1998.
* [19] Y. Yngvason. Thomas-Fermi theory for matter in a magnetic field as a limit of quantum mechanics. Letters in Mathemattcal Physics, 22:107–117, 1991.
|
XX 2022 ACML 2022
# Graph annotation generative adversarial networks
Yoann Boget<EMAIL_ADDRESS>
University of Geneva and Geneva School for Business administration HES-SO
Rue de la Tambourine 17
Carouge Switzerland Magda Gregorova<EMAIL_ADDRESS>
Center for Artificial Intelligence and Robotics (CAIRO) FHWS
Franz-Horn-Strasse 2
Würzburg-Schweinfurt Germany Alexandros Kalousis<EMAIL_ADDRESS>
Geneva School for Business administration HES-SO
Rue de la Tambourine 17
Carouge Switzerland
###### Abstract
We consider the problem of modelling high-dimensional distributions and
generating new examples of data with complex relational feature structure
coherent with a graph skeleton. The model we propose tackles the problem of
generating the data features constrained by the specific graph structure of
each data point by splitting the task into two phases. In the first it models
the distribution of features associated with the nodes of the given graph, in
the second it complements the edge features conditionally on the node
features. We follow the strategy of implicit distribution modelling via
generative adversarial network (GAN) combined with permutation equivariant
message passing architecture operating over the sets of nodes and edges. This
enables generating the feature vectors of all the graph objects in one go (in
2 phases) as opposed to a much slower one-by-one generations of sequential
models, prevents the need for expensive graph matching procedures usually
needed for likelihood-based generative models, and uses efficiently the
network capacity by being insensitive to the particular node ordering in the
graph representation. To the best of our knowledge, this is the first method
that models the feature distribution along the graph skeleton allowing for
generations of annotated graphs with user specified structures. Our
experiments demonstrate the ability of our model to learn complex structured
distributions through quantitative evaluation over three annotated graph
datasets.
###### keywords:
Graph, Annotation, Generative Model, GAN, adversarial.
††editors: Emtiyaz Khan and Mehmet Gonen
## 1 Introduction
Modern deep learning approaches for learning high-dimensional data
distributions and synthesizing new data examples have achieved great successes
in a number of domains. We focus here on the particularly challenging problem
of generating new data examples ${\mathbf{x}}_{i}\in\mathbb{R}^{m_{i}}$ with
different dimensionalities $m_{i}$ between individual instances and complex
relational structures in the feature spaces organized in graphs.
More concretely, we focus on learning the distributions and generating new
examples of features associated with nodes and edges of annotated graphs
conditioned on the graph structure.
We formulate the problem as that of modeling a conditional distribution, where
we learn to generate the node and edge features conditionally on the given
graph skeleton (non-annotated graph). In this respect, our paper complements
the existing tool-set of models for non-annotated graph generations (e.g. You
et al. (2018); Liao et al. (2019)). The non-annotated graph can be generated
by one of those or, more interestingly, can be provided by the user based on
the needs of a downstream task that requires a specific graph structure (e.g.
basis for scaffold-based _de novo_ drug discovery or molecular docking, types
and interactions of particles in high energy physic, individual features and
types of relations in a social network).
The graph annotation generative adversarial networks (GrannGAN) method that we
propose follows the strategy of implicit distribution modeling via adversarial
training Goodfellow et al. (2014), allowing us to sample new graphs from a
distribution $p$ approximating the true $p^{*}$ without explicitly formulating
the $p$ distribution function. In principle, the method can generate node of
edge features independently. In practice, we generate the node and edge
features in two phases depicted in Figure 1. In the first phase, our model
samples the node features conditioning on the graph skeleton. In the second
phase, it uses the generated node features as an additional conditional
variable to sample the edge features.
The entire model architecture relies on the permutation equivariant
transformations of message passing neural networks (MPNN) Gilmer et al. .
Thanks to these, we circumvent the difficulties related to the ordering of
graph objects in their representation. This critical property of the method
allows for efficient use of the model capacity, which needs to learn neither a
particular heuristic for unique representation ordering (as in the case of
linearized representations) nor the complete set of equivalent permutations.
In the following, we describe the newly proposed GrannGAN method and provide
some details of the technical implementation in sections 2. We then position
the method within the existing state of research and document its performance
competitive with the best of the well-established methods for graph
generations on a set of experiments in section 3 and 4. We conclude with a
discussion of possible future directions in section 5.
## 2 GrannGAN
Let ${\mathcal{G}}=\\{{\mathcal{V}},{\mathcal{E}},V,E\\}$ be an undirected
graph with a set of nodes (vertices) ${\mathcal{V}}$ and a set of edges
${\mathcal{E}}$ between pairs of nodes in the graph, $V$ and $E$ are the
corresponding node and edge features. Let $\nu_{i}$ denote a node $i$ and
${\mathbf{v}}_{i}\in\mathbb{R}^{d}$ the features associated with that node. An
edge $\epsilon_{ij}$ is connecting the pair of nodes $\nu_{i}$ and $\nu_{j}$
and ${\mathbf{e}}_{ij}\in\mathbb{R}^{c}$ are features corresponding to that
edge. We consider only the case where the edges are undirected so that
$\epsilon_{ij}=\epsilon_{ji}$ and ${\mathbf{e}}_{ij}={\mathbf{e}}_{ji}$, but
the model can be easily extended to the case with directed edges. We further
use the term _skeleton_ and the letter ${\mathcal{S}}$ to refer to the non-
annotated graph ${\mathcal{S}}=\\{{\mathcal{V}},{\mathcal{E}}\\}$
corresponding to ${\mathcal{G}}$.
### 2.1 Model factorization
In our approach, we model the underlying graph distribution in the following
factorization
$\displaystyle
p({\mathcal{G}})=p({\mathcal{S}},V,E)=p({\mathcal{S}})p(V\rvert{\mathcal{S}})p(E\rvert{\mathcal{S}},V)\enspace.$
(1)
Our model consists of modeling $p(V\rvert{\mathcal{S}})$ and
$p(E\rvert{\mathcal{S}},V)$ with conditional GAN. Note that we could
equivalently reverse the conditioning order and model
$p(E\rvert{\mathcal{S}})$ and $p(V\rvert{\mathcal{S}},E)$. In our experiments,
we found the first option yielding better results. So, we keep this ordering
for the following.
During training, we sample the graph skeleton from the data distribution
$p^{*}({\mathcal{S}})$. At inference, we can either sample the graph skeleton
or keep it fixed depending on the task.
### 2.2 Implicit data generation
We use the Wasserstein-GAN (WGAN) Arjovsky et al. (2017) formulation of
adversarial training with spectral normalization Miyato et al. (2018) in all
linear layers of the critic to enforce 1-lipschitzness. To sample new examples
of data ${\mathbf{x}}$ from a model distribution $p_{\theta}({\mathbf{x}})$
WGAN uses a generator $g_{\theta}$ mapping from a random latent variable
${\mathbf{z}}$ to the output ${\mathbf{x}}=g_{\theta}({\mathbf{x}})$. The
generator is learned to minimize the Wasserstein-1 (or Earth-Mover) distance
$W(p_{\theta}({\mathbf{x}}),p^{*}({\mathbf{x}}))$ between the implicit model
distribution and the true generative distribution through a min-max
optimization.
$\displaystyle\min_{\theta}\max_{\varphi}\ \mathbb{E}_{(y)\sim
p^{*}}f_{\varphi}({\mathbf{y}})-\mathbb{E}_{z\sim
p(z)}f_{\varphi}\big{(}g_{\theta}({\mathbf{z}})\big{)}\enspace,$ (2)
where $f_{\varphi}$ is the K-Lipschitz critic function mapping from the data
${\mathbf{y}}$ (real or generated) to a real-valued score
$f_{\varphi}({\mathbf{y}})\in\mathbb{R}$.
We use the conditional WGAN formulation twice in the GrannGAN pipeline
sketched out in figure 1. Following the factorization in equation (1) we first
model the conditional distribution $p(V\rvert{\mathcal{S}})$ in a _node-
annotation_ phase. Here, the method generates the node features
$\hat{V}=g_{V,\theta}({\mathcal{Z}},{\mathcal{S}})$ from the set of latent
noise variables ${\mathcal{Z}}$ and the skeleton ${\mathcal{S}}$. So, equation
2 becomes
$\displaystyle\min_{\theta}\max_{\varphi}\ \mathbb{E}_{(V,{\mathcal{S}})\sim
p^{*}}f_{V,\varphi}(V,{\mathcal{S}})-\mathbb{E}_{z\sim p(z),{\mathcal{S}}\sim
p^{*}}f_{V,\varphi}\big{(}g_{V,\theta}({\mathcal{Z}},{\mathcal{S}}),{\mathcal{S}}\big{)}\enspace.$
(3)
Similarly in the following _edge-annotation_ phase, we model the conditional
$p(E\rvert{\mathcal{S}},V)$. The edge features
$\hat{E}=g_{E,\theta}({\mathcal{Z}},V,{\mathcal{S}})$ are generated
conditionally on the previously generated node features by including these as
an additional input variable to the generator and critic. In this phase,
equation 2 becomes
$\displaystyle\min_{\theta}\max_{\varphi}\ \mathbb{E}_{(E,V,{\mathcal{S}})\sim
p^{*}}f_{E,\varphi}(E,V,{\mathcal{S}})-\mathbb{E}_{z\sim
p(z),(V,{\mathcal{S}})\sim
p^{*}}f_{E,\varphi}\big{(}g_{E,\theta}({\mathcal{Z}},V,{\mathcal{S}}),V,{\mathcal{S}}\big{)}\enspace,$
(4)
Note that the conditioning on the skeleton
${\mathcal{S}}=({\mathcal{V}},{\mathcal{E}})$ acts on the computational graph
of the critic $f$ and of the generator $g$, i.e. on the structure of the
Message Passing Neural Network (MPNN, explained in detail in the section 2.4).
This special way of conditioning through the computational graph is one of the
novelties of our method.
### 2.3 Implementation
In practice, we start by sampling the skeleton
${\mathcal{S}}=\\{{\mathcal{V}},{\mathcal{E}}\\}$ from the data. For all node
$\nu_{i}\in{\mathcal{V}}$, we sample a random noise vector ${\mathbf{z}}_{i}$.
So, we get ${\mathcal{Z}}=\\{{\mathbf{z}}_{i}\\}_{1}^{n}$ and, by construction
$|{\mathcal{V}}|=|{\mathcal{Z}}|$. Doing so, we obtain a graph with random
noise vectors as initial latent representation of the node features
${\mathcal{G}}_{0}=\\{{\mathcal{V}},{\mathcal{E}},{\mathcal{Z}}\\}$ (and
without edge feature). This graph is the input of the _node-annotation_
generator. Similarly, for the edge generation, we sample a graph from the data
without edge feature. We also sample
${\mathcal{Z}}=\\{{\mathbf{z}}_{i}\\}_{1}^{m}$, where $m=|{\mathcal{E}}|$. The
input of the _edge-annotation_ generator is the graph
${\mathcal{G}}_{0}=\\{{\mathcal{V}},{\mathcal{E}},V,{\mathcal{Z}}\\}$ with
latent random noise vectors as initial latent edge representation. The
generators are permutation-equivarient functions. They output node or edge
features following the order of the input.
The critics receive alternatively real or generated graphs (without edge
feature during the annotation step). The critics are permutation-invariant
functions. We provide details the generators and the critics in the next
subsection.
The two steps (the two conditional GANs) are trained independently. Using
teacher forcing, we use real data for the conditioning during training. For
inference, we generate the edges features by conditioning on the previously
generated node features. The Figure 1 illustrates the model architecture.
The software implementation of our method together with instructions for
replicating our experiments are available at
https://github.com/yoboget/GrannGAN. In this section we provide some details
of the implementation to help the reader understand important design
decisions.
Figure 1: GrannGAN architecture: grayscale boxes represent random noise,
colored boxes represent node and edge features.
### 2.4 Message passing neural network
In both the node- and the edge-annotation phases described in section 2.2 all
GrannGAN generator $g$ and critic $f$ functions are message passing neural
networks (MPNN) Gilmer et al. . As mentioned here-above, the node-
neighbourhood structure of the MPNNs is derived from the conditioning on the
graph skeleton $\\{{\mathcal{V}}{\mathcal{E}}\\}$. Therefore, the skeleton
acts on the $i$ and $j$ indices.
We use the following equations to perform $L$ update steps over the hidden
states ${\mathbf{h}}_{i}^{(l)}$ and ${\mathbf{r}}_{ij}^{(l)}$ of each node and
edge in the graph respectively
$\displaystyle{\mathbf{r}}_{ij}^{(l+1)}$
$\displaystyle=\phi_{r}^{(l)}([{\mathbf{h}}_{i}^{(l)},{\mathbf{h}}_{j}^{(l)},{\mathbf{r}}_{ij}^{(l)}])$
(5) $\displaystyle\widetilde{{\mathbf{h}}}_{i}^{(l+1)}$
$\displaystyle=\sum_{j\in{\mathcal{N}}_{i}}\frac{1}{\sqrt{d_{i}}{\sqrt{d_{j}}}}{\mathbf{r}}_{ij}^{(l+1)}$
(6) $\displaystyle{\mathbf{h}}_{i}^{(l+1)}$
$\displaystyle=\phi_{h}^{(l)}([{\mathbf{h}}_{i}^{(l)},\widetilde{{\mathbf{h}}}_{i}^{(l+1)}])\enspace.$
(7)
Here $\phi$ are learned differentiable functions (we use small feedforward
networks), ${\mathcal{N}}_{i}$ is the first order neighbourhood of node
${\mathbf{v}}_{i}$ as given by the graph skeleton ${\mathcal{S}}$, $d_{i}$ is
its degree, and $[{\bm{a}},{\bm{b}}]$ is the concatenation of vectors
${\bm{a}}$ and ${\bm{b}}$.
We outline the whole generative pipeline in figure 1. In the generator of the
node-annotation phase $g_{\theta_{\mathcal{V}}}$ we initiate the node hidden
states with a latent random noise vectors
${\mathbf{h}}_{i}^{(0)}={\mathbf{z}}_{i},\,\forall{\mathbf{v}}_{i}\in{\mathcal{V}}$
(and drop ${\mathbf{r}}_{ij}^{(0)}$ from inputs to $\phi_{r}^{(0)}$). The
generated node features are the hidden states of the last update step
$\widehat{{\mathbf{v}}}_{i}={\mathbf{h}}_{i}^{(L)}$. In contrast, in the edge-
annotation generator $g_{\theta_{\mathcal{E}}}$ the latent random noise is
used for initiating the edge hidden states
${\mathbf{r}}_{ij}^{(0)}={\mathbf{z}}_{ij},\,\forall{\mathbf{e}}_{ij}\in{\mathcal{E}}$
while the node hidden states are the node features generated from the previous
phase ${\mathbf{h}}_{i}^{(0)}=\widehat{{\mathbf{v}}}_{i}$. The generated edge
features are the hidden edge states of the last update step
$\widehat{\mathbf{e}}_{ij}={\mathbf{r}}_{ij}^{(L)}$.
The critic functions MPNNs are initiated from the real data examples as
${\mathbf{h}}_{i}^{(0)}={\mathbf{v}}_{i}$ and
${\mathbf{r}}_{ij}^{(0)}={\mathbf{e}}_{ij}$ or from the synthetic examples of
the generators respectively. To produce the critic scores, the last-step
update functions ($\phi_{h}^{(L)}$ for the node critic and $\phi_{r}^{(L)}$
for the edge critic) have scalar outputs that are averaged to enter the loss
in equation (2). Therefore, the critic evaluates each node, aggregating
information from all the nodes included in a radius of $2(L-1)$. While this
receptive field may not cover the whole graph, we assume that the node and
edge features can be evaluated locally.
It has been shown Arvind et al. (2020) that MPNN cannot capture graph
substructures other than forests of stars. To alleviate this issue, we further
embody the graph topology into the MPNNs by extending the node representation
by a set of skeleton-related features. Similarly to Bouritsas et al. (2020),
we complement the node hidden states at the initial step
${\mathbf{h}}_{i}^{(0)}$ of all generators and critics by the node degree
$d_{i}$ and the number of $k$-cycles (cycles of length $k$ the node is part
of) extracted from the graph skeleton.
### 2.5 Graph representations
An important property of graph representation ${\mathcal{G}}$ are their non-
uniqueness. There are in general $n!$ possible permutations $\pi$ determining
the ordering of the nodes. The particular choice of the representation
ordering from the complete set $\Pi=\\{\pi_{i}\\}_{i=1}^{n!}$ is therefore
another source of stochasticity so that
$\displaystyle
p({\mathcal{G}})=p\big{(}\bigcup_{\pi\in\Pi}\pi({\mathcal{G}})\big{)}=\sum_{\pi\in\Pi}p\big{(}\pi({\mathcal{G}})\big{)}\enspace.$
(8)
From equation (8) we observe that when relying on the ordered graph
representations ${\mathcal{G}}$, one shall in principle model the complete set
of distributions $p(\pi({\mathcal{G}}))$ for all the $n!$ permutations $\pi$
to capture the unordered-set graph distribution $p({\mathcal{G}})$. However,
operating over the individual permutations $\pi({\mathcal{G}})$ would lead to
inefficient use of the model capacity. As an alternative, previous methods
often fallback to operating over a single representation of each graph using
some heuristic to fix the canonical ordering (such as the minimum depth-first-
search Goyal et al. (2020), the unique breadth-first-search You et al. (2018)
or the further sequentiallized domain driven SMILES Daylight Chemical
Information Systems (2011)). In result, the model has to learn not only the
distribution of the graphs but also the ordering heuristic which is again
inefficient from the perspective of generating the unordered graph sets
${\mathcal{G}}$.
A fundamental property of GrannGAN is that it is completely insensitive to the
ordering $\pi({\mathcal{G}})$ that can in result be chosen arbitrarily. This
permuation equivariance111A function $f$ is equivariant with respect to
permutation $\pi$ if $f(\pi({\bm{x}}))=\pi(f({\bm{x}})).$ of the message
passing operations is pivotal for our method as it solves the problem of the
graph representation non-uniqueness while allowing for efficient use of the
model capacity.
## 3 Related work
GrannGAN is closely related to the broad category of graph generative models,
an area which has attracted significant attention of the research community
resulting in a flurry of papers in the last several years. A systematic review
of the major advancements in the field can be found for example in the
excellent survey Faez et al. (2021).
Unlike previous work on that field, our method focuses on the particular
problem of generating node and edge features conditionally over a given graph
skeleton. To the best of our knowledge, we are the first to investigate such
structural-based conditional graph feature generation.
The closest to our settings are methods modelling the distribution of
annotated graphs. These can be categorized into two large families: the ones
generating graphs sequentially and those generating in the whole graph in one
go.
### 3.1 Sequential graph generation
The first are methods generating the graphs sequentially, starting from small
structures to which they gradually connect new graph components (nodes, edges,
or complete sub-structures). Most of these focus on generating chemical
molecules and adopt various measures to improve the performance on this
particular generative problem. CharacterVAE Gómez-Bombarelli et al. (2018) and
GrammarVAE Kusner et al. (2017) rely directly on sequential SMILES
representation of the molecules. JT-VAE Jin et al. (2018) operates over hand-
crafted vocabulary of chemically valid sub-structures. MolecularRNN Popova et
al. (2019) and GraphAF Shi et al. (2019) generate the graphs by successive
node and edge sampling steps and ensure validity of the generated molecules by
valency checking and the possibility for resampling at each of these steps.
Thanks to these domain-motivated components, the sequential methods often
achieve excellent performance on molecular graphs datasets. However, those
directly relying on chemistry-specific data representations (e.g SMILES)
cannot be easily extended to beyond the molecular problems. On the other hand,
methods not using the SMILES representation need to establish their own
heuristic for the sequential traversal of the graph such as the breadth-first-
search in GraphRNN You et al. (2018) which in turn needs to be learned by the
generative model together with the graph data distribution. As any other
sequential models, the graph autoregressive generators need to be particularly
careful about capturing long-term dependencies, and suffer from slow sampling
process.
Different from the previous models NetGAN Bojchevski et al. (2018) uses GANs
to learn the distribution of random walks over one big graph.
### 3.2 Annotated graph generation in one go
The other large family of models, into which GrannGAN can also be related, are
those generating the graphs in one go, that is all the nodes and edges and
their features together. GraphVAE Simonovsky and Komodakis (2018) proposes to
sidestep the problem of graph linearization characteristic for sequential
models by relying on the variational auto-encoder framework Kingma and Welling
(2014). Due to modelling the graph in the ordered representation of annotation
and adjacency matrix, it needs to employ an expensive (though inexact) graph-
matching procedure in the training loss calculation, which significantly
hampers its scalability. The flow-based graphNVP Madhawa et al. (2019) adopts
the coupling strategy of Dinh et al. (2017) applying it to the rows of the
graph annotation and adjacency matrices and thus preserving the ordering of
the nodes and edges through the flow. GraphNVP is the only model for annotated
graph using permutation equivarient generative function. In general, annotated
graph generation in one go is still an open field of research. Our
contribution can be seen as a proposition to tackle this challenging issue.
Misc-GAN Zhou et al. (2019) aims to translate graphs from a source-domain
graph into a target-domain graph using a multiscale cycle-GAN. It operates on
unannotated graphs.
The closest to our GrannGAN is the MolGAN De Cao and Kipf (2018) model, which
uses GAN for molecule generation. Unlike in GrannGAN, the MolGAN generator is
a feed-forward network sampling the ordered annotation and adjacency matrices
of the graph representation. The authors of MolGAN observe that the model
tends to suffer from mode collapse resulting in insufficient variation in the
generated samples. We suppose that this issue comes from the loss of capacity
by learning to generate various permutations of the same graph and from an
additional Reinforcement Learning module encouraging the generator to produce
graphs with some specific properties. We never experienced mode collapse
during our experiments. As we show it in the experiment section, our model
presents excellent uniqueness and novelty rates. We compare to MolGAN and
other methods discussed here in our experiments in section 4.
## 4 Experiments
In this section, we present experiments using our model as a graph generative
model. There currently exists no other models conditioning on the graph
skeleton and, therefore, no other method to directly compare with. Instead, we
use generative models presented in the previous section as baselines. However,
we underline that these models do not have the same purpose. In the second
part of this section, we also present the results of the conditional
generation by fixing the skeleton.
We evaluate our method in a set of experiments over three datasets containing
(node- and edge-) annotated graphs222Unfortunately, we did not found other
public dataset with enough node- and edge-annotated graphs.. Two of the
datasets, QM9 Ramakrishnan et al. (2014) and ZINC Sterling and Irwin (2015),
are from the chemical domain that is frequently used as a test bed for
annotated graph modeling. The third is the fingerprint dataset Riesen and
Bunke (2008) included in the TUDataset collection Morris et al. (2020).
As it has recently been discussed, for example in O’Bray et al. (2021) and
Thompson et al. (2021), evaluating generative models of graph structures is
particularly challenging due to the impracticality of visual or perceptual
comparisons of the data examples. The comparison with existing graph
generative models is therefore difficult and can provide only crude indication
of the method capabilities.
For the chemical datasets the most common evaluation metrics are the following
* •
_validity_ is the proportion of chemically valid molecules in the total
generated examples and measures the ability of the method to understand the
chemical constraints differentiating general graphs from chemically valid
molecules
* •
_uniqueness_ is the proportion of unique samples within the _valid_ generated
samples and measures the variability of the generated data
* •
_novelty_ is the proportion of examples within the _valid_ and _unique_ set
that do not exist in the training dataset and measures the ability of the
method to go beyond data memorization
We complement these by an overall score of _valid-unique-novel_ molecules
calculated simply as the product of the three above metrics and measuring the
overall quality of the generated examples as a proportion of samples with all
three of these desirable properties in the set of generated data. So, the
overall score gives the rate of valid molecules that are neither in the
dataset nor already generated. Note that in most cases the overall score is
the metric of interest.
While these evaluation metrics may be useful indicators for chemical
downstream tasks, they do not provide any measure of the distance between the
distribution of reference and the generated distributions. Various indicators
calling on the framework of maximum mean discrepancy (MMD) Gretton et al.
(2012) have been proposed in the literature. These are used very
inconsistently and, as demonstrated in O’Bray et al. (2021), they are highly
sensitive to the specific choice of the graph statistics, the kernel and the
hyperparameters used for the MMD calculation. Instead of calculating the MMDs
on graphs statistics, we report directly the distance between the generated
feature distributions, using the Jenson-Shannon Distance for each feature. We
show that with respect to these metrics, our model outperforms by far GraphAF,
model considered as the state-of-the-art.
### 4.1 QM9
QM9 is one of the most commonly used datasets for testing models for annotated
graph generation. It consists of $\sim$134k stable organic molecules with up
to 9 atoms of four types. The 4 atom types and 4 bond types are encoded as
one-hot vectors in the node feature descriptions ${\mathbf{v}}$.
In table 1 we present an overview of the generative performance of our
GrannGAN method in comparison with a set of well-established graph generative
methods: MolGAN De Cao and Kipf (2018), GraphVAE Simonovsky and Komodakis
(2018), GraphNVP Madhawa et al. (2019), GraphAF Shi et al. (2019),
CharacterVAE Gómez-Bombarelli et al. (2018), GrammarVAE Kusner et al. (2017),
and JT-VAE Jin et al. (2018).
The GrannGAN results are calculated from 1000 new data examples generated by
conditioning on skeletons randomly sampled from the training data. The results
of the baseline methods are those reported by the authors in the original
papers333CharacterVAE and GrammarVAE results (not reported in the original
papers) are taken over from the method replications in GraphNVP.. Despite the
frailty of such comparisons due to inconsistencies in the original
experimental protocols (for example, some of the methods, such as GraphAF,
work over the _kekulized_ versions of the molecules reducing the number of
edge categories to three), our GrannGAN achieves excellent results. It
outperforms all one-go generative models and is on par with the best auto-
regressive model.
Table 1: QM9: performance comparison | Model | valid | unique | novel | overall
---|---|---|---|---|---
no chemistry | GrannGAN | 82.5 | 99.9 | 64.9 | 53.4
MolGAN (wo. RL) | 87.7 | 2.9 | 97.7 | 2.5
GraphVAE | 55.7 | 76.0 | 61.6 | 26.1
GraphNVP | 83.1 | 58.2 | 99.2 | 46.8
GraphAF (wo. validity) | 67.0 | 94.5 | 88.8 | 56.3
chemistry | MolGAN-RL | 99.8 | 2.3 | 97.9 | 2.2
CharacterVAE | 10.3 | 90.0 | 67.5 | 11.9
GrammarVAE | 60.2 | 80.9 | 9.3 | 11.9
GraphAF | 100 | 94.5 | 88.8 | 83.9
As explained above, the metrics in the table are very much domain specific.
Methods in the lower part of table 1 expressly focus on these introducing into
the models various expert-designed modules promoting chemical validity of the
generated molecular graphs (e.g., valid sub-substructures vocabulary, valency
constraints, rejection sampling, etc.). While these improve the targeted
metrics in table 1, they also bias the generative process and, therefore, the
model distribution towards the validity metric in exchange for the
distribution approximation goal. We list these methods in table 1 for
completeness despite them not being really comparable to our approach which
focuses on the distribution approximation of general graphs.
Figure 2: QM9: empirical distributions of real and generated features (samples of 1000 graphs) Table 2: Jenson-Shannon distances between real and generated distributions for node and edge features of the QM9 and ZINC datasets (the lower the better) Model | nodes - QM9 | edges - QM9 | nodes - ZINC | edges - ZINC
---|---|---|---|---
GrannGAN | $1.87\cdot 10^{-3}$ | $0.04\cdot 10^{-4}$ | $1.50\cdot 10^{-3}$ | $0.60\cdot 10^{-3}$
GraphAF | $15.80\cdot 10^{-3}$ | $11.5\cdot 10^{-4}$ | $15.5\cdot 10^{-3}$ | $3.39\cdot 10^{-3}$
Figure 2 documents the ability of the GrannGAN method to learn the feature
distribution by comparing the empirical real data distribution with the
distribution of the generated examples. We have replicated the results of the
GraphAF method, which is the most competitive with GrannGAN according to table
1), to extract similar statistics. GrannGAN matches the feature distributions
of the reference dataset almost exactly while we observe a less closer match
in the GraphAF444GraphAF uses somewhat different training set than we. For
example, by kekulizing the molecules it removes all aromatic edges and thus
reduces the number of edge types to 3. This is true also for the ZINC dataset.
generated data. This observation is numerically confirmed in table 2 listing
the Jensen-Shannon distances (JSD) between the real and generated data
samples, where GrannGAN is systematically better by one order of magnitude.
### 4.2 ZINC
We use here the ZINC250k version used previously for generative modeling
evaluations. It consists of 250k randomly selected molecules from the complete
ZINC set with up to 38 atoms of 9 types and 4 bond types.
Table 3 summarizes the generative performance of GrannGAN over 1000 newly
synthesized examples compared to results reported in the literature. MolGAN
and GraphVAE did not experiment (or did not provide the results) on this more
challenging dataset of larger graphs. GrannGAN scales to this dataset easily
and performs competitively with respect to the chemically motivated metrics
(keeping in mind the difficulties of such comparisons related to differences
in experimental protocols and evaluation procedures).
Figure 3 documents the excellent performance of GrannGAN in learning the
feature distributions, matching it closely for both the node and edge
features. The differences in the true and GraphAF generated data distributions
are more pronounced as is also clear from the numerical evaluation via the JSD
presented in table 2.
Table 3: ZINC: performance comparison | Model | valid | unique | novel | overall
---|---|---|---|---|---
no chemistry | GrannGAN | 56.5 | 100 | 100 | 56.5
MolGAN (wo. RL) | — | — | — | —
GraphVAE | 13.5 | — | — | —
GraphNVP | 42.6 | 94.8 | 100 | 40.4
GraphAF (wo. validity) | 68.0 | 99.1 | 100 | 67.3
chemistry | MolGAN-RL | — | — | — | —
CharacterVAE | 7.2 | 9.0 | 100 | 0.6
GrammarVAE | 0.7 | 67.5 | 100 | 0.5
JT-VAE | 100 | 100 | 100 | 100
GraphAF | 100 | 99.1 | 100 | 99.1
Figure 3: ZINC: empirical distributions of real and generated features
(samples of 1000 graphs)
### 4.3 Fingerprints
Fingerprint is a dataset of graphs representing the skeletonized regions of
interest in human fingerprints Riesen and Bunke (2008). The graphs consist of
up to 26 nodes. Both nodes and edges are described by 2 continuous features
related to their position and orientation in the 2d fingerprint image. We have
linearly re-scaled both node and edge features between -1 and 1.
We use the non-annotated graphs as the skeletons for new feature generations.
Each row displays graphs arranged according to the new features generated by
GrannGAN.
Figure 4: Fingerprints: empirical distributions ${\mathbf{v}}_{1}$,
${\mathbf{v}}_{2}$ and ${\mathbf{d}}$ sampled from our GrannGAN and from the
dataset (samples of 1000 graphs)
Figure 5: Fingerprints: empirical distributions ${\mathbf{e}}_{1}$ and
${\mathbf{e}}_{2}$ sampled from our GrannGAN and from the dataset (samples of
1000 graphs)
In table 4 we list the JS distance (with the logarithm in base 2) between the
empirical distributions of the training data and newly generated features
starting from 500 randomly selected skeletons. These have been calculated by
discretizing the features into 200 even sized bins. The columns in the table
correspond to the two node ${\mathbf{v}}$ and edge ${\mathbf{e}}$ features
respectively. Following the same methodology, we also report the JS distance
in the distribution of the euclidean distances between connected nodes
${\mathbf{d}}$. Unlike the feature distribution, this statistic consider the
relation between node features. In figures 4 and 5, we show the node and edge
features distribution for the data and newly generated examples from our model
as well as the distribution of distances between node ${\mathbf{d}}$555We
assume that the oscillation of the curve is just an artifact coming from the
relatively small data points by bin.
Table 4: Jenson-Shannon distances between real and generated distributions for node and edge features of the Fingerprint dataset (the lower the better) | v1 | v2 | d | e1 | e2
---|---|---|---|---|---
GrannGAN | $5.02\cdot 10^{-2}$ | $4.55\cdot 10^{-2}$ | $32.84\cdot 10^{-2}$ | $16.78\cdot 10^{-2}$ | $9.74\cdot 10^{-2}$
Data samples | $1.13\cdot 10^{-2}$ | $1.27\cdot 10^{-2}$ | $1.49\cdot 10^{-2}$ | $1.37\cdot 10^{-2}$ | $1.07\cdot 10^{-2}$
Again, our model shows excellent performance in closely matching the node and
edge features and their relations.
These experiments show the quality of our GrannGAN for the generative task. It
outperforms all previous one-go generative models. The sequential GraphAF is
the only model presenting slightly better results using the molecular metrics.
However, we show that our model is much better at capturing the feature
distributions.
### 4.4 Conditional generation
In this section, we show that our model can be used for conditional generation
by fixing the skeleton and still producing a high level of diversity.
We want to guarantee that the sample diversity does not come mainly from the
skeleton. To test the generative capacity of our GrannGAN, we randomly sampled
100 skeletons from the ZINC dataset. We generate 1000 new instances
conditioned on each sampled skeleton. We compute the average rate of
uniqueness, i.e. the rate of unique instances over all the valid molecules.
Table 5 reports the results of the experiment. As expected, the validity rate
is similar to the one by sampling a new skeleton for each instance. Even by
fixing the skeleton, we reached a rate of 92.5% unique instances on average.
On average, the conditional generation of 1000 instances over the same
skeleton produces 527.0 unique molecules. So, we show that the performance of
our model does not depend critically on the sampling of multiple skeletons and
can be properly used as a conditional generative model.
Table 5: Conditional generation of 1000 instances from the same skeleton (average) Model | Validity (%) | Uniqueness (%) | Valid and unique (‰)
---|---|---|---
GrannGAN | $57.0$ | $92.5$ | $527.0$
We have demonstrated that the GrannGAN is competitive with state-of-the-art
model for graph generation even though it is first meant for conditional
generation. GraphAF is slightly better on the molecular metric, but much worst
at matching the feature distribution. In addition, our model generates
instances in one go and is therefore much faster during inference. More
fundamentally, our model preserves its quality while generating new instances
conditioned on the skeleton, a task that no other model can do.
## 5 Conclusions
We have presented here a new method, GrannGAN, for implicit distribution
learning and generating features of nodes and edges conditioned on graph
skeletons. The promising results of the method, as documented in our
experiments, indicate that the method can learn the complex high-dimensional
distributions and generate new data examples with features coherent with the
underlying graph structure. When the generation of new skeletons is not
crucial, GrannGAN can be used as a full generative model by sampling the
skeleton from the data. It is also the first method for graph generation
conditioned on the skeleton.
These favorable results provide a solid starting point for multiple possible
extensions in future work. For example, in some applications (e.g. bio-
chemistry) a reasonable starting point for the feature generations may be a
partly annotated graph (in place of a completely non-annotated graph). Our
model, which is already conditional, could be easily adapted to this setting
by re-using the ideas of our second edge-annotation phase. Another interesting
extension is that of property optimization. An additional conditioning
variable could be included in the generators and critics to guide the model in
generating property-conditioned features.
The contribution of Magda Gregorova has been supported by High-Tech Agenda
Bayern.
## References
* Arjovsky et al. (2017) Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein Generative Adversarial Networks. In _Proceedings of the 34th International Conference on Machine Learning_ , pages 214–223. PMLR, July 2017. URL https://proceedings.mlr.press/v70/arjovsky17a.html. ISSN: 2640-3498.
* Arvind et al. (2020) V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On Weisfeiler-Leman invariance: Subgraph counts and related graph properties. _Journal of Computer and System Sciences_ , 113:42–59, nov 2020. ISSN 10902724.
* Bojchevski et al. (2018) Aleksandar Bojchevski, Oleksandr Shchur, Daniel Zügner, and Stephan Günnemann. NetGAN: Generating graphs via random walks. In Jennifer Dy and Andreas Krause, editors, _Proceedings of the 35th International Conference on Machine Learning_ , volume 80 of _Proceedings of Machine Learning Research_ , pages 610–619. PMLR, 10–15 Jul 2018. URL https://proceedings.mlr.press/v80/bojchevski18a.html.
* Bouritsas et al. (2020) Giorgos Bouritsas, Fabrizio Frasca, Stefanos Zafeiriou, and Michael M. Bronstein. Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , jun 2020. URL http://arxiv.org/abs/2006.09252.
* Daylight Chemical Information Systems (2011) Inc. Daylight Chemical Information Systems. Daylight Theory Manual, August 2011. URL https://www.daylight.com/dayhtml/doc/theory/.
* De Cao and Kipf (2018) Nicola De Cao and Thomas Kipf. MolGAN: An implicit generative model for small molecular graphs. _arXiv:1805.11973 [cs, stat]_ , May 2018. URL http://arxiv.org/abs/1805.11973. arXiv: 1805.11973.
* Dinh et al. (2017) Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real NVP. In _5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings_. OpenReview.net, 2017. URL https://openreview.net/forum?id=HkpbnH9lx.
* Faez et al. (2021) Faezeh Faez, Yassaman Ommi, Mahdieh Soleymani Baghshah, and Hamid R. Rabiee. Deep Graph Generators: A Survey. _IEEE Access_ , 9:106675–106702, 2021. ISSN 2169-3536. Conference Name: IEEE Access.
* (9) Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. In _Proceedings of the 34th International Conference on Machine Learning_ , pages 1263–1272. PMLR. URL https://proceedings.mlr.press/v70/gilmer17a.html. ISSN: 2640-3498.
* Goodfellow et al. (2014) Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. In _Advances in Neural Information Processing Systems_ , volume 27. Curran Associates, Inc., 2014. URL https://papers.nips.cc/paper/2014/hash/5ca3e9b122f61f8f06494c97b1afccf3-Abstract.html.
* Goyal et al. (2020) Nikhil Goyal, Harsh Vardhan Jain, and Sayan Ranu. GraphGen: A Scalable Approach to Domain-agnostic Labeled Graph Generation. 2020\. URL https://doi.org/10.1145/3366423.3380201.
* Gretton et al. (2012) Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander Smola. A Kernel Two-Sample Test. _Journal of Machine Learning Research_ , 13(25):723–773, 2012. ISSN 1533-7928. URL http://jmlr.org/papers/v13/gretton12a.html.
* Gómez-Bombarelli et al. (2018) Rafael Gómez-Bombarelli, Jennifer N. Wei, David Duvenaud, José Miguel Hernández-Lobato, Benjamín Sánchez-Lengeling, Dennis Sheberla, Jorge Aguilera-Iparraguirre, Timothy D. Hirzel, Ryan P. Adams, and Alán Aspuru-Guzik. Automatic Chemical Design Using a Data-Driven Continuous Representation of Molecules. _ACS Central Science_ , 4(2):268–276, February 2018. ISSN 2374-7943. URL https://doi.org/10.1021/acscentsci.7b00572. Publisher: American Chemical Society.
* Jin et al. (2018) Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction Tree Variational Autoencoder for Molecular Graph Generation. In _Proceedings of the 35th International Conference on Machine Learning_ , pages 2323–2332. PMLR, July 2018. URL https://proceedings.mlr.press/v80/jin18a.html. ISSN: 2640-3498.
* Kingma and Welling (2014) Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In Yoshua Bengio and Yann LeCun, editors, _2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings_ , 2014. URL http://arxiv.org/abs/1312.6114.
* Kusner et al. (2017) Matt J. Kusner, Brooks Paige, and José Miguel Hernández-Lobato. Grammar Variational Autoencoder. In _Proceedings of the 34th International Conference on Machine Learning_ , pages 1945–1954. PMLR, July 2017. URL https://proceedings.mlr.press/v70/kusner17a.html. ISSN: 2640-3498.
* Liao et al. (2019) Renjie Liao, Yujia Li, Yang Song, Shenlong Wang, Will Hamilton, David K Duvenaud, Raquel Urtasun, and Richard Zemel. Efficient Graph Generation with Graph Recurrent Attention Networks. In _Advances in Neural Information Processing Systems_ , volume 32. Curran Associates, Inc., 2019. URL https://proceedings.neurips.cc/paper/2019/hash/d0921d442ee91b896ad95059d13df618-Abstract.html.
* Madhawa et al. (2019) Kaushalya Madhawa, Katsuhiko Ishiguro, Kosuke Nakago, and Motoki Abe. GraphNVP: an Invertible Flow-based Model for Generating Molecular Graphs. September 2019. URL https://openreview.net/forum?id=ryxQ6T4YwB.
* Miyato et al. (2018) Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral Normalization for Generative Adversarial Networks. February 2018. URL https://openreview.net/forum?id=B1QRgziT-&foo=bar.
* Morris et al. (2020) Christopher Morris, Nils M. Kriege, Franka Bause, Kristian Kersting, Petra Mutzel, and Marion Neumann. Tudataset: A collection of benchmark datasets for learning with graphs. In _ICML 2020 Workshop on Graph Representation Learning and Beyond (GRL+ 2020)_ , 2020. URL www.graphlearning.io.
* O’Bray et al. (2021) Leslie O’Bray, Max Horn, Bastian Rieck, and Karsten Borgwardt. Evaluation Metrics for Graph Generative Models: Problems, Pitfalls, and Practical Solutions. September 2021. URL https://openreview.net/forum?id=tBtoZYKd9n.
* Popova et al. (2019) Mariya Popova, Mykhailo Shvets, Junier Oliva, and Olexandr Isayev. MolecularRNN: Generating realistic molecular graphs with optimized properties. _arXiv:1905.13372 [cs, q-bio, stat]_ , May 2019. URL http://arxiv.org/abs/1905.13372. arXiv: 1905.13372.
* Ramakrishnan et al. (2014) Raghunathan Ramakrishnan, Pavlo O. Dral, Matthias Rupp, and O. Anatole von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. _Scientific Data_ , 1(1):140022, August 2014. ISSN 2052-4463. URL https://www.nature.com/articles/sdata201422. Number: 1 Publisher: Nature Publishing Group.
* Riesen and Bunke (2008) Kaspar Riesen and Horst Bunke. IAM graph database repository for graph based pattern recognition and machine learning. In _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , volume 5342 LNCS, pages 287–297. Springer, Berlin, Heidelberg, 2008. ISBN 3540896880. URL http://www.iam.unibe.ch/fki/databases/iam-graph-database.
* Shi et al. (2019) Chence Shi, Minkai Xu, Zhaocheng Zhu, Weinan Zhang, Ming Zhang, and Jian Tang. GraphAF: a Flow-based Autoregressive Model for Molecular Graph Generation. September 2019. URL https://openreview.net/forum?id=S1esMkHYPr.
* Simonovsky and Komodakis (2018) Martin Simonovsky and Nikos Komodakis. GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders. _arXiv:1802.03480 [cs]_ , February 2018. URL http://arxiv.org/abs/1802.03480. arXiv: 1802.03480.
* Sterling and Irwin (2015) Teague Sterling and John J. Irwin. ZINC 15 – Ligand Discovery for Everyone. _Journal of Chemical Information and Modeling_ , 55(11):2324–2337, November 2015. ISSN 1549-9596. URL https://doi.org/10.1021/acs.jcim.5b00559. Publisher: American Chemical Society.
* Thompson et al. (2021) Rylee Thompson, Boris Knyazev, Elahe Ghalebi, Jungtaek Kim, and Graham W. Taylor. On Evaluation Metrics for Graph Generative Models. September 2021. URL https://openreview.net/forum?id=EnwCZixjSh.
* You et al. (2018) Jiaxuan You, Rex Ying, Xiang Ren, William Hamilton, and Jure Leskovec. GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models. In _Proceedings of the 35th International Conference on Machine Learning_ , pages 5708–5717. PMLR, July 2018. URL https://proceedings.mlr.press/v80/you18a.html. ISSN: 2640-3498.
* Zhou et al. (2019) Dawei Zhou, Lecheng Zheng, Jiejun Xu, and Jingrui He. Misc-gan: A multi-scale generative model for graphs. _Frontiers in Big Data_ , 2:3, 4 2019. ISSN 2624909X.
|
# The Canonical BV Laplacian on Half-Densities
Alberto S. Cattaneo Institut für Mathematik, Universität Zürich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland<EMAIL_ADDRESS>
###### Abstract.
This is a didactical review on the canonical BV Laplacian on half-densities.
I acknowledge partial support of the SNF Grant No. 200020_192080, of the
Simons Collaboration on Global Categorical Symmetries, and of the COST Action
21109 (CaLISTA). This research was (partly) supported by the NCCR SwissMAP,
funded by the Swiss National Science Foundation.
> _Das Bekannte überhaupt ist darum, weil es bekannt ist, nicht erkannt_
>
> GWFH
###### Contents
1. 1 Introduction
1. 1.1 Overview
2. 2 The standard BV Laplacian
3. 3 The BV bracket
4. 4 The BV Laplacian on functions
1. 4.1 Digression: further properties of the BV Laplacian
5. 5 The BV Laplacian on densities
1. 5.1 Digression: Darboux expression of $\Delta_{\mu}^{(s)}$
6. 6 Odd cotangent bundles
7. 7 Conclusion of the argument
8. A Some background details
1. A.1 The Lie derivative
2. A.2 Densities
9. B Applications of the BV formalism
10. C Historical remarks
## 1\. Introduction
The BV Laplacian on functions is a central ingredient of the
Batalin–Vilkovisky (BV) formalism. Its global definition is not canonical, yet
a remarkable fact observed by Khudaverdian [7, 8] is that its extension to
half-densities is so, and there are various ways to show it. In this review we
give a short, self-contained presentation which requires just a few simple
computations. It gives a special emphasis on the fact that half-densities have
a natural inner product.
This note has no pretence of originality. In Appendix C, we give a short
historical overview with the relevant references.
In Appendix A, we collect some technical background material. In Appendix B,
we recall why the BV formalism is important in applications.
### 1.1. Overview
We recall definitions and properties of BV Laplacians (on functions and
densities) on an oriented odd symplectic manifold with the goal of proving
that on half-densities it is canonically defined (whereas on other kinds of
densities, including functions, it requires the choice of a compatible
density). This goes as follows:
1. (1)
On functions we define $\Delta_{\mu}f=\frac{1}{2}{\mathrm{div}}_{\mu}X_{f}$,
where $\mu$ is an even, nowhere vanishing111Throughout the paper, nowhere
vanishing means nonvanishing at every point of the body after the nilpotents
are set to zero. An even, nowhere vanishing density $\mu$ is the same as a
basis for the module of densities. The $k$-density $\mu^{k}$ is then a basis
for the module of $k$-densities. density, and $X_{f}$ denotes the hamiltonian
vector field of $f$.
2. (2)
On half-densities we define
$\Delta^{(\frac{1}{2})}_{\mu}\sigma=\Delta_{\mu}(\sigma/\mu^{1/2})\mu^{1/2}$.
3. (3)
We check that the leading term is invariant under symplectomorphisms and does
not depend on the choice of $\mu$.
4. (4)
We check that $\Delta^{(\frac{1}{2})}_{\mu}$ is a symmetric operator. This
implies that in any Darboux chart only the zeroth-order term may depend on
$\mu$.
5. (5)
We show that for a compatible density $\mu$, i.e., for $\mu$ satisfying
$\Delta_{\mu}^{2}=0$, the zeroth-order term of the previous point must vanish.
This shows that $\Delta^{(\frac{1}{2})}_{\mu}$ does not depend on the choice
of the compatible density $\mu$.
6. (6)
We show, via the presentation of the given odd symplectic manifold as a
symplectomorphic odd cotangent bundle, that compatible densities always exist.
7. (7)
We conclude that the BV Laplacian on half-densities, with the property of
squaring to zero, is canonically defined.
8. (8)
As an aside, we also show how to define this canonical operator, denoted by
$\Delta$, directly on Darboux charts without any choice involved. Moreover, we
prove that the compatibility condition for a density $\mu$ can also be
expressed as $\Delta\mu^{\frac{1}{2}}=0$ and that the BV Laplacian on
functions can now be recovered as
$\Delta_{\mu}=\Delta(f\mu^{\frac{1}{2}})/\mu^{\frac{1}{2}}$.
We go through details, also checking signs carefully, to make this
presentation useful also as a reference. However, it should be observed that,
roughly, the above flow can be easily checked without going into sign
details—the only crucial point being (4), which requires some care.
###### Acknowledgment.
I thank A. Cabrera, D. Fiorenza, N. Moshayedi, M. Schiavina, and P. Ševera for
useful remarks. I am especially grateful to T. Voronov for several extremely
helpful exchanges.
## 2\. The standard BV Laplacian
On functions of odd variables $p_{1},\dots,p_{n}$ and even variables
$q^{1},\dots,q^{n}$, the standard BV Laplacian222This operator was introduced
to show that certain integrals are invariant under deformations of the
integration domain; see Appendix B. (a.k.a. BV Laplace operator or BV
operator) is defined as
$\triangle\coloneqq\partial_{i}\partial^{i},$
where we use Einstein’s summation convention and the shorthand notations
$\partial_{i}=\frac{\partial}{\partial
q^{i}}\quad\text{and}\quad\partial^{i}=\frac{\partial}{\partial p_{i}}.$
Another ingredient of the BV formalism is the BV bracket
(1) $(f,g)\coloneqq
f\overleftarrow{\partial{{}^{i}}}\,\overrightarrow{\partial{{}_{i}^{\phantom{i}}}}g-f\overleftarrow{\partial{{}_{i}^{\phantom{i}}}}\,\overrightarrow{\partial{{}^{i}}}g,$
where $f$ and $g$ are functions of homogeneous degree, and the arrows denote
the directions the derivatives are applied from. Explicitly,
$\overrightarrow{\partial{{}_{i}^{\phantom{i}}}}f=\partial_{i}f$,
$\overrightarrow{\partial{{}^{i}}}f=-\partial^{i}f$, and
$\begin{split}f\overleftarrow{\partial{{}_{i}^{\phantom{i}}}}&=\partial_{i}f,\\\
f\overleftarrow{\partial{{}^{i}}}&=-(-1)^{f}\partial^{i}f.\end{split}$
Here and in the following, when we put a function (or any other object) as an
exponent of $(-1)$ we mean its degree.333A more precise but more cumbersome
notation would be $(-1)^{|f|}$, where $|f|$ denotes the degree of $f$.
The standard BV Laplacian has the following properties which are easily proved
by direct computation:
(2a) $\displaystyle\triangle^{2}=0,$ (2b)
$\displaystyle\triangle(fg)=(\triangle f)g+(-1)^{f}f\triangle
g-(-1)^{f}(f,g),$ (2c) $\displaystyle\triangle\mathrm{e}^{S}=\left(\triangle
S-\frac{1}{2}(S,S)\right)\mathrm{e}^{S}.$
Here again $f$ and $g$ are functions of homogeneous degree, whereas the
function $S$ is assumed to be even. As an exercise, just using (2a) and (2b),
you can also show that
(3) $\triangle(f,g)=(\triangle f,g)-(-1)^{f}(f,\triangle g).$
The geometric interpretation of the content of this section is that the
$p_{i}$s and $q^{i}$s are Darboux coordinates for the odd symplectic form
$\omega={\mathrm{d}}p_{i}{\mathrm{d}}q^{i}$ with $(\ ,\ )$ its associated odd
Poisson bracket.444If the $q^{i}$s are coordinates on an open subset $U$ of
${\mathbb{R}}^{n}$, the functions of the $p$ and $q$ variables can be
identified with multivector fields on $U$. The BV bracket then gets identified
with the Schouten–Nijenhuis bracket. We will return to this, with a general
basis manifold, in Section 6; see footnote 15 on page 15. The main problem
with this, however, is that the standard BV Laplacian does not transform well
under symplectomorphisms, so it cannot be used as such to define an operator
on functions on an odd symplectic manifold. In the next sections, we will see
how to obviate this problem.
## 3\. The BV bracket
Let $(\mathcal{M},\omega)$ be an odd symplectic manifold. Given a function
$f$, we denote by $X_{f}$ its hamiltonian vector field defined via
$\iota_{X_{f}}\omega={\mathrm{d}}f.$
Given a second function $g$, we define the BV bracket as
$(f,g)\coloneqq(-1)^{f+1}X_{f}(g).$
This is an odd Poisson bracket, which in local Darboux coordinates agrees with
the one of the previous section:555This is a simple computation which however
requires checking signs. It is just needed to make sure that the conventions
we choose are compatible.
###### Lemma 3.1.
In Darboux coordinates, $\omega={\mathrm{d}}p_{i}{\mathrm{d}}q^{i}$, the BV
bracket agrees with the standard one defined in (1).
###### Proof.
Writing $X=X^{i}\partial_{i}+X_{i}\partial^{i}$, we get666We use the
convention of total degree, i.e., internal degree plus parity of the form
degree. This means that ${\mathrm{d}}$ is odd and that $\iota_{X}$ has parity
opposite to that of $X$. Moreover, $X^{i}$ has the same parity as $X$, whereas
$X_{i}$ has opposite parity. Finally, $f$ and $X_{f}$ have opposite parity to
each other.
$\iota_{X}\omega={\mathrm{d}}p_{i}\,X^{i}+X_{i}\,{\mathrm{d}}q^{i}={\mathrm{d}}p_{i}\,X^{i}-(-1)^{X}{\mathrm{d}}q^{i}\,X_{i}.$
From
${\mathrm{d}}f={\mathrm{d}}p_{i}\,\partial^{i}f+{\mathrm{d}}q^{i}\,\partial_{i}f$
we then get
(4a) $\displaystyle(X_{f})^{i}$
$\displaystyle=\partial^{i}f=(-1)^{f+1}f\overleftarrow{\partial{{}^{i}}}$ and
(4b) $\displaystyle(X_{f})_{i}$
$\displaystyle=(-1)^{f}\partial_{i}f=(-1)^{f}f\overleftarrow{\partial{{}_{i}^{\phantom{i}}}},$
so $(-1)^{f+1}X_{f}(g)$ is given by (1). ∎
The BV bracket satisfies several properties.
###### Proposition 3.2.
For all functions $f$, $g$, and $h$, we have
$\displaystyle[X_{f},X_{g}]$ $\displaystyle=X_{(f,g)},$ $\displaystyle(f,gh)$
$\displaystyle=(f,g)\,h+(-1)^{(f+1)g}g\,(f,h),$ $\displaystyle(g,f)$
$\displaystyle=-(-1)^{(f+1)(g+1)}(f,g),$ $\displaystyle(f,(g,h))$
$\displaystyle=((f,g),h)+(-1)^{(f+1)(g+1)}(g,(f,h)).$
The last three properties say that $(\ ,\ )$ is an odd Poisson bracket,777In
particular this means that $(\ ,\ )$ is a super Lie bracket with respect to
the opposite parity of its arguments. whereas the first says the the map
$f\mapsto X_{f}$ is a morphism of super Lie algebras.
###### Proof.
We prove only the first identity. The others are also easily obtained from the
definitions (or recovered from the identical identities for the standard BV
bracket which one obtains in each Darboux chart). We have
$\displaystyle\iota_{[X_{f},X_{g}]}\omega=[\mathsf{L}_{X_{f}},\iota_{X_{g}}]\omega=\mathsf{L}_{X_{f}}\iota_{X_{g}}\omega=\mathsf{L}_{X_{f}}{\mathrm{d}}g$
$\displaystyle=(-1)^{f+1}{\mathrm{d}}\mathsf{L}_{X_{f}}g=(-1)^{f+1}{\mathrm{d}}X_{f}(g)={\mathrm{d}}(f,g)=\iota_{X_{(f,g)}}\omega,$
where $\mathsf{L}_{X_{f}}$ denotes the Lie derivative with respect to $X_{f}$.
∎
## 4\. The BV Laplacian on functions
Let $(\mathcal{M},\omega)$ be an odd symplectic manifold and $\mu$ an even,
nowhere vanishing density on $\mathcal{M}$, which we assume to be orientable.
We then define the $\mu$-dependent BV Laplacian as
(5) $\Delta_{\mu}f\coloneqq\frac{1}{2}{\mathrm{div}}_{\mu}X_{f}.$
Recall (or see Appendix A.2) that the divergence operator of a vector field
$X$ with respect to an even, nowhere vanishing density $\mu$ is defined via
${\mathrm{div}}_{\mu}X\,\mu=\mathsf{L}_{X}\mu,$
where $\mathsf{L}_{X}$ denotes the Lie derivative.
Again we have to make sure that this definition of the BV Laplacian agrees
with the standard one in the appropriate case:
###### Proposition 4.1.
In Darboux coordinates, $\omega={\mathrm{d}}p_{i}{\mathrm{d}}q^{i}$, with
standard density
$\mu_{\text{stand}}\coloneqq{\mathrm{d}}^{n}q\,{\mathrm{d}}^{n}p$, we have
$\Delta_{\mu_{\text{stand}}}=\triangle.$
###### Proof.
With the standard density the divergence of a vector field
$X=X^{i}\partial_{i}+X_{i}\partial^{i}$ is given by (14a), i.e.,
${\mathrm{div}}_{\mu_{\text{stand}}}X=\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i}.$
If we now insert $X_{f}$ as calculated in (4), we get
${\mathrm{div}}_{\mu_{\text{stand}}}X_{f}=2\partial_{i}\partial^{i}f.$
∎
Next we check how the operator depends on the choice of density. Note that,
given two even, nowhere vanishing densities $\mu$ and $\widetilde{\mu}$, there
is a unique even, nowhere vanishing function $h$ such that
$\widetilde{\mu}=h\mu$.
###### Proposition 4.2.
We have
$\Delta_{h\mu}=\Delta_{\mu}+\frac{1}{2}\frac{1}{h}X_{h}.$
###### Proof.
The divergence operator depends on the choice of density as
${\mathrm{div}}_{h\mu}X_{f}={\mathrm{div}}_{\mu}X_{f}+\frac{1}{h}X_{f}(h)$
(see point (2) in Proposition A.4). However,
$X_{f}(h)=(-1)^{f+1}(f,h)=-(h,f)=X_{h}(f)$, since $h$ is even. ∎
We can always write $h=\pm\mathrm{e}^{S}$ with $S$ an even function. This way
the formula simplifies to
$\Delta_{h\mu}=\Delta_{\mu}+\frac{1}{2}X_{S},$
since $X_{\mathrm{e}^{S}}=\mathrm{e}^{S}X_{S}$. In particular, we have the
###### Lemma 4.3.
In Darboux coordinates with $\mu=\pm\mathrm{e}^{S}\mu_{\text{stand}}$, we have
(6) $\Delta_{\mu}=\triangle+\frac{1}{2}X_{S}=\triangle-\frac{1}{2}(S,\ ).$
This, together with Proposition 4.2, gives the
###### Proposition 4.4.
The BV Laplacian on functions is a second-order differential operator, and its
leading term is independent of the choice of density.
###### Remark 4.5.
Under the assumptions of Lemma 4.3, from (6), also using (3) and the Jacobi
identity, we get
$\Delta_{\mu}^{2}f=-\frac{1}{2}(F_{S},f)$
with
(7) $F_{S}\coloneqq\triangle S-\frac{1}{4}(S,S).$
We then see that $\Delta_{\mu}^{2}=0$ if and only if $F_{S}$ is constant. As
$F_{S}$ is odd, this happens if and only if $F_{S}=0$.888Note that this
condition can also be rephrased as $\triangle\mathrm{e}^{\frac{1}{2}S}=0$. The
appearance of the factor $\frac{1}{2}$ in the exponent will become clear when
we study half-densities (see Lemma 5.13). Therefore, in general the property
of equation (2a) does not extend to the global BV Laplacian.
###### Definition 4.6.
We say that an even, nowhere vanishing density $\mu$ is compatible with the
odd symplectic structure if $\Delta_{\mu}^{2}=0$.
###### Remark 4.7.
We will see that every odd symplectic manifold admits a compatible density.
###### Remark 4.8.
As we have seen in Remark 4.5, the property of equation (2a) does not extend
in general. One can try to obviate it by modifying the definition of the BV
operator. For example, in Darboux coordinates with
$\mu=\pm\mathrm{e}^{S}\mu_{\text{stand}}$, we could define
$\widetilde{\Delta}_{\mu}\coloneqq\Delta_{\mu}+F_{S}=\triangle+\frac{1}{2}X_{S}+\frac{1}{2}F_{S}$
with $F_{S}$ as in (7). It is an easy exercise (show first that
$\Delta_{\mu}F_{S}=0$ for every $S$) to show that
$\widetilde{\Delta}_{\mu}^{2}=0$ for every $\mu$. However, since
$\widetilde{\Delta}_{\mu}$ is not of the form in Lemma 4.3 (unless, of course,
$F_{S}=0$), it is not a BV Laplacian.
If $\Psi\colon\mathcal{M}\to\mathcal{N}$ is a diffeomorphism, we have the
pushforward
$\Psi_{*}\colon\operatorname{End}(C^{\infty}(\mathcal{M}))\to\operatorname{End}(C^{\infty}(\mathcal{N}))$
defined by $\Psi_{*}P=\Psi_{*}\circ P\circ\Psi_{*}^{-1}$, where on the right-
hand side $\Psi_{*}$ denotes the pushforward of functions. The BV Laplacian
transforms nicely under symplectomorphisms.
###### Proposition 4.9.
For every symplectomorphism $\Psi$,
$\Psi_{*}\Delta_{\mu}=\Delta_{\Psi_{*}\mu}.$
###### Proof.
The divergence operator changes under a diffeomorphism as
$\Psi_{*}{\mathrm{div}}_{\mu}X_{f}={\mathrm{div}}_{\Psi_{*}\mu}\Psi_{*}X_{f}$
(see point (1) in Proposition A.4). If $\Psi$ is a symplectomorphism, we also
have $\Psi_{*}X_{f}=X_{\Psi_{*}f}$. ∎
###### Remark 4.10.
In conjunction with Proposition 4.4, this implies that under a
symplectomorphism $\Psi\colon\mathcal{M}\to\mathcal{M}$, the leading term of
the BV Laplacian on functions is invariant under symplectomorphisms.
###### Remark 4.11.
The proposition also implies that a symplectomorphism sends compatible
densities to compatible densities.
### 4.1. Digression: further properties of the BV Laplacian
Even though we are not going to use this in the following, it is good to know
that the properties stated in equations (2b) and (2c) also hold for the global
BV Laplacian.
###### Proposition 4.12.
For every even, nowhere vanishing density $\mu$, every functions $f$ and $g$,
and every even function $S$, we have
(8a)
$\displaystyle\Delta_{\mu}(fg)=(\Delta_{\mu}f)g+(-1)^{f}f\Delta_{\mu}g-(-1)^{f}(f,g),$
(8b)
$\displaystyle\Delta_{\mu}\mathrm{e}^{S}=\left(\Delta_{\mu}S-\frac{1}{2}(S,S)\right)\mathrm{e}^{S}.$
###### Proof.
The identities may be proved by observing, thanks to (6), that in every
Darboux chart the BV Laplacian differs from $\triangle$ by a vector field.
They may also be proved directly using properties of the divergence operator
(see point (3) in Proposition A.4) and of hamiltonian vector fields. To
illustrate this, we prove the second identity. Since
$X_{\mathrm{e}^{S}}=\mathrm{e}^{S}X_{S}$, we have
${\mathrm{div}}_{\mu}X_{\mathrm{e}^{S}}={\mathrm{div}}_{\mu}(\mathrm{e}^{S}X_{S})=\mathrm{e}^{S}{\mathrm{div}}_{\mu}X_{S}+X_{S}(\mathrm{e}^{S}),$
which proves the identity once we observe that
$X_{S}(\mathrm{e}^{S})=-\mathrm{e}^{S}(S,S)$. ∎
###### Remark 4.13.
As we have seen in Remark 4.8, it is possible to define an operator
$\widetilde{\Delta}_{\mu}$ that squares to zero, no matter what $\mu$ is, so
as to satisfy the extension of the property of equation (2a). However, since
this is obtained by adding to $\Delta_{\mu}$ a multiplication operator, we see
that now (8a) is no longer satisfied (unless, of course, $F_{S}=0$).
## 5\. The BV Laplacian on densities
A differential operator $P$ defined on functions on a (super)manifold
$\mathcal{M}$ can be extended to sections of a trivial line bundle $L$ over
$\mathcal{M}$ once an even, nowhere vanishing section $\lambda$ of $L$ has
been chosen. For $\sigma\in\Gamma(L)$ there is a uniquely determined function
$f$ such that $\sigma=f\lambda$, and we set
$P^{(\lambda)}\sigma\coloneqq P(f)\lambda.$
###### Lemma 5.1.
The leading term of the differential operator $P^{(\lambda)}$ does not depend
on the choice of $\lambda$.
###### Proof.
If $\widetilde{\lambda}$ is also a nowhere vanishing section, then there is a
uniquely determined nowhere vanishing function $h$ with
$\widetilde{\lambda}=h\lambda$. For $\sigma=f\widetilde{\lambda}=fh\lambda$,
we have
$P^{(\widetilde{\lambda})}\sigma=P(f)\widetilde{\lambda}=P(f)h\lambda.$
On the other hand, we have
$P^{(\lambda)}\sigma=P(fh)\lambda=((P_{\text{leading}}f)\,h+\cdots)\lambda,$
where the dots denote terms where less than the maximum number of derivatives
hit $f$. ∎
We now want to extend the BV Laplacian $\Delta_{\mu}$ to $s$-densities (see
Appendix A.2 for a review). Since we already have an even, nowhere vanishing
section, $\mu$, of the density bundle, we can use it to get our reference
section, $\mu^{s}$, of the $s$-density bundle.999Since we are interested also
in nonintegral $s$, in particular $s=\frac{1}{2}$, it is essential to require
that $\mathcal{M}$ is oriented. That is,
###### Definition 5.2.
For an $s$-density $\sigma$, which we can uniquely write as $\sigma=f\mu^{s}$,
we set101010 We avoid the more pedantic notation $\Delta_{\mu}^{(\mu^{s})}$ on
the left-hand side.
$\Delta_{\mu}^{(s)}\sigma\coloneqq\Delta_{\mu}(f)\mu^{s}.$
Note that $\Delta_{\mu}^{(0)}$ is the same as $\Delta_{\mu}$.
###### Remark 5.3.
An immediate consequence of this definition is that $\Delta_{\mu}^{(s)}$
squares to zero on $s$-densities if and only if $\Delta_{\mu}$ does so on
functions (i.e., $\mu$ is compatible with the odd symplectic structure).
Proposition 4.4 and Lemma 5.1 immediately imply
###### Proposition 5.4.
The BV Laplacian on $s$-densities is a second-order differential operator, and
its leading term is independent of the choice of reference density.
Proposition 4.9 immediately implies
###### Proposition 5.5.
For every symplectomorphism $\Psi$ and for every $s$,
$\Psi_{*}\Delta^{(s)}_{\mu}=\Delta^{(s)}_{\Psi_{*}\mu}.$
###### Remark 5.6.
In particular, this implies that, for every $s$, under a symplectomorphism
$\Psi\colon\mathcal{M}\to\mathcal{M}$, the leading term of the BV Laplacian on
$s$-densities is invariant under symplectomorphisms.
Note that the product of an $s$-density $\sigma$ and a $(1-s)$-density $\tau$
yields a density, which can be integrated (if it has compact support). Given a
differential operator $P$ on $s$-densities, we define its transpose $P^{t}$ as
the differential operator on $(1-s)$-densities that satisfies
$\int_{\mathcal{M}}P\sigma\,\tau=(-1)^{P\,\sigma}\int_{\mathcal{M}}\sigma\,P^{t}\tau$
for all $s$-densities $\sigma$ with compact support (the upper index
“$P\,\sigma$” denotes the product of the degrees; see footnote 3 on page 3).
The case $s=\frac{1}{2}$ is special, since transposition becomes an
endomorphism on the space of differential operators.
###### Proposition 5.7.
We have
$(\Delta_{\mu}^{(s)})^{t}=\Delta_{\mu}^{(1-s)}$
In particular, $\Delta_{\mu}^{(\frac{1}{2})}$ is symmetric.
###### Proof.
Write $\sigma=f\mu^{s}$ and $\tau=g\mu^{1-s}$. Then
$\Delta_{\mu}^{(s)}\sigma\,\tau=\Delta_{\mu}f\,g\,\mu.$
But
$\displaystyle
2\Delta_{\mu}f\,g={\mathrm{div}}_{\mu}X_{f}\,g=(-1)^{(f+1)g}g\,{\mathrm{div}}_{\mu}X_{f}$
$\displaystyle=(-1)^{(f+1)g}({\mathrm{div}}_{\mu}(gX_{f})-(-1)^{(f+1)g}X_{f}(g))$
$\displaystyle=(-1)^{(f+1)g}{\mathrm{div}}_{\mu}(gX_{f})+(-1)^{f}(f,g),$
where we have used part (3) of Proposition A.4. Similarly,
$\sigma\,\Delta_{\mu}^{(1-s)}\tau=f\,\Delta_{\mu}g\,\mu,$
and
$\displaystyle
2f\,\Delta_{\mu}g=f\,{\mathrm{div}}_{\mu}X_{g}={\mathrm{div}}_{\mu}(fX_{g})-(-1)^{f(g+1)}X_{g}(f)$
$\displaystyle={\mathrm{div}}_{\mu}(fX_{g})+(-1)^{fg+f+g}(g,f)={\mathrm{div}}_{\mu}(fX_{g})+(f,g).$
Therefore,
$2\Delta_{\mu}f\,g-(-1)^{f}2f\,\Delta_{\mu}g=(-1)^{(f+1)g}{\mathrm{div}}_{\mu}(gX_{f})-(-1)^{f}{\mathrm{div}}_{\mu}(fX_{g}),$
which implies, thanks to part (4) of Proposition A.4, that111111Note the
different sign in (8a), which instead yields
$\int_{\mathcal{M}}(\Delta_{\mu}^{(s)}\sigma\,\tau+(-1)^{\sigma}\sigma\,\Delta_{\mu}^{(1-s)}\tau)=\int_{\mathcal{M}}(\Delta_{\mu}f\,g+(-1)^{f}f\,\Delta_{\mu}g)\,\mu=(-1)^{f}\int_{\mathcal{M}}(f,g)\,\mu.$
Subtracting the two identities yields the interesting formula
$\int_{\mathcal{M}}(f,g)\,\mu=2\int_{\mathcal{M}}f\,\Delta_{\mu}g\,\mu,$ which
in particular shows that the integral of the BV bracket of two functions (at
least one of which has compact support) against the density $\mu$ vanishes if
one of the two functions is in the kernel of $\Delta_{\mu}$.
$\int_{\mathcal{M}}(\Delta_{\mu}^{(s)}\sigma\,\tau-(-1)^{\sigma}\sigma\,\Delta_{\mu}^{(1-s)}\tau)=0.$
∎
###### Remark 5.8.
By inspection in the proof one sees that it is essential that we have used the
same density $\mu$ to define the BV Laplacian on functions and to extend it to
$s$-densities.
We now come to the fundamental consequence of this proposition.
###### Lemma 5.9.
In each Darboux chart, there is an odd function $G_{\mu}$, depending on $\mu$,
such that
$\Delta_{\mu}^{(\frac{1}{2})}\sigma=(\triangle
f+G_{\mu}f)\,\mu_{\text{stand}}^{\frac{1}{2}},$
where we have written $\sigma=f\,\mu_{\text{stand}}^{\frac{1}{2}}$.
###### Proof.
Since $\Delta_{\mu}^{(\frac{1}{2})}$ is a second-order differential operator
and its leading term is independent of $\mu$, we have
$\Delta_{\mu}^{(\frac{1}{2})}\sigma=(\triangle
f+Y_{\mu}(f)+G_{\mu}\,f)\,\mu_{\text{stand}}^{\frac{1}{2}},$
for some vector field $Y_{\mu}$ and some function $G_{\mu}$. Since $\triangle$
has constant coefficients, it is symmetric. Moroever, since $Y_{\mu}$ is a
vector field, we have $Y_{\mu}^{t}=-Y_{\mu}+$some function. Since
$\Delta_{\mu}^{(\frac{1}{2})}$ is symmetric, we get $Y_{\mu}=0$. ∎
This implies the following
###### Corollary 5.10.
For every compatible $\mu$, the BV Laplacian on half-densities has a canonical
form in every Darboux chart:
$\Delta_{\mu}^{(\frac{1}{2})}\sigma=\triangle
f\,\mu_{\text{stand}}^{\frac{1}{2}},$
where we have written $\sigma=f\,\mu_{\text{stand}}^{\frac{1}{2}}$. In
particular, $\Delta_{\mu}^{(\frac{1}{2})}$ is independent of the choice of the
compatible density $\mu$.
###### Proof.
Using $\triangle^{2}=0$, $G_{\mu}^{2}=0$, and (2b), we get
$0=(\Delta_{\mu}^{(\frac{1}{2})})^{2}\sigma=(\triangle
G_{\mu}\,f+(G_{\mu},f))\mu_{\text{stand}}^{\frac{1}{2}}.$
Therefore, for every function $f$, we have
$\triangle G_{\mu}\,f+(G_{\mu},f)=0.$
In particular, for $f=1$, we get $\triangle G_{\mu}=0$, so the condition
simplifies to $(G_{\mu},f)=0$ for every $f$. This implies that $G_{\mu}$ is
constant, but then it must vanish because it is odd.121212This argument fails
if one wants to work in families, as in this case odd constants are allowed. ∎
In every Darboux chart, we can consider the BV Laplacian
$\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$, which sends
$f\,\mu_{\text{stand}}^{\frac{1}{2}}$ to $\triangle
f\,\mu_{\text{stand}}^{\frac{1}{2}}$. If we move from this Darboux chart to
another one via the transition map $\phi$, the operator goes to
$\phi_{*}\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$, which, by Proposition
5.5, is $\Delta_{\phi_{*}\mu_{\text{stand}}}^{(\frac{1}{2})}$. However, since
$\mu_{\text{stand}}$ is a compatible density (and therefore, by Remark 4.11,
also $\phi_{*}\mu_{\text{stand}}$ is a compatible density), by the corollary
this is again $\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$ (in the new
chart). Therefore, we have the
###### Theorem/Definition 5.11.
There is a canonical BV operator $\Delta$ acting on half-densities, defined as
$\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$ in every Darboux chart.
Note that, by its very definition, the canonical BV operator is invariant
under symplectomorphisms and satisfies $\Delta^{2}=0$.
Corollary 5.10 may now be reformulated as
###### Theorem 5.12.
For every compatible $\mu$, the BV Laplacian on half-densities is equal to the
canonical BV operator:
$\Delta_{\mu}^{(\frac{1}{2})}=\Delta.$
By this theorem we have that the canonical BV operator is a BV Laplacian on
half-densities whenever the odd symplectic manifold admits compatible
densities,131313Unlike in the even symplectic case, where a canonical choice
of density is provided by the top exterior power of the symplectic form, there
is no canonical density in the odd case. which we have not proved yet.
On the other hand, what is remarkable is that $\Delta$ is canonically defined
(by Theorem/Definition 5.11) without choosing any density (in particular it
would exist even if there were no compatible densities).
The canonical BV operator gives another way of defining the BV Laplacian on
functions. Namely, given an even, nowhere vanishing density $\mu$, we define a
differential operator $\widehat{\Delta}_{\mu}$ on functions via
(9) $\widehat{\Delta}_{\mu}f=\Delta(f\mu^{\frac{1}{2}})\,\mu^{-\frac{1}{2}}.$
Note that $\widehat{\Delta}_{\mu}^{2}=0$ for every $\mu$. However, it is not a
BV Laplacian in general. We actually have the
###### Lemma 5.13.
For every $\mu$,
$\widehat{\Delta}_{\mu}=\widetilde{\Delta}_{\mu}$
in every Darboux chart, where $\widetilde{\Delta}_{\mu}$ is the operator
defined in Remark 4.8.
###### Proof.
Writing $\mu=\mathrm{e}^{S}\mu_{\text{stand}}$, we have
$\widehat{\Delta}_{\mu}f\,\mathrm{e}^{\frac{S}{2}}\mu_{\text{stand}}^{\frac{1}{2}}=\Delta(f\mathrm{e}^{\frac{S}{2}}\mu_{\text{stand}}^{\frac{1}{2}})=\triangle(f\mathrm{e}^{\frac{S}{2}})\,\mu_{\text{stand}}^{\frac{1}{2}}.$
One can easily compute
$\triangle(f\mathrm{e}^{\frac{S}{2}})=\left(\triangle
f-\frac{1}{2}(S,f)+\frac{1}{2}F_{S}\right)\mathrm{e}^{\frac{S}{2}}=\widetilde{\Delta}_{\mu}f\,\mathrm{e}^{\frac{S}{2}},$
with $F_{S}$ as in (7). ∎
This yields the following
###### Corollary 5.14.
Let $\mu$ be an even, nowhere vanishing density. Then
1. (1)
$\widetilde{\Delta}_{\mu}$ is globally defined;
2. (2)
$\mu$ is compatible if and only if $\widehat{\Delta}_{\mu}=\Delta_{\mu}$;
3. (3)
$\mu$ is compatible if and only if $\Delta\mu^{\frac{1}{2}}=0$.
###### Proof.
The first two properties are now trivial. We prove the third. Setting $f=1$ in
(9) yields
$\Delta(\mu^{\frac{1}{2}})=\widehat{\Delta}_{\mu}1\,\mu^{\frac{1}{2}}$. Lemma
5.13 then implies that $\Delta(\mu^{\frac{1}{2}})=0$ if and only if in every
Darboux chart $F_{S}=\widetilde{\Delta}_{\mu}1=0$. By Remark 4.5 this happens
if and only if $\mu$ is compatible. ∎
### 5.1. Digression: Darboux expression of $\Delta_{\mu}^{(s)}$
In a Darboux chart we have $\mu=\mathrm{e}^{S}\mu_{\text{stand}}$. If
$\sigma=f\,\mu_{\text{stand}}^{s}$, we then have
$\sigma=\mathrm{e}^{-sS}f\,\mu^{s}$, so
$\Delta_{\mu}^{(s)}\sigma=\Delta_{\mu}(\mathrm{e}^{-sS}f)\,\mu^{s}.$
By using (6) and the properties of $\triangle$ in (2), we get
$\Delta_{\mu}^{(s)}\sigma=\left(\triangle
f+\left(s-\frac{1}{2}\right)(S,f)-\left(s\triangle
S+\frac{1}{2}s(s-1)(S,S)\right)f\right)\mu_{\text{stand}}^{s}.$
One can now explictly check that
$(\Delta_{\mu}^{(s)})^{t}=\Delta_{\mu}^{(1-s)}$. Moreover, one explicitly sees
that the first-order term vanishes for every $S$ if and only if
$s=\frac{1}{2}$. Finally, for $\mu$ compatible, i.e., $\Delta
S=\frac{1}{4}(S,S)$, the zeroth-order term becomes $\frac{1}{4}(2s-1)s(S,S)$,
which vanishes for $s=\frac{1}{2}$ or $s=0$.
This formula, for $s=\frac{1}{2}$, yields a different way to prove Corollary
5.10 and therefore Theorem/Definition 5.11 and Theorem 5.12. However, proving
this formula is a bit more laborious than proving Proposition 5.7.
## 6\. Odd cotangent bundles
We now focus on an odd cotangent bundle $\Pi T^{*}M$,141414Here $\Pi$ denotes
the change-of-parity functor. It simply means that the fiber coordinates are
now regarded as odd. where $M$ is an ordinary manifold. We will see that it is
easy to show the existence of compatible densities in this case. Since every
odd symplectic manifold is symplectomorphic to the odd cotangent bundle of its
body [13], this implies that every odd symplectic manifold has a compatible
density and therefore that the canonical BV operator is indeed a BV Laplacian
on half-densities.
The first observation is that functions on $\Pi T^{*}M$ are the same as
multivector fields on $M$: $C^{\infty}(\Pi
T^{*}M)=\Gamma(\Lambda^{\bullet}TM)$ as super algebras.151515The interested
reader might also appreciate that $(C^{\infty}(\Pi T^{*}M),(\ ,\ ))$ and
$(\Gamma(\Lambda^{\bullet}TM),[\ ,\ ]_{\text{SN}})$ are naturally isomorphic
odd Poisson algebras, where $(\ ,\ )$ is the canonical BV bracket, associated
to the canonical symplectic form, and $[\ ,\ ]_{\text{SN}}$ is the
Schouten–Nijenhuis bracket.
The second observation is about the density bundle on $\Pi T^{*}M$. If
$\phi_{\alpha\beta}$ denotes a transition function on $M$, then the
corresponding transition function on $T^{*}M$ has a fiber part given by
$(({\mathrm{d}}\phi_{\alpha\beta})^{*})^{-1}$. Therefore, the Berezinian of
the corresponding transition function for $\Pi T^{*}M$ is
$({\mathrm{d}}\phi_{\alpha\beta})^{2}$. This means that densities on $\Pi
T^{*}M$ transform like $2$-densities on $M$. Since we assume $M$ to be
oriented, we also have that half-densities on $\Pi T^{*}M$ transform like top
forms on $M$. More precisely, we have a canonical isomorphism
$\Gamma(\operatorname{Dens}^{\frac{1}{2}}(\Pi
T^{*}M))\simeq\Gamma(\Lambda^{\bullet}TM)\otimes_{C^{\infty}(M)}\Omega^{n}(M),$
where $n$ is the dimension of $M$. The next observation is that we have a
canonical isomorphism (of $C^{\infty}(M)$-modules)
$\phi\colon\begin{array}[t]{ccc}\Gamma(\Lambda^{\bullet}TM)\otimes_{C^{\infty}(M)}\Omega^{n}(M)&\to&\Omega^{n-\bullet}(M)\\\
X\otimes v&\mapsto&\iota_{X}v\end{array}$
We can use this isomorphism to define an operator on half-densities:
$\mathcal{D}\coloneqq\phi^{-1}\circ{\mathrm{d}}\circ\phi,$
where ${\mathrm{d}}$ is the de Rham differential. One immediately sees that
$\mathcal{D}^{2}=0$ and that $\mathcal{D}v=0$ for every top form $v$ on $M$.
###### Proposition 6.1.
The canonical operator $\mathcal{D}$ is the same as the canonical BV operator
$\Delta$ on half-densities.
###### Proof.
We just have to check how $\mathcal{D}$ acts in Darboux coordinates. Let
$q^{1},\dots,q^{n}$ denote coordinates on a chart of $M$ and let
$p_{1},\dots,p_{n}$ be the corresponding odd fiber coordinates. By linearity
it is enough to consider a half-density $\sigma=g\,\mu_{\text{stand}}$ with
$g(p,q)$ of the form $f(q)\,p_{i_{1}}\cdots\,p_{i_{k}}$, with pairwise
distinct indices $i_{j}$. In the identification described above this
corresponds to
$\sigma=f(q)\,\partial_{i_{1}}\wedge\dots\wedge\partial_{i_{k}}\otimes{\mathrm{d}}^{n}q$,
so
$\phi(\sigma)=f(q)\,\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q.$
We then have
${\mathrm{d}}\phi(\sigma)={\mathrm{d}}f(q)\,\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q=\partial_{i}f(q)\,{\mathrm{d}}q^{i}\wedge\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q,$
since the form
$\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q$
has a constant coefficient, so it is closed. If the index $i$ is different
from all the $i_{j}$s, the corresponding term vanishes, since
${\mathrm{d}}q^{i}$ is already contained in ${\mathrm{d}}^{n}q$. Otherwise, we
have
$\begin{split}{\mathrm{d}}q^{i_{j}}\wedge\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q&=(-1)^{j-1}{\mathrm{d}}q^{i_{j}}\wedge\iota_{\partial_{i_{j}}}\iota_{\partial_{i_{1}}}\cdots\,\widehat{\iota}_{\partial_{i_{j}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q\\\
&=(-1)^{j-1}\iota_{\partial_{i_{1}}}\cdots\,\widehat{\iota}_{\partial_{i_{j}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q,\end{split}$
where the caret denotes omission. On the other hand,
$\partial_{i_{j}}(p_{i_{1}}\cdots\,p_{i_{k}})=(-1)^{j-1}p_{i_{1}}\cdots\,\widehat{p}_{i_{j}}\cdots\,p_{i_{k}}.$
Therefore,
$\phi(\triangle
g\,\mu_{\text{stand}})={\mathrm{d}}\phi(g\,\mu_{\text{stand}}),$
which shows that $\Delta=\mathcal{D}$. ∎
We then have $\Delta v=0$ for every top form $v$ on $M$ (regarded as a half-
density on $\Pi T^{*}M$). As a notation, we write $\mu_{v}$ for the even,
nowhere vanishing density $v^{2}$ on $\Pi T^{*}M$ associated to a volume form
$v$ on $M$. We then have
$\Delta\mu_{v}^{\frac{1}{2}}=0,$
which shows, by point (3) of Corollary 5.14, that $\mu_{v}$ is a compatible
density. We then have the
###### Proposition 6.2.
An odd cotangent bundle possesses compatible densities.
This implies, by Theorem 5.12, that $\Delta=\Delta^{(\frac{1}{2})}_{\mu_{v}}$.
Given a volume form $v$, we can now use (9) to get $\Delta_{\mu_{v}}$ on
functions. It is an easy exercise to see that
$\Delta_{\mu_{v}}=\phi_{v}^{-1}\circ{\mathrm{d}}\circ\phi_{v},$
where $\phi_{v}$ is the ($v$-dependent!) isomorphism
$\phi_{v}\colon\begin{array}[t]{ccc}C^{\infty}(\Pi
T^{*}M)=\Gamma(\Lambda^{\bullet}TM)&\to&\Omega^{n-\bullet}(M)\\\
X&\mapsto&\iota_{X}v\end{array}$
Note that, for a vector field $X$, we have
$\Delta_{\mu_{v}}X={\mathrm{div}}_{v}X$. For this reason, $\Delta_{\mu_{v}}$
is interpreted as the extension to multivector fields of the divergence
operator with respect to $v$.
## 7\. Conclusion of the argument
It is a theorem (see [13]) that every odd symplectic manifold $\mathcal{M}$
is, noncanonically, isomorphic to the odd cotangent bundle $\Pi T^{*}M$ of its
body $M$.161616This is a global Darboux theorem. If one is familiar with the
proof of Darboux’s theorem via Moser’s trick, the reason why this now works
globally is that it turns out that one has to integrate a vector field in the
odd directions, which has no problem of definition domain.
On $\Pi T^{*}M$ we have compatible densities (Proposition 6.2), e.g.,
$\mu_{v}$ for a volume form $v$ on $M$. If $\Psi\colon\Pi
T^{*}M\to\mathcal{M}$ is a symplectomorphism, which exists by the above
mentioned theorem, then $\Psi_{*}\mu_{v}$ is a compatible density on
$\mathcal{M}$ (Remark 4.11).
We therefore have—also using Theorem 5.12, equation (9), and point (2) of
Corollary 5.14—our final result:
###### Theorem 7.1.
Every odd symplectic manifold possesses compatible densities. For each
compatible density $\mu$, we have
$\Delta_{\mu}^{(\frac{1}{2})}=\Delta,$
with $\Delta$ the canonical BV operator on half-densities, and the BV
Laplacian $\Delta_{\mu}$ on functions may be recovered from
$\Delta_{\mu}f\,\mu^{\frac{1}{2}}=\Delta(f\mu^{\frac{1}{2}}).$
## Appendix A Some background details
### A.1. The Lie derivative
If $\Xi$ is an object (e.g., function, vector field, density) for which a
notion of pushforward $\Psi_{*}$ under a diffeomorphism $\Psi$ exists, we
define the Lie derivative with respect to a vector field $X$ as
$\mathsf{L}_{X}\Xi\coloneqq\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}(\Phi^{X}_{-t})_{*}\Xi.$
For an even vector field $X$, the diffeomorphism $\Phi^{X}_{t}$ is the flow of
$X$ at time $t$. If $X$ is odd, the variable $t$ is also odd (so the
evaluation at zero in the formula is redundant), and we define the morphism
$\Phi^{X}\colon\Pi{\mathbb{R}}\times\mathcal{M}\to\mathcal{M}$ via171717This
defines the flow of $X$ at odd time $t$ if and only if $[X,X]=0$. Otherwise,
the flow property $\Phi^{X}_{t+s}=\Phi^{X}_{t}\circ\Phi^{X}_{s}$ is not
satisfied, so in general $\Phi^{X}$ is not a flow.
$(\Phi^{X})^{*}\colon\begin{array}[t]{ccc}C^{\infty}(\mathcal{M})&\to&C^{\infty}(\Pi{\mathbb{R}}\times\mathcal{M})\\\
f&\mapsto&f+tX(f)\end{array}$
where $t$ is the odd coordinate on $\Pi{\mathbb{R}}$.181818Here $\Pi$ denotes
the change-of-parity functor: $\Pi{\mathbb{R}}$ is the superdomain with one
odd coordinate. Note that in local coordinates $z^{\mu}$, with
$X=X^{\mu}{\small\frac{\partial}{\partial z^{\mu}}}$, we have in both cases
(10)
$(\Phi^{X})^{\mu}\coloneqq(\Phi^{X})^{*}z^{\mu}=z^{\mu}+tX^{\mu}+O(t^{2}),$
where of course $O(t^{2})=0$ in the odd case. Also note that in both cases we
have
$\Psi_{*}(\Phi^{X})_{*}=(\Phi^{\Psi_{*}X})_{*}\Psi_{*}$
for every diffeomorphism $\Psi$.191919In the even case, one just observes how
solutions of ODEs are mapped under diffeomorphisms. In the odd case, it is
just an immediate application of the chain rule. This implies that
$\mathsf{L}_{\Psi_{*}X}\Psi_{*}\Xi=\Psi_{*}\mathsf{L}_{X}\Xi.$
If two objects $\Xi_{1}$ and $\Xi_{2}$ as above can be multiplied, we
immediately get
(11)
$\mathsf{L}_{X}(\Xi_{1}\Xi_{2})=\mathsf{L}_{X}\Xi_{1}\,\Xi_{2}+(-1)^{X\Xi_{1}}\Xi_{1}\,\mathsf{L}_{X}\Xi_{2}.$
On a function $f$ or on a vector field $Y$, the Lie derivative is readily
computed as
$\mathsf{L}_{X}f=X(f)\quad\text{and}\quad\mathsf{L}_{X}Y=[X,Y].$
### A.2. Densities
A density is an object that transforms in such a way that its integration may
be defined. For this we have to recall that, if $\Psi$ is a diffeomorphism
between two superdomains, then the change-of-variables formula for berezinian
integration involves the Berezinian $\operatorname{Ber}(\Psi)$ of the jacobian
matrix of $\Psi$ times the sign of the jacobian determinant of the reduction
to the body of $\Psi$. Since we assume throughout that the body of our
supermanifold is oriented, we will not have this factor. Therefore, given an
oriented atlas for $\mathcal{M}$ that has transition functions
$\phi_{\alpha\beta}$, we define the density bundle
$\operatorname{Dens}\mathcal{M}$ as the line bundle with transition functions
$\operatorname{Ber}(\phi_{\alpha\beta})^{-1}$. More generally, for any real
number $s$, we define the line bundle $\operatorname{Dens}^{s}\mathcal{M}$ of
$s$-densities with transition functions
$\operatorname{Ber}(\phi_{\alpha\beta})^{-s}$. Note that $1$-densities are the
same as densities, and $0$-densities are the same as functions.
With this definition the integral of a compactly supported202020The support is
defined in terms of the coefficient body functions. Namely, an object
(function, vector field, density,…) is compactly supported when in its
expansion in odd variables all the coefficients are compactly supported
functions on the body. density $\mu$ over $\mathcal{M}$ is defined as usual
(by choosing a partition of unity subordinated to the atlas and by berezinian
integration in each chart).
The pushforward of $s$-densities under diffeomorphisms is also defined. If we
go to charts, the pushforward is given by the pushforward of the representing
function times the Berezinian of the transformation to the power $-s$.
Armed with the pushforward, we can define the Lie derivative of an
$s$-density. In particular, we need the following
###### Lemma A.1.
In local coordinates $p_{1},\dots,p_{m}$ and $q^{1},\dots,q^{n}$ (odd and
even, respectively), the Lie derivative of the standard density
$\mu_{\text{stand}}={\mathrm{d}}^{n}q\,{\mathrm{d}}^{m}p$ is
$\mathsf{L}_{X}{\mu_{\text{stand}}}=(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})\,{\mu_{\text{stand}}},$
where we used the expansion $X=X^{i}\partial_{i}+X_{i}\partial^{i}$.
###### Proof.
Using (10), we see that212121By $\delta^{i}_{j}$ we mean the Kronecker delta,
which is equal to $1$ for $i=j$ and to $0$ otherwise.
$\begin{split}\operatorname{Ber}(\Phi^{X}_{-t})&=\operatorname{Ber}\begin{pmatrix}\delta_{i}^{j}-\partial_{i}(tX^{j})&0\\\
0&\delta^{i}_{j}-\partial^{i}(tX_{j})\end{pmatrix}+O(t^{2})\\\
&=-\partial_{i}(tX^{i})-\partial^{i}(tX_{i})+O(t^{2})\\\
&=-t\partial_{i}(X^{i})-(-1)^{X}t\partial^{i}(X_{i})+O(t^{2}),\end{split}$
using that the paritiy of $t$ is the same as that of $X$. Applying the
definitions of Lie derivative and of pushforward of a density yields the
result. ∎
Applying (11) to the case of densities, we get
(12) $\mathsf{L}_{X}(h\mu)=X(h)\,\mu+(-1)^{Xh}h\,\mathsf{L}_{X}\mu$
for every function $h$. In particular, we get the general formula for the Lie
derivative in local coordinates:
$\begin{split}\mathsf{L}_{X}\mu&=(X(h)+(-1)^{Xh}h(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i}))\,{\mu_{\text{stand}}}\\\
&=(X(h)+(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})h)\,{\mu_{\text{stand}}}\\\
&=X(h)\mu_{\text{stand}}+(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})\mu,\end{split}$
with $\mu=h\mu_{\text{stand}}$. This implies the following
###### Lemma A.2.
For every function $f$, vector field $X$, and density $\mu$, we have
$\mathsf{L}_{fX}\mu=f\mathsf{L}_{X}\mu+(-1)^{fX}X(f)\mu.$
###### Proof.
It is enough to check the formula in local coordinates. We have
$\begin{split}\mathsf{L}_{fX}\mu&=fX(h)\mu_{\text{stand}}+(\partial_{i}(fX^{i})-(-1)^{f+X}\partial^{i}(fX_{i}))\mu\\\
&=fX(h)\mu_{\text{stand}}+f\,(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})\mu+(\partial_{i}fX^{i}-(-1)^{f+X}\partial^{i}fX_{i})\mu\\\
&=f\mathsf{L}_{X}\mu+((-1)^{Xf}X^{i}\partial_{i}f-(-1)^{f+X}(-1)^{(f+1)(X+1)}X_{i}\partial^{i}f)\mu\\\
&=f\mathsf{L}_{X}\mu+(-1)^{fX}X(f)\mu.\end{split}$
∎
In the case of densities, the pushforward is related to the change-of-
variables formula. Namely, if $\Psi\colon\mathcal{M}\to\mathcal{N}$ is a
diffeomorphism (preserving the orientation of the bodies), then
(13) $\int_{\mathcal{N}}\Psi_{*}\mu=\int_{\mathcal{M}}\mu$
for every compactly supported density $\mu$. This leads to
###### Lemma A.3.
For every vector field $X$ and density $\mu$, one of which is compactly
supported, we have
$\int_{\mathcal{M}}\mathsf{L}_{X}\mu=0.$
###### Proof.
First consider the case when $\mu$ is compactly supported. In this case,
$(\Phi^{X}_{-t})_{*}\mu$ and hence $\mathsf{L}_{X}\mu$ are also compactly
supported. We then have
$\int_{\mathcal{M}}\mathsf{L}_{X}\mu=\int_{\mathcal{M}}\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}(\Phi^{X}_{-t})_{*}\mu=\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}\int_{\mathcal{M}}(\Phi^{X}_{-t})_{*}\mu=\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}\int_{\mathcal{M}}\mu=0,$
where we have also used (13).
If on the other hand $\mu$ is not compactly supported, but $X$ is, we replace
$\mu$ with $\widetilde{\mu}=\rho\mu$, where $\rho$ is a compactly supported
bump function which is identically equal to $1$ on the support of $X$. Since
$\mathsf{L}_{X}\mu=\mathsf{L}_{X}\widetilde{\mu}$, we get the result by the
previous case. ∎
If we have an even, nowhere vanishing density $\mu$, then every other density
can be written as $f\mu$ for a uniquely determined function $f$. Therefore, we
can define the divergence operator via the following formula:
${\mathrm{div}}_{\mu}X\,\mu=\mathsf{L}_{X}\mu.$
The properties of the Lie derivative immediately imply properties for the
divergence operator:
###### Proposition A.4.
Let $\mu$ be an even, nowhere vanishing density $\mu$ and $X$ a vector field.
Then
1. (1)
for every diffeomorphism $\Psi$,
$\Psi_{*}{\mathrm{div}}_{\mu}X={\mathrm{div}}_{\Psi_{*}\mu}\Psi_{*}X;$
2. (2)
for every even, nowhere vanishing function $h$,
${\mathrm{div}}_{h\mu}X={\mathrm{div}}_{\mu}X+\frac{1}{h}X(h);$
3. (3)
for every function $f$,
${\mathrm{div}}_{\mu}(fX)=f{\mathrm{div}}_{\mu}X+(-1)^{fX}X(f);$
4. (4)
under the additional assumption that $X$ is compactly supported,
$\int_{\mathcal{M}}{\mathrm{div}}_{\mu}X\,\mu=0.$
Moreover, from the local coordinate expression for the Lie derivative, we get
(14a) $\displaystyle{\mathrm{div}}_{\mu_{\text{stand}}}X$
$\displaystyle=\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i},$ and, rearranging
the terms, (14b) $\displaystyle{\mathrm{div}}_{\mu}X$
$\displaystyle=\frac{1}{h}(\partial_{i}(hX^{i})-(-1)^{X}\partial^{i}(hX_{i})),$
for $\mu=h\mu_{\text{stand}}$.
## Appendix B Applications of the BV formalism
We recall here the main reason why the BV formalism was introduced: to study
the invariance of certain integrals.
We start considering the case of Section 2, where we have Darboux coordinates
$p_{1},\dots,p_{n}$ and $q^{1},\dots,q^{n}$. For the sake of the argument, we
rearrange each pair $(p_{i},q^{i})$ into a new pair $(x_{i},y^{i})$, with
$x_{i}=p_{i}$ and $y^{i}=q^{i}$ for some $i$s and $x_{i}=q^{i}$ and
$y^{i}=p_{i}$ for the other $i$s. Note that $x_{i}$ has opposite parity to
$y^{i}$ (but we do not insist on $y^{i}$ being even). We now use the shorthand
notation
$\partial_{i}=\frac{\partial}{\partial
y^{i}}\quad\text{and}\quad\partial^{i}=\frac{\partial}{\partial x_{i}}.$
Note that the BV Laplacian still reads $\triangle=\partial_{i}\partial^{i}$.
If $\psi$ is an odd function of the variables $y$, it makes sense, in terms of
parity, to set $x_{i}=(-1)^{x_{i}}\partial_{i}\psi$ (we will explain the
reason for the signs in a moment). For a function $f$ in the $(x,y)$
variables, we define
$\int_{\mathcal{L}_{\psi}}f\coloneqq\int
f|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\;{\mathrm{d}}^{n}y,$
where $|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}$ means that we set each $x$
variable to the corresponding right-hand-side expression. The reason for the
notation is that $\mathcal{L}_{\psi}$ is a Lagrangian submanifold determined
by $\psi$. In fact,
$\omega={\mathrm{d}}x_{i}{\mathrm{d}}y^{i}={\mathrm{d}}(x_{i}{\mathrm{d}}y^{i})$
and
$(x_{i}{\mathrm{d}}y^{i})|_{x_{i}=(-1)^{x_{i}}\partial_{i}\psi}={\mathrm{d}}\psi.$
The first result is the BV lemma.
###### Lemma B.1.
Suppose that $f=\triangle g$ and $g$ is integrable. Then
$\int_{\mathcal{L}_{\psi}}f=0$ for every $\psi$.
###### Proof.
We have
$\begin{split}\sum_{i}\partial_{i}(\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}&=\sum_{i}(\partial_{i}\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\\\
&\phantom{=}\
+\sum_{ik}(\partial_{i}((-1)^{x_{k}}\partial_{k}\psi)\,\partial^{k}\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}.\end{split}$
Since
$\sum_{ik}(-1)^{x_{k}}(\partial_{i}\partial_{k}\psi\,\partial^{k}\partial^{i}g)=0$,
as one can easily see by exchanging the indices, we get
$(\triangle
g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}=\sum_{i}\partial_{i}(\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi},$
so $\int_{\mathcal{L}_{\psi}}f$ vanishes. ∎
This leads to the fundamental BV theorem.
###### Theorem B.2.
Let $\psi_{t}$ be a family of odd functions in the $y$ variables depending
smoothly on the even parameter $t$. If $f$ is integrable, on every
$\mathcal{L}_{\psi_{t}}$, and $\triangle f=0$, then
$I_{t}\coloneqq\int_{\mathcal{L}_{\psi_{t}}}f$
is constant.
###### Proof.
We have
$\dot{I}_{t}=\int\sum_{i}((-1)^{x_{i}}\partial_{i}\dot{\psi}\,\partial^{i}f)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\;{\mathrm{d}}^{n}y,$
where the dot denotes derivative with respect to $t$. From
$\triangle(\dot{\psi}\,f)=\sum_{i}\partial_{i}\partial^{i}(\dot{\psi}\,f)=\sum_{i}(-1)^{x_{i}}\partial_{i}(\dot{\psi}\,\partial^{i}f)=\sum_{i}(-1)^{x_{i}}\partial_{i}\dot{\psi}\,\partial^{i}f-\dot{\psi}\,\triangle
f,$
we get
$\dot{I}_{t}=\int\triangle(\dot{\psi}\,f)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\;{\mathrm{d}}^{n}y=\int_{\mathcal{L}_{\psi_{t}}}\triangle(\dot{\psi}\,f)=0,$
where we have used the previous lemma. ∎
The BV theorem is used in quantum gauge theories,222222In this application,
the odd symplectic manifold is actually infinite dimensional, so $\triangle$
is not defined. What one does is either to proceed formally or to regularize
the theory, e.g., by replacing the space of fields with a finite-dimensional
approximation. where the choice of $\psi$ (called the gauge-fixing fermion)
corresponds to the choice of a gauge fixing and the invariance under
deformations of this choice yields the independence of the theory from the
gauge fixing, assuming of course that $\triangle f=0$.
Using the results of Sections 4 and 5, one can extend the BV lemma and the BV
theorem globally. The main observation is that the restriction of a half-
density on an odd symplectic manifold to a Lagrangian submanifold defines a
density there [13].
###### Theorem B.3.
Let $\mathcal{M}$ be an odd symplectic manifold and $\sigma$ a half-density.
The following hold:
1. (1)
If $\sigma=\Delta\tau$ for some $\tau$, then
$\int_{\mathcal{L}}\sigma=0$
for every a Lagrangian submanifold $\mathcal{L}$ of $\mathcal{M}$ on which
$\sigma$ is integrable.
2. (2)
If $\Delta\sigma=0$ and $\mathcal{L}_{t}$ is a smooth family of Lagrangian
submanifolds of $\mathcal{M}$ on which $\sigma$ is integrable, then
$I_{t}\coloneqq\int_{\mathcal{L}_{t}}\sigma$
is constant.
In the case of an odd cotangent bundle $\Pi T^{*}M$, one can show that every
Lagrangian submanifold is a smooth deformation of an odd conormal bundle $\Pi
N^{*}C$, where $C$ is a submanifold of $M$. Moreover, one can show that
$\int_{\Pi N^{*}C}\sigma=\int_{C}\phi(\sigma)$
for every half-density $\sigma$. By Stokes’ theorem and the characterization
of the canonical BV operator as in Proposition 6.1 (i.e.,
$\phi\circ\Delta={\mathrm{d}}\circ\phi$), one gets that
$\int_{\Pi N^{*}C_{1}}\sigma=\int_{\Pi N^{*}C_{2}}\sigma$
if $C_{1}$ and $C_{2}$ are homologous and $\Delta\sigma=0$.
By Schwarz’ theorem one can then generalize the global BV theorem a bit.
###### Theorem B.4.
Let $\mathcal{M}$ be an odd symplectic manifold and $\sigma$ a half-density
satisfying $\Delta\sigma=0$. Then
$\int_{\mathcal{L}_{1}}\sigma=\int_{\mathcal{L}_{2}}\sigma$
whenever $\mathcal{L}_{2}$ can be obtained from $\mathcal{L}_{1}$ by a
combination of smooth deformations and homologous changes of the body,
assuming that $\sigma$ is integrable on each intermediate step.
## Appendix C Historical remarks
The BV formalism was developed by Batalin and Vilkovisky [1, 2] as a
generalization of the BRST formalism [3, 15], which in turn put gauge fixing
and the Faddeev–Popov determinant [5] into a cohomological setting. They
constructud the BV Laplacian (as in our Section 2) and proved their
fundamental theorem (in our notes, Theorem B.2). In addition, they showed
that, under suitable assumptions, a physical action can be extended to a BV
action $S$ satisfying the classical master equation $(S,S)=0$.
In [16] Witten recognized the relation between the BV operator on functions on
an odd cotangent bundle and the divergence operator on multivector fields on
the body (see the last paragraph of Section 6). The relation between the de
Rham differential and the divergence operator on multivector fields on an
ordinary manifold was already known to Soviet mathematicians in the ’80s (see,
e.g., [12]). Also note that Bernshtein and Leites [4] had already introduced
in 1977 the notion of “integral forms” on supermanifolds—where this notion
differs from that of differential forms—and defined the exterior differential
for them as the divergence operator.
Khudaverdian [6] was the first to give the global definition of the BV
Laplacian on functions, see our equation (5).
Khudaverdian was also the first to observe that an odd symplectic manifold
always admits global Darboux coordinates. This was used by Schwarz in [13],
where he also observed how to globalize and extend the BV theorem (in our
notes, Theorem B.3).
Finally, in [7, 8],232323Note that O. M. Khudaverdian and H. M. Khudaverdian
are just different spellings of the same name. Khudaverdian showed the
existence of a canonical BV Laplacian on half-densities (in our notes, this
corresponds to Theorem/Definition 5.11).
In a series of papers (among others, [9, 10, 11]), Khudaverdian and Voronov
further clarified the properties of the canonical BV Laplacian, essentially
covering all the constructions we present in these notes (and more). In [9]
they extended the construction of the odd Laplacian on half-densities to odd
Poisson manifolds. In [10] they discussed the principal and subprincipal
symbols of the BV Laplacians and their transposed operators (see our Section
5). In [11] they defined the canonical BV Laplacian on half-densities on an
odd cotangent bundle in terms of the de Rham differential of the corresponding
differential forms and then showed that this operator is invariant under all
symplectomorphisms—not just those coming from the base.
It is worth mentioning that Ševera [14] found a completely different
construction for the canonical BV Laplacian. The main observation is that the
complex of differential forms on $\mathcal{M}$ has two commuting coboundary
operators: the de Rham differential ${\mathrm{d}}$ and the operator
$\delta\coloneqq\omega\wedge\ $. It turns out that the associated spectral
sequence lives up to the $E_{2}$-term. More precisely, Ševera shows that
$E_{1}=H_{\delta}(\mathcal{M})$ is canonically isomorphic to
$\operatorname{Dens}^{\frac{1}{2}}(\mathcal{M})$ and that the induced
coboundary operator ${\mathrm{d}}_{1}$ vanishes, which implies that
$E_{2}=E_{1}=\operatorname{Dens}^{\frac{1}{2}}(\mathcal{M})$. Finally, Ševera
proves that the canonically induced coboundary operator ${\mathrm{d}}_{2}$ is
the canonical BV Laplacian and that all higher coboundary operators vanish.
## References
* [1] I. A. Batalin and G. A. Vilkovisky, “Relativistic S-matrix of dynamical systems with boson and fermion constraints,” Phys. Lett. B 69, 309–312 (1977).
* [2] I. A. Batalin and G. A. Vilkovisky, “Gauge algebra and quantization,” Phys. Lett. B 102, 27–31 (1981).
* [3] C. Becchi, A. Rouet, and R. Stora, “Renormalization of the abelian Higgs–Kibble model,” Commun. Math. Phys. 42, 127–162 (1975).
* [4] I. N. Bernshtein and D. A. Leites, “Integral forms and the Stokes formula on supermanifolds,” Funkts. Anal. Prilozhen. 11, 55-56 (1977).
* [5] L. D. Faddeev and V. Popov, “Feynman diagrams for the Yang–Mills field,” Phys. Lett. B 25, 29-30 (1967).
* [6] O. M. Khudaverdian, “Geometry of superspace with even and odd brackets,” J. Math. Phys. 32, 1934–1937, (1991).
* [7] O. M. Khudaverdian, “Delta-operator on semidensities and integral invariants in the Batalin–Vilkovisky geometry,” arXiv:math/9909117.
* [8] H. M. Khudaverdian, “Semidensities on odd symplectic supermanifolds,” Commun. Math. Phys. 247, 353–390 (2004).
* [9] H. M. Khudaverdian and T. Voronov, “On odd Laplace operators,” Lett. Math. Phys. 62, 127–142 (2002).
* [10] H. M. Khudaverdian and T. Voronov, “On odd Laplace operators. II,” arXiv:math/0212311.
* [11] H. M. Khudaverdian and T. Voronov, “Differential forms and odd symplectic geometry,” In: Geometry, Topology and Mathematical Physics. S. P. Novikov seminar: 2006-2007, (V. M. Buchstaber and I. M. Krichever, eds.) AMS Translations, Ser. 2, Vol. 224, Amer. Math. Soc., Providence, RI, 2008, 159–171, arXiv:math/0606560.
* [12] A. A. Kirillov, “Invariant operators on geometric quantities,” J. of Soviet Mathematics 18, 1–21 (1982).
* [13] A. Schwarz, “Geometry of Batalin–Vilkovisky quantization,”Commun. Math. Phys. 155, 249–260 (1993).
* [14] P. Ševera, “On the origin of the BV operator on odd symplectic supermanifolds,” Lett. Math. Phys. 78, 55–59 (2006).
* [15] I. V. Tyutin, “Gauge invariance in field theory and statistical physics in operator formalism,” Lebedev Physics Institute preprint 39 (1975), arXiv:0812.0580.
* [16] E. Witten, “A note on the antibracket formalism,” Mod. Phys. Lett. A 5, 487–494 (1990).
|
# Active vector models generalizing 3D Euler
and electron–MHD equations
Dongho Chae Department of Mathematics, Chung-Ang University, Heuk-Seok Ro 84,
Seoul, Republic of Korea 156-756. E-mail<EMAIL_ADDRESS>In-Jee Jeong
Department of Mathematical Sciences and RIM, Seoul National University,
Gwanak-gu, Gwanak 1, Seoul, Republic of Korea 08826. E-mail<EMAIL_ADDRESS>
###### Abstract
We introduce an active vector system, which generalizes both the 3D Euler
equations and the electron–magnetohydrodynamic equations (E–MHD). We may as
well view the system as singularized systems for the 3D Euler equations, in
which case the equations of (E–MHD) correspond to the order two more singular
one than the 3D Euler equations. The generalized surface quasi-geostrophic
equation (gSQG) can be also embedded into a special case of our system when
the unknown functions are constant in one coordinate direction. We investigate
some basic properties of this system as well as the conservation laws. In the
case when the system corresponds up to order one more singular than the 3D
Euler equations, we prove local well-posedness in the standard Sobolev spaces.
The proof crucially depends on a sharp commutator estimate similar to the one
used for (gSQG) in [11]. Since the system covers many areas of both physically
and mathematically interesting cases, one can expect that there are various
related problems to be investigated, parts of which are discussed here.
## 1 Introduction
### 1.1 The generalized SQG equations
The generalized surface quasi-geostrophic (gSQG) equation is an active scalar
system given in $\mathbb{R}^{2}$ as :
$\left\\{\begin{aligned} &\partial_{t}\theta+v\cdot\nabla\theta=0,\\\
&v=\nabla^{\perp}\Gamma\theta,\end{aligned}\right.$ (gSQG)
with $\theta(t,\cdot):\Omega\rightarrow\mathbb{R}$, where $\Gamma$ is a
Fourier multiplier. In the recent years, the model (gSQG) together with its
dissipative analogues has been intensively studied, with topics including
local and global well-posedness ([14, 13, 12, 11, 85, 86, 28, 31, 31, 33, 58,
80, 89, 69, 70, 63, 62, 36, 84, 50, 10, 16, 17, 57, 78, 83]), finite and
infinite time singularity formation ([73, 60, 74, 76, 46, 75, 92, 90, 47]),
rotating and traveling-wave solutions ([40, 59, 39, 61, 8, 9, 41, 6, 7, 54,
55, 56]), non-uniqueness ([64, 24, 3, 88]), and so on. For any $\Gamma$, the
system describes the transport of a scalar by an incompressible flow. The
primary motivation for studying the system (gSQG) comes from the fact that
(gSQG) interpolates the two physically important cases for
$\Gamma=(-\Delta)^{-1}$ and $\Gamma=(-\Delta)^{-\frac{1}{2}}$, where (gSQG)
corresponds to 2D Euler equations and the surface quasi-geostrophic (SQG)
equations, respectively. The SQG equations were derived in [35, 34] from the
system for non-homogeneous three-dimensional half-space rotating fluid under
the so-called quasi-geostrophic approximation. Various interesting phenomena
occur for solutions to (gSQG) with different choice of multipliers; see recent
surveys [28, 77, 72] as well as aforementioned references.
### 1.2 The active vector system
In this note, we introduce an active vector system defined in
$\mathbb{R}^{3}$, which not only generalizes two physically important systems,
namely the 3D Euler equations and the electron–MHD equations (also referred to
as the Hall equations) but also contains the entire family of (gSQG) as sub-
systems. The system we suggest is given by
$\left\\{\begin{aligned}
&\partial_{t}B+\nabla\times((\nabla\times\Gamma[B])\times B)=0,\\\
&\nabla\cdot B=0,\\\ \end{aligned}\right.$ (1.1)
where $B(t,\cdot):\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ and $\Gamma$ is a
Fourier multiplier whose symbol will be denoted by $\gamma=\gamma(|\xi|)\geq
0$. Since $B$ is divergence-free, (1.1) can be written as an active vector
form:
$\left\\{\begin{aligned} &\partial_{t}B+V\cdot\nabla B=B\cdot\nabla V,\\\
&V=-\nabla\times\Gamma[B],\\\ &\nabla\cdot B=0.\end{aligned}\right.$ (1.2)
The important special cases $\Gamma=(-\Delta)^{-1}$ and $\Gamma=1$ correspond
to the 3D Euler equations (with $B=\nabla\times V$ being the vorticity) and
the so-called electron magneto-hydrodynamics (E–MHD) system ($B$ being the
magnetic field), which can be obtained from the Hall–MHD system by setting the
fluid velocity to be zero ([1, 4, 81, 87]). That is, the equation for the
magnetic field reads
$\left\\{\begin{aligned} &\partial_{t}B+\nabla\times((\nabla\times B)\times
B)=0,\\\ &\nabla\cdot B=0.\\\ \end{aligned}\right.$ (E–MHD)
Recently, the Hall–MHD, E–MHD systems and their viscous counterparts have
received a lot of attention in the PDE community ([15, 18, 19, 20, 21, 42, 43,
44, 45]), not only because these equations are important in various physical
phenomena, but also they naturally appear in hydrodynamic limit problems ([1])
and raise new technical difficulties. Already at the level of proving well-
posedness, (E–MHD) can be considered as a transport equation for $B$ with
advecting velocity $V=\nabla\times B$, which is one order more singular than
$B$. This is in stark contrast with the three-dimensional incompressible Euler
equations ((1.2)), where $V=\nabla\times(-\Delta)^{-1}B$ is one order more
regular than $B$. The 3D Euler equations have solutions which blow up in
finite and infinite time ([52, 53, 23, 51, 25, 26]). Therefore, it will be
interesting to investigate the dynamics of solutions for the interpolating
systems, which was precisely the motivation for studying the generalized SQG
equations. Somewhat surprisingly, it turns out that if we consider the special
case of (1.2) with $B(t,x)=(0,0,\theta(t,x_{1},x_{2})),$ and
$V=\nabla^{\perp}\Gamma[\theta]$, then we obtain precisely (gSQG), see 2.2 for
details. That is, any gSQG equation can be embedded into our system (1.1), so
that (1.1) contains all the phenomena observed in the gSQG case. In the
current work, we shall focus on the multipliers $\Gamma$ which lie in between
those above two cases and discuss well/ill-posedness of the Cauchy problem for
(1.1).
###### Remark 1.1.
The term “active vector” was used in [29] to denote transport systems for
vector-valued functions associated with the incompressible Euler equations. We
generalize the notion of an active vector to allow a stretching term on the
right hand side.
### 1.3 Main result
Our first main result provides local well-posedness of smooth solutions for
(1.1) when the operator $\Gamma$ is up to one order more singular than the
case of the 3D Euler equations. The precise assumptions we impose on $\Gamma$
are given as follows.
Assumptions on $\Gamma$. Let $\Lambda=(-\Delta)^{\frac{1}{2}}$ be the Zygmund
operator. We shall assume that $\Gamma$ satisfies
* •
$\Gamma\lesssim\max\\{\Lambda^{-1},\Lambda^{-2}\\}$, $\Gamma$ is decreasing
with $\Lambda|\Gamma^{\prime}|\lesssim\Gamma$: to be precise, for any
$|\xi|>0$, we have
$\begin{split}\gamma(|\xi|)\leq{C(|\xi|^{-1}+|\xi|^{-2})}\end{split}$ (1.3)
and
$\begin{split}\gamma^{\prime}(|\xi|)\leq
0,\qquad|\xi||\gamma^{\prime}(|\xi|)|\leq C\gamma(|\xi|)\end{split}$ (1.4)
for some $C>0$.
* •
$\Gamma\gtrsim\min\\{\Lambda^{-2},\Lambda^{-1}\\}$: that is,
$\begin{split}\gamma(|\xi|)\geq
c\,{\min\\{|\xi|^{-2},|\xi|^{-1}\\}}\end{split}$ (1.5)
for all $|\xi|>0$ with some $c>0$. This assumption ensures that
$\nabla\times\Gamma[B]$ is well-defined as an $L^{2}$ function for
$B\in\dot{H}^{-1}\cap L^{2}(\mathbb{R}^{3})$.
Examples of $\Gamma$. The assumptions are clearly satisfied by
$\Gamma=\Lambda^{-a}$ (fractional powers of the negative Laplacian), with any
$1\leq a\leq 2$. It is not difficult to check that as long as $1<a<2$, one can
take less standard symbols $\Gamma=\Lambda^{-a}\log^{\alpha_{1}}(10+\Lambda)$,
$\Lambda^{-a}\log^{\alpha_{1}}(10+\Lambda)\log^{\alpha_{2}}(10+\log(10+\Lambda))$,
etc, for any $\alpha_{i}$.
We now introduce the function space $\mathcal{Y}^{1}(\mathbb{R}^{n})$ defined
by
$\mathcal{Y}^{1}(\mathbb{R}^{n})=\left\\{f\in\mathcal{D}^{\prime}(\mathbb{R}^{n})\,|\,\int_{\mathbb{R}^{n}}{(1+|\xi|)}|\hat{f}(\xi)|\,\mathrm{d}\xi=:\|f\|_{\mathcal{Y}^{1}}<+\infty\right\\},$
which is a slight variant of the $\mathcal{X}^{1}$ space used in [32, 79]. We
are now ready to state our main result.
###### Theorem A.
We have the following local well-posedness and blow-up criterion for (1.1).
* (i)
(Local well-posedness) For any $s>\frac{5}{2}$, the system (1.1) is locally
well-posed in $H^{s}(\mathbb{R}^{3})$: given a divergence-free initial data
$B_{0}\in H^{s}$, there exist $T>0$ and a unique solution $B\in
C([0,T];H^{s}(\mathbb{R}^{3}))$ with $B(t=0)=B_{0}$.
* (ii)
(Blow up criterion) Assume that $B$ is a solution to (1.1) belonging to
$C([0,T];H^{s}(\mathbb{R}^{3}))$. Then,
$\limsup_{t\to T}\|B\|_{H^{s}}=+\infty\quad\text{ if and only if
}\quad\int_{0}^{T}{\|B\|_{\mathcal{Y}^{1}}}\,\mathrm{d}t=+\infty.$
Therefore, the solution can be continued past $T$ if and only if
$\int_{0}^{T}{\|B\|_{\mathcal{Y}^{1}}}\,\mathrm{d}t<\infty$.
When the multiplier $\Gamma$ is more singular, we can prove local well-
posedness under the presence of a dissipative term. For concreteness, we focus
on the case $\Gamma=\Lambda^{-a}$ with $0\leq a<1$, whose local well-posedness
is not covered by Theorem A in the inviscid case.
###### Theorem B.
Consider the fractionally dissipative system
$\left\\{\begin{aligned}
&\partial_{t}B+\nabla\times((\nabla\times\Lambda^{-a}[B])\times
B)=-\Lambda^{b}B,\\\ &\nabla\cdot B=0.\end{aligned}\right.$ (1.6)
Then, for $0\leq a<1$ and $1-a<b$, the Cauchy problem for (1.6) is locally
well-posed in $C([0,T];H^{s}(\mathbb{R}^{3}))$ in the same sense as in Theorem
A, as long as $s>\frac{7}{2}-a$.
Sharpness of Theorems A and B. We emphasize that the local well-posedness
statements in Theorems A and B are not simple consequences of standard energy
estimates, even in the case of $\Gamma=\Lambda^{-a}$. To explain the sharpness
of Theorem A, one can note that already in the case of 3D Euler where
$\Gamma=\Lambda^{-2}$, $\nabla V$ in the right hand side of (1.2) is already
as singular as $B$, so that as soon as $\Gamma=\Lambda^{-a}$ with $a<2$,
$\nabla V$ is strictly more singular than $B$. Therefore, to close the a
priori estimate in Sobolev spaces in the range $a\in[1,2)$, a cancellation
structure and sharp commutator estimate must be used, which can be considered
as an extension of the one proved in [11] for local well-posedness of gSQG
equations in the “singular” regime. Furthermore, we believe that the threshold
$a=1$ is sharp for local well-posedness; in a recent work [65], strong ill-
posedness of the Cauchy problem for the electron–MHD system ((1.1) with
$\Gamma=1$) was established in any sufficiently regular Sobolev, Hölder and
even Gevrey spaces. From the paper [65], there are many reasons to believe
that the loss of derivative in the electron–MHD system is precisely equal to
1, suggesting ill-posedness of (1.1) in the whole range of multipliers
$\Gamma=\Lambda^{-a}$ with $0<a<1$. Formally, taking $\Gamma=\Lambda^{-a}$ and
plugging in the ansatz $j\simeq\Lambda^{-1}b^{z}$ (which is reasonable since
$b^{z}$ should be comparable with $\nabla j$), we have that
$\begin{split}\partial_{t}b^{z}+\nabla^{\perp}\Gamma b^{z}\cdot\nabla
b^{z}-\nabla^{\perp}\Lambda^{-1}b^{z}\cdot\nabla\Lambda^{1-a}b^{z}\simeq
0,\end{split}$
which is expected to be ill-posed, due to the singular term
$\nabla\Lambda^{1-a}b^{z}$. The methods developed in [65, 66, 16] should be
applicable to give ill-posedness, but the proof will be highly involved since
(1.1) is a three-dimensional system involving a non-local operator. Similarly,
we expect the restriction $1-a<b$ in Theorem B to be sharp; indeed, in the
electron–MHD case $a=0$, while Theorem B requires $b>1$, strong ill-posedness
was shown whenever $b\leq 1$ ([65]). Furthermore, it can be considered as an
extension of the local well-posedness results of [15, 21] in the case $b=2$.
Organization of the paper. Theorems A and B will be proved in Section 3 below,
after discussing a few basic properties in Section 2. Possible extensions and
open problems will be discussed in Section 4.
## 2 Basic properties of the active vector system
### 2.1 Conservation laws
We can also write (1.1) in another velocity formulation. Let us consider $u$,
which is the solution of $\nabla\times u=B$ satisfying the “gauge fixing
condition” $\nabla\cdot u=0$. We also assume $u$ decays sufficiently fast at
infinity. Then, since (1.1) is written as
$\nabla\times(\partial_{t}u+(\nabla\times u)\times\Delta\Gamma u)=0,$
there exists a scalar function $Q=Q(x,t)$ such that
$\left\\{\begin{aligned} &\partial_{t}u+(\nabla\times u)\times\Delta\Gamma
u=-\nabla Q,\\\ &\nabla\cdot u=0.\end{aligned}\right.$ (2.1)
We recall that in the case of Euler equations, $Q=p+\frac{|u|^{2}}{2}$, where
$p$ is the pressure. Comparing (2.1) with (1.1), we find the following
relation between the “two velocities”,
$V=\Delta\Gamma[u].$ (2.2)
We present the basic conservation laws for the active vector system. Among
others please note that the helicity defined below is independent of the
multiplier $\Gamma$ in (1.1).
###### Proposition 2.1 (Conservation laws).
For a sufficiently smooth and decaying solutions of (1.1), we have the
following conservations.
* (i)
(Energy) Let us define the energy:
$E(t)=\frac{1}{2}\int_{\mathbb{R}^{3}}|\Gamma^{\frac{1}{2}}B|^{2}\,\mathrm{d}x=\frac{1}{2}\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{1}{2}}\Gamma^{\frac{1}{2}}u|^{2}\,\mathrm{d}x.$
Then, $E(t)=E(0).$
* (ii)
(Helicity) We define the helicity:
$H(t)=\int_{\mathbb{R}^{3}}B\cdot\nabla\times(-\Delta)^{-1}B\,\mathrm{d}x=\int_{\mathbb{R}^{3}}B\cdot
u\,\,\mathrm{d}x.$
Then, $H(t)=H(0)$.
###### Proof.
* (i)
Taking the $L^{2}(\mathbb{R}^{3})$ inner product of (1.1) with $\Gamma B$ and
integrating by part, we have that
$\begin{split}\frac{1}{2}\frac{\mathrm{d}E(t)}{\mathrm{d}t}&=-\int_{\mathbb{R}^{3}}\nabla\times((\nabla\times\Gamma
B)\times B)\cdot\Gamma B\,\mathrm{d}x\\\
&=-\int_{\mathbb{R}^{3}}((\nabla\times\Gamma B)\times
B)\cdot(\nabla\times\Gamma B)\,\mathrm{d}x=0.\end{split}$
* (ii)
Using (1.1) and (2.1), and integrating by part we compute
$\displaystyle\frac{\mathrm{d}H(t)}{\mathrm{d}t}$
$\displaystyle=\int_{\mathbb{R}^{3}}B_{t}\cdot
u\,\,\mathrm{d}x+\int_{\mathbb{R}^{3}}B\cdot u_{t}\,\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{3}}\nabla\times\left((\nabla\times\Gamma[B])\times
B\right)\cdot u\,\,\mathrm{d}x-\int_{\mathbb{R}^{3}}B\cdot\left((\nabla\times
u)\times\Delta\Gamma[u]+\nabla Q\right)\,\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R}^{3}}\ (\nabla\times\Gamma[B])\times B\cdot
B\,\,\mathrm{d}x-\int_{\mathbb{R}^{3}}B\cdot
B\times\Delta\Gamma[u]\,\,\mathrm{d}x-\int_{\mathbb{R}^{3}}B\cdot\nabla
Q\,\mathrm{d}x$ $\displaystyle=0.$
This finishes the proof. ∎
We shall observe that the formulation (1.2), viewing $V$ as a velocity field,
is useful in the Lagrangian formulation of (1.1). Let us introduce the
particle trajectory mapping $X(\alpha,t)$ generated by $V$ defined by
$\frac{\partial X(t,\alpha)}{\partial t}=V(t,X(t,\alpha))\quad;\quad
X(0,\alpha)=\alpha.$
As an immediate consequence of (1.2) the following Cauchy formula
$B(t,X(t,\alpha))=B_{0}(\alpha)\cdot\nabla X(t,\alpha)$ (2.3)
holds. For details of the proof see e.g. [82, Lemma 1.4], which is obviously
applicable to our case. Let the initial closed curve $\mathcal{C}_{0}$ be an
integral curve of $B_{0}$, namely there exists
$\lambda(\cdot):[0,1)\to\mathbb{R}$ such that
$\mathcal{C}_{0}=\\{\eta(s)\in\mathbb{R}^{3}\,|\,s\in[0,1],\eta(0)=\eta(s),\,\eta^{\prime}(s)=\lambda(s)B_{0}(\eta(s))\\}.$
(2.4)
and define
$\mathcal{C}_{t}=X(t,\mathcal{C}_{0})=\\{X(t,\eta(s))\in\mathbb{R}^{3}\,|\,s\in[0,1],\eta(0)=\eta(s),\,\eta^{\prime}(s)=\lambda(s)B_{0}(\eta(s))\\}.$
(2.5)
Then, we claim $C_{t}$ is an integral curve of $B(\cdot,t)$, namely
$\frac{\partial X(t,\eta(s))}{\partial s}=\lambda(s)B(t,\eta(s))\quad\forall
s\in[0,1],\quad t>0.$ (2.6)
Indeed, using (2.3), we have
$\displaystyle\frac{\partial X(t,\eta(s))}{\partial s}$
$\displaystyle=\frac{d\eta(s)}{ds}\cdot\nabla
X(t,\alpha)=\lambda(s)B_{0}(\eta(s))\cdot\nabla X(t,\alpha)$
$\displaystyle=\lambda(s)B(t,\eta(s)).$
### 2.2 $2+\frac{1}{2}$-dimensional case
The system (1.1) formally satisfies translational invariance, and therefore we
may consider special solutions which are not dependent on the third
coordinate. Such a reduction is sometimes referred to as the
$2+\frac{1}{2}$–dimensional reduction ([82]). Both 3D Euler and Hall-MHD
systems have been extensively studied under this simplifying ansatz ([2, 48,
67, 65, 49, 91, 22]). From the $z$-independence and divergence-free condition,
we can write
$\begin{split}B=(-\partial_{y}j,\partial_{x}j,b^{z}).\end{split}$
For simplicity we set $G=\nabla\times\Gamma B$ and write
$G=(g^{x},g^{y},g^{z})$. From
$\begin{split}\partial_{t}B+B\cdot\nabla G-G\cdot\nabla B=0,\end{split}$
taking the $z$-component gives
$\begin{split}\partial_{t}b^{z}-G\cdot\nabla b^{z}+B\cdot\nabla
g^{z}=0.\end{split}$
We now note that
$\begin{split}B_{h}=\nabla^{\perp}j,\qquad G_{h}=-\nabla^{\perp}\Gamma
b^{z},\qquad g^{z}=\Gamma\Delta j.\end{split}$
($V_{h}$ stands for the vector consisting of the first two components of $V$.
Moreover, $\nabla^{\perp}=(-\partial_{y},\partial_{x})$.) Then, the equation
for $b^{z}$ can be written as
$\begin{split}\partial_{t}b^{z}+\nabla^{\perp}\Gamma b^{z}\cdot\nabla
b^{z}+\nabla^{\perp}j\cdot\nabla\Gamma\Delta j=0.\end{split}$
Next, from
$\begin{split}\partial_{t}b^{x}-G\cdot\nabla b^{x}+B\cdot\nabla
g^{x}=0,\end{split}$
noting that
$\begin{split}-G\cdot\nabla b^{x}+B\cdot\nabla
g^{x}&=-g^{x}\partial_{x}b^{x}-g^{y}\partial_{y}b^{x}+b^{x}\partial_{x}g^{x}+b^{y}\partial_{y}g^{x}\\\
&=\partial_{y}(g^{x}b^{y})-\partial_{y}(g^{y}b^{x})\end{split}$
(we have used the divergence-free conditions
$\partial_{x}b^{x}+\partial_{y}b^{y}=0$,
$\partial_{x}g^{x}+\partial_{y}g^{y}=0$), we have that
$\begin{split}\partial_{t}j+\nabla^{\perp}\Gamma b^{z}\cdot\nabla
j=0.\end{split}$
We have arrived at the closed system for $b^{z}$ and $j$:
$\left\\{\begin{aligned} &\partial_{t}b^{z}+\nabla^{\perp}\Gamma
b^{z}\cdot\nabla b^{z}+\nabla^{\perp}j\cdot\nabla\Gamma\Delta j=0,\\\
&\partial_{t}j+\nabla^{\perp}\Gamma b^{z}\cdot\nabla j=0.\end{aligned}\right.$
(2.7)
In particular, note that when $j_{0}=0$, then the second equation of (2.7)
implies that $j(\cdot,t)=0$ for $t>0$ as long as the smooth solution persists.
Then, we have simply
$\begin{split}\partial_{t}b^{z}+\nabla^{\perp}\Gamma b^{z}\cdot\nabla
b^{z}=0,\end{split}$
which is (gSQG). In this sense the (gSQG) systems are embedded into (1.1). In
the case of $\Gamma=\Delta^{-1}$, this is nothing but the 2D Euler equations.
## 3 Proof of the main theorem
We first prove the the following key commutator estimate.
###### Lemma 3.1.
Let $\Lambda=(-\Delta)^{\frac{1}{2}}$ and $\Gamma$ satisfy the assumptions
(1.3)–(1.5). For $g\in L^{2}(\mathbb{R}^{n})$ and ${b}\in\mathcal{Y}^{1}$, we
have
$\begin{split}\|{\mathbf{P}_{\geq
1}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},b]\nabla g}\|_{L^{2}}\leq
C\|{b}\|_{\mathcal{Y}^{1}}\|{g}\|_{L^{2}}\end{split}$ (3.1)
and
$\begin{split}\|{\Lambda^{\frac{1}{2}}\mathbf{P}_{<1}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},b]\nabla
g}\|_{L^{2}}\leq C\|{b}\|_{\mathcal{Y}^{1}}\|{g}\|_{L^{2}}.\end{split}$ (3.2)
Here, $\mathbf{P}_{\geq 1}$ is defined with Fourier multiplier
$\mathbf{1}_{|\xi|\geq 1}$ and $\mathbf{P}_{<1}=1-\mathbf{P}_{\geq 1}$.
###### Proof.
Taking the Fourier transform of
$f:=\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},b]\nabla g$, we have
$\begin{split}\widehat{f}(\xi)=|\xi|^{\frac{1}{2}}\int_{\mathbb{R}^{n}}\left(\gamma^{\frac{1}{2}}(|\xi|)-\gamma^{\frac{1}{2}}(|\eta|)\right)\widehat{b}(\xi-\eta)i\eta\widehat{g}(\eta)\,\mathrm{d}\eta.\end{split}$
We bound, using the assumptions for $\gamma$,
$\begin{split}|\gamma^{\frac{1}{2}}(|\xi|)-\gamma^{\frac{1}{2}}(|\eta|)|&=\frac{|\gamma(|\xi|)-\gamma(|\eta|)|}{\gamma^{\frac{1}{2}}(|\xi|)+\gamma^{\frac{1}{2}}(|\eta|)}{=}\frac{1}{\gamma^{\frac{1}{2}}(|\xi|)+\gamma^{\frac{1}{2}}(|\eta|)}\int_{|\xi|}^{|\eta|}-\gamma^{\prime}({\rho})\,\mathrm{d}{\rho}\\\
&\leq\int_{|\xi|}^{|\eta|}C\rho^{-1}\frac{\gamma(\rho)}{{\gamma^{\frac{1}{2}}(|\xi|)+\gamma^{\frac{1}{2}}(|\eta|)}}\,\mathrm{d}{\rho}\leq
C\int_{|\xi|}^{|\eta|}{\rho}^{-1}\gamma^{\frac{1}{2}}({\rho})\,\mathrm{d}{\rho}\\\
&\leq
C|\xi-\eta|\left(\frac{1}{(|\xi||\eta|)^{\frac{1}{2}}(|\xi|^{\frac{1}{2}}+|\eta|^{\frac{1}{2}})}+\frac{1}{|\xi||\eta|}\right)\\\
&\leq\frac{C|\xi-\eta|}{|\eta|}\left(|\xi|^{-\frac{1}{2}}+|\xi|^{-1}\right).\end{split}$
Here, we have assumed $|\xi|\leq|\eta|$, and used both of (1.4) and (1.3), but
the same inequality can be trivially deduced when $|\xi|>|\eta|$ as well.
Inserting this bound into the integral expression for $\widehat{f}(\xi)$, we
have that
$\begin{split}\left|\widehat{f}(\xi)\right|\leq
C{(1+|\xi|^{-\frac{1}{2}})}\int_{\mathbb{R}^{n}}|\xi-\eta||\widehat{b}(\xi-\eta)||\widehat{g}(\eta)|\,\mathrm{d}\eta.\end{split}$
By Young’s convolution inequality, we obtain
$\|{\mathbf{P}_{\geq 1}f}\|_{L^{2}}=C\|{\mathbf{1}_{|\xi|\geq
1}\widehat{f}}\|_{L^{2}}\leq C\|b\|_{\mathcal{Y}^{1}}\|g\|_{L^{2}}$
and
$\begin{split}\|{\Lambda^{\frac{1}{2}}\mathbf{P}_{<1}f}\|_{L^{2}}=C\|{|\xi|^{\frac{1}{2}}\mathbf{1}_{|\xi|<1}\widehat{f}}\|_{L^{2}}\leq
C\|b\|_{\mathcal{Y}^{1}}\|g\|_{L^{2}}.\end{split}$
This finishes the proof. ∎
When the operator $\Gamma$ is given precisely by $\Lambda^{-a}$, we can obtain
the following sharp inequality, with a straightforward modification of the
above proof.
###### Corollary 3.2.
For $g\in L^{2}(\mathbb{R}^{n})$ and
${b}\in\dot{\mathcal{Y}}^{1}(\mathbb{R}^{n})$, we have
$\begin{split}\|{\Lambda^{\frac{a}{2}}[\Lambda^{-\frac{a}{2}},b]\nabla
g}\|_{L^{2}}\leq
C\|{b}\|_{\dot{\mathcal{Y}}^{1}}\|{g}\|_{L^{2}},\quad\|{b}\|_{\dot{\mathcal{Y}}^{1}}:=\int_{\mathbb{R}^{n}}|\xi||\widehat{b}(\xi)|\,\mathrm{d}\xi.\end{split}$
(3.3)
###### Lemma 3.3.
The following embedding relations hold.
$H^{s}(\mathbb{R}^{n})\hookrightarrow\mathcal{Y}^{1}(\mathbb{R}^{n})\hookrightarrow\dot{W}^{1,\infty}(\mathbb{R}^{n})$
(3.4)
for $s>\frac{n}{2}+1.$
###### Proof.
For all $x\in\mathbb{R}^{n}$ the following estimates hold.
$\begin{split}|\nabla
f(x)|&\leq\int_{\mathbb{R}^{n}}|\xi||\hat{b}(\xi)|\mathrm{d}\xi{\leq}\|f\|_{\mathcal{Y}^{1}}\leq\int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{\frac{s}{2}}|\widehat{f}(\xi)|(1+|\xi|^{2})^{-\frac{s-1}{2}}\mathrm{d}\xi\\\
&\leq\left(\int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{s}|\widehat{f}(\xi)|^{2}\mathrm{d}\xi\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{-s+1}\mathrm{d}\xi\right)^{\frac{1}{2}}\leq
C\|{f}\|_{H^{s}},\end{split}$
where we used the fact
$\int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{-s+1}\,\mathrm{d}\xi<+\infty$ for
$s>\frac{n}{2}+1$. Hence, we have shown $\|\nabla
f\|_{L^{\infty}}\leq\|f\|_{\mathcal{Y}^{1}}\leq C\|f\|_{H^{s}}$, and the lemma
is proved. ∎
Lastly, we shall need a simple commutator estimate.
###### Lemma 3.4.
For two functions $f,g$ belonging to $H^{s}$, we have
$\begin{split}\|{[\Lambda^{s},f\cdot\nabla]g}\|_{L^{2}}\leq
C(\|{f}\|_{\mathcal{Y}^{1}}\|{g}\|_{H^{s}}+\|{g}\|_{\mathcal{Y}^{1}}\|{f}\|_{H^{s}}).\end{split}$
###### Proof.
Note that the Fourier transform of $H:=[\Lambda^{s},f\cdot\nabla]g$ is given
by
$\begin{split}\widehat{H}(\xi)=\int(|\xi|^{s}-|\eta|^{s})\widehat{f}(\xi-\eta)i\eta\widehat{g}(\eta)\,\mathrm{d}\eta.\end{split}$
Proceeding similarly with the proof of Lemma 3.1, we obtain that
$\begin{split}\left|\widehat{H}(\xi)\right|\leq
C\int(|\xi-\eta|^{s-1}+|\eta|^{s-1})|\xi-\eta||\widehat{f}(\xi-\eta)||\eta\widehat{g}(\eta)|\,\mathrm{d}\eta\end{split}$
which gives with Young’s convolution inequality that
$\begin{split}\|{H}\|_{L^{2}}\leq
C(\|{f}\|_{\mathcal{Y}^{1}}\|{g}\|_{H^{s}}+\|{g}\|_{\mathcal{Y}^{1}}\|{f}\|_{H^{s}}).\end{split}$
This gives the estimate. ∎
###### Proof of Theorem A.
In the proof below we shall find the following vector calculus identity
useful:
$\begin{split}\nabla\times(B\times F)=(\nabla\cdot
F+F\cdot\nabla)B-(\nabla\cdot B+B\cdot\nabla)F.\end{split}$ (3.5)
We compute
$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\Lambda^{s}B}\|_{L^{2}}^{2}&=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\Lambda^{s}\nabla\times((\nabla\times\Gamma
B)\times B)\,\mathrm{d}x\end{split}$
The main contribution on the right hand side comes from
$\begin{split}I_{1}:=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\nabla\times((\nabla\times\Gamma\Lambda^{s}B)\times
B)\,\mathrm{d}x\end{split}$
and
$\begin{split}I_{2}:=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\nabla\times((\nabla\times\Gamma
B)\times\Lambda^{s}B)\,\mathrm{d}x.\end{split}$
We first treat $I_{1}$: defining $F=\nabla\times\Lambda^{s}B$ for simplicity,
we have that
$\begin{split}I_{1}&=-\int_{\mathbb{R}^{3}}F\cdot(\Gamma F\times
B)\,\mathrm{d}x=\int_{\mathbb{R}^{3}}\epsilon_{ijk}\Gamma^{\frac{1}{2}}(F^{i}B^{k})\,\Gamma^{\frac{1}{2}}F^{j}\,\mathrm{d}x\\\
&=\int_{\mathbb{R}^{3}}\epsilon_{ijk}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}\,\Lambda^{-\frac{1}{2}}\Gamma^{\frac{1}{2}}F^{j}\,\mathrm{d}x,\end{split}$
where $\epsilon_{ijk}$ is the totally skew-symmetric tensor with the
normalization $\epsilon_{123}=1$, and we used
$\begin{split}\int_{\mathbb{R}^{3}}\epsilon_{ijk}B^{k}\Gamma^{\frac{1}{2}}(F^{i})\,\Gamma^{\frac{1}{2}}(F^{j})\,\mathrm{d}x=0\end{split}$
(repeated indices are being summed). We write
$\begin{split}\int_{\mathbb{R}^{3}}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}\,\Lambda^{-\frac{1}{2}}\Gamma^{\frac{1}{2}}F^{j}\,\mathrm{d}x&=\int_{\mathbb{R}^{3}}\mathbf{P}_{\geq
1}\left(\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}\right)\,\Lambda^{-\frac{1}{2}}\Gamma^{\frac{1}{2}}F^{j}\,\mathrm{d}x\\\
&\qquad+\int_{\mathbb{R}^{3}}\Lambda^{\frac{1}{2}}\mathbf{P}_{<1}\left(\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}\right)\,\Lambda^{-1}\Gamma^{\frac{1}{2}}F^{j}\,\mathrm{d}x\end{split}$
Applying Lemma 3.1, and using the assumption (1.3), we have that
$\begin{split}&\left|\int_{\mathbb{R}^{3}}\epsilon_{ijk}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}\,\Lambda^{-\frac{1}{2}}\Gamma^{\frac{1}{2}}F^{j}\,\mathrm{d}x\right|\\\
&\qquad\leq{C\sum_{i,j,k}\left(\|{\mathbf{P}_{\geq
1}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}}\|_{L^{2}}\|{\Lambda^{-\frac{1}{2}}\Gamma^{\frac{1}{2}}F^{j}}\|_{L^{2}}+\|{\Lambda^{\frac{1}{2}}\mathbf{P}_{<1}\Lambda^{\frac{1}{2}}[\Gamma^{\frac{1}{2}},B^{k}]F^{i}}\|_{L^{2}}\|{\Lambda^{-1}\Gamma^{\frac{1}{2}}F^{j}}\|_{L^{2}}\right)}\\\
&\qquad\leq
C{\|{B}\|_{\mathcal{Y}^{1}}}\|\Lambda^{s}B\|_{L^{2}}{\left(\|{\Lambda^{-\frac{1}{2}}\Gamma^{\frac{1}{2}}\nabla\times\Lambda^{s}B}\|_{L^{2}}+\|{\Lambda^{-1}\Gamma^{\frac{1}{2}}\nabla\times\Lambda^{s}B}\|_{L^{2}}\right)}\\\
&\qquad\leq C{\|{B}\|_{\mathcal{Y}^{1}}}\|{B}\|_{H^{s}}^{2}.\end{split}$
We now treat $I_{2}$: using (3.5),
$\begin{split}I_{2}&=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot(\Lambda^{s}B\cdot\nabla(\nabla\times\Gamma
B))\,\mathrm{d}x+\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot((\nabla\times\Gamma
B)\cdot\nabla)\Lambda^{s}B\,\mathrm{d}x\\\ &=I_{2,a}+I_{2,b}.\end{split}$
The term $I_{2,a}$ can be directly bounded by
$\begin{split}I_{2,a}\leq C\|{\nabla^{2}\Gamma
B}\|_{L^{\infty}}\|{B}\|_{H^{s}}^{2}\leq
C{\|{B}\|_{\mathcal{Y}^{1}}\|B\|_{{H^{s}}}^{2}},\end{split}$
using (1.3) and $s>\frac{5}{2}$. For the second term, we note that
$\begin{split}I_{2,b}=\int_{\mathbb{R}^{3}}((\nabla\times\Gamma
B)\cdot\nabla)\frac{|\Lambda^{s}B|^{2}}{2}\,\mathrm{d}x=-\int_{\mathbb{R}^{3}}(\nabla\cdot(\nabla\times\Gamma
B))\frac{|\Lambda^{s}B|^{2}}{2}\,\mathrm{d}x=0.\end{split}$
Finally, we treat the remainder. Using the identity (3.5), we observe that
$\begin{split}&I_{1}+I_{2}+\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\Lambda^{s}\nabla\times((\nabla\times\Gamma
B)\times B)\,\mathrm{d}x\\\
&\quad=\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\left\\{\Lambda^{s}(\nabla\times(G\times
B))-\nabla\times(\Lambda^{s}G\times
B)-\nabla\times(G\times\Lambda^{s}B)\right\\}\,\mathrm{d}x\\\
&\quad=\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\left\\{\Lambda^{s}(B\cdot\nabla
G-G\cdot\nabla B)-B\cdot\nabla\Lambda^{s}G+\Lambda^{s}G\cdot\nabla
B-\Lambda^{s}B\cdot\nabla G+G\cdot\nabla\Lambda^{s}B\right\\}dx\\\
&\quad=\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\\{[\Lambda^{s},B\cdot\nabla]G\\}dx-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\\{[\Lambda^{s},G\cdot\nabla]B\\}dx\\\
&\qquad\qquad-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\\{(\Lambda^{s}B\cdot\nabla)G\\}dx+\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\\{\Lambda^{s}G\cdot\nabla
B\\}dx\\\ &\quad:=K_{1}+K_{2}+K_{3}+K_{4},\end{split}$ (3.6)
where we set $G:=\nabla\times\Gamma B$. We note that
$\begin{split}\|{\Lambda^{s}G}\|_{L^{2}}\leq C\|{B}\|_{H^{s}},\qquad\|{\nabla
G}\|_{L^{\infty}}\leq C\|{B}\|_{{\mathcal{Y}}^{1}}.\end{split}$
To see the first inequality,
$\begin{split}\|{\Lambda^{s}G}\|_{L^{2}}^{2}&\leq
C\int|\xi|^{2s}|\xi\times\gamma(|\xi|)\hat{B}(\xi)|^{2}\mathrm{d}\xi\\\ &\leq
C\int|\xi|^{2s+2}(|\xi|^{-1}+|\xi|^{-2})^{2}|\hat{B}(\xi)|^{2}\mathrm{d}\xi\leq
C\|{B}\|_{H^{s}}^{2},\end{split}$
using the assumption (1.3) for $\gamma$. The proof of the other inequality is
similar.
Then, we can bound $K_{4}$ by
$\left|K_{4}\right|\leq
C\|{\Lambda^{s}B}\|_{L^{2}}\|{\Lambda^{s}G}\|_{L^{2}}\|{\nabla
B}\|_{L^{\infty}}\leq C\|B\|_{\dot{\mathcal{Y}}^{1}}\|{B}\|_{H^{s}}^{2}.$
Next, we estimate $K_{3}$ easily as follows:
$\begin{split}\left|K_{3}\right|\leq\int_{\mathbb{R}^{3}}|\Lambda^{s}B\cdot\\{\Lambda^{s}B\cdot\nabla
G\\}|\,\mathrm{d}x\leq C\|{\nabla G}\|_{L^{\infty}}\|{B}\|_{H^{s}}^{2}\leq
C\|B\|_{\dot{\mathcal{Y}}^{1}}\|{B}\|_{H^{s}}^{2}.\end{split}$
Lastly, using Lemma 3.4, we can directly estimate the first two terms by
$\begin{split}\left|K_{1}+K_{2}\right|&\leq\|\Lambda^{s}B\|_{L^{2}}\left(\|{[\Lambda^{s},B\cdot\nabla]G}\|_{L^{2}}+\|{[\Lambda^{s},G\cdot\nabla]B}\|_{L^{2}}\right)\\\
&\leq
C{(\|B\|_{\mathcal{Y}^{1}}+\|G\|_{\mathcal{Y}^{1}})}(\|{B}\|_{H^{s}}+\|{G}\|_{H^{s}})\|\Lambda^{s}B\|_{L^{2}}\\\
&\leq C\|B\|_{\mathcal{Y}^{1}}\|B\|_{H^{s}}^{2}.\end{split}$
This finishes the proof of
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{B}\|_{{\dot{H}^{s}}}\leq
C\|B\|_{\mathcal{Y}^{1}}\|{B}\|_{H^{s}}.\end{split}$
On the other hand, from (1.2), it is straightforward to obtain
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{B}\|_{L^{2}}\leq
C\|{\nabla^{2}\Gamma[B]}\|_{L^{\infty}}\|{B}\|_{L^{2}}\leq
C\|B\|_{\mathcal{Y}^{1}}\|{B}\|_{H^{s}}.\end{split}$
Combining the above estimates together with (3.4), we have
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{B}\|_{H^{s}}\leq
C\|{B}\|_{H^{s}}^{2}.\end{split}$
Hence, there exists $T>0$ depending only on $\|{B_{0}}\|_{H^{s}}$, such that
$\begin{split}\sup_{t\in[0,T]}\|{B(t,\cdot)}\|_{H^{s}}\leq
2\|{B_{0}}\|_{H^{s}}.\end{split}$ (3.7)
Given the a priori estimate, it is a simple matter to prove existence and
uniqueness of a solution belonging to $L^{\infty}([0,T];H^{s})$. For
uniqueness, one may assume that there are two solutions $B$ and $\tilde{B}$ to
(1.1) with the same initial data $B_{0}$, and that $B,\tilde{B}\in
L^{\infty}([0,T];H^{s})$ for some $T>0$. Proceeding similarly as in the proof
of (3.7), using the equation satisfied by the difference $B-\tilde{B}$, one
can prove
$\begin{split}\frac{d}{dt}\|{B-\tilde{B}}\|_{L^{2}}\leq
C(\|{B}\|_{L^{\infty}([0,T];H^{s})}+\|{\tilde{B}}\|_{L^{\infty}([0,T];H^{s})})\|{B-\tilde{B}}\|_{L^{2}}.\end{split}$
(See [17, 27] for details of this argument for closely related systems.) For
existence, we consider the regularized systems
$\left\\{\begin{aligned}
&\partial_{t}B+\nabla\times((\nabla\times\Gamma[B])\times B)-\nu\Delta B=0,\\\
&\nabla\cdot B=0,\\\ &B(t=0)=B_{0},\end{aligned}\right.$ (3.8)
for each $\nu>0$. Given $B_{0}\in H^{s}$, it is straightforward to show local
existence of a solution $B^{(\nu)}$ to (3.8) satisfying $B^{(\nu)}\in
L^{\infty}([0,T_{\nu}];H^{s})$ for some $T_{\nu}>0$, using the mild
formulation; one can follow the arguments of [15]. Since the a priori estimate
(3.7) applies to the viscous solutions $B^{(\nu)}$, the lifespan $T_{\nu}$ can
be bounded from below by some $T>0$ independent of $\nu>0$. There exists a
weak limit $B^{(\nu)}\to B$ in $L^{\infty}([0,T];H^{s})$ (by taking a sub-
sequence if necessary), and the limit is a solution to (1.1).
Lastly, we argue that the solution $B$ actually belongs to $C([0,T];H^{s})$.
Using time-translation and time-reversal symmetries, it suffices to prove
strong convergence $B(t)\to B_{0}$ for $t\to 0^{+}$. To begin with, from
uniform boundedness, we obtain weak convergence; this gives that
$\begin{split}\|{B_{0}}\|_{H^{s}}\leq\liminf_{t\to
0^{+}}\|{B(t)}\|_{H^{s}}.\end{split}$
On the other hand, from (3), we see that
$\begin{split}\|{B_{0}}\|_{H^{s}}-Ct\leq\|{B(t)}\|_{{H^{s}}}\leq\|{B_{0}}\|_{H^{s}}+Ct,\quad\forall
t\in[0,T_{0}]\end{split}$
for some $T_{0},C>0$ depending only on $\|{B_{0}}\|_{H^{s}}$. This gives norm
convergence $\|{B(t)}\|_{H^{s}}\to\|{B_{0}}\|_{H^{s}}$, which together with
weak convergence concludes strong convergence. The proof of (i) is now
complete.
The proof of (ii) immediately follows from the inequality
$\|B(t)\|_{H^{s}}\leq\|B_{0}\|_{H^{s}}e^{C\int_{0}^{t}\|B(s)\|_{\mathcal{Y}^{1}}ds},$
which is obtained from (3). ∎
###### Proof of Theorem B.
We shall simply establish an $H^{s}$–a priori estimate for (1.6), and omit the
proof of existence and uniqueness. Similarly as in the proof of Theorem A, we
compute
$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\Lambda^{s}B}\|_{L^{2}}^{2}+\|{\Lambda^{s+\frac{b}{2}}B}\|_{L^{2}}^{2}&=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\Lambda^{s}\nabla\times((\nabla\times\Lambda^{-a}B)\times
B)\,\mathrm{d}x.\end{split}$
The main terms on the right hand side are given by
$\begin{split}I_{1}:=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\nabla\times((\nabla\times\Lambda^{-a}\Lambda^{s}B)\times
B)\,\mathrm{d}x\end{split}$
and
$\begin{split}I_{2}:=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\nabla\times((\nabla\times\Lambda^{-a}B)\times\Lambda^{s}B)\,\mathrm{d}x.\end{split}$
With $F:=\nabla\times\Lambda^{s}B$, recall that
$\begin{split}I_{1}=\sum_{i,j,k}\int_{\mathbb{R}^{3}}\epsilon_{ijk}[\Lambda^{-\frac{a}{2}},B^{k}]F^{i}\,\Lambda^{-\frac{a}{2}}F^{j}\,\mathrm{d}x=\sum_{i,j,k}\int_{\mathbb{R}^{3}}\epsilon_{ijk}\Lambda^{\frac{1}{2}}[\Lambda^{-\frac{a}{2}},B^{k}]F^{i}\,\Lambda^{-\frac{a+1}{2}}F^{j}\,\mathrm{d}x,\end{split}$
which can be bounded by
$\begin{split}\left|I_{1}\right|\leq
C\|{\Lambda^{\frac{1}{2}}[\Lambda^{-\frac{a}{2}},B^{k}]\Lambda^{\frac{a+1}{2}}}\|_{L^{2}\to
L^{2}}\|{\Lambda^{-\frac{a+1}{2}}F^{j}}\|^{2}_{L^{2}}.\end{split}$
To bound the operator norm of the commutator, observe that
$\begin{split}\Lambda^{\frac{1}{2}}[\Lambda^{-\frac{a}{2}},B^{k}]\Lambda^{\frac{a+1}{2}}=\Lambda^{\frac{1-a}{2}}[B,\Lambda^{\frac{1+a}{2}}]-\Lambda^{\frac{1}{2}}[B,\Lambda^{\frac{1}{2}}]\end{split}$
and
$\begin{split}\|{\Lambda^{\frac{1-a}{2}}[B,\Lambda^{\frac{1+a}{2}}]}\|_{L^{2}\to
L^{2}}+\|{\Lambda^{\frac{1}{2}}[B,\Lambda^{\frac{1}{2}}]}\|_{L^{2}\to
L^{2}}\leq C\|{B}\|_{\dot{\mathcal{Y}}^{1}}.\end{split}$
This estimate can be proved easily using the Fourier transform and following
the arguments of Lemma 3.1 above. Therefore,
$\begin{split}\left|I_{1}\right|\leq
C\|{B}\|_{\dot{\mathcal{Y}}^{1}}\|{\Lambda^{s+\frac{1-a}{2}}B}\|_{L^{2}}^{2}\leq
C\|{B}\|_{H^{s}}\|{\Lambda^{s+\frac{1-a}{2}}B}\|_{L^{2}}^{2}.\end{split}$
Next, recall that with a cancellation, we have simply
$\begin{split}I_{2}&=-\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot(\Lambda^{s}B\cdot\nabla(\nabla\times\Lambda^{-a}B))\,\mathrm{d}x.\end{split}$
Then, it is not difficult to obtain the bound
$\begin{split}\left|I_{2}\right|\leq
C\|{\nabla^{2}\Lambda^{-a}B}\|_{L^{\infty}}\|{B}\|_{\dot{H}^{s}}^{2}\leq
C\|{B}\|_{H^{s}}^{3},\end{split}$
where we have used that $s>\frac{3}{2}+2-a$. Finally, proceeding in the same
way as (3.6), it is not difficult to prove the following estimate for the
remainder:
$\begin{split}\left|I_{1}+I_{2}+\int_{\mathbb{R}^{3}}\Lambda^{s}B\cdot\Lambda^{s}\nabla\times((\nabla\times\Lambda^{-a}B)\times
B)\,\mathrm{d}x\right|\leq C\|{B}\|_{H^{s}}^{3}.\end{split}$
Therefore, we conclude that
$\begin{split}\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|{\Lambda^{s}B}\|_{L^{2}}^{2}+\|{\Lambda^{s+\frac{b}{2}}B}\|_{L^{2}}^{2}&\leq
C\|{B}\|_{{H}^{s}}\|{\Lambda^{s+\frac{1-a}{2}}B}\|_{L^{2}}^{2}+C\|{B}\|_{H^{s}}^{3}\leq
C\|{B}\|_{{H}^{s}}^{1+2\alpha}\|{\Lambda^{s+\frac{b}{2}}B}\|_{L^{2}}^{2(1-\alpha)}+C\|{B}\|_{H^{s}}^{3}\end{split}$
for some $0<\alpha<1$ since $1-a<b$. Using Young’s inequality, we can absorb
the $\|{\Lambda^{s+\frac{b}{2}}B}\|_{L^{2}}$–term to the left hand side.
Together with an $L^{2}$ estimate for $B$, this finishes the proof of the a
priori estimate in $H^{s}$. ∎
## 4 Discussion and open problems
Extensions. The proof of local well-posedness can be adapted to three-
dimensional domains without boundary $\mathbb{R}^{k}\times\mathbb{T}^{3-k}$
for any $0\leq k<3$. Moreover, one may consider the systems with a
(fractional) dissipation term. It could be interesting to consider the
critically dissipative case, as an analogy with critically dissipative SQG
equations studied for instance in [5, 38, 37, 71].
Propagation of $\dot{H}^{-1}$. Given the formulation of the system in terms of
$u$ and scaling of the conservation law, it is natural to add the assumption
that $B\in\dot{H}^{-1}$. Indeed, $\|{B}\|_{\dot{H}^{-1}}$ is equivalent with
$\|{u}\|_{L^{2}}$, and it is easy to see using the equation for $u$ that as
long as $B\in L^{\infty}_{t}H^{s}$ with $s>\frac{5}{2}$,
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{B}\|_{\dot{H}^{-1}}\lesssim\|{B}\|_{\dot{H}^{-1}}.\end{split}$
That is, the assumption $B_{0}\in\dot{H}^{-1}$ propagates in time as long as
the solution remains smooth.
Blow-up criteria. We remark that the blow-up criterion can be replaced with
$\begin{split}\lim_{t\to
T}\int_{0}^{t}\|{B(\tau,\cdot)}\|_{H^{\frac{5}{2}}}\,\mathrm{d}\tau=+\infty.\end{split}$
(4.1)
To see this, we observe the logarithmic Sobolev estimate for $s>\frac{5}{2}$
$\begin{split}\|{B(\tau,\cdot)}\|_{\mathcal{Y}^{1}}\leq
C\|{B(\tau,\cdot)}\|_{H^{\frac{5}{2}}}\log(10+\|{B(\tau,\cdot)}\|_{H^{s}}).\end{split}$
(4.2)
Then, one may obtain the a priori estimate
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\|{B}\|_{H^{s}}\leq
C\|{B}\|_{H^{\frac{5}{2}}}\|{B}\|_{H^{s}}\log(10+\|{B}\|_{H^{s}}),\end{split}$
from which the blow-up criterion (4.1) follows by applying the Gronwall lemma
to the differential inequality
$y^{\prime}\leq ay\log y,\quad y=10+\|B(t)\|_{H^{s}},\quad
a=C\|B(t)\|_{H^{\frac{5}{2}}}.$
The inequality (4.2), in turn, can be verified as follows. We first estimate
$\displaystyle\int_{\mathbb{R}^{3}}|\xi||\hat{f}|\mathrm{d}\xi$
$\displaystyle\leq\int_{\\{|\xi|<R\\}}(1+|\xi|^{2})^{\frac{5}{4}}|\hat{f}|(1+|\xi|^{2})^{-\frac{3}{4}}\mathrm{d}\xi+\int_{\\{|\xi|\geq
R\\}}(1+|\xi|^{2})^{\frac{s}{2}}|\hat{f}|(1+|\xi|^{2})^{\frac{1-s}{2}}\mathrm{d}\xi$
$\displaystyle\leq\left(\int_{\\{|\xi|<R\\}}(1+|\xi|^{2})^{\frac{5}{2}}|\hat{f}|^{2}\mathrm{d}\xi\right)^{\frac{1}{2}}\left(\int_{\\{|\xi|<R\\}}(1+|\xi|^{2})^{-\frac{3}{2}}\mathrm{d}\xi\right)^{\frac{1}{2}}$
$\displaystyle\qquad+\left(\int_{\\{|\xi|\geq
R\\}}(1+|\xi|^{2})^{s}|\hat{f}|^{2}\mathrm{d}\xi\right)^{\frac{1}{2}}\left(\int_{\\{|\xi|\geq
R\\}}(1+|\xi|^{2})^{1-s}\mathrm{d}\xi\right)^{\frac{1}{2}}$ $\displaystyle\leq
C\|f\|_{H^{\frac{5}{2}}}\log(1+R)+C\|f\|_{H^{s}}R^{\frac{5}{2}-s}\quad\forall
R>0.$
Choosing $R=(\|f\|_{H^{s}}/\|f\|_{\frac{5}{2}})^{H^{\frac{2}{2s-5}}}$, we
obtain (4.2).
Issue of global well-posedness. Global well-posedness of smooth solutions in
the case of 3D Euler is a notoriously difficult open problem. In the
electron–MHD case, finite time singularity formation is available for smooth
and axisymmetric data ([21]) but unfortunately local well-posedness (more
precisely, uniqueness) is not known for the data used in the proof. It should
be interesting to investigate the possibility of singularity formation for the
interpolating models (1.1). Of course, when the multiplier $\Gamma$ becomes
“very regular”, e.g. $\Gamma=(1-\Delta)^{-N}$ with $N$ large, then global
well-posedness for smooth and decaying solutions can be proved using the
conservation of energy.
#### Acknowledgments
D. Chae was supported partially by NRF grant 2021R1A2C1003234. I.-J. Jeong was
supported by the NRF grant 2022R1C1C1011051. We sincerely thank the anonymous
referees for providing several helpful suggestions and pointing out errors as
well as relevant references.
#### Data availability statement
Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.
## References
* [1] Marion Acheritogaray, Pierre Degond, Amic Frouvelle, and Jian-Guo Liu, _Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system_ , Kinetic & Related Models, 2011, 4 (4) : 901–918. doi: 10.3934/krm.2011.4.901.
* [2] Claude Bardos and Edriss S. Titi, _Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations_, Discrete Contin. Dyn. Syst. Ser. S 3 (2010), no. 2, 185–197. MR 2610558
* [3] Tristan Buckmaster, Steve Shkoller, and Vlad Vicol, _Nonuniqueness of weak solutions to the SQG equation_ , Comm. Pure Appl. Math. 72 (2019), no. 9, 1809–1874. MR 3987721
* [4] D. Biskamp, E. Schwarz, and A. Zeiler, _Electron magnetohydrodynamic turbulence_ , Physics of Plasmas 6, 751 (1999); https://doi.org/10.1063/1.873312
* [5] Luis A. Caffarelli and Alexis Vasseur, _Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation_ , Ann. of Math. (2) 171 (2010), no. 3, 1903–1930. MR 2680400
* [6] Daomin Cao, Guolin Qin, Weicheng Zhan, and Changjun Zou, _On the global classical solutions for the generalized SQG equation_ , J. Funct. Anal. 283 (2022), no. 2, Paper No. 109503, 37 pp. MR 4406410
* [7] , _Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation_ , arXiv preprint.
* [8] Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, _Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations_ , Duke Math. J. 165 (2016), no. 5, 935–984. MR 3482335
* [9] Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, _Uniformly rotating analytic global patch solutions for active scalars_ , Ann. PDE 2 (2016), no. 1, Art. 1, 34. MR 3462104
* [10] Angel Castro, Diego Córdoba, Javier Gómez-Serrano, and Alberto Martín Zamora, _Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches_, Discrete Contin. Dyn. Syst. 34 (2014), no. 12, 5045–5059. MR 3223861
* [11] Dongho Chae, Peter Constantin, Diego Córdoba, Francisco Gancedo, and Jiahong Wu, _Generalized surface quasi-geostrophic equations with singular velocities_ , Comm. Pure Appl. Math. 65 (2012), no. 8, 1037–1066. MR 2928091
* [12] Dongho Chae, Peter Constantin, and Jiahong Wu, _Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations_ , Arch. Ration. Mech. Anal. 202 (2011), no. 1, 35–62. MR 2835862
* [13] , _Deformation and symmetry in the inviscid SQG and the 3D Euler equations_ , J. Nonlinear Sci. 22 (2012), no. 5, 665–688. MR 2982049
* [14] , _An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations_ , J. Math. Fluid Mech. 16 (2014), no. 3, 473–480. MR 3247363
* [15] Dongho Chae, Pierre Degond, and Jian-Guo Liu, _Well-posedness for Hall-magnetohydrodynamics_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, 555–565. MR 3208454
* [16] Dongho Chae, In-Jee Jeong, and Sung-Jin Oh, _Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities_ , in preparation.
* [17] , _Well-posedness for logarithmically singular surface quasi-geostrophic equations_ , in preparation.
* [18] D. Chae and J. Lee. _On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics_ , J. Differential Equations 256 (2014), no. 11, 3835–3858.
* [19] D. Chae and M. Schonbek. _On the temporal decay for the Hall-magnetohydrodynamic equations_ , J. Differential Equations 255 (2013) 3971–3982.
* [20] D. Chae, R. Wan, J. Wu. _Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion_ , J. Math. Fluid Mech. 17 (2015), no. 4, 627–638.
* [21] Dongho Chae and Shangkun Weng, _Singularity formation for the incompressible Hall-MHD equations without resistivity_ , Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 4, 1009–1022. MR 3519529
* [22] Dongho Chae and Jörg Wolf, _On partial regularity for the 3D nonstationary Hall magnetohydrodynamics equations on the plane_ , SIAM J. Math. Anal. 48 (2016), no. 1, 443–469. MR 3455137
* [23] Jiajie Chen and Thomas Y. Hou. _Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ velocity and boundary._ Comm. Math. Phys., 383(3):1559–1667, 2021.
* [24] X. Cheng, H. Kwon, and D. Li, _Non-uniqueness of steady-state weak solutions to the surface quasi-geostrophic equations_ , arXiv:2007.09591 (2021).
* [25] Kyudong Choi and In-Jee Jeong. _Filamentation near Hill’s vortex_ , Communications in Partial Differential Equations (2022), DOI: 10.1080/03605302.2022.2139721.
* [26] Kyudong Choi and In-Jee Jeong. _On vortex stretching for anti-parallel axisymmetric flows._ arXiv:2110.09079.
* [27] Young-Pil Choi and In-Jee Jeong, _Classical solutions for fractional porous medium flow_ , Nonlinear Analysis 210 (2021), 112393.
* [28] Peter Constantin, Analysis of Hydrodynamic Models, CBMS-NSF Regional Conference Series in Applied Mathematics, 90, (2017).
* [29] Peter Constantin, _An Eulerian-Lagrangian approach for incompressible fluids: local theory_ , J. Amer. Math. Soc. 14 (2001), no. 2, 263–278.
* [30] Peter Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1994), 73–98.
* [31] Peter Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc., 44 (2007), no. 4, 603–621.
* [32] Peter Constantin, Diego Córdoba, Francisco Gancedo, and Robert M. Strain, _On the global existence for the Muskat problem_ , J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 201–227. MR 2998834
* [33] Peter Constantin, Charles Fefferman and Andrew J. Majda, Geometric constraints on potential singularity formulation in the 3-D Euler equations, Comm. PDE, 21, (1996) no. 3-4, 559–571.
* [34] Peter Constantin, Andrew J. Majda, and Esteban Tabak, _Formation of strong fronts in the $2$-D quasigeostrophic thermal active scalar_, Nonlinearity 7 (1994), no. 6, 1495–1533. MR 1304437
* [35] Peter Constantin, Andrew J. Majda, and Esteban G. Tabak, _Singular front formation in a model for quasigeostrophic flow_ , Phys. Fluids 6 (1994), no. 1, 9–11. MR 1252829
* [36] Peter Constantin and Huy Quang Nguyen, _Global weak solutions for SQG in bounded domains_ , Comm. Pure Appl. Math. 71 (2018), no. 11, 2323–2333. MR 3862092
* [37] Peter Constantin, Andrei Tarfulea, and Vlad Vicol, _Long time dynamics of forced critical SQG_ , Comm. Math. Phys. 335 (2015), no. 1, 93–141. MR 3314501
* [38] Peter Constantin and Vlad Vicol, _Nonlinear maximum principles for dissipative linear nonlocal operators and applications_ , Geom. Funct. Anal. 22 (2012), no. 5, 1289–1321. MR 2989434
* [39] Antonio Córdoba, Diego Córdoba, and Francisco Gancedo, _Uniqueness for SQG patch solutions_ , Trans. Amer. Math. Soc. Ser. B 5 (2018), 1–31. MR 3748149
* [40] Diego Córdoba, Javier Gómez-Serrano, and Alexandru D. Ionescu, _Global solutions for the generalized SQG patch equation_ , Arch. Ration. Mech. Anal. 233 (2019), no. 3, 1211–1251. MR 3961297
* [41] Francisco de la Hoz, Zineb Hassainia, and Taoufik Hmidi, _Doubly connected V-states for the generalized surface quasi-geostrophic equations_ , Arch. Ration. Mech. Anal. 220 (2016), no. 3, 1209–1281. MR 3466846
* [42] M. Dai. _Local well-posedness of the Hall-MHD system in $H^{s}(\mathbb{R}^{n})$ with $s>n/2$_, Math. Nachr. 293 (2020), no. 1, 67–78.
* [43] M. Dai. _Local well-posedness for the Hall-MHD system in optimal Sobolev spaces_ , J. Differential Equations, 289 (2021), 159–181.
* [44] R. Danchin, J. Tan. _On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces_ , Comm. Partial Differential Equations 46 (2021), no. 1, 31–65.
* [45] R. Danchin, J. Tan. _The Global Solvability Of The Hall-magnetohydrodynamics System In Critical Sobolev Spaces._ https://doi.org/10.48550/arXiv.1912.09194
* [46] Sergey A. Denisov, _Infinite superlinear growth of the gradient for the two-dimensional Euler equation_ , Discrete Contin. Dyn. Syst. 23 (2009), no. 3, 755–764. MR 2461825
* [47] , _The centrally symmetric $V$-states for active scalar equations. Two-dimensional Euler with cut-off_, Comm. Math. Phys. 337 (2015), no. 2, 955–1009. MR 3339167
* [48] Ronald J. DiPerna and Andrew J. Majda, _Oscillations and concentrations in weak solutions of the incompressible fluid equations_ , Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643
* [49] Baoying Du, _Global regularity for the $2\frac{1}{2}$D incompressible Hall-MHD system with partial dissipation_, J. Math. Anal. Appl. 484 (2020), no. 1, 123701, 27. MR 4042622
* [50] Tarek Mohamed Elgindi, _Osgood’s lemma and some results for the slightly supercritical 2D Euler equations for incompressible flow_ , Arch. Ration. Mech. Anal. 211 (2014), no. 3, 965–990. MR 3158812
* [51] Tarek M. Elgindi. _Finite-time singularity formation for ${C}^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^{3}$_, Ann. of Math. (2) 194 (2021), no. 3, 647–727. MR 4334974
* [52] Tarek M. Elgindi and In-Jee Jeong. _Finite-time singularity formation for strong solutions to the axi-symmetric 3D Euler equations._ Ann. PDE, 5(2):Paper No. 16, 51, 2019.
* [53] Tarek M. Elgindi and In-Jee Jeong. _The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain._ Adv. Math., 393:Paper No. 108091, 63, 2021.
* [54] Tarek Mohamed Elgindi and In-Jee Jeong, _Symmetries and Critical Phenomena in Fluids_ , Comm. Pure Appl. Math. 73 (2020), no. 2, 257–316.
* [55] , _On Singular Vortex Patches, I: Well-posedness Issues_ , Mem. Amer. Math. Soc., to appear.
* [56] , _On Singular Vortex Patches, II: Long-time dynamics_ , Trans. Amer. Math. Soc. 373 (2020), no. 9, 6757–6775.
* [57] Susan Friedlander and Vlad Vicol, _On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations_ , Nonlinearity 24 (2011), no. 11, 3019–3042. MR 2844825
* [58] Francisco Gancedo, _Existence for the $\alpha$-patch model and the QG sharp front in Sobolev spaces_, Adv. Math. 217 (2008), no. 6, 2569–2598. MR 2397460
* [59] Javier Gómez-Serrano, _On the existence of stationary patches_ , Adv. Math. 343 (2019), 110–140. MR 3880855
* [60] Siming He and Alexander Kiselev, _Small scale creation for solutions of the SQG equation_ , Duke Math. J. 170 (2021), no. 5, 1027–1041. MR 4255049
* [61] Taoufik Hmidi and Joan Mateu, _Existence of corotating and counter-rotating vortex pairs for active scalar equations_ , Comm. Math. Phys. 350 (2017), no. 2, 699–747. MR 3607460
* [62] John K. Hunter, Jingyang Shu, and Qingtian Zhang, _Contour dynamics for surface quasi-geostrophic fronts_ , Nonlinearity 33 (2020), no. 9, 4699–4714. MR 4135092
* [63] , _Two-front solutions of the SQG equation and its generalizations_ , Commun. Math. Sci. 18 (2020), no. 6, 1685–1741. MR 4176355
* [64] Philip Isett and Vlad Vicol, _Hölder continuous solutions of active scalar equations_ , Ann. PDE 1 (2015), no. 1, Art. 2, 77. MR 3479065
* [65] In-Jee Jeong and Sung-Jin Oh, _On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: illposedness near degenerate stationary solutions_ , Ann. PDE 8 (2022), 15. MR 4456288
* [66] , _Strong ill-posedness for degenerate dispersive equations_ , in preparation.
* [67] In-Jee Jeong and Tsuyoshi Yoneda, _Vortex stretching and a modified zeroth law for the incompressible 3D Navier–Stokes equations_ , Math. Ann., (2021), 380(3), 2041–2072.
* [68] T. Kato and G. Ponce, _Commutator estimates and the Euler and Navier-Stokes equations,_ Commun. Pure Appl. Math. Vol. XLI, (1988), 891–907. MR 0951744
* [69] Calvin Khor and José L. Rodrigo, _Local existence of analytic sharp fronts for singular SQG_ , Nonlinear Anal. 202 (2021), 112116, 22. MR 4156974
* [70] , _On sharp fronts and almost-sharp fronts for singular SQG_ , J. Differential Equations 278 (2021), 111–145. MR 4199394
* [71] A. Kiselev, F. Nazarov, and A. Volberg, _Global well-posedness for the critical 2D dissipative quasi-geostrophic equation_ , Invent. Math. 167 (2007), no. 3, 445–453. MR 2276260
* [72] Alexander Kiselev, _Small scales and singularity formation in fluid dynamics_ , Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 2363–2390. MR 3966854
* [73] Alexander Kiselev and Fedor Nazarov, _A simple energy pump for the surface quasi-geostrophic equation_ , Nonlinear partial differential equations, Abel Symp., vol. 7, Springer, Heidelberg, 2012, pp. 175–179. MR 3289364
* [74] Alexander Kiselev, Lenya Ryzhik, Yao Yao, and Andrej Zlatoš, _Finite time singularity for the modified SQG patch equation_ , Ann. of Math. (2) 184 (2016), no. 3, 909–948. MR 3549626
* [75] Alexander Kiselev and Vladimir Šverák, _Small scale creation for solutions of the incompressible two-dimensional Euler equation_ , Ann. of Math. (2) 180 (2014), no. 3, 1205–1220. MR 3245016
* [76] Alexander Kiselev, Yao Yao, and Andrej Zlatoš, _Local regularity for the modified SQG patch equation_ , Comm. Pure Appl. Math. 70 (2017), no. 7, 1253–1315. MR 3666567
* [77] Alexander A. Kiselev, _Small scale creation in active scalars_ , Progress in mathematical fluid dynamics, Lecture Notes in Math., vol. 2272, Springer, Cham, [2020] ©2020, pp. 125–161. MR 4176667
* [78] Hyunju Kwon, _Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev space_ , J. Funct. Anal. 280 (2021), no. 7, 108822. MR 4211020
* [79] Zhen Lei and Fanghua Lin, _Global mild solutions of Navier-Stokes equations,_ Comm. Pure Appl. Math. 64, (2011), no. 9, 1297–1304. MR 2839302
* [80] Dong Li, _Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions_ , Nonlinearity 22 (2009), no. 7, 1639–1651. MR 2519682
* [81] M. J. Lighthill, _Studies on magneto-hydrodynamic waves and other anisotropic wave motions_ , Philos. Trans. Roy. Soc. London Ser. A 252 (1960), 397–430. MR 0148337
* [82] Andrew J. Majda and Andrea L. Bertozzi, _Vorticity and incompressible flow_ , Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
* [83] Fabien Marchand, _Existence and regularity of weak solutions to the quasi-geostrophic equations in the spaces $L^{p}$ or $\dot{H}^{-1/2}$_, Comm. Math. Phys. 277 (2008), no. 1, 45–67. MR 2357424
* [84] Huy Quang Nguyen, _Global weak solutions for generalized SQG in bounded domains_ , Anal. PDE 11 (2018), no. 4, 1029–1047. MR 3749375
* [85] Koji Ohkitani, _Growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids_ , Phys. Rev. E (3) 83 (2011), no. 3, 036317, 8. MR 2788267
* [86] , _Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations_ , Physics of Fluids 24 (2012), no. 095101.
* [87] Hans Pécseli, _Waves and oscillations in plasmas_ , Series in Plasma Physics (2012).
* [88] R. Shvydkoy, _Convex integration for a class of active scalar equations_ , J. Amer. Math. Soc. 24 (2011), no. 4, 1159–1174. MR 2813340
* [89] Henggeng Wang and Houyu Jia, _Local well-posedness for the 2D non-dissipative quasi-geostrophic equation in Besov spaces_ , Nonlinear Anal. 70 (2009), no. 11, 3791–3798. MR 2515299
* [90] Xiaoqian Xu, _Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow_ , J. Math. Anal. Appl. 439 (2016), no. 2, 594–607. MR 3475939
* [91] Kazuo Yamazaki, _Irreducibility of the three, and two and a half dimensional Hall-magnetohydrodynamics system_ , Phys. D 401 (2020), 132199, 21. MR 4034672
* [92] Andrej Zlatoš, _Exponential growth of the vorticity gradient for the Euler equation on the torus_ , Adv. Math. 268 (2015), 396–403. MR 3276599
|
11institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
22institutetext: Institut Universitaire de France, Paris, France
33institutetext: Aix Marseille Univ, CNRS, LIS, Marseille, France
44institutetext: INAF-IAPS, via del Fosso del Cavaliere 100, I-00133 Roma,
Italy 55institutetext: INAF – Astronomical Observatory of Capodimonte, via
Moiariello 16, I-80131, Napoli, Italy 66institutetext: Division of Science,
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
# Supervised machine learning on Galactic filaments
Revealing the filamentary structure of the Galactic interstellar medium
A. Zavagno 1122 F.-X. Dupé 33 S. Bensaid 1133 E. Schisano 44 G. Li Causi 44
M. Gray 11 S. Molinari 44 D. Elia 44 J.-C. Lambert 11 M. Brescia 55 D.
Arzoumanian 1166 D. Russeil 11 G. Riccio 55 S. Cavuoti , F.-X. Dupé and S.
Bensaid contributed equally to the work presented in this article.55
(Received May 25 2022; accepted November 20 2022)
###### Abstract
Context. Filaments are ubiquitous in the Galaxy, and they host star formation.
Detecting them in a reliable way is therefore key towards our understanding of
the star formation process.
Aims. We explore whether supervised machine learning can identify filamentary
structures on the whole Galactic plane.
Methods. We used two versions of UNet-based networks for image segmentation.
We used H2 column density images of the Galactic plane obtained with Herschel
Hi-GAL data as input data. We trained the UNet-based networks with skeletons
(spine plus branches) of filaments that were extracted from these images,
together with background and missing data masks that we produced. We tested
eight training scenarios to determine the best scenario for our astrophysical
purpose of classifying pixels as filaments.
Results. The training of the UNets allows us to create a new image of the
Galactic plane by segmentation in which pixels belonging to filamentary
structures are identified. With this new method, we classify more pixels (more
by a factor of 2 to 7, depending on the classification threshold used) as
belonging to filaments than the spine plus branches structures we used as
input. New structures are revealed, which are mainly low-contrast filaments
that were not detected before. We use standard metrics to evaluate the
performances of the different training scenarios. This allows us to
demonstrate the robustness of the method and to determine an optimal threshold
value that maximizes the recovery of the input labelled pixel classification.
Conclusions. This proof-of-concept study shows that supervised machine
learning can reveal filamentary structures that are present throughout the
Galactic plane. The detection of these structures, including low-density and
low-contrast structures that have never been seen before, offers important
perspectives for the study of these filaments.
###### Key Words.:
Methods: statistics – Stars: formation – ISM: general
## 1 Introduction
The Herschel infrared Galactic Plane Survey, Hi-GAL (Molinari et al. 2010),
revealed that the cold and warm interstellar medium (ISM) is organized in a
network of filaments in which star formation is generally observed above a
density threshold corresponding to $A_{\rm{V}}$=7 mag (André et al. 2014;
Könyves et al. 2020). The most massive stars are formed at the junction of the
densest filaments, called hubs (Kumar et al. 2020). Because filaments host
star formation and link the organization of the interstellar matter to the
future star formation, studying them is central to our understanding of all
properties related to star formation, such as the initial mass function, the
star formation rate, and the star formation efficiency. Filaments are
therefore extensively studied with observations at all wavelengths and
numerical simulations (André et al. 2010; Molinari et al. 2010; Arzoumanian et
al. 2011; Hacar et al. 2018; Arzoumanian et al. 2019; Shimajiri et al. 2019;
Clarke et al. 2020; Priestley & Whitworth 2022; Hacar et al. 2022, and
references therein). All these data reveal the complex structure of filaments
and show a changing morphology, depending on the way (resolution or tracers)
in which they are observed (Leurini et al. 2019). For example, high-resolution
molecular line observations of Galactic filaments with the Atacama Large
Millimeter Array (ALMA) show that they are made of fibers on a spatial scale ¡
0.1 pc (Shimajiri et al. 2019; Hacar et al. 2018). Their complex morphology
and dynamics are also revealed with 3D spectroscopic information (Mattern et
al. 2018; Hacar et al. 2020) and show their key role in the accretion process
from large (¿ 10 pc) to subparsec scales, funnelling material down to the
star-forming cores. However, the way filaments form and evolve in the ISM is
still debated (Hoemann et al. 2021; Hsieh et al. 2021). Recent results suggest
that compression from neutral (H i) and ionized (H ii) shells could play an
important role in forming and impacting the evolution of Galactic filaments
(Zavagno et al. 2020; Bracco et al. 2020). Their detection in nearby galaxies,
where they are also clearly linked to the star formation process, makes
studying filaments even more important and universal (Fukui et al. 2019).
Different algorithms are used to extract filaments from 2D images (Sousbie
2011; Schisano et al. 2014; Koch & Rosolowsky 2015; Zucker & Chen 2018;
Schisano et al. 2020; Men’shchikov 2021) and from 3D spectral data cubes
(Sousbie 2011; Chen et al. 2020). These algorithms often rely on a threshold
definition (for intensity or column density). Nonetheless, a close visual
inspection of 2D images and 3D cubes show that some filaments are missed by
all these detection algorithms, especially when the filaments have low column
density contrasts. This means that even large surveys of the Galactic plane
cannot deliver a complete (unbiased) view of the filaments present there.
Another limitation of these algorithms comes from their computation time,
which can make some of them too expensive to envision a complete run on large-
scale surveys data for multiple defined threshold and extraction parameters.
Because the multi-wavelength information available on our Galaxy on all
spatial scales is so rich, proposing another way of extracting filaments might
allow a leap forward for an unbiased census of these data. In this paper, we
explore the potential of supervised machine learning as a new way to reveal
filaments from 2D images. Using Hi-GAL data, Schisano et al. (2020) extracted
filaments from column density (N${}_{\rm{H_{2}}}$) images of the Galactic
plane. Based on these data, we study the possibility for convolutional UNet-
based networks (Fu & Mui 1981) to identify pixels as belonging to the filament
class, based on the input information given as previously identified filament
masks from Schisano et al. (2020). Except for the catalog of filament
candidates published in Schisano et al. (2020) where faint filaments present
in the Galactic plane are known to be missed, no complete extraction of
filamentary structures in the Galactic plane exists so far. This fact
motivates our work, in which we propose an alternative method that could allow
us to go beyond the current possibilities. However, this fact also indicates
that we worked with an incomplete ground truth (see Section 2.2) that renders
an absolute evaluation of the method performances proposed here impossible.
Nonetheless, we show that new filaments revealed by the UNet-based algorithm
and not detected before are confirmed through imaging at other wavelengths,
which gives us confidence that this new method can progress toward an unbiased
detection of filaments.
The paper is organized as follows: in Section 2 we describe the images and the
information on the filament locations we used in the supervised learning. The
supervised learning itself is described in Section 3. Results are presented in
Section 4 and are discussed in Section 5. Conclusions are given in Section 6.
## 2 Data
### 2.1 Hi-GAL catalog
Figure 1: Hi-GAL H2 column density image of the $l$=160-171° (top) and
$l$=349-356° (bottom) regions produced using Hi-GAL images as described in
Elia et al. (2013). The $l$=349-356° zone contains the bright star-forming
regions NGC 6334 and NGC 6357. These two regions are used in this paper to
illustrate the results. The dark pixels are saturated.
The _Herschel_ infrared Galactic Plane Survey, Hi-GAL (Molinari et al. 2010),
is a complete survey of the Galactic plane performed in five infrared
photometric bands centered at 70, 160, 250, 350, and 500 $\mu$m. H2 column
density (N${}_{\rm{H_{2}}}$) images were created for the whole Galactic plane
following the method described in Elia et al. (2013) and Schisano et al.
(2020). N${}_{\rm{H_{2}}}$ and dust temperature maps were computed from
photometrically calibrated images. The _Herschel_ data were convolved to the
500 $\mu$m resolution (36″), and a pixel-by-pixel fitting by a single-
temperature graybody was performed. An example of the column density image
covering the $l$=349-356° region is presented in Figure 1. This region
contains the bright and well-studied Galactic star-forming regions NGC 6334
and NGC 6357.
Schisano et al. (2020) analyzed the whole Galactic plane by extracting
filamentary structures from the H2 column density (N${}_{\rm{H_{2}}}$) maps.
In their work, a filament is defined as a two-dimensional, cylindric-like
structure that is elongated and shows a higher brightness contrast with
respect to its surroundings. The extraction algorithm is based on the Hessian
matrix $H(x,y)$ of the intensity map N${}_{\rm{H_{2}}}(x,y)$ to enhance
elongated regions with respect to any other emission. The algorithm performs a
spatial filtering and amplifies the contrast of small-scale structures in
which the emission changes rapidly. Further filtering allows identifying the
filamentary structures. Figure 2 shows an example of this filament extraction,
reproducing the figure 3 of Schisano et al. (2020). We chose this figure
because it shows the input we use in this work: the spine (blue line) and the
branches (red lines, both shown in the bottom left panel) associated with a
given filament. Schisano et al. (2020) defined a filament as traced by its
associated region of interest (RoI; bottom right), which covers a larger area
than the region that is defined with the spine plus branches. In this work we
use this spine plus branches structure to define a filament because the early
tests we made to train the networks with the input RoIs returned filamentary
structures that were too large compared to the structure that is observed in
the column density mosaics. This point is illustrated in Figure 21 and is
discussed in Sect. 4.3.
The analysis of the extracted structures from Schisano et al. (2020) resulted
in the publication of a first catalog of 32 059 filaments that were identified
over the entire Galactic plane. We used this published catalog of filaments
and their associated spine plus branches as ground truth of the filament class
for the training process (see Sect. 2.2). The method is described in Section
3.
Figure 2: Illustration of the filament extraction method from Schisano et al.
(2020, their Figure 3). The spine (blue line) and branches (red lines)
associated with a filament used for the supervised training process are shown
in the bottom left corner. The RoI (bottom right corner) is the zone used by
Schisano et al. (2020) to define a filament.
### 2.2 Data preprocessing
As methods based on deep learning strongly depend on the nature and on the
representativity of the input data, we took particular attention to the
construction of the data set. We used four input maps (see Fig. 3): 1) the
N${}_{\rm{H_{2}}}$ mosaics obtained as part of the Hi-GAL survey products
(Molinari et al. 2016; Schisano et al. 2020, e.g., Fig. 1), 2) the spines plus
branches of the detected filaments from Schisano et al. (2020, e.g., Fig. 2
bottom left corner), 3) a background map (localization of nonfilament pixels),
and 4) a missing-data map (see Appendix A). The origin and the ways in which
these maps were obtained are presented in Appendix A. An example of these maps
is given on Figure 3, illustrated for the two portions of the Galactic plane
that are located at 160-171° and 349-356°. We obtained results for the whole
Galactic plane, which we illustrate in two regions that we selected because
they represent the diversity of column density and filaments content observed
in the Galactic plane well. The 160-171° region samples a low column density
medium (up to 8$\times$1021 cm-2) in which only a few filaments are detected,
while the 349-356° region shows a rich content in filaments that are detected
in a high column density medium (up to 9$\times$1022 cm-2, see Figure 3). For
the training step, we merged all the individual mosaics (10°-long in longitude
direction) into one global map using the reproject module for astropy
(Robitaille et al. 2020, and Appendix A).
Figure 3: Illustration of the Galactic regions located at 160-171° (left) and
349-356° (right) of the four input maps used for the supervised learning. From
top to bottom, we show the N${}_{\rm{H_{2}}}$ column density map, the input
filament masks, a background localization map, and the missing-data map (0 in
purple, 1 in yellow). All these maps were obtained as explained in Appendix A.
The filament and background mask maps are multiplied by the missing-data map
before they were used in the training process. The red rectangle shown in each
column density map represents the region we extracted to compute the
performance of the training (see Sect. 4.1). Figure 4: Construction of patches
of size $p\times p$ using a sliding window.
As the four input maps are very large ($150000$ $\times$ $2000$ pixels), we
split them into many patches that constitute the original data set. As shown
in Figure 4, we split the maps into $p\times p$ patches. The size of $p=32$
pixels was chosen to preserve the information on the small filamentary
structures. This size is also the minimum size accepted by the UNet
architecture. The patches were generated by applying a sliding window of size
$p$ (the patch size) to the global mosaic of H2 column density
(N${}_{\rm{H_{2}}}$). To ensure the coherence between the four input maps, the
four patches (N${}_{\rm{H_{2}}}$, spine+branches, background, and missing
data) were taken using the same coordinates (see Figure 5). In order to avoid
any common information between patches, the construction was made without any
overlap between the patches.
Figure 5: Building the data set using the four input maps (on the left) into a
set of patches (on the right). On the left, the maps are the column density
(top left), filament spine+branches (top right), missing data (bottom left),
and background pixels (bottom right).
### 2.3 Data augmentation
Deep neural networks are greedy algorithms. In spite of the huge size of the
Galactic map, the final data set had merely $52.000$ patches after empty
patches were removed. Here, we refer as “empty patches” to patches that only
contain missed values (patches located on edges) or to patches that contain “0
pixels labeled data”. Fully unlabeled patches were removed from the patch data
set. As unlabeled pixels represent more than $80\%$ of the data set, this
resulted in the loss of many patches. The data augmentation is thus necessary
in order to increase the number of training and validation patches and to
thereby enable a sufficient convergence of the neural network during the
training step (Goodfellow et al. 2016). Two types of rotation were used:
rotation around the central pixel of the squared patch (0°(original), 90°,
180° , and 270°), and flipping the patch with respect to the $x$ \- and $y$
-axes. We also allowed a composition of both rotations. All possible
transformations are equally probable, that is, we selected the applied
transformation following a uniform distribution. To attenuate redundancy
issues, the augmentation was done on the fly, meaning that at each batch, we
produced a new set of patches using the augmentation process. With our
setting, we virtually increased the number of patches by a factor equal to 64.
Figure 6 shows some examples of the data augmentation process.
Figure 6: Example of data augmentation results.
## 3 Method
### 3.1 Segmentation pipeline
Our segmentation method relies on three components: a data preparation
procedure, a neural network with an architecture dedicated to the recognition
of filamentary structures, and a training procedure adapted to the
N${}_{\rm{H_{2}}}$ data. After the neural network was trained, we used it to
segment the N${}_{\rm{H_{2}}}$ map. The result of the segmentation process is
a map in which the pixels are classified into two classes: either a filament
pixel (identified as class 1), or a background pixel (identified as class 0).
With these two classes, we produced an intensity map with values of 0 and 1.
The values of the classification indicate whether a pixel belongs to the
filament class (the reverse map shows the classification value according to
which a pixel belongs to the background class).
#### 3.1.1 Segmentation with UNets
Automatic segmentation is a well-known issue in the artificial intelligence
community. Its origins lie in computer vision. It is a well-studied problem
today, especially where segmentation is mandatory for decision or prediction,
typically for medical or biology images (Fu & Mui 1981; Alzahrani & Boufama
2021). It has also been used for a long time in astrophysics, with recent
applications on galaxies (Zhu et al. 2019; Hausen & Robertson 2020; Bianco et
al. 2021; Bekki 2021). Most previous methods are based on classical machine-
learning methods, such as Support-Vector Machine (SVM) or Random Forest
(Hastie et al. 2001). These methods must extract an adapted set of features in
order to be sufficiently efficient: we have to be sure that the extracted
features represent the subject we wish to study well. These features are
usually given by expert knowledge of the problem.
Most successful machine-learning methods for image processing tasks today are
based on deep neural network methods (Goodfellow et al. 2016). These methods
have the particularity of learning both the task and a representation of the
data dedicated to the task itself. Thus, they are more powerful than methods
based on hand-tuned features. While the first methods were dedicated to
classification (e.g., AlexNet (Krizhevsky et al. 2012), LeNet (LeCun et al.
1989), or ResNets (He et al. 2016)), there are now many different
architectures depending on the targeted task. For segmentation, one of the
most promising neural networks is the UNet, which was introduced for medical
segmentation (Ronneberger et al. 2015). Many extensions exist, for instance,
UNet++ (Zhou et al. 2019) with layers to encode the concatenations, VNet
(Milletari et al. 2016), which is dedicated to 3D data, WNet (Xia & Kulis
2017), which has a double UNet architecture, and Attention-UNet (Oktay et al.
2018), which combines UNet with attention layers (Goodfellow et al. 2016).
Still, the UNet based architecture remains one of the most effective methods
for automatic segmentation.
In the context of astrophysical study, these neural networks have been
successfully used in different contexts. For example, Bekki (2021) used the
UNet to segment the spiral arms of galaxies. Bianco et al. (2021) used a UNet
based neural network called SegUNet to identify H ii regions during
reionization in 21 cm. Another variant based on UNet and inception neural
networks was used to predict localized primordial star formation (Wells &
Norman 2021). UNet was also used to segment cosmological filaments (Aragon-
Calvo 2019). We recommend Hausen & Robertson (2020) for a good introduction to
deep learning applied to astrophysical data.
Figure 7: Illustration of the UNet5 from Ronneberger et al. (2015).
UNet is a multiscale neural network based on convolutional and pooling layers,
as presented in Figure 7. In addition to its simple structure, the strength of
this network is an encoder-decoder-based architecture with skip connections.
First, the encoder extracts features from the input image down to a coarse
scale by using filters and max-pooling. Then, the decoder takes the
coefficients at the coarse level and combines them with those from each layer
of the encoder via the skip connections, in order to re-inject the details
that were lost in the down-sampling (max-pool) step and thereby build a better
semantic segmentation map. The final activation function of the network is
done by the sigmoid function (Goodfellow et al. 2016), as we wish to have
values in order to resolve a segmentation issue,
$s(x)=\frac{\exp(x)}{\exp(x)+1}\leavevmode\nobreak\ .$ (1)
This function guarantees an output between 0 and 1. Thus, the output of the
network can be read as a probability map for the class 1 filament (see
Goodfellow et al. 2016, Sect.6.2.2.2). However, in our case, both the
nonequilibrium between the two classes (filament and background) and the
incomplete ground-truth prevent the direct interpretation of the segmented map
values as probabilities (see Kull et al. 2017, about sigmoid output and
probabilities). In the following, we name the intensity value of the segmented
maps ”classification value”. In this study, the quality of the results is
assessed by comparing these classification thresholds with a given threshold
(see Sect. 3.1.5). This multiscale mirror-like structure makes the UNet very
suitable for image processing such as denoising (Batson & Royer 2019) or
segmentation (Ronneberger et al. 2015). Moreover, UNet belongs to the family
of fully convolutional networks. These networks are almost independent of the
size of the input images (Long et al. 2015). In UNet, the size of the output
image will be the same as that of the input if the input is large enough (the
minimum size is $32\times 32$ pixels).
Figure 8: Illustration of UNet++ from Zhou et al. (2019). The $X^{i,j}$ are
the same convolutional layers as for UNet. The difference between UNet and
UNet++ can be depicted in three main points: 1) convolution layers on skip
pathways (in green), which reduces the semantic gap between encoder and
decoder feature maps; 2) dense skip connections on skip pathways (in blue),
which improves the gradient flow; and 3) deep supervision (in red), which
enables model pruning (Lee et al. 2015).
A recent and more powerful extension of the UNet model, UNet++, was proposed
in (Zhou et al. 2019). This network belongs also to the fully convolutional
networks. As illustrated in Fig. 8, the plain skip connections of UNet are
replaced with a series of nested dense skip pathways in the UNet++ neural
network. The new design aims at reducing the semantic gap between the feature
maps of the encoder and decoder sub-networks that makes the learning task
easier to solve for the optimizer. In fact, the model captures more
efficiently fine-grained details when high-resolution feature maps from the
encoder are gradually enriched before fusion with the corresponding
semantically rich feature maps from the decoder. Note that these ”inner”
layers have also a mirror-like structure allowing a larger multi-scale
representation. However, it is worthy to note that UNet++ requires more data
than UNet as the latter has less parameters to tune (see Table 3 in Zhou et
al. 2019).
#### 3.1.2 Local normalization
Neural networks such as UNet are highly sensitive to the contrast inside the
input images (or patches). This sensitivity comes from the filters that belong
to the different convolution layers. In order to avoid this issue, input data
are usually normalized, generally by performing a global min-max normalization
(Goodfellow et al. 2016). This normalization allows us to temper the dynamic
of the contrast while keeping useful physical information about the structures
(morphology and gradient). However, in our case, the intensity of the
N${}_{\rm{H_{2}}}$ map presents a very high dynamical range, and a global
normalization would artificially weaken many filamentary structures. To avoid
this issue, we performed a local min-max normalization on each patch. As shown
in Figure 9, this normalization helps to deal with high-contrast variation in
nearby regions of the image. However, while this approach solves this issue,
the contrast still has high local variations in some cases, so that two nearby
patches may show different normalization.
Figure 9: The local min-max normalization of the patches helps to avoid
contrast issue allowing a better definition of the filaments.
#### 3.1.3 Training with UNet and UNet++
Training a neural network requires a loss function that computes the errors
between the model and the ground truth. For segmentation, a recommended
function is the binary cross-entropy (BCE), which casts the problem as a
classification problem (Jadon 2020). For the sake of clarity, we introduce
some notations before we give the expression of the loss function. Let
$\\{x_{i}\\}_{i}$ be the set of normalized N${}_{\rm{H_{2}}}$ patches. Let
$\\{y_{i}\\}_{i}$ be the set of segmentation target, that is, the set of
binary patches with 1 for filaments pixels and 0 for background pixels. Let
$\\{m_{i}\\}_{i}$ be the set of missing data patches, that is, the set of
binary patches with 0 for missing pixels and 1 elsewhere. For a given value of
$i$, $x_{i},y_{i}$, and $m_{i}$ share the same Galactic coordinates. The
cross-entropy loss for a set of $n$ patches is given by
$\begin{split}\mathcal{L}(\\{x_{i},y_{i},m_{i}\\}_{i};\theta)=\frac{1}{np^{2}}&\sum_{i=1}^{n}\sum_{k,l=1}^{p}m_{i}[k,l]\Big{(}y_{i}[k,l]\log(f_{\theta}(x_{i})[k,l])+\\\
&(1-y_{i}[k,l])\log(1-f_{\theta}(x_{i})[k,l])\Big{)}\end{split},$ (2)
where $f_{\theta}$ is the function that applies the forward propagation, and
$\theta$ are the weights of the neural network. By using
$\\{m_{i}\\}_{i\in[i\ldots n]}$, we ensure that only labeled data are used.
Figure 10: Five steps of an epoch during the training. For illustration
purposes, we reduced the batch to a set of one patch. $\theta_{t}$ represents
the weights of the neural network at epoch $t,$ and $\mu_{t}$ is the learning
rate at epoch $t$.
As we have many unlabeled pixels in the patches (see section 2.2), we have to
adapt the training step to avoid inconsistencies. We summarize the different
steps in Figure 10. First, in step (1), we take a set of patches (a batch) and
then apply the augmentation process (step (2)) on these patches. During this
step (2), we ensure that for a given $i$, the same transformation is applied
on $x_{i},y_{i}$, and $m_{i}$. The following steps are about computing the
prediction errors of the model on the patches and making the back-propagation
of the gradient of these errors in order to update the weights of the neural
networks (Goodfellow et al. 2016). Therefore, in step (3), we begin to apply
the network $f_{\theta_{t}}$ on the patches (forward propagation). Since this
step implies using the convolution layers in the network, we use both
unlabeled and labeled pixels. This is important as the neural network needs
the neighboring pixels to compute the value for one pixel. After we restrict
the result (step (4)) to labeled and nonmissing pixels using the mask $m_{i}$,
we can compute (step (5)) the errors on the restricted results compared to the
ground truth. Finally, in step (6), we update the weight of the network using
the back-propagation of the gradients of the errors. This is done by using a
stochastic gradient descent scheme (Goodfellow et al. 2016) with a learning
rate $\mu_{t}$ that changes during the training step following the epochs.
#### 3.1.4 Building the segmentation map
When the neural network has been trained, we can apply the model to segment an
image. As we described before, during the creation of the data set (Section
2.2), we must deal with the high dynamic contrast in images. Again we propose
to solve the issue by taking small patches and apply a local min-max
normalization. Moreover, since two closed patches may have a different
contrast, the normalization can lead to variation in the results when applying
the neural network. Therefore, in order to resolve this issue, we propose to
use an overlapping sliding window to obtain the patches: the segmentation
result is then the average between the output of the neural network applied on
the patches. These patches are distinct from those used for learning the
network (see Section 2.2). As the variance of contrast between patches
introduced variance inside the output results of the neural network, the
overlap and averaging operation (see Fig.11) allows us to decrease the
artifact that may appear (Pielawski & Wählby 2020).
Figure 11: Segmentation process. It takes patches from an observation (a),
then normalizes the patch and applies the segmentation model (b), the
segmented patch is positioned at the same coordinates (c), and is finally
weighted by coefficients (d) representing the number of patches in which each
pixel appears. Because a sliding window with overlap is used, a given pixel is
segmented several times (as long as it falls in the sliding window). Then, we
obtain several segmentation values for the same pixel. The final segmentation
value assigned to the pixel corresponds to the average of all the segmentation
values computed from the contributing sliding windows.
Thus, we apply the following segmentation procedure (illustrated in Figure
11). We browse the image using an overlapping sliding window that gives
patches (step (a)). Each patch is then normalized using a min-max
normalization (step (b)); here we avoid the missing data (around borders and
saturated areas). Then, we apply the trained neural network on the patch to
obtain the density map output and add on the output image at the coordinate of
the patches (step (c)). Since we use an overlapping sliding window, the
results are added to the output, and then we divide each pixel by a weight
representing the number of patches in which the pixel appears (this is done
using the weight map built in step (d)).
#### 3.1.5 Metrics
In supervised classification problems, the confusion matrix, also called error
matrix, is computed in order to assess the performance of the algorithm. We
refer to the filament and background classes as the positive (P) and the
negative (N) classes, respectively. We also refer to the correctly and
misclassified pixels as true (T) and false (F), respectively. In a binary
classification problem, the confusion matrix is thus expressed as in Table 1.
Table 1: Confusion matrix | Predicted
---|---
| | filament | background
Actual | filament | TP | FN
background | FP | TN
The confusion matrix is evaluated on the estimated filament masks that are
deduced from the segmented map at a classification threshold. It is important
to recall, however, that the evaluation set is restricted to labeled data. The
classification scores are thereafter derived from the confusion matrix. In
this work, the recall, precision, and dice index defined in 3, 4 and 5,
respectively, are used to evaluate the classifier performance. Maximizing
recall and precision amounts to minimizing false-negative and false-positive
errors, respectively, whereas maximizing the dice index amounts to finding the
optimal tradeoff between the two errors. Therefore, the closer to 1 these
scores, the better.
$\displaystyle\mathrm{Recall}$ $\displaystyle=\frac{TP}{TP+FN}$ (3)
$\displaystyle\mathrm{Precision}$ $\displaystyle=\frac{TP}{TP+FP}$ (4)
$\displaystyle\mathrm{Dice}$ $\displaystyle=\frac{2\ TP}{2\ TP+FP+FN}$ (5)
In addition to the recovery scores, we also calculate the rate of missed
structures (MS) in segmentation for the same set of thresholds. The MS score
can be defined as the ratio of the missed filament structures over all the
input filament structures,
$\mathrm{MS}=\frac{\text{number of missed structures}}{\text{total number of
filament structures}}$ (6)
Figure 12: Morphological reconstruction is a method that computes shapes from
marked pixels called seeds. (a) We first compute the seeds using the
intersection between the segmentation results and the ground truth. (b) We use
the intersection pixels as seeds (see the red seeds in the bottom left
corner). (c) We apply the reconstruction to obtain the filaments with at least
one seed.
The MS metric is based on morphological reconstruction (Vincent 1993; Soille
2003; Robinson & Whelan 2004). Figure 12 shows how we apply this method to
assess which known filaments are recovered. First, we compute the intersection
between the known filaments (ground truth) and the segmentation results
(Figure 12(a)). Then we use this intersection as the seed for the
reconstruction method: in Figure 12(b), the yellow elements are the seeds, and
the red elements are the part of filaments that is missed. The morphological
reconstruction takes the seeds to recover the known filaments by using a
shape-constrained growing process (Figure 12(c)). Only the filaments for which
at least one pixel is used as seed will be recovered (Robinson & Whelan 2004).
By subtracting the morphological reconstruction result from the ground truth,
we can identify the missed structures (see also Figure 17). Then we count the
number of missed structures by using a direct labeling of the pixels where two
neighboring pixels share the same label (Fiorio & Gustedt 1996). Then, we can
compute the MS metric and qualitatively assess the recovery of filaments in
terms of structures, rather than individual pixels.
### 3.2 Experimental setup
Taking the analysis of the normalization in Section 2.2 into consideration,
the patch size was fixed to the lowest value accepted by the UNet family,
$p=32$. In order to have proper training, validation, and test steps, we
randomly split our initial set of patches (Section 2.2) into three sets,
namely, the training, validation, and test sets, with proportions of $80\%$,
$10\%,$ and $10\%$, respectively. The random split ensures the presence of the
two classes (filament and background) in the three sets. The patches in
training and validation sets were then shuffled after each epoch to help
avoiding unwanted bias (Goodfellow et al. 2016, see). The total number of
epochs was set to $100$. UNet and UNet++ were trained using the Adam
optimization scheme with a multistep learning rate (Kingma & Ba 2015;
Ronneberger et al. 2015). During the first 30 epochs, the initial learning
rate value was divided by $10$ every five epochs. Four initial values of the
learning rate, $10^{-5},10^{-4},10^{-3},\text{and }10^{-2}$, were tested for
both networks. We denote by UNet[$\mathrm{lr}$] (UNet++[$\mathrm{lr}$]) the
UNet (UNet++) model learned with $\mathrm{lr}$ as the initial value of the
learning rate, where $\mathrm{lr}\in\\{10^{-5},10^{-4},10^{-3},10^{-2}\\}$. A
summary of the parameter values used in the training step is given in Table 2.
In the segmentation step, an overlapping sliding window of size $32\times 32$
was applied, where an overlap of $30$ pixels was used in order to limit edge
artifacts and to generate highly smooth segmentation.
In order to compare the performance of UNet with UNet++, two zones of the
global N${}_{\rm{H_{2}}}$ mosaic were excluded from the initial patch data
set. Constrained by the limited number of patches in the data set, small zones
were removed, namely, the zones 166.1-168.3° and 350.3-353.5°. The choice of
the removed regions was motivated by assessing the network performance in
regions with a highly diverse column density and filaments content. While
350.3-353.5°, removed from mosaic 349-356°, is dense and rich in filaments
($38541$ filament pixels), the region 166.1-168.3°, removed from the mosaic
160-171°, is sparser and contains fewer filaments ($1459$ filament pixels).
The filament density is always inferred based only on the incomplete ground
truth (labeled part of the data set). The two removed zones were then
segmented by the learned models, and the segmentation quality was assessed
using the evaluation scores described in Section 3.1.5. The evaluation scores
were also computed on the fully segmented Galactic plane in order to have a
global performance evaluation of models.
Table 2: Experimental setup
Parameter | Value
---|---
patch size ($p$) | 32 pixels
dataset split | {80%, 10%, 10%}
batch size | 64 patches
epochs | 100
initial learning rate | {$10^{-5},10^{-4},10^{-3},10^{-2}$}
111Parameters used in the experimental setup for UNet and UNet++ training
## 4 Results
### 4.1 Scores and segmented mosaic analysis
In order to evaluate the training performances, we discuss below the different
scores we obtained for the different tested scenarios. Two neural networks
were tested (UNet and UNet++) with four different initial learning rates for
each. For an input column density image, the segmentation process (see Section
3.1.4) returns a classification value mask from which it is possible to
identify pixels that likely belong to a filamentary structure. Nevertheless,
we did not attempt to extract the filaments like Schisano et al. (2020) did.
We postpone this physical analysis on the newly identified filaments to a
follow-up work. Here we present the method as a proof-of-concept and analyze
its performances and returned results (segmented map of the whole Galactic
plane). We illustrate these results on two portions of the Galactic plane that
were selected for their characteristics in terms of column density and
filament content (as known from the input data set of filament mask based on
the spine+branches).
To evaluate the performances of the different scenarios, Figure 14 presents
the BCE curves for the training set and the validation set during the first
$30$ epochs. For all models, these loss curves reach a plateau around the
tenth epoch, confirming the rapid convergence of UNet-based networks. The
performance of UNet++[$10^{-2}$] was removed from the displayed results due to
convergence issues. As shown in Figure 14 and confirmed by this analysis, the
models have similar performances and converge to a training error around
$0.01$, except for schemes with a starting learning-rate value of $10^{-5}$ ,
where higher errors are reported (more than $0.016$ and $0.012$ for
UNet[$10^{-5}$] and UNet++[$10^{-5}$], respectively). The similar performances
obtained for the two architectures and the different learning rates indicate
that the method is robust. Validation errors are slightly higher than training
errors, with a difference up to $0.0011$, except for UNet[$10^{-5}$] (around
$0.002$) and UNet++[$10^{-5}$] (around $0.0018$). The reported values indicate
that the learned models show low bias and low variance.
For both UNet and UNet++, the best performance is with an initial learning-
rate value of $10^{-3}$. Overall, the best model in terms of loss function is
the UNet++[$10^{-3}$]. For each scenario, the best-performing model in
validation was used to segment the test set and compute the underlying BCE
(see Table 3). BCEs calculated on the test set corroborate the previous
analysis: scenarios with initial learning rates in [$10^{-2}$, $10^{-3}$,
$10^{-4}$] result in similar test errors that do not exceed $0.009$, whereas
scenarios with initial learning rates of $10^{-5}$ show a higher error. The
lowest test error corresponds to the scenario UNet++[$10^{-3}$].
Figure 13: Dice curve evolution of schemes reported in Figure 14 are displayed
over the first $30$ epochs in training (continued lines) and validation
(dashed lines) steps, at classification threshold values $0.8$, $0.6$, $0.4,$
and $0.2$. The displayed curves are aligned with the results deduced from BCE
curves, where close performances were obtained with initial learning-rate
values of $10^{-4},10^{-3},\text{and }10^{-2},$ and a poorer performance is
obtained for schemes with a learning-rate value of $10^{-5}$. The highest dice
score is reported for UNet++[$10^{-3}$] (purple), and the lowest performance
corresponds to the UNet[$10^{-5}$] scheme, (red) especially at thresholds 0.2
and 0.4. Similarly to the BCE curves and for all displayed schemes, a plateau
regime is reached within the first ten epochs.
In Figure 13 the dice index curves for the training and validation sets is
plotted for different classification thresholds. The dice index was also
computed on a test set using the best-performing models in validation (the
models used in Table 3), and the corresponding results are given in Figure 7.
In the same way, the dice index results are in line with the BCE. These models
were used afterwards to segment the removed zones 166.1-168.3° and
350.3-353.5°.
Figure 14: BCE evolution over the first $30$ epochs in training (continued
lines) and validation (dashed lines) steps. UNet++[$10^{-2}$] schemes is
removed as its corresponding BCE diverged. The displayed schemes show similar
performances. Models with a learning-rate value of $10^{-5}$ resulted in
higher BCEs. The lowest error is reported for UNet++[$10^{-3}$] (purple), and
the highest error corresponds to the UNet[$10^{-5}$] scheme (red). A plateau
regime is reached within the first ten epochs for all models, which confirms
the rapid convergence of the UNet-based networks.
Figure 16 presents the results of the segmentation process for the different
scenarios tested and presented in Table 2. All segmented maps are presented
within the range $[0,1]$. Overall, the obtained images are in line with the
analyzed BCEs. The scenarios with initial learning rates of $10^{-2}$,
$10^{-3}$, $\text{and }10^{-4}$ return very similar segmented maps. A
noticeable difference is seen in maps that were segmented by both UNet and
UNet++ with an initial learning rate of 10-5. In these cases, the filamentary
structures are broader than those obtained for the other scenarios, especially
for UNet. Moreover, the intensity of the segmentation maps also varies for
these last two scenarios, where the low-value structures are better revealed
(intensity variation up to a factor of 10 in those zones). The ability of
these two last scenarios (and of the UNet in particular) to better reveal
structures with a lower classification threshold might be used to detect
structures that are not well seen on the original map, either due to their low
contrast and/or their low column density. A close visual inspection of the
column density images confirms that features revealed by UNet[$10^{-5}$] and
UNet++[$10^{-5}$] are low-contrast filaments that were present in the original
images, but absent from our input catalog of filaments that was used as ground
truth.
(a) PR curves for 350.3-353.5°
(b) PR curves for 166.1-168.3°
(c) PR curves for the fully segmented Galactic plane
Figure 15: P-R curves of the schemes reported in Figure 14, computed on the
segmented removed zones (top) and the full Galactic plane (bottom). (a) P-R
curves computed on the segmented 350.3-353.5° , which corresponds to the dense
region that was removed from the patches data set. (b) P-R curves computed on
the segmented 166.1-168.3° , which corresponds to the sparse region that we
removed from the patches data set. (c) P-R curves computed on the full
segmented Galactic plane. Unlike in Fig. 15(b), P-R curves obtained on the
latter are close to those obtained in Figure 15(a).
Figure 16: Segmented maps obtained for the models analyzed in Figure 14,
zooming in on a part of the two regions that was removed from the training
namely, the $l$=166.1-168.3° (top) and $l$=350.3-353.5° (bottom) (see the red
zones identified in Figure 3). The segmented images are displayed in the range
$[0,1]$ representing the classification value according to which a pixel
belongs to the filament class. For each region, the first row shows results of
the UNet segmentation, and the second row shows results of the UNet++
segmentation with initial learning-rate values of 103 (left), 10-4 to 10-5
(right). The UNet++ with a learning rate of 10-2 is not presented because of
diverging results (see Section 4.1). The N${}_{\rm{H_{2}}}$ is shown at this
position for each region. The regions are $0.45$°$\times 0.45$° wide.
In Figure 15, precision-recall curves (P-R curves) are shown for 350.3-353.5°
(top left) and 166.1-168.3° (top right). The P-R curve represents precision vs
recall for different threshold values. It is used to estimate the optimal
threshold that maximizes the dice index (a trade-off between precision and
recall). The more the P-R curve tends to the $(1,1)$ corner, the better the
model. In other terms, the larger the area under the curve, the better the
model. The objective is to estimate the optimal threshold that returns a
trade-off between precision and recall. For clarity sake, all curves are
zoomed in from $0-1$ to $0.7-1$ for precision and recall. In all figures,
black asterisks refer to the precision-recall values at the optimal threshold;
the values of the latter are reported in Tables 4.2, 4.2, and B. Different
approaches can be used to compute the optimal threshold, such as minimizing
the difference between the precision and recall, or minimizing the Euclidean
distance between the P-R curve and the optimal performance, corresponding to a
precision-recall of ($1,1$). Here, we computed the optimal threshold as the
one that maximized the dice index (trade-off between filament and background
recovery). In Tables 4.2 and 4.2, we report four samples from these P-R curves
corresponding to conservative ($0.8$), medium ($0.5$), relaxed ($0.2$) and
optimal (giving the best Dice index) thresholds. When investigating the dense
zone of 350.3-353.5°, we note that, for a given threshold, all the models give
results with similar performances (Dice indices $>85\%$), except for
UNet[$10^{-5}$] (74.79% at threshold 0.2). Close optimal threshold are also
obtained for all models, where values are situated between $0.35$ and $0.48$.
Note that at the conservative threshold 0.8, UNet[$10^{-5}$] and
UNet++[$10^{-5}$] result in low recall values compared with the remaining
scenarios (72.96% and 75.14%, respectively), confirming the results reported
with the segmented map where salient filaments are detected with lower values
in these two scenarios compared with the remaining ones. However, they tend to
be more performing when decreasing classification threshold, especially at
threshold 0.2 where they are performing better than the remaining scenarios
(the best recall is of 98.1% with UNet[$10^{-5}$], followed by
UNet++[$10^{-5}$] with a recall value of 97.26%). This can be explained by the
low value structures that are better revealed in these two scenarios as
noticed before in the segmented map. In Appendix B, precision, recall and Dice
index are computed on the fully-segmented Galactic plane, to infer the global
segmentation performance. The resulting global scores are inline with the ones
obtained with the dense zone where, for a given threshold, all models show
close performances, except UNet[$10^{-5}$] and UNet++[$10^{-5}$] slightly less
performing. Moreover, the optimal thresholds obtained with the global
segmentation are close to thresholds obtained for the dense mosaic, where
values range from $0.3$ to $0.44$. This result suggests that either the
training step is more driven by high density regions and/or that these regions
better represent the global properties observed on the Galactic plane.
Table 3: Binary Cross Entropy
Model | BCE Score
---|---
UNet$[10^{-2}]$ | $0.0085$
UNet$[10^{-3}]$ | $0.0084$
UNet$[10^{-4}]$ | $0.0088$
UNet$[10^{-5}]$ | $0.0161$
UNet++$[10^{-3}]$ | $0.0081$
UNet++$[10^{-4}]$ | $0.0088$
UNet++$[10^{-5}]$ | $0.0122$
222Binary Cross Entropy (BCE) evaluation on the test set for the schemes
reported in Figure 14. The lowest (highest) achieved BCE is given in blue
(red). BCE values in test are inline with performances in training and
validation steps with UNet++[$10^{-3}$] and UNet[$10^{-5}$] resulting in the
best and less performing schemes, respectively.
When examining the scores of the sparse zone of 166.1-168.3° in Table 4.2, all
the models result in similar performances in precision. However, the recall
performance per threshold has a higher contrast, where UNet++[$10^{-5}$] shows
lower recall values for thresholds 0.8, 0.5, and 0.2. While the background is
well recovered (the lowest precision is of $98.97\%$), lower recall values are
obtained compared to the dense zone, where we had to relax the threshold to
$0.2$ in order to improve filament recovery and obtain recall values higher
than $70\%$. Similarly to the dense zone, close optimal dice indices were
obtained for all models, where the difference between the best dice given by
UNet++[$10^{-4}$] ($99.38\%$) and the more poorly performing UNet[$10^{-2}$]
($98.8\%$) is less than 0.6%. Although interesting scores are obtained at the
optimal thresholds, it is very important to underline the very low values of
these thresholds in all scenarios. Optimal thresholds range from $0.01$ to
$<0.03$, except for UNet[$10^{-5}$] ($0.12$). The obtained values reflect the
difficulty of the different networks to reveal structures in this mosaic,
where almost $40\%$ to $60\%$ of the filament pixels are detected with
classification values lower than $0.5$ (see the segmented maps in Figure 16).
Moreover, we clearly note the discrepancy between scores computed on the
sparse zone and the global scores reported in Appendices B, which might
suggest that the trained models are more successful in revealing filaments in
dense than in lower density zones. It is difficult, however, to conclude about
the origin of this discrepancy because the number of labeled pixels (in both
filament and background) is very different between the sparse and the dense
zone. Limited labels in the sparse mosaic also imply that the computed scores
with higher uncertainties need to be considered. Nevertheless, visual
inspection of the segmented maps in Fig. 16 results in similar trends as
observed for the dense zone, where close performances are noted for all models
except for the models with an initial learning rate of $10^{-5}$. These models
reveal more details at moderate to low classification values.
### 4.2 Analysis of missed structures
In addition to pixel-level scores, the structure-level score was also computed
in order to evaluate filament recovery in terms of structures. When some
pixels of a given filament are missed, it does not automatically imply that
the whole structure is missed. In Tables 4.2 and 4.2, the MS rate is computed
at different classification thresholds for dense and sparse zones. As
expected, the higher the threshold, the more structures are missed. Overall,
we note that at conservative (0.8) and moderate (0.5) thresholds, low MS rates
are obtained for dense mosaics compared with the sparse mosaic. In the latter,
more strongly contrasting values are obtained across the classification
thresholds where MS rates range from 0 (all structures are revealed) to almost
60% (more than half of the structures are missed; see the MS rates in Table
4.2). In a region with a low concentration of filaments, missing (or
detecting) a structure would have more impact on the MS rate variation than in
a dense region. Similarly to pixel-level scores, the global MS rates reported
in Table B are closer to the values obtained with the dense zone than the
sparse zone, and this for the same reasons as we invoked for pixel-level
scores.
In order to learn more about the structure of missed filaments, a missed-
structure map at a classification threshold of $0.8$ was built. In this map,
any structure that was missed by any model is represented. Here, the map
intensity encodes the number of models that missed the structure, so that
values range from 0 (detected structure) to 7 models (missed by all models).
In Figure 17, representative portions from the sparse and dense regions are
displayed. We note the prevalence of structures that were missed by all the
models (yellow). In fact, $50\%$ of the missed structures in the whole
Galactic plane are the same for all the models. After a close visual
inspection of the missed structures, two categories are reported. The first
category consists of small structures, which is the most prevalent category.
These structures either correspond to small isolated filaments and/or to small
parts that are missed in larger filaments. The second category corresponds to
larger filaments that are misidentified as filaments in the ground truth. For
example, structures reported in Figure 17 (bottom right) at positions (349°,
1°) and (350.5°, $-$1°) are excluded from the filament class. There are also
isolated square-shaped structures that are mislabeled as filaments in the
ground truth, corresponding to saturated pixels (see Figure 17 (bottom right)
at position (350.8°, 1°). Unfortunately, trained models fail to reject these
saturation bins when they are entangled within a true large filamentary
structure.
Table 4: Segmentation scores (350.3-353.5°)
l—*7l Scores (%)Model UNet [10-2] UNet [10-3] UNet [10-4] UNet [10-5] UNet++
[10-3] UNet++ [10-4] UNet++ [10-5]
Precision at 0.8 99.67 99.76 99.84 99.89 99.71 99.76 99.65
Precision at 0.5 97.45 98.06 98.24 97.11 97.87 97.87 95.13
Precision at 0.2 88.10 89.75 88.79 60.42 90.12 87.59 78.58
Precision at thropt 96.01 95.63 96.45 96.38 96.06 96.48 94.32
Recall at 0.8 79.15 79.45 79.10 72.96 78.34 79.71 75.14
Recall at 0.5 91.15 90.86 91.63 91.75 90.45 91.73 90.97
Recall at 0.2 96.55 96.27 96.83 98.10 96.09 96.82 97.24
Recall at thropt 92.86 93.91 93.86 92.49 93.17 93.37 91.82
Dice at 0.8 88.24 88.45 88.27 84.33 87.74 88.62 85.67
Dice at 0.5 94.19 94.33 94.82 94.35 94.02 94.70 93.00
Dice at 0.2 92.13 92.89 92.64 74.79 93.01 91.98 86.92
Diceopt 94.41 94.76 95.14 94.40 94.59 94.90 93.05
MS at 0.8 4.25 4.96 4.96 7.79 4.96 5.31 6.02
MS at 0.5 1.06 1.24 1.24 1.42 1.59 1.42 1.06
MS at 0.2 0.35 0.18 0.35 0.18 0.35 0.18 0.18
MS at thropt 0.53 0.35 0.53 1.24 0.53 0.53 0.88
thropt 0.41 0.35 0.38 0.48 0.37 0.42 0.47
333Segmentation scores evaluated on the the dense zone of 350.3-353.5°
segmented by the models reported in Figure 14. Precision, recall, dice index,
and MS rate are evaluated at classification threshold values of $0.8$, $0.5$,
$0.2,$ and the optimal threshold. The latter refers to the threshold value
optimizing the dice index and estimated using P-R curves (see Figure 15(a)).
Blue (red) refers to the best (least performing) model in each row. The bold
scores correspond to the absolute best (if in blue) or lowest (if in red)
performance per score.
Table 5: Segmentation scores (166.1-168.3°)
l—*7l Scores (%)Model UNet [10-2] UNet [10-3] UNet [10-4] UNet [10-5] UNet++
[10-3] UNet++ [10-4] UNet++ [10-5]
Precision at 0.8 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Precision at 0.5 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Precision at 0.2 100.00 100.00 100.00 100.00 100.00 100.00 100.00
Precision at thropt 99.04 98.97 99.65 99.66 99.25 99.32 99.24
Recall at 0.8 25.91 28.38 25.22 25.29 33.45 30.64 20.84
Recall at 0.5 49.01 58.40 49.01 54.90 63.33 60.59 41.74
Recall at 0.2 77.11 87.32 81.49 94.59 87.18 87.73 72.86
Recall at thropt 98.56 98.83 98.90 99.04 99.31 99.45 99.04
Dice at 0.8 41.15 44.21 40.28 40.37 50.13 46.90 34.49
Dice at 0.5 65.78 73.73 65.78 70.88 77.55 75.46 58.90
Dice at 0.2 87.07 93.23 89.80 97.22 93.15 93.46 84.30
Diceopt 98.80 98.90 99.28 99.35 99.28 99.38 99.14
MS at 0.8 61.02 45.76 57.63 50.85 32.20 37.29 44.07
MS at 0.5 16.95 6.78 10.17 15.25 6.78 6.78 11.86
MS at 0.2 1.69 0.00 1.69 0.00 0.00 1.69 1.69
MS at thropt 0.00 0.00 0.00 0.00 0.00 0.00 0.00
thropt 0.03 0.03 0.02 0.12 0.01 0.02 0.02
444Same as in Table 4.2, but for the sparse zone of 166.1-168.3°
Figure 17: Input filaments missed by the segmentation process on the
$l$=160-171° (top) and $l$=349-356° (bottom) regions of the Galactic plane,
respectively. We show. the input filament mask (left) and the missed
structures at a classification threshold of $0.8$ in a cumulative way (for all
the scenarios) (right). The unit (color-coding) for the missed structures maps
corresponds to the number of tested scenarios (from 1 to 7) that missed a
given structure.
### 4.3 New possible filaments revealed by deep learning
Based on the performance analysis in Section 4.1, two groups of models can be
derived based on the initial learning rate: 1) models with an initial learning
rate of $10^{-4},10^{-3},\text{and }10^{-2}$ , and 2) models with an initial
learning rate of $10^{-5}$. In the following, we present the segmentation
results that we illustrate for the best model, UNet++[$10^{-3}$], on the two
selected submosaics.
Figure 18 presents the evolution of the segmented map for the two selected
regions as a function of the segmented map threshold.
Figure 18: Zoom-in on the evolution of segmentation results as a function of
the classification threshold showing the filamentary structures estimated by
UNet++ 10-3 in 160-171° (six top images) and 349-356° (six bottom images)
regions of the Galactic plane. The original H2 column density image (top left
in each group), the ground-truth input filament mask, and the corresponding
segmented image at different thresholds (0.8, 0.5, 0.2, and the optimal
threshold) are shown from top left to bottom right. The regions are
$2$°$\times 2$° wide.
In both regions, more pixels are classified as filaments by the training and
segmentation processes than in the input structures (input filament mask used
as the ground truth in the supervised training). The ratio (new filament
pixels to input filament pixels) varies between 2 to 7, depending on the
threshold value (from 0.8 to 0.3, the optimal threshold). The same conclusions
are drawn for the whole Galactic plane. In Figure 19, the distribution of
candidate filament pixels across the entire Galactic plane, estimated in bins
of 4.8° $\times$ 0.16°, is displayed for the ground truth and the segmentation
results at different classification thresholds. Even at a conservative
classification threshold of 0.8, more pixels are labeled as filaments than in
the ground truth used in the learning step. As expected, the more we decrease
the classification threshold, the more pixels are labeled as filament. A close
visual inspection of the segmented images indicates that structures observed
at thresholds lower than the optimal value are also seen on the original
column density image, but were not previously detected due to their low
contrast with respect to the surrounding background emission.
(a) Ground truth
(b) Segmentation at classification threshold of 0.8
(c) Segmentation at classification threshold of 0.5
(d) Segmentation at classification threshold of 0.2
(e) Ratio of 18(b) to 18(a)
Figure 19: Number density distribution of candidate filament pixels across the
entire Galactic plane, estimated in bins of 4.8° $\times$ 0.16° and comparing
the ground truth of Fig. 18(a) with the segmentation results of the model
UNet++ 10-3 at classification thresholds of 0.8 in Fig. 18(b), 0.5 in Fig.
18(c), and 0.2 in Fig. 18(d). In Fig. 18(e), we display the ratio of candidate
filament pixels in the segmentation at a classification threshold of 0.8 to
the ground truth.
The squared structures shown in Figure 18 (bottom) are saturated pixels
corresponding to bright sources located in filaments. These structures also
appear in the ground truth, and we therefore retrieve them when applying our
model. Because we lack information about the column density in these saturated
pixels, we left them as squares in the segmented maps.
Figure 20 presents the same result as Figure 18, but for the binarized version
of the segmented map.
Figure 20: Zoom-in on the evolution of segmentation results generated by
UNet++ 10-3 model in mosaics 160-171° (six top images) and 349-356° (six
bottom images). Here, the estimated binary filament masks are displayed at
classification thresholds 0.8, 0.5, 0.2, and the optimal threshold. The
original H2 column density image (image with the color bar) and the true input
filament mask are also displayed for comparison. The regions are $2$°$\times
2$° wide.
The filamentary structures identified at a given threshold are now represented
as 1 when the associated pixel belongs to the filament class 0 instead. This
representation allows a more direct comparison with the input filament mask
(spine plus branches; see Fig. 2). However, the classification value itself
(that indicates whether a pixel belongs to the filament class) no longer
appears in this representation. As for results presented in Figure 18, the
threshold decrease has two effects: i) more pixels are identified as belonging
to the filament class (new structures are detected, in particular, those with
a faint-to-low contrast that are barely visible in the original
N${}_{\rm{H_{2}}}$ image), and ii) a given structure becomes thicker. This
last effect can be identified as a lowering of the corresponding column
density threshold, where the highest threshold identifies the densest part of
the filament. It is interesting to note that the optimal threshold (as well as
lower threshold values) in both regions identifies the filamentary structures
as observed in the original column density map, down to their external
envelope emission, before reaching the background emission. This result is
important because it will allow a precise study of the filament-background
relation. The widening of the filamentary structures can also be seen as the
definition of the RoI given by Schisano et al. (2020) (see also Figure 2,
bottom right). Schisano et al. (2020) defined the RoI as the objects that
define filamentary candidates in their catalog. This point is important
because it implies that the comparison of our segmented map results with the
RoI would lower the factor we derived that represents the number of pixels
classified as filament as the RoI are always thicker than the spine plus
branches we used as input in this work to train the network to learn what a
filament is. In this work, we infer the filament mask as a ground truth from
the filament spine plus branches, as shown on Figure 2 (bottom left). However,
the RoI defines the filament in order to delineate its spread on the column
density map, and this corresponds to the observed widening of the structures
with the lowering of the threshold. The comparison of the different structures
is illustrated in Figure 21.
Figure 21: Comparison of the segmented map result using the UNet++ 10-3 with
the input spine plus branches (black contours) and the RoI (red contours).
Because we used spine plus branches as input to define a filament here, we
kept this input structure as a reference to compare with the result of the
segmentation process.
A key point in this work is to ascertain the nature of the filamentary
structures we reveal. Filaments are made of gas and dust. The filaments
detected by Schisano et al. (2020) are traced in dust emission maps. Dust
grains emit over a wide range of wavelengths, and they act as absorbers in the
optical, near-, and mid-infrared parts of the spectrum. Because of their dusty
composition, filaments are well visible as absorbing features at shorter
wavelengths (optical, near-, and mid-infrared) in the Galactic plane, and
these data can be used to ascertain the nature of the structures returned by
the training and segmentation processes. Only filaments that are visible in
absorption on a strong emission background can be detected at short
wavelengths. (Sub)millimeter emission of cold dust also reveals filaments
(Mattern et al. 2018; Leurini et al. 2019) and can be used to ascertain the
nature of the structures without encountering the extinction problem. This
empirical (data-based) validation of the results is a first step in the
analysis. In Figure 22, we illustrate the interest of this multiwavelength
analysis to ascertain the nature of new detected filaments on the Galactic
region G351.776-0.527. This region hosts a high-mass star-forming region
analyzed by Leurini et al. (2019).
Figure 22: Segmented map obtained for star-forming region G351.776-0.527
(bottom left) compared with the 2MASS $K$-band image (top left), where
filaments are observed in absorption, and the column density map (top right),
where filaments are observed in emission. The filamentary structures visible
in the image segmented by UNet++[10${-3}$] (bottom left) and displayed in the
[0,1] range are both visible in the 2MASS $K$ and N${}_{\rm{{H_{2}}}}$ column
density images, ensuring their nature. The G351.776-0.527 source is located at
the center of the images, connected to a filamentary network (hub). The
corresponding input filament mask is shown (bottom right).
Region G351.776-0.527 is located at the center of Figure 22 and appears as a
hub with filaments converging toward the saturated central point located at
$l$=351.77°, $b=-0.538$°. A filamentary structure observed in the segmented
map right of the bright central source that is not visible in the input
filament mask is seen in the 2MASS $K$-band image, confirming its nature. The
segmented image also suggests that the region might be located at the edge of
a bubble. A bright ionized region, G351.46-00.44, is located nearby and could
explain the high level of turbulence and the high-mass star formation observed
in this zone (Lee et al. 2012). The large-scale view of this region, revealed
by the segmented map with suggested multiple filament connections of the
central source with the surrounding medium, has to be confirmed with high-
sensitivity observations of dust emission that could be complemented by
spectroscopic data of dense gas molecular tracers, keeping in mind that, as
pointed by Hacar et al. (2022), filaments identified with Herschel data (i.e.,
using dust continuum emission) might be a different family of objects than
those detected in molecular line tracers.
## 5 Discussion and future prospects
The purpose of this work was to study the potential of supervised deep
learning as a new way to detect filaments in images of the Galactic
interstellar medium. At this stage, the filamentary structures are revealed,
but the filaments themselves are not extracted, with a measurement of their
physical properties, from the segmented images. While the first task requires
semantic segmentation, the second task consists of instance segmentation (Gu
et al. 2022). In the first, the task is limited to attributing a class to each
pixel, which is done in this paper, whereas in the latter, existing filaments
are in addition enumerated to allow a global statistical study. In this paper,
we used UNet-based networks which are the most modern methods in semantic
segmentation. The analyzed performance in Section 4 proves the efficiency of
these networks not only in revealing structures already existing in the
initial catalog, but also in adding new structures that have not been detected
before and that are confirmed through a detection at shorter and/or longer
wavelengths, namely, at near- and mid-infrared and/or (sub)millimeter
wavelengths, respectively. In astrophysics, several independent estimators can
be used to ascertain the true nature of the detected filamentary structures,
such as expert knowledge or a knowledge based on a large statistical
definition, such as the one used in citizen projects, of particular interest
for machine learning (Christy et al. 2022). Results of numerical simulations
and/or data obtained at other wavelengths can also be used. On the
multiwavelength data side, for example, filaments are clearly visible at other
wavelengths in the Galactic plane because they are composed of dust, and these
data can be used to ascertain the nature of the structures returned by the
segmentation process, as shown in Figure 22.
The UNet-based networks are supervised deep-learning algorithm. In spite of
the incomplete ground truth, these networks produced a good estimate of
filamentary structures. It is important to build a more enriched ground truth,
however, to solve more complex tasks such as instance segmentation. This might
be possible by combining several existing catalogs of filaments obtained on
the Galactic plane, for example, by combining the Hi-GAL catalog Schisano et
al. (2020) with getSF extractions made on several regions of the plane
(Men’shchikov 2021). Another possibility is to use filament segmentation by
UNets as a prestep and then consider the produced filaments mask as the ground
truth for the instance segmentation.
Another crucial step in filament segmentation using deep-learning algorithms
is data normalization. Filament detection depends not only on the intrinsic
column density of the structure, but also on the column density and the
structure of the background. By using local (per patch) normalization, the
filament contrast relative to the neighboring background is enhanced. As
illustrated in Figure 9, a classical local normalization method was used in
this work in order to enhance low-contrast filaments that are in turn well
integrated in the training process, allowing them to be representative.
Recently, more sophisticated multiscale normalization has been used in the Hi-
GAL image processing to highlight the faintest structures observed on the
Galactic plane (Li Causi et al. 2016). These normalized data are very
interesting as input for deep-learning networks. Unfortunately, no ground
truth exists for these normalized images so far, which does not allow their
use for the moment.
An important result of the segmentation for astrophysical purposes is to
determine the classification threshold (intensity of the segmentation map)
that allows for an optimal detection of filaments. While the optimal
thresholds reported in Table B allowed a good recovery of existing and new
filaments, filaments are still missed at these classification thresholds, some
because they were misclassified in the ground truth (see Sect. 4.2).
Another key point is to consider a region-specific optimal threshold rather
than a unique global one. According to results reported in Tables 4.2, 4.2,
and B, the optimal threshold of a given model is affected by the column
density of the studied zone. In this work, the optimal thresholds in Tables
4.2 and 4.2 were inferred in narrow zones, with a low number of labeled pixels
for the sparse zone. To obtain a robust estimate of a region-specific optimal
threshold, a split of the Galactic plane divided into large homogeneous zones
in terms of filament concentration is envisioned. The optimal thresholds can
then be inferred on these slices.
From the computational point of view, we trained the different networks on a
NVIDIA RTX 2080Ti. Table 6 gives the training time for the different scenarios
for 100 epochs. As UNet++ is a larger network than UNet, it takes slightly
more time to train: while UNet requires about $2.2$h, UNet++ asks for about
$2.8$h. As the patch data set is small (around 210 MB), these times are not
impacted by the loading of the data. When the training step is completed, the
neural networks are usually faster on CPU than on GPU Goodfellow et al. (2016)
because the transfer from CPU memory to GPU memory takes time. The
segmentation of the mosaics was therefore built on an Intel CPU machine
(i7-10610U). It took about $4$h per mosaic with an overlap of 30 pixels and
32$\times$32 patches. The total training and segmentation time is therefore
estimated to be $6.5$h per mosaic.
Table 6: Training time
Model | Training time [hour]
---|---
UNet$[10^{-2}]$ | $2.08$
UNet$[10^{-3}]$ | $2.23$
UNet$[10^{-4}]$ | $2.19$
UNet$[10^{-5}]$ | $2.17$
UNet++$[10^{-3}]$ | $3.22$
UNet++$[10^{-4}]$ | $2.79$
UNet++$[10^{-5}]$ | $2.73$
555Training time in hours for the schemes reported in Figure 14. The shortest
(longest) training time, in blue (red), is achieved by UNet[$10^{-2}$]
(UNet++[$10^{-3}$]).
From the method point of view, future works will include an improvement of the
segmentation process by using dedicated windows to build the patches as in
Pielawski & Wählby (2020). These tools may dramatically lower the
computational burden. We also investigate alternative ways of building a
larger set of patches while keeping good statistical properties. The method
used in this work guarantees a good preservation of the statistical
distribution, but leads to a small number of patches.
Although this method has some limitations, that is, the limited quantity of
patches for the training and the incomplete ground truth, and although it
reveals filament structures instead of extracting them, the net increase (a
factor between 2 and 7 on the whole segmented map) of the number of pixels
that belongs to the filament class and the robust detection of intrinsically
faint and/or low contrast ones offers important perspectives. We currently
explore the implementation of an augmented ground-truth data set using results
of numerical simulations on Galactic filaments. The extraction and separation
of filament pixels observed in the 2D segmented map is also ongoing, and we
add 3D spectroscopic data.
## 6 Conclusions
We explored whether deep-learning networks, UNet-5 and UNet++, can be used to
segment images of the whole Galactic plane in order to reveal filamentary
structures.
* •
Using molecular hydrogen column density image of the Galactic plane obtained
as part of the Hi-GAL survey and filaments previously extracted by Schisano et
al. (2020), we trained two different UNet-5 based networks with six different
scenarios based on a different initial learning rate.
* •
We showed the results and estimated the performances of the different
scenarios that we presented for two representative mosaics of the Galactic
plane selected for their low and high column density density and filament
content.
* •
We determined the best models for these mosaics based on machine-learning
metrics. We focused the training estimates on the recovery of input structures
(filaments and background) and defined for each mosaic and for the whole plane
an optimal classification threshold that ensured the best recovery of input
structures.
* •
We show that depending on the model and the selected threshold, new pixels
classified as filament candidates increase by a factor between 2 to 7
(compared to the input spine+branches structures used as ground truth). This
suggests that this new method has the potential of revealong filamentary
structures that may not be extracted by non-ML-based algorithms.
We point out the high potential of the produced database for future studies of
filaments (statistical analysis or follow-ups). We will use the results of the
numerical simulations to enrich the ground truth and assess the uncertainties
on segmented maps. The astrophysical analysis of the produced database is
ongoing and will be published in a separate paper.
###### Acknowledgements.
This project has received financial support from the CNRS through the MITI
interdisciplinary programs (Astroinformatics 2018, 2019). This project is part
of the ongoing project BigSF funded by the Aix Marseille Université A*Midex
Foundation (SB post-doctoral funding). AZ thanks the support of the Institut
Universitaire de France. We thank the anonymous referee for their constructive
comments that helped to improve the quality of the paper. This research has
made use of ”Aladin Sky Atlas” developed at CDS, Strasbourg Observatory,
France.
## References
* Alzahrani & Boufama (2021) Alzahrani, Y. & Boufama, B. 2021, SN Computer Science, 2, 1
* André et al. (2014) André, P., Di Francesco, J., Ward-Thompson, D., et al. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 27
* André et al. (2010) André, P., Men’shchikov, A., Bontemps, S., et al. 2010, A&A, 518, L102
* Aragon-Calvo (2019) Aragon-Calvo, M. A. 2019, Monthly Notices of the Royal Astronomical Society, 484, 5771
* Arzoumanian et al. (2011) Arzoumanian, D., André, P., Didelon, P., et al. 2011, A&A, 529, L6
* Arzoumanian et al. (2019) Arzoumanian, D., André, P., Könyves, V., et al. 2019, A&A, 621, A42
* Batson & Royer (2019) Batson, J. & Royer, L. 2019, in International Conference on Machine Learning, PMLR, 524–533
* Bekki (2021) Bekki, K. 2021, A&A, 647, A120
* Bernard et al. (2010) Bernard, J. P., Paradis, D., Marshall, D. J., et al. 2010, A&A, 518, L88
* Bianco et al. (2021) Bianco, M., Giri, S. K., Iliev, I. T., & Mellema, G. 2021, Monthly Notices of the Royal Astronomical Society, 505, 3982
* Bracco et al. (2020) Bracco, A., Bresnahan, D., Palmeirim, P., et al. 2020, A&A, 644, A5
* Chen et al. (2020) Chen, M. C.-Y., Di Francesco, J., Rosolowsky, E., et al. 2020, ApJ, 891, 84
* Christy et al. (2022) Christy, C. T., Jayasinghe, T., Stanek, K. Z., et al. 2022, PASP, 134, 024201
* Clarke et al. (2020) Clarke, S. D., Williams, G. M., & Walch, S. 2020, MNRAS, 497, 4390
* Elia et al. (2013) Elia, D., Molinari, S., Fukui, Y., et al. 2013, ApJ, 772, 45
* Fiorio & Gustedt (1996) Fiorio, C. & Gustedt, J. 1996, Theoretical Computer Science, 154, 165
* Fu & Mui (1981) Fu, K.-S. & Mui, J. 1981, Pattern recognition, 13, 3
* Fukui et al. (2019) Fukui, Y., Tokuda, K., Saigo, K., et al. 2019, ApJ, 886, 14
* Goodfellow et al. (2016) Goodfellow, I., Bengio, Y., & Courville, A. 2016, Deep learning (MIT press)
* Gu et al. (2022) Gu, W., Bai, S., & Kong, L. 2022, Image and Vision Computing, 120, 104401
* Hacar et al. (2020) Hacar, A., Bosman, A. D., & van Dishoeck, E. F. 2020, A&A, 635, A4
* Hacar et al. (2022) Hacar, A., Clark, S., Heitsch, F., et al. 2022, arXiv e-prints, arXiv:2203.09562
* Hacar et al. (2018) Hacar, A., Tafalla, M., Forbrich, J., et al. 2018, A&A, 610, A77
* Hastie et al. (2001) Hastie, T., Tibshirani, R., & Friedman, J. 2001, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer series in statistics (Springer)
* Hausen & Robertson (2020) Hausen, R. & Robertson, B. E. 2020, ApJS, 248, 20
* He et al. (2016) He, K., Zhang, X., Ren, S., & Sun, J. 2016, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 770–778
* Hoemann et al. (2021) Hoemann, E., Heigl, S., & Burkert, A. 2021, arXiv e-prints, arXiv:2104.04541
* Hsieh et al. (2021) Hsieh, C.-H., Arce, H. G., Mardones, D., Kong, S., & Plunkett, A. 2021, ApJ, 908, 92
* Jadon (2020) Jadon, S. 2020, 2020 IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB), 1
* Kingma & Ba (2015) Kingma, D. P. & Ba, J. 2015, in ICLR
* Koch & Rosolowsky (2015) Koch, E. W. & Rosolowsky, E. W. 2015, MNRAS, 452, 3435
* Könyves et al. (2020) Könyves, V., André, P., Arzoumanian, D., et al. 2020, A&A, 635, A34
* Krizhevsky et al. (2012) Krizhevsky, A., Sutskever, I., & Hinton, G. E. 2012, Communications of the ACM, 60, 84
* Kull et al. (2017) Kull, M., Silva Filho, T. M., & Flach, P. 2017, Electronic Journal of Statistics, 11, 5052
* Kumar et al. (2020) Kumar, M. S. N., Palmeirim, P., Arzoumanian, D., & Inutsuka, S. I. 2020, A&A, 642, A87
* LeCun et al. (1989) LeCun, Y., Boser, B. E., Denker, J. S., et al. 1989, Neural Computation, 1, 541
* Lee et al. (2015) Lee, C.-Y., Xie, S., Gallagher, P., Zhang, Z., & Tu, Z. 2015, in Proceedings of Machine Learning Research, Vol. 38, Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, ed. G. Lebanon & S. V. N. Vishwanathan (San Diego, California, USA: PMLR), 562–570
* Lee et al. (2012) Lee, E. J., Murray, N., & Rahman, M. 2012, ApJ, 752, 146
* Leurini et al. (2019) Leurini, S., Schisano, E., Pillai, T., et al. 2019, A&A, 621, A130
* Li Causi et al. (2016) Li Causi, G., Schisano, E., Liu, S. J., Molinari, S., & Di Giorgio, A. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, ed. H. A. MacEwen, G. G. Fazio, M. Lystrup, N. Batalha, N. Siegler, & E. C. Tong, 99045V
* Long et al. (2015) Long, J., Shelhamer, E., & Darrell, T. 2015, in Proceedings of the IEEE conference on computer vision and pattern recognition, 3431–3440
* Mattern et al. (2018) Mattern, M., Kauffmann, J., Csengeri, T., et al. 2018, A&A, 619, A166
* Men’shchikov (2021) Men’shchikov, A. 2021, A&A, 649, A89
* Milletari et al. (2016) Milletari, F., Navab, N., & Ahmadi, S.-A. 2016, in 2016 fourth international conference on 3D vision (3DV), IEEE, 565–571
* Molinari et al. (2016) Molinari, S., Schisano, E., Elia, D., et al. 2016, A&A, 591, A149
* Molinari et al. (2010) Molinari, S., Swinyard, B., Bally, J., et al. 2010, A&A, 518, L100
* Oktay et al. (2018) Oktay, O., Schlemper, J., Folgoc, L. L., et al. 2018, arXiv preprint arXiv:1804.03999
* Pielawski & Wählby (2020) Pielawski, N. & Wählby, C. 2020, PloS one, 15, e0229839
* Priestley & Whitworth (2022) Priestley, F. D. & Whitworth, A. P. 2022, arXiv e-prints, arXiv:2203.04367
* Robinson & Whelan (2004) Robinson, K. & Whelan, P. F. 2004, Pattern Recognition Letters, 25, 1759
* Robitaille et al. (2020) Robitaille, T., Deil, C., & Ginsburg, A. 2020, Astrophysics Source Code Library, ascl
* Ronneberger et al. (2015) Ronneberger, O., Fischer, P., & Brox, T. 2015, in International Conference on Medical image computing and computer-assisted intervention, Springer, 234–241
* Schisano et al. (2020) Schisano, E., Molinari, S., Elia, D., et al. 2020, MNRAS, 492, 5420
* Schisano et al. (2014) Schisano, E., Rygl, K. L. J., Molinari, S., et al. 2014, ApJ, 791, 27
* Shimajiri et al. (2019) Shimajiri, Y., André, P., Ntormousi, E., et al. 2019, A&A, 632, A83
* Soille (2003) Soille, P. 2003, Morphological Image Analysis: Principles and Applications, 2nd edn. (Berlin, Heidelberg: Springer-Verlag)
* Sousbie (2011) Sousbie, T. 2011, MNRAS, 414, 350
* Traficante et al. (2011) Traficante, A., Calzoletti, L., Veneziani, M., et al. 2011, MNRAS, 416, 2932
* Vincent (1993) Vincent, L. 1993, IEEE Transactions on Image Processing, 2, 176
* Wells & Norman (2021) Wells, A. I. & Norman, M. L. 2021, The Astrophysical Journal Supplement Series, 254, 41
* Xia & Kulis (2017) Xia, X. & Kulis, B. 2017, arXiv preprint arXiv:1711.08506
* Zavagno et al. (2020) Zavagno, A., André, P., Schuller, F., et al. 2020, A&A, 638, A7
* Zhou et al. (2019) Zhou, Z., Siddiquee, M. M. R., Tajbakhsh, N., & Liang, J. 2019, IEEE transactions on medical imaging, 39, 1856
* Zhu et al. (2019) Zhu, G., Lin, G., Wang, D., Liu, S., & Yang, X. 2019, Solar Physics, 294, 117
* Zucker & Chen (2018) Zucker, C. & Chen, H. H.-H. 2018, ApJ, 864, 152
## Appendix A Different input data maps
All the input maps used in the training process (the column density map, the
filament mask, the background mask, and the missing data mask maps; see
Section 2.2) were originally produced from the individual Hi-GAL mosaics
extending over 10° of longitude (see Schisano et al. (2020)). To facilitate
the use of the four input maps during the training process, we merged the
individual maps using the Python module
reproject666https://reproject.readthedocs.io/en/stable/index.html. The patches
were then cut as shown in Figure 4. The four input maps used in the supervised
training process are presented in Figure 3. These maps are described below.
### N${}_{\rm{H_{2}}}$ map
The column density maps N${}_{\rm{H_{2}}}$ were obtained as part of the
Herschel observations of the Galactic plane, Hi-GAL. The column density
N${}_{\rm{H_{2}}}$ maps were computed from the photometrically calibrated Hi-
GAL mosaics following the approach described in Elia et al. (2013). The
Herschel data were convolved to the 500 $\mu$m resolution ( 36$\arcsec$) and
rebinned on that map grid. Then a pixel-by-pixel fitting with a single-
temperature greybody function was performed, as described in Elia et al.
(2013); Schisano et al. (2020). We directly used the data derived and
presented in Schisano et al. (2020).
### Missing-data map
The N${}_{\rm{H_{2}}}$ map presents local degraded zones (noise, saturation,
and overlap issues) that we wished to exclude from the training process.
Examples of these degraded zones are shown in Figure 23.
(a)
(b)
(c)
(d)
Figure 23: Four sources of missing data. (a) Example of a complex boundary due
to satellite scanning. (b) Example of a noisy pattern inside the column
density map (in log scale) due to the satellite scan. (c) Saturated pixels.
(d) Example of structured artifacts built by the mapping process.
In Figure 23 we show the four sources of missing data. First, the boundaries
of the mosaics show a grid-like pattern, as shown in Figure 22(a). These
features are due to the scanning pattern followed by the Herschel satellite
while performing the observations of an Hi-GAL tile. The grid pattern
corresponds to the slewing phase of the scanning pattern, when the satellite
inverted the scan direction. Second, the Hi-GAL mosaics were built using the
Unimap mapmaking (Traficante et al. 2011) following a strategy to avoid large-
scale intensity gradients over the image. We refer to Appendix A of Schisano
et al. (2020) for a detailed description of how the mosaics were built. In
short, UNIMAP mapmaker was run to simultaneously process the data of two
adjacent Hi-GAL tiles, creating what the authors call a texel, spanning
4°$\times$ 2°. The UNIMAP processing is able to deal with the slewing region
along the side covered by observation of both Hi-GAL tiles. This produces an
image of better quality with respect to the simple mosaicking of the two
tiles. Multiple texels are combined together to produce the overall mosaic
that is 10° wide in longitude. While it is possible to directly process a
larger portion of the Galactic plane with UNIMAP, this approach introduces
large gradients in intensity over the entire mosaic. The UNIMAP mapmaker
produces maps with an average intensity level equal to zero. Therefore, the
texels require to be calibrated in flux (Bernard et al. 2010). In some cases,
there are discrepancies in the calibration level of adjacent texels that
introduce sharp variations in intensity in the overlapping region between
texels. An example is show in Figure 22(b) and Figure 22(d). These variations
are not physical and were masked out from the learning process.
The final full map of missing data was built by combining all the types of
possible missing data described above. They were then removed from the
training.
### Filament-mask map
Schisano et al. (2020) published a catalog of filaments detected in the
Galactic plane from N${}_{\rm{H_{2}}}$ images obtained with Hi-GAL photometric
data. The position of spines and branches is available through binary masks in
which pixels belonging to these structures are tagged for the 32059 filaments
published in the catalog (see Figure 2 and their figure 3). We used this
information as an input mask to define our ground truth for the filaments. The
ground truth was defined over the outputs of the Hi-GAL filament catalog, and
it depends on the completeness of that catalog. This implies that the ground
truth is not absolutely fully defined because it will miss the information
from any feature that may not have been detected by Schisano et al. (2020).
### Background map
On all pixels, the column density mosaic contains emission from both filaments
and background (Schisano et al. 2020). We examined each 10° mosaic to define a
background level as the lowest level of emission observed on the mosaic that
does not overlap any filament branches that were detected and used as ground
truth. This means that our definition of the background emission sets the
background class on a very low number of pixels, but ensures that these pixels
do not contain filaments. We work on a more precise definition of the
background to allow more pixels to be labeled in this class. We are primarily
interested in detecting filamentary structures here. Moreover, the local
normalization applied to all patches before the training process (see Figure
9) tends to limit the impact of the background on the detection of filaments.
This reduces the need for a precise definition of the background further.
## Appendix B Supplementary quantitative scores
Table 7: Dice index
Model | Dice index [$0.2$] | Dice index [$0.4$] | Dice index [$0.6$] | Dice index [$0.8$]
---|---|---|---|---
UNet$[10^{-2}]$ | $93.19$ | $94.34$ | $94.07$ | $92.06$
UNet$[10^{-3}]$ | $93.13$ | $94.37$ | $94.16$ | $92.43$
UNet$[10^{-4}]$ | $92.75$ | $94.09$ | $93.79$ | $91.9$
UNet$[10^{-5}]$ | $82.4$ | $89.6$ | $91.3$ | $87.66$
UNet++$[10^{-3}]$ | $93.46$ | $94.61$ | $94.28$ | $92.36$
UNet++$[10^{-4}]$ | $93.04$ | $94.07$ | $93.6$ | $91.57$
UNet++$[10^{-5}]$ | $90.24$ | $91.88$ | $91.02$ | $87.67$
777Dice index evaluation on the test set for the schemes reported in Figure 14
at classification threshold values of $0.8$, $0.6$, $0.4,$ and $0.2$. Blue
(red) refers to the best (less performing) scheme in each column. The bold
scores correspond to the absolute best (if in blue) or lowest (if in red) dice
index. The closer the dice index to $1$, the better. Dice index values in test
were aligned with performances in training and validation steps as close
performances are obtained for schemes with initial learning-rate values in
$[10^{-4},10^{-3},10^{-2}]$ and lower performance are noted for schemes with a
learning-rate value of $10^{-5}$. Moreover, UNet++[$10^{-3}$] and
UNet[$10^{-5}$] result in the best ($94.61\%$ at $0.4$) and least performing
($82.4\%$ at $0.2$) schemes, respectively.
Table 8: Segmentation scores
l—*7l Scores (%)Model UNet [10-2] UNet [10-3] UNet [10-4] UNet [10-5] UNet++
[10-3] UNet++ [10-4] UNet++ [10-5]
Precision at 0.8 99.71 99.74 99.75 99.73 99.74 99.74 99.72
Precision at 0.5 98.29 98.45 98.54 97.07 98.47 98.42 97.35
Precision at 0.2 91.43 92.08 91.75 64.85 92.45 91.17 86.16
Precision at thropt 95.65 95.85 95.88 94.95 95.75 95.94 93.61
Recall at 0.8 72.38 71.34 70.50 65.14 70.87 71.74 65.54
Recall at 0.5 90.13 89.90 89.13 89.25 89.78 89.78 85.59
Recall at 0.2 97.49 97.64 97.29 98.72 97.61 97.50 95.81
Recall at thropt 95.26 95.50 94.96 92.24 95.67 94.97 91.56
Dice at 0.8 83.88 83.19 82.62 78.81 82.87 83.46 79.10
Dice at 0.5 94.05 94.00 93.62 93.01 93.95 93.92 91.11
Dice at 0.2 94.41 94.82 94.49 78.31 95.01 94.27 90.76
Diceopt 95.46 95.68 95.42 93.58 95.71 95.45 92.57
MS at 0.8 14.36 14.81 16.51 17.58 15.28 15.10 17.69
MS at 0.5 4.61 4.40 5.25 4.07 4.62 4.55 5.83
MS at 0.2 0.83 0.77 0.81 0.28 0.85 0.82 1.10
MS at thropt 1.95 1.69 1.88 2.72 1.68 1.97 2.85
thropt 0.32 0.31 0.31 0.44 0.30 0.33 0.34
888Segmentation scores evaluated on the Galactic plane segmented by the models
reported in Figure 14. Precision, recall, dice index, and MS rate are
evaluated at classification threshold values of $0.8$, $0.5$, $0.2,$ and the
optimal threshold. The latter refers to the threshold value optimizing the
dice index and estimated using P-R curves (see Figure 15(c)). Blue (red)
refers to the best (least performing) scheme in each row. The bold scores
correspond to the absolute best (if in blue) or lowest (if in red) performance
per score. Overall, the segmentation scores obtained on the fully segmented
Galactic plane consolidate the results obtained with the zone of 350.3-353.5°
in Table 4.2
|
# FoPro: Few-Shot Guided Robust Webly-Supervised Prototypical Learning
Yulei Qin,1 Xingyu Chen,1 Chao Chen,1 Yunhang Shen,1 Bo Ren,1
Yun Gu,2 Jie Yang,2 Chunhua Shen3
###### Abstract
Recently, webly supervised learning (WSL) has been studied to leverage
numerous and accessible data from the Internet. Most existing methods focus on
learning noise-robust models from web images while neglecting the performance
drop caused by the differences between web domain and real-world domain.
However, only by tackling the performance gap above can we fully exploit the
practical value of web datasets. To this end, we propose a Few-shot guided
Prototypical (FoPro) representation learning method, which only needs a few
labeled examples from reality and can significantly improve the performance in
the real-world domain. Specifically, we initialize each class center with few-
shot real-world data as the “realistic” prototype. Then, the intra-class
distance between web instances and “realistic” prototypes is narrowed by
contrastive learning. Finally, we measure image-prototype distance with a
learnable metric. Prototypes are polished by adjacent high-quality web images
and involved in removing distant out-of-distribution samples. In experiments,
FoPro is trained on web datasets with a few real-world examples guided and
evaluated on real-world datasets. Our method achieves the state-of-the-art
performance on three fine-grained datasets and two large-scale datasets.
Compared with existing WSL methods under the same few-shot settings, FoPro
still excels in real-world generalization. Code is available at
https://github.com/yuleiqin/fopro.
## Introduction
The past decade has witnessed a revolution in computer vision with the advent
of large-scale labeled datasets (_e.g._ , ImageNet (Deng et al. 2009)).
However, a large collection of data are sometimes inaccessible, let alone the
time-consuming and expensive annotations. On the contrary, there are abundant
weakly labeled images on the Internet. Therefore, webly supervised learning
(WSL) has attracted growing attention from researchers (Krause et al. 2016;
Kaur, Sikka, and Divakaran 2017; Kolesnikov et al. 2019; Zhang et al. 2020; Tu
et al. 2020; Liu et al. 2021; Zhang et al. 2021).
Figure 1: Differences between web and real-world images. (a) Web dataset
noise. (b) Web dataset bias.
Queries and tags are directly used as weak labels without verification,
bringing about a considerable proportion of noises in web datasets (_e.g._ ,
20% in JMT-300M (Sun et al. 2017), 34% in WebVision (Li et al. 2017), and 32%
in WebFG496 (Sun et al. 2021)). As shown in Fig. 1(a), various noises include
label flipping errors, semantic ambiguity of polysemy queries, and outliers of
unknown categories. To alleviate their effect, prior knowledge such as
neighbor density (Guo et al. 2018), reference clean sets (Jiang et al. 2018;
Lee et al. 2018), and side information (Zhou et al. 2020; Cheng et al. 2020)
is explored for label correction and sample selection. Recently, Li et al.
(Li, Xiong, and Hoi 2020) develop a self-supervised method with representative
class prototypes (MoPro) to achieve satisfying performance.
Most existing WSL methods are merely concerned with noise reduction. They
ignore the model degradation in real-world scenarios because the performance
on web domain testing sets is emphasized in previous model assessments. Domain
gaps exist between images crawled from the web (_e.g._ , advertising photos,
artworks, and rendering) and those captured in reality (see Fig. 1(b)). In
this case, the better fitting of web images counteractively leads to worse
generalization on practical applications. Few studies try to tackle such
performance gaps by domain adaptation methods. For example, Xu et al. (Xu et
al. 2016) distill knowledge from the web domain to the real-world domain. Niu
et al. (Niu, Li, and Xu 2015) fine-tune pretrained models on real-world
datasets. However, both of them need plenty of labeled data in the target
domain, which impedes practicability.
Figure 2: (a) The t-SNE (Van der Maaten and Hinton 2008) of the low-
dimensional embeddings of web images substantiates that with the increase of
$K$, class prototypes ($\times$) are regularized to approach few shots
($\Delta$) with dense intra-class and isolated inter-class distribution. (b)
The diminished performance gap between the testing results of web
(WebVision1k) and real-world (ImageNet1k) images confirms that FoPro takes
full advantage of few shots to improve its generalization beyond the web
domain, making web data truly useful in learning representations for
actuality. (c) FoPro estimates noise-robust prototypes to pull instances
nearby closer. Noisy samples are filtered by assessing their relation with
prototypes. Only clean ones update prototypes in return. FoPro achieves better
interpretability, discriminability, and generalization. Best viewed magnified.
Unlike the methods above, our objective is to cost-efficiently mine web data
for real-world applications. We handle both the noise and domain gap by
resorting to a few human-labeled samples for guidance on whom to learn from
and what to learn. In our setting, clean labeled examples are too scarce to
train or fine-tune a deep model, and therefore alternative methods need to be
developed in response.
To this end, we propose a robust prototypical representation learning method
with noisy web data and a few clean examples (see Fig.2). Motivated by the
anchoring role of class prototypes (Li et al. 2020; Li, Xiong, and Hoi 2020),
we introduce Few-shot guided Prototypes, termed as FoPro, to effectively deal
with noise and domain gap. Technically, we project features of the penultimate
layer of a classification model to low-dimensional embeddings. The critical
problem is how to formulate a class-representative and domain-generalized
prototype in the embedding space without being deviated by the dominating
noises. Due to noise memorization (Arpit et al. 2017), simply averaging over
instances with high prediction confidence does not promise a noise-robust
estimation. Consequently, we first initialize each class prototype with
realistic few shots as the cluster center. Secondly, intra-class distance is
shortened by contrastive learning between web instances and prototypes. Then,
high-quality web examples are involved in polishing prototypes to improve
discriminability. Simultaneously, high similarity between prototypes and few
shots is regularized to maximize the interpretability and generalizability of
prototypes. Finally, we quantify the compatibility between instances and
prototypes by the proposed relation module for sample selection and
correction, which benefits prototype update in the next iteration.
Specifically, the relation module learns a flexible and transferable metric to
assess if a web image corresponds to its label. Besides, we set siamese
encoders (He et al. 2020) and prototypes are only updated by the momentum
encoder in a smooth and progressive way.
Our contributions can be summarized as follows:
* •
We propose a new few-shot learning setting in WSL with abundant noisy web
images and a few real-world images, which aims to improve the performance of
WSL for real-world applications in a cost-efficient way.
* •
We present a new method under the setting above called FoPro, which
simultaneously solves noise and data bias in an end-to-end manner.
Experimental results show that our method can significantly improve the
performance in real-world benchmark datasets.
* •
We propose a new relation module for label noise correction. It outperforms
existing methods that use fixed metrics (_e.g._ , cosine distance) by
evaluating instance-prototype similarity with a learnable metric.
* •
Extensive experiments on the fine-grained WebFG496 and the large-scale
WebVision1k datasets confirm the superiority of FoPro over the state-of-the-
art (SOTA) methods. Performance under the increasing $K$-shot settings
demonstrates that FoPro utilizes few shots wisely to bridge the gap towards
real-world applications.
## Related Work
### Webly Supervised Learning
WSL aims to leverage vast but weakly-annotated web resources. Previous works
utilize web images for tasks including classification (Bergamo and Torresani
2010; Wu et al. 2021; Yao et al. 2017, 2020), detection (Divvala, Farhadi, and
Guestrin 2014; Shen et al. 2020), and segmentation (Shen et al. 2018; Jin,
Ortiz Segovia, and Susstrunk 2017).
Recently, noise cleaning methods such as self-contained confidence (SCC) (Yang
et al. 2020) and momentum prototype (MoPro) (Li, Xiong, and Hoi 2020) are
proposed to improve representation learning in WSL. SCC balances two
supervision sources from web labels and predicted labels by introducing
instance-wise confidence. MoPro targets model pretraining for several down-
streaming tasks by combining self-supervised and webly-supervised techniques.
Specifically, MoPro is closely related to ours since the contrast between
instances and prototypes is used to learn discriminative features. Different
from MoPro, we formulate a brand-new setting where a few samples labeled by
experts are available. To assure that class prototypes are not misled by
noise, an implicit constraint on distribution is achieved by enforcing high
similarity between prototypes and few shots. Furthermore, we estimate the
relation score between instances and prototypes to correct labels and discard
out-of-distribution (OOD) samples.
### Learning from Noisy Labels
Labels in human-annotated datasets can still be noisy due to lack of expert
domain knowledge (Song et al. 2022). To prevent deep models from overfitting
noisy labels, several studies have been conducted and can be categorized as:
1) robust architecture (_e.g._ , noise transition layer (Chen and Gupta 2015)
and probability model (Xiao et al. 2015)); 2) regularization techniques
(_e.g._ , label smoothing (Pereyra et al. 2017) and mix-up (Zhang et al.
2018)); 3) robust losses (_e.g._ , MAE (Ghosh, Kumar, and Sastry 2017) and GCE
(Zhang and Sabuncu 2018)); 4) loss refinement (_e.g._ , reweighting (Wang,
Liu, and Tao 2017) and bootstrapping (Reed et al. 2015)); 5) sample selection
(_e.g._ , multi-model collaboration (Malach and Shalev-Shwartz 2017) and
iterative strategies (Li, Socher, and Hoi 2019)). Hybrid approaches are
designed in practice. For example, PeerLearn (Sun et al. 2021) develops a two-
stage framework with peer models. Each model chooses clean samples
independently and feeds them to train the other model. Different from the
existing methods, we do not assume that samples with small losses or high
confidence are clean. Instead, we maintain class prototypes and filter out
noise by comparing instances and prototypes in a non-linear metric. Moreover,
PeerLearn presumes that the percentage of noise is consistent across
categories, which contradicts our observation.
### Contrastive Representation Learning
Contrastive learning methods can be roughly categorized as: 1) context-
instance contrast, where the relationship of local parts with respect to
global context is learned (Kim et al. 2018); 2) instance-wise contrast, where
similar image pairs are pulled closer with dissimilar pairs pushed farther (He
et al. 2020; Chen et al. 2020a). Prototypical contrastive learning (PCL) (Li
et al. 2020) encourages each image embedding to be adjacent to its assigned
cluster prototype. However, their method is under an unsupervised setting
where k-means clustering is used to generate prototypes. Our model is
supervised by both numerous-yet-noisy web labels and limited-yet-clean few-
shot labels. Besides, in PCL, batch embeddings in the current epoch are
contrasted with the “outdated” prototypes in the previous epoch. FoPro keeps
modifying prototypes smoothly all the time so that clean samples can be
pinpointed by the latest features.
Figure 3: Overview of FoPro. The encoder, classifier, and projector are
trained to produce discriminative embeddings. Class prototypes are first
initialized by few shots and then polished with clean samples for contrastive
learning to regularize cluster distribution. Instance-wise contrastive loss is
optimized simultaneously. The relation module learns a distance metric between
an instance and its assigned class prototype. Finally, we adjust web labels
for confidence-weighted hybrid target learning.
## Method
In this section, a formal description of our few-shot WSL setting is
presented, followed by the detailed explanation of FoPro. Fig. 3 illustrates
the model architecture.
### Problem Statement
Existing WSL setting aims to train a deep model
$\mathcal{F}(\theta_{e};\theta_{c})$ with the optimal parameters of encoder
$\theta_{e}^{*}$ and classifier $\theta_{c}^{*}$ from the web dataset
$\mathit{D}^{w}=\\{(\mathbf{x}_{i}^{w},y_{i}^{w})\\}_{i=1}^{N^{w}}$. Here,
$\mathbf{x}_{i}^{w}$ denotes an image, $y_{i}^{w}\in\\{1,...,C\\}$ is its
class label. The number of classes and images are $C$ and $N^{w}$,
respectively. Due to noise issues, $y_{i}^{w}$ might not equal to the ground-
truth $y_{i}^{*}$. If $y_{i}^{w}\neq y_{i}^{*}$ and
$y_{i}^{*}\in\\{1,...,C\\}$, $(\mathbf{x}_{i}^{w},y_{i}^{w})$ is viewed as an
in-distribution (IND) sample with label-flipping error. If $y_{i}^{w}\neq
y_{i}^{*}$ and $y_{i}^{*}\notin\\{1,...,C\\}$, then
$(\mathbf{x}_{i}^{w},y_{i}^{w})$ is an out-of-distribution (OOD) sample.
We propose a new WSL setting that additional real-world images are available
with verified labels:
$\mathit{D}^{t}=\\{(\mathbf{x}_{i}^{t},y_{i}^{t})\\}_{i=1}^{N^{t}}$ and
$y_{i}^{t}=y_{i}^{*}$. The number of real-world samples is $N^{t}=K\cdot C$,
where $K$ denotes $K$-shot per class. Our FoPro aims to achieve two goals with
$\mathit{D}^{w}$: 1) to learn generalizable representations from high-quality
examples; 2) to correct IND samples and discard OOD samples.
### Model Architecture
The main components of FoPro include siamese encoder backbones, a classifier,
a projector, a reconstructor, an auxiliary classifier, and a relation module.
Our siamese encoder networks share the same architecture. Enlighted by MoCo
(He et al. 2020), we update parameters of the first encoder $\theta_{e}^{1}$
by back-propagation and employ momentum update for the second encoder
$\theta_{e}^{2}$:
$\theta_{e}^{2}=m_{e}\theta_{e}^{2}+(1-m_{e})\theta_{e}^{1},$ (1)
where $m_{e}$ is the momentum parameter. The plain and momentum encoders
respectively extract features $\mathbf{v}_{i}^{\\{w;t\\}}$ and
$\mathbf{v}_{i}^{\prime\\{w;t\\}}\in{\rm I\\!R}^{d_{e}}$ from inputs
$\mathbf{x}_{i}^{\\{w;t\\}}$ and their augmented counterparts
$\mathbf{x}_{i}^{\prime\\{w;t\\}}$. Note that our encoder is structure-
agnostic, and its choices are up to specific tasks. All layers except the last
fully connected (FC) layer are used.
A classifier is trained to map features $\mathbf{v}_{i}^{\\{w;t\\}}$ to the
predicted probabilities $\mathbf{p}_{i}^{\\{w;t\\}}$ over $C$ classes. It
consists of one FC layer with softmax activation.
A projector distills discriminative contents from features
$\mathbf{v}_{i}^{\\{w;t\\}}$ for low-dimensional embeddings
$\mathbf{z}_{i}^{\\{w;t\\}}\in{\rm I\\!R}^{d_{p}}$. It is composed of two FC
layers and one ReLU layer. We follow (Chen et al. 2020a, b) to perform
contrastive learning in the embedding space after projection.
$\ell_{2}$-normalization is involved for unit-sphere constraint on
$\mathbf{z}_{i}^{\\{w;t\\}}$.
A reconstructor recovers $\tilde{\mathbf{v}}_{i}^{\\{w;t\\}}$ from
$\mathbf{z}_{i}^{\\{w;t\\}}$, where $\tilde{\mathbf{v}}_{i}^{\\{w;t\\}}$
should be as close as possible to $\mathbf{v}_{i}^{\\{w;t\\}}$. Symmetric
structure is adopted for the projector and reconstructor.
An auxiliary classifier with one FC layer generates probabilities
$\mathbf{q}_{i}^{t}$ over $C$ classes based on embeddings
$\mathbf{z}_{i}^{t}$.
Our relation module compares each pair of one instance embedding
$\mathbf{z}_{i}^{\\{w;t\\}}$ and one class prototype $\mathbf{c}_{k}\in{\rm
I\\!R}^{d_{p}},k\in\\{1,...,C\\}$ for distance measurement. Given the
concatenated embeddings $[\mathbf{z}_{i}^{\\{w;t\\}},\mathbf{c}_{k}]$, it
learns to score their closeness $r_{ik}\in{\rm I\\!R}$ by two FC layers with
one ReLU layer.
### Training Strategy
FoPro employs a four-stage training strategy.
#### Stage 1: Preparation
In this early stage, we warm up the system by learning common, regular
patterns for the first $T_{1}$ epochs. As discovered by (Arpit et al. 2017),
easy examples are reliably learned with simple patterns before the model
overfits noise. We achieve this via training the encoder and classifier with
cross-entropy loss.
$\mathcal{L}_{i}^{cls}=-\log(\mathbf{p}_{i(y_{i})}^{\\{w;t\\}}).$ (2)
Since $\mathbf{v}_{i}^{\\{w;t\\}}$ might contain redundant features that make
outliers indistinguishable, we set a projector to only keep principal
components. The previous method PCL stems from the analogy of autoencoder to
PCA, and learns projection by minimizing the reconstruction loss for the
projector and reconstructor. In preliminary experiments, however, we find that
such optimization cannot give a good starting point for prototype
initialization because $\mathbf{z}_{i}^{\\{w;t\\}}$ is not necessarily class-
indicative. Therefore, an auxiliary classifier is applied on
$\mathbf{z}_{i}^{t}$ to bring back its representation capacity. Only few shots
are used here due to purity concerns.
$\mathcal{L}_{i}^{prj}=\|\tilde{\mathbf{v}}_{i}^{\\{w;t\\}}-\mathbf{v}_{i}^{\\{w;t\\}}\|^{2}_{2}-\log(\mathbf{q}_{i(y_{i})}^{t}).$
(3)
#### Stage 2: Incubation
Clean few shots play an anchoring role in “territory” enclosure in the
embedding space. Given extracted embeddings from the momentum encoder, we
initialize prototypes by averaging few shots in each class.
$\hat{\mathbf{c}}_{k}=\frac{1}{K}\sum_{y_{i}=k}{\mathbf{z}_{i}^{t}},\mathbf{c}_{k}=\frac{\hat{\mathbf{c}}_{k}}{\|\hat{\mathbf{c}}_{k}\|_{2}}.$
(4)
In this stage, we begin to pull instances within one class towards the
prototype for $T_{2}$ epochs. Besides, instance-level discrimination is
encouraged by contrastive losses (Chen et al. 2020a) to improve separation
across classes.
$\mathcal{L}_{i}^{pro}=-\log\frac{\exp((\mathbf{z}_{i}^{w;t}\cdot\mathbf{c}_{y_{i}}-\delta^{w;t})/\phi_{y_{i}})}{\sum_{k=1}^{C}\exp((\mathbf{z}_{i}^{w;t}\cdot\mathbf{c}_{k}-\delta^{w;t})/\phi_{k})},$
(5)
$\mathcal{L}_{i}^{ins}=-\log\frac{\exp(\mathbf{z}_{i}^{w;t}\cdot\mathbf{z}_{i}^{\prime
w;t}/\tau)}{\sum_{j=1}^{Q}\exp(\mathbf{z}_{i}^{w;t}\cdot\mathbf{z}_{j}^{\prime
w;t}/\tau)},$ (6)
where $\delta^{w;t}$ refers to the margin coefficient, and $Q$ is the length
of the memory bank for storing embeddings of visited instances. Temperature
coefficients can be fixed as $\tau$ or class-dependent as $\phi_{k}$. We put
constraints on learning representations with a high margin so that clean few
shots gather around prototypes tightly, ensuring better justification and
interpretability. Furthermore, to regularize the distribution of each class
cluster, adjustable temperature coefficients (Li et al. 2020) are estimated
based on concentration.
$\phi_{k}=\frac{\sum_{y_{i}=k}\|\mathbf{z}_{i}^{w;t}-\mathbf{c}_{k}\|_{2}}{N_{k}^{w;t}\log(N_{k}^{w;t}+\alpha)},$
(7)
where $N_{k}^{w;t}$ denotes the number of web and few-shot instances of class
$k$, and $\alpha$ is a smoother. Embeddings of large, loose clusters will be
penalized more to approach their prototypes, while those of small, tight
clusters will be relaxed.
#### Stage 3: Illumination
With parameters of the encoder and projector fixed, the relation module learns
to score the compatibility between one instance and each prototype. It sheds
light on whether the given label of a web image is correct. We select clean
samples $\mathit{D}^{r}$ for training the relation module.
$\mathit{D}^{r}=\mathit{D}^{t}\cup\\{(\mathbf{x}_{i}^{w},y_{i}^{w})|\sum_{j=1}^{C}|(\mathbf{z}_{i}^{w}-\mathbf{c}_{y_{i}})\cdot\mathbf{c}_{j}|\leq\sigma\\},$
(8)
where $\sigma$ is a threshold between 0 and 1. Such criterion comprehensively
considers both the cosine distance between instance and prototypes, and the
distance among prototypes. Then, the relation module is trained for $T_{3}$
epochs by:
$\mathcal{L}_{i}^{rel}=-\log\frac{\exp(r_{iy_{i}})}{\sum_{k=1}^{C}\exp(r_{ik})}.$
(9)
#### Stage 4: Verification
Armed with “pretrained” model, we start label correction, OOD removal,
prototype update, and continue noise-robust learning for $T_{4}$ epochs. Three
sources of prior knowledge are incorporated for cleaning: 1) self-prediction;
2) instance-prototype similarity; 3) relation score. Rules for adjusting
labels are detailed below:
$\mathbf{s}_{i}^{w}=\beta\mathbf{p}_{i}^{w}+(1-\beta)[\mathbf{c}_{1},...,\mathbf{c}_{C}]^{T}\cdot\mathbf{z}_{i}^{w}$
(10) $\hat{y}_{i}^{w}=\left\\{\begin{array}[]{lcl}y_{i}^{w}&&{\textrm{if}\
r_{iy_{i}}>\gamma,}\\\ \arg\max_{k}\mathbf{s}_{i(k)}^{w}&&{\textrm{else
if}\max_{k}\mathbf{s}_{i(k)}^{w}>\gamma,}\\\ y_{i}^{w}&&{\textrm{else if}\
\mathbf{s}_{i(y_{i})}^{w}>1/C,}\\\ \textrm{Null}\
(OOD)&&{\textrm{otherwise,}}\end{array}\right.$ (11)
where $\gamma$ is a threshold between 0 and 1. Since fine-grained categories
share highly similar visual patterns, the relation module is only used for
positive verification of the initial web label. Besides, we introduce an
alternative confidence measure from self-prediction and similarity for label
reassignment. When the first two conditions are not satisfied, an image will
be kept as hard example if its confidence is above average. Otherwise, it is
discarded as OOD. Note that the basic control flow above is inspired by MoPro.
We further improve it with the proposed relation module (Eq.11 cond. 1) to
better evaluate the compatibility between instances and class prototypes, and
thereafter enable accurate label-flipping-error correction and OOD removal
without ignoring hard examples by mistake (Eq.11 conds. 2–4).
After label adjustment, we exploit the predicted probabilities as pseudo-
labels for self-training (Tanaka et al. 2018; Han, Luo, and Wang 2019). Such
soft targets can be viewed as a regularizer on the classifier like label
smoothing (Müller, Kornblith, and Hinton 2019) and self-knowledge distillation
(Hinton et al. 2015). Instead of using a fixed coefficient, we follow (Yang et
al. 2020) to leverage confidence on the corrected label for weighting soft and
hard targets.
$\displaystyle\mathcal{L}_{i}^{cls}=$
$\displaystyle-\log(\mathbf{p}_{i(y_{i})}^{t})-\mathbf{s}_{i(\hat{y}_{i})}^{w}\log(\mathbf{p}_{i(\hat{y}_{i})}^{w})$
(12)
$\displaystyle-(1-\mathbf{s}_{i(\hat{y}_{i})}^{w})\sum_{k=1}^{C}\mathbf{p}_{i(k)}^{w}\log\mathbf{p}_{i(k)}^{w}.$
With the label-flipping errors and OOD reduced, class prototypes are updated
by embeddings of the remaining clean examples from the momentum encoder.
Exponential moving average (Li, Xiong, and Hoi 2020) is used for two reasons:
1) initialization by few shots remains to exert a profound anchoring effect.
2) smoothed transition is achieved to stabilize contrastive learning. For
class $k$, web images with $\hat{y}_{i}^{w}=k$ and few shot images with
$y_{i}^{t}=k$ are involved:
$\hat{\mathbf{c}}_{k}=m_{p}{\mathbf{c}}_{k}+(1-m_{p})\mathbf{z}_{i}^{w;t},\mathbf{c}_{k}=\frac{\hat{\mathbf{c}}_{k}}{\|\hat{\mathbf{c}}_{k}\|_{2}}.$
(13)
Note that reliable samples, which are selected by Eq. 11 per batch, also
participate in training the relation module. The criterion by Eq. 8 is only
used in stage 3.
## Experiments
We train FoPro on web datasets and evaluate it on real-world testing sets.
FoPro boosts $K$-shot performance and reaches the SOTA. Ablation study
validates the relation module.
### Datasets
Web Dataset | # Img. | # Cls. | Real-World
---|---|---|---
Web- FG496 | Bird | 18k | 200 | CUB200-2011
Air | 13k | 100 | FGVC-Aircraft
Car | 21k | 196 | Stanford Car
Web- Vision1k | | Web-
---
Vision1k
2.44M | 1k | ImageNet1k
Google500 | 0.61M | 500 | ImageNet500
Table 1: Statistics of web datasets.
##### WebFG496
(Sun et al. 2021) contains three fine-grained datasets sourced from Bing. The
testing sets of CUB200-2011 (Wah et al. 2011), FGVC-Aircraft (Maji et al.
2013), and Stanford Car (Krause et al. 2013) are used.
##### WebVision1k
(Li et al. 2017) is collected from Google and Flickr. The validation set of
ImageNet1k (Deng et al. 2009) is used. Besides, we also use Google500 (Yang et
al. 2020) where 500 out of 1k categories are randomly sampled with images only
from Google (see Table 1).
We randomly sample $K$ shots per class from the training sets of real-world
datasets. Classification accuracy (%) is adopted as the evaluation metric for
all experiments.
### Implementation Details
##### WebFG496
The B-CNN (Lin, RoyChowdhury, and Maji 2015) (VGG-16 (Simonyan and Zisserman
2014)) is used as encoder. We refer to (Sun et al. 2021) for the training
settings: optimizer is Adam with weight decay of $1\times 10^{-8}$; batch size
is 64; the learning rate is $1\times 10^{-4}$ and decays to 0 by cosine
schedule; a warm-up policy increases the learning rate linearly for 5 epochs
with the frozen encoders.
##### WebVision1k
The ResNet-50 (R50) (He et al. 2016) is used as encoder. We refer to (Yang et
al. 2020) for the training settings: batch size is 256; optimizer is SGD with
the momentum of 0.9 and weight decay of $1\times 10^{-4}$; the learning rate
is 0.01 and decays to 0 by cosine schedule.
We refer to MoPro to set $m_{e}=0.999$, $m_{p}=0.999$, $d_{p}=128$, and
$Q=8192$. In view of the dataset scale, we set $T_{1}=20$, $T_{2}=5$,
$T_{3}=20$, $T_{4}=175$ for WebFG496 and set $T_{1}=15$, $T_{2}=5$,
$T_{3}=10$, $T_{4}=30$ for WebVision1k/Google500. Preliminary experiments on
WebFG496 show that $\gamma=0.6$ and $\beta=0.5$ work better than $\gamma=0.2$
and $\beta=0,0.25,0.75,1$. A lower $\gamma$ means a more relaxed criterion on
clean sample selection, which might bring in noise and cause performance drop.
The balanced combination of self-prediction and similarity terms performs more
robust to noise than the biased cases. Other hyper-parameters are empirically
set as: $\delta^{w}=0$, $\delta^{t}=0.5$, $\tau=0.1$, $\alpha=10$,
$\sigma=20$. Their optimal values require meticulous fine-tuning, which is
beyond consideration of the present study.
Data augmentation includes random cropping and horizontal flipping. Strong
augmentation on the inputs to the momentum encoder (He et al. 2020)
additionally uses color jittering and blurring. Since birds might only differ
in color, random rotation in 45 degrees is used instead. Experiments are
conducted on a CentOS 7 workstation with an Intel 8255C CPU, 377 GB Mem, and 8
NVIDIA V100 GPUs.
### Results
Method | Back- | WebFG496
---|---|---
bone | Bird | Air | Car | Avg.
Vanilla | R50 | 64.43 | 60.79 | 60.64 | 61.95
MoPro† | R50 | 71.16 | 76.85 | 79.68 | 75.90
SCC† | R50-D | 61.10 | 74.92 | 83.49 | 73.17
Vanilla | B-CNN | 66.56 | 64.33 | 67.42 | 66.10
Decouple | B-CNN | 70.56 | 75.97 | 75.00 | 73.84
CoTeach | B-CNN | 73.85 | 72.76 | 73.10 | 73.24
PeerLearn | B-CNN | 76.48 | 74.38 | 78.52 | 76.46
PeerLearn† | B-CNN | 76.57 | 74.35 | 78.26 | 76.39
FoPro($K$=0) | B-CNN | 77.79 | 79.37 | 86.99 | 81.38
FoPro($K$=1) | B-CNN | 78.07 | 79.87 | 88.01 | 82.03
FoPro($K$=16) | B-CNN | 85.54 | 86.40 | 91.51 | 87.81
* $\dagger$
Results are reproduced by ourselves with the official codes.
Table 2: The SOTA results on fine-grained datasets.
##### Baselines
Our FoPro is compared with vanilla backbones and the SOTA methods including
SCC, MoPro, Decouple (Malach and Shalev-Shwartz 2017), CoTeach (Han et al.
2018), PeerLearn, MentorNet (Jiang et al. 2018), CurriculumNet (Guo et al.
2018), and CleanNet (Lee et al. 2018). Results of the SOTA methods that are
trained and evaluated on the same datasets are directly quoted here. We also
reproduce three closely-related methods of SCC, MoPro, and PeerLearn under
$K$-shot settings with the officially released codes. Their default hyper-
parameters are employed if the same web datasets are engaged. Otherwise, they
are set the same as ours. Additionally, we modify the proposed method only to
exhibit its applicability for 0-shot without specific design. In that case,
web images with predicted probability of the target class over $\gamma$ are
used to train the auxiliary classifier. In view of the dataset scale,
prototypes are initialized by randomly sampled 16 and 50 web images per class
from WebFG496 and WebVision1k/Google500, respectively.
Method† | Back- | ImageNet1k | ImageNet500
---|---|---|---
bone | Top 1 | Top 5 | Top 1 | Top 5
MentorNet | Inception ResNetV2 | 64.20 | 84.80 | – | –
Curriculum- Net | Inception V2 | 64.80 | 83.40 | – | –
Vanilla | R50-D | 67.23 | 84.09 | – | –
SCC | R50-D | 67.93 | 84.77 | 68.84 | 84.62
SCC† | R50-D | 67.57 | 85.74 | 64.40 | 81.56
Vanilla | R50 | 65.70 | 85.10 | 61.54 | 78.89
CoTeach | R50 | – | – | 62.18 | 80.98
CleanNet | R50 | 63.42 | 84.59 | – | –
MoPro | R50 | 67.80 | 87.00 | – | –
MoPro† | R50 | 66.05 | 85.66 | 58.68 | 78.39
PeerLearn† | R50 | 52.57 | 73.35 | 42.04 | 61.71
FoPro($K$=0) | R50 | 67.03 | 85.57 | 68.59 | 86.03
FoPro($K$=1) | R50 | 67.55 | 86.31 | 69.11 | 86.19
FoPro($K$=16) | R50 | 68.83 | 87.83 | 72.02 | 89.38
* $\dagger$
Results are reproduced by ourselves with the official codes.
Table 3: The SOTA results on large-scale datasets.
Table 2 confirms the superiority of the proposed method on fine-grained
datasets even under 0-shot. FoPro boosts the accuracy of vanilla backbones
more significantly than the SOTA methods with respect to their backbones.
FoPro reaches the optimal performance on large-scale datasets with $K$=16 (see
Table 3). The vanilla R50-D (He et al. 2019) performs better than R50.
Although FoPro is preceded by SCC and MoPro at first (0-shot), it rises
steadily after exploiting a few real-world examples efficiently.
Figure 4: The SOTA results under $K$-shot settings. Figure 5: One-shot real-
world examples and the web images sorted by their distance to class
prototypes. Best viewed magnified.
Under the degeneration circumstance ($K$=0), FoPro outperforms the SOTA
methods on WebFG496. The reason why FoPro ($K$=0) degrades slightly on
ImageNet1k/500 lies in the high percentage of noises in WebVision1k/Google500.
In that case, prototypes ($K$=0) are initialized and polished solely by noisy
web examples without intervention from clean shots, which may not be class-
representative. With few real-world examples ($K>0$) involved, FoPro regains
its advantage over the SOTA methods.
$K$ | WebFG496 Avg. | ImageNet1k | ImageNet500
---|---|---|---
Top 1 | Gap | Top 1 | Gap | Top 1 | Gap
0 | 81.38 | – | 67.03 | 5.57 | 68.59 | 3.85
1 | +0.65 | – | +0.52 | 5.22 | +0.52 | 3.63
2 | +0.85 | – | +0.67 | 5.20 | +1.35 | 3.29
4 | +2.17 | – | +0.32 | 4.60 | +1.50 | 2.91
8 | +4.10 | – | +0.85 | 4.64 | +2.06 | 2.90
16 | +6.43 | – | +1.80 | 3.91 | +3.43 | 2.19
16 | 87.81 | – | 68.83 | – | 72.02 | –
Ref. | 87.16† | – | 76.15‡ | – | 76.22‡ | –
* $\dagger$
Official results of the B-CNN trained on FGVC-Aircraft, CUB200-2011, and
Stanford Car are averaged.
* ‡
Official results of the R50 trained on ImageNet1k by PyTorch are quoted
respectively for 500 and 1k classes.
Table 4: FoPro gains of $K$-shot over 0-shot. Gap refers to the differences
between web and real-world testing results.
##### Effect of Few-Shots
We explore the potential of FoPro by varying the number of real-world examples
per class from 1 to 16. As shown in Fig. 4, FoPro achieves consistent
performance growth with $K$ on fine-grained datasets. It surpasses SCC and
PeerLearn by a large margin. On ImageNet1k, the abnormal case of $K$=4 is
mainly due to sampling jittering. Since ImageNet contains many unreal images,
it could not eliminate the possibility of sampling atypical images of certain
classes. However, as $K$ increases, FoPro starts to take the lead. We believe
more few shots directly refine the estimated prototypes for better
representation. Clean samples can be more appropriately selected to promote
discriminative feature learning. With amendment on cluster formation, FoPro
also enjoys a higher level of interpretability in class centers with
competitive performance.
Table 4 reports the performance gap between WebVision1k/ImageNet1k and
Google500/ImageNet500 when the testing sets of web domain are available. In
line with Fig. 2(b), the reduced gap reflects that we bridge the noisy web
domain and real-world domain with limited $K$ shots. FoPro approaches
reference benchmarks that are trained on real-world datasets, corroborating
its practical value that much labor of data collection and annotation can be
saved.
##### Effect of Relation Module
In Table 5, we study the effect of relation module for clean example
selection. We remove FC layers and directly compare an instance and each
prototype using cosine similarity. Results confirm that the proposed relation
module discovers clean examples more precisely than the pre-defined similarity
metric. By using a non-linear metric, we do not assume that element-wise
comparison could solely separate matching or mismatching pairs. Besides, such
a learnable metric is not sensitive to input variation and behaves better on
noisy samples.
$K$=1 | WebFG496 Avg. | ImageNet1k | ImageNet500
---|---|---|---
w/o RM | 81.59 | 65.22 | 64.69
w RM | 82.03 | 67.55 | 69.11
Table 5: Ablation Study on the Relation Module (RM).
##### Visualization
In Fig. 2(a), we visualize the low-dimensional embeddings with t-SNE for the
randomly chosen 10 categories in WebVision as a demonstration. For
convenience, all 16 real-world examples in each class are averaged and
displayed as one few-shot example. Differences in the cluster distribution
(from $K$=1 to $K$=16) are highlighted to show that: 1) the distance between
each prototype and the few-shot example becomes shorter; 2) the density of
class clusters is improved. From Fig. 5, we conclude the following insights:
1) Web images close to the estimated prototypes are clean and similar to real-
world photos with limited post-processing. Our FoPro learns to sort out noise
in web data for robust representation learning. 2) The proposed method
generalizes across various domains such as product close-up, computer
graphics, and screenshots. 3) Intra-class diversity (_e.g._ , wing postures of
the sooty albatross), uncaptured salient parts (_e.g._ , the yellowish patch
on the back head of the bobolink), and editing of tone curve (_e.g._ , colored
body of the painted bunting and yellow-breasted chat) are the possible reasons
why hard examples of 1-shot and clean web images still locate away from
prototypes.
## Conclusion
This paper introduces a new setting for webly-supervised learning, which
optimizes the learning system with a large quantity of noisy web images and a
few real-world images. Under this setting, we propose a few-shot guided
prototypical representation learning method called FoPro, which simultaneously
tackles noise and dataset bias in a cost-efficient manner. It is characterized
by the guidance from a few real-world domain images for learning noise-robust
and generalizable representations from web data. Experimental results
demonstrate that our method can effectively utilize few-shot images to improve
the performance of WSL on real-world benchmarks. Future work includes
investigation of side information from web datasets (_e.g._ , captions,
website titles, and user comments) and application extension to weakly-
supervised object detection and segmentation.
## References
* Arpit et al. (2017) Arpit, D.; Jastrzebski, S.; Ballas, N.; Krueger, D.; Bengio, E.; Kanwal, M. S.; Maharaj, T.; Fischer, A.; Courville, A.; Bengio, Y.; et al. 2017. A closer look at memorization in deep networks. In _International Conference on Machine Learning_ , 233–242. PMLR.
* Bergamo and Torresani (2010) Bergamo, A.; and Torresani, L. 2010. Exploiting weakly-labeled web images to improve object classification: a domain adaptation approach. _Advances in Neural Information Processing Systems_ , 23.
* Chen et al. (2020a) Chen, T.; Kornblith, S.; Norouzi, M.; and Hinton, G. 2020a. A simple framework for contrastive learning of visual representations. In _International Conference on Machine Learning_ , 1597–1607. PMLR.
* Chen et al. (2020b) Chen, X.; Fan, H.; Girshick, R.; and He, K. 2020b. Improved baselines with momentum contrastive learning. _arXiv preprint arXiv:2003.04297_.
* Chen and Gupta (2015) Chen, X.; and Gupta, A. 2015. Webly supervised learning of convolutional networks. In _Proceedings of the IEEE international conference on computer vision_ , 1431–1439.
* Cheng et al. (2020) Cheng, L.; Zhou, X.; Zhao, L.; Li, D.; Shang, H.; Zheng, Y.; Pan, P.; and Xu, Y. 2020. Weakly supervised learning with side information for noisy labeled images. In _European Conference on Computer Vision_ , 306–321. Springer.
* Deng et al. (2009) Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-Fei, L. 2009. ImageNet: A large-scale hierarchical image database. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 248–255. IEEE.
* Divvala, Farhadi, and Guestrin (2014) Divvala, S. K.; Farhadi, A.; and Guestrin, C. 2014. Learning everything about anything: Webly-supervised visual concept learning. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 3270–3277.
* Ghosh, Kumar, and Sastry (2017) Ghosh, A.; Kumar, H.; and Sastry, P. S. 2017. Robust loss functions under label noise for deep neural networks. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 31.
* Guo et al. (2018) Guo, S.; Huang, W.; Zhang, H.; Zhuang, C.; Dong, D.; Scott, M. R.; and Huang, D. 2018. CurriculumNet: Weakly supervised learning from large-scale web images. In _European Conference on Computer Vision_ , 135–150.
* Han et al. (2018) Han, B.; Yao, Q.; Yu, X.; Niu, G.; Xu, M.; Hu, W.; Tsang, I.; and Sugiyama, M. 2018\. Co-teaching: Robust Training of Deep Neural Networks with Extremely Noisy Labels. _Advances in Neural Information Processing Systems_ , 31.
* Han, Luo, and Wang (2019) Han, J.; Luo, P.; and Wang, X. 2019. Deep self-learning from noisy labels. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 5138–5147.
* He et al. (2020) He, K.; Fan, H.; Wu, Y.; Xie, S.; and Girshick, R. 2020. Momentum contrast for unsupervised visual representation learning. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 9729–9738.
* He et al. (2016) He, K.; Zhang, X.; Ren, S.; and Sun, J. 2016. Deep residual learning for image recognition. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 770–778.
* He et al. (2019) He, T.; Zhang, Z.; Zhang, H.; Zhang, Z.; Xie, J.; and Li, M. 2019. Bag of tricks for image classification with convolutional neural networks. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 558–567.
* Hinton et al. (2015) Hinton, G.; Vinyals, O.; Dean, J.; et al. 2015. Distilling the knowledge in a neural network. _arXiv preprint arXiv:1503.02531_ , 2(7).
* Jiang et al. (2018) Jiang, L.; Zhou, Z.; Leung, T.; Li, L.-J.; and Fei-Fei, L. 2018. MentorNet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In _International Conference on Machine Learning_ , 2304–2313. PMLR.
* Jin, Ortiz Segovia, and Susstrunk (2017) Jin, B.; Ortiz Segovia, M. V.; and Susstrunk, S. 2017. Webly supervised semantic segmentation. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 3626–3635.
* Kaur, Sikka, and Divakaran (2017) Kaur, P.; Sikka, K.; and Divakaran, A. 2017. Combining weakly and webly supervised learning for classifying food images. _arXiv preprint arXiv:1712.08730_.
* Kim et al. (2018) Kim, D.; Cho, D.; Yoo, D.; and Kweon, I. S. 2018. Learning image representations by completing damaged jigsaw puzzles. In _IEEE Winter Conference on Applications of Computer Vision_ , 793–802. IEEE.
* Kolesnikov et al. (2019) Kolesnikov, A.; Beyer, L.; Zhai, X.; Puigcerver, J.; Yung, J.; Gelly, S.; and Houlsby, N. 2019. Large scale learning of general visual representations for transfer. _arXiv preprint arXiv:1912.11370_ , 2(8).
* Krause et al. (2016) Krause, J.; Sapp, B.; Howard, A.; Zhou, H.; Toshev, A.; Duerig, T.; Philbin, J.; and Fei-Fei, L. 2016. The unreasonable effectiveness of noisy data for fine-grained recognition. In _European Conference on Computer Vision_ , 301–320. Springer.
* Krause et al. (2013) Krause, J.; Stark, M.; Deng, J.; and Fei-Fei, L. 2013. 3d object representations for fine-grained categorization. In _Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops_ , 554–561.
* Lee et al. (2018) Lee, K.-H.; He, X.; Zhang, L.; and Yang, L. 2018. CleanNet: Transfer Learning for Scalable Image Classifier Training with Label Noise. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 5447–5456.
* Li, Socher, and Hoi (2019) Li, J.; Socher, R.; and Hoi, S. C. 2019. DivideMix: Learning with Noisy Labels as Semi-supervised Learning. In _International Conference on Learning Representations_.
* Li, Xiong, and Hoi (2020) Li, J.; Xiong, C.; and Hoi, S. 2020. MoPro: Webly Supervised Learning with Momentum Prototypes. In _International Conference on Learning Representations_.
* Li et al. (2020) Li, J.; Zhou, P.; Xiong, C.; and Hoi, S. 2020. Prototypical Contrastive Learning of Unsupervised Representations. In _International Conference on Learning Representations_.
* Li et al. (2017) Li, W.; Wang, L.; Li, W.; Agustsson, E.; and Van Gool, L. 2017. WebVision database: Visual learning and understanding from web data. _arXiv preprint arXiv:1708.02862_.
* Lin, RoyChowdhury, and Maji (2015) Lin, T.-Y.; RoyChowdhury, A.; and Maji, S. 2015. Bilinear CNN models for fine-grained visual recognition. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 1449–1457.
* Liu et al. (2021) Liu, H.; Zhang, H.; Lu, J.; and Tang, Z. 2021. Exploiting Web Images for Fine-Grained Visual Recognition via Dynamic Loss Correction and Global Sample Selection. _IEEE Transactions on Multimedia_ , 24: 1105–1115.
* Maji et al. (2013) Maji, S.; Rahtu, E.; Kannala, J.; Blaschko, M.; and Vedaldi, A. 2013. Fine-grained visual classification of aircraft. _arXiv preprint arXiv:1306.5151_.
* Malach and Shalev-Shwartz (2017) Malach, E.; and Shalev-Shwartz, S. 2017. Decoupling” when to update” from” how to update”. _Advances in Neural Information Processing Systems_ , 30.
* Müller, Kornblith, and Hinton (2019) Müller, R.; Kornblith, S.; and Hinton, G. E. 2019. When does label smoothing help? _Advances in Neural Information Processing Systems_ , 32.
* Niu, Li, and Xu (2015) Niu, L.; Li, W.; and Xu, D. 2015. Visual recognition by learning from web data: A weakly supervised domain generalization approach. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2774–2783.
* Pereyra et al. (2017) Pereyra, G.; Tucker, G.; Chorowski, J.; Kaiser, Ł.; and Hinton, G. 2017. Regularizing neural networks by penalizing confident output distributions. _arXiv preprint arXiv:1701.06548_.
* Reed et al. (2015) Reed, S. E.; Lee, H.; Anguelov, D.; Szegedy, C.; Erhan, D.; and Rabinovich, A. 2015\. Training Deep Neural Networks on Noisy Labels with Bootstrapping. In _International Conference on Learning Representations (Workshop)_.
* Shen et al. (2018) Shen, T.; Lin, G.; Shen, C.; and Reid, I. 2018. Bootstrapping the performance of webly supervised semantic segmentation. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 1363–1371.
* Shen et al. (2020) Shen, Y.; Ji, R.; Chen, Z.; Hong, X.; Zheng, F.; Liu, J.; Xu, M.; and Tian, Q. 2020\. Noise-aware fully webly supervised object detection. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 11326–11335.
* Simonyan and Zisserman (2014) Simonyan, K.; and Zisserman, A. 2014. Very deep convolutional networks for large-scale image recognition. _arXiv preprint arXiv:1409.1556_.
* Song et al. (2022) Song, H.; Kim, M.; Park, D.; Shin, Y.; and Lee, J.-G. 2022. Learning from noisy labels with deep neural networks: A survey. _IEEE Transactions on Neural Networks and Learning Systems_.
* Sun et al. (2017) Sun, C.; Shrivastava, A.; Singh, S.; and Gupta, A. 2017. Revisiting unreasonable effectiveness of data in deep learning era. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 843–852.
* Sun et al. (2021) Sun, Z.; Yao, Y.; Wei, X.-S.; Zhang, Y.; Shen, F.; Wu, J.; Zhang, J.; and Shen, H. T. 2021. Webly supervised fine-grained recognition: Benchmark datasets and an approach. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 10602–10611.
* Tanaka et al. (2018) Tanaka, D.; Ikami, D.; Yamasaki, T.; and Aizawa, K. 2018. Joint optimization framework for learning with noisy labels. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 5552–5560.
* Tu et al. (2020) Tu, Y.; Niu, L.; Chen, J.; Cheng, D.; and Zhang, L. 2020. Learning from web data with self-organizing memory module. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 12846–12855.
* Van der Maaten and Hinton (2008) Van der Maaten, L.; and Hinton, G. 2008. Visualizing data using t-SNE. _Journal of Machine Learning Research_ , 9(11).
* Wah et al. (2011) Wah, C.; Branson, S.; Welinder, P.; Perona, P.; and Belongie, S. 2011. The caltech-ucsd birds-200-2011 dataset.
* Wang, Liu, and Tao (2017) Wang, R.; Liu, T.; and Tao, D. 2017. Multiclass learning with partially corrupted labels. _IEEE Transactions on Neural Networks and Learning Systems_ , 29(6): 2568–2580.
* Wu et al. (2021) Wu, Z.-F.; Wei, T.; Jiang, J.; Mao, C.; Tang, M.; and Li, Y.-F. 2021. NGC: a unified framework for learning with open-world noisy data. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 62–71.
* Xiao et al. (2015) Xiao, T.; Xia, T.; Yang, Y.; Huang, C.; and Wang, X. 2015. Learning from massive noisy labeled data for image classification. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2691–2699.
* Xu et al. (2016) Xu, Z.; Huang, S.; Zhang, Y.; and Tao, D. 2016. Webly-supervised fine-grained visual categorization via deep domain adaptation. _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 40(5): 1100–1113.
* Yang et al. (2020) Yang, J.; Feng, L.; Chen, W.; Yan, X.; Zheng, H.; Luo, P.; and Zhang, W. 2020. Webly supervised image classification with self-contained confidence. In _European Conference on Computer Vision_ , 779–795. Springer.
* Yao et al. (2020) Yao, Y.; Hua, X.; Gao, G.; Sun, Z.; Li, Z.; and Zhang, J. 2020. Bridging the web data and fine-grained visual recognition via alleviating label noise and domain mismatch. In _Proceedings of the ACM International Conference on Multimedia_ , 1735–1744.
* Yao et al. (2017) Yao, Y.; Zhang, J.; Shen, F.; Hua, X.; Xu, J.; and Tang, Z. 2017. Exploiting web images for dataset construction: A domain robust approach. _IEEE Transactions on Multimedia_ , 19(8): 1771–1784.
* Zhang et al. (2021) Zhang, C.; Bengio, S.; Hardt, M.; Recht, B.; and Vinyals, O. 2021. Understanding deep learning (still) requires rethinking generalization. _Communications of the ACM_ , 64(3): 107–115.
* Zhang et al. (2020) Zhang, C.; Yao, Y.; Liu, H.; Xie, G.-S.; Shu, X.; Zhou, T.; Zhang, Z.; Shen, F.; and Tang, Z. 2020. Web-supervised network with softly update-drop training for fine-grained visual classification. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, 12781–12788.
* Zhang et al. (2018) Zhang, H.; Cisse, M.; Dauphin, Y. N.; and Lopez-Paz, D. 2018. MixUp: Beyond Empirical Risk Minimization. In _International Conference on Learning Representations_.
* Zhang and Sabuncu (2018) Zhang, Z.; and Sabuncu, M. 2018. Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels. _Advances in Neural Information Processing Systems_ , 31.
* Zhou et al. (2020) Zhou, X.; Pan, P.; Zheng, Y.; Xu, Y.; and Jin, R. 2020. Large scale long-tailed product recognition system at alibaba. In _Proceedings of ACM International Conference on Information & Knowledge Management_, 3353–3356.
|
# Beyond Incompatibility: Trade-offs between Mutually Exclusive Fairness
Criteria in Machine Learning and Law
Meike Zehlike Zalando ResearchBerlinGermany<EMAIL_ADDRESS>, Alex
Loosley Zalando SEBerlinGermany<EMAIL_ADDRESS>, Håkan Jonsson
Zalando ResearchBerlinGermany<EMAIL_ADDRESS>, Emil Wiedemann
Institute of Applied Analysis, Ulm UniversityHelmholtzstr. 1889081 UlmGermany
<EMAIL_ADDRESS>and Philipp Hacker European New School of Digital
Studies, European University ViadrinaFrankfurt (Oder)Germany hacker@europa-
uni.de
###### Abstract.
Trustworthy AI is becoming ever more important in both machine learning and
legal domains. One important consequence is that decision makers must seek to
guarantee a ‘fair’, i.e., non-discriminatory, algorithmic decision procedure.
However, there are several competing notions of algorithmic fairness that have
been shown to be mutually incompatible under realistic factual assumptions.
This concerns, for example, the widely used fairness measures of ‘calibration
within groups’ and ‘balance for the positive/negative class’. In this paper,
we present a novel algorithm (FAir Interpolation Method: FAIM) for
continuously interpolating between these three fairness criteria. Thus, an
initially unfair prediction can be remedied to, at least partially, meet a
desired, weighted combination of the respective fairness conditions. We
demonstrate the effectiveness of our algorithm when applied to synthetic data,
the COMPAS data set, and a new, real-world data set from the e-commerce
sector. Finally, we discuss to what extent FAIM can be harnessed to comply
with conflicting legal obligations. The analysis suggests that it may
operationalize duties in traditional legal fields, such as credit scoring and
criminal justice proceedings, but also for the latest AI regulations put forth
in the EU, like the recently enacted Digital Markets Act.
## 1\. Introduction
As machine learning (ML) progressively permeates all sectors of society,
responsible and trustworthy AI development is more crucial than ever ((Dignum,
2019; Grodzinsky et al., 2011; Thiebes et al., 2021)). Increasingly, this
debate spills over from the ethics and machine learning communities to the
legal domain ((Laux et al., 2022)), where different regulatory frameworks for
ensuring trustworthy AI are emerging. In this context, discrimination against
protected groups in decisions facilitated by ML models remains a key challenge
((Wachter, 2022)). Non-discrimination legislation applying to biased AI
systems does exist in the US and the EU, but loopholes and enforcement
problems remain ((Barocas and Selbst, 2016; Hacker, 2018; Zuiderveen
Borgesius, 2020; Wachter et al., 2021b; Wachter, 2022)). Hence, the AI Act,
proposed by the EU Commission in 2021 and set to be enacted soon in the EU,
adds further duties for ML developers such as the duty to evaluate of training
data for potential biases (Article 10(2) AI Act), the duty to use
representative training data (Article 10(3) AI Act) and the duty to prevent
biased feedback loops (Article 15(3) AI Act) ((Veale and Borgesius, 2021;
Ebers et al., 2021; Hacker, 2021)). This is backed up by a framework for AI
liability proposed in September 2022 by the EU Commission ((Hacker, 2022a)).
Moreover, the recently enacted EU Digital Markets Act (DMA111Regulation (EU)
2022/1925 of the European Parliament and of the Council of 14 September 2022
on contestable and fair markets in the digital sector, OJ 2022, L265/1.) now
compels large online platforms to engage in ‘fair and non-discriminatory’
ranking (Article 6(5) DMA).
The laudable idea behind such approaches is to force developers to consider,
and potentially eliminate, biases during the design of ML models. Such non-
discriminatory models are called ‘fair’ in the ML community ((Mitchell et al.,
2021; Dwork et al., 2012)). However, foundational papers at the intersection
of machine learning and the law have shown that, under realistic assumptions,
several desirable fairness criteria are mutually incompatibility ((Kleinberg
et al., 2016)). This poses a significant challenge as the law, in general,
prohibits discrimination without explicitly favoring one metric over another.
Rather, a core tenet of legal research and jurisprudence is that conflicting
rights or interests need to be balanced, i.e., be partially realized to
satisfy each of the mutually incompatible demands to the extent that
conditions warrant ((De Vries, 2013; Morijn, 2006)). Lawmakers, courts and
regulators therefore need fairness frameworks that move beyond the mere
statement of incompatibility. Simultaneously, ML research may benefit from the
legal tradition of balancing by inducing decision makers and developers to
refine algorithms implementing a balance between the normative trade-offs
inherent in the mutually incompatible fairness criteria ((Hertweck and Räz,
2022)). Our paper develops one such algorithm allowing decision makers to
interpolate between three different incompatible fairness criteria singled out
for their importance in previous research. These three fairness criteria are
described below.
There are various ways in which unfairness, i.e., discrimination against
certain groups ((Friedman and Nissenbaum, 1996)), may enter an ML model
((Calders and Žliobaitė, 2013)). For example, the model may perpetuate
historical biases present in the training data; use non-representative
training data; or optimize for a target variable which unwittingly encodes
differential access of individuals or groups ((Barocas and Selbst, 2016)). The
result of such biased modeling may be that false positive or false negative
rates differ between groups of individuals. Specific fairness metrics such as
equalized odds, quantifying the difference of false positive and false
negative rates between groups ((Hardt et al., 2016)), are used to quantify how
much a classifier systematically rewards or punishes one group more than
another with respect to ground truth. Even if a classifier is fair in terms of
equalized odds, its prediction scores can lead to unfair outcomes if they are
uncalibrated, as decision makers are not able to reliably interpret its output
((Zadrozny and Elkan, 2001; Jiang et al., 2012)).222A classifier is calibrated
when, given a score $s$, $s$-fraction of all individuals receiving score $s$
are ground truth positive ((Dawid, 1982)). What is worse, if a classifier
emits scores that are differently calibrated between groups, identical scores
have different meanings for different protected groups, again leading to
unfair outcomes ((Kleinberg et al., 2016)).
Given that equality of false positive rates between groups, equality of false
negative rates between groups, and calibration between groups are all
important fairness criteria ((Corbett-Davies and Goel, 2018)), it would be
ideal if a classifier could satisfy all three. Unfortunately, previous work
shows that simultaneously fulfilling all three of these fairness criteria is
impossible, except under highly constrained circumstances such as when a
classifier achieves 100% accuracy with each instance receiving a prediction
score of exactly 0 or 1 ((Chouldechova, 2017; Kleinberg et al., 2016; Pleiss
et al., 2017)). Pleiss et al. (2017) attempt to find a relaxed notion of error
rate difference that can be simultaneously achieved alongside group
calibration. However, they demonstrate that one may only simultaneously
achieve group calibration with a relaxed version of one of either equalized
false positive or false negative rates. Furthermore, they show that
classifiers simultaneously satisfying two of these three fairness criteria
must engage in the equivalent of randomizing a certain percentage of
predictions. Thus, the authors suggest that practitioners should only attempt
to satisfy one of the three fairness criteria based on importance, instead of
trying to simultaneously fulfill a relaxed notion of two of them. Often,
however, all three fairness notions embody valid concerns of different groups
((Corbett-Davies and Goel, 2018)). Merely satisfying one of them may not do
justice to the various interests and rights of the groups, and, as discussed
above, might even amount to unjustified discrimination if the law demands a
balance rather than an absolute priority of one criterion.
Building on the assumption that a decision makers should determine, within the
constraints of the law, which combination of the three fairness criteria is
most crucial towards a fair result in practice, our work introduces a new
post-processing algorithm called FAir Interpolation Method (FAIM). It allows
decision makers to consciously and continuously interpolate between fairness
criteria. The algorithm is based on the notion that, given a classifier, a
representative evaluation data set, and the corresponding prediction scores
for each group, any one of the three fairness criteria could be satisfied by
applying a certain fairness criteria specific score mapping function mapping
scores from the distribution outputted by the model to a fair score
distribution. Given three such fairness criteria specific score distributions,
FAIM uses optimal transport to continuously interpolate between them. This
leads to a mapping function that can be applied to new classifier predictions,
post hoc, to achieve a desired, weighted combination of the respective
fairness criteria. FAIM cannot be used to fully satisfy all three fairness
criteria simultaneously, but it does give the practitioner a powerful tool to
make a continuous compromise along the fairness criteria simplex, and thus to
account for different concerns each fairness criterion encompasses.
The remainder of the article is organized as follows. First, related work is
outlined in detail. Next, the mathematical foundations regarding fairness
criteria incompatibility and optimal transport are addressed. Based on this,
FAIM is introduced, and tested empirically in three experiments. In the first
experiment, FAIM is applied to a synthetic classifier that produces two
distinct score distributions for two respective groups. Here the properties of
FAIM are elucidated under controlled conditions. In the second experiment,
FAIM is tested on the COMPAS data set, on which the COMPAS algorithm famously
scores individuals for recidivism risk. In the last experiment, FAIM is
applied to product rankings from the European e-commerce platform
Zalando.333www.zalando.de Here, we evaluate our model in terms of typical
fairness problems arising in e-commerce rankings. The discussion takes up the
results of the experiments and adds a specific legal perspective. We show how
FAIM may be harnessed to trade off conflicting legal requirements across a
wide variety of domains: credit scoring, criminal justice decisions, and fair
rankings according to the DMA.
Overall, the paper brings together technical and legal perspectives on
fairness criteria to take both discourses beyond incompatibility statements.
In this way, decision makers may flexibly adapt ML models to different use
cases in which varying interests of protected groups may necessitate different
trade-offs between the involved fairness criteria. Simultaneously, gradually
balancing the different fairness criteria, rather than prioritizing only one
of them, will often be conducive to fulfilling legal requirements, both in
established legal domains and in novel regulatory frameworks, such as the DMA.
## 2\. Related Work
The incompatibility of various different fairness criteria in algorithmic
decision making has been the subject of intense research in recent years.
Friedler et al. (2016) discuss the contrast between individual and group
fairness. While these results hint at an incompatibility between individual
and group fairness and relate these notions to two opposing worldviews, there
is no rigorous proof of such incompatibility, and no discussion of possible
intermediate worldviews. Some of the results in Feldman et al. (2015) and
Zehlike et al. (2020) can be viewed as proposals on how to continuously
navigate between the two extremes.
The recidivism prediction problem was treated in Kleinberg et al. (2016);
Chouldechova (2017), with the result that a prediction algorithm must
necessarily be unfair with regard to certain fairness criteria under realistic
factual assumptions. In particular, Kleinberg et al. (2016) show the mutual
incompatibility between calibration within groups and balance for the
positive/negative classes in the (typical) case of unequal base rates. These
fairness criteria also underlie the present contribution. The framework of
Kleinberg et al. (2016) is thoroughly reviewed below; let us remark here only
that the mentioned criteria of Kleinberg et al. (2016) acknowledge and accept
the possibility of empirically observed disparities between different groups,
but require the equal treatment of individuals in different groups conditional
on their similarity according to ‘ground truth’. The competing criterion of
calibration within groups concerns the accuracy of the algorithm and is shown
to conflict with the other two criteria.
The use of optimal transport methods in algorithmic fairness has become
increasingly popular ((Dwork et al., 2012; Feldman et al., 2015; Gordaliza et
al., 2019; Zehlike et al., 2020; Chiappa et al., 2020; Chiappa and Pacchiano,
2021)), applying optimal transport to trade-offs between individual vs group
fairness, and accuracy vs fairness. Optimal transport is a mathematical tool
that provides a very natural notion of distance of two probability
distributions (the _Wasserstein_ or earth mover distance) and, moreover, gives
an optimal (in a specific sense) way to translate between these distributions
by means of an _optimal transport map_ or at least an _optimal transport plan_
, the latter requiring some form of randomization. In (Zehlike et al., 2020),
particularly, an important role is played by the _Wasserstein-2 barycenter_ of
several probability distributions and the _displacement interpolation_ that
allows to continuously interpolate between various distributions. We make use
of these techniques in the present article.
The literature at the intersection of law and computer science has, in the
past few years, increasingly discussed how legal non-discrimination
requirements can be integrated into code, and vice versa ((Zehlike et al.,
2020; Hacker, 2018; Wachter et al., 2021a, b; Zuiderveen Borgesius, 2020;
Gerards and Borgesius, 2022; Wachter, 2020, 2022; Barocas and Selbst, 2016;
Raghavan et al., 2020; Hellman, 2020; Nachbar, 2020)). This strand of research
has focused particularly on remedying discrimination in criminal justice
proceedings ((Berk et al., 2021; Starr, 2014; Mayson, 2019; Selbst, 2017;
Katyal, 2019; Eaglin, 2017; Huq, 2019; Pruss, 2021)) and in other high-stakes
AI decisions ((Veale et al., 2018)), such as employment contexts ((Raghavan et
al., 2020)), credit scoring ((Bono et al., 2021)), or university admissions
((Zehlike et al., 2020)). More specifically, (Wachter et al. (2021a)) rightly
point out that the reliance on fairness metrics based on ground truth may
perpetuate biases if that ground truth itself is skewed against protected
groups. This is a concern, for example, in criminal justice settings ((Berk et
al., 2021; Bao et al., 2021)). We incorporate this constraint into the
application perimeter of our algorithm (see Part 6.3). Conversely, in the
e-commerce sector, numerous publications deal with the emerging competition
law framework for digital markets, for example the DMA ((Eifert et al., 2021;
Hacker, 2022b; Brouwer, 2021; Cabral et al., 2021; Laux et al., 2021; Podszun
and Bongartz, 2021)). However, they do not, to our knowledge, specifically
develop algorithms to solve legal questions arising in this field.
Existing publications often use US law as the relevant yardstick ((Barocas and
Selbst, 2016; Raghavan et al., 2020; Wachter, 2022; Selbst and Barocas, 2022;
Hellman, 2020; Gillis and Spiess, 2019; Kim, 2017, 2022)), but have lately
increasingly incorporated explicit discussions of EU law as well ((Zehlike et
al., 2020; Hacker, 2018; Wachter et al., 2021a, b; Zuiderveen Borgesius, 2020;
Gerards and Borgesius, 2022; Wachter, 2020, 2022; Veale and Binns, 2017)). Our
paper builds on and expands this strand of research by offering a novel method
to trade off and implement competing technical and legal fairness constraints.
In three case studies, it demonstrates the potential of FAIM to integrate
varying legal dimensions of fairness in ML settings, particularly in the
criminal justice and credit scoring domain. The third case study tackles an,
to our knowledge, entirely new field of law from an algorithmic fairness
perspective: the EU Digital Markets Act (DMA).
## 3\. Mathematical Theory and Algorithmic Implementation
This section introduces FAIM, our interpolation framework that allows a
continuous shift between three mutually exclusive fairness notions, as
presented in Kleinberg et al. (2016). We first provide a brief summary of the
findings from Kleinberg et al. (2016) to make the reader familiar with the
three fairness notions and establish necessary terminology. We then briefly
present mathematical preliminaries from optimal transport theory. Finally we
describe the mathematical theory of our algorithm as well as its
implementation in parallel, to help the reader translate between the formulas
and the code. We present the method in pseudocode in Algorithm 1.
For the simplicity of presentation, we consider a population of individuals
partitioned into two groups, indexed by $t=1,2$. Note first, that individuals
need not be human, but could also be products, companies, etc., and second,
that our method applies just as well to a setting with more than two groups.
Each individual, regardless of group membership, is either truly ‘positive’ or
‘negative’ with respect to some trait of interest; i.e., in criminal justice,
an individual would be ‘positive’ if they were to commit a certain type of
crime within a given period in the future, and negative if not. The attributes
‘positive’ and ‘negative’ do not carry any normative meaning and are thus
interchangeable, but we will refer to such a classification as ‘ground truth’:
It is assumed that the assignments of the positive/negative label reflect
reality (we will shortly discuss the limitations of this assumption).
Furthermore, each individual is assigned a score by some algorithm (in the
broadest sense). The score, which can take as its value any real number in the
interval $[0,1]$, is supposed to reflect the probability that the given
individual is positive. Therefore, if the ground truth were fully known to the
algorithm, it would assign value 1 to positive individuals and 0 to negative
ones. This unlikely scenario is known as _perfect prediction_.
A note on terminology: When we say that an individual is truly
negative/positive, we mean that the individual is negative/positive according
to ground truth, regardless of their predicted score. This should not be
confused with the common terminology in which a ‘true negative’ is an
individual who is negative in ground truth and who, in addition, receives a
negative prediction. In our setting, anyway, the predictions are not binary,
but in terms of a continuous score between zero and one.
### 3.1. Incompatibility of Fairness Criteria
The main result from Kleinberg et al. (2016) states that, in the setting
described above, three natural fairness criteria are mutually incompatible
except under trivial circumstances. We shortly summarize this incompatibility
theorem here through a simple proof. Eight quantities are of importance in
this context:
* •
$N_{t}$ is the total number of individuals in group $t$;
* •
$n_{t}$ is the number of (truly) positive individuals in group $t$;
* •
$x_{t}$ is the average score of the individuals in group $t$ that are (truly)
negative;
* •
$y_{t}$ is the average score of the individuals in group $t$ that are (truly)
positive.
For the purpose of deriving the incompatibility theorem, it need not be
assumed that the ground-truth dependent quantities $n_{t},x_{t},y_{t}$ be
known; rather, _whatever their values might be_ , the fairness criteria
proposed in the next paragraph are never met except in case of perfect
prediction or equal base rates. Here, by _base rate_ for group $t$ we mean the
ratio $n_{t}/N_{t}$, and _perfect prediction_ means that the score of each
individual is 1 when she is positive and 0 when she is negative.
The fairness criteria are as follows:
* A)
Calibration within groups: This measures the _accuracy of prediction_ of the
score. The criterion requires that _the average score in each group should
equal the ratio of (truly) positive individuals in that group_ , i.e.,
$\frac{x_{t}(N_{t}-n_{t})+y_{t}n_{t}}{N_{t}}=\frac{n_{t}}{N_{t}},\quad t=1,2.$
* B)
Balance for the negative class: _The average score of (truly) negative
individuals in group 1 should equal the average score of (truly) negative
individuals in group 2_ , i.e.,
$x_{1}=x_{2}=:x.$
* C)
Balance for the positive class: _The average score of (truly) positive
individuals in group 1 should equal the average score of (truly) positive
individuals in group 2_ , i.e.,
$y_{1}=y_{2}=:y.$
Let us illustrate the three requirements for the case of recidivism
prediction. The members of each group (say, black and white offenders) are
given a score that is intended to reflect the probability of the respective
individual to recidivate within a given time period. Criterion A) then
requires the following: The _average_ score within group $t$ should equal the
ratio of actual recidivists in that group; so if, say, half of the members of
the white group would turn out recidivist, then the average score given to
white individuals should be $0.5$. Concerning B), given a non-recidivist
individual, it is desirable that their score be independent of their group
membership. It would have to be considered highly unfair if a black person
received a higher score, and thus faced a higher likelihood of detention, than
a white person, given that both would actually not commit an offense if
released. This, in fact, is at the heart of the controversy around the COMPAS
algorithm. Note that B) only considers scores on an average level: The
_average_ score among all truly non-recidivist individuals should be equal
between groups. Of course, this tells nothing about higher order statistical
moments, such as the variance of the respective score distributions. The
discussion for criterion C) is analogous.
Suppose all the three requirements are satisfied, then this implies
(1) $\displaystyle x(N_{1}-n_{1})+yn_{1}$ $\displaystyle=n_{1},$
$\displaystyle x(N_{2}-n_{2})+yn_{2}$ $\displaystyle=n_{2}.$
Regarding $N_{t},n_{t}$ as given parameters, this is a linear system of two
equations for the two unknowns $x,y$. It has a unique solution if and only if
the determinant of the coefficient matrix is nonzero. This determinant is
computed as
(2) $(N_{1}-n_{1})n_{2}-(N_{2}-n_{2})n_{1},$
which is zero if and only if either (1) $n_{1}\neq 0\neq n_{2}$and
$\frac{N_{1}-n_{1}}{n_{1}}=\frac{N_{2}-n_{2}}{n_{2}}$, which is the case of
equal base rates; or (2) one of $n_{1}$ or $n_{2}$ is zero, in which case
(assuming $N_{1},N_{2}>0$) also the other one is zero, i.e., there are no
positive individuals at all. Here, the choice $x=0$ and $y$ arbitrary yields a
fair assignment according to Eq. 1. If there are no positive individuals, then
everyone should get score zero, which is thus an instance of _perfect
prediction_. For the first case, there are infinitely many solutions, one of
which is $x=y=\frac{n_{1}}{N_{1}}=\frac{n_{2}}{N_{2}}$ (as mentioned in
Kleinberg et al. (2016)). This is realizable by assigning the same score to
everyone. (A mathematically simpler solution would be $x=0,y=1$, which however
requires the algorithm to know who is positive and who isn’t and thus again
requires perfect prediction.)
Finally, let’s turn to the case when the determinant in Eq. 2 is nonzero, so
there exists a unique solution to Eq. 1. It thus suffices to give one solution
to rule out any other ones, and we can simply take $x=0,y=1$, i.e., perfect
prediction, which therefore is the only possibility.
In summary, we have thus verified Theorem 1.1, the main result, of Kleinberg
et al. (2016), which we paraphrase as follows:
###### Theorem 1.
For $t=1,2$, let $N_{t},n_{t}>0$. If a score function satisfies A), B), and
C), then
$\frac{n_{1}}{N_{1}}=\frac{n_{2}}{N_{2}}$
(equal base rates), or the positive individuals receive score 1 and the
negative individuals receive score 0 (perfect prediction).
###### Remark 2.
The formulation given here of criterion A) is weaker than the one stated in
(Kleinberg et al., 2016) Page 4, where the criterion is required individually
within each ‘bin’. As the computation here shows, the weaker requirement is
already sufficient for incompatibility.
### 3.2. Mathematical Preliminaries on Optimal Transport
Our goal is to design an algorithm that interpolates smoothly between the
three fairness criteria discussed, as they are typically not attainable at the
same time. The interpolation relies on the mathematical theory of optimal
transport, which we now wish to recall.
A probability measure $\nu$ on $[0,1]$ is called _absolutely continuous_ if it
is representable by an integrable probability density function
$f:[0,1]\to\mathbb{R}$, which means that $\nu((a,b))=\int_{a}^{b}f(x)dx$ for
any $a,b\in(0,1)$ with $a\leq b$. In words, the probability that a random
variable distributed according to $\nu$ has its value in the interval $(a,b)$
is given by the integral of the density function $f$ over said interval. For
this entire section, we will assume that all occurring probability measures
are absolutely continuous and have finite variance.
Given two probability measures $\nu$ and $\mu$ on $[0,1]$, a _transport map_
$T:[0,1]\to[0,1]$ from $\nu$ to $\mu$ is a map such that, for every interval
$(a,b)\subset[0,1]$,
$\mu((a,b))=\nu(T^{-1}(a,b)),$
where $T^{-1}(a,b)$ denotes the preimage of $(a,b)$ under $T$, i.e., the set
of all numbers that $T$ maps to the interval $(a,b)$. One often uses the
notation $\mu=\nu\circ T^{-1}$ in this situation. This means the following: If
a random variable $X$ with values in $[0,1]$ is distributed according to
$\nu$, then $T$ is a transport map from $\nu$ to $\mu$ if and only if the
random variable $T(X)$ is distributed according to $\mu$. As an example on
$\mathbb{R}$, consider the two normal distributions $\mathcal{N}(0,1)$ and
$\mathcal{N}(1,2)$. Then $T:\mathbb{R}\to\mathbb{R}$, $T(x)=2x+1$, would be a
transport map from $\mathcal{N}(0,1)$ to $\mathcal{N}(1,2)$. However, there
may be many transport maps: In this example, also $x\mapsto-2x+1$ would be a
transport map. Among the many transport maps, one (or potentially several) may
be _optimal_ in the following sense:
###### Definition 3 (Optimal transport map).
Let $\nu,\mu$ be two absolutely probability measures on $[0,1]$ with finite
variance. An _optimal transport map_ is a transport map between $\nu$ and
$\mu$ that minimises the cost functional
$C(\nu,\mu,T):=\int_{\mathbb{R}^{n}}|x-T(x)|^{2}d\nu(x)$
among all transport maps from $\nu$ to $\mu$.
Under the stated assumptions on $\nu$ and $\mu$, Brenier (1987, 1991) ensures
the existence and uniqueness444Uniqueness holds only $\nu$-almost everywhere;
this means that the optimal transport map is not determined on any values
outside the support of $\nu$. of an optimal transport map. This allows to
define the so-called quadratic _Wasserstein distance_ between $\nu$ and $\mu$,
given by
$W_{2}(\nu,\mu):=C(\nu,\mu,T)^{1/2}.$
The Wasserstein distance forms a metric on the space of all absolutely
continuous finite-variance probability measures on $[0,1]$. The notion of
optimal transport map allows to define a kind of continuous interpolation
between the measures $\mu$ and $\nu$:
###### Definition 4 (Displacement interpolation, cf. Remark 2.13 in (Ambrosio
and Gigli, 2013)).
Let $\nu,\mu$ be two probability measures on $[0,1]$ with unique optimal
transport map $T$. The _displacement interpolation_ between $\nu$ and $\mu$
with interpolation parameter $\theta\in[0,1]$ is defined by
$\nu^{\theta}((a,b))=\nu((T^{\theta})^{-1}(a,b)),$
where the map $T^{\theta}:[0,1]\to[0,1]$ is given by
$T^{\theta}(x)=(1-\theta)x+\theta T(x)$.
Clearly, $\nu^{0}=\nu$ and $\nu^{1}=\mu$. The measure $\nu^{\theta}$ thus is a
measure ‘in between’ $\nu$ and $\mu$ that is closer to $\nu$ the smaller
$\theta$ is chosen and closer to $\mu$ the larger it is.
Finally, we mention the notion of _barycenter_ of a family of probability
measures $\\{\nu_{k}\\}_{1,\ldots,N}$ with corresponding weights
$\\{w_{k}\\}_{1,\ldots,N}$:
###### Theorem 5 (Barycenter in Wasserstein space (Agueh and Carlier, 2011)).
Let
$\\{\nu_{k}\\}_{1,\ldots,N}$ be a family of absolutely continuous probability
with finite variance, and let $\\{w_{k}\\}_{k=1,\ldots,N}$ be positive weights
with $\sum_{k=1}^{N}w_{k}=1$. Then there exists a unique probability measure
$\nu$ on $[0,1]$ that minimizes the functional
$\nu\mapsto\sum_{k=1}^{N}w_{k}W^{2}_{2}(\nu_{k},\nu).$
This measure is called the _barycenter_ of $\\{\nu_{k}\\}_{1,\ldots,N}$ with
weights $\\{w_{k}\\}_{1,\ldots,N}$.
With two groups and respective score distributions $\nu_{1},\nu_{2}$ and
weights $w_{1},w_{2}$, the barycenter is given by the displacement
interpolation $\nu=\nu_{1}\circ(T^{w_{2}})^{-1}$, where $T$ is the optimal
transport map between $\nu_{1}$ and $\nu_{2}$ and $T^{w_{1}}$ is given in Def.
4.
### 3.3. The FAir Interpolation Method (FAIM)
The goal of this paper is to transform a given model output into one which
‘balances’ between the three fairness criteria to an extent that the user is
free to choose according to her specific situation. For instance, a model
(like COMPAS) that has been used for a while and has been evaluated to give
quite accurate predictions, but discriminate unfairly between groups, can be
transformed into an improved algorithm that respects group fairness to a
higher degree.
Before we delve into the mathematics, let us describe more clearly the
situation. Again we have two groups of individuals represented as (disjoint)
sets $X_{1},X_{2}$, and for each group a given scoring algorithm, which for
group $t$ ($t=1,2$) is simply a map $S_{t}:X_{t}\to[0,1]$. We assume this
algorithm has been used for a sufficient amount of time such that reasonably
reliable data is available on its performance; more precisely, we assume for
each group a map $[0,1]\to[0,1]$, $s\mapsto\lambda^{+}_{t}(s)$ to be given,
where $\lambda^{+}_{t}(s)$ reflects the proportion of truly positive555Recall
that ‘truly positive’ means positive according to ground truth, regardless of
the score received. individuals from group $t$ that were assigned score $s$.
Alternatively, one may interpret $\lambda^{+}_{t}(s)$ as the probability that
a randomly chosen individual from group $t$ whose score equals $s$ will be
truly positive. We also define $\lambda^{-}_{t}(s):=1-\lambda^{+}_{t}(s)$ for
all $s\in[0,1]$ as the proportion of truly negative instances among
individuals of group $t$ and score $s$. From historical data, we know the
score distributions in each group assigned by $S_{t}$, that is, we have two
probability measures $\nu_{1},\nu_{2}$ such that for any (measurable) set
$Q\subset[0,1]$, $\nu_{t}(Q)$ is the probability that an individual $x$
randomly chosen from group $t$ has a score in $Q$.
###### Remark 6.
In practice, the question arises how to collect the data encoded in
$\lambda_{t}^{+}$ and $\nu_{t}$. For the latter, it suffices to collect the
previous outcomes of the scoring algorithm $S_{t}$, since $\nu_{t}(s)$ is the
proportion of individuals from group $t$ that were assigned score $s$ in the
past. The integral of $\nu_{t}(s)$ over all possible scores will then equal
one, because the probability that a given individual was assigned _some_ score
is one. For $\lambda_{t}^{+}$, we need information on the _ground truth_ ,
which is always a difficult and controversial matter. In addition to the data
on past scores, we need data on the actual positivity or negativity of
individuals. Though such data may be difficult to collect reliably, we wish to
remark that such information is not just required for our method, but for
_any_ reasonable quality control of the algorithm in question. Without any (at
least approximate) determination of ground truth, there is generally no way to
evaluate the performance of a given algorithm.
In the first steps of Algorithm 1, we give for each of the criteria A), B), C)
a procedure to transform a given algorithm that does not satisfy the
respective criterion into one that does.
#### 3.3.1. Criterion A)
The original algorithm might not yet be correctly calibrated, i.e., the scores
produced by $S_{t}$ do not yet give an accurate representation of the actual
probability of being positive. To fix it, we compute a new map $S_{t}^{A}$
defined by
(3) $S_{t}^{A}(x):=\lambda^{+}_{t}(S_{t}(x)),$
which means that an individual from group $t$ who initially received score $s$
will now receive score $\lambda^{+}_{t}(s)$, which is precisely its
probability of being positive. In Algorithm 1, Lines 1–1 compute map
$S_{t}^{A}$, and Lines 1–1 yield the probability distributions $\mu^{A}_{t}$.
This probability, as discussed, is computed from historical data, which
approximately reflects ground truth. Thus, although the new score will not in
general be able to give a deterministic value (zero or one) of the particular
individual’s true positivity, the new algorithm does satisfy requirement A)
(even bin-wise, i.e., for each $s$).
#### 3.3.2. Criterion B)
To express the balance criterion, we need to determine the expected score of
an individual $x$ from group $t$ on the condition that $x$ is negative. These
conditional expectations for both groups should then be equal. Indeed, we can
find a new score map such that the entire probability distributions of the
score, conditional on negativity, coincide, thus giving statistical parity in
score between the negative instances in each group. In particular, the risk of
a false positive outcome will be equal for both groups.
We do this as follows (see again Algorithm 1): First (Lines 1–1), we compute
said conditional probabilities $\lambda_{t}^{-}$ and $\nu_{t}$ from the given
data. By Bayes’ Theorem, the probability distribution for the score of an
individual $x$ from group $t$ conditional on $x$ being negative is given as
(4)
$\sigma^{-}_{t}:=\frac{\lambda_{t}^{-}(s)\nu_{t}(ds)}{\int_{0}^{1}\lambda_{t}^{-}(s)d\nu_{t}(s)}.$
Secondly, we consider the Wasserstein-2 barycenter of
$\sigma^{-}_{1},\sigma^{-}_{2}$ on $[0,1]$ with weights
$\frac{N_{1}}{N_{1}+N_{2}}$ and $\frac{N_{2}}{N_{1}+N_{2}}$, respectively,
which we denote by $\bar{\sigma}^{-}$ (Line 1). Then there are two (optimal)
transport maps $T^{-}_{t}:[0,1]\to[0,1]$ such that
(5)
$\bar{\sigma}^{-}=\sigma^{-}_{1}\circ(T^{-}_{1})^{-1}=\sigma^{-}_{2}\circ(T^{-}_{2})^{-1}.$
Finally, set $S_{t}^{B}(x):=T^{-}_{t}(S_{t}(x))$ for $t=1,2$. Replacing
$\nu_{t}$ with $\nu_{t}^{B}:=\nu_{t}\circ(T^{-}_{t})^{-1}$, which is the score
distribution under the modified algorithm $S_{t}^{B}$, and replacing also
$\lambda^{-}_{t}$ with
$(\lambda^{-}_{t})^{B}:=\lambda_{t}^{-}\circ(T^{-}_{t})^{-1}$, which is the
new probability that an individual from group $t$ with modified score is
negative, we obtain via Eq. 4 (Line 3.3.2)
(6) $\mu_{t}^{B}:=\sigma^{-}_{t}\circ(T_{t}^{-})^{-1}=\bar{\sigma}^{-}$
as the probability distribution of the modified score conditional on being
negative. Since they agree for both groups, the new $S_{t}^{B}$ satisfies (a
much stronger version of) B).
input : $\texttt{rawScores[]}:$ an array with a score for each individual;
$\texttt{stepsize}:$ a float that specifies the bin width of the truncated
scores;
$\texttt{groundTruthLabels[]}:$ an array with a binary ground truth label for
each individual;
$\texttt{groups[]}:$ an array with a group membership label for each
individual;
$\texttt{thetas[[]]}:$ a matrix with rows
$\left[\theta^{A},\theta^{B},\theta^{C}\right]$ for each group $t$.
output : $\texttt{fairScores[]}:$ a new score array with a fair score for each
individual such that the fair score distributions map the fairness criteria
given by thetas.
1
// Normalize scores to $[0,1]$ and discretize
2 $\texttt{rawScores[]}=\texttt{truncateScores(0, 1, stepsize)}$
// Step 1: Compute score map $S^{A}_{t}$, yielding $\mu^{A}_{t}$ (Eq.3)
3 criterionAScores[] = zeros(rawScores[].length)
4 for _$t$ in groups[].uniqueValues()_ do
5 for _$s$ in rawScores[].uniqueValues()_ do
6 groupMask = groups[] == $t$; groundTruthPositivesMask = groundTruthLabels[]
== 1; scoreMask = rawScores[] == $s$;
7 $s^{A}_{t}$ = $\frac{\texttt{rawScores[groupMask \&\& scoreMask \&\&
groundTruthPositivesMask].length}}{\texttt{rawScores[groupMask \&\&
scoreMask].length}}$
// add key value pair to score map for criterion $A$.
8 $S^{A}_{t}$.add(($s,s^{A}_{t}$))
9 end for
// translate raw scores into fair scores w.r.t. criterion $A$: everybody with
raw score $s$ gets fair score $s^{A}_{t}$ assigned.
10 criterionAScores[groupMask] = rawScores[groupMask].translate($S^{A}_{t}$)
11 $\mu^{A}_{t}$ = criterionAScores[groupMask].histogram()
12
13 end for
// Step 2: compute $\mu^{B}_{t}$
14 groupSigmas = [[]]; groupSizesInPercent = [];
15 groundTruthNegativesMask = groundTruthLabels[] == 0
16 for _$t$ in groups[].uniqueValues()_ do
17 groupMask = groups[] == $t$
18 groupSizesInPercent.add(groupMask.length() / groups[].length())
// compute Eq. 4
19 $\sigma^{-}_{t}$ = zeros(rawScores.histogram().length())
20 for _$s$ in rawScores[].uniqueValues()_ do
21 scoreMask = rawScores[] == $s$
22 temp = $\frac{\texttt{rawScores[groupMask \&\& scoreMask \&\&
groundTruthNegativesMask].length}}{\texttt{rawScores[groupMask \&\&
groundTruthNegativesMask].length}}$
23 $\sigma^{-}_{t}$.add(temp)
24 end for
25 groupSigmas.add($\sigma^{-}_{t}$)
26 end for
// compute barycenter $\bar{\sigma}^{-}$ between groupSigmas
27 $\bar{\sigma}^{-}$ = wasserstein2barycenter(groupSigmas,
groupSizesInPercent)
28
Algorithm 1 Algorithm FAIM. First, criteria $A,B$ and $C$ are individually
fully satisfied, yielding distributions $\mu^{A}_{t},\mu^{B}_{t}$, and
$\mu^{C}_{t}$. Then, the barycenter $\bar{\mu}_{t}$ of
$\mu^{A}_{t},\mu^{B}_{t}$, and $\mu^{C}_{t}$ is computed, which yields optimal
transport map $T_{t}$. $T_{t}$ is finally used to map from the original score
distribution $\nu_{t}$ to the fair distribution $\bar{\mu}_{t}$. “emd” stands
for earth mover distance.
26 for _$t$ in groups[].uniqueValues()_ do
27 groupMask = groups[] == $t$
// use $\bar{\sigma}^{-}$ to get optimal transport maps for ground truth
negative individuals (Eq. 5)
28 $T^{-}_{t}$ = computeOptimalTransportMap(rawScores[groupMask &&
groundTruthNegativesMask].histogram(), $\bar{\sigma}^{-}$, kind=emd)
// For ground truth negatives, translate rawScores into $S^{B}_{t}$. The
histogram of $S^{B}_{t}$ forms $\mu_{t}^{B}$. Note, that $\mu^{B}_{t}$ is the
score distribution for group $t$ that contains all individuals (negatives and
positive), even though the translation is done for true negatives only.
29 $\mu_{t}^{B}$ = rawScores[groupMask &&
groundTruthNegativesMask].translate($T^{-}_{t}$).histogram()
30 end for
// Step 3: compute $\mu^{C}_{t}$. We omit this part of the algorithm for
brevity, since this is done the same way as $\mu^{B}_{t}$ (Lines 1\--3.3.2).
The only difference is that we would use a groundTruthPositivesMask =
groundTruthLabels[] == 1, instead of a groundTruthNegativesMask.
// Step 4: compute final barycenter $\bar{\mu}_{t}$ which incorporates
thetas[[]] and optimal transport maps $T_{t}$ for each group $t$. Then
translate rawScores of group into fair scores.
31 for _$t$ in groups[].uniqueValues()_ do
// compute $\bar{\mu}_{t}$ (Eq. 7)
32 $\bar{\mu}_{t}$ =
wasserstein2barycenter($\mu^{A}_{t},\mu^{B}_{t},\mu^{C}_{t}$, thetas[$t$])
33 groupMask = groups[] == $t$
34 $T_{t}$ = computeOptimalTransportMap(rawScores[groupMask].histogram(),
$\bar{\mu}_{t}$, kind=emd)
35 fairScores[groupMask] = rawScores[groupMask].translate($T_{t}$)
36 end for
return _fairScores[] _
Let us, however, not ignore a small issue here: The map $T^{-}_{t}$ is, as
mentioned above, only well-defined on the support of $\sigma_{t}^{-}$. For our
intended application, there might well be scores that have never been given to
a truly negative (or positive) individual: For example, if the original
algorithm was not completely misguided, then it would most probably never have
given score 1 to a truly negative individual, so that $\sigma_{t}^{-}$ would
vanish near 1. For such scores we are essentially free to define the map $T$.
The most natural and simple choice is to leave these scores unchanged, i.e.,
to set $T_{t}^{-}(x)=x$ for any $x$ outside the support of $\sigma_{t}^{-}$.
#### 3.3.3. Criterion C)
Replacing $-$ by $+$ everywhere in B), we obtain a new algorithm $S_{t}^{C}$
that satisfies C) and in particular assimilates the probabilities of a false
negative outcome of the evaluation for both groups.
#### 3.3.4. Combining the three procedures
Each of the modified score maps $S_{t}^{A},S_{t}^{B},S_{t}^{C}$ gives rise to
a corresponding score distribution $\mu_{t}^{A},\mu_{t}^{B},\mu_{t}^{C}$. For
each group $t=1,2$, we therefore obtain a triangle in Wasserstein space. Let
now $\theta_{t}^{A},\theta_{t}^{B},\theta_{t}^{C}\in[0,1]$ be three real
parameters that add up to $1$, which are given as input to the algorithm. We
then consider the weighted barycenter $\bar{\mu}_{t}$ in Wasserstein-2 space,
i.e., the unique probability measure $\bar{\mu}$ that minimizes the functional
(7)
$\mu\mapsto\theta_{t}^{A}W_{2}^{2}(\mu,\mu_{t}^{A})+\theta_{t}^{B}W_{2}^{2}(\mu,\mu_{t}^{B})+\theta_{t}^{C}W_{2}^{2}(\mu,\mu_{t}^{C}).$
This barycenter embodies a ‘compromise’ between our three incompatible
fairness criteria, where the parameters $\theta$ allow to continuously adjust
the weights that the decision-maker wishes to put on the respective criterion
(Line 3.3.2).
The fair score of an individual in group $t$, based on the original (‘unfair’)
score function $S_{t}$, is then determined as $T(S(x))$, where
$T:[0,1]\to[0,1]$ is the optimal transport map from the original score
distribution $\nu_{t}$ to the fair distribution $\bar{\mu}_{t}$ (Lines
3.3.2–3.3.2).
## 4\. Experiments
We conduct extensive experiments on three use cases (one with synthetic data
and two with real-world data) to show the effectiveness of FAIM to interpolate
between the three different fairness criteria described in Section 3.1. First,
we apply FAIM to synthetic data with two demographic groups and show how much
of each type of fairness can be achieved by tuning each fairness preference
$\theta^{A},\theta^{B}$ and $\theta^{C}$. As a motivating example, we consider
the synthetic data to represent credit applicants from two different gender
groups. Second, we apply the same analysis to the COMPAS data set, which
contains protected group variables for gender, race, and age. We are aware of
critical voices, such as (Bao et al., 2021), on using COMPAS to benchmark new
fairness methods, and discuss whether recidivism risk is an appropriate use
case for FAIM in Section 6.3. However, since Kleinberg et al. (2016), whose
terminology we adopt, build their argumentation on the COMPAS case, we want to
show how our method and findings compare to those of Kleinberg et al. (2016).
Last, we study how the application of FAIM affects rankings on the European
e-commerce platform Zalando, and whether it can be used to overcome the
problem of popularity bias ((Bellogín et al., 2017)).666Data and code
available under https://github.com/MilkaLichtblau/faim.git For each use case,
FAIM is tested with the following fairness objectives:
1. (1)
Fairness criterion A: Calibration within groups ($\theta^{A}=1$)
2. (2)
Fairness criterion B: Balance for the negative class ($\theta^{B}=1$)
3. (3)
Fairness criterion C: Balance for the positive class ($\theta^{C}=1$)
4. (4)
An equally weighted combination of fairness criteria A), B), and C)
($\theta^{A}=\theta^{B}=\theta^{C}=\frac{1}{3}$)
Note that any other weighted combination is possible, e.g., by setting
$\theta^{A}=0.5$ and $\theta^{B}=\theta^{C}=0.25$ if one cares about criteria
B) and C), but prefers a result that thoroughly calibrates scores. Note also,
that the $\theta$-array can be configured on a per-group-level, but for
simplicity of presentation we set the same $\theta$s for each group. For each
fairness objective, we calculate performance metrics (accuracy, precision, and
recall) and error rates including false negative rate (FNR) and false positive
rate (FPR). Results from using FAIM are compared to those produced by the base
model.
### 4.1. Experiments on Synthetic Data
Figure 1. Synthetic true (solid lines) and predicted (dashed lines) score
distributions for demographic two groups (orange and blue). The decision
boundary at $\text{score}=0$ determines ground truth positive and negative
labels based on the true scores, and predicted positive and negative labels
based on the predicted scores. This simulates a situation in which the scoring
algorithm overestimates the qualification of the advantaged group, and further
disadvantages the disadvantaged group.
As a motivating example for the synthetic data experiment, let us consider
credit scoring. Increasingly, scoring agencies are turning to ML to predict
creditworthiness ((Langenbucher, 2020; Langenbucher and Corcoran, 2021)). Now
imagine that a company predicts ML-based credit scores for candidates stemming
from two protected groups only, for example for men and women (no non-binary
persons applied). Candidates receiving a score equal or above 0 are labeled
offered a credit contract, while those below are rejected.
#### 4.1.1. Data set.
To model such a credit scoring situation, we generate a large synthetic data
sets with 100,000 individuals and two demographic groups (one advantaged,
e.g., male; and one disadvantaged, e.g., female) of approximately the same
size. Each individual is assigned two scores, a _true score_ , and a
_predicted score_. The true score is to be understood as an individual’s true
probability of being part of a desirable positive class (e.g., creditworthy
person), while the predicted score is meant to represent a score that has been
assigned to an individual by a (synthetic) scoring algorithm. They are sampled
from a multivariate normal distribution with covariance matrix
$[[1,0.8],[0.8,1]]$ and true score means $(1,-1)$ for the advantaged and
disadvantaged groups, respectively. The corresponding predicted score means
are $(2,-3)$. Additionally, we define a synthetic decision threshold to yield
ground truth and predicted labels: those with a true (resp. predicted) score
above zero are considered ground truth (resp. predicted) positive (i.e.,
creditworthy, and receive a credit offer). Conversely, those with a true
(resp. predicted) score below zero are considered ground truth (predicted)
negative (i.e., not creditworthy, and are rejected). Figure 1 depicts this
data set. Blue lines correspond to the advantaged group, with both predicted
scores (dashed blue line) and true scores (solid blue line) normally
distributed with peaks greater than zero. Orange lines correspond to the
disadvantaged group, with both predicted scores (dashed orange line) and true
scores (solid orange line) normally distributed with peaks less than zero.
This data set reflects a scenario of different base rates between the groups,
as the advantaged group has a higher true probability of being positive. In
our example, this is the male group. However, the scoring algorithm has
overestimated the capabilities of the blue, advantaged group (the predicted
score distribution is centered to the right of the true score distribution),
and underestimated the capabilities of the orange, disadvantaged group (the
predicted score distribution is centered to the left of the true score
distribution). Additionally, the algorithm does a better job in predicting the
true score distribution for the advantaged group, than it does for the
disadvantaged group (the blue distributions overlap more). To emphasize the
disparity between the two demographic groups, the advantaged group is 5.3
times more likely to be labeled positive based on the true scores, compared to
6250 times more likely to be labeled positive based on the predicted scores
from the (unfair) scoring algorithm.
These two particular properties of disparities in model quality across groups
are well studied in the algorithmic fairness literature ((Angwin et al., 2016;
Zafar et al., 2017; Hardt et al., 2016)), which is why we choose this
synthetic setting to present the functioning of FAIM. Note that we have
plotted the true score distributions only for demonstration purposes to give a
clear picture of the discriminatory scenario. They are usually not available,
which is why FAIM relies on _observable_ ground truth labels, as explained in
Section 3.3.
Data set | Parameters | Performance | Error Rates
---|---|---|---
| | Accur. ($\Delta$) | Precision ($\Delta$) | Recall ($\Delta$) | FPR ($\Delta$) | FNR ($\Delta$)
Synthetic | before FAIM | 0.852 | 0.853 | 0.852 | 0.138 | 0.157
blue | | 0.860 | 0.868 | 0.860 | 0.869 | 0.002
orange | | 0.844 | 0.869 | 0.844 | 0.000 | 0.990
Synthetic | $\theta^{A}=1$ | 0.885 (0.033) | 0.885 (0.033) | 0.884 (0.032) | 0.116 (-0.022) | 0.114 (-0.043)
blue | | 0.884 (0.024) | 0.873 (0.005) | 0.884 (0.024) | 0.560 (-0.309) | 0.032 (0.032)
orange | | 0.885 (0.041) | 0.874 (0.005) | 0.885 (0.041) | 0.032 (0.032) | 0.557 (-0.433)
Synthetic | $\theta^{B}=1$ | 0.879 (0.027) | 0.882 (0.029) | 0.879 (0.027) | 0.076 (-0.062) | 0.166 (0.009)
blue | | 0.877 (0.017) | 0.876 (0.008) | 0.877 (0.017) | 0.392 (-0.477) | 0.073 (0.073)
orange | | 0.882 (0.038) | 0.873 (0.004) | 0.882 (0.038) | 0.016 (0.016) | 0.666 (-0.324)
Synthetic | $\theta^{C}=1$ | 0.865 (0.013) | 0.873 (0.020) | 0.865 (0.013) | 0.208 (0.070) | 0.062 (-0.095)
blue | | 0.854 (-0.006) | 0.868 (0.000) | 0.854 (-0.006) | 0.918 (0.049) | 0.001 (0.001)
orange | | 0.877 (0.033) | 0.877 (0.008) | 0.877 (0.033) | 0.074 (0.074) | 0.389 (-0.601)
Synthetic | $\theta^{A}=\theta^{B}=\theta^{C}$ | 0.883 (0.031) | 0.883 (0.030) | 0.883 (0.031) | 0.137 (-0.001) | 0.098 (-0.059)
blue | | 0.884 (0.024) | 0.873 (0.005) | 0.884 (0.024) | 0.560 (-0.309) | 0.032 (0.032)
orange | | 0.881 (0.037) | 0.875 (0.006) | 0.881 (0.037) | 0.057 (0.057) | 0.450 (-0.540)
Table 1. We report the performance metrics and error rates of the synthetic
experiment after FAIM has been applied, and the corresponding relative
improvements or deteriorations (green or red values in parenthesis,
respectively). The first line always refers to the whole data set, whereas
“blue” and “orange” contain results disaggregated by group. The top row shows
metrics before FAIM has been applied.
#### 4.1.2. Results.
Figure 2 shows the resulting score distributions of the synthetic data set
after FAIM has been applied (left column), together with the corresponding
transport maps that convert raw into fair scores (right column). The transport
maps are to be read as follows: each raw score on the x-axis gets replaced by
its corresponding value on the y-axis. Note that the transport maps
necessarily differ by group, since the groups experience disparate treatment
by our synthetic model which underestimated the capabilities of the
disadvantaged group, and overestimated those of the advantaged. Each row of
sub-figures corresponds to one of the scenarios described earlier. Consider
Figures LABEL:fig:experiments:result:synthetic:condA and
LABEL:fig:experiments:result:synthetic:TransportMapcondA as an example: they
correspond to the experiment where we want to fully meet fairness criterion
A). We see from the transport map (Fig.
LABEL:fig:experiments:result:synthetic:TransportMapcondA), that the two score
distributions are brought towards each other, as the disadvantaged group
(orange) is assigned higher fair scores for the same raw scores. Figures
LABEL:fig:experiments:result:synthetic:condB and
LABEL:fig:experiments:result:synthetic:TransportMapcondB, as well as Figures
LABEL:fig:experiments:result:synthetic:condC and
LABEL:fig:experiments:result:synthetic:TransportMapcondC, correspond to the
experiments that fully implement the balance criteria B) and C), respectively.
Note that these two balance criteria are defined only on a subset of
individuals, i.e., the true negatives and the true positives, respectively.
Thus, FAIM uses only this subset of the true negative (resp. true positive)
individuals to calculate the transport maps and, in turn, the fair scores.
This is reflected in Figures LABEL:fig:experiments:result:synthetic:condB and
LABEL:fig:experiments:result:synthetic:condC: the fair score distributions
overlap _only for the truly negative (resp. positive) individuals_ , while the
scores from the true positive (resp. negative) class are left untouched.
Figures LABEL:fig:experiments:result:synthetic:compromise and
LABEL:fig:experiments:result:synthetic:TransportMapcompromise show the results
of using the fairness objective
$\theta^{A}=\theta^{B}=\theta^{C}=\frac{1}{3}$. We see that this setting
indeed yields a compromise between the three fairness criteria.
Table 1 shows how FAIM affects classification performance and fairness. We
understand fairness in terms of the three fairness objectives we seek to meet
and therefore report metric and error rate differences in order to judge
FAIM’s performance. The table reports the absolute values for each metric
after FAIM has been applied, together with the changes relative to when FAIM
is not applied (in parentheses). The top row shows the metrics before FAIM has
been applied.
Observe the results for our first experimental setting with $\theta^{A}=1$. We
expect an overall performance increase, as the algorithm calibrates the
predictions with respect to actual ground truth evaluation. We also expect the
effect to be more pronounced for the orange group, because of the larger error
between predicted and ground truth scores for that group. Both expectations
are confirmed in the results. The chance of the orange group to be labeled
positive is now 5.7% of the blue group, which marks an improvement of more
than two orders of magnitude.
Next, observe the results for the second and third fairness objectives where
$\theta^{B}=1$, and $\theta^{C}=1$, respectively (balance criteria B) and C),
respectively). When fulfilling criterion B), we expect the fair score
distributions for ground truth negative individuals from the two groups to
approximately match (recall Fig.
LABEL:fig:experiments:result:synthetic:condB). Thus, the scores of ground
truth negative blue and orange group individuals will decrease and increase,
respectively. This corresponds to an FPR decrease for the blue group, which
had a high base FPR including many false positive ground truth negative
individuals scoring just slightly above the decision boundary whom become true
negative after FAIM is applied. This also corresponds to either no or a slight
FPR increase for the orange group, which had a base FPR of zero and ground
truth negative individuals scoring far enough below the decision boundary that
either none or only a few are mapped by FAIM above the decision boundary to
become false positives. The results confirm our expectations. For the blue
group, FAIM achieves an FPR of 39.2%, which marks an improvement w.r.t. the
original FPR of 47.7%. We also see a slight increase of FPR for the orange
group (1.6%). After FAIM is applied the FNR shows an improvement of 32.4%.
Overall the probability of the orange group to receive a positive label is
5.3% of the blue group for $\theta^{B}=1$. When fulfilling criterion C), we
expect the fair score distributions for true positive individuals to match
(Fig. LABEL:fig:experiments:result:synthetic:condC), thus the false negative
rates should improve, particularly for the orange group since most blue
individuals were predicted positive by the original model. Again our
expectations are confirmed by the results and declines in performance and
error rates remain relatively small for both groups. Overall, the probability
of the orange group for a positive label is 7.7% of the blue group.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 2. (Best seen in color). Left side: distributions of scores for each
group after FAIM was applied under different fairness preferences (bold),
together with the original distributions for comparison (transparent). Right
side: transport maps that translate from the raw score (x-axis) to the
respective fair score (y-axis). The gray dashed line marks the identity
mapping. Note, that the two groups are assigned different fair scores for the
same raw score. and that raw scores were normalized to $[0,1]$.
Last, observe the last experimental setting
$\theta^{A}=\theta^{B}=\theta^{C}$, which corresponds to a compromise between
the three mutually exclusive fairness criteria. As is confirmed by Figures
LABEL:fig:experiments:result:synthetic:compromise and
LABEL:fig:experiments:result:synthetic:TransportMapcompromise, the results in
Table 1 show that FAIM yields a compromise between calibration, balance for
the true negatives, and balance for the true positives. It achieves similar
performance improvements as aiming for calibration only ($\theta^{A}=1$), but
better FPR and FNR improvements. Compared to the balance criteria, this
setting achieves better performance improvements, but only slightly worse FPR
and FNR. Overall, the probability of the orange group to receive a positive
label is 9.7% of the blue group for $\theta^{A}=\theta^{B}=\theta^{C}$.
(i) Accuracy rates disaggregated by decile score and gender.
(j) Accuracy rates disaggregated by decile score and race.
(k) Accuracy rates disaggregated by decile score and age categories. Figure 3.
Accuracy rates disaggregated by COMPAS score. Across all demographics,
accuracy rates show that the mid-range scores 4–7 do not provide very
meaningful inside on recidivism risk.
### 4.2. Experiments on the COMPAS data set by (Angwin et al., 2016)
#### 4.2.1. Data set
COMPAS (Correctional Offender Management Profiling for Alternative Sanctions)
is a commercial tool developed by Northpointe, Inc. to assess a criminal
defendant’s likelihood of recidivating within a certain period of time. Based
on the promise to enhance fairness in judicial decision making, COMPAS is used
in several US states as a decision aid for judges, e.g., in parole cases. In
2016, Angwin et al. (2016) published an analysis of the tool based on a data
set of criminal defendants from Broward County, Florida, in which they found
the tool to be biased against certain groups (more on this below).
From the data—as it was published by (Angwin et al., 2016)—we use decile_score
(integers $\\{1,2,...,10\\}$) as predicted scores, and two_year_recid
(boolean) as ground truth labels. Additionally, we construct groups based on
sex, race, and age category using the features sex, race, and age_cat. To
increase race group sizes, we merge races ‘Native American’ and ‘Asian’ into
‘Other’, leaving four race groups: ‘Caucasian,’ ‘African-American,’
‘Hispanic,’ and ‘Other.’ These three features, i.e., the predicted score, the
ground truth label, and the group, form the input for FAIM.
To assess the impact of FAIM on the predictive performance and the group error
rates, we perform an analysis similar to that carried out by Angwin et al.
(2016). We also translate the predicted score into a binary label of high and
low risk corresponding to predicted score $\geq 5$ and $<5$, respectively.
This binarization is applied to both the predicted scores from the COMPAS data
set and the resulting fair scores produced by FAIM. Additionally, we calculate
the probability of the disadvantaged groups to be assigned a high risk label
relative to the advantaged group while correcting for the seriousness of their
crime, previous arrests, and future criminal behavior.
#### 4.2.2. Performance Analysis of the Original Model
To better interpret the results of applying FAIM to the COMPAS data set, we
first provide a detailed performance analysis of the original COMPAS
algorithm. Figure 3 shows the accuracy rates of the original model
disaggregated by decile score and demographic group. Grey bars mark the
accuracy rates measured for the aggregated data set. A striking insight is how
poorly the model performs through the entire score range, but particularly for
intermediate scores: within the range of 4–7, the accuracy rate for any
demographic group hardly ever exceeds a level of 0.6. One could argue, of
course, that a triangular pattern of accuracy rates (high accuracy for edge
scores 0 and 10, lower accuracy for scores near the decision boundary of 5)
would be expected from a well-calibrated model. After all, for a calibrated
system, roughly 50% of individuals receiving a COMPAS score of 5
(corresponding to a normalized score of 0.5) would reoffend. Given that a
COMPAS score $\geq 5$ corresponds to a prediction that an individual will
reoffend, a calibrated system would achieve around 50% accuracy for the 5th
score decile. It is important to understand however, that if the model was
well-calibrated we would expect to see accuracy rates of _at least_ 0.9 for
scores 1 and 10, _at least_ 0.8 for scores 2 and 8, and so forth, requiring an
accuracy rate of _at least_ 0.5 for score 0.5. As shown, the model never
achieves any of these minimum performance rates for any demographic group.
Another interesting insight can be found by looking at the disparities of
accuracy distributions across different demographic groups. Observe Figure 3:
We see that women’s accuracy is higher than men’s in the lower scores, but
vice versa for higher scores. This means that women are predominantly
misclassified when they are assigned high scores, while men are predominantly
misclassified, when they are assigned low scores. In other words, women are
treated too harshly by the algorithm, while men are treated too gently (thus
confirming the finding of Angwin et al. (2016)). The same is true for
Hispanics in Figure 3, whose accuracy for the low scores is much higher than
the one for the high score range. Surprisingly, Figure 3 reveals that young
people, even though already being assigned relatively high scores, still seem
to be treated too gently by the algorithm.
#### 4.2.3. Experimental Results when Applying FAIM
Data set | Parameters | Performance | Error Rates
---|---|---|---
| | Accur. ($\Delta$) | Precision ($\Delta$) | Recall ($\Delta$) | FPR ($\Delta$) | FNR ($\Delta$)
COMPAS | before FAIM | 0.470 | 0.751 | 0.470 | 1.000 | 0.000
male | | 0.495 | 0.750 | 0.495 | 1.000 | 0.000
female | | 0.387 | 0.763 | 0.387 | 1.000 | 0.000
COMPAS | $\theta^{A}=1$ | 0.521 (0.051) | 0.751 (0.000) | 0.521 (0.051) | 0.739 (-0.261) | 0.185 (0.185)
male | | 0.495 (0.000) | 0.750 (0.000) | 0.495 (0.000) | 1.000 (0.000) | 0.000 (0.000)
female | | 0.613 (0.226) | 0.763 (0.000) | 0.613 (0.226) | 0.000 (-1.000) | 1.000 (1.000)
COMPAS | $\theta^{B}=1$ | 0.530 (0.060) | 0.751 (0.000) | 0.530 (0.060) | 0.000 (-1.000) | 1.000 (1.000)
male | | 0.505 (0.010) | 0.750 (0.000) | 0.505 (0.010) | 0.000 (-1.000) | 1.000 (1.000)
female | | 0.613 (0.226) | 0.763 (0.000) | 0.613 (0.226) | 0.000 (-1.000) | 1.000 (1.000)
COMPAS | $\theta^{C}=1$ | 0.530 (0.060) | 0.751 (0.000) | 0.530 (0.060) | 0.000 (-1.000) | 1.000 (1.000)
male | | 0.505 (0.010) | 0.750 (0.000) | 0.505 (0.010) | 0.000 (-1.000) | 1.000 (1.000)
female | | 0.613 (0.226) | 0.763 (0.000) | 0.613 (0.226) | 0.000 (-1.000) | 1.000 (1.000)
COMPAS | $\theta^{A}=\theta^{B}=\theta^{C}$ | 0.530 (0.060) | 0.751 (0.000) | 0.530 (0.060) | 0.000 (-1.000) | 1.000 (1.000)
male | | 0.505 (0.010) | 0.750 (0.000) | 0.505 (0.010) | 0.000 (-1.000) | 1.000 (1.000)
female | | 0.613 (0.226) | 0.763 (0.000) | 0.613 (0.226) | 0.000 (-1.000) | 1.000 (1.000)
Table 2. Evaluation of FAIM when applied to the COMPAS data set,
disaggregated by gender. These are results only for an original COMPAS score
of 5, because only individuals close to the decision boundary are reclassified
after a score reassignment through FAIM. As such, for score ranges 1–4, and
6–10, neither error rates nor performance metrics change at all. Note also
that, for one particular score, everybody is classified either positive or
negative and the per-group error rates are thus either 0 or 1.
Our experimental results using FAIM on the COMPAS data set mostly confirm an
important hypothesis: our algorithm cannot fix an inherently flawed model
(garbage in, garbage out). As described earlier, we assessed accuracy,
precision and recall, as well as false positive and false negative rates,
total and disaggregated by demographics (similarly to results shown in Table
1). Because the nominal changes were negligible, we performed this same
analysis disaggregated by COMPAS score (as in Fig. 3). Together with the
analysis of the transport maps, this revealed the following: to fulfill
fairness criteria B) and C), FAIM does not change the scores drastically. This
is understandable since FAIM does not correct for _accurate_ scores, but only
for _equal distributions_ , as soon as $\theta^{A}=0$. Thus, only few
individuals with original scores of 4 or 5 get reclassified from low to high
risk, and vice versa. Those with original scores 1–4, or 6–10, do not cross
the decision boundary between the low and high risk class and therefore, none
of the metrics changes for these score ranges.
Therefore, we exemplarily report results disaggregated by gender and _for
score 5 only_ in Table 2, to showcase the behavior of FAIM on the COMPAS data
set. We see that the algorithm behaves as expected and does, in fact, improve
performance and error rates the way we expect it to. However, since the
original model performance is so low (see first row section in Table 2), we
would recommend to rather abandon this model entirely. It seems futile to
change random guessing into fair random guessing.
### 4.3. E-commerce data set from Zalando
#### 4.3.1. Data set
For our last experiments, we use data from the e-commerce platform Zalando,
one of Europe’s largest fashion and lifestyle retailers, operating in 25
European countries. We collected a data set of 81,048 articles on 27th October
2021, which belong to four different women clothes categories with a similar
price range: skirts, jeans, trousers, and knitwear. Each data point contains
the following information: a brand, a ranking score, and the number of
impressions and clicks during the week after article collection. We look at
the ratio of clicks over impressions and when it is above a certain threshold,
assign a positive label to the article (as it got a positive weekly
performance), or a negative label if it is below the threshold (as this
article failed to gather enough clicks despite its visibility). Groups are
assigned as follows: we compute a brand’s visibility as the average number of
impressions per article for each brand and each category, and thus categorize
brands having either low or high visibility. When this average is below the
median, a brand is said to belong to the ‘low’ group, while if the number is
above the third quartile, a brand is set to belong to the ‘high’ group.
Articles of brands with visibility between the low and high group thresholds
are discarded, leaving us with 62,461 articles. 70.5% of them belong to high
visibility brands. Note, that the ranking scores are produced by a learning-
to-rank model trained daily to optimize a surrogate normalized Discounted
Cumulative Gain (nDCG) loss ((Järvelin and Kekäläinen, 2002)), but not to
classify articles according to the ground truth labeling we defined above.
However, because the nDCG relevance labels are based on customer interactions
(i.e., implicit feedback), it is reasonable to assume that a good ranking
model should be able to identify which article is likely to be clicked. Hence,
the scores of an accurate click-through-rate classifier should provide a
decent ranking. We observe in our data that, irrespective of whether those
high articles are ground truth positive or negative, they receive higher
scores than corresponding low articles. There are many reasons external to the
ranking algorithm why customers would prefer well-known and highly visible
brands: these brands have higher marketing budgets, they can afford a more
aggressive pricing strategy thanks to economies of scale, and they benefit
from the Matthew effect of accumulated advantage ((Perc, 2014)). Moreover, it
is in the commercial interest of the platform to highlight such best selling
items. At the same time, the platform may not want to reinforce the status quo
but rather provide a level playing field for all brands to compete fairly.
This makes it easier to attract new brands on the platform, offers a more
diverse assortment to a potentially larger customer base, and can be seen as a
good step to address the legal requirements discussed in 5.3.
Data set | Parameters | Performance | Error Rates
---|---|---|---
| | Accur. ($\Delta$) | Precision ($\Delta$) | Recall ($\Delta$) | FPR ($\Delta$) | FNR ($\Delta$)
Zalando | before FAIM | 0.624 | 0.623 | 0.192 | 0.079 | 0.808
high | | 0.586 | 0.622 | 0.238 | 0.122 | 0.762
low | | 0.717 | 0.747 | 0.012 | 0.002 | 0.988
Zalando | $\theta^{A}=1$ | 0.654 (0.030) | 0.585 (-0.039) | 0.518 (0.327) | 0.252 (0.173) | 0.482 (-0.327)
high | | 0.623 (0.038) | 0.585 (-0.037) | 0.604 (0.366) | 0.361 (0.239) | 0.396 (-0.366)
low | | 0.729 (0.012) | 0.576 (-0.171) | 0.186 (0.174) | 0.055 (0.053) | 0.814 (-0.174)
Zalando | $\theta^{B}=1$ | 0.621 (-0.003) | 0.625 (0.001) | 0.169 (-0.022) | 0.070 (-0.010) | 0.831 (0.022)
high | | 0.577 (-0.008) | 0.625 (0.003) | 0.189 (-0.050) | 0.095 (-0.027) | 0.811 (0.050)
low | | 0.725 (0.008) | 0.623 (-0.124) | 0.095 (0.083) | 0.023 (0.021) | 0.905 (-0.083)
Zalando | $\theta^{C}=1$ | 0.622 (-0.003) | 0.623 (-0.001) | 0.177 (-0.015) | 0.073 (-0.006) | 0.823 (0.015)
high | | 0.579 (-0.007) | 0.625 (0.003) | 0.196 (-0.043) | 0.099 (-0.023) | 0.804 (0.043)
low | | 0.725 (0.008) | 0.608 (-0.139) | 0.104 (0.092) | 0.027 (0.025) | 0.896 (-0.092)
Zalando | $\theta^{A}=\theta^{B}=\theta^{C}$ | 0.633 (0.009) | 0.617 (-0.007) | 0.259 (0.067) | 0.110 (0.031) | 0.741 (-0.067)
high | | 0.594 (0.009) | 0.618 (-0.004) | 0.294 (0.055) | 0.153 (0.031) | 0.706 (-0.055)
low | | 0.727 (0.010) | 0.601 (-0.146) | 0.124 (0.112) | 0.033 (0.031) | 0.876 (-0.112)
Table 3. We report the performance metrics and error rates of the experiment
on Zalando data after FAIM has been applied, and the corresponding relative
improvements or deteriorations (green or red values in parenthesis,
respectively). The first line always refers to the whole data set, whereas
“high” and “low” contain results disaggregated by group. The top row shows
metrics before FAIM has been applied.
#### 4.3.2. Results
From Table 3 we see that low visibility brands do indeed have a relative
disadvantage over high visibility brands. The low recall, and the high false
negative rate indicate that many relevant items from the low group are not
shown to the customer. This situation is improved by FAIM in all four
scenarios, albeit to varying degrees. Our results also show that the algorithm
yields expected improvements: When criterion A) is desired, accuracy and
recall is improved for both groups. Pursuing criterion B) or C) levels the
respective error rates. Finally, a compromise between all three criteria is
found when setting $\theta^{A}=\theta^{B}=\theta^{C}$.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4. (Best seen in color). Product rankings with low and high visibility
brands under different settings. Colors indicate group membership, stripes
indicate a positive label, i.e., these are the products customers find more
relevant. Fig. LABEL:fig:experiments:result:zalando:synthetic exemplifies an
optimal outcome, where all relevant products are in the top positions, and the
visibility of brands A and B is distributed according to their share of
relevant products. Fig. LABEL:fig:experiments:result:zalando:real shows the
distribution of relevant (and non-relevant) products from low and high brands
as shown by Zalando’s ranking algorithm by the time of data collection. We see
that famous brands do indeed have an advantage over less-known brands, as
there are almost no products from the low group in the first few bins. Fig.
LABEL:fig:experiments:result:zalando:condA—LABEL:fig:experiments:result:zalando:compromise
show how the ranking changes after FAIM is applied with different values for
$\theta^{A}$, $\theta^{B}$, and $\theta^{C}$. Depending on our preference for
calibration, balance for the positive class, or balance for the negative
class, we see varying degrees of visibility improvement for the low group.
For a better understanding on the effect of FAIM on the rankings, observe
Figure 4. In Figure LABEL:fig:experiments:result:zalando:synthetic we show a
fictive example for an ideal ranking: imagine a data set with two groups A and
B. A has 1,000 items: 200 positive and 800 negative. B has 500 items: 300
positive and 200 negative. If we assume a perfect ranker, we can sort those
items simply by their score. When we group them in bins of 50 items each, we
would expect the first 10 bins to contain only positive items, and the next 20
bins to have only negative items. Furthermore, the positive bins should
display a ratio of 300/200 for groups B and A respectively, while the negative
bins should display a ratio of 200/800. Figure
LABEL:fig:experiments:result:zalando:real shows the ranking as collected from
the Zalando website on 27th October 2021, which by virtue of being real is not
ideal. Since each bin contains more than 2,000 items, it is enough to focus on
the first bins, as other items are shown very rarely, unless customers are
actively looking for them. We see that the first five bins contain mostly high
brands, meaning that relevant products from low brands are rarely shown to the
customer, unless they start to use product filters or full text search. Again,
there are various reasons why a customer would prefer well-known over less-
known brands, but if we wanted to (or had to by legislation) mitigate the
visibility disparities between high and low brands, FAIM provides a convenient
approach (see Section 5.3 for a discussion on FAIM’s legal meaning for an
e-commerce scenario such as ours). Figures
LABEL:fig:experiments:result:zalando:condA–
LABEL:fig:experiments:result:zalando:compromise show rankings after FAIM has
been applied with different values for $\theta^{A}$, $\theta^{B}$, and
$\theta^{C}$. In all cases, FAIM distributes visibility better across both
brand groups, particularly for relevant products, which is the interesting
case for an e-commerce platform such as Zalando.
## 5\. Application of the Algorithm in Legal Scenarios
Given the _contextual agility_ ((Wachter et al., 2021b)) of EU regulations
with respect to the definition of fairness, it is advantageous to introduce a
similar flexibility in the operationalization of fairness concepts. FAIM has a
wide range of applications in scenarios in which the law compels the decision
maker to make decisions in both an accurate _and_ a non-discriminatory way.
Technically speaking, this may result in a trade-off between competing
fairness and accuracy measures embodying the contemplated criteria, i.e.,
calibration; balance for the negative class; balance for the positive class.
Importantly, the weighting of the different measures via the $\theta$
interpolation parameters is a deeply normative choice which reflects competing
visions of what is supposed to be considered fair, and legal, in a certain
situation. Depending on the domain of application, the achievement of
calibration, the balancing of false positives or of false negatives might be
considered most important. FAIM affords the advantage of allowing the decision
maker to flexibly adapt to these different scenarios by choosing different
$\theta$ weights.
The broad discussion of different fairness metrics both in computer science
and the law ((Wachter et al., 2021a; Hellman, 2020; Binns, 2018; Friedler et
al., 2021; Kim, 2017; Pessach and Shmueli, 2020; Mitchell et al., 2021; Ghoash
et al., 2021)) suggests that no single metric such as statistical parity,
error balance or equalized odds will fit all contexts. This also applies to
the fairness measures under discussion here ((Corbett-Davies and Goel, 2018)).
As pointed out before, an important prerequisite for those metrics is the
reliability of ground truth ((Bao et al., 2021)). If these data points are
collected in a way that systematically disfavors one protective group,
conditioning on ground truth data risks perpetuating imbalances included in
them ((Bao et al., 2021; Wachter et al., 2021a; Chouldechova, 2017)). Within
these constraints, ground truth measures like the ones underlying FAIM can
nevertheless be helpful tools in many situations to capture normative
desiderata, particularly if used alongside strategies to improve the
correctness and representativeness of ground truth.
Generally speaking, choosing, for example, between a stronger balance for the
negative or the positive class will depend on the respective consequences of
misclassification (false negative and false positive predictions). As a result
of prediction errors, individual and social costs arise, and legal norms may
be violated. These costs and norms will differ widely depending on the
deployment context. In the following, we review three examples in which
different normative considerations may lead a decision maker to adopt varying
weights for the respective accuracy and non-discrimination measures:
recidivism prediction, credit scoring, and fair ranking according to the
Digital Markets Acts (DMA).
### 5.1. Recidivism Prediction
In this paper, we consider recidivism prediction for technical and
argumentative reasons: the original papers by Kleinberg et al. (2016) and
Chouldechova (2017)showing incompatibility between fairness metrics did use
the COMPAS case, and we directly build on their work. Hence, we use COMPAS as
an example to clarify how FAIM works. However, when considering recidivism
prediction, the first questions that arise from a juridical perspective are
whether one legally may, or policy-wise should, use algorithmic tools such as
the COMPAS model to assess recidivism risk in Criminal Justice proceedings at
all (see also 6.3). We do not wish to make any claim as to the legitimacy of
the usage of these tools in this paper; there are, if anything, good reasons
to be quite skeptical about it. As mentioned, COMPAS is a proprietary model
used in pretrial settings in the US to determine whether potential offenders
should be detained until their trial, based on a prediction of recidivism
likelihood. Similar models are currently employed in Canada, the UK, and Spain
((Watch, 2019, p. 122)). While the use of such instruments has sparked
significant controversy, and critique, in the legal literature ((Starr, 2014;
Mayson, 2019; Selbst, 2017; Katyal, 2019; Eaglin, 2017; Huq, 2019; Pruss,
2021)), courts such as the Wisconsin Supreme Court have condoned their use
under certain safeguards,777 State v. Loomis, 881 N.W.2d 749 (Wis. 2016). and
their deployment is currently on the rise ((Hamilton, 2021); (Watch, 2019, p.
122)).
Within the scope of this paper, we cannot offer an in-depth discussion of the
promises and perils of algorithmic recidivism risk assessment (see, e.g.,
(Solow-Niederman et al., 2019; Kleinberg et al., 2018), and 6.3). Rather, we
would like to point out that, to the extent that such tools are used at all,
they must clearly fulfill minimum requirements seeking to safeguard normative
and legal principles of the jurisdictions in which they are deployed.
Importantly, the impossibility theorems concerning various fairness metrics
mentioned above apply equally in the case of algorithmic and human risk
assessments. While the influence of fairness considerations on human risk
predictions, for example by a judge, remains difficult to elucidate, FAIM
allows to make the relevant trade-offs transparent and to rank the involved
fairness metrics in varying degrees of priority.
In training the COMPAS model, the developers chose to prioritize calibration,
which inevitably led to differing false positive and false negative rates
given imperfect prediction and differing base rates between the involved
ethnic groups ((Dieterich et al., 2016; Larson et al., 2016)). While accuracy
remains significant in recidivism risk assessment—and is actually quite low
with the COMPAS algorithm ((Dressel and Farid, 2018))—the core normative
trade-off arguably occurs between the respective importance of equalizing
false positive and false negative predictions. On the one hand, one could
argue that the prevention of false positive outcomes should be prioritized
because, under the rule of law, it is of utmost importance not to detain any
person without sufficient reason. Under this reading, the greatest weight
should be put on aligning false positive rates between groups, so that the
burden of being unduly sent to prison is shared equally between the respective
groups. On the other hand, one could claim that matching false negative
predictions is crucial because they unduly spare individuals time in prison,
affording a significant individual advantage to them. Under this reading, that
undeserved benefit should not accrue to any protected group to a greater
extent.888 Note that the fact false negative predictions constitute a
particular risk to individuals and to society at large, with potentially
dangerous offenders roaming free, is likely irrelevant here: it should not
matter to victims, nor to society, by members of what protected group re-
offences are committed. Criterion B) balances the negative classes, but does
not (directly) reduce their size.
FAIM allows for the establishment of an intermediary position such that both
balances, for the positive and the negative class, are fulfilled to an equal
degree, even though only partially. To the extent that such instruments are
used at all, and that the data they are based on are considered adequate, FAIM
may therefore operationalize a policy compromise between factually
irreconcilable goals.
From a different perspective, however, balancing false positives may be
considered more important: losses loom larger than gains. Time in prison
constitutes a highly significant restriction of personal freedom, interrupting
private lives and careers, potentially endangering physical and mental health.
Hence, lawmakers or judges may come to the conclusion that it is more
important to share the burden of false positive predictions equally between
groups than the unwarranted benefit of false negative predictions. FAIM then
allows to prioritize the former criterion.
### 5.2. Credit Scoring
Entirely different normative considerations are present in the case of credit
scoring. Imagine a bank uses an ML-based scoring algorithm to assess the
creditworthiness of loan applicants, as in our synthetic experiment.
Increasingly, ML is indeed used for these purposes ((Langenbucher, 2020;
Langenbucher and Corcoran, 2021)). Here, accuracy facilitates so-called
responsible lending, i.e., loan decisions in which the credit institution
intends to ensure that the borrower is not overburdened by the repayment
obligations. After the financial crisis of 2008/09, responsible lending has
become a cornerstone of financial law. In the EU, for example, the obligation
to lend responsibly is enshrined in Article 8 of the Consumer Credit Directive
2008/48/EC, and other EU law instruments install a comprehensive compliance
and supervision regime demanding regular audits to ensure the accuracy of
credit scoring models (Art. 174 et seqq. of the Capital Requirements
Regulation 575/2013, CRR). Hence, accuracy should certainly receive
significant weight in the case of credit scoring. As Art. 174(1) CRR puts it,
statistical models used by banks need to have ‘good predictive power’.
However, with default base rates usually differing between protected groups,
high degrees of calibration will lead to an imbalance in the false positive or
false negative rates between the groups. A positive label, in credit scoring,
means that the loan request is denied, a negative label that it is granted
(because the risk of default is low enough). Clearly, false negative
predictions may inflict financial damage on the lender if the credit cannot be
repaid, but also potentially on the borrower, who may face financial
penalties, eviction and future encumbrance due to a negative credit record.
False positive predictions, on the other hand, give rise to opportunity costs:
the lender does not earn interest payments, and the borrower does not obtain
access to credit. Disparities concerning false negative or false positive
rates affect access to credit or default rates between protected groups, and
are therefore relevant for compliance with non-discrimination legislation.
In the EU, rules on indirect discrimination forbid even seemingly neutral
practices putting protected groups at a particular disadvantage, unless these
differences can be justified.999 See, e.g., Art. 2(b) of Directive
2004/113/EC; Art. 2(2)(b) of Directive 2000/43/EC. Higher false positive rates
clearly constitute a disadvantage for the affected group; even higher false
negative rates can be viewed as a burden on the more often wrongly classified
group, however, as they entail the concrete risk of default. Since accuracy
and error rates cannot be balanced between protected groups at the same time
under non-trivial conditions, the law will have to accept reasonable trade-
offs between these competing and mutually exclusive obligations. It cannot and
does not demand what is impossible (nemo ultra posse obligatur). From the
perspective of legal doctrine, this will arguably take the route of the
justification of a possible disadvantage. The damage inflicted must then be
proportionate, given the reasons which can be advanced for prioritizing the
other fairness criteria.
Given these preconditions, equalizing false negative credit predictions might
be considered more important in the case of high-stakes loans (e.g., large
sums and little collateral). Here, a default would be particularly disruptive,
and hence that burden should be shared equally between protected groups. As a
consequence, the $\theta$ weight of the balance for the positive class
(Criterion C) should be higher than that for the negative class (Criterion B).
Conversely, in the case of low-stakes loans (e.g., low credit volume;
sufficient collateral besides basic necessities of the borrower, e.g. beyond
the house a family lives in; or limited personal liability in case of
default), false positive predictions might be more deleterious than false
negative ones because the damage is limited if the borrower defaults on the
credit. Wrongfully denying access to credit may then have larger opportunity
costs. Under such circumstances, the $\theta$ weight of the balance for the
negative class (Criterion B) should be higher than that for the positive class
(Criterion C). As an example, calibration could be set to 0.5, balance for the
negative class to 0.35, and balance for the positive class to 0.15. Legally
speaking, the fact that an unequal rate of false negative predictions persists
will then arguably be justified by the overriding importance of aligning false
positive predictions and calibration between groups.
### 5.3. Fairness in E-Commerce Rankings
Finally, our method may be used to implement emerging legal notions of
fairness in e-commerce rankings. Particularly in EU law, a growing number of
provisions aim to safeguard the impartiality of rankings in online contexts.
At the most general level, these rules have shifted from the prohibition of
self-preferencing via a focus on transparency to, so far, under-researched
concepts of fairness in the Digital Markets Act (DMA). To the best of our
knowledge, our contribution constitutes the first attempt to operationalize
the fair ranking provisions of the DMA on both a technical and a legal basis.
#### 5.3.1. Competition Law
The oldest and best-known rule of fairness in e-commerce rankings is derived
from the prohibition, in general competition law, to abuse a dominant position
(Art. 102 TFEU in EU law). It has long been argued that many dominant online
undertakings, such as Google or Amazon, need to be scrutinized under this
provision due to their dual role as providers of online marketplaces – on
which third parties directly sell their goods to consumers – and as direct
sellers of goods on their platforms ((Padilla et al., 2020; Graef, 2019)).
Hence, such platforms may be considered competitors of the very business
customers they serve on their marketplaces (vertical integration). As a
consequence, dominant online platforms must not unduly preference their own
offers vis-à-vis those of their business customers in rankings they display as
a result of consumer search queries. This rule has led to some of the most
spectacular fines in recent EU competition law. For example, the General Court
of the European Union, in November 2021, affirmed the 2017 decision by the EU
Commission to fine Google with €2.4 billion for engaging in self-preferencing
in its online comparison shopping service.101010GCEU, Case T-612/17 (Google
Shopping). A similar issue is at stake in the Amazon buy box case, in which
the Italian Competition Authority, in December 2021, imposed a record fine of
€1.1 billion on several Amazon companies for tying access to the buy box to
the use of Amazon’s own logistics channel (Fulfillment by
Amazon).111111Italian Competition Authority, Case A528, Amazon, Press release,
9 December 2021.
The upshot of these rulings is that dominant platforms which serve a dual role
of marketplace and seller must abstain from systematically tweaking their
rankings to their own benefit. This rule against self-preferencing, deduced
from Art. 102 TFEU and pertinent case law, may be considered a first
substantive fairness element for the order of the ranking itself ((Podszun and
Bongartz, 2021)). However, it only applies to undertakings which dominate a
certain market.
#### 5.3.2. Transparency
Often, though, it is difficult for outsiders, even for the business customers,
to determine how these rankings come about. Two new provisions therefore
install explicit transparency provisions for rankings in EU law ((Hacker and
Passoth, 2022; Eifert et al., 2021)). First, Art. 5 of the so-called P2B
(Platform to Business) Regulation (EU) 2019/1150 obliges online intermediaries
and search engines, independent of their market power, to disclose the main
parameters of ranking and their relative importance. The provision has been in
effect since July 2020. It is supposed to foster the predictability and
understanding of the ranking for business users, and to foster competition
between different intermediaries with respect to the ranking parameters
(Recital 24 of the P2B Regulation). In a similar fashion, second, the new Art.
6a of the Consumer Rights Directive (CRD) has required online marketplaces
from the end of May 2022 on to divulge to consumers the main parameters for
rankings based on consumer search queries as well as their relative
importance. According to the new Art. 2(1)(n) of the Unfair Commercial
Practices Directive, ‘online marketplace’ denotes any software operated by or
on behalf of a trader which allows consumers to conclude distance contracts
with other traders or consumers. Therefore, the rule does not apply to
companies only selling goods directly to consumers, but is again independent
of market power.
Essentially, scholars have argued, these provisions introduce an obligation
for the global explanation of the ranking model ((Hacker and Passoth, 2022;
Grochowski et al., 2021)). It may be fulfilled, for example, by using
interpretable machine learning models (e.g., linear or logistic regression
((Rudin, 2019))) or by using global post-hoc explanations for black box models
(e.g., DNNs) such as an average over local SHAP values ((Lundberg and Lee,
2017)) or SpRAy ((Lapuschkin et al., 2019)). However, the provisions do not
add any specific, substantive fairness criteria concerning the order of the
ranking itself.
#### 5.3.3. The EU Digital Markets Act
The DMA, which will come into effect in the EU in 2023, takes these existing
provisions yet one step further by introducing a novel fairness condition for
rankings. Its importance is hard to overstate: failure to comply with the DMA
provisions may be sanctioned by fines of up to 10% of the global annual
turnover of the offender (Art. 30(1) DMA), and up to 20% in the case of a
repeated violation (Art. 30(2) DMA). The DMA includes a whole range of
provisions aimed at strengthening contestable markets and fair competition in
online environments by designing specific rules applicable to so-called
gatekeepers, in essence large online platforms.
##### Fairness in the DMA.
Art. 6(5) DMA specifically tackles rankings,121212 In the final version, it
applies, beyond rankings, also to indexing and crawling. including when
delivered via a virtual assistant (Recital 51 DMA; for example, a personal
voice assistant). Art. 6(5) DMA first reiterates the prohibition of self-
preferencing known from general antitrust law.131313 For differences to Art.
102 TFEU, see, e.g., (Eifert et al., 2021).
The second sentence of Art. 6(5) DMA starts by specifying that rankings need
to be transparent. As Recital 52 suggests, this could refer to the
transparency provisions of the P2B Regulation. Importantly, Art. 6(5) DMA goes
on to demand that gatekeepers must, in general, apply ‘fair and non-
discriminatory conditions’ to their rankings. In the legal space, this has
sparked a discussion on how these criteria may be interpreted ((Brouwer, 2021;
Hacker, 2022b)).141414For a discussion of non-discriminatory rankings in
general competition law, see Graef (2019). So far, however, no clear guidance
or consensus has emerged on what these additional fairness requirements for
rankings under the DMA could mean for gatekeepers.
This raises the question of how the obligation to apply fair and non-
discriminatory conditions to _rankings_ could be interpreted. We cannot
explore this conceptual issue in depth in this paper. If, however, courts
concluded in the future that fairness and non-discrimination under the DMA
clause was somehow linked to accuracy, false positives and false negatives, we
can smoothly operationalize the DMA fairness condition with FAIM. This is
significant insofar as previous discussions of fairness in algorithmic
contexts have invariably focused on non-discrimination law ((Zehlike et al.,
2017; Asudeh et al., 2019; Singh and Joachims, 2018; Zehlike et al., 2022b)),
and not on fairness conditions in e-commerce regulation, such as the
DMA.151515The DMA applies to a range of digital services, so-called core
platform services (Art. 2(2) DMA). These include social media and video-
sharing services, for example; online intermediation, typical for e-commerce,
is another core platform service according to Art. 2(2)(a) DMA.
##### DMA Ranking with FAIM.
FAIM allows to specifically and flexibly re-order gatekeeper rankings such
that certain conditions are prioritized, depending on the exact, and as of yet
unknown, binding interpretation of Art. 6(5) DMA. In the case of e-commerce
rankings, one hypothetical interpretation by courts could be that accuracy as
well as an equal false positive rate are the most important parameters among
the three conditions. Under such a reading, false negative predictions might
be considered less important than false positive predictions as most consumers
will not even see or engage with those items which are ranked at low
positions. If a platform, preemptively or as a result of a court or agency
order, wishes to avoid that criteria such as incumbent status play a role in
the top-ranked products, $\theta$ values can be set accordingly. They could,
for example, convey high priority to equalizing the false positive rate
(Criterion B) between high-visibility and low-visibility brands.161616To be
sure, there may be valid reasons for relegating certain brands to low-
visibility status, for example reasons relating to product quality.
Nonetheless, one may wish to equalize error rates between the groups. For
instance, platforms may seek to limit false negative rates in the low group
since even such brands may, eventually, produce some high-quality items; see
also next para.
A second hypothetical interpretation would focus on false negatives. False
negative predictions unduly demote products in the ranking and thus deprive
affected items of visibility and hence of transactions. If a regulatory agency
or court found that the false-negative rate needs to be balanced between
certain groups, FAIM can again be tuned to implement such a requirement by
choosing corresponding $\theta$ values (high values for Criterion C).
In future work, we will seek to identify further distinctions which may be
problematic under the DMA fairness clause and which may be remedied by FAIM.
Importantly, the fairness clause in the DMA provides an opportunity to reflect
upon fairness criteria in e-commerce rankings beyond general non-
discrimination law and the prohibition of self-preferencing in competition
law.
## 6\. Discussion and Limitations
### 6.1. Normative Choices and Applicability
To explicitly name the normative choices FAIM implies, we use the concepts by
Friedler et al. (2021) of two different metric spaces— _construct space_ and
_observable space_. Construct space denotes an idealized reality in which all
feature values are correctly registered, and a decision is based on those
features. However, decision makers usually do not have access to this realm of
an objective truth. Rather, they need to base their model on observable space,
which contains the feature values the decision maker has access to via
measurement., and the decisions based on them. Importantly, ground truth may
correctly reflect, or at least closely approximate, construct space—or it may
not. This implies a first limitation for our method. FAIM operates in
observable space, and does not undo any harm that might arise from a
potentially biased mapping between construct and observable space, e.g., by
systematic measurement errors. This is because we obtain our ground truth
labels from observing behavior of scored individuals, such as: “did someone
who was labeled creditworthy actually repay the loan?” Such observations
naturally incorporate historic and on-going discrimination, as they usually do
not ask: “If a person who was labeled creditworthy did not repay the loan, was
that because she can genuinely not handle money, or because she belongs to a
disadvantaged group?” To be sure, this question is crucial for the societal
mitigation of discrimination in the long term. However, with the data we have
at our disposition, we usually cannot answer it exhaustively. Hence, FAIM must
generally content itself with correcting unequal calibration or disparate
error rates, defined by a divergence from ground truth, that
disproportionately affect a certain group of individuals. As such disparities
often correlate with the membership of a disadvantaged group in society, it
may also correct discrimination caused by skewed mappings between construct
and observable space to a certain extend (as we have seen in our experimental
results). However, this is only a side-effect and depends on the nature of
discrimination that is present in the data. If ground truth is close to
reality, i.e., to construct space, then we can cure biased mappings between
construct and observable space; otherwise, not. For the same reasons, FAIM
addresses _technical_ and _emergent_ , but not _pre-existing_ bias, as defined
by Friedman and Nissenbaum (1996).
We therefore advise FAIM to be used in settings where ground truth
observations are sufficiently reliable (though this is a necessary condition
for any data-driven decision making), and _it can be assumed that groups can
be meaningfully compared by the respective measures of qualification or
quality_. As an example for a non-comparable qualification measure across
genders, consider the h-index ((Hirsch, 2005)), which suffers from strong
gender bias. Research shows that women, for the same amount and quality of
research, receive significantly lower h-index values because it takes them
longer to publish, and their work is less cited than men’s ((Caplar et al.,
2017)). In cases where groups cannot be meaningfully compared, a method that
implements statistical parity or other affirmative action policies independent
of ground truth (such as FA*IR by Zehlike et al. (2017, 2022a)) is an
appropriate choice (see also (Wachter et al., 2021a)).
### 6.2. Limitations of FAIM
##### Continuous Mathematical Framework.
As stated, we work with a _continuous_ scale of possible scores (in the real
interval $[0,1]$) and accordingly with continuous probability distributions.
In practice, of course, there will be only a discrete set of possible scores,
and a finite set of individuals under scrutiny. The reason we work in a
continuous setting is of a mathematical nature, in that it allows us to use
certain tools from the theory of optimal transport, which in a discrete
setting would complicate the theory considerably. It is common and reasonable
practice to approximate discrete models by continuous ones, as long as the
number of individuals considered is large and their scores are sufficiently
spread out over a finite scale. Thus, our framework would not be suitable for
a situation where there are only very few possible score values, e.g., only
three or four. Further discussion in this direction, for a related problem,
can be found in Zehlike et al. (2020).
##### Discontinuities in score distribution.
In criterion B) (we use this case as an example to explain the problem, but
the same applies to criterion C), we only balance the scores for the negative
class and ignore the true positives. This means that the scores of the true
positives remain unchanged by FAIM. This might become a problem for rare
scores of true negative individuals for whom there was no example in the data
when calculating the optimal transport (OT) matrix. Remember that our
algorithm calculates an optimal transport matrix to translate raw scores into
fair scores for _previously unseen_ individuals. This means that for new data
points, we do not know the ground truth label. Instead, FAIM consults the OT
matrix that corresponds to a certain group and setting of $\theta$ and checks
if it finds a raw score entry in that matrix. In such a case, the following
scenario might arise when trying to optimize for criterion B) only: Suppose a
true negative person’s score lies at the edge of the overall true negative
score distribution, i.e., only few others (in the world) have the same score
while being true negative. In addition, critically, no true negative person
had this score when the OT matrix was computed. In this case, the algorithm
does not have an entry for translation in the OT matrix. As a result, this
individual would be considered true positive, and their score would be left
untouched.
### 6.3. Critique of using COMPAS as an experimental scenario
Bao et al. (2021) raise the problem that Risk Assessment Instrument (RAI) data
sets such as COMPAS are used for benchmarking fairness methods in a purely
technical fashion without recognizing the domain- and data-specific concerns
of Criminal Justice (CJ). As a result, technical fairness method contributions
could be interpreted as substantive interventions in the CJ domain. However,
the authors rightly argue, the latter presupposes careful considerations of
the domain specifics that may invalidate such results. They raise several
serious concerns about COMPAS: outcome label bias due to construct invalidity;
measurement bias that cannot be calibrated due to lack of (reliable) ground
truth; protected attribute measurement bias; racial bias in covariates; and
distribution issues due to label selection bias. We recognize and acknowledge
the concerns raised by Bao et al. However, since our contribution is an
improvement on a previously published method presented on the COMPAS data set,
we deem it important to enable a comparison of our results with the original.
Thus, we publish the results from our experiments on the COMPAS data set only
as a technical comparison with the results in Kleinberg et al. (2016), and
under the caveat that reliable ground truth is not available for the COMPAS
data set. Hence, the results on applying FAIM to COMPAS are not intended as a
contribution to real-world outcomes in the criminal justice domain. Rather, we
discuss how FAIM could be used in a pretrial setting _if_ the model and data
were considered adequate.
### 6.4. Data Missingness
Data missingness is a validity concern for the study of any fairness
criterion, as noted by Goel et al. (2021), and is not unique for FAIM. As is
the case for many fairness criteria, FAIM assumes the availability of ground
truth data that is representative of the underlying distribution, and the
absence of systematic censorship ((Kallus and Zhou, 2018)) in the data
generation process. If systematic censorship occurs due to biased decision
making, then this prevents us from sampling from the ground truth distribution
needed for an unbiased fairness criterion, thus making evaluation problematic.
Concerning the three data sets used in this paper: as mentioned, we use the
COMPAS data set to allow a direct comparison with the metric proposed by
Kleinberg et al. (2016), and we thus do not want to modify the data set, for
example through recovery or imputation. As shown by Goel et al. (2021), it is
possible to recover parts of the distribution on certain assumptions about the
causal structure of the decision-making process, but making such assumptions
would not allow comparison with the results of Kleinberg et al. (2016) or
allow us to make any general conclusions about applying FAIM to COMPAS.
The e-commerce data set is subject to systematic censorship in the sense that
articles classified as ”low visibility” would receive less exposure and thus
do not have opportunities to gain more exposure. Modifying the exploration-
exploitation strategy for the learning to rank model would allow us to sample
from the missing data to a larger degree, but performing such experiments
would be prohibitively costly.
Finally, to further investigate the effect of data missingness on FAIM, we
could introduce synthetic missingness to the synthetic data set, but would
have to do so under the assumption of different causal structures, which are
in an undefined and infinite search space. Thus, it is unclear what the
relevant causal structures would be that would allow us to make meaningful
conclusions.
## 7\. Conclusion
In this paper, we have presented FAIM, a re-scoring algorithm that
interpolates between three important criteria from the literature on fairness
in machine learning: calibration, balance for the negative class, and balance
for the positive class. Previous research has shown that they are mutually
exclusive under reasonable factual assumptions. However, in many practical and
legal scenarios, it is unsatisfactory to fully prioritize one criterion while
neglecting the other two.
FAIM allows to find an optimal compromise between the three criteria through
three per-group parameters $\theta^{A},\theta^{B}$, and $\theta^{C}$ using
optimal transport theory. The parameter settings can be flexibly adapted to
various scenarios. In this way, decision makers can balance competing
interests and rights of protected groups; and they may adapt one model to
different legal requirements. We presented mathematical theory together with a
pseudocode implementation to help readers with different backgrounds to follow
our approach. The model was tested in extensive experiments on real-world data
sets from the criminal justice and the e-commerce domain. Our experiments have
shown that FAIM is effective under all scenarios, but that an inherently
flawed model or data set should not be “fixed” via fairness constraints. As
such, we want to stress that our model is not a means of ethical reasoning,
i.e., it can not determine whether a candidate should or should not receive
bail or credit, or whether a product should be included in a top-$k$ ranking.
Instead, it ensures an optimal compromise for mutually exclusive fairness
constraints according to the desires of a decision maker, expressed through
the choices of $\theta^{A},\theta^{B}$, and $\theta^{C}$.
However, we also stressed that any choice of $\theta$ values must, of course,
remain within the confines of the law. Hence, we discussed how FAIM may be
harnessed to comply with competing legal non-discrimination obligations in
three concrete settings corresponding to our experiments: credit scoring, bail
decisions, and emerging notions of fairness in e-commerce rankings. The
discussion shows that our model can handle both traditional legal fields and
novel obligations under the recently enacted DMA. To our knowledge, our paper
is the first to explore the DMA from both a technical and a legal perspective.
## 8\. Acknowledgments
We want to thank Geraud Le Falher and Pak-Hang Wong for their support and
insightful suggestions to improve the paper. The authors do not have any
funding to disclose other than their affiliation.
## References
* (1)
* Agueh and Carlier (2011) Martial Agueh and Guillaume Carlier. 2011. Barycenters in the Wasserstein space. _SIAM Journal on Mathematical Analysis_ 43, 2 (2011), 904–924.
* Ambrosio and Gigli (2013) Luigi Ambrosio and Nicola Gigli. 2013. A user’s guide to optimal transport. In _Modelling and optimisation of flows on networks_. Springer, 1–155.
* Angwin et al. (2016) Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. 2016. Machine bias. _ProPublica, May_ 23, 2016 (2016), 139–159.
* Asudeh et al. (2019) Abolfazl Asudeh, HV Jagadish, Julia Stoyanovich, and Gautam Das. 2019. Designing fair ranking schemes. In _Proceedings of the 2019 International Conference on Management of Data_. 1259–1276.
* Bao et al. (2021) Michelle Bao, Angela Zhou, Samantha Zottola, Brian Brubach, Sarah Desmarais, Aaron Horowitz, Kristian Lum, and Suresh Venkatasubramanian. 2021. It’s COMPASlicated: The Messy Relationship between RAI Datasets and Algorithmic Fairness Benchmarks. _arXiv preprint arXiv:2106.05498_ (2021).
* Barocas and Selbst (2016) Solon Barocas and Andrew D Selbst. 2016. Big data’s disparate impact. _Calif. L. Rev._ 104 (2016), 671.
* Bellogín et al. (2017) Alejandro Bellogín, Pablo Castells, and Iván Cantador. 2017\. Statistical biases in information retrieval metrics for recommender systems. _Information Retrieval Journal_ 20, 6 (2017), 606–634.
* Berk et al. (2021) Richard Berk, Hoda Heidari, Shahin Jabbari, Michael Kearns, and Aaron Roth. 2021. Fairness in criminal justice risk assessments: The state of the art. _Sociological Methods & Research_ 50, 1 (2021), 3–44.
* Binns (2018) Reuben Binns. 2018\. Fairness in Machine Learning: Lessons from Political Philosophy. In _Conference on Fairness, Accountability and Transparency_. 149–159.
* Bono et al. (2021) Teresa Bono, Karen Croxson, and Adam Giles. 2021\. Algorithmic fairness in credit scoring. _Oxford Review of Economic Policy_ 37, 3 (2021), 585–617.
* Brenier (1987) Yann Brenier. 1987\. Décomposition polaire et réarrangement monotone des champs de vecteurs. _CR Acad. Sci. Paris Sér. I Math._ 305 (1987), 805–808.
* Brenier (1991) Yann Brenier. 1991\. Polar factorization and monotone rearrangement of vector-valued functions. _Communications on pure and applied mathematics_ 44, 4 (1991), 375–417.
* Brouwer (2021) Dennis Jeroen Brouwer. 2021\. Towards a ban of discriminatory rankings by digital gatekeepers?: Reflections on the proposal for a Digital Markets Act. _Internet policy review_ (2021).
* Cabral et al. (2021) Luis Cabral, Justus Haucap, Geoffrey Parker, Georgios Petropoulos, Tommaso M Valletti, and Marshall W Van Alstyne. 2021. The EU digital markets act: a report from a panel of economic experts. _Publications Office of the European Union_ (2021).
* Calders and Žliobaitė (2013) Toon Calders and Indrė Žliobaitė. 2013\. Why unbiased computational processes can lead to discriminative decision procedures. In _Discrimination and privacy in the information society_. Springer, 43–57.
* Caplar et al. (2017) Neven Caplar, Sandro Tacchella, and Simon Birrer. 2017\. Quantitative evaluation of gender bias in astronomical publications from citation counts. _Nature astronomy_ 1, 6 (2017), 1–5.
* Chiappa et al. (2020) Silvia Chiappa, Ray Jiang, Tom Stepleton, Aldo Pacchiano, Heinrich Jiang, and John Aslanides. 2020\. A General Approach to Fairness with Optimal Transport.. In _AAAI_. 3633–3640.
* Chiappa and Pacchiano (2021) Silvia Chiappa and Aldo Pacchiano. 2021. Fairness with Continuous Optimal Transport. _arXiv preprint arXiv:2101.02084_ (2021).
* Chouldechova (2017) Alexandra Chouldechova. 2017\. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. _Big data_ 5, 2 (2017), 153–163.
* Corbett-Davies and Goel (2018) Sam Corbett-Davies and Sharad Goel. 2018. The measure and mismeasure of fairness: A critical review of fair machine learning. _arXiv preprint arXiv:1808.00023_ (2018).
* Dawid (1982) A Philip Dawid. 1982\. The well-calibrated Bayesian. _J. Amer. Statist. Assoc._ 77, 379 (1982), 605–610.
* De Vries (2013) Sybe A De Vries. 2013\. Balancing fundamental rights with economic freedoms according to the European Court of Justice. _Utrecht Law Review_ 9, 1 (2013), 169–192.
* Dieterich et al. (2016) William Dieterich, Christina Mendoza, and Tim Brennan. 2016\. COMPAS risk scales: Demonstrating accuracy equity and predictive parity. _Northpointe Inc._ (2016).
* Dignum (2019) Virginia Dignum. 2019\. _Responsible artificial intelligence: how to develop and use AI in a responsible way_. Springer Nature.
* Dressel and Farid (2018) Julia Dressel and Hany Farid. 2018. The accuracy, fairness, and limits of predicting recidivism. _Science advances_ 4, 1 (2018), eaao5580.
* Dwork et al. (2012) Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. 2012. Fairness through awareness. _Proceedings of the 3rd innovations in theoretical computer science conference_ , 214–226.
* Eaglin (2017) Jessica M. Eaglin. 2017\. Constructing Recidivism Risk. _Emory Law Journal_ 67, 1 (2017), 59–122.
* Ebers et al. (2021) Martin Ebers, Veronica RS Hoch, Frank Rosenkranz, Hannah Ruschemeier, and Björn Steinrötter. 2021\. The European commission’s proposal for an artificial intelligence act—a critical assessment by members of the robotics and AI law society (RAILS). _J_ 4, 4 (2021), 589–603.
* Eifert et al. (2021) Martin Eifert, Axel Metzger, Heike Schweitzer, and Gerhard Wagner. 2021. Taming the Giants: The DMA/DSA Package. _Common Market Law Review_ 58, 4 (2021), 987–1028.
* Feldman et al. (2015) Michael Feldman, Sorelle A Friedler, John Moeller, Carlos Scheidegger, and Suresh Venkatasubramanian. 2015\. Certifying and removing disparate impact. In _proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining_. 259–268.
* Friedler et al. (2016) Sorelle A. Friedler, Carlos Scheidegger, and Suresh Venkatasubramanian. 2016. On the (im)possibility of fairness. _arXiv preprint arXiv:1609.07236_ (2016).
* Friedler et al. (2021) Sorelle A. Friedler, Carlos Scheidegger, and Suresh Venkatasubramanian. 2021. The (Im)possibility of Fairness: Different Value Systems Require Different Mechanisms For Fair Decision Making. _Commun. ACM_ 64, 4 (2021), 136–143.
* Friedman and Nissenbaum (1996) Batya Friedman and Helen Nissenbaum. 1996. Bias in computer systems. _ACM Transactions on Information Systems (TOIS)_ 14, 3 (1996), 330–347.
* Gerards and Borgesius (2022) Janneke Gerards and Frederik Zuiderveen Borgesius. 2022\. Protected Grounds and the System of Non-Discrimination Law in the Context of Algorithmic Decision-Making and Artificial Intelligence. _Colo. Tech. LJ_ 20 (2022), 1.
* Ghoash et al. (2021) Bishwamittra Ghoash, Debabrota Basu, and Kuldeep S. Meel. 2021\. Algorithmic Fairness Verification with Graphical Models. _arXiv preprint arXiv:2109.09447_ (2021).
* Gillis and Spiess (2019) Talia B Gillis and Jann L Spiess. 2019. Big data and discrimination. _The University of Chicago Law Review_ 86, 2 (2019), 459–488.
* Goel et al. (2021) Naman Goel, Alfonso Amayuelas, Amit Deshpande, and Amit Sharma. 2021. The Importance of Modeling Data Missingness in Algorithmic Fairness: A Causal Perspective. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 35. 7564–7573.
* Gordaliza et al. (2019) Paula Gordaliza, Eustasio Del Barrio, Gamboa Fabrice, and Jean-Michel Loubes. 2019. Obtaining fairness using optimal transport theory. In _International Conference on Machine Learning_. PMLR, 2357–2365.
* Graef (2019) Inge Graef. 2019\. Differentiated treatment in platform-to-business relations: EU competition law and economic dependence. _Yearbook of European Law_ 38 (2019), 448–499.
* Grochowski et al. (2021) Mateusz Grochowski, Agnieszka Jabłonowska, Francesca Lagioia, and Giovanni Sartor. 2021\. Algorithmic Transparency and Explainability for EU Consumer Protection: Unwrapping the Regulatory Premises. _Critical Analysis of Law_ 8 (2021), 43–63.
* Grodzinsky et al. (2011) Frances S Grodzinsky, Keith W Miller, and Marty J Wolf. 2011\. Developing artificial agents worthy of trust:“Would you buy a used car from this artificial agent?”. _Ethics and information technology_ 13, 1 (2011), 17–27.
* Hacker (2018) Philipp Hacker. 2018\. Teaching fairness to artificial intelligence: existing and novel strategies against algorithmic discrimination under EU law. _Common Market Law Review_ 55, 4 (2018).
* Hacker (2021) Philipp Hacker. 2021\. A legal framework for AI training data—from first principles to the Artificial Intelligence Act. _Law, Innovation and Technology_ 13, 2 (2021), 257–301.
* Hacker (2022a) Philipp Hacker. 2022a. The European AI Liability Directives – Critique of a Half-Hearted Approach and Lessons for the Future. _arXiv preprint arXiv:2211.13960_ (2022).
* Hacker (2022b) Philipp Hacker. 2022b. KI und DMA – Zugang, Transparenz und Fairness für KI-Modelle in der digitalen Wirtschaft. _Gewerblicher Rechtsschutz und Urheberrecht_ 75 (2022), 1278–1285.
* Hacker and Passoth (2022) Philipp Hacker and Jan-Hendrik Passoth. 2022. Varieties of AI Explanations Under the Law. From the GDPR to the AIA, and Beyond. In _International Workshop on Extending Explainable AI Beyond Deep Models and Classifiers_. Springer, 343–373.
* Hamilton (2021) Melissa Hamilton. 2021\. Evaluating Algorithmic Risk Assessment. _New Criminal Law Review_ 24, 2 (2021), 156–211.
* Hardt et al. (2016) Moritz Hardt, Eric Price, and Nati Srebro. 2016. Equality of opportunity in supervised learning. _Advances in neural information processing systems_ 29 (2016), 3315–3323.
* Hellman (2020) Deborah Hellman. 2020\. Measuring Algorithmic Fairness. _Virginia Law Review_ 106 (2020), 811–866.
* Hertweck and Räz (2022) Corinna Hertweck and Tim Räz. 2022. Gradual (In) Compatibility of Fairness Criteria. In _AAAI Conference on Artificial Intelligence_.
* Hirsch (2005) Jorge E Hirsch. 2005\. An index to quantify an individual’s scientific research output. _Proceedings of the National academy of Sciences_ 102, 46 (2005), 16569–16572.
* Huq (2019) Aziz Z. Huq. 2019\. Racial Equity in Algorithmic Criminal Justice. _Duke Law Journal_ 68, 6 (2019), 1043–1134.
* Järvelin and Kekäläinen (2002) Kalervo Järvelin and Jaana Kekäläinen. 2002. Cumulated gain-based evaluation of IR techniques. _ACM Transactions on Information Systems (TOIS)_ 20, 4 (2002), 422–446.
* Jiang et al. (2012) Xiaoqian Jiang, Melanie Osl, Jihoon Kim, and Lucila Ohno-Machado. 2012. Calibrating predictive model estimates to support personalized medicine. _Journal of the American Medical Informatics Association_ 19, 2 (2012), 263–274.
* Kallus and Zhou (2018) Nathan Kallus and Angela Zhou. 2018. Residual unfairness in fair machine learning from prejudiced data. In _International Conference on Machine Learning_. PMLR, 2439–2448.
* Katyal (2019) Sonia K. Katyal. 2019\. Private Accountability in the Age of Artificial Intelligence. _UCLA Law Review_ 66, 1 (2019), 54–141.
* Kim (2022) Pauline Kim. 2022\. Race-aware algorithms: Fairness, nondiscrimination and affirmative action. _California Law Review, Forthcoming_ (2022).
* Kim (2017) Pauline T. Kim. 2017\. Auditing Algorithms for Discrimination. _University of Pennsylvania Law Review Online_ 166 (2017), 189–203.
* Kleinberg et al. (2018) Jon Kleinberg, Himabindu Lakkaraju, Jure Leskovec, Jens Ludwig, and Sendhil Mullainathan. 2018\. Human decisions and machine predictions. _The quarterly journal of economics_ 133, 1 (2018), 237–293.
* Kleinberg et al. (2016) Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan. 2016\. Inherent trade-offs in the fair determination of risk scores. _arXiv preprint arXiv:1609.05807_ (2016).
* Langenbucher (2020) Katja Langenbucher. 2020\. Responsible AI-based credit scoring–a legal framework. _European Business Law Review_ 31, 4 (2020).
* Langenbucher and Corcoran (2021) Katja Langenbucher and Patrick Corcoran. 2021. Responsible AI Credit Scoring–A Lesson from Upstart. com. In _Digital Finance in Europe: Law, Regulation, and Governance_. De Gruyter, 141–180.
* Lapuschkin et al. (2019) Sebastian Lapuschkin, Stephan Wäldchen, Alexander Binder, Grégoire Montavon, Wojciech Samek, and Klaus-Robert Müller. 2019\. Unmasking Clever Hans predictors and assessing what machines really learn. _Nature communications_ 10, 1 (2019), 1–8.
* Larson et al. (2016) Jeff Larson, Surya Mattu, Lauren Kirchner, and Julia Angwin. 2016\. How we analyzed the COMPAS recidivism algorithm. _ProPublica, Technical Report_ (2016).
* Laux et al. (2021) Johann Laux, Sandra Wachter, and Brent Mittelstadt. 2021\. Taming the few: Platform regulation, independent audits, and the risks of capture created by the DMA and DSA. _Computer Law & Security Review_ 43 (2021), 105613.
* Laux et al. (2022) Johann Laux, Sandra Wachter, and Brent Mittelstadt. 2022\. Trustworthy Artificial Intelligence and the European Union AI Act: On the Conflation of Trustworthiness and the Acceptability of Risk. _Available at SSRN 4230294_ (2022).
* Lundberg and Lee (2017) Scott M Lundberg and Su-In Lee. 2017. A unified approach to interpreting model predictions. _Advances in neural information processing systems_ 30 (2017).
* Mayson (2019) Sandra G. Mayson. 2019\. Bias in, Bias out. _Yale Law Journal_ 128, 8 (2019), 2218–2301.
* Mitchell et al. (2021) Shira Mitchell, Eric Potash, Solon Barocas, Alexander D’Amour, and Kristian Lum. 2021\. Algorithmic Fairness: Choices, Assumptions, and Definitions. _Annual Review of Statistics and Its Application_ 8 (2021), 141–163.
* Morijn (2006) John Morijn. 2006\. Balancing fundamental rights and common market freedoms in Union law: Schmidberger and Omega in the light of the European Constitution. _European Law Journal_ 12, 1 (2006), 15–40.
* Nachbar (2020) Thomas B Nachbar. 2020\. Algorithmic fairness, algorithmic discrimination. _Fla. St. UL Rev._ 48 (2020), 509.
* Padilla et al. (2020) Jorge Padilla, Joe Perkins, and Salvatore Piccolo. 2020\. Self-preferencing in markets with vertically-integrated gatekeeper platforms. _Available at SSRN 3701250_ (2020).
* Perc (2014) Matjaž Perc. 2014\. The Matthew effect in empirical data. _Journal of The Royal Society Interface_ 11, 98 (2014), 20140378\.
* Pessach and Shmueli (2020) Dana Pessach and Erez Shmueli. 2020. Algorithmic Fairness. _arXiv preprint arXiv:2001.09784_ (2020).
* Pleiss et al. (2017) Geoff Pleiss, Manish Raghavan, Felix Wu, Jon Kleinberg, and Kilian Q Weinberger. 2017. On Fairness and Calibration. In _Advances in Neural Information Processing Systems_ , I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (Eds.), Vol. 30. Curran Associates, Inc. https://proceedings.neurips.cc/paper/2017/file/b8b9c74ac526fffbeb2d39ab038d1cd7-Paper.pdf
* Podszun and Bongartz (2021) Rupprecht Podszun and Philipp Bongartz. 2021. The Digital Markets Act: Moving from Competition Law to Regulation for Large Gatekeepers. _Journal of European Consumer and Market Law_ 10, 2 (2021), 60–67.
* Pruss (2021) Dasha Pruss. 2021\. Mechanical Jurisprudence and Domain Distortion: How Predictive Algorithms Warp the Law. _Philosophy of Science_ 88, 5 (2021), 1101–1112.
* Raghavan et al. (2020) Manish Raghavan, Solon Barocas, Jon Kleinberg, and Karen Levy. 2020. Mitigating bias in algorithmic hiring: Evaluating claims and practices. In _Proceedings of the 2020 conference on fairness, accountability, and transparency_. 469–481.
* Rudin (2019) Cynthia Rudin. 2019\. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. _Nature Machine Intelligence_ 1, 5 (2019), 206–215.
* Selbst (2017) Andrew D. Selbst. 2017\. Disparate Impact in Big Data Policing. _Gerogia Law Review_ 52, 1 (2017), 109–196.
* Selbst and Barocas (2022) Andrew D Selbst and Solon Barocas. 2022. Unfair Artificial Intelligence: How FTC Intervention Can Overcome the Limitations of Discrimination Law. _UCLA School of Law, Public Law Research Paper_ 24 (2022).
* Singh and Joachims (2018) Ashudeep Singh and Thorsten Joachims. 2018. Fairness of exposure in rankings. In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. 2219–2228.
* Solow-Niederman et al. (2019) Alicia Solow-Niederman, YooJung Choi, and Guy Van den Broeck. 2019\. The institutional life of algorithmic risk assessment. _Berkeley Tech. LJ_ 34 (2019), 705.
* Starr (2014) Sonja B. Starr. 2014\. Evidence-Based Sentencing and the Scientific Rationalization of Discrimination. _Stanford Law Review_ 66 (2014), 803–872.
* Thiebes et al. (2021) Scott Thiebes, Sebastian Lins, and Ali Sunyaev. 2021. Trustworthy artificial intelligence. _Electronic Markets_ 31, 2 (2021), 447–464.
* Veale and Binns (2017) Michael Veale and Reuben Binns. 2017. Fairer machine learning in the real world: Mitigating discrimination without collecting sensitive data. _Big Data & Society_ 4, 2 (2017), 2053951717743530\.
* Veale and Borgesius (2021) Michael Veale and Frederik Zuiderveen Borgesius. 2021\. Demystifying the Draft EU Artificial Intelligence Act—Analysing the good, the bad, and the unclear elements of the proposed approach. _Computer Law Review International_ 22, 4 (2021), 97–112.
* Veale et al. (2018) Michael Veale, Max Van Kleek, and Reuben Binns. 2018\. Fairness and accountability design needs for algorithmic support in high-stakes public sector decision-making. In _Proceedings of the 2018 chi conference on human factors in computing systems_. 1–14.
* Wachter (2020) Sandra Wachter. 2020\. Affinity profiling and discrimination by association in online behavioral advertising. _Berkeley Tech. LJ_ 35 (2020), 367.
* Wachter (2022) Sandra Wachter. 2022\. The Theory of Artificial Immutability: Protecting Algorithmic Groups under Anti-Discrimination Law. _Tulane Law Review, Forthcoming_ (2022).
* Wachter et al. (2021a) Sandra Wachter, Brent Mittelstadt, and Chris Russell. 2021a. Bias preservation in machine learning: the legality of fairness metrics under EU non-discrimination law. _W. Va. L. Rev._ 123 (2021), 735.
* Wachter et al. (2021b) Sandra Wachter, Brent Mittelstadt, and Chris Russell. 2021b. Why fairness cannot be automated: Bridging the gap between EU non-discrimination law and AI. _Computer Law & Security Review_ 41 (2021), 105567.
* Watch (2019) Algorithm Watch. 2019\. Automating Society Report 2019.
* Zadrozny and Elkan (2001) Bianca Zadrozny and Charles Elkan. 2001. Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers. In _Icml_ , Vol. 1. Citeseer, 609–616.
* Zafar et al. (2017) Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez Rodriguez, and Krishna P Gummadi. 2017. Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment. In _Proceedings of the 26th international conference on world wide web_. 1171–1180.
* Zehlike et al. (2017) Meike Zehlike, Francesco Bonchi, Carlos Castillo, Sara Hajian, Mohamed Megahed, and Ricardo Baeza-Yates. 2017. Fa* ir: A fair top-k ranking algorithm. In _Proceedings of the 2017 ACM on Conference on Information and Knowledge Management_. 1569–1578.
* Zehlike et al. (2020) Meike Zehlike, Philipp Hacker, and Emil Wiedemann. 2020\. Matching code and law: achieving algorithmic fairness with optimal transport. _Data Mining and Knowledge Discovery_ 34, 1 (2020), 163–200.
* Zehlike et al. (2022a) Meike Zehlike, Tom Sühr, Ricardo Baeza-Yates, Francesco Bonchi, Carlos Castillo, and Sara Hajian. 2022a. Fair Top-k Ranking with multiple protected groups. _Information Processing & Management_ 59, 1 (2022), 102707\.
* Zehlike et al. (2022b) Meike Zehlike, Ke Yang, and Julia Stoyanovich. 2022b. Fairness in Ranking, Part I: Score-based Ranking. _ACM Computing Surveys (CSUR)_ (2022).
* Zuiderveen Borgesius (2020) Frederik J Zuiderveen Borgesius. 2020\. Strengthening legal protection against discrimination by algorithms and artificial intelligence. _The International Journal of Human Rights_ 24, 10 (2020), 1572–1593.
|
# Life on Titan May Signal Early Life in the Universe
Abraham Loeb Astronomy Department, Harvard University, 60 Garden St.,
Cambridge, MA 02138, USA
###### Abstract
The temperature of the cosmic microwave background (CMB) was equal to the
surface temperature of Saturn’s moon Titan, $94$K, at a redshift $z=33.5$,
after the first galaxies formed. Titan-like objects would have maintained this
surface temperature for tens of Myr irrespective of their distance from a
star. Titan has the potential for the chemistry of familiar life in its
subsurface water ocean, as well new forms of life in the rivers, lakes and
seas of liquid methane and ethane on its surface. The potential future
discovery of life on Titan would open the possibility that the earliest
lifeforms emerged in metal-rich environments of the earliest galaxies in the
universe, merely 100 Myr after the Big Bang.
## 1 Introduction
The cosmic microwave background (CMB) provides a universal heating source of
temperature (Fixsen et al., 1996), $T_{\rm cmb}=94{\rm K}\times[(1+z)/34.5]$
at a cosmological redshift $z$. Interestingly, this temperature matches the
surface temperature of Saturn’s largest moon, Titan at $z\sim 34$, about 90
Myr after the Big Bang. Hence, a Titan-like object at that early time would
have maintained this temperature for tens of Myr, sufficient for primitive
life to form in its liquid reservoirs or atmosphere, irrespective of its
distance from a star.
## 2 First Objects
The standard cosmological model predicts that the first generation of stars
and galaxies formed before $z\sim 34$ (Loeb & Furlanetto, 2013). Based on the
measured cosmological parameters (Planck Collaboration et al., 2020), the
first star-forming halos collapsed at $z\sim 71$ on our past light cone and at
$z\sim 77$ within the entire Hubble volume (Loeb, 2014), including the delay
from the streaming motion of baryons relative to dark matter (Fialkov et al.,
2012).
Hydrodynamical cosmological simulations predict that the first galaxies formed
population III stars that were predominantly massive (Loeb & Furlanetto,
2013). For massive stars that are dominated by radiation pressure and shine
near their Eddington luminosity $L_{\rm E}=1.3\times 10^{39}~{}{\rm
erg~{}s^{-1}}(M_{\star}/10M_{\odot})$, the lifetime is independent of stellar
mass $M_{\star}$ and is universally a few Myr, set by the nuclear efficiency
of converting rest-mass to radiation, 0.7$\%$, namely
$\sim(0.007M_{\star}c^{2})/L_{\rm E}=3~{}{\rm Myr}$ (Bromm et al., 2001).
Consequently, the subsequent delay in dispersing heavy elements from the first
stellar winds or pair-instability supernovae could have been as short as a few
Myr, only a few percent of the age of the Universe at $z\sim 34$. The
supernova ejecta could have produced high-metallicity islands that were not
fully mixed with the surrounding primordial gas, leading to efficient
formation of planets and moons within them.
Altogether, this suggests that massive stars and supernovae were able to
enrich the interstellar medium in the cores of the earliest galaxies with
heavy elements before $z\sim 34$, leading to metal-rich pockets of gas inside
of which the second generation of stars could have formed, accompanied by
Titan-like objects.
## 3 Prospects of Life on Titan
The temperature coincidence between Titan’s surface and the CMB at $z\sim 34$
raises the fascinating possibility of testing how early life could have arisen
in the Universe by studying Titan. In other words, the question of whether
Titan hosts life has cosmic implications.
In the Solar system, Titan is the only object besides Earth that has rivers,
lakes and seas on its surface, as well as a cycle of methane and ethane
liquids raining from clouds, flowing across its surface and evaporating back
into the atmosphere, similarly to Earth’s water cycle. Titan is also thought
to have a subsurface ocean of water. Its atmosphere is primarily nitrogen like
Earth’s, but with a $\sim 5\%$ contribution of methane. Titan’s landscape is
covered with dark dunes of hydrocarbon grains, primarily around the equatorial
regions.
Gravity measurements by the Cassini spacecraft revealed that Titan has an
underground ocean of liquid water, likely mixed with salts and ammonia (Lopes
et al., 2019). Radio signals detected by the Huygens probe in 2005 strongly
suggested the presence of an ocean 55-80 km below the icy surface, allowing
for the chemistry of life-as-we-know-it. In addition, Titan’s bodies of liquid
methane and ethane might serve as a foundation for the chemistry of life-as-
we-do-not-know-it on the moon’s surface.
Whether the physical conditions on Titan gave birth to these forms of life is
unknown. The realization that Titan’s atmosphere is rich in organic compounds
led to the proposal that it produced the chemical precursors of life. In
particular, stable cryogenic cell membranes could arise from compounds
observed in Titan’s atmosphere (Stevenson et al., 2015). The proposed chemical
base for these membranes is acrylonitrile, which was detected in Titan’s
atmosphere by Cassini and ALMA (Desai et al., 2017; Palmer et al., 2017).
Analysis of data from the Cassini-Huygens mission reported anomalies in the
atmosphere near the surface which could be consistent with the presence of
lifeform of methane-consuming organisms, but may alternatively be due to
abiotic chemical processes (Strobel, 2010; Clark et al., 2010).
Laboratory experiments (Hörst et al., 2010) indicate that when discharge power
is applied to a combination of gases like those in Titan’s atmosphere, they
make prebiotic molecules such as the five nucleotide bases of DNA and RNA as
well as amino-acids, among many other compounds.
## 4 Discussion
The thermal gradients needed for life can be supplied by geological variations
on the surface of early Titan-like objects. Examples for sources of free
energy are geothermal energy powered by the object’s gravitational binding
energy at formation and radioactive energy from unstable elements produced by
the earliest supernova. If life persisted at $z\lesssim 34$, it could have
also been transported to newly formed objects through panspermia (Ginsburg et
al., 2018).
Given the above considerations, the search for life on Titan could open the
possibility that life may have started at a redshift of $z\sim 34$ in the
standard cosmological model of our Universe.
In addition to studying Titan, the feasibility of life in the early universe
can be further tested by searching for planets with atmospheric bio-signatures
around low-metallicity stars in the Milky Way galaxy or its dwarf galaxy
satellites. Such stars represent the closest analogs to the first generation
of stars at early cosmic times.
## Acknowledgements
This work was supported in part by Harvard’s Black Hole Initiative, which is
funded by grants from JFT and GBMF.
## References
* Bromm et al. (2001) Bromm, V., Kudritzki, R. P., & Loeb, A. 2001, ApJ, 552, 464, doi: 10.1086/320549
* Clark et al. (2010) Clark, R. N., Curchin, J. M., Barnes, J. W., et al. 2010, Journal of Geophysical Research (Planets), 115, E10005, doi: 10.1029/2009JE003369
* Desai et al. (2017) Desai, R. T., Coates, A. J., Wellbrock, A., et al. 2017, ApJ, 844, L18, doi: 10.3847/2041-8213/aa7851
* Fialkov et al. (2012) Fialkov, A., Barkana, R., Tseliakhovich, D., & Hirata, C. M. 2012, MNRAS, 424, 1335, doi: 10.1111/j.1365-2966.2012.21318.x
* Fixsen et al. (1996) Fixsen, D. J., Cheng, E. S., Gales, J. M., et al. 1996, ApJ, 473, 576, doi: 10.1086/178173
* Ginsburg et al. (2018) Ginsburg, I., Lingam, M., & Loeb, A. 2018, ApJ, 868, L12, doi: 10.3847/2041-8213/aaef2d
* Hörst et al. (2010) Hörst, S., Yelle, R. V., Buch, A., et al. 2010, in AAS/Division for Planetary Sciences Meeting Abstracts, Vol. 42, AAS/Division for Planetary Sciences Meeting Abstracts #42, 36.20
* Loeb (2014) Loeb, A. 2014, International Journal of Astrobiology, 13, 337, doi: 10.1017/S1473550414000196
* Loeb & Furlanetto (2013) Loeb, A., & Furlanetto, S. R. 2013, The First Galaxies in the Universe
* Lopes et al. (2019) Lopes, R. M. C., Wall, S. D., Elachi, C., et al. 2019, Space Sci. Rev., 215, 33, doi: 10.1007/s11214-019-0598-6
* Palmer et al. (2017) Palmer, M. Y., Cordiner, M. A., Nixon, C. A., et al. 2017, Science Advances, 3, e1700022, doi: 10.1126/sciadv.1700022
* Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910
* Stevenson et al. (2015) Stevenson, J., Lunine, J., & Clancy, P. 2015, Science Advances, 1, 1400067, doi: 10.1126/sciadv.1400067
* Strobel (2010) Strobel, D. F. 2010, Icarus, 208, 878, doi: 10.1016/j.icarus.2010.03.003
|
# Efficient Stereo Matching on Embedded GPUs with Zero-Means Cross Correlation
Qiong Chang, Aolong Zha, Weimin Wang, Xin Liu, Masaki Onishi, Lei Lei, Meng
Joo Er, Tsutomu Maruyama
###### Abstract
Mobile stereo-matching systems have become an important part of many
applications, such as automated-driving vehicles and autonomous robots.
Accurate stereo-matching methods usually lead to high computational
complexity; however, mobile platforms have only limited hardware resources to
keep their power consumption low; this makes it difficult to maintain both an
acceptable processing speed and accuracy on mobile platforms. To resolve this
trade-off, we herein propose a novel acceleration approach for the well-known
zero-means normalized cross correlation (ZNCC) matching cost calculation
algorithm on a Jetson Tx2 embedded GPU. In our method for accelerating ZNCC,
target images are scanned in a zigzag fashion to efficiently reuse one pixel’s
computation for its neighboring pixels; this reduces the amount of data
transmission and increases the utilization of on-chip registers, thus
increasing the processing speed. As a result, our method is 2X faster than the
traditional image scanning method, and 26% faster than the latest NCC method.
By combining this technique with the domain transformation (DT) algorithm, our
system show real-time processing speed of 32 fps, on a Jetson Tx2 GPU for
1,280x384 pixel images with a maximum disparity of 128. Additionally, the
evaluation results on the KITTI 2015 benchmark show that our combined system
is more accurate than the same algorithm combined with census by 7.26%, while
maintaining almost the same processing speed.
## I INTRODUCTION
Stereo matching is a key algorithm for depth detection in computer vision, but
its usability is still limited because attaining high accuracy requires a very
high computational complexity. By achieving both high accuracy and processing
speed on mobile platforms, it can be used in many applications, including
auto-driving, autonomous robots, and so on.
Thus far, many researchers have focused upon accelerating stereo matching on
mobile platforms. Most of them focus on accelerating the two most
computationally intensive stages: cost calculation and cost aggregation.
During cost calculation, each pixel in the reference image is first matched
with several pixels in the target image one by one. Next, the similarity
between any two pixels is quantified by a numerical value (cost) calculated by
a matching method, such as the sum of absolute differences (SAD), census, or
convolution neural network (CNN). Then, to further improve the accuracy of the
matching system, the cost of each pixel within a certain region is expected to
be aggregated together to represent the similarity between any two regions;
this is called matching-cost aggregation. Many methods can be used to
determine the range of the matching regions, such as semi-global matching
(SGM) or domain transformation (DT). Various combinations of the above two
stages not only result in different matching accuracies, but also different
computational complexities; this leads to differing processing speeds.
Many researches implement their stereo-matching systems on FPGAs,
respectively. Wang [1] first combines a simplified SGM with the simple
absolute differences and census matching algorithms, processing 1,024x768
pixel images with 96 disparities at 67 fps. Mohammad [2], Zhang [3] and Kuo
[4] use the census algorithm to calculate their matching costs. Mohammad [2]
combines census with a cross-aggregation method to achieve a good error rate
of less than 9.22% and a high processing speed of faster than 200 fps on the
KITTI 2015 benchmark [5]. Zhang [3] uses a box filter to aggregate matching
costs and achieves a high processing speed of 60 fps for 1080p images. Kuo [4]
uses a two-pass aggregation method and achieves the same processing speed as
[3]. Oscar [6] uses SAD to calculate the matching cost and combines it with
SGM. Due to SGM’s high accuracy for even the simple SAD matching algorithm, it
can still achieve a lower error rate of 8.7%, except that its speed is reduced
to 50 fps. Additionally, Zhang [7] develops a special ASIC to accelerate the
implementation of SGM and achieves a processing speed of 30fps for 1080p
images. Here, due to the limitations of floating-point decimal calculation,
both the FPGA-based and the dedicated ASIC-based systems usually use methods
such as SAD and census to obtain integer cost values. Although this is
conducive to implementation on them, it also limits improvement in the
matching accuracy. Furthermore, hardware-based systems typically need long
development cycles and are also difficult to maintain. The recent advent of
embedded GPUs has allowed the development of many systems [8] [9]. Compared to
FPGAs, embedded GPU-based systems have short development cycles [10]. In
addition, they are easy to maintain and port on other platforms. Wang [11],
Smolyanskiy [12] and Tonioni [13] implement their systems on a Jetson Tx2
embedded GPU using the CNN; according to the evaluation results of KITTI 2015
[5] benchmark, their accuracies are high, with error rates between 3.2% and
6.2%. However, due to the significant calculations of CNN-based methods, their
processing speeds are only a few fps, far below the requirements for practical
applications. Daniel [14] constructs a fast stereo-matching system on a Jetson
Tx2 GPU. It also combines census algorithm with SGM to achieve an error rate
of 8.66% and a processing speed of 29 fps on KITTI 2015 benchmark. It is
currently the best system for balancing the accuracy and processing speed on
mobile GPUs; however, due to the census-matching method, this system is still
not accurate enough, even if implemented on GPUs that are good at floating-
point decimal calculations.
According to [15], the matching accuracy by normalized cross correlation (NCC)
is better than that by census because it has a higher ability to withstand
changes in gain and bias. Furthermore, zero-means NCC (ZNCC)–an improved
version of NCC–provides strong robustness because it also tolerates uniform
brightness variations [16][17]. However, ZNCC has not been widely used on the
mobile systems with limited hardware resources because of its higher
computational complexity.
In this paper, we accelerate ZNCC on a Jetson Tx2 embedded GPU and make it
possible to achieve a comparable processing speed to that of census with a
higher matching accuracy. The main contributions of this paper are as follows:
* •
We introduce a new calculation method, zigzag scanning based zero-means
normalized cross correlation (Z2-ZNCC) to reuse the computational results of a
pixel for the calculations of its neighbors. This makes it possible to reduce
data transfer between the global memory of the GPU and increase the processing
speed.
* •
We propose a strategy to make efficient use of registers during zigzag
scanning to achieve higher parallelism of GPU threads driven by GPU cores.
* •
We design GPU-implementation algorithms for two parallel summation methods
used in our Z2-ZNCC and comprehensively compare their performance.
* •
We create FastDT, an upgraded version of the GPU-based domain transformation
method presented in [18] by removing the cost-value shifting step and
increasing the flag code. Then, we combine it with Z2-ZNCC to construct a
real-time stereo-matching system on an embedded GPU.
The experimental results demonstrate that our method is 2X faster than the
traditional image-scanning method and 26% faster than the latest NCC method
[19]. Furthermore, our system achieves a processing speed of 32 fps and an
error rate of 9.26% for 1,242x375 pixel images when the maximum disparity is
128 on a KITTI 2015 dataset. It is one of the few embedded GPU-based real-time
systems, with an accuracy much higher than others.
This paper extends our previous work (short paper) [20] from the following
aspects.
* •
We design two parallel summation methods which maximize the processing speed
of Z2-ZNCC depending on the template size.
* •
We introduce an efficient two-step implementation technique for domain
transformation, which not only maintains a high processing speed, but also a
high accuracy of Z2-ZNCC.
* •
We conduct comprehensive experiments to examine the impact of various
conditions on the processing speed of Z2-ZNCC.
The rest of the paper is organized as follows. Section II introduces the
related works. Section III reviews ZNCC and DT calculation methods. Section IV
discusses the GPU implementation of Z2-ZNCC and FastDT. Section V shows the
evaluation results. Finally, Section VI presents the conclusions and our
future work.
## II Related works
Recently, many researchers have focused upon accelerating the performance of
NCC-based methods and applying them into stereo-matching systems.
Lin [21] proposes an optimization method for ZNCC calculation on a general
platform. This method divides the standard equation into four independent
parts and calculates the correlation coefficient efficiently using sliding
windows. Hence, the computational complexity of ZNCC could be reduced from the
original order. The computation time becomes constant for the window size;
however, a large memory is required to store the calculation results for
reuse. Although this method works nearly 10X faster than traditional ones, it
is not applicable to embedded GPUs because of their limited memory space.
Rui [22] implements a fast ZNCC-based stereo-matching system on GTX 970M GPU.
This work focuses on the use of integral images to calculate the mean and
standard deviation efficiently. Rui’s system runs approximately 9X faster than
a single-threading CPU implementation and about 2X faster than eight-
threading; however, according to our evaluation, this approach is inefficient
for a small-size window (less than 9x9) because obtaining an integral image
itself also requires calculation costs. These costs are mainly caused by data
transfer of the integration results, which cannot be ignored for an embedded
GPU with a high memory latency.
Han [19] implements an NCC-based stereo-matching system on a Jetson Tx2 GPU.
This method divides the equation into three parts, each with an identical
control flow but different data locations. All intermediate results are stored
on shared memory evenly so as to accelerate the calculation through reuse.
However, as mentioned in Section I, the heavy use of shared memory reduces the
parallelism of GPU blocks.
## III Algorithms and Optimizations
### III-A Zero-means Normalized Cross Correlation (ZNCC)
ZNCC is used to calculate matching costs between a reference pixel
$I_{R}(x,y)$ in the reference image and a series of target pixels
$I_{T}(x-d,y)$ in the target image. $d$ is called disparity, and its range is
$[0,D)$, where $D$ is a constant called maximum disparity. The function of
ZNCC is given as follows:
$C(x,y,d)=\frac{\displaystyle\sum_{(x,y)\in{W}}{\Delta{I_{R}}(x,y)\cdot{\Delta{I_{T}}(x-d,y)}}}{\sigma_{R}(x,y)\cdot{\sigma_{T}(x-d,y)}},$
(1)
where
$\begin{split}\sigma_{R}(x,y)&=\sqrt{\displaystyle\sum_{(x,y)\in{W}}{\Delta{I_{R}(x,y)^{2}}}},\\\
\sigma_{T}(x-d,y)&=\sqrt{\displaystyle\sum_{(x,y)\in{W}}{\Delta{I_{T}(x-d,y)^{2}}}},\\\
\end{split}$
and
$\begin{split}\Delta{I_{R}(x,y)}&=I_{R}(x,y)-\overline{I_{R}(x,y)},\\\
\Delta{I_{T}(x-d,y)}&=I_{T}(x-d,y)-\overline{I_{T}(x-d,y)}.\\\ \end{split}$
Here, $\overline{I_{R}(x,y)}$ and $\overline{I_{T}(x-d,y)}$ are the averages
of the pixel values in the matching windows $W$ surrounding $I_{R}(x,y)$ and
$I_{T}(x-d,y)$, respectively. $C(x,y,d)$ in (1) is the correlation coefficient
(i.e., the matching cost) between $I_{R}(x,y)$ and $I_{T}(x-d,y)$; its range
is $[0,1]$ (the closer to one, the more similar the two windows).
$\sigma_{R}(x,y)$ and $\sigma_{T}(x-d,y)$ are the standard deviations of the
pixel values in the two windows and are used to normalize the correlation
coefficient between them. Each reference pixel $I_{R}(x,y)$ needs to be
matched with $D$ target pixels $I_{T}(x-d,y)$; here, $\Delta{I_{R}(x,y)}$ and
$\Delta{I_{T}(x-d,y)}$ can be calculated in advance because the calculations
of these terms are closed in each image. As such, the total number of
calculations can be reduced. However, when the size of the matching window $W$
is $l^{2}$ (where $l=2r+1$ represents the side length of window $W$)
$l^{2}$-times the memory space is needed for each image because each window
has $l^{2}$ differences.
To further reduce the ZNCC’s computation complexity, (1) can be rewritten as
follows:
$\begin{split}C(x,y,d)=\frac{\displaystyle\sum_{(x,y)\in{W}}{\Pi_{RT}(x,y,d)}-l^{2}\cdot{\overline{\Pi_{RT}(x,y,d)}}}{\sigma_{R}(x,y)\cdot{\sigma_{T}(x-d,y)}},\end{split}$
(2)
where
$\begin{split}\Pi_{RT}(x,y,d)&=I_{R}(x,y)\cdot{I_{T}(x-d,y)},\\\
\overline{\Pi_{RT}(x,y,d)}&=\overline{I_{R}(x,y)}\cdot{\overline{I_{T}(x-d,y)}},\\\
\end{split}$
and
$\begin{split}\sigma_{R}(x,y)&=\sqrt{\displaystyle\sum_{(x,y)\in{W}}{I_{R}(x,y)^{2}}-l^{2}\cdot{\overline{I_{R}(x,y)}^{2}}},\\\
\sigma_{T}(x-d,y)&=\sqrt{\displaystyle\sum_{(x,y)\in{W}}{I_{T}(x-d,y)^{2}}-l^{2}\cdot{\overline{I_{T}(x-d,y)}^{2}}}.\\\
\end{split}$
In this calculation, $C(x,y,d)$, $\sigma_{R}(x,y)$, and $\sigma_{T}(x-d,y)$
are calculated from four values; $\overline{I_{R}(x,y)}^{2}$,
$\sum{I_{R}(x,y)^{2}}$, $\overline{I_{T}(x-d,y)}^{2}$, and
$\sum{I_{T}(x-d,y)^{2}}$, rather than from $\Delta{I_{R}(x,y)}$, and
$\Delta{I_{T}(x-d,y)}$ as shown in (1). These four values are only related to
their respective images and can all be calculated in advance. Thus, both
$\sigma_{R}(x,y)$ and $\sigma_{T}(x-d,y)$ can be calculated efficiently
without calculating $\Delta{I_{R}(x,y)}$ and $\Delta{I_{T}(x-d,y)}$ for each
pixel $l^{2}$ times. This transformation not only helps to reduce the total
calculation amount, but also reduce the required memory space.
Of course, ZNCC-based matching is performed in a fixed size window, which is
not accurate for irregular patterns in reality; therefore, the matching cost
$C(x,y,d)$ is usually combined with various aggregation methods to improve the
matching accuracy.
### III-B Domain Transformation (DT)
In cost aggregation step, the matching costs of all similar pixels in the same
area (e.g., the area within the pink-dashed line in Fig.1 (a)) are added
together. Domain Transformation (DT) [23] is an effective algorithms for use
at this stage. Unlike other algorithms, DT avoids over-fitting of cost
propagation by using the gradients of adjacent pixels to weight their costs in
different directions, rather than judging the boundary of each area in
advance. The advantage of DT is that there is no need to segment each area for
cost aggregation separately, and it is suitable for parallel processing to
increase the aggregation speed. In DT, the matching cost of each pixel is
aggregated from four different directions, while propagating its cost to four
neighboring pixels is done according to the following equations:
Figure 1: Domain transformation
$\displaystyle C_{L}(x,y,d)$
$\displaystyle=C(x,y,d)\small{+}C_{L}(x\small{-}1,y,d)\cdot{W_{L}(x,y)},$ (3)
$\displaystyle C_{R}(x,y,d)$
$\displaystyle=C_{L}(x,y,d)\small{+}C_{R}(x\small{+}1,y,d)\cdot{W_{R}(x,y)},$
(4) $\displaystyle C_{U}(x,y,d)$
$\displaystyle=C_{R}(x,y,d)\small{+}C_{U}(x,y\small{-}1,d)\cdot{W_{U}(x,y)},$
(5) $\displaystyle C_{D}(x,y,d)$
$\displaystyle=C_{U}(x,y,d)\small{+}C_{D}(x,y\small{+}1,d)\cdot{W_{D}(x,y)}.$
(6)
Here, $C_{L}$, $C_{R}$, $C_{U}$, and $C_{D}$ represent the aggregated costs
for each pixel from four different directions (left, right, up, and down). In
this aggregation, for example, the boundary condition for (3) is given by
$C_{L}(0,y,d)=C(0,y,d)$. $W_{L}$, $W_{R}$, $W_{U}$ and $W_{D}$ represent the
corresponding weights calculated by the gradient between adjacent pixel values
according to the following equations:
$W_{L}(x,y)=a^{1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x-1,y)|}},$
(7)
$W_{R}(x,y)=a^{1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x+1,y)|}},$
(8)
$W_{U}(x,y)=a^{1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x,y-1)|}},$
(9)
and
$W_{D}(x,y)=a^{1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x,y+1)|}},$
(10)
where
$a=\exp{(-\frac{1}{\sigma_{s}})}.$ (11)
In (7) to (11), $\sigma_{s}$ is a spatial parameter and $\sigma_{r}$ is an
intensity range parameter. Both are used to adjust the weight caused by
gradient changes in space and intensity.
To simplify the calculation, the weight equations can be further simplified
below (here, we only take (7) as an example):
$\displaystyle\ln{W_{L}}$
$\displaystyle=\ln{(a^{1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x-1,y)|}})}$
$\displaystyle=(1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x-1,y)|})\cdot{\ln{a}}$
$\displaystyle=(1+\frac{\sigma_{s}}{\sigma_{r}}\cdot{|I_{T}(x,y)-I_{T}(x-1,y)|})\cdot{(-\frac{1}{\sigma_{s}})}$
$\displaystyle=-\frac{1}{\sigma_{s}}-\frac{|I_{T}(x,y)-I_{T}(x-1,y)|}{\sigma_{r}},$
then
$\displaystyle W_{L}$
$\displaystyle=\exp(-\frac{1}{\sigma_{s}}-\frac{|I_{T}(x,y)-I_{T}(x-1,y)|}{\sigma_{r}})$
$\displaystyle=K\cdot{\exp(-\frac{|I_{T}(x,y)-I_{T}(x-1,y)|}{\sigma_{r}})},$
where
$\displaystyle K=\exp(-\frac{1}{\sigma_{s}}).$ (14)
$K$ is a constant coefficient used to ease the calculation of $W_{L}$.
Figure.1 shows an example of cost aggregation for pixel $I_{T}(x,y)$. Figure.1
(a) shows part of the reference image centered on $I_{T}(x,y)$. Figures.1 (b)
to 1 (e) show the cost propagation process from different directions. Three
curves with different colors represent the changes in the gradient, weights,
and propagated costs, respectively. According to (III-B), the weight is
calculated by the gradient and then used to weight the propagated cost value.
As shown in Figs.1 (b) to 1 (e), the weight changes in the opposite direction
to the change in gradient value, thereby ensuring that cost propagation can be
performed normally among non-edge pixels (Fig.1 (b)) and can also be
interrupted at edge pixels (Fig.1 (c)). When the propagation from down to up
is completed (i.e., when the final aggregation result $C_{D}(x,y,d)$ is
obtained), $C(x,y,d)$ is replaced by $C_{D}(x,y,d)$, and used in the following
stages.
### III-C Winner-Take-ALL (WTA)
After calculating matching cost for maximum disparity $D$ times, the target
pixel $I_{T}(x-d,y)$ that is most similar to reference pixel $I_{R}(x,y)$ is
determined as:
$D_{map}(x,y)=\mathop{\rm arg~{}min}\limits_{d}{(1-C(x,y,d))}.$ (15)
As shown in this equation, the value of $d$ that minimizes $(1-C(x,y,d))$ is
chosen as the disparity of the reference pixel $I_{R}(x,y)$.
## IV Implementation
Implementing ZNCC and DT on an embedded GPU is a key challenge for realizing a
fast and accurate mobile stereo-vision system. In this section, we first
introduce the architecture of Jetson Tx2 GPU; then, we describe the
acceleration approaches of ZNCC and DT, respectively.
### IV-A GPU Architecture and CUDA Programming Model
The Jetson Tx2 has 2 streaming multi-processors (SMs); each SM runs in
parallel using 128 cores (256 cores in total) and has two types of on-chip
memory: register memory and shared memory. Their sizes are limited, but their
access latencies are very low. This GPU also has a global memory (off-chip),
which is usually used to hold all data for processing. Due to the high latency
of access to the off-chip memory, the most important point for achieving high
performance on the GPU is to minimize the amount of data transfer between on-
chip and off-chip memory.
In our implementation, we use the GPGPU programming model CUDA [24]. CUDA
abstractly defines the GPU core, SM, and GPU itself as thread, block, and
grid, respectively. A grid is composed of blocks and a block is composed of
threads. Every 32 threads execute the same instruction, which is called a
warp. The warps are scheduled serially by the SMs. Users can define the number
of the abstract resources according to their requirements, which may exceed
the physical GPU resources; then, the CUDA driver schedules abstract resources
to work upon physical resources. Since the total number of registers and
shared-memory space are fixed, the amount of these resources allocated to each
thread and block determines how many of them can be active. The more
allocated, the fewer threads and blocks can be activated, which resulting in
reduced performance. By storing intermediate calculation results and reusing
them afterwards, the total amount of calculation can be reduced; however, more
hardware resources are needed to store these results, which limits the number
of active threads. On the other hand, by recalculating them each time, the
required amount of hardware resources can be reduced, and more threads can be
active. Balancing the hardware-resource usage and total amount of calculation
is a key point for achieving high performance, especially on embedded GPUs
with limited hardware resources.
### IV-B Implementation of ZNCC
Our ZNCC-acceleration approach includes two steps: (1) calculation of the
means and the sums of squares in matching windows, and (2) calculation of the
correlation coefficients of each pixel by zigzag scanning.
#### IV-B1 Summation
For the ZNCC-acceleration approach, the means and sums of squares in each
matching window are calculated in advance (Section III-A). Taking the example
of the pixel values in the reference image, we describe the two methods as
follows (for simplicity, we only describe the sum in the reference image):
Figure 2: Summation methods
Method 1: As shown in Fig.2 (a), $S_{R}(x,y)$, the sum of the pixel values in
each matching window $W$ is calculated as follows:
$\begin{split}S_{R}(x,y)&=\displaystyle\sum_{(x,y)\in W}I_{R}(x,y).\\\
\end{split}$ (16)
Here, $(2r+1)^{2}$ pixel values are simply added around the center pixel
$I_{R}(x,y)$.
The summation of each window is performed independently by different CUDA
threads. Since the two adjacent windows for two adjacent pixels share
$2r\times{(2r+1)}$ pixels, the data from $(2r+1)$ rows or columns are excepted
to be cached in the same shared memory. The allocated memory space grows as
the window size increases. The number of columns and rows processed by each
CUDA block at the same time depends upon the size of the shared memory
allocated; for smaller-sized windows, less hardware resources are required for
each thread, and higher parallelism can be expected; however, as the window
size increases, more hardware resources are required, and fewer threads can be
active. Therefore, this method is not suitable for the summation of large
windows.
Method 2: This method performs the summation using the integral image
$B_{R}(x,y)$:
$\displaystyle B_{R}(x,y)=\sum_{u=0}^{x}\sum_{v=0}^{y}I_{R}(u,v).$ (17)
As shown in Fig.2 (b), $S_{R}(x,y)$ is calculated using four points in the
integral image, regardless of the window size:
$\begin{split}S_{R}(x,y)&{\small=}B_{R}(x+r,y+r)+B_{R}(x{\small-}r{\small-}1,y{\small-}r{\small-}1)\\\
&{\small-}B_{R}(x+r,y{\small-}r{\small-}1){\small-}B_{R}(x{\small-}r{\small-}1,y+r).\end{split}$
(18)
In this calculation method, To calculate the sum for the pixels on row $y$,
only two rows ($y-r-1$ and $y+r$) of the integral image are needed. Thus, this
method is suitable for the summation of large windows.
Obtaining an integral image in parallel requires two steps:
1. 1.
integrate each row of $I_{R}$, defined as $B_{H}$,
2. 2.
integrate each column of $B_{H}$, defined as $B_{R}$.
Figure 3: Integration along the $x$-axis
To generate the integral image, all image data need to be loaded from global
memory to the shared memory. However, due to the limitation of the shared
memory, the pixels in one row are divided into several segments and integrated
partly as shown in Fig.3 (a). Considering the limitations of data sharing
between different GPU blocks, we only use one block (Block0 in Fig.3 (a)) for
the integration of one row, which means that each segment is integrated by the
same block one by one (Round1, Round2,…) rather than processing multiple
blocks in parallel. In each block, a two-layer parallel-integration strategy
based on the Kogge-Stone Adder algorithm is used as shown in Fig.3 (b) and
(c), since the threads work in the units of warp. The Kogge-Stone Adder method
is used because of its high computational efficiency and suitability for
thread-level parallel operations on GPUs. It is not necessary to check the
parity of the operand index for each stage.
Figure.3 (b) shows the parallel integration on the Warp layer. In Fig.3 (b),
“$\circ$” denotes the shared memory used in the current segment. Before
integration, all threads in each warp are initialized with a variable $S1$,
which represents the sum of the previous segment (step 1). For the first
segment in one row, $S1$ is initialized to 0\. Then, each warp performs the
integration independently and propagates its intermediate results by shared
memory according to the Kogge-Stone Adder method (step 2). In this step, the
result of each Warp $i$ is added to Warp $i+2^{t-1}\ (i+2^{t-1}<N)$ through
the shared memory, with $t$ representing the number of repetitions increasing
from 1 to $\lfloor{\log_{2}{N}\rfloor}$ and $N$ representing the number of
threads in each warp. Then, the sum of the current segment is updated by the
last thread (step 3); at the same time, the integrated result of each thread
is transferred to the global memory for the integration along the $y$-axis.
Figure 4: Stereo matching on the GPU
Figure.3 (c) shows the parallel integration on the Thread layer. After the
initialization shown in Fig.3 (b) step1, each thread loads the corresponding
pixel value from the global memory to the registers represented by
“$\bullet$”, and adds it to the initial value $S1$. Then, integration is
performed through the register shift among the threads in the same warp using
the method shown in Fig.3 (b) step 2, and the last result is stored in the
shared memory.
By repeating the above steps until the integration of the last segment ends,
the integral image of each row can be calculated and used to obtain the entire
integral image along the $y$-axis. Here, by transposing the matrix [25],
integration can be transformed from vertical to horizontal. However, this may
be less effective than a direct calculation when the vertical range is small,
because the memory overhead required by matrix transposition itself reduces
the parallelism of the GPU blocks. In this paper, we perform a sequential
column-wise integration rather than using a parallel-computing method, because
the height of the image set is less than 400 pixels.
#### IV-B2 Z2-ZNCC on Stereo Matching
After the summations above, the terms in (2) can be easily calculated with the
exception of $\sum_{(x,y)\in W}{\Pi_{RT}(x,y,d)}$ is omitted in the following
discussion to simplify the description. Here, we show that
$\sum{\Pi_{RT}(x,y,d)}$ can be calculated efficiently by scanning the image in
a zigzag fashion. Unlike the other summations, $\Pi_{RT}(x,y,d)$ represents a
3D-matching result between reference pixels and multiple target pixels under
different disparities. Efficient calculation of $\sum{\Pi_{RT}(x,y,d)}$ is the
most critical part of our implementation.
##### Task Assignment
As shown in Fig.4 (a), $I_{R}(x,y)$ is matched with $D$ pixels $I_{T}(x-d,y)$.
To calculate $\sum{\Pi_{RT}(x,y,d)}$ for each $I_{R}(x,y)$, $(2r+1)^{2}$
pixels around $I_{R}(x,y)$ and $(2r+1)\times{(D+2r)}$ pixels around
$I_{T}(x-d,y)\ (d\in[0,D))$ are required. For this matching, one block is
assigned because the pixel data loaded into the shared memory can be reused to
match adjacent pixels. In each block, $D$ threads are assigned to perform the
matching in parallel for each corresponding $d$, as shown in Fig.4 (b). Each
thread $i$ calculates $\sum{\Pi_{RT}(x,y,i)}$ using the pixels in the windows
$W$ (centered at $I_{R}(x,y)$) and windows $W^{\prime}$ (centered at
$I_{T}(x-i,y)$). Here, $\Pi_{RT}(x,y,i)$ is calculated element by element, and
their sum is calculated efficiently via our approach described below. After
calculating $\sum{\Pi_{RT}(x,y,d)}$ (as shown in Fig.4 (c)), the matching cost
$C(x,y,d)$ can be calculated according to (2) and then stored in the global
memory for use in the cost-aggregation stage. With this task assignment, the
data once loaded to the shared memory from the global memory can be
efficiently reused for the calculation of adjacent pixels when $D$ is
sufficiently large.
##### Zigzag Scanning
Figure 5: Zigzag scanning
In our approach, the image is scanned in a zigzag fashion, as shown in Fig.5
(a) along the $x$ and $y$ axis, while $\sum{\Pi_{RT}(x,y,d)}$ is calculated
for each pixel in parallel along the $d$-axis. $V_{Z}$ pixels in a column are
processed first from top to bottom; then, the same processing is repeated on
the next column. This scanning method is repeated from left to right. In this
zigzag scanning, the rows are segmented in the same way as in summation Method
2, and $(2r+V_{Z})\times{(2r+H_{Z})}$ pixels in the reference image and
$(2r+V_{Z})\times{(2r+H_{Z}+D)}$ pixels in the target image are loaded into
the shared memory respectively as shown in Fig.5 (b) (where $H_{z}$ is a
constant decided by the shared-memory size). With this zigzag scanning, after
$\sum{\Pi_{RT}(x,y,d)}$ was calculated, $\sum{\Pi_{RT}(x,y+1,d)}$ and
$\sum{\Pi_{RT}(x+1,y,d)}$ can be easily calculated as follows:
$\begin{split}\sum_{(x,y)\in
W}{\Pi_{RT}(x,y{\small+}1,d)}&{\small=}\displaystyle\sum_{\Delta{x}{\mbox{\scriptsize=}}-r}^{r}{\sum_{\Delta{y}{\mbox{\scriptsize=}}-r}^{r}{\Pi_{RT}(x{\small+}\Delta{x},y{\small+}\Delta{y},d)}}\\\
&{\small+}\displaystyle\sum_{\Delta{x}=-r}^{r}{\Pi_{RT}(x{\small+}\Delta{x},y{\small+}r{\small+}1,d)}\\\
&{\small-}\displaystyle\sum_{\Delta{x}=-r}^{r}{\Pi_{RT}(x{\small+}\Delta{x},y{\small-}r,d)},\\\
\end{split}$ (19) $\begin{split}\sum_{(x,y)\in
W}{\Pi_{RT}(x{\small+}1,y,d)}&{\small=}\displaystyle\sum_{\Delta{x}{\mbox{\scriptsize=}}-r}^{r}{\sum_{\Delta{y}{\mbox{\scriptsize=}}-r}^{r}{\Pi_{RT}(x{\small+}\Delta{x},y{\small+}\Delta{y},d)}}\\\
&{\small+}\displaystyle\sum_{\Delta{y}=-r}^{r}{\Pi_{RT}(x{\small+}r{\small+}1,y{\small+}\Delta{y},d)}\\\
&{\small-}\displaystyle\sum_{\Delta{y}=-r}^{r}{\Pi_{RT}(x{\small-}r,y{\small+}\Delta{y},d)}.\\\
\end{split}$ (20)
As shown in these two equations, the advantage of using the zigzag scanning
method is that as long as the sums of different rows and columns such as
$\sum_{\Delta{x}\in[-r,r]}$
${\Pi_{RT}(x{\small+}\Delta{x}}$,${y{\small-}r,d)}$ and
$\sum_{\Delta{y}\in[-r,r]}{\Pi_{RT}(x{\small-}r,y{\small+}\Delta{y},d)}$ can
be stored in the memory, they can be reused to efficiently calculate other
sums along both directions. However, storing these intermediate results along
the two directions requires a huge number of registers, which may reduce the
total efficiency.
##### Z2-ZNCC
To solve this problem, we propose a strategy for efficiently using registers.
Figure.6 shows the processing flow of the summation of $(2r+1)^{2}$ pixel
window. In this example, $V_{Z}=2$, meaning that two rows, $y$ and $y+1$, are
processed during one zigzag scanning. The calculation process is as follows:
* •
Step 1: $\sum{\Pi_{RT}(x,y,d)}$ is first calculated in order and stored in the
register $RS$; then, it can be used to calculate the matching cost $C(x,y,d)$.
During this step, in order to calculate $\sum{\Pi_{RT}(x,y+1,d)}$ efficiently,
the sum of $2r+1$ pixels on row $y-r$ is stored in the register $R0$.
* •
Step 2: The difference between $RS$ and $R0$ is calculated and stored in $RS$
to calculate $\sum{\Pi_{RT}(x,y+1,d)}$.
* •
Step 3: $\sum{\Pi_{RT}(x,y,d)}$ is still necessary for calculating
$\sum{\Pi_{RT}(x+1,y,d)}$, but its value of $RS$ was discarded in Step2. On
the other hand, the sum on row $y-r$ in $R0$ is no longer necessary. Thus, the
difference stored in $RS$ is added back to $R0$ to recalculate
$\sum{\Pi_{RT}(x,y,d)}$. This irregular procedure minimizes the number of
registers used for this calculation and makes more threads active.
* •
Step 4: The sum of row $y+r+1$ is calculated and added to $RS$. Then,
$\sum{\Pi_{RT}(x,y+1,d)}$ is obtained and used to calculate the matching cost
$C(x,y+1,d)$.
* •
Step 5: $\sum{\Pi_{RT}(x,y+1,d)}$ is stored in register $R1$ to calculate
$\sum{\Pi_{RT}(x+1,y+1,d)}$ in the same way.
At this point, $\sum{\Pi_{RT}(x,y,d)}$ and $\sum{\Pi_{RT}(x,y+1,d)}$ are
stored in $R0$ and $R1$ respectively, and these values are used to calculate
$\sum{\Pi_{RT}(x+1,y,d)}$ and $\sum{\Pi_{RT}(x+1,y+1,d)}$.
* •
Step 6,7: To calculate $\sum{\Pi_{RT}(x+1,y,d)}$ from $\sum{\Pi_{RT}(x,y,d)}$
in the same way, the sums of $2r+1$ pixels in columns $x-r$ and $x+r+1$ are
required. In our implementation, to make more threads active by reducing the
memory usage as much as possible, these sums are not stored in the memory
during the above calculations. Then, the difference between the pixels in
columns $x-r$ and $x+r+1$ is calculated and summed. In Step6, the difference
of the uppermost pixels is calculated and stored in $RS$, and in Step 7, the
differences of the other pixels are added to $RS$. Finally, $RS$ becomes the
difference between $\sum{\Pi_{RT}(x,y,d)}$ and $\sum{\Pi_{RT}(x+1,y,d)}$.
* •
Step 8: The accumulated difference is added to $R0$ and then
$\sum{\Pi_{RT}(x+1,y,d)}$ can be obtained, and $C(x+1,y,d)$ is calculated.
* •
Step 9: The difference stored in $RS$ is updated to calculate
$\sum{\Pi_{RT}(x+1,y+1,d)}$ by adding and subtracting $\Pi_{RT}$ on the four
corners.
* •
Step 10: The accumulated difference $RS$ is added to $R1$ and then
$\sum{\Pi_{RT}(x+1,y+1,d)}$ is obtained.
Using this method, we only need $V_{Z}+1$ registers for each thread to perform
the summation. In our implementation, only the intermediate results along the
$x$ axis are held on registers, while those along the $y$ axis are
recalculated. This strategy is chosen because $V_{Z}$ is smaller than $H_{Z}$.
By repeating the above calculation continuously, the overall processing speed
can be greatly improved by limiting the number of registers for each thread,
and by making more threads active.
Figure 6: Efficient zigzag scanning
### IV-C Implementation of DT
Since DT is performed based on the ZNCC computation results, the task
assignment is the same as that of ZNCC shown in Fig.4. In DT, because the cost
values must be aggregated in four directions (left to right, right to left, up
to down, and down to up), a large memory space is required. The size of on-
chip shared memory is too small for this purpose, and we need to use the off-
chip global memory. Here, because both the ZNCC ((1)) and weighting operations
($\eqref{equ:k1}$ to ${\eqref{equ:k4}}$) usually generate floating-point cost
values (32 bits), there is not only causes a great burden on data
transmission, but also a greater requirement for requires more shared-memory
space, which reduces the parallelism of multi-thread processing. To solve
these problems, it is usually good to use shorter integers (16 bits or even 8
bits) to represent the cost values instead of the floating-point data type.
However, a serious problem needs to be addressed here. The magnitude of the
aggregated cost values around the boundary of each area decreases due to the
use of weight. Hence, the magnitude is sufficiently small for 16-bit integers
but not for textureless regions. Figure.7 (a) shows an example of cost
aggregation in a large white wall with less texture. Let
$P(x^{\prime},y^{\prime})$ be a pixel that belongs to the wall in the image;
its cost values $C(x^{\prime},y^{\prime},d)$ are accumulated from all pixels
in this area according to (3) to (6). Due to the size of the wall (around
250x300 pixels) and the range of the ZNCC result ([0,1] as mentioned in
Section III-A), the accumulated floating-point cost values
$C(x^{\prime},y^{\prime},d)$ will largely exceed the upper limit on a 16-bit
integer, causing overflow. As such, these cost values cannot be simply
converted to integers by multiplying by a coefficient. In [18], we propose a
solution to this problem by shifting the matching cost of census to quickly
reduce the value range and compressing the 16-bit data into an 8-bit data with
a 1-bit flag code. However, the shifting method is not suitable for ZNCC
because of its decimal cost value; furthermore, a 1-bit flag code can only
specify two positions on a 16-bit integer, which may reduce accuracy.
Therefore, we upgrade the original solution by using a two-step strategy to
reduce the aggregated costs’ data width and burden of transmission:
* •
Use a cost-value normalization with a nearly zero-mean to represent the
original floating-point cost values with 16-bit short integers.
* •
Apply a data encoding & decoding method to further replace the normalized
16-bit short integers with 8-bit by using a 2-bit flag code.
The details of this strategy are as follows:
##### Cost-value Normalization with Nearly Zero-Mean
Figure.7 (b) shows the change in the cost values of $P(x^{\prime},y^{\prime})$
along the disparity (from 0 to $D-1$). Curve1 represents the change of
$C(x^{\prime},y^{\prime},d)$ obtained by (6) (where
$C_{D}(x^{\prime},y^{\prime},d)$ is used as final $C(x^{\prime},y^{\prime},d)$
as described above), and $C(x^{\prime},y^{\prime},d_{min})$ shows the minimum
value along the curve. The corresponding disparity $d_{min}$ is the result
obtained based on the magnitude relationship of $C(x^{\prime},y^{\prime},d)$
according to (15). Therefore, as long as the magnitude relationship remains
unchanged, changing the values of $C(x^{\prime},y^{\prime},d)$ will not affect
obtaining the correct $d_{min}$. Additionally, since the range of
$C(x^{\prime},y^{\prime},d)$ is narrow (as shown in Fig.7, Cost_gap, the
difference between the max and min of $C(x^{\prime},y^{\prime},d)$ is much
smaller than the values of $C(x^{\prime},y^{\prime},d)$ themselves. Since the
range of Cost_gap is narrow, each cost value can be nearly zero-mean
normalized by subtracting $C(x^{\prime},y^{\prime},d_{arb})$, where $d_{arb}$
is an arbitrary disparity ($d_{arb}\neq{d_{min}}$). By subtracting the median
value, the range of $C(x^{\prime},y^{\prime},d)$ can be minimized; in our
implementation, however, an arbitrary value $C(x^{\prime},y^{\prime},d_{arb})$
is used to simplify the calculation. Then (6) can be changed to:
$\displaystyle C^{\prime}_{D}(x,y,d)$
$\displaystyle=C^{\prime}_{U}(x,y,d)+C^{\prime}_{D}(x,y+1,d)\cdot{W_{D}(x,y)}$
$\displaystyle-C^{\prime}_{U}(x,y,d_{arb}),$ (21)
where $C^{\prime}$ represents the normalized cost value such that the mean
approaches zero. This effectively suppresses the increase in the aggregated
cost values. To further ensure that these values will not cause an overflow,
the cost-value normalization is extended in all directions. At the same time,
each accumulated floating-point cost value is scaled up to a 16-bit signed
integer via an integer coefficient $T$. The value of $T$ needs to be carefully
determined according to the actual situation. The larger the value, the higher
the accuracy but also the greater the risk of overflow. This approach not only
effectively reduces the burden of data transmission, but facilitates the
further reduction of data width using the following method.
Figure 7: Cost-value normalization with nearly zero-mean
##### Data Encoding & Decoding
After cost normalization, the value of
$C^{\prime}(x^{\prime},y^{\prime},d_{arb})$ is reduced to 0 and some values,
including $C^{\prime}(x^{\prime},y^{\prime},d_{min})$, become negative (when
$d_{arb}\neq{d_{min}}$). Therefore, it is possible to further reduce the data
width by focusing only upon the negative values while ignoring the positive
ones. Our method includes two stages: encoding and decoding. Figure.8 shows
the process of our method. In the encoding stage, the 16-bit cost value is
first compressed into an 8-bit code containing a 6-bit value and a 2-bit flag.
This 6-bit value represents the 6 significant bits of the 16-bit integer, and
the 2-bit flag shows its position. In the decoding stage, each 8-bit code is
decompressed into a 16-bit approximation by putting the 6-bit value in the
16-bit integer on the position specified by the 2-bit flag code. Between the
stages, four adjacent 8-bit codes are packed into a 32-bit integer for more
efficient transmission on a GPU.
The details of the process in Fig.8 can be described as follows:
* •
Encoding
1. 1.
As mentioned above, only negative values are used. Therefore, all positive
values are set to 0 by checking the sign bit.
2. 2.
Since two’s complement is used to represent the negative values, we first test
the 4-bit $Data_{16}[14:11]$ to find whether it is ${}^{\prime}1111^{\prime}$.
If not, ($Data_{16}[14:9]$) is chosen as the 6-bit code and
${}^{\prime}01^{\prime}$ is attached to it as the 2-bit flag to show the
position of the 6-bit code in the original 16-bit integer. Then, an 8-bit
dataset, $Data_{8}$, is constructed from the 16-bit integer.
3. 3.
If $Data_{16}$[14:11] is ${}^{\prime}1111^{\prime}$, the next 4-bit code
($Data_{16}[10:7]$) is checked in the same way. If it is not
${}^{\prime}1111^{\prime}$, $Data_{16}[10:5]$ is chosen as the 6-bit code and
the flag code ${}^{\prime}10^{\prime}$ is attached to show its position.
4. 4.
Finally, if $Data_{16}[14:11]$ and $Data_{16}[10:7]$ are both
${}^{\prime}1111^{\prime}$, $Data_{16}[6:1]$ is chosen as the 6-bit code (no
checking is necessary) and the 2-bit flag code ${}^{\prime}11^{\prime}$ is
attached.
* •
Decoding
1. 1.
We first check whether $Data_{8}$ is ${}^{\prime}0^{\prime}$ or not. If it is,
$Data_{16}$ is also set to ${}^{\prime}0^{\prime}$; if not, its flag code
$Data_{8}[1:0]$ is checked.
2. 2.
If the 2-bit flag code is ${}^{\prime}11^{\prime}$, the 6-bit code at
$Data_{8}[7:2]$ is copied to $Data_{16}[6:1]$; if the flag code is
${}^{\prime}10^{\prime}$, the 6-bit code is copied to $Data_{16}[10:5]$; if
the flag code is ${}^{\prime}01^{\prime}$, the 6-bit code is copied to
$Data_{16}[14:9]$.
3. 3.
Then, the bits on the left-hand side of the copied 6-bit code in $Data_{16}$
are set to ${}^{\prime}1^{\prime}$ and the bits on the right side are set to
${}^{\prime}0^{\prime}$.
Figure 8: Data encoding & decoding
This method uses the 2-bit flag code, which specifies four position; and the
position of the 6-bit code is not continuous on a 16-bit integer. While this
is more accurate than the 1-bit flag code which specifies two positions, our
approach still loses more information than general 16-bit to 6-bit data-width
reduction. However, according to our experiments, high speed processing is
possible without losing too much accuracy.
## V Evaluation
$3$$5$$7$$9$$11$$13$$15$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$Side LengthTime
(${ms}$)M1(BS=64x16)M1(BS=64x8)M1(BS=64x4)M2(BS=64)M2(BS=128)M2(BS=256) (a)
Regcount = 32
$3$$5$$7$$9$$11$$13$$15$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$Side
LengthM1(BS=64x16)M1(BS=64x8)M1(BS=64x4)M2(BS=64)M2(BS=128)M2(BS=256) (b)
Regcount = 48
$3$$5$$7$$9$$11$$13$$15$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$Side
LengthM1(BS=64x16)M1(BS=64x8)M1(BS=64x4)M2(BS=64)M2(BS=128)M2(BS=256) (c)
Regcount = 60
Figure 9: Processing speed comparison for summation
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side LengthTime
(ms)$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (a) Regcount=32, $H_{Z}$=32
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side
Length$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (b) Regcount=32,
$H_{Z}$=64
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$$65$Side
Length$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (c) Regcount=32,
$H_{Z}$=128
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side LengthTime
(ms)$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (d) Regcount=48, $H_{Z}$=32
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side
Length$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (e) Regcount=48,
$H_{Z}$=64
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side
Length$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (f) Regcount=48,
$H_{Z}$=128
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side LengthTime
(ms)$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (g) Regcount=60, $H_{Z}$=32
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side
Length$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (h) Regcount=60,
$H_{Z}$=64
$3$$5$$7$$9$$11$$13$$15$$10$$17$$24$$31$$38$$45$$52$Side
Length$V_{Z}$=2$V_{Z}$=3$V_{Z}$=4$V_{Z}$=5$V_{Z}$=6 (i) Regcount=60,
$H_{Z}$=128
Figure 10: Processing speed evaluation for Z2-ZNCC
$3$$5$$7$$9$$11$$13$$15$$10$$25$$40$$55$$70$$85$$100$Side LengthTime
(ms)Original(D=96)Original(D=128)Z2-ZNCC(D=96)Z2-ZNCC(D=128)RTNCC(D=90) Figure
11: Processing speed comparison for Z2-ZNCC
We implemented our acceleration approach on an embedded GPU Jetson Tx2 and
evaluated
1. 1.
summation,
2. 2.
Z2-ZNCC,
3. 3.
FastDT, and
4. 4.
the processing speed and matching accuracy of the stereo-matching system based
on the Z2-ZNCC, census, semi-global matching (SGM), and FastDT,
algorithms using the KITTI 2015 [5] benchmark.
### V-A Evaluation of Summation
The processing speeds of the two summation methods described in Section IV are
compared using 1,280x384 pixel images. In our evaluation, the maximum number
of registers (Regcount) for each thread is limited to 32, 48, and 60; then
comparisons are performed for different window sizes and GPU blocks.
Figure.9 compares the results of the two summation methods. In each graph, M1
and M2 represent the two methods and BS represent the GPU block size. The
$x$-axis represents matching windows’ side length from 3 to 15 and the
$y$-axis represents their corresponding processing times. In Method 1, each
block processes 64 columns and three sets of rows: 16, 8, and 4\. As the side
length increases, the processing time increases accordingly. This is occurs
not only due to the increase in the amount of the calculation, but also to the
increase in memory occupancy, which reduces the number of active threads. On
the other hand, in Method 2, each block processes three sets of columns: 64,
128, and 256. Because the sum is calculated based on the integral image, the
processing time does not change with side length. Here, we note that in all
cases, the processing time does not change significantly with the GPU-block
size because the parallelism of threads is not affected. For all three
different block sizes in Method 2, the processing times for the first two
steps of obtaining an integral image is about 355 $\mu{s}$ and 496 $\mu{s}$,
respectively, and the total time including the averaging calculation is close
to 1.4 ms. Due to the integration, the processing speed of Method 2 is not as
fast as that of Method 1 when the window size is smaller than 9x9. Therefore,
the methods can be chosen according to actual requirements. However, when BS =
64x16 and Regcount = 48 (or Regcount = 60), Method 1 becomes invalid when the
side length is larger than 9 (or 7); this is because the large number of
threads causes the number of allocated registers to exceed the allowable upper
limit. Therefore, for Method 1, a small GPU block should be chosen so as to
ensure that the summation can be performed correctly.
### V-B Evaluation of Z2-ZNCC
Figure.10 shows the evaluation results for Z2-ZNCC using the same 1,280x384
size images with a maximum disparity of 128. To show the performance of
Z2-ZNCC more clearly, Method 2 is used for the summation because of its
consistent processing speed. In addition to Regcount, the evaluation is
performed under various $V_{Z}$ and $H_{Z}$ conditions, as described in
Section.IV. $V_{Z}$ is changed from 2 to 6 and $H_{Z}$ is set to 32, 64, and
128. A large $V_{Z}$ means that the rate of data reuse is high; however, it
also requires a larger Regcount, which will affect the parallelism of threads.
Similarly, a large $H_{Z}$ suggests that a large number of $\Pi_{RT}(x,y,d)$
need to be calculated at the same time, which also requires many registers.
According to this figure, we note that in most cases, when $V_{Z}$ = 2,
Z2-ZNCC performs the worst because of its low data-reuse rate. Furthermore,
the larger the side length, the worse the performance, because step 7 in Fig.6
does not work effectively. For $V_{Z}$ = 5 or 6, the performance is still poor
when Regcount = 32 (because of the limited number of registers) or when
$H_{Z}$ = 128 (because of too much use of registers). Additionally, for the
case shown in Fig.10 (c), the larger the $V_{Z}$ value, the worse the
performance of Z2-ZNCC, with the exception of $V_{Z}$ = 2. Comparing Figs.10
(f) and (i) with (c), we can see that, by increasing the Regcount to 48 and 60
respectively, the performance can be improved accordingly. However, the
difference between (f) and (i) is not large, meaning that increasing Regcount
does not improve performance proportionally. For other cases, the processing
time increases with the side length without any obvious outliers. The results
show that increasing Regcount improves performance more than changing the
$H_{Z}$, as shown in Fig.10 (a), (d), and (g). Finally, when $V_{Z}$ = 4,
Z2-ZNCC always performs consistently. This means that it achieves a good
balance between hardware resources and calculations according to our
evaluations.
In addition, we also compared Z2-ZNCC to other methods under various maximum
disparity values. Figure.11 shows the results of five methods: two Z2-ZNCC
methods, two original progressive-scan methods, and RTNCC [19]. $D$ represents
the maximum disparity value and is set to 90, 96 and 128. Based on the
summation-speed comparison, we use Method 1 when the window sizes are smaller
than 9x9 and Method 2 when the window sizes are larger than 7x7. In the
Z2-ZNCC methods, $V_{Z}$ = 4, $H_{Z}$ = 32 and Regcount = 60. The proposed
Z2-ZNCC methods work faster than other methods where the disparity is 96 and
128. When the side length is 3 and $D$ = 128, the processing time is 18.05 ms
for Original and 11.31 ms for Z2-ZNCC: a 38% increase in speed. When the side
length is changed to 15, the processing time is 72.09 ms for Original and
34.35 ms for Z2-ZNCC, meaning that the processing time is reduced by more than
half. This is mainly because the Z2-ZNCC methods can reuse data in the
vertical direction but the original methods cannot. Furthermore, compared with
the latest RTNCC (its $D$ is only 90), our method for $D$ = 96 requires only
16.04 ms, which is 26% faster despite its computational complexity.
Figure 12: Matching Result(KITTI 2015): (a) Left image & ground truth; (b) S1:
Z2-ZNCC+SGM; (c) S2: Census+SGM; (d) S3: Census+FastDT; (e) S4: Z2-ZNCC+FastDT
$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$22$$24$203350901703306601200error rate
($\%$)Time (ms)AnyNet (10 fps)RTS2NET (6.3 fps)MADNet (3.8 fps)StereoDNN (0.1
fps)RTNCC (46 fps)S3:Census+FastDT(35 fps)S2:Census+SGM(29
fps)S1:Z2-ZNCC+SGM(28 fps)S4:Z2-ZNCC+FastDT(32 fps) Figure 13: Comparison with
other systems TABLE I: Comparison of Kernel Performance In Z2ZNCC.
Method | GT(G/s) | GE(%) | IPW | IPC
---|---|---|---|---
Original ZNCC | 0.92 | 67.09 | 1.06e+05 | 3.05
Z2ZNCC | 1.16 | 76.95 | 3.06e+05 | 3.15
* •
GT: gld_throughput GE: gld_efficiency IPW: inst_per_warp IPC:inst_per_cycle
Table.I shows the comparison between Z2ZNCC and the original ZNCC in terms of
kernel performance. We use gld_throughput and gld_efficiency to evaluate the
performance of our kernel in memory access, and use inst_per_warp and
inst_per_cycle for the computational efficiency. In particular, the number of
our IPW is roughly three times the original, which shows that our method can
effectively save on-chip resources to maintain a high degree of thread
parallelism.
### V-C Evaluation of FastDT
We evaluated FastDT in combination with Z2-ZNCC using the training set of the
KITTI 2015 [5] benchmark because the accuracy must be evaluated precisely. 200
pairs of images are included in the training set, and their sizes are close to
1,250x375 with a maximum disparity value of 128. In our evaluation, Regcount =
48, T = 21 and the side length of ZNCC is set to 3. The two parameters in
(III-B) are $\sigma_{s}$ = 5 and $\sigma_{r}$ = 52. $d_{arb}$ is set to 0 to
effectively reduce the range of cost values because $d_{min}$ is rarely equal
to zero in stereo matching.
TABLE II: Comparison of DT in KITTI 2015.
Direction | Float | Int32 | Int16 | Int8 (Ours) | Int16-NP
---|---|---|---|---|---
L2R∗(ms) | 13.10 | 13.76 | 13.63 | 13.68 | 13.59
R2L(ms) | 15.41 | 15.32 | 8.72 | 5.52 | 8.56
U2D(ms) | 15.12 | 15.05 | 8.74 | 7.22 | 8.65
D2U∗(ms) | 7.06 | 7.03 | 6.29 | 6.27 | 6.34
Frame Rate(fps) | 20 | 20 | 26 | 32 | 27
Error Rate(%) | 7.36 | 7.49 | 7.57 | 7.63 | 8.14
* •
L2R∗:Z2-ZNCC & left to right. R2L: right to left. U2D: up to down. D2U∗: down
to up & WTA. NP: non-normalization.
Table II shows the processing time for each step of DT under four different
methods, together with their accuracy. The four methods–Float, Int32, Int16,
and Int8–use floating-point, integer, short integer, and character
respectively during cost propagation. Because Z2-ZNCC calculates the cost
value for each pixel serially, it is combined with the cost propagation from
left to right (L2R). WTA is also combined with the cost propagation from down
to up (D2U) to calculate the disparity map directly without transferring cost
values back to the global memory. According to our observation, the cost
values aggregated during the L2R step are not usually large. Therefore, in our
implementation, cost-value normalization is performed only in the U2D steps of
Float, Int32, and Int16, and in the R2L step of Int8. The encoding is
performed at the end of the R2L step and the decoding is performed at the
beginning of the U2D step. Additionally, to verify the necessity of cost-value
normalization, we added the evaluation of Int16-NP, in which such
normalization is not performed. In the L2R step, the processing time is
roughly the same regardless of the data type because the transmission latency
is hidden by the calculation time of Z2-ZNCC. On the other hand, the
processing time of Int16 is only 8.72 ms in the R2L step and 8.74 ms in the
U2D step, which are roughly half of the values for Float and Int32. For
Int16-NP, since it is roughly the same as Int16 in terms of calculation and
data transmission, there is no obvious difference in processing time. Our Int8
shows the fastest processing speed of all methods, even though data encoding &
decoding is performed. This is because our method transfers data in 8-bit,
which greatly reduces the latency of data transfer. Table.III shows the
comparison between FastDT and the original DT in terms of global memory
accessing. As the results of gld_throughput and gld_efficiency shown, our
kernel significantly improves the efficiency of global memory access in both
R2L and U2D directions. This is mainly benefits from our encoder compression
method, which can improve transmission efficiency by bundling data. In our
implementation, Int8 achieved the requirement for real-time processing with 32
fps, which is 60% faster than the 20 fps attained by Float.
TABLE III: Comparison of Global Memory Efficiency In DT.
Method | L2R | R2L | U2D | D2U
---|---|---|---|---
Original DT | GT(G/s) | 0.42 | 0.56 | 5.18 | 38.7
GE(%) | 50.3 | 20.29 | 59.39 | 94.16
FastDT | GT(G/s) | 0.41 | 11 | 17.08 | 22.6
GE(%) | 50.13 | 99.2 | 88.9 | 88.9
* •
GT: gld_throughput GE: gld_efficiency.
TABLE IV: Comparison of matching accuracies in KITTI 2015
Error Rate (%) | | | D1-bg | | | | D1-fg | | | | D1-all |
---|---|---|---|---|---|---|---|---|---|---|---|---
| S1 | S2 | S3 | S4 | S1 | S2 | S3 | S4 | S1 | S2 | S3 | S4
All / All | 6.60 | 6.98 | 15.12 | 7.77 | 13.58 | 17.04 | 23.56 | 16.71 | 7.76 | 8.66 | 16.52 | 9.26
All / Est | 6.54 | 6.82 | 12.56 | 7.61 | 13.54 | 16.92 | 22.78 | 16.52 | 7.71 | 8.51 | 14.24 | 9.10
Noc / All | 5.22 | 5.48 | 14.33 | 6.92 | 11.38 | 14.83 | 21.88 | 15.01 | 6.24 | 7.03 | 14.48 | 8.26
Noc / Est | 5.20 | 5.44 | 11.73 | 6.87 | 11.38 | 14.82 | 21.38 | 14.99 | 6.22 | 6.99 | 13.38 | 8.21
* •
S1: Z2-ZNCC+SGM. S2: Census+SGM. S3: Census+FastDT. S4: Z2-ZNCC+FastDT.
In the accuracy evaluation, Float has the lowest error rate of 8.16%. As the
data width decreases, the error rate gradually increases. Compared with the
error rate of 8.37% for Int16, that for Int16-NP is higher (8.81%) because of
the overflow of the cost values. This shows that our cost normalization with
nearly zero-mean works very effectively. Int8 has an error rate of 8.41%, only
a 0.25% loss compared to Float. This means that our method can effectively
increase the processing speed even as it maintains a high accuracy in stereo
matching.
### V-D Evaluation of Stereo Matching
Finally, to further clarify the effectiveness of our Z2-ZNCC and FastDT for
stereo matching, we also combined them with the state-of-the-art algorithms
SGM and census, respectively. Then, we compared their accuracies and
processing speeds on a Jetson Tx2 GPU using the KITTI 2015 benchmark. SGM and
census were chosen because the GPU system in [14] uses them and has a good
performance (8.66% error rate, 29 fps) under the same conditions. Our
implementation of the SGM is almost the same as [14]. The difference is that
to clarify the role of Z2-ZNCC, we do not use the stream function. The two
parameters in SGM–$P1$ and $P2$–are set to 18 and 185, respectively, because
the results of Z2-ZNCC are multiplied by the coefficient $T$.
Figure.12 shows the comparisons of three systems based on use our proposed
methods, and one other system [14]. The four systems are Z2-ZNCC+SGM,
Census+SGM [14], Census+FastDT and Z2-ZNCC+FastDT; they are expressed as S1,
S2, S3, and S4, respectively. Four pairs of images are selected from the
training set to clearly show the difference in the results of these systems.
Figure.12 (a) shows the reference images and their ground truths. Figures.12
(b) to 12 (e) show the results of S1 to S4, respectively. The top half of each
figure shows disparity map and the bottom half the corresponding error image
compared with the ground truths. The color bar denotes the disparity range of
[0,128), with blue representing the farthest objects and red the closest. As
for the four sets of results No.000045, No.000076, No.000094, and No.000144,
S1 in Fig.12 (b) shows a clear advantage in terms of accuracy. Its error rates
of 3.61%, 2.01%, 2.85%, and 4.75% are obviously lower than those of other
systems. Compared with S2 in Fig.12 (c), S1 works better in photometric
distortions and weak-pattern areas, as shown by the red boxes in Figs.12 (c)
and (d). This means that ZNCC has stronger robustness than census. S3 shows a
disadvantage in accuracy. Its error rates are the highest among the four
systems. This is because, in addition to the less accurate matching by census,
DT also easily causes a fattening effect, making some details disappear, as
shown in the yellow boxes in Figs.12 (d) and (e). By replacing census with
ZNCC, the accuracy of S4 has been greatly improved, as shown in Fig.12 (e).
This means that as long as a high precision is achieved in the cost matching
stage, DT will not induce an excessive loss in accuracy. Table IV compares the
matching accuracies of the four systems using the testing set of KITTI 2015
benchmark. The result is consistent with the above, and the order of accuracy
is S1$>$S2$>$S4$>$S3. The use of ZNCC improves the accuracies of the census-
based systems by 0.9% (S1 vs. S2) and 7.26% (S4 vs. S3).
Figure.13 shows a comparison with other systems in terms of processing speed
and accuracy. The $x$-axis represents the error rate and the $y$-axis
represents the time required for processing in ms; thus, the closer the
evaluation results are to the origin, the higher the accuracy and the faster
the processing speed. All of the CNN-based systems (AnyNet [11], StereoDNN
[12], MADNet [13], and RTS2Net [26]) achieved high accuracy, but had no speeds
exceeding 10 fps. RTNCC [19] and our S3 (Census+FastDT) algorithm achieved
real-time processing speeds of 46 fps and 35 fps, respectively. However, their
error rates are larger than 16%, which also limits their usability. S2
(Census+SGM) [14] and our S1 (Z2-ZNCC+SGM) have almost the same processing
speed, but our accuracy is 0.5% lower. This shows that our method plays a
significant role in stereo matching. As mentioned above, our S4
(Z2-ZNCC+FastDT) improves the accuracy of S3 from 16.52% to 9.26%. Compared
with the systems based on SGM (S1 and S2), S4 still has a 1% lower accuracy,
but its processing speed is 17% faster, allowing it to truly achieve real-time
processing.
## VI Conclusion
In this paper, we proposed an acceleration method for the zero-means
normalized cross correlation (ZNCC) template-matching algorithm for stereo
vision on an embedded GPU. This method helps us reuse intermediate calculation
results efficiently without frequently transferring them among the hierarchy
of memories, leading to a higher processing speed. It also helps to improve
the matching accuracy because of its stronger robustness. We evaluated our
systems based on this method by using the KITTI 2015 benchmark and showed
their efficiency. Since the proposed method does not rely upon the GPU
experimental architecture, it can be used on different GPUs; furthermore, it
is not limited to GPUs and can be used on other hardware platforms like FPGAs.
To further improve out systems’ performance, we are planning to combine it
with other cost-aggregation algorithms. This will be done in future work.
## References
* [1] Wenqiang Wang, et al: ”Real-Time High-Quality Stereo Vision System in FPGA”. IEEE Transactions on Circuits and Systems for Video Technology. 25(10): 1696-1708 (2015)
* [2] Mohammad Dehnavi, Mohammad Eshghi: ”FPGA based real-time on-road stereo vision system,” Journal of Systems Architecture: Embedded Software Design, 81: 32-43 (2017)
* [3] Xuchong Zhang, Hongbin Sun, Shiqiang Chen, Lin Song, Nanning Zheng: NIPM-sWMF: ”Toward Efficient FPGA Design for High-Definition Large-Disparity Stereo Matching”. IEEE Transactions on Circuits and Systems for Video Technology. 29(5): 1530-1543 (2019)
* [4] Pin-Chen Kuo, et al: ”Stereoview to Multiview Conversion Architecture for Auto-Stereoscopic 3D Displays”. IEEE Transactions on Circuits and Systems for Video Technology. 28(11): 3274-3287 (2018)
* [5] Moritz Menze, Andreas Geiger: ”Object scene flow for autonomous vehiclesm” IEEE Conference on Computer Vision and Pattern Recognition(CVPR), 2015: 3061-3070
* [6] Oscar Rahnama, et al: ”Real-Time Highly Accurate Dense Depth on a Power Budget Using an FPGA-CPU Hybrid SoC,” IEEE Transactions on Circuits and Systems II: Express Briefs, 66-II(5): 773-777 (2019)
* [7] Xuchong Zhang, He Dai, Hongbin Sun, Nan-Ning Zheng: ”Algorithm and VLSI Architecture Co-Design on Efficient Semi-Global Stereo Matching”. IEEE Transactions on Circuits and Systems for Video Technology. 30(11): 4390-4403 (2020)
* [8] Christoph Hartmann, Ulrich Margull: GPUart - An application-based limited preemptive GPU real-time scheduler for embedded systems. Journal of Systems Architecture: Embedded Software Design. 97: 304-319 (2019)
* [9] Sparsh Mittal: A Survey on optimized implementation of deep learning models on the NVIDIA Jetson platform. Journal of Systems Architecture: Embedded Software Design. 97: 428-442 (2019)
* [10] Mustafa U. Torun, Onur Yilmaz, Ali N. Akansu: ”FPGA, GPU, and CPU implementations of Jacobi algorithm for eigenanalysis”. Journal of Parallel and Distributed Computing. 96: 172-180 (2016)
* [11] Yan Wang, et al: ”Anytime Stereo Image Depth Estimation on Mobile Devices,” IEEE International Conference on Robotics and Automation (ICRA), 2019: 5893-5900
* [12] Nikolai Smolyanskiy, Alexey Kamenev, Stan Birchfield: ”On the Importance of Stereo for Accurate Depth Estimation: An Efficient Semi-Supervised Deep Neural Network Approach,” IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPR), 2018: 1007-1015
* [13] Alessio Tonioni, Fabio Tosi, Matteo Poggi, Stefano Mattoccia, Luigi di Stefano: ”Real-Time Self-Adaptive Deep Stereo,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019: 195-204
* [14] Daniel Hernández Juárez, et al: ”Embedded Real-time Stereo Estimation via Semi-Global Matching on the GPU,” International Conference on Computational Science (ICCS), 2016: 143-153
* [15] Zhuoyuan Chen, et al: ”A Deep Visual Correspondence Embedding Model for Stereo Matching Costs,” IEEE International Conference on Computer Vision (ICCV), 2015: 972-980
* [16] Qiong Chang, Tsutomu Maruyama: ”Real-Time Stereo Vision System: A Multi-Block Matching on GPU,” IEEE Access 6: 42030-42046 (2018)
* [17] Martin Werner, Benno Stabernack, Christian Riechert: ”Hardware implementation of a full HD real-time disparity estimation algorithm,” IEEE Transactions on Consumer Electronics, 60(1): 66-73 (2014)
* [18] Qiong Chang, Aolong Zha, Masaki Onishi, Tsutomu Maruyama: ”A GPU Accelerator for Domain Transformation-Based stereo Matching,” International Conference on Algorithms, Computing and Artificial Intelligence (ACAI), 2019 (pp. 370-376).
* [19] Cui, Han, and Naim Dahnoun. ”Real-Time Stereo Vision Implementation on Nvidia Jetson TX2,” Mediterranean Conference on Embedded Computing (MECO), 2019. p. 1-5.
* [20] Qiong Chang, et al: Z2-ZNCC: ZigZag Scanning based Zero-means Normalized Cross Correlation for Fast and Accurate Stereo Matching on Embedded GPU. IEEE International Conference on Computer Design (ICCD), 2020: 597-600
* [21] Chuan Lin, Ya Li, Guili Xu, Yijun Cao: ”Optimizing ZNCC calculation in binocular stereo matching. Signal Processing: Image Communication, 52: 64-73 (2017)
* [22] Fan Rui, and Naim Dahnoun. ”Real-time implementation of stereo vision based on optimised normalised cross-correlation and propagated search range on a gpu,” IEEE International Workshop on Imaging Systems and Techniques (IST), 2017:pp.1-6.
* [23] Cuong Cao Pham and JaeWook Jeon. ”Domain Transformation-Based Efficient Cost Aggregation for Local Stereo Matching,” IEEE Transactions on Circuits and Systems for Video Technology, 2013: pp.1119-1130.
* [24] NVIDIA Corporation: ”NVIDIA CUDA C programming guide,” 2019.Version 10.1.243.
* [25] Qiong Chang, Tsutomu Maruyama: ”Real-Time High-Quality Stereo Matching System on a GPU,” IEEE International Conference on Application-specific Systems, Architectures and Processors (ASAP), 2018: 1-8
* [26] Pier Luigi Dovesi, et al: ”Real-Time Semantic Stereo Matching,” IEEE International Conference on Robotics and Automation (ICRA), 2020: 10780-10787.
|
# CUNI Non-Autoregressive System for the WMT 22 Efficient Translation Shared
Task
Jindřich Helcl
Charles University, Faculty of Mathematics and Physics
Institute of Formal and Applied Linguistics
Malostranské náměstí 25, 11800 Prague, Czech Republic
<EMAIL_ADDRESS>
###### Abstract
We present a non-autoregressive system submission to the WMT 22 Efficient
Translation Shared Task. Our system was used by Helcl et al. (2022) in an
attempt to provide fair comparison between non-autoregressive and
autoregressive models. This submission is an effort to establish solid
baselines along with sound evaluation methodology, particularly in terms of
measuring the decoding speed. The model itself is a 12-layer Transformer model
trained with connectionist temporal classification on knowledge-distilled
dataset by a strong autoregressive teacher model.
## 1 Introduction
In the past few years, non-autoregressive (NAR) models for neural machine
translation (NMT) attracted interest from the research community (Gu et al.,
2018; Lee et al., 2018). Given the conditional independence between the output
states, the decoding process can be parallelized across time steps. In theory,
this leads to higher decoding speeds.
Since efficient decoding is claimed to be the main motivation of non-
autoregressive models, the Efficient Translation Shared Task seems to be the
appropriate venue to provide fair comparison between these models and their
autoregressive counterparts. However, all submissions to this task were
autoregressive so far (Birch et al., 2018; Hayashi et al., 2019; Heafield et
al., 2020, 2021).
Recently, Helcl et al. (2022) pointed out common flaws in the evaluation
methodology of NAR models. We found that optimized autoregressive models still
achieve superior performance over NAR models. The only scenario where NAR
models showed some potential is GPU decoding with batch size of 1 (latency).
Nevertheless, optimized autoregressive models were still both faster and
better in terms of translation quality. The main purpose of this submission is
to provide a reasonable baseline to future non-autoregressive submissions.
## 2 Model
In our experiments, we use the non-autoregressive model proposed by Libovický
and Helcl (2018) based on Connectionist Temporal Classification (CTC; Graves
et al., 2006). We submit models that have been trained as a part of Helcl
(2022).
### Architecture.
The architecture is a 6-layer Transformer encoder, followed by a state-
splitting layer and another stack of 6 Transformer layers. The state-splitting
layer takes the encoder states, project them into $k$-times wider states using
an affine transformation, and then split the states into $k$-times longer
sequence while retaining the original model dimension. In the submitted model,
we set $k=3$. The latter 6 layers cross-attend to the states immediately after
state-splitting. We use Transformer model dimension of 1,024, 16 attention
heads and a dimension of 4,096 in the feed-forward sublayer.
The defining property of non-autoregressive models is that the decoding
process treats output states as conditionally independent. In this
architecture, we set the output sequence length to $k\times T_{x}$ where
$T_{x}$ is the length of the source sentence. To allow for shorter output
sequences, the any output state can produce an empty token. The training loss
is then computed using a dynamic programming algorithm as a sum of losses of
all possible empty token alignments which lead to the same output sentence.
The schema of the architecture is shown in Figure 1.
Input token
embeddingsEncoder$\mathbf{h}$$W_{\text{spl}}\mathbf{h}$$\mathbf{s}$DecoderConnectionist
Temporal
Classification$w_{1}$$w_{2}$$w_{3}$$\varnothing$$w_{4}$$\varnothing$$w_{5}$$w_{6}$$\varnothing$$\varnothing$$\varnothing$$w_{7}$$w_{8}$$\varnothing$$w_{9}$$\varnothing$Output
tokens / null symbols
Figure 1: The CTC-based model architecture. We show the original image from
Libovický and Helcl (2018).
### Training.
We train our model on the knowledge-distilled data generated by the provided
teacher (Chen et al., 2021). We use learning rate of 0.0001 in a inverse
square-root decay scheme with 8,000 warm-up and decay steps.
### Implementation.
We implement and train our model in the Marian toolkit (Junczys-Dowmunt et
al., 2018). For the CTC implementation, we use the warp-ctc
library111https://github.com/baidu-research/warp-ctc. We release our code at
https://github.com/jindrahelcl/marian-dev. The trained model (and a number of
different variants including models in opposite translation direction) can be
downloaded at https://data.statmt.org/nar.
## 3 Results
We refer the reader to the original paper for more details about the
evaluation and its results. The model we submitted is denoted in the paper as
“large”. A summary of the results follows.
### Translation Quality.
To summarize the main findings, the model achieves a competitive BLEU score
(Papineni et al., 2002) on the WMT 14 news test set (Bojar et al., 2014),
which serves as a comparison to other non-autoregressive models that use this
test set as the de facto standard benchmark. When evaluated on the WMT 19 news
test set, our model obtains BLEU of 47.8, and a COMET score (Rei et al., 2020)
of 0.1485. Compared to an similarly-sized autoregressive teacher model with
50.5 BLEU and COMET of 0.4110, we see a somewhat surprising gap between the
COMET scores while BLEU scores are relatively close. We hypothesize that the
errors that the non-autoregressive model makes are out of the training domain
of the COMET models, which makes them more sensitive towards this kind of
errors.
### Decoding Time.
We evaluated our models on the one million sentences benchmark used in the
previous editions of this task (Heafield et al., 2021), and we tried to
reproduce the official hardware setup to large extent. For CPU decoding, we
measured time to translate the test set on an Intel Xeon 6354 server from
Oracle Cloud, with 36 cores. We run the evaluation only in the batch decoding
mode, as the models were too slow to decode with a single sentence in batch.
With the submitted model, the translation on CPU took 7,434 seconds (using
batch of 16 sentences).
We used a single Nvidia A100 GPU for GPU decoding. In the latency setup, the
translation took 7,020 seconds, and the batched decoding ($b=128$) took 782
seconds. When compared with other submissions to this task, we find that the
smallest difference is indeed found in the GPU decoding latency setting.
However, the optimized models submitted to last year’s round still achieved
significantly better decoding times.
## 4 Conclusions
We submit a non-autoregressive system to the Efficient Translation Shared Task
to the WMT 22. The model is trained with connectionist temporal
classification, which allows the generation of empty tokens and thus making
generation of sentences of various length possible while retaining the
conditional independence among output tokens without explicit length
estimation.
The main motivation of this submission is to provide a reasonable baseline
system for future research. We believe that the sub-field of non-
autoregressive NMT cannot progress without controlled decoding speed
evaluation, which is exactly what the shared task organizers provide.
## Acknowledgements
This project received funding from the European Union’s Horizon Europe
Innovation programme under grant agreement no. 101070350 (HPLT). Our work has
been using data provided by the LINDAT/CLARIAH-CZ Research Infrastructure,
supported by the Ministry of Education, Youth and Sports of the Czech Republic
(Project No. LM2018101).
## References
* Birch et al. (2018) Alexandra Birch, Andrew Finch, Minh-Thang Luong, Graham Neubig, and Yusuke Oda. 2018\. Findings of the second workshop on neural machine translation and generation. In _Proceedings of the 2nd Workshop on Neural Machine Translation and Generation_ , pages 1–10, Melbourne, Australia. Association for Computational Linguistics.
* Bojar et al. (2014) Ondřej Bojar, Christian Buck, Christian Federmann, Barry Haddow, Philipp Koehn, Johannes Leveling, Christof Monz, Pavel Pecina, Matt Post, Herve Saint-Amand, Radu Soricut, Lucia Specia, and Aleš Tamchyna. 2014. Findings of the 2014 workshop on statistical machine translation. In _Proceedings of the Ninth Workshop on Statistical Machine Translation_ , pages 12–58, Baltimore, Maryland, USA. Association for Computational Linguistics.
* Chen et al. (2021) Pinzhen Chen, Jindřich Helcl, Ulrich Germann, Laurie Burchell, Nikolay Bogoychev, Antonio Valerio Miceli Barone, Jonas Waldendorf, Alexandra Birch, and Kenneth Heafield. 2021. The University of Edinburgh’s English-German and English-Hausa submissions to the WMT21 news translation task. In _Proceedings of the Sixth Conference on Machine Translation_ , pages 104–109, Online. Association for Computational Linguistics.
* Graves et al. (2006) Alex Graves, Santiago Fernández, Faustino Gomez, and Jürgen Schmidhuber. 2006. Connectionist temporal classification: Labelling unsegmented sequence data with recurrent neural networks. In _Proceedings of the 23rd International Conference on Machine Learning_ , pages 369–376, Pittsburgh, PA, USA. JMLR.org.
* Gu et al. (2018) Jiatao Gu, James Bradbury, Caiming Xiong, Victor O. K. Li, and Richard Socher. 2018\. Non-autoregressive neural machine translation. In _6th International Conference on Learning Representations, ICLR 2018_ , Vancouver, BC, Canada.
* Hayashi et al. (2019) Hiroaki Hayashi, Yusuke Oda, Alexandra Birch, Ioannis Konstas, Andrew Finch, Minh-Thang Luong, Graham Neubig, and Katsuhito Sudoh. 2019. Findings of the third workshop on neural generation and translation. In _Proceedings of the 3rd Workshop on Neural Generation and Translation_ , pages 1–14, Hong Kong. Association for Computational Linguistics.
* Heafield et al. (2020) Kenneth Heafield, Hiroaki Hayashi, Yusuke Oda, Ioannis Konstas, Andrew Finch, Graham Neubig, Xian Li, and Alexandra Birch. 2020. Findings of the fourth workshop on neural generation and translation. In _Proceedings of the Fourth Workshop on Neural Generation and Translation_ , pages 1–9, Online. Association for Computational Linguistics.
* Heafield et al. (2021) Kenneth Heafield, Qianqian Zhu, and Roman Grundkiewicz. 2021. Findings of the WMT 2021 shared task on efficient translation. In _Proceedings of the Sixth Conference on Machine Translation_ , pages 639–651, Online. Association for Computational Linguistics.
* Helcl et al. (2022) Jindřich Helcl, Barry Haddow, and Alexandra Birch. 2022. Non-autoregressive machine translation: It’s not as fast as it seems. In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1780–1790, Seattle, United States. Association for Computational Linguistics.
* Helcl (2022) Jindřich Helcl. 2022. _Non-Autoregressive Neural Machine Translation_. Ph.D. thesis, Charles University.
* Junczys-Dowmunt et al. (2018) Marcin Junczys-Dowmunt, Roman Grundkiewicz, Tomasz Dwojak, Hieu Hoang, Kenneth Heafield, Tom Neckermann, Frank Seide, Ulrich Germann, Alham Fikri Aji, Nikolay Bogoychev, André F. T. Martins, and Alexandra Birch. 2018. Marian: Fast neural machine translation in C++. In _Proceedings of ACL 2018, System Demonstrations_ , pages 116–121, Melbourne, Australia. Association for Computational Linguistics.
* Lee et al. (2018) Jason Lee, Elman Mansimov, and Kyunghyun Cho. 2018. Deterministic non-autoregressive neural sequence modeling by iterative refinement. In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 1173–1182, Brussels, Belgium. Association for Computational Linguistics.
* Libovický and Helcl (2018) Jindřich Libovický and Jindřich Helcl. 2018. End-to-end non-autoregressive neural machine translation with connectionist temporal classification. In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 3016–3021, Brussels, Belgium. Association for Computational Linguistics.
* Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002. Bleu: a method for automatic evaluation of machine translation. In _Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics_ , pages 311–318, Philadelphia, Pennsylvania, USA. Association for Computational Linguistics.
* Rei et al. (2020) Ricardo Rei, Craig Stewart, Ana C Farinha, and Alon Lavie. 2020. COMET: A neural framework for MT evaluation. In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 2685–2702, Online. Association for Computational Linguistics.
|
# Identifying Different Layers of Online Misogyny
Wienke Strathern
School of Social Sciences and Technology
Technical University of Munich
80333 Munich, Germany
<EMAIL_ADDRESS>
Juergen Pfeffer
School of Social Sciences and Technology
Technical University of Munich
80333 Munich, Germany
<EMAIL_ADDRESS>
###### Abstract
Social media has become an everyday means of interaction and information
sharing on the Internet. However, posts on social networks are often
aggressive and toxic, especially when the topic is controversial or
politically charged. Radicalization, extreme speech, and in particular online
misogyny against women in the public eye have become alarmingly negative
features of online discussions. The present study proposes a methodological
approach to contribute to ongoing discussions about the multiple ways in which
women, their experiences, and their choices are attacked in polarized social
media responses. Based on a review of theories on and detection methods for
misogyny, we present a classification scheme that incorporates eleven
different explicit as well as implicit layers of online misogyny. We also
apply our classes to a case study related to online aggression against Amber
Heard in the context of her allegations of domestic violence against Johnny
Depp. We finally evaluate the reliability of Google’s Perspective API—a
standard for detecting toxic language—for determining gender discrimination as
toxicity. We show that a large part of online misogyny, especially when
verbalized without expletive terms but instead more implicitly is not captured
automatically.
_Keywords_ Online misogyny $\cdot$ Hate speech detection
## 1 Introduction
In May 2016 actress, model, and activist Amber Heard went public and accused
her then-husband, actor Johnny Depp, of intimate partner violence. She
described a turbulent relationship and reported that "Johnny verbally and
physically abused me throughout our
relationship"111https://www.chicagotribune.com/entertainment/ct-johnny-depp-
amber-heard-statement-20160531- story.html. She publicly posted a picture of
injuries and filed for divorce. This sparked a firestorm on social media and
online news sites, with commentators offering wildly differing opinions as to
what happened and who was to blame. Of course, it is not possible for an
outsider to know exactly what happened in this incident or what the dynamics
were in the relationship. However, many were quick to make accusations and
blame one or the other.
In recent years more attention has been paid to the role of women in society,
unfortunately also because of cases of real hatred against
them.222https://onlineviolencewomen.eiu.com/ According to the Pew Research
Center report on online harassment (Vogels, 2021), women and men are similarly
often abused or threatened online. However, women are more likely than men to
report being sexually harassed (16% vs. 5%) or stalked (13% vs. 9%) online.
Young women are particularly often affected by sexual harassment on the
Internet—33% of women under 35 say they have been sexually harassed online.
With the constant growth of social media and microblogging platforms, hatred
of women is becoming more prevalent, creating numerous examples of how
misogyny can spread almost uncontrolled (Jane, 2017b; Ging and Siapera, 2018,
2019).
Misogyny, defined as hatred or prejudice against women is expressed
linguistically in a variety of ways, including social exclusion,
discrimination, hostility, threats of violence, and sexual objectification. A
study reveals the sheer scale and nature of online abuse faced by women and
provides a resource to researchers and engineers interested in exploring the
potential of machine learning in content
moderation.333https://decoders.amnesty.org/projects/troll-patrol/findings In
order to handle hateful content and protect people, automated systems are
being used extensively to identify potentially problematic content. But a
series of Failure-to-Act reports uncovers the dark side of social media
platforms, more often experienced by women who are active on social media:
"how harassment, violent threats, image-based sexual abuse can be sent by
strangers, at any time and in large volumes, directly into your DMs without
consent and platforms do nothing to stop
it"444https://counterhate.com/research/hidden-hate/. Machine learning
algorithms are deployed to scan content and flag it for human moderators. For
instance, the Perspective API developed by Google Jigsaw was used to flag
potentially toxic content for review on Wikipedia and in the New York Times
comments section.555https://perspectiveapi.com/case-studies/ One challenge is
to capture the linguistic specifics of hate speech, polarizing and offensive
statements. Udupa (2020) observed that users of online social media platforms
have managed to bypass automatic hate speech detection methods by using
creative indirect forms of linguistic expression. According to Strathern et
al. (2022) alternative methods to recognize moral slurs could be successfully
implemented.
Since hate is expressed in many different ways, automated methods can lack
context sensitivity when determining implicit hate. To shed light on this
discrepancy, we first examine which scientific theories and methods deal with
the topic of misogyny. In the second step, we examine more closely how, based
on theory and empirical work, classes of misogyny are built according to which
content of hate speech can be assigned. In this, we assume that, in addition
to a large amount of explicit hate speech, there is also a significant
proportion of implicit misogynistic hate. Consequently, another goal of our
study is to examine how well automated approaches to detect toxic language can
identify misogyny. We collected 240,000 tweets from 2019–2021 containing the
tweet handle @realamberheard and selected the top 5,000 most retweeted tweets
to label and score them according to the classes identified in the literature.
We then had these 5000 tweets analyzed by the Google Perspective API toxicity
metric. A major outcome of this study is that online misogyny cannot be
satisfactorily identified with this automated toxicity identification tool.
## 2 Review of Theories and Methods on Misogyny
Our study is motivated by work dealing with a) misogyny, its modeling and
detection, b) the classification of hate speech and c) the verification of
hate speech detectors.
### 2.1 Misogyny
According to Allen (2021) a definition of misogyny is disputed. Studies
examining online anti-feminist rhetoric have used alternative terms, including
"gender cyber hatred" (Jane, 2017a), "cyber harassment" (Citron, 2014),
"technological violence" (Ostini and Hopkins, 2015), "gender trolling"
(Mantilla, 2013), "e-bile" and "gender hate speech" (Jane, 2015). According to
Code (2003), misogyny is defined as any of the following acts or feelings:
sexual violence against women, physical violence against women, exclusion of
women, promotion of patriarchy, belittlement of women, and marginalization of
women. This framework is supplemented by specific forms of online misogyny by
Zuckerberg (2018). Jane (2015) identifies technological determinism as one
paradigm of ‘flaming’. The author argues that flaming is a product of the
digital medium, favoured by specific features of online communication systems
such as anonymity, invisibility, and disinhibition. Further research on
flaming did not show that online abuse is highly gender-specific (Lee, 2016).
In this analysis, reactions to ‘flaming’ or inflammatory messages refer to
expletives targeting women under the general heading of ‘insult’ rather than
being placed within the broader framework of structural misogyny. In contrast,
Herring and Martinson (2004) examined gender differences in communication
styles and feminist responses to ‘trolling’ and found that the ‘gender nature’
of online abuse messages and hate speech is paramount. Online misogyny has
offline effects that requires investigation and further research. Citron and
Norton (2011) hypothesize that the gendered nature of online harassment and
digital abuse is an important facet of women’s overall ‘digital citizenship’.
Megarry (2014) studies the psychological consequences of online misogyny, such
as pseudonymous involvement and withdrawal, noting that online misogyny limits
women’s voice and presence and reduces their digital engagement.
The case of Amber Heard was the subject of a study by Whiting et al. (2019).
They conducted their study from a psychological perspective on the subject of
domestic violence. The authors examined the commenting behavior of users on
various social media platforms. To better understand typical types of social
media reactions to allegations of domestic violence, the authors performed a
content analysis on Facebook. Five main categories were extracted, namely
victim blaming, perpetrator blaming, couple blaming, withholding judgment, and
mixed reactions to the process. The respective main topics also contain
subtopics on reactions to the allegations.
### 2.2 Modeling Misogyny
Determining and classifying misogyny in comments is a major challenge for
humans and computers. There are various definitions and approaches to modeling
this complex social and linguistic phenomenon. Fersini et al. (2018) developed
a machine learning classification approach to model misogyny. The main
categories are based on gender studies theory and contain classes that are
used to determine comments. The classes are: stereotyping and objectification,
dominance, derailment, sexual harassment, threats of violence, and
discrediting. The categorization starts after an a priori distinction of
whether a tweet is classified as misogynistic or not. In a study by Farrell et
al. (2019) a misogyny model was developed to examine the flow of extreme
language in online communities on Reddit. Based on feminist language
criticism, the author created nine lexica that capture specific misogyny
rhetoric (physical violence, sexual violence, hostility, patriarchy, stoicism,
racism, homophobia, disparagement, and inverted narrative), and used these
lexica to examine how language evolves within and between misogynist groups.
Recent work by Guest et al. (2021) presents a hierarchical taxonomy for online
misogyny and an expert-labeled data set that allows automatic classification
of misogyny content. The taxonomy consists of misogynistic content, broken
down into misogynistic pejoratives and treatment, misogynistic disparagement,
and gendered personal attacks.
### 2.3 Detecting Online Misogyny
In addition to modeling misogyny and detecting hate speech, we find studies
examining how politically and socially active women are treated in current
public debates. To gain insight into gender discrimination, various automated
methods are used. In a study by Rheault et al. (2019), the authors applied
machine learning models to predict rudeness directed at Canadian politicians
and US senators on Twitter. In particular, they test whether women in politics
are more affected by online abuse, as recent media reports suggest. Another
article by Beltran et al. (2021) examined gender insults towards Spanish
female politicians. In an analysis of tweets written by citizens, the authors
found evidence of gender slurs and note that mentions of appearance and
infantilizing words are disproportionately common in texts addressing female
politicians in Spain. The results show how citizens treat politicians
differently depending on their gender. Fuchs and Schäfer (2021) presented the
results of an exploratory analysis of misogynistic and sexist hate speech and
abuse against female politicians on Twitter, using computer-assisted corpus
linguistic tools and methods, supplemented by a qualitative in-depth study of
abuse by four prominent female politicians in Japan. Studies by Bauer (2015),
Ditonto et al. (2014), Herrnson et al. (2003), Lawless (2015) suggest that
voters evaluate candidates from the perspective of gender stereotypes and test
how this affects attitudes and voting behavior.
### 2.4 Hate Speech Classification
The annotation of hate speech is important for automated classification tasks.
The classification scheme and its underlying assumptions are crucial for
annotation. There are different approaches to this process such as predefined
word lists or more complex models. One of the main difficulties is the
definition of hate speech and its interpretation and therefore correct
application. Recently, the Gab Hate Corpus was published (Kennedy et al.,
2022), which uses a specially developed coding typology for annotating hateful
comments. It was developed based on a synthesis of hate speech definitions
drawn from legal precedents, previous hate speech coding typologies, and
definitions from psychology and sociology. In addition, it contains a
hierarchical clustering for dehumanizing and violent speech, as well as
indicators for target groups and rhetorical peculiarities. A study by Ben-
David and Fernández (2016) examined how overt hate speech and covert
discriminatory practices circulate on Facebook. They argue that hate speech
and discriminatory practices are not only explained by the motivations and
actions of the users, but also emerge through a network of connections between
the platform’s politics, its technological capabilities, and the communicative
actions of its users. The difficult task of capturing implicit and explicit
statements was addressed in a study by Gao et al. (2017). The authors proposed
a weakly supervised two-path bootstrapping approach for an online hate speech
detection model that uses big unlabeled data to address several limitations of
supervised hate speech classification methods, such as corpus bias and the
enormous cost of annotation. The implicitness of linguistic statements is also
the subject of a work by Frenda et al. (2022). The authors proposed a number
of statistical and computational analyses that support reflections on indirect
propositions that focus on the creative and cognitive aspects of implicitness.
In a more recent work by ElSherief et al. (2021), implicit statements were
used for machine learning tasks to introduce a theoretically based taxonomy of
hate speech. Wiegand et al. (2021) studies the detection of implicitly abusive
language, that is, abusive language that is not conveyed by abusive words. In
their position paper, they explain why existing datasets make learning
implicit abuse difficult and what needs to be changed in the design of such
datasets.
### 2.5 Bypassing Hate Speech Detection
Tricking or recalibrating automated methods results from the observation that
the underlying assumptions of common machine methods do not adequately define
group-specific hatred. That is, there seems to be a discrepancy between
methods for operationalization tasks and the complexity of social processes.
Against this background there are ways to trick hate speech detection methods
or to test them for their measurement accuracy and validation. Both, cultural
and associated linguistic peculiarities are thus taken into account. There are
studies that try to capture culture- and language-specific hatred, which
machines have difficulty recognizing. Zannettou et al. (2020) focused on
examining the spread of antisemitic content. The authors carried out a large-
scale quantitative analysis to discover abnormalities in language use. The
results show that there are several distinct facets of antisemitic language,
ranging from slurs to conspiracy theories, drawing on biblical literature and
narratives expressed differently in the language. Herein, antisemitism is
considered as one form of hate speech and the authors developed a method to
comply with this. Another study by Gröndahl et al. (2018), evaluates
empirically previously proposed models and datasets for classifying hate
speech. The results show that none of the pre-trained models performs well
when tested with a different dataset. The authors assert that the
characteristics indicative of hate speech are not consistent across different
datasets. The results show that the definitions of hate speech do not seem to
be consistent and that they need further differentiation and context
sensitivity. Another study by Hiruncharoenvate et al. (2015) examined ways to
circumvent the observation of the state in the Chinese language, which
suppresses free speech. In China, political activists use homophones (two
words that are written differently and have different meaning but sound the
same, e.g., brake/break) of censored keywords to avoid detection by keyword-
matching algorithms. The authors claim that it is possible to expand this idea
in a way that makes them difficult to counteract. One result of this work is
to mathematically (and almost optimally) change the content of a post by
replacing censored keywords with homophones. So, by tricking the system with
linguistic creativity, they bypass the derived rules for automatic speech
recognition on Weibo.
## 3 Overview of Misogyny Classes from the Literature
Based on the theories and methods discussed above, we have developed a
classification scheme for online misogyny that covers most of the aspects
discussed in the related literature. These classes include explicit and
implicit misogynistic language and are presented in the following. Some of
these classes are close to each other in their definitions and are not always
easy to distinguish. The case study in the second part of this manuscript will
show that they significantly overlap when used for coding real-world messages.
The goal of identifying misogyny classes was not to identify unambiguous
definitions, but to cover a wide variety of aspects of hate against women.
### 3.1 Explicit Misogyny
In explicit misogynistic statements users openly attack, insult, or even
threaten a woman (Waseem et al., 2017; Gao et al., 2017). Based on the
literature presented above, we have identified the following four
subcategories of explicit misogyny.
Call for action/violence. This class implies verbal threads that intend to
punish a target physically. Statements in which users call for deletion,
prison, boycott, or sending the target to a psychiatric institution (Fersini
et al., 2018).
Personal insult, denigration. Personal insults and denigration intend to cause
harm to a target verbally. Statements containing harmful wishes, demeaning,
threatening, denigrating, inciting, defaming, use of slur words (Fersini et
al., 2018; Guest et al., 2021; Farrell et al., 2019).
Gendered personal attack. Gendered personal attacks refer to stereotypes of
women. From these stereotypes, verbal (misogynistic) attacks derive.
Statements that contain misogynistic speech and swearwords, revenge porn, or
are sexually motivated because the target is a woman (Fersini et al., 2018;
Guest et al., 2021; Farrell et al., 2019).
Weakness of character, intellectual inferiority. Making negative judgments of
a woman’s moral and intellectual worth using explicit slur words. Statements
that call a woman controlling, psychotic, a liar, hypocritical, narcissistic,
or manipulative (Fersini et al., 2018; Guest et al., 2021; Farrell et al.,
2019).
### 3.2 Implicit Misogyny
Implicit statements of misogyny include cynicism and sarcasm, skepticism and
distrust, insinuation, accusations, speculation and questioning of
credibility, a demonstration of power, and taking a position (Waseem et al.,
2017; Gao et al., 2017; ElSherief et al., 2021; Frenda et al., 2022).
Cynicism, sarcasm. Cynicism and sarcasm represent a very derogatory attitude
of a person towards others. It is expressed in an indirect form and is
spiteful and bitter. Contextual knowledge is needed. Statements in which in a
subliminal way, a rejecting attitude is shown (Whiting et al., 2019).
Skeptical attitude, distrust. That includes “facts” or other details to
undermine a woman’s account. Doubtfulness toward a woman’s claims or
accusations. Questions whether the target had lied before and therefore cannot
be trusted (Whiting et al., 2019).
Imputation. Imputation is considered as assumptions that the target behavior
is motivated by flawed motivations. That includes statements that show a moral
judgment, containing comments where a woman is described as revenge-seeking,
vindictive, attention-seeking, monetarily driven (Whiting et al., 2019).
Allegation. The category implies actions in which the evidence and allegations
are challenged suggesting intentionally motivated actions. Statements of users
that offer facts that refute a woman’s account, evidence to the contrary
(Whiting et al., 2019).
Speculation, denying credibility. This category includes an investigative-
style attitude. Speculations and doubts about the target’s behavior.
Commenting on the case, e.g., of domestic violence and its severity, we find
claims about how this case might affect future reporting, users offering life
stories to undermine the target’s account, personal expertise, intent to prove
something, credibility from experience, claiming special predictive power
(Whiting et al., 2019).
Demonstration of power. The category implies a power relation between one
gender and the other. Statements in which support for the man is demonstrated
(Fersini et al., 2018).
Taking position. Taking position or ‘flipping the narrative’ encapsulates
terms and expressions that refer to the relationship between the target and
the perpetrator. Statements on who is the ‘perpetrator’ and who is the
‘victim’ (Fersini et al., 2018; Guest et al., 2021; Farrell et al., 2019).
### 3.3 Examples for Misogyny Classes
In order to study the prevalence of these misogynistic classes on social
media, we have collected and analyzed messages addressing Amber Heard’s
Twitter account @realamberheard in a case study in the next section. Here, in
Table 1, we show sample tweets to exemplify these classes. Since the content
contains explicit hate speech and profanity, we have redacted the texts.
Class | Example Tweet
---|---
Call for action/violence | Oh @realamberheard …. You ignorant witch. We ALL already know you’re the guilty one here.
| Johnny’s innocence has been proven. You’re just trying to buy time, before
you (hopefully)
| have you sit your scronny ass in a jail cell. You speak nothing but venomous
lies. #JohnnyDepp
Personal insult, denigration | Seriously, how fucking sick you have to be to pull a "prank" like this on someone ? What kind of
| gross bitch would think pooping in people’s bed is funny ? Well, apparently
@realamberheard does.
| #JusticeForJohnnyDepp
Gendered personal attack | Not a johnny Depp fan but @realamberheard claims have more holes than swiss cheese. I dont
| understand females who can’t make their own money and want to pocket off
someone elses. It’s
| hard to find a victim that no one sides with in todays world but I think we
all call bs on AH.
Weakness of character, | Look what headline just poped up on sky news! @realamberheard you dirty little Lier!
intellectual inferiority | #AmberHeardIsALiar #JusticeForJohnnyDepp
Cynicism, sarcasm | @realamberheard Yes, the excitement around #JusticeLeague was huge … definitely nothing to do
| with you though. Imagine being in a 4 hour movie for 5 minutes and being the
most insufferable
| part of it.
Skeptical attitude, distrust | I just noticed the ’actor/ activist’ claims in your biog @realamberheard !! Well, you certainly
| are an actress for real!! Only trouble is that the majority of your acting
seems to be done OFF
| stage!! And you have set ’activism’ back decades dear!! Ugh, you are some
piece of work!
Imputation | @realamberheard @realamberheard Put your hand down and stop exploiting Evan’s story to sway
| the public perception back in your favor. Don’t act like you didn’t break
bread and hang out with
| Marilyn Manson for years after his relationship with ERW/ your o
Refutation | Listen bitch, I just saw a video about you demanding Depp supporter info for some legal implica-
| tions!!If you want any info about me just DM me and I’ll be MORE than happy
to bring you upto
| speed!! @realamberheard I am allowed my opinion and you are scum (&u better
pay my airfare!)
Speculation, | @realamberheard You do not represent women nor survivors. I stand with Johnny Depp, Kate
denying credibility | James, Jennifer Howell, Lily-Rose Depp, Hilda Vargas, Samantha McMillen, Katherine Kendall,
| Trinity Esparza and ALL THE OTHER women and men who knows your true color
Demonstration of Power | Justice for Johnny Depp outside @wbpictures studio where @realamberheard is currently filming
| @aquamanmovie #JohnnyDepp #JusticeForJohnnyDepp #JOHNNY #AmberHeard
Taking a position | @realamberheard is not a victim, she is the perpetrator.
Table 1: Misogynic classes and example tweets
## 4 Case Study
To assess the importance of the misogyny classes presented in the manuscript,
we conducted a case study using Twitter data related to the celebrity domestic
violence abuse case between Amber Heard and Johnny Depp. In the following, we
describe the data and the annotation process as well as present quantitative
results showing the prevalence of our explicit and implicit misogyny classes
in the data.
Kennedy et al. (2022) documented that the annotation of hate speech has been
shown to lead to a high level of disagreement between the annotators, see also
Ross et al. (2016). According to Mostafazadeh Davani et al. (2020) this is due
to a combination of factors, including differences in understanding of the
definition of hate speech, interpretation of the annotated texts, or
assessment of the harm done to certain groups, i.e. inconsistent application
of the definition of hate speech to different social groups.
Data. By utilizing the Twitter Academic API (Pfeffer et al., 2023) we
collected 266,579 original tweets (excluding re-tweets) in January of 2022
that contained the account @realamberheard in the tweet texts. This resulted
in 266,579 tweets (2019: 64,334 tweets, 2020: 117,231 tweets, 2021: 85,014
tweets). For the annotation process, we extracted 5,000 tweets that have been
retweeted most often.
### 4.1 Annotation Process
For our case study we employed two annotators, a graduate student who is also
a co-author on this paper and was instrumental in developing the misogyny
classes (annotator $1$), as well as an undergraduate student who was new to
the topic (annotator $2$). The annotators were briefed with an introduction to
the topic in general and then presented with the misogyny classes. All the
information presented together with coding examples was also shared in a
coding manual. The manual also includes detailed descriptions of the
individual coding steps and further explanations of the definition of the
classes and the coding method according to the literature.
We analyzed the entire tweet at the sentence and word level, including the use
of emoticons and content on the websites following URLs appearing in tweets.
We looked at images, memes, or quotes, and watched linked videos. Each tweet
was rated by the annotators based on all of its content. If the tweet
contained statements supporting Amber Heard was neutral, or contained
advertising, we annotated this tweet as _other_ and ignored the tweet in the
subsequent analytical steps. We used the eleven misogyny classes for
annotation. After the annotation process, we created the explicit/implicit
annotation from the eleven classes following the categorization described
above. A single tweet could be annotated with multiple misogyny classes. If a
tweet contained multiple sentences where one was implicit and one was
explicit, we chose the explicit class due to the fact that a Tweet with
explicit misogynistic content will be perceived as being explicit in its
entirety.
Coding 11 classes with multiple overlapping definitions will lead to low
levels of completely identical annotations. However, when comparing the
explicit/implicit/other classes among the two annotators, the overall level of
agreement between the annotators was acceptable. We can report the following
values for Krippendorff’s alpha (Krippendorff, 2011): explicit 0.779, implicit
0.736, other 0.867.
### 4.2 Prevalence of the Misogyny Classes
For further analysis of this article, the annotator $1$ manually compared the
annotations from both annotators for all 5,000 tweets and harmonized the
annotations into a single mapping of tweets to misogyny classes. The
frequencies and proportions of the classes in the overall dataset as well as
in the misogynistic tweets can be seen in Table 2. Shockingly, two-thirds of
the most retweeted tweets addressing Amber Heard’s Twitter account have been
classified into explicit (35.6%) or implicit (30.3%) classes of misogyny.
While explicit and implicit classes can overlap within tweets, the meta-
classes explicit/implicit are mutually exclusive (see above).
| Misogyny Class | Frequency | All | Misogyny
---|---|---|---|---
Explicit | Call for Action | 681 | 13.6% | 20.4%
(35.6%) | Personal Insult | 1,649 | 33.0% | 49.5%
| Gendered Personal Attack | 730 | 14.6% | 21.9%
| Intellectual Inferiority | 1,325 | 26.5% | 39.8%
Implicit | Cynicism/Sarcasm | 367 | 7.3% | 11.0%
(30.3%) | Scepticism/Distrust | 461 | 9.2% | 13.8%
| Imputation | 556 | 11.1% | 16.7%
| Allegation | 546 | 10.9% | 16.4%
| Speculation | 305 | 6.1% | 9.2%
| Demonstration of Power | 459 | 9.2% | 13.8%
| Taking up a Position | 181 | 3.6% | 5.4%
N | | | 5,000 | 3,331
Table 2: Frequencies and proportions of misogyny classes in all 5,000
annotated tweets as well as proportions in 3,331 misogynistic tweets.
## 5 Comparing Misogyny Classes with Google’s Perspective API
Google’s Perspective API is one of the standards for identifying toxic
language on online platforms and is described as "the product of a
collaborative research effort by Jigsaw and Google’s Counter Abuse Technology
team exploring machine learning as a tool for better discussions
online."666https://www.perspectiveapi.com/research/. In this section, we will
test how well the toxicity scores of this API are capable of identifying
online misogyny as operationalized with our eleven classes to get an
understanding of how useful these approaches can be in automatically
identifying online misogyny.
We worked out the different attributes and evaluation methods of the API as
the first step for comparison. In the second step, we applied the API to the
same dataset of 5,000 tweets. For each tweet, the API specifies a range of
values for each of its categories. In the third step, we compared the values
using statistical methods and applied network analysis to show the co-
occurrence of classes and their average toxicity value reported by the
Perspective API.
### 5.1 Attributes of Perspective API
The Perspective API predicts the perceived impact of a comment on a
conversation by evaluating that comment with a set of emotional concepts
called attributes, namely toxicity, severe toxicity, identity attack, offense,
threat, and profanity. The returned values are in the range [0.1] and are an
indicator of the likelihood that something will be perceived as toxic. The
higher the score, the more likely it is that the patterns in the text are
similar to the patterns in comments that others have identified as toxic. The
values are intended to allow developers/users to set a threshold and ignore
values below that value. Values around $0.5$ indicate that the model does not
know if it is similar to toxic comments. The Google recommended threshold
setting is $0.7$. These thresholds are central to interpretation.
| Misogyny Class | Average Toxicity
---|---|---
Explicit (35.6%) | Call for Action | 0.504
ø= 0.572 | Personal Insult | 0.589
| Gendered Personal Attack | 0.619
| Intellectual Inferiority | 0.577
Implicit (30.3%) | Cynicism/Sarcasm | 0.356
ø= 0.493 | Scepticism/Distrust | 0.527
| Imputation | 0.557
| Allegation | 0.423
| Speculation | 0.572
| Demonstration of Power | 0.436
| Taking up a Position | 0.581
Other (34.1%) | Marketing/PR | 0.193
Table 3: Categories and Average Toxicity for Explicity and Implicity.
### 5.2 Measuring Toxicity for Misogyny Classes
To measure the average toxicity for the misogyny classes, we compare Google’s
probability score to our manual coding by summing up the codes divided by the
number of tweets in each meta-class. The results show that the average
toxicity score by Google for our category of explicit misogyny is $0.572$. For
our category of implicit misogyny, the average score by Google is $0.493$.
These numbers already are a strong indicator that toxicity, as identified with
the Perspective API, is a poor predictor of our variable of online misogyny,
and in particular of implicit hate against women. Table 3 reveals the average
toxicity scores for each class. In Figure 1 we can further see the density
distribution of toxicity scores for each of the meta-classes of tweets with
explicit or implicit misogyny as well as others.
Figure 1: Distribution of toxicity scores from Google’s Perspective API for
tweets with explicit or implicit misogyny as well as tweets without
misogynistic content.
In the _other_ sub-figure we can clearly see that there are almost no tweets
that have been identified by the Perspective API as toxic that we have not
also classified as misogynistic—consequently, the automated coding does not
create false positives. The explicit language used for the classes that we
have summarized with the meta-class _explicit_ can be identified by the
Perspective API to a certain degree, and the peak of the score distribution is
above the standard threshold of $0.7$. In other words, tweets coded with
explicit misogyny contain text patterns that are similar to the patterns in
comments that have been identified as toxic when the Perspective API models
have been trained.
Unfortunately, the picture looks different when looking at the distribution of
scores for the implicit misogyny classes. Here, the resulting toxicity scores
are almost evenly distributed, having more scores with very low values than
with very high values. Consequently, the tweets coded with implicit
misogynistic classes do not reflect text patterns that are similar to the
patterns that have been identified as toxic in the Perspective API’s training
data.
### 5.3 Co-Occurrence Network of Misogyny Classes
In addition to statistical analysis, we built a co-occurrence network that
maps manual coded classes and the average toxicity scores by the Perspective
API (3). Nodes represent the eleven classes and the edge value is the number
of co-occurrences, i.e., the co-occurrence of classes within a tweet. The edge
color is the edge value, and the node size is the proportion of the number per
code divided by the number of tweets. The node color is the average toxicity
value from the Perspective API where blue means low and red means high
toxicity values.
In the centre, we can find the dominant four explicit classes which are
identified to a certain degree as being toxic. The classes are well connected
with each other. Explicit abusive statements come with similar forms of
abusive language. For implicit statements, the picture looks different. In the
periphery, we can find the seven classes of our meta-class implicit. Implicit
misogynistic statements occur more with various forms of explicit abusive
language and less among each other. In many cases, something is said
implicitly, but it co-occurs with an explicit abusive statement. As mentioned
above, we decided to code a tweet as explicit if both classes occurred. But
the network analysis reveals the co-occurrence of explicit and implicit
abusive language against women within one statement. It offers a more
qualitative comparison of stereotypical hating: statements that contain a
demonstration of power come with inferiority and insults. A skeptical attitude
comes with abusive terms of inferiority, imputation, gendered personal
attacks, and insults. Statements of speculation and doubt come with sarcastic
and gender-attacking language. Despite the proximity of all classes, the
network reveals a distinction between explicit and implicit misogyny.
Figure 2: Co-Occurrences of Categories within a Tweet
### 5.4 Interpretation and Discussion
We asked how well an automated approach like Google’s Perspective API performs
in detecting misogyny. Based on our study, two things become apparent:
Google’s text model does recognize explicit misogyny in the text patterns as
toxic. However, the model does not recognize implicit misogyny in text
patterns as toxic. The interpretation of the following tweets underlines the
challenges of detecting and understanding implicit/indirect hate:
"@realamberheard It’s the way you think that posing this is going to change
public perception of you. We heard what you did in your own words. A failure
in the system isn’t uncommon, so thank you for proving that male victims will
never be taken seriously." A user recapitulates what has happened, draws
conclusions for men, and thanks the target person for that in a very calm
manner. But reading the tweet with contextual knowledge makes one understand
that the thankful gesture is a cynical one. No keyword of hate can be found
here; the words are all positive, but the underlying assumption is an
accusation against Amber Heard and against her gender. None of the scores
indicates harm in this tweet: Toxicity: 0.28, Severe Toxicity: 0.17, Identity
Attack: 0.26, Offense: 0.07, Threat: 0.21 and Profanity: 0.14.
In another tweet, a user comments on what has happened and concludes that this
behavior is not acceptable. The tweet contains a link to a screenshot in which
impressions of what happened are reflected. Again, there is no harmful word,
it all sounds positive in isolation, but clearly implies that this user is
rejecting the behavior of the woman and at the same time accusing her of what
she has done: "@realamberheard I had to translate to really understand where
you’re coming from. And no I wouldn’t encourage my daughter or sister to do
what you did (URL redacted)". But here as well, the scoring is very low.
Toxicity: 0.20, Severe Toxicity: 0.12, Identity Attack: 0.11, Offense: 0.07,
Threat: 0.16 and Profanity: 0.14.
The following example can exemplify how the toxicity score can be influenced
by a single word that is interpreted as negative, even though the tweet could
be interpreted as being funny: "@realamberheard @USNatArchives She will
forever be known as the lady who pooped on Johnny Depp’s bed." Toxicity: 0.69,
Severe Toxicity: 0.15, Identity Attack: 0.74, Offense: 0.65, Threat: 0.34, and
Profanity: 0.74.
There may be several reasons for this discrepancy to detect misogyny. One
reason could be that there was no misogynistic content in the training texts
for the human annotators. Or misogyny was never defined as an annotation
class, hence, annotator could not label it. Annotators could not be informed /
trained on the topic of misogyny and, therefore, could not recognize and
annotate it in the texts. Although we do not know how the data sets were
constructed and the model trained, we can summarize that Google’s Perspective
API struggles with identifying text patterns containing implicit misogynistic
statements.
## 6 Conclusion
In this manuscript, we have presented a classification scheme that
incorporates 11 classes of misogyny and have described a data set that
contains misogynistic content labels from Twitter. We have also provided a
detailed coding book and a data set with all of the labels. The data set
benefits from a detailed classification scheme based on the existing
literature on online misogyny. The involvement of trained annotators and an
adjudication process also ensures the quality of the labels.
We applied the classification scheme to a case related to online aggression
against Amber Heard in the context of her allegations of domestic violence
against Johnny Depp. For 5,000 tweets, we identified online misogyny
operationalized with our eleven classes for two-thirds of the tweets, one-
third as explicit misogyny, and one-third as implicit misogyny. Finally, we
evaluated the reliability of Google’s Perspective API for determining implicit
misogyny and found that this approach can identify explicit misogyny to a
certain extent, but fails with identifying implicit misogyny.
Ethical considerations and limitation. Ethical considerations must be taken
into account with regard to the training and supervision of the annotator. The
annotator was an undergraduate student who first read the typology and coding
manual and secondly performed a test of approximately 50 messages that had
been previously annotated and approved by one of the authors. Kennedy et al.
(2022) pointed out the pressing concern that annotators may experience trauma
or similar negative effects such as desensitization when annotating hate
speech. On the basis of our own annotation experiences, we would like to
highlight these thoughts. Although no research has examined the effects of
constant daily exposure to hate speech on human moderators, there is evidence
that exposure to linguistic and visual violence online can have negative
mental health effects, as shown by Kwan et al. (2020). We also provided the
annotator with Kennedy’s suggested written guide 777https://www.apa.org/ptsd-
guideline/ptsd.pdf to help detect changes in cognition and avoid secondary
trauma. It advises the user to take breaks and not imagine traumatic
situations. The annotator was asked to stay in contact with the author of the
study if she notices symptoms of PTSD, which are also listed in the guide. The
purpose of this guide is to normalize feelings that are negatively impacted by
work, provide trauma-specific education, recognize signs of traumatic stress
and provide a mechanism for support as a preventative measure against
secondary traumatic stress.
A limitation of this study is the fact that we do not know whether the
Perspective API’s text models contained misogynistic content and we do not
know whether the data sets contained implicit/indirect forms of hate.
Furthermore, we do not know whether the annotators were informed or trained on
the topic of misogyny or implicit/indirect forms of hate. However, our results
show that there may be a lack of information on misogyny according to existing
definitions.
Implications and Future Work. Given real-world online aggression against
women, it is probable that Google’s toxicity model would not identify it.
Thus, a huge fraction of implicit misogyny texts would stay visible and would
not be deleted or otherwise acted upon. Misogynous behavior and target
classification still remain a very challenging problem. One approach may be to
create lexicons capturing specific misogynistic rhetoric and improve
annotation scheme. Another challenge is to capture the peculiarities of
implicity or indirect forms of hate in language. Language is very context-
sensitive, and a negative tone can be expressed without a clear negative key
word. Moreover, implicit sentences depend decisively on the non-linguistic
accompanying signals. With our work, we would like to enhance existing
research on investigating linguistic distinction between implicit and group-
specific hate rhetoric. Furthermore, as we have seen from the network
perspective, aside from the technical solution questions arise on how and why
these different sub-classes are closely connected. From a gender perspective,
we ask why are these stereotypes so consistent over time?
Due to the still increasing number of users and posts in social media,
automated annotation based on machine learning is inevitable. There is no
other way to handle the sheer volume of text. At the same time, it becomes
apparent that the proportion of aggressive misogynistic speech is increasing
sharply. An assessment and, if necessary, the deletion of unacceptable
statements is imperative for the protection of people. Especially with regard
to women, their protection is of immense importance to enable participation in
public discourse and avoid withdrawing because of fear of being attacked or
marginalized. However, the key to better handling the problem is to better
understand the phenomenon of misogyny.
## 7 Acknowledgments
The authors gratefully acknowledge the financial support from the Technical
University of Munich - Institute for Ethics in Artificial Intelligence (IEAI).
Any opinions, findings and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the IEAI or its partners. The labeled dataset and the coding book will be
made publicly available.
## References
* (1)
* Allen (2021) Amy Allen. 2021\. Feminist Perspectives on Power. In _The Stanford Encyclopedia of Philosophy_ , Edward N. Zalta (Ed.). Metaphysics Research Lab, Stanford University.
* Bauer (2015) Nichole M. Bauer. 2015\. Emotional, Sensitive, and Unfit for Office? Gender Stereotype Activation and Support Female Candidates. _Political Psychology_ 36 (2015).
* Beltran et al. (2021) Javier Beltran, Aina Gallego, Alba Huidobro, Enrique Romero, and Lluís Padró. 2021. Male and female politicians on Twitter: A machine learning approach. _European Journal of Political Research_ 60 (2021), 239–251.
* Ben-David and Fernández (2016) Anat Ben-David and Ariadna Matamoros Fernández. 2016\. Hate Speech and Covert Discrimination on Social Media: Monitoring the Facebook Pages of Extreme-Right Political Parties in Spain. _International Journal of Communication_ 10 (2016).
* Citron (2014) Danielle Keats Citron. 2014\. Hate Crimes in Cyberspace - Introduction. _Harvard University Press_ (2014).
* Citron and Norton (2011) Danielle K. Citron and Helen Norton. 2011. Intermediaries and Hate Speech: Fostering Digital Citizenship for Our Information Age. _Boston University Law Review_ 91 (2011).
* Code (2003) Lorraine Code. 2003\. _Encyclopedia of Feminist Theories_. Routledge.
* Ditonto et al. (2014) Tessa M. Ditonto, Allison J. Hamilton, and David P. Redlawsk. 2014. Gender Stereotypes, Information Search, and Voting Behavior in Political Campaigns. _Political Behavior_ 36 (2014), 335–358.
* ElSherief et al. (2021) Mai ElSherief, Caleb Ziems, David Muchlinski, Vaishnavi Anupindi, Jordyn Seybolt, Munmun De Choudhury, and Diyi Yang. 2021\. Latent Hatred: A Benchmark for Understanding Implicit Hate Speech. In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_. Association for Computational Linguistics, 345–363.
* Farrell et al. (2019) Tracie Farrell, Miriam Fernandez, Jakub Novotny, and Harith Alani. 2019. Exploring Misogyny across the Manosphere in Reddit. In _Proceedings of the 10th ACM Conference on Web Science_.
* Fersini et al. (2018) E Fersini, P Rosso, and M Anzovino. 2018. Overview of the Task on Automatic Misogyny Identification at IberEval 2018. In _EVALITA@CLiC-it_.
* Frenda et al. (2022) Simona Frenda, Viviana Patti, and Paolo Rosso. 2022\. Killing me softly: Creative and cognitive aspects of implicitness in abusive language online. _Natural Language Engineering_ (2022). Publisher: Cambridge University Press.
* Fuchs and Schäfer (2021) Tamara Fuchs and Fabian Schäfer. 2021. Normalizing misogyny: hate speech and verbal abuse of female politicians on Japanese Twitter. _Japan Forum_ 33, 4 (2021).
* Gao et al. (2017) Lei Gao, Alexis Kuppersmith, and Ruihong Huang. 2017\. Recognizing Explicit and Implicit Hate Speech Using a Weakly Supervised Two-path Bootstrapping Approach. In _Proceedings of the Eighth International Joint Conference on Natural Language Processing_.
* Ging and Siapera (2018) Debbie Ging and Eugenia Siapera. 2018. Special issue on online misogyny. _Feminist Media Studies_ 18 (2018).
* Ging and Siapera (2019) Debbie Ging and Eugenia Siapera (Eds.). 2019. _Gender Hate Online: Understanding the New Anti-Feminism_. Springer International Publishing.
* Gröndahl et al. (2018) Tommi Gröndahl, Luca Pajola, Mika Juuti, Mauro Conti, and N. Asokan. 2018. All You Need is "Love": Evading Hate Speech Detection. In _Proceedings of the 11th ACM Workshop on Artificial Intelligence and Security_. 2–12.
* Guest et al. (2021) Ella Guest, Bertie Vidgen, Alexandros Mittos, Nishanth Sastry, Gareth Tyson, and Helen Margetts. 2021. An Expert Annotated Dataset for the Detection of Online Misogyny. In _Association for Computational Linguistics_. 1336–1350.
* Herring and Martinson (2004) Susan C. Herring and Anna Martinson. 2004. Assessing Gender Authenticity in Computer-Mediated Language Use: Evidence From an Identity Game. _Journal of Language and Social Psychology_ 23, 4 (2004). Publisher: SAGE Publications Inc.
* Herrnson et al. (2003) Paul S. Herrnson, J. Celeste Lay, and Atiya Kai Stokes. 2003\. Women Running “as Women”: Candidate Gender, Campaign Issues, and Voter-Targeting Strategies. _The Journal of Politics_ 65, 1 (2003).
* Hiruncharoenvate et al. (2015) Chaya Hiruncharoenvate, Zhiyuan Lin, and Eric Gilbert. 2015\. Algorithmically Bypassing Censorship on Sina Weibo with Nondeterministic Homophone Substitutions. _Proceedings of the International AAAI Conference on Web and Social Media_ 9 (2015), 150–158. Number: 1.
* Jane (2017a) Emma Jane. 2017a. Gendered cyberhate: A new digital divide? In _Theorizing Digital Divides_. Routledge.
* Jane (2017b) Emma Jane. 2017b. _Misogyny Online: A Short and (Brutish) History_. Sage Publications.
* Jane (2015) Emma A. Jane. 2015\. Flaming? What flaming? The pitfalls and potentials of researching online hostility. _Ethics and Information Technology_ 17 (2015), 65–87.
* Kennedy et al. (2022) Brendan Kennedy, Mohammad Atari, Aida Mostafazadeh Davani, Leigh Yeh, Ali Omrani, Yehsong Kim, Kris Coombs, Shreya Havaldar, Gwenyth Portillo-Wightman, Elaine Gonzalez, Joe Hoover, Aida Azatian, Alyzeh Hussain, Austin Lara, Gabriel Cardenas, Adam Omary, Christina Park, Xin Wang, Clarisa Wijaya, Yong Zhang, Beth Meyerowitz, and Morteza Dehghani. 2022\. Introducing the Gab Hate Corpus: defining and applying hate-based rhetoric to social media posts at scale. _Language Resources and Evaluation_ 56 (2022).
* Krippendorff (2011) Klaus Krippendorff. 2011\. _Computing krippendorff’s alpha-reliability_. Retrieved from: https://repository.upenn.edu/ascpapers/43.
* Kwan et al. (2020) Irene Kwan, Kelly Dickson, Michelle Richardson, Wendy MacDowall, Helen Burchett, Claire Stansfield, Ginny Brunton, Katy Sutcliffe, and James Thomas. 2020\. Cyberbullying and Children and Young People’s Mental Health: A Systematic Map of Systematic Reviews. _Cyberpsychology, Behavior and Social Networking_ 23, 2 (2020).
* Lawless (2015) Jennifer L. Lawless. 2015\. Female Candidates and Legislators. _Annual Review of Political Science_ 18 (2015).
* Lee (2016) Hangwoo Lee. 2016\. Behavioral Strategies for Dealing with Flaming in An Online Forum. _The Sociological Quarterly_ 46, 2 (2016), 385–403.
* Mantilla (2013) Karla Mantilla. 2013\. Gendertrolling: Misogyny Adapts to New Media. _Feminist Studies_ 39, 2 (2013), 563–571.
* Megarry (2014) Jessica Megarry. 2014\. Online incivility or sexual harassment? Conceptualising women’s experiences in the digital age. _Women’s Studies International Forum_ 47 (2014), 46–55.
* Mostafazadeh Davani et al. (2020) Aida Mostafazadeh Davani, Mohammad Atari, Brendan Kennedy, Shreya Havaldar, and Morteza Dehghani. 2020\. Hatred is in the Eye of the Annotator: Hate Speech Classifiers Learn Human-Like Social Stereotypes. In _31st Annual Conference of the Cognitive Science Society (CogSci)_.
* Ostini and Hopkins (2015) Jenny Ostini and Susan Hopkins. 2015. Online harassment is a form of violence. _The Conversation_ 8 (2015), 1–4. Publisher: The Conversation Media Trust.
* Pfeffer et al. (2023) Juergen Pfeffer, Angelina Mooseder, Jana Lasser, Luca Hammer, Oliver Stritzel, and David Garcia. 2023\. This Sample seems to be good enough! Assessing Coverage and Temporal Reliability of Twitter’s Academic API. _Proceedings of the International AAAI Conference on Web and Social Media_ (2023).
* Rheault et al. (2019) Ludovic Rheault, Erica Rayment, and Andreea Musulan. 2019\. Politicians in the line of fire: Incivility and the treatment of women on social media. _Research & Politics_ 6, 1 (2019).
* Ross et al. (2016) Björn Ross, Michael Rist, Guillermo Carbonell, Ben Cabrera, Nils Kurowsky, and Michael Wojatzki. 2016\. Measuring the Reliability of Hate Speech Annotations: The Case of the European Refugee Crisis. In _Proceedings of NLP4CMC III: 3rd Workshop on Natural Language Processing for Computer-Mediated Communication_ , Michael Beißwenger, Michael Wojatzki, and Torsten Zesch (Eds.). 6–9.
* Strathern et al. (2022) Wienke Strathern, Mirco Schoenfeld, Raji Ghawi, and Juergen Pfeffer. 2022. Identifying Lexical Change in Negative Word-of-Mouth on Social Media. _Social Network Analysis and Mining_ 59, 12 (2022).
* Udupa (2020) Sahana Udupa. 2020\. Artificial Intelligence and the Cultural Problem of Extreme Speech. _Social Science Research Council, (20 December 2020), online_ (2020).
* Vogels (2021) Emily A. Vogels. 2021\. The State of Online Harassment. _Pew Research Center_ (2021).
* Waseem et al. (2017) Zeerak Waseem, Thomas Davidson, Dana Warmsley, and Ingmar Weber. 2017. Understanding Abuse: A Typology of Abusive Language Detection Subtasks. In _Proceedings of the First Workshop on Abusive Language Online_. Association for Computational Linguistics, Vancouver, BC, Canada, 78–84.
* Whiting et al. (2019) Jason Whiting, Rachael Dansby Olufowote, Jaclyn Cravens-Pickens, and Alyssa Banford Witting. 2019\. Online Blaming and Intimate Partner Violence: A Content Analysis of Social Media Comments. _The Qualitative Report_ 24 (2019).
* Wiegand et al. (2021) Michael Wiegand, Josef Ruppenhofer, and Elisabeth Eder. 2021\. Implicitly Abusive Language – What does it actually look like and why are we not getting there?. In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_. Association for Computational Linguistics, 576–587.
* Zannettou et al. (2020) Savvas Zannettou, Joel Finkelstein, Barry Bradlyn, and Jeremy Blackburn. 2020. A Quantitative Approach to Understanding Online Antisemitism. In _Proceedings of the International AAAI Conference on Web and Social Media_ , Vol. 14. 786–797.
* Zuckerberg (2018) Donna Zuckerberg. 2018\. _Not All Dead White Men. Classics and Misogyny in the Digital Age_. Harvard University Press.
|
# A Survey of Mobile Edge Computing for the Metaverse: Architectures,
Applications, and Challenges
Yitong Wang
<EMAIL_ADDRESS>School of Computer Science and Engineering
Nanyang Technological University
Singapore
Jun Zhao
<EMAIL_ADDRESS>School of Computer Science and Engineering
Nanyang Technological University
Singapore
###### Abstract
Metaverse is an emerging virtual universe where humans can have real-time
interactions and solid social links like in the physical world, and it opens
up a new era of Internet and interactions. In Metaverse, an immersive and
photorealistic environment promotes social activities, including education,
meetings, and shopping of digital avatars based on critical technologies,
including 3D rendering, extended reality, digital twins, artificial
intelligence, and Blockchain. However, the limitations of computation,
storage, and energy resources restrict the development of Metaverse, and a
series of system issues (e.g., latency, security, and battery-life) continue
to arise. As a result, how to find corresponding measurements to mitigate
unsatisfactory influences becomes the focus. Mobile edge computing (MEC) as a
distributed computing paradigm offloads computation-intensive tasks to the
edge of the network. It brings the resources as close as possible to the end
devices, addressing the shortcomings mentioned above. In this paper, we
propose a comprehensive survey of the MEC-based Metaverse. Particular emphasis
is given to the technologies convergence, architectures, and application
scenarios, e.g., BoundlessXR and CloudXR. Significantly, we introduce the
potential future directions for developing Metaverse systems.
###### Index Terms:
Metaverse, Edge Computing, Extended Reality, Augmented Reality, Virtual
Reality, Blockchain.
## I INTRODUCTION
Metaverse, as a universe of ternary-world interactions (i.e., physical world,
human world, and digital world), enables humans as digital avatars in sub-
Metaverses to carry out various social activities in the physical world, such
as business meetings, learning, shopping, and digital assets trading[1].
Metaverse significantly revolutionises human manners of interactions with
machines, from conventional 2D web browsing to immersive real-time interaction
in 3D virtual environments, supported by high fidelity image technology, 3D
rendering, Blockchain, digital twins, artificial intelligence (AI) and
sophisticated sensor devices. Recently, Metaverse has been investigated by
lots of commercial and gaming companies. For example, Tencent and Facebook
have already invested in chat scenarios of Metaverse; even Facebook renamed
itself “Meta” in 2021.
The increasing number of mobile devices and users has caused the fact that
wireless communication networks constantly evolved to cope with the strict
requirement of high data rate, latency, and machine-to-machine(M2M) connection
density[2]. 5th generational communication network that consists of specific
attributes is the best choice to cater for the emerging Metaverse system. The
advent of 5G boosts the performance indicators of communication networks by a
multiple of 10 to 100 times, which meets the stringent requirements of various
services and applications. It acts as a bridge for the transmission of
original pose data and compressed image data generated by VR devices and edge
cloud servers, respectively, which perfectly enhances the rendering
performance of the Metaverse.
Extended reality (XR)[3] technology incorporates mixed reality (MR), augmented
reality (AR), virtual reality (VR), and any other technology that encompasses
the fusion of all reality and virtualization. XR devices collect data,
including biological data like eye-, hand-, and head-tracking, and the
accumulation of user data from other social media platforms[4]. XR
applications also provide multi-sensor immersiveness and real-time
interactions for users. Augmented reality(AR)[5], as a technology that
replicates the sensory perception of the real world in terms of time and space
domains, projects additional augmentations generated by computers upon natural
objects, combining real and virtual worlds to enhance people’s sensory
experience further. In the last few years, both KLM Royal Dutch Airlines and
British budget airline EasyJet have allowed passengers to check the size of
their suitcases for boarding through AR technology. AR is performed based on
realistic environments, and application services enrich reality with augmented
components[6]. On the opposite side, VR is a part of MR where its surrounding
environment is virtual, and people can fully have immersive experiences in
virtual worlds with appropriate wearable equipment (e.g., VR Helmets or
Glasses)[7]. To sum up, VR provides immersive interactivity in a digital
world. AR delivers authentic experiences to customers with digital holograms,
images, and videos based on real-world objects. MR acts as a transition,
providing a distinctive experience between VR and AR[8]. Further developments
in micro-sensors and XR technology are making XR equipment, such as helmet-
mounted displays (HMDs), promising to be the primary endpoint for experiencing
the Metaverse [9].
The exponential growth of mobile internet traffic has driven a dramatic
evolution in computing paradigms. Modern mobile applications have more
stringent requirements which cloud computing cannot satisfy, such as ultra-low
latency, ultra-high throughput, ultra-high stability, and high spectral
efficiency. Meanwhile, big data-associated Internet of Things devices causes
an immense growth in the total traffic flows. To break the limitations of
cloud computing and ensure better customer service performance, the emerging
computing paradigm of edge computing, especially mobile edge computing, has
gained the spotlight in both commercial and academic fields nowadays.
Following European Telecommunications Standard Institute (ETSI), Mobile Edge
Computing (MEC) is formally defined as a new platform with significant
computing resources, including IT and Cloud-computing and it gets closer to
the subscribers’ side [10]. Intending to extend 3GPP access scenarios such as
Wi-Fi, the original MEC concept has now transitioned to “Multi-access Edge
Computing” by ETSI. The remarkable characteristic of MEC is offloading
computation tasks from core data centres to distributed edge servers located
on the base stations. Thus, this typically distributed computing paradigm
significantly reduces the communication latency and provides a relatively
large amount of computation resources compared with conventional cloud
computing paradigms. Also, the deployment of MEC relies heavily on the
virtualized platforms[11].
Based on the evolution and combination of the various technologies, services,
and equipment mentioned above, Metaverse technology is rapidly gaining
attention and enhancement. However, there is a highly distant path to explore
to realise the Metaverse fully. Firstly, transmitting the surging data flows
together to cloud servers from end devices is contrary to what is being
sought. The reason is that it not only aggravates network latency on the
system but also causes network congestion and data loss when reaching cloud
services via the core network. Furthermore, data leakage and asset
authentication of the Metaverse are both obstacles encountered. Data security
leads to the loss of all personal information and the forgery of avatar
identities. As multiple administrators provide the assets and virtual
currencies in Metaverse, how to authenticate and trade among the diverse
assets in sub-Metaverses raises a concern.
The contributions of our survey are as follows:
* •
We first introduce the relevant background and basic concepts of Metaverse and
present a clear overview of the types of architectures, evolutionary and
eventual outcomes in Metaverse. Notably, we provide a nuanced introduction to
indicators and essential technologies of Metaverse. With these concise
introductions, readers can keep abreast of the ongoing technological
developments in Metaverse.
* •
We discuss the technological convergence of MEC and Metaverse and illustrate
the role that MEC plays in Metaverse. Unlike previous papers on MEC-based
Metaverse, we not only describe the contributions of MEC to various
performance indicators but also emphasize the application of collaborative
architectures of MEC with other computing paradigms in Metaverse. Specially,
we point out that MEC is crucial in solving the latency, privacy, and energy
challenges in Metaverse.
* •
We outline the research directions and challenges to pave the path toward
future research attempts to realize higher performance in MEC-based Metaverse
architectures. Furthermore, our survey aims to assist researchers in gaining a
thorough and in-depth understanding of the MEC-based Metaverse and grasping a
holistic view of the research in this area.
The remainder of this paper is organized as follows. Section II introduces the
present advanced developments of technologies in Metaverse, and the scenarios
of mobile edge computing applied to the XR domain. Section III interprets the
concepts, architectures, and features of the Metaverse and identifies some of
the current challenges. An introduction to mobile edge computing, including
its strengths and weaknesses, is presented in Section IV. Section V summarises
the enhancements and facilitations when mobile edge computing is applied to
the Metaverse and demonstrates the architectures of the Metaverse based on the
collaboration of mobile edge computing with cloud and fog computing. Future
research directions are shown in Section VI. Section VII provides the
conclusion of the paper.
Figure 1: The Metaverse architecture performs the interactions among virtual
area, human area, and physical area
## II RELATED WORK
We have witnessed many papers in terms of different aspects of Metaverse
technologies and application scenarios of mobile edge computing. In this
section, we discuss recent Metaverse and MEC research and show the main
distinction with our paper. Fernandez et al.[12] review an in-depth survey of
Metaverse from three main motivation perspectives, including privacy,
governance, and ethics design. Specifically, they analyze privacy from sensory
level, behaviour and communications, as well as users and bystanders. Also,
further challenges of these three factors and a novel modular-based framework
are presented in this paper. Xu et al.[13] explore the significance of using
specific spectrums in Metaverse and investigate the vision of Blockchain
technology, respectively. Cheng et al.[14] focus on some existing social VR
platforms that have the potential to evolve into the architectures of the
Metaverse and also conduct a test by looking in depth at the network operation
and capabilities of two typical platforms. Lee et al.[15] present several
innovations in the computation arts of Metaverse, proposing some research
agendas about democratizing computational arts, digital privacy and safety for
Metaverse artists, the identification of proprietary rights in digital
artworks, and the directions of technological developments. For MEC,
Siriwardhana et al.[16] emphasize the survey of mobile augmented reality (MAR)
based on MEC and discusses future vital application areas for MAR. Lim et
al.[17] introduce a study focused on edge-intelligence Metaverse in term of
virtual city development. A stochastic optimal resource allocation scheme
(SORAS) is further proposed byNg et al. based on stochastic integer
programming with the aim of optimizing the cost-effectiveness of virtual
service suppliers [18]. However, none of these recent papers focuses on the
convergence of the Metaverse and MEC paradigm in terms of unique perspectives
such as quality of experience, application scenarios, future challenges, and
so on.
## III METAVERSE
### III-A Metaverse Introduction
The term ’Metaverse’ first emerged in the 20th century in Neal Stephenson’s
science fiction novel named ’Snow Crash’[19], but did not receive much
attention at that time. Two primary factors have driven the Metaverse into
focus currently:
* •
Social factors: The Covid-19 pandemic has led to a dramatic shift in the
manners of people living and working via the Internet from outdoors to
indoors[20], which has caused an increased frequency of daily activities via
the Internet. Additionally, commercial enterprises’ strong focus on VR
resulted in the re-emergence of the Metaverse in the public arena;
* •
Technical factors: XR ecosystem developments (including AR, VR, MR) facilitate
the further enhancement of virtual scenarios and interactions between physical
and virtual worlds; Emerging Web 3.0 converts “server-centric” networks into
“user-centric” networks based on decentralized networks; MEC servers perform
data processing at the edge of the network (i.e., closer to the user) to
mitigate system latency; 5G networks with downlink (DL) speeds of 200Mbps meet
the constraints including low latency, high throughput, and reliability of the
Metaverse.
Through the ternary nature of the Metaverse definition, Fig. 1 shows the
architecture of the Metaverse, and the relationships between these three
worlds can be explained in detail as follows:
1) Physical Area: The physical area provides the essential infrastructures to
support the other two worlds, including computation, cache, transmission, and
sensors. Data from around the environment and the human body are collected
precisely by sensor infrastructures. Transmission infrastructures (e.g.
artificial satellites strictly) ensure the stability and continuity of the
entire network connection. Computationally intensive tasks are processed by
servers located at different positions via robust computation infrastructures,
depending on the type of these tasks. Also, cache infrastructures at various
locations (i.e., local cache, edge cache, and cloud cache) can store major
tasks and reduce the total latency of the entire task processing.
2) Virtual Area: In the Virtual world, the Metaverse engine facilitates and
manages large volumes of digital information from both the physical and human
worlds to enable large-scale Metaverse services[8].
* •
Engine Layer: Metaverse engine layer consists of various technologies to
provide a photorealistic environment, real-time interactions, non-fungible
token (NFT)[21] and translation, respectively. 3D rendering technology
provides a virtual environment with diverse spatiotemporal dimensions and
attributes. Digital twins (DT) is crucial to the behaviours of avatars in
Metaverse. Especially various human social activities are identified and
performed via DT technology. AI plays a vital role in guaranteeing the
reliability of Metaverse. By combining with other technologies, many machine
learning algorithms are used to solve challenging tasks, such as computational
resource allocation[22], predictions of human actions, and increased spectrum
utilisation.
* •
Virtual Layer: As illustrated in Fig. 1, the virtual layer is composed of the
virtual environment, digital avatars, virtual services/goods and digital
assets, which is supported by the critical technologies in the engine layer.
High-fidelity images are provided with 3D rendering and AI support by
collecting and processing real-world environmental data via XR sensor devices.
Digital avatars are capable of behaving exactly like humans in the real world
based on DT technology.
* •
Virtual Universe: In contrast to the concept of a unified Metaverse, the whole
Metaverse comprises a series of dispersed sub-Metaverses where avatars can be
served distinctly. Distinct sub-Metaverses can be freed from the constraints
of time and space. Due to the existence of multiple operators, creating
multiple sub-universes becomes a fact, but their interconnection holds the key
to future developments.
3) Human Area: In reality, people’s intricate social relationships and diverse
activities contribute to the functioning of society. Crucially, as in the real
world, people as avatars are at the centre of the virtual world and are
constantly creating virtual objects. Currently, by wearing devices (e.g. XR
HMDs, Wristband sensors), social activities such as business meetings, office
learning and entertainment concerts can be realised in the virtual world via
human-computer interaction (HCI) and especially brain-computer interaction
(BCI). Human activities can be accurately mapped onto the virtual world
through various Metaverse engines. And services in the virtual world can also
continuously influence human interactions and perceptions in the real world.
### III-B Metaverse Characteristics
Immersive: The ultimate goal of Metaverse development is to approximate the
user’s experience in the real world gradually. Supported by various
infrastructures and Metaverse engines, users can interact in a rendered and
high-fidelity virtual universe. Immersive means that the user can adapt to the
digital world and move around in the Metaverse without discomforts, such as
dizziness and nausea.
Interoperability: Multiple operating systems, such as Ethereum, Solana, and
Polygon, co-exist in the Metaverse. Accordingly, a variety of value tokens
that are derived from each technology continuously emerge. For example, the
Bitcoin Blockchain supports a token called Bitcoin, while the Ethernet-backed
tokens include SAND, MANA, AXS, and GALA. The interoperability of the
Metaverse is guaranteed by the accessibility of connections and transactions
between different sub-universes or currencies.
Multi-technology: A few technological advancements have contributed to the
emergence of the Metaverse. For example, 5G technology has improved peak
transmission rate, time delay, and reliability to meet the requirement that
transmissions must be completed in milliseconds or less, relieving users of
dizziness caused by high latency (>20ms). Also, because downlink speed is
proportional to the rendering resolution, higher downlink speeds can make
virtual environments more realistic. In summary, multi-technology provides an
immersive experience and high-fidelity virtual worlds based on augmented
reality technology, generates a city twin of the physical society with digital
twin technology [23], and creates an economic system based on Blockchain
technology [24].
### III-C Metaverse Technologies
XR: XR technology covers AR, VR, and MR. Wearable VR devices provide users
with a completely virtual scenario in which they can be fully immersive[25].
For example, Second life provides a virtual world in which players can
survive. In this world, players can control their avatars to perform
activities and enjoy personalized services. For AR/MR technology,
augmentations are overlaid in the real world[26], e.g., users can play Pokémon
GO on their mobile phones and catch virtual Pokemon. XR devices significantly
enhance the user’s immersion in the Metaverse.
Digital Twin: Digital twin (DT) projects humans, objects, and environments of
the real world into the virtual world, creating digital clones that are
visually indistinguishable from the real world in real-time[27]. By processing
the input data, DT is able to manage and optimize physical objects
periodically [28][29]. Significantly, the data flows between virtual objects
and the physical world are bidirectional. The physical object transmits the
collected data in multiple formats to the virtual twins, and the virtual twins
convey the processed feedback to the physical object[30], further accelerating
the intersection of the human and virtual worlds.
Blockchain: Blockchain is considered to be a decentralized technology [31].
Unlike traditional centralized systems, Blockchain technology can help
safeguard users’ data, identity information, and virtual assets with
certificates (i.e., NFTs) from leakage and theft in the event of a threat. At
the same time, Blockchain, as a distributed ledger, ensures that all data is
protected from tampering and modification. Hence, Blockchain not only ensures
data security and quality but also enables seamless data sharing and data
interoperability and integrity[32]. Furthermore, the convergence of Blockchain
and other key technologies can significantly empower the performence of the
Metaverse system [33].
AI: AI plays an indispensable role in creating and rendering large-scale
Metaverse. Conventional AI technologies are composed of supervised learning,
semi-supervised learning, unsupervised learning, as well as reinforcement
learning (RL)[34]. Nowadays, many advanced machine learning (ML) algorithms of
supervised and reinforcement learning have been used for different challenging
tasks, including automatic resource allocation, attack prevention, and network
fault detection[35][36]. Based on massive Metaverse multimodal input data,
high-quality Metaverse scenes can be created and rendered by AI via
considerable data interference[37]. Also, deep learning (DL) and ML algorithms
facilitate the provision of personalized services by making good decisions.
## IV MOBILE EDGE COMPUTING
### IV-A MEC Introduction
The motivation for investigating MEC computing paradigms is tackling the
tradeoff between computation and communication. As shown in Fig. 2, edge
servers are deployed on the edge of networks, which provides computational
resources. As a result, computational tasks are processed at the network’s
edge as proximate to the user as possible. Different from fog computing and
cloud computing, a MEC server is a node device which means that the
decentralization of mobile communication networks is achieved[38]. The
security and privacy of networks are enhanced greatly, and congestion during
peak transmission could also be tackled.
Nowadays, an increasing number of papers on mobile edge computing are
investigated, especially in the fields of system architecture design and task
offloading. Paper[39] proposes a task offloading strategy based on an improved
auction algorithm, divided into two main phases: task offloading and task
scheduling, considering the limited resources of smart devices and the
stronger computing and storage capabilities of edge servers. Furthermore, this
strategy provides the basis for offloading decisions in the decision phase by
considering the time cost of task execution locally or at the edge as well as
energy consumption to determine whether the computation task that needs to be
processed will be offloaded to the edge of the network. Once the task is
offloaded to the edge server, reducing computational latency and reducing
transmission energy consumption to achieve the global optimum are the main
objectives of the task scheduling phase. Paper [40] considers mobility at the
user end, classifies cellular networks in more detail and gives more accurate
state transfer probabilities. And this research models the VM migration
process as a Markov decision process and uses a policy iteration algorithm to
find the optimal solution, effectively reducing energy consumption and
latency.
### IV-B MEC Application
MEC is used as a key technology that guarantees the efficient realisation of
diverse services, especially in the practical application of task offloading
and resource allocation. In vehicular edge computing (VEC) networks[41],
vehicles can act as an edge server at the edge of the Internet of Things (IoT)
system to provide computing resources[42] and various services. Meanwhile, the
collaboration between vehicle edge servers (VES) and fixed edge servers (FES)
can provide a variety of offloads and computing options for network computing,
offering various edge computing resources depending on the tasks. Ren et al.
[43] demonstrate that edge servers can reduce the core network use by
elaborating on the system framework of web-based services with MEC technology
and further showing that MEC can address real-world Web AR deployments.
Figure 2: MEC computing paradigm features the low-latency and energy-
efficiency procession.
## V METAVERSE WITH MEC
It is theoretically feasible to perform all computing tasks on the local XR
devices or cloud servers, but this contradicts the desire to prolong battery
life, lighten device weight and maintain ultra-low latency. The convergence of
mobile edge computing and Metaverse results in a new generation of MEC
technology[44][45], which is being used to break the dependency of devices on
centralized cloud servers and to improve the high real-time performance and
security of the entire system. Primarily, splitting rendering is generated
after this convergence. As graphics are heavily rendered, on-device processes
are augmented by being partitioned to IoT devices and edge clouds. The
graphical rendering of the Metaverse system is enhanced by using photonic
processing for head tracking and motion tracking of latency-sensitive devices.
Furthermore, based on the design of stream processing technology, Metaverse
systems can also effectively decrease the latency and have excellent
performance as a high-performance distributed stream processing system
(DSPS)[46]. More studies are explained as follows.
Figure 3: Architecture of Mobile Edge Computing-Based Metaverse
### V-A MEC-Based Architecture
Conventional cloud-based Metaverse architecture is not beneficial to
implementing virtual functions and services via transmitting massive data such
as movements of avatars and physical factors of surroundings to the
centralized server. To mitigate negative influences caused by limitations,
innovative MEC-based Metaverse architectures have resulted in optimization and
a satisfactory performance:
1) Mobile Edge Computing-Based Metaverse: In the Metaverse of this
architecture, multiple dynamic edge nodes are incorporated to perform users’
instructions, which significantly decrease the delay caused by users’
movements[47]. As shown in Fig. 3, the private data generated by the customers
is transmitted to the specific edge server. Through the processing and
delivery of the edge servers, customers enjoy a low-latency service and an
immersive Metaverse experience. To cope with the data leakage when
transmitting users’ data, federated learning(FL) is regarded as a critical
technology. Consequently, parameters rather than private data from similar
local models are uploaded to the FL layer for further training.
2) Fog-Edge-Cloud Metaverse: This distributed architecture is hierarchical and
tiers different servers to eliminate the Metaverse fragmentation and
computational bottlenecks.Kechadi et al.[48] proposed this hybrid
computational architecture for Metaverse services. In cloud layers, Cloud
servers exist to simulate the virtual worlds. The edge layer servers are
dedicated to processing specific virtual buildings’ computation tasks and
providing virtual environment animation. For fog servers located on the fog
layer, servers are deployed to perform computation tasks of the virtual home
environment and users’ movements.
### V-B Improvement
User Side: In virtual worlds, accurate predictions and assessments of users’
behaviours, habitual preferences, and movement paths are crucial to providing
users with an immersive and real-time experience. Based on the attributes of
MEC, stable and reliable seamless services could be offered to users’ devices,
and real-time services are also provided as a result of the location of MEC
servers [49].
Furthermore, since edge servers have more computing capabilities than local
devices(e.g. VR devices, HMDs, and AR glasses), they can efficiently process
computation-intensive tasks and transfer feedback to mobile devices, which
will reduce latency and thus alleviate user discomfort such as dizziness
caused by latency.
System Side: The two main merits of applying MEC to Metaverse systems today
are enhanced resource utilisation and significantly reduced latency. Taking
advantage of the fact that MEC servers come with their own computational
resources[50], most Metaverse applications can be processed through these
servers. At the same time, the wearability and small size of most devices lead
to limitations on the battery life and computational resources, which is
tackled by the MEC-based Metaverse. Simultaneously, the dynamic resource
allocation[23] and edge intelligence-based technology[29] further improve the
performance of all aspects of the Metaverse substantially.
Figure 4: BoundlessXR consists of XR devices, 5G network, and edge-cloud.
### V-C Application Scenarios
On-device processing is an essential part of performing Metaverse
applications. In a stand-alone model, on-device processing is responsible for
dealing with entire XR processing assignments. While augmented by the edge
cloud, on-device processing provides rendering and tracking that are power-
efficiency, high-performance and latency-sensitive. Nowadays, Boundless XR
created by the Qualcomm uses split rendering to distribute the computation
tasks between the edge servers and end devices to deliver a genuinely
immersive XR experience over 5G. Specifically, the architecture is presented
in Fig. 4. Firstly, VR devices send 6-degrees-of-freedom head pose data to the
edge cloud over 5G. The edge cloud sends the compressed image to the devices
after processing this data and rendering a new image. Finally, VR devices
further perform on-device processing. Based on the 5G and MEC technologies,
the motion-to-render-to-photon latency is less than 20ms, which meets
Metaverse services’ requirements. NVIDIA’s CloudXR is also a novel solution
for streaming XR data from any openVR/XR application on a distant server,
which can also dynamically adjust network conditions and maximize image
quality while minimizing effective latency.
In addition to innovative applications by commercial companies, deep research
in edge computing is also evolving in academia. A remarkable FACT algorithm
was proposed in the paper[51] to balance three main factors, including network
latency, computation latency, and system accuracy in an edge-based mobile
augmented reality system. Applying this new paradigm to the MAR applications
significantly addressed the optimization problems as a result of multiple
mobile users and improved the performance of processing video frames in
Metaverse. In order to optimize the computation resources and AR
configurations on the edge servers, another optimization protocol called DARE
was presented in the paper[52]. When considering multiple coherence times in
the research of MEC-based MAR, the algorithm based on Lyapunov optimization in
the paper[53] can effectively improve the reliability and the quality of
augments (QoA) in Metaverse systems. Furthermore, when studied from the
perspective of energy consumption and the interactions between MAR system
parameters, the LEAF algorithm was studied by Wang et al.[54] in the specific
energy-aware MEC-based MAR system.
## VI FUTURE RESEARCH CHALLENGES
In this section, future challenges of MEC-based Metaverse are discussed from
the following aspects.
### VI-A Computer Resource Allocation
Currently, papers are primarily conducted on computation allocation. Tang et
al. [55] proposed an optimization algorithm by considering the computation
offloading strategies and resource allocation problems together and
effectively solved the interuser interference due to load. Nath et al. [56]
studied the cached content by four decision parameters were considered,
including whether a given task needs to be cached, how much transmission power
is appropriate to utilize during offloading, and how many MEC resources to
assign to perform a task to further improve collaboration issues between MEC
servers. However, these researches do not consider the dynamic nature of
Metaverse systems. Specifically, the computational resources of each node are
dynamic in real time rather than static. Therefore, idle computational
resources will be unused inefficiently when the MEC is processing
computationally intensive tasks after nodes are pre-determined. For further
research, how to solve the problem of dynamic allocation of computational
resources is a crucial challenge to be tackled.
### VI-B Mobility Management
Metaverse applications rely heavily on users’ states and previous behaviours,
meaning mobility is a key consideration when implementing applications
sensitive to continuity and real-time [57]. On the other hand, user mobility
and excellent mobility management ensure seamless service to enhance the user
experience and QoS further. The alternative approaches are caching[58] and SDN
controllers now. For caching, the main challenges are how the popularity of
the content to be cached can be accurately predicted and how we can achieve
collaborative caching across multiple cells to improve web caching
performance. [59] For SDN controllers, Shah et al. [60] propose an SDN-based
MEC-enabled 5G vehicular networks. They use four software modules developed by
themselves in the SDN controller to develop the rules of QoS further.
Furthermore, the SDN controller proposed can federate and coordinate the
allocation of MEC resources to further provide continuous service and seamless
coverage to mobile users. In future research directions, mobility management
can be combined with artificial intelligence. Through this technologies
convergence, the Metaverse system can learn the user’s behaviours on its own,
predict how the user will move, and transmit information about the user’s
state to the SDN controller in advance, thus providing a more reliable
service.
### VI-C User Experience
Digital avatars experience various activities in photorealistic sub-Metaverses
and can obtain perceptions through sensor devices. The final evolution of the
Metaverse is to integrate with the physical world, which means dark real-world
forces also exist in virtual life. For example, robberies and car accidents in
Metaverse badly influence humans in the real world and even cause physical
pain. Moreover, the unfettered behaviour of avatars, like sexual assault, also
causes psychological victimization of the user. Thus, establishing a code of
ethics to constrain the behaviour of avatars is a crucial challenge to
maintaining a safe community for users in virtual worlds[61]. Furthermore,
user addiction[62] is another harmful influence on emotional health in
Metaverse.
### VI-D Data
For obstacles of Metaverse in the future, two main factors are supposed to be
noticed: data leakage and data reliability. 1) data leakage: MEC-based
metaverse architecture is not the same as the decentralized bitcoin
architecture. Private and safety-sensitive data (such as avatar ID numbers and
passwords, digital assets, and currencies) are transmitted to the edge cloud
from end devices, which may suffer grievous attacks by potential adversaries.
Lee et al.[63] identify three main ad fraud threats, including blind-spot
tracking, gaze and controller cursor-jacking, and the misuse of assisted
display in content sharing. 2) data quality:The data quality levels directly
alter the metaverse applications’ QoS. High data quality significantly
improves the QoE of users, while low level stands on the opposite side. 3)
data reliability: Data sources and generation processes will cause the data to
be unreliable if the sources are fake IDs and jitters occur during
propagation. Hence, preventing leakage and improving the quality and
reliability of data in Metaverse are also essential requirements during the
development of realizing Metaverse.
### VI-E Delay
The main goal of the Metaverse is to deliver the processed data to the XR
device in real-time via the downlink and to render the virtual spatial
environment. For example, the delay generated by VR devices should not exceed
20ms; otherwise, it will cause spatial vertigo and vomiting to the user in the
Metaverse. Hence, the partitioning of computational tasks is a constant focus
of researchers[64]. Furthermore, balancing the tradeoff between task splitting
and delay is essential. For example, Lightweight and wearable XR devices can
be worn easily but with fewer computational resources.
### VI-F Privacy
Data privacy and user identity are the two significant aspects of Metaverse
security that must be ensured. For data privacy, users’ virtual assets, facial
and retinal data, and transaction records in the Metaverse are at risk of
leakage, especially during data processing and transmission. Moreover, after
one user’s behavioural data and interactions with other avatars have been
collected and analyzed by hackers[19], a fake avatar will be made that can
imitate the original avatar to carry out social activities in the Metaverse
and even commit crimes. Bystander privacy in the Metaverse is also at risk of
being acquired. Then, the protection of data and identity becomes the way
forward.
## VII CONCLUSION
In this paper, we have introduced an in-depth survey of Metaverse based on
mobile edge computing. We have presented an overview of Metaverse and
discussed its fundamentals, characteristics, core technologies, and existing
issues. Afterwards, fundamental research and application scenarios like
AR(Web-AR)/VR/MR of mobile edge computing are presented. Significantly, we
analyze the convergence of these two prominent technologies from aspects
including architectures, improved peculiarities, and application scenarios.
Furthermore, we show the future research challenges of mobile edge-based
Metaverse. We anticipate that this survey will shed light on the direction of
the MEC and Metaverse and spark even more impressive research in the future.
## Acknowledgement
This research is supported in part by Nanyang Technological University Startup
Grant; in part by the Singapore Ministry of Education Academic Research Fund
under Grant Tier 1 RG97/20, Grant Tier 1 RG24/20 and Grant Tier 2
MOE2019-T2-1-176.
## References
* [1] J. Zhong and Y. Zheng, “Empowering future education: Learning in the Edu-Metaverse,” in _2022 International Symposium on Educational Technology (ISET)_. IEEE, 2022, pp. 292–295.
* [2] P. Popovski, K. F. Trillingsgaard, O. Simeone, and G. Durisi, “5G Wireless Network Slicing for eMBB, URLLC, and mMTC: A Communication-Theoretic View,” _Ieee Access_ , vol. 6, pp. 55 765–55 779, 2018.
* [3] J. Guan, J. Irizawa, and A. Morris, “Extended Reality and Internet of Things for Hyper-Connected Metaverse Environments,” in _2022 IEEE Conference on Virtual Reality and 3D User Interfaces Abstracts and Workshops (VRW)_. IEEE, 2022, pp. 163–168.
* [4] J. Evans, “White Paper-The IEEE Global Initiative on Ethics of Extended Reality (XR) Report–Extended Reality (XR) Ethics in Medicine,” 2022.
* [5] N. R. D. Torres, G. A. L. Morteo, and R. G. Cruz-Flores, “Conceptualization of Augmented Digital Ecosystem (ADE) in context of Metaverse and The Augmented Dimension,” in _2022 IEEE Mexican International Conference on Computer Science (ENC)_. IEEE, 2022, pp. 1–5.
* [6] T. Braud, C. B. Fernández, and P. Hui, “Scaling-up AR: University Campus as a Physical-Digital Metaverse,” in _2022 IEEE Conference on Virtual Reality and 3D User Interfaces Abstracts and Workshops (VRW)_. IEEE, 2022, pp. 169–175.
* [7] M. Qu, Y. Sun, and Y. Feng, “Digital Media and VR Art Creation for Metaverse,” in _2022 2nd Asia Conference on Information Engineering (ACIE)_. IEEE, 2022, pp. 48–51.
* [8] Y. Wang, Z. Su, N. Zhang, D. Liu, R. Xing, T. H. Luan, and X. Shen, “A Survey on Metaverse: Fundamentals, Security, and Privacy,” _arXiv preprint arXiv:2203.02662_ , 2022.
* [9] M. Sugimoto, “Extended Reality (XR: VR/AR/MR), 3D Printing, Holography, AI, Radiomics, and Online VR Tele-Medicine for Precision Surgery,” in _Surgery and Operating Room Innovation_. Springer, 2021, pp. 65–70.
* [10] E. M. E. Computing, I. Initiative _et al._ , “Mobile-Edge Computing: Introductory Technical White Paper,” _ETSI: Sophia Antipolis, France_ , pp. 1–36, 2014.
* [11] Y. C. Hu, M. Patel, D. Sabella, N. Sprecher, and V. Young, “Mobile Edge Computing—A Key Technology towards 5G,” _ETSI white paper_ , vol. 11, no. 11, pp. 1–16, 2015.
* [12] C. B. Fernandez and P. Hui, “Life, the Metaverse and Everything: An Overview of Privacy, Ethics, and Governance in Metaverse,” _arXiv preprint arXiv:2204.01480_ , 2022.
* [13] H. Xu, Z. Li, Z. Li, X. Zhang, Y. Sun, and L. Zhang, “Metaverse Native Communication: A Blockchain and Spectrum Prospective,” _arXiv preprint arXiv:2203.08355_ , 2022.
* [14] R. Cheng, N. Wu, S. Chen, and B. Han, “Will Metaverse be NextG Internet? Vision, Hype, and Reality,” _arXiv preprint arXiv:2201.12894_ , 2022.
* [15] L.-H. Lee, Z. Lin, R. Hu, Z. Gong, A. Kumar, T. Li, S. Li, and P. Hui, “When Creators Meet the Metaverse: A Survey on Computational Arts,” _arXiv preprint arXiv:2111.13486_ , 2021.
* [16] Y. Siriwardhana, P. Porambage, M. Liyanage, and M. Ylianttila, “A Survey on Mobile Augmented Reality with 5G Mobile Edge Computing: Architectures, Applications, and Technical Aspects,” _IEEE Communications Surveys & Tutorials_, vol. 23, no. 2, pp. 1160–1192, 2021.
* [17] W. Y. B. Lim, Z. Xiong, D. Niyato, X. Cao, C. Miao, S. Sun, and Q. Yang, “Realizing the Metaverse with Edge Intelligence: A Match Made in Heaven,” _arXiv preprint arXiv:2201.01634_ , 2022.
* [18] W. C. Ng, W. Y. B. Lim, J. S. Ng, Z. Xiong, D. Niyato, and C. Miao, “Unified Resource Allocation Framework for The Edge Intelligence-Enabled Metaverse,” _arXiv preprint arXiv:2110.14325_ , 2021.
* [19] R. Di Pietro and S. Cresci, “Metaverse: Security and Privacy Issues,” in _2021 Third IEEE International Conference on Trust, Privacy and Security in Intelligent Systems and Applications (TPS-ISA)_. IEEE, 2021, pp. 281–288.
* [20] A. E. Bates, R. B. Primack, B. S. Biggar, T. J. Bird, M. E. Clinton, R. J. Command, C. Richards, M. Shellard, N. R. Geraldi, V. Vergara _et al._ , “Global COVID-19 Lockdown Highlights Humans as Both Threats and Custodians of The Environment,” _Biological conservation_ , vol. 263, p. 109175, 2021\.
* [21] R. García, A. Cediel, M. Teixidó, and R. Gil, “Semantics and Non-Fungible Tokens for Copyright Management on the Metaverse and Beyond,” _arXiv preprint arXiv:2208.14174_ , 2022.
* [22] P. Si, J. Zhao, H. Han, K.-Y. Lam, and Y. Liu, “Resource Allocation and Resolution Control in the Metaverse with Mobile Augmented Reality,” _arXiv preprint arXiv:2209.13871_ , 2022.
* [23] Y. Han, D. Niyato, C. Leung, C. Miao, and D. I. Kim, “A Dynamic Resource Allocation Framework for Synchronizing Metaverse with IoT Service and Data,” in _ICC 2022-IEEE International Conference on Communications_. IEEE, 2022, pp. 1196–1201.
* [24] H. Ning, H. Wang, Y. Lin, W. Wang, S. Dhelim, F. Farha, J. Ding, and M. Daneshmand, “A Survey on Metaverse: the State-of-the-art, Technologies, Applications, and Challenges,” _arXiv preprint arXiv:2111.09673_ , 2021\.
* [25] M. Xu, D. Niyato, J. Kang, Z. Xiong, C. Miao, and D. I. Kim, “Wireless Edge-Empowered Metaverse: A Learning-Based Incentive Mechanism for Virtual Reality,” in _ICC 2022-IEEE International Conference on Communications_. IEEE, 2022, pp. 5220–5225.
* [26] M. McGill, “The IEEE Global Initiative on Ethics of Extended Reality (XR) Report–Extended Reality (XR) and the Erosion of Anonymity and Privacy,” _Extended Reality (XR) and the Erosion of Anonymity and Privacy-White Paper_ , pp. 1–24, 2021.
* [27] C. A. Lynch, A. O. Adeyeye, and M. M. Tentzeris, “” Smart” Packaging of Self-Identifying and Localizable mmID for Digital Twinning and Metaverse Temperature Sensing Applications,” in _2022 IEEE 72nd Electronic Components and Technology Conference (ECTC)_. IEEE, 2022, pp. 87–92.
* [28] M. Aloqaily, O. Bouachir, F. Karray, I. Al Ridhawi, and A. El Saddik, “Integrating Digital Twin and Advanced Intelligent Technologies to Realize the Metaverse,” _IEEE Consumer Electronics Magazine_ , 2022.
* [29] D. Van Huynh, S. R. Khosravirad, A. Masaracchia, O. A. Dobre, and T. Q. Duong, “Edge Intelligence-based Ultra-Reliable and Low-Latency Communications for Digital Twin-enabled Metaverse,” _IEEE Wireless Communications Letters_ , 2022.
* [30] Y. Wu, K. Zhang, and Y. Zhang, “Digital twin networks: a survey,” _IEEE Internet of Things Journal_ , vol. 8, no. 18, pp. 13 789–13 804, 2021\.
* [31] Z. Wang, Q. Hu, M. Xu, and H. Jiang, “Blockchain-based Edge Resource Sharing for Metaverse.”
* [32] T. R. Gadekallu, T. Huynh-The, W. Wang, G. Yenduri, P. Ranaweera, Q.-V. Pham, D. B. da Costa, and M. Liyanage, “Blockchain for the Metaverse: A Review,” _arXiv preprint arXiv:2203.09738_ , 2022.
* [33] S. Badruddoja, R. Dantu, Y. He, M. Thompson, A. Salau, and K. Upadhyay, “Trusted AI with Blockchain to Empower Metaverse,” in _2022 Fourth International Conference on Blockchain Computing and Applications (BCCA)_. IEEE, 2022, pp. 237–244.
* [34] F. Lin, W. Ning, and H. Zhai, “ICDVAE: An Effective Self-supervised Reinforcement Learning Method for behavior decisions of Non-player Characters in Metaverse Games,” in _2022 IEEE 10th Joint International Information Technology and Artificial Intelligence Conference (ITAIC)_ , vol. 10. IEEE, 2022, pp. 1727–1731.
* [35] T. Huynh-The, Q.-V. Pham, X.-Q. Pham, T. T. Nguyen, Z. Han, and D.-S. Kim, “Artificial Intelligence for the Metaverse: A Survey,” _arXiv preprint arXiv:2202.10336_ , 2022.
* [36] X. Zhou, C. Liu, and J. Zhao, “Resource allocation of federated learning for the metaverse with mobile augmented reality,” _arXiv preprint arXiv:2211.08705_ , 2022.
* [37] O. Bouachir, M. Aloqaily, F. Karray, and A. Elsaddik, “AI-based Blockchain for the Metaverse: Approaches and Challenges,” in _2022 Fourth International Conference on Blockchain Computing and Applications (BCCA)_. IEEE, 2022, pp. 231–236.
* [38] F. Vhora and J. Gandhi, “A Comprehensive Survey on Mobile Edge Computing: Challenges, Tools, Applications,” in _2020 Fourth International Conference on Computing Methodologies and Communication (ICCMC)_. IEEE, 2020, pp. 49–55.
* [39] J. Sheng, J. Hu, X. Teng, B. Wang, and X. Pan, “Computation Offloading Strategy in Mobile Edge Computing,” _Information_ , vol. 10, no. 6, p. 191, 2019.
* [40] Q. Wang, R. Xie, and T. Huang, “Research on Mobility Management of Mobile Edge Computing,” _ZTE TECHNOLOGY JOURNAL_ , 2018.
* [41] Y. Liu, H. Yu, S. Xie, and Y. Zhang, “Deep Reinforcement Learning for Offloading and Resource Allocation in Vehicle Edge Computing and Networks,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 11, pp. 11 158–11 168, 2019.
* [42] K. Li, Y. Cui, W. Li, T. Lv, X. Yuan, S. Li, W. Ni, M. Simsek, and F. Dressler, “When Internet of Things Meets Metaverse: Convergence of Physical and Cyber worlds,” _arXiv preprint arXiv:2208.13501_ , 2022.
* [43] P. Ren, X. Qiao, J. Chen, and S. Dustdar, “Mobile Edge Computing–A Booster for The Practical Provisioning Approach of Web-Based Augmented Reality,” in _2018 IEEE/ACM Symposium on Edge Computing (SEC)_. IEEE, 2018, pp. 349–350.
* [44] W. Zhang, J. Chen, Y. Zhang, and D. Raychaudhuri, “Towards Efficient Edge Cloud Augmentation for Virtual Reality MMOGs,” in _Proceedings of the Second ACM/IEEE Symposium on Edge Computing_ , 2017, pp. 1–14.
* [45] S. Dhelim, T. Kechadi, L. Chen, N. Aung, H. Ning, and L. Atzori, “Edge-enabled Metaverse: The Convergence of Metaverse and Mobile Edge Computing,” _arXiv preprint arXiv:2205.02764_ , 2022.
* [46] J. Heo, K. Bhardwaj, and A. Gavrilovska, “Poster: Enabling Flexible Edge-Assisted XR,” in _2021 IEEE/ACM Symposium on Edge Computing (SEC)_. IEEE, 2021, pp. 465–467.
* [47] L. Chang, Z. Zhang, P. Li, S. Xi, W. Guo, Y. Shen, Z. Xiong, J. Kang, D. Niyato, X. Qiao _et al._ , “6G-Enabled Edge AI for Metaverse: Challenges, Methods, and Future Research Directions,” _arXiv preprint arXiv:2204.06192_ , 2022.
* [48] T. Kechadi, L. Chen, N. Aung, H. Ning, L. Atzori _et al._ , “Edge-enabled Metaverse: The Convergence of Metaverse and Mobile Edge Computing,” 2022.
* [49] L.-H. Lee, T. Braud, P. Zhou, L. Wang, D. Xu, Z. Lin, A. Kumar, C. Bermejo, and P. Hui, “All One Needs to Know about Metaverse: A Complete Survey on Technological Singularity, Virtual Ecosystem, and Research Agenda,” _arXiv preprint arXiv:2110.05352_ , 2021.
* [50] W. C. Ng, W. Y. B. Lim, J. S. Ng, Z. Xiong, D. Niyato, and C. Miao, “Unified Resource Allocation Framework for The Edge Intelligence-Enabled Metaverse,” in _ICC 2022-IEEE International Conference on Communications_. IEEE, 2022, pp. 5214–5219.
* [51] Q. Liu, S. Huang, J. Opadere, and T. Han, “An Edge Network Orchestrator for Mobile Augmented Reality,” in _IEEE INFOCOM 2018-IEEE Conference on Computer Communications_. IEEE, 2018, pp. 756–764.
* [52] Q. Liu and T. Han, “Dare: Dynamic Adaptive Mobile Augmented Reality with Edge Computing,” in _2018 IEEE 26th International Conference on Network Protocols (ICNP)_. IEEE, 2018, pp. 1–11.
* [53] J. Liu and Q. Zhang, “Edge Computing Enabled Mobile Augmented Reality with Imperfect Channel Knowledge,” in _European Wireless 2019; 25th European Wireless Conference_. VDE, 2019, pp. 1–6.
* [54] H. Wang and J. Xie, “User Preference Based Energy-Aware Mobile AR System with Edge Computing,” in _IEEE INFOCOM 2020-IEEE Conference on Computer Communications_. IEEE, 2020, pp. 1379–1388.
* [55] L. Tang and H. Hu, “Computation Offloading and Resource Allocation for The Internet of Things in Energy-Constrained MEC-Enabled HetNets,” _IEEE Access_ , vol. 8, pp. 47 509–47 521, 2020.
* [56] S. Nath and J. Wu, “Deep Reinforcement Learning for Dynamic Computation Offloading and Resource Allocation in Cache-Assisted Mobile Edge Computing Systems,” _Intelligent and Converged Networks_ , vol. 1, no. 2, pp. 181–198, 2020.
* [57] Z. Huang and V. Friderikos, “Mobility Aware Optimization in the Metaverse,” _arXiv preprint arXiv:2209.08899_ , 2022.
* [58] Y. Cai, J. Llorca, A. M. Tulino, and A. F. Molisch, “Joint Compute-Caching-Communication Control for Online Data-Intensive Service Delivery,” _arXiv preprint arXiv:2205.01944_ , 2022.
* [59] Y. Liu, J. Liu, A. Argyriou, L. Wang, and Z. Xu, “Rendering-Aware VR Video Caching over Multi-Cell MEC Networks,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 3, pp. 2728–2742, 2021.
* [60] S. D. A. Shah, M. A. Gregory, S. Li, R. Fontes, and L. Hou, “SDN-Based Service Mobility Management in MEC-Enabled 5G and Beyond Vehicular Networks,” _IEEE Internet of Things Journal_ , 2022.
* [61] J. Yu, A. Alhilal, P. Hui, and D. H. Tsang, “6G Mobile-Edge Empowered Metaverse: Requirements, Technologies, Challenges and Research Directions,” _arXiv preprint arXiv:2211.04854_ , 2022.
* [62] J. Zhu, P. Ni, and G. Wang, “Activity Minimization of Misinformation Influence in Online Social Networks,” _IEEE Transactions on Computational Social Systems_ , vol. 7, no. 4, pp. 897–906, 2020.
* [63] H. Lee, J. Lee, D. Kim, S. Jana, I. Shin, and S. Son, “AdCube: WebVR Ad Fraud and Practical Confinement of Third-Party Ads,” in _30th USENIX Security Symposium (USENIX Security 21)_ , 2021, pp. 2543–2560.
* [64] Y. Cai, J. Llorca, A. M. Tulino, and A. F. Molisch, “Compute-and Data-Intensive Networks: The Key to The Metaverse,” _arXiv preprint arXiv:2204.02001_ , 2022.
|
# Enabling Fast Unit Commitment Constraint Screening via Learning Cost Model
Xuan He1, Honglin Wen2, Yufan Zhang2 and Yize Chen1 1Hong Kong University of
Science and Technology (Guangzhou)
<EMAIL_ADDRESS><EMAIL_ADDRESS>2Shanghai Jiao Tong
University
{linlin00<EMAIL_ADDRESS>
###### Abstract
Unit commitment (UC) are essential tools to transmission system operators for
finding the most economical and feasible generation schedules and dispatch
signals. Constraint screening has been receiving attention as it holds the
promise for reducing a number of inactive or redundant constraints in the UC
problem, so that the solution process of large scale UC problem can be
accelerated by considering the reduced optimization problem. Standard
constraint screening approach relies on optimizing over load and generations
to find binding line flow constraints, yet the screening is conservative with
a large percentage of constraints still reserved for the UC problem. In this
paper, we propose a novel machine learning (ML) model to predict the most
economical costs given load inputs. Such ML model bridges the cost
perspectives of UC decisions to the optimization-based constraint screening
model, and can screen out higher proportion of operational constraints. We
verify the proposed method’s performance on both sample-aware and sample-
agnostic setting, and illustrate the proposed scheme can further reduce the
computation time on a variety of setup for UC problems.
## I Introduction
Solving large-scale optimization problems is one of the cornerstones for many
power system operation tasks, such as unit commitment (UC), optimal power flow
(OPF), and capacity expansion planning. Many such applications require to
solve the problem in a timely manner, as the solutions are essential to market
clearing, power grid reliability, and power grid operations [1]. And in
particular, as the DC power flow is often utilized in UC models, while the
generators’ ON/OFF statutes are often described as integer variables, UC
problems are usually formulated as mixed integer programming (MIP). There is a
lot of computation burden to find the most economical UC solutions for such
NP-hard problems on large-scale power networks.
The computational burden of the UC problem can be significant, especially for
large-scale transmission systems. This motivates the research on seeking
surrogate models that are much smaller than the original UC problems yet
ensure equivalence of constrains’ binding situation at optimal solution which
should be the same as the original one. On the one hand, the active
constraints can be kept with the constraint generation technique, where the
violated constraints are gradually added into the surrogate model until the
solution to the surrogate model is feasible to the original UC problem [2]. On
the other hand, the redundant constraints can be screened out by solving a
relaxed optimization for each constraint to identify whether the upper or
lower bound of each constraint is redundant [3]. After securely screening
constraints, the promise is to reduce the computation time significantly for
the reduced UC problems which only involve a subset of original physical
constraints in the UC problem.
Although the total number of UC constraints is theoretically prohibitive with
the above methods, empirical evidence shows that a vast majority of the
constraints are still redundant and only a smaller subset of constraints could
be binding (equality holds) given the region of load profiles [4]. Recent
efforts have explored the potential of using cost-driven [5] and data-driven
[6] ways to handle these issues. [5] adds an operational cost upper bound to
the standard relaxed optimization so as to further narrow the subset of
constraints; [6] proposes to use the k-nearest neighbor as a screening model
to classify the constraint binding statuses using historical samples, yet both
methods have tradeoffs in accuracy and efficiency.
Indeed, the availability of historical power system operation records can
provide rich information regarding operational decisions, constraint patterns
and load profiles [7, 8]. Note that the system operating pattern corresponds
to a unique region of load. The reexamination of the inactive or redundant
constraints is needed, once the system operating pattern changes due to the
change of the load. As such, [5] determines the cost upper bound for different
aggregated net demand, i.e., load level. [9] considers the constraint
screening for varying load ranges and results in a surrogate model that is
applicable in a long operation period. In [10], the spatial correlation
between nodal demands is taken into consideration for identifying the umbrella
constraints.
Figure 1: The schematic of our proposed method. We use the historical records
involving load vector and UC cost to train a neural network model serving as
cost model (a), and further use the trained model to get the upper bound of UC
cost so as to screen constraints and obtain a surrogate model (b).
However, the data-driven method used in [6, 7, 8] directly using machine
learning (ML) predictions to classify if constraints are redundant, which may
have simplicity but be hard to respond to the change of load region and
without guarantee of equivalence. Meanwhile, the cost-driven method proposed
in [5] can promise the solution accuracy, but it adds integer constraints to
the constraint screening problem, making the screening procedure cumbersome to
solve. Thus, it is required to come up with a method considering load region
to achieve the accuracy-simplicity balance for the constraints screening
problem.
In this work, we investigate the potential of designing cost-driven paradigms
to efficiently screen constraints in the unit commitment problem. We try to
bridge the insights given by machine learning algorithms and the standard
optimization-based constraint screening processes. Our approach also utilizes
the historical data, but here ML prediction is used to help optimization-based
method screen each constraint more efficiently. Specifically, we train a
neural network to predict the optimal costs given load inputs [11]. Then we
can conveniently upper-bound the search space of constraint screening problem
by integrating the cost level predicted and optimized via the trained neural
network model, where the formulation of the screening can be treated as a
simple linear programming problem. Besides, our method can be flexibly
integrated to screen constraints for either one specific load vector or for a
given region of load, which we term as _sample-aware_ [3] and _sample-
agnostic_ [9] constraint screening respectively. In sample-aware case, our
proposed method can screen out about 90.03% of the redundant constraints. In
sample-agnostic case, we can realize the UC cost prediction with relative
error less than 1% and remove the redundant constraints without cost error
along with saving the solution time.
## II Problem Setup
### II-A UC Problem Formulation
In this paper, we assume the system operators need to decide both the ON/OFF
statuses as well as dispatch level for all generators. As the realistic UC
problem requires to take start-up and shut-down costs and logic constraints as
well as ramp constraints into considerations, which make the analysis of
multi-step constraints more complicated, we firstly consider the single-period
UC problem as follows:
$\displaystyle J(\bm{\ell})=\min_{\mathbf{u},\mathbf{x},\mathbf{f}}\quad$
$\displaystyle\sum_{i=1}^{n}c_{i}x_{i}$ (1a) s.t. $\displaystyle
u_{i}\underline{x}_{i}\leq x_{i}\leq u_{i}\bar{x}_{i},\quad i=1,...,n$ (1b)
$\displaystyle-\overline{\mathbf{f}}\leq\mathbf{Kf}\leq\overline{\mathbf{f}}$
(1c) $\displaystyle\mathbf{x}+\overline{\mathbf{A}}\mathbf{f}=\bm{\ell}$ (1d)
$\displaystyle u_{i}\in\\{0,1\\},\quad i=1,...,n.$ (1e)
In the UC problem, we optimize over the generator statuses $\mathbf{u}$, the
generator dispatch $\mathbf{x}$ and the line power flow $\mathbf{f}$ to find
the least-cost solutions with cost denoted as $J(\bm{\ell})$ in the objective
function (1a). $c_{i}$ denotes the cost coefficient. Constraint (1b), (1c) and
(1d) denotes the generation bound, the flow bound and the nodal power balance
respectively. Note that the power flows are modeled as a DC approximation,
while the phase angles are absorbed into the fundamental flows
$\mathbf{f}\in\mathbb{R}^{n-1}$ [11, 12]; $K$ and $\bar{\mathbf{A}}$ map such
fundamental flows to flow constraints and nodal power balance respectively.
(1e) enforces the binary constraint of generator statuses, where $u_{i}=1$
indicates the generator is on.
### II-B Constraint Screening
Since there are many redundant line flow constraints when seeking the optimal
solution of UC problem with given load region or specific load vector, which
brings unnecessary computation burden, constraint screening for the line flow
constraints can be meaningful. Similar to [3], we relax the integer variables
$\mathbf{u}$ in (1) as continuous variables in $[0,1]$, and the screening
approach requires to iteratively solve the relaxed optimization problem to
find the upper and lower flow values on each transmission line. If the upper
and lower bound cannot be reached by the relaxed optimization problem, we can
safely screen out that line flow constraint.
For the case that the load region $\mathcal{L}$ is known, a _sample-agnostic
constraint screening problem_ can be formulated for a group of operating
scenarios, which can be given as follows,
$\displaystyle\max_{\mathbf{u},\mathbf{x},\mathbf{f},\bm{\ell}}/\min_{\mathbf{u},\mathbf{x},\mathbf{f},\bm{\ell}}\quad$
$\displaystyle f_{j}$ (2a) s.t. $\displaystyle u_{i}\underline{x}_{i}\leq
x_{i}\leq u_{i}\bar{x}_{i},\quad i=1,...,n$ (2b)
$\displaystyle-\overline{\mathbf{f}}_{\mathcal{F}/j}\leq\mathbf{K}_{\mathcal{F}/j}\tilde{\mathbf{f}}\leq\overline{\mathbf{f}}_{\mathcal{F}/j}$
(2c) $\displaystyle\mathbf{x}+\overline{\mathbf{A}}\mathbf{f}=\bm{\ell}$ (2d)
$\displaystyle 0\leq u_{i}\leq 1,\quad i=1,...,n$ (2e)
$\displaystyle\bm{\ell}\in\mathcal{L};$ (2f)
where $\mathcal{F}/j$ denotes all remaining entries of vectors or matrix which
excludes those correspond to $f_{j}$.
On the contrary, when the specific load vector is available, we can conduct
the following _sample-aware constraint screening_ :
$\displaystyle\max_{\mathbf{u},\mathbf{x},\mathbf{f}}/\min_{\mathbf{u},\mathbf{x},\mathbf{f}}\quad$
$\displaystyle f_{j}$ (3a) s.t. $\displaystyle u_{i}\underline{x}_{i}\leq
x_{i}\leq u_{i}\bar{x}_{i},\quad i=1,...,n$ (3b)
$\displaystyle-\overline{\mathbf{f}}_{\mathcal{F}/j}\leq\mathbf{K}_{\mathcal{F}/j}\tilde{\mathbf{f}}\leq\overline{\mathbf{f}}_{\mathcal{F}/j}$
(3c) $\displaystyle\mathbf{x}+\overline{\mathbf{A}}\mathbf{f}=\bm{\ell}$ (3d)
$\displaystyle 0\leq u_{i}\leq 1,\quad i=1,...,n;$ (3e)
where $\bm{\ell}$ is a known load vector for UC problem.
The above formulations are both optimization-based approaches, which seek to
find the limit of the flow while keeping all other flow and generation
constraints satisfied. However, this approach still allows some line flow
values causing unrealistic cost to reach the upper or lower bounds, and thus
there are more redundant constraints reserved[5].
Therefore, it is interesting to consider the economical goal in the original
UC problem, minimizing the system cost, to further safely screen out
constraints.
## III Learning to Predict UC Costs
### III-A Learning Cost Predictors for Unit Commitment Problem
As mentioned before, screening without cost objectives enlarge the possible
value range of load variables $f_{j}$, which leads to conservative screening
and keep more constraints as non-redundant. To close such gap, in this paper,
we investigate if it is possible to tighten the search space of constraint
screening by adding a cost constraint in the form of
$\sum_{i=1}^{n}c_{i}x_{i}\leq\bar{C}$, where $\bar{C}$ is the upper bound
whose value needs to be determined in the following sections.
To achieve such goal, the adopted method should approximate the map between
load input and system costs $J(\bm{\ell})$ well along with predicting system
costs efficiently. Thus, in this paper, we use a neural network (NN) to find
the upper bound. To train the NN model, we utilize the past record of UC
solutions and the training loss between the output of NN model and the actual
cost defined as follows,
$\displaystyle L:=\left\|f_{NN}(\bm{\ell})-J(\bm{\ell})\right\|_{2}^{2};$ (4)
where $f_{NN}(\bm{\ell})$ denotes the NN model given load inputs.
In the next subsection, we detail how to connect NN’s predicted costs to the
constraint screening problems.
### III-B Tightening the Search Space for Constraint Screening
Note that the ML model is not directly applied for making operation or
dispatch decisions, and alternatively, we are treating the ML prediction as a
constraint to reduce the search space of optimization-based constraint
screening problems (2) and (3). With such design, the resulting optimization
problem can still find feasible decisions for the original UC problem. We can
then add the neural network’s prediction as an additional constraint to the
original constraint screening problem to further restrict the search space for
each transmission’s flow bounds.
In sample-agnostic case, to ensure feasibility of the constraint screening
problem after adding the cost constraint for the whole load space along with
restricting the searching space effectively, we need to find a predicted cost
given by NN which can serve as the upper bound. Then projected gradient ascent
(PGA) algorithm can be adopted to achieve this goal. PGA can find the upper
bound iteratively by moving $\bm{\ell}$ in the gradient direction at each step
along with projecting it onto $\mathcal{L}$, and the details are listed in
Algorithm 1. Besides, in practice, the real load samples may be out of
distribution, and incur costs which are over the upper bound and thus causing
screening failure. Therefore, we use a relaxation parameter $\epsilon$ to
adjust the obtained upper bound $\texttt{PGA}(f_{NN}(\bm{\ell}))$ and
integrate it to (2). Then, we can get the following sample-agnostic screening
problem considering the cost constraint,
$\displaystyle\max_{\mathbf{u},\mathbf{x},\mathbf{f},\bm{\ell}}/\min_{\mathbf{u},\mathbf{x},\mathbf{f},\bm{\ell}}\quad$
$\displaystyle f_{j}$ (5a) s.t.
$\displaystyle\eqref{Screening:gen}\eqref{Screening:flow}\eqref{Screening:balance}\eqref{Screening:u}\eqref{Screening:load}$
(5b)
$\displaystyle\sum_{i=1}^{n}c_{i}x_{i}\leq\texttt{PGA}(f_{NN}(\bm{\ell}))(1+\epsilon).$
(5c)
0: Load distribution $\mathcal{L}$, trained NN model $f_{NN}(\bm{\ell})$, step
size $\beta$.
0: Upper bound $\texttt{PGA}(f_{NN}(\bm{\ell}))$.
0: Random load vector $\bm{\ell}^{(0)}\in\mathcal{L}$, $k=0$.
1: while $\bm{\ell}^{(k)}$ doesn’t converge do
2: Update: $\texttt{PGA}(f_{NN}(\bm{\ell}))\leftarrow f_{NN}(\bm{\ell}^{(k)})$
3: Calculate gradient $\nabla_{\bm{\ell}}f_{NN}(\bm{\ell})$
4: Update:
$\bm{\ell}^{(k+1)}\leftarrow\texttt{Proj}_{\mathcal{L}}(\bm{\ell}^{(k)}+\beta\nabla_{\bm{\ell}}f_{NN}(\bm{\ell}))$
5: $k\leftarrow k+1$
6: end while
7: Return $\texttt{PGA}(f_{NN}(\bm{\ell}))$
Algorithm 1 Projected Gradient Ascent Algorithm
In sample-aware case, we predict and still relax the UC cost for each specific
sample, as the predicted cost may be lower than the actual cost, which can
result in an infeasible adjusted screening problem for the investigated
sample. Then we add the relaxed cost to (3), and the adjusted sample-agnostic
screening problem can be formulated as follows,
$\displaystyle\max_{\mathbf{u},\mathbf{x},\mathbf{f}}/\min_{\mathbf{u},\mathbf{x},\mathbf{f}}\quad$
$\displaystyle f_{j}$ (6a) s.t.
$\displaystyle\eqref{Screening2:gen}\eqref{Screening2:flow}\eqref{Screening2:balance}\eqref{Screening2:u}$
(6b) $\displaystyle\sum_{i=1}^{n}c_{i}x_{i}\leq
f_{NN}(\bm{\ell})(1+\epsilon).$ (6c)
Note that the upper bound given by the NN model will be a constant given load
region or specific load vector, so the screening problems (5) and (6) can be
treated as linear programming problems which are efficient to solve.
## IV Case Study
Figure 2: The NN’s predicted costs of different load levels. Figure 3:
Percentage of the reduced constraints (upper) and relative solution time
(lower) of sample-agnostic screening on different load variation ranges.
To evaluate the performance of the proposed constraint screening algorithm, we
take the original optimization-based method as benchmark, and also compare our
method with KNN method in this section. The predicting accuracy of the
learning cost predictors, the computational efficiency and the solution
accuracy of the reduced UC problem are examined over a wide range of problem
settings. The details are given at
https://github.com/Hexuan085/UC_SCREENING_ML.
### IV-A Simulation Setups
We carry out the numerical experiments on IEEE 14-bus, IEEE 39-bus and IEEE
118-bus power systems. For each system, we consider the load with 0%, 25%,
50%, 75% and 100% variation which is defined as $r$ around the average nominal
values $\overline{\bm{\ell}}$. When investigating the sample-aware constraint
screening, the load level is known in our setting and is defined as
$\overline{L}$. Then the load region $\mathcal{L}$ considered here can be
represented as:
$\displaystyle(1-r)\overline{\bm{\ell}}\leq$
$\displaystyle\bm{\ell}\leq(1+r)\overline{\bm{\ell}}$ (7a)
$\displaystyle\sum_{i=1}^{n}$ $\displaystyle l_{i}=\overline{L}.$ (7b)
TABLE I: Comparisons of the relative cost error and relative solution time for IEEE 118-bus system | Total cost error (%) | Total solution time (%)
---|---|---
Method Range | 25 | 50 | 75 | 100 | 25 | 50 | 75 | 100
KNN5 | 5.3 | 9.8 | 3.3 | 4.5 | 18.3 | 16.5 | 16.4 | 16.7
KNN10 | 0.8 | 0 | 0.2 | 1.7 | 18.5 | 19.1 | 19.7 | 20.3
Benchmark | 0 | 0 | 0 | 0 | 31.4 | 36.1 | 40.8 | 45.6
Ours | 0 | 0 | 0 | 0 | 21.7 | 33.4 | 39.5 | 45.7
TABLE II: Percentage of average reduced constraints of Sample-aware screening | Num.Gen. | Num.Lines | Benchmark | Ours | Actual | $\epsilon$
---|---|---|---|---|---|---
14-bus | 5 | 20 | 92.5% | 97.5% | 97.5% | 0.01
39-bus | 10 | 46 | 84.7% | 86.9% | 92.4% | 0.03
118-bus | 54 | 186 | 81.5% | 85.7% | 97.3% | 0.01
To generate samples for training and validating the neural network model and
KNN model, we use uniform distribution to get different load vectors
$\bm{\ell}\in\mathcal{L}$ for sample-agnostic case or random $\bm{\ell}$ for
sample-aware case, and then solve (1) for all loads. The UC cost and the
binding situation of each line flow constraint are recorded. Under each
setting, we solve and collect 10,000 samples for each neural network with 20
percentage of generated data split as test samples, while for KNN we solve
2,000 samples only based on 118-bus system due to computation burden.
Moreover, when evaluating the screening performance of the benchmark, the
proposed method and KNN, we use the same validation data and consider 100
samples for each validation case. The used neural networks all have ReLU
activation units and 4 layers, and corresponding neurons on each hidden layer
are 50, 30, 30. We feed the load vector as input for the neural network and
the output is the corresponding UC cost, then we further use the cost to solve
(5) and (6).
All simulations have been carried out on a laptop with a 2.50 GHz processor
and 16G RAM. Specifically, all the optimization problems are modeled using
Python and solved with CVXPY[13] powered by GPLK_MI solver [14].
### IV-B Simulation results
To ensure the effectiveness and scalability of the proposed cost predictors,
we train the NNs for different load levels, and randomly select a specific
load vector from each load level to predict the corresponding cost. The
results are shown in Fig. 2, from which it can be seen that the predicted
costs are almost equal to the actual costs obtained by solving (1) with the
relative error less than 1%. Note that the predicted cost can be lower than
the actual cost, so it is reasonable to consider $\epsilon$ in (6c) to ensure
feasibility.
Using the NN models trained for the setting load levels and the PGA algorithm,
we can get the upper bounds in (5c) so as to conduct the sample-agnostic
constraint screening. Then, this method is compared with the benchmark and KNN
method, which is carried out on 118-bus system and the results are shown in
Table I and Fig. 3. According to Table I where the cost error and the solution
time of the reduced problems are relative to the result of the original
problem (1), KNN methods can reduce more solution time than other methods. The
total cost errors of the case K=5 are lower than that of the case K=10, while
the situation of the solution time is on the contrary. Though requiring more
solution time, the benchmark and our methods can promise the solution accuracy
without cost error.
Meanwhile, our method can screen more constraints and save more solution time
than the benchmark in all cases of investigated power systems with load
variation range from 0% to 50% according to Fig. 3. In the cases of 39-bus and
118-bus systems with 75% to 100% load range, the performance of the two
methods are very close. This may be due to the increasing patterns of non-
redundant constraints with larger load variation range, i.e., the percentage
of the redundant constraints decreases when widening the load variation as
shown in Fig. 3.
Furthermore, according to Table II, the average percentage of redundant
constraints can reach 92.4% to 97.3% for a specific load vector. Then, with
the sample-aware constraint screening, most of the redundant constraints can
be removed. Specifically, the benchmark method can screen 81.5% to 92.5% of
total constraints as redundant, while our method defined in (6) can screen
85.7% to 95.1% with setting $\epsilon$ properly.
The above results show the following positive effects of our method:
1. 1.
Capturing the mapping between load vector and UC cost well at different load
levels.
2. 2.
Realizing the trade-off between computational efficiency and solution accuracy
in the sample-agnostic case.
3. 3.
Achieving higher screening efficiency in the sample-aware case.
## V Conclusion and Future Works
In this paper, we introduce a novel usage of machine learning to help screen
redundant constraints. The neural networks are trained to predict UC cost so
as to integrate the cost constraints to original screening problem
efficiently. With the cost constraints, the search space of constraint
screening can be sufficiently tightened. Since our method does not necessarily
yield a minimal set of active constraints for the underlying UC problem, in
the future work we would like to seek theoretical understandings about the set
of constraints and investigate how the proposed techniques can be generalized
to multi-step UC problem with nonlinear constraints. We also plan to explore
the potential of making sample-agnostic case serve as the warm-start for the
sample-aware case.
## References
* [1] J. D. Glover, M. S. Sarma, and T. Overbye, _Power system analysis & design, SI version_. Cengage Learning, 2012\.
* [2] A. S. Xavier, F. Qiu, F. Wang, and P. R. Thimmapuram, “Transmission constraint filtering in large-scale security-constrained unit commitment,” _IEEE Transactions on Power Systems_ , vol. 34, no. 3, pp. 2457–2460, 2019.
* [3] Q. Zhai, X. Guan, J. Cheng, and H. Wu, “Fast identification of inactive security constraints in scuc problems,” _IEEE Transactions on Power Systems_ , vol. 25, no. 4, pp. 1946–1954, 2010.
* [4] A. J. Ardakani and F. Bouffard, “Identification of umbrella constraints in dc-based security-constrained optimal power flow,” _IEEE Transactions on Power Systems_ , vol. 28, no. 4, pp. 3924–3934, 2013.
* [5] Á. Porras, S. Pineda, J. M. Morales, and A. Jiménez-Cordero, “Cost-driven screening of network constraints for the unit commitment problem,” _arXiv preprint arXiv:2104.05746_ , 2021.
* [6] S. Pineda, J. M. Morales, and A. Jiménez-Cordero, “Data-driven screening of network constraints for unit commitment,” _IEEE Transactions on Power Systems_ , vol. 35, no. 5, pp. 3695–3705, 2020.
* [7] A. Ardakani and F. Bouffard, “Prediction of umbrella constraints,” in _2018 Power Systems Computation Conference (PSCC)_. IEEE, 2018, pp. 1–7.
* [8] Á. S. Xavier, F. Qiu, and S. Ahmed, “Learning to solve large-scale security-constrained unit commitment problems,” _INFORMS Journal on Computing_ , vol. 33, no. 2, pp. 739–756, 2021.
* [9] L. A. Roald and D. K. Molzahn, “Implied constraint satisfaction in power system optimization: The impacts of load variations,” in _2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)_. IEEE, 2019, pp. 308–315.
* [10] M. Awadalla and F. Bouffard, “Influence of stochastic dependence on network constraints screening for unit commitment,” _arXiv preprint arXiv:2208.12139_ , 2022.
* [11] Y. Chen, L. Zhang, and B. Zhang, “Learning to solve dcopf: A duality approach,” _Electric Power Systems Research_ , vol. 213, p. 108595, 2022\.
* [12] D. Bertsimas and J. N. Tsitsiklis, _Introduction to linear optimization_. Athena Scientific Belmont, MA, 1997, vol. 6.
* [13] S. Diamond and S. Boyd, “Cvxpy: A python-embedded modeling language for convex optimization,” _Journal of Machine Learning Research_ , vol. 17, no. 83, pp. 1–5, 2016. [Online]. Available: http://jmlr.org/papers/v17/15-408.html
* [14] A. Makhorin, “Glpk (gnu linear programming kit),” _http://www.gnu.org/s/glpk/glpk.html_ , 2008.
|
# CUNI Systems for the WMT 22 Czech-Ukrainian Translation Task
Martin Popel11footnotemark: 1 Jindřich Libovický11footnotemark: 1 Jindřich
Helcl11footnotemark: 1
Charles University, Faculty of Mathematics and Physics
Institute of Formal and Applied Linguistics
Malostranské náměstí 25, 118 00 Prague, Czech Republic
<EMAIL_ADDRESS>
###### Abstract
††footnotetext: ∗ The author order was determined by a coin toss.
We present Charles University submissions to the WMT 22 General Translation
Shared Task on Czech-Ukrainian and Ukrainian-Czech machine translation. We
present two constrained submissions based on block back-translation and tagged
back-translation and experiment with rule-based romanization of Ukrainian. Our
results show that the romanization only has a minor effect on the translation
quality. Further, we describe Charles Translator, a system that was developed
in March 2022 as a response to the migration from Ukraine to the Czech
Republic. Compared to our constrained systems, it did not use the romanization
and used some proprietary data sources.
## 1 Introduction
How fast can the machine translation (MT) community react to a sudden need of
a high-quality MT system which was previously under low demand? This question
motivated the new task at the WMT this year, which is Czech-Ukrainian
translation.
Both languages belong to the Slavic language family (Czech is western Slavic,
Ukrainian is eastern Slavic), and share some lexical and structural
characteristics. Unlike Czech, which uses the Latin script, Ukrainian uses its
variant of the Cyrillic alphabet.
We submit three systems to the WMT 22 General Translation Shared Task for this
language pair in each translation direction. The first system, CUNI-JL-JH,
implemented in Marian (Junczys-Dowmunt et al., 2018), uses tagged back-
translation and is a result of our experiments with romanization of Ukrainian.
Our second system, CUNI-Transformer, implemented in Tensor2Tensor (Vaswani et
al., 2018), uses block back-translation. Finally, we submit an unconstrained
system, Charles Translator, implemented in Tensor2Tensor, which has been
developed in spring 2022 as a response to the crisis caused by the Russian
invasion of Ukraine and the following migration wave.
## 2 Constrained WMT Submissions
We submitted two systems in each translation direction that use the same
parallel and monolingual data, but different techniques and different
toolkits. This section first describes the shared data processing steps and
then the specifics of each of the submissions in separate subsections.
### 2.1 Training Data
We use all parallel data allowed in the constrained task, along with 50
million Czech and 58 million Ukrainian sentences of monolingual data. In the
following paragraphs we describe the data cleaning steps when preparing the
training data. We further experiment with romanization of the Ukrainian
Cyrillic alphabet and with artificial noising of the data.
#### Parallel data.
The data for the constrained translation task consist of OPUS corpora
(Tiedemann, 2012) that have a Czech-Ukrainian part, WikiMatrix (Schwenk et
al., 2021) and the ELRC EU acts in Ukrainian.111 https://elrc-
share.eu/repository/search/?q=EU+acts+in+Ukrainian
We clean the parallel data using rule-based filtering in the following way:
1. 1.
Filter out non-printable and malformed UTF-8 characters.
2. 2.
Detect language using FastText (Grave et al., 2018), only keep Czech and
Ukrainian sentences on their respective source/target sides.
3. 3.
Only keep sentence pairs with character length ratio between $0.67$ and $1.5$
if longer than 10 characters.
4. 4.
Apply hand-crafted regular expressions to filter out the frequent errors, such
that the system does not attempt to translate e-mail addresses, currencies,
etc. In addition, regular expressions check translations of names of
Czech222https://uk.wikipedia.org/wiki/ Міста_Чехії and Ukrainian333
https://cs.wikipedia.org/wiki/Seznam_měst_na_Ukrajině municipalities
downloaded from Wikipedia.
We omit steps 2 and 3 for the XLEnt corpus, which seems to be very clean and
consist of short phrases (likely to get misclassified for language).
The sizes of the used parallel data sources before and after cleaning are
presented in Table 1.
Source | Original | Filtered
---|---|---
bible-uedin | 8 k | 8 k
CCMatrix | 3,992 k | 3,884 k
EUbookshop | 2 k | 1 k
GNOME | 150 | 81
KDE4 | 134 k | 64 k
MultiCCAligned | 1,607 k | 1,199 k
MultiParaCrawl | 1,773 k | 1,606 k
OpenSubtitles | 731 k | 273 k
QED | 161 k | 138 k
Tatoeba | 3 k | 2 k
TED2020 | 115 k | 106 k
Ubuntu | 0.2k | 0.2k
wikimedia | 2 k | 2 k
XLEnt | 695 k | 695 k
WikiMatrix | 105 k | 99 k
ELRC EU Acts | 130 k | 108 k
Total | 9,457 k | 8,186 k
Table 1: Sizes of parallel data sources (number of sentence pairs).
#### Monolingual data.
The overview of the monolingual data sources is in Table 2. For Czech, we use
the Czech monolingual portion of the CzEng 2.0 corpus (Kocmi et al., 2020).
For Ukrainian, we used all resources, available for WMT, i.e., the NewsCrawl,
the Leipzig Corpora (Biemann et al., 2007), UberText corpus (Khaburska and
Tytyk, 2019) and Legal Ukrainian Crawling by ELRC. The Uber corpus and the
Ukrainian Legal corpus are distributed tokenized with removed punctuation. We
automatically restored the punctuation and detokenized the models using a
lightweight Transformer model (Vaswani et al., 2017; Base model with 3 layers,
8k vocabulary) trained on the NewsCrawl corpus.
For Ukrainian, we only keep sentences shorter than 300 characters. For Czech,
we keep all sentence lengths from the CzEng corpus (up to 1400 characters).
For both languages, we remove non-printable and malformed UTF-8 characters.
Source | Original | Filtered
---|---|---
Czech | CzEng 2.0 | | 50.6 M
Ukrainian | NewsCrawl | 0 2.3 M | 0 2.0 M
Leipzig Corpora | 0 9.0 M | 0 7.6 M
UberText Corpus | 47.9 M | 41.2 M
ELRC Legal | 0 7.6 M | 0 7.2 M
Total | 66.8 M | 58.1 M
Table 2: Monolingual data sizes in number of sentences before and after
filtering.
#### Romanization.
We develop a reversible romanization than transcribes between the Ukrainian
and Czech alphabets. For example, Зараз у нас є 4-місячні миші is transcribed
to Zaraz u nas je 4-misjačni myši. This way the model can better exploit the
lexical similarities between the two languages (e.g. миші should be translated
to Czech as myši), while keeping all the necessary information to reconstruct
the original Cyrillic text. Note that the transcription of Cyrillic changes
when changing the target language, reflecting the phonology of that language
(e.g. ш transcribes to sh in English, but š in Czech). We introduce special
tags for words and characters that are written in Latin script found in
Cyrillic text. The romanization is specifically designed for Ukrainian (e.g. и
transcribes to y, not i as would be the case in Russian), so its reversibility
occasionally fails for Russian names.
#### Artificial noise.
We apply synthetic noise on the source side that should simulate the most
frequent deviations from the standard orthography (missing capitalization,
lower- or upper-casing parts of the sentences, missing or additional
punctuation).
All scripts for training data processing are available at
https://github.com/ufal/uk-cs-data-scripts. We use Flores 101 (Goyal et al.,
2022) development set for validation.
### 2.2 Tagged-back-translation-based System (CUNI-JL-JH)
The CUNI-JL-JH submission is a constrained system and uses the data described
in the paragraphs above. We train the system in 3 iterations of tagged back-
translation (Caswell et al., 2019) with greedy decoding. Each iteration, we
filter the back-translated data using Dual Cross-Entropy filtering (Junczys-
Dowmunt, 2018) when keeping $40,930,735$ synthetic sentences, i.e., 5$\times$
the size of clean authentic parallel data.
The first two back-translation iterations were done with the Cyrillic script
on the Ukrainian side. In the final back-translation iteration, we performed
romanization and noising of the source side. We train three models with random
initialization and submit the ensemble.
For all iterations, we used a Transformer Big model with tied embeddings and a
shared SentencePiece vocabulary size of $32$k (fitted on 5M randomly sampled
sentences; with sampling at the training time, $\alpha$=0.1; Kudo and
Richardson, 2018). We set the learning rate to $0.0003$ and use $8,000$ warm-
up steps. We initialize the models randomly in each back-translation
iteration.
For validation, we use greedy decoding. At test time, we decode with beam
search with beam size of 4 and length normalization of 1.0 (estimated on
validation data).
The system is implemented using Marian (Junczys-Dowmunt et al., 2018).
#### Negative results.
We experimented with Dual-Cross-Entropy filtering (Junczys-Dowmunt, 2018) for
parallel data selection and came to inconclusive results. Therefore, we used
all parallel data after rule-based filtering.444 Note that we use Dual-Cross-
Entropy for filtering the monolingual data, as described in the first
paragraph of this section, but we have not done any experiments with keeping
all the monolingual data.
Additionally, we experimented with MASS-style (Song et al., 2019) pre-training
using monolingual data only and continue with training on parallel data. We
were not able to find a hyper-parameter setting where the pre-trained model
would outperform the models trained from random initialization. Therefore, we
only use model trained from random initialization.
### 2.3 Block back-translation System (CUNI-Transformer)
The CUNI-Transformer submission is also constrained, trained on the same data
as CUNI-JL-JH. The system was trained in the same way as the sentence-level
English-Czech CUNI-Transformer systems submitted to previous years of WMT
shared tasks (Popel, 2018, 2020; Gebauer et al., 2021). It uses Block back-
translation (BlockBT) (Popel et al., 2020), where blocks of authentic (human-
translated parallel) and synthetic (backtranslated) training data are not
shuffled together, but checkpoint averaging is used to find the optimal ratio
of checkpoints from the authentic and synthetic blocks (usually 5:3). The
uk$\rightarrow$cs system was trained with a non-iterated BlockBT (i.e. cs-mono
data was translated with an authentic-only trained baseline). The
cs$\rightarrow$uk was trained with two iterations of BlockBT (i.e. the uk-mono
data was translated with the above mentioned uk$\rightarrow$cs non-iterated
BlockBT system). We had not enough time to train more iterations and apply
noised training and romanization. The system was implemented using
Tensor2Tensor (Vaswani et al., 2018).
#### Inline casing.
We experimented with Inline casing (InCa) pre-processing in the
cs$\rightarrow$uk direction. The main idea is to lowercase all training data
and insert special tags <titlecase> and <all-uppercase> before words in the
respective case, so that the original casing can be reconstructed (with the
exception of words like McDonald or iPhone, which use different casing
patterns than all-lowercase, all-uppercase and titlecase). We improved this
approach by remembering the most frequent casing variant of each (lowercased)
word in the training data. The most frequent variant does not need to be
prefixed with any tag, which makes the length of training sequences shorter.
We also introduced a third tag <all-lowercase> for encoding all-lowercased
words whose most frequent variant is different. For example, if the InCa
vocabulary includes only two items: iPhone and GB, sentence My iPhone 64GB and
iPod 64 GB or 32 gb will be encoded as <titlecase> my iphone <all-uppercase>
64gb and iPod 64 gb or 32 <all-lowercase> gb. Note that iPod was kept in the
original case because it was not included in the InCa vocabulary and it does
not match any of the three “regular” casing patterns. We applied InCa on both
the source and target side and experimented with training the InCa vocabulary
on the authentic data only or on authentic plus synthetic (monolingual
backtranslated).
Inline casing showed promising results in preliminary experiments (without
backtranslation), especially when combined with romanization and artificial
noise in training. Unfortunately, we had not enough time to train the
backtranslated model long enough, so we submitted it only as a contrastive run
and plan to explore it more in future.
## 3 Charles Translator for Ukraine
Charles Translator for Ukraine is a free Czech-Ukrainian online translation
service available for public at https://translator.cuni.cz and as an Android
app. It was developed at Charles University in March 2022 to help refugees
from Ukraine by narrowing the communication gap between them and other people
in Czechia. Similarly to CUNI-Transformer, it is based on Transformer and
iterated Block back-translation (Popel et al., 2020). The training used
source-side artificial noising, but no romanization and no inline casing. It
was trained on most (but not all) of the training data provided by WMT plus
about one million uk-cs sentences from the InterCorp v14 corpus (Čermák and
Rosen, 2012; Kotsyba, 2022), so this submission is unconstrained.
## 4 Results
In this section, we report BLEU scores on the Flores 101 development set that
we used to make our decisions about the system development and the final
automatic scores. Note that the validation set is very different from the test
set. The validation set consists of clean and rather complicated sentences
from Wikipedia articles, whereas the WMT 22 test set is noisy user-generated
text from the logs of the production deployment of Charles Translator.555 The
test set only contains sentences from users who provided their consent for
this usage and the sentences were pseudonymized.
#### Tagged BT systems.
Table 3 shows validation BLEU scores from the first three iterations of back-
translation. The second and third iteration did not bring substantial
improvements, so we decided not to further iterate.
Model | cs$\rightarrow$uk | uk$\rightarrow$cs
---|---|---
Authentic only | 20.91 | 22.95
BT iteration 1 | 21.69 | 23.70
BT iteration 2 | 21.87 | 23.98
BT iteration 3 (seed 1) | 21.53 | 23.76
Table 3: Validation BLEU scores for the first two iterations of BT for the
tagged BT systems.
Table 4 shows validation BLEU scores from the last (third) BT iteration –
three independently trained systems and their ensembles, and the Cyrillic and
romanized versions of the data. In general, ensembling only brings a small
improvement. Romanization does not bring a significant difference compared to
using the Cyrillic script. In the Czech-to-Ukrainian direction, the best
system was the ensemble of the romanized systems. However, in the Ukrainian-
to-Czech direction, the best system was one of the Cyrillic systems that used
accidentally 3 times higher batch size than the remaining ones. This result
suggests that the batch size has a much stronger effect than most of the
techniques that we experimented with and that we might have reached better
results if we opted for higher batch size.
Model | cs$\rightarrow$uk | uk$\rightarrow$cs
---|---|---
Cyrillic | Seed 1 | 21.53 | 23.76
Seed 2 | 22.28 | 25.10
Seed 3 | 21.96 | 24.39
Ensemble | 22.45 | 24.86
Romanized | Seed 1 | 21.42 | 23.99
Seed 2 | 21.76 | 23.91
Seed 3 | 22.37 | 24.18
Ensemble | 22.62 | 24.22
Table 4: Validation BLEU scores for the last (i.e., the third) iteration of BT comparing romanized and original script. | cs$\rightarrow$uk | uk$\rightarrow$cs
---|---|---
System | BLEU | chrF | COMET | BLEU | chrF | COMET
Best constrained (HuaweiTSC/AMU) | 36.0 | 62.6 | 0.994 | 37.0 | 60.7 | 1.048
CUNI-Transformer | 35.0 | 61.6 | 0.873 | 35.8 | 59.0 | 0.885
CUNI-JL-JH | 34.8 | 61.6 | 0.900 | 35.1 | 58.7 | 0.890
Best unconstrained (Lan-Bridge/Online-B) | 38.1 | 64.0 | 0.942 | 36.5 | 60.4 | 0.965
Charles Translator | 34.3 | 61.5 | 0.908 | 35.9 | 59.0 | 0.901
Table 5: Final automatic results on the WTM22 test data compared to the best
overall score achieved in each metric.
#### Results on WMT test.
Automatic evaluation on the WMT22 test set is presented in Table 5. Both the
constrained systems and Charles Translator show comparable results. The tagged
BT system reaches a slightly higher COMET score than the Block BT system,
however, Czech-Ukrainian was not in the training data of the COMET score,
which make the score unreliable for this particular language pair. For Czech-
to-Ukrainian, Charles Translator reaches a slightly higher COMET score and
slightly lower BLEU and chrF scores than both the constrained systems, but we
do not consider such small differences of automatic metrics relevant.
## 5 Conclusions
We presented Charles University submissions to the WMT 22 General Translation
Shared Task on Czech-Ukrainian and Ukrainian-Czech machine translation. We
present two constrained submissions based on block back-translation and tagged
back-translation and experiment with rule-based romanization of Ukrainian.
Further, we describe Charles Translator, a system that was developed in March
2022 as a response to the migration from Ukraine to the Czech Republic.
Compared to our constrained systems, it did not use the romanization and used
some proprietary data sources.
Our results show that the romanization only has a minor effect on the
translation quality, compared to machine-learning aspects that affect
translation quality. Block back-translation seems to deliver slightly better
results that tagged back-translation, however the differences are only small.
## Acknowledgements
This work has been supported by the Ministry of Education, Youth and Sports of
the Czech Republic, Project No. LM2018101 LINDAT/CLARIAH-CZ, by the Czech
Science Foundation (GACR) grant 20-16819X (LUSyD), and by the European
Commission via its Horizon 2020 research and innovation programme no. 870930
(WELCOME), Horizon Europe Innovation programme no. 101070350 (HPLT).
## References
* Biemann et al. (2007) Chris Biemann, Gerhard Heyer, Uwe Quasthoff, and Matthias Richter. 2007. The leipzig corpora collection-monolingual corpora of standard size. _Proceedings of Corpus Linguistic_ , 2007.
* Caswell et al. (2019) Isaac Caswell, Ciprian Chelba, and David Grangier. 2019. Tagged back-translation. In _Proceedings of the Fourth Conference on Machine Translation (Volume 1: Research Papers)_ , pages 53–63, Florence, Italy. Association for Computational Linguistics.
* Čermák and Rosen (2012) František Čermák and Alexandr Rosen. 2012. The case of InterCorp, a multilingual parallel corpus. _International Journal of Corpus Linguistics_ , 17(3):411–427.
* Gebauer et al. (2021) Petr Gebauer, Ondřej Bojar, Vojtěch Švandelík, and Martin Popel. 2021. CUNI systems in WMT21: Revisiting backtranslation techniques for English-Czech NMT. In _Proceedings of the Sixth Conference on Machine Translation_ , pages 123–129, Online. Association for Computational Linguistics.
* Goyal et al. (2022) Naman Goyal, Cynthia Gao, Vishrav Chaudhary, Peng-Jen Chen, Guillaume Wenzek, Da Ju, Sanjana Krishnan, Marc’Aurelio Ranzato, Francisco Guzmán, and Angela Fan. 2022. The Flores-101 evaluation benchmark for low-resource and multilingual machine translation. _Transactions of the Association for Computational Linguistics_ , 10:522–538.
* Grave et al. (2018) Edouard Grave, Piotr Bojanowski, Prakhar Gupta, Armand Joulin, and Tomas Mikolov. 2018. Learning word vectors for 157 languages. In _Proceedings of the Eleventh International Conference on Language Resources and Evaluation (LREC 2018)_ , Miyazaki, Japan. European Language Resources Association (ELRA).
* Junczys-Dowmunt (2018) Marcin Junczys-Dowmunt. 2018. Dual conditional cross-entropy filtering of noisy parallel corpora. In _Proceedings of the Third Conference on Machine Translation: Shared Task Papers_ , pages 888–895, Belgium, Brussels. Association for Computational Linguistics.
* Junczys-Dowmunt et al. (2018) Marcin Junczys-Dowmunt, Roman Grundkiewicz, Tomasz Dwojak, Hieu Hoang, Kenneth Heafield, Tom Neckermann, Frank Seide, Ulrich Germann, Alham Fikri Aji, Nikolay Bogoychev, André F. T. Martins, and Alexandra Birch. 2018. Marian: Fast neural machine translation in C++. In _Proceedings of ACL 2018, System Demonstrations_ , pages 116–121, Melbourne, Australia. Association for Computational Linguistics.
* Khaburska and Tytyk (2019) Anastasiia Khaburska and Igor Tytyk. 2019. Toward language modeling for the ukrainian. _Advances in Data Mining, Machine Learning, and Computer Vision. Proceedings_ , pages 71–80.
* Kocmi et al. (2020) Tom Kocmi, Martin Popel, and Ondrej Bojar. 2020. Announcing CzEng 2.0 parallel corpus with over 2 gigawords. _CoRR_ , abs/2007.03006.
* Kotsyba (2022) Natalia Kotsyba. 2022. Ukrainian-Czech part of InterCorp v14. https://intercorp.korpus.cz.
* Kudo and Richardson (2018) Taku Kudo and John Richardson. 2018. SentencePiece: A simple and language independent subword tokenizer and detokenizer for neural text processing. In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 66–71, Brussels, Belgium. Association for Computational Linguistics.
* Popel (2018) Martin Popel. 2018. CUNI transformer neural MT system for WMT18. In _Proceedings of the Third Conference on Machine Translation: Shared Task Papers_ , pages 482–487, Belgium, Brussels. Association for Computational Linguistics.
* Popel (2020) Martin Popel. 2020. CUNI English-Czech and English-Polish systems in WMT20: Robust document-level training. In _Proceedings of the Fifth Conference on Machine Translation_ , pages 269–273, Online. Association for Computational Linguistics.
* Popel et al. (2020) Martin Popel, Marketa Tomkova, Jakub Tomek, Łukasz Kaiser, Jakob Uszkoreit, Ondřej Bojar, and Zdeněk Žabokrtský. 2020. Transforming machine translation: a deep learning system reaches news translation quality comparable to human professionals. _Nature Communications_ , 11(4381):1–15.
* Schwenk et al. (2021) Holger Schwenk, Vishrav Chaudhary, Shuo Sun, Hongyu Gong, and Francisco Guzmán. 2021. WikiMatrix: Mining 135M parallel sentences in 1620 language pairs from Wikipedia. In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pages 1351–1361, Online. Association for Computational Linguistics.
* Song et al. (2019) Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu, and Tie-Yan Liu. 2019. MASS: masked sequence to sequence pre-training for language generation. In _Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA_ , volume 97 of _Proceedings of Machine Learning Research_ , pages 5926–5936. PMLR.
* Tiedemann (2012) Jörg Tiedemann. 2012. Parallel data, tools and interfaces in OPUS. In _Proceedings of the Eighth International Conference on Language Resources and Evaluation (LREC’12)_ , pages 2214–2218, Istanbul, Turkey. European Language Resources Association (ELRA).
* Vaswani et al. (2018) Ashish Vaswani, Samy Bengio, Eugene Brevdo, Francois Chollet, Aidan Gomez, Stephan Gouws, Llion Jones, Łukasz Kaiser, Nal Kalchbrenner, Niki Parmar, Ryan Sepassi, Noam Shazeer, and Jakob Uszkoreit. 2018. Tensor2Tensor for neural machine translation. In _Proceedings of the 13th Conference of the Association for Machine Translation in the Americas (Volume 1: Research Track)_ , pages 193–199, Boston, MA. Association for Machine Translation in the Americas.
* Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In _Advances in Neural Information Processing Systems 30_ , pages 6000–6010, Long Beach, CA, USA. Curran Associates, Inc.
|
# Real-Time High-Quality Stereo Matching System on a GPU
Qiong Chang, Tsutomu Maruyama
###### Abstract
In this paper, we propose a low error rate and real-time stereo vision system
on GPU. Many stereo vision systems on GPU have been proposed to date. In those
systems, the error rates and the processing speed are in trade-off
relationship. We propose a real-time stereo vision system on GPU for the high
resolution images. This system also maintains a low error rate compared to
other fast systems. In our approach, we have implemented the cost aggregation
(CA), cross-checking and median filter on GPU in order to realize the real-
time processing. Its processing speed is 40 fps for $1436\times{992}$ pixels
images when the maximum disparity is 145, and its error rate is the lowest
among the GPU systems which are faster than 30 fps.
## I Introduction
The aim of stereo vision systems is to reconstruct the 3-D geometry of a scene
from images taken by two separate cameras. The computational complexity of the
stereo vision is very high, and many acceleration systems with GPUs, FPGAs and
dedicated hardware have been developed [1][4]. All of them succeeded in real-
time processing of high resolution images, but their accuracy is not good
enough because they simplify the algorithms to fit the hardware architecture.
In this paper, we aim to construct a real-time GPU stereo vision system for
the high resolution image set ($1436\times{992}\times{145}$ disparities) in
the Middlebury Benchmark.
GPUs have more than hundred cores which run faster than 1GHz, and drastic
performance gain can be expected in many applications. However, the processing
speed of the stereo vision by GPUs is much slower than FPGAs such as [2][3].
This is caused by the fact that data for several lines are intensively
accessed at a time in the stereo vision, but the shared memory is too small to
cache those lines. Therefore, many memory accesses to the global memory are
required, and they limit the processing speed by GPUs. On the other hand, more
sophisticated algorithms can be implemented on GPUs than FPGAs, and lower
error rates have been achieved by many GPU systems[5][6]. The algorithms for
the stereo vision can be categorized into two groups: local and global. In the
local algorithms, only the local information around the target pixel is used
to decide the disparity of the pixel, while the disparities of all pixels are
decided considering the mutual effect of all pixels in the global algorithms.
Thus, in general, the global algorithms achieve lower error rates, but require
longer computation time. On many GPUs, algorithms which require only local
information in each step, but propagate the mutual effect gradually by
repeating the step many times are implemented. Their error rates are low
enough, but their processing speed is far behind of the real-time requirement.
We have implemented a local search algorithm, which is an improved version of
the algorithm that we have implemented on FPGA[9]. In our algorithm, AD
(absolute difference) and mini-Census transform [14] are used to calculate the
matching costs of the pixels in the two images, and they are aggregated along
the $x$\- and $y$-axes by using the Cross-based method [15][16], in order to
compare the pixels as a block of the similar color. The matching costs are
calculated twice using left and right image as the base, and two disparity
maps are generated. Then, the disparity map is improved using the disparities
which are common in both disparity maps. These operations are chosen to
realize low error rates without repetitive computation. However, when we
consider processing a high resolution image, its computational complexity is
too high to achieve real-time processing. In order to achieve high
performance, in this paper, we scale down the images to reduce the
computational complexity. As shown in Fig.1, to reduce the computational
complexity, we scale down the input images into a small size, and then scale
up the disparity map to the original size. The images are scaled down to 1/4
by reducing the width and height by half, and the disparities are calculated
on the scaled down images. The maximum disparity is also reduced to half,
which means that the total computational complexity can be reduced to 1/8.
This approach typically worsens the matching accuracy, but in our approach, an
bilateral interpolation method is performed during the scaling up step, and a
high matching accuracy can be maintained. This approach becomes possible
because of the high quality of the high resolution images.
Figure 1: Image Scaling
## II STEREO VISION
In the stereo vision systems, the matching of pixels in the two images (left
and right) taken by two separate cameras is searched to reconstruct the 3-D
geometry of the scene. When the two cameras are calibrated properly, epipolar
restriction can be used. Under this restriction, we can obtain a disparity map
$D_{map}$ by finding the matching of pixels on the epipolar lines of the two
images as shown in Fig.2. Fig.2 shows how to calculate a disparity map using
the left image as the base. A pixel in the left image $L(x,y)$ (or a window
centered by $L(x,y)$) is compared with $D$ pixels $R(x-d,y)d=[0,D-1]$ in the
right image (or $D$ windows centered by those pixels), and the most similar
pixel to $L(x,y)$ is searched. $D$ is the maximum disparity value. Suppose
that $R(x-d,y)$ is the most similar to $L(x,y)$. Then, this means that
$L(x,y)$ and $R(x-d,y)$ are the same point of an object, and the distance $Z$
to the object can be calculated from d and two parameters of the two cameras
using the following equation:
$Z=f{\frac{B}{d}}.$ (1)
Here, $f$ is the focus length of both cameras, and $B$ is the distance between
the two cameras. Smaller d means that the object is farther away from the
cameras, and $d=0$ means that the object is at infinity. The main problem in
the stereo vision is to find the matching pixels in the two images correctly.
Another important problem in the stereo vision is the occlusion. When an
object is taken by two cameras, some parts of the object appear in one image
but do not appear in another image, depending on the positions and angles
between the cameras and the object. These occlusions are the major source of
errors in the stereo vision systems. In order to avoid those errors, $D_{map}$
is calculated twice: once using $L$, the left image, as the base
($D^{L}_{map}$), and another using $R$, the right image, as the base
($D^{R}_{map}$). Then, by using the same matching found in both $D^{L}_{map}$
and $D^{R}_{map}$ as the reliable disparities (called the ground control
points, or GCPs), higher quality disparity maps can be obtained.
Figure 2: Local matching under the epipolar restriction
## III OUR ALGORITHM
In this section, we introduce the details of our algorithm. Our algorithm
consists of the following steps.
1. 1.
the two input images are gray-scaled
2. 2.
scaling down the two images
3. 3.
calculating matching cost of each pixel
4. 4.
cost aggregation along the $x$\- and $y$-axes
5. 5.
generating two disparity maps
6. 6.
detecting GCPs (ground control pixels) by cross-checking the two disparity
maps
7. 7.
refinement by a median filter and filling the non-GCPs by using a bilateral
estimation method
8. 8.
scaling up the disparity map
### III-A Scaling Down
In order to reduce the computational complexity, the two images are scaled
down linearly in both horizontal and vertical directions using the mean-
pooling method. Here, take the left image as an example:
$\displaystyle L(x,y)=$
$\displaystyle\frac{1}{(2m+1)^{2}}\times{\sum_{j=-m}^{m}\sum_{i=-m}^{m}L_{org}(K\cdot{x}+i,K\cdot{y}+j)}$
(2)
where $L_{org}(K\cdot{x}+i,K\cdot{y}+j)$ is the pixel in the original image,
and $K$ is the factor for the scaling down (in our implementation, $K=2$).
$L(x,y)$ is the pixel of the left scaled down image, and is smoothed by a
mean-filter the size of which is $(2m+1)^{2}$. By choosing the block size
carefully, we can avoid the loss of the matching accuracy, and can improve the
processing speed.
### III-B Matching cost between two pixels
The matching cost of each pixel is calculated using the absolute difference of
the brightness and the mini-census transform. When the left image $L(x,y)$ is
used as the base, the matching cost of the disparity $=d$ is given by
$C^{L}(x,y,d)=C^{L}_{AD}(x,y,d)+C^{L}_{MC}(x,y,d).$ (3)
$C^{L}_{AD}(x,y,d)$ is the cost by the absolute difference of the brightness
of the two pixels, and given by
$C^{L}_{AD}(x,y,d)=1-\exp(-\frac{|L(x,y)-R(x-d,y)|}{\lambda_{AD}})$ (4)
where $\lambda_{AD}$ is a constant, In the same way, $C^{L}_{MC}(x,y,d)$, the
cost by mini-census transform, is given by
$C^{L}_{MC}(x,y,d)=1-\exp(-\frac{MC(L(x,y),R(x-d,y))}{\lambda_{MC}})$ (5)
where $\lambda_{MC}$ is a constant, and MC($\alpha$, $\beta$) is the Hamming
distance between the mini-census transform of $\alpha$ and $\beta$. Mini-
census transform used in our approach is shown in Fig.3. The center pixel
$L(x,y)$ is compared with its six neighbors, and a six bit sequence is
generated as shown in Fig.3. This approach is based on the hypothesis that the
relative values of the brightness are kept in both images.
Figure 3: Mini-census Transform
When the right image $R(x,y)$ is used as the base, the matching cost is given
as follows.
$\begin{split}C^{R}(x,y,d)&=C^{R}_{AD}(x,y,d)+C^{R}_{MC}(x,y,d)\\\
&=1-\exp(-\frac{|R(x,y)-L(x+d,y)|}{\lambda_{AD}})\hskip 5.0pt+\\\ &\hskip
13.00005pt1-\exp(-\frac{MC(R(x,y),L(x+d,y))}{\lambda_{MC}})\\\
&=C^{L}(x+d,y,d).\end{split}$ (6)
This equation means that all $C^{R}(x,y,d)$ are already calculated when
$C^{L}(x,y,d)$ are calculated, and $C^{L}(x,y,d)$ can be reused as
$C^{R}(x-d,y,d)$.
Figure 4: A cost aggregation method
### III-C Cost Aggregation
The matching costs are aggregated as much as possible considering the
similarity of the brightness of the pixels to compare the pixels as a block of
the similar brightness. Fig.4 shows how the matching costs are aggregated.
First, the matching costs are aggregated along the $x$-axis.
$CA^{L}_{x}(x,y,d)=\sum^{+n}_{dx=-m}C^{L}(x+dx,y,d)$ (7)
Here, m and n are the number of the continuous pixels with the similar
brightness to $L(x,y)$ $(|L(x,y)-L(x+dx,y)|<\delta)$ on the left and right-
side of $L(x,y)$. For example, in Fig.4, $m=3$ and $n=3$, because all pixels
from $L(x-3,y)$ to $L(x+3,y)$ are similar to $L(x,y)$. Then,
$CA^{L}_{x}(x,y,d)$ are aggregated along the $y$-axis as
$CA^{L}(x,y,d)=\sum^{+N}_{dy=-M}CA^{L}_{x}(x,y+dy,d).$ (8)
Here, $M$ and $N$ are the number of the continuous pixels with similar
brightness to $L(x,y)$ on the upper and lower side of $L(x,y)$. In Fig.4, $M$
is 4 and $N$ is 4, because the pixels from $L(x,y-4)$ to $L(x,y+4)$ are
similar to $L(x,y)$.
Then, $d$ which minimizes $CA^{L}(x,y,d)$ is chosen as the disparity at
$L(x,y)$, and disparity map $D^{L}_{map}(x,y)$ is obtained.
$D^{L}_{map}(x,y)=\min_{d}{CA^{L}(x,y,d)}.$ (9)
By enlarging the range for summing up along the $x$\- and $y$\- axes, we can
obtain more accurate disparities, though it requires more computation time.
### III-D GCPs
In our approach, $D^{R}_{map}(x,y)$ is also calculated in the same way as
$D^{L}_{map}(x,y)$. Then, ground control points, or GCPs, are obtained by
comparing them [7]. Suppose that $D^{L}_{map}(x,y)=k$. This means that
$L(x,y)$ and $R(x-k,y)$ showed the best matching when the left image is used
as the base, and they are the same point of the object in the images.
Therefore, $D^{R}_{map}(x-k,y)$ should also be $k$. If this requirement
$D^{L}_{map}(x,y)=D^{R}_{map}(x-k,y)=k$ (10)
is satisfied, the point is called a GCP, and it is considered that GCPs have
higher reliability.
### III-E Bilateral Estimation
Ideally, all pixels except for those in the occluded regions should be GCPs,
however, in actuality more pixels become non-GCPs because of the slight change
of the brightness between the input images. To achieve more reliable
disparities of non-GCPs, two approaches are often used [10]. In both
approaches, for each non-GCP, the closest GCPs on the left and right hand-side
along the $x$-axis are searched first. Then, in the first approach, as shown
in Fig.5(a), the closer GCP in the distance is chosen as the disparity of the
non-GCP because the non-GCP and the closer GCP can be considered to belong to
the same object with a higher probability. In the second approach, as shown in
Fig.5(b), the smaller disparity is chosen as the disparity of the non-GCP
assuming that the non-GCP is caused by the occlusion. The disparity of the
occluded region is smaller than that of the foreground object because the
disparity of the closer object is larger, and the non-GCP should have a
smaller disparity. Both of these two methods can be easily implemented on GPU
because of their high parallelism. However, due to their single function, the
overall accuracy is not good enough.
Figure 5: Non-GCPs Filling
In our system, we proposed a bilateral estimation methods to fill the non-GCPs
as following steps:
1. 1.
Define the disparities of the GCP of $L(x-i,y)$ and $L(x+j,y)$ as $D(x-i,y)$
and $D(x+j,y)$.
2. 2.
If $|D(x-i,y)-D(x+j,y)|\leq{T}$, where $T$ is the threshold for the difference
of disparity, it can be considered that the disparity is changing continuously
in this range, and $D(x,y)$ is filled as:
$D(x-i,y)+i\cdot((D(x-i,y)-D(x+j,y))/(i+j)).$ (11)
3. 3.
If $|D(x-i,y)-D(x+j,y)|>T$, which means that the disparity changes rapidly in
this range, it can be considered that an edge exists in this range. Thus
$D(x,y)$ is chosen as the $D(x-i,y)$ if $L(x-i,y)$ is closer to $L(x,y)$ than
$L(x+j,y)$ in color, and otherwise, $D(x,y)$ is chosen as $D(x+j,y)$.
With this approach, we can fill the different areas in the different methods.
Then, an accurate disparity map can be expected.
## IV IMPLEMENTATION ON GPU
We have implemented the algorithm on Nvidia GTX780Ti. GTX780Ti has 15
streaming multi-processors (SMs). Each SM runs in parallel using 192 cores in
it (2880 cores in total). GTX780Ti has two-layered memory system. Each SM has
a 48KB shared memory, and one large global memory is shared among the SMs. The
access delay to the shared memory is very short, but that to the global memory
is very long. Therefore, the most important technique to achieve high
performance on GPU is how to cache a part of the data on the shared memory,
and to hide the memory access delay to the global memory. The shared and the
global memory have the restriction of the access to them. In the CUDA, which
is an abstracted architecture of Nvidia’s GPUs, 16 threads are managed as a
set. When accessing to the global memory, 16 words can be accessed in parallel
if the 16 threads access to continuous 16 words which start from 16 word-
boundary. Otherwise, the bank conflict happens, and several accesses to the
global memory happens. The shared memory consists of 16 banks, and in this
case, the 16 words can be accessed in parallel if they are stored in the
different memory banks (the addresses of the 16 words do not need to be
continuous). For reducing the memory accesses to the global memory, the order
of the calculation on GPU is different from the one described in the previous
section.
* •
Step1 transfer the input images onto the GPU and scale down them
* •
Step2 compare the brightness of the pixels along the $x$-axis
* •
Step3 calculate the matching costs and aggregate them along the $x$-axis
* •
Step4 compare the brightness of the pixels along the $y$-axis
* •
Step5 aggregate the cost along the $y$-axis, and generate two disparity maps
* •
Step6 find GCPs by cross-checking
* •
Step7 apply median filter to remove noises and estimate the disparity map
using bilateral method
* •
Step8 scale up the disparity map and transfer back to the CPU.
In the following discussion, $X_{org}\times{Y_{org}}$ is the image size
($X_{org}$ is the width, and $Y_{org}$ is the height), and $L_{org}[y][x]$ and
$R_{org}[y][x]$ are the pixels in the left and right images. $L[y][x]$ and
$R[y][x]$ are the pixels in the scaled-down images, and $X\times{Y}$ is the
image size of them ($X=X_{org}/2$, $Y=Y_{org}/2$). Fig.6 shows the task
assignment and input/output of each step. The details are discussed in the
following subsections.
Figure 6: Task assignment of each step
### IV-A Step1
The inputs to this step are $L_{org}[y][x]$ and $R_{org}[y][x]$, and they are
transferred onto the global memory of the GPU, and are processed in parallel
using 15 SMs.
1. 1.
$Y_{org}/15$ lines of both images are assigned to each SM as shown in Fig.7.
2. 2.
$X_{org}$ columns in the $Y_{org}/15$ lines are processed using $X^{\prime}$
($X_{org}\leq{X^{\prime}}$) threads in the SM ($X^{\prime}$ must be a multiple
of 64 because of the reason described below). When $X^{\prime}>X_{org}$,
$X^{\prime}-X_{org}$ threads work in the same way as the $X_{org}$ threads,
but generate no outputs.
3. 3.
When $X_{org}$ is larger than the maximum number of the threads in one SM
(1024), each thread processes more than one columns. In our implementation,
because the resolution of the input images are greater than 1024, each thread
processes 2 columns during the scaling-down step.
For each pixel $L_{org}(x_{even},y_{even})$, both of the vertical and
horizontal coordinates of which are even, all of the surrounding pixels
$L_{org}(x_{even}+dx,y_{even}+dy)$ ($dx\in[-1,1]$, $dy\in[-1,1]$) are added
together. Then, as the pixel value of the scaled-image, the average of the
summation is stored in the global memory.
Figure 7: Mapping pixels to the threads
### IV-B Step2
For $Y/15$ pixels in one column (let the pixels be
$L[y_{b}+k][x_{b}]$($k=0,14$)), each thread compares its pixel’s value with
its neighbors along the $x$-axis ($L[y_{b}+k][x_{b}+dx]$ ($dx=1,W_{x}$) and
$L[y_{b}+k][x_{b}-dx]$ ($dx=1,W_{x}$)). The data type of $L[y][x]$ is $8b$
(unsigned char). Therefore, four continuous pixels are packed in one $32b$
word, and stored in the same memory bank of the shared memory. This means that
these four continuous pixels can not be accessed in parallel owing to the
memory access restriction of the shared memory. In order to avoid the bank
conflict, Threadi processes column ($i/(X/4)+(i\times{4})\%X$) as shown in
Fig.7. In Fig.7, the first four pixels of each line ($L[y][0]$, $L[y][1]$,
$L[y][2]$, $L[y][3]$) are stored in the first bank, and next four pixels
($L[y][4]$, $L[y][5]$, $L[y][6]$, $L[y][7]$) in the second bank. Thread0
processes $Y/15$ pixels on $x=0$ ($L[y_{b}+dy][0]$ ($dy=0,14$)) sequentially,
and Thread1 processes $Y/15$ pixels on $x=4$ ($L[y_{b}+dy][4]$ ($dy=0,14$))
sequentially. By changing the order of the computation like this, bank
conflict can be avoided. In our algorithm, all pixels can be processed
independently, and the same results can be obtained regardless of the
computation order.
In this method, four continuous pixels are stored in the same bank, and 16
threads are executed at the same time in CUDA. Therefore, $X$ must be a
multiple of 64 ($4\times{16}$). When, $X$ is not the multiple of 64, larger
$X$ which is the multiple of 64 is chosen, and the computation results for the
extended part are discarded.
The outputs of this step are two integer values for each pixel, which show how
many pixels are similar to the center pixel to the plus/minus direction of the
$x$-axis. These values for $L[y][x]$ are stored in $W^{L}_{-}[y][x]$ and
$W^{L}_{+}[y][x]$, and those for $R[y][x]$ are stored in $W^{R}_{-}[y][x]$ and
$W^{R}_{+}[y][x]$. The data width of these arrays is $32b$, and the direct
access to these values causes no bank conflict. $L[y][x]$ and $R[y][x]$ are
transposed here, and stored in $L^{*}[x][y]$ and $R^{*}[x][y]$ respectively.
### IV-C Step3
In this step, first, two matching costs ($C^{L}(x,y,d)$ and $C^{R}(x,y,d)$)
are calculated, and then, they are aggregated along the $x$-axis using the
range information in $W^{L}_{-}[y][x]$, $W^{L}_{+}L[y][x]$, $W^{R}_{-}[y][x]$
and $W^{R}_{+}[y][x]$ to calculate $CA^{L}_{x}(x,y,d)$ and
$CA^{R}_{x}(x,y,d)$. Here, actually, we do not need to calculate
$C^{R}(x,y,d)$ as described in Section 3.C because $C^{L}(x+d,y,d)$ can be
used as $C^{R}(x,y,d)$. Therefore, all SMs are used to calculate
$C^{L}(x,y,d)$ as shown in Fig.6-$step3$, and each SM processes $Y/15$ lines
as follows.
1. 1.
For each of the Y/15 lines, repeat the following steps.
2. 2.
Set $d=0$.
3. 3.
Calculate $C^{L}(x,y,d)$ for all $x$ in the current line. $C^{L}(x,y,d)$ is
stored in $C[x]$ (an array in the shared memory). For this calculation, 3
lines of $L[y][x]$ and $R[y][x]$ are cached in the shared memory for
calculating the mini-census transform, and they are gradually replaced by the
next line as the calculation progresses.
4. 4.
Calculate $CA^{L}_{x}(x,y,d)$ as follows.
1. (a)
Set $CA_{x}[x]=C[x]$.
2. (b)
Add $C[x+dx]$ to $CA_{x}[x]$ starting from $dx=1$ to the position given by
$W^{L}_{+}[y][x]$.
3. (c)
Add $C[x-dx]$ to $CA_{x}[x]$ starting from $dx=1$ to the position given by
$W^{L}_{-}[y][x]$.
4. (d)
Store $CA_{x}[x]$ into $CA^{*L}_{x}[d][x][y]$ in the global memory (note that
this array is transposed).
5. 5.
Calculate $CA^{R}_{x}(x,y,d)$ as follows.
1. (a)
Set $CA_{x}[x]=C[x+d]$.
2. (b)
Add $C[x+d+dx]$ to $CA_{x}[x]$ starting from $dx=1$ to the position given by
$W^{R}_{+}[y][x]$.
3. (c)
Add $C[x+d-dx]$ to $CA_{x}[x]$ starting from $dx=1$ to the position given by
$W^{R}_{-}[y][x]$.
4. (d)
Store $CA_{x}[x]$ into $CA^{*R}_{x}[d][x][y]$ in the global memory (note that
this array is transposed).
6. 6.
Increment $d$ if $d<D$, and go to step 3.
In this step, $D$ arrays are stored in the global memory.
### IV-D Step4
$L[y][x]$ and $R[y][x]$ have been transposed and stored as $L^{*}[x][y]$ and
$R^{*}[x][y]$ in step2. By using these arrays, the brightness of the pixels
are compared efficiently along the $y$-axis. In this case, the pixel data (for
example $L^{*}[x][y]$) are compared horizontally (parallel memory accesses are
allowed only in this direction), and this means that $L^{*}[x][y]$ are
compared with $L^{*}[x][y+dy]$ ($dy=-W_{y},W_{y}$). The range of the similar
pixels are stored in $W^{*L}_{-}[x][y]$ and $W^{*L}_{+}[x][y]$ for
$L^{*}[x][y]$, and in $W^{*R}_{-}[x][y]$ and $W^{*R}_{+}[x][y]$ for
$R^{*}[x][y]$. $L^{*}[x][y]$ and $R^{*}[x][y]$ are processed in parallel as
shown in Fig.6-$step4$, and each SM processes $X/15$ columns. The $Y$ lines in
each column are assigned to $Y$ threads in the same way shown in Fig.7 though
$x$ and $y$ are transposed.
### IV-E Step5
In this step (Fig.6-$step5$), $CA^{*L}_{x}[d][x][y]$ and
$CA^{*R}_{x}[d][x][y]$ are aggregated along the $y$-axis in parallel using
$W^{*L}_{-}[x][y]$, $W^{*L}_{+}[x][y]$, $W^{*R}_{-}[x][y]$ and
$W^{*R}_{+}[x][y]$, and then $D^{L}_{map}[y][x]$ and $D^{R}_{map}[y][x]$
(disparity maps when $L[y][x]$ and $R[y][x]$ are used as the base) are also
generated as follows.
1. 1.
Each SM processes $X/15$ columns.
2. 2.
$Y$ threads in each SM processes $Y$ pixels in each of the $X/15$ columns.
3. 3.
Each thread repeats the following steps (in the following, only the steps for
the left image are shown).
1. (a)
Read $W^{*L}_{-}[x][y]$ and $W^{*L}_{+}[x][y]$ from the global memory.
2. (b)
Set $d=0$.
3. (c)
$Min[y]=$MAX_VALUE and $D_{map}[y]=0$.
4. (d)
Calculate $CA^{L}(x,y,d)$ as follow.
1. i.
Set $CA[y]=CA^{*L}[d][x][y]$.
2. ii.
Add $CA^{*L}_{x}[d][x][y+dy]$ to $CA[y]$ starting from $dy=1$ to the position
given by $W^{*L}_{+}[x][y]$.
3. iii.
Add $CA^{*L}_{x}[d][x][y-dy]$ to $CA[y]$ starting from $dy=1$ to the position
given by $W^{*L}_{-}[x][y]$.
5. (e)
If $CA[y]<Min[y]$ then $Min[y]=CA[y]$ and $D_{map}[y]=d$.
6. (f)
Increment $d$ if $d<D$, and go to step 3(c).
7. (g)
Store $D_{map}[y]$ in $D^{L}_{map}[y][x]$ in the global memory (note that his
array is re-transposed).
### IV-F Step6
In this step (Fig.6-$step6$), $D^{L}_{map}[y][x]$ and $D^{R}_{map}[y][x]$ are
read from the global memory line by line, and the condition for the GCP
(described in Section 3.D) is checked. In this step, each SM processes $Y/15$
lines. Threadx first accesses $D^{L}_{map}[y][x]$, and then
$D^{R}_{map}[y][+k]$ if $D^{L}_{map}[y][x]=k$. $k$ is different for each
thread, and bank conflict happens in this step.
### IV-G Step7
The median filter is applied using 15 SMs for the left image at first. Then,
the bilateral estimation is used to fill the non-GCPs. If $D^{L}_{map}[y][x]$
is not a GCP, $thread_{x}$ scans $D^{L}_{map}[y][x]$ to the $+/-$ direction of
the $x$-axis in order, and finds two GCPs (the GCPs closest on the left- and
right-hand side). Then, the difference of the disparities of the two GCPs is
calculated. If the difference is smaller than the threshold, the
$D^{L}_{map}[y][x]$ is filled linearly according to the disparities of the two
GCPs. If the different is larger than the threshold, the brightness of the two
GCP are compared with the target pixel, and the disparity of the target pixel
is replaced by the one which has similar brightness. Then, the improved
disparity map $D^{+L}_{map}[y][x]$ is stored in the global memory.
### IV-H Step8
Finally, the disparity map $D^{+L}_{map}[y][x]$ is scaled up by using the 15
SMs. In order to maintain a high accuracy, during the scaling-up along the
$x$-axis, the bilateral estimation method described above is used again. On
the other hand, the estimation along the $y$-axis is applied linearly. Then,
the final disparity map $D^{fL_{org}}[y][x]$ is transferred back to the CPU.
TABLE I: Error rate when the cost aggregation range is changed (average error rate (%)) $W_{L}$$\backslash$$W_{R}$ | $W_{y}=9$ | $W_{y}=11$ | $W_{y}=15$ | $W_{y}=21$ | $W_{y}=27$ | $W_{y}=31$
---|---|---|---|---|---|---
$W_{x}=5$ | 25.10 | 24.98 | 24.83 | 24.68 | 24.62 | 24.61
$W_{x}=9$ | 24.67 | 24.55 | 24.4 | 24.3 | 24.26 | 24.26
$W_{x}=21$ | 24.39 | 24.34 | 24.21 | 24.13 | 24.1 | 24.09
$W_{x}=41$ | 24.53 | 24.38 | 24.29 | 24.21 | 24.19 | 24.17
$W_{x}=61$ | 24.53 | 24.5 | 24.41 | 24.32 | 24.31 | 24.28
$W_{x}=141$ | 24.6 | 24.55 | 24.48 | 24.42 | 24.38 | 24.28
TABLE II: Execution Time For The Middlebury Benchmark Set (ms)
Image | Size | Dmax | SD | $W^{LR}_{\pm}$ | $W^{*LR}_{\pm}$ | $C+CA_{x}$ | CA | CC | Post | SU | Overall(GPU) | FPS
---|---|---|---|---|---|---|---|---|---|---|---|---
Adirondack(H) | $1436\times{992}$ | 145 | 0.035 | 0.149 | 0.317 | 10.443 | 13.943 | 0.437 | 0.181 | 0.09 | 25.595 | 40
Pipes(H) | $1482\times{994}$ | 128 | 0.038 | 0.135 | 0.289 | 19.8 | 11.44 | 0.506 | 0.238 | 0.09 | 32.536 | 31
Vintage(H) | $1444\times{960}$ | 380 | 0.038 | 0.143 | 0.315 | 26.142 | 36.394 | 0.092 | 0.221 | 0.087 | 63.432 | 16
Size: $W^{LR}_{\pm}=21$, $W^{*LR}_{\pm}=31$, $TC=13$ Dmax: Maximum Disparity
SD: Scaling-Down. $W^{LR}_{\pm}$: Edge detection along the $x$-axis.
$W^{*LR}_{\pm}$: Edge detection along the $y$-axis $C+CA_{x}$: cost
calculation & Aggregation along the $x$-axis. CA: Aggregation along the
$y$-axis. CC: Cross_check. Post: MedianFilter&Bilateral estimation. SU:
Scaling-up. Overall: The overall time taken on GPU.
TABLE III: COMPARISON WITH HIGH-SPEED STEREO VISION SYSTEMS System | Size | Dmax | Hardware | Benchmark | FPS | MDE/s
---|---|---|---|---|---|---
RT-FPGA[1] | $1920\times{1680}$ | 60 | Kintex 7 | Middlebury v2 | 30 | 5806
FUZZY[2] | $1280\times{1024}$ | 15 | Cyclone II | Middlebury v2 | 76 | 1494
Low-Power[13] | $1024\times{768}$ | 64 | Virtex-7 | Middlebury v2 | 30 | 1510
ETE[3] | $1242\times{375}$ | 256 | GTX TITAN X | KITTI 2015 | 29 | 3458
EmbeddedRT[8] | $640\times{480}$ | 128 | Tegra X1 | KITTI 2012 | 81 | 3185
MassP[4] | $1440\times{720}$ | 128 | GPU | Middlebury v3 | 128 | 3981
Our system | $1436\times{992}$ | 145 | GTX 780 Ti | Middlebury v3 | 40 | 7849
Figure 8: Processing results
## V experimental results
We have implemented the algorithm on Nvidia GTX780Ti. The error rate and the
processing speed are evaluated using Middlebury benchmark set [11].
In this evaluation, all parameters mentioned above affect the performance of
the accuracy. According to our tuning results, we first set
$\lambda_{AD}=0.3$, $\lambda_{MC}=2.3$ and $T=3$ to ensure a good accuracy. In
the cost aggregation step, lower error rates can be expected by adding more
cost along the $x$\- and $y$\- axes, though it requires more computation time,
and makes the system slower. The maximum range of the cost aggregation
$(W_{x},W_{y})$ can be changed when calculating $D^{L}_{map}$ and
$D^{R}_{map}$. By changing them, the different criteria are used for the left
and right image, and the GCPs can be more reliable. Table I shows the error
rate (%) when the cost aggregation range is changed. In Table I, $W_{x}$ are
the maximum aggregation range along the $x$-axis for the left and right
images, and $W_{y}$ is the maximum aggregation range along the $y$-axis
($W_{y}$ is common to the left and right images). As shown in Table I, by
enlarging $W_{y}$, the error rates can be improved when $W_{x}$ is small. We
have fixed $W_{x}=21$ and $W_{y}=31$. To our best knowledge, the accuracy of
our system is higher than other real-time systems (like [12]) which are listed
in Middlebury Benchmark [11]. Additionally, we also compared our error rates
with that obtained using the original size image set, which are processed
using larger window ranges $W_{x}=41$ and $W_{y}=61$. According to our
evaluation, the error rate (Bad 2.0) of our system (24.09%) is higher than
that by the original size images (32.82%). One of the reasons is that we
didn’t tuned the parameters $\lambda_{AD}$ and $\lambda_{MC}$ for the original
image set. The other one is that in the scaling down images, some information
that makes the matching difficult in the original size images, such as
repetition of patterns and a serious of similar pixels, are discarded, and
better matching becomes possible.
The processing speed of our system is almost proportional to the window size
and the maximum disparity. Therefore, the computation time for ’Vintage’
becomes the slowest. Table II shows the processing speed of our system and its
details. We ignore the time for CPU-GPU data transfers (less than 3% of the
total elapsed time) since it can be overlapped with the computation. As shown
in Table II, most of the computation time is used for the cost calculation and
its aggregation ($C+CA_{x}$ and $CA$). $W^{LR}_{\pm}$, $W^{*LR}_{\pm}$, $CC$,
$Post$, $SD$, $SU$ show the computation time for finding the cost aggregation
range along the $x$\- and $y$\- axes, cross checking, post-processing and the
image scaling. Unfortunately, for the ’Vintage’ set, its processing speed is
16fps due to the large disparity, and we cannot achieve the real-time
processing.
Table III compares the processing speed of our system with other hardware
systems. In Table III, all of the systems achieved a real-time processing as
shown in FPS field, but their target image size (Size) and disparity range
(Dmax) are different. According to the mega disparity evaluation per second
(MDE/S), it can be noted that our system is much faster than other systems.
Fig.7 shows the results of our system for the two benchmark sets: Adirondack,
Pipes and Vintage.
## VI Conclusion
In this paper, we have proposed a real-time stereo vision system for high
resolution images on GPU. Its processing speed is much faster than previous
one. At the same time, it also maintained a high accuracy. In our current
implementation, the processing speed is still limited by the access delay to
the global memory. To improve the processing speed so that the real-time
processing can be achieved slower GPUs is our main future work.
## References
* [1] Daolu Zha, Xi Jin, Tian Xiang: _A real-time global stereo-matching on FPGA_. Microprocessors and Microsystems - Embedded Hardware Design 47: 419-428 (2016)
* [2] Madaín Pérez Patricio and Abiel Aguilar-González and Miguel O. Arias-Estrada and Héctor-Ricardo Hernandez-de Leon and Jorge-Luis Camas-Anzueto and J. A. de Jesús Osuna-Coutiño: _An FPGA stereo matching unit based on fuzzy logic_. Microprocessors and Microsystems - Embedded Hardware Design 42: 87-99 (2016)
* [3] Alex Kendall, Hayk Martirosyan, Saumitro Dasgupta, Peter Henry, Ryan Kennedy, Abraham Bachrach, Adam Bry: _End-to-End Learning of Geometry and Context for Deep Stereo Regression_. CoRR abs/1703.04309 (2017)
* [4] Wenbao Qiao and Jean-Charles Créput: _Stereo Matching by Using Self-distributed Segmentation and Massively Parallel GPU Computing_. ICAISC (2) 2016: 723-733
* [5] Jure Zbontar, Yann LeCun: _Stereo Matching by Training a Convolutional Neural Network to Compare Image Patches_. Journal of Machine Learning Research 17: 65:1-65:32 (2016)
* [6] Xiaoqing Ye, Jiamao Li, Han Wang, Hexiao Huang, Xiaolin Zhang: _Efficient Stereo Matching Leveraging Deep Local and Context Information_. IEEE Access 5: 18745-18755 (2017)
* [7] Aaron F. Bobick, Stephen S: _Intille: Large Occlusion Stereo_. International Journal of Computer Vision 33(3): 181-200 (1999)
* [8] Daniel Hernández Juárez and Alejandro Chacón and Antonio Espinosa and David Vázquez and Juan Carlos Moure and Antonio M. López: _Embedded Real-time Stereo Estimation via Semi-Global Matching on the GPU_. ICCS 2016: 143-153
* [9] Minxi Jin, Tsutomu Maruyama: _Fast and Accurate Stereo Vision System on FPGA_. TRETS 7(1): 3:1-3:24 (2014)
* [10] Qingxiong Yang and Liang Wang and Ruigang Yang and Henrik Stewénius and David Nistér: _Stereo Matching with Color-Weighted Correlation, Hierarchical Belief Propagation, and Occlusion Handling_. IEEE Trans. Pattern Anal. Mach. Intell. 31(3): 492-504 (2009)
* [11] D.Scharstein and R.Szeliski: _http://vision.middlebury.edu/stereo/eval3/_.
* [12] Leonid Keselman, John Iselin Woodfill, Anders Grunnet-Jepsen, Achintya Bhowmik: _Intel RealSense Stereoscopic Depth Cameras_. CoRR abs/1705.05548 (2017)
* [13] Luca Puglia, Mario Vigliar, Giancarlo Raiconi: _Real-Time Low-Power FPGA Architecture for Stereo Vision_. IEEE Trans. on Circuits and Systems 64-II(11): 1307-1311 (2017)
* [14] Lu Zhang, Ke Zhang, Tian Sheuan Chang, Gauthier Lafruit, Georgi Krasimirov Kuzmanov, Diederik Verkest: _Real-time high-definition stereo matching on FPGA._ FPGA 2011: 55-64
* [15] Ke Zhang, Jiangbo Lu, Gauthier Lafruit: _Cross-Based Local Stereo Matching Using Orthogonal Integral Images._ IEEE Trans. Circuits Syst. Video Techn. 19(7): 1073-1079 (2009)
* [16] Edoardo Paone, Gianluca Palermo, Vittorio Zaccaria, Cristina Silvano, Diego Melpignano, Germain Haugou, Thierry Lepley: _An exploration methodology for a customizable OpenCL stereo-matching application targeted to an industrial multi-cluster architecture._ CODES+ISSS 2012: 503-512
|
# Principled Multi-Aspect Evaluation Measures of Rankings
Maria Maistro<EMAIL_ADDRESS>University of CopenhagenDenmark , Lucas Chaves
Lima<EMAIL_ADDRESS>University of CopenhagenDenmark , Jakob Grue Simonsen
<EMAIL_ADDRESS>University of CopenhagenDenmark and Christina Lioma
<EMAIL_ADDRESS>University of CopenhagenDenmark
(2021)
###### Abstract.
Information Retrieval evaluation has traditionally focused on defining
principled ways of assessing the relevance of a ranked list of documents with
respect to a query. Several methods extend this type of evaluation beyond
relevance, making it possible to evaluate different aspects of a document
ranking (e.g., relevance, usefulness, or credibility) using a single measure
(multi-aspect evaluation). However, these methods either are (i) tailor-made
for specific aspects and do not extend to other types or numbers of aspects,
or (ii) have theoretical anomalies, e.g. assign maximum score to a ranking
where all documents are labelled with the lowest grade with respect to all
aspects (e.g., not relevant, not credible, etc.).
We present a theoretically principled multi-aspect evaluation method that can
be used for any number, and any type, of aspects. A thorough empirical
evaluation using up to $5$ aspects and a total of $425$ runs officially
submitted to $10$ TREC tracks shows that our method is more discriminative
than the state-of-the-art and overcomes theoretical limitations of the state-
of-the-art.
Evaluation, ranking, multiple aspects, partial order
††journalyear: 2021††copyright: none††conference: Proceedings of the 30th ACM
International Conference on Information and Knowledge Management; November
1–5, 2021; Virtual Event, QLD, Australia††booktitle: Proceedings of the 30th
ACM International Conference on Information and Knowledge Management (CIKM
’21), November 1–5, 2021, Virtual Event, QLD, Australia††doi:
10.1145/3459637.3482287††isbn: 978-1-4503-8446-9/21/11††ccs: Information
systems Information retrieval††ccs: Information systems Evaluation of
retrieval results††ccs: Information systems Retrieval effectiveness
## 1\. Introduction
Multi-aspect evaluation is a task in Information Retrieval (IR) evaluation
where the ranked list of documents returned by an IR system in response to a
query is assessed in terms of not only relevance, but also other aspects (or
dimensions) such as credibility or usefulness. Generally, there are two ways
to conduct multi-aspect evaluation: (1) evaluate each aspect separately using
any appropriate single-aspect evaluation measure (e.g., AP, NDCG, F1), and
then aggregate the scores across all aspects into a single score; or (2)
evaluate all aspects at the same time using any appropriate multi-aspect
evaluation measure (Amigó et al., 2018b; Lioma et al., 2017; Tang and Yang,
2017). An advantage of the aggregating option (1) is that it is easy to
implement using evaluation measures that are readily available and well-
understood in the community. Its disadvantage is that it is not guaranteed
that all aspects will have similar distributions of labels, and aggregating
across wildly different distributions can give odd results (Lioma et al.,
2019). The second way of doing multi-aspect evaluation is to use a single
multi-aspect evaluation measure. The problem here is that few such evaluation
measures exist, and most of them are defined for specific aspects and do not
generalise to other types/numbers of aspects (see $\S$2).
Motivated by the above, we contribute a novel multi-aspect evaluation method
that works with any type and number of aspects, and avoids the above problems.
Given a ranked list, where documents are labelled with multiple aspects, our
method, Total Order Multi-Aspect (TOMA) evaluation, first defines a
preferential order (formally weak order relation) among documents with
multiple aspect labels, and then aggregates the document labels across aspects
to obtain a ranking of aggregated aspect labels, which can be evaluated by any
single-aspect evaluation measure, such as Normalized Discounted Cumulated
Gain (NDCG) or Average Precision (AP). Simply put, instead of evaluating each
aspect separately and then aggregating their scores, we first aggregate the
aspect labels and then evaluate the ranked list of documents. We do this in a
way that provides several degrees of freedom: our method can be used with any
number and type of aspects, can be instantiated with any binary or graded,
set-based or rank-based evaluation measure, and can accommodate any
granularity in the importance of each aspect or label, but still ensures, by
definition, that the preference order among multi-aspect documents is not
violated, and that the final measure score will meet some common requirements,
i.e., the minimum (worst) score being $0$ and the maximum (perfect) score
being $1$. We validate this empirically ($\S$4) and theoretically ($\S$4.2).
## 2\. Related Work
Multi-aspect evaluation measures for IR have been studied for different tasks
and aspects, starting from the INEX initiative with relevance and coverage
(Kazai et al., 2004). Since then, measures have been proposed to evaluate
relevance and novelty or diversity, such as $\alpha$-NDCG (Clarke et al.,
2008), MAP-IA (Agrawal et al., 2009) and IA-ERR (Chapelle et al., 2009);
relevance, novelty and the amount of user effort, such as nCT (Tang and Yang,
2017); relevance, redundancy and user effort, such as RBU (Amigó et al.,
2018b); relevance and understandability, such as uRBP (Zuccon, 2016) and the
Multidimensional Measure (MM) framework (Palotti et al., 2018); and relevance
and credibility, such as NLRE, NGRE, nWCS, Convex Aggregating Measure (CAM)
and WHAM (Lioma et al., 2017). All these measures have limitations; we
describe these next.
Firstly, except for RBU, none of the above measures are based on a formal
framework. They are defined as stand-alone tools to assess the effectiveness
of a ranked list of documents. This means that, even if the measure can assess
the effectiveness of an input ranking, the order induced by the measure over
the space of input rankings is not well-defined. Hence, there is no canonical
_ideal ranking_ 111An _ideal ranking_ is the best ranking of all assessed
documents for a given topic (Järvelin and Kekäläinen, 2002). that is well-
defined or easy to compute, e.g., for $\alpha$-NDCG, the computation of the
ideal ranking is equivalent to a minimal vertex covering problem (Clarke et
al., 2008), an NP-complete problem, while for CT and nCT, computing the ideal
ranking is equivalent to the minimum edge dominating set problem (Tang and
Yang, 2017), an NP-hard problem. Computationally better ways of comparing to
an ideal ranking can be devised using graded similarity—so-called
_effectiveness levels_ to an ideal ranking using Rank-Biased Overlap (Clarke
et al., 2020c, b, 2021), but this approach requires defining a (set of) ideal
ranking(s), which has not appeared for multi-aspect ranking prior to the
present paper.
Evaluation measures that do not compare against an ideal ranking may be harder
to interpret or problematic. DCG is not upper bounded by $1$, thus different
topics are not weighted equally and scores are not comparable. Failing to
compare against the ideal ranking is problematic in multi-aspect evaluation:
$\alpha$-NDCG allows systems to reach scores greater than $1$, which is
supposed to be the score of the perfect system. With NLRE and NGRE, a system
that retrieves no relevant or credible documents has error =$0$, i.e.,
achieves the best score, because the relative order of pairs of documents is
always correct (Lioma et al., 2019). Similarly, nWCS can reach the perfect
score of $1$, even if no relevant or credible documents are retrieved, since
the normalization is computed with a re-ranking of the input ranking, instead
of the ideal ranking.
Both uRBP and RBU have a different problem: to reach the perfect score of $1$,
a system must retrieve an infinite number of relevant and understandable
documents, even if those documents are not available in the collection. CAM
and WHAM use the weighted arithmetic and weighted harmonic mean of any IR
measure computed with respect to relevance and credibility independently.
Therefore, depending on the distribution of labels across the aspects, it can
be impossible for any system to reach the perfect score (see $\S$ 4.2).
Secondly, most of the above multi-aspect evaluation measures are defined for
specific contexts and with a limited set of aspects, e.g., novelty, diversity,
credibility and understandability, thus they cannot deal with a more general
scenario and a variable number of aspects. For RBU, even though a formal
framework is defined, its formulation specifies only diversity and redundancy
constraints, which cannot be applied to a general set of aspects. This
inability to generalise to more/other types of aspects means that, if a system
must be evaluated with respect to a new aspect, the measure needs to be
properly adapted. This can be easily done for some measures, e.g., CAM, WHAM,
and nWCS, but the lack of a formal framework behind them may lead to odd
results, e.g., extending NLRE to $3$ aspects returns a score distribution
compressed towards $0$, preventing the rankings to be evaluated in a fair way
(Lioma et al., 2019).
(a) Distance order.
(b) Euclidean distance.
(c) Manhattan distance.
(d) Chebyshev distance.
Figure 1. Example with two aspects $a_{1}$ and $a_{2}$. Each point is a tuple
of labels. The best label $\mathbf{l^{\star}}$ is in the top right. The
distance between tuples of labels and $\mathbf{l^{\star}}$ defines a weak
order relation. Blue lines connect tuples of labels at the same distance from
$\mathbf{l^{\star}}$.
## 3\. TOMA Framework
We formalize the problem and our proposed methodology: we explain why
reasoning in terms of multiple aspects leads to a partial order relation among
documents ($\S$ 3.1); how we complete the partial order relation with the
distance order ($\S$ 3.2); and how to use the distance order with state-of-
the-art IR evaluation measures ($\S$ 3.3).
### 3.1. Formalization of the Problem
Let $A=\\{a_{1},\ldots,a_{n}\\}$ be a set of _aspects_ ; each aspect $a\in A$
has a non-empty set of _labels_ $L_{a}=\\{l_{0}^{a},\ldots,l_{K_{a}}^{a}\\}$
and an order relation $\prec_{a}$ such that:
$l_{0}^{a}\prec_{a}l_{1}^{a}\prec_{a}\cdots\prec_{a}l_{K_{a}}^{a}$, e.g., we
may have $2$ aspects $A=\\{\text{relevance},\text{correctness}\\}$, with the
set $L_{r}=\\{\texttt{nr},\texttt{mr},\texttt{fr},\texttt{hr}\\}$ (non-
relevant, marginally relevant, fairly relevant, highly relevant) ordered as:
$\texttt{nr}\prec_{r}\texttt{mr}\prec_{r}\texttt{fr}\prec_{r}\texttt{hr}$; and
the set $L_{c}=\\{\texttt{nc},\texttt{pc},\texttt{c}\\}$ (non-correct,
partially correct, correct) ordered as:
$\texttt{nc}\prec_{c}\texttt{pc}\prec_{c}\texttt{c}$. Let $D$ be the set of
_documents_ and $T$ the set of _topics_. Each document $d\in D$ is mapped to a
_ground truth_ vector $\textrm{GT}(d,t)=(l_{1},\ldots,l_{n})\in
L_{a_{1}}\times\cdots\times L_{a_{n}}$ that contains the “true” label of $d$
for each aspect, e.g., a document may have $\textrm{GT}(d,t)=(\textrm{highly
relevant},\textrm{non-correct})$.
In IR, given a topic $t$, the objective is to rank documents in $D$ such that
for the documents $d^{\prime},d\in D$, if $d^{\prime}$ is ranked before $d$,
then $\textrm{GT}(d,t)\preceq_{*}\textrm{GT}(d^{\prime},t)$ for a given order
relation $\preceq_{*}$. When there is only one aspect $A=\\{a\\}$, one can use
$\prec_{a}$, the order on the set of labels $L_{a}$, to induce a weak order on
$D$ and decide if $d^{\prime}$ should be ranked before $d$. If only relevance
is assessed, we consider the relation induced by relevance labels, i.e.,
documents labelled “highly relevant” should be ranked before
“fairly/marginally relevant” and “non-relevant” documents. Applying this
approach to multiple aspects requires reasoning about orderings of tuples of
labels with different aspects, e.g., for documents $d^{\prime},d\in D$, such
that $\textrm{GT}(d^{\prime},t)=(\text{highly relevant},\text{correct})$ and
$\textrm{GT}(d,t)=($marginally relevant, correct$)$, it is reasonable to rank
$d^{\prime}$ before $d$.
Indeed, there is one _unequivocal_ way of deeming one document better than
another, and this is if document $d^{\prime}$ has better labels than document
$d$ for _every_ aspect: if for $\textrm{GT}(d,t)=(l_{1},\ldots,l_{n})$ and
$\textrm{GT}(d^{\prime},t)=(l_{1}^{\prime},\ldots,l_{n}^{\prime})$ we have
$l_{i}\preceq_{a_{i}}l_{i}^{\prime}$ for all $i\in\\{1,\ldots,n\\}$, then any
document labeled $(l_{1}^{\prime},\ldots,l_{n}^{\prime})$ is better or equal
than any document labelled $(l_{1},\ldots,l_{n})$ and should occur before it
in a “good” ranking. We denote this order relation by
$\textrm{GT}(d,t)\sqsubseteq\textrm{GT}(d^{\prime},t)$.
The order relation $\sqsubseteq$ leads to a _partial_ instead of a _total_
order, i.e., there are documents that are _not comparable_ 222A partial order
is reflexive, antisymmetric and transitive; a total order is a partial order
where all items are comparable; a weak order is a total order without
antisymmetry (Halmos, 1974)., e.g., if $d^{\prime}$ is now highly relevant and
partially correct, the final ranking is not clear: should one promote
$d^{\prime}$ (more relevant) or $d$ (more correct)? This is an example of
documents that are not comparable, so we have
$\textrm{GT}(d,t)\not\sqsubseteq\textrm{GT}(d^{\prime},t)$ and
$\textrm{GT}(d^{\prime},t)\not\sqsubseteq\textrm{GT}(d,t)$, and the choice of
whether $d^{\prime}$ is preferred to $d$ may lie on the intended application.
A partial order relation and the presence of not comparable documents imply
that it is not possible to univocally rank the documents in $D$. If we could
“complete” the partial order with a total order, or at least a weak order, we
could rank documents and define an ideal ranking, where for any
$d^{\prime},d\in D$, the order relation determines the rank position of
$d^{\prime}$ and $d$. So, before tackling the problem of evaluating a ranked
list of documents in a multi-aspect way, we build such an order relation. This
is detailed next.
### 3.2. The distance order
We now explain how to obtain a weak order relation from the partial order
relation $\sqsubseteq$. Consider the Cartesian product of all sets of labels
$L=L_{a_{1}}\times\cdots\times L_{a_{n}}$. An element $\mathbf{l}\in L$ is a
tuple of labels $\mathbf{l}=(l_{1},\dots,l_{n})$. The total order relation
will be denoted by $\preceq_{*}$ and it will be a weak order relation on $L$,
i.e., a total binary relation that is reflexive and transitive, but not
necessarily anti-symmetric (Ferrante et al., 2015, 2017, 2019b). This weak
order allows _all_ tuples of labels to be compared, i.e., for any two
$\mathbf{l},\mathbf{l}^{\prime}\in L$ we will have
$\mathbf{l}^{\prime}\preceq_{*}\mathbf{l}$ and/or
$\mathbf{l}\preceq_{*}\mathbf{l}^{\prime}$. Consequently, all documents will
be comparable through their tuple of labels.
We require that the weak order relation $\preceq_{*}$ respects the partial
order relation $\sqsubseteq$:
(1) $\forall\ \mathbf{l},\mathbf{l}^{\prime}\in L\textrm{ we have
}\mathbf{l}\sqsubseteq\mathbf{l}^{\prime}\Rightarrow\mathbf{l}\preceq_{*}\mathbf{l}^{\prime}$
This means that, for comparable documents, the partial order relation and the
weak order relation rank documents in the same way. Moreover the weak order
relation allows to rank even those documents that are not comparable with the
partial order relation.
To define $\preceq_{*}$, we embed the tuples of labels in the Euclidean space
and derive the weak order $\preceq_{*}$ using known distance functions. Let
$g$ be an embedding function that maps tuples of labels in Euclidean space
$\mathcal{L}=\mathbb{R}^{n}$:
$g(\mathbf{l})=g(l_{1},\ldots,l_{n})=(g_{a_{1}}(l_{1}),\ldots,g_{a_{n}}(l_{n}))$.
We assume that for each $a\in A$, $g_{a}$ is a non-decreasing map, i.e., for
any $l,l^{\prime}\in L_{a}$ if $l\preceq_{a}l^{\prime}$ then $g_{a}(l)\leq
g_{a}(l^{\prime})$. Intuitively, $g_{a}$ assigns a number to each label, which
allows to represent tuples of labels in the Euclidean space. We illustrate in
$\S$3.4 how the embedding function $g$ affects the final ranking of documents.
Through the embedding function $g$, each tuple of labels $\mathbf{l}$ is
represented by a point in the Euclidean space $\mathcal{L}$ denoted by
$\vec{l}=g(\mathbf{l})$. We define the _best label tuple_ as the tuple of
labels $\mathbf{l^{\star}}$ whose coordinates are the best label for each
aspect, $\mathbf{l^{\star}}=(l_{K_{a_{1}}},\ldots,l_{K_{a_{n}}})$. The idea is
to treat $\mathbf{l^{\star}}$ as the maximum element and use the distance from
this maximum element to define the desired weak order relation; e.g., for two
aspects $a_{1}$ and $a_{2}$, each tuple of labels is represented as a point in
the Euclidean plane, and the best label $\mathbf{l^{\star}}$ is represented by
the topmost and right-most point (see Fig. 1(a)). Then, given two documents
$d$ and $d^{\prime}$, $d$ is ranked before $d^{\prime}$ if $\textrm{GT}(d,t)$
is closer to the best label than $\textrm{GT}(d^{\prime},t)$.
We formally define the _distance order_ as the following relation:
(2) $\mathbf{l}\preceq_{*}\mathbf{l^{\prime}}\ \iff\
\text{Dist}(\vec{l},\vec{l^{\star}})\geq\text{Dist}(\vec{l^{\prime}},\vec{l^{\star}})$
where $\text{Dist}\colon\mathcal{L}\times\mathcal{L}\to[0,+\infty[$ is any
function such that $\text{Dist}(\vec{l}^{\star},\vec{l}^{\star})=0$333Distance
functions must be symmetric and satisfy the triangle inequality. Any such
distance function satisfies our condition on Dist, and so do our example
distances.. The relation $\preceq_{*}$ is a weak order: all $\mathbf{l}$,
$\mathbf{l^{\prime}}$ are comparable because
$\text{Dist}(\vec{l},\vec{l^{\star}})$ is defined for all $\mathbf{l}$, and as
$\geq$ is reflexive and transitive on $[0,+\infty[$, the relation
$\preceq_{*}$ is reflexive and transitive (but not necessarily antisymmetric).
Since the distance order is a weak order, it allows to deem items “equally
good” when it is impossible or undesirable to impose a strict total
order444This is the reason that _weak_ orders (that are not necessarily anti-
symmetric), rather than strict total orders, are typically used in the
literature (Yao, 1995; Ferrante et al., 2019b).. Thus we write:
(3) $\mathbf{l}=_{*}\mathbf{l^{\prime}}\ \iff\
\text{Dist}(\vec{l},\vec{l^{\star}})=\text{Dist}(\vec{l^{\prime}},\vec{l^{\star}})$
which means that $\vec{l}$ and $\vec{l^{\prime}}$ are at the same distance
from $\vec{l}^{\star}$.
Note that the distance order can be _tailored_ : we may instantiate Dist with
any valid distance function. We illustrate this in Fig. 1(b)-1(d) with
_Euclidean_ (order relation $\preceq_{2}$), _Manhattan_ (order relation
$\preceq_{1}$), and _Chebyshev_ (order relation $\preceq_{\infty}$). With
these choices of Dist, the distance order defined in Eq. (2)-(3) respects the
partial order $\sqsubseteq$, which means that it satisfies the requirement in
Eq. (1) because $g_{a}$ is a non decreasing map (a proof is provided in the
online
appendix555https://github.com/lcschv/TOMA/blob/1d562036c50f7ff0a6df00246195098d7282b1ac/CIKM2021_appendix.pdf).
### 3.3. Integration with IR measures
Next we integrate the distance order with known IR measures such as AP or
NDCG. The binary relation $=_{*}$ in Eq. (3) is an equivalence relation. Given
a tuple of labels $\mathbf{l}\in L$, its equivalence class $[\mathbf{l}]_{*}$
is the set of all tuples of labels with equal distance from the best label
$[\mathbf{l}]_{*}=\\{\mathbf{l^{\prime}}\in
L\colon\text{Dist}(\vec{l^{\prime}},\vec{l^{\star}})=\text{Dist}(\vec{l},\vec{l^{\star}})\\}$.
Inducing the relation defined in Eq. (2) on the set of documents $D$ allows to
rank documents by their membership to each equivalence class, which
corresponds to the distance of their tuple of labels to the best label. We
place closest to the top of the ranking documents whose equivalence class is
closest to the best label, and vice versa.
To combine the distance order with IR measures we map each equivalence class
(set of tuple of labels), to a non negative integer. This is similar to what
happens with single-aspect evaluation, where each label is mapped to a weight:
e.g., with $4$ relevance labels, we can compute NDCG with equi-spaced
relevance weights $\\{0,1,2,3\\}$ (Järvelin and Kekäläinen, 2002) or
exponential weights $\\{0,2,4,8\\}$ (Burges et al., 2005). We define a _weight
function_ $W\colon L\to\mathbb{N}_{0}^{+}$ as a map such that the order
relation $\preceq_{*}$ is preserved:
(4) $\forall\ \mathbf{l},\mathbf{l^{\prime}}\in
L\colon\mathbf{l}\preceq_{*}\mathbf{l^{\prime}}\implies W(\mathbf{l})\leq
W(\mathbf{l^{\prime}})$
where the constraint in Eq. (4) entails that $W$ is a non-decreasing function
with respect to the weak order $\preceq_{*}$ on the set of tuples of labels.
This means that $W$ can return different integers for each equivalence class,
but also the same integer for different equivalence classes, i.e., $0$ and
$1$, whenever we need to compute a binary single-aspect IR measure as AP.
To summarize, our TOMA method has $3$ steps:
1. (1)
We embed tuples of labels into elements of Euclidean space, and we derive the
weak order $\preceq_{*}$ using a distance function;
2. (2)
We define an adjustable weight function $W$ that preserves $\preceq_{*}$ and
maps each tuple of labels to a single integer weight (this allows to aggregate
tuple of labels so that better documents can be given greater weight);
3. (3)
Having such a weak order and the weight function $W$, any existing single-
aspect IR evaluation measure can be used to assess the quality. Thus, we
choose a single-aspect evaluation measure $\mu$ and compute the final
evaluation score as $M=\mu\circ W\colon
M(r_{t})=\mu(W(\textrm{GT}(d_{1},t)),\ldots,W(\textrm{GT}(d_{N},t)))$, where
$r_{t}$ is a ranked list of documents.
The above is compatible with any number and type of aspect.
### 3.4. Example
Table 1. Final ordering of tuples of labels embedded in the Euclidean space. Relevance labels are always embedded in the same mapping (under Relevance). We use different mappings for correctness labels (under Correctness). Tuples that are relevant and not correct (high-traffic fake news) are in red. Relevance | Correctness | Distance | Order among Tuples of Labels
---|---|---|---
$\\{0,1,2,3\\}$ | $\\{0,3/2,3\\}$ | Euclidean | $(3,3)\preceq_{*}(2,3)\preceq_{*}(3,3/2)\preceq_{*}(2,3/2)\preceq_{*}(1,3)\preceq_{*}(1,3/2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}\preceq_{*}(0,0)$
Manhattan | $(3,3)\preceq_{*}(2,3)\preceq_{*}(3,3/2)\preceq_{*}(1,3)\preceq_{*}(2,3/2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}\preceq_{*}(1,3/2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}\preceq_{*}(0,0)$
Chebyshev | $(3,3)\preceq_{*}(2,3)\preceq_{*}(3,3/2)=_{*}(2,3/2)\preceq_{*}(1,3)=_{*}(1,3/2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}=_{*}(0,0)$
$\\{0,1,2,3\\}$ | $\\{0,1,2\\}$ | Euclidean | $(3,2)\preceq_{*}(3,1)=_{*}(2,2)\preceq_{*}(2,1)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}=_{*}(1,2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}=_{*}(1,1)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}\preceq_{*}(0,0)$
Manhattan | $(3,2)\preceq_{*}(3,1)=_{*}(2,2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}=_{*}(2,1)=_{*}(1,2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}=_{*}(1,1)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}\preceq_{*}(0,0)$
Chebyshev | $(3,2)\preceq_{*}(3,1)=_{*}(2,1)=_{*}(2,2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}=_{*}(1,1)=_{*}(1,2)\preceq_{*}(0,0)$
$\\{0,1,2,3\\}$ | $\\{0,2,6\\}$ | Euclidean | $(3,6)\preceq_{*}(2,6)\preceq_{*}(1,6)\preceq_{*}(3,2)\preceq_{*}(2,2)\preceq_{*}(1,2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}\preceq_{*}(0,0)$
Manhattan | $(3,6)\preceq_{*}(2,6)\preceq_{*}(1,6)\preceq_{*}(3,2)\preceq_{*}(2,2)\preceq_{*}(1,2)=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}\preceq_{*}(0,0)$
Chebyshev | $(3,6)\preceq_{*}(2,6)\preceq_{*}(1,6)\preceq_{*}(3,2)=_{*}(2,2)=_{*}(1,2)\preceq_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(3,0)}}=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(2,0)}}=_{*}\mathbin{{\color[rgb]{0.86328125,0.078125,0.234375}\definecolor[named]{pgfstrokecolor}{rgb}{0.86328125,0.078125,0.234375}(1,0)}}=_{*}(0,0)$
We present an example on the role of different choices of embedding, distance,
and weight functions in TOMA with $4$ relevance labels
$\\{\texttt{nr},\texttt{mr},\texttt{fr},\texttt{hr}\\}$ and $3$ correctness
labels $\\{\texttt{nc},\texttt{pc},\texttt{c}\\}$. As in real scenarios (Lioma
et al., 2019), we assume that not relevant documents are not correct: as they
do not include information about the topic, they cannot be correct with
respect to that topic.
Tab. 1 shows $3$ different embeddings for correctness; the embedding for
relevance is fixed. Note that the distance functions are invariant under
translations and rotations, thus, rather than the actual values assigned from
the embedding function $g$, it is important to consider the relation between
different aspects. Independently of the choice of the embedding function and
due to the definition of the selected distance functions, we see that: (i)
Chebyshev generates the least number of equivalence classes and deems many
tuples of labels as equal, since by taking the maximum it considers just the
“furthest” or worst aspect to compute the distance; (ii) Manhatthan is somehow
in-between Chebyshev and Euclidean and generates the equivalence classes by
taking the sum across aspects; (iii) Euclidean generates the highest number of
equivalence classes as it differentiates among tuples more than Manhatthan and
is more sensitive to extreme cases, e.g., cases where one aspect has the best
label and all other aspects have the lowest label.
In the 1st scenario of Tab. 1 we map relevance and correctness to the same
interval $[0,3]$ (i.e., a highly relevant document is as “important” as a
correct document). All labels are equi-spaced in the given range (the
difference between a fairly relevant and a marginally relevant document is the
same as that between a highly relevant and a fairly relevant one). With the
Euclidean distance all relevant and not correct documents will be deemed worse
than all other documents, but will be placed before not relevant and not
correct documents. On the other hand, Chebyshev places relevant and not
correct documents in the same equivalence class as not relevant documents, so
those documents do not provide any contribution and can be simply filtered
out. Manhattan is a middle solution: highly relevant and not correct documents
are deemed better than marginally relevant and partially correct documents,
but worse than all other correct or partially correct documents.
In the 2nd scenario of Tab. 1 relevance and correctness are mapped to
different ranges, but all labels are equi-spaced with the same step of size
$1$. Here, relevance is more important than correctness. This is reflected on
the sorting of equivalence classes: for all distance functions, highly
relevant and not correct documents do not belong to the worst equivalence
classes, but they are somehow better than partially correct documents. Even
Chebyshev, which can be seen as the “strictest” distance function, places all
relevant and not correct documents in the same class, which is considered
better than the class of not relevant and not correct documents.
In the 3rd scenario of Tab. 1 correctness is mapped to a range twice the size
as the relevance range and we do not use equi-spaced labels for correctness.
We assign more importance to correctness than relevance, and among correctness
labels we penalize not correct and partially correct documents. The result is
that for all distance functions relevant and not correct documents are
considered among the worst equivalence classes. This particular setting
affects also the other equivalence classes: correctness is preferred over
relevance, e.g., correct documents should be always ranked before partially
correct documents, regardless of their relevance label.
Note that TOMA requires a weight function satisfying the requirement in Eq.
(4). If we wish to reward a system for sorting documents exactly as presented
by the equivalence classes in Tab. 1, then the weight function should assign a
different integer to each equivalence class. This choice of weight is similar
to the choice of weights for relevance labels and its impact on the evaluation
outcome is strictly tight to the evaluation measure used, as for example when
one considers NDCG with different weighting schemes (Järvelin and Kekäläinen,
2002).
Table 2. Experimental data. All aspects are labelled by TREC except popularity
($\dagger$ approximated by PageRank) and non-spamminess ($\ddagger$
approximated by Waterloo Spam Ranking). * means that the junk labels are
merged with non relevant.
| TREC tracks
---|---
| Web 2009 | Web 2010 | Web 2011 | Web 2012 | Web 2013 | Web 2014 | Task 2015 | Task 2016 | Decision 2019 | Misinfo2020
Collection | ClueWeb09 | ClueWeb12 | ClueWeb12-B13 | CommonCrawl News
Topics | 50 | 48 | 50 | 50 | 50 | 50 | 35 | 50 | 50 | 46
Submitted runs | 71 | 56 | 61 | 48 | 61 | 30 | 6 | 9 | 32 | 51
| relevance (4) | relevance (5*) | relevance (4*) | relevance (5*) | relevance (3*) | relevance (3) | relevance (2)
Aspects | popularity$\dagger$ (3) | popularity$\dagger$ (3) | popularity$\dagger$ (3) | popularity$\dagger$ (3) | usefulness (3) | credibility (2) | credibility (2)
(label grades) | non-spam$\ddagger$ (3) | non-spam$\ddagger$ (3) | non-spam$\ddagger$ (3) | non-spam$\ddagger$ (3) | popularity$\dagger$ (3) | correctness (2) | correctness (2)
| | | | | non-spam$\ddagger$ (3) | |
## 4\. Experimental Evaluation
We evaluate TOMA on $425$ rankings that were submitted as official runs to
$10$ TREC tracks (Voorhees and Ellis, 2016; Yilmaz et al., 2016; Lioma et al.,
2019; Clarke et al., 2009; Clarke and Cormack, 2011; Clarke et al., 2011;
Clarke et al., 2012; Collins-Thompson et al., 2014, 2015; Clarke et al.,
2020a) (see Tab. 2).
### 4.1. Experimental Setup
We use up to $5$ different aspects. All aspects are assessed by TREC assessors
as part of the corresponding track, except popularity and non-spamminess. We
approximate popularity by
PageRank666http://www.lemurproject.org/clueweb12/PageRank.php, and non-
spamminess by the Waterloo Spam
Ranking777https://www.mansci.uwaterloo.ca/~msmucker/cw12spam/. We discretize
the PageRank scores to generate $3$ grades of popularity (not popular, fairly
popular, highly popular), while simulating a power law distribution of popular
and not popular documents ($5\%$ highly popular, $10\%$ fairly popular, and
$85\%$ not popular). For non-spamminess, we generate $3$ grades of labels
(spam, fairly spam, not spam) from the Waterloo Spam Ranking. We treat any
document with score $<80$ as spam ($77\%$), documents with score in $[80,89]$
($14\%$) as fairly spam, and documents with score $\geq 90$ ($9\%$) as not
spam (Petersen et al., 2015).
For the Web 2010-2014 and Task 2015-2016 tracks, we merge the labels junk and
non relevant into non relevant, as was done by the TREC track organisers. For
Task 2015-2016, Decision 2019 and Misinformation 2020, _usefulness_ ,
_credibility_ , and _correctness_ were not assessed for not relevant
documents, thus not relevant documents are assumed to be not useful, not
credible, and not correct.
We evaluate $3$ versions of our method TOMA, with Euclidean, Manhattan, and
Chebyshev, as per the distance metric used in Eq. (2) (abbreviated as EUCL,
MANH, and CHEB henceforth). We compare these to two state-of-the-art
baselines, CAM (Lioma et al., 2017) and MM (Palotti et al., 2018).
Given a set of aspects $A$888CAM was originally formulated for two aspects
(Lioma et al., 2017)., CAM aggregates their scores through a weighted average:
(5) $\text{CAM}(r_{t})=\sum_{a\in A}{p_{a}\times\mu(\hat{r}_{t,a})}$
where $\mu(\cdot)$ is the evaluation measure (e.g., NDCG), $\hat{r}_{t,a}$ is
the ranking labelled with respect to aspect $a$, and $p_{a}$ is a parameter
controlling the importance of each aspect: $p_{a}\in[0,1]$ and $\sum_{a\in
A}p_{a}=1$.
MM (Palotti et al., 2018) aggregates the evaluation measure scores computed
for each aspect individually with a weighted harmonic mean:
(6) $\text{MM}(r_{t})=\frac{\sum_{a\in A}p_{a}}{\sum\limits_{a\in
A}\frac{p_{a}}{\mu(\hat{r}_{t,a})}}$
with the same notation as above.
Out of the other multi-aspect methods presented in $\S$2, we do not use WHAM
(Lioma et al., 2017) as baseline because it also uses the weighted harmonic
mean to aggregate the evaluation measure scores. However, WHAM is defined only
for relevance and credibility, and can therefore be seen as an instantiation
of MM restricted to two aspects. All other multi aspect measures in $\S$2 need
a predefined set and number of aspects, thus are not applicable in our
scenario.
We instantiate our method and the baselines using (1) NDCG (Kekäläinen and
Järvelin, 2002) and graded labels (when available); and using (2) AP (Buckley
and Voorhees, 2005) and binary labels (we convert all graded labels to binary
by treating all grades above zero as one, and grades equal/below zero as
zero). We consider all aspects equally important (all aspects are mapped to an
integer scale with one unit separating each grade). All source code is
publicly available999https://github.com/lcschv/TOMA.git.
### 4.2. Anomalies of CAM & MM
Next we discuss anomalies of CAM and MM that TOMA overcomes.
Table 3. CAM, MM and TOMA scores instantiated with AP & NDCG for all rankings
in $D$. The highest scores are in bold.
AP
---
Length $3$ | CAM | MM | EUCL | MANH | CHEB | Length $2$ | CAM | MM | EUCL | MANH | CHEB | Length $1$ | CAM | MM | EUCL | MANH | CHEB
$(d_{1},d_{2},d_{3})$ | $\mathbf{0.7917}$ | $\mathbf{0.3684}$ | $\mathbf{1}$ | $\mathbf{1}$ | $0.5$ | $(d_{1},d_{2})$ | $0.6250$ | $0.25$ | $\mathbf{1}$ | $\mathbf{1}$ | $0.5$ | $(d_{1})$ | $0.5$ | $0$ | $0.5$ | $0.5$ | $0$
$(d_{1},d_{3},d_{2})$ | $\mathbf{0.7917}$ | $\mathbf{0.3684}$ | $0.8333$ | $0.8333$ | $0.3333$ | $(d_{1},d_{3})$ | $0.6250$ | $0.25$ | $0.5$ | $0.5$ | $0$ | $(d_{2})$ | $0.25$ | $0$ | $0.5$ | $0.5$ | $\mathbf{1}$
$(d_{2},d_{1},d_{3})$ | $0.6667$ | $0.3125$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $(d_{2},d_{1})$ | $0.5$ | $0.25$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $(d_{3})$ | $0.25$ | $0$ | $0$ | $0$ | $0$
$(d_{2},d_{3},d_{1})$ | $0.6667$ | $0.25$ | $0.8333$ | $0.8333$ | $\mathbf{1}$ | $(d_{2},d_{3})$ | $0.5$ | $0$ | $0.5$ | $0.5$ | $\mathbf{1}$ | - | - | - | - | - | -
$(d_{3},d_{1},d_{2})$ | $0.6667$ | $0.3125$ | $0.5833$ | $0.5833$ | $0.3333$ | $(d_{3},d_{1})$ | $0.5$ | $0.25$ | $0.25$ | $0.25$ | $0$ | - | - | - | - | - | -
$(d_{3},d_{2},d_{1})$ | $0.6667$ | $0.25$ | $0.5833$ | $0.5833$ | $0.5$ | $(d_{3},d_{2})$ | $0.5$ | $0$ | $0.25$ | $0.25$ | $0.5$ | - | - | - | - | - | -
NDCG
Length $3$ | CAM | MM | EUCL | MANH | CHEB | Length $2$ | CAM | MM | EUCL | MANH | CHEB | Length $1$ | CAM | MM | EUCL | MANH | CHEB
$(d_{1},d_{2},d_{3})$ | $\mathbf{0.9073}$ | $0.4489$ | $0.9367$ | $0.9711$ | $0.8597$ | $(d_{1},d_{2})$ | $0.7682$ | $0.3491$ | $0.8080$ | $0.8147$ | $0.8597$ | $(d_{1})$ | $0.4728$ | $0.1491$ | $0.4290$ | $0.4693$ | $0.3801$
$(d_{1},d_{3},d_{2})$ | $0.8824$ | $0.4386$ | $0.8917$ | $0.9404$ | $0.7602$ | $(d_{1},d_{3})$ | $0.6483$ | $0.3145$ | $0.5914$ | $0.6667$ | $0.3801$ | $(d_{2})$ | $0.4682$ | $0.2258$ | $0.6006$ | $0.5475$ | $0.7602$
$(d_{2},d_{1},d_{3})$ | $0.9056$ | $\mathbf{0.4516}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $(d_{2},d_{1})$ | $0.7665$ | $0.3776$ | $0.8713$ | $0.8436$ | $\mathbf{1}$ | $(d_{3})$ | $0.2781$ | $0$ | $0.2574$ | $0.3129$ | $0$
$(d_{2},d_{3},d_{1})$ | $0.8801$ | $0.4319$ | $0.9775$ | $0.9795$ | $0.9502$ | $(d_{2},d_{3})$ | $0.6437$ | $0.2679$ | $0.7630$ | $0.7449$ | $0.7602$ | - | - | - | - | - | -
$(d_{3},d_{1},d_{2})$ | $0.8106$ | $0.3930$ | $0.8284$ | $0.8827$ | $0.6199$ | $(d_{3},d_{1})$ | $0.5765$ | $0.2801$ | $0.5281$ | $0.6089$ | $0.2398$ | - | - | - | - | - | -
$(d_{3},d_{2},d_{1})$ | $0.8100$ | $0.3827$ | $0.8509$ | $0.8929$ | $0.6697$ | $(d_{3},d_{2})$ | $0.5735$ | $0.1897$ | $0.6364$ | $0.6583$ | $0.4796$ | - | - | - | - | - | -
Figure 2. Box-plots for CAM and MM with NDCG on the Decision Track 2019. Topic
numbers are on the $x$-axis and measures scores on the $y$-axis. The maximum
value for CAM and MM is variable and depends on the topic and the aspects.
##### Problem 1: MM is ill-defined.
As the harmonic mean is not defined with zero values, MM is not defined if
$\exists a\in A$ such that $\mu(\hat{r}_{t,a})=0$, e.g., a ranking does not
retrieve any correct or relevant document. To compute MM even in these cases,
as the denominator in Eq. (6) tends to $+\infty$ if any $\mu(\hat{r}_{t,a})$
tends to zero, we set $\text{MM}(r_{t})=0$. For classification measures, this
problem is called the Strong Definiteness Axiom (Sebastiani, 2015). It
represents a serious issue for collections where there are a few documents
with a positive label for certain aspects. For example, for the Task Tracks,
since useful documents are very sparse, many systems are not able to retrieve
any useful document and they all have a $0$ score, independently of the number
of relevant documents they retrieve. TOMA does not have this problem, because
we first assign a weight to each tuple of labels and then compute a single-
aspect evaluation measure $\mu$, thus there is no division by $0$ and TOMA is
well defined.
##### Problem 2: CAM and MM can range in different intervals.
Given a set of documents $D$ and a set of aspects $A$, by definition CAM and
MM are multi-aspect evaluation measures $M\colon D^{*}\to[0,X]$, where $D^{*}$
is the set of rankings and $X\leq 1$. Depending on $D$ and $A$, there exist
cases with $X<1$.
To prove this claim, we need to show that when $M$ is CAM or MM, $\exists\
D,A$ such that:
(7) $\max_{r\in D^{*}}M(\hat{r}_{t})<1$
i.e., for each ranking of documents in $D^{*}$ the maximum measure score will
be less than $1$. To build such an example, the set $D$ needs to contain
documents with not comparable tuples of labels:
(8) $\exists d_{1},d_{2}\in
D\colon\textrm{GT}(d_{1})\not\sqsubseteq\textrm{GT}(d_{2})\textrm{ and
}\textrm{GT}(d_{2})\not\sqsubseteq\textrm{GT}(d_{1})\iff\\\ \exists
d_{1},d_{2}\in D\textrm{ and }\exists a_{1},a_{2}\in A\colon\\\
\textrm{GT}_{a_{1}}(d_{1})\prec_{a_{1}}\textrm{GT}_{a_{1}}(d_{2})\textrm{ and
}\textrm{GT}_{a_{2}}(d_{2})\prec_{a_{2}}\textrm{GT}_{a_{2}}(d_{1})$
In this case, CAM and MM cannot achieve a score equal to $1$ as illustrated by
the following example.
Consider the example in $\S$3.4 with $A=\\{$relevance, correctness$\\}$. Let
$D$ be a set with $3$ documents $D=\\{d_{1},d_{2},d_{3}\\}$ with labels
$\textrm{GT}(d_{1})=(\texttt{mr},\texttt{c})$,
$\textrm{GT}(d_{2})=(\texttt{hr},\texttt{pc})$ and
$\textrm{GT}(d_{3})=(\texttt{hr},\texttt{nc})$. Documents $(d_{1},d_{2})$ and
$(d_{1},d_{3})$ are not comparable and there is no unequivocal way of sorting
them, e.g., it is not clear if $d_{1}$ should be ranked before $d_{2}$ or
vice-versa.
Let us consider CAM and MM instantiated with AP and NDCG. For AP we use a
harsh mapping for relevance and correctness, i.e.,
$\\{\texttt{fr},\texttt{hr}\\}\mapsto 1$ and
$\\{\texttt{mr},\texttt{nr}\\}\mapsto 0$, and $\texttt{c}\mapsto 1$ and
$\\{\texttt{pc},\texttt{nc}\\}\mapsto 0$. For NDCG we map each category to a
different integer, for relevance we have: $\texttt{hr}\mapsto 15$,
$\texttt{fr}\mapsto 10$, $\texttt{mr}\mapsto 5$, $\texttt{nr}\mapsto 0$, and
for correctness we have: $\texttt{c}\mapsto 10$, $\texttt{pc}\mapsto 5$,
$\texttt{nc}\mapsto 0$. The NDCG ideal ranking (Järvelin and Kekäläinen, 2002)
for relevance is: $(\texttt{hr},\texttt{hr},\texttt{mr})$ and for correctness
is $(\texttt{c},\texttt{pc},\texttt{nc})$. The NDCG $\log$ base is set to $2$.
For TOMA we use the embedding of the first row in Tab. 1 and as weight
function we map each equivalence class to a different integer with step $1$.
We instantiate TOMA with AP and NDCG with log base $2$ (Tab. 3). Since AP does
not handle multi-graded weights, we map the top half of the equivalence
classes to $1$ and the rest to $0$. Tab. 3 shows CAM, MM, and TOMA scores
instantiated with AP NDCG for each possible ranking of documents in $D$.
In Tab. 3 none of the rankings in $D^{*}$ can achieve a score equal to $1$ for
CAM and MM, while TOMA has at least one ranking with score $1$. In CAM and MM
this happens because, any way we sort the documents, either we penalize
correctness, e.g., $(d_{2},d_{3},d_{1})$ or we penalize relevance, e.g.,
$(d_{1},d_{2},d_{3})$. TOMA does not have this problem, since it first defines
how to sort tuples of labels, then weights them accordingly and computes the
measure score. Thus, if we sort documents in the order induced by
$\preceq_{*}$, we obtain a score equal to $1$ (proof in the appendix).
Experiments on real data confirm this, as detailed next.
##### Estimating CAM and MM Upper Bound
With the following experiment we show that with real data CAM and MM can be
upper bounded by a value $X$ lower than $1$. To estimate the value $X$ with
real data, we generate different ideal rankings of documents with different
strategies. The intuition is that by ranking documents in the best possible
way, we should achieve a score equal to $1$, as it happens for any single-
aspect evaluation measure computed against the ideal ranking. Since CAM and MM
do not define how to sort documents, i.e., a total order relation
$\preceq_{*}$, we need to test different possible strategies to build these
ideal rankings.
First, we define the ideal rankings obtained with a recursive strategy: these
are the ideal rankings for each aspect when considered separately, e.g., for
$3$ aspects, $a_{1}$, $a_{2}$ and $a_{3}$, with a preference order where
$a_{1}$ is followed by $a_{2}$, followed by $a_{3}$: (1) we sort the documents
with decreasing label for $a_{1}$; (2) among the documents with the same label
for $a_{1}$, we sort the documents with decreasing label for $a_{2}$; (3)
among the documents with the same label for $a_{1}$ and $a_{2}$, we sort the
documents with decreasing label for $a_{3}$. We generate these ideal rankings
for each possible preference order among the aspects.
We also generate $3$ additional ideal rankings: (1) we sum the weights across
aspects and sort the documents by this sum; (2) we sum the squared weights
across aspects and sort the documents by this sum; (3) we consider the highest
weight across aspects and sort documents by their highest weight regardless of
the aspect.
Fig. 2 reports the distributions of CAM with AP scores for the ideal rankings
for the Decision Track 2019. These distributions depend on the aspect and the
topic. We see that the upper bound $X$ is variable and depends on the topic:
just for $2\%$ of topics it is equal to $1$ and for $26\%$ topics it is lower
than $0.9$. We obtain similar or even more extreme distributions of scores for
all the other tracks (except for Misinformation 2020, reported in the online
appendix).
##### Interpretability of CAM and MM scores
Problem $2$ is especially important because it affects the interpretability of
CAM and MM scores. When a measure is used to assess the quality of a single
ranking in isolation, it should be intuitively interpretable (Kumar and
Vassilvitskii, 2010), e.g., NDCG=$0.6$ has the intuitive interpretation that
the ranking can be further improved by $0.4$. If TOMA is instantiated with
NDCG, the intuitive interpretability of NDCG holds, but if CAM or MM are
instantiated with NDCG, the intuitive interpretability of NDCG is lost: by the
arguments above, CAM and MM may fail to obtain an optimal score of $1$, and
the optimal score depends on $A$ and $D$, hence it cannot in general be known
a priori.
This issue is important for MM, which is affected also by Problem $1$, and
therefore may have $X<<1$. Thus MM scores can be compressed towards $0$, and
this can lead to cases with many ties, where it is hard to distinguish between
different rankings.
### 4.3. Experimental Findings
Table 4. Kendall’s $\tau$ correlation between rankings of systems and
discriminative power (the higher, the better; best is in bold). Not all aspect
combinations occur in all tracks (marked grey).
| WEB2009 | WEB2010 | WEB2011 | WEB2012 | WEB2013 | WEB2014 | TASK15 | TASK16 | DECISION19 | MISINFO 2020
---|---|---|---|---|---|---|---|---|---|---
| NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP | NDCG | AP
| CORRELATION
EUCL - CAM | 0.25 | 0.16 | 0.18 | 0.12 | 0.21 | 0.11 | 0.31 | 0.26 | 0.22 | 0.12 | 0.30 | 0.23 | 0.68 | 0.54 | 0.63 | 0.55 | 0.76 | 0.60 | 0.72 | 0.51
EUCL - MM | 0.07 | 0.04 | 0.05 | 0.00 | 0.06 | 0.03 | 0.16 | 0.13 | 0.12 | 0.04 | 0.17 | 0.04 | 0.36 | 0.17 | 0.02 | -0.10 | 0.46 | 0.31 | 0.45 | 0.27
MANH - CAM | 0.22 | 0.16 | 0.21 | 0.12 | 0.16 | 0.11 | 0.34 | 0.26 | 0.31 | 0.12 | 0.41 | 0.23 | 0.62 | 0.54 | 0.60 | 0.55 | 0.69 | 0.60 | 0.72 | 0.51
MANH - MM | 0.06 | 0.04 | 0.01 | 0.01 | 0.06 | 0.03 | 0.16 | 0.13 | 0.11 | 0.04 | 0.15 | 0.04 | 0.28 | 0.17 | -0.06 | -0.10 | 0.41 | 0.31 | 0.45 | 0.27
CHEB - CAM | 0.01 | 0.02 | 0.06 | 0.05 | 0.02 | 0.02 | 0.19 | 0.15 | 0.11 | 0.00 | 0.19 | 0.07 | 0.26 | 0.16 | -0.13 | -0.18 | 0.27 | 0.27 | 0.29 | 0.24
CHEB - MM | 0.06 | 0.09 | 0.00 | 0.03 | 0.09 | 0.01 | 0.19 | 0.16 | 0.12 | 0.08 | 0.14 | 0.05 | 0.88 | 0.86 | 0.52 | 0.53 | 0.58 | 0.60 | 0.54 | 0.52
EUCL - MANH | 0.36 | 1.00 | 0.19 | 1.00 | 0.21 | 1.00 | 0.34 | 1.00 | 0.20 | 1.00 | 0.34 | 1.00 | 0.87 | 1.00 | 0.71 | 1.00 | 0.72 | 1.00 | 1.00 | 1.00
EUCL - CHEB | 0.01 | 0.02 | 0.10 | 0.03 | 0.01 | 0.03 | 0.32 | 0.11 | 0.22 | 0.04 | 0.30 | 0.12 | 0.33 | 0.19 | -0.21 | -0.21 | 0.28 | 0.21 | 0.26 | 0.20
MANH - CHEB | 0.01 | 0.02 | 0.03 | 0.03 | 0.02 | 0.03 | 0.21 | 0.11 | 0.10 | 0.04 | 0.18 | 0.12 | 0.32 | 0.19 | -0.24 | -0.21 | 0.25 | 0.21 | 0.26 | 0.20
CAM - MM | 0.10 | 0.05 | 0.05 | 0.01 | 0.11 | -0.01 | 0.23 | 0.13 | 0.16 | 0.03 | 0.26 | 0.04 | 0.30 | 0.18 | 0.09 | 0.00 | 0.51 | 0.41 | 0.51 | 0.42
| CORRELATION
Relevance - Popularity | 0.03 | 0.05 | 0.01 | 0.0 | 0.01 | 0.02 | 0.09 | 0.09 | 0.06 | 0.01 | 0.07 | 0.02 | 0.04 | 0.04 | -0.03 | 0.01 | | | |
Relevance - Non-spam | 0.05 | 0.03 | 0.02 | 0.0 | 0.03 | 0.01 | 0.07 | 0.05 | -0.02 | -0.01 | 0.07 | -0.01 | 0.25 | 0.17 | -0.07 | -0.08 | | | |
Popularity - Non-spam | 0.04 | 0.03 | 0.02 | -0.01 | 0.04 | 0.01 | 0.07 | 0.04 | -0.03 | -0.02 | 0.04 | 0.00 | 0.07 | 0.02 | 0.08 | 0.06 | | | |
Relevance - Usefulness | | | | | | | | | | | | | 0.75 | 0.75 | 0.75 | 0.74 | | | |
Usefulness - Popularity | | | | | | | | | | | | | 0.10 | 0.06 | -0.04 | 0.00 | | | |
Usefulness - Non-spam | | | | | | | | | | | | | 0.40 | 0.33 | -0.16 | -0.19 | | | |
Credibility - Correctness | | | | | | | | | | | | | | | | | 0.26 | 0.26 | 0.28 | 0.24
Relevance - Credibility | | | | | | | | | | | | | | | | | 0.33 | 0.33 | 0.29 | 0.25
Relevance - Correctness | | | | | | | | | | | | | | | | | 0.42 | 0.49 | 0.49 | 0.47
| DISCRIMINATIVE POWER OF MEASURES
CAM | 75.98 | 64.43 | 66.32 | 61.23 | 75.14 | 61.64 | 68.71 | 56.74 | 76.89 | 57.05 | 85.06 | 78.85 | 53.33 | 33.33 | 72.22 | 55.56 | 72.58 | 70.56 | 71.53 | 70.90
MM | 75.61 | 50.58 | 72.89 | 67.79 | 67.32 | 67.81 | 62.68 | 56.12 | 80.71 | 46.99 | 74.25 | 53.56 | 0.00 | 0.00 | 0.00 | 0.00 | 60.08 | 53.23 | 68.31 | 62.20
EUCL | 75.29 | 72.64 | 62.96 | 66.75 | 75.14 | 70.33 | 66.13 | 64.10 | 75.14 | 59.45 | 80.92 | 78.85 | 66.67 | 66.67 | 69.44 | 75.00 | 73.59 | 73.99 | 72.86 | 75.14
MANH | 76.66 | 72.68 | 63.59 | 67.14 | 77.32 | 70.38 | 66.05 | 64.18 | 76.67 | 59.34 | 86.44 | 79.08 | 66.67 | 53.33 | 75.00 | 75.00 | 73.79 | 73.79 | 73.02 | 74.98
CHEB | 50.18 | 6.32 | 59.82 | 51.49 | 73.06 | 50.11 | 61.08 | 39.36 | 77.10 | 49.34 | 75.17 | 66.21 | 0.00 | 0.00 | 0.00 | 0.00 | 42.54 | 29.84 | 65.41 | 53.33
Empirically, evaluation measures are commonly assessed in terms of their
correlation (Ferrante et al., 2019a), discriminative power (Sakai, 2007),
informativeness (Aslam et al., 2005), intuitiveness (Sakai, 2012), and
unanimity (Albahem et al., 2019). Out of these, we report only correlation and
discriminative power because the rest does not apply: the informativeness test
(Aslam et al., 2005) requires a precision recall-curve, which cannot be
defined for multi-aspect evaluation; the intuitiveness test (Sakai, 2012)
requires simple single-aspect measures (e.g. precision, recall), which do not
apply to multi-aspect evaluation; the unanimity test (Albahem et al., 2019),
which is defined for multi-aspect evaluation, requires that all the simple
measures agree over all aspects, which happened extremely rarely in our data,
especially as the number of aspects increased (see the low correlation among
aspects in Tab. 4).
#### 4.3.1. Correlation Analysis
We use Kendall’s $\tau$ (Kendall, 1945) to estimate TOMA’s correlation to MM
and CAM. Generally, if a new evaluation measure strongly correlates to an
existing one, it is likely to represent redundant information (Webber et al.,
2008). We use Kendall’s $\tau$ because it has better gross-error sensitivity
than the Pearson correlation coefficient (Croux and Dehon, 2010), and because
the Spearman correlation coefficient cannot handle ties. As per (Ferrante et
al., 2019a), we compute the correlation topic-by-topic. For each topic we
consider the Rankings of Submitted runs (RoS) corresponding to two different
measures (one ranking per measure) and then compute Kendall’s $\tau$ between
the two RoS. We report Kendall’s $\tau$ averaged across all topics. As per
(Voorhees, 1998, 2001), we consider two rankings equivalent if Kendall’s
$\tau$ is greater than $0.9$.
Tab. 4 shows the findings, which are summarised as follows:
* •
The RoS corresponding to EUCL - MANH are equivalent ($\tau=1$) at all times
for AP. This perfect correlation for AP happens because, by definition, when
the sets of equivalence classes from these approaches are mapped to binary
labels, they produce the exact same set of labels (see also Tab. 3). For NDCG,
$\tau=0.19-1$, where higher correlations correspond to tracks where some
aspects are not assessed for non relevant documents, thus there are less
extreme cases and EUCL is more similar to MANH.
* •
The RoS corresponding to (EUCL, MANH) - CHEB are very weakly correlated
($\tau=0.01-0.32$), this is due to Chebyshev distance being very harsh, since
many equivalence classes are considered equivalent to the class of non
relevant documents.
* •
The RoS corresponding to EUCL - CAM and MANH - CAM are very weakly correlated
($\tau=0.11-0.41$) for the Web tracks, but moderately correlated
($\tau=0.54-0.76$) for the Task, Decision and Misinformation tracks. This
happens because: (i) the runs submitted to the Web tracks were not designed to
account for multiple aspects and (ii) for the Task, Decision and
Misinformation tracks, usefulness, credibility and correctness are not
assessed for non relevant documents. Therefore, since some of the values are
missing, these methods generate a lower number of equivalence classes, which
make them more similar to CAM. Whereas, for the Web tracks, popularity and
non-spamminess are approximated for all documents, meaning that MANH and EUCL
can possibly generate all the different equivalence classes, even for non
relevant documents. This makes them less similar to CAM than on the Task or
Decision tracks.
* •
For the Task, Decision and Misinformation tracks, the RoS corresponding to MM
and CHEB are moderately correlated ($\tau=0.52-0.88$). The fact that, for
these tracks, usefulness, credibility and correctness are not assessed for non
relevant documents, means that all the documents that are mapped to a $0$
weight with CHEB, are also contributing as $0$ to MM.
To contextualise these findings, the middle part of Tab. 4 shows the $\tau$
values of the RoS corresponding to evaluating a single aspect only. Overall,
the resulting correlations are low to non-existent, meaning that considering
multiple aspects affects the final evaluation outcome. The two exceptions
where the correlation between RoS is not very low are:
* •
For Task 2015-2016, for relevance - usefulness, $\tau=0.74-0.75$. This happens
because: (1) usefulness is not assessed for non relevant documents, thus non
relevant documents are assumed to be not useful, and (2) usefulness is a very
sparse signal (1.75% of documents are useful).
* •
For the Decision and Misinformation Tracks, for all aspects, $\tau=0.24-0.49$.
Again here credibility and correctness are not assessed for non relevant
documents ($6.89\%$ of documents are credible and $9.75\%$ are correct for the
Decision Track; $13.73\%$ of documents are correct and $27.62\%$ are credible
for the Misinformation Track), so the correlation is not as high as for the
Task tracks.
Overall, the most correlated RoS correspond to: EUCL - MANH ($\tau$ up to
$1$), (EUCL, MANH)- CAM ($\tau$ up to $0.76$), and CHEB - MM ($\tau$ up to
$0.88$). Intuitively, EUCL and MANH may be more similar to CAM (mean), while
CHEB may be more similar to MM (harmonic mean). Thus TOMA proposes an
alternative evaluation framework, which overcomes CAM and MM anomalies (see
$\S$4.2). The fact that $\tau$ values between TOMA and the baselines are never
above $0.9$ means that there are noticeable differences between the RoS
generated by TOMA and by CAM or MM. Recall that all measures are instantiated
with NDCG or AP, meaning that differences between them are due to how multi-
aspect labels are treated.
#### 4.3.2. Discriminative Power
We use Bootstrap Hypothesis Test (Sakai, 2007) to estimate the discriminative
power of TOMA, CAM and MM. Given a set of topics and a set of runs, we first
generate subsets of topics by sampling with replacement the complete set of
topics. We set the number of bootstrap samples to $10\,000$. To assess whether
the measure scores for pairs of runs can be considered different at a given
confidence level, we use a Paired Bootstrap Hypothesis Test. The confidence
level is $1-\alpha$, where $\alpha$ is the Type I Error, i.e., the probability
to consider two systems different even if they are equivalent. We set
$\alpha=0.01$, requiring strong evidence for two systems to be different.
Tab. 4 (bottom part) displays the results of the discriminative power
analysis, where the higher the score, the more discriminative (i.e., the
better) the approach. We see that $16/20$ times either MANH ($12/20$) or EUCL
($6/20$)101010Ties are included in these counts. is best. The remaining $4$
times, MM is best $3$ times, and CAM once. We also see that CHEB is never
best, and for Task 2015-2016 it is actually zero. This is due to the very
small amount of positive labels for usefulness in that track. For the same
reason, MM is also zero for the same track. Overall, CHEB is the least
discriminative measure, followed by MM; this is due to how these methods treat
tuples of labels: the fact that if one aspect label is zero, then the whole
score is zero, practically means that many runs are considered equal purely on
that basis.
#### 4.3.3. Zero-aspect documents
Table 5. Number of times (%) that the labels of all aspects sum to $0$ for a
document that is ranked at position $1$-$5$ (column 1) in a run that has been
assessed as best per $\\{$topic, track, year$\\}$ separately with $\\{$CAM,
MM, EUCL, MANH, CHEB$\\}$ using a retrieval cutoff of 5. The lower, the
better.
Rank | CAM | MM | EUCL | MANH | CHEB
---|---|---|---|---|---
1 | 51 (1.18%) | 131 (3.02%) | 39 (0.90%) | 33 (0.76%) | 154 (3.55%)
2 | 65 (1.50%) | 159 (3.67%) | 50 (1.15%) | 48 (1.11%) | 179 (4.13%)
3 | 103 (2.38%) | 202 (4.66%) | 88 (2.03%) | 78 (1.80%) | 185 (4.17%)
4 | 102 (2.35%) | 173 (3.99%) | 86 (1.99%) | 74 (1.71%) | 183 (4.23%)
5 | 107 (2.47%) | 196 (4.53%) | 95 (2.19%) | 81 (1.87%) | 205 (4.73%)
1-5 | 428 (9.88%) | 861 (19.88%) | 358 (8.27%) | 314 (7.25%) | 906 (20.92%)
Our next analysis is motivated by the empirical trash$@k$ measure often used
in industry to mitigate the high cost of retrieving “trash” in high ranks. We
count how often the labels of all aspects sum to zero for a document that has
been ranked at position $1$-$5$ in a run that has been assessed as the best
run per track year, on a per query basis, using a retrieval cutoff of $5$,
separately with $\\{$CAM, MM, EUCL, MANH, CHEB$\\}$ when instantiated
separately with NDCG and AP. When the labels of all aspects sum to zero, this
means that the corresponding document is of the worst quality. Ideally, such
documents should not be retrieved, but when they do, they should not be in the
top 5.
In Tab. 5 we see that MANH is associated with the lowest amount of zero-aspect
documents, closely followed by EUCL. This happens because MANH is designed so
that the higher the sum of a document’s labels across aspects, the better that
document will be deemed. CHEB is overall worst, closely followed by MM. This
closeness between EUCL-MANH and CHEB-MM agrees with the previous correlation
and discriminative power analysis. Overall, MANH (and less so EUCL) penalise
low quality documents the best.
#### 4.3.4. Document quality @1-100
Table 6. Average sum of aspect labels for a document that is ranked at position $1$-$100$ (column 1) in a run that has been assessed as best per $\\{$topic, track, year$\\}$ separately with $\\{$CAM, MM, EUCL, MANH, CHEB$\\}$ using a retrieval cutoff of 100. The higher, the better. Ranks | CAM | MM | EUCL | MANH | CHEB
---|---|---|---|---|---
1-25 | 1.70 | 1.49 | 1.67 | 1.69 | 1.39
26-50 | 0.85 | 0.78 | 0.91 | 0.94 | 0.70
51-75 | 0.57 | 0.53 | 0.63 | 0.64 | 0.48
76-100 | 0.40 | 0.39 | 0.43 | 0.44 | 0.36
We look at the quality of documents that have been ranked at positions 1-100
in a run that has been assessed as best per $\\{$topic, track, year$\\}$
separately with $\\{$CAM, MM, EUCL, MANH, CHEB$\\}$, when instantiated
separately with NDCG and AP, using a retrieval cutoff of $100$. We split the
ranks $1$-$100$ into four sets ($1$-$25$, $26$-$50$, $51$-$75$, $76$-$100$).
For each document in each set, we sum the labels of its aspects, and we report
the average of these sums, which can be seen as an approximation of the
average document quality (the higher, the better).
As expected, we see that the numbers in Tab. 6, and hence document quality,
drop as we move down the ranking, at all times. Comparing across columns
however, we see that, for the runs that were assessed as best by MANH,
document quality is overall, albeit marginally, the best, at ranks 26-100.
This illustrates that the design of MANH (the higher the sum of a document’s
labels across aspects, the better that document will be considered) gives it
the practical advantage of, not only reducing the amount of low quality
documents in the top ranks (as seen in Tab. 5), but also of increasing the
quality of documents further down the ranking, as we see now. Again, as
previously, we observe that EUCL is a close second-best method, CHEB and MM
are overall worst, and CAM is in between (although best, together with MANH,
for the top ranks).
## 5\. Conclusion and Limitations
Multi-aspect evaluation is a special case of IR evaluation where the ranked
list of documents returned by an IR system in response to a query must be
assessed in terms of not only relevance, but also other aspects (or
dimensions) of the ranked documents, e.g., credibility or usefulness. We
presented a principled multi-aspect evaluation approach, called TOMA, that is
defined for any number and type of aspect, and that allows for (i) aspects
having different gradings, (ii) any relative importance weighting for
different aspects, and (iii) integration with any existing single-aspect
evaluation measure, such as NDCG. We showed that TOMA has better
discriminative power than prior approaches to multi-aspect evaluation, and
that it is better at rewarding high quality documents across the ranking.
One limitation of TOMA is represented by the arbitrary choices of the
embedding function, the distance function and the weight function. The
embedding function maps labels from a nominal or ordinal scale to an interval
or ratio scale. This calls for a in-depth investigation of the theoretical
properties of TOMA using the existing axiomatic treatments of IR effectiveness
measures (Amigó et al., 2018a; Maddalena and Mizzaro, 2014; Busin and Mizzaro,
2013; Amigó and Mizzaro, 2020; Ferrante et al., 2021). This also motivates a
deep analysis of the interactions between different aspects and/or documents
and how to handle them with TOMA, for example by defining a proper embedding
and distance function which account for aspects as diversity, novelty, and
redundancy. Moreover, the embedding function combined with the distance
function can generate a large number of tuples of labels, which can be mapped
to different integers through the weight function. This might be a problem for
gain based measures, thus a possible solution is to use TOMA to define the
ideal ranking and then use effectiveness measures based on similarity to ideal
rankings (Clarke et al., 2020c, b, 2021). Finally, the empirical impact of
varying both distance and weight functions should also be investigated, as
should the impact of employing further multi-graded measures as ERR (Chapelle
et al., 2009), and the alignment of our current approach with real user
preferences.
Acknowledgments. This paper is partially supported by the EU Horizon 2020
research and innovation programme under the MSCA grant No. 893667.
## References
* (1)
* Agrawal et al. (2009) R. Agrawal, S. Gollapudi, A. Halverson, and S. Ieong. 2009\. Diversifying Search Results. In _Proc. 2nd ACM International Conference on Web Searching and Data Mining (WSDM 2009)_ , R. Baeza-Yates, P. Boldi, B. Ribeiro-Neto, and B. B. Cambazoglu (Eds.). ACM Press, New York, USA, 5–14.
* Albahem et al. (2019) A. Albahem, , D. Spina, F. Scholer, and L. Cavedon. 2019. Meta-evaluation of Dynamic Search: How Do Metrics Capture Topical Relevance, Diversity and User Effort?. In _Advances in Information Retrieval. Proc. 41st European Conference on IR Research (ECIR 2019)_ , L. Azzopardi, B. Stein, N. Fuhr, P. Mayr, C. Hauff, and D. Hiemstra (Eds.). Lecture Notes in Computer Science (LNCS) 10772, Springer, Heidelberg, Germany, 607–620.
* Amigó et al. (2018a) E. Amigó, F. Giner, S. Mizzaro, and D. Spina. 2018a. A Formal Account of Effectiveness Evaluation and Ranking Fusion. In _Proc. 4th ACM SIGIR International Conference on the Theory of Information Retrieval (ICTIR 2018)_ , D. Song, T.-Y. Liu, L. Sun, P. Bruza, M. Melucci, F. Sebastiani, and G. Hui Yang (Eds.). ACM Press, New York, USA, 123–130.
* Amigó and Mizzaro (2020) E. Amigó and S. Mizzaro. 2020. On the Nature of Information Access Evaluation Metrics: a Unifying Framework. _Information Retrieval Journal_ 23, 3 (2020), 318–386. https://doi.org/10.1007/s10791-020-09374-0
* Amigó et al. (2018b) E. Amigó, D. Spina, and J. Carrillo-de Albornoz. 2018b. An Axiomatic Analysis of Diversity Evaluation Metrics: Introducing the Rank-Biased Utility Metric. In _Proc. 41th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2018)_ , K. Collins-Thompson, Q. Mei, B. Davison, Y. Liu, and E. Yilmaz (Eds.). ACM Press, New York, USA, 625–634.
* Aslam et al. (2005) J. A. Aslam, E. Yilmaz, and V. Pavlu. 2005. The maximum entropy method for analyzing retrieval measures. In _SIGIR 2005: Proceedings of the 28th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Salvador, Brazil, August 15-19, 2005_ , Ricardo A. Baeza-Yates, Nivio Ziviani, Gary Marchionini, Alistair Moffat, and John Tait (Eds.). ACM, 27–34. https://doi.org/10.1145/1076034.1076042
* Buckley and Voorhees (2005) C. Buckley and E. M. Voorhees. 2005. Retrieval System Evaluation. In _TREC. Experiment and Evaluation in Information Retrieval_ , D. K. Harman and E. M. Voorhees (Eds.). MIT Press, Cambridge (MA), USA, 53–78.
* Burges et al. (2005) C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender. 2005. Learning to Rank using Gradient Descent. In _Proc. 22nd International Conference on Machine Learning (ICML 2005)_ , S. Dzeroski, L. De Raedt, and S. Wrobel (Eds.). ACM Press, New York, USA, 89–96.
* Busin and Mizzaro (2013) L. Busin and S. Mizzaro. 2013. Axiometrics: An Axiomatic Approach to Information Retrieval Effectiveness Metrics. In _Proc. 4th International Conference on the Theory of Information Retrieval (ICTIR 2013)_ , O. Kurland, D. Metzler, C. Lioma, B. Larsen, and P. Ingwersen (Eds.). ACM Press, New York, USA, 22–29.
* Chapelle et al. (2009) O. Chapelle, D. Metzler, Y. Zhang, and P. Grinspan. 2009\. Expected Reciprocal Rank for Graded Relevance. In _Proc. 18th International Conference on Information and Knowledge Management (CIKM 2009)_ , D. W.-L. Cheung, I.-Y. Song, W. W. Chu, X. Hu, and J. J. Lin (Eds.). ACM Press, New York, USA, 621–630.
* Clarke et al. (2009) C. Clarke, N. Craswell, and I. Soboroff. 2009. Overview of the TREC 2009 Web Track. In _TREC_. National Institute of Standards and Technology (NIST).
* Clarke et al. (2020a) Charles Clarke, Maria Maistro, Mark Smucker, and Guido Zuccon. 2020a. Overview of the TREC 2020 Health Misinformation Track (to appear). In _Proceedings of the Twenty-Nine Text REtrieval Conference, TREC 2020, Gaithersburg, Maryland, USA, November 16-19, 2020_ _(NIST Special Publication)_ , Ellen M. Voorhees and Angela Ellis (Eds.), Vol. -. National Institute of Standards and Technology (NIST).
* Clarke and Cormack (2011) C. L. A. Clarke and G. V. Cormack. 2011. Overview of the TREC 2010 Web Track. In _TREC_. National Institute of Standards and Technology (NIST).
* Clarke et al. (2011) C. L. A. Clarke, N. Craswell, I. Soboroff, and E. M. Voorhees. 2011. Overview of the TREC 2011 Web Track. In _TREC_. National Institute of Standards and Technology (NIST).
* Clarke et al. (2012) C. L. A. Clarke, N. Craswell, and E. M. Voorhees. 2012\. Overview of the TREC 2012 Web Track. In _TREC_. National Institute of Standards and Technology (NIST).
* Clarke et al. (2008) C. L. A. Clarke, M. Kolla, G. V. Cormack, O. Vechtomova, A. Ashkan, S. Büttcher, and I. MacKinnon. 2008. Novelty and Diversity in Information Retrieval Evaluation. In _Proc. 31st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2008)_ , T.-S. Chua, M.-K. Leong, D. W. Oard, and F. Sebastiani (Eds.). ACM Press, New York, USA, 659–666.
* Clarke et al. (2020b) Charles L. A. Clarke, Mark D. Smucker, and Alexandra Vtyurina. 2020b. Offline Evaluation by Maximum Similarity to an Ideal Ranking. In _CIKM ’20: The 29th ACM International Conference on Information and Knowledge Management, Virtual Event, Ireland, October 19-23, 2020_ , Mathieu d’Aquin, Stefan Dietze, Claudia Hauff, Edward Curry, and Philippe Cudré-Mauroux (Eds.). ACM, 225–234.
* Clarke et al. (2020c) Charles L. A. Clarke, Alexandra Vtyurina, and Mark D. Smucker. 2020c. Offline Evaluation without Gain. In _ICTIR ’20: The 2020 ACM SIGIR International Conference on the Theory of Information Retrieval, Virtual Event, Norway, September 14-17, 2020_ , Krisztian Balog, Vinay Setty, Christina Lioma, Yiqun Liu, Min Zhang, and Klaus Berberich (Eds.). ACM, 185–192.
* Clarke et al. (2021) C. L. A. Clarke, A. Vtyurina, and M. D. Smucker. 2021\. Assessing Top-$k$ Preferences. arXiv:cs.IR/2007.11682
* Collins-Thompson et al. (2014) K. Collins-Thompson, P. Bennett, F. Diaz, C. L. A. Clarke, and E. M. Voorhees. 2014. TREC 2014 Web Track Overview. In _The Twenty-Second Text REtrieval Conference Proceedings (TREC 2013)_ , E. M. Voorhees (Ed.). National Institute of Standards and Technology (NIST), Special Publication 500-302, Washington, USA.
* Collins-Thompson et al. (2015) K. Collins-Thompson, P. Bennett, F. Diaz, C. L. A. Clarke, and E. M. Voorhees. 2015. TREC 2014 Web Track Overview. In _The Twenty-Third Text REtrieval Conference Proceedings (TREC 2014)_ , E. M. Voorhees and A. Ellis (Eds.). National Institute of Standards and Technology (NIST), Special Publication 500-308, Washington, USA.
* Croux and Dehon (2010) C. Croux and C. Dehon. 2010\. Influence Functions of the Spearman and Kendall Correlation Measures. _Statistical Methods & Applications_ 19 (2010), 497–515. https://doi.org/10.1007/s10260-010-0142-z
* Ferrante et al. (2021) M. Ferrante, N. Ferro, and N Fuhr. 2021. Towards Meaningful Statements in IR Evaluation. Mapping Evaluation Measures to Interval Scales. _CoRR_ abs/2101.02668 (2021). arXiv:2101.02668 https://arxiv.org/abs/2101.02668
* Ferrante et al. (2019a) M. Ferrante, N. Ferro, and E. Losiouk. 2019a. How do Interval Scales Help us with Better Understanding IR Evaluation Measures? _Information Retrieval Journal_ (2019). https://doi.org/10.1007/s10791-019-09362-z
* Ferrante et al. (2015) M. Ferrante, N. Ferro, and M. Maistro. 2015. Towards a Formal Framework for Utility-Oriented Measurements of Retrieval Effectiveness. In _Proceedings of the 2015 International Conference on The Theory of Information Retrieval_ _(ICTIR ’15)_. Association for Computing Machinery, New York, NY, USA, 21–30. https://doi.org/10.1145/2808194.2809452
* Ferrante et al. (2017) M. Ferrante, N. Ferro, and Silvia P. 2017. Are IR Evaluation Measures on an Interval Scale?. In _Proceedings of the ACM SIGIR International Conference on Theory of Information Retrieval_ _(ICTIR ’17)_. Association for Computing Machinery, New York, NY, USA, 67–74. https://doi.org/10.1145/3121050.3121058
* Ferrante et al. (2019b) M. Ferrante, N. Ferro, and S. Pontarollo. 2019b. A General Theory of IR Evaluation Measures. _IEEE Transactions on Knowledge and Data Engineering (TKDE)_ 31, 3 (2019), 409–422.
* Halmos (1974) P. R. Halmos. 1974\. _Naive Set Theory_. Springer-Verlag, New York, USA.
* Järvelin and Kekäläinen (2002) K. Järvelin and J. Kekäläinen. 2002. Cumulated Gain-Based Evaluation of IR Techniques. _ACM Trans. Inf. Syst._ 20, 4 (Oct. 2002), 422–446. https://doi.org/10.1145/582415.582418
* Kazai et al. (2004) G. Kazai, S. Masood, and M. Lalmas. 2004. A Study of the Assessment of Relevance for the INEX’02 Test Collection. In _Advances in Information Retrieval. Proc. 26th European Conference on IR Research (ECIR 2004)_ , S. McDonald and J. Tait (Eds.). Lecture Notes in Computer Science (LNCS) 2997, Springer, Heidelberg, Germany, 296–310.
* Kekäläinen and Järvelin (2002) J. Kekäläinen and K. Järvelin. 2002. Using Graded Relevance Assessments in IR Evaluation. _Journal of the American Society for Information Science and Technology (JASIST)_ 53, 13 (November 2002), 1120—1129.
* Kendall (1945) M. G. Kendall. 1945\. The Treatment of Ties in Ranking Problems. _Biometrika_ 33, 3 (November 1945), 239–251.
* Kumar and Vassilvitskii (2010) Ravi Kumar and Sergei Vassilvitskii. 2010. Generalized distances between rankings. In _Proceedings of the 19th International Conference on World Wide Web, WWW 2010, Raleigh, North Carolina, USA, April 26-30, 2010_ , Michael Rappa, Paul Jones, Juliana Freire, and Soumen Chakrabarti (Eds.). ACM, 571–580. https://doi.org/10.1145/1772690.1772749
* Lioma et al. (2019) C. Lioma, M. Maistro, M. D. Smucker, , and D. Zuccon. 2019. Overview of the TREC 2019 Decision Track. In _The Twenty-Eighth Text REtrieval Conference Proceedings (TREC 2019)_ , E. M. Voorhees and A. Ellis (Eds.). National Institute of Standards and Technology (NIST), Special Publication 500-331, Washington, USA.
* Lioma et al. (2017) C. Lioma, J. G. Simonsen, and B. Larsen. 2017. Evaluation Measures for Relevance and Credibility in Ranked Lists. In _Proc. 3rd ACM SIGIR International Conference on the Theory of Information Retrieval (ICTIR 2017)_ , J. Kamps, E. Kanoulas, M. de Rijke, H. Fang, and E. Yilmaz (Eds.). ACM Press, New York, USA, 91–98.
* Maddalena and Mizzaro (2014) E. Maddalena and S. Mizzaro. 2014. Axiometrics: Axioms of Information Retrieval Effectiveness Metrics. In _Proc. 6th International Workshop on Evaluating Information Access (EVIA 2014)_ , S. Mizzaro and R. Song (Eds.). National Institute of Informatics, Tokyo, Japan, 17–24.
* Palotti et al. (2018) J. Palotti, G. Zuccon, and A. Hanbury. 2018. MM: A New Framework for Multidimensional Evaluation of Search Engines. In _Proc. 27th International Conference on Information and Knowledge Management (CIKM 2018)_ , A. Cuzzocrea, J. Allan, N. W. Paton, D. Srivastava, R. Agrawal, A. Broder, M. J. Zaki, S. Candan, A. Labrinidis, A. Schuster, and H. Wang (Eds.). ACM Press, New York, USA, 1699–1702.
* Petersen et al. (2015) C. Petersen, C. Lioma, J. G. Simonsen, and B. Larsen. 2015\. Entropy and Graph Based Modelling of Document Coherence Using Discourse Entities: An Application to IR. In _Proceedings of the 2015 International Conference on The Theory of Information Retrieval_ _(ICTIR ’15)_. ACM, New York, NY, USA, 191–200. https://doi.org/10.1145/2808194.2809458
* Sakai (2007) T. Sakai. 2007. Evaluating Information Retrieval Metrics Based on Bootstrap Hypothesis Tests. _IPSJ Digital Courier_ 3 (2007), 625–642. https://doi.org/10.2197/ipsjdc.3.625
* Sakai (2012) T. Sakai. 2012. Evaluation with Informational and Navigational Intents. In _Proc. 21st International Conference on World Wide Web (WWW 2012)_ , A. Mille, F. L. Gandon, J. Misselis, M. Rabinovich, and S. Staab (Eds.). ACM Press, New York, USA, 499–508.
* Sebastiani (2015) F. Sebastiani. 2015\. An Axiomatically Derived Measure for the Evaluation of Classification Algorithms. In _Proc. 1st ACM SIGIR International Conference on the Theory of Information Retrieval (ICTIR 2015)_ , J. Allan, W. B. Croft, A. P. de Vries, C. Zhai, N. Fuhr, and Y. Zhang (Eds.). ACM Press, New York, USA, 11–20.
* Tang and Yang (2017) Z. Tang and G. H. Yang. 2017. Investigating per Topic Upper Bound for Session Search Evaluation. In _Proceedings of the ACM SIGIR International Conference on Theory of Information Retrieval_ _(ICTIR ’17)_. Association for Computing Machinery, New York, NY, USA, 185–192. https://doi.org/10.1145/3121050.3121069
* Voorhees (1998) E. M. Voorhees. 1998\. Variations in relevance judgments and the measurement of retrieval effectiveness. In _Proc. 21st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 1998)_ , W. B. Croft, A. Moffat, C. J. van Rijsbergen, R. Wilkinson, and J. Zobel (Eds.). ACM Press, New York, USA, 315–323.
* Voorhees (2001) E. M. Voorhees. 2001\. Evaluation by Highly Relevant Documents. In _Proc. 24th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2001)_ , D. H. Kraft, W. B. Croft, D. J. Harper, and J. Zobel (Eds.). ACM Press, New York, USA, 74–82.
* Voorhees and Ellis (2016) E. M. Voorhees and A. Ellis (Eds.). 2016. _Proceedings of The Twenty-Fifth Text REtrieval Conference, TREC 2016, Gaithersburg, Maryland, USA, November 15-18, 2016_. Vol. Special Publication 500-321. National Institute of Standards and Technology (NIST). http://trec.nist.gov/pubs/trec25/trec2016.html
* Webber et al. (2008) W. Webber, A. Moffat, J. Zobel, and T. Sakai. 2008\. Precision-at-ten Considered Redundant. In _Proceedings of the 31st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR ’08)_. ACM, New York, NY, USA, 695–696. https://doi.org/10.1145/1390334.1390456
* Yao (1995) Y. Y. Yao. 1995\. Measuring Retrieval Effectiveness Based on User Preference of Documents. _Journal of the American Society for Information Science (JASIS)_ 46, 2 (1995), 133–145. https://doi.org/10.1002/(SICI)1097-4571(199503)46:2<133::AID-ASI6>3.0.CO;2-Z
* Yilmaz et al. (2016) E. Yilmaz, M. Verma, R. Mehrotra, E. Kanoulas, B. Carterette, and N. Craswell. 2016\. Overview of the TREC 2015 Tasks Track. In _The Twenty-Fourth Text REtrieval Conference Proceedings (TREC 2015)_ , E. M. Voorhees and A. Ellis (Eds.). National Institute of Standards and Technology (NIST), Special Publication 500-319, Washington, USA.
* Zuccon (2016) G. Zuccon. 2016\. Understandability Biased Evaluation for Information Retrieval. In _Advances in Information Retrieval. Proc. 38th European Conference on IR Research (ECIR 2016)_ , N. Ferro, F. Crestani, M.-F. Moens, J. Mothe, F. Silvestri, G. M. Di Nunzio, C. Hauff, and G. Silvello (Eds.). Lecture Notes in Computer Science (LNCS) 9626, Springer, Heidelberg, Germany, 280–292.
*[IR]: Information Retrieval
*[AP]: Average Precision
*[NDCG]: Normalized Discounted Cumulated Gain
*[TOMA]: Total Order Multi-Aspect
*[INEX]: INitiative for the Evaluation of Retrieval
*[$\alpha$-NDCG]: α-Normalized Discounted Cumulated Gain
*[MAP-IA]: Intent Aware Mean Average Precision
*[IA-ERR]: Intent Aware Expected Reciprocal Rank
*[nCT]: Normalized Cube Test
*[RBU]: Rank-Biased Utility
*[uRBP]: Understandability-biased Rank-Biased Precision
*[MM]: Multidimensional Measure
*[NLRE]: Normalised Local Rank Error
*[NGRE]: Normalised Global Rank Error
*[nWCS]: Normalised Weighted Cumulative Score
*[CAM]: Convex Aggregating Measure
*[WHAM]: Weighted Harmonic Mean Aggregating Measure
*[DCG]: Discounted Cumulated Gain
*[RoS]: Rankings of Submitted runs
*[ERR]: Expected Reciprocal Rank
|
# Compositional Covariance Shrinkage and Regularised Partial Correlations
Suzanne Jin${}^{\leavevmode\nobreak\ ,2}$ Centre for Genomic Regulation (CRG),
Dr Aiguader, 88, 08003 Barcelona, Spain Cédric Notredame1, Universitat Pompeu
Fabra, Barcelona, Spain Ionas Erb1, Corresponding author<EMAIL_ADDRESS>
###### Abstract
We propose an estimation procedure for covariation in wide compositional data
sets. For compositions, widely-used logratio variables are interdependent due
to a common reference. _Logratio uncorrelated_ compositions are linearly
independent before the unit-sum constraint is imposed. We show how they are
used to construct bespoke shrinkage targets for logratio covariance matrices
and test a simple procedure for partial correlation estimates on both a
simulated and a single-cell gene expression data set. For the underlying
counts, different zero imputations are evaluated. The partial correlation
induced by the closure is derived analytically. Data and code are available
from GitHub.
#### MSC: G2F30, G2H20, G2P99
#### Keywords: Compositional Covariance Structure, Logratio Analysis, Partial
Correlation, James-Stein Shrinkage
## 1\. Introduction
The study of data variability is at the heart of data analysis. A simple way
of quantifying the variability around some central tendency is given by a
bilinear function defined for each pair of variables: the covariance. The
distribution that makes no assumptions beyond a finite covariance matrix is
the multivariate Gaussian, and it is thus optimal in a maximum-entropy sense.
Covariance matrices can capture the relevant aspects of variability in a wide
variety of scenarios Whittaker (1990), and the Gaussian doesn’t have to be the
“true” underlying distribution for this111For a discussion covering this
aspect of maximum-entropy approaches see van Nimwegen (2016). A more general
maximum-entropy approach to compositional data has recently been proposed
Weistuch et al. (2022), but it is difficult to obtain analytic results from
it..
Often the limitations of data acquisition make good covariance estimates
difficult, and regularisation techniques are adopted to deal with small data
sets. One such technique is covariance _shrinkage_ , which is a regularisation
technique that lets us invert singular covariance matrices which arise from
the presence of fewer samples than variables Schäfer and Strimmer (2005). It
has recently been applied in the context of _compositional_ data in genomics
Badri et al. (2020), where clear improvements over empirical measures were
shown with a comprehensive and reproducible benchmark. Compositional data are
constrained in that their samples sum to a constant, which causes biases that
can be alleviated by logratio transformations Aitchison (1986). Badri et al.
(2020) applied the standard shrinkage approach directly to logratio covariance
matrices, which does not take into account the interdependence among logratio
variables. Our main interest in this contribution is to provide and test an
improved shrinkage procedure that is custom-made for logratio-transformed
data. The properly regularized logratio covariance matrices can then be
inverted to estimate logratio-based partial correlations, which we have
proposed as a measure of interaction between compositional variables Erb
(2020).
On a more general note, we aim to contribute to finding a principled way of
quantifying interactions in data sets consisting of compositions or relative
counts. When inferring interactions between variables, a useful distinction
(see, e.g., Werhli et al. (2006)) is made between relevance networks, which
are based on pairwise interaction coefficients (such as Pearson correlation)
and graphical models, which make use of all the variables to infer pairwise
association (e.g., via partial correlation). While the former are susceptible
to inferring indirect interactions that are mediated by variables that are
unrelated to the pair of variables in question, the latter aim to infer the
conditional independence structure of their joint distribution Whittaker
(1990). This way they can in principle infer direct interactions between two
variables. The distinction is highly relevant for compositional data, where
variables are intrinsically connected due to the unit-sum constraint. Here,
simple relevance networks are inadequate because of the negative bias coming
from the constraint. This bias has been addressed by measures that use ratios
of variables, such as Aitchison’s logratio variance Aitchison (1986), as well
as other coefficients inferring proportionality Lovell et al. (2015); Erb and
Notredame (2016). These can be used to infer relevance networks in
compositional data, and Badri et al. (2020) have shown how shrinkage improves
estimates of the proportionality coefficient. However, to define graphical
models for compositions similar to Kurtz et al. (2015), logratio-based partial
correlations will be instrumental.
In what follows, in section 2 we briefly review how the covariance structure
and partial correlations are described in the logratio framework and outline
James-Stein shrinkage of general covariance matrices as proposed in Schäfer
and Strimmer (2005). In Section 3, the concept of logratio uncorrelated
compositions introduced by Aitchison is described, and it is shown how one can
make use of it to define shrinkage for logratio covariance matrices. Two
equivalent approaches are proposed: 1) shrinking a so-called _basis_
covariance matrix which is then transformed into a logratio covariance, and 2)
shrinking logratio covariance matrices directly with the help of logratio
uncorrelated targets. Due to its simplicity, we prefer approach 1, and both
the synthetic and the experimental data benchmark of Section 4 will be based
on it. In section 5, we discuss how the use of partial correlations relates to
normalisation techniques in genomics and provide an expression for the
remaining partial correlation in logratio-uncorrelated compositions. A summary
section and an Appendix containing some of the more technical details conclude
the paper.
## 2\. Preliminaries
###
2.1.
Compositional Data and Logratio Covariance
Let us denote $N$ compositional samples by vectors $\boldsymbol{p}_{i}$,
$i=1,\dots,N$, whose $D$ positive components (called parts) sum to 1.
Specifically, we consider data with a large number of parts $D$ compared with
the number of samples $N$. Such wide compositional data are of interest in
genomics Erb et al. (2020), where they appear mainly in the form of relative
counts coming from sequencing experiments Quinn et al. (2018). In this case we
would consider $\boldsymbol{p}$ the closed data obtained when dividing a count
sample by its total, i.e., the compositions are the empirical parameter
estimates of a multinomial model.
The counts play the role of a so-called _basis_ of the composition, a concept
that was introduced in Aitchison (1982), and which we will talk about in more
detail in section 2.4. Considering the counts as _relative_ data is justified
by assigning no importance to the variations in the count totals (which
correspond to the _size_ of the basis). Compared with real numbers, relative
counts constitute a _discrete_ basis, which implies some problems of its own,
especially when counts are small Lovell et al. (2020) or vanish entirely. Such
problems can be alleviated with suitable weights Law et al. (2014) or with
finite-size corrections to the frequencies that can again be obtained via
shrinkage Hausser and Strimmer (2009). The latter, known as frequency
shrinkage, has the advantage that it imputes count zeros naturally. The
promise that this has shown for compositional data Quinn and Erb (2021);
Greenacre et al. (2022) will be confirmed here in our benchmarks.
Let us now consider an $N\times(D-1)$ data matrix $\boldsymbol{X}$ with real-
valued elements. These elements are obtained from the log-ratios we evaluate
from our compositional samples $\boldsymbol{p}_{i}$ using the additive
logratio transformation (ALR, introduced in Aitchison (1982))
$x_{ij}=\log\frac{p_{ij}}{p_{iD}},\qquad j=1,\dots,D-1.$ (1)
The (unbiased) sample covariance matrix $\boldsymbol{S}$ is then defined by
its elements
$s_{ij}=\frac{1}{N-1}\sum_{k=1}^{N}(x_{ki}-\bar{x}_{i})(x_{kj}-\bar{x}_{j}),$
(2)
where $\bar{x}_{i}$ are the components of the $N$-dimensional vector
$\bar{\boldsymbol{x}}$ of column means of $\boldsymbol{X}$. In order to have a
criterion allowing us to judge the quality of our covariance estimate based on
$\boldsymbol{S}$, we need a distributional assumption. Let us assume that
$\boldsymbol{X}$ was sampled from a normal distribution with population
parameters $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$. This can be written
as
$\boldsymbol{x}\sim\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma}),$ (3)
where $\boldsymbol{x}$ denotes a $D-1$-dimensional random vector taking values
as in the rows of $\boldsymbol{X}$. In this case the random compositions
$\boldsymbol{p}$ are distributed according to Aitchison’s celebrated logistic
normal distribution Aitchison (1982). If we denote the density of the
compositions themselves by $f_{\mathcal{L}}$, the Gaussian density
$f_{\mathcal{N}}$ will be multiplied by an additional factor due to the
resulting Jacobian:
$f_{\mathcal{L}}(\boldsymbol{p}=\boldsymbol{p}_{i}|\boldsymbol{\mu},\boldsymbol{\Sigma})=\left(\prod_{j=1}^{D}p_{ij}\right)^{-1}f_{\mathcal{N}}(\boldsymbol{x}=\boldsymbol{x}_{i}|\boldsymbol{\mu},\boldsymbol{\Sigma}),$
(4)
where $\boldsymbol{x}_{i}$ is a function of $\boldsymbol{p}_{i}$ defined by
(1).
###
2.2.
Covariance Shrinkage
To improve estimates of $\boldsymbol{\Sigma}$ from data, adding a bit of bias
to the unbiased estimator $\boldsymbol{S}$ can reduce its overall error. This
is a consequence of the bias-variance decomposition of the mean-squared error
and can be achieved by convexly combining $\boldsymbol{S}$ with (a.k.a.
“shrinking it towards”) a suitable target matrix $\boldsymbol{T}$:
$\hat{\boldsymbol{\Sigma}}=\lambda\boldsymbol{T}+(1-\lambda)\boldsymbol{S},$
(5)
where the shrinkage intensity $\lambda$ is between zero and one.
$\hat{\boldsymbol{\Sigma}}$ is a so-called James-Stein type estimator James
and Stein (1961). If we forget for a moment that $\boldsymbol{X}$ was obtained
from logratios of compositions, a typical target matrix could be
$\mathrm{diag}(\boldsymbol{S})$. While (5) tells us nothing about suitable
values of $\lambda$, its optimal value can be estimated from the data via an
analytic expression that minimises a mean-squared error cost-function with
respect to the population covariance Ledoit and Wolf (2003). The derivation of
this expression makes use of the fact that $\boldsymbol{S}$ is unbiased. This
optimum is given by
$\lambda^{*}=\frac{\sum_{j=1}^{D}\sum_{i\neq
j}\left[\mathrm{var}(s_{ij})-\mathrm{cov}(s_{ij},\tau_{ij})\right]}{\sum_{j=1}^{D}\sum_{i\neq
j}\mathbbm{E}\left[(s_{ij}-\tau_{ij})^{2}\right]},$ (6)
where $\tau_{ij}$ denotes the matrix entries of $\boldsymbol{T}$. Note that
here, covariance and expectation of the matrix elements refer to parameters of
a distribution which can be approximated by their empirical estimates for
actual calculations, see Schäfer and Strimmer (2005).
To provide further background, we also show a principled interpretation of
covariance shrinkage in the Appendix. There, we sketch how covariance
shrinkage is equivalent to optimising the posterior probability of
$\boldsymbol{\Sigma}$ from data and prior information. Note that $\lambda^{*}$
is inferred from data, making this a quasi-empirical Bayes approach.
The shrunk covariance estimator is useful in itself as it often by far
outperforms the sample covariance unless the number of samples considerably
exceeds the number of variables. Regularisation becomes a necessity for
inverting the covariance matrix whenever the number of samples is smaller than
the number of variables. Therefore, shrinkage improves and extends the range
of applications of the analysis proposed in the next section.
###
2.3.
Logratio-based Partial Correlations
Partial correlations between two variables are evaluated by controlling for
the linear dependence on all the remaining variables, e.g., Whittaker (1990).
This is achieved by correlating the residuals of two variables, where the
residuals are with respect to the linear least squares predictors obtained
from the remaining variables. Intuitively, we only correlate those
contributions to our variables that are orthogonal to a subspace spanned by
the variables we control for. Partial correlations can therefore be used to
extract direct dependencies between variables, as opposed to the ones based on
Pearson correlation, which can be dominated by indirect interactions via other
variables. Partial correlations between logratio variables can be defined
independently of their reference part $D$, so they measure association
directly between parts, provided their reference is controlled for Erb (2020).
Let us recall the formula for the (logratio-based) partial correlation between
two parts $i,j$ of the random composition $\boldsymbol{p}$ via their ALR
covariance $\boldsymbol{\Sigma}$. Below, the backslash means taking the set
difference, so the variables we control for are all the ones not belonging to
the pair in question:
$r_{ij}(\boldsymbol{p}):=\mathrm{corr}(x_{i},x_{j}|\left\\{x_{1},\dots,x_{D-1}\right\\}\backslash\\{x_{i},x_{j}\\})=\frac{-\sigma^{(-1)}_{ij}}{\sqrt{\sigma^{(-1)}_{ii}\sigma^{(-1)}_{jj}}},$
(7)
where $i\neq j$ and $i,j=1,\dots,D-1$, and where $\sigma^{(-1)}_{ij}$ denotes
the elements of the inverse of $\boldsymbol{\Sigma}$. It may appear as if the
reference part $p_{D}$ is “sacrificed” here, and its partial correlations
cannot be determined. Note, however, that $r_{ij}$ is invariant under
permutation of the reference part: all $D$ choices of $\boldsymbol{\Sigma}$
lead to the same partial correlations (for the parts they have in common). It
follows that we can obtain the partial correlation of any pair of parts by
choosing an arbitrary reference part that doesn’t form part of the pair. We
have stated our definition in terms of the ALR covariance matrix because it
comes closest to what is known from unconstrained data. There is also a
(computationally) more convenient representation using a covariance matrix
involving all parts in form of centred logratios (the CLR covariance
introduced in section 2.4), from which the partial correlations can be
obtained via its pseudoinverse. The inverse can also be estimated with a
sparsity assumption, which has been done to create ecological networks of
microbes, see Kurtz et al. (2015).
###
2.4.
Linear Independence in Logratio and Basis Covariance Matrices
For covariance matrices of unconstrained data, vanishing off-diagonal elements
mean linear independence of the variables. Logratio covariance matrices cannot
be linearly independent in this sense. Although logratio transformations map
compositional data from the simplex to a real-valued space, they cannot remove
the dependence between the parts that is introduced by the loss of one
dimension. The notion coming closest to linear independence for logratios is
given by logratio uncorrelated compositions. In the following, we state some
results from Aitchison (1986), section 5.9, where this concept is introduced.
A composition $\boldsymbol{p}$ is said to be logratio uncorrelated (LU) if
$\mathrm{cov}\left(\log\frac{p_{i}}{p_{k}},\log\frac{p_{j}}{p_{l}}\right)=0$
(8)
for every selection of _four different_ indices $i,j,k,l$ from
$\\{1,\dots,D\\}$. This case is shown in Figure 1.
Figure 1: Dependence structure in a logratio uncorrelated composition
$\boldsymbol{u}$ with four parts. Transformed parts can be visualized as
centred vectors with sampled components (labels are in random-variable
notation), where length corresponds to standard deviation. The reference part
is indexed by $D=4$ and its basis $m_{4}=tu_{4}$ is usually unknown (resulting
in an unknown origin $O$). The vectors from the origin to the points indexed
by 1 to 4 are the log-transformed basis vectors, their squared lengths
correspond to the $\alpha_{k}$ defined in (9). As $\boldsymbol{u}$ is LU, they
are orthogonal to each other in four dimensions. In three dimensions, only the
logratio vectors of equal colour are orthogonal (e.g., the orange vectors
labelled with their logratios). These correspond to vectors where all indices
are different, see (8).
The covariance structure of LU compositions can be made explicit when defining
a $D$-component vector $\boldsymbol{\alpha}$ with elements
$\alpha_{k}=\mathrm{cov}\left(\log\frac{p_{i}}{p_{k}},\log\frac{p_{j}}{p_{k}}\right),$
(9)
with the three indices $i$, $j$, $k$ in $\\{1,\dots,D\\}$ and all different
from each other. For LU compositions, this definition makes sense because the
value of $\alpha_{k}$ is constant regardless of the (unequal) indices $i,j$.
The matrix $\boldsymbol{\Sigma}$ of an LU composition is now given by
$\sigma_{ij}=\left\\{\begin{array}[]{c@{\quad}l}\alpha_{i}+\alpha_{D}&\mbox{if
$i=j$,}\\\ \alpha_{D}&\mbox{if $i\neq j$.}\end{array}\right.\qquad
i,j=1,\dots,D-1.$ (10)
(we reproduce the proof of this in the Appendix). While these definitions may
seem overly abstract, they appear much more compelling in the light of
Aitchison’s concept of the _basis_ of a composition, see Aitchison (1986),
chapter 9. Intuitively, the basis is one of many ways our compositions can
look like before the closure operation is applied. It is defined by
$\boldsymbol{m}=t\boldsymbol{p}$ for some positive real $t$ which is called
the size of the basis and for which $t=\sum_{k=1}^{D}m_{k}$ holds (implying
that it can change for each data sample). Of course, infinitely many bases of
different size exist for a given unit-sum composition. Now the _basis
covariance_ matrix $\boldsymbol{\Omega}$ is defined to have the elements
$\omega_{ij}=\mathrm{cov}\left(\log m_{i},\log m_{j}\right),$ (11)
from which the covariance structure of a composition can be obtained
unambiguously. If we assume that $\boldsymbol{\Omega}$ is diagonal, i.e., the
logged components of the basis vector are mutually uncorrelated, we have the
result that $\boldsymbol{\alpha}$ coincides with
$\mathrm{var}(\log\boldsymbol{m})$:
$\alpha_{i}=\omega_{ii},\qquad i=1,\dots,D.$ (12)
This goes to show that $\boldsymbol{\alpha}$ can be seen as the variance
vector of an independent distribution in a sample space that has one
additional dimension (and in which the logratio sample space is embedded), see
Figure 1. As $\boldsymbol{\alpha}$ was defined for logratios, this statement
may appear confusing because of the apparent dependence of
$\mathrm{var}\log(t\boldsymbol{p})$ on the basis size $t$. However, the
$t$-dependence drops out because $\boldsymbol{\Omega}$ is diagonal (see the
proof of (12) in the Appendix).
For completeness, let us also mention an equivalent expression to (10) in
terms of the centred logratio transformation (CLR). Here the reference
involves all the parts of the composition. It transforms the compositional
data matrix as
$y_{ij}=\log\frac{p_{ij}}{g(\boldsymbol{p}_{i})},$ (13)
where $g(\boldsymbol{p}_{i})=\prod_{k=1}^{D}p_{ik}^{1/D}$ is the geometric
mean of the $i$-th composition. Note that the resulting data matrix
$\boldsymbol{Y}$ has a constraint $\sum_{k=1}^{D}y_{ik}=0$ acting on its rows.
This constraint leads to a singular CLR covariance matrix of rank $D-1$. The
covariance of $\boldsymbol{y}$ is denoted by $\boldsymbol{\Gamma}$. The two
forms of log-ratio covariance are equivalent and can be transformed into each
other (see the first two rows in Table 1). Using the transformation from
$\boldsymbol{\Sigma}$ to $\boldsymbol{\Gamma}$ on (10), we obtain the elements
of $\boldsymbol{\Gamma}$ for an LU composition as
$\gamma_{ij}=\left\\{\begin{array}[]{c@{\quad}l}\alpha_{i}-(2\alpha_{i}-\frac{1}{D}\sum_{k}\alpha_{k})/D&\mbox{if
$i=j$,}\\\ -(\alpha_{i}+\alpha_{j}-\frac{1}{D}\sum_{k}\alpha_{k})/D&\mbox{if
$i\neq j$.}\end{array}\right.\qquad i,j=1,\dots,D.$ (14)
$\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Gamma}$ | $\gamma_{ij}=\sigma_{ij}-\frac{1}{D}\sum_{k=1}^{D}\sigma_{ik}-\frac{1}{D}\sum_{k=1}^{D}\sigma_{kj}+\frac{1}{D^{2}}\sum_{k,l=1}^{D}\sigma_{kl}$
---|---
$\boldsymbol{\Gamma}\rightarrow\boldsymbol{\Sigma}$ | $\sigma_{ij}=\gamma_{ij}-\gamma_{iD}-\gamma_{Dj}+\gamma_{DD}$
3.5 $\boldsymbol{\Omega}\rightarrow\boldsymbol{\Sigma}$ | $\sigma_{ij}=\omega_{ij}-\omega_{iD}-\omega_{Dj}+\omega_{DD}$
$\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Omega}$ | $\omega_{ij}=\gamma_{ij}(\boldsymbol{\Sigma})+\beta_{i}+\beta_{j}$, where
| $\beta_{j}=\mathrm{cov}\left(\mathrm{clr}_{j}(\boldsymbol{p}),\log
g(\boldsymbol{m})\right)-\frac{1}{2}\mathrm{var}\left(\log
g(\boldsymbol{m})\right)$.
$\boldsymbol{\Omega}\rightarrow\boldsymbol{\Gamma}$ | $\gamma_{ij}=\omega_{ij}-\frac{1}{D}\sum_{k=1}^{D}\omega_{ik}-\frac{1}{D}\sum_{k=1}^{D}\omega_{kj}+\frac{1}{D^{2}}\sum_{k,l=1}^{D}\omega_{kl}$
$\boldsymbol{\Gamma}\rightarrow\boldsymbol{\Omega}$ | $\omega_{ij}=\gamma_{ij}+\beta_{i}+\beta_{j}$, see $\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Omega}$.
Table 1: Elementwise transformations between ALR, CLR, and basis covariance
matrices (collected from Aitchison (1986)). Here, indices of $\sigma_{ij}$ are
allowed to take the value $D$, in which case $\sigma_{ij}$ vanishes; clrj
stands for the $j$-th element of the CLR transform, and $g$ denotes the
geometric mean. All these transformations can also be expressed as simple
matrix operations that are not shown here.
## 3\. Logratio Covariance Shrinkage
The specific covariance structure of compositional data would suggest to use
log-ratio uncorrelated shrinkage targets instead of the diagonal targets used
for unconstrained data. We will show that it is straightforward to construct
such targets. However, it is usually more convenient in practice to work with
diagonal targets, for which we have to do the shrinkage on a basis of the
compositions. Once the shrinkage estimate for the basis covariance is
obtained, it can then be back-transformed to a logratio covariance matrix.
This strategy is described in the following section.
###
3.1.
Basis Covariance Shrinkage Using Diagonal Targets
A diagonal shrinkage target may often be preferred for practical reasons. As
an example, the corpcor R package Schäfer and Strimmer (2005) uses the
diagonal of the sample covariance as a target and does not allow for
alternative targets. However, we have seen that logratio covariance matrices
do not allow for this simple independence structure. Therefore, we now define
the shrinkage estimator directly for the _basis_ covariance matrix
$\boldsymbol{\Omega}$:
$\hat{\boldsymbol{\Omega}}=\lambda\boldsymbol{T}_{C}+(1-\lambda)\boldsymbol{C},$
(15)
where $\boldsymbol{C}$ is the empirical basis covariance matrix, and the
shrinkage target $\boldsymbol{T}_{C}$ could be simply its diagonal. Thus,
given some compositional data, our first step to construct an uncorrelated
shrinkage target is to obtain an empirical basis covariance matrix from the
data. For this, it is necessary to fix the size of the basis $t$. Note that
doing this specifies the variability of the totals across samples. Given that
we will back-transform our estimator to a logratio covariance matrix, this
variability is irrelevant. The logratio covariance structure will not contain
information about the basis size, so it is convenient to use $t_{i}=1$ for all
samples $i=1,\dots,N$, giving a basis
$\boldsymbol{m}_{i}=\log\boldsymbol{p}_{i}$. In the case of relative count
data, the counts themselves can provide the basis, and $\boldsymbol{C}$ can be
estimated from their logarithms (provided zeros are taken care of). Yet
another possibility is to start from an empirical logratio covariance matrix
and obtain $\boldsymbol{C}$ via a transformation given in Table 1. These
transformations simplify for a constant basis size.
The diagonal target has the additional advantage that the expression for the
optimal value of $\lambda$ simplifies Schäfer and Strimmer (2005) compared to
the more general expression (6). For diagonal targets we have
$\hat{\lambda}^{*}=\frac{\sum_{j=1}^{D}\sum_{i\neq
j}\widehat{\mathrm{var}}(c_{ij})}{\sum_{j=1}^{D}\sum_{i\neq j}c_{ij}^{2}}.$
(16)
To obtain the shrinkage estimator of a logratio covariance matrix, all we have
to do now is back-transform $\hat{\boldsymbol{\Omega}}$ to an appropriate
logratio covariance matrix as specified in Table 1. This will also be the
strategy we use for our benchmark in section 4.
###
3.2.
Logratio Uncorrelated Targets
While we will not use them in our benchmarks, it can be of general interest to
construct LU targets for direct logratio covariance shrinkage. A simple way of
constructing these targets would be as follows: pick the diagonal elements of
the empirical basis covariance $\boldsymbol{C}$ as our $\alpha_{i}$ and then
obtain the ALR target covariance by (10) and the CLR target covariance by
(14). However, we can also skip the construction of a basis covariance
entirely and express the targets directly in terms of the logratio sample
covariance matrices. For the ALR target given the empirical covariance
$\boldsymbol{S}$ we find
$\tau_{ij}=\left\\{\begin{array}[]{c@{\quad}l}s_{ii}-\frac{2}{D}\sum_{k=1}^{D-1}s_{ik}+\frac{2}{D^{2}}\sum_{k,l}s_{kl}&\mbox{if
$i=j$,}\\\ \frac{2}{D^{2}}\sum_{k,l}s_{kl}&\mbox{if $i\neq
j$.}\end{array}\right.\quad i,j=1,\dots,D-1.$ (17)
Similarly, using an empirical CLR covariance matrix $\boldsymbol{G}$, the
corresponding target elements $t_{ij}$ are given via the diagonal elements of
$\boldsymbol{G}$ by
$t_{ij}=\left\\{\begin{array}[]{c@{\quad}l}g_{ii}-(2g_{ii}-\frac{1}{D}\sum_{k}g_{kk})/D&\mbox{if
$i=j$,}\\\ -(g_{ii}+g_{jj}-\frac{1}{D}\sum_{k}g_{kk})/D&\mbox{if $i\neq
j$.}\end{array}\right.\qquad i,j=1,\dots,D.$ (18)
## 4\. A Benchmark of Logratio Covariance Shrinkage
We verify our estimators of covariance and partial correlation on both
simulated data as well as single-cell gene expression data. The latter are
subsampled from a “ground truth” of greater sample size (this strategy has
also been used for microbiome data in Badri et al. (2020)). Throughout, the
corpcor R package from Schäfer and Strimmer (2005) is used for covariance
matrix shrinkage. Note that we use the default implemented there, which
consists in separately shrinking the diagonal elements of the covariance
matrix with another $\lambda$ parameter, see Opgen-Rhein and Strimmer (2007).
The aim of the benchmark is the evaluation of our proposed procedure to first
shrink a basis covariance and then transform the result to a log-ratio
covariance. This is compared with a naive approach of direct log-ratio
covariance shrinkage (with a diagonal target containing variances that
inevitably mix parts) as well as no shrinkage at all. In the first part of
this section, the logratios of our compositions are taken for granted and we
just sample them from a normal distribution of rather moderate dimension
($D=40$). In this case we do not have to be concerned about zeros in the basis
samples. A scenario typical for genomics data sets is treated in the second
part of this section, where the basis of our compositions are high dimensional
relative counts taken from single-cell gene expression data ($D=500$). There
we also test various ways of dealing with zeros as well as a finite-size
correction for low counts in form of frequency shrinkage.
Figure 2: Mean squared error of partial correlation matrix for sample sizes
$N=200$, $N=40$, and $N=8$. Colours indicate different estimation procedures
(no shrinkage, naive shrinkage of CLR / ALR covariance and basis covariance
shrinkage). Each boxplot contains the results of 200 simulations. Whenever
estimates without shrinkage are not shown, their median value is given in the
legend.
###
4.1.
Synthetic Logistic Normal Data
To have reasonably realistic population parameters for the normal
distribution, they are inferred from a single-cell gene expression data set
Riba et al. (2022), where a subset of 240 genes that are non-zero everywhere
across 5637 cells are selected. From these, in each of the 200 repetitions of
the simulation, in step 1) $D=40$ parts are chosen randomly, from which ALR
and CLR covariance matrices as well as the partial correlation matrix (via the
pseudoinverse of the CLR covariance) are constructed across the 5K+ cells.
This is our ground truth. Then 2) the ALR mean vector $\boldsymbol{\mu}$ and
the ALR covariance matrix $\boldsymbol{\Sigma}$ are used to produce $N$
multivariate normal samples, where $N=8$, 40, and 200. 3) The $N$ samples are
backtransformed to compositions. From these, sample estimates of CLR, ALR and
(constant-size) basis covariance matrices as well as the partial correlation
estimates are produced. This is done in three different ways: without
shrinkage, with direct shrinkage of the logratio covariance matrices (using a
diagonal target for the logratios), and via shrinkage of the basis covariance
matrix (i.e., using a diagonal target for the log-transformed basis). 4) These
sample estimates are compared to the ground truth established in the first
step using an element-wise mean squared error. Note that after basis
covariance shrinkage, partial correlations are identical if the basis is
transformed to an ALR or a CLR covariance matrix (but it is convenient to use
the CLR to obtain them).
The results of this procedure are seen in Figure 2 for partial correlation and
Supplementary Figure 6 for covariance. There is no doubt that shrinkage leads
to substantial improvements when estimating covariation, even for $N=5D$. In
this regime, naive CLR shrinkage fares worse than using no shrinkage at all
when estimating partial correlations. Interestingly, the error is small for
the covariance matrix itself but seems to be compounded for inference of its
inverse. In all cases we see a clear advantage when using shrinkage of the
basis covariance, especially when comparing with naive ALR covariance
shrinkage. We conclude that we achieve substantial improvements of our
estimators when tailoring the shrinkage to the needs of logratio covariance
matrices.
Figure 3: Benchmark procedure for single-cell gene expression data.
###
4.2.
Single-cell gene expression data
Instead of simulating samples from a (realistic) covariance matrix, we now
draw our covariance matrices directly from the data. Since these are sample
covariance matrices, the benchmark now consists in recovering them as well as
possible with undersampled data. While this is not ideal (the test case could
in principle be as close or closer to the population covariance than the
sample covariance that serves as ground truth), it seems a reasonable
assumption that the degradation effect due to undersampling is greater than
the deviation of the full sample covariance from the population covariance. In
the previous section we could see that shrinking the covariance still has
positive effects even for $N=5D$. To avoid circularity, however, we prefer to
not shrink the ground truth covariance matrix here.
Figure 4: Mean squared error of partial correlation matrices computed on three
different sizes of subsamples ($N$=2500, 1000, 100) of single-cell gene
expression data. Boxplots contain 200 samples each using different estimation
procedures (no shrinkage, naive ALR/CLR shrinkage, basis covariance
shrinkage). Their colour indicates the type of zero imputation used.
Our benchmark procedure is summarized in Figure 3. Single-cell gene expression
data usually contain a very large number of zero counts, which would require
strategies (e.g., pooling several cells into a single sample) that are not
explored here. Instead, we concentrate on a core set of genes that are highly
expressed and thus yield only a few zero counts. These can still be used to
test a number of imputation strategies. We test two basic procedures
implemented in the R package zCompositions Palarea-Albaladejo et al. (2015)
(Count Zero Multiplicative replacement, or CZM, and Geometric Bayesian
Multiplicative replacement, or GBM), as well as one more sophisticated
strategy that takes covariance structure into account (lrSVD), which was added
to zCompositions recently Palarea-Albaladejo et al. (2022). We also test
frequency shrinkage as implemented in the R package entropy Hausser and
Strimmer (2009).
To separate the effects of shrinkage and imputation, we defined two scenarios:
1) imputation is avoided by using a large subset of cells with only nonzero
counts, and 2) the effect of shrinkage and imputation are jointly evaluated by
including all the cells. More precisely, we used 770 genes for which 3986
cells (out of a total of 5637 cells) have no zeros, from which a ground-truth
can be constructed. For each of the 200 resamplings, 500 genes are chosen
randomly from the 770 genes. Covariance and partial correlation matrices
computed on the 3986 $\times$ 500 nonzero matrix define our ground truth.
Then, in scenario 1, this matrix is subsampled to have $N$=2500, $N$=1000, and
$N$=100 cells. Covariance and partial correlations computed on each of these
matrices, and under the different shrinkage schemes are compared to the ground
truth using mean squared error. In the second scenario, the same ground truth
as defined before is used, while the subsampled data ($N$=2500, $N$=1000, and
$N$=100) are drawn from the extended matrix of 5637 cells x 500 genes. Here,
on average about 1500 zeros are imputed for the 2500 samples, 600 zeros are
imputed for the 1000 samples and 60 for the 100 samples. Having additional
samples that contain zeros is not ideal because these samples could in
principle induce a slightly different covariance structure. The benefit,
however, is that it provides a natural set of zero counts when otherwise zeros
would have to be artificially introduced. Now the subsampled matrix is first
imputed using four different methods, and then covariance and partial
correlations are again computed under the different shrinkage schemes.
The results for partial correlations are shown in Figure 4, and for the
covariance matrices in Supplementary Figure 7. Two main conclusions can be
drawn from the results for such high-dimensional data. First, the effect of
using the basis covariance shrinkage is strong compared with naive ALR
covariance shrinkage, but it is much weaker when comparing with naive CLR
covariance shrinkage. Naive shrinkage of the CLR covariance only leads to an
important deterioration for the partial correlations at higher sample sizes,
in concordance with what we observed in the synthetic benchmark. Apart from
that, the improvement is mainly observed when no zeros are imputed, which
hints at a flattening of the effect caused by the imputation. The advantage of
basis covariance shrinkage becomes entirely invisible when directly evaluating
the covariance matrices, whose inference seems to be less prone to error than
their inverse. A second main conclusion is that imputation of zeros leads to
considerable increase in mean squared error. All imputation schemes lead to a
comparable loss in accuracy, with only slight differences between methods.
## 5\. Logratio-based Partial Correlations for Wide Data
We have addressed some questions regarding covariance structure that remained
unanswered in Erb (2020), especially with respect to the application to wide
data sets like the ones occurring in genomics. We think it is beneficial to
also discuss some questions regarding the interpretation of logratio-based
partial correlations both in CoDA and genomics. In genomics, partial
correlation analysis is common, but it has rarely been based on logratio
analysis Kurtz et al. (2015). In CoDA, on the other hand, logratio analysis is
the dominant paradigm, but partial correlations have been introduced quite
recently Erb (2020). In the following sections we will discuss the
relationship between classical partial correlations and their logratio
counterparts.
###
5.1.
Normalisations to Avoid Compositional Bias
We can think of two strategies to avoid compositional bias when analyzing
covariance structure: trying to recover the original (absolute) signal (as
contained in one particular basis covariance $\boldsymbol{\Omega}$) or using
only that part of the signal that absolute and relative data have in common
(which is contained in the logratio covariance). The first (“normalisation”)
strategy relies on assumptions because the closure operation discards a part
of the signal and to recover the original totals up to a constant factor,
additional information is needed. The second (“CoDA”) strategy voluntarily
discards this information and doing so introduces dependencies between the
variables, but it is assumption-free. Interestingly, the distinction between
the two approaches is not all that clear-cut because the logratio
transformations Aitchison introduced can be used to pursue the “normalisation”
strategy of re-opening the data with additional assumptions. This was
discussed in (Quinn et al., 2018), see especially the Supplementary material.
In the following, we revisit these arguments from the perspective of logratio
covariance matrices and their basis covariance $\boldsymbol{\Omega}$. Let us
start with the ALR covariance $\boldsymbol{\Sigma}$. It can be obtained from
the basis covariance by
$\sigma_{ij}=\omega_{ij}-\omega_{iD}-\omega_{jD}+\omega_{DD}$ (19)
(see Table 1). Note that if the reference variable $m_{D}$ remains unchanged
across samples in the basis, the three last terms on the right-hand side
vanish, and the two covariances coincide. This is the reason behind the
widespread normalisation procedure in genomics where (absolutely) unchanged
reference genes are used to “normalize” the data to avoid compositional bias.
Similarly, for the CLR covariance $\boldsymbol{\Gamma}$ we have the
relationship
$\omega_{ij}=\gamma_{ij}+\beta_{i}+\beta_{j}$ (20)
(see again Table 1). From the dependence of $\beta_{i}$ on $g(\boldsymbol{m})$
we see that this implies that if the geometric mean of the basis remains
unchanged across samples, the $\beta$ terms vanish and the CLR covariance is
identical with the basis covariance. This is in agreement with the notion of
effective library size normalisation, where a reference that remains unchanged
on the absolute data is constructed to avoid compositional bias. The geometric
mean can remain unchanged across samples before closure when the majority of
genes is not subject to systematic change. Under this assumption the CLR
transformed data have an identical correlation structure as the log-
transformed absolute data.
We can see that the CoDA strategy has the advantage that, while it is on the
safe side if the assumptions aren’t met, it also recovers the original signal
if they happen to be fulfilled. In the latter case, however, a specific
reference and thus logratio covariance would be picked out because it recovers
the basis covariance. While this is necessary for evaluating both covariance
and correlation, it is unnecessary for partial correlation. The results remain
invariant under permuting the reference in $\boldsymbol{\Sigma}$ and they are
also identical when using the pseudoinverse of $\boldsymbol{\Gamma}$ for their
evaluation, see also Erb (2020).
These benefits of logratio-based partial correlation are lost when using more
naive approaches, like deriving partial correlations from the covariance
matrix of $\log\boldsymbol{p}$, or from a relative count basis using
$\log\boldsymbol{m}$ directly. Formally, partial correlations derived when not
specifying a reference do not coincide with their logratio counterparts.
Figure 5: A worst-case scenario for spurious logratio-based partial
correlation of a fixed pair of parts in artificially uncorrelated data (see
text). Successive increase of the number of parts from three (PC=0.69) to 770
(PC=0.001) shows how the closure-induced spurious signal tends to zero with
the number of parts. Inset: The case $D=3$. The vanishing correlation between
the first two parts could be recovered from the logratios with respect to the
third part (indicated in red) by correlating their projections into the plane
that is orthogonal to $\log m_{3}$. This would just be the correlation between
residuals when controlling for $\log m_{3}$. However, the information about
$\log m_{3}$ is lost, so partial correlation between the logratios remains
biased.
###
5.2.
What is the effect of the closure?
Partial correlations based on logratios can emulate their absolute
counterparts extremely well (a benchmark of which will be shown in another
contribution). However, the closure operation makes us lose one dimension
which cannot be recovered. So there remains a difference between partial
correlations derived from absolute (using logarithms only) versus relative
data (using logratios). Or, to put it differently, partial correlations can
change when we derive them from a logratio covariance as opposed to the
original (usually unknown) basis covariance. Using the relation between
$\boldsymbol{\Omega}$ and $\boldsymbol{\Gamma}$ of Table 1, the respective
covariance inverses are related by
$\boldsymbol{\Omega}^{-1}=\left(\boldsymbol{\Gamma}+\boldsymbol{B}\right)^{-1},$
(21)
where the $D\times D$ matrix $B$ has elements $\beta_{i}+\beta_{j}$. We see
that we can neatly separate the covariance signal introduced by the
variability of the basis size in the matrix $\boldsymbol{B}$, but taking the
inverse will mix these signals again. To further simplify the relationship of
the inverses, we need a more specific expression for the covariance. A simple
way to extract the partial correlation that is only due to the closure
operation is to consider a diagonal basis covariance, as it has no partial
correlation by definition. We know from section 2.4 that linear independence
corresponds to a logratio uncorrelated (LU) covariance in the embedded
logratio sample space (where one dimension is lost). In other words, the
closure applied to an uncorrelated basis leads to the dependence structure of
LU compositions222Dirichlet compositions are LU [Aitchison 1986], and this
(often undesired) property gave rise to Aitchsion’s introduction of the
logistic normal. The richer covariance structure of the latter enables
modelling that goes beyond unit-sum induced interactions. Of course, a
logistic normal with an LU covariance matrix also models LU compositions..
The elements of $\boldsymbol{\Sigma}$ for an LU composition are given by (10)
and (9). The fact that $\boldsymbol{\Sigma}$ has nonzero off-diagonal elements
for LU compositions immediately shows that the partial correlations of LU
compositions do not vanish. It is not difficult to determine the precise
expression. For $\boldsymbol{u}$ an LU composition, its (logratio-based)
partial correlations are given by
$r_{ij}(\boldsymbol{u})=\sqrt{\frac{\alpha^{-1}_{i}\alpha^{-1}_{j}}{\left(\sum_{k\neq
i}\alpha^{-1}_{k}\right)\left(\sum_{k\neq j}\alpha^{-1}_{k}\right)}},$ (22)
for all $i\neq j$, and where $\boldsymbol{\alpha}$ is given by (9) (see
Appendix for a proof).
To get an idea how strong this “spurious” signal is, we make the following
experiment. To obtain an (artificial) system of linearly independent
variables, we take the diagonal of the basis covariance matrix of the 770
genes considered in the previous section. This diagonal corresponds to
$\boldsymbol{\alpha}$, and we can calculate the logratio-based partial
correlations between all pairs using (22). For a worst-case scenario, we
select the pair of parts for which we obtain the strongest partial
correlation. With this pair fixed, we repeat calculating its partial
correlation after eliminating an arbitrary part (and thus one component from
the vector $\boldsymbol{\alpha}$). We repeat this process until we have only
one additional part left with our original pair. The result of this process is
shown in Figure 5 (going from right to left along the horizontal axis). We see
a similar “dilution” of the compositional effect as for the negative
correlations between multinomial counts, see Greenacre et al. (2022). In our
case, the spurious correlation turns out to be positive, but it is evaluated
between residuals of logratios, not compositional parts.
## 6\. Conclusion
We have proposed a simple way of obtaining shrinkage-based estimators for log-
ratio covariance matrices and their inverse. As shown in our benchmark, these
estimators clearly improve on the (still widely used) empirical estimators and
also outperform estimators that shrink log-ratio covariance matrices towards a
diagonal target as proposed in Badri et al. (2020). The use of Aitchison’s
notion of log-ratio uncorrelated compositions is crucial to obtain the
improved estimates.
With our contribution we also aim to promote the use of partial correlations
for the analysis of compositional data sets. We see three advantages in using
them: 1) They take all parts into account for the evaluation of pairwise
interaction (and thus can factor out indirect interactions). 2) Their
evaluation is reference-independent. 3) Negative correlations can be
meaningfully evaluated with respect to the set of variables that are
controlled for. While partial correlations change under taking
subcompositions, this is expected because it reflects the effect of the
removal of variables that are partialled out. However, we have also shown that
a “spurious” positive signal remains that is due to the loss of the $D$-th
dimension in compositional data sets. We have derived the expression for this
remaining partial correlation induced by the closure operation and have shown
how it “dilutes out” with a growing number of parts. In this sense, partial
correlations between parts are expected to show little spurious signal
whenever there are sufficiently many parts, a common occurrence in genomic
data.
For genomic data sets, we have also drawn some connections with the problem of
their normalisation. Sequencing data are relative counts, and as such their
analysis within the scale-free logratio framework can be beneficial. The use
of various normalisation schemes to control for sequencing depth can
essentially be circumvented in this way. We see much promise in the use of
logratio-based partial correlations especially for situations when
normalisation assumptions fail.
## 7\. Appendix
###
7.1.
Bayesian Interpretation of Covariance Shrinkage
In the Bayesian view of statistical inference, the parameters of a
distribution are considered to follow distributions themselves. The posterior
distribution of the parameter in question incorporates information from both
the data (via the likelihood) and the prior distribution of the parameter.
Shrinkage estimators also can incorporate prior information, and while a
suitable empirical estimator can maximise likelihood, optimization of
shrinkage is related to maximizing the posterior of the parameter333As an
example, in the case of shrinkage estimates of multinomial frequencies, the
target frequencies are renormalized pseudocounts to the count data, and the
shrinkage intensity $\lambda$ is the prior sample size of the conjugate
Dirichlet prior that has the pseudocounts as its parameters Hausser and
Strimmer (2009).. In the following we want to make this more precise for
Gaussian covariance shrinkage.
Let us start with the Gaussian conjugate prior distribution of the precision
matrix $\boldsymbol{\Sigma}^{-1}$ in the case of a fixed mean
$\boldsymbol{\mu}$. This is known as the Wishart (or multivariate Gamma)
distribution with density
$f_{\mathcal{W}}(\boldsymbol{\Sigma}^{-1}|\boldsymbol{V},\nu)=\frac{|\boldsymbol{\Sigma}^{-1}|^{(\nu-D)/2}e^{-\frac{1}{2}\mathrm{tr}\left(\boldsymbol{\boldsymbol{V}^{-1}\Sigma}^{-1}\right)}}{2^{\nu(D-1)/2}|\boldsymbol{V}|^{\nu/2}\Gamma_{D-1}(\frac{\nu}{2})},$
(23)
where the number of degrees of freedom $\nu$ is a positive integer, the
parametric matrix $\boldsymbol{V}$ is positive definite of order
$(D-1)\times(D-1)$, and $\Gamma$ denotes the multivariate Gamma function. The
particular form of this density can be understood better when decomposing the
precision matrix as
$\boldsymbol{\Sigma}^{-1}=\boldsymbol{U}\boldsymbol{U}^{T},$ (24)
where $\boldsymbol{U}$ is of order $(D-1)\times\nu$. If the $\nu$ vectors in
$\boldsymbol{U}$ are independently drawn from a normal distribution with mean
zero and covariance $\boldsymbol{V}$ (where $\nu\geq D-1$), then the precision
matrix follows a Wishart distribution with density (23).
To estimate a Gaussian covariance matrix, we need more than a single sample of
data. Let us now consider a data matrix $\boldsymbol{X}$ where the $N$ rows
were sampled according to (3). The joint likelihood of these samples can be
written in terms of their sample parameter estimates $\boldsymbol{S}$ and
$\bar{\boldsymbol{x}}$ as
$f_{\mathcal{N}}(\boldsymbol{X}|\boldsymbol{\mu},\boldsymbol{\Sigma})=\prod_{i=1}^{N}f_{\mathcal{N}}(\boldsymbol{x}_{i}|\boldsymbol{\mu},\boldsymbol{\Sigma})=\\\
\left(\frac{(2\pi)^{D-1}}{|\boldsymbol{\Sigma}|}\right)^{\frac{N}{2}}\mathrm{exp}\left\\{-\frac{1}{2}\mathrm{tr}\left[\boldsymbol{\Sigma}^{-1}\left((N-1)\boldsymbol{S}+N(\bar{\boldsymbol{x}}-\boldsymbol{\mu})^{T}(\bar{\boldsymbol{x}}-\boldsymbol{\mu})\right)\right]\right\\}.$
(25)
Let us now assign the conjugate priors as follows:
$\displaystyle\boldsymbol{\mu}\leavevmode\nobreak\ |\leavevmode\nobreak\
\boldsymbol{\Sigma}^{-1}$ $\displaystyle\sim$
$\displaystyle\mathcal{N}\left(\boldsymbol{\mu}_{0},\frac{1}{\kappa}\boldsymbol{\Sigma}\right),$
(26) $\displaystyle\boldsymbol{\Sigma}^{-1}$ $\displaystyle\sim$
$\displaystyle\mathcal{W}(\boldsymbol{V},\nu).$ (27)
The joint posterior density then factorizes as
$P(\boldsymbol{\mu}|\boldsymbol{\Sigma}^{-1})P(\boldsymbol{\Sigma}^{-1})$,
where the marginal $P(\boldsymbol{\Sigma}^{-1})$ is again Wishart. More
precisely
$P(\boldsymbol{\Sigma}^{-1}|\boldsymbol{S},\bar{\boldsymbol{x}},\boldsymbol{\mu}_{0},\kappa,\boldsymbol{V},\nu)=f_{\mathcal{W}}(\boldsymbol{\Sigma}^{-1}|\boldsymbol{V}^{*},\nu+N),$
(28)
where
$\boldsymbol{V}^{*}=(N-1)\boldsymbol{S}+\boldsymbol{V}+\frac{\kappa
N}{\kappa+N}(\bar{\boldsymbol{x}}-\boldsymbol{\mu}_{0})^{T}(\bar{\boldsymbol{x}}-\boldsymbol{\mu}_{0})$
(29)
(see DeGroot (2004), p.178)444For the sake of completeness, it should perhaps
be mentioned that $P(\boldsymbol{\mu}|\boldsymbol{\Sigma}^{-1})=\\\
f_{\mathcal{N}}\left((\kappa\boldsymbol{\mu}_{0}+N\bar{\boldsymbol{x}})/(\kappa+N),\boldsymbol{\Sigma}/(\kappa+N)\right)$..
It follows that under the Bayesian model, an improved covariance estimator
will involve an additive correction to the sample covariance. We can now write
(29) in terms of the shrinkage estimator when taking into account the
posterior sample size:
$(\nu+N)\hat{\boldsymbol{\Sigma}}=\boldsymbol{V}^{*}.$ (30)
Identifying (30) with (5), we obtain the identities
$\displaystyle(1-\lambda)$ $\displaystyle=$ $\displaystyle(N-1)/(\nu+N),$ (31)
$\displaystyle\lambda\boldsymbol{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\nu+N}\left(\boldsymbol{V}+\frac{\kappa
N}{\kappa+N}(\bar{\boldsymbol{x}}-\boldsymbol{\mu}_{0})^{T}(\bar{\boldsymbol{x}}-\boldsymbol{\mu}_{0})\right),$
(32)
from which it follows that
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle\frac{\nu+1}{\nu+N},$
(33) $\displaystyle\boldsymbol{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\nu+1}\left(\boldsymbol{V}+\frac{\kappa
N}{\kappa+N}(\bar{\boldsymbol{x}}-\boldsymbol{\mu}_{0})^{T}(\bar{\boldsymbol{x}}-\boldsymbol{\mu}_{0})\right).$
(34)
Optimization of $\lambda$ thus boils down to a tuning of the prior degrees of
freedom $\nu$ relative to the sample size $N$, while the target
$\boldsymbol{T}$ defines a prior covariance structure.
###
7.2.
Proof of (10) for LU compositions
We start with the case $i\neq j$ (see [Aitchison 1986], section 5.9):
$\sigma_{ij}=\mathrm{cov}\left(\log\frac{p_{i}}{p_{D}},\log\frac{p_{j}}{p_{D}}\right)=\mathrm{cov}\left(\log\frac{p_{i}}{p_{k}}+\log\frac{p_{k}}{p_{D}},\log\frac{p_{j}}{p_{D}}\right)\\\
\ =0+\sigma_{kj}$ (35)
because of (8). It is therefore clear that $\sigma_{ij}$ is constant for
unequal indices and the definition (9) makes sense for LU compositions. So
$\sigma_{ij}=\alpha_{D}$ if $i\neq j$. Similarly, in the case $i=j$ we have
$\sigma_{ii}=\mathrm{var}\left(\log\frac{p_{i}}{p_{D}}\right)\\\
=\mathrm{cov}\left(\log\frac{p_{i}}{p_{D}},\log\frac{p_{i}}{p_{D}}\right)=\mathrm{cov}\left(\log\frac{p_{i}}{p_{k}}+\log\frac{p_{k}}{p_{D}},\log\frac{p_{i}}{p_{D}}\right)\\\
=\mathrm{cov}\left(\log\frac{p_{k}}{p_{i}},\log\frac{p_{D}}{p_{i}}\right)+\mathrm{cov}\left(\log\frac{p_{k}}{p_{D}},\log\frac{p_{i}}{p_{D}}\right)=\alpha_{i}+\alpha_{D},$
(36)
which concludes the proof.
###
7.3.
Proof of (12) for an Uncorrelated Basis
Let the composition $\boldsymbol{u}$ have a basis $t\boldsymbol{u}$ with
$\mathrm{cov}(\log(tu_{i}),\log(tu_{j}))=0$ for all $i\neq j$. We show that it
follows that
$\mathrm{var}\left(\log(tu_{k})\right)=\mathrm{cov}\left(\log\frac{u_{i}}{u_{k}},\log\frac{u_{j}}{u_{k}}\right)$
(37)
where none of the indices $i$, $j$, $k$ are equal. We write the left-hand side
as
$\mathrm{cov}\left(\log(tu_{k}),\log(tu_{k})\right)=\mathrm{cov}\left(\log\frac{tu_{k}}{tu_{i}}+\log(tu_{i}),\log\frac{tu_{k}}{tu_{j}}+\log(tu_{j})\right)\\\
=\mathrm{cov}\left(\log\frac{tu_{k}}{tu_{i}},\log\frac{tu_{k}}{tu_{j}}\right)+\mathrm{cov}\left(\log\frac{tu_{k}}{tu_{i}},\log(tu_{j})\right)\\\
+\mathrm{cov}\left(\log(tu_{i}),\log\frac{tu_{k}}{tu_{j}}\right)+\mathrm{cov}\left(\log(tu_{i}),\log(tu_{j})\right)\\\
=\mathrm{cov}\left(\log\frac{u_{i}}{u_{k}},\log\frac{u_{j}}{u_{k}}\right)+\mathrm{cov}\left(\log(tu_{k})-\log(tu_{i}),\log(tu_{j})\right)+\dots{}$
(38)
The terms after the first one are all zero because of the condition that
$\mathrm{cov}(\log(tu_{i}),\log(tu_{j}))=0$ for any indices $i\neq j$, which
concludes the proof.
###
7.4.
Proof of (22) for LU compositions
Let $\boldsymbol{\Sigma}$ be given by the elements (10). The following are
expressions for its determinant (here given without proof):
$|\boldsymbol{\Sigma}|=\prod_{i=1}^{D}\alpha_{i}\sum_{i=1}^{D}\frac{1}{\alpha_{i}}=\sum_{i=1}^{D}\prod_{k\neq
i}\alpha_{k},$ (39)
where the second equality is found by evaluating the greatest common
denominator of the $\alpha_{i}^{-1}$. We show that the matrix inverse of
$\boldsymbol{\Sigma}$ is given by the elements
$\sigma_{ij}^{(-1)}=\frac{1}{\sum_{k=1}^{D}\prod_{l\neq
k}\alpha_{l}}\left\\{\begin{array}[]{c@{\quad}l}\sum_{k\neq i}\prod_{l\neq
k,i}\alpha_{l}&\mbox{if $i=j$,}\\\ -\prod_{k\neq i,j}\alpha_{k}&\mbox{if
$i\neq j$.}\end{array}\right.\quad i,j=1,\dots,D-1.$ (40)
That this is the inverse can be easily proven when multiplying the resulting
matrix with the matrix whose elements are given by (10). The off-diagonal
elements (where $i\neq j$) of the result are
$\sum_{m=1}^{D-1}\sigma_{im}\sigma^{(-1)}_{mj}=\sum_{m\neq
i,j,D}\sigma_{im}\sigma_{mj}^{(-1)}+\sigma_{ii}\sigma_{ij}^{(-1)}+\sigma_{ij}\sigma_{jj}^{(-1)}=\\\
\frac{1}{\sum_{k=1}^{D}\prod_{l\neq k}\alpha_{l}}\left[-\alpha_{D}\sum_{m\neq
i,j,D}\prod_{k\neq m,j}\alpha_{k}-(\alpha_{i}+\alpha_{D})\prod_{k\neq
i,j}\alpha_{k}+\alpha_{D}\sum_{k\neq j}\prod_{l\neq k,j}\alpha_{l}\right]$
(41)
The square bracket evaluates to
$-\alpha_{D}\sum_{m\neq j,D}\prod_{k\neq m,j}\alpha_{k}-\alpha_{i}\prod_{k\neq
i,j}\alpha_{k}+\alpha_{D}\sum_{k\neq j}\prod_{l\neq k,j}\alpha_{l}\\\
=-\alpha_{D}\sum_{m\neq j,D}\prod_{k\neq m,j}\alpha_{k}-\alpha_{i}\prod_{k\neq
i,j}\alpha_{k}+\alpha_{D}\sum_{k\neq j,D}\prod_{l\neq
k,j}\alpha_{l}+\alpha_{D}\prod_{l\neq D,j}\alpha_{l}=0.$ (42)
Thus the off-diagonal elements vanish. For the diagonal elements we have
$\sum_{m=1}^{D-1}\sigma_{im}\sigma^{(-1)}_{mi}=\sum_{m\neq
i,D}\sigma_{im}\sigma_{mi}^{(-1)}+\sigma_{ii}\sigma_{ii}^{(-1)}=\\\
\frac{1}{\sum_{k=1}^{D}\prod_{l\neq k}\alpha_{l}}\left[-\alpha_{D}\sum_{m\neq
i,D}\prod_{k\neq m,i}\alpha_{k}+(\alpha_{i}+\alpha_{D})\sum_{k\neq
i}\prod_{l\neq k,i}\alpha_{l}\right].$ (43)
The square bracket evaluates to
$-\alpha_{D}\sum_{m\neq i,D}\prod_{k\neq m,i}\alpha_{k}+\alpha_{i}\sum_{k\neq
i}\prod_{l\neq k,i}\alpha_{l}+\alpha_{D}\sum_{k\neq i,D}\prod_{l\neq
k,i}\alpha_{l}+\alpha_{D}\prod_{l\neq D,i}\alpha_{l}\\\ =\alpha_{i}\sum_{k\neq
i}\prod_{l\neq k,i}\alpha_{l}+\alpha_{D}\prod_{l\neq
D,i}\alpha_{l}=\sum_{k\neq i}\prod_{l\neq k}\alpha_{l}+\prod_{l\neq
i}\alpha_{l}=\sum_{k}\prod_{l\neq k}\alpha_{k}.$ (44)
Inserting this back into (43) shows that the diagonal elements are 1, proving
the inverse. We can now evaluate the partial correlation coefficient (7) for
an LU composition $\boldsymbol{u}$ by inserting the expression for the inverse
(40) and (to obtain the third equality below) making use of the second
equality in (39):
$r_{ij}(\boldsymbol{u})=\frac{-\sigma^{(-1)}_{ij}}{\sqrt{\sigma^{(-1)}_{ii}\sigma^{(-1)}_{jj}}}=\frac{\prod_{k\neq
i,j}\alpha_{k}}{\sqrt{\left(\sum_{k\neq i}\prod_{l\neq
k,i}\alpha_{l}\right)\left(\sum_{k\neq j}\prod_{l\neq
k,j}\alpha_{l}\right)}}\\\ =\frac{\prod_{k\neq
i,j}\alpha_{k}}{\sqrt{\left(\prod_{l\neq i}\alpha_{l}\sum_{k\neq
i}\alpha_{l}^{-1}\right)\left(\prod_{l\neq j}\alpha_{l}\sum_{k\neq
j}\alpha_{l}^{-1}\right)}}\\\ =\frac{\prod_{k\neq
i,j}\alpha_{k}}{\left(\prod_{k\neq
i,j}\alpha_{k}\right)\sqrt{\left(\alpha_{j}\sum_{k\neq
i}\alpha_{l}^{-1}\right)\left(\alpha_{i}\sum_{k\neq
j}\alpha_{l}^{-1}\right)}}\\\
=\sqrt{\frac{\alpha^{-1}_{i}\alpha^{-1}_{j}}{\left(\sum_{k\neq
i}\alpha^{-1}_{k}\right)\left(\sum_{k\neq j}\alpha^{-1}_{k}\right)}},$ (45)
which is the expression given in (22).
###
7.5.
Supplementary Figures
Figure 6: Mean squared error of CLR and ALR covariance matrices for different
sample sizes ($N$=200, 40, 8) and estimation procedures (no shrinkage, naive
shrinkage of CLR/ALR covariance matrix, and basis covariance shrinkage)
computed on synthetic data. Each boxplot contains the results of 200
simulations. Whenever estimates without shrinkage are not shown, their median
value is given in the legend. Figure 7: Mean squared error of CLR and ALR
covariance matrices for different sample sizes ($N$=2500, 1000, 100) and
estimation procedures (no shrinkage, naive shrinkage of CLR/ALR covariance
matrix, and basis covariance shrinkage) computed on single-cell gene
expression data. Each boxplot contains the results of 200 resamplings from
data. Colours indicate the type of zero imputation used.
###
7.6.
Code and Data Availability
All the code used to perform the benchmark and reproduce the results of this
paper, as well as the subset of single-cell gene expression data, are
available on GitHub under https://github.com/suzannejin/pcor.bshrink.git.
The R package propr Quinn et al. (2017) enables compositional data analysis on
relative gene expression data. Originally designed for efficient calculations
of proportionality indices, it has been updated several times, e.g., to
include differential proportionality across groups Erb et al. (2017). Until
the full implementation of partial correlations in propr, an R implementation
for the computation of partial correlations with basis shrinkage is available
on the repository above (bShrink function in file bin/rlib/shrink.R).
## References
* Whittaker (1990) Whittaker, J: Graphical models in applied multivariate statistics. Wiley (1990)
* van Nimwegen (2016) van Nimwegen E (2016). Inferring Contacting Residues within and between Proteins: What Do the Probabilities Mean? PLoS Comput Biol 12(5), e1004726.
* Weistuch et al. (2022) Weistuch, C, Zhu, J, Deasy, JO and Tannenbaum, AR (2022). The maximum entropy principle for compositional data. BMC Bioinformatics 23:449.
* Schäfer and Strimmer (2005) Schäfer, J and Strimmer, K (2005). A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics. Statistical Applications in Genetics and Molecular Biology 4(1), 32.
* Badri et al. (2020) Badri, M, Kurtz, ZD, Bonneau, R, Müller, CL (2020). Shrinkage improves estimation of microbial associations under different normalization methods, NAR Genomics and Bioinformatics 2(4), lqaa100.
* Aitchison (1986) Aitchison, J: The statistical analysis of compositional data. Chapman and Hall (1986)
* Erb (2020) Erb, I (2020). Partial correlations in compositional data analysis Applied Computing and Geosciences 6, 100026.
* Werhli et al. (2006) Werhli, AV, Grzegorczyk, M, Husmeier, D (2006). Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical gaussian models and bayesian networks, Bioinformatics, 22(20), 2523–2531.
* Lovell et al. (2015) Lovell, D. et al. (2015). Proportionality: a valid alternative to correlation for relative data. PLoS Comp Biol 11, e1004075.
* Erb and Notredame (2016) Erb, I and Notredame, C (2016). How should we measure proportionality on relative gene expression data? Theory in Biosciences 135(1-2), 21–36.
* Kurtz et al. (2015) Kurtz, ZD, Müller, CL, Miraldi, ER, Littman, DR, Blaser, MJ, Bonneau, RA (2015) Sparse and Compositionally Robust Inference of Microbial Ecological Networks. PLoS Comput Biol 11(5):e1004226.
* Erb et al. (2020) Erb, I, Gloor, GB, Quinn, TP (2020). Editorial: Compositional data analysis and related methods applied to genomics – a first special issue from NAR Genomics and Bioinformatics. NAR Genomics and Bioinformatics 2(4) lqaa103.
* Quinn et al. (2018) Quinn, TP, Erb, I, Richardson, MF, Crowley, TM (2018). Understanding sequencing data as compositions: an outlook and review. Bioinformatics 34(16), 2870–2878.
* Aitchison (1982) Aitchison, J (1982). The Statistical Analysis of Compositional Data. J Royal Stat Soc B 44(2), 139–160.
* Lovell et al. (2020) David R Lovell, Xin-Yi Chua, Annette McGrath (2020). Counts: an outstanding challenge for log-ratio analysis of compositional data in the molecular biosciences, NAR Genomics and Bioinformatics, 2(2), lqaa040.
* Law et al. (2014) Law, CW, Chen, Y, Shi, W, and Smyth, GK (2014). voom: precision weights unlock linear model analysis tools for RNA-seq read counts Genome Biology, 15:R29.
* Hausser and Strimmer (2009) Hausser, J and Strimmer, K (2009). Entropy Inference and the James-Stein Estimator, with Application to Nonlinear Gene Association Networks Journal of Machine Learning Research 10, 1469-1484.
* Quinn and Erb (2021) Quinn, TP, Erb, I (2021). Amalgams: data-driven amalgamation for the dimensionality reduction of compositional data NAR Genomics and Bioinformatics 2(4) Iqaa076.
* Greenacre et al. (2022) Greenacre, M Grunsky, E Bacon-Shone, J, Erb, I and Quinn, T (2022). Aitchison’s Compositional Data Analysis 40 years On: A Reappraisal. arXiv:2201.05197
* James and Stein (1961) James, W and Stein, C. (1961) Estimation with quadratic loss. In: Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1 361–379, Univ. California Press
* Ledoit and Wolf (2003) Ledoit, O. and Wolf, M (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance 10, 603–621.
* Opgen-Rhein and Strimmer (2007) Opgen-Rhein, R and Strimmer, K (2007). Accurate Ranking of Differentially Expressed Genes by a Distribution-Free Shrinkage Approach. Statistical Applications in Genetics and Molecular Biology 6(1).
* Palarea-Albaladejo et al. (2015) Palarea-Albaladejo, J, Josep Antoni MartÃn-Fernández, JA (2015). zCompositions – R package for multivariate imputation of left-censored data under a compositional approach, Chemometrics and Intelligent Laboratory Systems 143, 85–96.
* Palarea-Albaladejo et al. (2022) Palarea-Albaladejo, J, Antoni MartÃn-Fernández, J, Ruiz-Gazen, A, Thomas-Agnan, C. (2022). lrSVD: An efficient imputation algorithm for incomplete high-throughput compositional data. Journal of Chemometrics e3459.
* Riba et al. (2022) Riba, A., Oravecz, A., Durik, M. et al. (2022). Cell cycle gene regulation dynamics revealed by RNA velocity and deep-learning. Nat Commun 13, 2865.
* DeGroot (2004) DeGroot, MH: Optimal Statistical Decisions, Wiley (2004)
* Quinn et al. (2017) Quinn, TP, Richardson, MF, Lovell, D and Crowley, TL (2017). propr: An R-package for Identifying Proportionally Abundant Features Using Compositional Data Analysis. Scientific Reports 7(1) 16252.
* Erb et al. (2017) Erb, I, Quinn, TP, Lovell, D and Notredame, C (2017). Differential Proportionality – A Normalization-Free Approach To Differential Gene Expression. Proceedings of CoDaWork 2017, The 7th Compositional Data Analysis Workshop; available under bioRxiv, 134536.
|
# HIKE: High Intensity Kaon Experiments at the CERN SPS
Matthew Moulson for the HIKE Collaboration INFN Laboratori Nazionali di
Frascati, 00044 Frascati RM, Italy<EMAIL_ADDRESS>
###### Abstract
The availability of high intensity kaon beams at the CERN SPS North Area gives
rise to unique possibilities for sensitive tests of the Standard Model in the
quark flavor sector. Precise measurements of the branching ratios for the
flavor-changing neutral current decays $K\to\pi\nu\bar{\nu}$ can provide
unique constraints on CKM unitarity and, potentially, evidence for new
physics. Building on the success of the NA62 experiment, plans are taking
shape at CERN for a comprehensive program that will include experimental
phases to measure the branching ratio for $K^{+}\to\pi^{+}\nu\bar{\nu}$ to
$\sim$5% and to $K_{L}\to\pi^{0}\nu\bar{\nu}$ to $\sim$20% precision. These
planned experiments would also carry out lepton flavor universality tests,
lepton number and flavor conservation tests, and perform other precision
measurements in the kaon sector, as well as searches for exotic particles in
kaon decays. We overview the physics goals, detector requirements, and project
status for the next generation of kaon physics experiments at CERN.
## 1 Introduction
Rare kaon decays provide information on the unitary triangle, as illustrated
in Figure 1. These are flavor-changing neutral current processes (FCNC) that
probe the $s\to d\nu\bar{\nu}$ or $s\to d\ell^{+}\ell^{-}$ transitions. They
are highly GIM-suppressed and their SM rates are very small. The
$K\to\pi\nu\bar{\nu}$ decays are the least affected by long-distance physics.
The branching ratios (BRs) for the $K\to\pi\nu\bar{\nu}$ decays are among the
observable quantities in the quark-flavor sector most sensitive to new
physics.
Figure 1: Determination of the unitary triangle with rare kaon decays.
In the SM, the uncertainties on the $K\to\pi\nu\bar{\nu}$ rates are entirely
dominated by the uncertainties on the CKM matrix elements $|V_{ub}|$ and
$|V_{cb}|$ and the angle $\gamma$. Using values for these parameters from the
analysis of tree-level observables, Buras et al. obtain [1]
$\begin{split}{\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})&=(3.4\pm 0.6)\times
10^{-11},\\\ {\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})&=(8.4\pm 1.0)\times
10^{-11}.\end{split}$ (1)
Assuming no new-physics effects in $\epsilon_{K}$ and $\sin{2\beta}$ from
$B\to J/\psi K_{S}$, the BRs can be determined independently of $|V_{cb}|$ as
${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})=(2.94\pm 0.15)\times 10^{-11}$ and
${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=(8.60\pm 0.42)\times 10^{-11}$ [2]. The
intrinsic theoretical uncertainties, if the CKM parameters are taken to be
exact, are about 1.5% and 3.5%, respectively.
Because the amplitude for $K^{+}\to\pi^{+}\nu\bar{\nu}$ has both real and
imaginary parts, while the amplitude for $K_{L}\to\pi^{0}\nu\bar{\nu}$ is
purely imaginary, the decays have different sensitivity to new sources of $CP$
violation. Measurements of both BRs would therefore be extremely useful not
only to uncover evidence of new physics in the quark-flavor sector, but also
to distinguish between new physics models. More generally, measurement of all
of the FCNC kaon decays would allow the unitarity triangle to be
overconstrained as illustrated in Figure 1, potentially providing evidence of
new physics independently of and in comparison to results from $B$ and $D$
meson decays and providing important information about the flavor structure of
that physics.
## 2 The NA62 experiment
NA62 is a fixed-target experiment at the CERN SPS, the goal of which is to
measure ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ to within 10% [3]. The
secondary positive beam, consisting of 6% $K^{+}$ with a total rate of 750
MHz, is derived from the 400-GeV primary proton beam from the SPS incident at
zero angle on a beryllium target at a rate of $3\times 10^{12}$ protons per
pulse (ppp). The $K^{+}\to\pi^{+}\nu\bar{\nu}$ decay is detected in flight.
The signature is a $K^{+}$ entering the experiment and a $\pi^{+}$ leaving the
decay vertex, with missing momentum at the vertex and no other particles
observed in the final state. Principal backgrounds include those from the
abundant decays $K^{+}\to\mu^{+}\nu$ and $K^{+}\to\pi^{+}\pi^{0}$, as well as
backgrounds from upstream decays and interactions. The signal decay is
identified via selection in the $(p_{\pi},m^{2}_{\rm miss})$ plane to exclude
the abundant two-body decays, where $p_{\pi}$ is the momentum of the candidate
pion track and $m^{2}_{\rm miss}$ is the squared missing mass at the vertex.
The distinguishing features of the experiment include high-rate, precision
tracking for both the beam and secondary particles, redundant particle
identification systems and muon vetoes, and hermetic photon vetoes, including
a high-performance EM calorimeter.
Between 2016 and 2018, NA62 observed more than $4\times 10^{12}$ $K^{+}$
decays in its fiducial volume, with the expectation of observing 10 signal
events and 7 background events, principally from upstream decays and
interactions. A total of 20 events were observed, establishing the
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay with $3.4\sigma$ significance and
providing the most precise measurement to date for the branching ratio [4]:
${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})=(10.6^{+4.0}_{-3.4}|_{\rm stat}\pm
0.9_{\rm syst})\times 10^{-11}.$
NA62 resumed data taking in July 2021 with a number of key modifications to
the beamline and detector to reduce background from upstream decays and
interactions and to allow data to be taken at the full nominal beam intensity.
The experiment is approved for data taking up to LHC Long Shutdown 3 (LS3),
currently foreseen for the end of 2025, and is expected to measure ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ to 10% precision by then. In the longer
term, fixed-target runs in the SPS North Area are foreseen at least through
2040. There is therefore an opportunity at the SPS for an integrated program
to pin down new physics in the kaon sector via measurement of all rare kaon
decay modes—both charged and neutral.
## 3 The HIKE physics program
The HIKE program (High Intensity Kaon Experiments at the CERN SPS) [5] is
foreseen to include three experimental phases for the comprehensive, high-
precision study of rare kaon decays during the period from the end of LS3 to
the FCC era:
* •
Phase 1: A $K^{+}$ experiment running at four times the intensity of NA62
($1.2\times 10^{13}$ ppp) to measure ${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$
to 5% precision. Phase 1 will also focus on studies of lepton
universality/number/flavor violation through observables such as
$R_{K}=\Gamma(K^{+}\to e^{+}\nu)/\Gamma(K^{+}\to\mu^{+}\nu)$ and
$K^{+}\to\pi^{+}\ell\ell$, and searches for decays such as
$K^{+}\to\pi^{-}\ell^{+}\ell^{+}$ and $K^{+}\to\pi^{+}\mu e$. The experiment
will also make precision measurements of leptonic and semileptonic, radiative,
and Dalitz decays and chiral parameters of the kaon system.
* •
Phase 2: An experiment with a neutral beam and tracking and PID systems
optimized for the measurement of decays like $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$
and $K_{L}\to\mu^{+}\mu^{-}$, as well as studies of lepton
universality/number/flavor violation in $K_{L}$ decays, radiative $K_{L}$
decays and precision measurements, and measurements of $K_{L}$, $n$, and
$\Lambda$ fluxes in the neutral beam and halo to prepare for the final phase.
* •
Phase 3: An experiment to measure ${\rm BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ to
20%, also known as KLEVER [6].
In addition, periodic runs will be taken in beam-dump mode, with the target
out and the collimator in the secondary beam line closed, to allow sensitive
searches for the decays of exotic, long-lived particles produced in the decays
of mesons from interactions in the beam dump, with the goal of collecting
$10^{19}$ protons on target (pot) in dump mode in Phase 1 and up to $5\times
10^{19}$ pot by the end of Phase 3.
The experimental setup for all three phases will rely to the maximum extent
possible on the reuse of the same detectors in different configurations. In
particular, when a detector for HIKE is newly built or extensively upgraded,
if at all possible, it is designed to meet the performance requirements for
all successive phases.
### 3.1 HIKE Phase 1
Figure 2: Experimental setup for HIKE Phase 1
The four-fold increase in primary intensity needed for the measurement of
${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ to 5% will require major upgrades of
the primary and secondary beamlines, as discussed in [7]. From the standpoint
of the experiment, the success of NA62 has validated the measurement technique
and proves that the background can be handled. The key challenge from the
intensity increase is that the time resolution of the detectors must be
improved across the board by a factor of four in order to maintain the loss of
events from accidental coincidence (random veto) to acceptable levels
($\lesssim 25\%$), which must be achieved while maintaining other key
performance specifications such as space-time reconstruction performance, low
material budget, high single photon efficiencies, etc. The experimental setup,
shown in Figure 2, is not very different from that of NA62, but most detectors
will need to be rebuilt or extensively overhauled to secure the needed
performance.
Of particular interest, the NA62 beam tracker (Gigatracker, GTK), consisting
of three stations of silicon pixel detectors, will need to be upgraded to
track at 3 GHz. A time resolution of better than 50 ps will be required, and
the detector will have to be able to handle rates of 8 MHz/mm2 and be
radiation resistant up for particle fluences of more than $2\times 10^{15}$
$n$ eq/cm2/yr. An excellent candidate technology is provided by the timeSPOT
project [8, 9], which is developing hybrid 3D-trenched pixels in which the
pixel electrode geometry is optimized for timing performance.
The experiment’s rate capability for secondary particles must be improved as
well. New straw-tube designs are being developed at CERN, based on past
collaboration with Dubna, that will allow straw chambers for use in vacuum to
be developed with 5-mm diameter straws with wall thickness of 20 $\mu$m.
It is natural to inquire whether the NA48 liquid-krypton (LKr) calorimeter
[10] used in NA62 can be reused for any of the HIKE phases. NA48-era studies
suggest and NA62 experience confirms that the photon detection efficiency is
sufficient for at least Phases 1 and 2. The time resolution, however, is
insufficient for the high-intensity program and would require improvement by
at least a factor of four. Two directions are being pursued. The first is to
examine whether the LKr can be used for HIKE Phase 1. In addition to necessary
consolidation work, this would require upgrades to make the calorimeter
faster, including an increase in the operating voltage to increase the drift
velocity and faster signal shaping and digitizers for the readout system. For
the $K_{L}$ phases, the diameter of the LKr inner bore limits the solid angle
of the beam that can be used, so a new calorimeter would be necessary in any
case. The new calorimeter could also be used in Phase 1, if it is ready in
time. An ideal choice of technologies appears to be the fine-sampling shashlyk
design used for the KOPIO and PANDA calorimeters [11], which has been shown to
provide excellent energy and time resolution. For HIKE, PID capability could
be added by including independently read out, 1-cm-thick “spy tiles” at key
points in the shashlyk stack (for example, at the front of the calorimeter for
charged-particle identification, at shower maximum, and deep in the stack).
Prototypes with this design have been tested at Protvino and DESY. Another
option under investigation is to construct the calorimeter with new-
generation, nanocomposite scintillators [12], which offer high light output,
fast response, and good radiation robustness.
### 3.2 HIKE Phase 2
Figure 3: Experimental setup for HIKE Phase 2
For HIKE Phase 2, a new neutral beamline is required. The baseline design is
the original 120-m beamline for KLEVER [13], featuring four collimation
stages, including an active final collimator that is incorporated into the
experiment and an oriented-crystal-metal photon converter at the center of the
dump collimator to reduce the flux of prompt photons in the beam [14]. The
collimation system defines a beam opening angle of 0.4 mrad. The neutral beam
is produced at $\theta=2.4$ mrad; $K_{L}$ mesons in the beam have an average
momentum of 79 GeV, while those decaying in the fiducial volume (FV) have an
average momentum of 46 GeV. Relative to the Phase-1 experiment, in addition to
the changes to the beamline and the removal of the beamline detectors for
charged particles, the RICH is removed and the spectrometer is moved further
downstream to increase the acceptance for decays such as
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$. The beamline and experimental setup are
illustrated in Figure 3.
At a primary intensity of $2\times 10^{13}$ ppp (a six-fold increase with
respect to NA62), nearly $2\times 10^{14}$ kaon decays will be observed in the
FV in five years of running. This will allow single-event sensitivities for
$K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ to be improved by two orders of magnitude
with respect to the current best limits from KTeV [15, 16]. There are
topologically identical backgrounds to these channels from
$K_{L}\to\gamma\gamma\ell^{+}\ell^{-}$, with BRs that are 3–4 orders of
magnitude greater than for $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ [17]. Suppression
of these backgrounds relies on the excellent energy resolution of the HIKE EM
calorimeter for the reconstruction of the $\pi^{0}$ mass peak for the signal
decay. HIKE Phase 2 will be well positioned to make the first observation of
the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ decay, as well as to measure ${\rm
BR}(K_{L}\to\mu^{+}\mu^{-})$ with a statistical precision of 0.2%, improving
on the current best result from BNL-E871 [18], and should also achieve BR
sensitivities of $O(10^{-12})$ for a broad range of rare and forbidden $K_{L}$
decays, such as lepton-flavor violating processes, representing improvement on
current best limits from BNL-E871 by up to a factor of 60.
### 3.3 HIKE Phase 3
Figure 4: Experimental setup for HIKE Phase 3 (KLEVER)
HIKE Phase 3 is a dedicated experiment, known as KLEVER, to measure ${\rm
BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$ to 20%. Specifically, with a total exposure
of $6\times 10^{19}$ pot in five years at an intensity of $2\times 10^{13}$
ppp, the KLEVER goal is to detect 60 signal events at the Standard Model
branching ratio, with a signal-to-background ratio $S/B\sim 1$. The experiment
is complementary to KOTO, in the sense that the beam energy is significantly
higher. As a result, photons from $K_{L}$ decays receive a significant boost,
which makes photon vetoing easier. On the other hand, the length of the
experiment is much greater, and a very long beamline is required to allow
$\Lambda$ baryons and $K_{S}$ mesons to decay upstream of the fiducial volume.
Relative to the Phase-2 beamline, the KLEVER beamline needs to be extended by
an additional 150 m from target to final collimator. This in turn requires a
downstream extension of the ECN3 hall by a similar amount. For KLEVER running,
the production angle for the neutral beam will be increased from 2.4 to 8.0
mrad: this decreases the $n/K_{L}$ and $\Lambda/K_{L}$ ratios for the beam and
softens the momentum spectra so that the average $K_{L}$ momentum is 39 GeV at
production (26 GeV for $K_{L}$ mesons that decay in the FV). This also
decreases the absolute $K_{L}$ flux, which is further reduced by the need to
collimate the beam more tightly ($\Delta\theta=0.256$ mrad) for the extended
beamline. The need for additional construction to extend the ECN3 hall is a
major factor in scheduling KLEVER towards the end of the HIKE program. Cost
estimates are in progress.
From the standpoint of the experiment, the layout of the vacuum tank and
fiducial volume is roughly the same as for the other HIKE phases. The
spectrometer will be removed and the number of large-angle photon veto
detectors will be increased from 12 to 25 to extend the polar angle coverage
out to 100 mrad (from 50 mrad for Phases 1 and 2). The HIKE large-angle vetoes
themselves will be fine-segmented lead/scintillating tile detectors similar to
the VVS detectors for the planned but never realized CKM experiment [19]. One
particularly challenging detector for KLEVER is the small-angle calorimeter
(SAC), which sits in the neutral beam at the downstream end of the experiment
and which must reject photons from background decays such as
$K_{L}\to\pi^{0}\pi^{0}$ that would otherwise escape via the beam exit. The
SAC must have good photon detection efficiency, especially for high energy
photons (e.g., the inefficiency must be $<10^{-4}$ for photons with $E>30$
GeV) while being as insensitive as possible to the accidental coincidence of
nearly 600 MHz of neutral hadrons ($n$ and $K_{L}$) in the beam. The SAC must
also have $\sigma_{t}<100$ ps, be able to separate two pulses a few ns apart,
and be radiation hard to $10^{14}$ $n$/cm2 and $10^{5}$–$10^{6}$ Gy. Our
proposed solution is an ultra-fast, high-$Z$ crystal calorimeter based on a
Cerenkov radiator like PbF2 or an ultra-fast scintillator such as PWO-UF [20],
which has a dominant emission component with $\tau<0.7$ ns. The SAC will have
transverse and longitudinal segmentation for $\gamma/n$ discrimination. From
an engineering standpoint, it will be very similar to the CRILIN calorimeter
[21], and R&D is proceeding in concert between the HIKE and CRILIN
collaborations. An additional possibility under investigation is to exploit
the coherent interactions of high-energy photons in oriented crystals to
stimulate early pair conversion, allowing the calorimeter to be realized with
reduced thickness, increasing the transparency to neutral hadrons while
maintaining high detection efficiency for photons [22].
## 4 Conclusions
The HIKE project consists of a three-phase experimental program for the
comprehensive study of flavor physics in the kaon sector. The experimental
apparatus changes over time with a staged approach, allowing HIKE to evolve
and adapt its physics scope, an important feature for a project that embraces
a time scale of more than decade, during which the physics landscape could
change. Thanks to the successful experience of NA62 and its predecessor NA48,
the experimental techniques are well established and robust expectations of
sensitivity can be obtained from the extrapolation of existing data. The HIKE
Letter of Intent was submitted to the CERN SPSC at the beginning of November
2022 [5]. A formal proposal is in preparation for submission in fall 2023.
## References
## References
* [1] Buras A J, Buttazzo D, Girrbach-Noe J and Knegjens R 2015 JHEP 11 033 (Preprint 1503.02693)
* [2] Buras A J and Venturini E 2021 Acta Phys. Polon. B 53 A1 (Preprint 2109.11032)
* [3] Cortina Gil E et al. (NA62) 2017 JINST 12 P05025 (Preprint 1703.08501)
* [4] Cortina Gil E et al. (NA62) 2021 JHEP 06 093 (Preprint 2103.15389)
* [5] Cortina Gil E et al. (HIKE) 2022 HIKE, High Intensity Kaon Experiments at the CERN SPS: Letter of Intent Tech. Rep. SPSC-2022-031/SPSC-I-257 CERN (Preprint 2211.16586) URL https://cds.cern.ch/record/2839661
* [6] Ambrosino F et al. (KLEVER Project) 2019 (Preprint 1901.03099)
* [7] Gatignon L, Bernhard J, Brugger M, Doble N et al. 2018 Report from the Conventional Beams Working Group to the Physics Beyond Collider Study and to the European Strategy for Particle Physics Tech. rep. CERN Geneva URL https://cds.cern.ch/record/2650989
* [8] Lai A 2018 A system approach towards future trackers at high luminosity colliders: the timespot project (Sydney, NSW, Australia: IEEE) pp 1–3 ISBN 978-1-5386-8495-5 ISSN 1082-3654
* [9] Lai A et al. 2022 Timespot project testbeam results and simulations talk given at ”1st MONOLITH Workshop on silicon sensors for timing and their applications”, Geneve, September 2022 URL https://indico.cern.ch/event/1179742/contributions/4961955/attachments/2502408/4298914/ Timing_ALai_UniGe.pdf
* [10] Fanti V et al. (NA48) 2007 Nucl. Instrum. Meth. A574 433–471
* [11] Atoian G S et al. 2008 Nucl. Instrum. Meth. A584 291–303 (Preprint 0709.4514)
* [12] Gandini M et al. 2020 Nat. Nanotechnol. 15 462–468
* [13] Van Dijk M et al. 2019 The K12 beamline for the KLEVER experiment URL doi.org/ 10.18429/JACoW-IPAC2019-THPGW061
* [14] Soldani M et al. 2022 (Preprint 2203.07163)
* [15] Alavi-Harati A et al. (KTeV) 2004 Phys. Rev. Lett. 93 021805 (Preprint hep-ex/0309072)
* [16] Alavi-Harati A et al. (KTEV) 2000 Phys. Rev. Lett. 84 5279–5282 (Preprint hep-ex/0001006)
* [17] Greenlee H B 1990 Phys. Rev. D 42 3724–3731
* [18] Ambrose D et al. (E871) 2000 Phys. Rev. Lett. 84 1389–1392
* [19] Ramberg E, Cooper P and Tschirhart R 2004 IEEE Trans. Nucl. Sci. 51 2201–2204
* [20] Korzhik M et al. 2022 Nucl. Instrum. Meth. A 1034 166781
* [21] Ceravolo S et al. 2022 JINST 17 P09033 (Preprint 2206.05838)
* [22] Bandiera L, Haurylavets V and Tikhomirov V 2019 Nucl. Instrum. Meth. A 936 124–126
|
# Study of radio transients from the quiet Sun during an extremely quiet time
Surajit Mondal Center for Solar Terrestrial Research, New Jersey Institute of
Technology, Newark, New Jersey, 07102, United States of America Divya Oberoi
National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, Pune-411007, India Ayan Biswas National Centre for Radio
Astrophysics, Tata Institute of Fundamental Research, Pune-411007, India
(Received TBD ; Revised ; Accepted TBD )
###### Abstract
In this work we study a class of recently discovered metrewave solar
transients referred to as Weak Impulsive Narrowband Quiet Sun Emission
(WINQSEs, Mondal et al., 2020). Their strength is a few percent of the quiet
Sun background and are characterised by their very impulsive, narrow-band and
ubiquitous presence in quiet Sun regions. Mondal et al. (2020) hypothesised
that these emissions might be the radio counterparts of the nanoflares and
their potential significance warrants detailed studies. Here we present an
analysis of data from an extremely quiet time and with an improved methodology
over the previous work. As before, we detect numerous WINQSEs, which we have
used for their further characterisation. Their key properties, namely, their
impulsive nature and ubiquitous presence on the quiet Sun are observed in
these data as well. Interestingly, we also find some of the observed
properties to differ significantly from the earlier work. With this
demonstration of routine detection of WINQSEs, we hope to engender interest in
the larger community to build a deeper understanding of WINQSEs.
Quiet solar corona (1992); Solar corona (1483); Solar coronal radio emission
(1993); Solar coronal heating (1989)
††facilities: Murchison Widefield Array (MWA) (Lonsdale et al., 2009; Tingay
et al., 2013)††software: astropy (Astropy Collaboration et al., 2013, 2018),
matplotlib (Hunter, 2007), Numpy (Harris et al., 2020), SciPy (Virtanen et
al., 2020), CASA (McMullin et al., 2007)
## 1 Introduction
Radio transients from the Sun were discovered at very the dawn of solar radio
astronomy (e.g. Wild et al., 1963; Wild & Smerd, 1972, etc.). Since then
numerous studies have analysed and studied their various properties ranging
from emission mechanisms, spectro-temporal characteristics, relationships with
other solar phenomenon, signatures in other wavebands and so on (e.g.
Takakura, 1967; Sturrock, 1964; Melrose, 1980; McLean & Labrum, 1985; Pick &
Vilmer, 2008; Kontar et al., 2017; Reid & Kontar, 2017; Reid, 2020, and many
others). Most of these transients, particularly those happening in the meter
wavelengths regime, are associated with active regions and violent solar
activity like X-ray flares and coronal mass ejections (e.g. Pick & Vilmer,
2008; Bain & Fletcher, 2009; Krucker et al., 2011; Reid, 2020; Mondal &
Oberoi, 2021, etc.). However, there are hypotheses which suggest that such
radio transients, albeit much weaker, should also be present in the quiet
solar corona (Che, 2018). The primary reasoning behind this expectation comes
from the Parker’s nanoflare hypothesis (Parker, 1988) which states that the
solar corona maintains its million K temperature because of the energy
continuously deposited by numerous tiny nanoflares happening everywhere in the
corona. If this hypothesis is true, it implies that small scale reconnections
are happening everywhere and all the time in the corona. These small scale
reconnections, like their larger counterparts, will also accelerate electrons,
which in turn, on their interaction with the thermal plasma, should emit
plasma emission, observable in the radio bands (Che, 2018). To the best of our
knowledge, the first detection of the ubiquitous presence of these radio
transients from the quiet sun came from Mondal et al. (2020) (henceforth
referred to as M20). They found that these transients are highly impulsive
(duration $\lesssim 1$s) and narrowband in nature. Based on these properties
they also concluded that these emissions are generated by nonthermal
processes. They also found these emissions to be present throughout the quiet
sun. Mondal (2021) has demonstrated, although in a single instance, that the
energy associated with a group of these transients is $\sim 10^{25}$ergs.
These observations suggest the possibility that these transients are the radio
counterparts of the long hypothesised nanoflares. Henceforth we refer to these
emissions as Weak Impulsive Narrowband Quiet Sun Emissions (WINQSEs).
Due to the large potential significance of this discovery, it is crucial to
verify the presence of WINQSEs in other datasets obtained under different
solar conditions, using independent detection techniques and especially during
quiet Sun times. As a part of this exercise, recently Sharma et al. (2022)
(henceforth referred to as S22) have reported detection of WINQSEs using a
completely independent technique and analysis pipeline using a different
dataset. Interestingly S22 also identified a very weak type III solar radio
burst like drifting emission feature, which supports the hypothesis that
WINQSEs are the weaker cousins of the stronger type III bursts. Continuing
with the same motivation, here we present an analysis of data from an
extremely quiet time on 20th June, 2020. The solar activity levels on this day
are much lower than those prevailing during the observations of M20 and S22.
We also take this opportunity to improve the analysis methodology beyond what
was used in M20.
This paper is structured as follows. In Section 2 we present the observation
details, Section 3 discusses the details of data analysis including the
improvements over M20. In Section 4 we present the results, in Section 5 the
discussion and finally Section 6 gives the concluding remarks.
## 2 Observation details
The data presented here come from a solar observation with the Murchison
Widefield Array Phase II (MWA, Lonsdale et al., 2009; Tingay et al., 2013;
Wayth et al., 2018). These data were acquired on 20 June, 2020 between
03:35:00–04:40:00 UT under the project ID G0002. The MWA was in its extended
configuration, with a maximum baseline of $\sim 5$km. The observing frequency
spanned the band from 119.68–150.40 MHz. The data was acquired with a time and
frequency resolution of 0.5 s and 10 kHz respectively. Here we present the
analysis at four selected spectral slices, each of width 160 kHz. These slices
are centered around 120.22, 127.9, 135.58 and 139.9 MHz.
The Sun was extremely quiet on this
day111https://solarmonitor.org/index.php?date=20200620. No X-ray flares were
reported by the Geostationary Operational Environmental Satellites (GOES) and
no events of any kind were reported by the Space Weather Prediction Center
event list for the day. The Learmonth Spectrograph operating between 25–180
MHz also did not report any radio flare on this day. No NOAA active region was
present on the visible solar disc.
## 3 Data analysis
Making the large number of images needed for this investigation necessarily
requires a robust automated pipeline. Such a tool has recently been developed
and christened the Automated Imaging Routine for Compact Arrays for Radio Sun
(AIRCARS; Mondal et al., 2019). The credentials of this self-calibration based
pipeline have already been established in multiple recent works requiring high
quality spectroscopic snapshot imaging (e.g. Mohan et al., 2019; Mondal et
al., 2020; Mohan, 2021; Mondal & Oberoi, 2021, etc.). It is known from past
experience that the presence of strong compact sources on the Sun makes
calibration easier and the data corresponding to a large but featureless quiet
Sun is harder to calibrate (Mondal et al., 2019). While the imaging of the
data used here was done using AIRCARS, to improve the accuracy of calibration
for these quiet Sun data, an improved strategy from what was followed in M20
was employed. For this work, the calibration solutions were determined using
data averaged over 9 s. This time integration helps reduce the thermal noise
substantially and is especially useful in improving the signal-to-noise of
visibilities from longer baselines. The calibration solutions obtained were
linearly interpolated in time and applied to the 0.5 s resolution data which
were then imaged. The analysis for each of the four frequencies presented here
was carried out independently following this procedure. This approach relies
on the assumption that the solar variability over the integration period is
small enough to be ignored. We find this to be a good assumption for this
extremely quiet time and is evidenced by the higher imaging dynamic ranges of
the images produced using solutions derived from time-averaged visibilities.
On its own, AIRCARS does not ensure a common flux density scale across
independent observations. For the present analysis, this can lead to
discontinuities in the fluxscale at the boundaries of observations, typically
of a duration of 4-5 minutes, though this issue has now been resolved in its
next incarnation, named, Polarimetry using Automated Imaging Routine for
Compact Arrays for the Radio Sun (P-AIRCARS; Kansabanik et al., 2022b). A
precise absolute flux density is not required for this work and we have
adopted the strategy described below for relative flux density calibration
between all of the observations at a given frequency. It also takes care of
the amplitude variability of the order of 4-5% of the MWA instrumental
response which otherwise is not possible to take care of by using the method
described in Kansabanik et al. (2022a). While under most circumstances, this
level of variability is too small to be of any consequence, in the present
context it is important to correct for it as it is of a magnitude comparable
to the strengths of WINQSEs.
A correction for the instrumental primary beam is first applied to the solar
maps and then it is enforced that the median disc integrated flux density of
the Sun be the same for all five-minute chunks of data used. The underlying
assumption is that as the Sun was very quiet during our observing period, its
disc-integrated flux density should also be stable in this period. The
characteristics of WINQSEs - flux densities of $\sim$ mSFU (as compared to a
few SFU from the Sun) and short durations ($\lesssim$1 s) - imply that they
are too weak to influence the disc-integrated flux density in any significant
manner. We also found that the point-spread-function (psf) corresponding to
different time slices for a given frequency can change slightly across time.
Flux densities in radio maps are measured in units of $Jy/beam$, where $beam$
refers to the area of the psf. Hence, slight changes in the psf area can give
rise to low-level jumps in flux density estimates. To avoid contamination due
to this, we have smoothed all of the images used here to a common coarser
resolution of $280^{"}\times 280^{"}$.
A key step for identifying WINQSEs in M20 was to calculate the median image,
which requires accurate alignment of images relative to each other. M20
achieved this by aligning the radio peak of the active region present in their
dataset across images. Absence of a clearly identifiable compact feature
forces us to look for other alternatives to achieve this objective. We compute
a center of intensity ($\hat{X}$, $\hat{Y}$) for each 0.5 s solar image,
defined as:
$\hat{X}=\frac{\sum_{i}x_{i}}{\sum_{i}1},\quad\hat{Y}=\frac{\sum_{i}y_{i}}{\sum_{i}1},$
(1)
where ($x_{i}$, $y_{i}$) are the coordinates of the pixels which exceed the
threshold chosen visually such that no noise contours appear in the image and
the region it demarcates is reasonably symmetric. The centers of intensity for
all the images were aligned to lie at the same pixel. The aligned maps were
used to generate the median map, one for each five-minute data chunk and
frequency. An example 0.5 s map at 127.9 MHz and the corresponding median map
are shown in Fig. 1.
Figure 1: Left panel: An example 0.5 s map at 127.9 MHz. Right panel: The
corresponding median map at 127.9 MHz.
## 4 Results
For detection of WINQSEs in the large number of images used in this work, we
have modified the technique used in M20. First, all 0.5 s solar images were
smoothed using a circular kernel of size 280". Let the images prior and post
this smoothing operation be denoted by $I_{init}$ and $I_{flux}$ respectively.
As mentioned in the previous section, this smoothing operation ensured that
the numerical value of each pixel in the radio image $I_{flux}$ is equal to
the flux density for the resolution element corresponding to the circular
smoothing kernel centered on that pixel in $I_{init}$. The median map obtained
using $I_{init}$ images was used to obtain $I_{median,flux}$ using the same
technique as that used to obtain $I_{flux}$. $I_{median,flux}$ was then
subtracted from each $I_{flux}$ to obtain the residual maps for each 0.5 s
timeslice, henceforth referred to as $I_{res}$. The pixel values in $I_{res}$
is equivalent to $\Delta F_{i,\nu,t}=F_{i,\nu,t}-<F_{i,\nu}>$, where
$F_{i,\nu,t}$ is the flux density of a psf sized region $i$ at frequency $\nu$
some time $t$ and $<F_{i,\nu}>$ is the median flux density for region $i$ at
frequency $\nu$. A WINQSE is said to be present when the pixel value in
$I_{res}$ exceeds a given threshold. In this work, the threshold is set to
$3\sigma$, where $\sigma$ is the rms in the corresponding $I_{flux}$ in a
region away from the Sun. To ensure that only independent pixels are searched
for WINQSEs, the search algorithm steps through the pixels in steps of 7
pixels in both directions which translates to $280^{"}$ in each direction and
is equal to the effective image resolution. Figure 2 shows the contours of an
example residual image with a bright WINQSE, overlaid on a AIA 193Å image. The
compact feature seen in the image is the WINQSE detected in this snapshot and
it is evident that it has been detected with good signal-to-noise. The
following subsections discuss the various properties of the WINQSEs detected
in these data.
Figure 2: Radio contours of an example WINQSE at 136 MHz are overlaid on a 193
Å image from AIA/SDO. The contour levels are at 0.7, 0.85 and 0.99 times the
peak of the median subtracted map.
### 4.1 Flux density distribution
For convenience and consistency with M20, for a given frequency $\nu$, we
refer to the collection of $\Delta F_{i,\nu,t}$ for all psf sized regions $i$
s and all times $t$ s as $\Delta F$ and similarly to $<F_{i,\nu}>$ as $F$. We
define $\Delta F/F$ to be the ratio of the flux density in residual images
($I_{res}$) to that in the corresponding median image ($I_{median,flux}$) at
the same pixel location. Figure 3 shows the histogram of $\Delta F/F$ in
detected WINQSEs. These distributions are strikingly different from those
found by M20. In Sec. 5.2 we try to understand and reconcile these
differences.
The red line in Fig. 3 shows a lognormal fit to the data shown with blue
points. It is evident that the lognormal distribution is a good fit to the
data. However, it is worth noting that the part of the distribution where
$\Delta F/F\lesssim 0.1$ is affected by incompleteness due to a combination of
multiple factors including sensitivity, and spatial and temporal resolution.
These factors are not expected to significantly affect the data at higher
$\Delta F/F$ values and hence this part of the histogram is robust. The fact
that such lognormal distributions have been observed in earlier works when
studying weak impulsive emissions, though at EUV wavelengths and with rather
different temporal and spatial sampling (Pauluhn & Solanki, 2007; Tajfirouze &
Safari, 2012; Upendran & Tripathi, 2021), suggests that WINQSEs too might
follow a lognormal distribution. Testing this hypothesis in a more robust
manner requires an instrument with better sensitivity and resolution. We plan
to explore this with the next phase of the MWA (Phase III) which will offer a
factor of two larger effective collecting area.
Figure 3: Histograms of $\Delta F/F$ are shown using the blue circles. The red
line shows the lognormal fit to these data. The observation frequencies are
mentioned in each panel.
### 4.2 Temporal width distribution
The temporal width of a WINQSE is defined as the time duration for which the
value of a pixel in $I_{res}$ continuously exceeds the 3$\sigma$ threshold.
Figure 4 shows the temporal width distribution of the detected WINQSEs. The
powerlaw index of the distributions is close to -2, very similar to that found
in M20. The bursts are highly impulsive with a steep powerlaw like
distribution. The peak of the distribution lies at the instrumental resolution
of 0.5 s, implying that most of the WINQSEs are unresolved in time.
Figure 4: Temporal width distribution of the detected WINQSEs.
### 4.3 Wait time distribution
The wait time between WINQSEs is defined as the time interval between the
successive times when the pixel value at a given pixel in $I_{res}$ exceeds
the 3$\sigma$ threshold. The wait time distribution of detected WINQSEs is
shown in Fig. 5. The wait time distribution is very similar to that found in
M20. During the interpretation of the wait time distribution, a key point was
however missed by M20. Kivelä & Porter (2015) pointed out that the sharp
cutoff at high wait times, like what is observed in Fig. 5, can arise because
of data gaps and hence is not reliable. To take this into account, in Fig. 5,
we draw red and black dashed lines to indicate the 10% and 20% error
probabilities respectively. This implies that the data is sufficient to say
that at low wait times the distribution follows a powerlaw, but the nature of
the distribution at higher wait times cannot be constrained with these data.
The powerlaw nature of wait time distribution is observed for solar flares and
has been explained using a non-stationary Poisson model for solar flares
(Aschwanden & McTiernan, 2010). It is possible that similar behaviour is also
present for WINQSEs. M20 mention this aspect and alluded to the possibility
that the observed wait time distribution might be severely affected by a lack
of data. Based on the analysis here we conclude that this is indeed the case
for the high wait time regime.
Figure 5: Wait time distribution of the detected WINQSEs. The red and black
lines show the wait time till which the uncertainties due to incomplete
sampling is less than 10% and 20% respectively.
### 4.4 Spatial distribution
A key result of M20 was that the WINQSEs are ubiquitous in nature. Figure 6
shows the fractional occupancy of the detected WINQSEs over the radio Sun. The
fractional occupancy is defined as the fraction of observation time for which
WINQSEs were detected in a resolution element. There is no patch on the Sun
where WINQSEs are absent.
Figure 6: Fractional occupancy distribution of the detected WINQSEs. Contours
of the median map at respective frequencies has been overlaid. The contour
levels correspond to 0.2, 0.4, 0.6, 0.8, 0.9, 0.95 times the peak in the
median map.
## 5 Discussion
### 5.1 Improving the data analysis technique over M20
We have taken this opportunity to improve the analysis methodology to account
for two low-level effects which were realized during the course of this work
and are listed below:
1. 1.
In M20 each five minute datachunk was processed independently, this can lead
to small differences in absolute fluxscale between data chunks. At a few
percent level, these differences are smaller than the uncertainty in the
absolute flux density and usually too small to be of any consequence. In the
present context, however, it is important to correct for them as they are
comparable in strength to flux densities of WINQSEs. As described in Sec. 3,
an additional step of forcing the median disc integrated solar flux density to
a constant value has been introduced in the calibration process to ensure a
constant fluxscale throughout the observing duration.
2. 2.
During the calibration process, a varying number of antennas can get flagged
for different time slices even at the same frequency. This can lead to small
differences between the psfs for different images. To ensure that this does
not affect the flux density estimated, all of the images were smoothed to the
same coarser resolution for this work, as described in Sec. 3.
### 5.2 Comparison with M20
The results from this improved methodology confirm the key properties of
WINQSEs reported by M20 – weak flux densities, impulsive nature and ubiquitous
presence on the quiet Sun during this period of very low solar activity.
However, there are some significant quantitative differences which need some
discussion.
#### 5.2.1 $\Delta F/F$ distribution of WINQSEs
This work highlights that the distribution of $\Delta F/F$, can change
significantly between observations. At the low $\Delta F/F$ end, the
distributions found by M20 tend to flatten out rather than drop, as seen in
Fig. 3. At the high $\Delta F/F$ end they show clear power law tails spanning
more than an order of magnitude which are clearly missing in this work.
For a given frequency, M20 plotted all $\Delta F/F$ values for all times and
all the psf sized regions when presenting these histograms, irrespective of
the signal-to-noise of the detection. Hence the low end of the $\Delta F/F$
distribution was dominated primarily by noise fluctuations. In contrast, we
use an image noise based threshold on $\Delta F$ to limit ourselves only to
reliable detections of WINQSEs. This filters out the low signal-to-noise
detections and explains the discrepancy between these works at the low end of
the $\Delta F/F$ histograms. To substantiate this, we plot in Fig. 7 the data
from an example frequency from M20 with the same thresholding technique as
used here. It is evident that the low $\Delta F/F$ end of the histogram is now
similar to the ones shown in Fig. 3, though it retains the powerlaw behaviour
shown in M20 at the large $\Delta F/F$ end. The powerlaw index estimated is
consistent with that reported in M20. Similar results were obtained at other
frequencies as well.
Though M20 had attempted to avoid contamination from the only active region
present on the Sun during their observations, we find that despite their
efforts the high $\Delta F/F$ tail observed in the histograms obtained in M20
arises due to this lone active region. To substantiate this, the left panel of
Fig. 8 shows the location of all WINQSEs detected in M20 at 132 MHz, overlaid
on an example radio image at the same frequency. The regions are colored blue
(red) if they host at least one WINQSE with $\Delta F/F>0$ ($>1$). It is
evident that while WINQSEs are distributed all over the Sun, those with
$\Delta F/F>1$ are clustered around the active region and not a single one of
them is present far away from it.
Active regions have very complex magnetic fields and are sites of continuous
magnetic reconnections (e.g. Mondal & Oberoi, 2021). The complexity of the
magnetic field is expected to decrease smoothly as the distance from the
active region increases. Consequently, the strength and the number of magnetic
reconnections taking place would also decrease in a similar smooth manner. We
hypothesise this to be the reason for regions showing the presence of strong
WINQSEs in M20 to lie close to the active region. Additionally, such a region
is expected not only to produce strong WINQSEs, but also to give rise to a
larger number of weaker WINQSEs, as compared to regions far away from the
active region. Hence we hypothesise this to be the reason behind the presence
(absence) of powerlaw behaviour in the $\Delta F/F$ distribution of the
detected WINQSEs in M20 (this work). To corroborate this hypothesis, the right
panel of Fig. 8 presents the $\Delta F/F$ distribution after removing the
contribution of the regions shown in red in the left panel. The red line shows
the straight line fitted to the same range of $\Delta F/F$ used for the fit in
Fig. 7. It is evident that the powerlaw is no longer a good representation of
the distribution. It is also observed that there is hump-like feature around
$\Delta F/F\approx 0.5$. The signature of this hump is also observed at a
similar location in Fig. 7 in this work and Fig 4 of M20. Understanding the
exact details of this feature is beyond the scope of the present work. This
observation of the transition from a powerlaw behaviour towards a more
lognormal-like nature suggests that as the solar activity increases, more
active regions are observed, leading to a larger number of stronger WINQSEs.
Hence with increasing solar activity a transition from a lognormal nature to a
powerlaw behaviour in the $\Delta F/F$ distribution can be expected. This
prediction should be tested in future with data spanning a large part of a
solar cycle.
Note that the peaks in the $\Delta F/F$ histograms lie at lower values in the
current data as compared to M20. We attribute this to a combination of
exceptionally quiet solar conditions and lower image noise characteristics
delivered by the improved calibration employed here.
Figure 7: The data used here correspond to the 132 MHz data presented in M20.
The $\Delta F/F$ histogram using data from M20 following the same procedure as
that followed here. The red line shows the powerlaw fit to the data beyond the
peak. The powerlaw index is mentioned in the figure.
Figure 8: Left panel: The blue filled circles show the locations of all the
psf sized regions used by M20 with WINQSEs detections. The regions which were
determined by M20 to be affected by active region emission and excluded from
their analysis have been excluded here as well. The red filled circles show
the locations where the $\Delta F/F$ of the detected WINQSEs exceeds 1. The
contours correspond to 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64 times the peak
of an example image at 132 MHz. Histogram made using the same data as
presented in Fig. 7 after removing contribution of regions which show at least
one $\Delta F/F>1$.
#### 5.2.2 Fractional occupancy of WINQSEs
The median fractional occupancies obtained from the data presented here are
0.017, 0.021, 0.024 and 0.045 at 120, 128, 136 and 146 MHz respectively. In
comparison, at similar frequencies of 120 and 132 MHz, M20 had found higher
median fractional occupancies of 0.07 and 0.06 respectively. There are at
least four known factors which contribute to a higher fractional occupancy for
M20. The first is the improved analysis methodology used here which rigorously
rejects low signal-to-noise detections. This was not the case with M20 and
hence some spurious noise peaks can get counted as WINQSEs inflating the true
fractional occupancy number. Secondly, M20 computed a single median for the
entire duration, in contrast to the piece-wise median estimated here. The
appendix shows the results obtained using a single median and it is evident
that the results thus obtained are closer to that obtained by M20 (3-5%). The
third important reason is the more active state of the Sun during M20, some
evidence and arguments for contributions from which have already been
presented.
### 5.3 Comparison with S22
As mentioned earlier, S22 have recently reported the detection of WINQSEs
using a very different technique, namely the visibility subtraction technique.
For completeness, we briefly describe the technique first and then present a
comparison of their technique and the results with the present work.
Similar to the technique used here, S22 also assumes that there is a slowly
varying background solar radio emission on which are the superposed the
impulsive WINQSEs. While we separate the slowly varying and impulsive parts of
emission in the image domain by estimating and subtracting a median image, S22
does so in the visibility domain. For each baseline, they compute the median
complex visibility over a 15 second running window and assume that the median
represents the visibility from the slowly varying Sun for that baseline for
the instant centered on that time interval. In principle, subtraction in the
visibility domain and subtraction in the image domain should be identical due
to the Fourier relationship between them. However, in practice, there can be
differences due to non-linear nature of deconvolution which is used to produce
the radio images. The subtraction in visibility domain is expected to perform
better if there are significant deconvolution errors in the images. A
disadvantage of the visibility subtraction technique, however, is that the
Fourier component sampled by a given baseline changes as a function of time.
For a source with the complex morphology like the Sun, the emission morphology
can itself give rise to variations in the observed visbilities on a given
baseline even if the source were time invariant. To limit the magnitudes of
these intrinsic changes in measured visibilities, S22 used a rather short
duration of 15 s over which to compute the median, while our technique was
able to use the full 4 min duration of the scan to compute the median image.
An implication is that S22 would miss the longer duration events. While this
will bias any studies of the temporal width distribution done using this
technique, given the very steep slope of this distribution, however, the
number of events missed will be a very small fraction of the total number.
The solar images used in this work were of high quality and no deconvolution
errors or artefacts were evident. In addition to the excellent uv-coverage,
precise calibration performed using AIRCARS (Mondal et al., 2019) and the
absence of any strong source on the solar disc, all have contributed to
ensuring a high fidelity images. This has allowed us to use the image
subtraction technique with a high confidence and enabled the detection of
these WINQSEs.
### 5.4 Need for a better WINQSE detection technique
A shortcoming of both this work and M20 stems from the way WINQSEs are
detected. We have searched for WINQSEs in discrete psf-sized regions and
classified the transients as true detections if their flux density exceeds
some threshold. If a WINQSE is larger than the psf, then it will contribute
flux density to multiple regions and hence can lead to over-counting of the
number of WINQSEs. On the other hand, since the psf is circular/elliptical in
nature, the regions used here and in M20 do not form a closed packed
structure. Hence some part of the Sun is not covered by these regions and
could lead to some WINQSEs being missed. Additionally, when the spatial
location of a WINQSE does not coincide with the centre of some psf sized
region, the observed increase in flux density of the region due to the WINQSE
is lower than its true flux density. This will lead to instances when the
observed flux density will fall short of the detection threshold due to this
artificial reduction. Due to these effects, it is non-trivial to estimate the
true number of WINQSEs present in the data. To address this shortcoming, we
have recently developed a machine learning based technique to reliably
identify and characterize WINQSEs in solar radio images. This work will be
presented in an upcoming paper (Bawaji et al., 2022).
An additional shortcoming of both these works stems from the way the median is
computed. M20 had computed a single median map for the entire time duration
studied, whereas here we have computed a median map for every five minutes of
data. Both of these methods have their own strengths and weaknesses. For
example, while calculating an independent median for every five minutes of
data, can reduce some systematic calibration artefacts, if present, it can
also produce spurious WINQSEs at the five minute data boundaries if for that
data chunk and pixel, the lightcurve has a smoothly decreasing or increasing
trend. We have repeated the entire analysis using a single median map and, as
expected, we find that the number of WINQSEs found in this analysis is
significantly larger than when done using independent medians for each data
chunk. We are not aware of any systematic effects in the data which can change
the flux densities in different regions of the Sun while maintaining the
integrated flux at all times. Choosing to err on the side of caution, we have
presented the results corresponding to the medians computed every five minutes
in the preceding sections. The statistical properties of WINQSEs obtained from
the single median analysis are very similar and are presented in the Appendix.
While the results are consistent, we realize the need to improve WINQSEs
detection strategy beyond what has been employed here. Work on this front is
already underway.
### 5.5 A dynamic middle corona
Radio observations at the small range of frequencies used here probe coronal
height of $\sim 1.4\ (2.0)\ R_{\odot}$, under the standard assumption that the
emission arises due to plasma emission mechanism at the fundamental (harmonic)
and using well established coronal electron density models. The observations
in M20, Sharma et al. (2022) and this work imply that energetically weak
reconnections are ever-present even in the quiet Sun regions at these coronal
heights. These reconnections accelerate electrons which then emit in the radio
band. Due to instrumental limitations at high energies (EUV and X-rays), these
coronal heights are typically accessible only at metrewave radio frequencies.
The bulk of the radio studies in the past have focused on strong events, again
primarily due to the instrumental limitations in radio bands. So there have
been rather few investigations of weak or level emissions from the middle
coronal heights. As more capable instruments are becoming available, the
evidence for middle coronal heights to also be a dynamic place is slowly
building. Recently, gradual bulk flows originating in the middle corona have
been observed using the Solar Ultraviolet Imager (SUVI) onboard the
Geostationary Operational Environmental Satellite (Seaton et al., 2021). The
authors suggest that this can happen due to magnetic reconnection processes
higher in the corona. While a direct comparison between these EUV emissions
and WINQSEs is beyond the scope of this work, these disparate observations
paint a consistent picture of a highly dynamic middle solar corona even during
extremely quiet times.
## 6 Conclusion
M20 was the first work to present ubiquitous detection of WINQSEs. Due to the
significant potential implications of this work on coronal heating, and
presence of nonthermal particles in the solar corona in general, it is very
important to verify their presence under a variety of solar conditions, take a
closer look at the analysis and characterise them better. With this
motivation, this work forms the next step in this progression. We have
examined data from extremely quiet solar conditions, improved on the
methodology used in M20 and presented robust detection of large numbers of
WINQSEs. This work corroborates the key findings of M20 – their ubiquity and
impulsive nature. While significant differences between the distribution of
$\Delta F/F$ presented here and M20 are observed, we show that they can be
justifiably attributed to differences in methodology and variations in the
level of solar activity. Parallel to this effort, independent techniques have
been developed to identify and characterise WINQSEs (Bawaji et al., 2022;
Sharma et al., 2022) and they all paint a consistent picture. Together these
works place the presence of WINQSEs on a firm pedestal. We hope that this will
provide the motivation to the wider community to understand them in greater
detail.
This scientific work makes use of the Murchison Radio-astronomy Observatory,
operated by CSIRO. We acknowledge the Wajarri Yamatji people as the
traditional owners of the Observatory site. Support for the operation of the
MWA is provided by the Australian Government (NCRIS), under a contract to
Curtin University administered by Astronomy Australia Limited. We acknowledge
the Pawsey Supercomputing Centre which is supported by the Western Australian
and Australian Governments. We thank the anonymous referee for their comments,
which have improved this paper significantly. SM acknowledges partial support
by USA NSF grant AGS-1654382 to the New Jersey Institute of Technology. AB and
DO acknowledge support of the Department of Atomic Energy, Government of
India, under the project no. 12-R&D-TFR-5.02-0700. As discussed in Sec. 5.4,
the number of WINQSEs detected depends on the choice of the median map used.
Results presented in the paper are based on the use of an independent median
computed for each five minute dataset. For completeness and comparison, here
we present results (analogs of Figs. 3–6) obtained using a single median
computed over the entire observing duration as was done by Mondal et al.
(2020). As expected the number of WINQSEs detected here are larger than using
a median computed every five minutes.
Figure 9: Histogram of $\Delta F/F$ using a median computed over the entire
duration of the data is shown using the blue circles. The red line shows the
lognormal fit to these data. The observation frequencies are written in each
panel. Figure 10: Temporal width distribution of the detected WINQSEs using a
median computed over the entire duration of the data. Figure 11: Wait time
distribution of the detected WINQSEs using a median computed over the entire
duration of the data. The red and black lines show the wait time till which
the uncertainties due to incomplete sampling is less than 10% and 20%
respectively. Figure 12: Fractional occupancy distribution of the detected
WINQSEs using a median computed over the entire duration of the data.
## References
* Aschwanden & McTiernan (2010) Aschwanden, M. J., & McTiernan, J. M. 2010, ApJ, 717, 683, doi: 10.1088/0004-637X/717/2/683
* Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
* Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
* Bain & Fletcher (2009) Bain, H. M., & Fletcher, L. 2009, A&A, 508, 1443, doi: 10.1051/0004-6361/200911876
* Bawaji et al. (2022) Bawaji, S., Alam, U., Mondal, S., & Oberoi, D. 2022, in Astronomical Society of the Pacific Conference Series, Vol. 532, Astronomical Society of the Pacific Conference Series, ed. J. E. Ruiz, F. Pierfedereci, & P. Teuben, 211. https://arxiv.org/abs/2103.05371
* Che (2018) Che, H. 2018, in Journal of Physics Conference Series, Vol. 1100, Journal of Physics Conference Series, 012005, doi: 10.1088/1742-6596/1100/1/012005
* Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
* Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
* Kansabanik et al. (2022a) Kansabanik, D., Mondal, S., Oberoi, D., Biswas, A., & Bhunia, S. 2022a, ApJ, 927, 17, doi: 10.3847/1538-4357/ac4bba
* Kansabanik et al. (2022b) Kansabanik, D., Oberoi, D., & Mondal, S. 2022b, ApJ, 932, 110, doi: 10.3847/1538-4357/ac6758
* Kivelä & Porter (2015) Kivelä, M., & Porter, M. A. 2015, Phys. Rev. E, 92, 052813, doi: 10.1103/PhysRevE.92.052813
* Kontar et al. (2017) Kontar, E. P., Yu, S., Kuznetsov, A. A., et al. 2017, Nature Communications, 8, 1515, doi: 10.1038/s41467-017-01307-8
* Krucker et al. (2011) Krucker, S., Kontar, E. P., Christe, S., Glesener, L., & Lin, R. P. 2011, ApJ, 742, 82, doi: 10.1088/0004-637X/742/2/82
* Lonsdale et al. (2009) Lonsdale, C. J., Cappallo, R. J., Morales, M. F., et al. 2009, Proceedings of the IEEE, 97, 1497, doi: 10.1109/JPROC.2009.2017564
* McLean & Labrum (1985) McLean, D. J., & Labrum, N. R. 1985, Solar radiophysics : studies of emission from the sun at metre wavelengths
* McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & D. J. Bell, 127
* Melrose (1980) Melrose, D. B. 1980, Space Sci. Rev., 26, 3, doi: 10.1007/BF00212597
* Mohan (2021) Mohan, A. 2021, A&A, 655, A77, doi: 10.1051/0004-6361/202142029
* Mohan et al. (2019) Mohan, A., Mondal, S., Oberoi, D., & Lonsdale, C. J. 2019, ApJ, 875, 98, doi: 10.3847/1538-4357/ab0ae5
* Mondal (2021) Mondal, S. 2021, Sol. Phys., 296, 131, doi: 10.1007/s11207-021-01877-3
* Mondal et al. (2019) Mondal, S., Mohan, A., Oberoi, D., et al. 2019, The Astrophysical Journal, 875, 97, doi: 10.3847/1538-4357/ab0a01
* Mondal & Oberoi (2021) Mondal, S., & Oberoi, D. 2021, ApJ, 920, 11, doi: 10.3847/1538-4357/ac1076
* Mondal et al. (2020) Mondal, S., Oberoi, D., & Mohan, A. 2020, The Astrophysical Journal, 895, L39, doi: 10.3847/2041-8213/ab8817
* Parker (1988) Parker, E. N. 1988, ApJ, 330, 474, doi: 10.1086/166485
* Pauluhn & Solanki (2007) Pauluhn, A., & Solanki, S. K. 2007, Astronomy and Astrophysics, 462, 311, doi: 10.1051/0004-6361:20065152
* Pick & Vilmer (2008) Pick, M., & Vilmer, N. 2008, A&A Rev., 16, 1, doi: 10.1007/s00159-008-0013-x
* Reid (2020) Reid, H. A. S. 2020, Frontiers in Astronomy and Space Sciences, 7, 56, doi: 10.3389/fspas.2020.00056
* Reid & Kontar (2017) Reid, H. A. S., & Kontar, E. P. 2017, A&A, 606, A141, doi: 10.1051/0004-6361/201730701
* Seaton et al. (2021) Seaton, D. B., Hughes, J. M., Tadikonda, S. K., et al. 2021, Nature Astronomy, 5, 1029, doi: 10.1038/s41550-021-01427-8
* Sharma et al. (2022) Sharma, R., Oberoi, D., Battaglia, M., & Krucker, S. 2022, ApJ, 937, 99, doi: 10.3847/1538-4357/ac87fc
* Sturrock (1964) Sturrock, P. A. 1964, Type III Solar Radio Bursts, Vol. 50, 357
* Tajfirouze & Safari (2012) Tajfirouze, E., & Safari, H. 2012, ApJ, 744, 113, doi: 10.1088/0004-637X/744/2/113
* Takakura (1967) Takakura, T. 1967, Sol. Phys., 1, 304, doi: 10.1007/BF00151359
* Tingay et al. (2013) Tingay, S. J., Goeke, R., Bowman, J. D., et al. 2013, Publications of the Astronomical Society of Australia, 30, doi: 10.1017/pasa.2012.007
* Upendran & Tripathi (2021) Upendran, V., & Tripathi, D. 2021, ApJ, 916, 59, doi: 10.3847/1538-4357/abf65a
* Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
* Wayth et al. (2018) Wayth, R. B., Tingay, S. J., Trott, C. M., et al. 2018, PASA, 35, e033, doi: 10.1017/pasa.2018.37
* Wild & Smerd (1972) Wild, J. P., & Smerd, S. F. 1972, ARA&A, 10, 159, doi: 10.1146/annurev.aa.10.090172.001111
* Wild et al. (1963) Wild, J. P., Smerd, S. F., & Weiss, A. A. 1963, ARA&A, 1, 291, doi: 10.1146/annurev.aa.01.090163.001451
|
[a,b]Kai Yi
# Recent CMS results on exotic resonances
Jingqing Zhang and on behalf of the CMS Collaboration
###### Abstract
Many exotic resonances have been recently observed at the LHC and other
experiments. In this report, CMS studies of exotic multiquark states are
reported using the data collected in pp collisions at $\sqrt{s}$ = 13 TeV.
## 1 Selected CMS contributions to heavy exotic states
Quantum chromodynamics (QCD) is an important part of the standard model (SM)
in particle physics and has gotten much support from experimental results as
has the rest of the SM. In the QCD framework, hadrons beyond the quark
configurations of $q\bar{q}$ and $qqq$ ($\bar{q}\bar{q}\bar{q}$) are ‘exotic’
and are allowed in the QCD theory. Experimental studies on the exotic hadrons
will help deepen our understanding of QCD.
The CMS experiment [1] at the LHC has performed many important studies in
hadron spectroscopy and the exotic hadron sector [2, 3, 4, 5, 6, 7, 8, 9],
which include the measurement of the $X(3872)$ production cross section [2],
the confirmation of the $Y(4140)$ in $B^{\pm}\rightarrow J/\psi\phi K^{\pm}$
decays [4], and observation of the $B^{0}_{s}\rightarrow X(3872)\phi$ decay
[9] —all in proton-proton collisions.
Here we present recent results on exotic resonances from the CMS experiment:
evidence for the $X(3872)$ in heavy-ion collisions [10]; observation of the
$B^{0}_{s}\rightarrow\psi(2S)K^{0}_{S}$ and the
$B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ decays [11], and observation
of new structures in the $J/\psi J/\psi$ mass spectrum [12].
## 2 Evidence for $X(3872)$ in PbPb collisions
The $X(3872)$ was first observed by the Belle Collaboration [13]. Although its
quantum numbers have been determined to be $J^{PC}=1^{++}$ by the LHCb
collaboration [14], its nature is still not fully understood. The production
and survival of the $X(3872)$ in relativistic heavy ion collisions is expected
to depend on the $X(3872)$’s internal structure [15, 16]. Therefore, study of
the $X(3872)$ production in relativistic heavy-ion collisions provides new
opportunities to probe the nature of the $X(3872)$.
The CMS Collaboration performed a study of $X(3872)$ production in Pb-Pb
collisions at $\sqrt{S_{NN}}=5.02$ TeV using 1.7 nb-1 sample collected in 2018
[10]. The candidates for the $X(3872)$ and $\psi(2S)$ are reconstructed via
their decays into $J/\psi\pi^{+}\pi^{-}$, where the $J/\psi$ decays into
$\mu^{+}\mu^{-}$. Figure 1 shows the observed $m_{\mu\mu\pi\pi}$ distribution
for the $X(3872)$ and $\psi(2S)$ candidates, where the upper plot shows the
inclusive sample and the bottom one shows the b-enriched (nonprompt dominated,
i.e. transverse decay length $l_{xy}>0.1$ mm) sample. The significance of the
inclusive $X(3872)$ signal is 4.2 standard deviations. The prompt $X(3872)$ to
$\psi(2S)$ yield ratio is found to be $1.08\pm 0.49(stat)\pm 0.52(syst)$,
whereas the typical value is around 0.1 in pp collisions.
Figure 1: Invariant mass distribution of $m_{\mu\mu\pi\pi}$ in Pb-Pb
collisions, for the inclusive (upper) and $b$-enriched (bottom) samples [10].
The vertical lines on points represent statistical uncertainties in the data.
The results of the unbinned maximum-likelihood fits for the signal +
background, and background alone, are also shown by the solid and dashed
lines, respectively. The pull distribution is represented by the shaded bars.
The $X(3872)$ peak mass resolution, $\sigma_{X(3872)}$, calculated at the
half-maximum of the signal peak, is also listed for reference.
## 3 Observation of the $B^{0}_{s}\rightarrow\psi(2S)K^{0}_{S}$ and the
$B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ decays
Multibody decays of the B mesons are well suited to the search for, and study
of, exotic resonances. For example, the discovery of $X(3872)$ was in
$B\rightarrow KJ/\psi\pi\pi$ decays [13], and that of the first charged
tetraquark candidate, $Z(4430)^{+}$, was in $B\rightarrow\psi(2S)K\pi^{\pm}$
[17].
The CMS experiment performed the first measurement of the
$B^{0}_{s}\rightarrow\psi(2S)K^{0}_{S}$ and
$B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ decays, using a data sample
of proton-proton collisions at $\sqrt{s}=13$ TeV, and an integrated luminosity
of 103 fb-1, collected in 2017 and 2018 [11]. The $\psi(2S)$ and $K^{0}_{S}$
mesons are reconstructed using their decays into $\mu^{+}\mu^{-}$ and
$\pi^{+}\pi^{-}$, respectively. The observed invariant mass distribution of
$\psi(2S)K^{0}_{S}$ (left) and $\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ (right) are
shown in Fig. 2. Using the $B^{0}\rightarrow\psi(2S)K^{0}_{S}$ as a reference
channel, the relative branching fractions of
$B^{0}_{s}\rightarrow\psi(2S)K^{0}_{S}$ and
$B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ decays are measured to be
$\mathcal{B}(B^{0}_{s}\rightarrow\psi(2S)K^{0}_{S})/\mathcal{B}(B^{0}\rightarrow\psi(2S)K^{0}_{S})=(3.33\pm
0.69(stat)\pm 0.11(syst)\pm 0.34(f_{s}/f_{d}))\times 10^{-2}$, and
$\mathcal{B}(B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-})/\mathcal{B}(B^{0}\rightarrow\psi(2S)K^{0}_{S})=0.480\pm
0.013(stat)\pm 0.032(syst)$, where the last uncertainty in the first ratio
corresponds to the uncertainty in the ratio of the production cross sections
of $B^{0}_{s}$ and $B^{0}$ mesons. With the currently available, statistics-
limited data, the 2- and 3- body invariant mass distributions of the
$B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ decay products do not show
significant exotic narrow structures in addition to the known light meson
resonances.
Figure 2: Measured mass distribution of $\psi(2S)K^{0}_{S}$ (left) and
$\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ (right) candidates [11].
## 4 Observation of new structures in the $J/\psi J/\psi$ mass spectrum in pp
collisions
The $X(3872)$ and many other exotic candidates contain two heavy quarks
($c\bar{c}$). An analogue to heavy quarkonia would be fully heavy tetraquarks,
which have been explored in theoretical models and are expected to be
experimentally observable. The recent observation of the $X(6900)$ decaying
into $J/\psi J/\psi$ has been reported by the LHCb Collaboration [18].
The CMS experiment performed a study of the low-mass region of the $J/\psi
J/\psi$ mass spectrum in pp collisions, using a data sample corresponding to
an integrated luminosity of 135 fb-1 at a center-of-mass energy of 13 TeV
[12]. The two $J/\psi$ candidates are reconstructed using their
$\mu^{+}\mu^{-}$ mode, and the final $J/\psi J/\psi$ mass distribution is
shown in Fig. 3, where three signal Breit-Wigner structures and a background
component are used to fit the distribution. The statistical significance of
the three structures are $6.5\sigma$, $9.4\sigma$, and $4.1\sigma$ for
$X(6600)$, $X(6900)$ and $X(7300)$, respectively. The measured masses and
widths of three structures are summarized in Table 1.
Figure 3: The CMS $J/\psi J/\psi$ mass spectrum with a fit consisting of three signal BW functions and a background model [12]. The left plot shows the fit over the full mass range, and on the right is the same fit expanded by only displaying masses below 9 GeV. | BW1 | BW2 | BW3
---|---|---|---
$m$ | $6552\pm 10\pm 12$ | $6927\pm 9\pm 5$ | $7287\pm 19\pm 5$
$\Gamma$ | $124\pm 29\pm 34$ | $122\pm 22\pm 19$ | $95\pm 46\pm 20$
$N$ | $474\pm 113$ | $492\pm 75$ | $156\pm 56$
Table 1: Summary of the fit results of the CMS $m(J/\psi J/\psi)$
distribution: the mass $m$ and natural width $\Gamma$, in MeV, and the signal
yields $N$ are given for three signal structures [12]. The first uncertainties
are statistical and the second systematic.
Our $X(6900)$ parameters are in a good agreement with LHCb’s non-interference
result, while the $X(6600)$ and $X(7300)$ are new structures. In order to
remove potential model dependencies in a comparison of the $X(6900)$ results,
we also apply the principal two LHCb fit models to the CMS data, but using
CMS-specific background shapes. Figure 4 shows the application of LHCb’s Model
I (left, non-interference) and Model II (right, non-resonant single parton
scattering (NRSPS) interfering with a Breit-Wigner structure – the $X(6700)$
in our application). LHCb’s Model I consists of the $X(6900)$ signal, NRSPS,
non-resonant double parton scattering (NRDPS) and two more BWs – around 6300
(BW0) and 6500 MeV (BW1) – to account for the threshold enhancement. LHCb’s
Model II consists of the $X(6900)$ signal, NRDPS, and the interference
contribution of a Breit-Wigner structure $X(6700)$ and NRSPS.
In both models, the $X(6900)$ parameters are in a good agreement with LHCb’s
measurements, while our $X(6700)$ in Model II has a much larger amplitude and
width compared to the LHCb’s interfering Breit-Wigner, and none of the LHCb
models provide a satisfactory description of our data.
Figure 4: The CMS $J/\psi J/\psi$ mass spectrum [12]. The data are fit using
LHCb models: Model I (non-interference) on the left and Model II
(interference) on the right.
## 5 Summary
In summary, recent CMS results are presented, including the first evidence for
the $X(3872)$ production in heavy ion collisions, the first observation of the
$B^{0}\rightarrow\psi(2S)K^{0}_{S}\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow\psi(2S)K^{0}_{S}$, and the observation of new structures
in $J/\psi J/\psi$ mass spectrum in pp collisions.
## References
* [1] CMS collaboration, S. Chatrchyan et al., _The CMS Experiment at the CERN LHC_ , _JINST_ 3 (2008) S08004.
* [2] CMS collaboration, S. Chatrchyan et al., _Measurement of the $X$(3872) Production Cross Section Via Decays to $J/\psi\pi^{+}\pi^{-}$ in $pp$ collisions at $\sqrt{s}$ = 7 TeV_, _JHEP_ 04 (2013) 154 [1302.3968].
* [3] CMS collaboration, S. Chatrchyan et al., _Search for a New Bottomonium State Decaying to $\Upsilon(1S)\pi^{+}\pi^{-}$ in pp Collisions at $\sqrt{s}$ = 8 TeV_, _Phys. Lett. B_ 727 (2013) 57 [1309.0250].
* [4] CMS collaboration, S. Chatrchyan et al., _Observation of a Peaking Structure in the $J/\psi\phi$ Mass Spectrum from $B^{\pm}\to J/\psi\phi K^{\pm}$ Decays_, _Phys. Lett. B_ 734 (2014) 261 [1309.6920].
* [5] CMS collaboration, A. M. Sirunyan et al., _Search for the X(5568) state decaying into $\mathrm{B}^{0}_{\mathrm{s}}\pi^{\pm}$ in proton-proton collisions at $\sqrt{s}=$ 8 TeV_, _Phys. Rev. Lett._ 120 (2018) 202005 [1712.06144].
* [6] CMS collaboration, A. M. Sirunyan et al., _Study of the ${\mathrm{B}}^{+}\to\mathrm{J}/\psi\overline{\Lambda}\mathrm{p}$ decay in proton-proton collisions at $\sqrt{s}$ = 8 TeV_, _JHEP_ 12 (2019) 100 [1907.05461].
* [7] CMS collaboration, A. M. Sirunyan et al., _Observation of the $\Lambda_{\mathrm{b}}^{0}\to$ J/$\psi\Lambda\phi$ decay in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Phys. Lett. B_ 802 (2020) 135203 [1911.03789].
* [8] CMS collaboration, A. M. Sirunyan et al., _Measurement of the $\Upsilon$(1S) pair production cross section and search for resonances decaying to $\Upsilon$(1S)$\mu^{+}\mu^{-}$ in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Phys. Lett. B_ 808 (2020) 135578 [2002.06393].
* [9] CMS collaboration, A. M. Sirunyan et al., _Observation of the B ${}^{0}_{\mathrm{s}}\to$X(3872)$\phi$ decay_, _Phys. Rev. Lett._ 125 (2020) 152001 [2005.04764].
* [10] CMS collaboration, A. M. Sirunyan et al., _Evidence for X(3872) in Pb-Pb Collisions and Studies of its Prompt Production at $\sqrt{s_{NN}}$=5.02 TeV_, _Phys. Rev. Lett._ 128 (2022) 032001 [2102.13048].
* [11] CMS collaboration, A. Tumasyan et al., _Observation of B 0$\to$$\psi$(2S)K${}^{0}_{\mathrm{S}}\pi^{+}\pi^{-}$ and B${}^{0}_{\mathrm{S}}$$\to$$\psi$(2S)K${}^{0}_{\mathrm{S}}$ decays_, _Eur. Phys. J. C_ 82 (2022) 499 [2201.09131].
* [12] CMS collaboration, _Observation of new structures in the $\mathrm{J}/\psi\mathrm{J}/\psi$ mass spectrum in $\mathrm{p}\mathrm{p}$ collisions at $\sqrt{s}=13$ TeV_, _CMS-PAS-BPH-21-003_ (2022) .
* [13] Belle collaboration, S. K. Choi et al., _Observation of a narrow charmonium-like state in exclusive $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$ decays_, _Phys. Rev. Lett._ 91 (2003) 262001 [hep-ex/0309032].
* [14] LHCb collaboration, R. Aaij et al., _Determination of the X(3872) meson quantum numbers_ , _Phys. Rev. Lett._ 110 (2013) 222001 [1302.6269].
* [15] H. Zhang, J. Liao, E. Wang, Q. Wang and H. Xing, _Deciphering the Nature of X(3872) in Heavy Ion Collisions_ , _Phys. Rev. Lett._ 126 (2021) 012301 [2004.00024].
* [16] B. Wu, X. Du, M. Sibila and R. Rapp, _$X(3872)$ transport in heavy-ion collisions_, _Eur. Phys. J. A_ 57 (2021) 122 [2006.09945].
* [17] Belle collaboration, S. K. Choi et al., _Observation of a resonance-like structure in the $\pi^{\pm}\psi^{\prime}$ mass distribution in exclusive $B\to K\pi^{\pm}\psi^{\prime}$ decays_, _Phys. Rev. Lett._ 100 (2008) 142001 [0708.1790].
* [18] LHCb collaboration, R. Aaij et al., _Observation of structure in the $J/\psi$ -pair mass spectrum_, _Sci. Bull._ 65 (2020) 1983 [2006.16957].
|
# Subquadratic Weighted Matroid Intersection Under Rank Oracles††thanks: To
appear in the _33rd International Symposium on Algorithms and Computation
(ISAAC 2022)_.
Ta-Wei Tu Department of Computer Science and Information Engineering, National
Taiwan University, Taipei, Taiwan. Email<EMAIL_ADDRESS>
###### Abstract
Given two matroids $\mathcal{M}_{1}=(V,\mathcal{I}_{1})$ and
$\mathcal{M}_{2}=(V,\mathcal{I}_{2})$ over an $n$-element integer-weighted
ground set $V$, the weighted matroid intersection problem aims to find a
common independent set $S^{*}\in\mathcal{I}_{1}\cap\mathcal{I}_{2}$ maximizing
the weight of $S^{*}$. In this paper, we present a simple deterministic
algorithm for weighted matroid intersection using
$\widetilde{O}(nr^{3/4}\log{W})$ rank queries, where $r$ is the size of the
largest intersection of $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ and $W$ is the
maximum weight. This improves upon the best previously known
$\widetilde{O}(nr\log{W})$ algorithm given by Lee, Sidford, and Wong
[FOCS’15], and is the first subquadratic algorithm for polynomially-bounded
weights under the standard independence or rank oracle models. The main
contribution of this paper is an efficient algorithm that computes shortest-
path trees in weighted exchange graphs.
## 1 Introduction
#### Matroid Intersection.
A matroid is an abstract structure that models the notion of independence on a
given ground set $V$. In particular, a subset $S\subseteq V$ is either
_independent_ or _dependent_ , such that the family of independent sets is
well-structured (see Section 2 for a complete definition). Matroids model many
fundamental combinatorial objects, and examples of independent sets of a
matroid include acyclic subgraphs of an undirected graph and linearly
independent rows of a matrix. One of the most important optimization problems
related to matroids is _matroid intersection_ : Given two matroids, we would
like to find a set with the largest cardinality that is independent in both
matroids. Similarly, in the weighted case, each element in the ground set is
associated with an integer weight, and the weighted matroid intersection
problem is to find the maximum-weight common independent set. These problems
have been extensively studied in the past since they capture many
combinatorial optimization problems such as bipartite matching and colorful
spanning trees.
#### Oracle Model.
Since we are dealing with general matroids without additional constraints, we
have to specify a way of reading the description of the two matroids. One way
is to express them directly by reading the truth table of independence.
However, that would require an exponentially-sized input. Instead, we are
given oracle access to the matroids, which gives us information about a
queried set $S\subseteq V$. Standard oracles include the _independence oracle_
, which returns whether $S$ is independent in $O(\mathcal{T}_{\mathsf{ind}})$
time, and the _rank oracle_ , which returns the rank, i.e., the size of the
largest independent subset, of $S$ in $O(\mathcal{T}_{\mathsf{rank}})$ time.
In this paper, we focus on the stronger rank oracle model.
#### Prior Work.
Polynomial-time algorithms for both the weighted and unweighted matroid
intersection problems have long been designed and improved. For the unweighted
case, Edmonds [Edm10, Edm01, Edm79], Lawler [Law75], and also Aigner and
Dowling [AD71] gave algorithms that run in
$O(nr^{2}\cdot\mathcal{T}_{\mathsf{ind}})$ time. Here, $n$ denotes the number
of elements in $V$ and $r$ denotes the size of the largest intersection of the
two matroids. Cunningham [Cun86] obtained an
$O(nr^{3/2}\cdot\mathcal{T}_{\mathsf{ind}})$ algorithm using the “blocking-
flow” idea. Lee, Sidford, and Wong [LSW15] gave quadratic algorithms using the
cutting-plane method, running in
$\widetilde{O}(nr\cdot\mathcal{T}_{\mathsf{rank}}+n^{3})$ and
$\widetilde{O}(n^{2}\cdot\mathcal{T}_{\mathsf{ind}}+n^{3})$ times111 For
function $f(n)$, $\widetilde{O}(f(n))$ denotes
$O(f(n)\operatorname{polylog}{f(n)})$. , respectively. This also gives rise to
the “quadratic barrier” of matroid intersection: most previous algorithms
involve building exchange graphs that contain $\Theta(nr)$ edges explicitly
and therefore cannot go beyond quadratic time. Chakrabarty, Lee, Sidford,
Singla, and Wong [CLS+19] were the first to partially break the barrier. They
obtained a $(1-\epsilon)$-approximation algorithm running in
$\widetilde{O}(n^{3/2}/\epsilon^{3/2}\cdot\mathcal{T}_{\mathsf{ind}})$ time
and an $\widetilde{O}(n\sqrt{r}\cdot\mathcal{T}_{\mathsf{rank}})$ exact
algorithm. One of the major components of Chakrabarty et al.’s improvements is
to show that edges in exchange graphs can be efficiently discovered using
binary search (this was discovered independently by Nguyễn [Ngu19]). This
technique also allows them to obtain improved
$\widetilde{O}(nr\cdot\mathcal{T}_{\mathsf{ind}})$ exact algorithms. Combining
the approximation algorithm and a faster augmenting-path algorithm, Blikstad,
van den Brand, Mukhopadhyay, and Nanongkai [BvdBMN21] broke the quadratic
barrier completely by giving an
$\widetilde{O}(n^{9/5}\cdot\mathcal{T}_{\mathsf{ind}})$ exact algorithm. This
result was later optimized by Blikstad [Bli21] to
$\widetilde{O}(nr^{3/4}\cdot\mathcal{T}_{\mathsf{ind}})$ by improving the
approximation algorithm to run in
$\widetilde{O}(n\sqrt{r}/\epsilon\cdot\mathcal{T}_{\mathsf{ind}})$ time.
For the weighted case, the blocking flow idea does not seem to apply anymore.
Frank [Fra81] obtained an $O(nr^{2}\cdot\mathcal{T}_{\mathsf{ind}})$ algorithm
by characterizing the optimality of a common independent set using weight
splitting. Fujishige and Zhang [FZ95] improved the running time to
$\widetilde{O}(nr^{3/2}\log{W}\cdot\mathcal{T}_{\mathsf{ind}})$ by solving a
more general _independent assignment_ problem using a scaling framework. The
same bound was achieved by Shigeno and Iwata [SI95] and also by Gabow and Xu
[GX96]. Lee, Sidford, and Wong’s [LSW15] algorithms work for the weighted case
as well, albeit with an extra factor of $\operatorname{polylog}{W}$, in
$\widetilde{O}(n^{2}\log{W}\cdot\mathcal{T}_{\mathsf{ind}}+n^{3}\operatorname{polylog}{W})$
and
$\widetilde{O}(nr\log{W}\cdot\mathcal{T}_{\mathsf{rank}}+n^{3}\operatorname{polylog}{W})$
times. Huang, Kakimura, and Kamiyama [HKK16] obtained a generic framework that
transforms any algorithm that solves the unweighted case into one that solves
the weighted case with an extra $O(W)$ factor. Plugging in the state-of-the-
art algorithms of [CLS+19] and [Bli21], we get $\widetilde{O}(n\sqrt{r}\cdot
W\cdot\mathcal{T}_{\mathsf{rank}})$ and $\widetilde{O}(nr^{3/4}\cdot
W\cdot\mathcal{T}_{\mathsf{ind}})$ algorithms. Chekuri and Quanrud [CQ16] also
gave an $\widetilde{O}(n^{2}/\epsilon^{2}\cdot\mathcal{T}_{\mathsf{ind}})$
approximation algorithm which, according to [BvdBMN21], can be improved to
subquadratic by applying more recent techniques. A similar
$\widetilde{O}(nr^{3/2}/\epsilon\cdot\mathcal{T}_{\mathsf{ind}})$
approximation algorithm was obtained independently by Huang et al. [HKK16].
#### Our Result.
The question of whether weighted matroid intersection can be solved in
subquadratic time with polylogarithmic dependence on $W$ under either oracle
model remained open. We obtain the first subquadratic algorithm for exact
weighted matroid intersection under rank oracles. The formal statement of
Theorem 1.1 is presented as Theorem 2.4 in Section 2.
###### Theorem 1.1.
Weighted matroid intersection can be solved in
$\widetilde{O}(nr^{3/4}\log{W}\cdot\mathcal{T}_{\mathsf{rank}})$ time.
Our algorithm relies on the framework of Fujishige-Zhang [FZ95] and Shigeno-
Iwata [SI95], where they first obtain an approximate solution by adjusting
weights of some elements (similar to the “auction“ algorithms for bipartite
matching [OA92]) and then refine it by augmenting the solution iteratively.
We obtain efficient algorithms for these two phases, leading to the final
subquadratic algorithm.
## 2 Preliminaries
#### Notation.
For a set $S$, let $|S|$ denote the cardinality and $2^{S}$ the power set of
$S$. Let $S\setminus R$ consist of elements of $S$ which are not in $R$. Let
$e=(u,v,w)$ denote a weighted directed edge directing from $u$ to $v$ with
weight $w=w(e)$ and $(u,v)$ be its unweighted counterpart. Let
$\operatorname{head}(e)=v$ and $\operatorname{tail}(e)=u$. For an edge set
$E$, let $\operatorname{head}(E)=\\{\operatorname{head}(e)\mid e\in E\\}$ and
$\operatorname{tail}(E)=\\{\operatorname{tail}(e)\mid e\in E\\}$. For
functions $f,g$ mapping from a set $V$ to $\mathbb{R}$, let $f+g$, $f-g$, and
$f+c$ for $c\in\mathbb{R}$ denote functions from $V$ to $\mathbb{R}$ with
$(f+g)(x)=f(x)+g(x)$, $(f-g)(x)=f(x)-g(x)$, and $(f+c)(x)=f(x)+c$ for each
$x\in V$. We often abuse notation and use $f$ to denote the function from
$2^{V}$ to $\mathbb{R}$ with $f(S)=\sum_{x\in S}f(x)$ for each $S\subseteq V$.
#### Matroid.
Let $V$ be a finite set and $w:V\to\mathbb{Z}$ be a given weight function. For
$S\subseteq V$, let $\overline{S}=V\setminus S$. Let $n=|V|$ and $W=\max_{x\in
V}|w(x)|$. An ordered pair $\mathcal{M}=(V,\mathcal{I})$ with _ground set_ $V$
and a non-empty family $\emptyset\in\mathcal{I}\subseteq 2^{V}$ is a _matroid_
if
M1.
for each $S\in\mathcal{I}$ and $R\subseteq S$, it holds that
$R\in\mathcal{I}$, and
M2.
for each $R,S\in\mathcal{I}$ with $|R|<|S|$, there exists an $x\in S\setminus
R$ such that $R\cup\\{x\\}\in\mathcal{I}$.
Sets in $\mathcal{I}$ are _independent_ ; sets not in $\mathcal{I}$ are
_dependent_. A _basis_ is a maximal independent set. A _circuit_ is a minimal
dependent set. It is well-known from the definition of matroid that all bases
are of the same cardinality. For an independent set $S$ and $x\not\in S$,
$S\cup\\{x\\}$ contains at most one circuit $C$ and if it does, then $x\in C$
(see [Pri15, Lemma 1.3.3]). The _rank_ of $S\subseteq V$, denoted by
$\mathsf{rank}(S)$, is the size of the largest $S^{\prime}\subseteq S$ such
that $S^{\prime}\in\mathcal{I}$. The rank of $\mathcal{M}$ is the rank of $V$,
i.e., the size of the bases of $\mathcal{M}$. Given two matroids
$\mathcal{M}_{1}=(V,\mathcal{I}_{1})$ and
$\mathcal{M}_{2}=(V,\mathcal{I}_{2})$ over the same ground set, the _weighted
matroid intersection_ problem is to find an
$S^{*}\in\mathcal{I}_{1}\cap\mathcal{I}_{2}$ maximizing $w(S^{*})$. Let
$r=\max_{S\in\mathcal{I}_{1}\cap\mathcal{I}_{2}}|S|$. In this paper, the two
matroids are accessed through _rank oracles_ , one for each matroid.
Specifically, let $\mathsf{rank}_{1}(\cdot)$ and $\mathsf{rank}_{2}(\cdot)$
denote the rank functions of $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$,
respectively. We assume that given pointers to a linked list containing
elements of $S$ (see, e.g., [CLSW17]), the rank oracles compute
$\mathsf{rank}_{1}(S)$ and $\mathsf{rank}_{2}(S)$ in
$O(\mathcal{T}_{\mathsf{rank}})$ time. With the
$O(n\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ unweighted matroid
intersection algorithm of Chakrabarty et al. [CLS+19], we also assume that
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are of the same rank and share a
common basis $S^{(0)}$ of size $r$ by adjusting the given rank oracles
properly.222We can compute $r$ via the unweighted matroid intersection
algorithm and regard all sets of size greater than $r$ as dependent. By adding
$r$ zero-weight elements to $V$, we may also assume that each common
independent set $S\in\mathcal{I}_{1}\cap\mathcal{I}_{2}$ is contained in a
common basis of the same weight.333In particular, let
$Z=\\{z_{1},\ldots,z_{r}\\}$ be the set of newly added zero-weight elements.
For each $i\in\\{1,2\\}$, instead of working with $\mathcal{M}_{i}$, we now
work with $\widetilde{\mathcal{M}}_{i}=(V\cup Z,\widetilde{\mathcal{I}}_{i})$
such that for each $\widetilde{S}\subseteq V\cup Z$,
$\widetilde{S}\in\widetilde{\mathcal{I}}_{i}$ if and only if
$|\widetilde{S}|\leq r$ and $\widetilde{S}\setminus Z\in\mathcal{I}_{i}$. This
change is reflected in the new rank function
$\widetilde{\mathsf{rank}}_{i}(\widetilde{S})=\min(\mathsf{rank}_{i}(\widetilde{S}\setminus
Z)+|\widetilde{S}\cap Z|,r)$, which can be implemented via the given oracle
$\mathsf{rank}_{i}$. Therefore, it suffices to find a common basis $S^{*}$
maximizing $w(S^{*})$. Note that elements with negative weights can be safely
discarded from $V$.
#### Weight-Splitting.
For weight function $f:V\to\mathbb{R}$, a basis $S$ of matroid $\mathcal{M}$
is _$f$ -maximum_ if $f(S)\geq f(R)$ holds for each basis $R$ of
$\mathcal{M}$. Let $w^{\epsilon}=(w^{\epsilon}_{1},w^{\epsilon}_{2})$ with
$w^{\epsilon}_{i}:V\to\mathbb{R}$ being a weight function for each
$i\in\\{1,2\\}$. Let
$w^{\epsilon}(x)=w^{\epsilon}_{1}(x)+w^{\epsilon}_{2}(x)$. We say that
$w^{\epsilon}$ is an _$\epsilon$ -splitting_ (see, e.g., [SI95]) of $w$ with
$\epsilon>0$ if $w(x)\leq w^{\epsilon}(x)\leq w(x)+\epsilon$ holds for each
$x\in V$. If $w^{\epsilon}$ is an $\epsilon$-splitting of $w$ and $S_{i}$ is a
$w^{\epsilon}_{i}$-maximum basis of $\mathcal{M}_{i}$ for each
$i\in\\{1,2\\}$, then we call $(w^{\epsilon},S)$ with $S=(S_{1},S_{2})$ an
_$\epsilon$ -partial-solution_ of $w$. Note that by M1., $S_{1}\cap S_{2}$ is
a common independent set. If $S_{1}=S_{2}$, then $(w^{\epsilon},S)$ is an
_$\epsilon$ -solution_ of $w$. In this case, we may abuse notation and refer
to $S_{1}$ as simply $S$.
#### Matroid Algorithms.
The unweighted version of the following lemma was shown in [CLS+19] (it is
also mentioned in [Ngu19]), and it was extended to the weighted case
implicitly in [BvdBMN21].
###### Lemma 2.1 ([CLS+19, Lemma 13], [Ngu19], and [BvdBMN21]).
For $i\in\\{1,2\\}$, given $S\in\mathcal{I}_{i}$, $B\subseteq S$
(respectively, $B\subseteq\overline{S}$), $x\in\overline{S}$ (respectively,
$x\in S$), and weight function $f:V\to\mathbb{R}$, it takes
$O(\log{|B|}\cdot\mathcal{T}_{\mathsf{rank}})$ time to either obtain a $b\in
B$ minimizing/maximizing $f(b)$ such that
$(S\setminus\\{b\\})\cup\\{x\\}\in\mathcal{I}_{i}$ (respectively,
$(S\setminus\\{x\\})\cup\\{b\\}\in\mathcal{I}_{i}$) or report that such an
element does not exist in $B$.
The main idea of Lemma 2.1 is to perform binary search on $B$ ordered by $f$.
Throughout this paper, we will maintain such an ordered set in a balanced
binary search tree where each element holds pointers to its successor and
predecessor and each node holds pointers to the first and the last elements in
its corresponding subtree. This allows us to perform binary search on the tree
and obtain pointers to the linked list containing elements in a consecutive
range efficiently.
The following greedy algorithm for finding a maximum-weight basis is folklore.
###### Lemma 2.2 (See, e.g., [Edm71]).
It takes $O(n\log{n}+n\mathcal{T}_{\mathsf{rank}})$ time to obtain a
$f$-maximum basis $S$ of a given matroid $\mathcal{M}$ and weight function
$f:V\to\mathbb{R}$.
### 2.1 The Framework
The core of our algorithm is the following subroutine.
###### Theorem 2.3.
Given a $2\epsilon$-solution $(w^{2\epsilon},S^{\prime})$ of $w$, it takes
$O(nr^{3/4}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time to obtain an
$\epsilon$-solution $(w^{\epsilon},S)$.
With Theorem 2.3, the weighted matroid intersection algorithm follows from the
standard weight-scaling framework (see, e.g, [FZ95, SI95]). Recall that our
goal is to find a maximum-weight common basis.
###### Theorem 2.4 (Weighted Matroid Intersection).
Given two matroids $\mathcal{M}_{1}=(V,\mathcal{I}_{1})$ and
$\mathcal{M}_{2}=(V,\mathcal{I}_{2})$, it takes
$O(nr^{3/4}\log{n}\log{(rW)}\cdot\mathcal{T}_{\mathsf{rank}})$ time to obtain
an $S^{*}\in\mathcal{I}_{1}\cap\mathcal{I}_{2}$ maximizing $w(S^{*})$.
###### Proof.
Let $w^{W}=(w^{W}_{1},w^{W}_{2})$ with $w^{W}_{i}(x)=\frac{W}{2}$ for each
$x\in V$ and the initial common basis $S^{(0)}$ obtained via the unweighted
matroid intersection algorithm be a $W$-solution of $w$. Repeatedly apply
Theorem 2.3 for $O(\log{rW})$ iterations to obtain a $\frac{1}{2r}$-solution
$(w^{\frac{1}{2r}},S^{*})$. For each $S\in\mathcal{I}_{1}\cap\mathcal{I}_{2}$,
we have
$w(S)\leq w^{\frac{1}{2r}}(S)\leq w^{\frac{1}{2r}}(S^{*})\leq
w(S^{*})+r\cdot\frac{1}{2r}<w(S^{*})+1.$
Since $w(S)$ and $w(S^{*})$ are integers, $S^{*}$ is a maximum-weight common
basis. The algorithm runs in
$O(nr^{3/4}\log{n}\log{(rW)}\cdot\mathcal{T}_{\mathsf{rank}})$ time. The
theorem is proved. ∎
The rest of the paper proves Theorem 2.3.
## 3 The Algorithm
As in [FZ95] and [SI95], the algorithm of Theorem 2.3 consists of the
following two parts.
### 3.1 Weight Adjustment
The first part of the algorithm is the following subroutine which computes two
bases $S_{1}$ and $S_{2}$ with a large enough intersection. This part is
essentially the same as Shigeno and Iwata’s algorithm [SI95], except that we
replace the fundamental (co-)circuit queries in it with calls to Lemma 2.1.
###### Lemma 3.1.
Given a $2\epsilon$-solution $(w^{2\epsilon},S^{\prime})$ and a parameter
$1\leq k\leq r$, it takes $O(nk\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time
to obtain an $\epsilon$-partial-solution $(w^{\epsilon},S)$ with $|S_{1}\cap
S_{2}|\geq\left(1-\frac{O(1)}{k}\right)r$.
Since the algorithm and analysis are essentially the same as in [SI95], here
we only describe how we can obtain $S_{1}$ and $S_{2}$ in the desired time
bound. Please refer to [SI95] or Lemma A.1 in Appendix A for the proof of
$|S_{1}\cap S_{2}|\geq\left(1-\frac{O(1)}{k}\right)r$.
#### Algorithm of Lemma 3.1.
Let $w^{\epsilon}=(w^{2\epsilon}_{1},w-w^{2\epsilon}_{1}+\epsilon)$ be the
initial $\epsilon$-splitting and $S_{i}$ be the $w^{\epsilon}_{i}$-maximum
basis of $\mathcal{M}_{i}$ obtained by Lemma 2.2 in
$O(n\log{n}+n\mathcal{T}_{\mathsf{rank}})$ time for each $i\in\\{1,2\\}$. Let
$p(x)=0$ for each $x\in V$. Repeat the following _weight adjustment_ for an
arbitrary $x\in S_{1}\setminus S_{2}$ with $p(x)<k$ until such an $x$ becomes
non-existent.
* •
If $w^{\epsilon}(x)=w(x)+\epsilon$, then set $w^{\epsilon}_{1}(x)\leftarrow
w^{\epsilon}_{1}(x)-\epsilon$. Apply Lemma 2.1 to obtain a $y\in V\setminus
S_{1}$ maximizing $w^{\epsilon}_{1}(y)$ such that
$(S_{1}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{1}$. If
$w^{\epsilon}_{1}(x)<w^{\epsilon}_{1}(y)$, then set
$S_{1}\leftarrow(S_{1}\setminus\\{x\\})\cup\\{y\\}$.
* •
Otherwise, set $p(x)\leftarrow p(x)+1$ and $w^{\epsilon}_{2}(x)\leftarrow
w^{\epsilon}_{2}(x)+\epsilon$. Apply Lemma 2.1 to obtain a $y\in S_{2}$
minimizing $w^{\epsilon}_{2}(y)$ such that
$(S_{2}\setminus\\{y\\})\cup\\{x\\}\in\mathcal{I}_{2}$. If
$w^{\epsilon}_{2}(x)>w^{\epsilon}_{2}(y)$, then set
$S_{2}\leftarrow(S_{2}\setminus\\{y\\})\cup\\{x\\}$.
Since $p(x)$ is only incremented when $x\in S_{1}\setminus S_{2}$, we have
$p(x)\leq k$ for each $x\in V$ when the procedure terminates. Apparently,
$w^{\epsilon}(x)$ oscillates between $w(x)$ and $w(x)+\epsilon$, and thus the
number of weight adjustments for $x$ is bounded by $2p(x)$. We also have that
$S_{i}$ remains $w^{\epsilon}_{i}$-maximum for each $i\in\\{1,2\\}$ due to the
potential exchange of $x$ and $y$ after the adjustment. Each weight adjustment
takes $O(\mathcal{T}_{\mathsf{rank}}\log{n})$ time by Lemma 2.1, hence the
total running time is $O(nk\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$.
### 3.2 Augmentation
With $S_{1}$ and $S_{2}$ obtained from Lemma 3.1, we then run “few”
augmentations to make these two bases equal. To do so, we need the following
notion of exchange graphs, which is slightly different compared to previous
algorithms for unweighted matroid intersection (e.g., [BvdBMN21, CLS+19,
Cun86, Law75]).
#### Exchange Graph.
Let $(w^{\epsilon},S)$ be an $\epsilon$-partial-solution of $w$ with
$S_{1}\neq S_{2}$. The _exchange graph_ with respect to $(w^{\epsilon},S)$ is
a weighted directed multi-graph $G_{w^{\epsilon},S}=(V\cup\\{s,t\\},E)$ with
$s,t\not\in V$ and $E=E_{1}\cup E_{2}\cup E_{s}\cup E_{t}$, where
$\displaystyle E_{1}$
$\displaystyle=\\{(x,y,w^{\epsilon}_{1}(x)-w^{\epsilon}_{1}(y))\mid x\in
S_{1},y\not\in
S_{1},\;\text{and}\;(S_{1}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{1}\\},$
$\displaystyle E_{2}$
$\displaystyle=\\{(y,x,w^{\epsilon}_{2}(x)-w^{\epsilon}_{2}(y))\mid x\in
S_{2},y\not\in
S_{2},\;\text{and}\;(S_{2}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{2}\\},$
$\displaystyle E_{s}$ $\displaystyle=\\{(s,x,0)\mid x\in S_{1}\setminus
S_{2}\\},\;\text{and}$ $\displaystyle E_{t}$ $\displaystyle=\\{(x,t,0)\mid
x\in S_{2}\setminus S_{1}\\}.$
Since $S_{i}$ is $w^{\epsilon}_{i}$-maximum for each $i\in\\{1,2\\}$, all edge
weights are non-negative. Note that this definition of exchange graph is a
simplified version of the _auxiliary graph_ defined by Fujishige and Zhang
[FZ95] to solve the more generalized _independent assignment_
problem444Specifically, given a bipartite graph $G=(V_{1}\cup V_{2},E)$ with
$V_{1}$ and $V_{2}$ being copies of $V$ and two matroids
$\mathcal{M}_{1}=(V,\mathcal{I}_{1})$, $\mathcal{M}_{2}=(V,\mathcal{I}_{2})$
on $V$, the independent assignment problem aims to find the largest
$S_{1}\in\mathcal{I}_{1}$ and $S_{2}\in\mathcal{I}_{2}$ such that $G$ admits a
perfect matching between $S_{1}\subseteq V_{1}$ and $S_{2}\subseteq V_{2}$.
Analogously, the weighted version of the problem wants to find $S_{1}$ and
$S_{2}$ such that the weight of the maximum-weight perfect matching between
$S_{1}$ and $S_{2}$ is maximized. Clearly, the (weighted) matroid intersection
problem is a special case of the (weighted) independence assignment problem
with $E=\\{(v,v)\mid v\in V\\}$.. We have the following properties of the
exchange graph, for which we also provide simplified and more direct proofs
for self-containedness in Appendix A.
###### Lemma 3.2 ([FZ95]; See Appendix A).
$G_{w^{\epsilon},S}$ admits an $st$-path.
Let $d(x)$ be the $sx$-distance in $G_{w^{\epsilon},S}$ for each $x\in V$ (set
$d(x)$ to a large number if $x$ is unreachable from $s$; see Section 3.3 for
the exact value) and $P$ be the shortest $st$-path with the least number of
edges.
###### Lemma 3.3 ([FZ95]; See Appendix A).
$\widehat{S}_{1}=(S_{1}\setminus\operatorname{tail}(P\cap
E_{1}))\cup\operatorname{head}(P\cap E_{1})$ and
$\widehat{S}_{2}=(S_{2}\setminus\operatorname{head}(P\cap
E_{2}))\cup\operatorname{tail}(P\cap E_{2})$ are a
$\widehat{w}^{\epsilon}_{1}$-maximum and $\widehat{w}^{\epsilon}_{2}$-maximum
basis of $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, respectively, where
$\widehat{w}^{\epsilon}_{1}(x)=w^{\epsilon}_{1}(x)+d(x)$ and
$\widehat{w}^{\epsilon}_{2}(x)=w^{\epsilon}_{2}(x)-d(x)$ for each $x\in V$. In
particular, $(\widehat{w}^{\epsilon},\widehat{S})$ with
$\widehat{w}^{\epsilon}=(\widehat{w}^{\epsilon}_{1},\widehat{w}^{\epsilon}_{2})$
and $\widehat{S}=(\widehat{S}_{1},\widehat{S}_{2})$ is an $\epsilon$-partial-
solution. Moreover, we have $|\widehat{S}_{1}\cap\widehat{S}_{2}|>|S_{1}\cap
S_{2}|$.
With the above properties and Lemma 3.1, we finish our algorithm with the
following shortest-path procedure. Note that in order to make the algorithm
subquadratic, we do not construct the exchange graphs explicitly.
Nevertheless, we show that a partial construction suffices to compute the
shortest-path trees in them.
###### Lemma 3.4.
It takes $O(n\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time to obtain
$d(x)$ for each $x\in V$ and the shortest $st$-path with the least number of
edges in $G_{w^{\epsilon},S}$.
We are now ready to prove Theorem 2.3.
###### Proof of Theorem 2.3.
Apply Lemma 3.1 with $k=r^{3/4}$ to obtain an $\epsilon$-partial-solution
$(w^{\epsilon},S)$ of $w$ such that $|S_{1}\cap S_{2}|\geq r-O(r^{1/4})$ in
$O(nr^{3/4}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time. For $O(r^{1/4})$
iterations, apply Lemmas 3.4 and 3.3 to obtain
$(\widehat{w}^{\epsilon},\widehat{S})$ with $\widehat{S}_{1}$ and
$\widehat{S}_{2}$ having a larger intersection than $S_{1}$ and $S_{2}$ do,
and set $(w^{\epsilon},S)\leftarrow(\widehat{w}^{\epsilon},\widehat{S})$ until
$S_{1}=S_{2}$. This takes overall
$O(nr^{3/4}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time as well. Note that
$w^{\epsilon}(x)=w^{\epsilon}_{1}(x)+w^{\epsilon}_{2}(x)$ remains the same,
completing the proof. ∎
The remainder of this section proves Lemma 3.4. For the ease of notation, we
abbreviate $G_{w^{\epsilon},S}$ as $G$.
Intuitively, we would like to run Dijkstra’s algorithm on $G$ to build a
shortest-path tree. However, naïve implementation takes $O(nr)$ time since we
might need to relax $O(nr)$ edges. This is unlike the BFS algorithm of
Chakrabarty et al. [CLS+19] for the unweighted case, where we can immediately
mark all out-neighbors of the current vertex as “visited”, leading to a near-
linear running time. To speed things up, note that using Lemma 2.1, for a
vertex $x$, we can efficiently find the vertex which is “closest” to $x$. Let
$F$ denote the set of visited vertices whose exact distances are known. The
closest unvisited vertex to $F$ must be closest to some $x\in F$. Therefore,
in each iteration, it suffices to only relax the “shortest” edge from each
$x\in F$. This can be done efficiently by maintaining a set of “recently
visited” vertices $B$ of size roughly $\sqrt{n}$ and computing the distance
estimate from $F\setminus B$ to all unvisited vertices555In the actual
algorithm, we maintain two buffers instead of one to further improve the
running time to $\widetilde{O}(n\sqrt{r})$ from $\widetilde{O}(n\sqrt{n})$.
This makes our weighted matroid intersection algorithm $o(nr)$ as opposed to
just $o(n^{2})$.. In each iteration, we relax the shortest edge from each
$x\in B$, and now the vertex with the smallest distance estimate is closest to
$F$ and therefore we include it into $B$ (and thus $F$). When $B$ grows too
large, we clear $B$ and recompute the distance estimates from $F$ in
$\widetilde{O}(n)$ queries. This leads to a subquadratic algorithm. We now
prove the lemma formally.
###### Proof of Lemma 3.4.
The algorithm builds a shortest-path tree of $G$ using Dijkstra’s algorithm.
We maintain a distance estimate $\widehat{d}(x)$ for each $x\in
V\cup\\{s,t\\}$. Initially, $\widehat{d}(x)=0$ for each $x\in(S_{1}\setminus
S_{2})\cup\\{s\\}$ and $\widehat{d}(x)=\infty$ for other vertices. Edge set
$E_{t}$ is only for the convenience of defining an $st$-path and thus we may
ignore it here. Let $F$ be the set of _visited_ vertices whose distance
estimates are correct, i.e., $d(x)=\widehat{d}(x)$ holds for each $x\in F$.
Initially, $F=\\{s\\}$. The algorithm runs in at most $n$ iterations, and in
the $t$-th iteration, we visit a new vertex $v_{t}$ such that
$d(v_{t})=\widehat{d}(v_{t})$ and $d(v_{t})\leq d(v)$ for each $v\not\in F$.
We maintain two _buffers_ $B_{1}\subseteq F\cap S_{1}$ and $B_{2}\subseteq
F\cap\overline{S_{2}}$ containing vertices in $S_{1}$ and $\overline{S_{2}}$
that are visited “recently”. That is, after the $t$-th iteration, we have
$B_{1}=\\{v_{i},\ldots,v_{t}\\}\cap S_{1}$ or $B_{1}=\emptyset$ and
$B_{2}=\\{v_{j},\ldots,v_{t}\\}\cap\overline{S_{2}}$ or $B_{2}=\emptyset$ for
some $i,j\leq t$. Recall that $E_{1}$ and $E_{2}$ are the edges in $G$ that
correspond to exchange relations in $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$,
respectively. For each $i\in\\{1,2\\}$ and edge $e=(x,y)$, let
$w_{i}(x,y)=|w^{\epsilon}_{i}(x)-w^{\epsilon}_{i}(y)|$ be the edge weight of
$e$ in $E_{i}$ if $e\in E_{i}$ and $w_{i}(x,y)=\infty$ otherwise. Let
$E({B_{i}})=\\{(x,y)\in E_{i}\mid x\in B_{i}\;\text{and}\;y\not\in F\\}$ be
edges in $E_{i}$ directing from $B_{i}$ to $V\setminus F$ and
$E(B)=E({B_{1}})\cup E({B_{2}})$. Let $E({F})=\\{(x,y)\in E_{1}\cup E_{2}\mid
x\in F\;\text{and}\;y\not\in F\\}$. For $v\in V\setminus F$ and edge set
$E^{\prime}$ such that $\operatorname{tail}(E^{\prime})\subseteq F$, let
$\widetilde{d}(v,E^{\prime})=\min_{(x,v)\in E^{\prime}}\\{d(x)+w(x,v)\\}$ be
the shortest distance to $v$ “relaxed” by edges in $E^{\prime}$ (recall that
$w(x,v)$ is the weight of the edge $(x,v)$). We maintain the following
invariants after each iteration of the algorithm except the last one.
1. (i)
$d(v)\leq\widehat{d}(v)\leq\widetilde{d}(v,E(F)\setminus E(B))$ holds for each
$v\in V\setminus F$.
2. (ii)
There exists a $v\in V\setminus F$ such that
$\widehat{d}(v)=d^{*}_{F}:=\min_{u\in V\setminus
F}\\{\widetilde{d}(u,E(F))\\}$.
Intuitively, Invariant (i) asserts that all edges in $E(F)\setminus E(B)$ are
“relaxed” while Invariant (ii) ensures that the distance estimate of the
target vertex, i.e., one with the shortest distance from $s$, is correct.
Initially, both invariants are satisfied since $\widehat{d}(x)=0$ holds for
each $x\in S_{1}\setminus S_{2}$. We maintain a priority queue $Q$ containing
vertices in $V\setminus F$ ordered by $\widehat{d}(\cdot)$. In the $t$-th
iteration, let $v_{t}$ be the vertex $v$ with the smallest $\widehat{d}(v)$.
By Invariants (i) and (ii), we have $\widehat{d}(v_{t})=d^{*}_{F}$ and thus
$d(v_{t})\leq d(v^{\prime})$ holds for each $v^{\prime}\in V\setminus F$
according to Dijkstra’s algorithm. As such, we push $v_{t}$ into $F$ and
update $B_{1}$, $B_{2}$ appropriately by checking if $v_{t}$ belongs to
$S_{1}$ and $\overline{S_{2}}$. Now, we would like to modify $\widehat{d}(v)$
for some $v\in V\setminus F$ so that both invariants remain true. For each
$i\in\\{1,2\\}$, depending on the size of $B_{i}$, we perform one of the
following.
1. 1.
If $|B_{i}|\geq\sqrt{r}$, then we compute
$\widetilde{d}_{i}(v)=\widetilde{d}(v,E(B_{1}))$ and set
$\widehat{d}(v)\leftarrow\min(\widehat{d}(v),\widetilde{d}_{i}(v))$ for each
$v\in V\setminus F$ using Lemma 3.5 below. For $i=1$, by definition of $G$,
$\operatorname{head}(E_{1})\subseteq V\setminus S_{1}$ and thus we only need
to compute $\widetilde{d}_{i}(v)$ for $v\in V\setminus S_{1}$, and therefore
Lemma 3.5 takes $O(n\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time. For $i=2$,
similarly, $\operatorname{head}(E_{2})\subseteq S_{2}$ and thus we only need
to compute $\widetilde{d}_{i}(v)$ for $v\in S_{2}$, taking
$O(r\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time. Then, we set
$B_{i}\leftarrow\emptyset$, and the above modification ensures that Invariant
(i) holds since $\widetilde{d}(v,E(F))=\min(\widetilde{d}(v,E(F)\setminus
E(B)),\widetilde{d}(v,E(B)))$.
2. 2.
If $|B_{i}|<\sqrt{r}$, then we do not clear $B_{i}$ and therefore Invariant
(i) trivially holds. For each $b\in B_{i}$, we find a $v_{b}\in V\setminus F$
minimizing $d(b)+w_{i}(b,v_{b})$ via Lemma 2.1 as follows. If $i=1$, then we
have $w_{1}(b,v_{b})=w^{\epsilon}_{1}(b)-w^{\epsilon}_{1}(v_{b})$, and thus we
find the $v_{b}$ maximizing $w^{\epsilon}_{1}(v_{b})$ such that
$(S_{1}\setminus\\{b\\})\cup\\{v_{b}\\}\in\mathcal{I}_{1}$. If $i=2$, then
$w_{2}(b,v_{b})=w^{\epsilon}_{2}(v_{b})-w^{\epsilon}_{2}(b)$, and thus we find
the $v_{b}$ minimizing $w^{\epsilon}_{2}(v_{b})$ such that
$(S_{2}\setminus\\{v_{b}\\})\cup\\{b\\}\in\mathcal{I}_{2}$. Then, we set
$\widehat{d}(v_{b})\leftarrow\min(\widehat{d}(v_{b}),d(b)+w_{i}(b,v_{b}))$ and
update $v_{b}$’s position in $Q$ appropriately. This takes
$O(\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time.
In both cases, as argued above, Invariant (i) holds. We argue that Invariant
(ii) holds after the iteration as well. Let $B_{1}^{(t)}$ be $B_{1}$ after the
$t$-th iteration and define $B_{2}^{(t)}$ and $F^{(t)}$ similarly. Let
$E(B^{(t)})$ denote $E(B_{1}^{(t)})\cup E(B_{2}^{(t)})$. Let
$v^{*}=\operatorname*{arg\,min}_{v\in V\setminus
F^{(t)}}\\{\widetilde{d}(v,E(F^{(t)}))\\}$ be an unvisited vertex after the
$t$-th iteration with the smallest distance from $s$ and let $e^{*}=(u,v^{*})$
be the edge such that $u\in F^{(t)}$ and $d(v^{*})=d(u)+w(e^{*})$. That is,
$e^{*}$ is the edge connecting $v^{*}$ and its parent in the shortest-path
tree. If $e^{*}\in E(F^{(t-1)})\setminus E(B^{(t-1)})$, then Invariant (ii)
trivially follows from the end of the $(t-1)$-th iteration. Otherwise, we must
have either $e^{*}\in E(B_{1}^{(t-1)})$ or $e^{*}\in E(B_{2}^{(t-1)})$.
Without loss of generality, let’s assume $e^{*}\in E(B_{1}^{(t-1)})$. If
$|B_{1}^{(t-1)}|+1\geq\sqrt{r}$ (i.e., Case 1), then after setting
$B_{1}^{(t)}\leftarrow\emptyset$, Invariant (ii) follows from the fact the
Invariant (i) holds for $v^{*}$ and $\widetilde{d}(v^{*},E(F^{(t)})\setminus
E(B^{(t)}))\leq d(u)+w(e^{*})$ since $e^{*}\in E(F^{(t)})\setminus
E(B^{(t)})$. If $|B_{1}^{(t-1)}|+1<\sqrt{r}$ (i.e., Case 2), then there must
exists a $b\in B_{1}^{(t)}$ such that
$d(b)+w_{1}(b,v^{*})=\min_{v}\\{d(b)+w_{1}(b,v)\\}$ and thus we have at least
one $v_{b}\in V\setminus F^{(t)}$ such that $\widehat{d}(v_{b})\leq
d(b)+w_{1}(b,v_{b})=d(v^{*})$. This shows that Invariant (ii) indeed holds
after the $t$-th iteration. The correctness of the algorithm follows from the
two invariants and the analysis of Dijkstra’s algorithm.
To bound the total running time, observe that for $B_{1}$, Case 1 happens at
most $O(r/\sqrt{r})=O(\sqrt{r})$ times since $|S_{1}|=r$. Thus, it takes
$O(n\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time in total.
Similarly, for $B_{2}$, Case 1 happens at most $O(n/\sqrt{r})$ time, taking
$O(n/\sqrt{r}\cdot
r\log{n}\cdot\mathcal{T}_{\mathsf{rank}})=O(n\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$
time in total as well. For Case 2, each iteration takes
$O(\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time, contributing a
total of $O(n\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time. As a
result, the algorithm runs in
$O(n\sqrt{r}\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time, as claimed.
Finally, it is easy to maintain balanced binary search trees of elements
ordered by $u_{1}$, $u_{2}$, $\widehat{d}+u_{1}$, and $\widehat{d}-u_{2}$ in
$O(n\log{n})$ time throughout the procedure so that Lemma 2.1 can be applied
without overhead. The shortest $st$-path can also be easily recovered by
maintaining the optimal parent in the shortest-path tree for each vertex. This
proves the lemma. ∎
###### Lemma 3.5.
For each $i\in\\{1,2\\}$, given $B_{i}\subseteq F$ and $R\subseteq V\setminus
F$, it takes takes $O(|R|\log{n}\cdot\mathcal{T}_{\mathsf{rank}})$ time to
compute $\widetilde{d}(v,E(B_{i}))$ for all $v\in R$.
###### Proof.
For $i=1$ and $e=(b,v)\in E(B_{i})$, we have
$w(e)=w^{\epsilon}_{1}(b)-w^{\epsilon}_{1}(v)$. Therefore, $d(v,E(B_{1}))$ can
be computed by finding the $b\in B$ with the smallest
$d(b)+w^{\epsilon}_{1}(b)$ such that
$(S_{1}\setminus\\{b\\})\cup\\{v\\}\in\mathcal{I}_{1}$ via Lemma 2.1.
Similarly, for $i=2$, we have $w(e)=w^{\epsilon}_{2}(v)-w^{\epsilon}_{2}(b)$,
and thus $d(v,E(B_{2}))$ can be computed by finding the $b\in B$ with the
smallest $d(b)-w^{\epsilon}_{2}(b)$. The lemma simply follows by calling Lemma
2.1 once for each $v\in R$. ∎
### 3.3 Bounding the Numbers
Finally, to conclude the analysis of our algorithm, we argue that the numbers
such as $w^{\epsilon}_{1}(x)$ and $w^{\epsilon}_{2}(x)$ are bounded by
$\widetilde{O}(\operatorname{poly}(nW))$ so that the number of bits needed to
store them and the time for a single arithmetic operation only grow by a
constant factor. In the weight adjustment stage, each number is adjusted at
most $O(r)$ times and each adjustment changes the number by at most $O(W)$
since $\epsilon$ is at most $W$. Therefore, the accumulative change to a
number via weight adjustments is at most $O(\operatorname{poly}(nW))$. For
growth incurred by augmentations, we first assume that all vertices are
reachable from $s$ in $G_{w^{\epsilon},S}$. Consider a single run of Lemma 3.4
and fix an $x\in V$. Let $P_{x}=\\{s,v_{1},\ldots,v_{k}\\}$ with $v_{k}=x$ be
the shortest $sx$-path in $G_{w^{\epsilon},S}$. Suppose that
$(v_{1},v_{2}),(v_{k-1},v_{k})\in E_{1}$, then by definition, we have
$\displaystyle d(x)$
$\displaystyle=w^{\epsilon}_{1}(v_{1})-w^{\epsilon}_{1}(v_{2})+w^{\epsilon}_{2}(v_{3})-w^{\epsilon}_{2}(v_{2})+\cdots+w^{\epsilon}_{1}(v_{k-1})-w^{\epsilon}_{1}(v_{k})$
$\displaystyle\leq
w^{\epsilon}_{1}(v_{1})-w(v_{2})+(w(v_{3})+\epsilon)+\cdots+(w(v_{k-1})+\epsilon)-w^{\epsilon}_{1}(v_{k})$
$\displaystyle\leq
w^{\epsilon}_{1}(v_{1})-w^{\epsilon}_{1}(v_{k})+\left(\sum_{i=2}^{k-1}(-1)^{i+1}w(v_{i})\right)+nW.$
Since $\widehat{w}^{\epsilon}_{1}(x)=w^{\epsilon}_{1}(x)+d(x)$ and
$\widehat{w}^{\epsilon}_{2}(x)=w^{\epsilon}_{2}(x)-d(x)$ as defined in Lemma
3.3, we have
$\displaystyle|\widehat{w^{\epsilon}}_{1}(x)|\leq|w^{\epsilon}_{1}(v_{1})|+2nW$
and
$\displaystyle|\widehat{w}^{\epsilon}_{2}(x)|\leq|w^{\epsilon}_{1}(v_{1})|+2nW.$
(1)
Similarly, if $(v_{k-1},v_{k})\in E_{2}$, then
$\displaystyle d(x)$
$\displaystyle=w^{\epsilon}_{1}(v_{1})-w^{\epsilon}_{1}(v_{2})+w^{\epsilon}_{2}(v_{3})-w^{\epsilon}_{2}(v_{2})+\cdots+w^{\epsilon}_{2}(v_{k})-w^{\epsilon}_{2}(v_{k-1})$
$\displaystyle\leq
w^{\epsilon}_{1}(v_{1})-w(v_{2})+(w(v_{3})+\epsilon)+\cdots+(w(v_{k-2})+\epsilon)-w(v_{k-1})+w^{\epsilon}_{2}(v_{k})$
$\displaystyle\leq
w^{\epsilon}_{1}(v_{1})+w^{\epsilon}_{2}(v_{k})+\left(\sum_{i=2}^{k-1}(-1)^{i+1}w(v_{i})\right)+nW,$
implying (1) as well. The case when $(v_{1},v_{2})\in E_{2}$ holds similarly,
except now we have
$\displaystyle|\widehat{w}^{\epsilon}_{1}(x)|\leq|w^{\epsilon}_{2}(v_{1})|+2nW$
and
$\displaystyle|\widehat{w}^{\epsilon}_{2}(x)|\leq|w^{\epsilon}_{2}(v_{1})|+2nW.$
(2)
Since the number of augmentations is $\widetilde{O}(r^{1/4})$, we indeed have
that
$|w^{\epsilon}_{1}(x)|=|w^{\epsilon}_{2}(x)|=O(\operatorname{poly}(nW))=\Theta((nW)^{k})$
for some constant $k$. For the case where some vertex $x$ is not reachable
from $s$, we can simply set $d(x)$ to some $c(nW)^{k+1}$ for a large enough
constant $c$ and the desired bound still holds.
## 4 Concluding Remarks
We present a simple subquadratic algorithm for weighted matroid intersection
under the rank oracle model, providing a partial yet affirmative answer to one
of the open problems raised by Blikstad et al. [BvdBMN21]. Whether the same is
achievable under the independence oracle model remains open. It seems that our
techniques for computing shortest-path trees do not solely result in a
subquadratic augmenting-path algorithm under the independence oracle. Removing
the dependence on $\log{W}$ and making the algorithm run in strongly-
polynomial time is also of interest. Finally, as noted in [BvdBMN21], there
were very few non-trivial lower bound results for matroid intersection. It
would be helpful to see if there is any super-linear lower bound on the number
of queries for these problems or even for computing shortest-path trees in the
exchange graphs under either oracle model.
## Acknowledgements
I would like to thank Prof. Hsueh-I Lu for advising the project and helpful
suggestions on the writing and notation, Brian Tsai for proof-reading an
initial draft of this paper, and the anonymous reviewers of ISAAC 2022 for
their useful comments.
## References
* [AD71] Martin Aigner and Thomas A Dowling. Matching theory for combinatorial geometries. Transactions of the American Mathematical Society, 158(1):231–245, 1971. doi:10.2307/1995784.
* [Bli21] Joakim Blikstad. Breaking o(nr) for matroid intersection. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, volume 198, pages 31:1–31:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. doi:10.4230/LIPIcs.ICALP.2021.31.
* [BvdBMN21] Joakim Blikstad, Jan van den Brand, Sagnik Mukhopadhyay, and Danupon Nanongkai. Breaking the quadratic barrier for matroid intersection. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 421–432. ACM, 2021. doi:10.1145/3406325.3451092.
* [CLS+19] Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, Sahil Singla, and Sam Chiu-wai Wong. Faster matroid intersection. In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, pages 1146–1168. IEEE Computer Society, 2019. doi:10.1109/FOCS.2019.00072.
* [CLSW17] Deeparnab Chakrabarty, Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. Subquadratic submodular function minimization. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 1220–1231. ACM, 2017. doi:10.1145/3055399.3055419.
* [CQ16] Chandra Chekuri and Kent Quanrud. A fast approximation for maximum weight matroid intersection. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 445–457. SIAM, 2016. doi:10.1137/1.9781611974331.ch33.
* [Cun86] William H. Cunningham. Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput., 15(4):948–957, 1986. doi:10.1137/0215066.
* [Edm71] Jack Edmonds. Matroids and the greedy algorithm. Math. Program., 1(1):127–136, 1971. doi:10.1007/BF01584082.
* [Edm79] Jack Edmonds. Matroid intersection. Annals of Discrete Mathematics, 4:39–49, 1979. doi:10.1016/S0167-5060(08)70817-3.
* [Edm01] Jack Edmonds. Submodular functions, matroids, and certain polyhedra. In Combinatorial Optimization - Eureka, You Shrink!, Papers Dedicated to Jack Edmonds, 5th International Workshop, Aussois, France, March 5-9, 2001, Revised Papers, volume 2570 of Lecture Notes in Computer Science, pages 11–26. Springer, 2001. doi:10.1007/3-540-36478-1\\_2.
* [Edm10] Jack Edmonds. Matroid partition. In 50 Years of Integer Programming 1958-2008 - From the Early Years to the State-of-the-Art, pages 199–217. Springer, 2010. doi:10.1007/978-3-540-68279-0\\_7.
* [Fra81] András Frank. A weighted matroid intersection algorithm. J. Algorithms, 2(4):328–336, 1981. doi:10.1016/0196-6774(81)90032-8.
* [FZ95] Satoru Fujishige and Xiaodong Zhang. An efficient cost scaling algorithm for the independent assignment problem. Journal of the Operations Research Society of Japan, 38(1):124–136, 1995. doi:10.15807/jorsj.38.124.
* [GX96] Harold N. Gabow and Ying Xu. Efficient theoretic and practical algorithms for linear matroid intersection problems. J. Comput. Syst. Sci., 53(1):129–147, 1996. doi:10.1006/jcss.1996.0054.
* [HKK16] Chien-Chung Huang, Naonori Kakimura, and Naoyuki Kamiyama. Exact and approximation algorithms for weighted matroid intersection. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pages 430–444. SIAM, 2016. doi:10.1137/1.9781611974331.ch32.
* [Law75] Eugene L. Lawler. Matroid intersection algorithms. Math. Program., 9(1):31–56, 1975. doi:10.1007/BF01681329.
* [LSW15] Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, pages 1049–1065. IEEE Computer Society, 2015. doi:10.1109/FOCS.2015.68.
* [Ngu19] Huy L. Nguyen. A note on cunningham’s algorithm for matroid intersection. CoRR, abs/1904.04129, 2019. URL: http://arxiv.org/abs/1904.04129, arXiv:1904.04129.
* [OA92] James B. Orlin and Ravindra K. Ahuja. New scaling algorithms for the assignment and minimum mean cycle problems. Math. Program., 54:41–56, 1992. doi:10.1007/BF01586040.
* [Pri15] Christopher Price. Combinatorial algorithms for submodular function minimization and related problems. Master’s thesis, University of Waterloo, 2015. URL: http://hdl.handle.net/10012/9356.
* [S+03] Alexander Schrijver et al. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003.
* [SI95] Maiko Shigeno and Satoru Iwata. A dual approximation approach to weighted matroid intersection. Oper. Res. Lett., 18(3):153–156, 1995. doi:10.1016/0167-6377(95)00047-X.
## Appendix A Omitted Proofs
### A.1 Proofs of Lemmas in Section 3.1
For self-containedness, we include the proof that the two bases obtained in
the weight adjustment phase have a large intersection by Shigeno and Iwata
[SI95] here.
###### Lemma A.1 ([SI95]).
Let $S_{1}$ and $S_{2}$ be obtained from the procedure described in Lemma 3.1.
Then, $|S_{1}\cap S_{2}|\geq\left(1-\frac{O(1)}{k}\right)r$.
###### Proof.
Let $p(S)$ denote $\sum_{x\in S}p(x)$. Observe that an element is never moved
to $S_{2}\setminus S_{1}$ during weight adjustments, and therefore we have
$p(S_{2}\setminus S_{1})=0$ and $p(S_{1}\setminus S_{2})=p(S_{1})-p(S_{2})$.
Recall that $S^{\prime}$ is a common basis such that
$(w^{2\epsilon},S^{\prime})$ is a $2\epsilon$-solution. Since $p(x)$ equals
the number of adjustments of $w^{\epsilon}_{2}(x)$ and each such adjustment is
preceded by an adjustment of $w^{\epsilon}_{1}(x)$, we have
$p(x)\cdot\epsilon=w^{\epsilon}_{2}(x)-(w(x)-w^{2\epsilon}_{1}(x))\leq
w^{2\epsilon}_{1}(x)-w^{\epsilon}_{1}(x)$
for each $x\in V$. Thus,
$\displaystyle p(S_{1}\setminus S_{2})\cdot\epsilon$
$\displaystyle=\left(p(S_{1})-p(S_{2})\right)\cdot\epsilon$
$\displaystyle\leq\left(w^{2\epsilon}_{1}(S_{1})-w^{\epsilon}_{1}(S_{1})\right)-\left(w^{\epsilon}_{2}(S_{2})-w(S_{2})+w^{2\epsilon}_{1}(S_{2})\right)$
$\displaystyle\stackrel{{\scriptstyle\text{(a)}}}{{\leq}}w^{2\epsilon}_{1}(S_{1})-w^{\epsilon}_{1}(S^{\prime})-w^{\epsilon}_{2}(S^{\prime})+w(S_{2})-w^{2\epsilon}_{1}(S_{2})$
$\displaystyle\stackrel{{\scriptstyle\text{(b)}}}{{\leq}}w^{2\epsilon}_{1}(S_{1})-w(S^{\prime})+w(S_{2})-w^{2\epsilon}_{1}(S_{2}),$
where (a) is because $S_{i}$ is $w^{\epsilon}_{i}$-maximum for each
$i\in\\{1,2\\}$ and (b) is because $w(S^{\prime})\leq
w^{\epsilon}(S^{\prime})$ as $w^{\epsilon}$ is an $\epsilon$-splitting. Since
$(w^{2\epsilon},S^{\prime})$ is a $2\epsilon$-solution,
$w^{2\epsilon}_{2}(S)-2\epsilon r\leq w(S)-w^{2\epsilon}_{1}(S)\leq
w^{2\epsilon}_{2}(S)$ holds for each basis $S$. This combined with the fact
that $S^{\prime}$ is $w^{2\epsilon}_{i}$-maximum for each $i\in\\{1,2\\}$
implies
$p(S_{1}\setminus S_{2})\cdot\epsilon\leq 2\epsilon
r-w^{2\epsilon}_{2}(S^{\prime})+w^{2\epsilon}_{2}(S_{2})\leq 2\epsilon
r\implies p(S_{1}\setminus S_{2})\leq 2r.$
When the algorithm terminates, we have $p(x)=k$ for all $x\in S_{1}\setminus
S_{2}$, implying
$p(S_{1}\setminus S_{2})=|S_{1}\setminus S_{2}|\cdot k\leq
2r\implies|S_{1}\setminus S_{2}|\leq\frac{2r}{k}.$
As a result,
$|S_{1}\cap S_{2}|=r-|S_{1}\setminus
S_{2}|\geq\left(1-\frac{O(1)}{k}\right)r.$
∎
### A.2 Proofs of Lemmas in Section 3.2
In this section, we prove the properties of the exchange graphs. The proofs
for the more generalized auxiliary graph given by Fujishige and Zhang can be
found in [FZ95].
To prove Lemma 3.2, it would be more convenient to refer to the following
definition of a directed bipartite graph based on exchange relationships,
which is heavily used in unweighted matroid intersection algorithms. For
$S\in\mathcal{I}_{1}\cap\mathcal{I}_{2}$, let
$\widetilde{G}_{S}=(V\cup\\{s,t\\},\widetilde{E})$ with $s,t\not\in V$ denote
the directed graph with
$\widetilde{E}=\widetilde{E}_{1}\cup\widetilde{E}_{2}\cup\widetilde{E}_{s}\cup\widetilde{E}_{t}$,
where
$\displaystyle\widetilde{E}_{1}$ $\displaystyle=\\{(x,y)\mid x\in S,y\not\in
S,\;\text{and}\;(S\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{1}\\},$
$\displaystyle\widetilde{E}_{2}$ $\displaystyle=\\{(y,x)\mid x\in S,y\not\in
S,\;\text{and}\;(S\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{2}\\},$
$\displaystyle\widetilde{E}_{s}$ $\displaystyle=\\{(s,x)\mid
S\cup\\{x\\}\in\mathcal{I}_{1}\\},\;\text{and}$
$\displaystyle\widetilde{E}_{t}$ $\displaystyle=\\{(x,t)\mid
S\cup\\{x\\}\in\mathcal{I}_{2}\\}.$
###### Lemma A.2 ([Law75]).
$\widetilde{G}_{S}$ for $|S|<r$ admits an $st$-path.
We will use the existence of an $st$-path in $\widetilde{G}_{\widetilde{S}}$
to prove that such a path exists in $G_{w^{\epsilon},S}$, for
$\widetilde{S}=S_{1}\cap S_{2}$. The following claims certify that
$\widetilde{G}_{\widetilde{S}}$ and $G_{w^{\epsilon},S}$ are almost the same.
###### Claim A.3.
Let $\mathcal{M}=(V,\mathcal{I})$ be a matroid, $S\subseteq
S^{\prime}\in\mathcal{I}$, $x\in S$, and $y\not\in S^{\prime}$ such that
$(S\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}$ but
$S\cup\\{y\\}\not\in\mathcal{I}$, then
$(S^{\prime}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}$.
###### Proof.
Let $C$ be the unique circuit in $S\cup\\{y\\}$. Since $C\subseteq
S^{\prime}\cup\\{y\\}$ and $S^{\prime}\cup\\{y\\}$ has only one circuit, $C$
is the unique circuit in $S^{\prime}\cup\\{y\\}$ as well. Moreover,
$(S\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}$ if and only if $x\in C$ and
therefore $(S^{\prime}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}$. ∎
###### Claim A.4.
Let $\mathcal{M}=(V,\mathcal{I})$ be a matroid, $S\subseteq
S^{\prime}\in\mathcal{I}$ where $S^{\prime}$ is a basis of $\mathcal{M}$, and
$x\not\in S^{\prime}$ such that $S\cup\\{x\\}\in\mathcal{I}$. Then, there
exists a $y\in S^{\prime}\setminus S$ such that
$(S^{\prime}\setminus\\{y\\})\cup\\{x\\}\in\mathcal{I}$.
###### Proof.
Let $S^{\prime\prime}$ be an arbitrary basis of $\mathcal{M}$ that contains
$S\cup\\{x\\}$. Since $x\in S^{\prime\prime}\setminus S^{\prime}$, by the
strong exchange property (see, e.g., [S+03, Theorem 39.6]) of bases, there
exists a $y\in S^{\prime}\setminus S^{\prime\prime}\subseteq
S^{\prime}\setminus S$ such that
$(S^{\prime}\setminus\\{y\\})\cup\\{x\\}\in\mathcal{I}$, completing the proof.
∎
We are now ready to prove Lemma 3.2.
###### Proof of Lemma 3.2.
Let $\widetilde{P}=\\{s,v_{1},\ldots,v_{k},t\\}$ be the shortest $st$-path in
$\widetilde{G}_{\widetilde{S}}$ for $\widetilde{S}=S_{1}\cap S_{2}$. The
existence of such a path is guaranteed by Lemma A.2. We have
$\widetilde{S}\cup\\{v_{i}\\}\not\in\mathcal{I}_{1}$ and
$\widetilde{S}\cup\\{v_{i}\\}\not\in\mathcal{I}_{2}$ for each $1<i<k$ since
$\widetilde{P}$ is the shortest path. For an odd $1\leq i<k$, we have
$v_{i}\not\in S$ and $v_{i+1}\in S$. If $v_{i}\not\in S_{2}\setminus S_{1}$,
then by A.3, we have $(v_{i},v_{i+1})\in E(G_{w^{\epsilon},S})$. Similarly,
for an even $1\leq i<k$, if $v_{i+1}\not\in S_{1}\setminus S_{2}$, then we
have $(v_{i},v_{i+1})\in E(G_{w^{\epsilon},S})$. Suppose that $v_{1}\not\in
S_{1}\setminus S_{2}$, then by A.4, we can find a $v_{0}\in S_{1}\setminus
S_{2}$ such that
$(S_{1}\setminus\\{v_{0}\\})\cup\\{v_{1}\\}\in\mathcal{I}_{1}$. Similarly, if
$v_{k}\not\in S_{2}\setminus S_{1}$, then we can find a $v_{k+1}\in
S_{2}\setminus S_{1}$ such that
$(S_{2}\setminus\\{v_{k+1}\\})\cup\\{v_{k}\\}\in\mathcal{I}_{2}$. Therefore,
without loss of generality, we may assume that there exists the last vertex
$v_{i}\in S_{1}\setminus S_{2}$ and the first vertex $v_{j}\in S_{2}\setminus
S_{1}$ after $v_{i}$. Now, for each $i<k<j$, we have
$v_{k}\not\in(S_{1}\setminus S_{2})\cup(S_{2}\setminus S_{1})$. Therefore,
$(v_{k},v_{k+1})\in E(G_{w^{\epsilon},S})$ holds for each $i\leq k<j$, and we
obtain an $st$-path in $G_{w^{\epsilon},S}$ as
$P=\\{s,v_{i},\ldots,v_{j},t\\}$. This concludes the proof. ∎
Finally, to prove Lemma 3.3, we need the following results.
###### Lemma A.5 ([Pri15, Proposition 2.4.1]).
Given a matroid $\mathcal{M}=(V,\mathcal{I})$ and an $S\in\mathcal{I}$.
Suppose that $(a_{1},\ldots,a_{p})\subseteq V\setminus S$ and
$(b_{1},\ldots,b_{p})\subseteq S$ are two sequences satisfying the following
conditions:
1. 1.
$(S\setminus\\{b_{i}\\})\cup\\{a_{i}\\}\in\mathcal{I}$ for each $1\leq i\leq
p$ and
2. 2.
$(S\setminus\\{b_{j}\\})\cup\\{a_{i}\\}\not\in\mathcal{I}$ for each $1\leq
j<i\leq p$.
Then,
$(S\setminus\\{b_{1},\ldots,b_{p}\\})\cup\\{a_{1},\ldots,a_{p}\\}\in\mathcal{I}$
holds.
###### Lemma A.6 ([Pri15, Lemma 2.4.2]).
Let $\mathcal{M}$, $S$, $(a_{1},\ldots,a_{p})$, and $(b_{1},\ldots,b_{p})$ be
the same as in Lemma A.5. Let
$S^{\prime}=(S\setminus\\{b_{1},\ldots,b_{p}\\})\cup\\{a_{1},\ldots,a_{p}\\}$.
For $x\in S^{\prime}$ and $y\in V\setminus S^{\prime}$, if
$(S^{\prime}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}$ but either $y\in S$ or
$(S\setminus\\{x\\})\cup\\{y\\}\not\in\mathcal{I}$, then there exists
$1\leq\ell\leq k\leq p$ such that
$(S\setminus\\{x\\})\cup\\{a_{k}\\}\in\mathcal{I}$ and either $b_{\ell}=y$ or
$(S\setminus\\{b_{\ell}\\})\cup\\{y\\}\in\mathcal{I}$.
In essence, Lemma A.5 captures the validity of an augmentation while Lemma A.6
models the condition in which new exchange relationships emerge in the
augmented independent set. The following three claims imply Lemma 3.3. Recall
that $P=\\{s,v_{1},\ldots,v_{k},t\\}$ is the shortest $st$-path with the least
number of edges and $d(x)$ is the $sx$-distance in $G_{w^{\epsilon},S}$.
###### Claim A.7.
$|\widehat{S}_{1}\cap\widehat{S}_{2}|>|S_{1}\cap S_{2}|$ holds.
###### Proof.
If $k=2$, then either $v_{1}\in\widehat{S}_{1}\cap\widehat{S}_{2}$ or
$v_{k}\in\widehat{S}_{1}\cap\widehat{S}_{2}$ must hold, depending on whether
$(v_{1},v_{2})\in E_{1}$ or $(v_{1},v_{2})\in E_{2}$, and the claim trivially
holds in this case. Thus, in the following, we assume that $k>2$. Since $P$ is
the shortest, we may assume that $v_{i}\in(S_{1}\cap
S_{2})\cup(\overline{S_{1}}\cap\overline{S_{2}})$ holds for each $1<i<k$.
Also, for $1<i<k-1$, if $v_{i}\in S_{1}\cap S_{2}$, then $v_{i+1}$ must be in
$\overline{S_{1}}\cap\overline{S_{2}}$ due to the way $G_{w^{\epsilon},S}$ is
constructed. Similarly, if $v_{i}\in\overline{S_{1}}\cap\overline{S_{2}}$,
then we must have $v_{i+1}\in S_{1}\cap S_{2}$. Let
$P_{\text{mid}}=\\{v_{2},\ldots,v_{k-1}\\}$, $I=S_{1}\cap S_{2}$, and
$O=\overline{S_{1}}\cap\overline{S_{2}}$. Clearly, we have
$|\widehat{S}_{1}\cap\widehat{S}_{2}|-|S_{1}\cap S_{2}|=|P_{\text{mid}}\cap
O|-|P_{\text{mid}}\cap I|+\llbracket
v_{1}\in\widehat{S}_{1}\cap\widehat{S}_{2}\rrbracket+\llbracket
v_{k}\in\widehat{S}_{1}\cap\widehat{S}_{2}\rrbracket.$ (3)
We prove the claim by considering the following four possible cases.
$k$ is even and $v_{2}\in I$.
We have $(v_{1},v_{2})\in E_{2}$, $(v_{k-1},v_{k})\in E_{2}$, and
$|P_{\text{mid}}\cap I|=|P_{\text{mid}}\cap O|$. Also,
$v_{1}\in\operatorname{tail}(P\cap E_{2})$ and therefore
$v_{1}\in\widehat{S}_{1}\cap\widehat{S}_{2}$.
$k$ is even and $v_{2}\in O$.
We have $(v_{1},v_{2})\in E_{1}$, $(v_{k-1},v_{k})\in E_{1}$, and
$|P_{\text{mid}}\cap I|=|P_{\text{mid}}\cap O|$. Also,
$v_{k}\in\operatorname{head}(P\cap E_{1})$ and therefore
$v_{k}\in\widehat{S}_{1}\cap\widehat{S}_{2}$.
$k$ is odd and $v_{2}\in I$.
We have $(v_{1},v_{2})\in E_{2}$, $(v_{k-1},v_{k})\in E_{1}$, and
$|P_{\text{mid}}\cap I|=|P_{\text{mid}}\cap O|+1$. Also,
$v_{1}\in\operatorname{tail}(P\cap E_{2})$, $v_{k}\in\operatorname{head}(P\cap
E_{1})$ and therefore $v_{1},v_{k}\in\widehat{S}_{1}\cap\widehat{S}_{2}$.
$k$ is odd and $v_{2}\in O$.
We have $(v_{1},v_{2})\in E_{1}$, $(v_{k-1},v_{k})\in E_{2}$, and
$|P_{\text{mid}}\cap I|=|P_{\text{mid}}\cap O|-1$.
In all cases, we have $|\widehat{S}_{1}\cap\widehat{S}_{2}|>|S_{1}\cap S_{2}|$
via Equation 3, concluding the proof. ∎
We prove the following claims for $i=1$. The proofs for $i=2$ follow
analogously.
###### Claim A.8.
$\widehat{S}_{i}\in\mathcal{I}_{i}$ holds for each $i\in\\{1,2\\}$.
###### Proof.
Let $P_{1}=P\cap E_{1}=\\{(b_{1},a_{1}),(b_{2},a_{2}),\ldots(b_{p},a_{p})\\}$,
where $(S_{1}\setminus\\{b_{i}\\})\cup\\{a_{i}\\}\in\mathcal{I}_{1}$ holds for
each $1\leq i\leq p$. Since $P$ is the shortest path,
$d(b_{i})+u_{1}(b_{i})-u_{1}(a_{i})=d(a_{i})\implies
d(b_{i})+u_{1}(b_{i})=d(a_{i})+u_{1}(a_{i})$
holds for each $i$. Reorder $P_{1}$ so that $d(b_{1})+u_{1}(b_{1})\leq
d(b_{2})+u_{1}(b_{2})\leq\cdots\leq d(b_{p})+u_{1}(b_{p})$. Moreover, if
$d(b_{i})+u_{1}(b_{i})=d(b_{j})+u_{1}(b_{j})$ for some $i,j$, then
$(b_{i},a_{i})$ precedes $(b_{j},a_{j})$ in $P_{1}$ if and only if
$(b_{i},a_{i})$ precedes $(b_{j},a_{j})$ in $P$. It follows that for each
$1\leq j<i\leq p$, it holds that
$(S_{1}\setminus\\{b_{j}\\})\cup\\{a_{i}\\}\not\in\mathcal{I}_{1}$ since
otherwise we would have
$d(b_{j})+u_{1}(b_{j})-u_{1}(a_{i})\geq d(a_{i})\implies
d(b_{j})+u_{1}(b_{j})\geq d(a_{i})+u_{1}(a_{i}).$ (4)
Because $j<i$, (4) must take equality, but this would contradict with the fact
the $P$ has the least number of edges since the edge $(b_{j},a_{i})$ “jumps”
over vertices $a_{j},b_{j+1},\ldots,b_{i}$ in $P$ and has the same weight as
the subpath $b_{j},a_{j},\ldots,b_{i},a_{i}$. As such, by Lemma A.5, the claim
is proved. ∎
###### Claim A.9.
$\widehat{S}_{i}$ is $\widehat{w}^{\epsilon}_{i}$-maximum for each
$i\in\\{1,2\\}$.
###### Proof.
Let $P_{1}=P\cap E_{1}=\\{(b_{1},a_{1}),\ldots,(b_{p},a_{p})\\}$ be ordered
the same way as in the proof of A.8. It suffices to show that
$\widehat{w}^{\epsilon}_{1}(x)\geq\widehat{w}^{\epsilon}_{1}(y)$ holds for
each $x\in\widehat{S}_{1}$ and $y\not\in\widehat{S}_{1}$ with
$(\widehat{S}_{1}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{1}$. Consider the
following two cases.
1. 1.
$(S_{1}\setminus\\{x\\})\cup\\{y\\}\in\mathcal{I}_{1}$: Since $(x,y)\in
E_{1}$, it follows that
$d(x)+u_{1}(x)-u_{1}(y)\geq
d(y)\implies\widehat{w}^{\epsilon}_{1}(x)=d(x)+u_{1}(x)\geq
d(y)+u_{1}(y)=\widehat{w}^{\epsilon}_{1}(y).$
2. 2.
$(S_{1}\setminus\\{x\\})\cup\\{y\\}\not\in\mathcal{I}_{1}$: By Lemma A.6,
there exists $1\leq\ell\leq k\leq p$ such that (1)
$(S_{1}\setminus\\{x\\})\cup\\{a_{k}\\}\in\mathcal{I}_{1}$ and either (2.1)
$b_{\ell}=y$ or (2.2)
$(S_{1}\setminus\\{b_{\ell}\\})\cup\\{y\\}\in\mathcal{I}_{1}$. (1) implies
that $\widehat{w}^{\epsilon}_{1}(x)\geq\widehat{w}^{\epsilon}_{1}(a_{k})$. If
(2.1) holds, then
$\widehat{w}^{\epsilon}_{1}(x)\geq\widehat{w}^{\epsilon}_{1}(a_{k})\geq\widehat{w}^{\epsilon}_{1}(b_{\ell})=\widehat{w}^{\epsilon}_{1}(y)$.
If (2.2) holds, then
$\widehat{w}^{\epsilon}_{1}(x)\geq\widehat{w}^{\epsilon}_{1}(a_{k})\geq\widehat{w}^{\epsilon}_{1}(b_{\ell})\geq\widehat{w}^{\epsilon}_{1}(y)$.
The claim is proved. ∎
|
Magnetic storms during the space age: Occurrence and relation to varying solar activity
Kalevi Mursula1, Timo Qvick1, Lauri Holappa1, Timo Asikainen1
1Space Climate Group, Space Physics and Astronomy Research Unit, University of Oulu, Finland
Kalevi<EMAIL_ADDRESS>
* We explain the occurrence of magnetic storms in space age by their relation to the varying solar activity and solar magnetic structure.
* The occurrence of large, moderate and weak HSS/CIR storms follows the decrease of the HCS tilt in the declining phase of solar cycle.
* Maxima of HSS/CIR storms have shifted from late declining phase in cycles 20-22 to earlier times in cycles 23-24 due to recent HCS widening.
We study the occurrence of magnetic storms in space age (1957–2021) using Dst and Dxt indices.
We find 2526/2743 magnetic storms in the Dxt/Dst index, out of which 45% are weak, 40% moderate, 12% intense and 3% major storms.
Occurrence of storms in space age follows the slow decrease of sunspot activity and the related change in solar magnetic structure.
We quantify the sunspot - CME storm relation in the five cycles of space age.
We explain how the varying solar activity changes the structure of the heliospheric current sheet (HCS), and how this affects the HSS/CIR storms.
Space age started with a record number of storms in 1957–1960, with roughly one storm per week.
Solar polar fields attained their maximum in cycle 22, which led to an exceptionally thin HCS, and a space age record of large HSS/CIR storms in 1990s.
In the minimum of cycle 23, for the only time in space age, CME storm occurrence reduced below that predicted by sunspots.
Weak sunspot activity since cycle 23 has weakened solar polar fields and widened the HCS, which has decreased the occurrence of large and moderate HSS/CIR storms.
Because of a wide HCS, the Earth has spent 50% of its time in slow solar wind since cycle 23.
The wide HCS has also made large and moderate HSS/CIR storms occur in the early declining phase in recent cycles, while in the more active cycles 20–22 they occurred in the late declining phase.
§ INTRODUCTION
Magnetic storms <cit.>for a review see, e.g.,>[]Dessler_Parker_1959, Gonzalez_1994,Daglis_1999, Daglis_2003, Daglis_SSR_2006 are the largest disturbances of the near-Earth space, driven by enhanced interaction between the solar wind (SW), including the heliospheric magnetic field (HMF; also called the interplanetary magnetic field, IMF), and the Earth's magnetosphere.
Certain structures in the solar wind and the HMF contain favourable conditions for enhanced interaction that last typically from one to several days.
The two most important structures leading to storms are the interplanetary manifestations of coronal mass ejections (CME) <cit.> and the corotating interaction regions (CIR) related to high-speed solar wind streams (HSS) <cit.>.
CMEs arise from the eruption of large coronal flux tubes and lead to increased solar wind density and pressure <cit.>.
They often have the internal structure of a magnetic cloud which includes a systematic rotation of magnetic field lines and a long period of southward directed magnetic field <cit.>.
This leads to enhanced reconnection at the dayside magnetopause, producing a magnetic storm.
Coronal flux tubes that precede CMEs arise from the solar convection layer and have a close connection with sunspots.
Accordingly, the occurrence of CMEs varies closely in phase with sunspot activity, reflecting the appearance of new magnetic flux on the solar surface <cit.>.
High-speed solar wind streams emanate from large solar coronal holes <cit.>, where the magnetic field is typically unipolar and experiences less of super-radial expansion than in the neighborhood of closed field regions <cit.>.
Accordingly, solar wind flow can more freely escape from large coronal holes and is accelerated to higher speeds than elsewhere.
On the other hand, the wind originating at the streamer belt and neighboring regions is slower but more dense.
Since the streamer belt also carries the heliospheric current sheet (HCS) and its center, the magnetic neutral line (NL), the slow wind is commonly equated with the HCS.
When the HCS is inclined with respect to the solar equator, the slow and fast solar wind regions can be at the same heliographic latitude and be emitted successively in the same direction.
When the fast stream catches the slow wind ahead, an interaction region called the corotating interaction region, also called the stream interaction region (SIR), is formed <cit.>.
Due to compressed plasma density and increased magnetic field intensity, as well as to the following fast solar wind, the HSS/CIR structure is very effective in producing weak and moderate storms <cit.>.
Solar magnetic fields experience a dramatic structural change over the sunspot cycle.
During solar minima the global solar magnetic field is mainly dipolar, with a few active regions around the solar equator and large coronal holes of unipolar field with opposite polarities around the two solar poles <cit.>.
As activity increases in the ascending phase of the cycle, surges of magnetic flux with polarity opposite to the prevailing polar field are transported to each pole <cit.>.
These surges reduce the size of the old-polarity coronal holes and, eventually, reverse the polarity of the polar field around solar maxima.
Subsequent surges in the declining phase increase the new-polarity field at the poles and extend the area of polar coronal holes.
Moreover, surges can form contiguous unipolar regions from the low-latitude origin of surges to the solar pole.
This leads to the formation of longitudinally asymmetric extensions of polar coronal holes, so called elephant trunks, that can cover a wide range of latitudes and emit fast solar wind even at low latitudes <cit.>.
Such extensions of polar coronal holes to low latitudes are an important source of high-speed streams reaching the Earth.
Smaller-scale coronal holes can be formed between active regions at any latitude and at any time of the cycle, especially at solar maxima.
Related HSS streams can affect (e.g., accelerate) CMEs bursting from neighboring active regions <cit.>.
Later in the declining phase, as the emergence of magnetic flux and related surge production subside, the global field slowly returns back to its minimum-time dipolar structure with a polarity structure opposite to the previous minimum.
The lower boundary of polar coronal holes becomes more symmetric, which reduces the tilt of the HCS and the occurrence of HSSs at the Earth.
In addition to this solar cycle evolution of solar magnetic fields, there are also longer-term changes, e.g., in the overall sunspot activity, leading to the varying height of sunspot cycles.
It is known that sunspot activity in the 20th century reached a record level at least for a couple of thousand years <cit.>, culminating at the maximum of solar cycle 19 in 1957.
This highly active time of the mid-20th century is commonly called the Modern Grand Maximum (MGM).
After solar cycle 19 (to be called SC19), solar activity remained at a fairly high level for 3–4 cycles whereafter, since the maximum of SC23, solar activity has considerably subsided.
This is evidenced by an exceptionally long and deep minimum between SC23 and SC24, and a considerably low cycle 24.
We note that, curiously, the start of space age, as marked by the flight of the first satellite in 1957, coincides with the maximum of the MGM.
Thus, the space age, at least until the recent times, is characterized and affected by slowly, but unsteadily reducing solar activity.
The varying level of sunspot activity during the space age directly affects, e.g., the occurrence of CMEs and magnetic storms produced by CMEs.
In addition, there is also a related long-term change in the solar magnetic field structure, which affects coronal holes and, thereby, the occurrence of HSS/CIRs and magnetic storms produced by them.
In fact, while in the earlier, active cycles there were only small and short-lived coronal holes at low latitudes <cit.>, the declining phase of cycle 23 was characterized by rather large, persistent low-latitude coronal holes <cit.>, which led to a space age record activity of HSSs and geomagnetic activity in 2003 <cit.>.
Another long-term change is the recent weakening of solar polar fields <cit.> as the result of reduced emergence of new flux on the solar surface.
These changes have important consequences to the long-term occurrence of magnetic storms.
The enhanced solar wind-magnetosphere interaction during CMEs and HSS/CIRs accelerates particles and drives several current systems that lead to magnetic disturbances in different parts of the Earth's surface.
However, rather than measuring the overall disturbance level like, e.g., the Kp index of geomagnetic activity, magnetic storms are defined and quantified in terms of one current system only, the ring current <cit.>.
The storm-time ring current consists mainly of (positively charged) hydrogen, helium and oxygen ions of some 10–300 keV energy, drifting around the Earth at the distance of about 3–7 Earth radii <cit.>.
Drifting westward around the Earth, they produce a negative deflection in the horizontal magnetic field on the ground, which is a direct measure of the energy content of the ring current <cit.>.
This deflection has been measured from 1957 onward by four ground-based magnetometer stations located at low latitudes, roughly equidistantly in longitude.
A dedicated recipe was developed <cit.> in order to remove the secular, seasonal and daily quiet-time variations from the locally observed magnetic field, and to quantify the ring current in terms of an index called the Dst index.
Year 1957 was the International Geophysical Year (IGY), when several international programs on solar-terrestrial research were started, developing both ground-based and flying instrumentation.
Thus, it is not surprising that the Dst index is available since the start of space age.
However, as noted above, it is curious, but also quite appropriate, that the Sun reached its all-time maximum activity during the International Geophysical Year.
Since continuous monitoring of space by satellites started soon after the IGY, we also have satellite observations of the solar wind and the HMF available for most of the time of the Dst index.
This allows us to study the solar wind drivers of magnetic storms almost over the whole space age.
During the several years of storm studies some errors and inconsistencies have been found in the Dst index <cit.>.
Therefore Karinen_Mursula_2005 recalculated the Dst index using the original recipe but correcting the noticed problems.
This revised Dst index is called the Dxt index.
We will use here both the Dst and the Dxt index in order to study magnetic storms during the space age in 1957–2021.
Even though there are considerable differences between the two ring-current indices leading, e.g., to somewhat different numbers of magnetic storms, we find that the storm occurrences and their implications about the long-term change of the Sun, remain the same.
This paper is organized as follows.
In Section <ref>, we present the Dst and Dxt indices and discuss their differences.
Section <ref> describes the solar wind classification into the three main structures (flow types).
Section <ref> explains how storms are derived from the two indices.
Section <ref> gives the total numbers and mean intensities of storms of different intensity for the whole space age, and Section <ref> presents their yearly numbers.
We assign the storms into their solar wind drivers in Section <ref> and present their yearly numbers in Section <ref>.
In Section <ref> we discuss large and moderate CME storms and their relation to sunspots, separately during the five solar cycles.
Section <ref> presents the connection between HSS/CIR storms and the structure of the heliospheric current sheet, and discusses the implications of our results to the long-term evolution of the Sun.
Finally, in Section <ref> we discuss the obtained results and give our conclusions.
§ DXT AND DST INDICES
As noted above, in our quest for long-term homogeneity, we have found earlier that the Dst index depicts some errors and inconsistencies.
Correcting these problems eventually led to the recalculation of the index and to the development of the corrected Dst index, the Dxt index <cit.>.
The first error found was an erroneous diurnal variation of the Dst index.
While the Dst index includes a rather small diurnal UT variation,
it is artificially enlarged in 1971 <cit.>.
We suggested <cit.> that this erroneous UT variation in 1971 is most likely due to an erroneous treatment of the diurnal variation of SJG station data.
Unfortunately, the WDC-C2 at Kyoto does not provide the individual disturbances of the four Dst stations, i.e., the local Dst indices, which would clarify this problem.
Therefore the cause of this error remains without final clarification.
Nonetheless, the Dst index is erroneous in 1971 and the Dxt index corrects this error.
The Dst index is some 2 nT more negative, on an average, than the Dxt index <cit.>.
However, there are four consecutive years in 1963–1966 when the Dst index is considerably above (less negative than) the Dxt index <cit.>see Figure 5 in>[]Karinen_Mursula_2005.
Moreover, it was found that the year 1965 is unique in that it is the only year for which the annual Dst index is positive, far above any other year.
In comparison, annual means of the Dxt index remain negative in all years, even in 1965.
Again, at the lack of local Dst indices, the cause of the exceptionally high Dst index in 1963–1966 remains without final explanation, but the great consistency of global and local Dxt indices suggests that the Dst index is slightly flawed in these years.
The latitudinal normalization of the Dst index is questionable.
Originally the mean of local disturbances was normalized by the cosine of the mean of geomagnetic latitudes <cit.>.
However, later the mean of local disturbances was normalized by the mean of the cosines of geomagnetic latitudes <cit.>.
Alas, since the Dst stations are at different latitudes, each local disturbance should be normalized by the cosine of the respective geomagnetic latitude in order to find the local disturbance of the same equatorial (horizontal) electric current.
Only then the different stations measure the same ring current intensity, and their normalized disturbances (local Dst indices) can be averaged to find the global mean of the ring current <cit.>.
Note that this physically correct way of normalization has never been adopted in the Dst index recipe, nor can the Dst index be properly normalized because of the lack of local disturbances.
Note also that, although later studies have shown that other magnetospheric current systems <cit.> and induced earth-currents also contribute to the Dst index, their contribution is not subtracted from the Dst index when identifying storms or estimating their intensity.
Here we mostly follow the Dxt reconstruction presented in Karinen and Mursula (2005).
However, meanwhile, we have adopted a few new practices that slightly modify the original Dst recipe and further improve the Dxt index.
In order to reduce the effect of seasonal variation and solar activity upon the baseline, we now calculate the secular variation using local midnight values (23 and 00 in local time) of the international quiet days.
We also use a more sophisticated smoothing of the annual values by a 5-point quadratic Savitzky-Golay filter <cit.>, instead of using the second-order polynomial of the original recipe.
We now calculate the hourly values of the baseline by interpolating the smoothed annual values to hourly resolution by a piecewise cubic Hermite interpolating polynomial (PCHIP).
We also smooth the daily quiet-time curves using a 60-day Gaussian window, which removes the steps between successive monthly values that exist in the Dst index and in the earlier version of the Dxt index.
The top and middle panels of Figure <ref> show the hourly and annual values of the Dxt and Dst indices, and the bottom panel depicts the Dxt-Dst difference of annual indices.
One can see that the two indices differ even at annual resolution where their mean difference (mean absolute difference) is about 2.5 nT (3.3 nT, respectively).
Note that the Dxt-Dst difference depicts a systematic long-term variation, with larger positive differences (relatively more disturbed Dst) during the active times from 1970s to early 2010s, and larger negative differences (relatively more disturbed Dxt) during the less active times (mainly in 1960s).
This inconsistent long-term difference between the two indices is mainly due to the inconsistent quiet-time level of the Dst index, which follows the varying level of solar activity, being too low in the weak decade of 1960s and too high in the subsequent, more active decades.
While a more complete analysis of the differences between the two indices will be presented elsewhere, here we will only discuss the practical consequences of these differences to the number and strength of magnetic storms.
In particular, because the Dst index is, on an average, 2.5 nT lower than the Dxt index, it produces a somewhat larger number of magnetic storms than the Dxt index.
However, as we will show in this paper, despite these differences, our results on storm occurrences and their implications about the long-term change of the Sun, remain the same for the two indices.
Hourly (top panel) and yearly (middle panel) values of the Dxt (red) and Dst (blue) indices, and the Dxt-Dst difference (bottom panel) of yearly values.
We would also like to note that there are currently a few slightly different versions of the Dst index in the different servers and databases.
E.g., the Dst index in the ISGI data server and in the OMNI2 database of the NSSDC differ from the Kyoto WDC Dst index and from each other during long time intervals in the last few years, at least by the time of this writing (April 2022).
This is most likely due to different (provisional or final) versions of the Dst index of WDC being implemented in the other databases at different times.
(We have notified the respective institutions of these differences.
A detailed list of these differences can be obtained from the authors of this paper at request.)
Finally, we note that there are a number of more recent developments aiming to an improved estimate of the ring current, either using a novel reconstruction method <cit.>, improved temporal accuracy <cit.> or increased spatial accuracy by an extended station network <cit.>.
While all of these developments are motivated and have their own applications, we will use here only the Dst/Dxt indices since they are the only indices to cover the whole space age.
§ SOLAR WIND FLOW TYPES
Richardson_JGR_2000,Richardson_etal_2002a and Richardson_GA_2012 developed an hourly list of solar wind flow types using the measured values of the near-Earth solar wind parameters and some auxiliary data, like the sudden storm commencements, magnetospheric energetic particles, and cosmic rays.
They classified the solar wind into three different main flow types:
* CME-related flows that consist of interplanetary CMEs, including their possible upstream shocks and sheath regions
* high-speed streams and the related corotating interaction regions
* slow solar wind that can be related with the streamer belt and the HCS.
The hourly list of solar wind flow types is very extensive but not complete, covering about 91.9% of all hours in 1964–2021.
The annual coverages of solar wind flow type data are given in the bottom panel of Figure <ref>.
Flow type data coverage is almost complete since 1995 when the ACE and WIND satellites started operation.
It was very good also in 1965–1970 and 1973–1981, when several satellites were flying in the solar wind.
However, in 1982–1994, when solar wind was measured mostly by only one Earth-orbiting satellite (IMP-8), gaps in the flow type data cover almost one third of time.
Note also that, due to the auxiliary data, the overall coverage of solar wind flow type data is clearly better than the overall coverage of solar wind data (about 76.5%).
Here we use an updated version of the flow type list extending from November 1963 to January 2022.
(For more details on flow types and auxiliary data, see Richardson_GA_2012).
The top panel of Figure <ref> shows the annual fractions of the three solar wind types.
Here the fractions are calculated as a ratio to all available flow type values in the respective year.
Thus, they exclude the data gaps, and could, therefore, also be called relative flow type fractions.
The overall (relative) coverage of CME fraction is about 18.0%.
The top panel of Figure <ref> shows that the CME flow fraction varies closely in phase with the sunspot cycle.
The correlation coefficient between yearly sunspots and CME fractions is excellent, cc = 0.88, with an almost vanishing p-value of $1.7*10^{-19}$.
Even the overall level of CME fraction during each cycle closely follows the changing height of sunspot cycles, although the exact timing of cycle peaks of CME fractions may differ from sunspot maxima by 1–2 years.
The good agreement between CME fractions and sunspot activity is also seen, e.g., in the considerably lower CME fraction during the low sunspot cycle 24, compared to the level of CME fractions during the earlier, higher sunspot cycles.
Top: Annual fractions of CME (red), HSS/CIR (blue) and slow solar wind (cyan) solar wind flows in 1964–2021.
Scaled yearly sunspot numbers are depicted as a shaded area.
Bottom: Annual coverages of the solar wind flow data.
The HSS/CIR flow fraction maximizes in the declining phase of the sunspot cycle, when solar polar coronal holes extend to low latitudes.
This timing difference leads to a significant negative correlation between yearly sunspots and HSS/CIR fractions (cc = -0.42; p = 0.0012).
Note that the (relative) HSS/CIR fraction is, on an average, 41.3%, i.e., twice larger than the average CME fraction.
In fact, the HSS/CIR fraction is larger than the CME fraction in almost all years except for a few sunspot maximum years.
Interestingly, the cycle maxima of the HSS/CIR fraction seem to shift earlier in the sunspot cycle during the time interval included in Fig. <ref>.
The HSS/CIR maxima are found in the pre-minimum to late declining phase in SC20–SC22, but in the early to mid-declining phase in SC23–SC24.
Frequent high-speed streams in 1974 were found to come from persistent polar coronal hole extensions <cit.>, while in 2003 most HSSs originated from low-latitude coronal holes <cit.>.
These differences reflect the systematic long-term evolution in the structure of solar magnetic fields (to be discussed later in more detail).
Note also that, similarly to the CME fraction, the HSS/CIR fraction is smaller during SC24 than in earlier cycles.
The slow wind fraction depicts a small solar cycle variation with maxima typically during or soon after sunspot minima, and minima around sunspot maxima.
Accordingly, there is a significant negative correlation between yearly sunspots and slow wind fractions (cc = -0.59; p = $1.1*10^{-6}$).
However, the most prominent feature in the slow wind fraction is its recent increase.
Until 2002 the average slow wind fraction was 36.0%, twice larger than the CME fraction but clearly smaller than the HSS/CIR fraction.
However, after 2003 the average slow wind fraction is 50.5%, making the slow solar wind the most common solar wind flow type in the last two decades.
This recent increase in slow wind fraction during weakening sunspot activity contributes to the above-mentioned negative correlation between the two parameters.
The two highest peaks in the slow wind fraction occur around the two sunspot minima in 2009 and 2019, and attain values slightly above and below 70%, respectively.
This increase in the slow wind fraction during the last two decades reflects the widening of the streamer belt due to the weakening of solar polar magnetic fields <cit.>.
Polar fields, again, are slowly weakening since cycle 22 as a consequence of the slow decrease in the overall solar activity that characterizes the whole space age.
§ STORM IDENTIFICATION
We use here a fairly similar procedure to identify geomagnetic storms as adopted earlier by Yakovchouk_2012.
We apply this procedure here separately both to the Dxt index and to the Dst index in order to study how the storms of different classes differ between these two indices.
According to the storm identification procedure, we first locate all local minima in the Dxt/Dst index in 1964–2021.
A storm is then identified as the deepest index minimum within a 2-day (48-hour) interval from any other minimum.
Accordingly, all minima within a 2-day interval are counted to belong to the same storm as separate (smaller) intensifications.
This definition correctly joins together to one storm, e.g., the two possible intensifications of a CME storm, one due to the sheath and the other due to the core (ejecta), which typically have a smaller time separation of about 12–24 hours.
However, using this definition we miss one storm in those cases where two separate CMEs, each of which exceeds the storm threshold, follow each other within two days.
While a more detailed estimate will be made in a separate study later, we have made a preliminary estimate that this leads to an underestimate of storms by less than about 10–15%.
This is too small to have an effect on our main results and, therefore, no change due to possibly omitted storms is made here.
However, we note that a shorter time interval of 2 days was applied here, in difference to the 3-day interval used by Yakovchouk_2012, because the shorter separation alleviates the problem of possibly omitted storms.
This also leads to somewhat higher numbers of storms here than in Yakovchouk_2012.
Figure <ref> depicts the Dxt index (red line) and the Dst index (blue line) in March-April 1980, as well as the storms identified by the Dxt index (magenta squares) and the Dst index (blue dots).
The Dst index identifies six storms during this time interval, but the Dxt index only five storms.
As discussed above, the Dxt index is, on an average, slightly higher than the Dst index and misses the weak storm in April 7 (min Dxt = -24 nT; min Dst = -32 nT).
Overall, the selected index minima are quite appropriate to denote the peaks of storm main phases.
Dxt index (red line) and Dst index (blue line) in March–April 1980, as well as storms identified by the Dxt index (magenta dots) and the Dst index (blue dots).
We have classified the storms identified by the Dxt index (the Dxt storms) and, separately, by the Dst index (the Dst storms) according to the minimum value of the respective index into four storm classes or categories using the following definition:
* Weak: -50 nT $<$ Dxt, Dst $\leq$ -30 nT
* Moderate: -100 nT $<$ Dxt, Dst $\leq$ -50 nT
* Intense: -200 nT $<$ Dxt, Dst $\leq$ -100 nT
* Major: Dxt, Dst $\leq$ -200 nT.
§ TOTAL STORM NUMBERS DURING SPACE AGE
Table <ref> shows the storm numbers in 1957–2021 both in total and when classified into the four intensity categories, separately for the two indices.
There are in total 2526 geomagnetic storms according to the Dxt index and 2743 storms, i.e., some 8.6% more, according to the Dst index.
When dividing these by the number of years (65), one finds that there have been 39/42 storms per year according to the Dxt/Dst index.
Thus, as a rule of thumb, one can say that there have been, on an average, three storms per solar rotation during the space age.
The largest category of storms, about 45% of all storms are weak storms, while 40% are moderate storms, 12% intense storms and 3% major storms.
Almost the same percentages are found for both indices, which gives evidence for the robustness of these results.
So, roughly speaking, almost a half of the storms were weak storms and three quarters of the rest were moderate storms.
As seen in Table <ref>,
There are 8.0% more of weak storms, 10.1% more of moderate storms, 6.9% more of intense storms, and 4.4% more of major storms in the Dst index than in the Dxt index.
Table <ref> also lists the means of the storm minimum Dxt/Dst (storm peak) values (mean intensities), for all storms and separately for the four intensity categories.
One can see that the two indices give very closely similar storm peak mean values of about -38 nT, -68 nT, -131 nT and -277/-276 nT for the weak, moderate, intense and major storms.
Even though the Dst index produces more storms than the Dxt index, the distribution of storm numbers as a function of storm intensity is very similar for the two indices.
Therefore the storm mean intensities also remain quite similar.
The mean storm peak value for all storms is almost the same as for the moderate storms.
Storm numbers (with respective percentages) and mean intensities for four storm classes in 1957–2021.
Weak Moderate Intense Major All
Dxt storms 1142 (45.2%) 1012 (40.1%) 304 (12.0%) 68 (2.7%) 2526
Dst storms 1233 (45.0%) 1114 (40.6%) 325 (11.8%) 71 (2.6%) 2743
Mean Dxt-min -38.1 nT -67.5 nT -130.5 nT -277.4 nT -67.5 nT
Mean Dst-min -38.3 nT -67.7 nT -130.6 nT -275.8 nT -67.3 nT
§ YEARLY STORM NUMBERS DURING SPACE AGE
Figure <ref> depicts the yearly numbers of Dxt storms (red lines and dots) and Dst storms (blue lines and dots) during the space age.
The top panel of Fig. <ref> shows the yearly numbers of all storms, with yearly sunspots depicted as gray background to indicate sunspot cycles.
One can see that in most years the number of Dst storms is slightly larger than Dxt storms.
Only in 8 years the Dxt index includes more storms than the Dst index.
Yearly numbers of Dxt and Dst storms. Top: all storms with yearly sunspots depicted as gray background; Second: weak storms; Third: moderate storms; Bottom: yearly sums of intense and major storms.
Despite these differences in the total number of storms, the two indices agree on the two years when the number of storms is largest: 1960 and 2003, as well as on the two years when it is smallest: 1965 and 2009.
The two indices also depict almost the same number of storms in these four extreme years.
The storm numbers experienced a dramatic decline in the declining phase of SC23 from 60/61 storms in 2003 to 7/8 storms in 2009.
A similar but slightly less dramatic decline occurred in the declining phase of SC19 from 58/58 storms in 1960 to 11/17 storms in 1965.
After 1960, the two storm number series agree quite well with each other until the declining phase of SC20.
The differences between Dxt and Dst storm numbers are particularly large in the declining phases of cycles 20 and 21 and around the maxima of cycles 22 and 23.
They agree very well again during the last two, rather quiet decades since 2003.
The larger differences in yearly storm numbers during the more active cycles correspond well to the time evolution of the Dxt-Dst difference depicted in the bottom panel of Figure <ref>.
The second panel of Fig. <ref> shows the yearly numbers of weak storms during the space age.
Weak storms follow a fairly systematic solar cycle variation with maxima in the declining phase of the cycle.
During SC19–SC22 these maxima are located in the late declining phase or even at sunspot minimum, but during the last two cycles SC23–SC24 they have shifted to the early to mid-declining phase of the cycle.
We will study this shift later in this paper when discussing the long-term evolution of solar magnetic fields and storm drivers.
During the active cycles SC20–SC22 the number of weak storms had cycle minima typically around sunspot maxima, while during SC19 and the last two cycles the weak storm minima were around sunspot minima.
Years 1965 and 2009, which are exceptionally low in the number of all storms (see top panel of Fig. <ref>), are the lowest also in weak storms but do not stand out so dramatically among other cycle minima of weak storms.
Moreover, the number of weak storms has been at a quite high level even during the weak cycle 24, around the minimum thereafter and in the first years of cycle 25.
The third panel of Fig. <ref> shows the yearly numbers of moderate storms, which have a fairly similar long-term evolution as the number of all storms.
Here the minimum of 2009 also stands out quite dramatically.
Cycle maxima of moderate storms are mostly found slightly earlier in the declining phase than for weak storms.
We will discuss this difference in more detail later in the connection with storm drivers.
Note that, excluding some years like 2003, there is a rather steady long-term decline in the number of moderate storms since the maximum in SC21.
The latest minimum between SC24 and SC25 also remained quite low in the number of moderate storms.
The bottom panel of Fig. <ref> shows the yearly numbers of intense and major storms combined together (to be called here large storms), because the yearly number of major storms is very low and vulnerable to be dominated by random fluctuations.
The cyclic evolution of large storms, as well as their cycle maximum numbers, follow the sunspot cycles and their heights very well.
This is due to the fact that most large storms are caused by CMEs, which follow sunspots <cit.>see, e.g.,>[]Webb_and_Howard_1994, Gopalswamy_2004, Cremades_2007, Robbrecht_ASR_2006, Webb_LRSP_2012.
Note also the steady decline in the cycle maxima of large storms since the maximum of SC22 in 1989.
Moreover, the 4-year period in 2007–2010 with no large storms is unique in the 65-year time interval, masking out even the latest minimum with two years (2019–2020) devoid of large storms.
Figure <ref> also gives information on the differences between Dxt and Dst storm numbers.
There are 36 years when the number of weak Dst storms is larger than the number of weak Dxt storms.
However, there are also 19 years when the number of weak Dxt storms is larger than the number of weak Dst storms.
This leads to the perhaps somewhat surprising result that the relative difference between the Dst and Dxt storm numbers is not in weak storms but in moderate storms.
Note also that the large storm numbers are the same for Dxt and Dst in 41 years.
§ DRIVER CONTRIBUTIONS TO STORM NUMBERS
We have identified the solar wind driver (flow type) for each storm in 1964–2021 (when there is solar wind flow type data available) by comparing the time of the storm peak (Dxt/Dst minimum) with the corresponding time in the list of solar wind flow types.
A storm is assigned as a CME, HSS/CIR or slow wind storm, if the storm peak was inside the corresponding solar wind stream.
Since, as discussed above, Richardson's classification has gaps, some storms remain unclassified.
We call the latter no-SW storms.
(Corresponding hours are called “unclear” in Richardson list.)
The number of storms for each intensity category and solar wind driver, separately for Dxt and Dst storms, are summarized in Table <ref>.
Out of a total of 2202/2416 storms in 1964–2021, 2059/2244 storms (93.5%$/$92.9%) could be assigned to a solar wind driver.
This large fraction guarantees that the results on the relative fractions of storms of different solar wind drivers are representative and reliable.
Using the storm numbers of Table <ref> we have also calculated various fractions of storms and listed them in Table <ref> and Table <ref>.
Table <ref> gives the fraction (in percentage) of storms of a certain intensity category and solar wind driver out of all solar wind-classified storms, separately for Dxt and Dst storms.
Accordingly, the sum of all numbers in the four central columns (or only in the 'All' column) of Table <ref> add up to 100%.
Note that the relative fraction of all weak storms has increased from Table <ref> due to the shorter time interval and the neglect of the active cycle 19 which has relatively more of moderate and large storms than in the whole time interval.
On the other hand, Table <ref> lists the fraction (in percentage) of Dxt/Dst storms of certain intensity and driver out of all storms of certain intensity.
Thus, the sum of each column in Table <ref> is 100%.
Dxt/Dst storm numbers in 1964–2021 classified according to storm intensity and storm driver.
Weak Moderate Intense Major All
CME 162/154 371/377 204/218 48/51 785/800
HSS 591/640 386/452 35/37 0/0 1012/1129
Slow 203/237 56/76 3/2 0/0 262/315
No SW 75/98 63/66 5/8 0/0 143/172
All 1031/1129 876/971 247/265 48/51 2202/2416
Fractions (in percentage) of Dxt/Dst storms of certain intensity and driver out of all Dxt/Dst storms in 1964–2021 covered by solar wind classification.
Weak Moderate Intense Major All
CME 7.9%/6.9% 18.0%/16.8% 9.9%/9.7% 2.3%/2.3% 38.3%/35.7%
HSS 28.7%/28.5% 18.7%/20.1% 1.7%/1.6% 0.0%/0.0% 49.2%/50.3%
Slow 9.9%/10.6% 2.7%/3.4% 0.1%/0.1% 0.0%/0.0% 12.7%/14.0%
All 46.5%/46.0% 39.4%/40.3% 11.7%/11.4% 2.3%/2.3% 100%/100%
Fractions (in percentage) of Dxt/Dst storms of certain intensity and driver out of all Dxt/Dst storms of certain intensity in 1964–2021.
Accordingly, the sum of each column is 100%.
Weak Moderate Intense Major
CME 16.9%/14.9% 45.6%/41.7% 84.3%/84.8% 100%/100%
HSS 61.8%/62.1% 47.5%/49.9% 14.5%/14.4% 0%/0%
Slow 21.3%/23.0% 6.9%/8.4% 1.2%/0.8% 0%/0%
Tables <ref> and <ref> show that the HSS/CIR streams produced 1012/1129 storms according to the Dxt/Dst index, almost exactly one half (49.2%$/$50.3%) of all solar wind-classified storms in 1964–2021.
As a rule of thumb, HSS/CIR streams produced some 1.5 storms per solar rotation.
There were 785/800 CME storms, making a good third (38.3%$/$35.7%) of all solar wind-classified storms.
This is very close to one CME storm per solar rotation, on an average.
Finally, there were 262/315 (12.7%$/$14.0%) slow wind storms, roughly one slow wind storm in a couple of rotations.
The three solar wind streams contributed very differently to the three storm intensity categories.
This is most notable for the largest storms, with CME streams producing all 48/51 major Dxt/Dst storms and 204/218 (84.3%/84.8%) out of the 242/257 solar wind-classified intense storms.
Altogether, CME streams were responsible for 87% of all large (intense and major) solar wind classified storms, while HSS/CIR streams produced only 12% and slow wind streams only 1% of them.
This solar wind driver division (87%$/$12%$/$1%) is the same for the large storms of both indices.
The strongest storm in this 65-year interval occurred on 14 March 1989.
It was, naturally, driven by a CME and reached the maximum intensity of -594/-589 nT according to Dxt/Dst index.
This storm was by far larger than all other storms, since the second strongest storm on 20 November 2003 reached the intensity of only -427/ -422 nT, and all other storms remained above -400 nT.
No HSS/CIR storm reached even close to the threshold of a major storm, since the strongest HSS/CIR storm on 13 September 1993 had the intensity only of -155/-161 nT.
As seen in Tables <ref> and <ref>, there were altogether 35/37 (14.5%/14.4%) intense HSS/CIR driven storms.
Slow wind streams produced only 3/2 (1.2%/0.8%) intense storms.
§ DRIVER CONTRIBUTIONS YEARLY
Figures <ref> and <ref> show the yearly numbers of Dxt/Dst magnetic storms of different categories in 1964–2021, separately for the three solar wind drivers: CME streams (red line and dots), HSS/CIR streams (blue) and slow solar wind streams (cyan).
Figures <ref> and <ref> depict all storms (second panel), weak storms (third panel), moderate storms (fourth panel) and large (intense and major) storms (fifth panel) related to the three drivers.
Yearly and monthly sunspot numbers are depicted in top panel for reference.
Top panel: Monthly (blue) and yearly (red) sunspot numbers for reference.
Panels 2–5: Yearly storm numbers of all (panel 2), weak (panel 3), moderate (panel 4) and large (intense and major; panel 5) storms of the Dxt index,
separated into CME (red), HSS/CIR (blue), and slow solar wind (cyan) storms.
Same as Figure <ref>, but using storms of the Dst index.
The third panel of Figures <ref> and <ref> verifies that the weak storms are mainly driven by HSS/CIR streams.
The number of weak HSS/CIR storms greatly exceeds the number of weak CME storms or weak slow wind storms in most years, and maximizes in the declining phase of each cycle.
Cycle maxima of the yearly number of weak HSS/CIR storms vary from 24 in SC20 to 12 in SC24.
There is an interesting long-term shift in the occurrence of weak HSS storms and in the location of their cycle maxima.
While during SC20, SC21 and SC22 most weak HSS/CIR storms are found
in the late declining to minimum phase,
during SC23 and SC24 they occur clearly earlier in the cycle, with cycle maxima in the early to mid-declining phase.
This shift agrees with a similar shift in the location of cycle maxima of all weak storms (see Fig. <ref>), and proves that this shift is related to the evolution of HSS/CIR storms.
In fact, a similar shift is also seen in the fraction of HSS/CIR streams depicted in Figure <ref>.
On the other hand, in accordance with the close relation between sunspots and CMEs <cit.>see, e.g.,>[]Webb_and_Howard_1994, Gopalswamy_2004, Cremades_2007, Robbrecht_ASR_2006, Webb_LRSP_2012, the number of weak CME storms also maximizes in sunspot maximum years, but remains mostly below the number of weak HSS/CIR storms even then.
As shown in Table <ref>, slow wind streams produce more weak storms than CME streams.
In fact, as seen in Figures <ref> and <ref>, the yearly numbers of weak slow wind storms even surpass the corresponding numbers of weak HSS/CIR storms in a couple of years.
The number of weak slow wind storms has notably increased during cycle 24.
The average annual number of weak slow wind storms since 2010 is some 70% larger than before 2000.
This increase is in agreement with the increasing fraction of slow wind streams in solar wind seen in Figure <ref>.
The occurrence of large CME storms (see bottom panel of Figs. <ref> and <ref>) follows closely the sunspot cycle.
Note, however, that the number of large CME storms decreases in SC24 relatively faster than, e.g., the number of weak or moderate CME storms.
As discussed above (see Table <ref>), only 12% of large storms are HSS/CIR storms.
Interestingly, the number of large HSS/CIR storms maximized in the declining phase of cycle 22, in early to mid-1990s, when their number even exceeded the number of large CME storms in four successive years (1993–1996).
In fact, about 43% of all large HSS/CIR storms occurred in six years from 1991–1996.
This is very similarly reproduced in both Dxt and Dst storms.
We will discuss this development in Section <ref> in more detail.
Moderate storms (fourth panel in Figs. <ref> and <ref>) are caused roughly equally by CME and HSS/CIR streams.
In fact, as shown in Table <ref>, out of the 813/905 moderate Dxt/Dst storms with solar wind data, 371/377 are driven by CMEs and 386/452 by HSS/CIRs.
Moderate CME storms peak around the sunspot maximum, while moderate HSS/CIR storms maximize in the declining phase.
As in the case of weak HSS/CIR storms, moderate HSS/CIR storms peak somewhat earlier in the declining phase in the two recent, less active solar cycles (SC23 and SC24) than in the earlier, more active cycles.
The all-time maximum number of moderate HSS/CIR storms occurred in 2003, in the mid-declining phase of cycle 23.
This year also marks the all-time maximum of all HSS/CIR storms during the 65-year time interval, as seen in the second panel of Figures <ref> and <ref>.
This year was the third most frequent year in high-speed streams (see Fig. <ref>) that mostly emanated from persistent, isolated low-latitude coronal holes <cit.>.
§ CME STORMS DURING SPACE AGE
We have studied magnetic storms during the space age, which is characterized by the decline of the Modern Grand Maximum when solar activity has been slowly but unsteadily decreasing.
As discussed above, large storms (bottom panel in Figs. <ref> and <ref>) are almost exclusively CME storms, whose occurrence is related to the new magnetic flux emerging on the solar surface.
Accordingly, the temporal evolution of the yearly numbers of large storms closely follows the corresponding evolution of sunspot numbers.
Figure <ref> shows that the first years 1957–1960 include the largest average number of large storms in any sequence of four consecutive years since 1957.
This agrees very well with the fact these are the years of the maximum and early declining phase of SC19, and form the stormy peak of the MGM.
The yearly numbers of large storms follow not only the sunspot cycle but also the longer-term evolution of sunspot activity during the whole space age.
Large storms are, similarly to sunspots, greatly depleted from SC19 to SC20.
Then their number increases until SC22 and, thereafter, systematically decreases during cycles 23 and 24.
The variation of sunspot cycle amplitudes (see top panel of Fig. <ref>) roughly corresponds to the variation of cycle maxima of large storms, with the exception that the peak of SC22 is slightly higher in large storms than SC21, while the two cycles are roughly equally high in sunspots.
In order to better quantify the relation between large storms and sunspots, we have calculated the correlation between yearly sunspot numbers and yearly numbers of large storms using different (yearly) lags between the two parameters.
The maximum correlation
is obtained at zero lag and has a correlation coefficient of 0.74 (p $< 2*10^{-10}$).
Accordingly, the relation between sunspots and large storms is extremely significant, and sunspots explain more than a half of the variability of yearly numbers of large storms.
This result supports the intimate connection between the two parameters.
A much higher correlation between sunspots and large storms is not even expected since, in addition to sunspots, also other, much longer-lived active regions and filaments can produce CMEs <cit.>.
We also note that the correlation between sunspots and large storms is considerably reduced already at one-year lag.
Assuming that the fractions of Table <ref> are valid for the whole time period of Figure <ref> (so, also for the first 7 years 1957–1963 not covered by solar wind flow type data), a slightly smaller fraction than one half of all moderate storms are CME storms.
On the other hand, almost exactly one half of moderate storms are HSS/CIR storms.
As discussed above, the moderate HSS/CIR storms have their cycle maxima in the early to mid-declining phase of the solar cycle.
Accordingly, we find that the maximum correlation between the yearly numbers of sunspots and all moderate storms is obtained at one-year delay.
This correlation (cc = 0.58) is lower than for large storms but is still extremely significant (p $< 5*10^{-6}$).
§.§ Large CME storms in five separate solar cycles
Scatter plot of yearly sunspots and yearly numbers of large CME storms in 1965–2019, together with the corresponding best fit line (black thick line) and the two 95% CL lines (black dashed lines).
Years of the five sunspot cycles and their separate best fit lines with sunspots are marked with separate colors and markers as indicated in legend.
Slopes, intercepts, correlation coefficients and p-values for the correlation between yearly sunspots and large CME storm numbers for five solar cycles (years in second column) and for all years in 1965–2019.
Cycle Years slope intercept cc p
SC20 1965–1975 0.040 0.30 0.72 0.013
SC21 1976–1986 0.024 2.94 0.55 0.082
SC22 1987–1996 0.067 -1.16 0.97 2.3$*10^{-6}$
SC23 1997–2008 0.047 1.31 0.79 0.002
SC24 2009–2019 0.029 0.67 0.53 0.092
All 1965–2019 0.044 0.68 0.76 2.3$*10^{-9}$
Using data from the time interval (1964–2021) that is covered with information on solar wind flow types,
we find a slightly higher correlation coefficient (cc = 0.75) between the yearly numbers of sunspots and the (flow type certified) large CME storms.
However, due to the slightly smaller number of years, the significance is slightly reduced but remains extremely high (p $<4*10^{-10}$).
In view of conducting here a study of individual, complete solar cycles, we limit the time to 1965–2019, from the start year of SC20 to the end year of SC24.
Calculating the same correlation for these years (1965–2019), we find a similar change: the correlation coefficient is slightly increased to cc = 0.76, and the p-value is reduced to p $<2.3*10^{-9}$.
Thus, sunspots explain, on an average, about 60% of the variability of the yearly number of large CME storms.
We have depicted these latter values, as well as the slope and intercept of the best fit line of the corresponding linear correlation in the last row of Table <ref>.
The decline in the yearly number of large CME storms from SC22 to SC24 depicted in the bottom panel of Figures <ref> and <ref> is indeed quite dramatic and may even look exceptional compared to the corresponding decline in sunspot activity.
In order to study the relation between sunspots and large CME storms we have plotted the scatter of all yearly sunspots and yearly numbers of large CME storms in 1965-2019 in Figure <ref>.
We have included there the corresponding best fit line of linear correlation between the two parameters and the 95% CL lines of the fit.
Note that the intercept of this best fit line is close to zero (0.68), indicating that sunspots are, statistically, prerequisite to the occurrence of large CME storms.
At intermediate sunspot activity with yearly mean of about 100, the number of large CME storms is about 5 and at higher sunspot activity of 200, twice larger.
However, the 95% CL band is quite wide, and allows the storm numbers to vary roughly between 1 and 9 for intermediate sunspot activity and between 6 and 13 for high sunspot activity.
We have marked in Figure <ref> the years of each of the five solar cycles (SC20–SC24) with a specific color and marker.
We used the sunspot minimum months given at the Solar Influences Data analysis Center (SIDC) (see http://sidc.oma.be/silso/cyclesminmax) to separate the cycles.
(The minimum years were allocated to the cycle which included a larger fraction of the respective year.)
One can see that there are "outliers" (points beyond the 95% CL lines; stricly speaking they are not outliers but we use this word for lack of a more suitable term) in Fig. <ref> from each of the five solar cycles.
Cycle 21 has two outliers, all others one, including cycle 24, which does not deviate from the other cycles in this relation.
In order to study if there was a change in the relation between sunspots and large CME storms during SC24 or, in fact, during any of the five cycles 20–24, we have correlated yearly sunspots and large CME storm numbers separately for each of the five solar cycles.
We have plotted the corresponding five best fit lines also in Fig. <ref> using the same cycle-specific color scheme as for the yearly dots.
The slopes, intercepts, correlation coefficients and p-values of the best fit lines for the five cycles are given in Table <ref>.
Figure <ref> and Table <ref> show that the intercepts and even the slopes of the five best fit lines vary considerably from cycle to cycle.
The best fit lines of cycle 20 and cycle 23 have almost the same slope as the overall best fit line of all years.
However, the intercepts of SC20 and SC23 best fit lines are quite different, leading to SC23 producing typically one large CME storm more for the same amount of sunspot activity than SC20.
Still, both of these lines remain well within the 95% CL lines.
Correlation coefficients for SC20 and SC23 are also quite close to the correlation coefficient of the overall fit, and correlation is significant for both of these two cycles.
Cycles 21 and 24 have lower slopes than the slope of the overall best fit line.
The intercept of SC21 is larger than the intercept of SC24, in fact larger than any other intercept.
Correlation coefficients for SC21 and SC24 are smaller than for other cycles, explaining only some 30% of variation.
In fact, correlation between yearly sunspots and large CME storms is not even significant for cycles 21 and 24 at 95% confidence level, but the fact that the respective p-values are below 0.10 indicates marginal significance.
Figure <ref> shows that SC21 has two outlier points, one of them farthest below the lower 95% CL line.
Curiously, for both SC21 and SC24, the outlier points correspond to the highest sunspot activity years of the respective cycles.
As seen in Figures <ref> and <ref>, there are two peaks in the number of large CME storms during these cycles, separated by a dramatic dropout exactly at the respective sunspot maxima (years 1979 and 1980 in SC21 and year 2014 in SC24).
This evolution, which essentially deteriorates the studied correlation in these cycles, closely reminds the structure of the Gnevyshev gap <cit.>.
A simultaneous Gnevyshev gap in sunspots is clearly visible (although far less strong than in storm number) only in SC24, but not in SC21 where sunspots even reach the highest yearly value among the five cycles studied.
Cycle 22 deviates from the other cycles in all aspects.
As seen in Table <ref>, the slope of SC22 is the largest, more than twice the slopes of SC21 and SC24, and some 50% larger than the slope in SC20 and SC23.
Most dramatically, there is an almost perfect correlation between the yearly sunspots and large CME storms in SC22, with sunspots explaining almost 95% of the annual variation of large CME storms.
The close relation between sunspots and large CME storms in SC22 can also be seen (see Figs. <ref> and <ref>) in that both sunspots and large CME storms have SC22 maximum in 1989 and another, slightly lower peak in 1991.
This is another demonstration of a Gnevyshev gap that is simultaneous in both parameters.
§.§ Moderate CME storms in five separate solar cycles
Same as Figure <ref> but for moderate CME storms.
Slopes, intercepts, correlation coefficients and p-values for the correlation between yearly sunspots and moderate CME storm numbers for five solar cycles (years in second column) and for all years in 1965–2019.
Cycle Years slope intercept cc p
SC20 1965–1975 0.040 2.96 0.72 0.012
SC21 1976–1986 0.052 3.20 0.78 0.005
SC22 1987–1996 0.067 0.27 0.88 0.0007
SC23 1997–2008 0.045 2.43 0.64 0.025
SC24 2009–2019 0.072 0.91 0.76 0.006
All 1965–2019 0.055 1.84 0.78 1.8$*10^{-12}$
Figure <ref> depicts similar correlations for moderate CME storms as Figure <ref> for large CME storms.
The corresponding parameters of correlations are given in Table <ref>.
The slope of the correlation (0.055) between sunspots and all moderate CME storms (bottom row of Table <ref>) is larger than in the case of all large CME storms (0.044), reflecting the overall larger number of moderate CME storms (see Table <ref> and Figs. <ref> and <ref>).
The correlation coefficient for all moderate CME storms (0.78) is almost the same as for large CME storms (0.76), but the p-value is three orders of magnitude smaller, indicating improved significance due to larger statistics.
Comparing the best fit correlation lines of the five cycles in Figure <ref> to the corresponding lines in Figure <ref>, one can see that they are more coherently aligned between themselves and also with the overall best fit line in Figure <ref>.
As Tables <ref> and <ref> show, the slopes vary from 0.040 to 0.072 (by 0.032) for moderate storms and from 0.024 to 0.067 (by 0.043) for large storms.
Similarly, the spread of the five intercepts is smaller for moderate than large CME storms.
Accordingly, the improved significance of the correlation between all moderate CME storms and sunspots is due to the five cycles having more similar correlations for moderate CME storms than for large CME storms.
This is most likely due to the fact that the yearly number of moderate CME storms is larger than the number of large CME storms, which reduces statistical fluctuations between the five cycles for moderate CME storms.
Note also that
the correlation between sunspots and moderate CME storms is significant for all five cycles.
Curiously, for some cycles, the correlation parameters remain surprisingly similar for moderate and large CME storms.
This is most clearly valid for SC20 for which the slopes and correlation coefficients are exactly the same.
Even the p-values are almost the same.
The slopes of moderate CME storms remain almost the same as for large CME storms also in SC22 and SC23, but the significance of correlation is higher for large CME storms in these two cycles.
Still, correlation is highly significant in SC22, where the correlation coefficient for moderate storms (as for large storms) is highest among all cycles, with sunspots explaining almost 80% of the annual variation of moderate CME storms.
SC23 has the weakest correlation and the largest number of outlier points for moderate CME storms.
The slopes and correlation coefficients in SC21 and SC24 are larger for moderate than large CME storms.
Correlations are also 95% CL significant for moderate CME storms (but only marginally significant for large CME storms) in these two cycles.
The slope for moderate CME storms is largest in SC24, even slightly larger than the corresponding slope in SC22.
§.§ CME storms at solar minima
Scatter plot of three-year means of sunspots and CME storms calculated for the minimum years of the four solar cycles SC20–SC22 and SC24 (blue circles),
together with the corresponding best fit line (black thick line) and the two 95% CL lines black dashed lines).
Red square denotes SC23 which is not included in the fit.
Blue solid line denotes the best fit line between yearly sunspots and all CME storm numbers.
A notable feature in the yearly number of all CME storms (see the second panel of Figs. <ref> and <ref>) is the low number of CME storms around the sunspot minimum between cycles 23 and 24 (to be called minimum of SC23).
In fact, there were no CME storms at all in two successive years of 2007–2008.
There was only one other year (1964) with no CME storms within the whole 58-year time interval (1964–2021) with solar wind flow data.
Moreover, since the solar wind flow data was very partial in 1964 (see Fig. <ref>), the lack of CME storms in 1964 is not as solidly founded as in 2007–2008 when the coverage was almost 100%.
Even the latest solar minimum after the weak cycle SC24 had at least two CME storms in each year.
Moreover, during the four-year time interval in 2006–2009, there were altogether only 5 CME storms, out of which only one was large.
In comparison, there were 12 (17) CME storms in the second-weakest (third-weakest) 4-year minimum of 2018–2021 (1993–1996, respectively) in 1964–2021.
Note also that these results on the minimum-time storm numbers are similarly reproduced in both the Dxt and Dst indices, which gives strong support on their validity.
In order to study the relation between yearly sunspots and yearly CME storm numbers during solar minima, we calculated the mean of yearly CME storm numbers in three successive years around each solar minimum when the number of CME storms was smallest.
Figure <ref> shows these three-year mean CME storm numbers for the five cycle minima as a function of the corresponding three-year mean sunspot numbers.
(Cycles 20–22 and 24 are depicted with blue circles, cycle 23 is depicted as a separate red square).
We have also plotted there the best fit line (and the 95% CL lines) for the correlation between the sunspots and the three-year CME storm numbers for the four minima (cycles 20–22 and 24), leaving the minimum of SC23 out of the fit.
The correlation coefficient is fairly high (cc = 0.95) and correlation is significant but, because of the small number of points, the p-value is only 0.047.
Interestingly, Figure <ref> shows that the three-year mean CME activity at the minimum of SC23 is below the best fit correlation line of the four other cycles.
This gives additional evidence for the view that CME activity in the minimum after cycle 23 was exceptionally weak, and underlines the uniqueness of this minimum among all other solar minima during the space age.
We have also included in Figure <ref> the best fit line for the correlation between yearly sunspots and yearly numbers of all CME storms.
This correlation is better (cc = 0.855; p $< 10^{-16}$) than between yearly sunspots and large CME storms (see Fig. <ref> and Table <ref>) or sunspots and moderate CME storms (Fig. <ref> and Table <ref>).
The best fit line of the all-CME correlation is above the 95% confidence lines of the minimum time correlation.
However, since the slope (0.123) of the all-CME best fit line is slightly smaller than the slope for the minimum time correlation, the all-CME line will fall within the 95% CL limit at sunspot value of about 55–60 and will cross the three-year minimum best fit line at about 95.
This indicates a small nonlinearity in the relation between sunspots and CME storms at small sunspot numbers.
§ HSS/CIR STORMS AND HCS EVOLUTION
WSO coronal source surface synoptic maps (radial model with source surface distance at 3.25 solar radii) for one (June–July) month in (left column) 1993 (CR1870), 1994 (CR1884), 1995 (CR1897), 1996 (CR1910) and in (right column) 2015 (CR2165), 2016 (CR2178), 2017 (CR2192), 2018 (CR2205).
Maps are redrawn using WSO data and depict the coronal source surface field for one Carrington rotation in longitude (x-axis from 0$^\circ$ to 360$^\circ$) - latitude (y-axis; from -70$^\circ$ to +70$^\circ$) grid.
Light gray background color denotes the positive polarity region, dark gray color the negative polarity region. (Both minima are positive polarity times).
Neutral line is marked in black thick line and corresponds to the center of the heliospheric current sheet.
Curves on both sides of the neutral line (above NL: blue line; below NL: red line) mark isolines of the source surface field intensity at $\pm0.5, \pm1, \pm2.5$, and $\pm5$ $\mu$T.
As noted in Section <ref>, about 43% of all large HSS/CIR storms occurred in the six years from 1991–1996 (see bottom panel of Figs. <ref> and <ref>).
Such an exceptional occurrence of intense HSS storms at this time is most likely due to the fact that the HCS was exceptionally thin during the declining phase of SC22 and the subsequent minimum <cit.>, allowing the Earth to reach higher heliomagnetic latitudes and more of high-speed streams than during the corresponding time of other solar cycles.
Large HSS/CIR storms had their all-time maximum of four storms per year in two consecutive years in 1993–1994.
However, interestingly, the moderate HSS/CIR storms (see the fourth panel of Figs. <ref> and <ref>) had their cycle maximum one year later, in 1995, and the weak HSS/CIR storms (the third panel of Figs. <ref> and <ref>) in 1996.
(The latter was the all-time maximum for Dst and second highest peak for Dxt).
A similar temporal ordering can be found in the peak number of large, moderate and weak HSS/CIR storms during all cycles of the space age.
Moderate HSS/CIR storms tend to peak not earlier than large HSS/CIR storms, and weak HSS/CIR storms not earlier than moderate HSS/CIR storms.
During the declining phase of SC23, the peak number of large HSS/CIR storms (3/4 in Dxt/Dst) was found in 2002, while the peak of moderate and weak HSS/CIR storms was one year later in 2003.
During SC24, one large HSS/CIR storm occurred in 2013 and also in 2015, according to both indices.
Moderate HSS/CIR storms had their maximum in 2015, and weak storms in 2015 and 2017 in Dxt and only in 2017 in Dst.
During the earlier cycles, the peak numbers and timings are less certain because of the larger number of data gaps in solar wind flow classification, especially in the 1980s (see bottom panel of Fig. <ref>).
Still, the same ordering is followed during the declining phase of SC20, with the peak of large and moderate HSS/CIR storms occurring in 1973, and weak HSS/CIR storms in 1974.
In cycle 21, according to Dxt, large and moderate HSS/CIR storms had their peak in 1984 and weak storms in 1985.
According to Dst, they occurred in 1984, 1983 and 1987, respectively.
Note that the results on the temporal ordering of HSS/CIR storm numbers do not only apply to storm peaks but also to the temporal ordering of the bulk of corresponding HSS/CIR storms.
For all cycles and for both storm indices, the mean of the distribution of moderate HSS/CIR storms occurs earlier in the declining phase than the mean of weak HSS/CIR storms.
This temporal ordering in the occurrence of HSS/CIR storms of different intensity is related to a systematic evolution of the heliospheric magnetic field, in particular of the structure of the heliospheric current sheet, in the declining phase of the solar cycle.
Figure <ref> depicts coronal (source surface) synoptic maps of the Wilcox Solar Observatory (WSO) for one solar rotation in four years in the mid-1990s (1993–1996), in the declining to minimum phase of SC22 (left column), as well as in four years in the late 2010s (2015–2018), in the declining phase of SC24 (right column).
We have selected a rotation including the June month of each year, because the Earth is then close to the solar equator, viewing both hemispheres roughly equally.
The coronal (source surface) magnetic field is calculated from photospheric field observations at WSO with a potential field source surface (PFSS) model <cit.>.
We have selected here the radial model, which assumes that the photospheric field is radial, and the distance of the source surface is at 3.25 solar radii.
This model needs no polar correction and depicts the structure of the HCS very clearly.
Neutral line, the solar magnetic equator and the center of the heliospheric current sheet, is denoted as a thick black line in each map of Figure <ref>.
Neutral line divides the coronal (source surface) magnetic field into two opposite polarity regions, which for both SC22 and SC24 are ordered in the same sense of positive (negative) polarity field dominating in the northern (southern, respectively) hemisphere.
(Both are minima of positive solar polarity).
Positive polarity field is depicted in the synoptic maps with light gray color and negative polarity field with dark gray.
The white region between these two opposite-polarity regions is the heliospheric current sheet (streamer belt), whose center is the neutral line.
Synoptic maps also include other curves which tend to roughly follow the neutral line structure.
These curves are colored in blue above the neutral line and in red below it,
and mark the isolines of the coronal (source surface) field intensity at $\pm0.5, \pm1, \pm2.5$, and $\pm5\,\mu$T.
The $\pm1\,\mu$T lines (second blue and second red lines) define the HCS region and their latitudinal separation is used as an approximate width of the HCS.
Accordingly, the closer the isolines are to each other, the larger the magnetic field gradients around the neutral line are, and the closer to the NL the polar coronal holes and the related high-speed streams are.
Figure <ref> shows that the estimated HCS is rather narrow, about 20–25$^\circ$ wide in heliographic latitude during all the years of SC22 (1993–1996), while it is considerably wider, about 35–40$^\circ$ in SC24 (2015–2018).
This is in agreement with the above noted exceptionally thin HCS during the declining phase of SC22 and the subsequent minimum <cit.>.
The thin current sheet allows a better access for the fast solar wind from polar coronal holes to low heliolatitudes.
Therefore, the Earth is, on an average, more exposed to the high-speed solar wind streams in the 1990s than, e.g., in the 2010s.
The occurrence and temporal duration of high-speed solar wind streams at the Earth depends not only on the thickness of the HCS, but also on its curviness, which is mainly determined by the solar dipole tilt angle.
The dipole tilt angle has a systematic variation along the solar cycle.
It has its maximum of 90$^\circ$ at the time of polarity reversal, which happens close to the sunspot maximum.
The tilt angle decreases slowly but unsteadily during the declining phase to its minimum (typically close to 0$^\circ$) at or soon after the sunspot minimum.
One can see in Figure <ref> the typical decrease of the HCS tilt (curviness) from 1993 to 1996 (left maps) and from 2015 to 2018 (right maps).
Note that these maps are from slightly different parts of the declining phase of the two cycles, reflecting the different evolution of polar fields.
This also leads to the different timing of HSS/CIR storms during these two cycles, as discussed above.
In SC24, the first map in 2015 is from the early declining phase, while the first year of SC22 (1993) is already in the mid-declining phase.
Accordingly, the tilt is slightly larger in 2015 than in 1993 (see top row of Fig. <ref>).
Anyway, in both years, the HCS is quite curved and regions of positive magnetic polarity from the northern hemisphere intrude into the southern hemisphere, and vice versa.
This is clear, e.g, in 1993 when the northern field intrudes into the southern hemisphere in longitude range of about 120–210$^\circ$ and about 270–360$^\circ$, or in 2015 when the southern field intrudes strongly into the northern hemisphere in the longitude range of about 110–220$^\circ$.
During such intrusions the Earth is subjected to the effect of HSS streams from coronal holes (and related CIRs developing during the passage from the Sun to the Earth).
Since the solar wind speed increases with heliomagnetic latitude, i.e., with the distance from the neutral line, the further the intrusion takes the neutral line from the solar equator (or, more exactly, from the ecliptic), the faster the solar wind measured at the Earth is and the longer the HSS stream lasts.
Thus, since the tilt is decreasing during the declining phase of the solar cycle, the intrusions most effectively produce large HSS/CIR storms in the early to mid-declining phase.
Moreover, since the HMF intensity has its cycle maximum in the early declining phase, the southward HMF component, which controls the reconnection in the dayside magnetosphere, is, on an average, also slightly stronger at this time.
These two effects lead to the fact that the cycle maxima of large HSS/CIR storms are typically seen in the early to mid-declining phase, before the corresponding maxima of moderate or weak HSS/CIR storms.
In 1994 and in 2016 (second row of Fig. <ref>), the HCS structure is quite dipolar, but the tilt is already smaller than one year earlier.
Although the Earth's largest distance from the NL is now smaller (Earth reaches slightly lower heliomagnetic latitudes) than one year earlier, the Earth is still during a large fraction of time outside of the HCS.
The tilt is further reduced in 1995 and 2017 (third row of Fig. <ref>), and again in 1996 and 2018 when the Earth was mostly within the HCS region, at least in June.
As seen in Figures <ref> and <ref>, the number of moderate HSS/CIR storms in SC24 systematically declines from the maximum in 2015 onwards, as the tilt is decreasing.
During SC22, the maximum of large HSS/CIR storms was found in 1993–1994 when the tilt was still fairly large, while the moderate HSS/CIR storms peaked in 1995.
Note that Figure <ref> depicts the HCS in June when the Earth is at low heliographic latitudes.
However, during the high-latitude periods in Spring and Fall, when the Earth is below or, respectively, above the solar equator, the HCS thickness plays a crucial role.
Then, if the HCS is narrow, as in the declining phase of SC22, the Earth can be exposed to fast solar wind streams around equinoxes.
For example, in 1996, at the maximum year of weak HSS/CIR storms of SC22, some 75% of weak storms occurred in Spring or Fall, and only 25% in Summer and Winter.
Accordingly, the evolution of the properties (in particular thickness and tilt) of the HCS and solar magnetic fields over the solar cycle and over the whole space age can explain all the observed changes in the occurrence of HSS/CIR storms noted in earlier Sections.
First of all, the temporal ordering of the cycle maxima of large, moderate and weak HSS/CIR storms (in this order) in the declining phase of each solar cycle reflects the systematic decrease of the HCS tilt (curviness) during the respective phase of each solar cycle.
This decrease, again, follows the regular dynamics of solar magnetic fields that also control the solar dipole tilt.
As explained in the Introduction, surges of new magnetic flux create extensions of polar coronal holes to lower latitudes and, thereby, affect the dipole tilt.
When activity subsides in the declining phase, less surges appear and the dipole tilt decreases.
Since large (weak) HSS/CIR storms require a larger (smaller, respectively) tilt, their occurrence maxima follow the decreasing tilt during the declining phase.
We note that the HCS affects the HSS/CIR storm occurrence even at other times of the solar cycle, but the effect is far less clear and systematic because the solar wind speed does not have the same latitudinal ordering at other times (especially around solar maxima) as in the declining to minimum phase of the cycle.
Secondly, the change in overall solar activity during the space age and its effect upon the structure of solar magnetic fields and the HCS can explain the shift in the location of cycle maxima of HSS/CIR storms.
As noted in Section <ref>, during SC20–SC22 the cycle maxima of large and moderate HSS/CIR storms are located in the late declining phase, but during the last two cycles SC23–SC24 they have shifted to the early to mid-declining phase of the cycle (see two lowest panels of Figs. <ref> and <ref>).
The same shift is also seen in the timing of cycle maxima of all weak storms (see second panel of Fig. <ref>).
After the reversal of the solar dipole, surges of new-polarity magnetic flux strengthen the polar fields.
In the case of weak cycles, there are fewer surges, whence the polar fields remain weaker, polar coronal holes smaller and the HCS region wider.
Also, extensions of polar coronal holes only occur in the early to mid-declining phase of the cycle, leading to early maxima of HSS/CIR storms in weak cycles.
Moreover, because of the thick HCS, the Earth stays within the slow wind of the HCS region from quite early on in the declining phase, which reduces the occurrence even of weak HSS/CIR storms.
On the other hand, during strong solar cycles, there is more of new magnetic flux emerging on the solar surface in the form of sunspots and other active regions.
Accordingly, there are more surges that strengthen the polar fields and form coronal hole extensions, which can occur even in the late declining phase of the cycle.
These changes lead to a later maximum of HSS/CIR storms in these cycles.
Strong polar fields also push the HCS region thinner, which allows the occurrence of weak HSS/CIR storms even around sunspot minimum.
This evolution culminated during the extreme cycle of SC22, when the solar polar fields and the HMF attained their maximum value during the space age <cit.>.
Note that the field intensity isolines reach $\pm5 \mu$T in 1990s (left plots of Fig. <ref>) but only $\pm2 \mu$T in 2010s (right plots of Fig. <ref>).
The exceptionally strong polar fields in SC22 produced an exceptionally narrow HCS during the subsequent declining phase and minimum <cit.>, which allowed the HSS streams to occasionally reach the Earth even during the sunspot minimum.
Quite appropriately, the only cycle when the maximum of weak HSS/CIR storms was on a minimum year, was cycle 22.
After SC22, the weakening sunspot activity produced weaker solar polar fields <cit.> and smaller polar coronal holes <cit.> in SC23 and SC24, which led to a thicker HCS during the respective declining phases and minima <cit.>.
This is also seen in the fact that, during this millennium, the Earth has spent an increasing fraction, roughly half of the time within the HCS/streamer belt region.
Also, the slow wind of the streamer belt has become a more important source of weak magnetic storms than ever during the space age.
§ DISCUSSION AND CONCLUSIONS
We have studied in this paper the occurrence of magnetic storms of different intensities during the whole space age from 1957 until 2021.
We have used both the original storm index, the Dst index, and its recalculated and corrected version, the Dxt index.
These two indices have differences, e.g., in overall normalization and quiet-time levels.
Therefore the yearly storm numbers in the four intensity categories extracted from these two indices slightly differ between each other.
However, even despite a small systematic long-term trend in the ratio of the two indices, there is no significant difference in the relative occurrence of storms of different intensities according to the two indices.
Rather, we find that all of the main results on the long-term occurrence of magnetic storms and their implications about the evolution of solar magnetic fields and the solar wind are the same using either of the two indices.
Moreover, the changes in the Sun during the space age leading to a varying number of magnetic storms are much larger than the differences in the storm numbers between the two indices.
There were in total 2526 magnetic storms during the space age according to the Dxt index and 2743 storms, i.e., some 8.6% more, according to the Dst index.
This implies that there were, on an average, 39/42 storms per year, i.e., roughly three storms per solar rotation.
About 45% of all storms were weak storms (-50 nT$<$Dxt/Dst$\leq$-30 nT), 40% moderate storms (-100 nT$<$Dxt/Dst$\leq$-50 nT), 12% intense storms (-200 nT$<$Dxt/Dst$\leq$-100 nT) and 3% major storms (Dxt/Dst$\leq$-200 nT).
So, roughly speaking, almost a half of storms were weak storms and three quarters of the rest were moderate storms.
Almost exactly the same percentages are found for both indices.
The two indices gave also very closely similar storm peak mean values of about -38 nT, -68 nT, -131 nT and -277/-276 nT for weak, moderate, intense and major storms.
We also used solar wind flow type information <cit.> in order to assign magnetic storms occurring in 1964-2021 to their three main solar wind drivers, the coronal mass ejections (CME), the high-speed solar wind streams (HSS/CIR) and the slow wind region related to the streamer belt and the heliospheric current sheet (HCS).
The HSS/CIR streams produced a bit more than one thousand (1012/1129) solar wind-classified Dxt/Dst storms, almost exactly a half of all solar wind-classified storms in 1964–2021.
There were 785/800 CME storms, making a good third (38.3%$/$35.7%) of all solar wind-classified storms, and roughly one CME storm per solar rotation, on an average.
Slow solar wind produced some 300 (262/315) storms, making 12.7%$/$14.0% of all solar wind-classified storms, roughly one slow wind storm in a couple of rotations.
The three solar wind streams contributed very differently to the different storm intensity categories.
CME streams were the cause of all the 48/51 major storms (Dxt/Dst$\leq$ -200 nT) occurring in 1964-2021.
Although CME storms may also include an effect of a HSS, this result proves that the generation of a major storm without a CME was extremely unlikely during the space age.
CMEs produced 84.3%/84.8% of intense (-200 nT$<$Dxt/Dst$\leq$-100 nT) storms, HSS/CIR streams produced 14.5%/14.4% of them and slow wind streams only 1.2%/0.8%.
Moderate storms were caused slightly more often by HSS/CIR streams (47.5%/49.9%) than by CME (45.6%/41.7%) streams, or by slow wind (6.9%/8.4%).
HSS/CIR streams (61.8%/62.1%) were clearly the dominant source of weak storms, but even the slow wind streams (21.3%/23.0%) produced more weak storms than CMEs (16.9%/14.9%).
The whole solar-terrestrial environment during the space age is characterized by a slow, but unsteadily evolving decrease of solar magnetic activity after the maximum of solar cycle 19, the peak of the Modern Grand Maximum (MGM).
This long-term evolution also affects the sources, occurrence frequencies, intensities and other properties of magnetic storms.
Note also that, since the three main solar wind drivers of magnetic storms depend on different solar parameters and vary at different time scales, the long-term decrease in solar activity affects differently on storms of different origin or different intensity.
Coronal mass ejections are mainly produced by fairly new magnetic flux emerging on the solar surface.
This emergence is evidenced and traditionally even quantified by sunspots.
It has been shown that the occurrence of CMEs follows the sunspot cycle <cit.>see, e.g.,>[]Webb_and_Howard_1994, Gopalswamy_2004, Cremades_2007, Robbrecht_ASR_2006, Webb_LRSP_2012.
Here we verified that CMEs producing moderate and, separately, large (intense or major) magnetic storms vary closely with the changing sunspot activity, not only over the solar cycle but even at longer time scales.
Large storms, which are almost exclusively produced by CMEs, occurred most frequently in the four years (1957–1960) of the maximum and early declining phase of SC19, the peak of the MGM.
CME storms mainly follow sunspots within a year, thus being produced by fairly newly emerged flux, rather than distributed flux.
Sunspots explain typically 60% of the variation of the yearly number of large and moderate CME storms.
On an average, the yearly number of large CME storms is zero if no sunspots exist, suggesting that active regions without sunspots are not effective in producing large CME storms.
The yearly number of large CME storms increases to 4–5 storms for an intermediate sunspot number of 100 (version 2.0, <cit.>).
On the other hand, about two moderate CME storms in a year can occur even if no sunspots exist.
This suggests that, e.g., active regions without sunspots, filaments and other forms of distributed (not newly-emerged) solar magnetic fields can produce CMEs that are sufficiently strong for moderate storms.
Increasing yearly sunspot number to 100 (200) increases the yearly number of moderate CME storms, on an average, to about 7 (13) storms.
We studied the correlation between sunspots and yearly number of CME storms separately for each of the five full solar cycles (SC20–SC24) included within the space age.
Most cycles had significant correlations between sunspots and the yearly number of large and, separately, of moderate CME storms.
For large CME storms in SC21 and SC24 correlation was only marginal, which
is due to the fact that large CME storms have two separate peaks around the respective sunspot maxima, with a deep minimum in between.
This evolution reminds of the so-called Gnevyshev gap which is quite a common feature in the solar cycle evolution of several solar and heliospheric parameters <cit.>.
Since sunspots typically depict only a small decrease during a possible Gnevyshev gap (and hardly any in SC21), this leads to a couple of years where storm numbers are small but sunspot numbers high, which deteriorates the overall correlation.
Because such a two-peak structure does not seem to be common to all cycles, it likely results from random fluctuations of rather small numbers of yearly large CME storms.
This will be studied later in more detail.
For moderate CME storms the storm numbers are larger, which reduces the effect of random fluctuations.
Moreover, as discussed above, moderate CME storms can be produced even without sunspots.
These facts may explain the larger number of moderate storms even at the corresponding Gnevyshev gap minimum, which makes the correlation between sunspots and moderate CME storms more significant even for SC21 and SC24, although a (less deep) Gnevyshev gap is seen also in the number of moderate storms for both cycles.
The best correlation between sunspots and large and, separately, moderate CME storms was found during cycle 22.
During SC22, sunspots explain 94% and 77%, i.e., far more than the average 60%, of the variation of yearly number of large and moderate CME storms, respectively.
Cycle 22 also marks the highest peak in the number of large CME storms during the solar wind flow covered period (SC20–SC24).
Sunspots depict a clear (but not very deep) Gnevyshev gap during this cycle, and large CME storms follow this evolution closely, without an excessive decrease between the two peaks.
Thus, the cycle evolution of sunspots and large CME storms is very similar during SC22, leading to the extremely high correlation.
On the other hand, the Gnevyshev gap of moderate CME storms in SC22 is relatively deeper than in large CME storms.
This reduces the respective correlation with sunspots in SC22 below that for large storms, but is still the best for moderate storms among all cycles.
The parameters of the linear correlation between sunspots and large and moderate CME storms also vary considerably from cycle to cycle.
For large CME storms, SC20 and SC23 have almost the same slope as the overall best fit line for all years.
SC21 and SC24 have considerably lower slopes, while SC22 has a slope more than twice higher than SC21 and SC24.
For moderate CME storms, the five slopes deviate less from each other and from the common mean, probably because of better statistics due to a larger number of storms.
However, despite some difference in correlation parameters, the best fit lines of all cycles remained within the 95% CL boundaries of the overall fit.
Thus, interestingly, no cycle clearly deviated from the other cycles in the relation between sunspots and large or moderate CME storms.
This applies also to the low cycle 24.
However, when studying the relation between sunspots and all CME storms around sunspot minima (using 3-year means in order to increase statistics), we found that the minimum between cycles 23 and 24 deviates from the other four minima by remaining below the 95% CL correlation boundary.
This minimum was very weak in sunspot activity, although still slightly more active than the following minimum.
However, it was exceptionally weak in CME storms, with no CME storms at all in two years 2007–2008.
Although this minimum breaks the common correlation between sunspot activity and CME storm occurrence at sunspot minima, the remaining years of the two annexing cycles, SC23 and SC24, recover this relation for full cycles and make it agree with all other cycles.
We have shown in this paper that there is an intimate connection between the occurrence of HSS/CIR storms and the structure of the heliospheric current sheet, in particular its latitudinal width (thickness) and the tilt (curviness).
A thin HCS makes large gradients of solar wind properties around the neutral line (center of the HCS region), whence the Earth is more often affected by high-speed solar wind streams.
On the other hand, the tilt of the HCS determines how high (northern or southern) heliomagnetic latitudes the Earth can reach.
A large tilt in the HCS can produce an intrusion of high-speed solar wind to low heliographic latitudes and to the ecliptic.
The tilt angle of the HCS decreases fairly systematically from very high (about $90^\circ$) to very low tilt angles during the declining phase of the solar cycle.
Three-year running mean numbers of CME (red line) and HSS/CIR (blue line) Dxt storms.
Panels depict (from top to bottom) all storms, weak storms, moderate storms and large storms.
Vertical lines denote the locations of cycle maxima of HSS/CIR storms of respective intensity.
Annual sunspots are depicted as gray background in each panel.
This decrease of the HCS tilt angle controls the occurrence of HSS/CIR storms of different intensity in the declining phase of all solar cycles.
Due to this decrease, intense HSS/CIR storms tend to occur not later than moderate HSS/CIR which, again, tend to occur not later than weak HSS/CIR storms.
We could see this ordering already in the yearly number of HSS/CIR storms depicted in Figures <ref> and <ref>.
However, since random fluctuations may have some effect on the ordering of peaks and, thereby, cast doubt on our results, we have calculated three-year running means of the yearly CME and HSS/CIR storm numbers, separately for weak, moderate, large and all Dxt storms, and depicted them in Figure <ref>.
Three-year running means weight the distribution of storms more widely and, therefore, are less vulnerable to random fluctuations of yearly peaks.
We have included in Figure <ref> also vertical lines to indicate the cycle maxima of HSS/CIR storms of each intensity class.
(In case two years had the same mean number, the first was marked; this only applies to large HSS/CIR storms where numbers are small).
The vertical lines of Figure <ref> clearly demonstrate the above discussed ordering of the cycle maxima of HSS/CIR storms.
During all the five cycles SC20–SC24 of space age, the cycle peak of large HSS/CIR storms did not occur later than the corresponding peak of moderate HSS/CIR storms which, again, was not later than the peak of weak HSS/CIR storms in the same cycle.
This ordering is most clear in the declining phase of SC22, where the three-year mean maxima of large, moderate and weak HSS/CIR storms follow each other in steps of one year from 1993 to 1994 and 1995.
As noted earlier, cycle 22 is also exceptional in the number of large HSS/CIR storms, which is dramatically clearly seen in the bottom panel of Figure <ref>.
More than 40% of all large HSS/CIR storms in 1964–2021 occurred in the six years of 1991–1996.
Although neither moderate nor weak HSS/CIR storms attained their space age peak during SC22, the three-year running mean number of all HSS/CIR storms maximized in 1994, in the late declining phase of SC22.
These results are due to the exceptionally thin HCS in the declining phase of SC22 <cit.>, which resulted from exceptionally strong solar polar and heliospheric fields in the declining phase of this cycle <cit.>.
Note also that the solar magnetic polarity is positive around the minimum between cycles 22 and 23, which enhances geomagnetic activity at high heliolatitudes (in Spring and Fall) by the Russell-McPherron (RMP) mechanism <cit.>.
While this mechanism alone does not cause the exceptional storminess of cycle 22, the thin current sheet also increases the occurrence of dominant HMF sectors at high heliolatitudes, (the so-called Rosenberg-Coleman effect <cit.>), which enhances the effectivity of the RMP mechanism during cycle 22.
Thus, the increased HSS activity and the enhanced RMP effectivity, both due to the exceptionally thin HCS, lead to the exceptional occurrence of large HSS/CIR storms during the declining phase of cycle 22.
Similarly as for CME storms, the long-term decrease of sunspot activity during the space age also affects the occurrence of HSS/CIR storms.
However, the decrease during the space age was not steady or continuous, neither in sunspot activity nor in magnetic storms.
The total number of storms experienced two dramatic dropouts, one in the declining phase of SC19, from the record storm level in 1957–1960 until the second-lowest minimum in 1965, and the other in the declining phase of SC23, which continued to the low cycle 24 (see Fig. <ref>).
Between these two dropouts, both sunspot and storm activity first increased from SC20 on until cycle 22, and then slightly decreased in SC23.
The increase in sunspot activity from SC20 on strengthened the intensity of polar magnetic fields, which culminated in SC22 <cit.>.
Strong polar fields pressed the HCS region exceptionally thin in the declining phase of SC22, producing a record of large HSS/CIR storms during the space age.
Since SC22, polar fields have continued to decline, which has widened the HCS region.
During a wide HCS, the Earth can reach the fast solar wind only at times when the HCS tilt angle is large.
Therefore, the maxima of HSS/CIR storms occur earlier in the declining phase of weak cycles of SC23 and SC24 than during the earlier, stronger cycles.
This can be seen in the yearly storm numbers of Figures <ref> and <ref>, and it can even more clearly be seen in the three-year running mean storm numbers of Figure <ref>.
The shift of cycle peaks of HSS/CIR storms from the late declining phase in the more active cycles SC20–SC22 to the early declining phase in the recent, more quiescent cycles SC23 and SC24 is, very appropriately, most clearly visible for the large HSS/CIR storms.
In SC20–SC22, the maxima of large HSS/CIR storms are found some 3–4 years after the respective sunspot maximum, but in SC23 only 1–2 years later and in SC24 the two maxima occur in the same year.
Since the cycle maxima of moderate and weak HSS/CIR storms occur, as discussed above, later (or not at least earlier) than large HSS/CIR storm peaks, this long-term shift is less clearly seen for the weaker HSS/CIR storms.
For moderate storms, this shift is still seen even in the smoothed numbers of Figure <ref> (peaks in SC20–SC22 are 4–6 years, in SC23 2–3 years and in SC24 2 years after the respective sunspot maximum).
Note, first, that the 3-year smoothing moves the weak storm peak in SC23 to 2007, while for yearly storm numbers it was in 2003 (see Figs. <ref> and <ref>).
Year 2003 was the peak storm year for both weak and moderate (and, naturally, all) HSS/CIR storms, and only one year after the peak in large HSS/CIR storms.
This year is known to be the record year of geomagnetic activity in space age <cit.> and one of the record years of HSS occurrence (see Fig. <ref>).
In this year, the HSSs mainly originated from low-latitude coronal holes <cit.>.
Accordingly, the occurrence of HSS/CIR storms at this time was less strongly controlled by solar polar coronal holes than in the other cycles.
Secondly, the occurrence of weak HSS/CIR storms in SC24 (and partly even in the declining phase of SC23) was increasingly affected by the widening of the HCS.
As noted earlier, the yearly number of weak HSS/CIR storms in the minimum of SC24, and later in the early ascending phase of SC25, is not much less than in the prior declining phase (see Figs. <ref> and <ref>).
This leads to the fairly constant (smoothed) number of weak HSS/CIR storms in Figure <ref>) since the maximum of SC24.
Thirdly, since the HCS has been exceptionally wide for about 15–20 years, the Earth stays, for the same tilt angle, longer inside the HCS region.
Thus, the decrease of solar activity and solar polar fields since SC22 have led to the fact that the Earth stays, since the declining phase of SC23, more than 50% of the time within the HCS and the related slow solar wind (streamer belt).
This has greatly reduced the number of intense and moderate HSS/CIR storms, and significantly raised the relative fraction of slow solar wind as the source of weak magnetic storms.
Concluding, we have studied here the occurrence of magnetic storms of different intensity over the space age (1957–2021).
Space age is characterized by a slow, unsteady decrease of solar activity, which directly affects the CME storms and, indirectly, via its effects to the structure of the heliospheric current sheet, also the HSS/CIR storms.
We find that the variations in sunspot activity closely explain the variation in the yearly number of CME storms over the whole space age.
All cycles separately comply to this common, rather tight connection between sunspots and CME storms.
However, during the weak solar minimum between cycles 23 and 24 the number of CME storms is smaller than predicted by the common relation.
During four years from 2006–2009, there were only five CME storms, out of which only one was large.
In comparison, there were 12 CME storms in the second-weakest four-year time interval of 2018–2021.
Increasing sunspot activity from SC20 to SC22 increased solar polar fields, which led to an exceptionally thin HCS during the declining phase of SC22, producing a record of large HSS/CIR storms.
Subsequent decrease in sunspot activity has weakened solar polar fields and widened the HCS region.
These long-term changes in HCS width have affected the occurrence of HSS/CIR storms during the solar cycle so that HSS/CIR storms occur, since SC23, in the early to mid-declining phase of the solar cycle, while during the earlier, more active cycles 20–22, they maximized in the late declining phase.
Widening HCS has also increased the significance of the HCS and the related slow solar wind (streamer belt) for the Earth.
The Earth spends now, since the declining phase of SC23, i.e., roughly from the start of the new millennium, about 50% of the time, more than ever before during the space age, in the slow solar wind.
This has reduced the number of large and moderate HSS/CIR storms and increased the fraction of slow wind as the source of weak magnetic storms.
The authors acknowledge financial support by the Academy of Finland
to the Postdoctoral Researcher project of L. Holappa (No. 322459) and to the PROSPECT project (No. 321440) of T. Asikainen.
T. Qvick was supported by University of Oulu Graduate School.
We thank I. Richardson for the updated version of solar wind flow data <cit.>.
The Dxt index is maintained by the University of Oulu at http://dcx.oulu.fi/ <cit.>.
The Dst index is provided by the WDC for Geomagnetism, Kyoto, Japan at http://wdc.kugi.kyoto-u.ac.jp/wdc/Sec3.html <cit.>.
Sunspot data are from the World Data Center SILSO, Royal Observatory of Belgium, Brussels, on-line catalogue at https://www.sidc.be/silso/datafiles.
The WSO source surface maps are available at http://wso.stanford.edu/synsourcel.html.
Altschuler, MD.
Newkirk, G.
Magnetic Fields and the Structure of the Solar Corona.
I: Methods of Calculating Coronal Fields Magnetic Fields and the
Structure of the Solar Corona. I: Methods of Calculating Coronal
Sol. Phys.9131-149.
, Maliniemi
Asikainen, T.
, Maliniemi, V.
Mursula, K.
Modeling the contributions of ring, tail, and
magnetopause currents to the corrected Dst index Modeling the
contributions of ring, tail, and magnetopause currents to the corrected Dst
J. Geophys. Res.115A1412203.
Balogh, A.
, Bothmer, V.
, Crooker, NU.
, Forsyth, RJ.
, Gloeckler, G.
, Hewish, A.
Mikić, Z.
The Solar Origin of Corotating Interaction Regions and
Their Formation in the Inner Heliosphere The solar origin of corotating
interaction regions and their formation in the inner heliosphere.
Space Sci. Rev.89141-178.
, Sittler
, Mariani
Burlaga, L.
, Sittler, E.
, Mariani, F.
Schwenn, R.
Magnetic loop behind an interplanetary shock: Voyager,
Helios, and IMP 8 observations Magnetic loop behind an interplanetary
shock: Voyager, Helios, and IMP 8 observations.
J. Geophys. Res. (Space)86A86673–6684.
, McPherron
Burton, RK.
, McPherron, RL.
Russell, CT.
An Empirical Relationship Between Interplanetary
Conditions and Dst An empirical relationship between interplanetary
conditions and Dst.
J. Geophys. Res.804204–4214.
St. Cyr
St. Cyr
Cremades, H.
St. Cyr, OC.
Coronal mass ejections: Solar cycle aspects Coronal
mass ejections: Solar cycle aspects.
Adv. Space Res.4071042-1048.
Crooker, NU.
Cliver, EW.
Postmodern View of M-regions Postmodern view of
J. Geophys. Res.99A1223383–23390.
Daglis, IA.
Ring Current Dynamics Ring Current
Space Sci. Rev.1241-4183-202.
Daglis, IA.
, Kozyra, JU.
, Kamide, Y.
, Vassiliadis, D.
, Sharma, AS.
, Liemohn, MW.
Lu, G.
Intense space storms: Critical issues and open
disputes Intense space storms: Critical issues and open
J. Geophys. Res. (Space)108A51208.
, Thorne
, Baumjohann
Daglis, IA.
, Thorne, RM.
, Baumjohann, W.
Orsini, S.
The terrestrial ring current: Origin, formation, and
decay The terrestrial ring current: Origin, formation, and decay.
Rev. Geophys.37407-438.
Dessler, AJ.
Parker, EN.
Hydromagnetic theory of geomagnetic storms
Hydromagnetic theory of geomagnetic storms.
J. Geophys. Res.64122239-2252.
, Tokumaru
, Hayashi
, Satonaka
Fujiki, K.
, Tokumaru, M.
, Hayashi, K.
, Satonaka, D.
Hakamada, K.
Long-term Trend of Solar Coronal Hole Distribution from
1975 to 2014 Long-term trend of solar coronal hole distribution from 1975
to 2014.
Astrophys. J. Lett.8272L41.
Gannon, JL.
Love, JJ.
USGS 1-min Dst index USGS 1-min Dst
J. Atm. Sol.-Terr. Phys.73323-334.
Gibson, SE.
, Kozyra, JU.
, de Toma, G.
, Emery, BA.
, Onsager, T.
Thompson, BJ.
If the Sun is so quiet, why is the Earth ringing? A
comparison of two solar minimum intervals If the Sun is so quiet, why is
the Earth ringing? A comparison of two solar minimum intervals.
J. Geophys. Res. (Space)114A13A09105.
Gnevyshev, MN.
Essential features of the 11-year solar cycle
Essential features of the 11-year solar cycle.
Sol. Phys.51175-183.
Gonzalez, WD.
, Joselyn, JA.
, Kamide, Y.
, Kroehl, HW.
, Rostoker, G.
, Tsurutani, BT.
Vasyliunas, VM.
What is a geomagnetic storm? What is a geomagnetic
J. Geophys. Res. (Space)995771-5792.
, Nunes
, Yashiro
Gopalswamy, N.
, Nunes, S.
, Yashiro, S.
Howard, R.
Variability of solar eruptions during cycle 23
Variability of solar eruptions during cycle 23.
Adv. Space Res.342391-396.
Solar Variability and Climate Change
Gosling, JT.
The Solar Flare Myth The solar flare myth.
J. Geophys. Res. (Space)9818937.
, Asbridge
, Bame
Gosling, JT.
, Asbridge, JR.
, Bame, SJ.
Feldman, WC.
Solar Wind Speed Variations: 1962–1974 Solar wind
speed variations: 1962–1974.
J. Geophys. Res.815061–5070.
Gosling, JT.
Pizzo, VJ.
Formation and Evolution of Corotating Interaction
Regions and their Three Dimensional Structure Formation and Evolution of
Corotating Interaction Regions and their Three Dimensional
Space Sci. Rev.8921-52.
Gosling, JT.
Riley, P.
The acceleration of slow coronal mass ejections in the
high-speed solar wind The acceleration of slow coronal mass ejections in
the high-speed solar wind.
Geophys. Res. Lett.23212867-2870.
Häkkinen, LVT.
, Pulkkinen, TI.
, Pirjola, RJ.
, Nevanlinna, H.
, Tanskanen, EI.
Turner, NE.
Seasonal and diurnal variation of geomagnetic activity:
Revised Dst versus external drivers Seasonal and diurnal variation of
geomagnetic activity: Revised Dst versus external drivers.
J. Geophys. Res. (Space)108A21060.
, Asikainen
Hamada, A.
, Asikainen, T.
Mursula, K.
A Uniform Series of Low-Latitude Coronal Holes in
1973–2018 A Uniform Series of Low-Latitude Coronal Holes in
Sol. Phys.296240.
Harvey, JW.
Sheeley, NR.
Coronal Holes and Solar Magnetic Fields Coronal holes
and solar magnetic fields.
Space Sci. Rev.23139–158.
Harvey, KL.
Recely, F.
Polar Coronal Holes During Cycles 22 and 23 Polar
Coronal Holes During Cycles 22 and 23.
Sol. Phys.21131-52.
, Wilcox
Hoeksema, JT.
, Wilcox, JM.
Scherrer, PH.
The Structure of the Heliospheric Current Sheet The
structure of the heliospheric current sheet.
J. Geophys. Res.889910.
Karinen, A.
Mursula, K.
A new reconstruction of the Dst index for 1932–2002
A new reconstruction of the Dst index for 1932–2002.
Ann. Geophys.23475-485.
, Mursula
, Takalo
Karinen, A.
, Mursula, K.
, Takalo, J.
Ulich, T.
An erroneous Dst Index in 1971 An erroneous Dst
Index in 1971.
H. Sawaya-Lacoste (), Solspa 2001, Proceedings of
the Second Solar Cycle and Space Weather Euroconference Solspa 2001,
proceedings of the second solar cycle and space weather euroconference ( 477, 443-446).
, Koskinen
Kilpua, E.
, Koskinen, HEJ.
Pulkkinen, TI.
Coronal mass ejections and their sheath regions in
interplanetary space Coronal mass ejections and their sheath regions in
interplanetary space.
Liv. Rev. Solar Phys.1415.
, Pesnell
, Young
Hess Webber
Kirk, MS.
, Pesnell, WD.
, Young, CA.
Hess Webber, SA.
Automated detection of EUV Polar Coronal Holes during
Solar Cycle 23 Automated detection of EUV Polar Coronal Holes during
Solar Cycle 23.
Sol. Phys.25799-112.
Klein, LW.
Burlaga, LF.
Interplanetary magnetic clouds at 1 AU
Interplanetary magnetic clouds at 1 AU.
J. Geophys. Res. (Space)87A2613-624.
, Timothy
Krieger, AS.
, Timothy, AF.
Roelof, EC.
A Coronal Hole and Its Identification as the Source of a
High Velocity Solar Wind Stream A coronal hole and its identification as
the source of a high velocity solar wind stream.
Sol. Phys.29505–525.
Love, JJ.
Gannon, JL.
Revised Dst and the epicycles of magnetic disturbance:
1958–2007 Revised Dst and the epicycles of magnetic disturbance:
Ann. Geophys.273101-3131.
, Holappa
Mursula, K.
, Holappa, L.
Karinen, A.
Correct normalization of the Dst index Correct
normalization of the Dst index.
Astrophys. Space Sci. Transact.441-45.
, Holappa
Mursula, K.
, Holappa, L.
Karinen, A.
Uneven weighting of stations in the Dst index
Uneven weighting of stations in the Dst index.
J. Atm. Sol.-Terr. Phys.73316-322.
, Holappa
Mursula, K.
, Holappa, L.
Lukianova, R.
Seasonal solar wind speeds for the last 100 years:
Unique coronal hole structures during the peak and demise of the Grand Modern
Maximum Seasonal solar wind speeds for the last 100 years: Unique coronal
hole structures during the peak and demise of the Grand Modern
Geophys. Res. Lett.44130-36.
, Lukianova
Mursula, K.
, Lukianova, R.
Holappa, L.
Occurrence of High-speed Solar Wind Streams over the
Grand Modern Maximum Occurrence of High-speed Solar Wind Streams over the
Grand Modern Maximum.
Astrophys. J.80130.
Newell, PT.
Gjerloev, JW.
SuperMAG-based partial ring current indices
SuperMAG-based partial ring current indices.
J. Geophys. Res. (Space)117A5A05215.
, Iyemori
, Sugiura
Nose, M.
, Iyemori, T.
, Sugiura, M.
Kamei, T.
Geomagnetic Dst index. Geomagnetic Dst index.
Web page
() of the
World Data Center for Geomagnetism (WDC-C2), Kyoto, Japan, accessed April 01,
O'Brien, T.
McPherron, R.
An empirical phase space analysis of ring current
dynamics: Solar wind control of injection and decay An empirical phase
space analysis of ring current dynamics: Solar wind control of injection and
J. Geophys. Res.105A47707-7719.
Petrie, GJD.
Solar Magnetism in the Polar Regions Solar
Magnetism in the Polar Regions.
Liv. Rev. Solar Phys.1215.
IG. Richardson
IG. Richardson
Richardson, IG.
Solar wind stream interaction regions throughout the
heliosphere Solar wind stream interaction regions throughout the
Liv. Rev. Solar Phys.1511.
IG. Richardson
IG. Richardson
Richardson, IG.
Cane, HV.
Near-earth solar wind flows and related geomagnetic
activity during more than four solar cycles (1963–2011) Near-earth solar
wind flows and related geomagnetic activity during more than four solar
cycles (1963–2011).
J. Space Weather Space Clim.2A02.
IG. Richardson
, Cane
IG. Richardson
Richardson, IG.
, Cane, HV.
Cliver, EW.
Sources of geomagnetic activity during nearly three
solar cycles (1972–2000) Sources of geomagnetic activity during nearly
three solar cycles (1972–2000).
J. Geophys. Res.1071187.
IG. Richardson
, Cliver
IG. Richardson
Richardson, IG.
, Cliver, EW.
Cane, HV.
Sources of geomagnetic activity over the solar cycle:
Relative importance of coronal mass ejections, high-speed streams, and slow
solar wind Sources of geomagnetic activity over the solar cycle: Relative
importance of coronal mass ejections, high-speed streams, and slow solar
J. Geophys. Res.105A818,203-18,213.
JD. Richardson
JD. Richardson
Richardson, JD.
Paularena, KI.
Streamer Belt Structure at Solar Minima Streamer belt
structure at solar minima.
Geophys. Res. Lett.241435–1438.
, Berghmans
van der Linden
Robbrecht, E.
, Berghmans, D.
van der Linden, RAM.
Objective CME detection over the solar cycle: A first
attempt Objective CME detection over the solar cycle: A first
Adv. Space Res.383475-479.
Rosenberg, RL.
Coleman, PJ., Jr.
Heliographic latitude dependence of the dominant
polarity of the interplanetary magnetic field. Heliographic latitude
dependence of the dominant polarity of the interplanetary magnetic
J. Geophys. Res.745611-5622.
Russell, CT.
McPherron, RL.
Semiannual variation of geomagnetic activity
Semiannual variation of geomagnetic activity.
J. Geophys. Res.7892–108.
Savitzky, A.
Golay, MJE.
Smoothing and differentiation of data by simplified
least squares procedures Smoothing and differentiation of data by
simplified least squares procedures.
Anal. Chem.361627-1639.
, Wilcox
Schatten, KH.
, Wilcox, JM.
Ness, NF.
A Model of Interplanetary and Coronal Magnetic Fields
A model of interplanetary and coronal magnetic fields.
Sol. Phys.6442–455.
Sckopke, N.
A general relation between the energy of trapped
particles and the disturbance field near the Earth A general relation
between the energy of trapped particles and the disturbance field near the
J. Geophys. Res.71133125-3130.
SILSO World Data Center
SILSO World Data Center
SILSO World Data Center.
The International Sunspot Number The International
Sunspot Number.
International Sunspot Number Monthly Bulletin and
online catalogue.
Smith, EJ.
The global heliospheric magnetic field The global
heliospheric magnetic field.
A. Balogh, LJ. Lanzerotti ST. Suess (), The Heliosphere through the Solar Activity Cycle
The heliosphere through the solar activity cycle ( 79).
Smith, EJ.
Balogh, A.
Decrease in heliospheric magnetic flux in this solar
minimum: Recent Ulysses magnetic field observations Decrease in
heliospheric magnetic flux in this solar minimum: Recent Ulysses magnetic
field observations.
Geophys. Res. Lett.35L22103.
, Usoskin
, Kromer
, Schüssler
Solanki, SK.
, Usoskin, IG.
, Kromer, B.
, Schüssler, M.
Beer, J.
Unusual activity of the Sun during recent decades
compared to the previous 11,000 years Unusual activity of the Sun during
recent decades compared to the previous 11,000 years.
Storini, M.
, Bazilevskaya, GA.
, Fluckiger, EO.
, Krainev, MB.
, Makhmutov, VS.
Sladkova, AI.
The GNEVYSHEV gap: A review for space weather The
gnevyshev gap: A review for space weather.
Adv. Space Res.31895-900.
Subramanian, P.
Dere, KP.
Source Regions of Coronal Mass Ejections Source
Regions of Coronal Mass Ejections.
Astrophys. J.5611372-395.
Sugiura, M.
Hourly values of equatorial Dst for the IGY Hourly
values of equatorial Dst for the IGY.
Ann. Int. Geophys. Year359.
Sugiura, M.
IAGA Resolution 2 IAGA Resolution 2.
IAGA Bulletin 27 IAGA Bulletin 27 ( 123).
Madrid, Spain.
Sugiura, M.
Kamei, T.
Equatorial Dst index 1957–1986 Equatorial
Dst index 1957–1986.
A. Berthelier M. Menvielle (), IAGA
Bulletin 40. IAGA Bulletin 40.
Saint-Maur-des-Fossés, FranceISGI Publ. Office.
Takalo, J.
Mursula, K.
A model for the diurnal universal time variation of the
Dst index A model for the diurnal universal time variation of the Dst
J. Geophys. Res. (Space)106A610905-10914.
Takalo, J.
Mursula, K.
On the diurnal UT variation of the Dst index On the
diurnal UT variation of the Dst index.
H. Sawaya-Lacoste (), Solspa 2001, Proceedings of
the Second Solar Cycle and Space Weather Euroconference Solspa 2001,
proceedings of the second solar cycle and space weather euroconference ( 477, 483-486).
Tsurutani, BT.
, Gonzalez, WD.
, Gonzalez, ALC.
, Guarnieri, FL.
, Gopalswamy, N.
, Grande, M.
Vasyliunas, V.
Corotating solar wind streams and recurrent geomagnetic
activity: A review Corotating solar wind streams and recurrent
geomagnetic activity: A review.
J. Geophys. Res. (Space)111A7.
Virtanen, II.
Mursula, K.
Asymmetry of solar polar fields and the southward shift
of HCS observed by Ulysses Asymmetry of solar polar fields and the
southward shift of HCS observed by Ulysses.
J. Geophys. Res.115A14A09110.
Virtanen, II.
Mursula, K.
North-South Asymmetric Solar Cycle Evolution:
Signatures in the Photosphere and Consequences in the Corona North-South
Asymmetric Solar Cycle Evolution: Signatures in the Photosphere and
Consequences in the Corona.
Astrophys. J.781299.
, Nash
Wang, YM.
, Nash, AG.
Sheeley, NR., Jr.
Evolution of the Sun's Polar Fields During Sunspot
Cycle 21 - Poleward Surges and Long-Term Behavior Evolution of the Sun's
polar fields during sunspot cycle 21 - poleward surges and long-term
Astrophys. J.347529–539.
, Robbrecht
Wang, YM.
, Robbrecht, E.
Sheeley, NR., Jr.
On the Weakening of the Polar Magnetic Fields during
Solar Cycle 23 On the Weakening of the Polar Magnetic Fields during Solar
Cycle 23.
Astrophys. J.7071372-1386.
Wang, YM.
Sheeley, J., N. R.
Solar Wind Speed and Coronal Flux-Tube Expansion
Solar Wind Speed and Coronal Flux-Tube Expansion.
Astrophys. J.355726.
On Dst index (description in the IAGA Bulletin No 40).
On Dst index (description in the IAGA Bulletin no 40).
Web page
() of the
World Data Center for Geomagnetism (WDC-C2), Kyoto, Japan, accessed April 01,
Webb, DF.
Howard, RA.
The solar cycle variation of coronal mass ejections and
the solar wind mass flux The solar cycle variation of coronal mass
ejections and the solar wind mass flux.
J. Geophys. Res. (Space)994201-4220.
Webb, DF.
Howard, TA.
Coronal Mass Ejections: Observations Coronal mass
ejections: Observations.
Liv. Rev. Solar Phys.913.
, Mursula
, Holappa
, Veselovsky
Yakovchouk, OS.
, Mursula, K.
, Holappa, L.
, Veselovsky, IS.
Karinen, A.
Average properties of geomagnetic storms in 1932–2009
Average properties of geomagnetic storms in 1932–2009.
J. Geophys. Res. (Space)117A03201.
Zhou, X.
Smith, EJ.
Solar cycle variations of heliospheric magnetic flux
Solar cycle variations of heliospheric magnetic flux.
J. Geophys. Res.114A133106.
Altschuler, MD.
Newkirk, G.
Magnetic Fields and the Structure of the Solar Corona.
I: Methods of Calculating Coronal Fields Magnetic Fields and the
Structure of the Solar Corona. I: Methods of Calculating Coronal
Sol. Phys.9131-149.
, Maliniemi
Asikainen, T.
, Maliniemi, V.
Mursula, K.
Modeling the contributions of ring, tail, and
magnetopause currents to the corrected Dst index Modeling the
contributions of ring, tail, and magnetopause currents to the corrected Dst
J. Geophys. Res.115A1412203.
Balogh, A.
, Bothmer, V.
, Crooker, NU.
, Forsyth, RJ.
, Gloeckler, G.
, Hewish, A.
Mikić, Z.
The Solar Origin of Corotating Interaction Regions and
Their Formation in the Inner Heliosphere The solar origin of corotating
interaction regions and their formation in the inner heliosphere.
Space Sci. Rev.89141-178.
, Sittler
, Mariani
Burlaga, L.
, Sittler, E.
, Mariani, F.
Schwenn, R.
Magnetic loop behind an interplanetary shock: Voyager,
Helios, and IMP 8 observations Magnetic loop behind an interplanetary
shock: Voyager, Helios, and IMP 8 observations.
J. Geophys. Res. (Space)86A86673–6684.
, McPherron
Burton, RK.
, McPherron, RL.
Russell, CT.
An Empirical Relationship Between Interplanetary
Conditions and Dst An empirical relationship between interplanetary
conditions and Dst.
J. Geophys. Res.804204–4214.
St. Cyr
St. Cyr
Cremades, H.
St. Cyr, OC.
Coronal mass ejections: Solar cycle aspects Coronal
mass ejections: Solar cycle aspects.
Adv. Space Res.4071042-1048.
Crooker, NU.
Cliver, EW.
Postmodern View of M-regions Postmodern view of
J. Geophys. Res.99A1223383–23390.
Daglis, IA.
Ring Current Dynamics Ring Current
Space Sci. Rev.1241-4183-202.
Daglis, IA.
, Kozyra, JU.
, Kamide, Y.
, Vassiliadis, D.
, Sharma, AS.
, Liemohn, MW.
Lu, G.
Intense space storms: Critical issues and open
disputes Intense space storms: Critical issues and open
J. Geophys. Res. (Space)108A51208.
, Thorne
, Baumjohann
Daglis, IA.
, Thorne, RM.
, Baumjohann, W.
Orsini, S.
The terrestrial ring current: Origin, formation, and
decay The terrestrial ring current: Origin, formation, and decay.
Rev. Geophys.37407-438.
Dessler, AJ.
Parker, EN.
Hydromagnetic theory of geomagnetic storms
Hydromagnetic theory of geomagnetic storms.
J. Geophys. Res.64122239-2252.
, Tokumaru
, Hayashi
, Satonaka
Fujiki, K.
, Tokumaru, M.
, Hayashi, K.
, Satonaka, D.
Hakamada, K.
Long-term Trend of Solar Coronal Hole Distribution from
1975 to 2014 Long-term trend of solar coronal hole distribution from 1975
to 2014.
Astrophys. J. Lett.8272L41.
Gannon, JL.
Love, JJ.
USGS 1-min Dst index USGS 1-min Dst
J. Atm. Sol.-Terr. Phys.73323-334.
Gibson, SE.
, Kozyra, JU.
, de Toma, G.
, Emery, BA.
, Onsager, T.
Thompson, BJ.
If the Sun is so quiet, why is the Earth ringing? A
comparison of two solar minimum intervals If the Sun is so quiet, why is
the Earth ringing? A comparison of two solar minimum intervals.
J. Geophys. Res. (Space)114A13A09105.
Gnevyshev, MN.
Essential features of the 11-year solar cycle
Essential features of the 11-year solar cycle.
Sol. Phys.51175-183.
Gonzalez, WD.
, Joselyn, JA.
, Kamide, Y.
, Kroehl, HW.
, Rostoker, G.
, Tsurutani, BT.
Vasyliunas, VM.
What is a geomagnetic storm? What is a geomagnetic
J. Geophys. Res. (Space)995771-5792.
, Nunes
, Yashiro
Gopalswamy, N.
, Nunes, S.
, Yashiro, S.
Howard, R.
Variability of solar eruptions during cycle 23
Variability of solar eruptions during cycle 23.
Adv. Space Res.342391-396.
Solar Variability and Climate Change
Gosling, JT.
The Solar Flare Myth The solar flare myth.
J. Geophys. Res. (Space)9818937.
, Asbridge
, Bame
Gosling, JT.
, Asbridge, JR.
, Bame, SJ.
Feldman, WC.
Solar Wind Speed Variations: 1962–1974 Solar wind
speed variations: 1962–1974.
J. Geophys. Res.815061–5070.
Gosling, JT.
Pizzo, VJ.
Formation and Evolution of Corotating Interaction
Regions and their Three Dimensional Structure Formation and Evolution of
Corotating Interaction Regions and their Three Dimensional
Space Sci. Rev.8921-52.
Gosling, JT.
Riley, P.
The acceleration of slow coronal mass ejections in the
high-speed solar wind The acceleration of slow coronal mass ejections in
the high-speed solar wind.
Geophys. Res. Lett.23212867-2870.
Häkkinen, LVT.
, Pulkkinen, TI.
, Pirjola, RJ.
, Nevanlinna, H.
, Tanskanen, EI.
Turner, NE.
Seasonal and diurnal variation of geomagnetic activity:
Revised Dst versus external drivers Seasonal and diurnal variation of
geomagnetic activity: Revised Dst versus external drivers.
J. Geophys. Res. (Space)108A21060.
, Asikainen
Hamada, A.
, Asikainen, T.
Mursula, K.
A Uniform Series of Low-Latitude Coronal Holes in
1973–2018 A Uniform Series of Low-Latitude Coronal Holes in
Sol. Phys.296240.
Harvey, JW.
Sheeley, NR.
Coronal Holes and Solar Magnetic Fields Coronal holes
and solar magnetic fields.
Space Sci. Rev.23139–158.
Harvey, KL.
Recely, F.
Polar Coronal Holes During Cycles 22 and 23 Polar
Coronal Holes During Cycles 22 and 23.
Sol. Phys.21131-52.
, Wilcox
Hoeksema, JT.
, Wilcox, JM.
Scherrer, PH.
The Structure of the Heliospheric Current Sheet The
structure of the heliospheric current sheet.
J. Geophys. Res.889910.
Karinen, A.
Mursula, K.
A new reconstruction of the Dst index for 1932–2002
A new reconstruction of the Dst index for 1932–2002.
Ann. Geophys.23475-485.
, Mursula
, Takalo
Karinen, A.
, Mursula, K.
, Takalo, J.
Ulich, T.
An erroneous Dst Index in 1971 An erroneous Dst
Index in 1971.
H. Sawaya-Lacoste (), Solspa 2001, Proceedings of
the Second Solar Cycle and Space Weather Euroconference Solspa 2001,
proceedings of the second solar cycle and space weather euroconference ( 477, 443-446).
, Koskinen
Kilpua, E.
, Koskinen, HEJ.
Pulkkinen, TI.
Coronal mass ejections and their sheath regions in
interplanetary space Coronal mass ejections and their sheath regions in
interplanetary space.
Liv. Rev. Solar Phys.1415.
, Pesnell
, Young
Hess Webber
Kirk, MS.
, Pesnell, WD.
, Young, CA.
Hess Webber, SA.
Automated detection of EUV Polar Coronal Holes during
Solar Cycle 23 Automated detection of EUV Polar Coronal Holes during
Solar Cycle 23.
Sol. Phys.25799-112.
Klein, LW.
Burlaga, LF.
Interplanetary magnetic clouds at 1 AU
Interplanetary magnetic clouds at 1 AU.
J. Geophys. Res. (Space)87A2613-624.
, Timothy
Krieger, AS.
, Timothy, AF.
Roelof, EC.
A Coronal Hole and Its Identification as the Source of a
High Velocity Solar Wind Stream A coronal hole and its identification as
the source of a high velocity solar wind stream.
Sol. Phys.29505–525.
Love, JJ.
Gannon, JL.
Revised Dst and the epicycles of magnetic disturbance:
1958–2007 Revised Dst and the epicycles of magnetic disturbance:
Ann. Geophys.273101-3131.
, Holappa
Mursula, K.
, Holappa, L.
Karinen, A.
Correct normalization of the Dst index Correct
normalization of the Dst index.
Astrophys. Space Sci. Transact.441-45.
, Holappa
Mursula, K.
, Holappa, L.
Karinen, A.
Uneven weighting of stations in the Dst index
Uneven weighting of stations in the Dst index.
J. Atm. Sol.-Terr. Phys.73316-322.
, Holappa
Mursula, K.
, Holappa, L.
Lukianova, R.
Seasonal solar wind speeds for the last 100 years:
Unique coronal hole structures during the peak and demise of the Grand Modern
Maximum Seasonal solar wind speeds for the last 100 years: Unique coronal
hole structures during the peak and demise of the Grand Modern
Geophys. Res. Lett.44130-36.
, Lukianova
Mursula, K.
, Lukianova, R.
Holappa, L.
Occurrence of High-speed Solar Wind Streams over the
Grand Modern Maximum Occurrence of High-speed Solar Wind Streams over the
Grand Modern Maximum.
Astrophys. J.80130.
Newell, PT.
Gjerloev, JW.
SuperMAG-based partial ring current indices
SuperMAG-based partial ring current indices.
J. Geophys. Res. (Space)117A5A05215.
, Iyemori
, Sugiura
Nose, M.
, Iyemori, T.
, Sugiura, M.
Kamei, T.
Geomagnetic Dst index. Geomagnetic Dst index.
Web page
() of the
World Data Center for Geomagnetism (WDC-C2), Kyoto, Japan, accessed April 01,
O'Brien, T.
McPherron, R.
An empirical phase space analysis of ring current
dynamics: Solar wind control of injection and decay An empirical phase
space analysis of ring current dynamics: Solar wind control of injection and
J. Geophys. Res.105A47707-7719.
Petrie, GJD.
Solar Magnetism in the Polar Regions Solar
Magnetism in the Polar Regions.
Liv. Rev. Solar Phys.1215.
IG. Richardson
IG. Richardson
Richardson, IG.
Solar wind stream interaction regions throughout the
heliosphere Solar wind stream interaction regions throughout the
Liv. Rev. Solar Phys.1511.
IG. Richardson
IG. Richardson
Richardson, IG.
Cane, HV.
Near-earth solar wind flows and related geomagnetic
activity during more than four solar cycles (1963–2011) Near-earth solar
wind flows and related geomagnetic activity during more than four solar
cycles (1963–2011).
J. Space Weather Space Clim.2A02.
IG. Richardson
, Cane
IG. Richardson
Richardson, IG.
, Cane, HV.
Cliver, EW.
Sources of geomagnetic activity during nearly three
solar cycles (1972–2000) Sources of geomagnetic activity during nearly
three solar cycles (1972–2000).
J. Geophys. Res.1071187.
IG. Richardson
, Cliver
IG. Richardson
Richardson, IG.
, Cliver, EW.
Cane, HV.
Sources of geomagnetic activity over the solar cycle:
Relative importance of coronal mass ejections, high-speed streams, and slow
solar wind Sources of geomagnetic activity over the solar cycle: Relative
importance of coronal mass ejections, high-speed streams, and slow solar
J. Geophys. Res.105A818,203-18,213.
JD. Richardson
JD. Richardson
Richardson, JD.
Paularena, KI.
Streamer Belt Structure at Solar Minima Streamer belt
structure at solar minima.
Geophys. Res. Lett.241435–1438.
, Berghmans
van der Linden
Robbrecht, E.
, Berghmans, D.
van der Linden, RAM.
Objective CME detection over the solar cycle: A first
attempt Objective CME detection over the solar cycle: A first
Adv. Space Res.383475-479.
Rosenberg, RL.
Coleman, PJ., Jr.
Heliographic latitude dependence of the dominant
polarity of the interplanetary magnetic field. Heliographic latitude
dependence of the dominant polarity of the interplanetary magnetic
J. Geophys. Res.745611-5622.
Russell, CT.
McPherron, RL.
Semiannual variation of geomagnetic activity
Semiannual variation of geomagnetic activity.
J. Geophys. Res.7892–108.
Savitzky, A.
Golay, MJE.
Smoothing and differentiation of data by simplified
least squares procedures Smoothing and differentiation of data by
simplified least squares procedures.
Anal. Chem.361627-1639.
, Wilcox
Schatten, KH.
, Wilcox, JM.
Ness, NF.
A Model of Interplanetary and Coronal Magnetic Fields
A model of interplanetary and coronal magnetic fields.
Sol. Phys.6442–455.
Sckopke, N.
A general relation between the energy of trapped
particles and the disturbance field near the Earth A general relation
between the energy of trapped particles and the disturbance field near the
J. Geophys. Res.71133125-3130.
SILSO World Data Center
SILSO World Data Center
SILSO World Data Center.
The International Sunspot Number The International
Sunspot Number.
International Sunspot Number Monthly Bulletin and
online catalogue.
Smith, EJ.
The global heliospheric magnetic field The global
heliospheric magnetic field.
A. Balogh, LJ. Lanzerotti ST. Suess (), The Heliosphere through the Solar Activity Cycle
The heliosphere through the solar activity cycle ( 79).
Smith, EJ.
Balogh, A.
Decrease in heliospheric magnetic flux in this solar
minimum: Recent Ulysses magnetic field observations Decrease in
heliospheric magnetic flux in this solar minimum: Recent Ulysses magnetic
field observations.
Geophys. Res. Lett.35L22103.
, Usoskin
, Kromer
, Schüssler
Solanki, SK.
, Usoskin, IG.
, Kromer, B.
, Schüssler, M.
Beer, J.
Unusual activity of the Sun during recent decades
compared to the previous 11,000 years Unusual activity of the Sun during
recent decades compared to the previous 11,000 years.
Storini, M.
, Bazilevskaya, GA.
, Fluckiger, EO.
, Krainev, MB.
, Makhmutov, VS.
Sladkova, AI.
The GNEVYSHEV gap: A review for space weather The
gnevyshev gap: A review for space weather.
Adv. Space Res.31895-900.
Subramanian, P.
Dere, KP.
Source Regions of Coronal Mass Ejections Source
Regions of Coronal Mass Ejections.
Astrophys. J.5611372-395.
Sugiura, M.
Hourly values of equatorial Dst for the IGY Hourly
values of equatorial Dst for the IGY.
Ann. Int. Geophys. Year359.
Sugiura, M.
IAGA Resolution 2 IAGA Resolution 2.
IAGA Bulletin 27 IAGA Bulletin 27 ( 123).
Madrid, Spain.
Sugiura, M.
Kamei, T.
Equatorial Dst index 1957–1986 Equatorial
Dst index 1957–1986.
A. Berthelier M. Menvielle (), IAGA
Bulletin 40. IAGA Bulletin 40.
Saint-Maur-des-Fossés, FranceISGI Publ. Office.
Takalo, J.
Mursula, K.
A model for the diurnal universal time variation of the
Dst index A model for the diurnal universal time variation of the Dst
J. Geophys. Res. (Space)106A610905-10914.
Takalo, J.
Mursula, K.
On the diurnal UT variation of the Dst index On the
diurnal UT variation of the Dst index.
H. Sawaya-Lacoste (), Solspa 2001, Proceedings of
the Second Solar Cycle and Space Weather Euroconference Solspa 2001,
proceedings of the second solar cycle and space weather euroconference ( 477, 483-486).
Tsurutani, BT.
, Gonzalez, WD.
, Gonzalez, ALC.
, Guarnieri, FL.
, Gopalswamy, N.
, Grande, M.
Vasyliunas, V.
Corotating solar wind streams and recurrent geomagnetic
activity: A review Corotating solar wind streams and recurrent
geomagnetic activity: A review.
J. Geophys. Res. (Space)111A7.
Virtanen, II.
Mursula, K.
Asymmetry of solar polar fields and the southward shift
of HCS observed by Ulysses Asymmetry of solar polar fields and the
southward shift of HCS observed by Ulysses.
J. Geophys. Res.115A14A09110.
Virtanen, II.
Mursula, K.
North-South Asymmetric Solar Cycle Evolution:
Signatures in the Photosphere and Consequences in the Corona North-South
Asymmetric Solar Cycle Evolution: Signatures in the Photosphere and
Consequences in the Corona.
Astrophys. J.781299.
, Nash
Wang, YM.
, Nash, AG.
Sheeley, NR., Jr.
Evolution of the Sun's Polar Fields During Sunspot
Cycle 21 - Poleward Surges and Long-Term Behavior Evolution of the Sun's
polar fields during sunspot cycle 21 - poleward surges and long-term
Astrophys. J.347529–539.
, Robbrecht
Wang, YM.
, Robbrecht, E.
Sheeley, NR., Jr.
On the Weakening of the Polar Magnetic Fields during
Solar Cycle 23 On the Weakening of the Polar Magnetic Fields during Solar
Cycle 23.
Astrophys. J.7071372-1386.
Wang, YM.
Sheeley, J., N. R.
Solar Wind Speed and Coronal Flux-Tube Expansion
Solar Wind Speed and Coronal Flux-Tube Expansion.
Astrophys. J.355726.
On Dst index (description in the IAGA Bulletin No 40).
On Dst index (description in the IAGA Bulletin no 40).
Web page
() of the
World Data Center for Geomagnetism (WDC-C2), Kyoto, Japan, accessed April 01,
Webb, DF.
Howard, RA.
The solar cycle variation of coronal mass ejections and
the solar wind mass flux The solar cycle variation of coronal mass
ejections and the solar wind mass flux.
J. Geophys. Res. (Space)994201-4220.
Webb, DF.
Howard, TA.
Coronal Mass Ejections: Observations Coronal mass
ejections: Observations.
Liv. Rev. Solar Phys.913.
, Mursula
, Holappa
, Veselovsky
Yakovchouk, OS.
, Mursula, K.
, Holappa, L.
, Veselovsky, IS.
Karinen, A.
Average properties of geomagnetic storms in 1932–2009
Average properties of geomagnetic storms in 1932–2009.
J. Geophys. Res. (Space)117A03201.
Zhou, X.
Smith, EJ.
Solar cycle variations of heliospheric magnetic flux
Solar cycle variations of heliospheric magnetic flux.
J. Geophys. Res.114A133106.
More Information and Advice:
Math coded inside display math mode
\[ ...\]
will not be numbered, e.g.,:
\[ x^2=y^2 + z^2\]
Math coded inside
\begin{equation} and \end{equation}
be automatically numbered, e.g.,:
\begin{equation}
x^2=y^2 + z^2
\end{equation}
\begin{eqnarray}
x_{1} & = & (x - x_{0}) \cos \Theta \nonumber \\
&& + (y - y_{0}) \sin \Theta \nonumber \\
y_{1} & = & -(x - x_{0}) \sin \Theta \nonumber \\
&& + (y - y_{0}) \cos \Theta.
\end{eqnarray}
|
2022 XXX 1st December
Hot & Cold QCD plans for 2023 Long Range Plan for Nuclear Science
The ALICE-USA Collaboration††thanks: See Appendix A for the list of
collaboration members ALICE-USA Collaboration
## 1 Executive Summary
The ALICE experiment was built to study many-body Quantum Chromo-Dynamics
(QCD) at high temperature and effectively zero baryon density, using
relativistic heavy-ion collisions at the Large Hadron Collider (LHC). These
collisions form the Quark Gluon Plasma (QGP), a state of matter where quarks
and gluons are no longer confined inside hadrons. The ALICE program centers
around the key questions related to QGP phenomena. These include the
macroscopic and microscopic properties of the QGP, and the details of the QGP
phase transition to hadrons, that is believed to have taken place in the early
Universe. At the same time, ALICE’s versatile setup allows for the study of pp
collisions, p–Pb collisions, and ultra-peripheral collisions. The associated
studies can provide some of the most stringent tests of QCD and Beyond
Standard Model searches. They serve as deep probes of the properties of cold
nuclear matter, and allow for investigations of stellar and interstellar
phenomena. The ALICE-USA collaboration consists of 13 institutes, representing
about 6% of the total ALICE authorship. It has, and continues to play, an
essential role in all areas of ALICE leadership: physics, instrumentation, and
management. ALICE-USA has been involved in $\sim$25% of ALICE’s $\sim$400
ALICE papers since 2009, including many of ALICE’s most cited and high profile
publications. We, the ALICE-USA collaboration, will provide two
recommendations for the U.S. 2023 Long Range Plan for Nuclear Science period,
and beyond. These recommendations are essential for maintaining the continued
success and development of the U.S. Nuclear Science community for both Hot and
Cold QCD. The recommendations will be accompanied by details of the vital
ALICE-USA scientific priorities. We will also describe the broader impact and
support needed for U.S. involvement in ALICE.
### Recommendation 1: Continue and broaden the contribution from U.S.
institutes for the ALICE program in Runs 3&4.
ALICE has just completed a number of major upgrades. ALICE-USA has made vital
contributions to the new Inner Tracking System (ITS) and Time Projection
Chamber (TPC) readout. ALICE-USA will now utilize these upgrades for a
comprehensive physics program. This ‘ALICE 2’ phase also provides a unique
opportunity for Hot and Cold QCD studies between the expected times when RHIC
discontinues taking data in 2025, and when the EIC begins taking data around
2035. The new detector setup began taking data in Run 3 (2022-2025), and will
continue to operate in Run 4 (2029-2032). During these periods, ALICE will
collect data from collisions of: Pb–Pb at $\sqrt{s_{\mathrm{NN}}}=5.36$ TeV,
pp at $\sqrt{s}=13.6$ TeV, pp at the same energy as Pb–Pb, p–Pb at
$\sqrt{s_{\mathrm{NN}}}=8$ TeV, p–O at $\sqrt{s_{\mathrm{NN}}}=9.9$ TeV, and
O–O at $\sqrt{s_{\mathrm{NN}}}=7$ TeV. The new data set of Pb–Pb collisions
($\sim$13 nb-1) represents an increase of two orders of magnitude over Runs
1&2\. These increased capabilities in Runs 3&4 will allow for:
* •
The elucidation of the microscopic parton dynamics underlying QGP properties
using hard processes of lower cross section, such as heavy-flavor jets, over a
large range of transverse momenta;
* •
The characterization of the macroscopic QGP properties with extended
precision, including the exploration of unknown dynamical QGP transport
parameters at LHC energies, such as the the baryon diffusion coefficient;
* •
Deeper studies of the hadronization of heavy-flavor baryons and mesons
produced in high temperature QCD matter;
* •
Multiple observables with an increased precision to contribute to the
development of a unified picture of particle production and QCD dynamics from
small (pp) to large (nucleus–nucleus);
* •
Unique explorations of parton densities in protons and nuclei in a broad ($x$,
$Q^{2}$) kinematic range, reaching to $x\sim 10^{-6}$.
In relation to the last point, ALICE-USA is one of the key proponents of the
Forward Calorimeter (FoCal). This a new detector that will collect data in Run
4. The development, installation, and operation of the FoCal will occur during
the LRP duration, and prior to the start of the EIC data taking period. The
FoCal is designed to provide unique capabilities to probe the structure of
nucleons and nuclei in an unexplored regime of $x$ and $Q^{2}$. This kinematic
region, of $x\sim 10^{-6}$, will probe the regime where gluon saturated matter
is expected to be dominant. Such a small-$x$ region is not accessible at the
EIC. Global fits of this lower $x$ data, with new high precision EIC data,
will make an unprecedented step to understanding the full evolution of Parton
Distributions Functions (PDFs) in protons and nuclei. The R&D and design phase
of the EIC detectors have well defined synergies with the FoCal
instrumentation work. We also stress the additional benefits of U.S.
involvement in ALICE’s instrumentation R&D for the ITS3 sensors that will be
part of the Run 4 setup. These have various applications for the EIC detector
projects, as recognized in multiple forums. Support for both of these
endeavors is of vital importance.
Beyond the scientific and technological implications discussed, the active
participation of U.S. institutions in the ALICE collaboration in the coming
years would greatly benefit the training of the next generation of scientists.
It is vital that PhD students, postdoctoral researchers, and early career
faculty, have the opportunity to develop their data analysis, modeling, and
interpretation expertise using the largest and most complex datasets of QCD
phenomena ever garnered. These younger members of the community will also have
the opportunity to participate in the full cycle of detector operation and
development in the 2020s. Such experience is essential for maintaining the
U.S. as a leader in nuclear science. Therefore, continued support for ALICE
research is crucial. We urge a modest expansion of the ALICE efforts to allow
groups currently focused at RHIC to also participate prior to EIC operations.
### Recommendation 2: Begin participation in ALICE-3 R&D and construction.
ALICE-USA fully supports the ALICE 3 detector, a next generation detector that
is designed to operate in Runs 5&6 (2035 and beyond). It is currently in the
R&D phase, and construction of the ALICE 3 detector is expected to begin 2028.
It represents a unique direction to pursue Hot QCD studies at the highest
temperatures possible into and beyond the 2030s. ALICE 3 will have the largest
acceptance and most precise tracking ever achieved for a heavy-ion experiment.
It is expected to record $\sim$20 times more heavy-ion data compared to Runs
3&4\. The general purpose detector design of ALICE 3 will provide many physics
opportunities, which are expected to lead to new discoveries. It will also
provide a level of data quality required to transform our understanding of the
QGP, in terms of first principles QCD calculations. The key topics unique to
ALICE 3, and under the umbrella of current ALICE-USA interests, include:
* •
Jet hadrochemistry over an unprecedented kinematic range for transverse
momentum and rapidity, to provide new ways to investigate QGP jet energy loss
and the microscopic structure of the QGP;
* •
Measurements of multi-heavy-flavor and exotic charm states to provide unique
information on hadronization and hadronic interactions at high temperatures;
* •
Beyond Standard Model physics with searches for axion-like particles in ultra-
peripheral collisions, which will leverage the high collision rates and large
acceptance.
The design of ALICE 3 will also enable measurements of high-precision, multi-
differential of electromagnetic radiation from the QGP to probe its early
evolution and the restoration of chiral symmetry through the coupling of
vector and axial-vector mesons. Measurements of net-quantum number
fluctuations over a wide rapidity range will constrain the susceptibilities of
the QGP, and test the realization of the cross-over phase transition expected
at LHC energies. Finally, for the first time, measurements of the production
of ultra-soft photons will be possible. These can quantitatively test the
infrared limits of quantum field theories such as QED and QCD. In order to
meet these physics goals, support for both the R&D and construction efforts is
critically required from the U.S. community. This needs to begin before the
end of the current 2023 Long Range Plan for Nuclear Science. Based on its
expertise, ALICE-USA is currently exploring contributions to the silicon and
calorimeter detectors. These parallel and independent efforts will also
benefit the EIC achieving its construction milestones.
## 2 Overview of ALICE upgrades and U.S. impact
ALICE is a large acceptance experiment, with world leading hadron
identification capabilities. Unique to the LHC experiments, ALICE’s primary
focus, design, and proposed upgrades, enable extremely accurate
characterizations of the QGP using a multitude of observables. It has
extensive capabilities in exploring few-body hadronic and nuclear
interactions. The first major upgrade for LHC Runs 3&4 enables an increase of
ALICE’s recorded luminosities by two orders of magnitude via the introduction
of a continuous readout system in the TPC. The second involves a greatly
improved tracking performance using new inner trackers - ITS2 in Run 3, and
ITS3 in Run 4. The ITS2 system has a reduction of a factor three in radiation
lengths, and factor two improvement in pointing resolution compared to the ITS
used in Runs 1&2\. Further factor of three improvements of the pointing
resolution for ITS3 (compared to ITS2) will occur via the introduction of
wafer-scale ultra-thin silicon strips, which are also being pursued by the
EPIC detector at the EIC. Details of these upgrades and others can be found
elsewhere [1]. For Run 4, the Forward Calorimeter detector (FoCal) upgrade,
with both electromagnetic and hadronic calorimeters, will provide unique
capabilities for both Cold and Hot QCD studies at forward rapidities of
$3.4<\eta<5.8$ [2].
For Runs 5&6, a completely new detector has been proposed, named ALICE 3 [3].
Designed by heavy-ion physicists for heavy-ion physics, it is a next
generation detector with a main tracking system that covers a large
pseudorapidity range ($-4<\eta<4$), and can reconstruct charged tracks down to
$p_{\rm T}\sim 100$ MeV/c. The light weight and small radiation length design
will greatly reduce the background and improve tracking resolution for
electromagnetic and heavy-flavor probes, compared to ALICE 2. The pointing
resolution at midrapidity is projected to be about three times better than
that of the ITS3, which is in part achieved by placing a highly novel vertex
detector 5mm from the beamline. The tracking system is placed in a
superconducting solenoid with a field of up to $B=2$ T, to obtain a momentum
resolution of 1–2% over a broad pseudorapidity and momentum range. This
tracking is complemented by multiple sub-detector systems for particle
identification; two TOF detectors and a RICH detector. These systems have the
ability to identify leptons and photons in the entire thermal emission range
of $p_{\rm T}\lesssim 3$ GeV/c, which is inaccessible for other LHC
experiments. Charged hadron identification on the $3\sigma$ level is possible
up to $p_{\rm T}\sim 14$ GeV/c, and decay hadrons can be reconstructed much
more efficiently and cleanly at low-$p_{\rm T}$ compared to ALICE 2. The fast
readout systems will be able to record all of the expected heavy-ion
luminosity provided by the LHC. The ALICE 3 program aims to collect an
integrated luminosity of about 35 nb-1 with Pb–Pb collisions and 18 fb-1 with
pp collisions at top LHC energy. The potential to further increase the
luminosity for ion running in the LHC by using smaller ions, e.g. 84Kr or
128Xe, as well as further runs with small collision systems, is being
explored.
Figure 1: Event display of one of the first LHC Run 3 Pb–Pb
$\sqrt{s_{\mathrm{NN}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.36$ TeV
collisions in November 18th 2022, using many detectors ALICE-USA has
contributed toward the construction and upgrade.
As of September 2022, the ALICE-USA groups consist of teams from two national
laboratories: Lawrence Berkeley (LBNL) and Oak Ridge (ORNL), and nine
universities: California (Berkeley), CalPoly, Chicago State, Creighton,
Houston, Kansas, Ohio State, Tennessee (Knoxville), Texas (Austin), Wayne
State, and Yale. While the majority of the program is supported by DOE, there
are two NSF supported university teams. ALICE-USA has been responsible for two
completed DOE funded projects - the EMCal/DCal in Runs 1&2 [4], and the most
recently completed Barrel Tracking Upgrade (BTU) for Runs 3&4 111See the BTU
website at https://sites.google.com/site/alicebtusite for further details. The
BTU project involved critical U.S. contributions to both the TPC and ITS
upgrades. A key example of the success of all these projects can be observed
in Fig. 1, which shows one of the first ALICE Run 3 event-displays of Pb–Pb
collisions using many detectors where ALICE-USA has made essential
contributions. Thanks to the long and successful experience building and
operating calorimeters, ALICE-USA is leading the FoCal project [5]. It also
oversees a major computing project providing the necessary U.S. contribution
to ALICE’s data processing infrastructure, under the umbrella of the Worldwide
LHC Computing GRID (WLCG).
In terms of the scientific output, the contributions from ALICE-USA span a
wide range of physics topics within the collaboration, and have been an
essential part of the ALICE physics program. Although ALICE-USA members are
present at an about 6% level in terms of authorship within the collaboration,
they have provided leadership in terms of paper committee contributions well
beyond the average contributions. ALICE-USA members were involved in 1 of 2
Nature articles published by ALICE [6], and 1 of 2 Nature Physics articles
[7]. ALICE-USA members were also involved in 5 of the top 7 most cited physics
publications from ALICE [8, 9, 10, 11, 7], and numerous other high-profile
publications. These papers span both the hard and soft physics using heavy-ion
and pp collisions. ALICE has published just over 400 papers since 2009, with
the overall U.S. involvement on the paper committee level being about 25%.
In addition to publications, members of the ALICE-USA have held numerous
leadership roles across all seniority levels of the collaboration regarding
physics output. These include: Deputy Physics Coordinator (two times),
multiple Physics Working Group (PWG) Conveners, and Physics Analysis Group
Coordinators. Of particular note, out of the 8 ALICE PWGs, currently ALICE-USA
members co-convene 4 (50%) of the those PWGs. Over the years members of ALICE-
USA were copiously represented in the governance of ALICE – currently with
members of the collaboration holding functions as members of the Conference
Committee and the Management Board, and as one of the two Deputy
Spokespersons. Moreover, in the past ALICE-USA members served as Chair of the
Collaboration Board and co-chair of the Editorial Board. Finally, for the
$\sim 100$ ALICE-USA members, roughly half are PhD students. At this rate,
ALICE-USA institutions have graduated $\sim 50$ PhD students over a 5 year
period. Many of our students have achieved leading roles in research positions
in our field and beyond, as well as in industry and teaching jobs.
## 3 A brief summary of ALICE accomplishments from Runs 1&2
The observations ALICE has made during Runs 1&2 (2009-2019) have profoundly
changed the landscape of QCD at high temperatures and collision energies [12].
They have been carried out in conjunction with other LHC experiments, and with
the continuation of the RHIC program - where huge advances in luminosities and
centre-of-mass energy coverage have been achieved. ALICE explorations of high
temperature QCD have continued to reveal emergent behavior in many-body
interactions at the highest possible temperatures in the laboratory. In heavy-
ion collisions, ALICE has measured significant yield suppression for a wide
range of hadrons and reconstructed jets in both inclusive and coincidence
channels, showing that in-medium energy loss occurs at the partonic level, and
quantifying its magnitude. ALICE measurements of heavy-flavor yield
suppression provide insight into details of this process, via the QCD “dead-
cone” effect. This manifests itself in a larger energy loss observed for charm
compared to bottom quarks. It has also, for the time, provided direct evidence
of the dead-cone effect for charm quarks in pp collisions, using state of the
art jet re-clustering algorithms. ALICE jet substructure measurements indicate
preferential suppression of wide-angle radiation in the jet shower within the
QGP, which is sensitive to color coherence and the space-time structure of
jets.
In the soft sector, ALICE measurements of identified hadron spectra and
anisotropic flow demonstrate that a QGP formed at LHC energies undergoes the
most rapid expansion ever observed for a many-body system in the laboratory.
The hydrodynamic description of a huge variety of such data has been tested in
heavy-ion collisions at the LHC, offering an environment far beyond the usual
application in fluid dynamics. It has shown a QGP is strongly coupled at
scales of the QGP’s temperature, on the order of a few hundred MeV. ALICE
measurements also imply thermalisation effects for charm quarks in a QGP, and
when coupled with transport model calculations, the ALICE data demonstrate
microscopically how equilibrium can occur on extremely small time scales. In
addition, ALICE results have demonstrated that a QGP transitions into
chemically equilibrated hadrons. ALICE has the most extensive set of
measurements ever achieved regarding identified particle production, including
for the first time a statistical description in the charm sector. ALICE
investigations into the hadron-gas phase, via resonance and femtoscopic
measurements, indicate this phase is prolonged, and that the decoupling of
particles from the expanding hadron gas is likely to be a continuous process.
Using RHIC and LHC Run 1&2 data, comprehensive efforts at global fitting for
precise determination of QGP properties and dynamics are currently underway,
utilizing the powerful approach of Bayesian Inference for rigorous comparisons
between theory and experiments. Such analyses are more advanced in the soft
sector, and are in their infancy in the hard probes sector. ALICE has played a
critical role in this endeavor by providing the experimental data, and
developing theory-experiment synergies. Along with other RHIC and LHC
experiments, ALICE has discovered QGP-like signatures in high multiplicity pp
and p-Pb collisions, which probe the thresholds of QGP formation. Such
findings have ignited a debate of whether pp and p-Pb collisions at LHC and
RHIC energies could create small QGP droplets.
ALICE has investigated few-body hadronic interactions on the soft and hard
scales to an unprecedented precision. In pp collisions, it has demonstrated
charmed hadron fragmentation functions are not universal with respect to e+e-
collisions, revealing unique hadronisation mechanisms that enhance charmed
baryons compared to mesons. It also has provided world leading constraints
regarding how rarely produced hadrons interact with stable nuclear matter,
whose behavior have broad implications for understanding various features in
the Universe, such as the equation of state of neutron stars and the
interstellar composition of dark matter. Finally, ALICE has been a leader in
the study of ultra-peripheral heavy-ion collisions since Run 1, exploring cold
QCD matter at the highest possible photon-induced energies. ALICE has observed
photoproduction of vector mesons on proton and ion targets, namely, the
$J/\psi$, $\psi(2S)$, and $\rho$ mesons. $J/\psi$ photoproduction is
moderately suppressed on Pb targets compared to free protons, in line with
nuclear shadowing effects for gluon parton distribution functions at $x\approx
10^{-3}$. Coherent $\rho$ photoproduction measurements also demonstrate
nuclear shadowing for both the Xe and Pb nuclei.
## 4 Key physics questions addressed for LHC Runs 3&4 and beyond from ALICE-
USA
Despite the many extraordinary scientific findings by the ALICE Collaboration,
many open questions remain, and will be addressed using data to be collected
in Runs 3&4 at the LHC (2022-2032). This section details such questions, with
a description of the leadership ALICE-USA groups intend to provide addressing
them. Opportunities using the ALICE 3 detector in the Run 5&6 period (2035 and
beyond) will also be discussed.
1. 1.
What is the gluonic structure in protons & nucleons at small-$x$?
We propose to maximize the physics potential for small-$x$ physics in ALICE
with the installation a high-granularity forward calorimeter (FoCal) in the
LHC Run 4 (2029–2032). The FoCal will explore the small-$x$ parton structure
of nucleons and protons in a kinematic domain only accessible at the LHC [5].
It is designed to provide unique capabilities to investigate Parton
Distribution Functions (PDFs) in the unexplored regime of $x\sim 10^{-6}$ and
four momentum transfer $Q$ of a few GeV. In this regime, it is expected that
the hadronic structure requires non-linear dynamics due to the high gluon
densities, leading to gluon saturation. Gluon saturation is a necessary
consequence of the non-Abelian nature of QCD. Its discovery would represent a
landmark in our understanding of the strong interaction. The primary
objectives of the FoCal include: high-precision inclusive direct photon and
jets measurements in pp and p–Pb collisions, photon-jet and jet-jet
measurements in pp and p–Pb collisions, and measurements of vector mesons
photoproduction in ultra-peripheral p–Pb and Pb–Pb collisions. These
measurements by the FoCal would constitute an essential part of a
comprehensive small-$x$ program at the LHC, with a broad array of
complementary probes. While the FoCal has some complementarity to future
measurements by the EIC [13] and the LHCb experiment, it can reach
considerably lower $x$ values than the EIC.
Vector meson photoproduction and electroweak boson studies have provided
unique constraints on the gluon parton distribution function for nuclei using
Runs 1& 2 data [14, 15, 16]. They provide clear evidence that the partonic
structure of nuclei is different compared to free protons, with nuclear
shadowing effects increasing with decreasing longitudinal momentum fractions
$x$. In addition, no evidence of saturation of the gluon PDF in the proton is
observed between HERA and LHC energies, down to $x\sim 10^{-5}$ [17, 18]. For
a Pb target, the $J/\psi$ photoproduction cross-section is compatible with
moderate gluon shadowing, in line with the leading twist approximation and the
central values of the EPS09 and EPPS16 nuclear parameterizations [19, 20, 21,
22]. The ALICE data has smaller uncertainties than these parameterizations,
pointing to the utility of UPC data for constraining parton distributions
[23]. The comparison of the energy dependence of photoproduced vector mesons
measured in p–Pb and Pb–Pb collisions is expected to significantly reduce the
theoretical uncertainties [24], thus a systematic program of UPC vector meson
measurements is needed.
The FoCal will enable measurements of photoproduced J/$\rm\psi$ and $\psi$(2S)
mesons to reach a $x\sim 10^{-6}$ and $Q^{2}\approx 3-4$ GeV2. This is a
region that cannot be explored by the EIC. The energy dependence of the ratio
between $\psi(2S)$ and J/$\rm\psi$ is particularly sensitive to the difference
between linear and non-linear gluon evolution [25], hence saturation effects.
Photoproduction data in Runs 3&4 will allow ALICE to go beyond measurements of
vector meson cross-sections. Studies of open charm [26] and/or dijets [27] are
sensitive to gluon distributions with fewer theoretical assumptions than for
vector mesons. The study of angular correlations of diffractive dijets [28] is
also sensitive to the gluon nuclear density, particularly for low-momentum
track-based jets where ALICE would have a competitive advantage.
Measurements of $d\sigma/dt$ for coherent and incoherent photoproduction of
vector mesons are respectively sensitive to the transverse distribution of
gluons in a nucleus (similar to a GPD) [29, 30], and the event-by-event
fluctuations in the nuclear configuration, including gluonic hotspots [31, 32,
33]. ALICE will collect large enough data samples to make these measurements
for a number of different mesons [24], including the $J/\psi$ and the $\rho$.
The new ALICE streaming DAQ system will be important for increasing the vector
meson sample sizes, and reducing systematic uncertainties due to the trigger
requirements. The FoCal detector will allow ALICE to extend these measurements
to even lower $x$ values, providing enhanced sensitivity for gluon saturation.
The future UPC measurements using the FoCal will also benefit from having
other ALICE detectors. It is anticipated that the Zero Degree Calorimeters can
be utilized to distinguish the photon direction in ultra-peripheral Pb–Pb
collisions [34]. This will allow for the exploration of the lowest possible
$x$ values in Pb. ALICE measurements can also be carried out combining the
FoCal together with the central barrel or the forward muon detector. In this
way, a larger kinematic range can be explored, which is vital for gluon
saturation searches. More details of the physics prospects in ultra-peripheral
heavy-ion collisions using the FoCal detector can be found elsewhere [35].
Gluon saturation at low-$x$ has also been postulated to be the source of
quantum entanglement effects. These could lead to instant thermalization based
on the potential equivalence of entanglement and thermodynamic entropy, and a
determination of final state particle production in the context of minimal
decoherence effects. In order to study this type of parton-hadron duality,
ALICE can complement the EIC measurements in ep and eA systems via
measurements in pp, pA and AA. The survival of the coherent state as a
function of system size can be measured through particle multiplicities and
quantum tomographic correlation functions of particles from low-$x$ processes.
Quantum entanglement can also be studied by reconstructing the density state
operator using quantum tomography [36]. In this direction, the study of
exclusive four $\pi$ photoproduction will provide a search for exited $\rho$
states [37].
2. 2.
What are the large scale nuclear structures and QGP transport parameters?
Large scale nuclear structures, such as the nuclear deformation and radius,
can be precisely investigated using measurements of anisotropic flow. This is
achieved courtesy of the QGP hydrodynamic response, where initial state
features are imprinted on final state anisotropic flow observables. These
nuclear structures are often accessible via low energy nuclear experiments,
but are limited to the electrically charged proton profiles. Measurements in
heavy-ion collisions can in principle access the entire nuclear matter
profile. These profiles are also inaccessible for DIS nuclear collisions by
the virtue of their design to probe sub nucleon scales. Such measurements have
already been used to determine 129Xe deformation, with values of the
quadrapole deformation $\beta_{2}=0.18\pm 0.02$ obtained in Run 2 from ALICE
data [38]. These have never been measured previously. Similar studies have
proved equally successful constraining the 238U and 197Au deformations from
RHIC data [39]. Firstly, we propose to carry out a suite of anisotropic flow
measurements in 16O–16O and p-16O collisions. These will occur in Run 3.
Hydrodynamic calculations have shown such measurements are sensitive to
$\alpha$ clustering in the 16O nucleus [40], a longstanding pursuit in the low
energy nuclear community. The clustering effects are calculable in Lattice
QCD. Clustering affects anisotropic flow measurements more strongly at LHC
energies than at RHIC.
Anisotropic flow and flow fluctuations measurements in central large nuclei
A-A collisions are particularly sensitive to the nuclear profile. In addition
to Xe–Xe collisions at the LHC, this has also been demonstrated with data from
the RHIC Isobar run coupled with hydrodynamic calculations, where these
profile parameters have been highly constrained [41]. Anisotropic flow
measurements of $v_{2}$ and $v_{3}$ in very central Pb–Pb collisions at the
LHC have also revealed the “ultra-central problem” - that is a deviation from
the hydrodynamic description where it is expected to be most applicable (for a
recent review see here [42]). Proposals to explore this further, such as the
introduction of an octupole deformation ($\beta_{3}$) for the Pb nucleus [43],
which has been predicted for the doubly magic Pb208 ground state, require
measurements of $v_{3}\\{4\\}/v_{3}\\{2\\}$ well beyond the accuracy of those
achieved in Runs 1&2 by ALICE [11]. There are also predictions for an
quadrapole deformation of $\beta_{2}=0.05$ for Pb208 [44], which would lead to
finite value of $v_{2}\\{4\\}$. This has yet to be observed in within
statistic precision for previous ALICE data in very central Pb-Pb collisions
[45]. Therefore, the greater statistics provided by Runs 3-6 will be crucial
in unraveling these nuclear properties. Their constraints are also critical
for the determination of QGP transport parameters obtained by the hydrodynamic
framework, in the regime where hydrodynamics is most applicable. These
transport parameters include the famous shear viscosity over entropy ratio
$\eta/s$. Such nuclear profile constraints could be particularly important for
ALICE 3, as non double magic nuclei such as 84Kr or 128Xe (with larger
deformations compared to the double magic 208Pb nucleus) are being explored
for heavy-ion running.
The hydrodynamic framework used to describe collective flow involves numerous
transport parameters, which characterize the coupling of the QGP. To date,
only two of those, the shear and bulk viscosity over entropy ratios ($\eta/s$
and $\zeta/s$), have benefited from extensive constraints using heavy-ion
data. Those constraints provide evidence the QGP is the most strongly coupled
system ever studied in the lab. The corresponding relaxation times, of the
order of 1 fm/c or less, demonstrate how thermalization can be achieved on the
most rapid timescales ever observed for a many body system. Comparisons of
these extracted transport parameters to fundamental descriptions offer a
unique test to Holographic models [46]. These include AdS/CFT, which predicts
$\eta/s=1/4\pi$ in the infinite coupling limit. Such approaches assume a
correspondence of 5 dimensional strong gravitational fields in a black hole
with a 4 dimensional high temperature QCD system. These transport parameters
are also predictable in Lattice QCD, but often have much larger uncertainties
[47].
Two-particle correlations with net baryons can be used to explore additional
transport parameters of the hydrodynamic evolution. The baryon diffusion
constant $D_{B}$ is an example of such a parameter beyond $\eta/s$ and
$\zeta/s$. It characterizes the mobility of baryon number, and is predicted to
be finite at the LHC, despite the fact that $\mu_{B}\sim 0$. A two-particle
correlation function has been proposed to constrain $D_{B}$ [48]. It explores
correlations of net-baryon fluctuations as a function of azimuthal and
rapidity separations. Such an analysis has yet to be carried out from Run 1 or
2 data, since it is statistically challenging. It will be greatly aided by the
increase of two orders of magnitude in the Pb–Pb integrated luminosity
foreseen for Runs 3&4\. The ALICE detector is particularly suited to this
task, given its world leading particle identification capabilities. Finally,
the recently published measurements of balance functions (BFs) of identified
charged hadrons $(\pi,\rm K,\rm p)$ in Pb–Pb collisions from Runs 1&2 play an
important role in constraining the charge diffusion coefficient $D_{e}$ for
quarks [49]. However, the central tracking acceptance of $-1<\eta<1$ for Runs
1&2 leads to large uncertainties in $D_{e}$ from ALICE BF data [50]. In turn,
this leads to values between $0.5D_{Latt}<D_{e}<4D_{Latt}$ being permissible,
where $D_{Latt}$ represents the Lattice QCD prediction. These uncertainties
will be significantly reduced when the same measurements are performed using
the ALICE 3 setup, as the acceptance where identified charged hadron
measurements can be made will increase to $-4<\eta<4$. The reduction in
$D_{e}$ uncertainties due to increases in the $\Delta\eta$ coverage are
expected to be at least a factor 4.
3. 3.
How does the QGP affect hard probes?
The energetic partons produced in heavy-ion collisions from hard scatterings
in the initial collision undergo successive branching. This results in a
parton shower that can be modified in the presence of the QGP. While the
produced particles are highly collimated about the direction of the initial
parton, they also cover a range of different momentum scales. The properties
of these collimated sprays of particles, known as jets, and how they emerge
from QCD calculations, have been extensively studied in high-energy physics.
Jets are a primary tool for uncovering the details of interactions of partons
with the QGP medium. This is because jets lose energy and are modified as they
traverse the medium, a phenomenon called jet quenching. ALICE is pursuing a
multi-messenger approach that systematically studies different features of jet
quenching to explore the microscopic structure of the QGP and its governing
degrees of freedom.
The inclusive jet and hadron production, as well as their correlations from
low- to high-$p_{\rm T}$, allow for an assessment of one of the hallmarks of
jet quenching - jet energy loss to the medium. In practice, although energy is
conserved, the yields of jets with finite jet resolution parameter $R$ are
suppressed in heavy-ion collisions due to jet-medium interactions. The
suppression of the leading hadrons and fully reconstructed jets in Pb–Pb
collision at the LHC broadened the observations found at RHIC. That is the QGP
is opaque to jets over a large energy range [51, 52, 53]. Recent measurements
of hadron-jet correlations [54] show a potential explanation for the fate of
the lost energy in the jet quenching process - in both the momentum and
angular scale. The energy is distributed over large angles, and recovered in
low-$p_{\rm T}$ (below 20 GeV/c) jets. Additionally, new results on inclusive
jet suppression [55], using Machine Learning techniques [56], show that larger
$R$ jets (up to $R=0.6$) are suppressed relative to smaller $R$ jets at
$p_{\rm T}$ = 40 GeV/c, and the energy is not yet recovered. In Runs 3&4,
ALICE will continue measurements of jet energy loss with inclusive jet
suppression and semi-inclusive coincidences of hadron-jet [57], photon-hadron,
and photon-jet. The substantial increase in statistics and new experimental
techniques, such as mixed events and Machine Learning [56], will provide
increased precision and an extended kinematic range for jets, and high-$p_{\rm
T}$ hadrons and photons.
At the same time, the hadron-jet coincidences are sensitive to the _jet-
correlated_ response of the medium to the presence of energetic probes [57].
This flux of medium energy induced by the propagating jet is one of the most
sought after consequences of jet-medium interactions [58]. A medium composed
of scattering centers - quasi-particles - ought to, with some finite
probability, induce large angle Moliere scatterings of the propagating partons
[59, 60]. Such a jet deflection should manifest itself in the modified
acoplanarity of the di-jet pair in Pb–Pb compared to pp collisions. Utilizing
hadron-jet coincidences, ALICE has investigated this effect at the lowest jet
$p_{\rm T}$ possible and with a large jet $R$ [54]. Additionally, jet
deflection should manifest itself in large momentum kicks to the hard core of
the jet [61, 59]. ALICE also uses the groomed hardest $k_{T}$ to probe this by
looking at differences between pp and Pb–Pb collisions at larger groomed
$k_{\rm T}$ values [55]. Likewise, in this area, Runs 3&4 will provide further
precision to put improved quantitative constraints on the probability of this
process within the QGP.
Jet fragmentation patterns and modification of the jet structure are also key
to understanding of the interactions of the medium with the jet. ALICE has for
the first time measured fully corrected angular and momentum sub-jet structure
of jets [62, 63], and has made advances towards measurements of other
quantities such as jet angularities, jet-axes, the Lund Plane, and intrajet
hadron correlations [64, 54, 65, 66]. These measurements are compared to both
Monte Carlo models and analytical calculations. They offer a new and more
stringent view of the jet-medium interactions. In particular, new measurements
show the largest angle splittings within a groomed jet are suppressed as
compared to small-angle splittings, which can be connected to a characteristic
aperture scale at which the jet fragments/prongs interact with the medium
incoherently. Moreover, the new measurements of leading subjets suggest
sensitivity of the predicted flavor dependence of jet quenching that should be
quantified with more precise data. Concurrently, the new measurements of jet-
axes disfavor the intra-jet $p_{\rm T}$ broadening, as prescribed by the BDMPS
formalism as the main mechanism of energy loss in the QGP [67]. The Run 3&4
data will be instrumental in providing the necessary precision to further
investigate such observations. However, most importantly, the new high
statistics data will enable new differential measurements [24]. These will
study different regions of the parton shower phase-space related to rare/hard
vs. multiple-soft medium-induced radiation. This can be achieved via
simultaneous measurements of angular and momentum distributions of jet
structure such as the Jet Lund Plane [68].
The modification of jets in the medium is expected to depend on the path the
jet traveled in the medium. ALICE traditionally measures this in two ways.
They involve using correlations and measuring the jet yield suppression with
respect to the event-plane angle. Measurements by ALICE of hadron-jet
correlations show no dependence on the angle of the jet with respect to the
event plane angle within the uncertainties [69]. On the other hand,
measurements of the jet $v_{2}$ demonstrate a significant azimuthal anisotropy
[70]. ALICE has continued to explore these effects using higher statistics
data from Run 2. For example, a technique that utilizes Event Shape
Engineering [71] to select events based on their anisotropy within a
centrality class, demonstrates an increased suppression of out-of-plane yields
for more anisotropic events [72]. This technique can be further pursued using
the higher statistics Run 3&4 data to investigate the path length dependence
of jet substructure modification. Finally, a first look at photon-jet
correlation measurements with ALICE was investigated [73]. These studies
demonstrated the ability to measure photon-tagged jets at lower $p_{\rm
T}^{\gamma}$ (20 GeV/$c$). This measurement is statistically limited, thus
future ALICE photon-jet measurements will be crucial for measuring the
absolute energy loss, since the photon provides an estimate of the initial
$p_{\rm T}$ of the jet.
ALICE has also performed measurements of the heavy-flavor hadrons as an
additional probe of parton-medium interactions, with examples found in [74,
75, 76]. Heavy flavor hadrons offer a unique opportunity to study the medium
effects because of the well established probe - heavy quarks. These are
created early in the hard scatterings, and maintain their flavor identity
throughout the evolution of the medium. A new set of measurements focusing on
the leading heavy-flavor particles will take advantage of this theoretical
control. It will also profit from improved experimental control, such as
better discrimination of ”combinatorial jets” - the uncorrelated background to
hard scattering flux of energy, over the light flavor measurements. This
applies to both measurements of the heavy-flavor jet $p_{\rm T}$ spectra and
jet substructure observables. One of the striking findings by ALICE is the
observed mass dependence of energy loss. This manifests itself in smaller
nuclear modification factors for prompt $D$ mesons (formed from charm)
compared to non-prompt $D$ mesons (formed from bottom quarks) at $p_{\rm T}$
$>8$ GeV/$c$. This onset is interpreted as the dead cone effect [6]. In future
runs, ALICE will take advantage of its precision tracking to study this effect
in more detail, and contrast it with the impact of gluon splitting processes
within the parton shower [77].
In general, with the newly upgraded ALICE detector, the data from Runs3&4 are
expected to improve the quantification of how light and heavy quarks propagate
within the medium. The new high-statistics measurements will bring additional
constraints to the extraction of the transverse and longitudinal transport
coefficients. These include the jet transport parameter $\hat{q}$ [78, 79],
which quantifies the transverse diffusion. The longitudinal drag $\hat{e}$ and
diffusion $\hat{e_{2}}$ coefficients [80], as well as the heavy-flavor
diffusion coefficient $D_{s}$ [81], are also expected to be better
constrained. These coefficients characterize the coupling of such processes to
the medium, and the ultimate task is to perform a dedicated set of
measurements that will allow further connections to the emergence of the QGP
on a microscopic level. Runs 3&4 will also allow for a revisiting of the jet
substructure measurements for the heavy-flavor induced jets e.g. angular
structure of groomed and ungroomed jets. This can be achieved over a broad
kinematic regime, that is largely unique to ALICE (from $p_{\rm T}$
$\approx\leavevmode\nobreak\ 0$ of heavy-flavor hadrons to hundreds of GeV/$c$
jets). The new data will enable ALICE to fully explore the momentum and the
angular correlations within the jet substructure with a systematic study of
the Lund Plane (see considerations in [24]). With this new data, and new
techniques, the flavor dependence of the Lund Plane can be extended beyond the
charm sector [6] to the beauty sector [82].
For Runs 5&6, ALICE 3 will provide an environment for precision jet
measurements over much larger kinematic ranges than previously accessible at
the LHC. Specifically, jet hadrochemistry will become possible over such a
range with the excellent PID capabilities for identifying hadrons inside jets.
This is valuable to study in vacuum to provide inputs to hadronization models
[83], and also in the QGP since the interactions with the medium could change
the chemistry make-up of a jet [84]. Additionally, ALICE 3 will provide
increased resolution, purity, and statistics for heavy-flavor jets. The
increased psuedorapidity coverage will allow for measurements of heavy-flavor
jet correlations or photon-heavy-flavor jet correlations with unprecedented
purity at low transverse momentum scales, providing insight into the
microscopic properties of in-medium energy loss, including its path length
dependence. Finally, ALICE 3 allows for the possibility to resolve heavy-
flavor hadron pairs inside of jets to probe gluon splittings, which can be
used to study the space-time picture of jet quenching [85], and access a pure
gluon sample of jets.
Finally, while we are in the midst of fully understanding the measurements
that need to be done, the continued development of analysis techniques is
essential for jets in a complex environment. This is related to both the
careful selection of the most useful measurements [86], but also to the
development of the appropriate correction strategies allowing for meaningful
theoretical comparisons. These include fully corrected substructure
measurements [87, 88], and background mitigated hadron-jet correlations down
to lowest jet momenta with large jet resolution parameters. At the same time,
it is important to encourage the divulgence of uncertainty correlations for
the new generation of measurements. Such an activity will take advantage of
the newly developed Bayesian inference methods [78, 89]. This will also aid
Machine Learning projects that will drive the development of the modeling of
jet-medium interactions in the coming years [90, 91, 86, 92].
4. 4.
How do rarely produced hadrons form and interact?
A detailed program by ALICE for Runs 3-6 will shed further light on more
exotic heavier flavor states in order to unambiguously determine the
hadronization mechanisms, and thus the creation of matter, as a function of
flavor. In this regard, the measurements of multi-heavy-flavour hadrons and
exotic states offer unprecedented sensitivity. To achieve high statistics for
multi-heavy-flavour particles, a novel experimental approach is needed to
track all their decay products, typically including hyperons, before they
decay. This calls very high tracking/vertexing precision very close to the
interaction point, particle identification capabilities over a wide transverse
momentum range, and high readout rates. Moreover, large acceptance is
required, not only for reasons of statistics, but also in order to investigate
the dependence of the production of multi-heavy-flavour hadrons on the
variation of the heavy quark density with rapidity.
Recent flavor dependent hadronization measurements have become a unifying
theme for the bulk and jet sector of QCD. The hadronization of a final state
parton cannot be described adopting a perturbative approach, and is usually
modelled through a phase of string breaking and/or cluster formation, which is
considered to be independent of the surrounding parton density [93]. However,
as also confirmed by measurements of e.g. baryon yields in pp collisions at
the LHC [7, 94], additional hadronization mechanisms may exist, whereby quarks
that are close in phase space can combine into colourless hadrons. Dynamic
modeling of these processes, in particular in the strange and charm sector,
has led to many novel approaches that were implemented in established event
generator models, such as PYTHIA. Color reconnection (CR) and multiple parton
interactions (MPI) were attempts to parameterize features of the initial state
and final state into the actual formation process of hadrons. Furthermore, it
was shown that initial state gluon saturation might lead to quantum
entanglement effects, which impact the final state hadron multiplicities [95].
Experimental advances in the strange sector include the detailed mapping of
strangeness enhancement in proton-proton collisions as a function of
centrality [7], and the potential flavor hierarchy in the light/strange hadron
formation during the QCD crossover [96]. These led to refinements of the
transport codes that were then followed up by more detailed measurements of
baryon to meson formation in the strange sector. Such advancements have been
studied in the charm sector, as a function of system size [94]. In heavy-ion
collisions, where partons may travel freely over distances much larger than
the typical hadron sizes, and a dense system of partons close to thermal
equilibrium is formed, recombination mechanisms become more dominant. These
make the production of baryons and other heavy hadrons more favourable than in
pp collisions [97]. First measurements of baryon/meson ratios in the charm
sector also indicate a low-$p_{\rm T}$ enhancement consistent with such a
picture that has dominated similar measurements in the light and strange
sector associated with the soft underlying event [98].
A comprehensive campaign of precision measurements of charm baryon production
yields across collision systems is planned for Runs 3&4\. This starts with
baryons containing only one charm quark, but several light and strange quark
combinations. The evolution of particle production as a function of increasing
strangeness or charm content is one of the main open questions regarding
hadron formation, which is a key feature of the non-perturbative sector of
QCD. For example, LHCb has found a series of tetra- and penta-quark states in
the charm sector. A detailed study of strange penta- and hexa-quark states in
ALICE found no evidence in the strange sector. The measured upper limits are
several orders of magnitude below estimates from statistical hadronization
models (SHM), which have done extremely well on postulating the yields of
composite objects in the light sector [99]. SHM predictions assume full
equilibration of all flavors up to charm. The comparison of the measured
yields to these predictions would provide a very sensitive measure of the
degree of equilibration of charm quarks in the medium, including its system-
size dependence.
Ultimately, it must be possible to relate the equilibration level to the
transport properties of the QGP via kinetic transport modeling in order to
provide a consistent description of the entire phenomenology. Complementing
these opportunities, relativistic heavy-ion collisions provide the only
available tool to study the formation and interaction of exotic hadronic
states inside a medium filled with deconfined color charges. This puts new
constraints on the properties of the states, the characterization of their
binding potential, and the details of their hadronization mechanisms. Nuclear
collisions at the LHC energies, in particular, offer a unique opportunity to
test these mechanisms in the presence of large charm-quark densities. In this
environment, while the screening of color charges may reduce the yield of
certain bound states, medium-induced or medium enhanced mechanisms may enhance
their production in the low-$p_{\rm T}$ region, as predicted by statistical
hadronization models. The interest of these observations is twofold. On one
hand, the rates of formation and dissociation of bound states depend on their
binding energy and size, and the measurement of the production rates and
anisotropic flow will allow one to distinguish compact multi-quark
configurations from molecular states. On the other hand, states whose nature
can be determined by other means, for example by measuring interaction
potentials through final state interactions, could be exploited as a
calibrated probe in nuclear collisions. This would shed light on the
underlying mechanisms and timescales driving the hadronization in a deconfined
medium, and test the properties of the deconfined medium itself.
Predictions for light flavor tetraquark states have existed for over 40 years
[100], but the existence of these states has up to now not been confirmed. To
begin to address this, ALICE has published three papers on tetraquark studies
of the $a_{0}(980)$ resonance, a tetraquark candidate, using kaon-kaon
correlations from Runs 1&2 [101, 102, 103] for pp and Pb–Pb collisions. In
Runs 3&4, ALICE will continue these two-meson correlation studies for other
low lying tetraquark candidates, the $K^{*}_{0}(700)$ and $f_{0}(500)$. The
high rate capabilities in Run 3&4, will also allow access to rare phenomena in
the hadronic phase formed after the phase transition. These include the
production of nuclei, hypernuclei and, combined with excellent heavy-flavour
capabilities, possibly the as-yet undiscovered supernuclei (nuclei in which a
nucleon is replaced by a charmed baryon). Wide acceptance, again combined with
outstanding heavy-flavour performance at low transverse momenta, will allow to
extend the studies of hadron-hadron potentials via two-particle correlations
to the charm sector. This will provide a powerful tool to investigate the
structure of the newly discovered charmed exotic states. In Runs 5&6, the
measurement of the production yields of multi-charm baryons, which can only be
produced by combination of uncorrelated charm quarks, would provide a
qualitatively new handle on the production of heavy-flavour hadrons.
Measurements of multi-charm hadrons, such as the $\Xi^{+}_{cc}$ (ccd),
$\Xi^{++}_{cc}$ (ccu), $\Omega^{+}_{cc}$ (ccs), $\Omega^{++}_{ccc}$ (ccc), and
exotic states such as the newly discovered T${}^{+}_{cc}$ (ccud) [104, 105],
would provide a direct window on hadron formation from the QGP. In fact, the
yields of multi-charm baryons relative to the number of produced charm quarks
are predicted to be significantly enhanced in AA relative to pp collisions
[106]. Enhancements are expected by as much as a factor 100 for the recently
observed $\Xi_{cc}$ baryon [107], and even by as much as a factor 1000 for the
as yet undiscovered $\Omega_{ccc}$ baryon. The observation and precise
quantification of such effects, would represent a major discovery for the
study of the properties of deconfined matter.
In addition, LHC heavy-ion collisions also have a discovery potential for more
exotic hadronic states. This is because of the long-lived deconfined medium
and the large cross sections for heavy quarks available at the highest
energies. These provide an ideal playground for the production of exotic
states with e.g. multiple b quarks. The most notable case is the
T${}^{-}_{bb}$ bb, whose experimental detection would profit from the long
predicted lifetime (c$\tau$ $\sim$ 2.3 mm) due to the stability of the state
with respect to strong decays [108] (contrary to the recently discovered,
shortly-lived T${}^{+}_{cc}$ state) . Discovery opportunities also exist for
compact bound hidden-charm hexaquarks, also predicted to be stable with
respect to strong interaction decays [109], and molecular states composed of
three D mesons. Studying exotic QCD states in nuclear collisions is therefore
of central importance for QCD physics. The feasibility of such studies has
been demonstrated by a recent measurement of $\chi_{c1}$(3872) production in
Pb–-Pb collisions by the CMS collaboration in the range $p_{\rm T}$ $>$ 10
GeV/c [110]. An immediate goal for ALICE 3 is to measure the production of
$\chi_{c1}$(3872) down to the $p_{\rm T}$ region $<$5-6 GeV/c, where a
significant enhancement of the yield was predicted [111]. This is not
accessible by other LHC experiments. For hadrons containing charm and beauty
quarks, scattering experiments are not feasible to determine the interaction
strength between hadrons, therefore the only way to access the information is
the femtoscopy technique. This technique consists of the measurement of
correlations in momentum space for hadron-hadron pairs, and can be used to
extract the corresponding scattering parameters [112]. The ALICE 3 upgrade
will allow the measurement of several hadron combinations including DD∗,
$\Lambda_{c}^{+}\Sigma_{c}^{0;+;++}$ and BB∗ in pp, p–Pb and Pb–Pb collisions
and thereby shed light on the nature of many exotic hadrons. In particular,
using the same method as mentioned above in studying tetraquark states with
meson-meson correlations, the ALICE 3 upgrade will allow the study of the
$f_{2}(2010)$ and $\psi(3770)$ tetraquark candidates with $\phi\phi$ and
$D^{+}D^{-}$ correlations, respectively.
5. 5.
What are the mechanisms that lead to QGP-like signals in small systems?
The formation of the QGP in heavy-ions has been confirmed via numerous
measurements. A key example is the simultaneous observation of collective flow
phenomena traced to a strongly coupled liquid, and jet quenching. Measurements
in small collision systems, such as pp and p–Pb at the LHC, have shown
particle correlations and strangeness yields in high-multiplicity collisions
resemble observations associated with the creation of the QGP [113, 9, 114,
7]. On the other hand, jet quenching in small systems has not been observed.
The resolution of this enigma remains one of the most important questions for
our field. ALICE has contributed with a stringent limit on energy loss within
small collision systems [115], and plans to improve on those measurements with
the high-statistics data in the upcoming LHC runs [24]. ALICE can also use
machine learning methods to design and measure new jet observables that are
maximally modified by traversing small systems [86, 116].
Anisotropic flow measurements in high-multiplicity small system collisions at
RHIC and the LHC are consistent with hydrodynamic predictions [117, 118]. This
indicates these collisions may create the smallest possible QGP droplets.
However, these measurements from Runs 1&2 suffer from large uncertainties due
to ambiguities in the non-flow subtraction methods pursued at the LHC [119].
For Run 4, the introduction of the FoCal will enable $\Delta\eta$ separations
of up to 9 units for two-particle correlation anisotropic flow measurements of
$v_{n}$, with such separations being critical in reducing non-flow. The broad
increase from the ALICE 3 acceptance will also reduce non-flow contributions
in two-particle correlation $v_{n}$ measurements by a factor of 4 for Runs 5&6
[120]. Four particle $v_{n}$ measurements enjoy a reduction of a factor 64,
and this increases exponentially with the number of particles used to
determine $v_{n}$. Such reductions are vital in the transition of anisotropic
flow measurements in small systems from being qualitative to truly
quantitative. Measurements in Run 4 and ALICE 3 are therefore essential for
precise quantitative constraints of the studies of the applicability of
hydrodynamics at its limits in small systems [121]. The improvements in non-
flow suppression and increased statistics are also essential for identified
hadron $v_{n}$ measurements vs. $p_{\rm T}$ in small systems. These test
hydrodynamic mass ordering at low-$p_{\rm T}$, with arises due to a common
radial flow profile. Such measurements, using ALICE’s highly competitive
particle identification capacities, will be key for all future precision tests
of hydrodynamics in small systems. These tests can also be expanded for the
unmeasured $\Xi$ and $\Omega$ $v_{n}$ vs. $p_{\rm T}$, which have proved
statistically challenging in Runs 1&2.
Finally, the increased statistics for extremely high multiplicity pp
collisions in Runs 3&4 will allow for measurements of the $\Omega/\pi$ ratio
at $\rm{d}N_{ch}/\rm{d}\eta\sim 96$. This is six times larger than
corresponding measurements from Runs 1&2, and fully overlap with ALICE’s Pb–Pb
measurements [7]. This will in turn allow for a critical test regarding
particle production mechanisms and the interpretation of strangeness
enhancment in pp collisions [24]. If this ratio saturates and reaches the
thermal limit in Pb–Pb collisions, this would support the idea the QGP can be
created in pp collisions. On the other hand, if this ratio continues to
increase beyond the Pb–Pb values, this perhaps would favor non-QGP mechanisms,
such as color ropes implemented in the DIPSY model [122], that predict a
continuing increase.
6. 6.
What are the connections and broad impacts of ALICE measurements to other
fields of physics?
One of the recent major discoveries are gravitational wave signals from
neutron star mergers. The tidal deformation during approach and the ring-down
of the frequency spectrum after the merger hold potentials clues on the core
composition of neutron stars. In particular, this applies for stars with a
large mass to radius ratio. Model predictions range from simple neutron
matter, to hyperon matter, and deconfined quark matter [123]. Theoretically,
the main required ingredient is an equation of state for the system as a
function of density and temperature. Experimentally, relativistic heavy-ion
experiments, and in particular ALICE with its high luminosity and data rate
beyond Run 3, will be able to map out the production of hyperons, hypermatter
and quark matter. Since deconfined matter and hyperon production have been
discussed in the previously, the focus here is on hyper nuclei production and
the measurement of the balance of attractive and repulsive forces in the
interaction between hyperons and protons/neutrons through femtoscopic
measurements. During Runs 1&2 these studies have shown great promise as
documented in a recent Nature article by ALICE [112]. Certain di-baryon
correlations in the light and strange sector show attractive potentials, which
should lead to bound hexa-quark states. These results should also be viewed in
the context of hyper nuclei production. In previous campaigns, ALICE has
successfully reconstructed the lightest hypernucleus, the hypertriton and its
anti-particle [124].
In Run 4, we hope to extend the statistical sample in Pb–Pb collisions to a
level that might make the reconstruction of A=4 hypernuclei, in particular the
${}^{4}_{\Lambda}$H. Their measurement is interesting for precision tests of
particle production models [125] and to constrain hyperon-nucleon potentials
[126], and thus the formation probability of hyper-matter in dense stellar
objects. Regarding Runs 5&6, anti-nuclei and anti-hyper-nuclei with A$>$4 such
as anti-${}^{5}_{\Lambda}$He or anti-6Li have yet to be discovered, and may
well be in reach of ALICE 3. The ALICE 3 apparatus is ideally suited for the
observation of the $A=4$ or $A=5$ hyper-nuclei like ${}^{4}_{\Lambda}$H or
${}^{5}_{\Lambda}$He. The measurement of $A=6$ nuclei would provide precision
tests for the formation of bound clusters thanks to the special nature of 6He
and 6Li. 6He is the lightest known (anti-)halo-nucleus. Its production is
therefore expected to be suppressed in coalescence models with respect to
thermal-statistical models, due to its much larger size. 6 Li is a stable
isotope with a spin of $J=1$. With respect to the helium isotopes 4He and 6He
with $J=0$, 6Li production is therefore expected to be enhanced by the
degeneracy factor g of its spin-substates; $g=2J+1=3$. The expected production
yields $d$N/$dy$ for A=5 hyper-nuclei and A=6 nuclei are in the 10-9 to 10-11
range in Pb–Pb collisions.
7. 7.
What can ALICE achieve regarding Beyond Standard Model (BSM) Physics?
Although studying BSM physics is not ALICE’s primary purpose, there are areas
where it can make extremely useful contributions, with the benefit of the
upgrades and increased statistics in future runs. One such area is the study
of light-by-light scattering, $\gamma\gamma\rightarrow\gamma\gamma$, using
UPCs. Light-by-light scattering occurs via a box diagram, which includes
contributions from all (standard model or BSM) electrically charged particles.
In addition, it can be mediated by axion-like particles,
$\gamma\gamma\rightarrow A\rightarrow\gamma\gamma$, which leads to a diphoton
invariant mass spectrum that is peaked at the axion mass. By virtue of its
sensitivity to lower $p_{\rm T}$ particles, ALICE (especially ALICE 3) can
probe lower axion masses than ATLAS or CMS. ALICE is also sensitive to two-
photon production of $\tau^{+}\tau^{-}$. Such studies of this reaction can be
used to put limits on BSM phenomena such as a $\tau$ anomalous magnetic
moment.
The discovery of stable massive particles (SMPs) beyond the Standard Model
(BSM), such as monopoles, gluinos, heavy leptons, etc., would address a number
of important questions in modern physics, including the nature of dark matter,
and the unification of fundamental forces. The majority of predictions suggest
that SMPs are far too massive to be produced in a foreseeable accelerator
[127]. However, there are suggestions some of them, e.g. magnetic monopoles,
could appear in a mass range accessible at the LHC [128], in particular in
Pb–Pb collisions. Detection of such particles is a very challenging due to
their highly ionizing nature, which leads to saturation and even malfunction
of electronics of e.g. silicon-based detectors [129]. The newly upgraded GEM
(Gas Electron Multiplier) based Time Projection Chamber (TPC) of ALICE is free
from this challenge, and is capable of measuring very large energy deposits
anticipated from monopoles.
An exciting avenue to study dark matter signals lies with cosmic-ray
antinuclei, such as antihelium. This is considered another promising signature
of the existence of weakly-interactive mass particles (WIMP), which represent
an important candidate for dark matter [130]. Since the background from
hadronic interactions of primary rays are negligible [131], the preliminary
evidence of a handful of anti-3He events collected by the AMS collaboration,
could be due to a previously neglected process. Namely, these could be the
production of $\Lambda_{b}^{0}$ baryons in dark-matter annihilation, and their
subsequent decay into anti-3He [132]. However, the decay rates of
$\Lambda_{b}^{0}$ to anti-nuclei are not experimentally measured, and serve as
crucial inputs to understand the AMS data. The large beauty-quark production
cross section at LHC energies, and the identification capabilities for nuclei
over practically the full phase space, make ALICE 3 the ideal experiment for
such studies.
## References
* [1] ALICE Collaboration, “ALICE upgrade physics performance studies for 2018 Report on HL/HE-LHC physics”,. https://cds.cern.ch/record/2661798.
* [2] ALICE Collaboration, “A Forward Calorimeter (FoCal) in the ALICE experiment”, Tech. Rep. ALICE-PUBLIC-2019-005, Oct, 2019. https://cds.cern.ch/record/2696471.
* [3] ALICE Collaboration, “Letter of intent for ALICE 3: A next-generation heavy-ion experiment at the LHC”, arXiv:2211.02491 [physics.ins-det].
* [4] ALICE Collaboration, “Letter of intent for ALICE 3: A next generation heavy-ion experiment at the LHC”, tech. rep., CERN, Geneva, 2022\. https://cds.cern.ch/record/2803563.
* [5] ALICE Collaboration, “Performance of the ALICE Electromagnetic Calorimeter”, arXiv:2209.04216 [physics.ins-det].
* [6] ALICE Collaboration, S. Acharya et al., “Direct observation of the dead-cone effect in quantum chromodynamics”, Nature 605 no. 7910, (2022) , arXiv:2106.05713 [nucl-ex]. [Erratum: Nature 607, E22 (2022)].
* [7] ALICE Collaboration, J. Adam et al., “Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions”, Nature Phys. 13 (2017) , arXiv:1606.07424 [nucl-ex].
* [8] ALICE Collaboration, K. Aamodt et al., “Elliptic flow of charged particles in Pb-Pb collisions at 2.76 TeV”, Phys. Rev. Lett. 105 (2010) , arXiv:1011.3914 [nucl-ex].
* [9] ALICE Collaboration, B. B. Abelev et al., “Long-range angular correlations of $\rm\pi$, K and p in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV”, Phys. Lett. B 726 (2013) , arXiv:1307.3237 [nucl-ex].
* [10] ALICE Collaboration, K. Aamodt et al., “Centrality dependence of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV”, Phys. Rev. Lett. 106 (2011) , arXiv:1012.1657 [nucl-ex].
* [11] ALICE Collaboration, K. Aamodt et al., “Higher harmonic anisotropic flow measurements of charged particles in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV”, Phys. Rev. Lett. 107 (2011) , arXiv:1105.3865 [nucl-ex].
* [12] ALICE Collaboration, “The ALICE experiment - A journey through QCD”, arXiv:2211.04384 [nucl-ex].
* [13] R. Abdul Khalek et al., “Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report”, Nucl. Phys. A 1026 (2022) , arXiv:2103.05419 [physics.ins-det].
* [14] J. G. Contreras and J. D. Tapia Takaki, “Ultra-peripheral heavy-ion collisions at the LHC”, Int. J. Mod. Phys. A 30 (2015) .
* [15] S. Klein and P. Steinberg, “Photonuclear and Two-photon Interactions at High-Energy Nuclear Colliders”, Ann. Rev. Nucl. Part. Sci. 70 (2020) , arXiv:2005.01872 [nucl-ex].
* [16] S. Klein et al., “New opportunities at the photon energy frontier”, arXiv:2009.03838 [hep-ph].
* [17] ALICE Collaboration, B. B. Abelev et al., “Exclusive $\mathrm{J/}\psi$ photoproduction off protons in ultra-peripheral p-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV”, Phys. Rev. Lett. 113 no. 23, (2014) , arXiv:1406.7819 [nucl-ex].
* [18] ALICE Collaboration, S. Acharya et al., “Energy dependence of exclusive $\mathrm{J}/\psi$ photoproduction off protons in ultra-peripheral p–Pb collisions at $\sqrt{s_{\mathrm{\scriptscriptstyle NN}}}=5.02$ TeV”, Eur. Phys. J. C 79 no. 5, (2019) , arXiv:1809.03235 [nucl-ex].
* [19] ALICE Collaboration, B. Abelev et al., “Coherent $J/\psi$ photoproduction in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV”, Phys. Lett. B 718 (2013) , arXiv:1209.3715 [nucl-ex].
* [20] ALICE Collaboration, E. Abbas et al., “Charmonium and $e^{+}e^{-}$ pair photoproduction at mid-rapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\rm NN}}$=2.76 TeV”, Eur. Phys. J. C 73 no. 11, (2013) , arXiv:1305.1467 [nucl-ex].
* [21] ALICE Collaboration, S. Acharya et al., “Coherent $J/\psi$ and $\psi^{\prime}$ photoproduction at midrapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.02$ TeV”, Eur. Phys. J. C 81 no. 8, (2021) , arXiv:2101.04577 [nucl-ex].
* [22] CMS Collaboration, V. Khachatryan et al., “Coherent $J/\psi$ photoproduction in ultra-peripheral PbPb collisions at $\sqrt{s_{NN}}=$ 2.76 TeV with the CMS experiment”, Phys. Lett. B 772 (2017) , arXiv:1605.06966 [nucl-ex].
* [23] K. J. Eskola, C. A. Flett, V. Guzey, T. Löytäinen, and H. Paukkunen, “Exclusive $J/\psi$ photoproduction in ultraperipheral Pb+Pb collisions at the LHC to next-to-leading order perturbative QCD”, arXiv:2203.11613 [hep-ph].
* [24] Z. Citron et al., “Report from Working Group 5: Future physics opportunities for high-density QCD at the LHC with heavy-ion and proton beams”, CERN Yellow Rep. Monogr. 7 (2019) , arXiv:1812.06772 [hep-ph].
* [25] M. Hentschinski et al., “White Paper on Forward Physics, BFKL, Saturation Physics and Diffraction”, arXiv:2203.08129 [hep-ph].
* [26] S. R. Klein, J. Nystrand, and R. Vogt, “Heavy quark photoproduction in ultraperipheral heavy ion collisions”, Phys. Rev. C 66 (2002) , arXiv:hep-ph/0206220.
* [27] ATLAS Collaboration, ATLAS Collaboration, “Photo-nuclear dijet production in ultra-peripheral Pb+Pb collisions.” ATLAS-CONF-2017-011, 2017\.
* [28] CMS Collaboration, “Azimuthal correlations within exclusive dijets with large momentum transfer in photon-lead collisions”, arXiv:2205.00045 [nucl-ex].
* [29] STAR Collaboration, L. Adamczyk et al., “Coherent diffractive photoproduction of $\rho^{0}$mesons on gold nuclei at 200 GeV/nucleon-pair at the Relativistic Heavy Ion Collider”, Phys. Rev. C 96 no. 5, (2017) , arXiv:1702.07705 [nucl-ex].
* [30] STAR Collaboration, S. R. Klein, “Using coherent dipion photoproduction to image gold nuclei”, SciPost Phys. Proc. 8 (2022) , arXiv:2107.10447 [nucl-ex].
* [31] J. Cepila, J. G. Contreras, and J. D. Tapia Takaki, “Energy dependence of dissociative $\mathrm{J/}\psi$ photoproduction as a signature of gluon saturation at the LHC”, Phys. Lett. B 766 (2017) , arXiv:1608.07559 [hep-ph].
* [32] J. Cepila, J. G. Contreras, M. Krelina, and J. D. Tapia Takaki, “Mass dependence of vector meson photoproduction off protons and nuclei within the energy-dependent hot-spot model”, Nucl. Phys. B 934 (2018) , arXiv:1804.05508 [hep-ph].
* [33] S. R. Klein and H. Mäntysaari, “Imaging the nucleus with high-energy photons”, Nature Rev. Phys. 1 no. 11, (2019) , arXiv:1910.10858 [hep-ex].
* [34] M. Broz, J. G. Contreras, and J. D. Tapia Takaki, “A generator of forward neutrons for ultra-peripheral collisions: $\textbf{n}^{\textbf{O}}_{\textbf{O}}\textbf{n}$”, Comput. Phys. Commun. 253 (2020) , arXiv:1908.08263 [nucl-th].
* [35] A. Bylinkin, J. Nystrand, and D. Tapia Takaki, “Vector meson photoproduction in UPCs with FoCal”, arXiv:2211.16107 [nucl-ex].
* [36] J. C. Martens, J. P. Ralston, and J. D. Tapia Takaki, “Quantum tomography for collider physics: Illustrations with lepton pair production”, Eur. Phys. J. C 78 no. 1, (2018) , arXiv:1707.01638 [hep-ph].
* [37] M. Klusek-Gawenda and J. D. Tapia Takaki, “Exclusive Four-pion Photoproduction in Ultra-peripheral Heavy-ion Collisions at RHIC and LHC Energies”, Acta Phys. Polon. B 51 no. 6, (2020) , arXiv:2005.13624 [hep-ph].
* [38] ALICE Collaboration, S. Acharya et al., “Anisotropic flow in Xe-Xe collisions at $\mathbf{\sqrt{s_{\rm{NN}}}=5.44}$ TeV”, Phys. Lett. B 784 (2018) , arXiv:1805.01832 [nucl-ex].
* [39] G. Giacalone, J. Jia, and C. Zhang, “Impact of Nuclear Deformation on Relativistic Heavy-Ion Collisions: Assessing Consistency in Nuclear Physics across Energy Scales”, Phys. Rev. Lett. 127 no. 24, (2021) , arXiv:2105.01638 [nucl-th].
* [40] N. Summerfield, B.-N. Lu, C. Plumberg, D. Lee, J. Noronha-Hostler, and A. Timmins, “16O 16O collisions at energies available at the BNL Relativistic Heavy Ion Collider and at the CERN Large Hadron Collider comparing $\alpha$ clustering versus substructure”, Phys. Rev. C 104 no. 4, (2021) , arXiv:2103.03345 [nucl-th].
* [41] J. Jia, G. Giacalone, and C. Zhang, “Separating the impact of nuclear skin and nuclear deformation on elliptic flow and its fluctuations in high-energy isobar collisions”, arXiv:2206.10449 [nucl-th].
* [42] A. V. Giannini, M. N. Ferreira, M. Hippert, D. D. Chinellato, G. S. Denicol, M. Luzum, J. Noronha, T. N. da Silva, and J. Takahashi, “Assessing the ultra-central flow puzzle in the Bayesian era”, arXiv:2203.17011 [nucl-th].
* [43] P. Carzon, S. Rao, M. Luzum, M. Sievert, and J. Noronha-Hostler, “Possible octupole deformation of 208Pb and the ultracentral $v_{2}$ to $v_{3}$ puzzle”, Phys. Rev. C 102 no. 5, (2020) , arXiv:2007.00780 [nucl-th].
* [44] S. Raman, C. W. G. Nestor, Jr, and P. Tikkanen, “Transition probability from the ground to the first-excited 2+ state of even-even nuclides”, Atom. Data Nucl. Data Tabl. 78 (2001) .
* [45] ALICE Collaboration, B. B. Abelev et al., “Multiparticle azimuthal correlations in p -Pb and Pb-Pb collisions at the CERN Large Hadron Collider”, Phys. Rev. C 90 no. 5, (2014) , arXiv:1406.2474 [nucl-ex].
* [46] S. I. Finazzo, R. Rougemont, H. Marrochio, and J. Noronha, “Hydrodynamic transport coefficients for the non-conformal quark-gluon plasma from holography”, JHEP 02 (2015) , arXiv:1412.2968 [hep-ph].
* [47] C. Ratti, “Lattice QCD and heavy ion collisions: a review of recent progress”, Rept. Prog. Phys. 81 no. 8, (2018) , arXiv:1804.07810 [hep-lat].
* [48] S. Floerchinger and M. Martinez, “Fluid dynamic propagation of initial baryon number perturbations on a Bjorken flow background”, Phys. Rev. C 92 no. 6, (2015) , arXiv:1507.05569 [nucl-th].
* [49] ALICE Collaboration, S. Acharya et al., “General balance functions of identified charged hadron pairs of ($pi$,K,p) in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV”, Phys. Lett. B 833 (2022) , arXiv:2110.06566 [nucl-ex].
* [50] S. Pratt and C. Plumberg, “Charge balance functions for heavy-ion collisions at energies available at the CERN Large Hadron Collider”, Phys. Rev. C 104 no. 1, (2021) , arXiv:2104.00628 [nucl-th].
* [51] ALICE Collaboration, K. Aamodt et al., “Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb-Pb Collisions at $\sqrt{s_{NN}}=$ 2.76 TeV”, Phys. Lett. B 696 (2011) , arXiv:1012.1004 [nucl-ex].
* [52] ALICE Collaboration, S. Acharya et al., “Transverse momentum spectra and nuclear modification factors of charged particles in pp, p-Pb and Pb-Pb collisions at the LHC”, JHEP 11 (2018) , arXiv:1802.09145 [nucl-ex].
* [53] ALICE Collaboration, S. Acharya et al., “Measurements of inclusive jet spectra in pp and central Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV”, Phys. Rev. C 101 no. 3, (2020) , arXiv:1909.09718 [nucl-ex].
* [54] Reynier Cruz Torres for the ALICE Collaboration, “Jet acoplanarity and energy flow within jets in pb-pb and pp collisions with alice.” url: https://indico.cern.ch/event/895086/contributions/4742231/, 4, 2022. Quark Matter 2022 Conference.
* [55] Hannah Bossi for the ALICE Collaboration, “Measurements of the $r$-dependence of inclusive jet suppression and groomed jet splittings in heavy-ion collisions with alice.” url: https://indico.cern.ch/event/895086/contributions/4736731/, 4, 2022. Quark Matter 2022 Conference.
* [56] R. Haake and C. Loizides, “Machine Learning based jet momentum reconstruction in heavy-ion collisions”, Phys. Rev. C 99 no. 6, (2019) , arXiv:1810.06324 [nucl-ex].
* [57] ALICE Collaboration, J. Adam et al., “Measurement of jet quenching with semi-inclusive hadron-jet distributions in central Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV”, JHEP 09 (2015) , arXiv:1506.03984 [nucl-ex].
* [58] S. Cao and X.-N. Wang, “Jet quenching and medium response in high-energy heavy-ion collisions: a review”, Rept. Prog. Phys. 84 no. 2, (2021) , arXiv:2002.04028 [hep-ph].
* [59] F. D’Eramo, K. Rajagopal, and Y. Yin, “Molière scattering in quark-gluon plasma: finding point-like scatterers in a liquid”, JHEP 01 (2019) , arXiv:1808.03250 [hep-ph].
* [60] J. a. Barata, Y. Mehtar-Tani, A. Soto-Ontoso, and K. Tywoniuk, “Revisiting transverse momentum broadening in dense QCD media”, Phys. Rev. D 104 no. 5, (2021) , arXiv:2009.13667 [hep-ph].
* [61] J. a. Barata, Y. Mehtar-Tani, A. Soto-Ontoso, and K. Tywoniuk, “Medium-induced radiative kernel with the Improved Opacity Expansion”, JHEP 09 (2021) , arXiv:2106.07402 [hep-ph].
* [62] A Large Ion Collider Experiment, ALICE Collaboration, S. Acharya et al., “Measurement of the groomed jet radius and momentum splitting fraction in pp and Pb$-$Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV”, Phys. Rev. Lett. 128 no. 10, (2022) , arXiv:2107.12984 [nucl-ex].
* [63] ALICE Collaboration, “Measurement of inclusive and leading subjet fragmentation in pp and Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV”, arXiv:2204.10270 [nucl-ex].
* [64] James Mulligan for the ALICE Collaboration, “Jet angularity and fragmentation measurements in heavy-ion collisions with alice.” url: https://indico.cern.ch/event/895086/contributions/4715856/, 4, 2022. Quark Matter 2022 Conference.
* [65] ALICE Collaboration, “Measurements of the groomed jet radius and momentum splitting fraction with the soft drop and dynamical grooming algorithms in pp collisions at $\sqrt{s}=5.02$ TeV”, arXiv:2204.10246 [nucl-ex].
* [66] ALICE Collaboration, “Physics Preliminary Summary: Measurement of the primary Lund plane density in pp collisions at $\sqrt{s}=\rm{13}$ TeV with ALICE”,.
* [67] F. Ringer, B.-W. Xiao, and F. Yuan, “Can we observe jet $P_{T}$-broadening in heavy-ion collisions at the LHC?”, Phys. Lett. B 808 (2020) , arXiv:1907.12541 [hep-ph].
* [68] A. Lifson, G. P. Salam, and G. Soyez, “Calculating the primary Lund Jet Plane density”, JHEP 10 (2020) , arXiv:2007.06578 [hep-ph].
* [69] ALICE Collaboration, S. Acharya et al., “Jet-hadron correlations measured relative to the second order event plane in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV”, Phys. Rev. C 101 no. 6, (2020) , arXiv:1910.14398 [nucl-ex].
* [70] ALICE Collaboration, J. Adam et al., “Azimuthal anisotropy of charged jet production in $\sqrt{s_{\rm NN}}$ = 2.76 TeV Pb-Pb collisions”, Phys. Lett. B 753 (2016) , arXiv:1509.07334 [nucl-ex].
* [71] J. Schukraft, A. Timmins, and S. A. Voloshin, “Ultra-relativistic nuclear collisions: event shape engineering”, Phys. Lett. B 719 (2013) , arXiv:1208.4563 [nucl-ex].
* [72] Caitlin Beattie for the ALICE Collaboration, “Study of path-length dependent energy loss of jets in p-pb and pb-pb collisions with alice.” url: https://indico.cern.ch/event/895086/contributions/4715857/, 4, 2022. Quark Matter 2022 Conference.
* [73] Alwina Liu for the ALICE Collaboration, “Isolated-photon production and photon-jet correlations in pp and pb-pb collisions at $\sqrt{s_{\rm NN}}=$ 5.02 tev in alice.” url: https://indico.cern.ch/event/895086/contributions/4715853/, 4, 2022. Quark Matter 2022 Conference.
* [74] ALICE Collaboration, S. Acharya et al., “Prompt D0, D+, and D∗+ production in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV”, JHEP 01 (2022) , arXiv:2110.09420 [nucl-ex].
* [75] ALICE Collaboration, S. Acharya et al., “Measurement of beauty production via non-prompt ${\rm D}^{0}$ mesons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV”, arXiv:2202.00815 [nucl-ex].
* [76] ALICE Collaboration, S. Acharya et al., “Measurement of electrons from semileptonic heavy-flavour hadron decays at midrapidity in pp and Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV”, Phys. Lett. B 804 (2020) , arXiv:1910.09110 [nucl-ex].
* [77] M. Voutilainen, “Heavy quark jets at the LHC”, Int. J. Mod. Phys. A 30 no. 31, (2015) , arXiv:1509.05026 [hep-ex].
* [78] JETSCAPE Collaboration, S. Cao et al., “Determining the jet transport coefficient from inclusive hadron suppression measurements using Bayesian parameter estimation”, Phys. Rev. C 104 no. 2, (2021) , arXiv:2102.11337 [nucl-th].
* [79] JETSCAPE Collaboration, R. Ehlers et al., “Bayesian analysis of QGP jet transport using multi-scale modeling applied to inclusive hadron and reconstructed jet data”, in 29th International Conference on Ultra-relativistic Nucleus-Nucleus Collisions. 8, 2022. arXiv:2208.07950 [hep-ph].
* [80] S. Cao, A. Majumder, G.-Y. Qin, and C. Shen, “Drag Induced Radiation and Multi-Stage Effects in Heavy-Flavor Energy Loss”, Phys. Lett. B 793 (2019) , arXiv:1711.09053 [nucl-th].
* [81] G. D. Moore and D. Teaney, “How much do heavy quarks thermalize in a heavy ion collision?”, Phys. Rev. C 71 (2005) , arXiv:hep-ph/0412346.
* [82] F. A. Dreyer, G. Soyez, and A. Takacs, “Quarks and gluons in the Lund plane”, JHEP 08 (2022) , arXiv:2112.09140 [hep-ph].
* [83] ATLAS Collaboration, “Dependence of the Jet Energy Scale on the Particle Content of Hadronic Jets in the ATLAS Detector Simulation”,.
* [84] S. Sapeta and U. A. Wiedemann, “Jet hadrochemistry as a characteristics of jet quenching”, Eur. Phys. J. C 55 (2008) , arXiv:0707.3494 [hep-ph].
* [85] M. Attems, J. Brewer, G. M. Innocenti, A. Mazeliauskas, S. Park, W. van der Schee, and U. A. Wiedemann, “The medium-modified $g\to c\bar{c}$ splitting function in the BDMPS-Z formalism”, arXiv:2203.11241 [hep-ph].
* [86] Y. S. Lai, J. Mulligan, M. Płoskoń, and F. Ringer, “The information content of jet quenching and machine learning assisted observable design”, JHEP 10 (2022) , arXiv:2111.14589 [hep-ph].
* [87] A. Andreassen, P. T. Komiske, E. M. Metodiev, B. Nachman, and J. Thaler, “OmniFold: A Method to Simultaneously Unfold All Observables”, Phys. Rev. Lett. 124 no. 18, (2020) , arXiv:1911.09107 [hep-ph].
* [88] J. Mulligan and M. Ploskon, “Identifying groomed jet splittings in heavy-ion collisions”, Phys. Rev. C 102 no. 4, (2020) , arXiv:2006.01812 [hep-ph].
* [89] G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, “Bayesian analysis of heavy ion collisions with the heavy ion computational framework Trajectum”, Phys. Rev. C 103 no. 5, (2021) , arXiv:2010.15134 [nucl-th].
* [90] Y.-T. Chien and R. Kunnawalkam Elayavalli, “Probing heavy ion collisions using quark and gluon jet substructure”, arXiv:1803.03589 [hep-ph].
* [91] L. Apolinário, N. F. Castro, M. Crispim Romão, J. G. Milhano, R. Pedro, and F. C. R. Peres, “Deep Learning for the classification of quenched jets”, JHEP 11 (2021) , arXiv:2106.08869 [hep-ph].
* [92] L. Liu, J. Velkovska, and M. Verweij, “Identifying quenched jets in heavy ion collisions with machine learning”, arXiv:2206.01628 [hep-ph].
* [93] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, “Parton Fragmentation and String Dynamics”, Phys. Rept. 97 (1983) .
* [94] ALICE Collaboration, S. Acharya et al., “$\Lambda^{+}_{c}$ Production and Baryon-to-Meson Ratios in pp and p-Pb Collisions at $\sqrt{s_{NN}}$=5.02 TeV at the LHC”, Phys. Rev. Lett. 127 no. 20, (2021) , arXiv:2011.06078 [nucl-ex].
* [95] Z. Tu, D. E. Kharzeev, and T. Ullrich, “Einstein-Podolsky-Rosen Paradox and Quantum Entanglement at Subnucleonic Scales”, Phys. Rev. Lett. 124 no. 6, (2020) , arXiv:1904.11974 [hep-ph].
* [96] F. A. Flor, G. Olinger, and R. Bellwied, “Flavour and Energy Dependence of Chemical Freeze-out Temperatures in Relativistic Heavy Ion Collisions from RHIC-BES to LHC Energies”, Phys. Lett. B 814 (2021) , arXiv:2009.14781 [nucl-ex].
* [97] ALICE Collaboration, S. Acharya et al., “Constraining hadronization mechanisms with $\rm\Lambda_{\rm c}^{+}$/D0 production ratios in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV”, arXiv:2112.08156 [nucl-ex].
* [98] ALICE Collaboration, S. Acharya et al., “Production of $\Lambda$ and $K^{0}_{s}$ in jets in p–Pb collisions at $\sqrt{s_{NN}}$=5.02 TeV and pp collisions at $\sqrt{s}$=7 TeV”, Phys. Lett. B 827 (2022) , arXiv:2105.04890 [nucl-ex].
* [99] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, “Decoding the phase structure of QCD via particle production at high energy”, Nature 561 no. 7723, (2018) , arXiv:1710.09425 [nucl-th].
* [100] R. L. Jaffe, “Multi-Quark Hadrons. 1. The Phenomenology of (2 Quark 2 anti-Quark) Mesons”, Phys. Rev. D 15 (1977) .
* [101] ALICE Collaboration, S. Acharya et al., “Measuring K${}^{0}_{\rm S}$K± interactions using Pb-Pb collisions at ${\sqrt{s_{\rm NN}}=2.76}$ TeV”, Phys. Lett. B 774 (2017) , arXiv:1705.04929 [nucl-ex].
* [102] ALICE Collaboration, S. Acharya et al., “Measuring K${}^{0}_{\rm S}$K± interactions using pp collisions at $\sqrt{s}=7$ TeV”, Phys. Lett. B 790 (2019) , arXiv:1809.07899 [nucl-ex].
* [103] ALICE Collaboration, S. Acharya et al., “Measuring K${}^{0}_{\rm S}$K± interactions using pp collisions at $\sqrt{s}=5$ and $\sqrt{s}=13$ TeV”, Phys. Lett. B 833 (2022) , arXiv:2111.06611 [nucl-ex].
* [104] LHCb Collaboration, R. Aaij et al., “Observation of an exotic narrow doubly charmed tetraquark”, Nature Phys. 18 no. 7, (2022) , arXiv:2109.01038 [hep-ex].
* [105] LHCb Collaboration, R. Aaij et al., “Study of the doubly charmed tetraquark $T_{cc}^{+}$”, Nature Commun. 13 no. 1, (2022) , arXiv:2109.01056 [hep-ex].
* [106] A. Andronic, P. Braun-Munzinger, M. K. Köhler, A. Mazeliauskas, K. Redlich, J. Stachel, and V. Vislavicius, “The multiple-charm hierarchy in the statistical hadronization model”, JHEP 07 (2021) , arXiv:2104.12754 [hep-ph].
* [107] LHCb Collaboration, R. Aaij et al., “Observation of the doubly charmed baryon $\Xi_{cc}^{++}$”, Phys. Rev. Lett. 119 no. 11, (2017) , arXiv:1707.01621 [hep-ex].
* [108] E. Hernández, J. Vijande, A. Valcarce, and J.-M. Richard, “Spectroscopy, lifetime and decay modes of the $T^{-}_{bb}$ tetraquark”, Phys. Lett. B 800 (2020) , arXiv:1910.13394 [hep-ph].
* [109] Z. Liu, H.-T. An, Z.-W. Liu, and X. Liu, “Where are the hidden-charm hexaquarks?”, Phys. Rev. D 105 no. 3, (2022) , arXiv:2112.02510 [hep-ph].
* [110] CMS Collaboration, A. M. Sirunyan et al., “Evidence for X(3872) in Pb-Pb Collisions and Studies of its Prompt Production at $\sqrt{s_{NN}}$=5.02 TeV”, Phys. Rev. Lett. 128 no. 3, (2022) , arXiv:2102.13048 [hep-ex].
* [111] A. Andronic, P. Braun-Munzinger, M. K. Köhler, K. Redlich, and J. Stachel, “Transverse momentum distributions of charmonium states with the statistical hadronization model”, Phys. Lett. B 797 (2019) , arXiv:1901.09200 [nucl-th].
* [112] ALICE Collaboration, A. Collaboration et al., “Unveiling the strong interaction among hadrons at the LHC”, Nature 588 (2020) , arXiv:2005.11495 [nucl-ex]. [Erratum: Nature 590, E13 (2021)].
* [113] ALICE Collaboration, B. Abelev et al., “Long-range angular correlations on the near and away side in $p$-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV”, Phys. Lett. B 719 (2013) , arXiv:1212.2001 [nucl-ex].
* [114] ALICE Collaboration, J. Adam et al., “Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV”, Phys. Lett. B 753 (2016) , arXiv:1506.08032 [nucl-ex].
* [115] ALICE Collaboration, S. Acharya et al., “Constraints on jet quenching in p-Pb collisions at $\mathbf{\sqrt{s_{NN}}}$ = 5.02 TeV measured by the event-activity dependence of semi-inclusive hadron-jet distributions”, Phys. Lett. B 783 (2018) , arXiv:1712.05603 [nucl-ex].
* [116] K. Lee, J. Mulligan, M. Płoskoń, F. Ringer, and F. Yuan, “Machine learning-based jet and event classification at the Electron-Ion Collider with applications to hadron structure and spin physics”, arXiv:2210.06450 [hep-ph].
* [117] PHENIX Collaboration, C. Aidala et al., “Creation of quark–gluon plasma droplets with three distinct geometries”, Nature Phys. 15 no. 3, (2019) , arXiv:1805.02973 [nucl-ex].
* [118] B. Schenke, C. Shen, and P. Tribedy, “Running the gamut of high energy nuclear collisions”, Phys. Rev. C 102 no. 4, (2020) , arXiv:2005.14682 [nucl-th].
* [119] S. H. Lim, Q. Hu, R. Belmont, K. K. Hill, J. L. Nagle, and D. V. Perepelitsa, “Examination of flow and nonflow factorization methods in small collision systems”, Phys. Rev. C 100 no. 2, (2019) , arXiv:1902.11290 [nucl-th].
* [120] S. A. Voloshin, A. M. Poskanzer, and R. Snellings, “Collective phenomena in non-central nuclear collisions”, Landolt-Bornstein 23 (2010) , arXiv:0809.2949 [nucl-ex].
* [121] J. L. Nagle and W. A. Zajc, “Small System Collectivity in Relativistic Hadronic and Nuclear Collisions”, Ann. Rev. Nucl. Part. Sci. 68 (2018) , arXiv:1801.03477 [nucl-ex].
* [122] C. Bierlich and J. R. Christiansen, “Effects of color reconnection on hadron flavor observables”, Phys. Rev. D 92 no. 9, (2015) , arXiv:1507.02091 [hep-ph].
* [123] E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, “Evidence for quark-matter cores in massive neutron stars”, Nature Phys. 16 no. 9, (2020) , arXiv:1903.09121 [astro-ph.HE].
* [124] ALICE Collaboration, S. Acharya et al., “Hypertriton Production in p-Pb Collisions at $\sqrt{s_{NN}}$=5.02 TeV”, Phys. Rev. Lett. 128 no. 25, (2022) , arXiv:2107.10627 [nucl-ex].
* [125] F. Bellini and A. P. Kalweit, “Testing production scenarios for (anti-)(hyper-)nuclei with multiplicity-dependent measurements at the LHC”, Acta Phys. Polon. B 50 (2019) , arXiv:1907.06868 [hep-ph].
* [126] J. Haidenbauer and I. Vidana, “Structure of single-$\Lambda$ hypernuclei with chiral hyperon-nucleon potentials”, Eur. Phys. J. A 56 no. 2, (2020) , arXiv:1910.02695 [nucl-th].
* [127] M. Fairbairn, A. C. Kraan, D. A. Milstead, T. Sjostrand, P. Z. Skands, and T. Sloan, “Stable Massive Particles at Colliders”, Phys. Rept. 438 (2007) , arXiv:hep-ph/0611040.
* [128] R. Bruce et al., “New physics searches with heavy-ion collisions at the CERN Large Hadron Collider”, J. Phys. G 47 no. 6, (2020) , arXiv:1812.07688 [hep-ph].
* [129] CMS Tracker Collaboration, W. Adam et al., “The effect of highly ionising particles on the CMS silicon strip tracker”, Nucl. Instrum. Meth. A 543 no. 2-3, (2005) .
* [130] F. Donato, N. Fornengo, and P. Salati, “Anti-deuterons as a signature of supersymmetric dark matter”, Phys. Rev. D 62 (2000) , arXiv:hep-ph/9904481.
* [131] R. Duperray, B. Baret, D. Maurin, G. Boudoul, A. Barrau, L. Derome, K. Protasov, and M. Buenerd, “Flux of light antimatter nuclei near Earth, induced by cosmic rays in the Galaxy and in the atmosphere”, Phys. Rev. D 71 (2005) , arXiv:astro-ph/0503544.
* [132] M. W. Winkler and T. Linden, “Dark Matter Annihilation Can Produce a Detectable Antihelium Flux through $\bar{\Lambda}_{b}$ Decays”, Phys. Rev. Lett. 126 no. 10, (2021) , arXiv:2006.16251 [hep-ph].
## Appendix A The ALICE-USA Collaboration
N. Alizadehvandchali9, N. Apadula5, M. Arslandok13, C. Beattie13, R.
Bellwied9, J.T. Blair8, F. Bock6, H. Bossi13, A. Bylinkin10, H. Caines13, I.
Chakaberia5, M. Cherney3, T.M. CormierI,6, R. Cruz-Torres5, P. Dhankher4, D.U.
Dixit4, R.J. Ehlers5,6, W. Fan5, M. Fasel6, F. Flor9, A.N. Flores8, D.R.
Gangadharan9, E. Garcia-Solis2, A. Gautam10, E. Glimos11, V. Gonzalez12, A.
Hamdi5, R. Hannigan8, J.W. Harris13, A. Harton2, H. Hassan6, L.B. Havener13,
C. Hughes11, T.J. Humanic7, A. Hutson9, T. Isidori10, B. Jacak4,5, P.M.
Jacobs5, F. Jonas6, A. Khatun10, M. Kim4, J.L. Klay1, S. Klein5, A.G. Knospe9,
Y.S. Lai5, E.D. Lesser4, I. Likmeta9, A. Liu4, C. Loizides6, C. Markert8, J.L.
Martinez9, A.S. Menon9, J.D. Mulligan5, A.I. Nambrath4, C. Nattrass11, N.
Novitzky6, A.C. Oliveira Da Silva11, M.H. Oliver13, L. Pinsky9, M. Płoskoń5,
M.G. Poghosyan6, C.A. Pruneau12, R.E. Quishpe9, S. Ragoni3, K.F. Read6,11,
O.V. Rueda9, D. Sarkar12, M.H.P. Sas13, J. Schambach6, N.V. Schmidt6, A.R.
Schmier11, J.E. Seger3, O. Sheibani9, N. Smirnov13, J. Song9, P.J.
Steffanic11, J.D. Tapia Takaki10, C. Terrevoli9, D. Thomas8, A.R. Timmins9,
S.A. Voloshin12, S.L. Weyhmiller13, J.R. Wright8
## Affiliation Notes
I Deceased
## Collaboration Institutes
1 California Polytechnic State University, San Luis Obispo, California
2 Chicago State University, Chicago, Illinois
3 Creighton University, Omaha, Nebraska
4 Department of Physics, University of California, Berkeley, California
5 Lawrence Berkeley National Laboratory, Berkeley, California
6 Oak Ridge National Laboratory, Oak Ridge, Tennessee
7 Ohio State University, Columbus, Ohio
8 The University of Texas at Austin, Austin, Texas
9 University of Houston, Houston, Texas
10 The University of Kansas, Lawrence, Kansas
11 University of Tennessee, Knoxville
12 Wayne State University, Detroit, Michigan
13 Yale University, New Haven, Connecticut
|
S-Dual of Maxwell Chern-Simons Theory
Adi Armoni
Department of Physics, Faculty of Science and Engineering
Swansea University, SA2 8PP, UK
<EMAIL_ADDRESS>
Abstract
We discuss the dynamics of three dimensional Maxwell theory coupled to a level
$k$ Chern-Simons term. Motivated by S-duality in string theory we argue that
the theory admits an S-dual description. The S-dual theory contains a non-
gauge 1-form field, previously proposed by Deser and Jackiw [1] and a level
$k$ $U(1)$ Chern-Simons term, ${\cal Z}_{\rm MCS}={\cal Z}_{\rm DJ}{\cal
Z}_{\rm CS}$. The couplings to external electric and magnetic currents and
their string theory realisations are also discussed.
## 1 Introduction
Chern-Simons theory is vastly used in mathematical physics, in condensed
matter physics and in string theory [2]. It was studied intensively in the
past three decades, yet the dynamics of Yang-Mills Chern-Simons theory is not
fully understood at the strong coupling regime.
Four dimensional S-duality is an exact duality between two ${\cal N}=4$ super
Yang-Mills theories, enabling us to calculate quantities in the strong
coupling regime using a dual weakly coupled theory. In the Abelian case it
reduces to the old electric-magnetic duality which swaps electric and magnetic
fields
$F\longleftrightarrow*F\,.$ (1.1)
In 3d Abelian S-duality relates the electric field to a dual scalar
$f\longleftrightarrow*d\phi\,.$ (1.2)
The purpose of this note is to extend S-duality to 3d Maxwell Chern-Simons
(MCS) theory, with either a compact or non-compact $U(1)$ gauge group. It
should hold on any spin manifold. The Lagrangian of the theory is given by
$L=-\frac{1}{2g^{2}}da_{e}\wedge*da_{e}+\frac{k}{4\pi}a_{e}\wedge da_{e}\,.$
(1.3)
MCS theory contains a vector boson of mass $M=\frac{g^{2}k}{2\pi}$. At low
energies the kinetic term is irrelevant and the theory flows to a pure level
$k$ Chern-Simons theory. As explained in section (3) the theory admits a
global $\mathbb{Z}_{k}$ 1-form symmetry generated by
$G\equiv\exp\left(i\oint(a_{e}-\frac{1}{M}*da_{e})\right)\,.$ (1.4)
When the theory is compactified on a torus, the global ${\mathbb{Z}_{k}}$
1-form symmetry is spontaneously broken, resulting in $k$ degenerate vacua.
Several attempts were made to find the S-dual of (1.3). In [1] Deser and
Jackiw proposed a ’self-dual model’ (SDM) which describes a massive vector.
While SDM describes a massive vector it does not admit a ${\mathbb{Z}_{k}}$
1-form symmetry neither does it flow to a pure Chern-Simons theory at low
energies, hence it cannot be an exact dual of MCS theory.
A closely related problem concerns the open string dynamics on a certain
Hanany-Witten brane configuration. It is well known [3, 4] that MCS theory
lives on the left brane configuration of fig.(1). Type IIB S-duality maps the
left configuration into the right configuration. Thus, knowing the field
theory that lives on the right configuration will solve the problem of finding
the S-dual. In early attempts [3, 4, 5] the authors found gauge theories with
a fractional level Chern-Simons term. While the theories they found are
classically equivalent to MCS, it cannot be the full answer, as it does not
admit the symmetries nor the same dynamics as the electric theory.
Fig. 1: The electric theory on the left brane configuration is Maxwell Chern-
Simons. The magnetic theory, obtained by type IIB S-duality, lives on the
right brane configuration.
## 2 Derivation of the duality
We may use 4d S-duality between Maxwell theories to derive the 3d duality. In
4d a pure Maxwell theory with a coupling $g$ is dual to a pure Maxwell theory
with a coupling $1/g$.
Consider the following partition function
${\cal Z}=\int DF_{m}DA_{e}\exp
i\int(-\frac{g^{2}}{2}F_{m}\wedge*F_{m}+F_{m}\wedge dA_{e})$ (2.1)
$A_{e}$ is the ’electric’ gauge field, $F_{m}$ is a ’magnetic’ gauge invariant
2-form. $g$ is the ’electric’ gauge coupling.
Upon integrating over $F_{m}$ we obtain the electric theory
${\cal Z}=\int DA_{e}\exp
i\int\left(-\frac{1}{2g^{2}}dA_{e}\wedge*dA_{e}\right)\,.$ (2.2)
If instead we integrate over $A_{e}$ we obtain
${\cal Z}=\int DF_{m}\delta(dF_{m})\exp
i\int\left(-\frac{g^{2}}{2}F_{m}\wedge*F_{m}\right)\,,$ (2.3)
hence it can be written in terms of $A_{m}$ such that $F_{m}=dA_{m}$.
${\cal Z}=\int DA_{m}\exp
i\left(-\frac{g^{2}}{2}dA_{m}\wedge*dA_{m}\right)\,.$ (2.4)
This is the magnetic theory dual to the electric theory.
Let us use dimensional reduction of (2.1) in order to derive the 3d duality.
Upon reducing to 3d the 4d 2-form $F_{m}$ becomes a 3d 2-form $f_{m}$ and
1-form $a_{m}$. The 4d gauge field $A_{e}$ becomes a 3d gauge field $a_{e}$
and a scalar $\phi_{e}$. The the 2-form $f_{m}$ and the scalar $\phi_{e}$
decouple from the rest of the action and admit
${\cal Z}=\int Df_{m}D\phi_{e}\exp
i\int(-\frac{g^{2}}{2}f_{m}\wedge*f_{m}+f_{m}\wedge d\phi_{e})\,,$ (2.5)
which leads to the well known S-duality
$d\hat{a}_{m}\longleftrightarrow*d\phi_{e}\,,$ (2.6)
where $f_{m}=d\hat{a}_{m}$.
Let us focus on the duality between $a_{e}$ and $a_{m}$, which is the prime
purpose of this note. We add to the action a Chern-Simons term111The Chern-
Simons term can be obtained by a dimensional reduction of a space dependent
theta term $\int d^{4}x\,\theta(x)F_{e}\wedge F_{e}$, such that
$\theta(x)=\frac{k}{4\pi}H(x^{3})$, where $H(x^{3})$ is the Heaviside step
function.. Our proposal is the following ’master’ partition function
${\cal Z}=\int Da_{m}Da_{e}\exp
i\int(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}+a_{m}\wedge
da_{e}+\frac{k}{4\pi}a_{e}\wedge da_{e})$ (2.7)
Note that $a_{m}$ is a gauge invariant 1-form. Upon integration over $a_{m}$
we obtain the electric theory
${\cal Z}=\int Da_{e}\exp
i\int(-\frac{1}{2g^{2}}da_{e}\wedge*da_{e}+\frac{k}{4\pi}a_{e}\wedge
da_{e})\,,$ (2.8)
namely Maxwell Chern-Simons theory.
In order to derive the magnetic theory we should use (2.7) and integrate over
$a_{e}$. This is a subtle point. Instead, let us use a change of variables
$a_{e}=b-\frac{2\pi}{k}a_{m}$, to obtain the following partition function
${\cal Z}=\int Da_{m}Db\exp
i\int(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}-\frac{\pi}{k}a_{m}\wedge
da_{m}+\frac{k}{4\pi}b\wedge db).$ (2.9)
Equation (2.9) is our proposal for the S-dual of Maxwell Chern-Simons theory.
The partition function of the magnetic theory is a product of the Deser-Jackiw
theory and a level $k$ Chern-Simons term
${\cal Z}_{\rm MCS}={\cal Z}_{\rm DJ}{\cal Z}_{\rm CS}\,.$ (2.10)
Note that $a_{m}$ is not a gauge field and therefore the term
$\frac{\pi}{k}a_{m}\wedge da_{m}$ is not ill-defined.
Both the electric and the magnetic theories describe a massive vector of mass
$M=\frac{g^{2}k}{2\pi}$ and a decoupled level $k$ Chern-Simons theory. Both
theories exhibit a 1-form ${\mathbb{Z}_{k}}$ symmetry.
Let us now provide another argument in favour of our proposal (2.9). We begin
with the magnetic brane configuration of fig.(1). It was argued by Gaiotto and
Witten [6] that the theory which lives on the intersection of the 3-brane and
the tilted 5-brane (without a D5 brane) is
${\cal Z}=\int DaDc\exp i\int(\frac{1}{2\pi}a\wedge dc+\frac{k}{4\pi}c\wedge
dc)\,.$ (2.11)
In order to understand what happens when we add a D5 brane, let us assume that
the terms that we need to add to the action are $k$ independent. Indeed, the
information about $k$ is encoded in the tilted fivebrane, not in the
threebrane. Let us use $k=0$, since in this case the duality is well
understood: the electric theory is pure Maxwell and the magnetic (mirror)
theory is a massless scalar. The brane realisation of the duality was provided
in the seminal work of Hanany and Witten [7].
We may write the theory of a free massless scalar as follows
${\cal Z}=\int DaDc\exp i\int(a\wedge*a+\frac{1}{2\pi}a\wedge dc)\,,$ (2.12)
with $a$ being a gauge invariant 1-form. The equation of motion for $c$ is
$*da=0$, namely that $a=d\chi$. Thus, for $k=0$ we obtain a theory of a free
scalar $(d\chi)^{2}$, as expected.
We found that adding a term $a\wedge*a$ to the action yields a theory that
describes the correct dual of Maxwell theory. We propose that $a\wedge*a$ is
the missing term in (2.11), namely that by adding it to (2.11) we obtain the
dual of MCS for any $k$. Note that
${\cal Z}=\int DaDc\exp i\int(a\wedge*a+\frac{1}{2\pi}a\wedge
dc+\frac{k}{4\pi}c\wedge dc)\,$ (2.13)
is almost identical to (2.7). An important difference is that Gaitto and
Witten introduced a gauge field $a$, whereas in (2.13) we added a term that
breaks gauge invariance. We may re-introduce gauge invariance in (2.13) by
transforming the fixed gauge vector $a$ into a gauge invariant term by adding
a scalar $\eta$ as follows
${\cal Z}=\int DaDcD\eta\exp
i\int((a-d\eta)\wedge*(a-d\eta)+\frac{1}{2\pi}a\wedge dc+\frac{k}{4\pi}c\wedge
dc)\,\,,$ (2.14)
such that under a gauge transformation $a\rightarrow
a+d\lambda,\eta\rightarrow\eta+\lambda$, with $a$ a $U(1)$ gauge field.
Eq.(2.13) may be viewed as the fixed gauge version of eq.(2.14) with
$d\eta=0$.
Our proposal (2.9) passes all the requirements from a dual theory: it admits a
${\mathbb{Z}_{k}}$ global symmetry, it flows to pure Chern-Simons theory in
the IR, it contains a massive vector of mass $M$ and, finally, when $k=0$ it
agrees with the results of Hanany and Witten [7]. As we shall see the brane
realisations of both electric and magnetic theories predict the existence of
$k$ degenerate vacua.
We summarize this section by writing the precise map between the electric and
magnetic variables using (2.7)
$\displaystyle-g^{2}a_{m}=*da_{e}$ (2.15) $\displaystyle
b=a_{e}-\frac{1}{M}*da_{e}$ (2.16)
or
$a_{e}=b-\frac{2\pi}{k}a_{m}$ (2.17)
## 3 Comments on ${\mathbb{Z}_{k}}$
Let us introduce a Wilson loop in MCS theory. We wish to measure the
${\mathbb{Z}_{k}}$ charge of the loop, namely the number of fundamental
strings, $n$, that pass through a a certain contour $C$. We will define an
operator $G$ such that
$GW_{n}=\exp(i\frac{2\pi n}{k})W_{n}\,,$ (3.1)
with $W_{n}$ a Wilson loop of charge $n$, $W_{n}=\exp(in\oint a_{e})$.
In order to the define $G$, let us consider the equation of motion in MCS
$d(\frac{1}{g^{2}}*da_{e}-\frac{k}{2\pi}a_{e})=j_{e}\equiv dJ_{e}\,,$ (3.2)
where $J_{e}$ is the integral of the electric current $j_{e}$ over a disc $D$
such that $C=\partial D$. The setup is depicted in fig.(2). By integrating
(3.2) we learn that
$\frac{1}{g^{2}}*da_{e}-\frac{k}{2\pi}a_{e}=J_{e}\,.$ (3.3)
Fig. 2: A Wilson loop passing through a domain $D$ (shaded region) whose
boundary is $C=\partial D$.
We can therefore define a generator of a ${\mathbb{Z}_{k}}$ symmetry as
follows
$G=\exp\left(i\frac{2\pi}{k}\oint_{C}(\frac{k}{2\pi}a_{e}-\frac{1}{g^{2}}*da_{e})\right)=\exp\left(i\oint_{C}b\right).$
(3.4)
Note that the implication of ${\mathbb{Z}_{k}}$ symmetry is a symmetry
$n\rightarrow n+k$, namely that a collection of $k$ strings is topologically
isomorphic to a singlet, namely to no strings at all. This is supported by
string theory: suppose that we attempt to place the endpoints of $k$
coincident strings on the worldvolume of the D3 brane. The collection of $k$
fundamental strings can transform itself into an anti D-string and a $(k,1)$
string. Instead of ending on the worldvolume of the D3 brane, the D-string can
end on an NS5 brane and a $(k,1)$ string can end on a $(1,k)$ fivebrane. Thus
string theory predicts that a collection of $k$ strings can be removed from
the worldvolume of the 3d gauge theory. A similar phenomenon happens in the
magnetic dual if we attempt to introduce $k$ coincident D-strings in the
worldvolume of the magnetic theory.
When the theory is defined on the torus the ${\mathbb{Z}_{k}}$ symmetry is
broken, resulting in $k$ vacua [8][9]. An intuitive explanation is as follows:
The level $k$ $U(1)$ Chern-Simons theory is equivalent (using level-rank
duality) to a level $1$ $SU(k)$ theory, that admits a ${\mathbb{Z}_{k}}$
centre symmetry. When it is defined on the torus the $SU(k)$ theory
deconfines, resulting in $k$ degenerate vacua, parameterized by the
eigenvalues of the ’t Hooft loop.
The $k$ vacua manifests themselves in both the electric and magnetic brane
configurations as follows: the D3 brane may end on any of the $k$ constituents
of the fivebranes. Each one of the $k$ choices corresponds to a vacuum.
## 4 Coupling to external sources, Wilson and magnetic loops
Consider the coupling of the electric gauge field to a source $j_{e}$, namely
$a_{e}j_{e}$. It translates to the coupling $(b-\frac{\pi}{k}a_{m})j_{e}$ in
the magnetic side. We therefore suggest that the Wilson loop
$W_{e}=\exp i\oint a_{e}$ (4.1)
in the electric side is mapped into a magnetic loop of the form
$M_{m}=\exp i\oint(b-\frac{2\pi}{k}a_{m})$ (4.2)
in the magnetic side.
We may use the above map between the Wilson loop in the electric side and its
magnetic counterpart to study the dynamics of 3d QED-CS. Using the worldline
formalism [10] we can write the partition function of MCS theory coupled to
$N_{f}$ massless fields as follows
${\cal Z}_{\rm QED-CS}=\int Da_{e}\exp(iS_{\rm
MCS})\sum_{n}\frac{(N_{f}\Gamma_{e})^{n}}{n!}$ (4.3)
with
$\Gamma_{e}=\int\frac{dt}{t^{\frac{5}{2}}}\int
Dx\exp(-\int_{0}^{t}d\tau(\dot{x})^{2})\exp i\oint a_{e}$ (4.4)
The duality transformation yields the following partition function
${\cal Z}_{\rm magnetic}=\int Da_{m}Db\exp(iS_{\rm DJ-
CS})\sum_{n}\frac{(N_{f}\Gamma_{m})^{n}}{n!}$ (4.5)
with
$\Gamma_{m}=\int\frac{dt}{t^{\frac{5}{2}}}\int Dx\exp(-\int
d\tau(\dot{x})^{2})\exp i\oint(b-\frac{2\pi}{k}a_{m})\,.$ (4.6)
It suggests that the dynamics of QED with $N_{f}$ massless flavours is
captured by a dual DJ-CS theory coupled to $N_{f}$ massless ’monopoles’. The
precise coupling of $a_{m}$ and $b$ to the monopoles is given by (4.6). We may
write the dual magnetic theory in a more ’standard’ form
${\cal Z}=\int Da_{m}DbD\bar{\psi}_{m}D\psi_{m}\exp iS_{\rm magnetic}\,,$
(4.7)
with $S_{magnetic}$ given by
$S_{magnetic}=\int\left(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}-\frac{\pi}{k}a_{m}\wedge
da_{m}+\frac{k}{4\pi}b\wedge
db+\bar{\psi}_{m}\gamma\wedge\star(i\partial+b-\frac{2\pi}{k}a_{m})\psi_{m}\right)\,.$
(4.8)
It is interesting to note that the QED-CS theory is mapped to a theory of
interacting massless magnetic ’monopoles’, with a coupling $1/gk$. Thus, when
the electrons couple strongly to $a_{e}$, the ’monopoles’ couple weakly to
$a_{m}$ and we may use perturbation in theory in the magnetic side to study
the strongly coupled electric theory.
Following Itzhaki[11] let us define a magnetic (“disorder”) loop in the
electric theory
$M_{e}=\exp\left(i\oint_{C}(ka_{e}-\frac{2\pi}{g^{2}}*da_{e})\right)\,,$ (4.9)
which is mapped into the electric loop in the magnetic side
$W_{m}=\exp\left(ik\oint_{C}b\right).$ (4.10)
The magnetic loop in the electric side and the electric (Wilson) loop in the
magnetic side are trivial [11].
Fig. 3: Rectangular Wilson loops can be realised in string theory by ending a
pair of F-string and anti F-string on the the threebrane. Similarly
Rectangular ’t Hooft loops can be realised by ending a pair of D-string and
anti D-string on the three brane. The end of the F-string represents a heavy
quark, whereas the end of the D-string represents a heavy monopole.
We suggest that a rectangular Wilson loop (or magnetic loop) should be
identified with the end points of an F-string and anti F-string (or a D-string
and anti D-string) that end on the threebranes of fig.(3).
A D-string can end on an NS5 brane instead of a threebrane, hence the magnetic
loop in the electric theory should be trivial. Similarly, a F-string can end
on a D5 brane instead of a threebrane, hence a Wilson loop in the magnetic
theory should be trivial. This is consistent with our definitions of the
magnetic loop (4.9) and the Wilson loop (4.10).
## 5 Summary
The purpose of this note is to find the S-dual of MCS theory. We found that
the dual theory (2.9) contains a non-gauge vector of mass $M$ and a decoupled
pure TQFT. The magnetic theory nicely captures the dynamics of the electric
theory: a theory with a mass gap that flows in the IR to a TQFT. The duality
we uncovered in this note is a precise manifestation of the duality between a
topological insulator and a topological superconductor outlined in ref.[12].
It will be interesting to find the S-dual of the non-Abelian $U(N)$ theory
that lives on a collection of $N$ coincident $D3$ branes, suspended between
tilted fivebranes. The master field of that theory may be obtained by
replacing the Abelian 1-forms of (2.7) by non-Abelian 1-forms as follows222I
thank Shigeki Sugimoto for suggesting that.
${\cal Z}=\int Da_{m}Da_{e}\exp i\,{\rm
tr}\int(-\frac{g^{2}}{2}a_{m}\wedge*a_{m}+a_{m}\wedge(da_{e}+a_{e}\wedge
a_{e})+\frac{k}{4\pi}(a_{e}\wedge da_{e}+\frac{2}{3}a_{e}\wedge a_{e}\wedge
a_{e}))$ (5.1)
together with $a_{e}=b-\frac{2\pi}{k}a_{m}$. Other dualities that involve
$SO/Sp$ (and an orientifold in string theory) could also be derived. The
generalisation to supersymmetric QED/QCD theories with a CS term [5] is also
interesting and can be written down using the worldline formalism, as in
section (4). The duality found in this letter is useful to study the strong
coupling regime of those theories.
Finally, it is well known that MCS theory admits Seiberg duality. The
manifestation of the duality using an exchange of fivebranes in the magnetic
theory, might teach us about fivebranes dynamics.
### Acknowledgements
A.A. would like to thank the Yukawa institute for theoretical physics, where
part of this work was done, for warm hospitality. I would also like to thank
Mohammad Akhond and Shigeki Sugimoto for numerous discussions and
collaboration. I am grateful to Anton Kapustin and Zohar Komargodski for
reading and commenting on a draft version of this paper.
## References
* [1] S. Deser and R. Jackiw, “’Selfduality’ of Topologically Massive Gauge Theories,” Phys. Lett. B 139 (1984), 371-373 doi:10.1016/0370-2693(84)91833-1
* [2] G. V. Dunne, “Aspects of Chern-Simons theory,” [arXiv:hep-th/9902115 [hep-th]].
* [3] T. Kitao and N. Ohta, “Spectrum of Maxwell-Chern-Simons theory realized on type IIB Brane configurations,” Nucl. Phys. B 578 (2000), 215-238 doi:10.1016/S0550-3213(99)00715-4 [arXiv:hep-th/9908006 [hep-th]].
* [4] O. Bergman, A. Hanany, A. Karch and B. Kol, “Branes and supersymmetry breaking in three-dimensional gauge theories,” JHEP 10 (1999), 036 doi:10.1088/1126-6708/1999/10/036 [arXiv:hep-th/9908075 [hep-th]].
* [5] A. Kapustin and M. J. Strassler, “On mirror symmetry in three-dimensional Abelian gauge theories,” JHEP 04 (1999), 021 doi:10.1088/1126-6708/1999/04/021 [arXiv:hep-th/9902033 [hep-th]].
* [6] D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,” Adv. Theor. Math. Phys. 13 (2009) no.3, 721-896 doi:10.4310/ATMP.2009.v13.n3.a5 [arXiv:0807.3720 [hep-th]].
* [7] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nucl. Phys. B 492 (1997), 152-190 doi:10.1016/S0550-3213(97)00157-0 [arXiv:hep-th/9611230 [hep-th]].
* [8] C. L. Ho and Y. Hosotani, “Operator algebra in Chern-Simons theory on a torus,” Phys. Rev. Lett. 70 (1993), 1360-1363 doi:10.1103/PhysRevLett.70.1360 [arXiv:hep-th/9210103 [hep-th]].
* [9] D. Tong, Lectures notes, https://www.damtp.cam.ac.uk/user/tong/gaugetheory/83d.pdf
* [10] M. J. Strassler, “Field theory without Feynman diagrams: One loop effective actions,” Nucl. Phys. B 385 (1992), 145-184 doi:10.1016/0550-3213(92)90098-V [arXiv:hep-ph/9205205 [hep-ph]].
* [11] N. Itzhaki, “Anyons, ’t Hooft loops and a generalized connection in three-dimensions,” Phys. Rev. D 67 (2003), 065008 doi:10.1103/PhysRevD.67.065008 [arXiv:hep-th/0211140 [hep-th]]. [12]
* [12] J. Murugan and H. Nastase, JHEP 05 (2017), 159 doi:10.1007/JHEP05(2017)159 [arXiv:1606.01912 [hep-th]].
|
11institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
22institutetext: Observatoire de Paris, PSL University, Sorbonne Université,
CNRS UMR 8112, LERMA, 61 Avenue de l’Observatoire, 75014 Paris, France
33institutetext: Departamento de Astronomia de Chile, Universidad de Chile,
Santiago, Chile 44institutetext: Instituto de Radioastronomia y Astrofisica,
Universidad Nacional Autónoma de México, P.O. Box 3-72, 58090, Morelia,
Michoacán, México
# Modeling the CO outflow in DG Tau B:
Swept-up shells versus perturbed MHD disk wind.
A. de Valon Modeling the CO outflow in DG Tau B: Swept-up shells versus
perturbed MHD disk wind.Modeling the CO outflow in DG Tau B: Swept-up shells
versus perturbed MHD disk wind. C. Dougados Modeling the CO outflow in DG Tau
B: Swept-up shells versus perturbed MHD disk wind.Modeling the CO outflow in
DG Tau B: Swept-up shells versus perturbed MHD disk wind. S. Cabrit Modeling
the CO outflow in DG Tau B: Swept-up shells versus perturbed MHD disk wind.
and Modeling the CO outflow in DG Tau B: Swept-up shells versus perturbed MHD
disk wind.Modeling the CO outflow in DG Tau B: Swept-up shells versus
perturbed MHD disk wind. and Modeling the CO outflow in DG Tau B: Swept-up
shells versus perturbed MHD disk wind. F. Louvet Modeling the CO outflow in DG
Tau B: Swept-up shells versus perturbed MHD disk wind. and Modeling the CO
outflow in DG Tau B: Swept-up shells versus perturbed MHD disk wind.Modeling
the CO outflow in DG Tau B: Swept-up shells versus perturbed MHD disk wind.
and Modeling the CO outflow in DG Tau B: Swept-up shells versus perturbed MHD
disk wind. L. A. Zapata Modeling the CO outflow in DG Tau B: Swept-up shells
versus perturbed MHD disk wind.Modeling the CO outflow in DG Tau B: Swept-up
shells versus perturbed MHD disk wind. D. Mardones Modeling the CO outflow in
DG Tau B: Swept-up shells versus perturbed MHD disk wind.Modeling the CO
outflow in DG Tau B: Swept-up shells versus perturbed MHD disk wind.
###### Abstract
Context. The origin of outflows and their exact impact on disk evolution and
planet formation remain crucial open questions. DG Tau B is a Class I
protostar associated with a rotating conical CO outflow and a structured disk.
Hence it is an ideal target to study these questions.
Aims. We aim to characterize the morphology and kinematics of the DG Tau B
outflow in order to elucidate its origin and potential impact on the disk.
Methods. Our analysis is based on Atacama Large Millimeter Array (ALMA)
12CO(2-1) observations of DG Tau B at 0.15′′ (20 au) angular resolution. We
developed a tomographic method to recover 2D (R,Z) maps of vertical velocity
$V_{\rm Z}$ and specific angular momentum $j=R\times V_{\phi}$. We created
synthetic data cubes for parametric models of wind-driven shells and disk
winds, which we fit to the observed channel maps.
Results. Tomographic analysis of the bright inner conical outflow shows that
both $V_{\rm Z}$ and $j$ remain roughly constant along conical surfaces,
defining a shear-like structure. We characterize three different types of
substructures in this outflow (arches, fingers, and cusps) with apparent
acceleration. Wind-driven shell models with a Hubble law fail to explain these
substructures. In contrast, both the morphology and kinematics of the conical
flow can be explained by a steady conical magnetohydrodynamic (MHD) disk wind
with foot-point radii $r_{0}\simeq 0.7-3.4$ au, a small magnetic level arm
parameter ($\lambda\leq 1.6$), and quasi periodic brightness enhancements.
These might be caused by the impact of jet bow shocks, source orbital motion
caused by a 25 MJ companion at 50 au, or disk density perturbations accreting
through the wind launching region. The large CO wind mass flux (four times the
accretion rate onto the central star) can also be explained if the MHD disk
wind removes most of the angular momentum required for steady disk accretion.
Conclusions. Our results provide the strongest evidence so far for the
presence of massive MHD disk winds in Class I sources with residual infall,
and they suggest that the initial stages of planet formation take place in a
highly dynamic environment.
###### Key Words.:
stars: formation – protoplanetary disk – ISM : jets and outflows – stars :
individual: DG Tau B
## 1 Introduction
Understanding the origin of protostellar flows is a key element to our full
comprehension of the star formation process. Protostellar flows come in two
components: high speed collimated jets and slower, often less collimated winds
and outflows. We focus here on the slow molecular outflows which are
traditionally associated with the earlier stages of star formation. However,
they have also recently been detected around more evolved Class II systems
(e.g., Pety et al. 2006; Louvet et al. 2018; Fernández-López et al. 2020).
Despite their ubiquity, the exact origin of molecular outflows, their link to
the high-velocity jets, and their impact on the young forming star and disk
are still crucial open questions.
Two main paradigms are currently considered. The first traditional model
describes these slow outflows as swept-up material, tracing the interaction
between an inner jet or a wide-angle wind with the infalling envelope or
parent core. These models have been mainly used for interpreting outflows from
Class 0 and I stars which are still surrounded by massive envelopes (Zhang et
al. 2019; Shang et al. 2020; Lee et al. 2000). However, on small scales (less
than a few 1000 au), recent observations have revealed rotating molecular
outflows to originate from well within the disk at all evolutionary stages
from Class 0 to Class II (e.g., Launhardt et al. 2009; Zapata et al. 2015;
Bjerkeli et al. 2016; Tabone et al. 2017; Hirota et al. 2017; Louvet et al.
2018; Zhang et al. 2018; Lee et al. 2018; de Valon et al. 2020; Lee et al.
2021). These observations suggest an alternative paradigm by which these slow
molecular outflows, at least at their base, would trace matter directly
ejected from the disk, by thermal or magnetic processes. In support of this
interpretation, the flow rotation signatures are consistent with an origin
from disk radii $r_{0}\simeq 1-50$ au (see references above), where Panoglou
et al. (2012) have shown that magnetic disk winds could remain molecular.
These two paradigms imply different evolutions for the disk. Jet- and wind-
driven shell models predict that an important mass is swept up from the
envelope, impacting the reservoir of matter infalling onto the disk. Disk wind
models predict an extraction of mass from the disk and, in the case of
magnetohydrodynamic (MHD) models, an extraction of angular momentum which
could drive disk accretion (Bai et al. 2016). Evaluating the contributions of
each of these mechanisms to the slow molecular outflow emission requires high-
angular resolution studies of the molecular outflow base. The flux of angular
momentum extracted by the rotating molecular wind is estimated in only two
sources so far, HH30 and HH212, and it is found sufficient to drive disk
accretion at the current observed rate in both cases (Louvet et al. 2018;
Tabone et al. 2020).
Recent high-angular resolution observations also reveal striking signatures of
multiple CO shells in a few sources (Zhang et al. 2019; Fernández-López et al.
2020). Under the classical paradigm where CO outflows trace swept-up shells,
they would require short episodic wind and jet outbursts every few 100 yrs.
Characterizing and understanding the origin of these variabilities could bring
critical insights into the star formation dynamics.
We present here an analysis of the DG Tau B CO outflow, based on recent
Atacama Large Millimeter Array (ALMA) observations at 0.15′′ resolution by de
Valon et al. (2020) (hereafter DV20). DG Tau B is a Class I 1.1 M⊙ protostar
located in the Taurus cloud ($\approx$ 140 pc) and is associated with a
bipolar atomic jet (Mundt & Fried 1983) and a strongly asymmetric CO outflow
first mapped by Mitchell et al. (1997). The bright redshifted CO outflow lobe
displays a striking bright and narrow conical shape at its base. Zapata et al.
(2015) detect rotation signatures in the same sense as the disk. The ALMA
observations by DV20 clearly confirm rotation in the bright inner conical
redshifted lobe and show that it is surrounded by a wider and slower outflow.
Residual infall signatures are detected at opening angles $\geq 70^{\circ}$,
almost tangent to the disk surface. In addition, DV20 report striking
substructures in the CO channel maps at different line-of-sight velocities,
reminiscent of the nested layers recently identified in HH46/47 by Zhang et
al. (2019) and suggesting variability or interaction processes. The exquisite
levels of detail provided by these new ALMA observations provide a prime
opportunity to distinguish between swept-up and disk wind origins.
In Sect. 2, we recall the main properties of the DG Tau B outflow and
characterize the three types of substructures visible in the channel maps. In
Sect. 3 we present a model-independent analysis of the inner conical outflow
component, which allowed us to retrieve 2D maps of the expansion velocity
$V_{\rm Z}$ and specific angular momentum $j=RV_{\phi}$. We compare these
overall properties with parametric models of wind-driven shells (in Sect. 4)
and disk winds (in Sect. 5). We discuss our results and their implications for
the origin of the CO outflows in DG Tau B in Sect. 6. Section 7 summarizes our
conclusions.
## 2 Summary of outflow structure
Figure 1: Substructures in the DG Tau B redshifted CO outflow. Left panels:
12CO channel maps at selected line-of-sight velocities. The white dashed line
traces the $\theta=17^{\circ}$ outer limiting cone of the inner outflow as
defined by DV20. Right panels: Transverse PV diagrams across the flow averaged
over a slice of $\Delta Z=0.2^{\prime\prime}$. Black dotted lines indicate the
outer limits of the conical outflow at the specified height. The triangle (a)
and the two circle symbols (c and d) highlight the on-axis height of one arch
and two different cusps, respectively. The square symbol (b) is located in the
extended outer flow. The symbols are represented both on the channel maps and
PV diagrams.
Figure 1 summarizes the main properties of the DG Tau B redshifted CO outflow
as identified in DV20. A narrow, limb brightened conical outflow is visible in
the channel maps at $(V-V_{\rm sys})>2.0$ km s-1. Its opening angle decreases
from 17∘ at ($V-V_{\rm sys})=2.8$ km s-1 until 12∘ at $(V-V_{\rm sys})\geq
5.0$ km s-1. The sheer-like velocity gradient across the conical layer is best
seen in transverse position-velocity (hereafter PV) cuts (Fig. 1), where the
flow width clearly narrows down at higher velocity, up $(V-V_{\rm sys})>5.0$
km s-1. The conical outflow is surrounded by a slower and wider outflowing
component visible at $(V-V_{\rm sys})<2$ km s-1. This outer flow is visible in
PV diagrams as an extended pedestal with a shallower velocity gradient (see
Fig. 1).
Another striking property of the DG Tau B redshifted outflow, revealed by the
ALMA observations in DV20, are brightness enhancements visible in channel
maps. These various substructures, illustrated in Fig. 1, can be classified in
three types:
Bow-shaped intensity enhancements are visible at low velocities $(V-V_{\rm
sys})=0.88-1.51$ km s-1 (see Fig. 1, left panels). We refer to these
substructures as arches. The radial extent of the biggest arch is larger than
the inner conical outflow, implying that this arch is at least partially
formed outside of the conical outflow. The arch seems to increase in height
with increasing velocity although this phenomenon is difficult to quantify due
to the limited spectral sampling.
At intermediate velocities, from $(V-V_{\rm sys})=2.46-3.42$ km s-1, thin
quasi vertical lines (see white arrow in Fig. 1) are visible inside the
conical outflow, close to the edge. They are almost vertical at $(V-V_{\rm
sys})=2.46$ km s-1 and more open at higher projected velocity until they
become almost tangent to the edge of the conical outflow at $(V-V_{\rm
sys})\leavevmode\nobreak\ =\leavevmode\nobreak\ 3.42$ km s-1. We refer to
these substructures as fingers.
At high velocities, from $(V-V_{\rm sys})=3.1$ to almost 7 km s-1, multiple
U-shaped structures are visible inside the conical outflow. We refer to these
substructures as cusps. The contrast of these cusps is maximal at $(V-V_{\rm
sys})=4.37$ km s-1 and decreases with increasing velocity. The cusps show a
signature of apparent acceleration: their projected distance from the source
increases with increasing projected velocity.
Table 1: Characteristics of the observed arches and cusps
Arches
---
Name | position at | Radial extension at | Aspect Ratio at
| $V_{\rm A}$aa$a$$(V-V_{\rm sys})=1.19$ km s-1(”) | $V_{\rm A}$aa$a$$(V-V_{\rm sys})=1.19$ km s-1(”) | $V_{\rm A}$aa$a$$(V-V_{\rm sys})=1.19$ km s-1
A0 | 14.9$\pm$ 0.2 | 9.7$\pm$ 0.3 | 1.4
A1 | 9.4$\pm$ 0.2 | ? | ?
A2 | 6.2$\pm$ 0.2 | 1.5$\pm$ 0.3 | 1.2
A3 | 3.7$\pm$ 0.2 | 1$\pm$ 0.2 | 1.4
Cusps
Name | position at | $N_{\rm chan}$ | derivative
| $V_{\rm U}$bb$b$$(V-V_{\rm sys})=4.37$ km s-1(”) | | (”/km s-1)
U0 | 11.9 $\pm$ 0.2 | (5) | 3.2 $\pm$ 0.5
U1 | 8.2 $\pm$ 0.2 | (6) | 2.2 $\pm$ 0.3
U2 | 6.5 $\pm$ 0.1 | (4) | 1.4 $\pm$ 0.5
U3 | 5.0 $\pm$ 0.1 | (6) | 0.7 $\pm$ 0.4
U4 | 3.6 $\pm$ 0.1 | (3) | 0.5 $\pm$ 0.2
U5 | 2.2 $\pm$ 0.1 | (4) | 0.4 $\pm$ 0.2
111
Table 1 lists the characteristics of the main arches and cusps. We identify
four arches (A0 to A3) and six cusps (U0 to U5). On the channel map at
$(V-V_{\rm sys})=1.19$ km s-1 we derived the maximal height of each arch on
axis (at $\delta x=0$) and the maximal radial extension. We divided these two
values to derive the arch aspect ratio. The cusps are also characterized from
the channel maps at $3.73\boldsymbol{\leq}(V-V_{\rm sys})\leq 5.32$ km s-1
(See Fig. 22). At higher velocities, the cusps could not be characterized
because the outflow signal-to-noise ratio (S/N) decreases drastically.
Moreover, the region at $\delta z<2.2^{\prime\prime}$ was not studied because
the cusps locations are complex to identify due to overlapping structures. We
derived the cusp reference height on-axis on the channel map at ($V-V_{\rm
sys}$) = 4.37 km s-1. The apparent acceleration of each cusp, in (′′)/km s-1
listed in Table 1, was obtained by measuring the average spatial shift of the
cusp between two consecutive channel maps (taking as error bar the rms
dispersion between measurements in different channels).
Internal discrete structures are also visible in transverse PV-diagrams as
pseudo-ellipses (see Fig. 1). The top and bottom of the ellipses seem to match
with respectively the top of some arches and bottom of some cusps (see Fig.
1). This potentially indicates that cusps and arches are linked to the same
phenomenon. We present a model-dependent study of these ellipses in Sect. 5.
## 3 Tomography of the inner conical outflow
In this section, we develop a model-independent method that allows us to
recover the dynamics and the morphology of the inner conical outflow
component. This method assumes that the outflow is axisymmetric. We later
discuss possible departures from axisymmetry and their implications on the
analysis conducted here.
### 3.1 Method
We followed Louvet et al. (2018) who modeled the outflow of HH30 at a given
vertical offset by an emitting ring with radius R and extended their method to
take into account the inclination of the outflow. For this purpose, we defined
the outflow and the observer reference systems (see Fig. 2). On the outflow
reference system, $\@vec{Z}$ is defined by the outflow axis and $\@vec{X}$ is
tangent to the plane of sky. The observer reference system is defined by
$\@vec{\delta z}$ the projection of the outflow axis onto the plane of the
sky, $\@vec{\delta y}$, the line-of-sight direction, and $\@vec{\delta
x}=\@vec{X}$, in the plane of the sky. The inclination of the outflow $i$ is
then defined by the angle between $\@vec{\delta y}$ and $\@vec{Z}$. In the
case of edge-on disks such as HH30, the two reference systems are identical.
We modeled one layer of the outflow at a specified height Z by an emitting
ring of radius R and azimuthal angle $\phi$ (see Fig. 2) with $X=R\cos{\phi}$
and $Y=R\sin{\phi}$. For each ring the velocity components are defined in
cylindrical coordinates with: $V_{\rm R}(R,Z)$, $V_{\rm Z}(R,Z)$ and
$V_{\phi}(R,Z)$ (see Fig. 2). Hence, an emitting ring is defined by 5
parameters: $Z$, $R$, $V_{\rm R}(R,Z)$, $V_{\rm Z}(R,Z)$ and $V_{\phi}(R,Z)$.
Figure 2: Principle of tomographic reconstruction method. Top panel: 3D
representation of the two reference systems used: the outflow $(X,Y,Z)$ in
black and the observer $(\delta x,\delta y,\delta z)$ in yellow. The colored
circles illustrate the emitting rings defined by five parameters (see text).
The colored dots trace the locations at $\phi=0$ and $\phi=\pi$ along the
emitting rings. Bottom panels: Transverse PV diagrams at $\delta
z=2.5^{\prime\prime}$ across the flow axis and averaged over a slice of
$\Delta z=0.2^{\prime\prime}$. In the left panel is shown the schematic
projection of the colored rings and corresponding colored dots in the PV
diagram. Because of the flow inclination, the rings are in fact projected at
slightly different heights. Their real projection is studied in Sect. D.1. The
white dots in the right panel illustrate the outer limits of the PV diagram.
The red curve shows the polynomial fitting of the two edges.
The observational coordinates on a position-position-velocity (PPV) data cube
are defined by the projection of the outflow on the plane of sky
($\@vec{\delta x},\@vec{\delta z}$) and the projected velocities on the line
of sight $V_{los}=-\@vec{V}\cdot\@vec{e_{\rm y}}$ with redshifted velocities
considered as positive. This depends on $R$,$Z$ and $\phi$ as:
$\displaystyle\delta x$ $\displaystyle=$ $\displaystyle R\cos{\phi}$ (1)
$\displaystyle\delta z$ $\displaystyle=$ $\displaystyle
Z\sin{i}-R\sin{\phi}\cos{i}$ (2) $\displaystyle V_{\rm los}$ $\displaystyle=$
$\displaystyle-V_{\rm z}\cos{i}-V_{\phi}\cos{\phi}\sin{i}-V_{\rm
R}\sin{\phi}\sin{i}.$ (3)
A transverse PV diagram corresponds to a pseudo-slit of the data cube
perpendicular to the flow axis. This corresponds to a solution of Eqs. 1,2,3
with $\delta z=cst$. In the case of edge-on flows, a ring traces a perfect
ellipse in the PV diagram. A fit of these ellipses give complete information
about the morphology and dynamics of the outflow as shown by Louvet et al.
(2018).
In the case of an inclined outflow such as DG Tau B, different rings overlap
on the transversal PV diagrams (see Fig. 2 left). Hence it was not possible to
fit them individually. However constraints on some of the ring parameters
could be derived from characterizing the outer limits of the PV diagram. The
radius of the ring corresponds on the first order to $\delta x_{max}=\delta
x(\phi\approx 0,\pi)$. In addition, the projected velocities at the edge of
the ellipses $V_{\rm los}(\delta x_{\rm max})$ allow one to recover both
$V_{\rm Z}(R,Z)$ and $V_{\phi}(R,Z)$ from the following equations:
$\displaystyle Z$ $\displaystyle=$ $\displaystyle\frac{\delta z}{\sin{i}}$ (4)
$\displaystyle R$ $\displaystyle=$ $\displaystyle\delta x_{\rm max}$ (5)
$\displaystyle V_{\rm Z}$ $\displaystyle\simeq$ $\displaystyle\frac{V_{\rm
los}(\delta x_{\rm max})+V_{\rm los}(-\delta x_{\rm
max})}{-\leavevmode\nobreak\ 2\cos{i}}$ (6) $\displaystyle V_{\phi}$
$\displaystyle\simeq$ $\displaystyle\frac{V_{\rm los}(\delta x_{\rm
max})-V_{\rm los}(-\delta x_{\rm max})}{2\sin{i}}.$ (7)
By consequence, characterizing the outer limits of the PV diagram along
multiple heights allowed us to recover a 2D map of the expansion velocity
$V_{\rm Z}$ and specific angular momentum $j=RV_{\phi}$. In the following, we
used the inclination derived from DV20 at $i=117^{\circ}\pm 2$. To
characterize the outer shape of the transverse PV diagram at a given $\delta
z$, we derived the maximal projected velocity for each value of $\pm\delta x$.
Numerically, we computed the gradient of the emission profile at a fixed
$\delta x$ and localized its maximum. We also derived an uncertainty on
$V_{\rm los}$ which is found to vary in the range 0.05 to 0.2 km s-1. We used
in the following a mean value of 0.1 km s-1. Figure 2 illustrates our method
to determine $V_{\rm los}$ on the edges of the PV diagram.This procedure
failed at low radii (or high velocities) because of the low S/N and the almost
vertical profile that generated high uncertainties on the velocity estimate.
The determination of the velocity was also limited by our spectral resolution
of 0.3 km s-1. We fit the two edges for each PV diagram with a polynomial
curve (see Fig. 2). We also applied a Gaussian filter with $\delta z$ using a
standard deviation of 0.16′′.
### 3.2 Results
Figure 3: Tomographic maps of $V_{\rm Z}$ (left panel) and specific angular
momentum $j$ (right panel) in the outflow referential. The black dashed line
traces the conical fit of the region $V_{\rm Z}=10-11$ km s-1. The white
(resp. black) contours in the left (resp. right) panel show $V_{\rm Z}$
contours. The red dashed lines indicate the height of the two extrema in
specific angular momentum.
Using Eqs. 5, 4, 6, and 7, a tomographic map of $V_{\rm Z}(R,Z)$ and
$V_{\rm\phi}(R,Z)$ could be recovered. We show the specific angular momentum
$RV_{\rm\phi}(R,Z)$ instead of the rotation alone as this is more meaningful
in the understanding of the dynamics. Figure 3 shows the resulting tomographic
map of $V_{\rm Z}$ and $RV_{\phi}$ in the outflow referential. The tomography
efficiently traces the conical shape visible on the channel maps. Curves of
constant $V_{\rm Z}$ trace conical surfaces with semi-opening angles varying
from 12∘ for the highest velocities to 17∘ for the lowest velocities. $V_{\rm
Z}$ radially decreases from $\approx$ 14 km s-1 to 5 km s-1. This range of
velocities is conserved until at least $Z=1200$ au.
Figure 4: Specific angular momentum $j$ along curves of constant $V_{\rm Z}$.
The corresponding range of $V_{\rm Z}$ is shown in the box. The uncertainty of
specific angular momentum was obtained by propagating the $V_{\rm proj}$
uncertainty. The red dashed line corresponds to the median value of each
curve.
The specific angular momentum derived from the tomographic study varies from 0
to 140 au km s-1 and is consistently in the same sense as the disk rotation.
At $Z<500$ au the specific angular momentum increases with radius from
$\approx 30$ au km s-1 in the inner radius to $\approx 100$ au km s-1 on the
outer radius. The specific angular momentum is also roughly constant on
conical lines of constant $V_{\rm Z}$ until $Z\approx 500$ au (see Fig. 4).
Our average value around 70 au km s-1 is consistent with the previous estimate
of DV20 of $\approx$ 65 au km s-1.
Two extrema in the specific angular momentum map can be observed at
$Z=550-800$ au and $900-1100$ (see Fig. 4). At the lowest altitude, the
specific angular momentum reaches zero while at the highest altitude the
specific angular momentum increases up to $j>140$ au km s-1. These
irregularities are also visible on channel maps. They correspond to regions
where bumps in the cones are observed: toward $\delta x<0$ at $\approx
4.5^{\prime\prime}$, $\delta x>0$ at $\approx 6.5^{\prime\prime}$ (see the
channel map $(V-V_{\rm sys})=4.37$ km s-1 on Fig. 1). These bumps may be due
to local radial displacements of the outflow axis, due for example to
wiggling. We study the impact of small amplitude wiggling in 5.2.
### 3.3 Limitations and biases
In this section, we discuss the different biases and limitations of this
tomographic study. Firstly, this study could not recover the radial velocity
component $V_{\rm R}$ as it impacts mostly the size of the ellipse at $x=0$
where all the ellipses are stacked. A model-dependent study to characterize
this radial velocity will be achieved in Sect. 5. Furthermore, in order to
apply this method, it is critical that the centers of the rings are not
significantly displaced from $\delta x=0$. Such displacements can be induced
by a poorly estimated outflow position axis (PA) or by outflow axis wiggling.
We derived the PA of the redshifted outflow in Sect. A at $\rm
PA=295^{\circ}\pm 1^{\circ}$. This value is in very good agreement with the
disk rotation axis $\rm PA=115.7^{\circ}\pm 0.3^{\circ}$ determined by DV20.
We also determined an upper limit of 0.5∘ for the wiggling of the CO outflow
axis. We discuss in Appendix 5.2 the impact of possible low-amplitude wiggling
on our results.
The different biases were also computed. We show in Appendix D.1 that assuming
that the maximal radial extent corresponded to $\phi=0,\pi$ was partially
inaccurate, and could introduce a bias in the estimate of $R$, $Z$, $V_{\rm
Z}(R,Z)$ and $V_{\phi}(R,Z)$. The effect of ellipse stacking and its effect on
the estimate of the velocities was studied in Appendix D.2. We evaluate at
$\lesssim$ 20% the potential bias in our estimate of the conical outflow
dynamics. Our estimates of $V_{\rm Z}$ and specific angular momentum are
overestimated and underestimated respectively (see Appendix D.2). We estimate
the bias on $R$ and $Z$ to be respectively $<1.5\%$ and $<3\%$, resulting in
an error $<3\%$ on our estimate of the opening angle $\theta$.
The highly asymmetric pedestal emission visible on the two sides of the PV
diagram at $(V-V_{\rm sys})<2$ km s-1 traces the outer region (see Fig. 1).
DV20 show that this region is outflowing and surrounds the conical outflow. We
did not attempt to apply the tomographic method to this region. From its
morphology in the channel maps, we derived for this component an opening angle
$>30^{\circ}$. Such a large opening angle produces a bias of $\approx 60\%$ in
the estimation of $V_{\rm Z}$ using our reconstruction method (see Appendix
D). Moreover, this pedestal may potentially be explained by the top of one
large ellipse, with the extremal region located at velocities
$<\leavevmode\nobreak\ 1$ km s-1, absorbed by the medium. In that case, our
reconstruction method is not applicable. By consequence, we did not apply our
method for line-of-sight velocities $(V-V_{\rm sys})<2$ km s-1. In the
following, we investigate to which extent wind-driven shells and disk winds
can account for both the conical velocity stratification determined here and
the striking substructures (arches, fingers, cusps) identified in Sect. 2.
## 4 Wind-driven shell modeling
The traditional interpretation proposed for CO molecular outflow cavities
around young stars is that they trace shells of ambient material swept up by a
wide-angle wind or by jet bow shocks (see Cabrit et al. 1997; Lee et al. 2000;
Arce et al. 2007, for reviews). In this section, we investigate the simplest
and most widely used model to interpret CO outflow observations, namely the
wind-driven shell (hereafter WDS) solution of Lee et al. (2000) where the
shell is a parabola that expands radially in all directions with a velocity
proportional to the local distance from the source (hereafter referred to as
the Hubble law). Such a shell structure is predicted under a set of specific
conditions in the wind and ambient medium222It is obtained when a wide-angle
wind with velocity varying with angle as $V_{w}\propto\cos\theta$ and density
varying as $\propto 1/(r^{2}\sin^{2}\theta)$ sweeps-up a static, flattened
isothermal core with density $\propto\sin^{2}\theta/r^{2}$, and they mix
instantly in the shell. We note that the radial shell expansion results from
instant mixing, while the Hubble law derives from the identical radial fall-
off of wind and ambient density (both $\propto 1/r^{2}$), which yields a shell
speed that is constant over time (Shu et al. 1991). Finally, the parabolic
shell shape derives from the combined $\theta$-dependencies of the densities
and wind speed (Lee et al. 2001).(see Sect. 6 for details). This simple WDS
model is recently shown by Zhang et al. (2019) to reproduce several features
of the multiple CO shell structures at the base of the HH46/47 molecular
outflow. Therefore it is natural to investigate whether the same WDS model can
also reproduce the morphology and kinematics of the DG Tau B outflow, on
smaller spatial scales.
Following Lee et al. (2000), the parabolic morphology and the radial Hubble-
law kinematics of the shell can be empirically described by two parameters,
$C$ and $\tau$, through:
$\centering Z=C\times R^{2}\qquad\qquad V_{\rm Z}=\frac{Z}{\tau}\qquad\qquad
V_{\rm R}=\frac{R}{\tau},\@add@centering$ (8)
where $\tau$ is the age of the shell, $C$ is the inverse size of the parabola
at $\theta=45\degr$ (where $Z=R=1/C$), and the product $\tau C$ defines the
shell expansion speed at each polar angle $\theta=\arctan(R/Z)$ through:
$\displaystyle V_{\rm Z}(\theta)$ $\displaystyle=\frac{1}{\tau
C}\left(\frac{Z}{R}\right)^{2}=\left({\tau C\tan^{2}\theta}\right)^{-1}$ (9)
$\displaystyle V_{\rm R}(\theta)$ $\displaystyle=\frac{1}{\tau
C}\left(\frac{Z}{R}\right)=\left(\tau C\tan\theta\right)^{-1}.$ (10)
The above equations always produce ellipses in both channel maps and
transverse PV diagrams (Lee et al. 2000). This is a direct result of the
assumed Hubble law, where the shell velocity vector is proportional to the
position vector. A channel map at a given line-of-sight velocity is then
equivalent to making a cut through the shell at a given depth along the line
of sight and this cut is shaped as an ellipse. Similarly, a transverse PV
diagram has the same (elliptical) shape as a cut through the shell at the
corresponding projected height. On channel maps, the ellipse is projected at
increasing distances from the source with increasing velocity, due to the
Hubble law. Similarly, on transverse PV diagrams, the mean velocity of the
ellipse increases with the distance of the PV cut from the source (see Figs.
24 and 26 in Lee et al. 2000). In Appendix E, we derived analytical formulae
for the center of the ellipse in channel maps as well as for its aspect ratio.
Interestingly, we find that the ellipse aspect ratio only depends on the
inclination $i$ and is equal to $1/|\cos{i}|$ for the classical model of Eq. 8
(see Appendix E). In the following, we attempt to fit with this model first
the low-velocity outer flow component, and then the bright conical outflow and
its discrete structures.
### 4.1 Low-velocity outer flow
Figure 5: Comparison of the low-velocity outer CO outflow with a classical
WDS model. Left panels: 12CO individual channel maps at different line-of-
sight velocities tracing the low-velocity outer CO outflow. The white contours
trace the model of a WDS of parabolic shape defined by $C=10^{-3}$ au-1,
dynamical age $\tau=6000$ yr and specific angular momentum $j=250\pm 50$ au km
s-1 with an inclination of $i=117^{\circ}$. Right panel: Transverse PV cut at
$\delta z=7.5^{\prime\prime}$ averaged over a slice of $\Delta
Z=0.2^{\prime\prime}$. The red ellipse traces the WDS model. ($V-V_{\rm sys}$)
units are km s-1.
The wide and low-velocity outflow at $(V-V_{\rm sys})<2$ km s-1 shows several
properties suggestive of a ”classical” parabolic WDS with a Hubble law. Its
outer border in channel maps has a parabolic shape, and it exhibits a larger
offset from the origin at higher line-of-sight velocities (see Fig.1 left
panels). Although such WDS models do not usually consider rotation, we
included rotation to properly fit the large left-right asymmetry observed in
the channel maps. To reduce the number of free parameters, we considered that
the specific angular momentum $j$ is the same at all positions of the swept-up
shell. $C$ was then fixed by the global parabolic shape of the cavity, $\tau$
by its spatial shift between the channel maps, and $j$ by its global left-
right asymmetry.
Figure 5 (left panels) shows that the outer contour of the low-velocity
outflow and its increased spatial offset with velocity are well fit by a WDS
obeying Eq. 8 with parameters $C=10^{-3}$ au-1, $\tau$ = 6000 years and
$j=250\pm 50$ au km s-1. The rightmost panel in Fig. 5 shows that this shell
model reproduces well the most extended, lowest velocity emission of the broad
pedestal in transverse PV cuts; the predicted blue-shifted emission from the
front side of the shell falls very close to systemic velocity, consistent with
its nondetection in our data. On the other hand, our assumption of a thin
parabolic shell does not match the observed outflow thickness at high
altitudes.This discrepancy is visible at $\delta z\approx 15^{\prime\prime}$
on Fig. 5 where the observed width of the emissive outer layer is $\approx
3^{\prime\prime}$, significantly larger than predicted by our model. The shell
should actually have a thickness $\simeq 3^{\prime\prime}\simeq 500$ au.
### 4.2 Conical outflow and discrete structures
Figure 6: Dynamical times ($\tau=Z/V_{\rm Z}$) derived from the tomographic
map of $V_{\rm Z}$ are shown in color. The white lines correspond to a
parabolic curve with $1/C$ varying from 30 to 135 au, in steps of 15 au.
In this section, we attempt to model the conical outflow and its discrete
structures (arches, cusps, fingers, described in Sect. 2) by a stacking of
several parabolic wind-driven shells with a Hubble velocity law. Several
qualitative features are suggestive of such a model: the loop shapes of the
arches at low velocity are reminiscent of the ellipses predicted in channel
maps (see Appendix E), the apparent acceleration of the cusps (increased
altitude with increasing velocity) is reminiscent of the predicted Hubble law
dynamics, and finally, in the conical flow studied by tomography, contours of
constant $\tau=Z/V_{\rm Z}$ follow quasi-parabolic curves above $Z\simeq 400$
au (see Fig. 6). Hence we investigate below whether the conical outflow could
be made of successive nested parabolic wind-driven shells, where the apparent
continuous aspect of the tomography would be an artifact of our limited
spatial and spectral sampling, and the discrete structures (arches, cusps,
fingers) would trace a few individual shells brighter than average.
Contrary to the slow outer flow modeled in Sect. 4.1, the left-right asymmetry
in these faster flow regions is small. Therefore, we neglected rotation when
fitting WDS models to the channel maps. We set the WDS axis inclination equal
to the large-scale disk inclination derived from ALMA studies
($i=63^{\circ}\pm 2^{\circ}$, DV20), leading to $i=180\degr-63\degr=117\degr$
in the redshifted lobe.
Here, we find that the ”classical” WDS model of Lee et al. (2000) encounters a
major problem, as shown in the top row of Fig. 7: the predicted aspect ratio
of ellipses in channel maps, $A=1/|\cos{i}|$ (see Appendix E) is too large
($\simeq 2.2$). In order to match the observed aspect ratio of the arches
($\approx 1.4$), the inclination of the shell axis should be $i\approx
135\degr$ instead of $i=117\degr$. In the WDS model, however, the direction of
shell elongation is not arbitrary but must follow the direction of both
highest wind density (traced by the axial jet) and lowest ambient density
(traced by core flattening). Proper motions of jet knots in DG Tau B imply a
jet axis inclination of $i\geq 65^{\circ}$ for the blue-shifted lobe
(Eislöffel & Mundt 1998), hence $i\leq 115^{\circ}$ for the redshifted lobe.
This limit agrees within 2° with the large-scale disk inclination determined
by ALMA ($i=63\degr\pm 2\degr$, DV20), which should follow the core
flattening. Therefore, we can exclude a shell axis at $i\approx 135\degr$ as a
solution to the ellipse aspect-ratio problem of the WDS model of Lee et al.
(2000).
Figure 7: Comparison of 12CO channel maps at three different line-of-sight
velocities (color maps) with predicted ellipses for parabolic WDS (white
contours). The model used on each row is sketched in the left-most panel. Top
row: Classical model with radial Hubble expansion (see Eq. 8 and Lee et al.
2000). Middle and bottom rows: Modified model with ”collimated” expansion (see
Eq. 11). Green, blue, and yellow contours highlight specific height ranges,
indicated in the first column. White dotted lines outline the radial boundary
defined by the shell ellipses with increasing velocity. Model parameters are
listed in Table 2. $\delta x$ and $\delta z$ units are arcseconds.
Since the ellipse aspect ratio in channel maps does not depend on $\tau$ nor
$C$ (see Appendix E) the only possibility to reduce it without changing the
shell parabolic shape is to modify the shell dynamics. The maximum height of
the ellipse is reached on-axis ($\delta x=0$) where the projected velocity is
greatly affected by the radial velocity component $V_{\rm R}$ (see Eq. 3 with
$\phi=\pm\frac{\pi}{2}$). To keep a small number of model parameters, we thus
chose to add an ad hoc free parameter $\eta$ that modified the radial velocity
as:
$\centering Z=CR^{2}\qquad\qquad V_{\rm Z}=\frac{Z}{\tau}\qquad\qquad V_{\rm
R}=\eta\frac{R}{\tau}.\@add@centering$ (11)
In Appendix E, we show that the ellipse aspect ratio in this modified model is
set at $A=(\eta\leavevmode\nobreak\ \tan^{2}{i}+1)|\cos{i}|$. Since we wanted
to reduce the aspect ratio, we needed $\eta<1$. In other words, we needed a
velocity vector that is more collimated (forward-directed) than the radial
shell expansion in the original WDS model of Lee et al. (2000). We refer to
this ad hoc model as ”modified collimated WDS.”
As shown in the second row of Fig. 7, the shape of the largest arch A0 at
$(V-V_{\rm sys})=1.19$ km s-1 is well fit by a modified collimated WDS with
$\eta=0.6$. The smallest two arches A2,A3 and the smallest cusp U5 are also
well fit by a collimated WDS with $\eta=0.5$ (Fig. 7, bottom row). The
parameters of these best-fit solutions are listed in Table 2. The inferred
shell dynamical times have typical intervals of $\Delta\tau\simeq 300-750$
yrs, similar to those inferred by Zhang et al. (2019) in HH46-47. The fit
velocity values $V_{\rm Z}$ at $\theta=14\degr$ (in the region of the conical
flow) are also listed333We note that $\tan{14\degr}\sim 1/4$ hence $V_{\rm
Z}(14\degr)\simeq 16/(\tau C)$, cf. Eq. 9. Not surprisingly, their range of
$\simeq 7.5-16$ km s-1 is similar to our ”model-independent” tomographic
results for $V_{\rm Z}$ in the conical flow region.
Table 2: Parameters of parabolic wind-driven shells with collimated Hubble
law fit to arches and cusps in Fig. 7.
Name | $C$ | $\tau$ | $\eta$ | $V_{\rm Z}(\theta=14\degr)$aafootnotemark: $a$
---|---|---|---|---
of model | (au-1) | (yr) | | (km s${}^{-1})$
WDS-A0 | 0.003 | 1600 | 0.6 | 16
WDS-A2 | 0.01 | 850 | 0.5 | 9
WDS-A3 | 0.02 | 500 | 0.5 | 7.5
WDS-U5 | 0.04 | 220 | 0.5bbfootnotemark: $b$ | 8.5
444The collimated parabolic WDS model is described by Eq. 11: $C$ is the
inverse characteristic parabola size, $\tau$ the shell age along $Z$, $\eta$
the $V_{\rm R}$ reduction factor ($\eta=1$ for a radial flow).
aafootnotemark: $a$ $V_{\rm Z}(\theta)=1/(\tau C\tan^{2}\theta)$ (see Eq. 9).
bbfootnotemark: $b$In the case of WDS-U5, we do not have any constraint on the
aspect ratio, and set $\eta$ at 0.5.
Figure 8: Kinematic evolution along the flow axis. Panels a) Position-
velocity diagrams averaged over a slice of width $\Delta Z=0.2^{\prime\prime}$
at three different $\delta z$ positions along the flow. The red contours trace
solutions of collimated WDS used to fit A2, A3, and U5 (third line on Fig. 7).
Panel b) In black is represented the maximal velocity of emission of the PV
diagram. In red is shown the center velocity of the ellipses on the PV
diagrams. Panel c) The gray region highlights the conical outflow domain. The
black lines represent the parabolic morphology of the three solutions. The red
dots show for each solution the region where the projected velocity reach
$V_{\rm max}$.
The Hubble law in the WDS model predicts an ever-increasing shell speed at
higher altitudes, until it reaches the polar wind speed, which is $\simeq 125$
km s-1 according to the redshifted jet speed in DG Tau B (Eislöffel & Mundt
1998). In contrast, the maximum line-of-sight velocity with detectable
emission in our transverse PV cuts (averaged between the left and right sides)
is found to stay roughly constant with altitude at $V_{\rm max}\simeq 8\pm 1$
km s-1 (see Fig. 8b). Therefore, all successive wind-driven shells should be
truncated, or have their CO emission strongly suppressed, above the point
where they reach $V_{\rm Z}=V_{\rm max}/\cos{i}\simeq 18$ km s-1. This
velocity limit is close to the molecule dissociation limit $\simeq$ 20 km s-1
in dense hydrodynamical shocks (see e.g., Wilgenbus et al. 2000). Therefore,
the disappearance of CO emission above a certain speed might be explained by
shock-dissociation of ambient CO. Figure 8c shows that the corresponding
truncation region for the best-fit WDS models in Table 2 has a rough conical
shape with $\theta\simeq 9\degr$. However, our ad hoc modified collimated WDS
model meets two serious issues, detailed below.
The model predicts a full ellipse in each channel map (white contours in Fig.
7), which is not observed. In contrast, discrete structures highlight only a
portion of ellipse, depending on the velocity range (see Fig. 7 and Sect. 2):
the ellipse top at low-velocities (arches), ellipse flanks at mid-velocities
(fingers), and ellipse bottom at high-velocities (cusps). We find that a
transition from arches at low velocity to cusps at high velocity can only be
obtained if emission is restricted to a range of heights from $z_{\rm min}$ to
$z_{\rm max}$, as illustrated by the colored contours in Fig. 7. Serious
discrepancies still remain with observations, however: In the broadest shell,
WDS-A0, the extents of Arch A0 and Cusp U1 require inconsistent ranges of
emitting heights, and the predicted ”fingers” at intermediate velocity are
much wider than observed (see blue and green contours in middle row of Fig.
7). In the smaller inner WDS, full ellipses are still predicted in
intermediate velocity channels, which are not observed (see yellow contours in
bottom row of Fig. 7). The same problems remain even if we adopt conical
shapes for the shells instead of parabolae, hence the above discrepancies
appear intrinsically linked to the assumed Hubble-law dynamics.
Another serious issue is that the best-fitting value of $\eta$ in our modified
collimated WDS models is always close to 0.5 (see Table 2). A ratio $V_{\rm
R}/V_{\rm Z}=0.5R/Z$ corresponds to a velocity vector locally tangent to the
parabola. Hence the shell is not expanding but stationary. The physical
justification for the Hubble law in the WDS model, namely a shell expanding at
constant speed over time (Shu et al. 1991), is then no longer applicable. If
the emitting material is moving parallel to the shell, a velocity increasing
in proportion to distance would require, instead, a constant accelerating
force of unknown nature operating out to z=3000 au, which is totally
unphysical.
In summary, we find that only the outer faint, low-velocity flow in DG Tau B
can be reproduced with the parabolic WDS model with radial Hubble law proposed
by Lee et al. (2000). In contrast, the bright conical outflow at mid to high
velocity, although reminiscent of WDS models because of the apparent
acceleration of its discrete structures, cannot be explained by such models,
even when ad hoc modifications to the kinematics, emissivity range, and shape
are introduced. The model faces important issues which seem intrinsically
linked to the Hubble-law dynamics. We discuss in Sect. 6.1 the implications of
these results in the context of more general wind or jet-driven shell
scenarios.
## 5 Disk-wind modeling
In this section, and alternatively to the WDS models considered in Sect. 4, we
investigate a simple kinematical disk wind model for the DG Tau B redshifted
outflow where the conical morphology visible in Fig. 3 would trace the
trajectory of CO molecules ejected from the disk. Although we cannot derive
$V_{\rm R}$ in a model independent way, from the external contours of
transverse PV diagrams (see Sect. 3), we showed in Sect. 2 the existence of
brighter elliptical structures visible on transverse PV diagrams. Assuming
that they trace each a specific layer of the flow along which $V_{\rm Z}$ and
$V_{\rm R}$ stay roughly constant with height, we fit these ellipses to derive
both the $V_{\rm R}$ and $V_{\rm Z}$ components of the velocity (see Appendix
C). Figure 23 shows that the derived velocity directions are parallel to the
conical contours of constant $V_{Z}$. This comforts our hypothesis that the
trajectory of the outflow follow lines of constant $V_{\rm Z}$. From this
hypothesis, we derived the collimation and kinematics of the streamlines using
the tomography and created a synthetic data cube of the conical disk wind
outflow.
### 5.1 Steady disk-wind model
We made the assumption that the flow is axisymmetric and that the matter has
reached its terminal velocity and has a constant poloidal velocity along its
trajectory. We fit this trajectory by a conical surface defined by an angle
$\theta$ from the Z-axis and an anchoring radius $r_{0,\rm geo}$. We extracted
from the tomography the specific angular momentum, $j=R\times V_{\phi}$, along
curves of constant $V_{\rm Z}$ (see Fig. 4) and derived a median value for
each streamline. We defined the uncertainty of this value as the standard
deviation of specific angular momentum. We also computed the poloidal velocity
$V_{\rm P}=V_{\rm Z}/\cos{\theta}$.
Figure 9: Disk-wind properties derived from the tomography along lines of
constant $V_{\rm Z}$. Red dashed curves show fits as a function of the
poloidal velocity of the streamline $V_{\rm p}$: the anchoring radius of the
streamline $r_{0,\rm con}$ was fit by a third-order polynomial, the angle of
the streamline with the flow axis $\theta$ by a power law and the product
$V_{\rm P}\times j$ by a constant value of 570 au km${}^{2}s^{-2}$. Blue
dotted lines represent the extrapolation used to model the high-velocity
component that could not be mapped by tomography due to its low S/N.
Figure 9 represents the derived values of $r_{0,\rm geo}$, $\theta$ and
$V_{\rm P}\times j$ for each streamline of constant $V_{\rm P}$. We fit the
variation of anchoring radius $r_{0,\rm con}$ with the poloidal velocity by a
polynomial law. The variation of ejection angle $\theta$ was fit by a power
law. $V_{\rm P}\times j$ was taken constant at 570$\pm$ 50 au km2 s-2 for all
the streamlines. The fits were achieved using nonlinear least squares, The
equations are as follow:
$\displaystyle r_{0}$ $\displaystyle\simeq$ $\displaystyle 0.18-5.33V_{\rm
p}+44.29V_{\rm p}^{2}-73.23V_{\rm p}^{3}$ (12) $\displaystyle\theta$
$\displaystyle\simeq$ $\displaystyle 12.63+348.57V_{\rm p}^{-2.55}$ (13)
$\displaystyle j\times V_{\rm p}$ $\displaystyle\simeq$ $\displaystyle 570.$
(14)
Figure 10: Comparison of observations with steady disk-wind model. Left
panels: Individual 12CO channel maps computed from the global disk-wind model
at selected line-of-sight velocities (top row) are compared to observations
(bottom row). The color scale is the same for all the channel maps. Right
panels: Transverse PV diagrams at two positions $\delta z$ along the flow and
averaged over a slice of width $\Delta Z=0.2^{\prime\prime}$. The background
grayscale image shows the observations and the red contours trace the
predictions from the disk-wind model. $(V-V_{\rm sys})$ units are km s-1.
Here $r_{0}$ is in au, $V_{\rm P}$ in km s-1, $\theta$ in degrees, and $j$ in
au km s-1. We modeled the disk wind with axisymmetric conical streamlines with
the dynamics and morphology laws derived in Fig. 9 and created a synthetic
data cube of the conical outflow. We set the external and slower layer at
$V_{\rm p}=6$ km s-1, corresponding to the smallest value that could be mapped
with our tomography (see Sect. 3. We set the internal, faster velocity at 20
km s-1. The parameters in the velocity range $V_{\rm p}=14-20$ km s-1, not
covered by the tomography, were determined from an extrapolation of our fits
(blue dotted line in Fig. 9). This extrapolation was done in order to describe
the almost-vertical high-velocity component not described by the tomography
due to insufficient S/N. For each layer, we set the initial value of
$V_{\phi}(R)$ assuming $V_{\rm p}\times j=570$ au km2 s-2 (see Fig. 9). We
assumed optically thin emission throughout the outflow, which is justified by
the observed ratios of 13CO/12CO (see DV20). We did not consider a variation
of emissivity with radius of ejection nor with height (see Sect. F). Proper
modeling would require CO chemistry and temperature profiles, which is well
beyond the scope of this paper. Projection and beam convolution effects were
also taken into account.
Figure 10 shows synthetic channel maps and PV diagrams for our model compared
with observations. The global morphology of the outflow at $(V-V_{\rm sys})>2$
km s-1 as well as its variation with line-of-sight velocity are well recovered
as expected since we use the tomographic results to constrain the wind
collimation and kinematics. This model does not attempt to describe the
extended outflow surrounding the cone at low velocities $(V-V_{\rm sys})=1.15$
km s-1 (see Sect. 2, Fig. 10). To describe completely this extended low-
velocity component, we would need to extrapolate the disk-wind model at larger
ejection radii. However, as this component falls at absorbed cloud velocities,
we are not able to derive model-independent constraints on the dynamics of
this component. Proper modeling would require time-consuming and uncertain
parameter space exploration. We choose therefore to focus in the following on
the conical outflow; nonetheless, the disk wind could be more extended
radially than we describe with our current modeling.
Although effective to describe the global morphology of the outflow, our
simple axisymmetric and steady disk-wind model does not reproduce the
different substructures identified in our observations: cusps and arches and
the local deviations of specific angular momentum at $Z\approx 600$ and 1000
au. In the following subsections, we discuss two small perturbations of our
disk-wind models which could explain the various substructures observed.
### 5.2 Wiggling of the flow axis
Although we do not detect a clear signature of wiggling in our data, we cannot
exclude a small amplitude wiggling of the CO axis $<0.5^{\circ}$ (see Appendix
A). A wiggling of the outflow axis could explain the variations observed on
the specific angular momentum tomographic map. Indeed, in order to determine
the specific angular momentum using Eq. 7, we assumed that the center of the
layer is located at $\delta x=0$ at all heights. If the center is shifted
toward $\delta x>0$ or $\delta x<0$, the specific angular momentum computed
with our method will be respectively higher or lower than the true value. This
effect is more critical if the PV diagram shows a strong velocity gradient,
which is the case for DG Tau B. In this section, we investigate this effect,
and show that small amplitude wiggling may also create the substructures
observed (cusps, fingers and arches).
Figure 11: Synthetic data cubes computed from the generic disk-wind model with
flow axis precession: with a precession angle of 0.2∘ (left panels) and 0.5∘
(right panels) and assuming a constant precession period $\tau_{\rm p}=400$
yrs (top row) or a constant precession spatial wavelength $\Lambda_{\rm
p}=800$ au (bottom row). For each model are shown both a channel map at
$(V-V_{\rm sys})=4.37$ km s-1 and the corresponding tomography of $j$ derived
with the method described in Sect. 3. The color scale of $j$ ranging from 0 to
140 au km s-1 is identical to the color scale of Fig. 3.
We modifed the disk-wind model presented in Sect. 5.1 to add a precession of
the outflow axis. Each conical layer of the outflow precesses with an angle
$\alpha$ and a precession period $\tau_{\rm p}$. This is an extension of the
model developed by Masciadri & Raga (2002) for jets, modified to take into
account the conical morphology of the outflow and inclination to the plane of
the sky. We modeled both a prograde and retrograde precession. However, due to
the small value of $\alpha$ in our models, the two models give very similar
results. We present here results for the prograde model only. We first assumed
that all disk-wind layers precess with the same $\alpha$ and the same
precession period $\tau_{\rm p}$. Due to the velocity shear across the
outflow, the spatial period $\Lambda_{\rm p}=\tau_{\rm p}\times V_{\rm p}$
then varies between layers according to the poloidal velocity. We also
investigated a precession model where the spatial period $\Lambda_{\rm p}$ is
constant across all streamlines. A constant $\Lambda_{\rm p}$ corresponds to a
variation of precession period as $\tau_{\rm p}\propto V_{\rm p}^{-1}\propto
r_{0}^{0.5}$. We visually fit $\tau_{\rm p}$ and $\Lambda_{\rm p}$ to best
reproduce the location of the two extrema variations in the specific angular
momentum map separated by $\simeq$ 400 au. For each model, we computed
synthetic data cubes and derived the specific angular momentum map using the
same method used in Sect. 3 for the observations.
Fig. 11 shows the resulting channel maps and the specific angular momentum
maps for the two precession models (constant $\tau_{\rm p}$ and constant
$\Lambda_{\rm p}$) with two different precession angles (0.2 and 0.5∘)
compatible with the upper limit derived for the CO outflow axis wiggling in
Annex A. Precession models with constant $\tau_{\rm p}=400$ yrs successfully
reproduce the channel maps morphology, in particular the cusps at high-
velocity and arches at low velocities (see Fig. 28). A best match to the
intensity contrast is obtained for $\alpha=0.5^{\circ}$. However, the
resulting map of specific angular momentum $j$ is not fully consistent with
our observations. Indeed, as $\Lambda_{\rm p}$ is different for each layer,
the perturbations of specific angular momentum are not localized at one
specific height, as in the observations.
The modified model with constant $\Lambda_{\rm p}=800$ au for all layers
better reproduce the positions of the two extrema at $Z=600$ and $Z=1000$ au
in the specific angular momentum map. However, the cusps have a lower
intensity contrast than observed, even with the maximum allowed wiggling angle
of 0.5∘. In addition, this model predicts clear detectable wiggling on the
edges of the cone in channel maps, which is not seen in the observations. A
model in between these two extremes, that is to say with $\alpha\simeq
0.5\degr$ and a precession period $\tau_{\rm p}$ increasing more slowly than
$r_{0}^{0.5}$ may better account for all observational properties.
A remaining discrepancy with observations is that none of the wiggling models
reproduce the short spacing of $\simeq 1.5^{\prime\prime}=200$ au between the
inner cusps, as well as the apparent increase of cusp separation with distance
from the source (see Table 1, Fig. 28), although this latter effect is mostly
seen in the farthest cusp A0 and may result from a lack of sensitivity. The
constant $\Lambda_{\rm p}$ model predicts a projected separation between the
cusps of $\Lambda_{\rm p}\sin(i)$, corresponding to $\simeq 5^{\prime\prime}$,
while the constant $\tau_{\rm p}$ model a twice smaller separation typically.
However, we stress that our wiggling models are probably too simplistic as
they do not take into account the (magneto)-hydrodynamical interactions
between the layers. Masciadri & Raga (2002) have shown that simulations depart
rapidly from analytical solutions in the case of jet wiggling due to
precession. This difference is potentially even greater with a shearing
outflow. Dedicated numerical simulations are required to fully test this
scenario. Nonetheless, this model is a promising candidate to explain the
variation of specific angular momentum along the DG Tau B outflow. We discuss
in Sect. 6 possible wiggling mechanisms and their implications.
### 5.3 Emissivity enhancements in the disk wind
In this section, we investigate an alternative model where cusps and arches
are created by localized emissivity enhancements in the conical disk wind. We
first derived their location from the tomography, assuming that they are
axisymmetric, and then created a synthetic data cube to compare with our
observations.
Figure 12: Line-of-sight velocity on-axis (at $\delta x=0$ ) of the model
disk-wind flow surfaces as a function of their poloidal velocity. Orange
circles show the front side ($\phi=\frac{\pi}{2}$), and green circles show the
back side ($\phi=-\frac{\pi}{2}$) of the disk-wind flow surfaces. In gray, we
show the line-of-sight velocity range where the arches and cusps are observed
(see Annex B). In light gray, is represented the low S/N domain where the
cusps could not be characterized.
Figure 12 shows the projected velocity on-axis at $\delta x=0$ for the front
side and the back side ($\phi=\pm\frac{\pi}{2}$) of each conical wind layer in
our model, predicted from Eq. 3. We also represent the domain of line-of-sight
velocities where arches and cusps positions could be measured in channel maps
as described in Sect. 2. The cusps observed at $(V-V_{\rm sys})>5$ km s-1
could not be characterized in Sect. 2 because of low S/N.
Figure 12 shows that if the observed substructures are due to axisymmetric
emissivity enhancements in the conical outflow, the cusps come from the back
side ($\phi\approx-\frac{\pi}{2}$) and the arches from the front side
($\phi\approx+\frac{\pi}{2}$) of the enhanced ring. Figure 12 also shows that
wind layers with poloidal velocities $V_{\rm p}>9$ km s-1 are not located on
the arch domain, and are located on the low S/N cusp domain, which would
require weaker emissivity enhancements in the fastest internal layers.
Each cusp was characterized by its projected height at $\delta x=0$ at a
specific projected line-of-sight velocity $V_{\rm los}$. The projected height
of the cusp corresponds to an equation $Z(R)$ derived from Eq. 2, with
$\phi=-\frac{\pi}{2}$. Similarly, the projected velocity of the cusps
corresponds to a poloidal velocity as shown in Fig. 12 and assuming $V_{\rm
R}$, a conical line of constant $V_{\rm Z}$ in the tomography. As a result, we
can associate the cusp observed in a given channel map with a $(R,Z)$ location
on the $V_{\rm Z}$ tomography map. This location would be the intersection
between the $Z(R)$ equation derived from the projected height, and the conical
line derived from the projected velocity.
Figure 13: Tomographic map of $V_{\rm Z}$ from Fig. 3 extrapolated from
$Z=1200$ au to $Z=3000$ au using the conical wind model from Fig. 9 . Gray
hatched regions named SU0-5 represent the locus of emissivity enhancements
required to explain the cusps on-axis positions at $V-V_{\rm sys}=3.73-5.32$
km s-1 (see Fig. 22). White dots represent the solutions for the cusps at
$V-V_{\rm sys}=4.37$ km s-1. Red dotted lines indicate the height of the two
extrema in specific angular momentum (from Fig. 3). The gray points located on
the $R=0$ axis represent the positions of jet knots observed by Podio et al.
(2011). The red dots indicate the positions with associated errors of the
ellipses identified in transverse PV diagrams (see Sect. C) and indicate that
emissivity enhancements extend to the inner streamlines.
Figure 13 shows the tomography of the conical outflow using equations from
Fig. 9 for extrapolation at $Z>1200$ au. The white dots correspond to the
solutions (named SU0 to SU5) for the cusps heights on axis identified in the
channel map at $(V-V_{\rm sys})=4.37$ km s-1. We were able to follow six
different cusps in up to six different channel maps (see Fig. 22), which
allowed us to reconstruct the shape for some of these enhancements, shown as
hatched areas in Fig. 13. As mentioned in Appendix B, the cusps are also
visible at higher line-of-sight velocities, but could not be characterized
reliably due to their lower S/N. Consequently, the hatched area shown in Fig.
13 should be extended toward the inner streamlines. Moreover, density
enhancements could also be present at $\delta z<2.2^{\prime\prime}$ but were
not identified by our procedure.The apparent acceleration of the cusps seen in
channel maps can be readily reproduced, in this disk-wind model, by
axisymmetric emissivity enhancements that cross obliquely the flow
streamlines. The upward shift of each cusp at higher velocity (apparent
acceleration) is simply a result of the velocity shear across flow
streamlines. It does not require a Hubble-law dynamics in the underlying flow.
Interestingly enough, the density enhancements SU3-4 are located close to the
two extrema of specific angular momentum, at $Z=600$ and 1000 au, suggesting a
possible link. We could not derive the shape of the emissivity enhancements
for $V_{\rm p}$ ¿ 9 km s-1, due to the low S/N of the cusps at the
corresponding projected velocities $(V-V_{\rm sys})>5.32$ km s-1 (see Fig.
12).
We computed the dynamical age of these emissivity enhancements with $\tau_{\rm
w}=\frac{Z}{V_{\rm Z}}$. This value would give the true dynamical age of the
enhancement if it is created by variability of ejection from the disk surface.
The derived values of $\tau_{\rm w}$ for cusps visible on multiple channel
maps are shown in Fig. 14. The dynamical age is almost constant within each
cusp with only a slight decrease with increasing poloidal velocities,
especially for U3 & U4. The difference in dynamical ages between two
successive cusps varies from 190 to 490 years and roughly increases linearly
with the age.
Figure 14: Dynamical evolution of the enhancements.Left panel: Dynamical age
$\tau=\frac{Z}{V_{\rm Z}}$ of the enhancements SU0-5 as a function of $V_{\rm
P}$. The red dotted line represents the mean value. Right panel: Dynamical age
versus timescale between adjacent cusps ($\tau_{i}-\tau_{i+1}$). The
uncertainties are determined by propagating the errors on the cusp positions.
These solutions were determined using only the properties of the cusps on-axis
(at $\delta x=0$), corresponding to $\phi=-\frac{\pi}{2}$. In order to check
if the reconstructed enhancements were consistent with the full cusp
morphology in all channel maps, we developed a 3D model where we modified the
emissivity profile of the synthetic disk-wind model presented in Sect. 5.1. We
multiplied the underlying DW emissivity profile by Gaussian components
representing the emissivity enhancement. We used the mean $\tau_{i}$ values
derived from Fig. 14 to determine the location of the enhancements along each
wind streamline. In order to reproduce the observed width and intensities in
the channel maps, we set the full width at half maximum of the Gaussian
component at 23 years and the maximal emissivity enhancement at $G_{\rm i}=3$.
We introduced these enhancements only in the external layers of the outflow
($V_{\rm P}<9$ km s-1), since we did not have constraints on their location at
higher velocities. Figure 15 shows computed and observed channel maps at
different line-of-sight velocities. The contour of the modeled enhancement SU0
is also represented on top of the observations. This model successfully
reproduces the morphology of the cusps as well as their apparent offset from
source with increasing velocity. Interestingly enough, the model used to
reproduce the cusps also matches the arches at low velocity, with the
locations of the apexes consistent with our observations (see Fig. 28). The
intensity contrast is also roughly recovered with our $G_{\rm i}$ value,
meaning that the outflow brightness is locally multiplied by three.
Figure 15: 12CO channel maps at selected line-of-sight velocities for the
steady conical disk-wind model (bottom row) and observations (top row). White
contours highlight the predicted emission from the modeled density enhancement
SU0. The intensity color scale is the same for all channel maps.
The channel map at ($V-V_{\rm sys})=1.51$ km s-1 shows that only the top of
the arches is reproduced. Indeed, the flanks of the largest arches are wider
than the conical flow region modeled by our disk wind in Fig. 9. In order to
fit completely the arches, we would need to extend the disk-wind model to
larger radii and opening angles and lower poloidal velocities as suggested in
Sect. 5.1, and extend the density enhancements to these regions. However, due
to cloud absorption at low velocities $<2$ km s-1, we do not have model-
independent tomographic constraints on the streamline shape and kinematics in
this slow external flow. Therefore we cannot determine a reliable solution for
the emissivity enhancements producing these arch flanks.
## 6 Discussion
In this section, we use the results of our parametric modeling of the flow
(Sect. 4,5) and tomographic study to critically discuss two possible origins
for the small-scale redshifted CO outflow in DG Tau B: 1) a stacking of
multiple shells swept-up by an inner wind (or jet), without any contribution
from an extended disk wind (Sect. 6.1), 2) an extended disk wind (Sect. 6.2),
with internal perturbations causing the observed substructures (Sect. 6.3). We
find that our measurements of rotation put stringent constraints on each of
these scenarios, and we discuss physical implications for the ejection process
and its relation to the disk accretion process.
### 6.1 Stacking of wind-driven shells
Figure 16: Specific angular momentum map (in color) for the model of ballistic
rotating infall (Ulrich 1976) with $R_{\rm d}=700$ au and
$M_{\star}=1.1M_{\odot}$. The angular momentum varies from 25 au km s-1 to 900
au km s-1. The green hatched and red filled regions represent respectively the
limits of the conical outflow derived from the tomography in Sect. 3 and the
parabolic shape of the classical Hubble-law WDS model fit to the external CO
outflow.The white and red contours outline the infalling streamlines with a
specific angular momentum similar to the conical outflow $j=40-100$ au km s-1
(white), and to the WDS model $j=200-300$ au km s-1 (red). The black hatched
region outline the streamlines with initial $\theta_{0}=70\pm 5^{\circ}$
reproducing the infall signatures seen in DG Tau B (DV20).
We discuss here a scenario where the redshifted CO outflow in DG Tau B can be
accounted for by a stacking of multiple swept-up shells resulting from the
interaction of an episodic wide-angle wind (or jet) with the ambient medium.
Such a scenario is recently proposed by Zhang et al. (2019) to reproduce the
multiple shell structures at the base of the HH46/47 outflow. They find good
agreement with the WDS model of Lee et al. (2000, 2001), namely a parabolic
layer undergoing radial expansion following a Hubble law
$\@vec{V}\propto\@vec{r}$. Below we summarize the key successes and failures
encountered in Sect. 4 by the same WDS model when applied to the various flow
components in DG Tau B, and we show that our rotation measurements raise
additional issues for this model in terms of cavity refilling.
#### 6.1.1 Outer flow component
In Sect. 4.1, we find that the morphology and kinematics of the low-velocity
outflow can be well reproduced by the simple WDS model of Lee et al. (2000).
We now examine whether a WDS origin is physically consistent with the large
specific angular momentum inferred from our modeling, $j_{\rm outer}\simeq
250\pm 50$ au km s-1, in the same sense as the disk (see Sect. 4.1).
Studies of rotation signatures in pre-stellar and proto-stellar cores show
that specific angular momentum stops decreasing with radius below scales $\leq
3000$ au and becomes roughly constant (Ohashi et al. 1997; Gaudel et al.
2020). This ”plateau” is interpreted as the region where infall motions start
to dominate, and specific angular momentum is roughly conserved along
streamlines. Depending on the object, the specific angular momentum in the
”plateau” is $\approx$ 40–400 au km s-1 ($0.2-2\times 10^{-3}$ km s-1 pc,
Gaudel et al. 2020). Our estimate for the outer flow in DG Tau B, $j_{\rm
outer}\simeq 250\pm 50$ au km s-1 falls well within this range. In addition,
infall signatures are identified around DG Tau B at large polar angles
$\theta\simeq 70\degr$ (DV20). No such signatures are seen at smaller polar
angles, but rotational flattening predicts lower envelope densities there
(Ulrich 1976), hence they might be too faint for detection. It thus appears
promising to consider that infalling material might dominate the rotation in
the outer CO layer.
Strictly speaking, the Hubble law assumed in the WDS model of Lee et al.
(2000) is only valid for a static ambient medium with a $1/r^{2}$ radial
density decrease555A static medium ensures that, after full mixing, the shell
expands in the same radial direction as the wind, while the $1/r^{2}$ ambient
density decrease ensures that the expansion speed is constant over time (the
density ratio between the wind and ambient medium being independent of
radius); both properties then together yield the ”Hubble law”
$\@vec{V}=\@vec{r}/\tau$ (Shu et al. 1991; Lee et al. 2001). A rotating
infalling envelope, in contrast, has a non-radial motion and a flatter radial
density law $\propto 1/r^{1.5}$ (Ulrich 1976). However, the calculations of
López-Vázquez et al. (2019) for such an ambient medium show that the WDS
expansion remains quasi radial, except close to the mid-plane, and with almost
constant speed after 200 yrs. Our simple model in Sect. 4.1 thus remains
roughly valid if the wind expands into an infalling envelope.
Using the shell rotation speeds computed by López-Vázquez et al. (2019), we
expect a shell specific angular momentum close to that of the infalling
material immediately ahead of it. Therefore, we consider in Fig. 16 the
spatial distribution of specific angular momentum in a rotating, free-falling
envelope from Ulrich (1976), with a centrifugal radius $R_{d}=700$ au and
central mass $M_{\star}=1.1M_{\odot}$ appropriate to DG Tau B (DV20). The
predicted $j$ values along the fit parabolic outer flow boundary are very
similar to the observed one, $j_{\rm outer}$. This detailed comparison
confirms that an infalling envelope in DG Tau B, if present up to polar angles
$\theta\simeq 30\degr$, could provide enough specific angular momentum to
explain the rotation in the outer CO layer.
However, this analysis is highly simplistic. First, we consider here that the
specific angular momentum of the envelope is fully transferred to the
entrained layer. This assumption gives an upper limit for the shell rotation
velocity, as turbulent mixing would decrease its specific angular momentum.
Secondly, the spherical and ballistic infall model used here (Ulrich 1976)
does not take into account the effects of pressure gradients or the magnetic
field. Pressure gradients could potentially increase the specific angular
momentum of the infalling envelope at large polar angles, through ”pushing”
outer infalling streamlines toward the axis, while the magnetic field would
decrease it due to magnetic breaking. Dedicated numerical simulations of the
interaction of an infalling material with an inner wind component and taking
into account all these effects are needed to fully test the entrainment
scenario for the outer CO layer.
A serious issue with this interpretation, however, is the young inferred shell
age, $\tau=V_{\rm Z}/Z=6000$ yrs (Sect. 4.1), much younger than the true age
of DG Tau B. A first way out of this ”short age problem” would be that the
interface between wind and envelope in DG Tau B is not expanding, but static.
A static shell may form when mixing between the wind and the ambient material
is not instantaneous, as assumed in most WDS models, but very gradual. Shocked
ambient material is then slowly entrained along the shell surface by the
shocked wind in a thin turbulent mixing-layer. The static shell shape and
mixing-layer properties were recently computed in the case of a free-falling
rotating envelope by Liang et al. (2020). Using again $R_{d}=700$ au and
$M_{\star}=1.1M_{\odot}$ for DG Tau B (DV20), the specific angular momentum in
the mixing-layer is predicted to be $j\approx 0.15\sqrt{GM_{\star}R_{d}}$
$\simeq 120$ au km s-1, twice lower than estimated in the outer flow.
Therefore, a static wind/envelope interface does not seem able to explain the
rotating outer flow.
A second way out of the short age problem would be that the outer flow
component does not trace the first wind encounter with the infalling envelope,
but a more recent wind outburst from 6000 yrs ago. To provide its high angular
momentum material to the young shell, however, the infalling envelope should
somehow manage to penetrate and ”refill” all the older shells created by
previous (unobserved) wind outbursts. Whether such an efficient cavity
refilling by the envelope is physically possible on $\leq 6000$ yr timescales
is a difficult open question, well outside the scope of the present paper. As
shown below, the issue of cavity ”refilling” becomes even more acute when the
WDS scenario is applied to the inner conical flow.
#### 6.1.2 Inner conical outflow
In contrast to the outer flow component, we find in Sect. 4.2 that the
morphology and kinematics of the inner conical outflow and its bright
substructures (arches, fingers and cusps) cannot be reproduced by the model of
parabolic WDS with radial Hubble law used by Lee et al. (2000) and Zhang et
al. (2019), even after several ad hoc modifications. We identify two serious
issues: (1) The observed aspect ratio of arches in channel maps at low
velocity is significantly shorter than predicted by the original model
(ellipse aspect ratio = $1/\cos{i}$); it can be reproduced by a more
collimated WDS model where the flow is parallel to the parabola; but that is
no longer physically consistent with a Hubble-law velocity field, which
requires an expanding shell (Shu et al. 1991). (2) The shell models fitting
the arches at low-velocity and cusps at high velocity do not agree with the
observed emission morphology in channel maps at mid-velocities, predicting
fingers that are broader than the cone, or full ellipses that are not seen.
We find that assuming a conical shell instead of a parabolic one, but keeping
a Hubble law, still creates the same problems. They appear intrinsically
caused by the Hubble-law velocity field, regardless of the shell detailed
morphology. Therefore, any model where the shell is expanding quasi-radially
and at nearly constant speed over time will fail to reproduce our
observations. This, in particular, discards all models where the wide-angle
wind and the ambient medium share a similar power-law in $r$, and where they
instantly mix in the shell (e.g., the models of López-Vázquez et al. 2019;
Shang et al. 2006).
Simulations including magnetic field (Wang et al. 2015; Shang et al. 2020) and
stationary solutions (Liang et al. 2020) show the formation of a shear layer
along the shell more in line with DG Tau B. However the first model predicts a
shell anchoring radius increasing with time, while the second has a shell
anchored near the centrifugal radius $R_{d}\simeq 700$ au (DV20). This is
inconsistent with the small observed anchoring radius of the DG Tau B conical
outflow ($\leq 50$ au, DV20). Alternative models of swept-up shells exist
involving an infalling sheet (Cunningham et al. 2005) or a jet instead of a
wide angle wind (e.g., in Downes & Cabrit 2007), but they have no analytical
solutions. Therefore, dedicated numerical simulations would be required to
test them in DG Tau B.
In the following, we show that the specific angular momentum measured in the
conical outflow by tomography, $j_{\rm cone}\simeq 40-100$ au km s-1 (see Fig.
3), raises additional issues for the swept-up shell scenario. We first note
that a wide-angle ”X-wind” cannot explain the observed rotation in the conical
outflow; with a launching radius $\simeq 0.05-0.1$ au from the central
protostar and a wind magnetic lever arm parameter $\lambda=j_{X}/j_{\rm
Kep}\simeq 3$ (Shang et al. 1998), its specific angular momentum is predicted
at $j_{X}\approx 20-30$ au km s-1, a factor two to three times less than
observed. In addition, the wind cannot dominate the swept-up shell mass unless
it is slower than twice the shell speed666ram pressure equilibrium between the
reverse shock in the wind and the forward shock in the static ambient medium
imposes $\rho_{\rm w}(V_{\rm w}-V_{\rm s})^{2}=\rho_{a}V_{\rm s}^{2}$, where
$V_{\rm w}$ and $V_{\rm s}$ are the wind and shell speeds, and $\rho_{\rm w}$
and $\rho_{a}$ are the wind and ambient density. The mass-flux entering the
shell from the wind side will then dominate over the swept-up mass if $(V_{\rm
w}-V_{\rm s})<V_{\rm s}$., which in the present case would require $V_{\rm
w}\leq 15$ km s-1 (see Table 2). This is inconsistent with a wind originating
from close to the protostar. The ”X-wind” model, for example, has $V_{\rm
w}\simeq 150$ km s-1 (Shang et al. 1998).
The low observed expansion speeds $\simeq 6-14$ km s-1 in the conical flow
(see Fig.3) is more consistent with jet bow shocks dominating the shell mass.
A jet magnetic lever arm parameter $\lambda\simeq 9$ would then provide enough
angular momentum. Such a scenario, however, cannot explain why regions of
lower speed in the conical flow have inversely higher specific angular
momentum (see Fig. 3). In a jet bow shock, lower speeds arise where more
ambient mass has been swept-up. Assuming the ambient medium provides no
angular momentum, the jet angular momentum would get more diluted, and the
shell specific angular momentum would drop, instead of increasing. We conclude
that the observed rotation in the conical flow cannot come from an inner wind
or jet.
To reproduce the observed conical flow rotation in the swept-up shell
scenario, we thus need an external medium with an important angular momentum.
The infalling rotating envelope is an obvious candidate. However, the observed
specific angular momentum in the conical flow is twice larger than predicted,
in the same region, by the Ulrich infalling envelope model (see Fig. 16). In
addition, it is unclear how infalling matter could penetrate and ”refill” the
space between the closely spaced shells producing the cone substructures,
especially when the region immediately outside the conical flow is instead in
outflow motion (see Fig. 5).
An alternative would be that the swept-up material originates from the
rotating disk atmosphere, at radii $R_{0}\simeq j^{2}/(GM_{\star})\simeq$ 2-9
au. The problem is then to refill the cleared cavities between shells with a
”new” static disk atmosphere. We note that our three shell models fitting the
substructures in the conical flow have remarkably identical expansion speeds
within 1 km s-1 (see last column of Table 2 for WDS-A2, WDS-A3 and WDS-U5).
Assuming that the corresponding wind and jet outbursts were of similar
strength, it implies that they met an identical ambient density ahead of them,
hence the atmosphere refilling process should be extremely efficient. This
appears difficult to achieve unless a large-scale disk wind is present.
Realistic simulations of the interaction between an episodic inner wind or jet
with an infalling rotating envelope and the disk atmosphere will be required
to definitely exclude that swept-up shells with enough angular momentum and
appropriate kinematics could be generated. However, at this stage and taking
into account all the above-mentioned difficulties, we do not favor this
scenario as the origin of the small-scale rotating CO outflow in DG Tau B. In
the next section, we discuss an alternative scenario where this rotating
outflow traces a (perturbed) extended disk wind.
### 6.2 The disk-wind scenario: constraints on the driving mechanism
We therefore favor the scenario in which the inner CO conical outflow traces
matter directly ejected from the disk. We show in Sect. 5 that the stratified
kinematical structure derived for the conical outflow is suggestive of a
quasi-steady disk-wind. In this section, we discuss constraints on the driving
mechanism. Disk winds come in different flavors, depending on the main
physical mechanism responsible for driving the flow: pure thermal effects in
photo-evaporated disk winds (PDW) (Alexander et al. 2014), cold magneto-
centrifugal ejection (Pudritz et al. 2007) or a combination of the two
processes in the so-called warm or magneto-thermal disk winds (Casse &
Ferreira 2000; Bai et al. 2016). In the following we refer to the two last
classes of magnetized disk winds as MHD disk winds (hereafter MHD DW).
Figure 17: Launching radius of the streamline as a function of its poloidal
velocity assuming: conical extrapolation for the geometrical radius (green
circles), steady thermal ejection conserving angular momentum (orange circles)
and steady cold magneto-centrifugal ejection (blue circles). See text for more
details. We do not derive the launching radius for the largest poloidal
velocities due to the large uncertainties.
Figure 18: Specific angular momentum ($j=R\times V_{\phi}$) versus poloidal
velocity $V_{\rm P}$ for steady and axisymmetric MHD disk winds. The black
symbols represent values derived from the tomography and averaged along lines
of constant $V_{\rm Z}$, with their uncertainty. Red curves show the expected
relation from self-similar cold magneto-centrifugal disk winds with $r_{0}$
varying from 0.5 to 100 au and $\lambda$ between 1.52 and 2.3 from Ferreira et
al. (2006). The green dot corresponds to a warm MHD disk wind solution from
Casse & Ferreira (2000) with $r_{0}=1$ au and $\lambda=1.9$. The green arrow
shows the path followed by the solution with an increase of $r_{0}$. The gray
box represents the estimated specific angular momentum for the outer low
velocity layer on Sect. 4.1, taking a maximal poloidal velocity corresponding
to a height of $Z=3300$ au in the outflow referential.
#### 6.2.1 Photo-evaporated disk winds
In photo-evaporated disk winds, the high energy radiation (UV and X-rays) of
the central accreting protostar heats the surface layers of the disk to high
temperatures (103-104 K) up to significant radial distances. Beyond the
gravitational radius $r_{\rm g}=(GM_{\star})/c_{\rm s}^{2}$, where $c_{\rm s}$
is the sound speed in the upper disk surface layers, thermal energy exceeds
the gravitational binding energy. Numerical simulations show that significant
mass loss starts before $r_{\rm g}$, at the critical radius $r_{\rm c}\simeq
0.1-0.2r_{\rm g}$. Consequently, matter is ejected and reaches terminal
velocities of typically two to three times $c_{\rm s}$ (Alexander et al. 2014,
and references therein). The exact properties of the wind depends on the
dominant source of high-energy irradiation. Extreme-UV (EUV) heating creates
an isothermal ionized layer on the disk surface with temperature $T\simeq
10^{4}$ K ($c_{\rm s}=10$ km s-1) which drives a fast wind ($\sim 35$ km s-1)
with mass-loss rates 10-9-10-10 M⊙ yr-1 (Font et al. 2004; Wang & Goodman
2017). X-ray irradiation results in cooler and slower flows ($c_{\rm s}\simeq
3-5$ km s-1, $v\simeq 15-20$ km s-1) but penetrates at higher densities and
therefore can drive mass-loss rates up to a few 10-8 M⊙ yr-1 (Picogna et al.
2019). Non ionizing far-UV (FUV) heating mostly drives slow mass-loss
($v\simeq 18$ km s-1) from the outer disk regions.
The conical morphology of the DG Tau B CO outflow matches expectations from
both self-similar PDW models by Clarke & Alexander (2016) and hydrodynamical
simulations by Owen et al. (2011) and Wang & Goodman (2017). In such disk-wind
solutions, the specific angular momentum is conserved along the streamlines
and is equal to the Keplerian value at the foot-point radius. The observed
values of $j$ in the CO conical outflow imply footpoint radii in the range
$r_{0}=1.6$ to 8.2 au. Figure 17 shows that these foot-point radii are lower
(by a factor $\simeq$ three) than the ones derived from a straight
extrapolation of the conical morphology on large scales. This requires a
larger opening angle of the streamlines at the base, consistent with the self-
similar models of Clarke & Alexander (2016).
The values of $r_{0}$ are in line with expectations from EUV dominated PDW
models. Indeed, for a 1 M⊙ star and $c_{\rm s}\simeq 10$ km s-1, expected in
EUV heated PDW, the gravitational radius $r_{\rm g}\simeq 10$ au and
significant mass loss starts at $r_{\rm c}\simeq 1-2$ au. On the other hand,
observed terminal velocities of $V_{\rm p}=4-16$ km s-1 require sound speeds
$\leq$ 2-8 km s-1 at radii $r_{0}\leq 10$ au, excluding EUV-driven models.
The large derived mass flux of $1.7-2.9\times 10^{-7}$ M⊙ yr-1 for the CO
conical flow (DV20) within $r_{0}\leq 10$ au however excludes FUV driven
winds, which fail by at least one order of magnitude (Wang & Goodman 2017). We
show below that it is also inconsistent with the latest X-ray driven models.
Indeed, recent X-ray driven photo-evaporation models of Picogna et al. (2019)
predict mass loss rates up to 10-7 M⊙ yr-1 for stellar X-ray luminosities
$L_{\rm X}\geq 10^{31}$ erg s-1. However, these high $L_{\rm X}$ models also
predict a stronger contribution of the outer disk regions to the total mass
flux due to the increased penetration of X-ray photons. Figure 9 in Picogna et
al. (2019) shows that only 10 % of the total mass flux originates from disk
foot-point radii below 10 au. In summary, current PDW models fail to account
for the combination of large mass flux and small foot-point radii of
$r_{0}=1.6-8.2$ au, derived for the DG Tau B CO conical flow.
Last but not least, the survival of CO molecules in such a wind is
problematic. The full thermo-chemical computation of Wang & Goodman (2017)
shows that in their fiducial models, CO survives only at the very base of the
wind in an intermediate layer on scales $z/r\leq 0.6$. However, the models of
Wang & Goodman (2017) are EUV dominated and hence result in warm and fully
ionized winds. Similar problems are expected in thermally driven winds
launched from the inner disk, which require base temperature greater than 2000
K. Therefore, we conclude that pure thermal processes appear highly unlikely
as the main driving mechanism of the DG Tau B CO conical wind.
#### 6.2.2 Magnetic disk winds
Disk winds driven by magnetic forces require a large scale poloidal magnetic
field anchored in the disk. This large-scale field exerts a torque on the
rotating disk that both ejects matter and removes angular momentum from the
disk (Blandford & Payne 1982; Pudritz et al. 2007). The strength of this
torque is characterized by the magnetic lever arm parameter
$\lambda\simeq(r_{A}/r_{0})^{2}$, where $r_{A}$ is the poloidal Alfven radius
and $r_{0}$ the disk foot-point radius of the streamline. In principle, such
disk winds can produce at the same time fast and collimated jets originating
from the inner streamlines and much slower and less collimated winds
originating from outer disk radii. The full kinematics and morphology of these
solutions also depend on whether thermal effects are important in the
launching regions. Numerical simulations show that the mass loss can be
significantly increased when thermal effects are taken into account (Casse &
Ferreira 2000; Bai et al. 2016). Such magneto-thermal winds have low to
moderate $\lambda$ values but can extract significant mass and angular
momentum from the disk.
Terminal velocities depend on both the foot-point radii $r_{0}$, $\lambda$ and
thermal effects. Under the assumption of steady magnetically driven ejection,
the asymptotic values of $V_{\rm p}$ and $V_{\phi}$ are given by (Eqs. 4&5 in
Ferreira et al. 2006):
$\displaystyle RV_{\phi}$ $\displaystyle=$ $\displaystyle\lambda R_{\rm
0}V_{\phi}(R_{\rm 0})$ (15) $\displaystyle V_{\rm p}$ $\displaystyle=$
$\displaystyle V_{\phi}(R_{\rm 0})\sqrt{2\lambda-3+\beta},$ (16)
where $\beta$ encompasses all pressure effects, including thermal and
turbulent Alfvén waves (see Ferreira et al. 2006). The streamline foot-point
radius can be estimated from these equations assuming cold MHD ejection, ie.
negligible thermal effects ($\beta\simeq 0$) following (Anderson et al. 2003).
Figure 18 traces the relationship between the mean values of $V_{\rm P}$ and
$j=RV_{\phi}$ for the various conical layers of constant $V_{\rm Z}$ derived
from the tomography. In this figure is also represented the parameter space
($r_{0}$,$\lambda$) predicted by cold magneto-centrifugal disk-wind models
(Ferreira et al. 2006). The mean poloidal velocities and specific angular
momentum coincide with a line of constant $\lambda_{\rm cold}\simeq 1.58$ with
foot-point radii $r_{0,\rm cold}=0.7-3.4$ au. If thermal effects play a
dynamical role at the base of the wind, Eqs. 15,16 show that the values of
$\lambda$ and $r_{0}$ derived under the cold assumption are respectively upper
and lower limits. This effect is illustrated in Fig. 18 where we plot $V_{\rm
P}$ and $j$ from the warmest solution of Casse & Ferreira (2000) with
$\lambda=1.9$, at $r_{0}=1$ au. We see that using the cold MHD curves would
lead to overestimate $\lambda\simeq 2.5$ and underestimate $r_{0}\simeq 0.7$
au. The cold assumption is only valid if $\beta\ll 2\lambda-3\simeq 0.2$,
which would require a cold disk atmosphere and no substantial wind heating. On
the other hand, the low derived upper limit on $\lambda\leq 1.6$ is consistent
with warm MHD DW models (Casse & Ferreira 2000; Bai et al. 2016; Wang et al.
2019) or cold MHD DW from weakly magnetized disks (Jacquemin-Ide et al. 2019).
The derived minimum foot-point radius of $r_{\rm in}\geq 0.7$ au for the CO
wind streamlines is in good agreement with the thermo-chemical predictions of
Panoglou et al. (2012), who demonstrate that CO molecules magnetically
launched from foot-point radii $\geq 1$ au survive in the case of accretion
rates in the disk $\geq 10^{-7}$ M⊙ yr-1. Similar results are obtained for
warm magneto-thermal wind solutions (Wang et al. 2019).
The streamline foot-point radii derived from the kinematics are significantly
smaller than the radii obtained from direct conical extrapolation (Fig. 17),
suggesting wider opening angle of the streamlines at their base. This is
indeed expected in MHD DW solutions where streamlines originally follow a
conical trajectory and recollimate on larger scales due to the hoop stress
provided by the azimuthal B-field. The constant opening angle of the
streamlines observed out to $Z=3000$ au suggests that the magnetic hoop stress
drops rapidly above $Z\simeq 50$ au.
Contrary to pure thermal disk winds, MHD DW also account for the observed
large mass flux in the DG Tau B conical CO flow. Wind mass loss rates in the
range 10-8-10-7 M⊙ yr-1 are predicted by the magneto-thermal wind solutions of
Wang et al. (2019), on the same order as the accretion rate in the underlying
disk and increasing with disk magnetization.
We estimated the local ejection efficiency $\xi$ defined as $\dot{M}_{\rm
acc}(r)\propto r^{\xi}$ (Ferreira et al. 2006). We estimated $\xi$ from
estimates of the disk accretion rate at the inner launching radius of the CO
outflow. Indeed, from the mass conservation across the disk region launching
the conical CO outflow ($\dot{M}_{\rm acc}(r_{\rm in})=\dot{M}_{\rm
acc}(r_{\rm out})-\dot{M}_{\rm DW}$) and the definition of $\xi$, we derived
the following expression:
$\frac{\dot{M}_{\rm DW}}{\dot{M}_{\rm acc}(r_{\rm in})}=\bigg{(}\frac{r_{\rm
out}}{r_{\rm in}}\bigg{)}^{\xi}-1.$ (17)
We estimated the accretion rate onto the central star by taking 10 % of the
jet mass flux (Ellerbroek et al. 2013). Podio et al. (2011) estimate the red
jet mass flux at $6.4\times 10^{-9}$ M⊙ yr-1, giving a mass accretion rate
onto the star of $\simeq 6\times 10^{-8}$ M⊙ yr-1. We took this value as a
lower limit to $\dot{M}_{\rm acc}(r_{\rm in})$. From the measured mass flux in
the conical CO flow $\dot{M}_{\rm DW}=2.3\times 10^{-7}$ M⊙ yr-1 and the disk-
wind launching zone $r_{\rm out}/r_{\rm in}\simeq 5$, we then derived an upper
limit on $\xi\leq 1$. The mass flux in the conical wind is $\sim 4$ times the
estimated accretion rate onto the star, implying that 80% of the mass
accreting at $r_{\rm out}$ is being ejected before reaching the star.
If the transport of angular momentum in the disk is entirely provided by the
torque exerted by the MHD DW, one expects in steady state the following
relationship: $\xi=1/(2(\lambda-1))$. The upper limit derived above on
$\xi\leq 1$ translates into $\lambda\geq 1.5$. This condition is compatible
with our upper limit on $\lambda\leq 1.58$ derived from the kinematics. Thus
the CO mass flux combined with the constraints on launching radii and magnetic
lever arm appear compatible with an MHD DW extracting all angular momentum
required for the disk to accrete from $r_{\rm out}$ to $r_{\rm in}$.
If the inner conical outflow is tracing a disk wind, then the outer parabolic
outflow cannot be explained by the interaction between the envelope and an
inner jet or X-wind. However, the outer outflow could be tracing the
interaction between the envelope and outer disk-wind streamlines located
outside of the conical outflow ($r_{0}>4$ au). The arches located at least
partially outside of the conical outflow as well as the continuous aspect in
the PV diagrams suggests an ”intermediate” outflow located between the conical
outflow and the outer parabolic surface. Alternatively, the outer flow could
also be tracing directly the outer disk-wind streamlines. The derived $j$ and
maximal velocity in the outer layer would indicate launching radii $\geq 30$
au and a similar low $\lambda$ value as for the inner cone (see Fig. 18).
Moreover, Bai et al. (2016) show that some MHD DW solutions can accelerate
until R $\sim$ 100 $r_{0}$, possibly explaining the apparent acceleration of
this component seen in our channel maps. The global outflow would then be a
continuous MHD DW originating from 0.7 au to $\geq 30$ au. Unfortunately, the
morphology and dynamics of the potential disk wind originating from $r_{0}>4$
au could not be studied in detail due to our limited spectral sampling and the
absorption by the surrounding cloud or envelope. However, in that scenario,
the origin for the difference of emissivity between the bright conical outflow
and the faint outer flow is not clear but could reflect the radial
distribution of magnetic field strength or surface density in the underlying
disk.
### 6.3 Disk-wind scenario: origin of perturbations
We discuss here the merits of different models for the origin of the
substructures (arches, fingers, and cusps) in the conical flow, observed in
the channel maps.
#### 6.3.1 Perturbation by jet bow shocks
A first potential explanation for the bright substructures seen in the conical
outflow is that the steady disk wind is perturbed by nested bow shock wings
created by the propagation of the variable axial jet (see Fig. 20, scenario
B). Perturbation of the inner streamlines of a rotating disk wind by a large
jet bow shock is recently reported in the much younger system of HH 212 by Lee
et al. (2021). In DG Tau B, this interpretation is supported by the similar
spatial spacings between the inferred locations of the perturbations producing
the substructures in the CO conical wind, and the axial jet knots identified
in optical images, as shown in Fig. 13 (for z=500-3000 au along the flow axis
spacings range between 200-1300 au for the jet knots, 300-700 au for the over-
densities). Indeed, in the jet-wind interaction scenario, the over-densities
trace the point of contact between each bow shock and the outer disk wind so
they propagate along the interface at the bow shock propagation speed, which
is similar to the inner jet knot propagation speed.
Although the optical knot observations are not synchronous with our ALMA CO
observations, and jet knots move away from the source at the jet speed $\simeq
150$ km s-1, the general pattern of knot spacing as a function of distance is
set by the underlying jet variability properties, and thus will tend to remain
similar at different epochs in a given jet. If, in addition, the jet undergoes
low-amplitude wiggling (as frequently seen in young stars), perturbations to
the disk wind caused by jet bow shocks would be slightly nonaxisymmetric,
possibly explaining the apparent distortions in specific angular momentum
along disk-wind streamlines using tomography (see Fig. 4). Finally, this
scenario might also explain the lack of recollimation of the conical disk-wind
streamlines, due to the additional internal pressure created by the jet driven
bow shocks wings.
Hydrodynamical simulations of the interaction of a variable inner jet with a
slower outer disk wind have been recently performed by Tabone et al. (2018).
These simulations show the formation of a dense stationary conical layer
closing down at the source, created by the stacking of jet bow shock wings in
the disk wind. This shell exhibits local over-densities at the positions of
individual jet bow shocks, illustrated by the red regions in Fig. 10 (right
panels) in Tabone et al. (2018) . These over-densities globally reproduce the
observed shapes of the perturbations in the DG Tau B conical layer: for the
perturbations closer-in, the density map is dominated by the regions close to
the bow shock apex which bend inward, while for the perturbations farther out
the density map is dominated by the bow shock wings which bend outward. This
simulation also shows that the conical dense shell mostly retains the velocity
of the surrounding disk wind, because the shock is weak and oblique.
Therefore, the simulations in Tabone et al. (2018) does not show the
characteristic stratification in $V_{\rm Z}$ observed in the DG Tau B inner
conical flow. The numerical simulations of Tabone et al. (2018) also reproduce
the observed trend of similar spacings between the axial jet knots and the
over-densities in the contact region (shown in red in their Fig. 10, right
panels). However, the simulations are made in a simplistic configuration where
the outer disk wind is assumed to have a constant vertical velocity of 40% of
the jet speed, uniform density, and no rotation motion. More realistic
simulations taking into account the velocity and density gradients across the
outer disk wind, and including rotation and magnetic fields, are strongly
needed to reliably test this scenario.
#### 6.3.2 Possible origin of CO outflow axis wiggling
As discussed in Sect. 5.2, wiggling of the wind ejection axis is the only
model explored here that can reproduce simply the variations of angular
momentum observed along the DG Tau B outflow. From this analysis, we derived
estimates of the wiggling period $\tau\simeq 400$ yrs and semi-amplitude
wiggling angle $\alpha\simeq 0.5^{\circ}$.
Wind axis wiggling can originate from the precession of the underlying disk
angular momentum vector. Disk axis precession can be induced by a mis-aligned
companion. In that scenario, orbital periods are expected to be significantly
shorter than the disk axis precession period (Terquem et al. 1998), hence the
companion would be located well within the disk. A companion in a mis-aligned
orbit can open a gap in the disk, separating the dynamical evolution of the
inner and outer disks (Zhu 2019). The inner disk then starts to precess with a
period $\tau_{\rm p}$ related to the orbital period of the companion by
$\tau_{o}/\tau_{\rm p}\simeq 0.37\mu/\sqrt{1-\mu}$, where
$\mu=m_{2}/(m_{1}+m_{2})$ is the ratio of the companion mass to the total mass
of the binary system, and assuming a small mis-alignment $i_{\rm p}$ as
suggested by the maximal wind precession angle $\alpha=0.5\degr$ (see Eq. 27
in Zhu 2019). An additional constraint can be obtained by requiring that the
semi-amplitude wiggling angle $\alpha$ is dominated by the precession motion,
which translates into the condition $V_{0}\leq V_{\rm Z}\tan(\alpha)$, that is
$V_{0}\leq 0.1$ km s-1where $V_{0}$ is the orbital velocity of the flow
source. Combining these two constraints, we derive a companion mass ratio
$\mu\leq 2.5\times 10^{-3}$ and binary separation: $a\leq 0.5$ au. This
separation is smaller than the launching radius of the CO disk wind
($r_{0}\geq 0.7$ au, see Fig. 17). So the precessing disk launching the CO
flow would be outside the orbit of the planetary mass companion, which is
inconsistent with the scenario investigated here. Indeed, in such a scenario
the outer disk is precessing on much longer timescales than the ones given by
the formula above.
Nixon et al. (2013) and Facchini et al. (2018) investigate circumbinary disk
precession around an inner mis-aligned binary system. In such circumstances
the inner rim of the circumbinary disk can break from the outer disk and
precess. However, large mis-alignments or massive enough companions are
necessary for this situation to occur (Facchini et al. 2018). We show that
this is not likely in the DG Tau B case. The inner binary system would
truncate the disk at 1.5-1.7 times the separation $a$ of the binary (Facchini
et al. 2018). If we take an upper limit of $r_{\rm in}=0.7$ au for the inner
circumbinary disk rim, to allow the launching of the CO flow, we get an upper
limit of $a\leq 0.5$ au for the binary separation, corresponding to an orbital
period $\tau_{0}\leq 0.35$ yrs using Third Kepler’s law and a total mass of 1
M⊙ for the binary system (DV20). With a small misalignment (less than a few
degrees) between the outer disk and the binary orbital planes, suggested by
the small wiggling angle of the CO outflow, a mass ratio of the companion
$\mu\leq 10^{-2}$ would be required to get a precession period of 400 yrs (see
Eq. 4 in Facchini et al. 2018). Equation 2 in Facchini et al. (2018) then
shows that such a low mass companion combined with a small misalignment will
not break the inner circumbinary disk. Therefore this second precession
scenario can be ruled out to explain the CO outflow wiggling.
So far we have investigated only precession as the origin of the wiggling of
the CO flow axis. However, wiggling of the disk-wind ejection axis can be also
induced by the orbital motion of the CO outflow source in a binary system. The
equations of motion of the ejected gas will be equivalent to the precession
solution investigated so far. We follow the formulation developed by
(Masciadri & Raga 2002; Anglada et al. 2007) assuming an orbital plane
perpendicular to the outflow axis. From the third Kepler’s law of motion,
$a^{3}=M_{\rm tot}\tau_{o}^{2}$, with $\tau_{o}=400$ yrs and $M_{\rm tot}=1$
M⊙ we can derive the mean separation of the companion at $a\simeq 50$ au. On
the other hand, from the semi-amplitude of the wiggling ($\alpha\simeq
0.5^{\circ}$) the orbital velocity of the CO outflow source is constrained at:
$V_{0}=V_{\rm CO}\times\tan(\alpha)=0.1$ km s-1, using an average velocity of
$\simeq 10$ km s-1 for the CO outflow. This in turn gives the orbital radius
of the outflow source around the center of mass of the binary $r_{0}=1.3$ au
and the ratio $\mu$ between the mass of the companion and the total mass of
the system: $\mu=r_{0}/a=0.025$. Thus a brown dwarf or massive planetary mass
companion located at $\simeq 50$ au ($=0.35^{\prime\prime}$) separation would
be required to account for the observed wiggling of the CO outflow in the
orbital scenario. Such a low mass companion could have escaped direct
detection so far (Rodríguez et al. 2012). Strikingly the predicted companion
separation is very close to an emission bump at $r=62$ au detected in the
continuum emission profile of the disk at millimetric wavelengths (de Valon et
al. 2020; Garufi et al. 2020). However no clear gap is detected in the disk
emission at this position which would be expected for such a massive
companion.
Therefore, the precession scenario is excluded to account for the observed
wiggling of the CO flow while the orbital scenario requires a brown dwarf or
massive planetary mass companion at 50 au separation, which signature we do
not clearly see in the disk yet. We also recall that our wiggling models
cannot account for the observed variable separations between cusps in channel
maps. So we conclude that although attractive to explain some of the
substructures observed in the DG Tau B CO outflow, the interpretation of the
wiggling scenario faces some difficulties. We discuss below the alternative
model where substructures arise from axisymmetric brightness enhancements in
the disk wind.
#### 6.3.3 A variable disk wind
We show in Sect. 5.3 that the cusps, fingers, and a section of the arches can
be explained by brightness or density enhancements in the conical outflow. The
timescales between the density enhancements are typically a few hundred years
(see Fig. 14). Unfortunately, these timescales cannot be directly compared to
the ones observed in the DG Tau B jet due to the larger jet velocity and its
fading brightness at large distances. However, such timescales are observed on
younger molecular outflows. The cluster W43-MM1 in Nony et al. (2020), CARMA-7
in Plunkett et al. (2015) as well as HH46/47 in Zhang et al. (2019) show
signatures of variability in molecular outflows with timescales between
episodic events typically of a few hundred years. We may be witnessing similar
variability in the DG Tau B CO outflow.
In the following, we discuss the possibility that these density enhancements
are created by variability at the source in the disk-wind launching regions.
For the model presented in Fig. 15, we considered for the sake of simplicity
that the density bursts take place simultaneously at all radii in the disk
(that is, $\tau_{i}=cst$ for all layers). A more physical assumption would be
to assume that the density burst propagates radially across the disk with a
velocity $V_{\rm prop}$. If we consider that the burst takes place close to
the mid-plane ($Z\approx 0$), the expression of the travel time for the
density launched from a fixed radius $r_{0}$ is :
$\displaystyle\tau(\rm r_{0})$ $\displaystyle=$ $\displaystyle t-t_{\rm
eject}(r_{0})$ (18) $\displaystyle t_{\rm eject}(r_{0})$ $\displaystyle=$
$\displaystyle t_{\rm eject}(0)+\frac{r_{0}}{V_{\rm prop}}$ (19)
$\displaystyle\tau(\rm r_{0})$ $\displaystyle=$
$\displaystyle\tau(0)-\frac{\rm r_{0}}{\rm V_{\rm prop}},$ (20)
where $t$ is the current time and $t_{\rm eject}(r_{0})$ is the epoch of
density ejection. Here, $V_{\rm prop}$ is positive when the density burst
moves from the inner to the outer disk regions.
Figure 19: Dynamical timescales of the density enhancements responsible for
the high-velocity cusps in the channel maps, as a function of the launching
radius $r_{0}$ of the streamline under the cold MHD disk-wind hypothesis
(assuming $\lambda=1.58$). For each cusp, the red dashed line represents the
best linear fit of $\tau(r_{0})$ and its corresponding velocity. The one
$\sigma$ domain is shown in gray and the uncertainties of the fits are noted
on the right.
Figure 14 indicates that $\tau$ decreases with increasing poloidal velocity,
which is consistent with a density enhancement moving from the outer to the
inner regions of the disk. Figure 19 is a modification of Fig. 14 in which we
transformed the poloidal velocity into the corresponding radius of ejection
under the MHD disk-wind hypothesis, using Eq. 8 from Ferreira et al. (2006)
with $\lambda_{\phi}=1.58$ and $\beta=0$. We did not take into account the
uncertainty on the estimation of $r_{0}$ ($\simeq 20\%$, see Fig. 17) as these
uncertainties would make impossible the estimation of $V_{\rm prop}$.
Nonetheless, this study gave an estimate of the propagation velocity. For each
cusp, we traced the profile $\tau(\rm r_{0})$ derived from the observed
positions of the cusp apex in the different channel maps, as described in
Sect. 5.3. Linear fits to these profiles with associated $\boldsymbol{\tau}$
uncertainties are also shown. The slopes of these profiles is directly linked
to $1/V_{\rm prop}$, and hence give the radial propagation velocities of the
density enhancement at the origin. The average velocity over all cusps is
$V_{\rm prop}\approx-0.2$ km s-1. The same study could be achieved with PDW
models. In that case, the radius of ejection were multiplied by $\lambda^{2}$
($=2.5$) and therefore $V_{\rm prop}$ values were also multiplied by 2.5,
giving an average radial velocity $V_{\rm prop}\approx-0.5$ km s-1. From
uncertainties on the slopes, we derived uncertainties on $V_{\rm prop}$ values
in the case of U3, U4, and U5. In the case of U0,1,2, a horizontal solution
($V_{\rm prop}=\infty$) could not be excluded. For these last three fits, we
derived the minimal radial velocities for the two opposite directions of
propagation.
Fig. 19 seems to indicate that the density bursts propagate from the outer to
the inner regions of the disk at $\approx|V_{\rm prop}|=0.2-0.5$ km s-1. The
accreting density must also propagate toward the inner regions of the conical
flow ($r_{0}<2$ au) where the density enhancements were visible but not
characterized. This propagation velocity corresponds to $\approx
0.01-0.04\leavevmode\nobreak\ V_{\rm Kep}(r=2-5\leavevmode\nobreak\ \rm au)$.
Such accretion velocities match expectations for radial surface velocities in
MHD wind-driven accretion (Riols et al. 2020). From this velocity and our
estimate of the burst duration of $\simeq$ 23 years, we can constraint the
radial extent of the burst propagating along the disk at $\Delta R\approx 1$
au. Figure 20 illustrates the proposed scenario.
Episodic density bursts propagating inward are obviously reminiscent of Fu Ori
and Ex Ori type variable accretion events. Some Fu Ori have burst durations
$<30$ years (e.g., V1515 Cyg, or V1714 Cyg in Hartmann & Kenyon 1996). But
periods are usually assumed to be $\approx$ 10${}^{4}-10^{5}$ years. On the
other extreme, Ex Ori have typically bursts of a few months and periods of few
years. DG Tau B variability seems to be located between these two extrema.
However, the mass accretion rate increase in Fu Ori type events is typically
three to four orders of magnitude higher than suggested in the DG Tau B
outflow by the moderate factor three emissivity enhancement during the bursts.
Moreover, models of Fu Ori events predict a global accretion affecting the
whole vertical structure of the disk during the high state. In magnetically
accreting disk models, most of the mass is concentrated in the mid-plane,
where the radial accretion velocity is $\simeq V_{\rm Kep}/10^{3}$ (Riols et
al. 2020). The high propagation velocity combined with the moderate emissivity
enhancements derived in DG Tau B suggest that the accretion burst takes place
locally on the disk surface and is less extreme than in typical Fu Ori
phenomena. Such moderate accretion bursts could be due for example to residual
infalling envelope material creating a shock wave when infalling into the disk
(Hennebelle et al. 2017). It is important to note, however, that this
axisymmetric model does not reproduce the local deviations observed in the
specific angular momentum map (Fig. 3). Nonaxisymmetric perturbations would be
required.
Figure 20: Schematic scenarios describing the different components of the DG
Tau B redshifted outflow in the disk-wind paradigm (see text).
## 7 Conclusions
We present a detailed analysis and modeling of the ALMA 12CO(2-1) observations
of the DG Tau B redshifted outflow published in de Valon et al. (2020), with
the aim to constrain its origin. Our main conclusions are as follows:
* •
We identify three classes of discrete structures visible on the 12CO channel
maps: arches at low velocities, fingers at medium velocities, and cusps at
high velocities. Both cusps and arches show apparent acceleration in channel
maps.
* •
We derived the 2D kinematics of the inner conical outflow using a tomographic
method, assuming only axisymmetry of the outflow. We reconstructed 2D maps for
both the expansion velocity $V_{\rm Z}$ and specific angular momentum
$j=R\times V_{\phi}$. The inner outflow shows a striking $V_{\rm Z}$ shear
with faster material closer to the flow axis. Lines of constant $V_{\rm Z}$
are conical from $Z\simeq 50$ au out to $\approx 1200$ au. Specific angular
momentum is roughly constant along those lines (except in two localized
regions), and it increases outward inversely with $V_{\rm Z}$ from $\simeq 40$
to 100 au km s-1.
* •
The lower velocity external CO outflow shows a parabolic morphology, apparent
acceleration, and a large specific angular momentum of $j=250\pm 50$ au km
s-1. This suggests that it is tracing either a swept-up infalling and rotating
envelope, or an extended disk wind launched from $\sim 30$ au.
* •
The conical outflow and the discrete structures could not be described by
wind-driven shells with radial Hubble velocity laws. Such models fail to
reproduce at the same time the morphologies of the observed structures
(arches, fingers, and cusps) in the channel maps. Numerical simulations of the
interaction between an episodic jet- and wide-angle wind with an infalling
envelope are, however, required to confirm these conclusions.
* •
Instead, the conical outflow global morphology and kinematics appear
consistent with matter directly ejected from the disk. Constraints on the
disk-wind foot-point radii were derived at $r_{0}=1.6-8.2$ au (resp. 0.7 - 3.4
au) in the limiting cases where thermal (resp. magneto-centrifugal) processes
dominate. However, none of the current photo-evaporated wind models can
reproduce the large observed mass flux ( $2.3\times 10^{-7}$ M⊙ yr-1 ) ejected
from $r_{0}<10$ au. In contrast, an MHD disk-wind model with a constant
magnetic level arm parameter $\lambda\leq 1.58$ can reproduce – at the same
time – the flow velocity and angular momentum, as well as the large mass flux
if it extracts most of the angular momentum for disk accretion across the wind
launching region. The low lambda value is consistent with recent models of
warm or weakly magnetized MHD DW (Bai et al. 2016; Jacquemin-Ide et al. 2019).
* •
The wiggling of the flow axis may explain both the localized deviations of
specific angular momentum and the morphology of the substructures in the
conical flow. Orbital motion of the flow source in a binary of separation
$\simeq 50$ au with companion mass $2.5\times 10^{-3}$ M⊙ can explain the
inferred wiggling period and amplitude. Such a low mass companion could have
escaped direct detection so far, but it should produce a gap signature in the
continuum dust disk emission, which is not currently detected. In addition,
the wiggling scenario fails to account for the variable separation between
cusps in channel maps.
* •
Alternatively, the substructures observed in the CO channel maps can be
explained by a series of mild (a factor three) density perturbations in the
wind launching region, propagating inward at a radial velocity of $\simeq
0.2-0.5$ km s-1, consistent with the surface accretion flow predicted in MHD
wind-driven accretion. We derived a typical perturbation width of $\sim 1$ au
and intervals of 200-500 years between perturbations. Alternatively the
conical morphology and local density enhancements might be explained by the
interaction of inner jet bow shocks with the disk wind (Tabone et al. 2018);
although, further numerical simulations are required to fully test this
hypothesis.
The discrete structures that are increasingly observed in Class 0 and Class I
outflows on larger scales have been usually interpreted in terms of nested
shells swept up by an episodic inner wind. In contrast, we have shown in this
paper that the substructures in the DG Tau B outflow appear best explained by
density enhancements at the disk surface and that they propagate in a shear-
like MHD disk wind. If confirmed, these results would directly demonstrate the
link between accretion and ejection processes in embedded sources.
DG Tau B is a Class I protostar with a structured disk, infalling flows, and
possible variable disk wind. Structures in Class I disks are assumed to trace
early stages of planetary formation processes. Therefore, our results suggest
that planetary formation is taking place in a very dynamic environment. The
impact of such outflowing and infalling flows on the disk and its evolution
remains an open question. Additional models and simulations of variable disk
wind are needed in order to fully comprehend its impact on disk evolution and
planet formation.
###### Acknowledgements.
The authors would like to thank the referee, whose comments helped improve the
quality of the paper. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2015.1.01108.S, ADS/JAO.ALMA#2017.1.01605.S. ALMA is a
partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile.The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. This work was supported by
the Programme National de Physique Stellaire (PNPS) and the Programme National
de Physique et Chimie du Milieu Interstellaire (PCMI) of CNRS/INSU co-funded
by CEA and CNES. FL acknowledges the support of the Marie Curie Action of the
European Union (project MagiKStar, Grant agreement number 841276).
## References
* Alexander et al. (2014) Alexander, R., Pascucci, I., Andrews, S., Armitage, P., & Cieza, L. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 475
* Anderson et al. (2003) Anderson, J. M., Li, Z.-Y., Krasnopolsky, R., & Blandford, R. D. 2003, ApJ, 590, L107
* Anglada et al. (2007) Anglada, G., López, R., Estalella, R., et al. 2007, AJ, 133, 2799
* Arce et al. (2007) Arce, H. G., Shepherd, D., Gueth, F., et al. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 245
* Bai et al. (2016) Bai, X.-N., Ye, J., Goodman, J., & Yuan, F. 2016, ApJ, 818, 152
* Bjerkeli et al. (2016) Bjerkeli, P., van der Wiel, M. H. D., Harsono, D., Ramsey, J. P., & Jørgensen, J. K. 2016, Nature, 540, 406
* Blandford & Payne (1982) Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883
* Cabrit et al. (1997) Cabrit, S., Raga, A., & Gueth, F. 1997, in Herbig-Haro Flows and the Birth of Stars, ed. B. Reipurth & C. Bertout, Vol. 182, 163–180
* Casse & Ferreira (2000) Casse, F. & Ferreira, J. 2000, A&A, 353, 1115
* Clarke & Alexander (2016) Clarke, C. J. & Alexander, R. D. 2016, MNRAS, 460, 3044
* Cunningham et al. (2005) Cunningham, A., Frank, A., & Hartmann, L. 2005, ApJ, 631, 1010
* de Valon et al. (2020) de Valon, A., Dougados, C., Cabrit, S., et al. 2020, A&A, 634, L12 (DV20)
* Downes & Cabrit (2007) Downes, T. P. & Cabrit, S. 2007, A&A, 471, 873
* Eislöffel & Mundt (1998) Eislöffel, J. & Mundt, R. 1998, AJ, 115, 1554
* Ellerbroek et al. (2013) Ellerbroek, L. E., Podio, L., Kaper, L., et al. 2013, A&A, 551, A5
* Facchini et al. (2018) Facchini, S., Juhász, A., & Lodato, G. 2018, MNRAS, 473, 4459
* Fernández-López et al. (2020) Fernández-López, M., Zapata, L. A., Rodríguez, L. F., et al. 2020, AJ, 159, 171
* Ferreira et al. (2006) Ferreira, J., Dougados, C., & Cabrit, S. 2006, A&A, 453, 785
* Font et al. (2004) Font, A. S., McCarthy, I. G., Johnstone, D., & Ballantyne, D. R. 2004, ApJ, 607, 890
* Garufi et al. (2020) Garufi, A., Podio, L., Codella, C., et al. 2020, A&A, 636, A65
* Gaudel et al. (2020) Gaudel, M., Maury, A. J., Belloche, A., et al. 2020, A&A, 637, A92
* Guilloteau et al. (2011) Guilloteau, S., Dutrey, A., Piétu, V., & Boehler, Y. 2011, A&A, 529, A105
* Hartmann & Kenyon (1996) Hartmann, L. & Kenyon, S. J. 1996, ARA&A, 34, 207
* Hennebelle et al. (2017) Hennebelle, P., Lesur, G., & Fromang, S. 2017, A&A, 599, A86
* Hirota et al. (2017) Hirota, T., Machida, M. N., Matsushita, Y., et al. 2017, Nature Astronomy, 1, 0146
* Jacquemin-Ide et al. (2019) Jacquemin-Ide, J., Ferreira, J., & Lesur, G. 2019, MNRAS, 490, 3112
* Launhardt et al. (2009) Launhardt, R., Pavlyuchenkov, Y., Gueth, F., et al. 2009, A&A, 494, 147
* Lee et al. (2018) Lee, C.-F., Li, Z.-Y., Hirano, N., et al. 2018, ApJ, 863, 94
* Lee et al. (2000) Lee, C.-F., Mundy, L. G., Reipurth, B., Ostriker, E. C., & Stone, J. M. 2000, ApJ, 542, 925
* Lee et al. (2001) Lee, C.-F., Stone, J. M., Ostriker, E. C., & Mundy, L. G. 2001, ApJ, 557, 429
* Lee et al. (2021) Lee, C.-F., Tabone, B., Cabrit, S., et al. 2021, ApJ, 907, L41
* Liang et al. (2020) Liang, L., Johnstone, D., Cabrit, S., & Kristensen, L. E. 2020, ApJ, 900, 15
* López-Vázquez et al. (2019) López-Vázquez, J. A., Cantó, J., & Lizano, S. 2019, ApJ, 879, 42
* Louvet et al. (2018) Louvet, F., Dougados, C., Cabrit, S., et al. 2018, A&A, 618, A120
* Masciadri & Raga (2002) Masciadri, E. & Raga, A. C. 2002, ApJ, 568, 733
* Mitchell et al. (1997) Mitchell, G. F., Sargent, A. I., & Mannings, V. 1997, ApJ, 483, L127
* Mundt et al. (1987) Mundt, R., Brugel, E. W., & Buehrke, T. 1987, ApJ, 319, 275
* Mundt & Fried (1983) Mundt, R. & Fried, J. W. 1983, ApJ, 274, L83
* Mundt et al. (1991) Mundt, R., Ray, T. P., & Raga, A. C. 1991, A&A, 252, 740
* Nixon et al. (2013) Nixon, C., King, A., & Price, D. 2013, MNRAS, 434, 1946
* Nony et al. (2020) Nony, T., Motte, F., Louvet, F., et al. 2020, A&A, 636, A38
* Ohashi et al. (1997) Ohashi, N., Hayashi, M., Ho, P. T. P., et al. 1997, ApJ, 488, 317
* Owen et al. (2011) Owen, J. E., Ercolano, B., & Clarke, C. J. 2011, MNRAS, 412, 13
* Panoglou et al. (2012) Panoglou, D., Cabrit, S., Pineau Des Forêts, G., et al. 2012, A&A, 538, A2
* Pety et al. (2006) Pety, J., Gueth, F., Guilloteau, S., & Dutrey, A. 2006, A&A, 458, 841
* Picogna et al. (2019) Picogna, G., Ercolano, B., Owen, J. E., & Weber, M. L. 2019, MNRAS, 487, 691
* Plunkett et al. (2015) Plunkett, A. L., Arce, H. G., Mardones, D., et al. 2015, Nature, 527, 70
* Podio et al. (2011) Podio, L., Eislöffel, J., Melnikov, S., Hodapp, K. W., & Bacciotti, F. 2011, A&A, 527, A13
* Pudritz et al. (2007) Pudritz, R. E., Ouyed, R., Fendt, C., & Brandenburg, A. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 277
* Riols et al. (2020) Riols, A., Lesur, G., & Menard, F. 2020, A&A, 639, A95
* Rodríguez et al. (2012) Rodríguez, L. F., Dzib, S. A., Loinard, L., et al. 2012, Rev. Mexicana Astron. Astrofis., 48, 243
* Shang et al. (2006) Shang, H., Allen, A., Li, Z.-Y., et al. 2006, ApJ, 649, 845
* Shang et al. (2020) Shang, H., Krasnopolsky, R., Liu, C.-F., & Wang, L.-Y. 2020, ApJ, 905, 116
* Shang et al. (1998) Shang, H., Shu, F. H., & Glassgold, A. E. 1998, ApJ, 493, L91
* Shu et al. (1991) Shu, F. H., Ruden, S. P., Lada, C. J., & Lizano, S. 1991, ApJ, 370, L31
* Tabone et al. (2017) Tabone, B., Cabrit, S., Bianchi, E., et al. 2017, A&A, 607, L6
* Tabone et al. (2020) Tabone, B., Cabrit, S., Pineau des Forêts, G., et al. 2020, A&A, 640, A82
* Tabone et al. (2018) Tabone, B., Raga, A., Cabrit, S., & Pineau des Forêts, G. 2018, A&A, 614, A119
* Terquem et al. (1998) Terquem, C., Papaloizou, J. C. B., Nelson, R. P., & Lin, D. N. C. 1998, ApJ, 502, 788
* Ulrich (1976) Ulrich, R. K. 1976, ApJ, 210, 377
* Wang et al. (2019) Wang, L., Bai, X.-N., & Goodman, J. 2019, ApJ, 874, 90
* Wang & Goodman (2017) Wang, L. & Goodman, J. 2017, ApJ, 847, 11
* Wang et al. (2015) Wang, L.-Y., Shang, H., Krasnopolsky, R., & Chiang, T.-Y. 2015, ApJ, 815, 39
* Wilgenbus et al. (2000) Wilgenbus, D., Cabrit, S., Pineau des Forêts, G., & Flower, D. R. 2000, A&A, 356, 1010
* Zapata et al. (2015) Zapata, L. A., Lizano, S., Rodríguez, L. F., et al. 2015, ApJ, 798, 131
* Zhang et al. (2019) Zhang, Y., Arce, H. G., Mardones, D., et al. 2019, ApJ, 883, 1
* Zhang et al. (2018) Zhang, Y., Higuchi, A. E., Sakai, N., et al. 2018, ApJ, 864, 76
* Zhu (2019) Zhu, Z. 2019, MNRAS, 483, 4221
## Appendix A Determination of the redshifted outflow PA
We show in Sect. 3 that the velocity difference between two radially symmetric
positions, at a given altitude z in the flow, is directly linked to the
rotation velocity of the outflow. However, this derivation is highly sensitive
to the determination of the outflow radial center position. Because of the
steep gradient in projected velocity versus radius observed in transverse PV
diagrams, a small variation in radial position would cause an important
variation in the velocity difference. Therefore, the determination of the
position axis is a key concern in the characterization of the specific angular
momentum of the outflow.
To accurately derive the flow center position as a function of the projected
altitude, we adopted the following method, illustrated in Fig. 21. At each
projected altitude above the disk $\delta z$, we integrated the emissivity
from $(V-V_{\rm sys})=6.64$ km s-1 to $(V-V_{\rm sys})=9.77$ km s-1. This gave
the radial emissivity profile of the high-velocity emission tracing the
limiting inner cone. Indeed as shown in DV20, at large projected velocities,
the emission traces an inner cone with almost constant radius. We derived the
radial positions of the two peaks tracing the edges of the high-velocity
component (See Fig. 21) using Gaussian fitting. From the median value of these
two positions, we derived the radial center position and its associated
uncertainty. We applied this method after rotating the data cube with three
different values of outflow PA (sampled around the disk PA), and the
transverse PV diagram was obtained with a slice perpendicular to this axis.
The effect of the different PV cut is completely negligible, as the considered
variation of outflow P.A is small (less than 2 degrees).
Figure 21 shows the derived radial offset as a function of the projected
height for three values of the flow position angle (PA). With a PA of
$296^{\circ}$ and $294^{\circ}$, the radial offset increases consistently with
a miss-estimation of the outflow PA of $\pm 1^{\circ}$. The global offset is
minimized with a PA of $295\pm 1^{\circ}$. This is consistent with the atomic
redshifted jet PA of 296∘ derived by Mundt et al. (1987) and the PA of the
projected disk in the plane of the sky, determined at $25.7\pm 0.3^{\circ}$ by
DV20 and $24\pm 1^{\circ}$ by Guilloteau et al. (2011).
The outflow rotation could potentially introduce a bias in this method.
Indeed, at a given projected velocity, the two sides of a rotating ring fall
at different projected radial offsets from the axis, inducing an artificial
shift of the position centroid of the flow axis. However, since a rotating
ring produces a tilted ellipse in the transverse PV diagram, centered at
$V_{\rm Z}\times\cos(i)$, the radial shift has an opposite sense at velocities
above and below $V_{\rm Z}\times\cos(i)$. By integrating emission over a broad
range of velocity, we thus averaged out this effect. Moreover, if the
variation of radial offset was due to flow rotation, the measured radial
offsets should decrease with distance from the source, as the rotation
velocity decreases (due to the conservation of angular momentum along conical
streamlines). This is not consistent with our results in Fig. 21 where the
radial offsets stay constant (at our nominal PA) or increase linearly (at non-
nominal PAs) with height. Fig. 21 also shows no clear signature of wiggling in
the high-velocity component of DG Tau B. This is consistent with the absence
of wiggling in the atomic jet observed by Mundt et al. (1991). From the
maximum error bars observed, we derived an upper limit for the wiggling angle
of $\theta\leq 0.5^{\circ}$. We consider in the following that the redshifted
outflow PA is constant for all layers (velocities) at $\rm PA=295^{\circ}$,
corresponding to the PA derived here for the high-velocity emission.
Figure 21: Determination of the outflow axis PA. Top panels: Determination of
the high-velocity component radial center as a function of projected heights
above the disk for different outflow axis PA orientations. The black dotted
lines show an angle of $\pm 1^{\circ}$ with respect to the central axis.
Middle panel: 12CO PV diagram perpendicular to the outflow axis. $(V-V_{\rm
sys})$ unit is km s-1. Bottom panel: The black dashed profile shows the
emissivity integrated from the PV diagram between $(V-V_{\rm sys})=6.64$ km
s-1 to $(V-V_{\rm sys})=9.77$ km s-1, this domain is indicated by the red
dashed lines in the middle panel. The red curves show Gaussian fits used to
derive the radial positions of the two edge peaks. The vertical black dotted
lines in the bottom and middle panels indicate the positions of these peaks.
The average of these two radial positions give the radial center at this
height.
## Appendix B Characterization of the cusps
Figure 22: Characterization of the cusps. Left panels: Individual 12CO channel
maps at different line-of-sight velocities illustrating the spatial evolution
of the cusps. Six cusps, labeled U0-U5, are clearly identified (U4 is only
visible on three channel maps). We identify the projected heights of the
bottom of the cusps (red circles) and their evolution in the channel maps
(white dashed lines). Right panel: Longitudinal PV diagram at $\delta
x=0^{\prime\prime}$ averaged over a slice of $\Delta Z=0.2^{\prime\prime}$.
The velocity and height locations of the cusps derived in the channel maps are
shown as red dots. The four orange crosses indicate the maximum on-axis
projected velocity (back side) of the top of the fit four bright elliptical
structures identified in transverse PV diagrams at the corresponding heights
(see Fig. 23).
## Appendix C Ellipse fits in the transverse PV diagrams
In this section, we discuss an alternative method to derive the radial
component of the velocity, not constrained by our tomographic method. At a few
positions along the flow, transverse PV diagrams clearly show elliptical
structures nested inside the main emission (see Fig. 23). We assumed that each
of these ellipses traces one emitting layer of the outflow. If the outflow
velocity does not vary drastically with height, one layer of the outflow of
fixed radius will be projected as an inclined elliptical structure in the PV
diagram (Louvet et al. 2018). The velocity width of the ellipse at $x=0$ is
directly linked to the radial velocity component of the layer. We fit ellipses
by the naked eye to the structures observed. From these fits, we recovered the
radial velocity $V_{\rm R}$ as well as $V_{\rm Z}$ at a few specific positions
along the outflow.
Figure 23 shows the $V_{\rm Z}$ profile of the outflow reconstructed with the
tomographic technique and extrapolated until Z $\approx$ 2700 au. The red
arrows trace the poloidal vectors derived from individual elliptical fits. The
angle of the arrow is determined by the ratio between $V_{\rm Z}$ and $V_{\rm
R}$. The length and colors of the arrows correspond to $V_{\rm Z}$. The
$V_{\rm Z}$ values derived from the elliptical fits appear consistent with the
estimates from the tomographic study. The poloidal velocity direction is also
consistent with the conical lines of constant $V_{\rm Z}$ in the tomography.
This suggests that the flow is indeed aligned with these conical lines.
Figure 23: Constraints on the radial velocity component $V_{\rm R}$. Left
panels: Transverse PV diagrams averaged over a slice of width $\Delta
Z=0.2^{\prime\prime}$ at selected positions $\delta z$ along the flow. In red
are shown the ellipse fits. We show side by side the PV diagram with and
without the fit for more visibility. Right panel: $V_{\rm Z}$ tomography of
the outflow in the outflow referential, extrapolated beyond $Z>1200$ au. The
red arrows represent the velocity directions determined from the ellipse fits.
The arrow colors trace the $V_{\rm Z}$ values determined from the ellipse fit.
$R$ and $Z$ units are in au.
We studied the relation between the ellipses visible in the transverse PV
diagrams and the cusps visible in the channel maps. The on-axis maximal
velocity of the fit ellipses was compared to the cusp location in the on-axis
longitudinal PV diagram (see Fig. 22). The (R,Z) positions of the fit ellipses
were also compared to the cusp locations in the tomographic $V_{Z}$ map (see
Fig. 13). The ellipse located at $\delta z=1.1^{\prime\prime}$ comes from a
lower altitude region, not included in our cusp analysis because of crowding.
The ellipses at $\delta z=2.1^{\prime\prime}$ and $2.9^{\prime\prime}$ seem to
extend the U5 cusp, at higher and lower line-of-sight velocities respectively.
The ellipse at $\delta z=7.5^{\prime\prime}$ is not so clearly associated with
a cusp extension in Fig. 22. It could be the high velocity extension of the U3
cusp, or associated with a fainter cusp located between U2 and U3, not
included in our analysis. Moreover, Fig. 24 indicates that the tops of the
ellipses located at $\delta z=2.9^{\prime\prime}$ and $\delta
z=7.5^{\prime\prime}$ are possibly consistent with a cusp structure which was
not included in our study, due to their low S/N. Similarly, Fig. 24 shows that
the cusps U1 and U2 also coincide with internal structures in the transverse
PV diagram, but the stacking of ellipses in the PV diagram makes the
identification more confusing than in the channel maps.
The cusps could be precisely located only at moderate projected velocities
$<5.3$ km s-1, where they are sufficiently bright (see Fig. B.1), hence they
probe outer streamlines; in contrast elliptical structures in transverse PV
diagrams are best distinguished at high projected velocities $>5.3$ km
s-1where they have less overlap with each other (see Fig. C.1), hence most of
them probe inner faster flow streamlines. However, we see clear
correspondences between faint cusps in channel maps and the on-axis high-
velocity portion of some ellipses in transverse PV cuts, and vice-versa, which
demonstrates that they trace different portions of the same underlying
substructures, extending across the whole conical outflow. Therefore, assuming
a radial flow to deproject the cusp apparent positions seems fully justified.
Figure 24: Illustration of the correspondences between ellipses and cusps.
Left panels: transverse PV diagrams at $\delta z=2.9^{\prime\prime}$ and
$7.5^{\prime\prime}$ (from Fig 23), with red dots indicating the on-axis
projected velocities of cusps U1 and U2 at $\delta z=7.5^{\prime\prime}$. Each
of these two cusps corresponds in position and velocity to the maximum
velocity (back side) of an ellipse in the PV cut. Right panels: channel maps
at the same line-of-sight velocity as the orange crosses in the left panels,
showing that elliptical structures seem to correspond to fainter cusps in
position-position space, but falling outside of the velocity domain where cusp
characterization was possible.
## Appendix D Biases in the tomographic reconstruction
We studied possible biases in the method presented in Sect. 3 to reconstruct
poloidal maps of $V_{\rm Z}$ and $j=R\times V_{\phi}$. We first studied the
bias introduced by projection effects considering one single layer of the
outflow using the analytical solutions from Eqs. 1,2,3. To study the impact of
beam convolution and multiple layers of the outflow, we also applied our
tomographic reconstruction method to the synthetic data cube presented in
Sect. 10 and estimate the difference between the reconstructed $V_{\rm Z}$ and
$j$ and the initial values of the model.
### D.1 Single shell
Figure 25: Biases in the tomographic reconstruction for one single conical
shell. Top panels: Black curves show computed transverse PV diagrams for one
conical layer with constant $V_{\rm Z}$, $V_{\rm R}$, and $j$ seen at
different inclinations $i$ and with $\theta_{V}=\theta=17^{\circ}$. The red
dashed curve traces an elliptical fit. Bottom Panel: Computed relative bias in
$V_{\rm Z}$ (green curves) and $j$ (orange curves) for one conical layer as a
function of the inclination and for different combinations of $\theta$ and
$\theta_{v}$ : solid curves: $\theta=\theta_{v}=17^{\circ}$, dotted curves:
$\theta=17^{\circ}$ & $\theta_{v}=30^{\circ}$, and dashed curves:
$\theta=\theta_{v}=30^{\circ}$.
In our tomographic study, we assumed that one shell will be projected as an
ellipse in transverse PV diagrams. We also assumed that the projected
velocities at the extrema radii allowed us to recover $V_{\rm Z}$ and
$V_{\phi}$. We determine in this section biases introduced by these two
assumptions.
We assumed a conical shell of radius R(Z) in the outflow referential and with
local opening angle $\theta$, such as $R(Z)=R_{0}+Z\tan{\theta}$. For the DG
Tau B inner conical outflow, the opening angle $\theta$ of the layers vary
between 17∘ and 12∘, increasing with decreasing velocities. The estimated
opening angle of the lower velocity emission contributing to the pedestal is
$\theta\simeq 30^{\circ}$. We defined the parameter $\theta_{v}$ corresponding
to the angle of the velocity vector with the Z axis in the poloidal plane
($V_{\rm R}=V_{\rm Z}\tan{\theta_{V}}$). For a flow parallel to the conical
surface, $\theta=\theta_{V}$. We also assumed a constant specific angular
momentum ($V_{\phi}=j/R(\phi)$) as well as constant $V_{\rm Z}$ and $V_{\rm
R}$ over the transverse slit width. Solving Eq. 2 with constant $\delta z$
gave:
$R(\phi)=\frac{\delta
z\tan{\theta}+R_{0}\sin{i}}{\sin{i}-\sin{\phi}\cos{i}\tan{\theta}}.$ (21)
In the case $i=\theta$ and $i<\theta$, the cut of the conical outflow will be
respectively a parabola and an hyperbola. Consequently, $R(\phi)$ will tend
toward infinite. This is not the case for the DG Tau B conical outflow. We
then implemented this solution into Eqs. 3,1 for multiple values of the
inclination. The resulting transverse PV diagrams are represented in Fig. 25
for $\theta_{V}=\theta=17^{\circ}$. The PV diagrams were generated using Eqs.
3, 2 with the solution from Eq. 21. The difference with an ellipse increases
with the inclination but is expected to be small for the DG Tau B case
($i=117^{\circ}$).
We assume in Sect. 3 that the radial edges of the ellipse correspond to
$\phi=0-\pi$. However, this is an approximation. The radial extrema of the
ellipse correspond to solutions of the equation: $\frac{\partial\delta
x}{\partial\phi}=0$. For a conical layer, the solution is obtained for:
$\sin{\phi}=\frac{\tan{\theta}}{\tan{i}}.$ (22)
This difference is small in our situation. However, this leads to bias in the
estimation of $R$, $Z$, $V_{\rm Z}$ and $V_{\phi}$. We determined the
estimated value of $V_{\rm est,Z}$ and $V_{\rm est,\phi}$ using Eqs. 6,7. We
then computed the relative differences with the real values $R_{\rm real}$,
$Z_{\rm real}$, $V_{\rm real,Z}$, and $V_{\rm real,\phi}$:
$\frac{R_{\rm est}-R_{\rm real}}{R_{\rm
real}}=\sqrt{1-\frac{\tan^{2}{\theta}}{\tan^{2}{i}}}-1$ (23) $\frac{Z_{\rm
est}-Z_{\rm real}}{Z_{\rm real}}=-\frac{\tan^{2}\theta}{\tan^{2}i}$ (24)
$\frac{V_{\rm est,Z}-V_{\rm real,Z}}{V_{\rm
real,Z}}=\tan{\theta}\times\tan{\theta_{V}}$ (25) $\frac{V_{\rm
est,\phi}-V_{\rm real,\phi}}{V_{\rm
real,\phi}}=\sqrt{1-\frac{\tan^{2}{\theta}}{\tan^{2}{i}}}-1.$ (26)
Hence, $R$, $Z$, and $j$ are systematically underestimated, while $V_{\rm Z}$
is overestimated. Figure 25 shows the relative biases due to inclination and
projection effects in the estimation of $V_{\rm Z}$ and $V_{\phi}$ for
different values of $\theta$ and $\theta_{v}$. In the conical outflow, where
$\theta_{V}=\theta\leq 17^{\circ}$, the inclination bias is expected to be
$\leq 10\%$ in $V_{\rm Z}$ and $Z$, $\leq$ 1 % in $j$ and $R$.
### D.2 Multiple layers
Figure 26: Biases in the tomographic reconstruction for a stacking of conical
shells. Left panels: The gray area shows the limits of the transverse PV
diagram predicted for a stacking of conical shells with a shearing factor
$f_{\rm sh}$ (see text). In black is shown the ellipse corresponding to a
single layer of maximal radius $R$. The blue and red dots correspond
respectively to the velocity of the ellipse at $\pm R$ and the maximal
velocity of the PV diagram at $\pm R$. The red dots are the ones used in our
method, while the blue dots correspond to the true measurements. The
differences in velocities will lead to a bias in our $j$ and $V_{\rm Z}$
estimation. The red line shows the slope of the PV diagram $\frac{\partial
V_{\rm los}(\pm R)}{\partial R}$, see text for more details. Right panel: Plot
of the relative bias in $V_{\rm Z}$ (green curves) and $j$ (orange curves) due
to the shearing aspect of the PV diagram as a function of the $x$ parameter
(see text). The solid, dashed, and dotted lines correspond to different
configurations of $\theta$ and $\theta_{v}$. The black histogram, labeled
C.O., shows the distribution of the $x$ parameter derived at different heights
Z and radii R in the conical outflow. The relative biases corresponding to the
a and b ellipses shown in the left panels are represented. The orange (resp.
green) colored contours trace the relative biases derived from applying the
tomographic method to the synthetic data cube. Contours outline 30 to 90 % of
the distributions in step of 15 %.
In the previous section, we estimated the bias in the estimation of $V_{\rm
Z}$, $V_{\rm\phi}$ due to projection effects for one single conical layer.
However, the DG Tau B transverse PV diagrams shows a clear shear-like velocity
structure suggesting a stacking of layers. We modeled this effect directly in
the transverse PV diagrams by stacking the elliptical projections for conical
layers of increasing radii at origin $R_{0}$ with the same opening angle
$\theta$. The velocity shear in $V_{\rm Z}$ of the conical layers was defined
by a shearing parameter $f_{\rm sh}=\frac{\partial V_{\rm Z}}{\partial R}$.
The specific angular momentum of each layer, $j$, was assumed constant with
$Z$ and vary between 20 and 90 au km s-1, increasing with increasing radius,
such as $j\times V_{\rm P}$ was kept constant to mimic the DG Tau B
observations.
Due to the stacking, the maximal velocity at $\pm R$ does not perfectly
describe the velocity of the ellipse of radius R. This effect is larger when
$f_{\rm sh}$ is small. Using a similar procedure as before, we derived the
relative difference between the estimation of $V_{Z}$ and $j$ with the
tomographic method and the input theoretical values. This was achieved for a
range of $f_{\rm sh}$ values between 0.2 to 20 km s-1 au-1 and with the three
($\theta$, $\theta_{V}$) configurations studied in the previous subsection.
In order to study efficiently this bias, we defined the a-dimensional
parameter $x=\frac{[V_{\rm los}(\pm R)]}{R}$ $\times$ ($\frac{\partial[V_{\rm
los}(\pm R)]}{\partial R}$ )-1, where we defined $[V_{\rm los}(\pm
R)]=1/2(V_{\rm los}(R)+V_{\rm los}(-R))$. This a-dimensional parameter can be
derived from the observations. We show in Fig. 26 the predicted relative
biases in $V_{\rm Z}$ and $j$ as a function of this parameter $x$. Biases
increase with increasing $x$ values illustrating the effect of the velocity
shear. We also show the distribution of observed x values computed at each
(R,Z) position in the conical outflow. The x values are concentrated around
$\simeq 0.5$, suggesting moderate biases in both $V_{\rm Z}$ and $j$ for the
conical outflow where $\theta_{V}=\theta\leq 17^{\circ}$ .
However, this modeling did not include the effect of beam smearing nor the
impact of our polynomial fitting method to describe the shape of the PV
diagram. In order to study these effects, we directly applied our tomographic
method to the synthetic data cube presented in Sect. 10 and determined the
relative differences between the computed and the input $V_{\rm Z}$ and $j$
values at each (R,Z) position along the conical flow. We show in Fig. 26 the
derived relative biases in $V_{\rm Z}$ and $j$ as a function of the $x$
parameter. Their distributions are broader and flatter than predicted,
especially in $j$, likely due to the combination of velocity shear and beam
smearing effects.
From this study, we estimate that in the conical part of the outflow (at
$(V-V_{\rm sys})\geq 2$ km s-1), the tomographic method suffers from a
relative bias $\leq 15$% in the estimation of $V_{\rm Z}$ and $\leq 20$% in
the estimation of $j$. The X values extend up to $\approx 1.3$ in the pedestal
region. With an opening angle $\theta$, $\theta_{v}>30^{\circ}$, a tomographic
study of the low velocity component would suffer from a relative bias larger
than 60% and 30% in the estimation of $V_{\rm Z}$ and $j$ respectively.
## Appendix E Wind-driven shell analytical solutions
In this section, using Eqs. 1, 2,3 and 11, we derive an analytical solution
for predicted channel maps in the case of the generalized WDS model introduced
in Sect. 4. The WDS model is defined by three parameters: $C$, $\tau$ and
$\eta$, see Eqs. 11. The projection on the plane of the sky ($\delta x$,
$\delta z$) for the emissivity map at $(V-V_{\rm sys})=V_{\rm CM}$ could be
recovered by solving Eq. 3 with $V_{\rm los}=V_{\rm CM}$ and using Eqs. 11 for
$V_{\rm Z}$ and $V_{\rm R}$:
$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle\frac{4V_{\rm CM}\tau
C\cos{i}}{\eta^{2}\sin^{2}{i}}$ (27) $\displaystyle R$ $\displaystyle=$
$\displaystyle-\eta\frac{\tan{i}}{2C}(\sin{\phi}\pm\sqrt{\sin^{2}{\phi}-\zeta})$
(28) $\displaystyle\delta x$ $\displaystyle=$ $\displaystyle R\cos{\phi}$ (29)
$\displaystyle\delta z$ $\displaystyle=$ $\displaystyle
CR^{2}\sin{i}-R\sin{\phi}\cos{i}.$ (30)
Equation 28 has no solution in the case $\zeta>1$ ($V_{\rm
CM}\cos{i}>\frac{\eta^{2}\sin^{2}{i}}{4\tau C}$). This corresponds to the case
where no emission is predicted at $V_{\rm los}=V_{\rm CM}$. Similarly, in the
situation $0\geq\zeta\geq 1$, only a fraction of $\phi$ will be projected such
as $\sin^{2}{\phi}>\zeta$. Developing Eq. 30, we obtained the following:
$\displaystyle\delta z$ $\displaystyle=$
$\displaystyle\eta\frac{\sin^{2}{\phi}\sin{i}}{2C}(\eta\tan^{2}{i}+1)-V_{CM}\tau\tan{i}$
(32)
$\displaystyle\pm\eta\frac{\sin{\phi}\sin{i}}{2C}(\eta\tan^{2}{i}+1)\sqrt{\sin^{2}{\phi}-\zeta}.$
We reformulated $\delta z$ as $z_{0}+\Delta Z\sin{\beta}$ with:
$\displaystyle z_{0}$ $\displaystyle=$
$\displaystyle\eta\frac{\sin{i}}{2C}(\eta\tan^{2}{i}+1)-V_{CM}\tau\tan{i}$
(33) $\displaystyle\Delta Z$ $\displaystyle=$
$\displaystyle\eta\frac{\sin{i}}{2C}(\eta\tan^{2}{i}+1)\sqrt{1-\zeta}$ (34)
$\displaystyle\sin{\beta}$ $\displaystyle=$
$\displaystyle\frac{-1+\sin^{2}{\phi}\pm\sin{\phi}\sqrt{\sin^{2}{\phi}-\zeta}}{\sqrt{1-\zeta}}.$
(35)
Similarly, manipulating Eq. 29, $\delta x$ could be reformulated as $\delta
x=\Delta X\cos{\beta}$, with the following:
$\Delta X=\mp\eta\frac{\tan{i}}{2C}\sqrt{1-\zeta}.$ (36)
Hence ($\delta x$, $\delta z$) trace an ellipse of center (0, $z_{0}$) and
aspect ratio:
$\Bigl{|}\frac{\Delta Z}{\Delta X}\Bigr{|}=(\eta\tan^{2}{i}+1)|\cos{i}|.$ (37)
Therefore, in the classical WDS models with radial velocity vectors
($\eta=1$), the aspect ratio of the ellipse on the channel maps only depends
on the inclination. We represent on the channel maps in Figs. 5, 7, and 8 the
two limiting ellipses computed at $V-\delta V/2$ and $V+\delta V/2$ to take
into account the width of the channel map.
## Appendix F Global model
Figure 27: Schematic representation of the method used to create synthetic
data cube.
In order to comprehend the impact of projection or convolution effects on the
observations, we developed a locally axisymmetric code that allows us to
simulate optically thin observations of simple outflow models. We took
advantage of the axisymmetric hypothesis that permits to reduce the complexity
of the model by defining at each height the radius $R(Z)$ and the cylindrical
velocities $V_{\rm z}$, $V_{\rm r}$ and $V_{\phi}$ (See the schematic view of
Fig. 2). The dependency between the height and the radius or the velocities
vary with the model used. We then created at each height an emitting ring with
azimuth parametrized with $\phi$. As the morphology is axisymmetric, neither
the radius nor velocities depend on $\phi$. Under the assumption of optically
thin emission, the emissivity at position ($Z$,$\phi$) is proportional to the
elementary volume $dV=R(Z)d\phi dRdZ$. We added an additional variation of the
emissivity with the height and radius as a power-law with parameters $\alpha$
and $\beta$ respectively. Proper modeling of the emissivity would require the
temperature and chemistry to be solved, which is well beyond the scope of this
model. The positions of the outflow emission on the data cube ($\delta
x$,$\delta y$, $V_{\rm proj}$) were then defined by Eqs. 1,2,3.
We then created a data cube with the same spectral and spatial resolution than
our observations, and placed on each point ($\delta x$,$\delta y$, $V_{\rm
proj}$) the emissivity I. Under the assumption of optically thin emission, we
summed each emissivity corresponding to the same positions on the data cube.
We set a step size of 1∘ for $\phi$ and a fraction of the spatial pixel for Z.
We then convolved the data cube by a 2D Gaussian matching the spatial beam
characteristics in order to fully simulate the ALMA observations. The code,
written in Python 3, is publicly online777https://github.com/Alois-
deValon/Axoproj.
## Appendix G Channel maps of disk-wind models
Figure 28: 12CO observed (first column) and synthetic disk-wind channel maps
at selected line-of-sight velocities. The second and third columns correspond
to the two disk-wind models with axis precession presented in Sect. 5.2. The
last column corresponds to a disk-wind model with axisymmetric density
enhancements, as presented in Sect. 5.3. $\delta x$ and $\delta z$ units are
arcseconds.
|
ON A FAMILY OF $2$-AUTOMATIC SEQUENCES
DERIVED FROM ULTIMATELY PERIODIC SEQUENCES AND
GENERATING ALGEBRAIC CONTINUED FRACTIONS IN $\mathbb{F}_{2}((1/t))$
(Suites $2$-automatiques à colone vertébrale ultimement périodique)
by A. Lasjaunias
Warning:This note is not intended to be officially published. The matter
exposed here grew from numerous exchanges during the past two months with
Yining Hu (at a very and too large distance !). My aim is to report in a first
draft, and in a very private and personal way, on a curious mathematical
structure.
We consider a large family $\Large{\mathcal{F}}$ of infinite sequences over a
finite alphabet $\mathcal{A}=\left\\{a_{1},a_{2},...,a_{k}\right\\}$. This
family includes a celebrated example of a 2-automatic sequence on the set
$\left\\{a,b\right\\}$, called Period-doubling sequence, which has been
studied in a previous article [1].
Each sequence ${\bf{s}}$ in $\large{\mathcal{F}}$ is built in the following
way, from another sequence
${\boldsymbol{\varepsilon}}=(\varepsilon_{i})_{i\geq 0}$ over
$\left\\{a_{1},a_{2},...,a_{k}\right\\}$, this last one being ultimately
periodic.
Starting from the empty word $W_{0}$, we consider the sequence of words
$(W_{n})_{n\geq 0}$ such that for $n\geq 0$ we have
$W_{n+1}=W_{n},\varepsilon_{n},W_{n}$ (note the coma is for concatenation and
it will be omitted when it is suitable). Hence, we have
$W_{1}=\varepsilon_{0}$,
$W_{2}=\varepsilon_{0},\varepsilon_{1},\varepsilon_{0}$ etc…Observe by
construction that $W_{n+1}$ starts by $W_{n}$ and therefore we may consider
$W_{\infty}$ the inductive limit of these words (i.e. the word begining by
$W_{n}$ for all $n\geq 0$). Note that, for all $n\geq 1$, $W_{n}$ is a
palindrome, centered in $\varepsilon_{n}$, of length $2^{n}-1$.
This $W_{\infty}$ represents the sequence ${\bf{s}}$, which we may denote by
$\bf{s}(\boldsymbol{\varepsilon})$. Hence, we have :
$\textbf{s}(\boldsymbol{\varepsilon})=\varepsilon_{0}\varepsilon_{1}\varepsilon_{0}\varepsilon_{2}\varepsilon_{0}\varepsilon_{1}\varepsilon_{0}\varepsilon_{3}\varepsilon_{0}\varepsilon_{1}\varepsilon_{0}\varepsilon_{2}\varepsilon_{0}....$
In this family $\Large{\mathcal{F}}$, each sequence $\textbf{s}=(s_{i})_{i\geq
0}$ , over $\left\\{a_{1},a_{2},...,a_{k}\right\\}$, generates an infinite
continued fraction $\alpha$ in $\mathbb{F}_{2}((1/t))$, denoted $CF(\bf{s})$,
by replacing the letters $a_{i}$ by non-constant polynomials in
$\mathbb{F}_{2}[t]$ (this choice is arbitrary, hence there is a $CF(\bf{s})$
for each choice but it is considered as unique in the sequel). For basic
information on continued fractions, particularly in power series fields, the
reader is refered to [2]. Hence we will write :
$\alpha=CF({\bf{s}})=[s_{0},s_{1},....,s_{n},....]$
where the $s_{i}$ are the partial quotients in $\mathbb{F}_{2}[t]$. We recall
that the sequence of convergents to $\alpha$ is denoted $(x_{n}/y_{n})_{n\geq
1}$. For $n\geq 1$, we have
$x_{n}/y_{n}=[s_{0},...,s_{n-1}]=s_{0}+1/(s_{1}+1/....)$ , hence
$x_{1}=s_{0}$, $y_{1}=1$ and $x_{2}=s_{0}s_{1}+1$, $y_{2}=s_{1}$ etc…
To be more precise about periodic sequences, we introduce the following
definition.
Definition. _Let $l\geq 0$ and $d\geq 1$ be two integers. An ultimately
periodic sequence $\boldsymbol{\varepsilon}$ is called of type $(l,d)$ if :
1) $l=0$ and $\boldsymbol{\varepsilon}$ is purely periodic, with period of
length $d$, this being denoted by
$\boldsymbol{\varepsilon}=(\varepsilon_{0},\varepsilon_{1},...,\varepsilon_{d-1})^{\infty}$.
2) $l>0$ and $\boldsymbol{\varepsilon}$ is ultimately periodic, with a prefix
of length $l$ and a period of length $d$, this being denoted by
$\boldsymbol{\varepsilon}=\varepsilon_{0},\varepsilon_{1},...,\varepsilon_{l-1},(\varepsilon_{l},\varepsilon_{l+1},...,\varepsilon_{l+d-1})^{\infty}$.
Here all the $\varepsilon_{i}$’s are in $\mathcal{A}$ (assuming that $k\geq
l+d$ to allow different values to the terms of the sequence
$\boldsymbol{\varepsilon}$)._
Let us illustrate the construction of $\bf{s}(\boldsymbol{\varepsilon})$ in
two basic cases :
1) $\boldsymbol{\varepsilon}=a,b,(c)^{\infty}$ then
$\textbf{s}(\boldsymbol{\varepsilon})=a,b,a,c,a,b,a,c,....=(abac)^{\infty}$
Note that here $\boldsymbol{\varepsilon}$ is ultimately constant (of type
$(3,1)$) and $\bf{s}(\boldsymbol{\varepsilon})$ is periodic. Consequently a
basic property on continued fractions shows that
$\alpha=CF(\bf{s}(\boldsymbol{\varepsilon}))$ is quadratic over
$\mathbb{F}_{2}(t)$.
2) $\boldsymbol{\varepsilon}=(a,b)^{\infty}$ then
$\textbf{s}(\boldsymbol{\varepsilon})=abaaabababaaaba....$
Here $\boldsymbol{\varepsilon}$ is of type $(0,2)$ and
$\bf{s}(\boldsymbol{\varepsilon})$ is the celebrated sequence mentionned above
and called Period-doubling. It has been proved that
$\alpha=CF(\bf{s}(\boldsymbol{\varepsilon}))$ satisfies an algebraic equation
of degree 4 with coefficients in $\mathbb{F}_{2}[t]$ (see [1]).
As it happens in these two simple cases, we are going to prove in the
following theorem that all sequences $\bf{s}(\boldsymbol{\varepsilon})$ in
$\large{\mathcal{F}}$ generate a continued fraction $\alpha$ which is
algebraic over $\mathbb{F}_{2}(t)$. During the proof, the algebraic equation
satisfied by $\alpha$ will appear explicitely.
Let $\boldsymbol{\varepsilon}$ be a sequence of type $(l,d)$ then we denote by
$\mathbb{F}_{2}(\boldsymbol{\varepsilon})$ the subfield of $\mathbb{F}_{2}(t)$
generated by the vector $(\varepsilon_{0},...,\varepsilon_{l+d-1})$ whose
$l+d$ coordinates belong to $\mathbb{F}_{2}[t]$.
Theorem. _Let $l\geq 0$ and $d\geq 1$ be integers. Let
$\boldsymbol{\varepsilon}$ be an ultimately periodic sequence of type $(l,d)$.
Let $\bf{s}(\boldsymbol{\varepsilon})$ the sequence in $\mathbb{F}_{2}[t]$ and
$\alpha=CF(\bf{s}(\boldsymbol{\varepsilon}))$ the continued fraction in
$\mathbb{F}_{2}((1/t))$, both be defined as above. Then there is a polynomial
$P$ in $\mathbb{F}_{2}(\boldsymbol{\varepsilon})[x]$ such that
$\deg_{x}(P)=2^{d}$ and $P(\alpha)=0$. To be more precise, setting
$\beta=1/\alpha$, there are $d+1$ elements in
$\mathbb{F}_{2}(\boldsymbol{\varepsilon})$, $A$ and $B_{k}$ for $0\leq k\leq
d-1$, such that_
$\beta^{2^{d}}=A+\sum_{0\leq k\leq d-1}B_{k}\beta^{2^{k}}.$
Note that the case $d=1$ is trivial as we saw in case 1) above. Indeed, in
that case $\boldsymbol{\varepsilon}$ is ultimately constant. Hence
$\textbf{s}(\boldsymbol{\varepsilon})=W,\varepsilon_{l},W,\varepsilon_{l},...=(W,\varepsilon_{l})^{\infty}$
where $W$ is a finite (or empty) word and therefore $\alpha$ is quadratic. In
the sequel, we may assume that $d\geq 2$.
The proof of the Theorem lies on the existence of a particular subsequence of
convergents to $\alpha$. These particular convergents are linked to the
structure of the word $\textbf{s}(\boldsymbol{\varepsilon})$. Indeed, it is
natural to consider the truncation of the continued fraction $\alpha$
containing the partial quotients from $s_{0}$ up to $s_{2^{n}-2}$ for $n\geq
1$, thus corresponding to the finite word $W_{n}$, of length $2^{n}-1$,
mentionned above. Hence, for $n\geq 1$, we set
$u_{n}/v_{n}=[s_{0},...,s_{2^{n}-2}]$. We have
$(u_{1},v_{1})=(s_{0},1)\quad\text{and}\quad(u_{2},v_{2})=(s_{0}s_{1}s_{2}+s_{0}+s_{2},s_{1}s_{2}+1).$
The first step of the proof was introduced in our previous work concerning the
particular case of the Period-doubling sequence. In [1,p. 4 Lemma 3.2.], using
basic properties on continuants, we could prove that the pair $(u_{n},v_{n})$
satisfies a simple recurrence relation.
Indeed, for $n\geq 1$, we have
$(R)\qquad u_{n+1}=\varepsilon_{n}u_{n}^{2}\quad\text{and}\quad
v_{n+1}=\varepsilon_{n}u_{n}v_{n}+1,$
with $(u_{1},v_{1})=(\varepsilon_{0},1)$. From $(R)$ we get immediately
$v_{n+1}/u_{n+1}=v_{n}/u_{n}+1/u_{n+1}$
and therefore we obtain
$v_{n}/u_{n}=\sum_{1\leq i\leq n}1/u_{i}\quad\text{ for}\quad n\geq 1.$
Let us consider $\beta=1/\alpha$. Then $\beta=lim_{n}(v_{n}/u_{n})$ and
consequently we have
$\beta=\sum_{n\geq 1}1/u_{n}.\qquad eq(0)$
From $eq(0)$, we will show that $\beta$ is algebraic in the following way. By
successive elevation to the power 2, for $0\leq i\leq d$, we can define
inductively $d+1$ sequences, $(\varepsilon(n,i))_{n\geq 0}$, as follows:
$\varepsilon(n,0)=1\quad\text{and}\quad\varepsilon(n,i+1)=\varepsilon(n,i)^{2}\varepsilon_{n+i}\quad\text{for}\quad
0\leq i\leq d-1.$
Note that, for $n\geq 0$, we have $\varepsilon(n,1)=\varepsilon_{n}$. Then we
observe that, by elevating $eq(0)$ to the power 2, we get
$\beta^{2}=\sum_{n\geq 1}1/u_{n}^{2}=\sum_{n\geq
1}\varepsilon_{n}/u_{n+1}=\sum_{n\geq 1}\varepsilon(n,1)/u_{n+1}.\qquad eq(1)$
Moreover, by successive elevation to the power 2, starting from $eq(0)$ and
introducing the sequences $(\varepsilon(n,i))_{n\geq 0}$, we also get, for
$0\leq i\leq d$,
$\beta^{2^{i}}=\sum_{n\geq 1}\varepsilon(n,i)/u_{n+i}.\qquad eq(i)$
Remark. _For all $1\leq i\leq d$ the sequence $(\varepsilon(n,i))_{n\geq 0}$
is ultimately periodic of type $(l,d)$.
Proof by induction. This is true for $i=1$. If $(\varepsilon(n,i))_{n\geq 0}$
is a periodic sequence of type $(l,d)$, then we have, for $n\geq l$,
$\varepsilon(n+d,i)=\varepsilon(n,i)$ and consequently
$\varepsilon(n+d,i+1)=\varepsilon(n+d,i)^{2}\varepsilon_{n+d+i}=\varepsilon(n,i)^{2}\varepsilon_{n+i}=\varepsilon(n,i+1)$._
Now we introduce a partition of the set of positive integers into $d+1$
subsets: first the finite set $F=\\{k\quad|\quad 1\leq k\leq l+d-1\\}$ and the
$d$ subsets
$E_{j}=\\{md+l+j\quad|\quad m\geq 1\\}\quad\text{for}\quad 0\leq j\leq d-1.$
$\mathbb{N}^{*}=\bigcup_{j=0}^{d-1}E_{j}\bigcup F.$
Linked to this partition, we introduce $d$ elements, $\beta_{j}$ for $0\leq
j\leq d-1$, in $\mathbb{F}_{2}((1/t))$, defined by
$\beta_{j}=\sum_{n\in E_{j}}1/u_{n}.\qquad(B)$
Combining $eq(0)$ and $(B)$, and defining
$z_{0}\in\mathbb{F}_{2}(\boldsymbol{\varepsilon})$ by $\sum_{k\in F}1/u_{k}$,
we can write,
$\beta=z_{0}+\sum_{0\leq j\leq d-1}\sum_{n\in E_{j}}1/u_{n}=z_{0}+\sum_{0\leq
j\leq d-1}\beta_{j}.\qquad Eq(0)$
Using the above remark, concerning the periodicity of the sequences
$(\varepsilon(n,i))_{n\geq 0}$, for $0\leq j\leq d-1$ and for $1\leq i\leq d$,
we can write ,
$\varepsilon(d+l+j-i,i)\beta_{j}=\sum_{m\geq
1}\varepsilon(md+l+j-i,i)/u_{md+l+j}$
$\varepsilon(d+l+j-i,i)\beta_{j}=\sum_{n+i\in E_{j}}\varepsilon(n,i)/u_{n+i}.$
Consequently, we observe that there exists
$z_{i}\in\mathbb{F}_{2}(\boldsymbol{\varepsilon})$, a finite sum of the first
terms in the series appearing in $eq(i)$, such that, for $1\leq i\leq d$,
$eq(i)$ becomes the following equality
$\beta^{2^{i}}=z_{i}+\sum_{0\leq j\leq
d-1}\varepsilon(d+l+j-i,i)\beta_{j}.\qquad Eq(i)$
(Note that these quantities $z_{i}$ have a different form depending on the
triplet $(l,d,i)$. See the three examples below.)
Now, let us introduce the following square matrix of order $d$ :
$M(d)=(m_{i,j})_{0\leq i,j\leq d-1}\quad\text{where}\quad
m_{i,j}=\varepsilon(d+l+j-i,i).$
Introducing two column vectors $B$ and $C$, we observe that the $d$ equations
$Eq(i)$, for $0\leq i\leq d-1$, can be summed up introducing the following
linear system $(S):\quad M(d).B=C\quad$, where
$B=\begin{bmatrix}\beta_{0}\\\ \beta_{1}\\\ \vdots\\\
\beta_{d-1}\end{bmatrix}\quad\text{and}\quad C=\begin{bmatrix}\beta+z_{0}\\\
\beta^{2}+z_{1}\\\ \vdots\\\ \beta^{2^{d-1}}+z_{d-1}\end{bmatrix}.$
We introduce the determinant, $\Delta(d)$, of the matrix $M(d)$ and also the
determinant $\Delta(j,d)$ obtained from $\Delta(d)$ by replacing the column
vector of rank $j$ by the column vector $C$. Hence, applying Cramer’s rule for
solving the linear system $(S)$, we get
$\beta_{j}=\Delta(j,d)/\Delta(d)\quad\text{for}\quad 0\leq j\leq d-1.$
Finally, reporting these values for $\beta_{j}$ in $Eq(d)$, we obtain
$\beta^{2^{d}}=z_{d}+(\sum_{0\leq j\leq
d-1}\varepsilon(l+j,d)\Delta(j,d))/\Delta(d).\quad Eq(*)$
We observe that $\Delta(d)$ belongs to
$\mathbb{F}_{2}(\boldsymbol{\varepsilon})$. While, $\Delta(j,d)$ belongs to
$\mathbb{F}_{2}(\boldsymbol{\varepsilon})[\beta]$. Indeed, developping the
determinant $\Delta(j,d)$ along the column of rank $j$, we get
$\Delta(j,d)=c_{j}+\sum_{0\leq k\leq d-1}b_{k,j}\beta^{2^{k}}$. Consequently
$Eq(*)$ can be witten as expected ( with coefficients in
$\mathbb{F}_{2}(\boldsymbol{\varepsilon})$) :
$\beta^{2^{d}}=A+\sum_{0\leq k\leq d-1}B_{k}\beta^{2^{k}}.\qquad Eq(**)$
So the proof of the theorem is complete.
We present, here below, three examples. In order to avoid unnecessary
complications with the subscripts, we use $(a,b,c)$ for the letters
$(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2})$.
Example 1: Type (0,2). Period-Doubling sequence . Let us consider the case 2,
mentioned above, where
$\boldsymbol{\varepsilon}=(a,b)^{\infty}\quad\text{and}\quad\beta=1/CF(\textbf{s}(\boldsymbol{\varepsilon})).$
We have $(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2})=(a,b,a)$ and
$(z_{0},z_{1},z_{2})=(1/a,0,1)$.
$\mathrm{\Delta(2)=}\begin{vmatrix}1&1\\\
\varepsilon_{1}&\varepsilon_{2}\end{vmatrix}=a+b,$
$\mathrm{\Delta(0,2)=}\begin{vmatrix}\beta+z_{0}&1\\\
\beta^{2}+z_{1}&\varepsilon_{2}\end{vmatrix}\quad\textrm{ and
}\quad\mathrm{\Delta(1,2)=}\begin{vmatrix}1&\beta+z_{0}\\\
\varepsilon_{1}&\beta^{2}+z_{1}\end{vmatrix}.$
Hence we get
$\Delta(0,2)=\beta^{2}+a\beta+1\quad\text{and}\quad\Delta(1,2)=\beta^{2}+b\beta+b/a.$
Since $\varepsilon(0,2)=\varepsilon_{0}^{2}\varepsilon_{1}=a^{2}b$ and
$\varepsilon(1,2)=\varepsilon_{1}^{2}\varepsilon_{2}=b^{2}a$, $Eq(*)$ becomes
$(a+b)\beta^{4}=a+b+(\beta^{2}+a\beta+1)a^{2}b+(\beta^{2}+b\beta+b/a)b^{2}a.$
From this, we get (as expected, see [1, p 2, Th 1.1] ) :
$\beta^{4}=1+b(a+b)+ab(a+b)\beta+ab\beta^{2}.\qquad Eq(**)$
Example 2: Type (1,2). Here we have :
$\boldsymbol{\varepsilon}=a,(b,c)^{\infty}\quad\text{and}\quad\beta=1/CF(\textbf{s}(\boldsymbol{\varepsilon})).$
We have
$(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})=(a,b,c,b)$
and $(z_{0},z_{1},z_{2})=(1/a+1/ba^{2},1/a^{2},0)$.
$\mathrm{\Delta(2)=}\begin{vmatrix}1&1\\\
\varepsilon_{2}&\varepsilon_{3}\end{vmatrix}=b+c$
$\mathrm{\Delta(0,2)=}\begin{vmatrix}\beta+z_{0}&1\\\
\beta^{2}+z_{1}&\varepsilon_{3}\end{vmatrix}\quad\textrm{ and
}\quad\mathrm{\Delta(1,2)=}\begin{vmatrix}1&\beta+z_{0}\\\
\varepsilon_{2}&\beta^{2}+z_{1}\end{vmatrix}.$
Hence we get
$\Delta(0,2)=\beta^{2}+b\beta+b/a\quad\text{and}\quad\Delta(1,2)=\beta^{2}+c\beta+1/a^{2}+c/a+c/ba^{2}.$
We have $\varepsilon(1,2)=\varepsilon_{1}^{2}\varepsilon_{2}$ and
$\varepsilon(2,2)=\varepsilon_{2}^{2}\varepsilon_{3}$. Consequently, $Eq(*)$
becomes
$(b+c)\beta^{4}=(\beta^{2}+b\beta+b/a)b^{2}c+(\beta^{2}+c\beta+1/a^{2}+c/a+c/ba^{2})c^{2}b.$
From this, we get :
$\beta^{4}=bc(b+c)/a+c^{2}/a^{2}+bc(b+c)\beta+bc\beta^{2}.\qquad Eq(**)$
(Note that changing $c$ into $a$, we have
$\boldsymbol{\varepsilon}=a,(b,c)^{\infty}=a,(b,a)^{\infty}=(a,b)^{\infty}$
and we regain the previous example and the same algebraic equation for $\beta$
as above.)
Example 3: Type (0,3). Here we have :
$\boldsymbol{\varepsilon}=(a,b,c)^{\infty}\quad\text{and}\quad\beta=1/CF(\textbf{s}(\boldsymbol{\varepsilon})).$
We have
$(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})=(a,b,c,a,b)$
and
$(z_{0},z_{1},z_{2},z_{3})=(1/a+1/ba^{2},1/a^{2},0,1).$
$\mathrm{\Delta(3)=}\begin{vmatrix}1&1&1\\\
\varepsilon_{2}&\varepsilon_{3}&\varepsilon_{4}\\\
\varepsilon_{1}^{2}\varepsilon_{2}&\varepsilon_{2}^{2}\varepsilon_{3}&\varepsilon_{3}^{2}\varepsilon_{4}\end{vmatrix}=ba^{2}(a+c)+ac^{2}(b+c)+cb^{2}(a+b)$
$\mathrm{\Delta(0,3)=}\begin{vmatrix}\beta+z_{0}&1&1\\\ \beta^{2}+z_{1}&a&b\\\
\beta^{4}+z_{2}&c^{2}a&a^{2}b\end{vmatrix}=(a+b)\beta^{4}+(a^{2}b+ac^{2})\beta^{2}+ab(a^{2}+c^{2})\beta+\delta_{0}$
$\mathrm{\Delta(1,3)=}\begin{vmatrix}1&\beta+z_{0}&1\\\ c&\beta^{2}+z_{1}&b\\\
b^{2}c&\beta^{4}+z_{2}&a^{2}b\end{vmatrix}=(a+b)\beta^{4}+(a^{2}b+ac^{2})\beta^{2}+ab(a^{2}+c^{2})\beta+\delta_{1}$
$\mathrm{\Delta(2,3)=}\begin{vmatrix}1&1&\beta+z_{0}\\\ c&a&\beta^{2}+z_{1}\\\
b^{2}c&c^{2}a&\beta^{4}+z_{2}\end{vmatrix}=(a+c)\beta^{4}+(ac^{2}+b^{2}c)\beta^{2}+ac(c^{2}+b^{2})\beta+\delta_{2}$
together with
$\delta_{0}=ab(a^{2}+c^{2})z_{0}+(a^{2}b+c^{2}a)z_{1}+(a+b)z_{2}$
$\delta_{1}=cb(a^{2}+b^{2})+(a^{2}b+b^{2}c)z_{1}+(a+b)z_{2}$
$\delta_{2}=ac(c^{2}+b^{2})+(c^{2}a+b^{2}c)z_{1}+(b+c)z_{2}.$
Here, $Eq(*)$ becomes
$\beta^{8}=z_{3}+(\sum_{0\leq j\leq 2}\varepsilon(j,3)\Delta(j,3))/\Delta(3)$
and we also have
$\varepsilon(0,3)=a^{4}b^{2}c,\qquad\varepsilon(1,3)=b^{4}c^{2}a\quad\text{and}\quad\varepsilon(2,3)=c^{4}a^{2}b.$
Finally combining these values and the four values for the determinants given
above, from $Eq(*)$, we get the desired outcome :
$\beta^{8}=A+\sum_{0\leq k\leq 2}B_{k}\beta^{2^{k}}.\qquad Eq(**)$
At last, remarkably enough, we can check that the four coefficients in this
last equation do not only belong to $\mathbb{F}_{2}(\boldsymbol{\varepsilon})$
but are indeed elements in $F_{2}[t]$ and we have
$A=a^{3}b^{2}c+a^{2}b^{2}c^{2}+ab^{3}c^{2}+b^{4}c^{2}+ab^{2}c^{3}+abc^{4}+a^{2}bc+ab^{2}c+abc^{2}+c^{4}+1$
$B_{0}=a^{4}b2c+a^{3}b^{2}c^{2}+a^{2}b^{3}c^{2}+ab^{4}c^{2}+a^{2}b^{2}c^{3}+a^{2}bc^{4}$
$B_{1}=a^{3}b^{2}c+a^{2}b^{2}c^{2}+ab^{3}c^{2}+a^{2}bc^{3}$
and
$B_{2}=a^{2}bc+ab^{2}c+abc^{2}.$
An important and last point need to be discussed. Indeed, the reader will
probably ask the following question : are the sequences, belonging to the
family $\Large{\mathcal{F}}$, $2$-automatic as it is indicated in the title of
this note ?
There are different ways to characterize automatic sequences. A direct way is
to consider the letters of the infinite world as elements in a finite field
$\mathbb{F}_{q}$ of characteristic $p$. If a power series $\gamma$ in
$\mathbb{F}_{q}((1/t))$ is algebraic over $\mathbb{F}_{q}(t)$, then the
sequence of its coefficients is $p$-automatic (Christol’s theorem).
Concerning the sequences $\bf{s}(\boldsymbol{\varepsilon})$ described above,
in the general case the automaticity will result from a conjecture. First we
assume that the $l+d$ elements defining the sequence are in a finite field
$\mathbb{F}_{q}$ of characteristic $2$ with $q=2^{s}\geq l+d$, and
consequently we may consider the power series $\gamma\in\mathbb{F}_{q}((1/t))$
associated to this sequence. Beginning by the trivial case $d=1$, we have
observed that $\bf{s}(\boldsymbol{\varepsilon})$ is utimately periodic and
therefore $p$-automatic for all $p$. Note that the power series $\gamma$,
associated to it, is rational and consequently it satisfies a polynomial of
degree $1$ over $\mathbb{F}_{q}(t)$. We make the following conjecture :
Conjecture. _Let $l\geq 0$ and $d\geq 2$ be integers. Let
$\boldsymbol{\varepsilon}$ be an ultimately periodic sequence of type $(l,d)$.
Let $\bf{s}(\boldsymbol{\varepsilon})$ $=(s_{n})_{n\geq 0}$ be the sequence
defined above. Then there exists a finite field $\mathbb{F}_{q}$ of
characteristic $2$, containing $l+d$ elements identified with the terms of
this sequence so that we may consider $\gamma=\sum_{n\geq 0}s_{n}t^{-n}$ in
$\mathbb{F}_{q}((1/t))$ and there is a polynomial $P$ in
$\mathbb{F}_{q}(t)[x]$ such that $\deg_{x}(P)=2^{d-1}$ and $P(\gamma)=0$._
In the simpler case $(l,d)=(0,2)$ and
$\boldsymbol{\varepsilon}=(a,b)^{\infty}$, already considered several times
above, the resulting sequence $\bf{s}(\boldsymbol{\varepsilon})$ (called
Period-doubling) is well known to be $2$-automatic. More precisely, the above
conjecture is true. Indeed, if we identify the pair $(a,b)$ with the pair
$(0,1)$ in $\mathbb{F}_{2}$ then $\gamma=\sum_{n\geq
0}s_{n}t^{-n}\in\mathbb{F}_{2}((1/t))$ satisfies
$\gamma^{2}+t\gamma=t^{2}/(t^{2}+1)$.
## References
* [1] Y. Hu and A. Lasjaunias. Period-doubling continued fractions are algebraic in characteristic 2. arXiv:2204.01068, 3 Apr 2022.
(to appear in Annales de l’Institut Fourier)
* [2] A. Lasjaunias. Continued fractions. arXiv:1711.11276, 30 Nov 2017.
September 2022.
|
# Adaptive Safety Evaluation for Connected and Automated Vehicles with Sparse
Control Variates
Jingxuan Yang * —, Haowei Sun * —, Honglin He * —, Yi Zhang * —, ,
Shuo Feng * —, and Henry X. Liu * — This work is supported by National Key
Research and Development Program under Grant 2021YFB2501200 and National
Natural Science Foundation of China under Grant 62133002. (Corresponding
author: Shuo Feng.)Jingxuan Yang and Honglin He are with the Department of
Automation, Tsinghua University, Beijing 100084, China (email: {yangjx20,
hehl21}@mails.tsinghua.edu.cn).Haowei Sun and Henry X. Liu are with the
Department of Civil and Environmental Engineering, University of Michigan, Ann
Arbor, MI 48109, USA (e-mail: {haoweis, henryliu}@umich.edu).Yi Zhang is with
the Department of Automation, Beijing National Research Center for Information
Science and Technology (BNRist), Tsinghua University, Beijing 100084, China
(e-mail: [email protected]).Shuo Feng is with the Department of Automation,
Tsinghua University, Beijing 100084, China and the University of Michigan
Transportation Research Institute, Ann Arbor, MI 48109, USA (e-mail:
[email protected]).
###### Abstract
Safety performance evaluation is critical for developing and deploying
connected and automated vehicles (CAVs). One prevailing way is to design
testing scenarios using prior knowledge of CAVs, test CAVs in these scenarios,
and then evaluate their safety performances. However, significant differences
between CAVs and prior knowledge could severely reduce the evaluation
efficiency. Towards addressing this issue, most existing studies focus on the
adaptive design of testing scenarios during the CAV testing process, but so
far they cannot be applied to high-dimensional scenarios. In this paper, we
focus on the adaptive safety performance evaluation by leveraging the testing
results, after the CAV testing process. It can significantly improve the
evaluation efficiency and be applied to high-dimensional scenarios.
Specifically, instead of directly evaluating the unknown quantity (e.g., crash
rates) of CAV safety performances, we evaluate the differences between the
unknown quantity and known quantity (i.e., control variates). By leveraging
the testing results, the control variates could be well designed and optimized
such that the differences are close to zero, so the evaluation variance could
be dramatically reduced for different CAVs. To handle the high-dimensional
scenarios, we propose the sparse control variates method, where the control
variates are designed only for the sparse and critical variables of scenarios.
According to the number of critical variables in each scenario, the control
variates are stratified into strata and optimized within each stratum using
multiple linear regression techniques. We justify the proposed method’s
effectiveness by rigorous theoretical analysis and empirical study of high-
dimensional overtaking scenarios.
###### Index Terms:
Adaptive safety evaluation, connected and automated vehicles, sparse control
variates, high-dimensional scenarios
## I Introduction
Testing and evaluation of safety performance are major challenges for the
development and deployment of connected and automated vehicles (CAVs). One
proposed way is to test CAVs in the naturalistic driving environments (NDE)
through a combination of software simulation, test tracks, and public roads,
observe their performances, and make statistical comparisons with human
drivers. Due to the rarity of safety-critical events in NDE, however, hundreds
of millions of miles and sometimes hundreds of billions of miles would be
required to demonstrate CAVs’ safety performance at the human-level [1], which
is intolerably inefficient. To improve the efficiency and accelerate the
evaluation process, the past few years have witnessed increasingly rapid
advances in the field of testing scenario library generation (TSLG) [2, 3, 4,
5, 6, 7, 8, 9, 10], where safety-critical testing scenarios are usually
purposely generated utilizing prior knowledge of CAVs such as surrogate models
(SMs) of CAVs. However, due to the high complexity and black-box properties of
CAVs, there exist significant performance dissimilarities between SMs and CAVs
under test, which could severely compromise the effectiveness of the generated
testing scenarios and decrease the evaluation efficiency.
Towards addressing this problem, several adaptive testing and evaluation
methods have been proposed [11, 12, 13, 14]. The basic idea of existing
methods is to adaptively generate the testing scenarios during the testing
process of CAVs. With more testing results of CAVs, more posteriori knowledge
of CAVs can be obtained, and therefore the testing scenarios can be more
customized and optimized for the CAVs under test. However, most existing
methods can only be applied to relatively simple scenarios, and how to handle
high-dimensional scenarios remains an open question. For example, Mullins et
al. [11] proposed an adaptive sampling method that uses Gaussian process
regression (GPR) and $k$-nearest neighbors to discover performance boundaries
of the system under test and then updates the SM with new testing results
obtained near the performance boundaries. Koren et al. [12] put forward an
adaptive stress testing method that uses deep reinforcement learning to find
the most-likely failure scenarios. Feng et al. [13] proposed an adaptive
testing scenario library generation method using Bayesian optimization
techniques with classification-based GPR and acquisition functions to select
subsequent testing scenarios and then update the SMs with new testing results.
Sun et al. [14] presented an adaptive design of experiments method to detect
safety-critical scenarios, which uses supervised machine learning models as
SMs to approximate the testing results and devises acquisition functions for
updating the SMs.
The challenge for adaptively generating high-dimensional scenarios comes from
the compounding effects of the “Curse of Rarity” (CoR) and the “Curse of
Dimensionality” (CoD) [15]. The CoR refers to the concept that, due to rarity
of safety-critical events, the amount of data needed to obtain sufficient
information grow dramatically, while the CoD refers to the dimensionality of
variables to represent realistic scenarios, which makes the computation cost
increase exponentially with the growth of scenario dimensions. Most existing
scenario-based testing approaches can only handle short scenario segments with
limited background road users, where the decision variables are low-
dimensional, which cannot represent the full complexity and variability of the
real-world driving environment [16, 17, 18, 19, 20]. Towards addressing this
challenge, the naturalistic and adversarial driving environment (NADE) method
has been developed in our previous work [21], which can generate high-
dimensional highway driving scenarios. However, the NADE did not consider the
performance gap between CAVs and SMs, which could also slow down the testing
process. To the best of the authors’ knowledge, there is no existing work that
can handle the adaptive testing and evaluation problem in high-dimensional
scenarios, and the goal of this paper is to fill this gap.
Figure 1: Illustration of the adaptive testing and evaluation framework. The
focus of this study is the adaptive evaluation method for high-dimensional
scenarios, where the sparse control variates method is proposed.
In general, the adaptive testing and evaluation methods can be categorized
into two types including adaptive testing scenario generation and adaptive
testing result evaluation, which are complementary to each other as shown in
Fig. 1. Most existing studies focus on the former one, while in this study, we
focus on the latter one and propose an adaptive evaluation framework that can
handle high-dimensional scenarios. We note that how to realize the former one
in high-dimensional scenarios also remains unsolved, which we leave for future
study. In the proposed framework, we apply the NADE method to generate high-
dimensional testing scenarios, where combinations of multiple SMs are utilized
to improve the robustness of the generated scenarios for different CAVs under
test. Then we propose a sparse control variate (SCV) method to adjust the
testing results and evaluate CAVs’ performance adaptively. Essentially, the
SCV method could reduce the estimation variance for the CAV under test and
thus reduce the required number of tests, accelerating the evaluation process
adaptively.
In the following paragraphs, we further explain the major idea of the proposed
SCV method. The control variates (CV) method [22] is a popular variance
reduction technique applied in research areas such as deep learning [23] and
reinforcement learning [24]. Suppose we want to estimate
$\mu\triangleq\mathbb{E}_{p}[f(X)]$ by Monte Carlo sampling [25], where $p$ is
the probabilistic distribution of the random variable $X$ and $f$ is the
performance index of interest. Instead of directly estimating the unknown
quantity $\mu$, the control variates method estimates the differences between
the unknown quantity and known quantity as
$\mu^{\prime}\triangleq\mathbb{E}_{p}[f(X)-h(X)+\theta]$, where $h(X)$ is the
control variate and $\theta\triangleq\mathbb{E}_{p}[h(X)]$ is a known value.
Then, if $h(X)$ correlates with the performance index $f(X)$ (hence can
provide some information about $f(X)$), the estimation variance of
$\mu^{\prime}$ will always be less than directly estimating $\mu$ [26]. For
testing and evaluation of CAVs, the control variate $h(X)$ can be designed by
utilizing the prior knowledge of CAVs (e.g., different SMs). $h(X)$ usually
contains adjustable control parameters, which can be optimized by leveraging
the testing results. In such way, the information about the CAV under test
could be incorporated, which makes the adaptive evaluation possible. However,
due to the CoD, the computation cost of optimal control parameters will
increase exponentially with the growth of scenario dimensions, so directly
applying the ordinary CV method in high-dimensional scenarios is problematic.
Figure 2: Illustration of the sparse control variates method. The SCV are
constructed by only considering critical variables (represented as red dots in
testing scenarios). The testing results are stratified into strata according
to the number of critical variables and then adjusted by SCV within each
stratum. Finally, the performance index are obtained by summing up these
evaluation results with proportion weights.
To address this problem, we propose the sparse control variates (SCV) method,
as shown in Fig. 2. The key idea is to construct the SCV by only considering
the sparse but critical variables (e.g., behaviors of principal other vehicles
at critical moments), following the similar idea from [21] that handles the
CoD. However, the number of critical variables varies in different testing
scenarios, which cannot be handled by ordinary CV method. To address this
issue, in the SCV method, we stratify the testing scenarios into strata
according to the number of critical variables. Then the control parameters can
be optimized by multiple linear regression (MLR) [27] within each stratum, and
the final evaluation results are obtained by summing up those evaluation
results in each stratum with the proportion weights. Since the number of
critical variables is much less than the dimension of testing scenarios, the
computation cost of optimal control parameters for SCV could be greatly
reduced, overcoming the CoD challenge.
To verify the proposed method, we theoretically analyze its accuracy,
efficiency, and optimality. The theorems show that our method is unbiased, and
its estimation variance is nearly proportional to the best one that all the
SMs used for generating testing scenarios could have. Moreover, under certain
assumptions about the SMs, our method can provide a zero-variance estimator.
To validate our method, the high-dimensional overtaking scenarios with large-
scale naturalistic driving data are investigated. Simulation results show that
our method can further accelerate the evaluation process by about one order of
magnitude for different types of CAV models, comparing with the estimation
efficiency in NADE.
Compared with our previously published conference paper about SCV [28], the
new contributions of this paper are listed as follows. First, we significantly
extend our methodology into high-dimensional scenarios and establish the
theoretical analysis for the accuracy, efficiency, and optimality of the
proposed method with rigorous proofs. Second, a more realistic overtaking case
study with large-scale naturalistic driving data is investigated to
systematically validate the performances of our method.
The remainder of this paper is organized as follows. Section II provides
preliminary knowledge for the generation of NDE and NADE. Section III
formulates the adaptive testing and evaluation problem and elaborates the
challenges of applying ordinary CV for adaptive safety evaluation. To address
these challenges, in Section IV, the SCV method is proposed. Then Section V
and VI verify and validate the accuracy and efficiency of the proposed method
from the theoretical and experimental perspectives, respectively. Finally,
Section VII concludes the paper and discusses future research.
## II Preliminaries
### II-A Naturalistic Driving Environment Testing
As discussed above, the prevailing approach for CAV evaluation is to test CAVs
in the naturalistic driving environments (NDE) [29], observe their
performances, and make statistical comparisons with human drivers. In NDE, one
of the vehicles is the automated vehicle (AV) under test and the others are
background vehicles (BVs), which can be formulated as Markov games [30]. A
Markov game for $N$ agents (i.e., BVs) is defined by a set of states
$\mathcal{S}$ describing the positions and velocities of all vehicles and a
collection of action (i.e., acceleration) sets
$\mathcal{A}_{1},\dots,\mathcal{A}_{N}$, one for each agent in NDE. The total
action space is denoted as
$\mathcal{A}=\mathcal{A}_{1}\times\cdots\times\mathcal{A}_{N}$. Then a
scenario is defined as the time series of the states of all vehicles and the
actions of all agents, i.e.,
$x=(s_{0},a_{0},\dots,s_{T},a_{T})\in\mathcal{X},$ (1)
where $x$ represents the scenario, $\mathcal{X}$ is the set of all feasible
scenarios, $s_{t}\in\mathcal{S}$ is the state of all vehicles at time $t$,
$a_{t}\in\mathcal{A}$ is the action of all agents at time $t$, and $T$ is the
time horizon.
Let $\Omega=\mathcal{X}$ be the sample space incorporating all feasible
scenarios. Consider the probability space $(\Omega,\mathcal{F},\mathbb{P})$,
where $\mathcal{F}\triangleq 2^{\Omega}$ is the power set of $\Omega$ and
$\mathbb{P}$ is a probability measure on $\mathcal{F}$. Let $X:x\mapsto x$,
$\forall x\in\mathcal{X}$ be the random variable of scenarios. For testing and
evaluation of CAVs, the crash event is usually of most interest, which can be
defined as $A=\\{x\in\mathcal{X}:s_{T}\in\mathcal{S}_{c}\\}$, where
$\mathcal{S}_{c}$ is the set of all crash states. Then the crash rate is
selected as the performance index, which can be computed as
$\mu=\mathbb{P}(A)=\mathbb{E}_{p}[\mathbb{I}_{A}(X)]=\sum_{x\in\mathcal{X}}\mathbb{P}(A|x)p(x),$
(2)
where $\mathbb{I}_{A}$ is the indicator function of $A$, and $p$ is the
naturalistic joint distribution of $x$. The essence of testing AV in NDE is to
estimate the performance index $\mu$ by Monte Carlo simulation, i.e.,
$\hat{\mu}_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathbb{P}(A|X_{i}),\quad X_{i}\sim
p.$ (3)
### II-B Naturalistic and Adversarial Driving Environment Generation
The NDE faces the CoR, making its estimation catastrophically inefficient. To
improve the estimation efficiency, the importance sampling (IS) technique [19,
17, 18] has been used to sample testing scenarios from the importance function
$q$, which puts more weights on crash-prone scenarios. In IS, the performance
index can be estimated as
$\hat{\mu}_{q}=\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbb{P}(A|X_{i})p(X_{i})}{q(X_{i})},\quad
X_{i}\sim q.$ (4)
However, the IS method faces the CoD if the testing scenarios are high-
dimensional [31]. To address both the CoR and the CoD, the naturalistic and
adversarial driving environment (NADE) [21] has been proposed to only sample
critical variables of testing scenarios from importance functions, while other
variables remain their naturalistic distributions.
Denote $x=(x_{c},x_{-c})$, where $x_{c}=\\{x_{c_{1}},\dots,x_{c_{l}}\\}$ is
the set of critical variables, $c_{1},\dots,c_{l}$ are called the critical
moments, $l=0,1,\dots,L$ is the number of control steps (i.e., the number of
critical variables in $x_{c}$), and $x_{-c}$ is the set of other variables.
Let $X_{c}:x\mapsto x_{c}$ be the random variable of critical variables and
$X_{-c}:x\mapsto x_{-c}$ be the random variable of other variables, then we
have $X=(X_{c},X_{-c})$. The importance function can then be formulated as
$q(x)=q(x_{c})p(x_{-c})$, and therefore the performance index can be estimated
in NADE as
$\tilde{\mu}_{q}=\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbb{P}(A|X_{i})p(X_{c,i})}{q(X_{c,i})},\quad
X_{i}\sim q,$ (5)
where $X_{c,i}$ is the random variable of critical variables of $X_{i}$.
## III Problem Formulation
### III-A Adaptive Testing and Evaluation
Due to the black-box property and various types of CAVs, how to adaptively
test and evaluate CAVs remains a major challenge. One way of adaptive testing
and evaluation is adaptively generating testing scenarios. For example, we can
minimize the estimation variance by optimizing the importance function, i.e.,
$\min_{q\in\mathcal{Q}}~{}\mathrm{Var}_{q}\left(\frac{\mathbb{P}(A|X)p(X)}{q(X)}\right),$
(6)
where $\mathcal{Q}$ is the function space of $q$. Better importance functions
can be found by leveraging the posteriori knowledge of CAVs obtained from
testing results. Then the testing scenarios can be adaptively generated by
sampling from updated importance functions.
In this paper, we focus on another way of adaptive testing and evaluation,
i.e., adaptively evaluating weighted testing results. Specifically, the
control variates (CV) method is adopted. This problem can be formulated as
$\min_{h\in\mathcal{H}}~{}\mathrm{Var}_{q}\left(\frac{\mathbb{P}(A|X)p(X)}{q(X)}-h(X)\right),$
(7)
where $h:\mathcal{X}\to\mathbb{R}$ is the control variate and $\mathcal{H}$ is
the function space of $h$. The goal is to further reduce the estimation
variance by optimizing $h$ in $\mathcal{H}$, leveraging the testing results.
### III-B Control Variates
Control variates are widely used as a basic variance reduction technique in
Monte Carlo simulation. They can be usefully combined with the mixture
importance sampling, where individual importance functions can serve as CV. In
mixture IS, the scenarios $X_{i},i=1,\dots,n$ are sampled from the mixture
importance function $q_{\alpha}=\sum_{j=1}^{J}\alpha_{j}q_{j}$, where
$\alpha_{j}\geqslant 0$, $\sum_{j=1}^{J}\alpha_{j}=1$ and the $q_{j}$ are
importance functions. One commonly used way is to construct CV by using the
linear combination of individual importance functions as
$h_{\beta}(X)=\sum_{j=1}^{J}\beta_{j}\left[\frac{q_{j}(X)}{q_{\alpha}(X)}-1\right],$
(8)
where $\beta=(\beta_{1},\dots,\beta_{J})^{\top}$ is the control vector,
$\beta_{j}\in\mathbb{R}$ are control parameters, and $q_{j}/q_{\alpha}-1$ are
individual control variate. Combining the control variate $h_{\beta}$ with
mixture IS gives the estimation
$\hat{\mu}_{q_{\alpha},\beta}=\frac{1}{n}\sum_{i=1}^{n}\left[\frac{\mathbb{P}(A|X_{i})p(X_{i})}{q_{\alpha}(X_{i})}-h_{\beta}(X_{i})\right]$
(9)
for $X_{i}\sim q_{\alpha}$.
The unbiasedness of $\hat{\mu}_{q_{\alpha},\beta}$ is guaranteed since
$\mathbb{E}_{q_{\alpha}}[\hat{\mu}_{q_{\alpha},\beta}]=\mathbb{E}_{q_{\alpha}}\left[\frac{\mathbb{P}(A|X)p(X)}{q_{\alpha}(X)}-h_{\beta}(X)\right]=\mu,$
(10)
where the second equality is obtained from the unbiasedness of IS and
$\mathbb{E}_{q_{\alpha}}[h_{\beta}(X)]=0$. The variance of
$\hat{\mu}_{q_{\alpha},\beta}$ can be compared to that of IS with individual
importance functions $q_{j}$. We have the following lemma.
###### Lemma 1
Let $\beta^{*}$ be any minimizer over $\beta$ of
$\mathrm{Var}_{q_{\alpha}}(\hat{\mu}_{q_{\alpha},\beta})$, then
$\mathrm{Var}_{q_{\alpha}}(\hat{\mu}_{q_{\alpha},\beta^{*}})\leqslant\min_{1\leqslant
j\leqslant J}\frac{\sigma_{q_{j}}^{2}}{n\alpha_{j}},$ (11)
where $\sigma_{q_{j}}^{2}$ is the asymptotic variance of $\hat{\mu}_{q_{j}}$,
i.e.,
$\sigma_{q_{j}}^{2}=\mathrm{Var}_{q_{j}}\left(\frac{\mathbb{P}(A|X)p(X)}{q_{j}(X)}\right),~{}j=1,\dots,J.$
(12)
###### Proof:
This is the Theorem 2 in [32]. ∎
It can be seen from Lemma 1 that the variance of
$\hat{\mu}_{q_{\alpha},\beta}$ will be zero if any one of the $q_{j}$ is
optimal. This is a significant feature because we can nearly omit the
influence of all other worse-performed importance functions. In applications,
using only one SM to test CAVs is usually under huge risk, because the
performance gap between the SM and various types of CAVs may be too large to
give a good estimation efficiency. Therefore, to ensure the robustness, we can
combine multiple SMs to test the CAVs. However, there often exist some poor-
performed SMs that will compromise the overall estimation efficiency. Using
mixture IS with CV provides an effective way to ensure both good estimation
efficiency and robustness to various types of CAVs.
In practice, the optimal control vector $\beta^{*}$ is usually unknown, and
its estimation $\hat{\beta}$ can be obtained by multiple linear regression
(MLR). Denote the weighted testing results as
$Y_{i}=\mathbb{P}(A|X_{i})p(X_{i})/q_{\alpha}(X_{i})$, $i=1,\dots,n$, and the
individual control variate as $Z_{ij}=q_{j}(X_{i})/q_{\alpha}(X_{i})-1$,
$i=1,\dots,n$, $j=1,\dots,J-1$. Then the $\hat{\beta}$ is given as the vector
of coefficients obtained from MLR of $Y_{i}$ on $Z_{ij}$. In essence, this
process is to search for the best control variate defined in Eq. (8) in the
function space spanned by individual control variate $q_{j}/q_{\alpha}-1$.
However, challenges of estimating optimal control parameters arise when the
testing scenarios are high-dimensional.
### III-C CoD of Control Variates
Considering the Markov chain structure of scenarios with $T+1$ time steps, the
mixture importance function is given by
$q_{\alpha}(x)=q_{\alpha}(s_{0})\prod_{t=0}^{T}q_{\alpha}(a_{t}|s_{t}),~{}\forall
x\in\mathcal{X},$ (13)
where $q_{\alpha}(s)=\sum_{j=1}^{J}{\alpha_{j}q_{j}(s)},~{}\forall
s\in\mathcal{S}$, and
$q_{\alpha}(a|s)=\sum_{j=1}^{J}{\alpha_{j}q_{j}(a|s)},~{}\forall
a\in\mathcal{A},~{}s\in\mathcal{S}$. It can be found that $q_{\alpha}(x)$ is
the product of $T+2$ individual importance functions and thus is also the
summation of $J^{T+2}$ combinations of different importance functions at each
time step. Specifically, these individual importance functions are
$q_{j_{0},\dots,j_{T+1}}(x)=q_{j_{0}}(s_{0})q_{j_{1}}(a_{0}|s_{0})\cdots
q_{j_{T+1}}(a_{T}|s_{T}),$ (14)
where $j_{0},\dots,j_{T+1}=1,\dots,J$. Then the individual control variate are
given by $q_{j_{0},\dots,j_{T+1}}/q_{\alpha}-1$.
To find the estimation of optimal control parameters, we have to conduct MLR
of $n$ weighted testing results on $J^{T+2}$ individual control variate. The
number $J^{T+2}$ will increase exponentially with the dimension of scenarios,
leading to the CoD of MLR. For example, if we have $J=10$ individual
importance functions and the testing scenarios last for 10 seconds at a
frequency of 10 Hz, then the number of individual control variate will be
10102. This means that a matrix with dimension 10102 should be inverted in
MLR, which is not tractable. Moreover, the situation will get even worse if
the duration of scenarios grows to several hours, which are common in daily
driving yet far from being tractable. The following section aims to address
this challenge.
## IV Adaptive Safety Evaluation with Sparse Control Variates
In this section, we will address the CoD discussed above and show how to
estimate the optimal control parameters.
### IV-A Sparse Control Variates
We propose the sparse control variates (SCV) method to address the CoD of
applying CV in high-dimensional scenarios. Specifically, the SCV are
constructed by only considering the importance functions of only sparse and
critical variables in high-dimensional testing scenarios. The number of
critical variables is usually much less than the dimension of scenarios in
NADE. Therefore, the number of SCV is also much less than the number of
ordinary CV, which could greatly address the CoD. However, as the number of
SCV varies in different testing scenarios, we can not directly apply SCV to
the weighted testing results. Towards addressing this issue, we propose to
stratify the testing scenarios into strata according to the number of critical
variables and then apply SCV within each stratum.
Let $\mathcal{X}_{l}=\\{x\in\mathcal{X}:|x_{c}|=l\\}$, $l=0,1,\dots,L$ be the
stratum of scenarios that are controlled $l$ steps, satisfying
$\bigcup_{l=0}^{L}\mathcal{X}_{l}=\mathcal{X}$. Using mixture importance
function $q_{\alpha}$, the estimation of the performance index in NADE is
$\tilde{\mu}_{q_{\alpha}}=\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbb{P}(A|X_{i})p(X_{c,i})}{q_{\alpha}(X_{c,i})},\quad
X_{i}\sim q_{\alpha}.$ (15)
The performance index of scenarios in stratum $\mathcal{X}_{l}$ can be written
as
$\mu_{l}\triangleq\mathbb{E}_{p}[\mathbb{I}_{A}(X)\mathbb{I}_{\mathcal{X}_{l}}(X)]$,
$l=0,1,\dots,L$, then we have
$\mu=\sum_{l=0}^{L}\mathbb{E}_{p}[\mathbb{I}_{A}(X)\mathbb{I}_{\mathcal{X}_{l}}(X)]=\sum_{l=0}^{L}\mu_{l}.$
(16)
Similar to Eq. (15), the estimation of $\mu_{l}$ is given by
$\tilde{\mu}_{l,q_{\alpha}}=\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbb{P}(A|X_{i})\mathbb{I}_{\mathcal{X}_{l}}(X_{i})p(X_{c,i})}{q_{\alpha}(X_{c,i})},$
(17)
and then we have
$\displaystyle\tilde{\mu}_{q_{\alpha}}$
$\displaystyle=\sum_{l=0}^{L}\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbb{P}(A|X_{i})\mathbb{I}_{\mathcal{X}_{l}}(X_{i})p(X_{c,i})}{q_{\alpha}(X_{c,i})}=\sum_{l=0}^{L}\tilde{\mu}_{l,q_{\alpha}}.$
(18)
Let $q_{j_{1},\dots,j_{l}}(x)=p(x_{-c})q_{j_{1}}(x_{c_{1}})\cdots
q_{j_{l}}(x_{c_{l}})$ be the importance functions that sample $x_{-c}$ from
$p$ and sample $x_{c_{1}},\dots,x_{c_{l}}$ from $q_{j_{1}},\dots,q_{j_{l}}$
respectively, where $j_{1},\dots,j_{l}=1,\dots,J$, $l=1,\dots,L$. Then the
individual importance functions of critical variables are given by
$q_{j_{1},\dots,j_{l}}(x_{c})$. Denote the linear combination of these
individual importance functions as
$\tilde{h}_{l}(x)\triangleq\sum_{j_{1},\dots,j_{l}}\beta_{l,j_{1},\dots,j_{l}}q_{j_{1},\dots,j_{l}}(x),~{}l=1,\dots,L,$
(19)
where $\beta_{l,j_{1},\dots,j_{l}}\in\mathbb{R}$ are associated control
parameters. Then the SCV are given by
$h_{l}(x_{c})=\frac{\tilde{h}_{l}(x_{c})\mathbb{I}_{\mathcal{X}_{l}}(x_{c})}{q_{\alpha}(x_{c})}-\theta_{l},~{}l=1,\dots,L,$
(20)
where
$\theta_{l}\triangleq\mathbb{E}_{q_{\alpha}}\big{[}\tilde{h}_{l}(X)\mathbb{I}_{\mathcal{X}_{l}}(X)/q_{\alpha}(X)\big{]}$.
Therefore, the estimation $\tilde{\mu}_{l,q_{\alpha}}$ in Eq. (17) can be
evaluated with SCV as
$\displaystyle\tilde{\mu}_{l,q_{\alpha},\beta_{l}}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left[\frac{\mathbb{P}(A|X_{i})\mathbb{I}_{\mathcal{X}_{l}}(X_{i})p(X_{c,i})}{q_{\alpha}(X_{c,i})}-h_{l}(X_{c,i})\right]$
(21)
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbb{P}(A|X_{i})p(X_{c,i})-\tilde{h}_{l}(X_{c,i})}{q_{\alpha}(X_{c,i})}\mathbb{I}_{\mathcal{X}_{l}}(X_{i})+\theta_{l}$
for $l=1,\dots,L$, where $\beta_{l}=\mathrm{vec}(\beta_{l,j_{1},\dots,j_{l}})$
is the vector of control parameters, and $\mathrm{vec}(\cdot)$ is the
vectorization operator that flattens a tensor into a long vector. Note that
there is no critical variable for $l=0$, and thus we set $\beta_{0}\triangleq
0$. In summary, the performance index estimated by the proposed SCV method is
given by
$\tilde{\mu}_{q_{\alpha},\beta}=\sum_{l=0}^{L}\tilde{\mu}_{l,q_{\alpha},\beta_{l}},$
(22)
where $\beta=\\{\beta_{l}\\}_{l=0}^{L}$ is the set of all control vectors.
### IV-B Optimal Control Parameters
To estimate the optimal control parameters that minimize the estimation
variance, multiple linear regression (MLR) technique is applied in each
stratum. Let
$\mathbb{X}_{l}\triangleq\\{X_{i}|X_{i}\in\mathcal{X}_{l},i=1,\dots,n\\}$ be
the set of sampled scenarios with $l$ controlled steps,
$n_{l}\triangleq\sum_{i=1}^{n}\mathbb{I}_{\mathcal{X}_{l}}(X_{i})$ be the
number of tests with $l$ controlled steps and $d_{l}\triangleq J^{l}$ be the
number of SCV, $l=1,\dots,L$. Denote the vector of testing results as
$Y_{l}\triangleq\left[\frac{\mathbb{P}(A|X_{i})p(X_{i})}{q_{\alpha}(X_{i})}~{}\mathrm{for}~{}X_{i}\in\mathbb{X}_{l}\right]\in\mathbb{R}^{n_{l}},$
(23)
the individual SCV as
$h^{\prime}_{j_{1},\dots,j_{l}}(x_{c})=\frac{q_{j_{1},\dots,j_{l}}(x_{c})}{q_{\alpha}(x_{c})}-\sum_{x_{c}\in\mathcal{X}_{l}}q_{j_{1},\dots,j_{l}}(x_{c}),$
(24)
for $l=1,\dots,L$. Then the matrix of individual SCV can be formulated as
$H_{l}\triangleq\left[\mathrm{vec}\left(h^{\prime}_{j_{1},\dots,j_{l}}(X_{c,i})\right)~{}\mathrm{for}~{}X_{i}\in\mathbb{X}_{l}\right]\in\mathbb{R}^{n_{l}\times
d_{l}},$ (25)
for $l=1,\dots,L$. Then the regression formula is given by
$Y_{l}\approx\eta_{l}+H_{l}\beta_{l}$. The MLR of $Y_{l}$ on $H_{l}$ is to
find the optimal solution of the following optimization problem, i.e.,
$\min_{\eta_{l},\beta_{l}}~{}f(\eta_{l},\beta_{l})=\|Y_{l}-\eta_{l}-H_{l}\beta_{l}\|_{2}^{2}.$
(26)
Letting the partial derivatives of $f$ with respect to $\eta_{l}$ and
$\beta_{l}$ both equal zero, we have $\hat{\eta}_{l}=1^{\top}Y_{l}/n_{l}$ and
$\hat{\beta}_{l}=(H_{l}^{\top}H_{l})^{-1}H_{l}^{\top}Y_{l}$, assuming that the
control matrix $M_{l}\triangleq H_{l}^{\top}H_{l}\in\mathbb{R}^{d_{l}\times
d_{l}}$ is invertible. Then the estimated performance index is
$\hat{\mu}_{l}=n_{l}\hat{\eta}_{l}/n$. In practice the control matrix may
often not be invertible, then we use singular value decomposition (SVD) [33]
to compute the regression coefficients $\hat{\beta}_{l}$, and the rank of the
control matrix is
$\mathrm{rank}(M_{l})=\mathrm{rank}(H_{l})\leqslant\min\\{n_{l},d_{l}\\}.$
(27)
If $n_{l}<d_{l}$, then the control matrix $M_{l}$ will be singular and has
utmost $n_{l}$ nonzero singular values. As the number of tests $n_{l}$ in
$\mathbb{X}_{l}$ will not grow exponentially with the number of control steps
$l$, the rank of the control matrix will also not, albeit the dimension
$d_{l}=J^{l}$ of the control matrix increases exponentially with $l$. In
conclusion, solving the optimal control parameters for SCV is tractable and
will not face the CoD challenge. We will further demonstrate this in
Subsection VI-E.
## V Theoretical Analysis
This section theoretically justifies the accuracy, efficiency and optimality
of the proposed SCV method.
### V-A Accuracy Analysis
We first prove that the estimation is unbiased.
###### Theorem 1
Let $\tilde{\mu}_{q_{\alpha},\beta}$ be given by Eq. (22) where $q_{\alpha}>0$
whenever $\mathbb{P}(A|x)p(x)>0$, then
$\mathbb{E}_{q_{\alpha}}[\tilde{\mu}_{q_{\alpha},\beta}]=\mu$.
###### Proof:
To establish unbiasedness, write
$\displaystyle\mathbb{E}_{q_{\alpha}}[\tilde{\mu}_{q_{\alpha},\beta}]$
$\displaystyle=\mathbb{E}_{q_{\alpha}}\left[\sum_{l=0}^{L}\tilde{\mu}_{l,q_{\alpha},\beta_{l}}\right]$
(28)
$\displaystyle=\sum_{l=0}^{L}\mathbb{E}_{q_{\alpha}}\left[\tilde{\mu}_{l,q_{\alpha}}-\frac{\tilde{h}_{l}(X)}{q_{\alpha}(X)}\mathbb{I}_{\mathcal{X}_{l}}(X)+\theta_{l}\right]$
$\displaystyle=\sum_{l=0}^{L}(\mu_{l}-\theta_{l}+\theta_{l})=\mu.$
∎
###### Remark 1
This theorem indicates that the estimation is unbiased if the control
parameters $\beta$ are independent of the sample data. It’s worth noting that
in practice the control parameters are usually estimated by the sample data,
which would bring a bias. However, that bias is ordinarily negligible (please
see Section 8.9 in [26] for more discussions).
### V-B Efficiency Analysis
Next, we evaluate the efficiency of the SCV method. The variance of the
estimation $\tilde{\mu}_{q_{\alpha},\beta}$ is
$\mathrm{Var}_{q_{\alpha}}(\tilde{\mu}_{q_{\alpha},\beta})=\sigma_{q_{\alpha},\beta}^{2}/n$,
where $\sigma_{q_{\alpha},\beta}^{2}$ is the asymptotic variance of
$\tilde{\mu}_{q_{\alpha},\beta}$, i.e.,
$\sigma_{q_{\alpha},\beta}^{2}=\mathrm{Var}_{q_{\alpha}}\left(\sum_{l=0}^{L}\frac{\mathbb{P}(A|X)p(X)-\tilde{h}_{l}(X)}{q_{\alpha}(X)}\mathbb{I}_{\mathcal{X}_{l}}(X)\right)$
(29)
for $X\sim q_{\alpha}$. Denote
$Z_{l}\triangleq\frac{\mathbb{P}(A|X)p(X)-\tilde{h}_{l}(X)}{q_{\alpha}(X)}\mathbb{I}_{\mathcal{X}_{l}}(X),~{}l=0,\dots,L,$
(30)
then the asymptotic variance $\sigma_{q_{\alpha},\beta}^{2}$ can be expressed
as
$\sigma_{q_{\alpha},\beta}^{2}=\mathrm{Var}_{q_{\alpha}}\left(\sum_{l=0}^{L}Z_{l}\right)=\mathbb{E}_{q_{\alpha}}\left[\left(\sum_{l=0}^{L}\Big{[}Z_{l}-\mathbb{E}_{q_{\alpha}}[Z_{l}]\Big{]}\right)^{2}\right].$
(31)
Let $L^{\prime}=L+1$, then by convexity of quadratic function and Jensen’s
inequality, we have
$\displaystyle\sigma_{q_{\alpha},\beta}^{2}$
$\displaystyle\leqslant\mathbb{E}_{q_{\alpha}}\left[L^{\prime}\sum_{l=0}^{L}\Big{(}Z_{l}-\mathbb{E}_{q_{\alpha}}[Z_{l}]\Big{)}^{2}\right]$
(32) $\displaystyle=L^{\prime}\sum_{l=0}^{L}\mathrm{Var}_{q_{\alpha}}(Z_{l}).$
Denote
$\sigma_{l,q_{\alpha},\beta_{l}}^{2}\triangleq\mathrm{Var}_{q_{\alpha}}(Z_{l})$
and the asymptotic variance of $\tilde{\mu}_{l,q}$ over $\mathcal{X}_{l}$ as
$\sigma_{l,q}^{2}$, i.e.,
$\sigma_{l,q}^{2}\triangleq\sum_{x\in\mathcal{X}_{l}}\left(\frac{\mathbb{P}(A|x)p(x)}{q(x)}-\mu_{l}\right)^{2}q(x),~{}l=1,\dots,L,$
(33)
then we have the following theorem.
###### Theorem 2
If $\beta^{*}$ is any minimizer of $\sigma_{q_{\alpha},\beta}^{2}$, then
$\displaystyle\sigma_{q_{\alpha},\beta^{*}}^{2}$ $\displaystyle\leqslant
L^{\prime}\sigma_{0,p,\beta_{0}}^{2}$ (34)
$\displaystyle\quad+L^{\prime}\sum_{l=1}^{L}\min_{j_{1},\dots,j_{l}}\left\\{\frac{\sigma_{l,q_{j_{1},\dots,j_{l}}}^{2}}{\prod_{\ell=1}^{l}\alpha_{j_{\ell}}}+3\left(\frac{\mu_{l}}{\prod_{\ell=1}^{l}\alpha_{j_{\ell}}}\right)^{2}\right\\}.$
###### Proof:
Take $\sigma_{1,q_{\alpha},\beta_{1}}^{2}$ as an example. Following the proof
in [32], we consider the particular vector $\beta_{1}$ having $\beta_{1,1}=0$
and $\beta_{1,j}=-\mu_{1}\alpha_{j}/\alpha_{1}$ for $j>1$. Let
$r_{1}(x)\triangleq[\mathbb{P}(A|x)p(x)-\mu_{1}q_{1}(x)]\mathbb{I}_{\mathcal{X}_{1}}(x)$,
then we have $\sum_{x\in\mathcal{X}}r_{1}(x)=\mu_{1}(1-\xi_{1})$, where
$\xi_{1}\triangleq\sum_{x\in\mathcal{X}_{1}}q_{1}(x)$, $\xi_{1}\in[0,1]$.
Substituting these values, we find that for this $\beta_{1}$,
$\displaystyle Z_{1}$
$\displaystyle=\frac{\mathbb{P}(A|X)p(X)-\tilde{h}_{1}(X)}{q_{\alpha}(X)}\mathbb{I}_{\mathcal{X}_{1}}(X)$
(35)
$\displaystyle=\frac{\mathbb{P}(A|X)p(X)-\mu_{1}q_{1}+\mu_{1}q_{1}-\tilde{h}_{1}(X)}{q_{\alpha}(X)}\mathbb{I}_{\mathcal{X}_{1}}(X)$
$\displaystyle=\frac{r_{1}(X)}{q_{\alpha}(X)}+\frac{\mu_{1}}{\alpha_{1}}\mathbb{I}_{\mathcal{X}_{1}}(X),$
and $\mathbb{E}_{q_{\alpha}}[Z_{1}]=\mu_{1}\alpha_{1,1}/\alpha_{1}$, where
$\alpha_{1,1}\triangleq\alpha_{1}+\sum_{j=2}^{J}\alpha_{j}\linebreak\sum_{x\in\mathcal{X}_{1}}q_{j}(x)$,
$\alpha_{1,1}\in[0,1]$. Therefore, we have
$\displaystyle\sigma_{1,q_{\alpha},\beta_{1}}^{2}$
$\displaystyle=\mathbb{E}_{q_{\alpha}}\left[\Big{(}Z_{1}-\mathbb{E}_{q_{\alpha}}[Z_{1}]\Big{)}^{2}\right]$
(36)
$\displaystyle=\sum_{x\in\mathcal{X}}\left[\frac{r_{1}(x)}{q_{\alpha}(x)}+\frac{\mu_{1}}{\alpha_{1}}\Big{(}\mathbb{I}_{\mathcal{X}_{1}}(x)-\alpha_{1,1}\Big{)}\right]^{2}q_{\alpha}(x)$
$\displaystyle\triangleq V_{1,1}+V_{1,2}+V_{1,3},$
where
$\displaystyle V_{1,1}$
$\displaystyle\triangleq\sum_{x\in\mathcal{X}}\frac{r_{1}^{2}(x)}{q_{\alpha}(x)}=\sum_{x\in\mathcal{X}}\frac{[\mathbb{P}(A|x)p(x)-\mu_{1}q_{1}(x)]^{2}}{q_{\alpha}(x)}\mathbb{I}_{\mathcal{X}_{1}}(x)$
(37)
$\displaystyle\leqslant\sum_{x\in\mathcal{X}_{1}}\frac{[\mathbb{P}(A|x)p(x)-\mu_{1}q_{1}(x)]^{2}}{\alpha_{1}q_{1}(x)}=\frac{\sigma_{1,q_{1}}^{2}}{\alpha_{1}},$
$\displaystyle V_{1,2}$
$\displaystyle\triangleq\sum_{x\in\mathcal{X}}\frac{2\mu_{1}r_{1}(x)(\mathbb{I}_{\mathcal{X}_{1}}(x)-\alpha_{1,1})}{\alpha_{1}}$
(38)
$\displaystyle=\frac{2\mu_{1}^{2}(1-\xi_{1})(1-\alpha_{1,1})}{\alpha_{1}}\leqslant
2\left(\frac{\mu_{1}}{\alpha_{1}}\right)^{2},$
and
$\displaystyle V_{1,3}$
$\displaystyle\triangleq\sum_{x\in\mathcal{X}}\left[\frac{\mu_{1}(\mathbb{I}_{\mathcal{X}_{1}}(x)-\alpha_{1,1})}{\alpha_{1}}\right]^{2}q_{\alpha}(x)$
(39)
$\displaystyle\leqslant\sum_{x\in\mathcal{X}}\left(\frac{\mu_{1}}{\alpha_{1}}\right)^{2}q_{\alpha}(x)=\left(\frac{\mu_{1}}{\alpha_{1}}\right)^{2}.$
Therefore, we conclude that
$\sigma_{1,q_{\alpha},\beta_{1}^{*}}^{2}\leqslant\sigma_{1,q_{\alpha},\beta_{1}}^{2}\leqslant\frac{\sigma_{1,q_{1}}^{2}}{\alpha_{1}}+3\left(\frac{\mu_{1}}{\alpha_{1}}\right)^{2}.$
(40)
By making similar arguments for $j=2,\dots,J$, we have
$\sigma_{1,q_{\alpha},\beta_{1}^{*}}^{2}\leqslant\min_{j}\left\\{\frac{\sigma_{1,q_{j}}^{2}}{\alpha_{j}}+3\left(\frac{\mu_{1}}{\alpha_{j}}\right)^{2}\right\\}.$
(41)
It’s straightforward to extend the proof for $l=2,\dots,L$, then Eq. (34) is
established. ∎
###### Remark 2
For $l=1$, we expect to get approximately $n_{1}\alpha_{j}$ scenarios in
$\mathcal{X}_{1}$ from the importance function $q_{j}$. The quantity
$\sigma_{1,q_{j}}^{2}/\alpha_{j}$ in Eq. (41) is the variance we would obtain
from $n_{1}\alpha_{j}$ such scenarios alone. It is hard to imagine that we
could do better in general, because when $\sigma_{1,q_{j}}^{2}=\infty$ for all
but one of the mixture components it is guaranteed that those bad components
do not make the estimation worse than what we would have had from the one good
importance function. Moreover, if there exists an optimal importance function
in $q_{j}$, then the minimum value of $\sigma_{1,q_{j}}^{2}/\alpha_{j}$ will
be zero, which will greatly reduce the estimation variance. It should be noted
that the upper bound for variance in Eq. (41) contains a residual term
$3(\mu_{1}/\alpha_{j})^{2}$, which is the cost for stratifying the scenarios.
### V-C Optimality Analysis
Under the following assumptions, the estimation variance of the SCV method can
be zero.
###### Assumption 1
The scenarios in $\mathcal{X}_{0}$ will not be sampled by $q_{\alpha}$, i.e.,
$q_{\alpha}(x)=0$, $\forall x\in\mathcal{X}_{0}$.
###### Assumption 2
The control policy satisfies $|x_{c}|=1$, i.e., the number of critical
variable of all sampled scenarios is 1.
###### Assumption 3
There exists an optimal control policy such that
$\mathbb{P}(A|x_{c})=\mathbb{P}(A|x)$, which means that the critical variable
$x_{c}$ can totally dominate the crash probability.
###### Assumption 4
There exists an optimal importance function among $q_{j}$. Without loss of
generality, let $q_{1}$ be the optimal importance function, i.e.,
$q_{1}(x_{c})\triangleq\mathbb{P}(A|x_{c})p(x_{c})/\mu$.
###### Theorem 3
Under Assumptions 1, 2, 3 and 4, if $\beta^{*}$ is any minimizer of
$\sigma_{q_{\alpha},\beta}^{2}$, then $\sigma_{q_{\alpha},\beta^{*}}^{2}=0$.
###### Proof:
From Assumptions 1 and 2, we know that all sampled scenarios will only be
controlled once, i.e., $\mathcal{X}=\mathcal{X}_{1}$ and $\mu=\mu_{1}$, then
$Z_{1}=\frac{r_{1}(X)}{q_{\alpha}(X)}+\frac{\mu_{1}}{\alpha_{1}}\mathbb{I}_{\mathcal{X}_{1}}(X)=\frac{r_{1}(X)}{q_{\alpha}(X)}+\frac{\mu_{1}}{\alpha_{1}},$
(42)
and
$\mathbb{E}_{q_{\alpha}}[Z_{1}]=\mu_{1}\alpha_{1,1}/\alpha_{1}=\mu_{1}/\alpha_{1}$.
Therefore, the asymptotic variance $\sigma_{1,q_{\alpha},\beta_{1}}^{2}$ is
$\displaystyle\sigma_{1,q_{\alpha},\beta_{1}}^{2}$
$\displaystyle=\mathbb{E}_{q_{\alpha}}\left[\Big{(}Z_{1}-\mathbb{E}_{q_{\alpha}}[Z_{1}]\Big{)}^{2}\right]$
(43)
$\displaystyle=\sum_{x\in\mathcal{X}}\frac{r_{1}^{2}(x)}{q_{\alpha}(x)}\leqslant\frac{\sigma_{1,q_{1}}^{2}}{\alpha_{1}}.$
By Assumptions 3 and 4, we have $\mathbb{P}(A|x_{c})=\mathbb{P}(A|x)$ and
$q_{1}(x_{c})=\mathbb{P}(A|x_{c})p(x_{c})/\mu$, then
$\displaystyle\sigma_{1,q_{1}}^{2}$
$\displaystyle=\sum_{x\in\mathcal{X}_{1}}\left(\frac{\mathbb{P}(A|x)p(x)}{q_{1}(x)}-\mu_{1}\right)^{2}q_{1}(x)$
(44)
$\displaystyle=\sum_{x\in\mathcal{X}}\left(\frac{\mathbb{P}(A|x_{c})p(x_{c})}{q_{1}(x_{c})}-\mu\right)^{2}q_{1}(x)=0.$
Therefore, we conclude that
$\sigma_{q_{\alpha},\beta^{*}}^{2}=\sigma_{1,q_{\alpha},\beta_{1}}^{2}=0$. ∎
###### Remark 3
Assumption 1 suggests that the scenarios in $\mathcal{X}_{0}$ should not be
sampled. Since there are no crash in these scenarios, they can not make any
contribution to the estimation. Assumption 2 requires that the number of
critical variable is 1, because stratifying scenarios into different strata
leads to some residual terms (e.g., $3(\mu_{1}/\alpha_{j})^{2}$ in Eq. (41))
in estimation variance that can not be eliminated. Assumption 3 indicates that
the critical variables should dominate the crash probability, since otherwise
we may lose some critical information about the scenarios and obtain the
suboptimal testing results. Assumption 4 requires that one of the importance
functions should be optimal, together with Assumption 3 further reducing the
asymptotic variances to zero. Although in practice these assumptions may not
be fully satisfied, they could provide useful guidance for us to implement the
SCV method.
###### Remark 4
The theorems in this section hold regardless of the specifics of SMs, which
may be constructed by traditional traffic models or by neural networks.
## VI Overtaking Case Study
### VI-A Overtaking Scenarios
Figure 3: Illustration of the overtaking scenarios.
The overtaking scenarios are shown in Fig. 3, where the leading vehicle (LV)
runs at the left lane, the background vehicle (BV) follows LV and the
automated vehicle (AV) runs at the right lane. If BV cuts in to the right
lane, then AV will follow BV and may rear-end BV, resulting in a crash. The
state of the overtaking scenarios can be formulated as
$s\triangleq\big{(}v_{\mathrm{BV}},R_{1},\dot{R}_{1},R_{2},\dot{R}_{2}\big{)},$
(45)
where $R_{1}\triangleq x_{\mathrm{LV}}-x_{\mathrm{BV}}$,
$\dot{R}_{1}\triangleq v_{\mathrm{LV}}-v_{\mathrm{BV}}$, $R_{2}\triangleq
x_{\mathrm{BV}}-x_{\mathrm{AV}}$, and $\dot{R}_{2}\triangleq
v_{\mathrm{BV}}-v_{\mathrm{AV}}$. The $x_{\mathrm{BV}}$, $x_{\mathrm{LV}}$,
$x_{\mathrm{AV}}$ are the positions and $v_{\mathrm{BV}}$, $v_{\mathrm{LV}}$,
$v_{\mathrm{AV}}$ are the velocities of BV, LV and AV, respectively. The
action of the overtaking scenario is defined as the actions of LV and BV,
i.e., $a\triangleq(a_{\mathrm{LV}},a_{\mathrm{BV}})$. We note that the
overtaking scenarios are more stochastic and complicated than simple scenarios
such as cut-in scenarios and car-following scenarios, since the BV in
overtaking scenarios may have many chances to cut in, resulting in different
cut-in scenarios and car-following scenarios between BV and AV. This is the
reason why overtaking scenarios are always much more high-dimensional than
cut-in scenarios.
### VI-B Generation of NDE
The essence of NDE is to provide a driving environment where all BVs travel
like humans. To generate NDE, the probability distributions of the behaviors
of all BVs should be consistent with the naturalistic driving data (NDD) [29].
In this paper, the probability distributions of free-driving, car-following,
and cut-in behaviors are extracted from the NDD of the Safety Pilot Model
Deployment (SPMD) [34] program and Integrated Vehicle-Based Safety System
(IVBSS) [35] at the University of Michigan, Ann Arbor. The initial state is
set as
$s_{0}=[v_{\mathrm{BV},0},R_{1,0},\dot{R}_{1,0},R_{2,0},\dot{R}_{2,0}],$ (46)
where $v_{\mathrm{BV},0}$, $R_{1,0}$, $\dot{R}_{1,0}$ are sampled from the
naturalistic distributions of car-following scenarios,
$R_{2,0}\sim\mathcal{U}(20~{}\text{m},100~{}\text{m})$,
$\dot{R}_{2,0}\sim\mathcal{U}(-5~{}\text{m/s},-10~{}\text{m/s})$, where
$\mathcal{U}$ is the uniform distribution. After sampling the initial state,
all vehicles select actions independently and simultaneously for each time
step (0.1 s). The cut-in maneuver of BV is set completed within one time step.
The car-following maneuver of AV is controlled by the intelligent driver model
(IDM)[36]. The simulation continues until AV rear-ends BV or maximum
simulation time (20 s) reached. Typically, the dimension of overtaking
scenarios will exceed 1400 (201 time steps, each with 5 state variables and 2
action variables), leading to the high-dimensionality challenge.
### VI-C Generation of NADE
The goal of NADE is to generate high-dimensional testing scenarios where the
behaviors of BVs are adjusted only at critical moments, while keeping
naturalistic distributions as in NDE at other time steps [21]. To construct
the importance function, the maneuver criticality of BV is evaluated at each
time step, which is defined as the multiplication of the exposure frequency
and the maneuver challenge. The exposure frequency represents the probability
of each action given current state in NDE. The maneuver challenge measures the
probability of crash between AV and BV given current state and action. Since
the AV models are usually black-boxes, the surrogate models (SMs) are adopted
to approximate the maneuver challenge. In this paper, we use IDM and full
velocity difference model (FVDM) [36] as SMs with different parameters: (1)
IDM, denoted as SM-I; (2) FVDM with $a_{\min}=-1$ m/s2, denoted as SM-II; (3)
FVDM with $a_{\min}=-6$ m/s2, denoted as SM-III. Then the importance functions
can be obtained from the maneuver criticalities estimated by these SMs.
Readers can find more technical details in [21].
### VI-D Application of SCV
Input: $p$, $q_{\alpha}$, $X_{c,i}$, and $\mathbb{P}(A|X_{i})$, $i=1,\dots,n$
Output: $\tilde{\mu}_{q_{\alpha},\hat{\beta}}$,
$\mathrm{Var}_{q_{\alpha}}(\tilde{\mu}_{q_{\alpha},\hat{\beta}})$
1 initialize $Y_{l}$ and $H_{l}$ as empty arrays, $l=0,\dots,L$;
2 initialize $n_{l}=0$, $l=0,\dots,L$;
3 for _$i\leftarrow 1$ to $n$_ do
4 $l\leftarrow$ number of control steps of $X_{c,i}$;
5 $n_{l}\leftarrow n_{l}+1$;
6 if _$l=0$_ then
7 append $Y_{l}$ with $\mathbb{P}(A|X_{i})$;
8 append $H_{l}$ with 0;
9
10 else
11 append $Y_{l}$ with $\mathbb{P}(A|X_{i})p(X_{c,i})/q_{\alpha}(X_{c,i})$;
12 append $H_{l}$ with
$\mathrm{vec}(q_{j_{1},\dots,j_{l}}(X_{c,i})/q_{\alpha}(X_{c,i}))$,
$j_{1},\dots,j_{l}=1,\dots,J-1$;
13
14 end if
15
16 end for
17for _$l\leftarrow 0$ to $L$_ do
18 $H_{l}\leftarrow H_{l}-\mathrm{average}(H_{l})$;
19 $\mathrm{MLR}\leftarrow$ multiple linear regression of $Y_{l}$ on $H_{l}$;
20 $\hat{\beta}_{l}\leftarrow$ estimated coefficients from $\mathrm{MLR}$;
21 $\hat{\eta}_{l}\leftarrow$ estimated intercept from $\mathrm{MLR}$;
22 $\tilde{\mu}_{l,q_{\alpha},\hat{\beta}_{l}}\leftarrow
n_{l}\hat{\eta}_{l}/n$, $Z_{l}\leftarrow Y_{l}-H_{l}\hat{\beta}_{l}$;
23
24 end for
25$Z\leftarrow[Z_{0},\dots,Z_{L}]$;
26
$\tilde{\mu}_{q_{\alpha},\hat{\beta}}\leftarrow\sum_{l=0}^{L}\tilde{\mu}_{l,q_{\alpha},\hat{\beta}_{l}}$,
$\mathrm{Var}_{q_{\alpha}}(\tilde{\mu}_{q_{\alpha},\hat{\beta}})\leftarrow\mathrm{var}(Z)$;
27 return $\tilde{\mu}_{q_{\alpha},\hat{\beta}}$,
$\mathrm{Var}_{q_{\alpha}}(\tilde{\mu}_{q_{\alpha},\hat{\beta}})$;
Algorithm 1 Adaptive safety evaluation with sparse control variates by
multiple linear regression
As shown in Algorithm 1, the SCV method can be applied to adjust the testing
results and reduce estimation variance after testing AV in NADE. The key is to
use importance functions of only sparse and critical variables to construct
SCV, and then apply MLR of weighted testing results on SCV in each stratum.
Finally, the estimated performance index is given by the summation of weighted
intercepts obtained from MLR in all strata.
### VI-E Evaluation Results
We validate the accuracy and efficiency of AV evaluation in NDE and NADE by
the simulation of overtaking scenarios. The simulation is parallel conducted
using 100 threads on a computer equipped with AMD® EPYC™ 7742 CPU and 512 GB
RAM. Fig. 4 shows the crash rates of AV in NDE and NADE, respectively. The
crash rate in NDE is presented as the black line in Fig. 4, with the bottom
$x$-axis as its number of tests. The blue line in Fig. 4 represents the crash
rate in NADE, and the top $x$-axis is the number of tests. The light shadow
gives the 90% confidence interval. It can be seen that the crash rates in NDE
and NADE converge to the same value, while NADE requires a much smaller number
of tests. To measure the estimation precision of the crash rate, the relative
half-width (RHW) [19] is adopted as the metric. The threshold of RHW is set to
0.3. To reach this threshold, NADE requires 6.76 $\times$ 106 number of tests,
while NDE requires 1.21 $\times$ 108 number of tests, as shown in Fig. 5. It
can be found that NADE can accelerate the evaluation by about 17.90 times
compared with NDE. We note that the acceleration ratio is smaller than that in
[21], because combinations of multiple various SMs are applied in this paper,
which improves the robustness yet decreases the efficiency. The goal of the
adaptive evaluation is to improve the efficiency while keeping the robustness.
Figure 4: Crash rates of AV in NDE and NADE, where the dashed line is the
crash rate estimated by NDE. Figure 5: RHW of AV evaluation in NDE and NADE,
where the dashed line represents the RHW threshold (0.3). Figure 6: Crash rate
of AV using NADE and SCV for (a) $n=5.92\times 10^{5}$, (b) $n=1.29\times
10^{6}$, (c) $n=4\times 10^{6}$, (d) $n=7\times 10^{6}$ and (e) $n=1\times
10^{7}$, where $n$ is the total number of tests and the dashed line is the
crash rate estimated by NDE; (f) RHW of AV evaluation using NADE and SCV,
where the dashed line in black represents the RHW threshold (0.3) and 5 dashed
lines in orange correspond to (a)-(e).
To investigate the performance of the SCV method, the accuracy and efficiency
of AV evaluation in NADE with and without SCV are compared. It can be seen in
Fig. 6 (a)-(e) that the crash rates of NADE and SCV converge to the same value
for different number of tests. Fig. 6 (f) shows that the required numbers of
tests of NADE and SCV for reaching the RHW threshold are 6.76 $\times$ 106 and
5.92 $\times$ 105, respectively, resulting in a further acceleration ratio of
11.42. The weighted testing results before and after being adjusted by SCV
with different number of control steps are compared in Fig. 7 (a)-(i), and
Fig. 7 (j) shows the total 107 adjusted testing results. It can be seen that
the SCV method is able to adjust the testing results into a much narrower
interval, especially for relatively large number of control steps (e.g.,
$l\geqslant 4$), resulting in a considerable reduction of the estimation
variance.
Figure 7: Adjusted testing results by NADE and SCV for (a)-(i) the number of
control steps (#CS) from 1 to 9 and (j) total 107 testing results. Figure 8:
Number of tests, number of SCV and maximum rank of control matrices for
different number of control steps.
The detailed regression processes of the SCV method are also investigated.
Fig. 8 shows the number of tests, the number of SCV and the maximum rank of
the control matrices for the number of control steps $l=1,\dots,9$,
respectively. Note that for $l\geqslant 10$, we only use the first 9 control
steps to construct the SCV. It can be seen that the maximum number of tests
appears at $l=6$ and then the number of tests decreases to a relatively low
level. As shown in Eq. (27), the maximum rank of the control matrices is the
minimum value between the number of tests and the number of SCVs, and hence
will not grow exponentially with the number of control steps, although the
number of SCVs will do. Therefore, the SVD of control matrices is always
tractable in each stratum and the optimal control parameters can be found to
minimize the estimation variance.
Since the scenario generation processes are stochastic, the testing and
evaluation results are usually not the same in different experiments.
Therefore, to find the average performances, we shuffle the testing results
200 times to bootstrap them and obtain the frequency distributions of the
required number of tests (RNoT) in NDE and NADE. The average RNoT of NDE and
NADE are 1.20 $\times$ 108 and 8.71 $\times$ 106, respectively. Therefore, the
average acceleration ratio (AAR) of NADE with respect to NDE is 13.78. The
testing results of SCV are also bootstrapped by 200 times. For cases with
maximum RHW below 0.3, we use the RNoT when the maximum RHW is reached. The
average RNoT of SCV is 1.29 $\times$ 106, resulting in an AAR of 6.76 times
compared with NADE.
### VI-F Generalizability Analysis
TABLE I: AARs of SCV where AV admits IDMs with different $\alpha$ values, and the rightmost column corresponds to the VT-IDM. $\alpha$ | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | VT-IDM
---|---|---|---|---|---|---|---
AAR | 11.52 | 9.02 | 7.87 | 6.76 | 7.73 | 10.90 | 7.30
$\alpha$ | 3.5 | 4.0 | 4.5 | 5.0 | 5.5 | 6.0 |
AAR | 13.44 | 11.95 | 11.12 | 10.61 | 10.45 | 10.05 |
In the above experiments, we have set the AV model the same as SM-I, i.e.,
they are both IDMs with same parameters. To investigate the generalizability
of the SCV method for different AV models, the IDMs with a series of
parameters $\alpha=0.5,1.0,\dots,6.0$ are chosen as AV models. The AARs of SCV
compared with NADE are shown in Table I. The testing results of all AV models
are shuffled 200 times to obtain the AARs. It can be seen that the minimum AAR
appears at $\alpha=2.0$, where the AV model is the same as SM-I, while the
maximum AAR appears at $\alpha=3.5$. The mean AAR for different AV models is
10.12. Therefore, the SCV method can further accelerate the evaluation process
by about one order of magnitude for various types of AV models. Moreover, the
AARs of SCV with AV models different from SM-I are always greater than that of
AV model the same as SM-I. The reason is that although using AV models
different from SM-I will do harm to both the estimation efficiency of NADE and
SCV, the damage to NADE is more than to SCV.
In addition, we also select the calibrated IDM in [37] (denoted as VT-IDM) as
the AV model to further validate the generalization performance of the SCV
method. The testing results shuffled 200 times give an AAR of 7.30 for SCV
compared with NADE, which is shown at the rightmost column in Table I.
Therefore, the SCV method can also increase the evaluation efficiency
considerably for AV model with completely different calibrated parameters.
This is not a surprising result because the only requirement for the SCV
method to work is that the SMs and the AV model have some correlation, and
more correlation contributes to more variance reduction. Although the VT-IDM
and IDM have totally different parameters, they are still correlated to some
extent.
## VII Conclusion
In this paper, we propose an adaptive safety evaluation framework for CAVs in
high-dimensional scenarios with a newly developed sparse control variates
(SCV) method. To address the CoD, the SCV are constructed by only considering
the sparse and critical variables of testing scenarios and stratified into
strata accordingly. By optimizing the SCV leveraging the testing results
within each stratum, the estimation variance is significantly reduced for
different CAVs adaptively, accelerating the evaluation process. The accuracy,
efficiency and optimality of the proposed method are verified and validated by
both theoretical analysis and empirical studies. Comparing with the evaluation
efficiency in NDE and NADE, our method is always more efficient particularly
for CAVs that are different from SMs. It has been noted that adaptive testing
scenario generation and adaptive testing result evaluation are two
complementary approaches for adaptive testing and evaluation of CAVs. How to
develop the former in high-dimensional scenarios deserves further
investigation.
## References
* [1] N. Kalra and S. M. Paddock, “Driving to safety: How many miles of driving would it take to demonstrate autonomous vehicle reliability?” _Transportation Research Part A: Policy and Practice_ , vol. 94, pp. 182–193, 2016.
* [2] A. Li, S. Chen, L. Sun, N. Zheng, M. Tomizuka, and W. Zhan, “Scegene: Bio-inspired traffic scenario generation for autonomous driving testing,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
* [3] J. Wang, A. Pun, J. Tu, S. Manivasagam, A. Sadat, S. Casas, M. Ren, and R. Urtasun, “Advsim: Generating safety-critical scenarios for self-driving vehicles,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 9909–9918.
* [4] T. Menzel, G. Bagschik, and M. Maurer, “Scenarios for development, test and validation of automated vehicles,” in _2018 IEEE Intelligent Vehicles Symposium (IV)_. IEEE, 2018, pp. 1821–1827.
* [5] Y. Tian, K. Pei, S. Jana, and B. Ray, “Deeptest: Automated testing of deep-neural-network-driven autonomous cars,” in _Proceedings of the 40th International Conference on Software Engineering_ , 2018, pp. 303–314.
* [6] D. Rempe, J. Philion, L. J. Guibas, S. Fidler, and O. Litany, “Generating useful accident-prone driving scenarios via a learned traffic prior,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 17 305–17 315.
* [7] L. Li, W.-L. Huang, Y. Liu, N.-N. Zheng, and F.-Y. Wang, “Intelligence testing for autonomous vehicles: A new approach,” _IEEE Transactions on Intelligent Vehicles_ , vol. 1, no. 2, pp. 158–166, 2016.
* [8] L. Li, Y.-L. Lin, N.-N. Zheng, F.-Y. Wang, Y. Liu, D. Cao, K. Wang, and W.-L. Huang, “Artificial intelligence test: A case study of intelligent vehicles,” _Artificial Intelligence Review_ , vol. 50, no. 3, pp. 441–465, 2018.
* [9] L. Li, X. Wang, K. Wang, Y. Lin, J. Xin, L. Chen, L. Xu, B. Tian, Y. Ai, J. Wang _et al._ , “Parallel testing of vehicle intelligence via virtual-real interaction,” _Science Robotics_ , 2019.
* [10] S. Riedmaier, T. Ponn, D. Ludwig, B. Schick, and F. Diermeyer, “Survey on scenario-based safety assessment of automated vehicles,” _IEEE access_ , vol. 8, pp. 87 456–87 477, 2020.
* [11] G. E. Mullins, P. G. Stankiewicz, R. C. Hawthorne, and S. K. Gupta, “Adaptive generation of challenging scenarios for testing and evaluation of autonomous vehicles,” _Journal of Systems and Software_ , vol. 137, pp. 197–215, 2018\.
* [12] M. Koren, S. Alsaif, R. Lee, and M. J. Kochenderfer, “Adaptive stress testing for autonomous vehicles,” in _2018 IEEE Intelligent Vehicles Symposium (IV)_. IEEE, 2018, pp. 1–7.
* [13] S. Feng, Y. Feng, H. Sun, Y. Zhang, and H. X. Liu, “Testing scenario library generation for connected and automated vehicles: an adaptive framework,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 23, no. 2, pp. 1213–1222, 2022.
* [14] J. Sun, H. Zhou, H. Xi, H. Zhang, and Y. Tian, “Adaptive design of experiments for safety evaluation of automated vehicles,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
* [15] H. X. Liu and S. Feng, ““curse of rarity” for autonomous vehicles,” _arXiv preprint arXiv:2207.02749_ , 2022.
* [16] S. Feng, Y. Feng, X. Yan, S. Shen, S. Xu, and H. X. Liu, “Safety assessment of highly automated driving systems in test tracks: A new framework,” _Accident Analysis & Prevention_, vol. 144, p. 105664, 2020.
* [17] S. Feng, Y. Feng, C. Yu, Y. Zhang, and H. X. Liu, “Testing scenario library generation for connected and automated vehicles, part i: Methodology,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 22, no. 3, pp. 1573–1582, 2021.
* [18] S. Feng, Y. Feng, H. Sun, S. Bao, Y. Zhang, and H. X. Liu, “Testing scenario library generation for connected and automated vehicles, part ii: Case studies,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 22, no. 9, pp. 5635–5647, 2021.
* [19] D. Zhao, H. Lam, H. Peng, S. Bao, D. J. LeBlanc, K. Nobukawa, and C. S. Pan, “Accelerated evaluation of automated vehicles safety in lane-change scenarios based on importance sampling techniques,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 18, no. 3, pp. 595–607, 2016.
* [20] D. Zhao, X. Huang, H. Peng, H. Lam, and D. J. LeBlanc, “Accelerated evaluation of automated vehicles in car-following maneuvers,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 19, no. 3, pp. 733–744, 2017.
* [21] S. Feng, X. Yan, H. Sun, Y. Feng, and H. X. Liu, “Intelligent driving intelligence test for autonomous vehicles with naturalistic and adversarial environment,” _Nature Communications_ , vol. 12, no. 1, pp. 1–14, 2021.
* [22] R. Y. Rubinstein and R. Marcus, “Efficiency of multivariate control variates in monte carlo simulation,” _Operations Research_ , vol. 33, no. 3, pp. 661–677, 1985.
* [23] W. Grathwohl, D. Choi, Y. Wu, G. Roeder, and D. Duvenaud, “Backpropagation through the void: Optimizing control variates for black-box gradient estimation,” in _International Conference on Learning Representations_ , 2018\.
* [24] C.-A. Cheng, X. Yan, and B. Boots, “Trajectory-wise control variates for variance reduction in policy gradient methods,” in _Conference on Robot Learning_. PMLR, 2020, pp. 1379–1394.
* [25] A. Shapiro, “Monte carlo sampling methods,” _Handbooks in operations research and management science_ , vol. 10, pp. 353–425, 2003.
* [26] A. B. Owen, _Monte Carlo theory, methods and examples_. Stanford, 2013.
* [27] D. J. Olive, “Multiple linear regression,” in _Linear Regression_. Springer, 2017, pp. 17–83.
* [28] J. Yang, H. He, Y. Zhang, S. Feng, and H. X. Liu, “Adaptive testing for connected and automated vehicles with sparse control variates in overtaking scenarios,” in _IEEE International Intelligent Transportation Systems Conference (ITSC)_. IEEE, 2022.
* [29] X. Yan, S. Feng, H. Sun, and H. X. Liu, “Distributionally consistent simulation of naturalistic driving environment for autonomous vehicle testing,” _arXiv preprint arXiv:2101.02828_ , 2021.
* [30] R. Lowe, Y. I. Wu, A. Tamar, J. Harb, O. Pieter Abbeel, and I. Mordatch, “Multi-agent actor-critic for mixed cooperative-competitive environments,” _Advances in neural information processing systems_ , vol. 30, 2017.
* [31] S.-K. Au and J. Beck, “Important sampling in high dimensions,” _Structural safety_ , vol. 25, no. 2, pp. 139–163, 2003.
* [32] A. Owen and Y. Zhou, “Safe and effective importance sampling,” _Journal of the American Statistical Association_ , vol. 95, no. 449, pp. 135–143, 2000\.
* [33] M. E. Wall, A. Rechtsteiner, and L. M. Rocha, “Singular value decomposition and principal component analysis,” in _A practical approach to microarray data analysis_. Springer, 2003, pp. 91–109.
* [34] D. Bezzina and J. Sayer, “Safety pilot model deployment: Test conductor team report,” _Report No. DOT HS_ , vol. 812, no. 171, p. 18, 2014.
* [35] J. Sayer, D. LeBlanc, S. Bogard, D. Funkhouser, S. Bao, M. L. Buonarosa, A. Blankespoor _et al._ , “Integrated vehicle-based safety systems field operational test: Final program report,” United States. Joint Program Office for Intelligent Transportation Systems, Tech. Rep., 2011.
* [36] J. W. Ro, P. S. Roop, A. Malik, and P. Ranjitkar, “A formal approach for modeling and simulation of human car-following behavior,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 19, no. 2, pp. 639–648, 2017.
* [37] J. Sangster, H. Rakha, and J. Du, “Application of naturalistic driving data to modeling of driver car-following behavior,” _Transportation research record_ , vol. 2390, no. 1, pp. 20–33, 2013.
|
1
# Unexpected Scaling in Path Copying Trees
Ilya Kokorin ITMO UniversityRussia<EMAIL_ADDRESS>, Alexander
Fedorov IST AustriaAustria<EMAIL_ADDRESS>, Trevor Brown University
of Waterloo, CanadaCanada<EMAIL_ADDRESS>and Vitaly Aksenov ITMO
University, RussiaRussia<EMAIL_ADDRESS>
###### Abstract.
Although a wide variety of handcrafted concurrent data structures have been
proposed, there is considerable interest in universal approaches (henceforth
called Universal Constructions or UCs) for building concurrent data
structures. These approaches (semi-)automatically convert a sequential data
structure into a concurrent one. The simplest approach uses locks that protect
a sequential data structure and allow only one process to access it at a time.
The resulting data structures use locks, and hence are blocking. Most work on
UCs instead focuses on obtaining non-blocking progress guarantees such as
obstruction-freedom, lock-freedom, or wait-freedom. Many non-blocking UCs have
appeared. Key examples include the seminal wait-free UC by Herlihy, a NUMA-
aware UC by Yi et al., and an efficient UC for large objects by Fatourou et
al.
We borrow ideas from persistent data structures and multi-version concurrency
control (MVCC), most notably path copying, and use them to implement
concurrent versions of sequential persistent data structures. Despite our
expectation that our data structures would not scale under write-heavy
workloads, they scale in practice. We confirm this scaling analytically in our
model with private per-process caches.
††conference: ACM SIGPLAN Conference on Programming Languages; January 01–03,
2018; New York, NY, USA††journalyear: 2018††doi: ††copyright: none††ccs:
Software and its engineering General programming languages††ccs: Social and
professional topics History of programming languages
## 1\. Introduction
Although a wide variety of handcrafted concurrent data structures have been
proposed, there is considerable interest in universal approaches (henceforth
called _Universal Constructions_ or UCs) for building concurrent data
structures. These approaches (semi-)automatically convert a sequential data
structure into a concurrent one. The simplest approach uses locks (Herlihy et
al., 2020; Lamport, 1987) that protect a sequential data structure and allow
only one process to access it at a time. The resulting data structures use
locks, and hence are blocking. Most work on UCs instead focuses on obtaining
non-blocking progress guarantees such as _obstruction-freedom_ , _lock-
freedom_ or _wait-freedom_. Many non-blocking UCs have appeared. Key examples
include the seminal wait-free UC (Herlihy, 1991) by Herlihy, a NUMA-aware UC
(Yi et al., 2021) by Yi et al., and an efficient UC for large objects
(Fatourou et al., 2020) by Fatourou et al.
In this work, we consider the simpler problem of implementing persistent (also
called functional) data structures, which preserve the old version whenever
the data structure is modified (Okasaki, 1999). Usually this entails copying a
part of the data structure, for example, the path from the root to a modified
node in a tree (Kaplan, 2018), so that none of the existing nodes need to be
changed directly.
We borrow ideas from persistent data structures and multi version concurrency
control (MVCC) (Sun et al., 2019), most notably path copying, and use them to
implement concurrent versions of sequential persistent data structures. Data
structures implemented this way can be highly efficient for searches, but we
expect them to not scale in write-heavy workloads. Surprisingly, we found that
a concurrent treap implemented in this way obtained up to 2.4x speedup
compared to a sequential treap (Seidel and Aragon, 1996) with 4 processes in a
write-heavy workload. We present this effect experimentally, and analyze it in
a model with private per-processor caches: informally, as the number of
processes grows large, speedup in our treap of size $N$ tends to $\Omega(\log
N)$.
## 2\. Straightforward Synchronization for Persistent Data Structures
In the following discussion, we focus on rooted data structures, but one could
imagine generalizing these ideas by adding a level of indirection in data
structures with more than one entry point (e.g., one could add a dummy root
node containing all entry points).
We store a pointer to the current version of the persistent data structure
(e.g., to the root of the current version of a persistent tree) in a Read/CAS
register called Root_Ptr.
Read-only operations (queries) read the current version and then execute
sequentially on the obtained version. Note that no other process can modify
this version, so the sequential operation is trivially atomic.
Modifying operations are implemented in the following way: 1) read the current
version; 2) obtain the new version by applying the sequential modification
using path copying (i.e., by copying the root, and copying each visited node);
3) try to atomically replace the current version with the new one using CAS;
if the CAS succeeds, return: the modifying operation has been successfully
applied; otherwise, the data structure has been modified by some concurrent
process: retry the execution from step (1). This approach clearly produces a
lock-free linearizable data structure.
We expect read-only operations to scale extremely well. Indeed, two processes
may concurrently read the current version of the persistent data structure and
execute read-only persistent operations in parallel.
However, modification operations seemingly afford no opportunity for scaling.
When multiple modifications contend, only one can finish successfully, and the
others must retry. For example, consider concurrent modification operations on
a set: 1) process P calls insert(2) and fetches the current pointer RP; 2)
process Q calls remove(5) and fetches the current pointer RP; 3) P constructs
a new version $\texttt{RP}_{\texttt{P}}$ with key 2; 4) Q constructs a new
version $\texttt{RP}_{\texttt{Q}}$ without key 5; 5) P successfully executes
CAS(&Set.Root_Pointer, RP, $\texttt{RP}_{\texttt{P}}$); 6) Q executes CAS from
RP to $\texttt{RP}_{\texttt{Q}}$ but fails; thus, Q must retry its operation.
Successful modifications are applied sequentially, one after another.
Intuitively, this should not scale at all in a workload where all operations
must perform successful modifications. As we will see in Section 4, this
intuition would be incorrect.
## 3\. Analysis
The key insight is that failed attempts to perform updates load data into
processor caches that may be useful on future attempts. To better understand,
consider the binary search tree modification depicted in Fig. 1. Suppose we
want to insert two keys: 5 and 75. We compare how these insertions are
performed sequentially and concurrently.
At first, we consider the sequential execution. We insert key 5 into the tree.
It should be inserted as a left child of 10. Thus, we traverse the tree from
the root to the leaf 10. On the way, we fetch nodes {40, 30, 20, 10} into the
processor’s cache. Note this operation performs four uncached loads.
Now, we insert 75. It should be inserted as the right child of 70. Our
traversal loads four nodes: {40, 50, 60, 70}. Node 40 is already cached, while
three other nodes must be loaded from memory. Thus, we perform three uncached
loads, for a total of seven uncached loads.
Now, we consider a concurrent execution with two processes, in which P inserts
5 and Q inserts 75. Initially, both processes read Root_Ptr to load the
current version. Then, 1) P traverses from the root to 10, loading nodes {40,
30, 20, 10}, and 2) Q traverses from the root to 70, loading nodes {40, 50,
60, 70}.
Each process constructs a new version of the data structure, and tries to
replace the root pointer using CAS. Suppose P succeeds and Q fails. Q retries
the operation, but on the new version (Fig. 1). Note that the new version
shares most nodes with the old one.
Figure 1. The new version (green) of the tree shares its nodes with the old
version (white)
Q inserts 75 into the new version. Again, the key should be inserted as the
right child of 70. Q loads four nodes {40, 50, 60, 70} from the new version of
the tree. Crucially, nodes {50, 60, 70} are already cached by Q. This retry
only incurs one cache miss!
Thus, there are only five serialized loads in the concurrent execution,
compared to seven in the sequential execution.
### 3.1. High-level analysis
We use a simple model that allows us to analyze this effect. (The full proof
appears in Appendix A.) In this model, the processes are synchronous, i.e.,
they perform one primitive operation per tick, and each process has its own
cache of size $M$. We show that for a large number of processes $P$, the
speedup is $\Omega(\log N)$, where $N$ is the size of the tree.
Now, we give the intuition behind the proof. To simplify it, we suppose that
the tree is external and balanced, i.e., each operation passes though $\log N$
nodes. We also assume that the workload consists of successful modification
operations on keys chosen uniformly at random. We first calculate the cost of
an operation for one process: $(\log N-\log M)\cdot R+\log M$ where
$M=O(N^{1-\varepsilon})$ is the cache size and $R$ is the cost of an uncached
load. This expression captures the expected behaviour under least-recently-
used caching. The process should cache the first $\log M$ levels of the tree,
and thus, $\log M$ nodes on a path are in the cache and $\log N-\log M$ are
not.
To calculate the throughput in a system with $P$ processes, we suppose that
$P$ is quite large ($\approx\Omega(min(R,\log N))$). Thus, each operation
performs several unsuccessful attempts, ending with one successful attempt,
and all successful attempts (over all operations) are serialized. Since the
system is synchronous, each operation attempt $A$ loads the version of the
data structure which is the result of a previous successful attempt
$A^{\prime}$. The nodes evicted since the beginning of $A$ are those created
by $A^{\prime}$. One can show that in expectation only two nodes on the path
to the key are uncached. Finally, the successful attempt of an operation
incurs cost $2\cdot R+(\log N-2)$. Since successful attempts are serialized,
the expected total speedup is $\frac{(\log N-\log M)\cdot R+\log M}{2\cdot
R+(\log N-2)}$ giving $\Omega(\log N)$ with $R=\Omega(\log N)$.
## 4\. Experiments
We implemented a lock-free treap and ran experiments comparing it with a
sequential treap in Java on a system with an 18 core Intel Xeon 5220. Each
data point is an average of 15 trials. We highlight the following two
workloads. (More results appear in Appendix B.)
### 4.1. Batch inserts and batch removes
Suppose we have $P$ concurrent processes in the system. Initially the set
consists of $10^{6}$ random integer keys. Processes operate on mutually
disjoint sets of keys. Each process repeatedly: inserts all of its keys, one
by one, then removes all of its keys. Since the key sets are disjoint, each
operation successfully modifies the treap. We report the speedup for our treap
over the sequential treap below.
### 4.2. Random inserts and removes
In this workload, we first insert $10^{6}$ random integers in [$-10^{6}$;
$10^{6}$], then each process repeatedly generates a random key and tries to
insert/remove it with equal probability. Some operations do not modify the
data structure (e.g., inserting a key that already exists).
Workload | Seq Treap | UC 1p | UC 4p | UC 10p | UC 17p
---|---|---|---|---|---
Batch | $451\,940$ | 0.89x | 1.23x | 1.47x | 1.47x
Random | $419\,736$ | 1.48x | 2.38x | 3.07x | 3.19x
## References
* (1)
* Fatourou et al. (2020) Panagiota Fatourou, Nikolaos D Kallimanis, and Eleni Kanellou. 2020. An efficient universal construction for large objects. _arXiv_ (2020).
* Herlihy (1991) Maurice Herlihy. 1991\. Wait-free synchronization. _ACM Transactions on Programming Languages and Systems (TOPLAS)_ 13, 1 (1991), 124–149.
* Herlihy et al. (2020) Maurice Herlihy, Nir Shavit, Victor Luchangco, and Michael Spear. 2020. _The art of multiprocessor programming_. Newnes.
* Kaplan (2018) Haim Kaplan. 2018\. Persistent data structures. In _Handbook of Data Structures and Applications_. Chapman and Hall/CRC, 511–527.
* Lamport (1987) Leslie Lamport. 1987\. A fast mutual exclusion algorithm. _ACM Transactions on Computer Systems (TOCS)_ 5, 1 (1987), 1–11.
* Okasaki (1999) Chris Okasaki. 1999\. _Purely functional data structures_. Cambridge University Press.
* Seidel and Aragon (1996) Raimund Seidel and Cecilia R Aragon. 1996. Randomized search trees. _Algorithmica_ 16, 4 (1996), 464–497.
* Sun et al. (2019) Y Sun, G Blelloch, W Lim, and A Pavlo. 2019\. On supporting efficient snapshot isolation for hybrid workloads with multi-versioned indexes. _VLDB_ 13, 2 (2019).
* Yi et al. (2021) Z Yi, Y Yao, and K Chen. 2021. A Universal Construction to implement Concurrent Data Structure for NUMA-muticore. In _50th ICPP_. 1–11.
## Appendix A Mathematical model
### A.1. Sequential execution
Let us estimate how much time is spent on executing $T$ operations
sequentially on a binary search tree. Suppose our binary search tree is
_external_ , i.e., data is contained only in leaves, while internal nodes
maintain only routing information. Suppose tree contains $N$ keys and the tree
is balanced, therefore the tree height is $O(\log N)$. We suppose _uniform
workload_ : all keys from the tree are accessed uniformly at random.
Suppose the cache size is $M=O(N^{1-\varepsilon})$, therefore, approximately
upper $\log M$ levels of the tree are cached, while $\log N-\log M$ lower
levels of the tree are not (Fig. 2).
Figure 2. Upper levels of the tree are cached, while lower levels reside in
RAM
Each operation first loads $\log M$ nodes from the cache, spending $1$ time
unit per each cache fetch. After that, it loads $\log N-\log M$ nodes from the
RAM, spending $R$ time units per RAM fetch.
Thus, the sequential execution will take
$T\cdot\left(\log M+R\cdot\left(\log N-\log M\right)\right)$ time units to
finish, where $T$ is the number of operations.
### A.2. Concurrent execution
Suppose we have $P$ concurrent processes $\\{t_{i}\\}_{i=1}^{P}$ executing
operations concurrently, while each process has its own cache of size larger
than $\log N$.
In our model we assume that each successful try of a modifying operation
causes $p-1$ unsuccessful tries of modifying operations on other processes
(Fig. 3).
Figure 3. Each successful try of an operation causes unsuccessful tries of
$p-1$ operations
We also assume that operation completion events are distributed among
processes in a round-robin pattern: first process $t_{1}$ executes its
successful try of the operation, then process $t_{2}$ executes its successful
try of the operation, and so on. Finally, $t_{P}$ manages to complete its
operation, the next process to get its successful try is yet again $t_{1}$
(Fig. 4).
Figure 4. Nearly each successful modifying operation consists of $P$ retries:
$P-1$ unsuccessful and one successful
As follows from the diagram, almost each successful try of an operation is
preceded by $P-1$ unsuccessful retries (except for $P-1$ first successful
operation, which are preceded by the lower number of unsuccessful retries).
Let us estimate, how long the first retry takes to execute. We must load $\log
N$ nodes, none of which might be cached. Thus, we spend $R\cdot\log N$ time
units on the first retry.
Let us estimate now how much time we spend on subsequent retries. We begin
with estimating, how many nodes on the path to the requested leaf have been
modified (Fig. 5).
Figure 5. The number of modified nodes on the path to the requested node
Consider the successful modifying operation $op$, that led to a latest failure
of our CAS and made us retry our operation the last time. Remember, that
arguments of operations are chosen uniformly at random, therefore:
* •
There is $\frac{1}{2}$ probability that $op$ modified some leaf from
Root->Right subtree, thus, the number of modified nodes on our path is $1$;
* •
Similarly, there is $\frac{1}{4}$ probability that the number of modified
nodes on our path is $2$;
$\ldots$
* •
Similarly, there is $\frac{1}{2^{k}}$ probability that the number of modified
nodes on our path is $k$.
Thus, we can calculate the expected number of modified nodes on our path
$\sum\limits_{k=1}^{\log
N}\frac{k}{2^{k}}\leq\sum\limits_{k=1}^{\infty}\frac{k}{2^{k}}=2$. Thus, the
expected number of modified nodes on our path is not greater than $2$.
Modified nodes were created by another process, thus they do not exist in our
process cache. Therefore, they should be loaded out-of-cache, while all the
remaining nodes reside in the local cache and can be loaded directly from it.
Therefore, we spend $2\cdot R$ time on average to load all the necessary
nodes. In addition, we spend $\log N-2$ time on average to load all the
necessary nodes from the the local cache. Therefore, we spend $2\cdot R+\log
N-2$ time to fetch all the nodes required for a last operation retry.
An operation execution consists of the first retry, executed in $R\cdot\log N$
and $P-1$ subsequent retries executed in $(P-1)\cdot(2\cdot R+\log N-2)$.
Thus, a single operation is executed in $R\cdot\log N+(P-1)\cdot(2\cdot R+\log
N-2)$.
Therefore, we execute $T$ operations in
$\frac{T\cdot R\cdot\log N+T\cdot(P-1)\cdot(2\cdot R+\log N-2)}{P}$ time,
since we execute these operations in parallel on $P$ processes.
To measure the speedup we simply divide the sequential execution time by
parallel execution time:
$\frac{T\cdot\left(\log M+R\cdot\left(\log N-\log
M\right)\right)}{\frac{T\cdot R\cdot\log N+T\cdot(P-1)\cdot(2\cdot R+\log
N-2)}{P}}=P\cdot\frac{\log M+R\cdot\left(\log N-\log M\right)}{R\cdot\log
N+(P-1)\cdot(2\cdot R+\log N-2)}$. This gives us $\Omega(\log N)$ speedup when
$P=\Omega(min(R,\log N))$ and $R=\Omega(\log N)$.
## Appendix B Experiments on other processors
We did the same experiments on Intel Xeon Platinum 8160 with 24 cores and AMD
EPYC 7662 with 64 cores.
Workload | Seq Treap | UC 1p | UC 6p | UC 12p | UC 23p
---|---|---|---|---|---
Batch | $638\,600$ | 0.93x | 1.31x | 1.37x | 1.08x
Random | $487\,161$ | 1.24x | 3.23x | 3.55x | 2.8x
Table 1. Results for Intel Xeon Platinum 8160.
Workload | Seq Treap | UC 1p | UC 8p | UC 16p | UC 32p | UC 63p
---|---|---|---|---|---|---
Batch | $459\,580$ | 0.96x | 1.7x | 1.91x | 1.55x | 1.02x
Random | $396\,898$ | 1.36x | 3.63x | 2.41x | 2.81x | 2.3x
Table 2. Results for AMD EPYC 7662.
Unfortunately, one can see that the results are not so impressive when the
number of processes is large enough. We suggest that the bottleneck for our
benchmarks occurs in Java memory allocator.
|
11institutetext: Center of Ubiquitous Computing
Faculty of Information Technology and Electrical Engineering
University of Oulu, Finland
# The Limits of Learning and Planning:
Minimal Sufficient Information Transition Systems
Basak Sakcak Vadim Weinstein Steven M. LaValle This work was supported by a
European Research Council Advanced Grant (ERC AdG, ILLUSIVE: Foundations of
Perception Engineering, 101020977), Academy of Finland (projects PERCEPT
322637, CHiMP 342556), and Business Finland (project HUMOR 3656/31/2019).
(e-mail: [email protected]).
###### Abstract
In this paper, we view a policy or plan as a transition system over a space of
information states that reflect a robot’s or other observer’s perspective
based on limited sensing, memory, computation, and actuation. Regardless of
whether policies are obtained by learning algorithms, planning algorithms, or
human insight, we want to know the limits of feasibility for given robot
hardware and tasks. Toward the quest to find the best policies, we establish
in a general setting that minimal information transition systems (ITSs) exist
up to reasonable equivalence assumptions, and are unique under some general
conditions. We then apply the theory to generate new insights into several
problems, including optimal sensor fusion/filtering, solving basic planning
tasks, and finding minimal representations for feasible policies.
###### keywords:
planning, sensing uncertainty, information spaces, theoretical foundations
## 1 Introduction
Robotics increasingly appears as an application area for other fields. It is a
frequent target for designing and testing machine learning algorithms,
planning algorithms, sensor fusion methods, control laws, and so on. This may
lead many to believe that robotics itself does not have its own, unique
theoretical core (on this, we agree with Koditschek [6]). Thus, are we missing
something? Surely robotics is not a pure algorithmic problem or pure nonlinear
control problem. Could there be a theory that plays a similar role to Turing
machines for computer science, or ${\dot{x}}=f(x,u)$ over differentiable
manifolds for control theory, and yet is distinct from both? Can we formulate
and potentially answer questions such as: Does a solution even exist to the
problem? What are the minimal necessary components to solve it? What should
the best learning approach imaginable produce as a representation? Such
questions would be analogous to existence and uniqueness in control and
dynamical systems, or decidability and complexity (especially Kolomogorov) in
theoretical computer science.
| |
---|---|---
(a) | | (b)
Figure 1: (a)The internal robot brain is defined as an ITS that interacts with
the external world (robot body and environment). (b)Coupled internal and
external systems mathematically capture sensing, actuation, internal
computation, and the external world.
This paper proposes a robotics theory that is built from the input-output
relationships between a programmable mechanical system (robot) and its
environment via sensing and actuation; see Figure 1(a). The key is to focus on
necessary and sufficient conditions that a robot’s internal processor (i.e.,
“brain”) must maintain to solve required tasks, such as collision-free
navigation or coverage. We assume that the robot hardware and actuation model
are fixed, and that for a given set of tasks in a space of environments, we
must determine the weakest amount of sensing, actuation, and computation that
would be sufficient for solving tasks. We will call such conditions minimal
sufficient: If you take away anything from a minimal sufficient system, the
tasks will become unsolvable.
We introduce the notion of an information transition system (ITS) to formally
model the robot’s brain (as well as any other system observers). The
“information” part of an ITS is inspired by von Neumann’s definitions in the
context of sequential games with hidden information (and not Shannon’s later
notion of information theory). This inspired the development of information
spaces [8] (Chapter 11) as a foundation of planning with imperfect state
information due to sensing uncertainty. The concept of sufficient information
mappings appears therein. It is generalized in this paper, and the state space
of each ITS will in fact be an information space.
In our work, the ITS and its underlying information space serve as the domain
over which a plan or policy can be expressed and analyzed. Note that prior
work in planning usually assumes that the space one plans over is fixed, as in
a configuration space or state (phase) space based on the robot’s mobility.
Even the information spaces described in [8] remain fixed in the planning
phase. A notable exception is by O’Kane and Shell [11], in which information
spaces for passive filtering are reduced algorithmically, and is closely
connected to this paper. All such spaces will be considered here as potential
information spaces, and we intend to reduce or collapse them as much as
possible in the development of an information-feedback plan. This is perhaps
closer to the goals of machine learning, in which candidate representations
are determined through optimization of discrepancies with respect to input-
output data. In this paper, we in fact consider both model-free and model-
based ITSs, in alignment with the choices commonly found in machine learning
[2].
The robot’s ITS is coupled to the physical world, which is itself modeled as a
transition system. Note that the physical world model is not “given” to the
robot; we will formalize notions of “who gets what” information in Section 3.
The coupled system is inspired by neuroscience models (for example, [4]). Many
of the concepts in this paper build upon [12], in which we recently proposed
an enactivist-oriented model of cognition based on information spaces. By
enactivist [5], it is meant that the necessary brain structures emerge from
sensorimotor interaction and do not necessarily have predetermined, meaningful
components as in the classical representationalist sense.
Section 2 provides a mathematical formulation of robot-environment interaction
as transition systems. Section 3 develops notions of sufficiency and
minimality over the space of possible ITSs. Section 4 applies the general
concepts to address what it means to solve both passive (filtering) and active
(planning/control) tasks minimally and Section 5 provides simple examples.
## 2 Mathematical Models of Robot-Environment Systems
#### 2.0.1 Internal and external systems
In this paper, we will consider a robot embedded in an environment and
describe this system as two subsystems, named internal and external, connected
through symmetric input-output relations. External refers to the system
describing the physical world, and internal is the complement of it. This
interaction is shown in Figure 1(b). In this sense, the states of the external
and internal systems are similar to the use of the term in control theory and
computer science, respectively.
External system corresponds to the totality of the environment and the robot
body within it. Let $X$ denote the set of states of this system; a state could
be for example, the configuration of the robot in a known environment (or
within a set of possible environments) or its phase. There are no restrictions
on $X$; it may be discrete, an $n$-dimensional manifold, a function space, and
so on. Next, let $U$ be the set of control inputs (also referred to as
actions) such that when applied at state $x\in X$ causes it to change
according to a state transition function $f:X\times U\rightarrow X$. The set
$U$ can also be anything: a finite or infinite discrete set, a compact or non-
compact manifold, and so on. Similarly, at each state $x$, $y=h(x)$ is the
output in which $h:X\rightarrow Y$ is a state-based sensor mapping and $Y$ is
the set of all possible observations.
The internal system (robot’s brain) observes the external system through a
sensor mapping and interacts with it through a selection of actions with
respect to a policy (alternatively, we can call it a plan or a strategy).
Therefore, the input to the internal system is an observation and its output
is an action. The states of this system correspond to the retained information
gathered through the outcomes of actions in terms of sensor observations. To
this end, the basis of our mathematical formulation of the internal is the
notion of __information space_ (I-space)_ [8]. We will use the term
__information state_ (I-state)_ to refer to the state of the internal system
and denote it with ${\iota}$, and ${\cal I}$ will denote the set of all
I-states, that is, the _I-space_. Note that the notions I-space and I-state
are not exclusive to the internal system and we will use them in a more
general setting in the following sections. Similar to the external system, the
internal system evolves with each $y\in Y$ according to the information
transition function $\phi:{\cal I}\times Y\rightarrow{\cal I}$. The output
then corresponds to the control command given by an information feedback
policy $\pi:{\cal I}\rightarrow U$.
Finally, we can write the coupled dynamical system composed of these two
subsystems defined as external and internal as
$\displaystyle\centering x^{\prime}\@add@centering$
$\displaystyle=f(x,\pi({\iota}))$ $\displaystyle y=$ $\displaystyle h(x)$
$\displaystyle{\iota}^{\prime}$ $\displaystyle=\phi({\iota},h(x))$
$\displaystyle u=$ $\displaystyle\pi({\iota}).$ (1)
Whereas the equations on the left side describe the evolution of this coupled
system, the ones on the right show the respective outputs of each subsystem.
Given an initial state $(x_{1},{\iota}_{1})\in X\times{\cal I}$, there exists
a unique state-trajectory.
Suppose the system evolves in discrete stages. For the external system,
starting from an initial state $x_{1}$, each stage $k$ corresponds to applying
an action $u_{k}$ which then yields the next stage $k+1$ and the next state
$x_{k+1}=f(x_{k},u_{k})$. As the system evolves through stages,
${\tilde{x}}_{k}=(x_{1},x_{2},\dots,x_{k})$,
${\tilde{u}}_{k-1}=(u_{1},u_{2},\dots,u_{k-1})$,
${\tilde{y}}_{k}=(y_{1},y_{2},\dots,y_{k})$, correspond to the state, action
and observation histories up to stage $k$, respectively. Note that applying
the action $u_{k}$ at stage $k$ would result in a transition to state
$x_{k+1}$ and the corresponding sensor reading $y_{k+1}=h(x_{k+1})$. The same
applies for the internal system, we can describe its evolution, starting from
an initial I-state ${\iota}_{0}$ following the state transition equation
${\iota}_{k}=\phi({\iota}_{k-1},y_{k})$. At stage $k$, $\pi({\iota}_{k})$
would produce the action $u_{k}$. Note that the stage index of the I-state
starts from $0$, this corresponds to any prior information the internal system
might have regarding the external; ${\iota}_{1}$ is then obtained using
${\iota}_{0}$ and $y_{1}$. Furthermore, $\pi({\iota}_{0})=u_{0}=()$ for all
policies $\pi$, meaning that no action is outputted at this stage.
#### 2.0.2 Generalizing to transition systems
Without loss of generality, we can describe the internal and external
subsystems as transition systems of the form $(S,\Lambda,T)$ in which $S$ is
the set of states, $\Lambda$ is the set of names for the outgoing transitions,
and $T\subset S\times\Lambda\times S$ is a ternary relation describing the
transitions. If for each $(s,\lambda)\in S\times\Lambda$ there is a unique
$s^{\prime}\in S$ such that $(s,\lambda,s^{\prime})\in T$, then we will write
this system as $(S,\Lambda,\tau)$ in which $\tau:S\times\Lambda\rightarrow S$
is a function, and call the system an automaton. This corresponds to a
deterministic system. Note that our definition of an automaton differs from
the one usually used in computer science in the sense that ours do not
necessarily have a start state and a set of accepting states, and it is not
necessarily finite. Suppose ${\cal T}:S\times\Lambda\rightarrow{\rm pow}(S)$,
in which ${\rm pow}(\cdot)$ denotes the power set. Then, the transition system
$(S,\Lambda,{\cal T})$ is a nondeterministic automaton.
In [12], we have used the notion of state-relabeled transition systems to
model the internal and external systems. A state-relabeled transition system
is the quintuple $(S,\Lambda,T,\sigma,L)$ in which $\sigma:S\rightarrow L$ is
a labeling function and $(S,\Lambda,T)$ is a transition system. Preimages of a
labeling function $\sigma$ induce a partitioning of the state space $S$. Let
$S/\sigma$ be the set of equivalence classes $[s]_{\sigma}$ induced by
$\sigma$ such that $S/\sigma=\\{[s]_{\sigma}\mid s\in S\\}$ and
$[s]_{\sigma}=\\{s^{\prime}\in S\mid\sigma(s^{\prime})=\sigma(s)\\}$. Then, we
can define a new transition system $(S/\sigma,\Lambda,T/\sigma)$ called the
_quotient_ of $(S,\Lambda,T)$ by $\sigma$, in which
$T/\sigma=\\{\left([s]_{\sigma},\lambda,[s^{\prime}]_{\sigma}\right)\mid(s,\lambda,s^{\prime})\in
T\\}$. Note that $(S/\sigma,\Lambda,T/\sigma)$ is a reduced version of
$(S,\Lambda,T)$. We might be interested in finding a labeling function
$\sigma$ such that the corresponding quotient transition system is as simple
as possible while ensuring that it is still useful. In the following sections,
we will provide motivations for a reduction and discuss in more detail the
requirements on $\sigma$ for the quotient system to be useful.
Considering the deterministic case and the description of an automaton given
above, external and internal systems can be written as state-relabeled
automata $(X,U,f,h,Y)$ and $({\cal I},Y,\phi,\pi,U)$, respectively, in which
$h$ and $\pi$ are considered as labeling functions. Interpreting the labels as
the output of a transition system, coupled internal-external system can be
described in terms of the state-relabeled transition systems formulation too
such that output of one transition system is an input for another. Described
this way, coupling of two transition systems result in unique paths in either
automaton initialized at a particular state.
## 3 Sufficient Information Transition Systems
### 3.1 Information transition systems
In the general setting, an I-state corresponds to the available (stored)
information at a certain stage with respect to the action and observation
histories. Consequently, an I-space refers to the collection of all possible
I-states. We will use the term __information transition system_ (ITS)_ to
refer to a transition system whose state space is an I-space. We have already
used the notion of I-space while modeling the internal system representing the
robot brain, which makes it an ITS. Here, we extend the notion of an ITS to
include different perspectives from which the external and the coupled system
is viewed. In particular, we identified three perspectives corresponding to 1)
a plan executor which corresponds to the robot brain 2) a planner, and 3) an
(independent) observer. With a slight abuse of previously introduced notation
and terminology, we will use the term “internal” to refer to any system that
is not the external and we will use ${\cal I}$ to denote a generic I-space. We
describe an ITS in a robot-centric way such that an observation will refer to
a sensor-reading, that is, $y$. However, an independent observer defined over
the coupled system can observe, at stage $k$, both the action taken $u_{k-1}$
and the corresponding sensor-reading $y_{k}$.
Recall that the information regarding the external is obtained through the
sensor-mapping and any potential prior knowledge. Suppose that no policy is
fixed over the I-space. Then, the corresponding internal system can be modeled
as an ITS of the form $({\cal I},U\times Y,\phi$), in which $\phi:{\cal
I}\times(U\times Y)\rightarrow{\cal I}$ is a state (information) transition
function, if it is deterministic. We will then use the term _deterministic
information transition system_ (DITS) to refer to them. Otherwise, it is
called a _nondeterministic information transition system_ (NITS) and described
as $({\cal I},U\times Y,\Phi)$, in which $\Phi\subseteq{\cal I}\times(U\times
Y)\times{\cal I}$ is the transition relation. This formulation corresponds to
the perspectives other than the plan executor such that it is possible to take
any action from an I-state as it is not constrained by a policy, hence the
outgoing transitions are determined by the elements of $U\times Y$.
A plan executor corresponds to the internal system (robot’s brain) described
in the previous section. The only information regarding the external is gained
through manipulating the state of the external system through actions and
obtaining the corresponding sensor readings. Recall the representation used in
the previous section, that is, $({\cal I},Y,\phi)$, and $\pi:{\cal
I}\rightarrow U$ a labeling function. This can be considered as a constrained
version of the DITS described in the previous paragraph such that the
transitions are restricted to those that can be realized under $\pi$. To show
that, we augment the definition of internal system corresponding to the robot
brain such that the transitions now also correspond to labels. Let $({\cal
I},U\times Y,\Phi)$ be the augmented transition system describing the internal
such that
$\Phi=\\{({\iota},(u,y),{\iota}^{\prime})\in{\cal I}\times(U\times
Y)\times{\cal I}\mid u=\pi({\iota})\land{\iota}^{\prime}=\phi({\iota},y)\\},$
(2)
by construction, this augmented ITS is also deterministic111We could use the
same approach for the external system too. In that case, let $(X,Y\times U,F)$
be this augmented transition system corresponding to the external, in which
$F$ is the set of transitions such that $F=\\{(x,(y,u),x^{\prime})\in
X\times(Y\times U)\times X\mid y=h(x)\land x^{\prime}=f(x,u)\\}$. Further
creating bipartite graphs (for either system) such that transitions from a
state correspond either to an observation $y\in Y$ or to an action $u\in U$
allows us to describe the coupling as a form of intersecting two automata.
However, because it is not central to this paper we will not elaborate on this
topic.. Suppose $({\cal I},U\times Y,\Phi^{\prime})$ is the DITS that is not
constrained by a policy. Then, $\Phi\subseteq\Phi^{\prime}$.
### 3.2 History information spaces
An I-space constitutes the state space of an ITS. Therefore, we describe the
basic I-space named _history I-space_ denoted as ${\cal I}_{hist}$. It will be
used to derive other I-spaces as well. A _history I-state_ at stage $k$
corresponds to all the information that is gathered through sensing (and
potentially also through actions) up to stage $k$ assuming perfect memory. Let
${\eta}_{k}$ denote the history I-state at stage $k$, that is
${\eta}_{k}=({\eta}_{0},{\tilde{u}}_{k-1},{\tilde{y}}_{k})$, in which
${\eta}_{0}$ is the initial condition. Recall that ${\tilde{u}}_{0}$ is
assumed to be the null-tuple, hence, ${\tilde{u}}_{k}$ starts with $u_{1}$ for
any $k>1$.
Let ${\cal I}_{0}$ be the set of initial conditions whose description varies
with the available prior information. We defer the descriptions of possible
${\cal I}_{0}$ to the following paragraph. The history information space at
stage $k$ is expressed as ${\cal I}_{k}={\cal
I}_{0}\times{\tilde{U}}_{k-1}\times{\tilde{Y}}_{k}$. In general, the number of
stages that the system will go through is not fixed. Therefore, we can define
history I-space as the union over all $k\in\mathbb{N}$, that is, ${\cal
I}_{hist}=\bigcup_{k\in\mathbb{N}}{\cal I}_{k}$. The DITS corresponding to
${\cal I}_{hist}$ becomes $({\cal I}_{hist},U\times Y,{\phi_{hist}})$, in
which
${\eta}_{k}={\phi_{hist}}({\eta}_{k-1},u_{k-1},y_{k})={\eta}_{k-1}{}^{\frown}u_{k-1}{}^{\frown}y_{k}$
and ⌢ is the concatenation operation that adds an element at the end of a
sequence.
We consider two categories of initial conditions depending on whether
information regarding the state space $X$ of the external system is available
or not. Suppose $X$ or any information regarding $X$ is not given. Then, an
I-state at stage $k$ simply is
${\eta}_{k}=({\tilde{u}}_{k-1},{\tilde{y}}_{k})$, that is, the concatenation
of action and observation histories up till stage $k$. We call this type of
history I-space, the _model-free history I-space_, and respectively call the
corresponding ITS, _model-free history ITS_. In this case, we can treat
${\eta}_{0}$ as ${\eta}_{0}=()$. Thus, ${\cal I}_{0}=\\{()\\}$. For the second
category of initial conditions, full or partial information regarding $X$,
against which the actions and observations can be interpreted, is given. We
will then use the terms _model-based history I-space_ and _model-based history
ITS_ to refer to the respective I-space and ITS. The initial condition
${\eta}_{0}$ could be (i) a known state $x_{1}\in X$ such that ${\cal
I}_{0}=X$, (ii) a set of possible initial states $X_{1}\subset X$ such that
${\cal I}_{0}={\rm pow}(X)$ or (iii) a probability distribution $P(x_{1})$
over $X$ such that ${\cal I}_{0}\subseteq\mathcal{P}(X)$, in which
$\mathcal{P}(X)$ is the set of all probability distributions over $X$.
### 3.3 Sufficient state-relabeling
In [12] we have introduced a notion of _sufficiency_ that substantially
generalizes the definition in Chapter 11 of [8] and is presented here for
completeness.
###### Definition 3.1 (Sufficient state-relabeling).
Let $(S,\Lambda,T)$ be a transition system. A labeling function
$\sigma:S\rightarrow L$ defined over the states of a transition system is
sufficient if and only if for all $s,q,s^{\prime},q^{\prime}\in S$ and all
$\lambda\in\Lambda$, the following implication holds:
$\sigma(s)=\sigma(q)\land(s,\lambda,s^{\prime})\in
T\land(q,\lambda,q^{\prime})\in
T\implies\sigma(s^{\prime})=\sigma(q^{\prime}).$
If $\sigma$ is defined over the states of an automaton $(S,\Lambda,\tau)$,
then $\sigma$ is sufficient iff for all $s,q\in S$ and all
$\lambda\in\Lambda$, $\sigma(s)=\sigma(q)$ implies that
$\sigma(\tau(s,\lambda))=\sigma(\tau(q,\lambda))$.
Consider the stage-based evolution of external system $(X,U,f,h,Y)$ with
respect to the action (control input) sequence
${\tilde{u}}_{k-1}=(u_{1},\dots,u_{k-1})$. This corresponds to the state and
observation histories till stage $k$, that are
${\tilde{x}}_{k}=(x_{1},\dots,x_{k})$ and
${\tilde{y}}_{k}=(y_{1},\dots,y_{k})$. Recall that applying $u_{k}$ at stage
$k$ would result in a transition to $x_{k+1}$ and the corresponding
observation $y_{k+1}=h(x_{k+1})$. Hence, in this context, sufficiency of $h$
implies that given the label $y_{k}=h(x_{k})$ and the action $u_{k}$, it is
possible to determine the label $y_{k+1}=h(x_{k+1})$. One interpretation of
sufficiency of $h$ is that the respective quotient system sufficiently
represents the underlying system up to the induced equivalence classes. This
notion is similar to minimal realization of a system, that is, the minimal
state space description that models the given input-output measurements (see
for example [7]). Second interpretation is in a predictive sense. Suppose the
quotient system is known. Then, the label $y_{k+1}=h(x_{k+1})$ can be
determined before the system gets to $x_{k+1}$, using the current label
$y_{k}$ and the action to be applied $u_{k}$. Furthermore, under a fixed
policy, complete observation-trajectory can be determined from the initial
observation by induction.
Now, consider an internal system with a labeling function ${\kappa}:{\cal
I}\rightarrow{\cal I}^{\prime}$, that is, $({\cal I},U\times
Y,\phi,\kappa,{\cal I}^{\prime})$, and its evolution with respect to the
observation history $\tilde{y}=(y_{1},\dots,y_{k})$. At stage $k$, the state
of the automaton is ${\iota}_{k}$ and with $(u_{k},y_{k+1})$ the system
transitions to ${\iota}_{k+1}=\phi({\iota}_{k},u_{k},y_{k+1})$. Sufficiency of
${\kappa}$ implies that given ${\kappa}({\iota}_{k})$, $u_{k}$, and $y_{k+1}$,
we can determine ${\kappa}({\iota}_{k+1})$. This is equivalent to the
definition introduced in Chapter 11 of [8] and makes it a special case for
Definition 3.1.
### 3.4 Derived information transition systems
Even though it seems natural to rely on a history ITS, dimension of a history
I-space increases linearly with each stage, making it impractical in most
cases. Thus, we are interested in defining a reduced ITS that is more
manageable.Furthermore, this would largely simplify the description of a
policy for a planner or a plan executor.
Recall the quotient of a transition system by a a labeling function. We
rewrite $({\cal I}_{hist},U\times Y,\phi_{hist})$ as $({\cal I}_{hist},U\times
Y,\Phi_{hist})$, in which
$\Phi_{hist}=\\{({\eta},(u,y),\phi_{hist}({\eta},u,y))\in{\cal
I}_{hist}\times(U\times Y)\times{\cal I}_{hist}\\}.$ (3)
We can introduce an _information mapping_ (I-map) ${\kappa}:{\cal
I}_{hist}\rightarrow{\cal I}_{der}$ that categorizes the states of ${\cal
I}_{hist}$ into equivalence classes through its preimages. In this case,
${\kappa}$ serves as a labeling function and a reduction can be obtained in
terms of the quotient of $({\cal I}_{hist},U\times Y,\Phi)$ by ${\kappa}$,
that is, $({\cal I}_{hist}/{\kappa},U\times Y,\Phi/{\kappa})$.
It is crucial that the derived ITS is a DITS so that the labels can be
determined using only the derived ITS without making reference to the history
ITS.Considering the quotient system derived from $({\cal I}_{hist},U\times
Y,\phi)$, which is a DITS by definition, by ${\kappa}$, we can not always
guarantee that the resulting ITS is deterministic. This depends on the I-map
used for state-relabeling as stated in the following proposition.
###### Proposition 3.2.
For all non-empty $U$ and $Y$, and for the corresponding ${\cal I}_{hist}$,
there exists a labeling function ${\kappa}$ such that the quotient of $({\cal
I}_{hist},U\times Y,\phi)$ by $\kappa$, that is, $({\cal
I}_{hist}/{\kappa},U\times Y,\Phi/{\kappa})$, in which $\Phi$ is defined as in
(3), is not a DITS.
###### Proof 3.3.
Let ${\kappa}:{\cal I}_{hist}\rightarrow\\{l_{1},l_{2}\\}$ such that
${\kappa}^{-1}(l_{1})=\\{{\eta}_{k}=({\tilde{u}}_{k-1},{\tilde{y}}_{k})\in{\cal
I}_{hist}\mid{\tilde{u}}_{k-1}=(u_{i})_{i=1,\dots,k-1},u_{i}=u,\forall
i=1,\dots,k-1\\}$ is the set of histories that correspond to applying the same
action for $k-1$ times and ${\kappa}^{-1}(l_{2})$ is its complement, that is,
${\kappa}^{-1}(l_{2})={\cal I}_{hist}\setminus{\kappa}^{-1}(l_{1})$. Then,
there exist sequences ${\eta}_{k-2}=({\tilde{u}}_{k-3},{\tilde{y}}_{k-2})$ and
${\eta}_{k-1}=({\tilde{u}}_{k-2},{\tilde{y}}_{k-1})$ such that
${\eta}_{k-2}={\eta}_{k-1}{}^{\frown}(u,y)$ and
${\eta}_{k}={\eta}_{k-1}{}^{\frown}(u,y)$ for which
${\kappa}({\eta}_{k-2})={\kappa}({\eta}_{k-1})=l_{2}$ and
${\kappa}({\eta}_{k})=l_{1}$. Thus,
$\\{([{\eta}_{k-2}]_{\kappa},(u,y),[{\eta}_{k-1}]_{\kappa}),([{\eta}_{k-1}]_{\kappa},(u,y),[{\eta}_{k}]_{\kappa})\\}\in\Phi/{\kappa}.$
Since $[{\eta}_{k-2}]_{\kappa}=[{\eta}_{k-1}]_{\kappa}$ and
$[{\eta}_{k-1}]_{\kappa}\neq[{\eta}_{k}]_{\kappa}$, the transition
corresponding to $([{\eta}_{k-1}]_{\kappa},(u,y))$ is not unique; thus,
$({\cal I}_{hist}/{\kappa},U\times Y,\Phi/{\kappa})$ is not deterministic.∎
Note that Proposition 3.2 holds also in the case of a generic ITS $({\cal
I},U\times Y,\phi)$, with non-history I-states, if $\exists
s,s^{\prime},q,q^{\prime}\in{\cal I}$ such that
$\\{(s,(u,y),s^{\prime}),(q,(u,y),q^{\prime})\\}\in\Phi$, in which $\Phi$ is
defined using $\phi$ as in (3). Then, any I-map ${\kappa}$ such that
${\kappa}(s)={\kappa}(q)$ and ${\kappa}(s^{\prime})\neq{\kappa}(q^{\prime})$
results in a quotient system that is not a DITS.
For the quotient system derived from $({\cal I}_{hist},U\times Y,\phi)$ to be
a DITS depends on the sufficiency of ${\kappa}$. In [12] it is shown that the
quotient of a transition system $(S,\Lambda,T)$ by a labeling function
$\sigma$ is an automaton (recall our definition) if and only if
$(S,\Lambda,T)$ is full222A transition system $(S,\Lambda,T)$ is full, if
$\forall s\in S,\lambda\in\Lambda$ there exists at least one $s^{\prime}\in S$
with $(s,\lambda,s^{\prime})\in T$. and $\sigma$ is sufficient. As
$\phi_{hist}$ is a function with domain ${\cal I}_{hist}\times(U\times Y)$, it
is full, then, the following follows from [12] as a special case.
###### Proposition 3.4.
Let $({\cal I}_{hist}/{\kappa},U\times Y,\Phi_{hist}/{\kappa})$ be the
quotient of $({\cal I}_{hist},U\times Y,\phi_{hist})$ by ${\kappa}$, in which
$\Phi$ is defined as in (3), then $({\cal I}_{hist}/{\kappa},U\times
Y,\Phi/{\kappa})$ is a DITS if and only if ${\kappa}$ is sufficient.
For an I-map ${\kappa}:{\cal I}_{hist}\rightarrow{\cal I}_{der}$, $({\cal
I}_{hist}/{\kappa},U\times Y,\Phi_{hist}/{\kappa})$ is isomorphic to $({\cal
I}_{der},U\times Y,\Phi_{der})$, in which
$\Phi_{der}=\\{({\kappa}({\eta}),(u,y),{\kappa}({\eta}^{\prime}))\mid({\eta},(u,y),{\eta}^{\prime})\in\Phi_{hist}\\}$
[12]. Thus, we can use the labels introduced by ${\kappa}$ as the new
(derived) I-space and the corresponding quotient system as the derived ITS.
Suppose, ${\kappa}$ is sufficient. Then, the derived ITS is a DITS, meaning
that given the I-state ${\iota}_{k-1}\in{\cal I}_{der}$, and ($u_{k-1}$,
$y_{k}$), ${\iota}_{k+1}\in{\cal I}_{der}$ can be uniquely determined.
Consequently, we can write the derived ITS as $({\cal I}_{der},U\times
Y,\phi_{der})$ in which $\phi_{der}:{\cal I}_{der}\times(U\times
Y)\rightarrow{\cal I}_{der}$ is the new information transition function.
Therefore, we no longer need to rely on the full histories and the history ITS
and can rely solely on the derived ITS. This is shown in the first two rows of
the following diagram:
${{\cal I}_{hist}}$${{\cal I}_{hist}}$${{\cal I}_{hist}}$${{\cal
I}_{hist}}$${{\cal I}_{hist}}$ ${{\cal I}_{der}}$${{\cal I}_{der}}$${{\cal
I}_{der}}$${{\cal I}_{der}}$${{\cal I}_{der}}$ ${{\cal I}_{min}}$${{\cal
I}_{min}}$${{\cal I}_{min}}$${{\cal I}_{min}}$${{\cal I}_{min}}$ ${{\cal
I}_{task}}$${{\cal I}_{task}}$${{\cal I}_{task}}$${{\cal I}_{task}}$${{\cal
I}_{task}.}$$\scriptstyle{u_{1},y_{2}}$$\scriptstyle{{\kappa}}$$\scriptstyle{u_{2},y_{3}}$$\scriptstyle{{\kappa}}$$\scriptstyle{u_{3},y_{4}}$$\scriptstyle{{\kappa}}$$\scriptstyle{u_{4},y_{5}}$$\scriptstyle{{\kappa}}$$\scriptstyle{{\kappa}}$$\scriptstyle{u_{1},y_{2}}$$\scriptstyle{{\kappa}^{\prime}}$$\scriptstyle{u_{2},y_{3}}$$\scriptstyle{{\kappa}^{\prime}}$$\scriptstyle{u_{3},y_{4}}$$\scriptstyle{{\kappa}^{\prime}}$$\scriptstyle{u_{4},y_{5}}$$\scriptstyle{{\kappa}^{\prime}}$$\scriptstyle{{\kappa}^{\prime}}$$\scriptstyle{u_{1},y_{2}}$$\scriptstyle{{\kappa}^{\prime\prime}}$$\scriptstyle{u_{2},y_{3}}$$\scriptstyle{{\kappa}^{\prime\prime}}$$\scriptstyle{u_{3},y_{4}}$$\scriptstyle{{\kappa}^{\prime\prime}}$$\scriptstyle{u_{4},y_{5}}$$\scriptstyle{{\kappa}^{\prime\prime}}$$\scriptstyle{{\kappa}^{\prime\prime}}$
(4)
Note that we can similarly define an I-map that maps any derived I-space to
another. An example is given in (4) as the mappings ${\kappa}^{\prime}:{\cal
I}_{der}\rightarrow{\cal I}_{min}$ and ${\kappa}^{\prime\prime}:{\cal
I}_{min}\rightarrow{\cal I}_{task}$. The corresponding quotient system is
deterministic for ${\kappa}^{\prime}$, indicating that it is sufficient.
However, the quotient system by ${\kappa}^{\prime\prime}$ derived from ${\cal
I}_{min}$ is not deterministic, hence, ${\kappa}^{\prime\prime}$ is not
sufficient, as the next I-state can not be uniquely determined. Note that an
I-map whose domain is ${\cal I}_{hist}$ can also be defined as composition of
the mappings along the column of the diagram. For instance,
${\kappa}_{min}:{\cal I}_{hist}\rightarrow{\cal I}_{min}$ is the composition
of ${\kappa}$ and ${\kappa}^{\prime}$, that is,
${\kappa}_{min}={\kappa}^{\prime}\circ{\kappa}$ (same for
${\kappa_{task}}:{\cal I}_{hist}\rightarrow{\cal I}_{task}$).
### 3.5 Lattice of information transition systems
We fix ${\cal I}_{hist}$, which corresponds to fixing the set of initial
states ${\cal I}_{0}$. Then, each I-map ${\kappa}$ defined over ${\cal
I}_{hist}$ induces a partition of ${\cal I}_{hist}$ through its preimages,
denoted as ${\cal I}_{hist}/{\kappa}$. An I-map ${\kappa}^{\prime}$ is a
refinement of ${\kappa}$, denoted as ${\kappa}^{\prime}\succeq{\kappa}$, if
$\forall A\in{\cal I}_{hist}/{\kappa}^{\prime}$ there exists a $B\in{\cal
I}_{hist}/{\kappa}$ such that $A\subseteq B$. Let $K({\cal I}_{hist})$ denote
the set of all partitions over ${\cal I}_{hist}$. Refinement induces a partial
ordering since not all partitions of ${\cal I}_{hist}$ are comparable. The
partial ordering given by refinements form a lattice of partitions over ${\cal
I}_{hist}$, denoted as $(K({\cal I}_{hist}),\succeq)$.
At the top of the lattice, there is the partition induced by an identity I-map
(or equivalently, by a bijection), ${\kappa}_{id}:{\cal
I}_{hist}\rightarrow{\cal I}_{hist}$, since all of its elements are singletons
(all equivalence classes contain exactly one element), making it the maximally
distinguishable case. Conversely, we can define a constant mapping
${\kappa}_{const}:{\cal I}_{hist}\rightarrow{\cal I}_{const}$ for which ${\cal
I}_{hist}/{\kappa}_{const}$ is a singleton, that is, ${\cal
I}_{const}=\\{{\iota}_{const}\\}$, which then will be at the bottom of the
lattice. In turn, ${\kappa}_{const}$ yields the minimally distinguishable case
as all histories now belong to a single equivalence class. This idea is
similar to the notion of the _sensor lattice_ defined over the partitions of
$X$ [9, 13]. Indeed, if we take ${\cal I}_{0}=X$ and consider
${\kappa}_{est}:{\cal I}_{hist}\rightarrow X$, the ordering of partitions of
${\cal I}_{hist}$ such that ${\cal I}_{hist}/{\kappa}_{est}$ is the least
upper bound gives out the sensor lattice.
As motivated in previous sections, we are interested in finding a sufficient
I-map such that the quotient ITS derived from the history ITS is still
deterministic. Notice that the constant I-map ${\kappa}_{const}$ is sufficient
by definition since for all $(u,y)\in U\times Y$, and all
${\eta},{\eta}^{\prime}\in{\cal I}_{hist}$, we have that
${\kappa}_{const}({\eta})={\kappa}_{const}({\eta}^{\prime})$ and
${\kappa}_{const}(\phi_{hist}({\eta},(u,y))={\kappa}_{const}(\phi_{hist}({\eta}^{\prime},(u,y)).$
On the other hand, in certain cases it is crucial to differentiate certain
histories from the others. This will become clear in the next section when we
describe the notion of a task. Suppose ${\kappa}$ is a labeling that
partitions ${\cal I}_{hist}$ into equivalence classes that are of importance
and suppose that ${\kappa}$ is not sufficient. Then, we want to find a
refinement of ${\kappa}$ that is sufficient. This will serve as a lower bound
on the lattice of partitions over ${\cal I}_{hist}$ since for any partition
such that ${\cal I}_{hist}/{\kappa}$ is a refinement of it, the classes of
histories that are deemed crucial will not be distinguished. The following
defines the refinement of ${\kappa}$ that ensures sufficiency and a minimal
number of equivalence classes.
###### Definition 3.5.
Let $({\cal I}_{hist},U\times Y,{\phi_{hist}})$ be a history ITS and
${\kappa}$ an I-map. A _minimal sufficient refinement_ of ${\kappa}$ is a
sufficient I-map ${\kappa}^{\prime}$ such that there does not exist a
sufficient I-map ${\kappa}^{\prime\prime}$ that satisfy
${\kappa}^{\prime}\succ{\kappa}^{\prime\prime}\succeq{\kappa}$.
###### Remark 3.6.
It is shown in [12] that the minimal sufficient refinement of ${\kappa}$
defined over the states of an automaton $(S,\Lambda,\tau)$ is unique.
## 4 Solving Tasks Minimally
#### 4.0.1 Definition of a task
We now connect the general ITS concepts to the accomplishment of particular
tasks. We have two categories: 1) active, which corresponds to planning and
executing an information-feedback policy that forces a desirable outcome in
the environment, and 2) passive, which means only to observe the environment
without being able to effect changes. Recall from Section 3.4 that there may
be model-free or model-based formulations. In the model-free case, tasks are
specified using a logical language over ${\cal I}_{hist}$ which will result in
a labeling and derived I-space ${\cal I}_{task}$ and associated I-map
${\kappa_{task}}$ that corresponds to the “resolution” at which the tasks are
specified. Various logics are allowable, such as propositional or a temporal
logic. The resulting sentences of the language involve combinations of
predicates that may assign true or false values to subsets of ${\cal
I}_{hist}$. Solving an active task (or tasks) requires that a sentence of
interest becomes true during execution of the policy. This is called
satisfiability. For example, the task may be to simply reach some goal set
$G\subset{\cal I}_{hist}$, causing a predicate in-goal$({\cal I}_{hist})$ to
become satisfied (in other words, be true). Using linear temporal logic, more
complex requirements, such as cycling through a finite sequence of subsets
forever while avoiding others, can be specified [3].
Solving a passive task only requires maintaining whether a sentence is
satisfied, rather than forcing an outcome; this corresponds to filtering.
Whether the task is active or passive, if satisfiability is concerned with a
single, fixed sentence, then a task-induced labeling (or task labeling for
short), that is, ${\kappa_{task}}$, over ${\cal I}_{hist}$ assigns two labels:
Those I-states that result in true and those that result in false. A task
labeling may also be assigned for a set of possible sentences by assigning a
label to each set of the common refinement of the partition of ${\cal
I}_{hist}$ induced by each possible sentence. In the model-based case, tasks
are instead specified using a language over $X$, and sentence satisfiability
must be determined by an I-map that converts history I-states into expressions
over $X$.
#### 4.0.2 Problem families
It is assumed that the state-relabeled transition system $(X,U,f,h,Y)$
describing the external system is fixed, but it is unknown or partially known
to the observer (a robot or other observer).
Filtering (passive case) requires maintaining the label of an I-state
attributed by ${\kappa_{task}}$. Since ${\kappa_{task}}$ is not necessarily
sufficient, we can not guarantee that the quotient system by ${\kappa_{task}}$
is a DITS (Propositions 3.2 and 3.4). Thus, relying solely on the quotient
system by ${\kappa_{task}}$, we can not determine the class that the current
history belongs to (see the last row in (4)) and, hence can not determine
whether a sentence describing the task is satisfied (or which sentences are
satisfied).
Suppose the sets $U$ and $Y$ are specified, and at each stage $k$, $u_{k-1}$
is known and $y_{k}$ is observed. The following describes the problem for a
passive task given a state-relabeled (history) ITS $({\cal I}_{hist},U\times
Y,\phi_{hist},{\kappa_{task}},{\cal I}_{task})$, in which
${\kappa_{task}}:{\cal I}_{hist}\rightarrow{\cal I}_{task}$ is a task labeling
that is not sufficient, and ${\cal I}_{task}$ is the corresponding I-space.
###### Problem 4.1 (Find a sufficient I-space filter).
Find a sufficient refinement of ${\kappa_{task}}$.
Note that ${\cal I}_{hist}/{\kappa_{task}}$ determines a lower bound on the
partitioning of ${\cal I}_{hist}$ which is interpreted as the crucial
information that can not be lost. Consequently, histories belonging to
different equivalence classes with respect to ${\kappa_{task}}$ must always be
distinguished from each other. However, Problem 4.1 does not an upper bound.
At the limit, a bijection from ${\cal I}_{hist}$ is always a sufficient
refinement of ${\kappa_{task}}$. As stated previously, using history ITS can
create computational obstructions in solving problems. This motivates the
following problem.
###### Problem 4.2 (Filter minimization).
Find a minimal sufficient refinement of ${\kappa_{task}}$.
We now consider a basic planning problem, for which ${\cal
I}_{task}=\\{0,1\\}$, such that ${\kappa_{task}}^{-1}(1)\subset{\cal
I}_{hist}$ is the set of histories that achieve the goal, and
${\kappa_{task}}^{-1}(0)\subset{\cal I}_{hist}$ is its complement. Most
planning problems refer to finding a labeling function $\pi$ such that, when
used to label the states of the internal system, guarantees task
accomplishment. Then $\pi$ is called a feasible policy, which is defined in
the following. Consider an external system $(X,f,U,h,Y)$. Let
$\mathcal{R}_{X}({\cal I}_{task})\subseteq X$ be the set of initial states for
which there exist a $k$ and histories ${\tilde{x}}_{k}$, ${\tilde{u}}_{k-1}$,
and ${\tilde{y}}_{k}$, such that $x_{i+1}=f(x_{i},u_{i})$ and $y_{i}=h(x_{i})$
for all $0<i<k$, and ${\eta}_{k}\in{\kappa_{task}}^{-1}(1)$, in which
${\eta}_{k}$ is the history I-state corresponding to ${\tilde{u}}_{k-1}$ and
${\tilde{y}}_{k}$.
###### Definition 4.3 (Feasible policy for ${\cal I}_{task}$).
Let $({\cal I},Y,\phi,\pi,U)$ and $(X,f,U,h,Y)$ be the state-relabeled
transition systems corresponding to internal and external systems,
respectively. Labeling function $\pi$ defined over ${\cal I}$ is a feasible
policy for ${\cal I}_{task}$ if for all $x\in\mathcal{R}_{X}({\cal
I}_{task})$, the history corresponding to the coupled internal-external system
initialized at $({\iota}_{0},x)$ belongs to ${\kappa_{task}}^{-1}(1)$.
Most problems in the planning literature consider a fixed DITS and look for a
feasible policy for ${\cal I}_{task}$. This yields the following problem.
Typically, the I-space considered is $X$ which corresponds to state
estimation. Note that a DITS, in other words, the robot brain, can be seen as
an I-space filter itself.
###### Problem 4.4 (Find a feasible policy).
Given $({\cal I},Y,\phi)$, an internal system (robot brain), find a labeling
function $\pi:{\cal I}\rightarrow U$ that is a feasible policy for ${\cal
I}_{task}$.
We can further extend the planning problem to consider an unspecified internal
system which refers to finding a DITS, and a policy such that the resulting
histories satisfy the task description, that is, he problem of jointly finding
an I-space-filter and a policy defined over its states. Recall from (2) that a
policy constrains the transitions of an ITS to the ones that are realizable
under that policy.
###### Problem 4.5 (Find a DITS and a feasible policy).
Given $({\cal I}_{hist},U\times Y,\phi_{hist})$, for which
${\kappa_{task}}:{\cal I}_{hist}\rightarrow{\cal I}_{task}$ is a task labeling
and ${\cal I}_{task}$ is the corresponding I-space, find a sufficient I-map
${\kappa}:{\cal I}_{hist}\rightarrow{\cal I}$ and a feasible policy $\pi$ for
${\cal I}_{task}$ as a labeling function for the resulting quotient system by
${\kappa}$.
Note that ${\kappa_{task}}$ can already be sufficient, so that it is the
minimal sufficient refinement of itself, however, this does not necessarily
imply the existence of a feasible policy defined over ${\cal I}_{task}$.
Therefore, we are not looking for a refinement of ${\kappa_{task}}$ while
describing the DITS over which the policy is defined. On the other hand, we
can still talk about a notion of minimality. Let $({\cal I},Y,\phi,\pi,U)$ be
a state-relabeled DITS that solves Problem 4.5 (or similarly Problem 4.4)
which is the quotient system of history ITS by ${\kappa}$, then, $({\cal
I},Y,\phi,\pi,U)$ is _minimal for $\pi$_ if there does not exist a sufficient
I-map ${\kappa}^{\prime}$ with ${\kappa}\succ{\kappa}^{\prime}$ for which
there exists a $\pi^{\prime}$ for the quotient system by ${\kappa}^{\prime}$
that satisfies $\pi({\iota})=\pi^{\prime}({\kappa}^{\prime}({\iota}))$.
#### 4.0.3 Learning a sufficient ITS
Although learning and planning overlap significantly, some unique issues arise
in pure learning (see also [12]). This corresponds to the case when ${\cal
I}_{task}$ is not initially given but needs to be revealed through
interactions with the external system, that is, respective action and
observation histories. It is assumed that whether the sentence (or sentences)
describing the task is satisfied or not can be assessed at a particular
history I-state.
We can address both filtering and planning problems defined previously within
this context, considering model-free and model-based cases. In the model-free
case, the task is to compute a minimal sufficient ITS that is consistent with
the actions and observations. Variations include lifelong learning, in which
there is a single, ‘long’ history I-state, or more standard learning in which
the system can be restarted, resulting in multiple trials, each yielding a
different history I-state. In the model-based case, partial specifications of
$X$, $f$, and $h$ may be given, and unknown parameters are estimated using the
history I-state(s). Different results are generally obtained depending on what
assumptions are allowed. For example, do identical history I-states imply
identical state trajectories? If not, then set-based, nondeterministic models
may be assumed, or even probabilistic models based on behavior observed over
many trials and assumptions on probability measure, priors, statistical
independence, and so on.
## 5 Illustrative Examples
In this section we provide some simple examples to show how the ideas
presented in this paper apply to filtering and planning problems. All problems
can be posed as well in a machine learning context for which ${\cal I}_{task}$
is not given but it is revealed through interactions between the internal and
external as the input-output data.
Figure 2: (a) State-relabeled history ITS described in Example 5.1, and
labeling function ${\kappa_{task}}$ yellow colored states correspond to states
that satisfy the task description. (b) Equivalence classes induced by
${\kappa}^{\prime}$; the minimal sufficient refinement of ${\kappa_{task}}$.
(c) Quotient of the history ITS by ${\kappa}^{\prime}$. (d)DITS describing the
internal system solving the planning problem described in Example 5.2. (e)
Environment used in Examples 5.1,5.2, the obstacle (an open disk) is shown in
black. (f) L-shaped corridor; $l_{1},l_{2}\leq l$.
Let $E\subseteq{\mathbb{R}}^{2}$ be a bounded planar environment (see Figure
2(e)) that is partitioned into regions separated by gates Each gate is either
green or red whose color can be detected by the robot’s color sensor and
follows the rule that each region shares a boundary with exactly two gates;
one green and one red. The set of possible observations are $Y=\\{r,g\\}$.
###### Example 5.1.
This example considers a filtering problem from the perspective of an
independent observer. Suppose the actions taken by the robot are not
observable and the only information about the system is the history of
readings coming from the robot’s color sensor; for example, $(r,r,r,g,r,g)$.
Then, the history I-space is the set of all finite length sequences of
elements of $Y$, that is, ${\cal I}_{hist}=Y^{*}$, which refers to the free
monoid generated by the elements of $Y$ (or the Kleene star of $Y$). Hence,
the history ITS can be represented as an infinite binary tree. The task is to
determine whether the robot crosses the gates consistently (in a clockwise or
counterclockwise manner) or not. Hence, the preimages of
${\kappa_{task}}:{\cal I}_{hist}\rightarrow{\cal I}_{task}$ partition ${\cal
I}_{hist}$ into two subsets: one which the condition is satisfied and the
others. The labeling induced by ${\kappa_{task}}$ is shown in Figure 2(a).
Clearly, ${\kappa_{task}}$ is not sufficient since there exist I-states
${\eta},{\eta}^{\prime}$ such that
${\kappa_{task}}({\eta})={\kappa_{task}}({\eta}^{\prime})$ and there exists a
$y$ for which
${\kappa_{task}}({\phi_{hist}}({\eta},y))\neq{\kappa_{task}}({\phi_{hist}}({\eta}^{\prime},y))$;
for example consider ${\eta}=(r,g)$, ${\eta}^{\prime}=(r,g,r)$ and $y=g$. A
sufficient refinement of ${\kappa_{task}}$ can be obtained (equivalence
classes shown in 2(b)), denote as ${\kappa}^{\prime}$, for which the quotient
DITS is shown in Figure 2(d). Furthermore, ${\kappa}^{\prime}$ is a minimal
sufficient refinement of ${\kappa_{task}}$ since it follows from Proposition
4.2 that if a labeling is not minimal then there is a minimal one that is
strictly coarser and is still sufficient. However, neither of the subsets that
belong to ${\cal I}_{hist}/{\kappa}^{\prime}$ can be merged, since merging
${\iota}_{nt}$ (colored gray) with anything else violates the condition that
${\kappa}^{\prime}$ is a refinement of ${\kappa_{task}}$ and any pairwise
merge of the others violate sufficiency.
Suppose the robot has a boundary detector, and executes a bouncing motion
using the two motion primitives $U=\\{u_{1},u_{2}\\}$, in which $u_{1}$ is
move forward and $u_{2}$ is rotate in place. Let a basic motion be move
forward and bounce off the walls, which can be implemented using the elements
of $U$. We assume that the boundary detector and color sensor readings do not
arrive simultaneously.
###### Example 5.2.
We now consider a planning problem (that belongs to the class described in
Problem 4.5) for which the goal is to ensure that the robot crosses the gates
consistently. The history I-space of the planner is ${\cal I}_{hist}=(U\times
Y)^{*}$ and the preimages of ${\kappa_{task}}$ partition ${\cal I}_{hist}$
into two sets; the histories that satisfy the predicate and the ones that do
not. Then, a DITS with only three states (see Figure 2(d)) can be derived
using the mapping ${\kappa}:{\cal I}_{hist}\rightarrow{\cal I}$, in which
${\cal I}=\\{i_{0},i_{1},i_{2}\\}$ such that for $\pi({\iota}_{1})$ boundary
with the red gate is set as a wall, $\pi({\iota}_{2})$ the boundary with the
green gate is set as a wall, and for $\pi({\iota}_{0})$ no boundary with gates
are considered as a wall. We assume that a bouncing motion can be determined
using the motion primitives so that the resulting trajectory will strike every
open interval in the boundary of every region infinitely often, with non-zero,
non-tangential velocities [1].
Consider a robot in an L-shaped planar corridor (Figure 2(f)). Let
$\mathcal{E}_{l}$ be the set of all such environments such that
$l_{1},l_{2}\leq l$, in which $l_{1}$ and $l_{2}$ are the dimensions of the
corridor bounded by $l$. We assume that the minimum length/width is larger
than the robot radius, that is, $1$. The state space $X$ is defined as the set
of all pairs $(q,E_{i})$, in which $(q_{1},q_{2})\in E_{i}$, and
$E_{i}\in\mathcal{E}_{l}$. The action set is $U=\\{0,1\\}\times\\{0,1\\}$
which corresponds to moving one step in one of the 4 cardinal directions; if
the boundary is reached, the state does not change. The robot has a sensor
that reports $1$ if the motion is blocked.
###### Example 5.3.
Consider a model-based history ITS with ${\eta}_{0}\subset X$ that specifies
the initial position as $q_{0}=(0,0)$ but does not specify the environment so
that it can be any $E_{i}\in{\cal{E}}_{l}$. Let ${\cal I}_{hist}$ be its set
of states and ${\kappa}_{ndet}:{\cal I}_{hist}\rightarrow{\rm pow}(X)$ be an
I-map that maps ${\eta}_{k}$ to a subset of $X_{k}\subseteq X$. Since
$(X,U,f,h,Y)$, and $X_{0}={\eta}_{0}$ are known, transitions for the quotient
system can be described by induction as
$X_{k+1}=\hat{X}_{k+1}(X_{k},u_{k})\cap H(y_{k+1})$, in which
$\hat{X}_{k+1}=\bigcup_{x_{k}\in X_{k}}f(x_{k},u_{k})$ and
$H(y_{k+1})\subseteq X$ is the set of all states that could yield $y_{k+1}$.
By construction, ${\kappa}_{ndet}$ is sufficient. Suppose
${\kappa_{task}}:{\cal I}_{hist}/{\kappa}_{ndet}\rightarrow{\cal I}_{task}$ is
a task labeling for localization that assigns each singleton a unique label
and all the other subsets are labeled the same. Since the transition
corresponding to $([X^{\prime}]_{{\kappa_{task}}},(u,y))$ in which
$X^{\prime}\subseteq X$ is not a singleton can lead to multiple labels
$[x^{\prime}]_{\kappa_{task}}$, in which $x^{\prime}$ is a singleton,
${\kappa_{task}}$ is not sufficient. Furthermore, ${\kappa}_{ndet}$ is a
minimal sufficient refinement of ${\kappa_{task}}$ because it is sufficient
and because there does not exist a sufficient ${\kappa}$ such that
${\kappa}_{ndet}\succ{\kappa}\succeq{\kappa_{task}}$. Suppose ${\kappa}$
exists, that would mean some equivalence classes can be merged. However, this
is not possible because merging any of the non-singleton subsets violates
sufficiency (as shown for ${\kappa_{task}}$) and merging singletons with
others violates that it is a refinement. A policy can be described over ${\cal
I}_{hist}/{\kappa}_{ndet}$; $u=(1,0)$ starting from $X_{0}$ until $y_{k}=1$ is
obtained and applying $u=(0,1)$ starting from $X_{k}$ until $y_{n}=1$ is
obtained, then it is found that $q=(k,n)$ and $E$ is the corridor with
$l_{1}=k$, $l_{2}=n$.
## 6 Discussion
We have introduced a mathematical framework for determining minimal feasible
policies for robot systems, by comparing ITSs over I-spaces. The uniqueness
and minimality results are quite general: $X$ and $U$ could be discrete,
typical configuration spaces, or more exotic, such as the power set of all
functions from an infinite-dimensional Hilbert space into an infinite-
dimensional Banach space.
Nevertheless, there are opportunities to expand the general theory. For
example, we assumed that $u$ is both the output of a policy and the actuation
stimulus in the physical world; more generally, we should introduce a mapping
from an action symbol $\sigma\in\Sigma$ to a control function
${\tilde{u}}\in{\tilde{U}}$ so that plans are expressed as $\pi:{\cal
I}\rightarrow\Sigma$ and each $\sigma=\pi(\iota)$ produces energy in the
physical world via a mapping from $\Sigma$ to ${\tilde{U}}$. Another extension
is to consider stochastic models, that amounts to an ITS with a probabilistic
I-space, and in which ways it ties to the representations used in the
literature such as _predictive state representations_ (PSRs) [10]. Other
potential extensions include continuous-time “transitions” and novel logics to
consider satisfiability questions over spaces of robot systems and tasks. An
interesting direction is to consider the hardware and actuation models are
variables, and fix other model components.
A grand challenge remains: The results here are only a first step toward
producing a more complete and unique theory of robotics that clearly
characterizes the relationships between common tasks, robot systems,
environments, and algorithms that perform filtering, planning, or learning. We
should search for lattice structures that play a role similar to that of
language class hierarchies in the theory of computation. This includes the
structures of the current paper and the sensor lattices of [9, 13]. Many
existing filtering, planning, and learning methods can be formally
characterized within this framework, which would provide insights into
relative complexity, completeness, minimality, and time/space/energy
tradeoffs.
## References
* [1] L. Bobadilla, O. Sanchez, J. Czarnowski, K. Gossman, and S. M. LaValle. Controlling wild bodies using linear temporal logic. In Proceedings Robotics: Science and Systems, 2011.
* [2] Axel Brunnbauer, Luigi Berducci, Andreas Brandstätter, Mathias Lechner, Ramin M. Hasani, Daniela Rus, and Radu Grosu. Latent imagination facilitates zero-shot transfer in autonomous racing. https://arxiv.org/abs/2103.04909v2.
* [3] G. E. Fainekos, A. Girard, H. Kress-Gazit, and G. J. Pappas. Temporal logic motion planning for dynamic mobile robots. Automatica, 45(2):343–352, 2009.
* [4] K. Friston. The free-energy principle: A unified brain theory? Nature Reviews Neuroscience, 11(2):127–138, 2010.
* [5] D. D. Hutto and E. Myin. Radicalizing enactivism: Basic minds without content. MIT Press, 2012.
* [6] D. E. Koditschek. What is robotics? Why do we need it and how can we get it? Annual Review of Control, Robotics, and Autonomous Systems, 4:1–33, May 2021.
* [7] Ü. Kotta, C. H. Moog, and M. Tõnso. Minimal realizations of nonlinear systems. Automatica, 95:207–212, 2018.
* [8] S. M. LaValle. Planning Algorithms. Cambridge University Press, Cambridge, U.K., 2006. Also available at http://lavalle.pl/planning/.
* [9] S. M. LaValle. Sensing and Filtering: A Fresh Perspective Based on Preimages and Information Spaces, volume 1:4 of Foundations and Trends in Robotics Series. Now Publishers, Delft, The Netherlands, 2012.
* [10] Michael Littman and Richard S Sutton. Predictive representations of state. Advances in neural information processing systems, 14, 2001.
* [11] J. M. O’Kane and D. A. Shell. Concise planning and filtering: Hardness and algorithms. IEEE Transactions on Automation Science and Engineering, 14(4):1666–1681, 2017.
* [12] V. Weinstein, B. Sakcak, and S. M. LaValle. An enactivist-inspired mathematical model of cognition. under review, 2022.
* [13] Y. Zhang and D. A. Shell. Lattices of sensors reconsidered when less information is preferred. In IEEE International Conference on Robotics and Automation, 2021\.
*[I-space]: information space
*[I-state]: information state
*[ITS]: information transition system
*[DITS]: deterministic information transition system
*[NITS]: nondeterministic information transition system
*[ I-space]: information space
*[ I-state]: information state
*[ ITS]: information transition system
*[I-map]: information mapping
*[PSRs]: predictive state representations
|
# Numerical simulations of inflationary dynamics: slow-roll and beyond
Siddharth S. Bhatt Swagat S. Mishra Soumen Basak and Surya N. Sahoo
###### Abstract
Cosmic inflation is a period of rapid accelerated expansion of space in the
very early universe. During inflation, vacuum quantum fluctuations are
amplified and stretched to cosmological scales which seed the fluctuations in
the cosmic microwave background as well as the large-scale structure of our
universe. Large quantum fluctuations may lead to the formation of primordial
black holes (PBHs) in the post-inflationary universe. Numerical simulations of
the inflationary dynamics are presented here for a single canonical scalar
field minimally coupled to gravity. We spell out the basic equations governing
the inflationary dynamics in terms of cosmic time $t$ and define a set of
dimensionless variables convenient for numerical analysis. We then provide a
link to our simple numerical Python code on GitHub that can be used to
simulate the background dynamics as well as the evolution of linear
perturbations during inflation. The code computes both scalar and tensor power
spectra for a given inflaton potential $V(\phi)$. We discuss a concrete
algorithm to use the code for various purposes, especially for computing the
enhanced scalar power spectrum in the context of PBH formation. We intend to
extend the framework to simulate the dynamics of a number of different
quantities, including the computation of scalar-induced second-order tensor
power spectrum in the revised version of this manuscript in the near future.
## 1 Introduction
Cosmic inflation has emerged as the leading scenario for describing the very
early universe prior to the commencement of the radiative hot Big Bang Phase
[1, 2, 3, 4, 5, 6]. According to the inflationary paradigm, a transient epoch
of at least 60-70 e-folds of rapid accelerated expansion suffices in setting
natural initial conditions for the background space-time in the form of
spatial flatness as well as statistical homogeneity and isotropy on large
angular scales [2, 3, 4, 7]. Additionally, (and more significantly,) quantum
fluctuations during inflation naturally generate a spectrum of almost scale-
invariant initial scalar fluctuations which seed the temperature and
polarisation fluctuations in the Cosmic Microwave Background (CMB) Radiation,
and later, the formation of structure in the universe [8, 9, 10, 11, 7]. In
addition to scalar perturbations, quantum fluctuations during inflation also
create a spectrum of almost scale-invariant tensor perturbations which later
become gravitational waves [12, 13].
The simplest models of inflation comprising of a single scalar field, called
the ‘inflaton’, which is minimally coupled to gravity, makes several distinct
predictions [14] (i.e an almost scale-invariant, nearly Gaussian, and
adiabatic spectrum of scalar fluctuations) most of which have received
spectacular observational confirmation, particularly from the latest CMB
missions [15].
However, as mentioned earlier, inflation also generates tensor perturbations
that later constitute the relic gravitational wave background (GW) which
imprints a distinct signature on the CMB power spectrum in the form of the
B-mode polarization [15]. The amplitude of these relic GWs provides us
information about the inflationary energy scale while their spectrum enables
us to access general properties of the epoch of reheating, being exceedingly
sensitive to the post-inflationary equation of state [13, 16]. The amplitude
of inflationary tensor fluctuations, relative to that of scalar fluctuations,
is usually characterised by the tensor-to-scalar ratio $r$. Different models
of inflation predict different values for $r$ which is sensitive to the
gradient of the inflaton potential $V_{,\phi}(\phi)=\frac{dV(\phi)}{d\phi}$
relative to its height $V(\phi)$. Convex potentials predict large values for
$r$, while concave potentials predict relatively small values of $r$. While
the spectrum of inflationary tensor fluctuations has not yet been observed,
current CMB observations are able to place an upper bound on the tensor-to-
scalar ratio on large angular scales. In particular, the latest CMB
observations of BICEP/Keck [17], combined with those of the PLANCK mission
[15], place the strong upper bound $r\leq 0.036$ (at $95\%$ confidence).
This most recent upper bound on $r$ has important consequences for single
field canonical inflation. In particular, given $r\leq 0.036$, all
monotonically increasing convex potentials, including the whole family of
monomial potentials $V(\phi)\propto\phi^{p}$, are completely ruled out in the
canonical framework. Among these strongly disfavoured models are the simplest
classic inflaton potentials $\frac{1}{2}m^{2}\phi^{2}$ and $\lambda\phi^{4}$.
Instead, the observational upper bound on $r$ appears to favour
asymptotically-flat potentials possessing one or two plateau-like wings; see
[19, 18]. Current observational data lead to a scenario in which the inflaton
$\phi$ slowly rolls down a shallow potential $V(\phi)$ thereby giving rise to
a quasi-de Sitter early stage of near-exponential expansion. A thorough
analysis of the inflationary phase-space dynamics $\\{\phi,\dot{\phi}\\}$ for
plateau potentials shows [20] that a large range of initial conditions leads
to adequate inflation in these models.
However, it is important to stress that the CMB window constitutes only a tiny
part of the observationally available field space between the Hubble-exit of
the largest scales in the sky and the smallest scale at the end of inflation.
Consequently, a substantial period of the inflationary dynamics corresponding
to potentially interesting small-scale primordial physics (which accounts
roughly to the last 40–50 e-folds of accelerated expansion during inflation)
remains observationally unexplored, being inaccessible to the CMB and LSS
observations. Any departure from the slow-roll regime, that might be triggered
by a change in the dynamics of the inflaton field, would lead to interesting
observational consequences on small-scales. In particular, the presence of a
feature at intermediate field values that might lead to large enough
amplification of the small-scale scalar fluctuations, would facilitate the
formation of Primordial Black Holes upon the Hubble re-entry of these modes
during the post-inflationary epochs.
Primordial Black Holes (PBHs) are extremely interesting compact objects which
might have been formed from the collapse of large density fluctuations in the
early universe [21, 22, 23, 24] and they constitute a potential candidate for
dark matter [27, 26, 25, 28, 29, 30]. Seeds for such large fluctuations can be
generated during inflation, as mentioned above. For instance, a feature in the
inflaton potential in the form of a flat inflection point can further slow
down the already slowly rolling inflaton field substantially, leading to an
enhancement of the primordial scalar power $P_{\zeta}$. A number of different
features for enhancing small-scale power during inflation have been proposed
in the recent years [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49]. Hence PBHs (and the associated induced relic GWs) are
excellent probes of the small-scale primordial physics.
In this preliminary version of our paper, we discuss a simple code developed
for numerical simulations of the inflationary dynamics. We introduce the
relevant dimensionless variables used in our numerical analysis and provide a
link to the code in our GitHub account. We also discuss how to use the code in
various scenarios, which include phase-space analysis of inflationary initial
conditions, inflationary background dynamics and determining scalar and tensor
power spectra both under slow-roll approximation and beyond. The latter case
has important implications for PBH formation and we discuss how to use the
code to simulate the Mukhanov-Sasaki equation mode by mode. We also discuss a
number of important future directions that are to be included in the
forthcoming version of our paper. The primary version of our numerical
framework is quite simple and less compact. It is intended to provide a
pedagogical guideline for researchers who are relatively new to numerical
simulations of inflation. In the forthcoming version of our work, we will
introduce a much more compact numerical framework that we are currently
working on which will incorporate additional new features.
This paper is organised as follows: we begin with a brief introduction of the
inflationary scalar field dynamics in section 2 and quantum fluctuations in
section 3. We then proceed to discuss numerical simulations of the background
dynamics in section 4. This also includes studying the scalar and tensor
fluctuations under the slow-roll approximations. Section 5 is dedicated to
studying the inflationary quantum fluctuations by numerically solving the
Mukhanov-Sasaki equation, and its application to inflaton potentials
possessing a slow-roll violating feature. We also mention a number of future
extensions of our numerical set-up in section 6, before concluding with a
discussion section.
We work in the units $c,\hbar=1$. The reduced Planck mass is defined to be
$m_{p}\equiv 1/\sqrt{8\pi G}=2.43\times 10^{18}~{}{\rm GeV}$. We assume the
background universe to be described by a spatially flat Friedmann-Lemaitre-
Robertson-Walker (FLRW) metric with signature $(-,+,+,+)$.
## 2 Inflationary Dynamics
The Action for a scalar field which couples minimally to gravity has the
following general form
$S[\phi]=\int d^{4}x\,\sqrt{-g}\;{\cal L}(F,\phi),$ (2.1)
where the Lagrangian density ${\cal L}(\phi,F)$ is a function of the field
$\phi$ and the kinetic term
$F=\frac{1}{2}\partial_{\mu}\phi\;\partial^{\mu}\phi.$ (2.2)
Varying (2.1) with respect to $\phi$ results in the equation of motion
$\frac{\partial{\cal
L}}{\partial\phi}-\left(\frac{1}{\sqrt{-g}}\right)\partial_{\mu}\left(\sqrt{-g}\frac{\partial{\cal
L}}{\partial\left(\partial_{\mu}\phi\right)}\right)=0.$ (2.3)
The energy-momentum tensor associated with the scalar field is
$T^{\mu\nu}=\left(\frac{\partial{\cal L}}{\partial
F}\right)\,\left(\partial^{\mu}\phi\;\partial^{\nu}\phi\right)-g^{\mu\nu}\,{\cal
L}~{}.$ (2.4)
Specializing to a spatially flat FRW universe and a homogeneous scalar field,
one gets
$ds^{2}=-dt^{2}+a^{2}(t)\left[dx^{2}+dy^{2}+dz^{2}\right]\,,$ (2.5)
$T^{\mu}_{\;\>\;\nu}=\mathrm{diag}\left(-\rho_{{}_{\phi}},p_{{}_{\phi}},p_{{}_{\phi}},p_{{}_{\phi}}\right),$
(2.6)
where the energy density, $\rho_{{}_{\phi}}$, and pressure, $p_{{}_{\phi}}$,
are given by
$\displaystyle\rho_{{}_{\phi}}$ $\displaystyle=\left(\frac{\partial{\cal
L}}{\partial F}\right)\,(2\,F)-{\cal L}\,,$ (2.7) $\displaystyle
p_{{}_{\phi}}$ $\displaystyle={\cal L}\,,$ (2.8)
and $F=-({\dot{\phi}}^{2}/2)$. The evolution of the scale factor $a(t)$ is
governed by the Friedmann equations:
$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}\equiv H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3m_{p}^{2}}\,\rho_{{}_{\phi}},$ (2.9)
$\displaystyle\frac{\ddot{a}}{a}\equiv\dot{H}+H^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{6m_{p}^{2}}\,\left(\rho_{{}_{\phi}}+3\,p_{{}_{\phi}}\right),$
(2.10)
where $H\equiv\dot{a}/a$ is the Hubble parameter and $\rho_{{}_{\phi}}$
satisfies the conservation equation
${\dot{\rho}_{{}_{\phi}}}=-3\,H\left(\rho_{{}_{\phi}}+p_{{}_{\phi}}\right)~{}.$
(2.11)
In the standard single field inflationary paradigm, inflation is sourced by a
minimally coupled canonical scalar field $\phi$ with a suitable potential
$V(\phi)$ (see figure 1). For such a canonical scalar field
${\cal L}(F,\phi)=-F-V(\phi),$ (2.12)
Figure 1: This figure schematically depicts a prototype inflation potential
$V(\phi)$, plotted in solid green curve. The ‘CMB Window’ represents field
values corresponding to the Hubble-exit epochs of scales
$k\in\left[0.0005,0.5\right]~{}{\rm Mpc}^{-1}$ that are observable by the
latest CMB missions.
Substituting (2.12) into (2.7) and (2.8), we find
$\displaystyle\rho_{{}_{\phi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\dot{\phi}}^{2}+\;V(\phi),$ $\displaystyle
p_{{}_{\phi}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\dot{\phi}}^{2}-\;V(\phi),~{}~{}$ (2.13)
consequently the two Friedmann equations (2.9), (2.10) and the equation (2.11)
become
$\displaystyle H^{2}\equiv\frac{1}{3m_{p}^{2}}\,\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{3m_{p}^{2}}\left[\frac{1}{2}{\dot{\phi}}^{2}+V(\phi)\right]\,,$
(2.14) $\displaystyle\dot{H}\equiv\frac{\ddot{a}}{a}-H^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m_{p}^{2}}\,\dot{\phi}^{2}\,,$ (2.15)
$\displaystyle{\ddot{\phi}}+3\,H{\dot{\phi}}+V_{,\phi}(\phi)$ $\displaystyle=$
$\displaystyle 0\,.$ (2.16)
The epoch of inflation at any time $t<t_{\rm end}$ is conveniently marked by
the number of e-folds before the end of inflation
$N_{e}=\log_{e}{\frac{a_{\rm end}}{a(t)}}=\int_{t}^{t_{\rm
end}}H(t^{\prime})dt^{\prime}\,,$ (2.17)
where $H(t)$ is the Hubble parameter during inflation. $a(t)$ and $a_{\rm
end}$ denote the scale factor at time $t$ and at the end of inflation
respectively. Typically a period of quasi-de Sitter inflation lasting for at
least 60-70 e-folds is required in order to address the problems of the
standard hot Big Bang model. We denote $N_{*}$ as the number of e-folds
(before the end of inflation) when the CMB pivot scale
$k_{*}=(aH)_{*}=0.05~{}\rm Mpc^{-1}$ left the comoving Hubble radius during
inflation. For convenience, we have chosen $N_{*}=60$ for the most part of
this work, although the exact value of $N_{*}$ depends upon the particular
detail of reheating history.
The quasi-de Sitter like phase corresponds to the inflaton field rolling
slowly down the potential $V(\phi)$. This slow-roll regime111It is well known
that the slow-roll phase of the inflation is actually a local attractor for
many different models of inflation, see [50, 20] and the references therein.
of inflation, ensured by the presence of the Hubble friction term in the
equation (2.16), is usually characterised by the first two kinematical Hubble
slow-roll parameters $\epsilon_{H}$, $\eta_{H}$, defined by [7]
$\displaystyle\epsilon_{H}$ $\displaystyle=$
$\displaystyle-\frac{\dot{H}}{H^{2}}=\frac{1}{2m_{p}^{2}}\,\frac{\dot{\phi}^{2}}{H^{2}},$
(2.18) $\displaystyle\eta_{H}$ $\displaystyle=$
$\displaystyle-\frac{\ddot{\phi}}{H\dot{\phi}}=\epsilon_{H}+\frac{1}{2\epsilon_{H}}\,\frac{d\epsilon_{H}}{dN_{e}}~{},$
(2.19)
where the slow-roll regime of inflation corresponds to
$\epsilon_{H},~{}\eta_{H}\ll 1~{}.$ (2.20)
The slow-roll regime is also often characterised by the dynamical potential
slow-roll parameters [7], defined by
$\epsilon_{{}_{V}}=\frac{m_{p}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}~{},~{}~{}\eta_{{}_{V}}=m_{p}^{2}\,\left(\frac{V_{,\phi\phi}}{V}\right)~{}.$
(2.21)
For small values of these parameters $\epsilon_{H},\,\eta_{H}\ll 1$, one finds
$\epsilon_{H}\simeq\epsilon_{{}_{V}}$ and
$\eta_{H}\simeq\eta_{{}_{V}}-\epsilon_{{}_{V}}$. Using the definition of
Hubble parameter, $H=\dot{a}/a$, we have
$\ddot{a}/a=\dot{H}+H^{2}=H^{2}(1+\dot{H}/H^{2})$. From the expression for
$\epsilon_{H}$ in (2.18), it is easy to see that
$\frac{\ddot{a}}{a}=\big{(}1-\epsilon_{H}\big{)}\,H^{2}~{}.$ (2.22)
Which implies that the universe accelerates, ${\ddot{a}}>0$, when
$\epsilon_{H}<1$. Using equation (2.14), the expression for $\epsilon_{H}$ in
(2.18) reduces to $\epsilon_{H}\simeq\frac{3}{2}\frac{\dot{\phi}^{2}}{V}$ when
${\dot{\phi}}^{2}\ll V$.
Figure 2: This figure schematically illustrates the evolution of the comoving
Hubble radius $(aH)^{-1}$ with scale factor. During inflation $(aH)^{-1}$
decreases which causes physical scales to exit the Hubble radius. After
inflation ends $(aH)^{-1}$ increases, and physical scales begin to re-enter
the Hubble radius. The CMB pivot scale, as used by the Planck mission, is set
at $k_{*}=0.05~{}{\rm Mpc}^{-1}$ and has been depicted by the dashed blue
line. The ‘CMB Window’ corresponds to the scales
$k\in\left[0.0005,0.5\right]~{}{\rm Mpc}^{-1}$ that are observable by the
latest CMB missions. Note that $(aH)^{-1}\propto a$ during the radiative
regime and $(aH)^{-1}\propto a^{-1}$ during inflation.
## 3 Quantum fluctuations during inflation
In the standard scenario of a minimally coupled single canonical scalar field
$\phi$ as the inflaton, two gauge-independent massless fields, one scalar, and
one transverse traceless tensor, get excited during inflation and receive
quantum fluctuations that are correlated over super-Hubble scales [51] at late
times.
### Scalar fluctuations
The evolution of the scalar degree of freedom, called the curvature
perturbation222Note that the comoving curvature perturbation ${\cal R}$ is
related to the curvature perturbation on uniform-density hypersurfaces,
$\zeta$, and both are equal during slow-roll inflation as well as on super-
Hubble scales, $k\ll aH$, in general (see [7]). $\zeta$ is described by the
following quadratic Action [7, 52]
$\boxed{S_{(2)}[\zeta]=\frac{1}{2}\int{\rm d}\tau{\rm
d}x^{3}\,z^{2}\,\left[\,(\zeta^{\prime})^{2}-(\partial_{i}\zeta)^{2}\,\right]}~{},$
(3.1)
which upon the change of variable
$v\equiv z\,{\zeta}~{},\quad\mbox{with}\quad
z=am_{p}\sqrt{2\epsilon_{H}}=a\,\frac{\dot{\phi}}{H}\,,$ (3.2)
takes the form
$S_{(2)}\left[v\right]=\frac{1}{2}\int{\rm d}\tau{\rm
d}x^{3}\left[\left(v^{\prime}\right)^{2}-\left(\partial_{i}v\right)^{2}+\frac{z^{\prime\prime}}{z}v^{2}\right]~{},$
(3.3)
where the $(^{\prime})$ denotes derivative with respect to conformal time
$\tau=\int\frac{dt}{a(t)}$ ($\simeq\frac{-1}{aH}$ for quasi-de Sitter
expansion). The variable $v$, which itself is a scalar quantum field like
$\zeta$, is called the Mukhanov-Sasaki variable in literature. Its Fourier
modes $v_{k}$ satisfy the famous Mukhanov-Sasaki equation given by [53, 54]
$\boxed{v^{\prime\prime}_{k}\,+\,\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0}~{},$
(3.4)
where the effective mass term is given by the following exact expression [55]
$\displaystyle\boxed{\frac{z^{\prime\prime}}{z}=(aH)^{2}\left(2-\epsilon_{1}+\frac{3}{2}\epsilon_{2}+\frac{1}{4}\epsilon_{2}^{2}-\frac{1}{2}\epsilon_{1}\epsilon_{2}+\frac{1}{2}\epsilon_{2}\epsilon_{3}\right)}~{},$
(3.5)
$\displaystyle\Rightarrow\boxed{\frac{z^{\prime\prime}}{z}=(aH)^{2}\left[2+2\epsilon_{H}-3\eta_{H}+2\epsilon_{H}^{2}+\eta_{H}^{2}-3\epsilon_{H}\eta_{H}-\frac{1}{aH}\,\eta^{\prime}_{H}\right]}~{},$
(3.6)
with $\epsilon_{1}=\epsilon_{H}$ and where
$\epsilon_{n+1}=-\frac{d\ln{\epsilon_{n}}}{dN_{e}}~{}$ (3.7)
are the ‘Hubble flow’ parameters. Given a mode $k$, at sufficiently early
times when it is sub-Hubble i.e $k\gg aH$, we can assume $v$ to be in the
Bunch-Davies vacuum [56] satisfying
$v_{k}\rightarrow\frac{1}{\sqrt{2k}}e^{-ik\tau}~{}.$ (3.8)
During inflation as the comoving Hubble radius falls (see figure 2), modes
start becoming super-Hubble i.e $k\ll aH$ and equation (3.4) dictates that
$|v_{k}|\propto z$ and hence $\zeta_{k}$ approaches a constant value. By
solving the Mukhanov-Sasaki equation we can estimate the dimensionless
primordial power spectrum of $\zeta$ using the following relation [51]
$\boxed{P_{\zeta}\equiv\frac{k^{3}}{2\pi^{2}}\,|{\zeta}_{k}|^{2}\Big{|}_{k\ll
aH}=\frac{k^{3}}{2\pi^{2}}\,\frac{|v_{k}|^{2}}{z^{2}}\Big{|}_{k\ll aH}}~{}.$
(3.9)
During slow-roll inflation, the factor
$\frac{z^{\prime\prime}}{z}=\frac{\nu^{2}-0.25}{\tau^{2}}$ with $\nu\approx
1.5+\epsilon_{H}+\frac{\dot{\epsilon}_{H}}{2H\epsilon_{H}}$. Solving the
Mukhanov-Sasaki equation with suitable Bunch-Davies vacuum conditions leads to
the famous slow-roll approximation formula [7]
$\boxed{P_{\zeta}=\frac{1}{8\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\frac{1}{\epsilon_{H}}}~{}.$
(3.10)
Note that one could also directly try to solve for the fourier modes of the
comoving curvature perturbation $\zeta$ (instead of the Mukhanov-Sasaki
variable $v$) which satisfies the equation
$\boxed{{\zeta}^{\prime\prime}_{k}+2\left(\frac{z^{\prime}}{z}\right){\zeta}^{\prime}_{k}+k^{2}{\zeta}_{k}=0}$
(3.11)
and implement the corresponding Bunch-Davies initial conditions for
${\zeta}_{k}$. The friction term in equation (3.11) is given by
$\boxed{\frac{z^{\prime}}{z}=aH\,(1+\epsilon_{H}-\eta_{H})}~{}.$ (3.12)
Before moving forward, we stress that the slow-roll regime of inflation
necessarily requires both the slow-roll parameters to be small i.e
$\epsilon_{H}\ll 1$ and $\eta_{H}\ll 1$. Violation of either of these
conditions invalidates the above analytical treatment. When either of the
slow-roll conditions is violated, which is the situation in the context of
primordial black hole formation, a more accurate determination of $P_{\zeta}$
is provided by solving the Mukhanov-Sasaki equation (3.4) numerically. The
computation of power spectrum when the slow-roll approximation is violated
will be our primary focus.
### Tensor fluctuations
The corresponding quadratic Action for tensor fluctuations is given by [7, 52]
$\boxed{S_{(2)}[\gamma_{ij}]=\frac{1}{2}\int{\rm d}\tau{\rm
d}^{3}x\,a^{2}\left(\frac{m_{p}}{2}\right)^{2}\left[(\gamma_{ij}^{\prime})^{2}-(\partial\gamma_{ij})^{2}\right]}\,.$
(3.13)
The Mukhanov-Sasaki variable for tensor fluctuations are defined as
$\frac{m_{p}}{2}\,a\,\gamma_{ij}\equiv\left(\begin{smallmatrix}h_{+}&h_{\times}&0\\\
h_{\times}&-h_{+}&0\\\ 0&0&0\end{smallmatrix}\right)$ (3.14)
or,
$\frac{m_{p}}{2}\,a\,\gamma_{ij}\equiv\sum_{s=+,\times}\Pi_{ij}^{s}\ h_{s}\,,$
(3.15)
where $\Pi^{+}$ and $\Pi^{\times}$ are the 2 polarization modes, written as
$\Pi^{+}_{ij}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix}\quad\quad{\rm
and}\quad\quad\Pi^{\times}_{ij}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}\,.$ (3.16)
Thus,
$\gamma_{ij}=\frac{2}{m_{p}}\sum_{s=+,\times}\Pi_{ij}^{s}\,\frac{h_{s}}{a}\,.$
(3.17)
The evolution equation for the mode functions (by dropping the ‘s’ subscript
and remembering that it is summed over for 2 polarization states) is given by
$\boxed{h_{k}^{\prime\prime}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)h_{k}=0}\,.$
(3.18)
The subsequent computation of tensor power spectrum
$P_{T}(k)\equiv
2\times\frac{k^{3}}{2\pi^{2}}\,\frac{|h_{k}|^{2}}{a^{2}}\Big{|}_{k\ll aH}\,,$
(3.19)
under quasi-de Sitter approximation leads to [7]
$\boxed{P_{T}(k)=\frac{2}{\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}}\,.$ (3.20)
Note that, unlike the Mukhanov-Sasaki equation (3.4) for scalar fluctuations,
the tensor mode equation (3.18) does not depend upon $z$, rather it depends
only upon the scale factor $a$. Hence, as long as the quasi-de Sitter
approximation is valid, i.e $\epsilon_{H}\ll 1$, power spectrum of tensor
fluctuations does not get affected by an appreciable amount even if slow-roll
is violated. Although, this statement is true only at linear order in
perturbation theory. Tensor fluctuations at second order in perturbation
theory can be induced by large first-order scalar fluctuations [57, 58, 59,
60] (also see [61] and references therein).
### 3.1 Large scale primordial fluctuations
On large cosmological scales which are accessible to CMB observations, the
scalar power spectrum typically takes the form of a power law represented by
$P_{\zeta}(k)=A_{{}_{S}}\left(\frac{k}{k_{*}}\right)^{n_{{}_{S}}-1},$ (3.21)
where $A_{{}_{S}}=P_{\zeta}(k_{*})$ is the amplitude of the scalar power
spectrum at the pivot scale $k=k_{*}$, given by333Note that in general, $k$
may correspond to any observable CMB scale in the range
$k\in\left[0.0005,0.5\right]~{}{\rm Mpc}^{-1}$. However, in order to derive
constraints on the inflationary observables $\\{n_{{}_{S}},r\\}$, we mainly
focus on the CMB pivot scale, namely $k\equiv k_{*}=0.05~{}{\rm Mpc}^{-1}$.
$\boxed{A_{{}_{S}}=\frac{1}{8\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\frac{1}{\epsilon_{H}}\,\bigg{|}_{\phi=\phi_{*}}}~{},$
(3.22)
where $\phi_{*}$ is the value of the inflaton field at the epoch of Hubble
exit of the CMB pivot scale $k_{*}$. The scalar spectral tilt $n_{{}_{S}}$, in
the slow-roll regime is given by [7]
$\displaystyle\boxed{n_{{}_{S}}-1\equiv\frac{d\,\mathrm{ln}P_{\zeta}}{d\,\mathrm{ln}k}=2\eta_{H}-4\epsilon_{H}}~{}.$
(3.23)
Similarly the tensor power spectrum, in the slow-roll limit, is represented by
$P_{T}(k)=A_{{}_{T}}\left(\frac{k}{k_{*}}\right)^{n_{{}_{T}}},$ (3.24)
with the amplitude of tensor power spectrum at the CMB pivot scale is given by
[7, 51]
$\boxed{A_{{}_{T}}\equiv
P_{T}(k_{*})=\frac{2}{\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\bigg{|}_{\phi=\phi_{*}}}~{},$
(3.25)
and the tensor spectral index (with negligible running) is given by
$\boxed{n_{{}_{T}}=-2\,\epsilon_{H}}~{}.$ (3.26)
The tensor-to-scalar ratio $r$ is defined by
$\boxed{r\equiv\frac{A_{{}_{T}}}{A_{{}_{S}}}=16\,\epsilon_{H}}~{},$ (3.27)
yielding the single field consistency relation
$\boxed{r=-8\,n_{{}_{T}}}~{}.$ (3.28)
Hence the slow-roll parameters $\epsilon_{H}$ and $\eta_{H}$ play an important
role in characterising the power spectra of scalar and tensor fluctuations
during inflation. Before going forward, we briefly discuss the implications of
the latest CMB observations for the slow-roll parameters as well as for other
relevant inflationary observables. In order to relate the CMB observables to
the inflaton potential $V(\phi)$, we work with the potential slow-roll
parameters defined in equation (2.21).
Consider a canonical scalar field minimally coupled to gravity and having the
potential
$V(\phi)=V_{0}\,f\left(\frac{\phi}{m_{p}}\right)~{}.$ (3.29)
The potential slow-roll parameters (2.21) are given by
$\displaystyle\epsilon_{{}_{V}}=\frac{m_{p}^{2}}{2}\left(\frac{f_{,\phi}}{f}\right)^{2}~{},$
(3.30)
$\displaystyle\eta_{{}_{V}}=m_{p}^{2}\left(\frac{f_{,\phi\phi}}{f}\right)~{}.$
(3.31)
In the slow-roll limit $\epsilon_{{}_{V}},\,\eta_{{}_{V}}\ll 1$, the scalar
power spectrum is given by the expression (3.21) with the amplitude of scalar
power at the CMB pivot scale $k\equiv k_{*}=0.05~{}{\rm Mpc}^{-1}$ expressed
as [7]
$A_{{}_{S}}\equiv
P_{\cal\zeta}(k_{*})\simeq\frac{1}{24\pi^{2}}\frac{V_{0}}{m_{p}^{4}}\frac{f\left(\phi_{k}\right)}{\epsilon_{{}_{V}}(\phi_{k})}\bigg{|}_{k=k_{*}}~{},$
(3.32)
and the scalar spectral index (with negligible running) is given by
$n_{{}_{S}}\simeq
1+2\,\eta_{{}_{V}}(\phi_{*})-6\,\epsilon_{{}_{V}}(\phi_{*})~{},$ (3.33)
Similarly the amplitude of tensor power spectrum at the CMB pivot scale is
given by
$A_{{}_{T}}\equiv
P_{T}(k_{*})=\frac{2}{\pi^{2}}\left(\frac{H}{m_{p}}\right)^{2}\bigg{|}_{\phi=\phi_{*}}\simeq\frac{2}{3\pi^{2}}\frac{V_{0}}{m_{p}^{4}}f\left(\phi_{*}\right)~{},$
(3.34)
and the tensor spectral index (3.26) becomes
$n_{{}_{T}}\simeq-2\,\epsilon_{{}_{V}}(\phi_{*})~{},$ (3.35)
and the tensor-to-scalar ratio (3.27) can be written as
$r\simeq 16\,\epsilon_{{}_{V}}(\phi_{*})~{},$ (3.36)
satisfying the single field consistency relation (3.28). From the CMB
observations of Planck 2018 [15], we have
$A_{{}_{S}}=2.1\times 10^{-9}~{},$ (3.37)
while the $2\sigma$ constraint on the scalar spectral index is given by
$n_{{}_{S}}\in[0.957,0.976]~{}.$ (3.38)
Similarly the constraint on the tensor-to-scalar ratio $r$, from the latest
combined observations of Planck 2018 [15] and BICEP/Keck [17], is given by
$r\leq 0.036~{},$ (3.39)
which translates into $A_{{}_{T}}\leq 3.6\times 10^{-2}\,A_{{}_{S}}$. Equation
(3.34) helps place the following upper bound on the inflationary Hubble scale
$H^{\rm inf}$ and the energy scale during inflation $E_{\inf}$
$\displaystyle H^{\rm inf}\leq 4.7\times 10^{13}~{}{\rm GeV}~{},$ (3.40)
$\displaystyle E_{\inf}\equiv\left[\sqrt{3}\,m_{p}\,H^{\rm
inf}\right]^{1/2}\leq 1.4\times 10^{16}~{}{\rm GeV}~{}~{}.$ (3.41)
Similarly the CMB bound on $r$ when combined with (3.36) translates into an
upper bound on the first slow-roll parameter
$\epsilon_{H}\simeq\epsilon_{{}_{V}}\leq 0.00225~{},$ (3.42)
rendering the tensor tilt from equation (3.35) to be negligibly small
$|n_{{}_{T}}|\leq 0.0045~{}.$ (3.43)
Given the upper limit on $\epsilon_{{}_{V}}$, using the CMB bound on
$n_{{}_{S}}$ from (3.38) in (3.33), we infer that the second slow-roll
parameter is negative and obtain interesting upper and lower limits on its
magnitude, given by
$|\eta_{H}|\in[0.0075,0.0215]~{}.$ (3.44)
The EOS $w_{\phi}$ of the inflaton field is given by
$w_{\phi}=\frac{\frac{1}{2}\dot{\phi}^{2}-V(\phi)}{\frac{1}{2}\dot{\phi}^{2}+V(\phi)}\simeq-1+\frac{2}{3}\epsilon_{{}_{V}}(\phi)\,,$
(3.45)
Therefore one finds from (3.42) the following constraint on the inflationary
EOS at the pivot scale
$w_{\phi}\leq-0.9985\,,$ (3.46)
implying that the expansion of the universe during inflation was near
exponential (quasi-de Sitter like).
Figure 3: This figure shows the Starobinsky potential (3.48) with CMB pivot
scale $k_{*}=0.05~{}{\rm Mpc}^{-1}$ by a blue color star as well as the CMB
window $k_{\rm CMB}\in\left[0.0005,0.5\right]~{}{\rm Mpc}^{-1}$ in grey color
shade in the field space. From this figure, it is clear that CMB window
constitutes only a tiny portion of the available field space in between
$\phi_{\rm CMB}$ and end of inflation $\phi_{e}$.
The CMB observations, in the context of single field slow-roll inflationary
paradigm, favours asymptotically-flat potentials (featuring either one or two
plateau wings) with $n_{{}_{S}}\simeq 0.965$ and $r\leq 0.036$. A typical
plateau-potential is demonstrated in figure 1 and this is the standard/vanilla
scenario. Given that power spectrum is almost scale-invariant with slightly
red tilt, i.e $n_{{}_{S}}-1\lesssim 0$, large-scale fluctuations are more
important while nothing drastic is expected to happen on smaller cosmological
scales that are super-Hubble at the end of inflation.
Before proceeding further to discuss small-scale inflationary fluctuations,
let us make our nomenclature concrete (which is consistent with the standard
nomenclature in the inflationary literature).
• Quasi-de Sitter inflation corresponds to the condition $\epsilon_{H}\ll 1$.
• Slow-roll inflation corresponds to $\epsilon_{H},\,\eta_{H}\ll 1$.
This distinction will be important for the rest of the discussions in this
paper. Under either of the aforementioned assumptions, the expression for
conformal time is given by
$\displaystyle\boxed{-\tau\simeq\frac{1}{aH}}\,.$ (3.47)
### 3.2 Small-scale primordial fluctuations
As mentioned above, the recent CMB observations support the scenario of the
inflaton field rolling slowly down an asymptotically-flat potential at the
time when the observable CMB scales made their Hubble exit during inflation.
However, the current CMB and LSS observations probe only about 7-8 e-folds of
inflation around the Hubble exit time of the CMB pivot scale. We explicitly
mention that CMB observations probe primordial fluctuations with comoving
scales $k_{\rm CMB}\in\left[0.0005,0.5\right]~{}{\rm Mpc}^{-1}$ (which
includes pivot scale $k_{*}=0.05~{}{\rm Mpc}^{-1}$) corresponding to multipole
$l\in\left[2,2500\right]$ in the angular sky. Additionally, Lyman-$\alpha$
forest observations enforce constraints on the primordial power spectrum upto
$k\simeq{\cal O}(1)~{}{\rm Mpc}^{-1}$ (see [29]).
Hence a large portion of evolution during the inflationary phase that accounts
roughly to about 50 e-folds of expansion, corresponding to scales smaller than
those probed by CMB, remains observationally inaccessible at present.
Consequently the associated dynamics of the inflaton field also remains
unprobed. For example, figure 3 demonstrates that the CMB window constitutes
only a tiny part of the observationally available field space between the
largest scales in the sky and the smallest scale at the end of inflation for
Starobinsky potential [1, 62]
$\boxed{V(\phi)=V_{0}\,\left(1-e^{-\frac{2}{\sqrt{6}}\,\frac{\phi}{m_{p}}}\right)^{2}}\,.$
(3.48)
Any deviation from the quasi-de Sitter expansion and/or departure from the
slow-roll regime $\epsilon_{{}_{H}},\eta_{{}_{H}}\ll 1$ that might be
triggered by a change in the dynamics of the inflaton field, would lead to
interesting observational consequences on small scales. In particular if the
inflaton potential possesses a near inflection point-like broad feature at
some intermediate field values, then the scalar quantum fluctuations
corresponding to scales becoming super-Hubble around the time when the
inflaton rolls past such features, might receive enough amplification to
facilitate the formation of Primordial Black Holes (PBHs) upon their Hubble
re-entry during the radiative epoch.
Figure 4: This figure schematically depicts a prototype plateau potential,
plotted in solid green curve. The ‘CMB Window’ represents field values
corresponding to the Hubble-exit epochs of scales
$k\in\left[0.0005,0.5\right]~{}{\rm Mpc}^{-1}$ that are observable by the
latest CMB missions. The potential exhibits a small-scale feature (shown in
the salmon colour shading) in the form of a flat inflection point-like segment
which results in ultra slow-roll (USR) inflation. After exiting the first
slow-roll phase (SR-I) near the CMB window, the inflaton enters into an USR
phase, during which the second slow-roll condition is violated, namely
$\eta_{H}\simeq+3$. This leads to an enhancement of power spectrum at small-
scales. Later, the inflaton emerges out of the USR to another slow-roll phase
(SR-II) before the end of inflation.
We note that, there are a plethora of possible features that would lead to
deviation from standard scale-invariant power spectrum [63]. However, in this
paper we will only focus on potentials with a tiny local bump/dip like feature
[44] in order to illustrate the efficiency of our numerical analysis. Our code
can be used to simulate a number of different types of features in the
inflaton potential.
PBH formation requires the enhancement of the inflationary power spectrum by
roughly a factor of $10^{7}$ within less than 40 e-folds of expansion (on
scales smaller than the pivot scale $k_{*}$) as illustrated in figure 5.
Therefore the quantity $\Delta\ln{\epsilon_{H}}/\Delta N$, and hence also
$|\eta_{H}|$, can grow to become of order unity, thereby violating the second
slow-roll condition [64]. In fact the second Hubble slow-roll parameter
$|\eta_{H}|$ becomes larger than unity even though $\epsilon_{H}$ itself
remains much smaller than unity. As a result, equation (3.22) can no longer be
trusted to compute the power spectrum and one must determine $P_{\cal R}$ by
numerically integrating the Mukhanov-Sasaki equation (3.4).
We proceed as follows. We first discuss the simulations of inflationary
background dynamics in section 4, where we also discuss how to generate phase-
space portrait $\\{\phi,\dot{\phi}\\}$ during inflation. In section 5.1, we
introduce our numerical scheme for studying quantum fluctuations during slow-
roll inflation. We work with convex as well as asymptotically-flat potentials.
In section 5.2, we apply our numerical scheme to potentials featuring a local
bump/dip like feature that facilitates the amplification of scalar power on
small primordial scales. We demonstrate that slow-roll formula (3.22)
underestimates both the location as well as the height of scalar power
spectrum ${\cal P}_{\zeta}(k)$ for both type of aforementioned features and
hence one must solve the Mukhanov-Sasaki equation (3.4) numerically to
estimate the power spectrum accurately. We also demonstrate that the growth of
the power spectrum obeys the steepest growth bounds discussed in [65, 66, 67,
68].
Figure 5: This figure schematically illustrates the typical amplification of
inflationary primordial power spectrum at smaller length scales required for
PBH formation.
## 4 Numerical analysis of inflationary background dynamics
A complete analysis of the inflationary background dynamics can be obtained
from the evolution of $\phi$, $\dot{\phi}$ and $H$. All of these quantities
can be simulated by numerically solving equations (2.14), (2.15) and (2.16).
The evolution of the scale factor follows directly from $H=\dot{a}/a$. Our
system is defined by the following set of equations (as a function of cosmic
time $t$)
$\displaystyle H^{2}$
$\displaystyle=\frac{1}{3m_{p}^{2}}\left[\frac{1}{2}{\dot{\phi}}^{2}+V(\phi)\right]\,,$
(4.1) $\displaystyle\dot{H}$ $\displaystyle=-\frac{\dot{\phi}^{2}}{2m_{p}}\,,$
(4.2) $\displaystyle\ddot{\phi}$
$\displaystyle=-3H\dot{\phi}-V_{,\phi}(\phi)\,,$ (4.3)
where the functional form of the potential $V(\phi)$ is given by the specific
inflationary model. However, the rest of the algorithm is largely model-
independent. We can re-write the potential as
$V(\phi)=V_{0}\,f(\phi)\,.$ (4.4)
In order to carry out numerical simulations, it is convenient to write down
the dynamical equations in terms of dimensionless variables (which also
ensures that we do not need to worry about keeping track of units).
Furthermore, it is important to re-scale the time variable by a factor $S$
which can be suitably chosen according to the energy scale of the
dynamics444Depending upon the potential, we usually choose the value of $S$ to
be in the range $S\in[10^{-5},\,10^{-3}]$.. Our primary dimensionless
variables are defined as
$\displaystyle T$ $\displaystyle=\left(t\,m_{p}\right)\,S\,,$ (4.5)
$\displaystyle x$ $\displaystyle=\frac{\phi}{m_{p}}\,,$ (4.6) $\displaystyle
y$ $\displaystyle=\left(\frac{\dot{\phi}}{m_{p}^{2}}\right)\,\frac{1}{S}\,,$
(4.7) $\displaystyle z$
$\displaystyle=\left(\frac{H}{m_{p}}\right)\,\frac{1}{S}\,,$ (4.8)
$\displaystyle A$ $\displaystyle=\left(a\,m_{p}\right)\,S\,.$ (4.9)
In terms of these variables, the dynamical equations (to be simulated) take
the form
$\displaystyle\frac{{\rm d}x}{{\rm d}T}$ $\displaystyle=y\,,$ (4.10)
$\displaystyle\frac{{\rm d}y}{{\rm d}T}$
$\displaystyle=-3\,z\,y-\frac{v_{0}}{S^{2}}\,f_{,x}(x)\,,$ (4.11)
$\displaystyle\frac{{\rm d}z}{{\rm d}T}$
$\displaystyle=-\frac{1}{2}\,y^{2}\,,$ (4.12) $\displaystyle\frac{{\rm
d}A}{{\rm d}T}$ $\displaystyle=A\,z\,.$ (4.13)
We can also define the dimensionless potential to be
$\frac{V(\phi)}{m_{p}^{4}}\equiv v_{0}\,f(x)=\frac{V_{0}}{m_{p}^{4}}\,f(x)\,.$
(4.14)
We can solve the aforementioned set of equations with appropriate initial
conditions. In our analysis, we use the odeint function provided in the
scipy.integration package. By incorporating initial conditions
$\\{x_{i},\,y_{i},\,z_{i},\,A_{i}\\}$ for the primary dynamical variables
$\\{x,\,y,\,z,\,A\\}$, we simulate their time evolution during inflation.
Accordingly, we determine the crucial derived/secondary (dimensionless)
dynamical variables from the primary ones by555Note that the observables
$A_{{}_{S}},\,A_{{}_{T}},\,n_{{}_{S}},\,n_{{}_{T}},\,{\rm and}\,r$ are related
to inflationary scalar and tensor fluctuations and the expressions given here
are under slow-roll approximations, i.e $\epsilon_{H},|\eta_{H}|\ll 1$, during
which they can be determined purely from the dynamics of background quantities
such as $H,\,\epsilon_{H},\,\eta_{H}$. Computation of inflationary power
spectra when slow-roll is violated is described in section 5.2.
$\displaystyle N$ $\displaystyle=\log{\frac{A}{A_{i}}}$ (4.15)
$\displaystyle\epsilon_{H}$
$\displaystyle=\frac{1}{2}\,\frac{y^{2}}{z^{2}}~{},$ $\displaystyle\eta_{H}$
$\displaystyle=-\frac{1}{yz}\frac{{\rm d}y}{{\rm d}T}\,,$ (4.16)
$\displaystyle A_{{}_{S}}$
$\displaystyle=\frac{1}{8\pi^{2}}\,\frac{\left(Sz\right)^{2}}{\epsilon_{H}}~{},$
$\displaystyle A_{{}_{T}}$
$\displaystyle=\frac{2}{\pi^{2}}\,\left(Sz\right)^{2}\,,$ (4.17)
$\displaystyle n_{{}_{S}}$ $\displaystyle=1+2\,\eta_{H}-4\,\epsilon_{H}~{},$
$\displaystyle n_{{}_{T}}$ $\displaystyle=-2\,\epsilon_{H}\,,$ (4.18)
$\displaystyle r$ $\displaystyle=16\,\epsilon_{H}\,.$ (4.19)
We define $N_{T}$ to be the number of e-folds of accelerated expansion
realised in between an arbitrary initial time and the end of inflation, which
is marked by $\epsilon_{H}=1$. We then define the more important quantity
$N_{e}=N_{T}-N$ as the number of e-folds before the end of inflation. Note
that $N_{e}=0$ at the end of inflation while $N_{e}>0$ at early times. This
will be our primary time variable against which we will be plotting the
dynamics of different inflationary observables. In order to realise adequate
amount of inflation, i.e. $N_{T}>60$, initial value of scalar field must be
large enough (and is model dependent). For most large field potentials, this
value is of the order $\phi_{i}\lesssim{\cal O}(10)\,m_{p}$.
Composing the code involves typing down the dimensionless equations in the
appropriate syntax and solving them by using an ODE solver. See our
supplementary Python code 666https://github.com/bhattsiddharth/NumDynInflation
for details. For a particular model of interest, we need to input the inflaton
potential in the form $\frac{V(\phi)}{m_{p}^{4}}=v_{0}\,f(x)$. Since the slow-
roll parameters and the duration $N_{T}$ do not strongly depend on $v_{0}$, we
can initially set its value to roughly $v_{0}=10^{-10}$. We can later adjust
the value of $v_{0}$ to yield the correct CMB normalised value of scalar power
spectrum (3.37) at the pivot scale $N=N_{*}$. One can proceed in the following
step-by-step algorithm.
Figure 6: Time evolution of the number of e-folds (scale factor in the
logarithm scale) of expansion of the universe is shown for Starobinsky
potential (3.48). For the most part of inflation, the expansion is almost
exponential (quasi-de Sitter) i.e $a\sim e^{Ht}$, leading to a rapid growth in
the number of e-folds within a small amount of time. While after the end of
inflation, the expansion is decelerated, leading to a much slower growth in
the scale factor.
1. 1.
After setting the parameters of the potential and defining the function
$f(x)$, we need to incorporate initial conditions for the four primary
variables $\\{x,\,y,\,z,\,A\\}$. We enter appropriate initial conditions
$x_{i},\,y_{i}$ and $A_{i}$ in the following way. $A_{i}$ can be set
arbitrarily in a spatially flat universe, however depending upon the energy
scale of inflation, one can provide an appropriate value. We suggest a typical
$A_{i}=1\times 10^{-3}$, although its precise value does not affect the
dynamics. In regard to the initial value of $x$, we need to ensure that
$x_{i}$ is large enough (or small enough if we are working with symmetry
breaking hilltop type potentials) to yield adequate amount of inflation, i.e
$N_{T}\geq 70$. As mentioned before, the typical value for large field models
is $x_{i}\lesssim{\cal O}(10)$.
Since we will be mostly working with potentials that exhibit slow-roll
behaviour at initial times, and given that slow-roll trajectory is an
attractor in relatively large field models [20], we can safely set $y_{i}=0$,
as long as $x_{i}$ is large enough. One can also incorporate slow-roll initial
conditions from the beginning, namely
$y_{i}=-\frac{v_{0}}{3}\,\frac{f_{,x}}{Sz}$, as is usually done in
practice777Note that for phase-space analysis, we need to incorporate
arbitrary values of $x_{i},\,y_{i}$ (consistent with fixed initial $z_{i}$)
which may be away from the slow-roll trajectory as discuss below.. Finally,
the initial value of $z$ can be incorporated in terms of $x_{i},\,y_{i}$ using
the dimensionless Friedmann equation
$\displaystyle
z_{i}=\sqrt{\frac{1}{6}\,y_{i}^{2}+\frac{1}{3}\,v_{0}\,f(x_{i})\,\frac{1}{S^{2}}}\,.$
(4.20)
2. 2.
We then proceed to solve the system of equations by taking adequately small
time steps $T$ in the appropriate range $T\in[T_{i}=0,\,T_{f}]$. We then plot
$N$ vs $T$ as given in figure 6 for Starobinsky potential. Typically, $N$
grows linearly with $T$ during near exponential inflation and a substantial
decrease in the rate of growth of $N$ indicates the end of inflation.
3. 3.
In order to concretely determine the value of $N_{T}$, we plot $\epsilon_{H}$
vs $N$, and note the value of $N$ after which $\epsilon_{H}\geq 1$. By
definition, initially $N=0$. If $N_{T}<70$, then we repeat this step by
increasing the value of $x_{i}$, until we get $N_{T}\geq 70$. (Alternatively,
if inflation has not ended888Note that if one simulates the cosmological
equations in terms of number of e-folds $N$, rather than cosmic time $t$, this
step can usually be avoided by simulating the system from $N=0$ to $N=70$.
However, one has to adjust the value of $x_{i}$ in order to get enough
inflation., i.e $\epsilon_{H}<1$, at the end of our simulation, then either
one can increase the value of $T_{f}$ or decrease $x_{i}$. We suggest the
latter.) We can then define the number of e-folds before the end of inflation
to be $N_{e}=N_{T}-N$.
Figure 7: This figure describes the evolution of inflaton field $\phi$, and
its speed $\dot{\phi}$ in the left panel, while the Hubble parameter $H$ in
the right panel as a function of the number of e-folds before the end of
inflation $N_{e}$ for Starobinsky potential (3.48). Note that during slow-roll
inflation, $\dot{\phi}$ and $H$ are nearly constant, while $\phi$ changes
quite slowly. However, $\phi$ and $H$ begin to change rapidly towards the end
of inflation. After inflation ends, $\phi$ and $\dot{\phi}$ start oscillating
around the minimum of the potential (which is not shown in this figure).
Figure 8: Evolution of the slow-roll parameters $\epsilon_{H}$ and $\eta_{H}$
is shown as a function of the number of e-folds before the end of inflation
$N_{e}$ for Starobinsky potential (3.48). From this plot, it is easy to notice
that at early times when $N_{e}\gg 1$, the slow-roll conditions are satisfied
i.e $\epsilon_{H},\,|\eta_{H}|\ll 1$. However, the slow-roll conditions are
violated towards the end of inflation (marked by $N_{e}=0$ and
$\epsilon_{H}=1$).
4. 4.
The pivot scale can then be fixed to an appropriate value, for example
$N_{e}=60$, as used in this work. Figure 7 describes the evolution of
$\phi,\,\dot{\phi},\,{\rm and}\,H$, while figure 8 illustrates the dynamics of
slow-roll parameters $\epsilon_{H},\,|\eta_{H}|$ for Starobinsky potential, as
determined from our code. As mentioned before, we usually plot the dynamics of
inflation as a function of $N_{e}$.
5. 5.
In order to accurately fix the value of $v_{0}$, we need to impose
$A_{S}=2.1\times 10^{-9}$ at the pivot scale $N_{e}=60$. If $A_{S}$ is lower
than expected for the given value of $v_{0}$, then we increase the value of
$v_{0}$ or vice versa until we arrive at the correct value of $A_{S}$, and fix
the corresponding value of $v_{0}$.
Following the aforementioned algorithm, we can easily simulate the
inflationary background dynamics and investigate the evolution of relevant
quantities of our interest. Before going forward, we would like to stress that
many of the aforementioned steps (in the present version of our code) are
rather meant to be carried out manually by the user. While we are already
developing an automated version of this code (which will be presented in the
revised version of our paper), we believe that the present version will help
the user to understand the inflationary dynamics much better.
### 4.1 Phase-space analysis
Phase-space analysis of inflationary dynamics is usually carried out to
determine the set of initial conditions that results in adequate amount of
inflation, and hence it is important to access the generality of initial
conditions for inflation [69, 50, 20]. For a spatially flat background, the
phase-space portrait consists of trajectories of $\\{\phi,\,\dot{\phi}\\}$ for
different initial conditions, with fixed $H_{i}$. The standard algorithm to
generate such a plot is the following.
1. 1.
The initial energy scale of inflation is kept constant by fixing the value of
initial Hubble parameter in the phase-space portrait simulations. A typical
value often used is $H_{i}\leq m_{p}$ (see [20] for detail). Hence, the user
is expected to incorporate an appropriate value of $z_{i}$.
2. 2.
One can then input a suitable value of $x_{i}$ and determine the value of
$y_{i}$ for a given potential function $f(x)$ from the dimensionless Friedmann
equation (4.20) as
$y_{i}=\pm\sqrt{6}\,\sqrt{z_{i}^{2}-\frac{1}{3}\,v_{0}\,f(x_{i})\,\frac{1}{S^{2}}}~{}.$
(4.21)
3. 3.
With these initial conditions, one can then simulate the system of
dimensionless differential equations for $\\{x,\,y,\,z,\,A\\}$ from $T_{i}=0$
till an appropriate $T_{f}$. One can then repeat the same step by
incorporating a number of different values of of $x_{i}$ in order to generate
the phase-space portrait for the given potential. We provide the GitHub link
to our phase-space portrait framework here
999https://github.com/bhattsiddharth/NumDynInflation/blob/main/inf_dyn_phase.py.
The phase-space portraits for Starobinsky potential (3.48), and quadratic
potential $V(\phi)\propto\phi^{2}$ are illustrated in the left and right
panels of figure 9 respectively.
Figure 9: The phase-space portrait $\\{\phi,\,\dot{\phi}\\}$ of the inflaton
field has been illustrated for Starobinsky potential (3.48) in the left panel,
and for quadratic potential $V(\phi)=\frac{1}{2}\,m^{2}\,\phi^{2}$ in the
right panel corresponding to different initial conditions
$\\{\phi_{i},\,\dot{\phi_{i}}\\}$ (plotted in solid black colour) with a fixed
initial scale $H_{i}$. The figure demonstrates that trajectories commencing
from a large class of initial field values (including those with large initial
velocities $\dot{\phi_{i}}$) quickly converge towards the slow-roll attractor
separatrix $\dot{\phi}=-V_{,\phi}/3H\simeq\mathrm{const.}$ (plotted in green
colour) as can be seen from the rapid decline in the inflaton speed until they
meet the green colour curve. After the end of inflation, the inflaton begins
to oscillate around the minimum of the potential.
In order to determine the degree of generality of inflation, we need to define
a measure for the distribution of $\\{\phi_{i},\,\dot{\phi_{i}}\\}$. The
correct choice for the measure might depend on the quantum theory of gravity.
However, a uniform measure is usually considered in the literature. Interested
readers are referred to [69, 20] for further detail.
### 4.2 Quantum fluctuations under the slow-roll approximation
In section 3, we described the expressions for a number of inflationary
observables such as $A_{{}_{S}},\,A_{{}_{T}},\,n_{{}_{S}},\,n_{{}_{T}},\,{\rm
and}\,r$ associated with the scalar and tensor power spectra which can be
determined purely from the dynamics of background quantities such as
$H,\,\epsilon_{H},\,\eta_{H}$ under the slow-roll approximation. Hence they
can be conveniently determined from our background dynamics code as discussed
earlier in section 4. For example, the scalar and tensor power spectra for
Starobinsky inflation have been plotted in figure 10 as a function $N_{e}$.
Similarly, one can plot the spectral indices101010In the standard literature,
one usually plots $r$ vs $n_{{}_{S}}$ for a given inflaton potential for a
range of possible values of $N_{*}\in[50,60]$ which can also be done easily
using our code. $n_{{}_{S}}-1$ and $n_{{}_{T}}$ and determine their values at
the pivot scale $N_{*}$. The spectral indices for Starobinsky potential have
been plotted in figure 11.
Figure 10: The power spectra of scalar and tensor quantum fluctuations
(computed using the slow-roll formulae (3.10) and (3.25) respectively) are
shown for comoving modes exiting the Hubble radius at different number of
e-folds $N_{e}$ before the end of inflation for Starobinsky potential (3.48).
The CMB window (shown in grey shaded region) corresponds to comoving modes in
the range $k_{\rm CMB}\in[0.0005,0.5]~{}{\rm Mpc}^{-1}$ that are being probed
by the current CMB missions. Fluctuations over larger scales (shown in red
shaded region) are outside the observable universe at present and those over
smaller scales remain to be (potentially) probed by a plethora of upcoming
missions, from GW observatories to PBHs.
Figure 11: The scalar and tensor spectral indices $n_{{}_{S}}$ and
$n_{{}_{T}}$ are shown as a function of $N_{e}$ for Starobinsky potential
(3.48) as determined by their slow-roll approximated formulae (3.23) and
(3.26) respectively. Around the pivot scale, they take the approximate values
$n_{{}_{S}}\simeq 0.967\text{ and }n_{{}_{T}}\simeq-0.0004$. Note that we have
plotted $n_{{}_{S}}-1$ (instead of $n_{{}_{S}}$) since it is the correct
scalar spectral index. The tensor-to-scalar ratio is given by
$r\simeq-8\,n_{{}_{T}}$.
## 5 Numerical analysis for quantum fluctuations during inflation
In the previous section we used slow-roll approximated formulae to study the
spectra of inflationary fluctuations in terms of background quantities such as
$H,\,\epsilon_{H},\,\eta_{H}$. Hence, we only had to simulate the background
dynamics for a given potential in order to plot the relevant inflationary
observables. However, if we want to analyze the behaviour of quantum
fluctuations more accurately, especially in situations where one or both the
slow-roll conditions (2.20) are violated, we need to numerically solve the
Mukhanov-Sasaki equation (3.4) corresponding to each comoving scale $k$.
For this purpose, we first rewrite the Mukhanov-Sasaki equation (3.4) in
cosmic time as
$\frac{{\rm d}^{2}v_{k}}{{\rm d}t^{2}}+H\frac{{\rm d}v_{k}}{{\rm
d}t}+\left[\frac{k^{2}}{a^{2}}-\frac{1}{a^{2}}\frac{z^{\prime\prime}}{z}\right]v_{k}=0\,.$
(5.1)
Note that here $z$ is not the dimensionless Hubble parameter used in our
numerical code, rather it is the variable $z=am_{p}\sqrt{2\epsilon_{H}}$ in
the Mukhanov-Sasaki equation (3.4). The effective mass term
$z^{\prime\prime}/z$ in (3.6) can be re-written as
$\frac{z^{\prime\prime}}{z}=a^{2}\left[\frac{5}{2}\frac{\dot{\phi}^{2}}{m_{p}^{2}}+2\frac{\dot{\phi}\ddot{\phi}}{Hm_{p}^{2}}+2H^{2}+\frac{1}{2}\frac{\dot{\phi}^{4}}{H^{2}m_{p}^{4}}-V_{,\phi\phi}(\phi)\right]\,.$
(5.2)
Since $v_{k}$ is a complex valued function, it is convenient to split it into
its real and imaginary parts to study their evolution separately for the
numerical analysis. While both will follow the same evolution equation, they
will be supplied with different initial conditions in the form of the real and
imaginary parts of the Bunch-Davies vacuum (3.8). Writing the Mukhanov-Sasaki
equation for scalar fluctuations in terms of dimensionless variables, we
obtain
$\boxed{\frac{{\rm d}^{2}v_{k}}{{\rm d}T^{2}}+z\,\frac{{\rm d}v_{k}}{{\rm
d}T}+\left[\frac{k^{2}}{A^{2}}-\frac{5}{2}\,y^{2}+2\,\frac{y}{z}\left(3\,z\,y+\frac{v_{0}}{S^{2}}\,f_{,x}\right)-2\,z^{2}-\frac{1}{2}\,\frac{y^{4}}{z^{2}}+\frac{v_{0}}{S^{2}}\,f_{,xx}\right]v_{k}=0}\,.$
(5.3)
Figure 12: Evolution of scalar power $\frac{k^{3}}{2\pi^{2}}|\zeta_{k}|^{2}$
is plotted by numerically solving the Mukhanov-Sasaki equation (5.3) for a
mode exiting the Hubble radius at about 60 e-folds before the end of inflation
(for Starobinsky potential). At early times when the mode is sub-Hubble, i.e.
$k\gg aH$, the power decreases as ${\cal P}_{\zeta}\sim(aH)^{-2}$ as expected.
After the Hubble-exit, the power freezes to a constant in the super-Hubble
regime when $k\ll aH$. We note down its value after the mode-freezing as the
super-Hubble scale power corresponding to that mode. Repeating the procedure
for a range of scales $k$ yields us the power spectrum of scalar fluctuations.
The same numerical analysis can be carried out for tensor fluctuations.
Our primary goal in this section is to numerically solve equation (5.3) for
the Fourier modes $v_{k}$ corresponding to each comoving scale $k$ and plot
the frozen value of the scalar power spectrum of $\zeta_{k}$ given by (3.9)
after the mode becomes super-Hubble. We can conveniently relate a comoving
scale $k$ to its Hubble-exit epoch by $k=aH$. Since we are only interested in
the super-Hubble power spectra, we only need to simulate the system to evolve
$v_{k}$ for a small duration of time around the Hubble-exit of scale $k$
(which should be sufficiently early enough to impose Bunch-Davies initial
conditions and sufficiently late enough for the mode to be frozen outside the
Hubble radius).
In the following, we discuss the algorithm to solve the Mukhanov-Sasaki
equation (5.3) and determine the scalar power spectrum (3.9) numerically. We
also discuss how to solve the corresponding equation (3.18) for the tensor
power spectrum (at linear order in perturbation theory)111111Second-order
tensor fluctuations which are induced by first-order scalar fluctuations will
be discussed in the revised version of our manuscript.. The dimensionless
Mukhanov-Sasaki equation for tensor fluctuations is given by
$\boxed{\frac{{\rm d}^{2}h_{k}}{{\rm d}T^{2}}+z\,\frac{{\rm d}h_{k}}{{\rm
d}T}+\left[\frac{k^{2}}{A^{2}}+\frac{1}{2}\,y^{2}-2\,z^{2}\right]h_{k}=0}\,.$
(5.4)
We explicitly write down the Mukhanov-Sasaki equations for scalar and tensor
fluctuations in terms of dimensionless variables (as used in our code) in the
following way
$\displaystyle v_{k,T}$ $\displaystyle=\frac{{\rm d}v_{k}}{{\rm d}T}\,,$ (5.5)
$\displaystyle\frac{{\rm d}v_{k,T}}{{\rm d}T}$
$\displaystyle=-z\,v_{k,T}-\left[\frac{k^{2}}{A^{2}}-\frac{5}{2}\,y^{2}+2\,\frac{y}{z}\left(3\,z\,y+\frac{v_{0}}{S^{2}}\,f_{,x}\right)-2\,z^{2}-\frac{1}{2}\,\frac{y^{4}}{z^{2}}+\frac{v_{0}}{S^{2}}\,f_{,xx}\right]v_{k}\,;$
(5.6) $\displaystyle h_{k,T}$ $\displaystyle=\frac{{\rm d}h_{k}}{{\rm d}T}\,,$
(5.7) $\displaystyle\frac{{\rm d}h_{k,T}}{{\rm d}T}$
$\displaystyle=-z\,h_{k,T}-\left(\frac{k^{2}}{A^{2}}+\frac{1}{2}\,y^{2}-2\,z^{2}\right)\,h_{k}\,.$
(5.8)
In our numerical set up, we split $v_{k}$ and $h_{k}$ into their real and
imaginary parts and simulate them separately with appropriate Bunch-Davies
initial conditions. We begin with a discussion of numerical simulations of the
Mukhanov-Sasaki equations (5.3) and (5.4) for a purely slow-roll potential
(which we choose to be the Starobinsky potential (3.48) as usual), before
moving forward to discuss the same for a potential with a slow-roll violating
feature. This latter case is of the primary focus of our paper. In particular,
we will illustrate our numerical scheme for the case of a base slow-roll
potential possessing a tiny local bump feature, which was proposed in [44] in
the context of PBH formation.
Figure 13: The super-Hubble power spectra of scalar fluctuations (in green
colour) and tensor fluctuations (in red colour) are plotted for modes exiting
the Hubble radius at different number of e-folds $N_{e}$ before the end of
inflation for Starobinsky potential (3.48). The solid curves represent power
spectra computed under the slow-roll approximation (3.10), while the dotted
curves represent the power computed by numerically solving the Mukhanov-Sasaki
equation (5.3). We conclude that for Starobinsky model, since slow-roll
conditions $\epsilon_{H},\,|\eta_{H}|\ll 1$ are easily satisfied for most part
of inflation, the power spectra computed under the slow-roll approximation
match quite well with their numerically determined counterparts.
### 5.1 Numerical analysis for slow-roll potentials
1. 1.
As the first step, we numerically solve the background dynamics for a given
potential, determine the values of all relevant parameters of the potential
and the evolution of relevant primary dynamical variables
$\\{x,\,y,\,z,\,A\\}$ as well as the derived quantities such as
$\\{N_{e},\,\epsilon_{H},\,\eta_{H}\\}$ (as discussed in section 4).
2. 2.
We then proceed to identify different comoving scales $k$. This can be done by
determining their Hubble-exit epochs in the following way. For example, we
plot $aH$ (in log scale) against $N_{e}$ and identify the value of $aH$ at
$N_{e}=N_{*}$ to be the CMB pivot scale $k_{p}$. As mentioned before, we take
$N_{*}=60$ in all our analysis. Similarly, we associate a corresponding value
of $N_{e}$ to each comoving scale $k$ by the value of $aH$ at its Hubble-exit
epoch. This step ensures that we have a one-to-one correspondence between $k$
and $N_{e}$ in our analysis and we can use them interchangeably.
3. 3.
We intend to impose Bunch-Davies initial conditions for a given mode $v_{k}$
at an epoch when it is sub-Hubble. As it tuns out, for most potentials, the
Bunch-Davies initial conditions can be safely imposed as long as $k\geq
100\,aH$. Hence, rather than simulating the Mukhanov-Sasaki equation for each
mode (making Hubble-exit at the corresponding value of $N_{e}$) all through
the inflationary history (starting from $\phi_{i}>\phi_{*}$), we actually
impose the initial conditions from the background solutions for
$\\{x,\,y,\,z,\,A\\}$ at around $5$ e-folds before the Hubble-exit of that
mode. This step greatly reduces the running-time of the code. We then
incorporate the initial value of scale factor $A_{i}$ at the same epoch,
namely $A_{i}\exp{(N_{T}-N_{e}-5)}$ and do the same for the initial values of
the field $x_{i}$, and its derivative $y_{i}$. The initial conditions for the
mode functions $v_{k}$ and their derivatives $\dot{v_{k}}$ can then be safely
taken to be of Bunch-Davies type.
4. 4.
We solve the set of cosmological equations with these initial conditions for a
period of time $T=T_{i}\to T=T_{f}$ such that the mode becomes super-Hubble
and its power ($k^{3}|\zeta_{k}|^{2}/2\pi^{2}$) is frozen to a constant value
(see figure 12), which is typically within $5$ e-folds after Hubble-exit in
the kind of models we are interested in. We note down this frozen value as the
value of the power spectrum of that mode. While we have been discussing about
scalar fluctuations mostly, the same can be done for tensor fluctuations which
we have incorporated in our code.
5. 5.
We then select another mode that leaves the Hubble radius at some epoch
$N_{e}$ and repeat the procedure until we have collected the frozen super-
Hubble power spectra of a range of scales that we are interested in (see
figure 13).
Figure 14: This is a schematic plot of an asymptotically-flat inflationary
potential with a tiny bump/dip feature (5.9) near intermediate field values
$\phi\simeq\phi_{\rm PBH}$ that leads to an enhancement of the scalar power
spectrum. The full potential asymptotes to the base (slow-roll) potential near
the CMB window $\phi\simeq\phi_{*}$, thus satisfying observational constraints
on large cosmological scales. Slow-roll is violated around the feature, whose
position $\phi_{\rm PBH}$ dictates the range of moving scales $k$ that receive
amplification of power (which accordingly determines the mass and abundance of
formed PBHs). Note that the feature has been greatly exaggerated for
illustration purpose. In most realistic models, both the height and the width
of the feature are too small to be seen (without zooming-in considerably).
From the above numerical analysis of featureless vanilla potentials which
exhibit slow-roll dynamics until close to the end of inflation, we observe
that the power spectra of scalar and tensor fluctuations are nearly scale-
invariant (with small red-tilt) and their behaviour (as obtained from
numerically solving the Mukhanov-Sasaki equation) matches quite well with the
analytical predictions under the slow-roll approximations (see figure 13).
However, for potentials exhibiting a small-scale feature at intermediate field
values $\phi<\phi_{*}$, there might exist a short period of slow-roll
violating phase before the end of inflation during which slow-roll
approximations break down. In particular, as we will see, while the first
slow-roll parameter remains small $\epsilon_{H}\ll 1$, the second slow-roll
parameter might become $\eta_{H}\sim{\cal O}(1)$. Hence, a numerical analysis
of the Mukhanov-Sasaki equation is desired in order to determine the scalar
power spectrum more accurately. This will be the main focus of discussion in
the next subsection.
Figure 15: Evolution of the field value $\phi$ for the KKLT potential with a
tiny bump (5.10) is shown in solid green curve as a function of number of
e-folds $N_{e}$ before the end of inflation. We note that at CMB scales,
$\phi$ is much smaller than the corresponding value in the base model (KKLT)
(shown in dashed black curve). At intermediate scales when the inflaton
evolves across the bump feature in the potential, we gain a lot of extra
e-folds of expansion $\Delta N_{e}\simeq 15$ with little change in the field
value. After crossing the feature, evolution of $\phi$ mimics its
corresponding value in the base model. Figure 16: Evolution of the slow-roll
parameters $\epsilon_{H}$ and $\eta_{H}$ is shown in solid green and solid red
curves respectively for the KKLT potential with a tiny bump (5.10). Both
$\epsilon_{H}$ and $\eta_{H}$ are close to their corresponding values for the
base KKLT potential (shown in dashed curves) at early times near the CMB
window. At $N_{e}\simeq 30$, the value of $\epsilon_{H}$ starts decreasing
rapidly leading to an increase in $\eta_{H}$ from $|\eta_{H}|\ll 1$ to a
higher and positive value $\eta_{H}\simeq+3.3$ (almost USR phase). Thereafter,
the inflaton enters a phase of constant-roll inflation (where
$\eta_{H}\simeq-0.37$) before returning to the final slow-roll phase.
### 5.2 Numerical analysis for potentials with a local bump/dip feature
In order to facilitate PBH formation, we need a large amplification of scalar
power spectrum at smaller scales during inflation. This can be achieved by
introducing a small-scale feature in the potential which leads to a transient
period of slow-roll violating phase (including a short almost-USR phase).
Adequate amplification in the super-Hubble scalar power spectrum results in a
large density contrast in the post-inflationary universe (upon the Hubble-
entry of the corresponding modes) which in turn can collapse to form PBHs.
Usually, such a PBH feature in the potential leads to an increase in the value
of the second slow-roll parameter $\eta_{H}$ from near-zero to a positive
value of $\eta_{H}\sim{\cal O}(1)$.
A number of models with different types of features have been proposed in the
recent literature (as mentioned before) that facilitate the amplification of
scalar power spectrum at small scales. The most common amongst them is an
inflection point-like feature. However, we choose the model proposed in [44]
in which the base inflaton potential $V_{b}(\phi)$ possesses a tiny local bump
or dip $\pm\varepsilon(\phi,\phi_{0})$ at an intermediate field value
$\phi_{0}$ of the form
$V(\phi)=V_{b}(\phi)\,\left[1\pm\varepsilon(\phi,\phi_{0})\right]\,,$ (5.9)
where we assume $V_{b}(\phi)$ to be a symmetric or an anti-symmetric
asymptotically-flat potential in order to satisfy CMB constraints at large
cosmological scales. Such a potential has been schematically illustrated in
figure 14. To be specific, in this paper we choose the base potential to be
the D-brane KKLT potential [70, 71, 72, 73] with a tiny Gaussian bump of the
form [44]
$\boxed{V(\phi)=V_{0}\,\frac{\phi^{2}}{m^{2}+\phi^{2}}\,\left[1+A\,\exp{\left({-\frac{1}{2}\,\frac{(\phi-\phi_{0})^{2}}{\sigma^{2}}}\right)}\right]}~{},$
(5.10)
where $m$ is a mass scale in the KKLT model, while $A$ and $\sigma$ represent
the height and the width of the tiny bump respectively. We use this particular
model to demonstrate our numerical framework because of its simplicity and
efficiency. However, one can choose any model of their interest. Values of all
the parameters appearing in (5.10), which we use in our numerical analysis,
have been explicitly shown in figure 17.
The height and the width of the (bump/dip) feature required to facilitate
adequate amount of power amplification are quite small, and hence the feature
is tiny and local (in contrast to inflection point-like features). This
ensures that the feature does not significantly affect the CMB observables.
However, since we gain a lot of extra e-folds of expansion $\Delta N_{e}\simeq
15$ with little change in the field value (as shown in figure 15) when the
inflaton crosses the feature, the CMB pivot scale gets shifted towards smaller
values as compared to the same for the base potential.
Figure 17: The super-Hubble power spectra of scalar fluctuations are plotted
for modes exiting the Hubble radius at different number of e-folds $N_{e}$
before the end of inflation for KKLT potential with a tiny bump (5.10). The
solid green curve represents the power spectrum computed under the slow-roll
approximation (3.10), while the dotted red curve represents the power computed
by numerically solving the Mukhanov-Sasaki equation (5.3). Since the second
slow-roll condition is violated due to the presence of the bump (leading to
$\eta_{H}>1$), the slow-roll approximation underestimates the value of power
spectrum near the peak (as well as the position of the peak) for modes exiting
the Hubble radius near the epoch when the inflaton crosses the local maximum
around the bump feature. This leads to an incorrect estimation of the mass
fraction as well as the central mass of the PBHs formed in this scenario.
Since slow-roll is violated in these models (as shown in figure 16), we need
to solve the Mukhanov-Sasaki equation numerically in order to accurately
compute the scalar power spectrum. This has been explicitly demonstrated in
figure 17. Note that the inflationary dynamics in such models contains a
number of phases that include an early slow-roll phase SR-I near the CMB
window, a transition T-I from the early SR-I to the subsequent almost ultra
slow-roll phase USR and a transition T-II back to the next slow-roll phase SR-
II after passing through an intermediate constant-roll phase CR (shown in
figure 16).
Figure 18: Evolution of the effective mass term in the Mukhanov-Sasaki
equation is shown here in solid green curve for the KKLT potential with a tiny
bump (5.10). Important transient phases of the scalar field dynamics are also
highlighted following the behaviour of the second slow-roll parameter
$\eta_{H}$ (plotted in dashed blue curve). There is a sharp dip in the
effective mass term when the field transitions from the first slow-roll phase
(SR-I) to an almost ultra slow-roll phase (USR). The inflaton later makes a
transition to a phase of constant-roll inflation with $\eta_{H}\simeq-0.37$,
before reaching a final slow-roll phase until the end of inflation.
The effective mass term $z^{\prime\prime}/z$ in the Mukhanov-Sasaki equation
(3.4), which primarily governs the dynamics of scalar fluctuations, has been
shown in figure 18, and the resultant Hubble-exit behaviour of different modes
is described in figure 19. As the inflaton approaches the PBH feature,
$\eta_{H}$ starts to increase rapidly. This is accompanied by an initial sharp
dip in the effective mass term $z^{\prime\prime}/z$, which then increases to a
higher plateau as $\eta_{H}$ approaches its maximum in the USR type phase. The
modes that leave the Hubble radius slightly before the transition already
start receiving power amplification on super-Hubble scales as shown by the
orange color curve in figure 19.
It is worth noting that the power spectrum exhibits a dip which corresponds to
very narrow range of scales $k\simeq k_{\rm dip}$ that leave the Hubble radius
a few e-folds before the USR phase (shown by the blue color curve in figure
19). The maximum rate of growth observed in this model is consistent with the
steepest growth bound discussed in [65]. Near the USR phase when
$\eta_{H}\gtrsim 3$, $z^{\prime\prime}/z$ saturates to a constant value. Modes
leaving the Hubble radius around this USR epoch receive maximal amplification
in their super-Hubble power spectrum.
Figure 19: This figure demonstrates the horizon exit behaviour of different
modes i.e the evolution of $\sqrt{\cal{P}_{\zeta}}$ for different modes as
they cross the Hubble radius in KKLT model with a Gaussian bump (5.10). The
dot on each curve corresponds to its Hubble-exit epoch. The sharp dip in the
power spectrum (figure 17) corresponds to the mode $k_{\mathrm{dip}}$ (plotted
in blue color) that exits the Hubble radius a few e-folds before the
commencement of the USR phase. The mode $k_{\mathrm{PBH}}$ which exits the
Hubble radius during the USR phase receives a maximal amplification of power
(plotted in green color).
As the field crosses the maximum of the bump feature, $\eta_{H}$ decreases to
a constant negative value. It stays in this constant-roll phase until the
inflaton meets the base potential eventually and approaches the final slow-
roll phase in its dynamics before the end of inflation. The dynamics of scalar
fluctuations in the aforementioned phases are quite rich, and interesting.
However, since the main aim of this paper is to illustrate how to use our
numerical code with an example, we do not discuss these phases and their
impact on the power spectrum (some of which have been explicitly shown in
figures 18, 19, 20), and refer the interested readers to [63] for more detail.
Figure 20: The super-Hubble power spectrum of scalar fluctuations obtained by
numerically solving the Mukhanov-Sasaki equation (5.3) is plotted here for the
KKLT potential with a tiny bump (5.10). At large scales (around the CMB
window), the power spectrum matches that of the base KKLT potential. The power
spectrum (after exhibiting a sharp dip) receives a large amplification at
intermediate scales $k\sim k_{\rm PBH}$ around the USR phase. The maximum rate
of growth observed in this model is consistent with the steepest growth bound
discussed in [65]. After reaching the peak, the power then decreases at a
steady rate during the constant-roll phase before finally asymptoting towards
its base slow-roll value towards the end of inflation.
## 6 Future extension of our numerical framework
In preceding sections, we described the relevant cosmological equations
governing the inflationary dynamics in terms of dimensionless variables. We
also introduced our numerical code (written in terms of cosmic time) that can
easily simulate the inflationary dynamics both at the background level in
section 4 and at linear order in perturbation theory in section 5. However,
with minimal to moderate extension, our numerical code can be used to simulate
a number of different scenarios associated with scalar field dynamics both
during inflation as well as in the post-inflationary universe. We have already
started working on some of these aspects which will appear in the revised
version of our manuscript. In the following we discuss some of the important
future extensions of our code that we intend to include in our revised
version.
1. 1.
As stressed in section 4, the present version of our numerical code, although
quite fast and neat, contains segments that require the user to carry out a
number of tasks manually. While we believe that the present version will
definitely help a user (who is relatively new to the field) to understand the
inflationary dynamics much better, we are already developing an automated
version of this code that is much more compact and requires substantially less
manual involvement of the user. We also plan to make the code even faster. We
will present the updated version of our code in the revised version of our
paper and refer to it in the same GitHub link
121212https://github.com/bhattsiddharth/NumDynInflation.
2. 2.
In section 5, we briefly discussed how to solve the evolution equation for the
tensor fluctuations at linear order in perturbation theory. However, first-
order scalar fluctuations induce tensor fluctuations at second order in
perturbation theory which might be significant when slow-roll is violated,
especially in the scenario where scalar power spectrum is largely amplified in
order to source PBH formation. We will incorporate the computation of such
scalar-induced Gravitational Waves in the updated version of our code.
3. 3.
In our analysis, we used the Mukhanov-Sasaki variable $v_{k}$ with its
corresponding evolution equation (5.1) in order to simulate scalar
fluctuations at linear order. Our code runs quite quickly and generates the
power spectrum without requiring a large computational time. However, one can
study the scalar power spectrum by using other variables proposed in the
literature. We are particularly interested in two such variables. The first
one is the curvature perturbation $\zeta_{k}$ itself using the corresponding
evolution equation (3.11). For example, it was claimed in [74] that numerical
simulations are more stable in terms of $\zeta_{k}$ since it is explicitly
frozen well outside the Hubble radius. Similarly, authors of [75] have
suggested a change in the variable $v_{k}$ in the form
$g_{k}\equiv\frac{v_{k}}{z}\,e^{ik\tau}$ that is supposed to make the
simulations much more stable since it removes the early time
oscillations131313We are thankful to Christian Byrnes for bringing this to our
attention..
In the revised version of our work, we plan on carrying out a comprehensive
numerical analysis to compare both stability as well speed of the numerical
simulations using all three of the aforementioned variables $v_{k}$,
$\zeta_{k}$, and $g_{k}$.
4. 4.
It is easy to extend our numerical analysis to incorporate the dynamics of
more than one scalar fields during inflation, at least in the background
level. In the future, we are going to provide an updated numerical code to
study both the background dynamics as well as quantum fluctuations in two-
field inflationary dynamics (where the second field might also source
inflation, or act as a spectator field).
5. 5.
Additionally, our code can be extended to simulate the post-inflationary
dynamics of the inflaton field as well as to study parametric resonance by
simulating the evolution of different Fourier modes of the inflaton
fluctuations. During the post-inflationary oscillations, it is usually
advisable to make a change in the Mukhanov-Sasaki variable of the form
${\tilde{v}_{k}}=a^{1/2}\,v_{k}$ as suggested in [76, 77]. The code can also
be extended to study the dynamics of scalar field dark matter and quintessence
by suitably redefining the dimensionless variables as per the energy scale of
the dynamics.
## 7 Discussion
In the present version of our manuscript, we introduced our numerical approach
to simulate the cosmological equations in order to study the inflationary
dynamics at the level of both background as well as linear-order in
perturbation theory. We provided the link to our open-source GitHub repository
where we have supplied a Python-based simple numerical code to simulate
inflationary dynamics in terms of cosmic time $t$. We explicitly demonstrated
how to use the code to study the inflationary background dynamics in section 4
that includes plotting the phase-space portrait of inflation as well as to
characterise quantum fluctuations during inflation using the simulations of
the background dynamics.
Section 5 was dedicated to study quantum fluctuations during inflation
(without using slow-roll approximated expressions) by numerically solving the
mode function equations of scalar and tensor fluctuations. For a featureless
slow-roll inflaton potential, the difference between the results obtained
numerically and those obtained under slow-roll approximations were negligible
until close to the end of inflation, as expected. We used the Starobinsky
potential (3.48) as an example to illustrate our analysis. Our primary focus
was the numerical evaluation of the scalar power spectrum ${\cal
P}_{\zeta}(k)$ for potentials that exhibit a slow-roll violating feature. In
particular, we used the example of an asymptotically-flat base inflationary
potential that possesses a tiny local bump feature (5.9) to illustrate our
numerical scheme in section 5.2. By suitably choosing the parameters of the
potential (5.10), one can achieve a large enough amplification of the scalar
power spectrum in order to facilitate the formation of PBHs in the post-
inflationary epoch.
In our numerical analysis for the case of potentials with a slow-roll
violating feature, we explicitly chose the parameters in order to amplify the
small-scale scalar power by a factor of $\sim 10^{7}$ with respect to the
corresponding power at large cosmological scales, as is usually assumed in the
literature to facilitate the formation of of PBHs in the subsequent radiation
dominated epoch. However, it is important to stress that a significant growth
in the scalar power spectrum at small-scales might engender the dynamics to
enter into non-perturbative regime. For example, a careful computation of loop
corrections to the two-point scalar fluctuations demonstrates [78, 79] that
contribution from $1$-loop effects becomes of the same order as the tree-level
computation (which we carried out in this work) if ${\cal P}_{\zeta}(k)\sim
10^{-2}$ indicating a breakdown of the perturbative analysis.
Moreover, the mechanism of PBH formation in the context of single field models
of inflation involves additional intricacies that demand for a non-
perturbative analysis of primordial fluctuations. Firstly, a sharp drop in the
classical drift speed of the inflaton due to the presence of the PBH-producing
feature often incites the system to enter into a phase where stochastic
quantum diffusion effects become non-negligible, at times even significant.
More importantly, since PBHs form from rare extreme peaks, and hence are
determined by the tail of the probability distribution function (PDF)
$P[\zeta]$ of the primordial fluctuations, perturbative computations based on
only power-spectrum lead to an inaccurate estimation of the PBH mass fraction.
Consequently, determination of the full primordial PDF becomes crucial, which
can be computed non-perturbatively using the stochastic inflation framework
[80, 81, 82, 83, 84, 85, 86, 88, 87, 89, 90, 91, 92, 94, 95, 93, 96, 97, 98,
99, 100, 101, 102, 103, 104, 105], which usually predicts a non-Gaussian
exponential tail [86, 90]. Tail of the primordial PDF can also be computed by
using semi-classical techniques discussed in [106]. Determining the tail of
the primordial PDF is an important and active topic of research [90, 106, 107,
108, 109, 110, 111] at present, which is beyond the scope of our perturbative
analysis presented here.
Before concluding, let us mention that we also stressed upon various important
future extensions of our numerical scheme in section 6 that will result in
making our code more efficient, and enable us to simulate the scalar field
dynamics in a number of interesting scenarios. This includes updating our code
to make it more compact and automated, as well as extending it to study the
spectrum of scalar-induced gravitational waves, inflaton dynamics in the post-
inflationary epoch, multi-field inflationary dynamics, and even scalar field
models of dark matter and dark energy. We will incorporate most of these
additional features in the revised version of our manuscript. In the meantime,
we welcome constructive comments and suggestions as well as queries from
interested readers which will help us in improving the quality of our
numerical work and presentation in the updated version of the paper.
## 8 Acknowledgements
S.S.M thanks Satadru Bag, Shabbir Shaikh, and Varun Sahni for crucial inputs
during the early stages of development of this code. The authors are grateful
to Parth Bhargava and Sanket Dave for stimulating discussions on various
topics related to numerical dynamics discussed in this paper. S.S.M. is
supported as a postdoctoral Research Associate at the School of Physics and
Astronomy, University of Nottingham by the STFC funded consolidated grant, UK.
S.S.B was supported by the INSPIRE scholarship of the Department of Science
and Technology (DST), Govt. of India during his Master’s thesis work during
which a significant portion of this work was carried out.
## References
* [1] A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91, 99-102 (1980).
* [2] A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23, 347 (1981).
* [3] A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems”, Phys. Lett. B 108, 389 (1982).
* [4] A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220 (1982).
* [5] A. D. Linde, “Chaotic Inflation,” Phys. Lett. B 129, 177-181 (1983).
* [6] A.D. Linde, Particle Physics and Inflationary Cosmology, Harwood, Chur, Switzerland (1990).
* [7] D. Baumann, “TASI Lectures on Inflation”, [arXiv:0907.5424].
* [8] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532 (1981).
* [9] S. W. Hawking, Phys. Lett. B 115, 295 (1982).
* [10] A. A. Starobinsky, Phys. Lett. B 117, 175 (1982).
* [11] A. H. Guth and S. -Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).
* [12] A.A. Starobinsky, JETP Lett., 30 , 682 (1979).
* [13] V. Sahni, Phys. Rev. D42, 453 (1990).
* [14] M. Tegmark, “What does inflation really predict?,” JCAP 04 (2005), 001 [arXiv:astro-ph/0410281 [astro-ph]].
* [15] Y. Akrami et al. [Planck], “Planck 2018 results. X. Constraints on inflation” Astron. Astrophys. 641, A10 (2020) [arXiv:1807.06211 [astro-ph.CO]].
* [16] S. S. Mishra, V. Sahni and A. A. Starobinsky, “Curing inflationary degeneracies using reheating predictions and relic gravitational waves,” JCAP 05, 075 (2021) [arXiv:2101.00271 [gr-qc]].
* [17] P. A. R. Ade et al. [BICEP and Keck], “Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,” Phys. Rev. Lett. 127, no.15, 151301 (2021) [arXiv:2110.00483 [astro-ph.CO]].
* [18] S. S. Mishra and V. Sahni, “Canonical and Non-canonical Inflation in the light of the recent BICEP/Keck results,” [arXiv:2202.03467 [astro-ph.CO]].
* [19] R. Kallosh and A. Linde, “BICEP/Keck and cosmological attractors,” JCAP 12, no.12, 008 (2021) [arXiv:2110.10902 [astro-ph.CO]].
* [20] S. S. Mishra, V. Sahni and A. V. Toporensky, “Initial conditions for Inflation in an FRW Universe,” Phys. Rev. D 98, no.8, 083538 (2018) [arXiv:1801.04948 [gr-qc]].
* [21] S. Hawking, “Gravitationally collapsed objects of very low mass,” Mon. Not. Roy. Astron. Soc. 152 (1971), 75
* [22] B. J. Carr and S. W. Hawking, “Black holes in the early Universe,” Mon. Not. Roy. Astron. Soc. 168 (1974), 399-415
* [23] B. J. Carr, “The Primordial black hole mass spectrum,” Astrophys. J. 201 (1975), 1-19
* [24] M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, “Primordial black holes—perspectives in gravitational wave astronomy,” Class. Quant. Grav. 35 (2018) no.6, 063001 [arXiv:1801.05235 [astro-ph.CO]].
* [25] G. F. Chapline, “Cosmological effects of primordial black holes,” Nature 253 (1975) no.5489, 251-252
* [26] P. Meszaros, “Primeval black holes and galaxy formation,” Astron. Astrophys. 38 (1975), 5-13
* [27] P. Ivanov, P. Naselsky and I. Novikov, “Inflation and primordial black holes as dark matter,” Phys. Rev. D 50 (1994), 7173-7178
* [28] B. Carr, F. Kuhnel and M. Sandstad, “Primordial Black Holes as Dark Matter,” Phys. Rev. D 94 (2016) no.8, 083504 [arXiv:1607.06077 [astro-ph.CO]].
* [29] A. M. Green and B. J. Kavanagh, “Primordial Black Holes as a dark matter candidate,” J. Phys. G 48, no.4, 043001 (2021) [arXiv:2007.10722 [astro-ph.CO]].
* [30] B. Carr and F. Kuhnel, “Primordial Black Holes as Dark Matter: Recent Developments,” Ann. Rev. Nucl. Part. Sci. 70 (2020), 355-394 [arXiv:2006.02838 [astro-ph.CO]].
* [31] E. Bugaev and P. Klimai, “Large curvature perturbations near horizon crossing in single-field inflation models,” Phys. Rev. D 78 (2008), 063515 [arXiv:0806.4541 [astro-ph]].
* [32] C. Germani and T. Prokopec, “On primordial black holes from an inflection point,” Phys. Dark Univ. 18 (2017), 6-10 [arXiv:1706.04226 [astro-ph.CO]].
* [33] J. M. Ezquiaga, J. Garcia-Bellido and E. Ruiz Morales, “Primordial Black Hole production in Critical Higgs Inflation,” Phys. Lett. B 776 (2018), 345-349 doi:10.1016/j.physletb.2017.11.039 [arXiv:1705.04861 [astro-ph.CO]].
* [34] J. Garcia-Bellido and E. Ruiz Morales, “Primordial black holes from single field models of inflation,” Phys. Dark Univ. 18 (2017), 47-54 [arXiv:1702.03901 [astro-ph.CO]].
* [35] I. Dalianis, A. Kehagias and G. Tringas, “Primordial black holes from $\alpha$-attractors,” JCAP 01 (2019), 037 [arXiv:1805.09483 [astro-ph.CO]].
* [36] O. Özsoy, S. Parameswaran, G. Tasinato and I. Zavala, “Mechanisms for Primordial Black Hole Production in String Theory,” JCAP 07 (2018), 005 [arXiv:1803.07626 [hep-th]].
* [37] M. Cicoli, V. A. Diaz and F. G. Pedro, “Primordial Black Holes from String Inflation,” JCAP 06 (2018), 034 [arXiv:1803.02837 [hep-th]].
* [38] M. P. Hertzberg and M. Yamada, “Primordial Black Holes from Polynomial Potentials in Single Field Inflation,” Phys. Rev. D 97 (2018) no.8, 083509 [arXiv:1712.09750 [astro-ph.CO]].
* [39] G. Ballesteros and M. Taoso, “Primordial black hole dark matter from single field inflation,” Phys. Rev. D 97 (2018) no.2, 023501 [arXiv:1709.05565 [hep-ph]].
* [40] H. Di and Y. Gong, “Primordial black holes and second order gravitational waves from ultra-slow-roll inflation,” JCAP 07 (2018), 007 [arXiv:1707.09578 [astro-ph.CO]].
* [41] H. Motohashi, S. Mukohyama and M. Oliosi, “Constant Roll and Primordial Black Holes,” JCAP 03 (2020), 002 [arXiv:1910.13235 [gr-qc]].
* [42] R. Mahbub, “Primordial black hole formation in inflationary $\alpha$-attractor models,” Phys. Rev. D 101 (2020) no.2, 023533 [arXiv:1910.10602 [astro-ph.CO]].
* [43] N. Bhaumik and R. K. Jain, “Primordial black holes dark matter from inflection point models of inflation and the effects of reheating,” JCAP 01 (2020), 037 [arXiv:1907.04125 [astro-ph.CO]].
* [44] S. S. Mishra and V. Sahni, “Primordial Black Holes from a tiny bump/dip in the Inflaton potential,” JCAP 04, 007 (2020) [arXiv:1911.00057 [gr-qc]].
* [45] K. Kefala, G. P. Kodaxis, I. D. Stamou and N. Tetradis, “Features of the inflaton potential and the power spectrum of cosmological perturbations,” Phys. Rev. D 104 (2021) no.2, 023506 [arXiv:2010.12483 [astro-ph.CO]].
* [46] H. V. Ragavendra, P. Saha, L. Sriramkumar and J. Silk, “Primordial black holes and secondary gravitational waves from ultraslow roll and punctuated inflation,” Phys. Rev. D 103 (2021) no.8, 083510 [arXiv:2008.12202 [astro-ph.CO]].
* [47] O. Özsoy and Z. Lalak, “Primordial black holes as dark matter and gravitational waves from bumpy axion inflation,” JCAP 01 (2021), 040 [arXiv:2008.07549 [astro-ph.CO]].
* [48] R. Zheng, J. Shi and T. Qiu, “On Primordial Black Holes and secondary gravitational waves generated from inflation with solo/multi-bumpy potential,” [arXiv:2106.04303 [astro-ph.CO]].
* [49] K. Inomata, E. McDonough and W. Hu, “Primordial black holes arise when the inflaton falls,” Phys. Rev. D 104 (2021) no.12, 123553 [arXiv:2104.03972 [astro-ph.CO]].
* [50] R. Brandenberger, “Initial conditions for inflation — A short review,” Int. J. Mod. Phys. D 26 (2016) no.01, 1740002 [arXiv:1601.01918 [hep-th]].
* [51] D. Baumann, PoS TASI 2017, 009 (2018) [arXiv:1807.03098 [hep-th]].
* [52] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 05, 013 (2003) [arXiv:astro-ph/0210603 [astro-ph]].
* [53] M. Sasaki, “Large Scale Quantum Fluctuations in the Inflationary Universe,” Prog. Theor. Phys. 76, 1036 (1986).
* [54] V. F. Mukhanov, “Quantum Theory of Gauge Invariant Cosmological Perturbations,” Sov. Phys. JETP 67, 1297 (1988) [Zh. Eksp. Teor. Fiz. 94N7, 1 (1988)].
* [55] H. Motohashi, A. A. Starobinsky and J. Yokoyama, “Inflation with a constant rate of roll,” JCAP 1509, 018 (2015) [arXiv:1411.5021 [astro-ph.CO]].
* [56] T. S. Bunch and P. C. W. Davies, “Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting,” Proc. Roy. Soc. Lond. A 360, 117 (1978).
* [57] S. Matarrese, S. Mollerach and M. Bruni, “Second order perturbations of the Einstein-de Sitter universe,” Phys. Rev. D 58 (1998), 043504
* [58] D. Baumann, P. J. Steinhardt, K. Takahashi and K. Ichiki, “Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations,” Phys. Rev. D 76 (2007), 084019 [arXiv:hep-th/0703290 [hep-th]].
* [59] N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso and A. Riotto, “Primordial Black Hole Dark Matter: LISA Serendipity,” Phys. Rev. Lett. 122 (2019) no.21, 211301 [arXiv:1810.12218 [astro-ph.CO]].
* [60] R. g. Cai, S. Pi and M. Sasaki, “Gravitational Waves Induced by non-Gaussian Scalar Perturbations,” Phys. Rev. Lett. 122 (2019) no.20, 201101 [arXiv:1810.11000 [astro-ph.CO]].
* [61] G. Domènech, “Scalar Induced Gravitational Waves Review,” Universe 7 (2021) no.11, 398 [arXiv:2109.01398 [gr-qc]].
* [62] B. Whitt, “Fourth Order Gravity as General Relativity Plus Matter,” Phys. Lett. B 145 (1984), 176-178
* [63] A. Karam, N. Koivunen, E. Tomberg, V. Vaskonen and H. Veermäe, “Anatomy of single-field inflationary models for primordial black holes,” [arXiv:2205.13540 [astro-ph.CO]].
* [64] H. Motohashi and W. Hu, “Primordial Black Holes and Slow-Roll Violation,” Phys. Rev. D 96 (2017) no.6, 063503 doi:10.1103/PhysRevD.96.063503 [arXiv:1706.06784 [astro-ph.CO]].
* [65] C. T. Byrnes, P. S. Cole and S. P. Patil, “Steepest growth of the power spectrum and primordial black holes,” JCAP 06, 028 (2019) [arXiv:1811.11158 [astro-ph.CO]].
* [66] P. Carrilho, K. A. Malik and D. J. Mulryne, “Dissecting the growth of the power spectrum for primordial black holes,” Phys. Rev. D 100, no.10, 103529 (2019) [arXiv:1907.05237 [astro-ph.CO]].
* [67] O. Özsoy and G. Tasinato, “On the slope of the curvature power spectrum in non-attractor inflation,” JCAP 04, 048 (2020) [arXiv:1912.01061 [astro-ph.CO]].
* [68] P. S. Cole, A. D. Gow, C. T. Byrnes and S. P. Patil, “Steepest growth re-examined: repercussions for primordial black hole formation,” [arXiv:2204.07573 [astro-ph.CO]].
* [69] V. A. Belinsky, I. M. Khalatnikov, L. P. Grishchuk and Y. B. Zeldovich, “INFLATIONARY STAGES IN COSMOLOGICAL MODELS WITH A SCALAR FIELD,” Phys. Lett. B 155 (1985), 232-236.
* [70] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, “De Sitter vacua in string theory,” Phys. Rev. D 68, 046005 (2003) [arXiv:hep-th/0301240 [hep-th]].
* [71] S. Kachru, R. Kallosh, A. D. Linde, J. M. Maldacena, L. P. McAllister and S. P. Trivedi, “Towards inflation in string theory,” JCAP 0310, 013 (2003) [hep-th/0308055].
* [72] R. Kallosh and A. Linde, “CMB targets after the latest Planck data release,” Phys. Rev. D 100, no.12, 123523 (2019) [arXiv:1909.04687 [hep-th]].
* [73] J. Martin, C. Ringeval and V. Vennin, “Encyclopædia Inflationaris,” Phys. Dark Univ. 5-6 (2014), 75-235 [arXiv:1303.3787 [astro-ph.CO]].
* [74] W. J. Handley, A. N. Lasenby, H. V. Peiris and M. P. Hobson, “Bayesian inflationary reconstructions from Planck 2018 data,” Phys. Rev. D 100 (2019) no.10, 103511 [arXiv:1908.00906 [astro-ph.CO]].
* [75] S. Rasanen and E. Tomberg, “Planck scale black hole dark matter from Higgs inflation,” JCAP 01 (2019), 038 [arXiv:1810.12608 [astro-ph.CO]].
* [76] F. Finelli and R. H. Brandenberger, “Parametric amplification of gravitational fluctuations during reheating,” Phys. Rev. Lett. 82 (1999), 1362-1365 [arXiv:hep-ph/9809490 [hep-ph]].
* [77] K. Jedamzik, M. Lemoine and J. Martin, “Collapse of Small-Scale Density Perturbations during Preheating in Single Field Inflation,” JCAP 09 (2010), 034 [arXiv:1002.3039 [astro-ph.CO]].
* [78] K. Inomata, M. Braglia and X. Chen, “Questions on calculation of primordial power spectrum with large spikes: the resonance model case,” [arXiv:2211.02586 [astro-ph.CO]].
* [79] J. Kristiano and J. Yokoyama, “Ruling Out Primordial Black Hole Formation From Single-Field Inflation,” [arXiv:2211.03395 [hep-th]].
* [80] A. A. Starobinsky, “STOCHASTIC DE SITTER (INFLATIONARY) STAGE IN THE EARLY UNIVERSE,” Lect. Notes Phys. 246, 107-126 (1986) doi:10.1007/3-540-16452-9_6
* [81] D. S. Salopek and J. R. Bond, “Stochastic inflation and nonlinear gravity,” Phys. Rev. D 43, 1005-1031 (1991) doi:10.1103/PhysRevD.43.1005
* [82] A. A. Starobinsky and J. Yokoyama, “Equilibrium state of a selfinteracting scalar field in the De Sitter background,” Phys. Rev. D 50, 6357-6368 (1994) [arXiv:astro-ph/9407016 [astro-ph]].
* [83] T. Fujita, M. Kawasaki, Y. Tada and T. Takesako, “A new algorithm for calculating the curvature perturbations in stochastic inflation,” JCAP 12, 036 (2013) [arXiv:1308.4754 [astro-ph.CO]].
* [84] T. Fujita, M. Kawasaki and Y. Tada, “Non-perturbative approach for curvature perturbations in stochastic $\delta N$ formalism,” JCAP 10, 030 (2014) [arXiv:1405.2187 [astro-ph.CO]].
* [85] V. Vennin and A. A. Starobinsky, Eur. Phys. J. C 75, 413 (2015) [arXiv:1506.04732].
* [86] C. Pattison, V. Vennin, H. Assadullahi and D. Wands, JCAP 10, 046 (2017) [arXiv:1707.00537].
* [87] J. M. Ezquiaga and J. García-Bellido, “Quantum diffusion beyond slow-roll: implications for primordial black-hole production,” JCAP 08 (2018), 018 [arXiv:1805.06731 [astro-ph.CO]].
* [88] M. Biagetti, G. Franciolini, A. Kehagias and A. Riotto, “Primordial Black Holes from Inflation and Quantum Diffusion,” JCAP 07 (2018), 032 [arXiv:1804.07124 [astro-ph.CO]].
* [89] C. Pattison, V. Vennin, H. Assadullahi and D. Wands, “Stochastic inflation beyond slow roll,” JCAP 07, 031 (2019) [arXiv:1905.06300 [astro-ph.CO]].
* [90] J. M. Ezquiaga, J. García-Bellido and V. Vennin, “The exponential tail of inflationary fluctuations: consequences for primordial black holes,” JCAP 03 (2020), 029 [arXiv:1912.05399 [astro-ph.CO]].
* [91] H. Firouzjahi, A. Nassiri-Rad and M. Noorbala, “Stochastic Ultra Slow Roll Inflation,” JCAP 01 (2019), 040 [arXiv:1811.02175 [hep-th]].
* [92] V. Vennin, “Cosmological Inflation: Theoretical Aspects and Observational Constraints,” (PhD Thesis) tel-01094199.
* [93] G. Ballesteros, J. Rey, M. Taoso and A. Urbano, “Stochastic inflationary dynamics beyond slow-roll and consequences for primordial black hole formation,” JCAP 08 (2020), 043 [arXiv:2006.14597 [astro-ph.CO]].
* [94] K. Ando and V. Vennin, “Power spectrum in stochastic inflation,” JCAP 04, 057 (2021) [arXiv:2012.02031 [astro-ph.CO]].
* [95] A. De and R. Mahbub, “Numerically modeling stochastic inflation in slow-roll and beyond,” Phys. Rev. D 102, no.12, 123509 (2020) [arXiv:2010.12685 [astro-ph.CO]].
* [96] D. G. Figueroa, S. Raatikainen, S. Rasanen and E. Tomberg, “Non-Gaussian Tail of the Curvature Perturbation in Stochastic Ultraslow-Roll Inflation: Implications for Primordial Black Hole Production,” Phys. Rev. Lett. 127, no.10, 101302 (2021) [arXiv:2012.06551 [astro-ph.CO]].
* [97] D. Cruces and C. Germani, Phys. Rev. D 105, no.2, 023533 (2022) [arXiv:2107.12735].
* [98] G. Rigopoulos and A. Wilkins, JCAP 12, no.12, 027 (2021) [arXiv:2107.05317].
* [99] C. Pattison, V. Vennin, D. Wands and H. Assadullahi, “Ultra-slow-roll inflation with quantum diffusion,” JCAP 04, 080 (2021) [arXiv:2101.05741 [astro-ph.CO]].
* [100] E. Tomberg, “A numerical approach to stochastic inflation and primordial black holes,” J. Phys. Conf. Ser. 2156, no.1, 012010 (2021) [arXiv:2110.10684 [astro-ph.CO]].
* [101] D. G. Figueroa, S. Raatikainen, S. Rasanen and E. Tomberg, “Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,” [arXiv:2111.07437 [astro-ph.CO]].
* [102] Y. Tada and V. Vennin, “Statistics of coarse-grained cosmological fields in stochastic inflation,” JCAP 02, no.02, 021 (2022) [arXiv:2111.15280 [astro-ph.CO]].
* [103] R. Mahbub and A. De, “Smooth coarse-graining and colored noise dynamics in stochastic inflation,” [arXiv:2204.03859 [astro-ph.CO]].
* [104] N. Ahmadi, M. Noorbala, N. Feyzabadi, F. Eghbalpoor and Z. Ahmadi, “Quantum Diffusion in Sharp Transition to Non-Slow-Roll Phase,” JCAP 08 (2022), 078 [arXiv:2207.10578 [gr-qc]].
* [105] J. H. P. Jackson, H. Assadullahi, K. Koyama, V. Vennin and D. Wands, “Numerical simulations of stochastic inflation using importance sampling,” JCAP 10 (2022), 067 [arXiv:2206.11234 [astro-ph.CO]].
* [106] M. Celoria, P. Creminelli, G. Tambalo and V. Yingcharoenrat, “Beyond perturbation theory in inflation,” JCAP 06, 051 (2021) doi:10.1088/1475-7516/2021/06/051 [arXiv:2103.09244 [hep-th]].
* [107] S. Hooshangi, M. H. Namjoo and M. Noorbala, “Rare events are nonperturbative: Primordial black holes from heavy-tailed distributions,” Phys. Lett. B 834 (2022), 137400 [arXiv:2112.04520 [astro-ph.CO]].
* [108] Y. F. Cai, X. H. Ma, M. Sasaki, D. G. Wang and Z. Zhou, “One small step for an inflaton, one giant leap for inflation: A novel non-Gaussian tail and primordial black holes,” Phys. Lett. B 834 (2022), 137461 [arXiv:2112.13836 [astro-ph.CO]].
* [109] J. M. Ezquiaga, J. García-Bellido and V. Vennin, “Could ”El Gordo” be hinting at primordial quantum diffusion?,” [arXiv:2207.06317 [astro-ph.CO]].
* [110] Y. F. Cai, X. H. Ma, M. Sasaki, D. G. Wang and Z. Zhou, “Highly non-Gaussian tails and primordial black holes from single-field inflation,” [arXiv:2207.11910 [astro-ph.CO]].
* [111] A. D. Gow, H. Assadullahi, J. H. P. Jackson, K. Koyama, V. Vennin and D. Wands, “Non-perturbative non-Gaussianity and primordial black holes,” [arXiv:2211.08348 [astro-ph.CO]].
|
# EBHI-Seg: A Novel Enteroscope Biopsy Histopathological Haematoxylin and
Eosin Image Dataset for Image Segmentation Tasks
Liyu Shi Xiaoyan Li<EMAIL_ADDRESS>Weiming Hu Haoyuan Chen
Jing Chen Zizhen Fan Minghe Gao Yujie Jing Guotao Lu Deguo Ma Zhiyu Ma
Qingtao Meng Dechao Tang Hongzan Sun Marcin Grzegorzek Shouliang Qi
Yueyang Teng Chen Li<EMAIL_ADDRESS>Microscopic Image and Medical
Image Analysis Group, College of Medicine and Biological Information
Engineering, Northeastern University, Shenyang 110169, China Department of
Pathology, Cancer Hospital of China Medical University, Liaoning Cancer
Hospital and Institute, Shengyang 110042, China Shengjing Hospital, China
Medical University, Shenyang 110122, China Institute of Medical Informatics,
University of Luebeck, Luebeck 23538, Germany Department of Knowledge
Engineering, University of Economics in Katowice, Bogucicka 3, 40287 Katowice,
Poland
###### Abstract
Background and Purpose: Colorectal cancer is a common fatal malignancy, the
fourth most common cancer in men, and the third most common cancer in women
worldwide. Timely detection of cancer in its early stages is essential for
treating the disease. Currently, there is a lack of datasets for
histopathological image segmentation of rectal cancer, which often hampers the
assessment accuracy when computer technology is used to aid in diagnosis.
Methods: This present study provided a new publicly available _Enteroscope
Biopsy Histopathological Hematoxylin and Eosin Image Dataset for Image
Segmentation Tasks_ (EBHI-Seg). To demonstrate the validity and extensiveness
of EBHI-Seg, the experimental results for EBHI-Seg are evaluated using
classical machine learning methods and deep learning methods.
Results: The experimental results showed that deep learning methods had a
better image segmentation performance when utilizing EBHI-Seg. The maximum
accuracy of the Dice evaluation metric for the classical machine learning
method is 0.948, while the Dice evaluation metric for the deep learning method
is 0.965.
Conclusion: This publicly available dataset contained 5,170 images of six
types of tumor differentiation stages and the corresponding ground truth
images. The dataset can provide researchers with new segmentation algorithms
for medical diagnosis of colorectal cancer, which can be used in the clinical
setting to help doctors and patients. EBHI-Seg is publicly available at:
https://doi.org/10.6084/m9.figshare.21540159.v1
###### keywords:
Colorectal Histopathology , Enteroscope Biopsy , Image Dataset , Image
segmentation
††journal: Journal of LaTeX Templates
## 1 Introduction
Colon cancer is a common deadly malignant tumor, the fourth most common cancer
in men, and the third most common cancer in women worldwide. Colon cancer is
responsible for $10\%$ of all cancer cases [1]. According to prior research,
colon and rectal tumors share many of the same or similar characteristics.
Hence, they are often classified collectively. The present study categorized
rectal and colon cancers into one colorectal cancer category [2].
Histopathological examination of the intestinal tract is both the gold
standard for the diagnosis of colorectal cancer and a prerequisite for disease
treatment [3].
The advantage of using the intestinal biopsy method to remove a part of the
intestinal tissue for histopathological analysis, which is used to determine
the true status of the patient, is that it considerably reduces damage to the
body and rapid wound healing [4]. The histopathology sample is then sectioned
and processed with Hematoxylin and Eosin (H&E). Treatment with H&E is a common
approach when staining tissue sections to show the inclusions between the
nucleus and cytoplasm and highlight the fine structures between tissues [5,
6]. When a pathologist performs an examination of the colon, they first
examine the histopathological sections for eligibility and find the location
of the lesion. The pathology sections are then examined and diagnosed using a
low magnification microscope. If finer structures need to be observed, the
microscope is adjusted to use high magnification for further analysis.
However, the following problems usually exist in the diagnostic process: the
diagnostic results become more subjective and varied due to different doctors
reasons; doctors can easily overlook some information in the presence of a
large amount of test data; it is difficult to analyze large amounts of
previously collected data [7]. Therefore, it is a necessary to address these
issues effectively.
With the development and popularization of computer-aided diagnosis (CAD), the
pathological sections of each case can be accurately and efficiently examined
with the help of computers [8]. Now, CAD is widely used in many biomedical
image analysis tasks, such as microorganism image analysis [9, 10, 11, 12, 13,
14, 15, 16, 17], COVID-19 image analysis [18], histopatholgical image analysis
[19, 20, 21, 22, 23, 24, 25, 26], cytopathological image analysis [27, 28, 29,
30] and sperm video analysis [31, 32]. Therefore, the application of computer
vision technology for colorectal cancer CAD provides a new direction in this
research field [33].
One of the fundamental features of CAD is the aspect of image segmentation,
the results of which can be used as key evidence in the pathologists’
diagnostic processes. Along with the rapid development of medical image
segmentation methodology, there is a wide demand for its application to
identify benign and malignant tumors, tumor differentiation stages, and other
related fields [34]. Therefore, a multi-class image segmentation method is
needed to obtain high segmentation accuracy and good robustness [35].
The present study presents a novel _Enteroscope Biopsy Histopathological H_
&_E Image Dataset for Image Segmentation Tasks_ (EBHI-Seg), which contains
5,710 electron microscopic images of histopathological colorectal cancer
sections that encompass six tumor differentiation stages: normal, polyp, low-
grade intraepithelial neoplasia, high-grade intraepithelial neoplasia,
serrated adenoma, and adenocarcinoma. The segmentation coefficients and
evaluation metrics are obtained by segmenting the images of this dataset using
different classical machine learning methods and novel deep learning methods.
## 2 Related Work
The present study analyzed and compared the existing colorectal cancer biopsy
dataset and provided an in-depth exploration of the currently known research
findings. The limitations of the presently available colorectal cancer dataset
were also pointed out.
The following conclusions were obtained in the course of the study. For
existing datasets, the data types can be grouped into two major categories:
Multi and Dual Categorization datasets. Multi Categorization datasets contain
tissue types at all stages from Normal to Neoplastic. In [36], a dataset
called “Collection of textures in colorectal cancer histology” is described.
It includes 5,000 patches of size 74 $\mu$m $\times$ 74 $\mu$m and contains
seven categories. However, because there were only 10 images, it is too small
for a data sample and lacked generalization capability. In [22], a dataset
called “NCT-CRC-HE-100K” is proposed. This is a set of 100,000 non-overlapping
image patches of histological human colorectal cancer (CRC) and normal tissue
samples stained with (H&E) that was presented by the National Center for Tumor
Diseases (NCT). These image patches are from nine different tissues with an
image size of 224 $\times$ 224 pixels. The nine tissue categories are adipose,
background, debris, lymphocytes, mucus, smooth muscle, normal colon mucosa,
cancer-associated stroma, and colorectal adenocarcinoma epithelium. This
dataset is publicly available and commonly used. However, because the image
sizes are all 224 $\times$ 224 pixels, the dataset underperformed in some
global details that need to be observed in individual categories. Two datasets
are utilized in [37]: one containing colonic H&E-stained biopsy sections (CRC
dataset) and the other consisting of prostate cancer H&E-stained biopsy
sections (PCa dataset). The CRC dataset contains 1,133 colorectal biopsy and
polypectomy slides grouped into three categories and labelled as non-
neoplastic, low-grade and high-grade lesions. In [38], a dataset named “MICCAI
2016 gland segmentation challenge dataset (GlaS)” is used. This dataset
contained 165 microscopic images of H&E-stained colon glandular tissue
samples, including 85 training and 80 test datasets. Each dataset is grouped
into two parts: benign and malignant tumors. The image size is 775 $\times$
522 pixels. Since this dataset has only two types of data and the number of
data is too little, so that it performs poorly on some multi-type training.
Dual Categorization datasets usually contain only two types of tissue types:
Normal and Neoplastic. In [39], a dataset named “FFPE” is proposed. This
dataset obtained its images by extracting 328 Formalin-fixed Paraffin-embedded
(FFPE) whole-slide images of colorectal polyps classified into two categories
of : hyperplastic polyps (HPs) and sessile serrated adenomas (SSAs). This
dataset contained 3,125 images with an image size of 224 $\times$ 224 pixels
and is small in type and number. In [40], two datasets named “UHCW” and “TCGA”
are proposed. The first dataset is a colorectal cancer biopsy sequence
developed at the University Hospital of Coventry and Warwickshire (UHCW) for
internal validation of the rectal biopsy trial. The second dataset is the
Cancer Genome Atlas (TCGA) for external validation of the trial. This dataset
is commonly used as a publicly available cancer dataset and stores genomic
data for more than 20 types of cancers. The two dataset types are grouped into
two categories: Normal and Neoplastic. The first dataset contains 4,292
slices, and the second dataset contained 731 slices with an image size of 224
$\times$ 224 pixels.
All of the information for the existing datasets is summarized in Table 1. The
issues associated with the dataset mentioned above included fewer data types,
small amount of data, inaccurate dataset ground truth, etc. The current study
required an open-source multi-type colonoscopy biopsy image dataset.
Table 1: A dataset for the pathological classification of colorectal cancer.
Dataset Name | Multi Categorization | Amount | Size | Year
---|---|---|---|---
Collection of textures in colorectal cancer histology | lymphoid follicles, mucosal glands, debris, adipose, tumor epithelium simple stroma, complex stroma, background patches with no tissue | 5000 | 74$\mu$m $\times$ 74$\mu$m (0.495 micrometre per pixel) | 2016
HE-NCT-CRC-100K | MUS, NORM, STR, TUM ADI, BACK, DEB, LYM, MUC | 100000 | 224$\times$224 pixels | 2016
MICCAI’16 gland seg- mentation challenge dataset | benign tumors, malignant tumors | 85 | 775$\times$522 pixels | 2017
CRC dataset | non-neoplastic, low-grade, high-grade lesions | 1133 | 512$\times$512 pixels | 2021
Dataset Name | Dual Categorization | Amount | Size | Year
FFPE | HPs, SSAs | 3152 | 224$\times$224 pixels | 2021
The Cancer Genome Atlas dataset | Normal, Neoplastic | 731 | 224$\times$224 pixels | 2021
University Hospitals Coventry and Warwick- shire dataset | Normal, Neoplastic | 4292 | 224$\times$224 pixels | 2021
## 3 Basic Information for EBHI-Seg
### 3.1 Dataset Overview
The dataset in the present study contained 5,710 histopathology images,
including 2,855 histopathology section images and 2,855 ground truth images.
The basic information for the dataset is described in detail below. EBHI-Seg
is publicly available at: https://doi.org/10.6084/m9.figshare.21540159.v1
In the present paper, H&E-treated histopathological sections of colon tissues
are used as data for evaluating image segmentation. The dataset is obtained
from two histopathologists at the Cancer Hospital of China Medical University
(proved by “Research Project Ethics Certification” (No. 202229) ). It is
prepared by 12 biomedical researchers according to the following rules:
Firstly, if there is only one differentiation stage in the image and the rest
of the image is intact, then the differentiation stage became the image label;
Secondly, if there is more than one differentiation stage in the image, then
the most obvious differentiation is selected as the image label; In general,
the most severe and prominent differentiation in the image was used as the
image label.
Intestinal biopsy was used as the sampling method in this dataset. The
magnification of the data slices is $400\times$, with an eyepiece
magnification of $10\times$ and an objective magnification of $40\times$. A
Nissan Olympus microscope and NewUsbCamera acquisition software are used. The
image input size is $224\times 224$ pixels, and the format is *.png. The data
are grouped into five types described in detail in section 2.2.
### 3.2 Data Type Description
Normal: Colorectal tissue sections of the standard category are made-up of
consistently ordered tubular structures and that does not appear infected when
viewed under a light microscope [41]. Section images with the corresponding
ground truth images are shown in Figure 1(a).
Polyp: Colorectal polyps are similar in shape to the structures in the normal
category, but have a completely different histological structure. A polyp is a
redundant mass that grows on the surface of the body’s cells. Modern medicine
usually refers to polyps as unwanted growths on the mucosal surface of the
body [42]. The pathological section of the polyp category also has an intact
luminal structure with essentially no nuclear division of the cells. Only the
atomic mass is slightly higher than that in the normal category. The polyp
category and corresponding ground truth images are shown in Figure 1(b).
Intraepithelial neoplasia: Intraepithelial neoplasia (IN) is the most critical
precancerous lesion. Compared to the normal category, its histological images
show increased branching of adenoid structures, dense arrangement, and
different luminal sizes and shapes. In terms of cellular morphology, the
nuclei are enlarged and vary in size, while nuclear division increases [43].
The standard Padova classification currently classifies intraepithelial
neoplasia into low-grade and high-grade INs. High-grade IN demonstrate more
pronounced structural changes in the lumen and nuclear enlargement compared to
low-grade IN. The images and ground truth diagrams of high-grade and low-grade
INs are shown in Figure 1(c)(d).
Adenocarcinoma: Adenocarcinoma is a malignant digestive tract tumor with a
very irregular distribution of luminal structures. It is difficult to identify
its border structures during observation, and the nuclei are significantly
enlarged at this stage [44]. An adenocarcinoma with its corresponding ground
truth diagram is shown in Figure 1(e).
Serrated adenoma: Serrated adenomas are uncommon lesions, accounting for 1% of
all colonic polyps [45]. The endoscopic surface appearance of serrated
adenomas is not well characterized but is thought to be similar to that of
colonic adenomas with tubular or cerebral crypt openings [46]. The image of a
serrated adenoma with a corresponding ground truth diagram is shown in Figure
1(f).
Figure 1: An example of histopathological images database: (a) Normal and
ground truth, (b) Polyp and ground truth, (c) High-grade Intraepithelial
Neoplasia and ground truth, (d) Low-grade Intraepithelial Neoplasia and ground
truth, (e) Adenocarcinoma and ground truth, (f) Serrated adenoma and ground
truth.
## 4 Evaluation of EBHI-Seg
### 4.1 Image Segmentation Evaluation Metric
Six evaluation metrics are commonly used for image segmentation tasks. The
Dice ratio metric is a standard metric used in medical images that is often
utilized to evaluate the performance of image segmentation algorithms. It is a
validation method based on spatial overlap statistics that measures the
similarities between the algorithm segmentation output and ground truth [47].
The Dice ratio is defined in Eq. (1).
$\centering\rm DiceRatio=\frac{2\left|X\cap
Y\right|}{\left|X\right|+\left|Y\right|}.\@add@centering$ (1)
In Eq. (1), for a segmentation task, $X$ and $Y$ denote the ground truth and
segmentation mask prediction, respectively. The range of the calculated
results is [0,1], and the larger the result the better.
The Jaccard index is a classical set similarity measure with many practical
applications in image segmentation. The Jaccard index measures the similarity
of a finite set of samples: the ratio between the intersection and
concatenation of the segmentation results and ground truth [48]. The Jaccard
index is defined in Eq. (2).
$\rm JaccardIndex=\frac{\left|X\cap Y\right|}{\left|X\cup Y\right|}.$ (2)
The range of the calculated results is [0,1], and the larger the result the
better.
Recall and precision are the recall and precision rates, respectively. The
range of the calculated results is [0,1]. A higher output indicates a better
segmentation result. Recall and precision are defined in Eq. (3) and Eq. (4),
$\rm Precison=\frac{TP}{TP+FP},$ (3)
$\rm Recall=\frac{TP}{TP+FN},$ (4)
where TP, FP, TN, and FN are defined in table 2.
Table 2: Confusion Matrix Ground truth | Predict mask
---|---
Positive | Negative
Positive | TP | TN
Negative | FP | FN
The conformity coefficient (Confm Index) is a consistency coefficient, which
is calculated by putting the binary classification result of each pixel from
[$-\infty$,1] into continuous interval [$-\infty$,1] to calculate the ratio of
the number of incorrectly segmented pixels to the number of correctly
segmented pixels to measure the consistency between the segmentation result
and ground truth. The conformity coefficient is defined in Eq. (5)(6),
$\rm ConfmIndex=(1-\frac{\theta_{AE}}{\theta_{TP}}),\theta_{TP}>0,$ (5) $\rm
ConfmIndex=Failure,\theta_{TP}=0,$ (6)
where $\theta_{AE}$= $\theta_{FP}$+$\theta_{FN}$ represents all errors of the
fuzzy segmentation results. $\theta_{TP}$ is the number of correctly
classified pixels. Mathematically, ConfmIndex can be negative infinity if
$\theta_{TP}$=0. Such a segmentation result is definitely inadequate and
treated as failure without the need of any further analysis.
### 4.2 Classical Machine Learning Methods
Image segmentation is one of the most commonly used methods for classifying
image pixels in decision-oriented applications [49]. It groups an image into
regions high in pixel similarity within each area and has a significant
contrast between different regions [50]. Machine learning methods for
segmentation distinguish the image classes using image features. (1) $k$-means
algorithm is a classical division-based clustering algorithm, where image
segmentation means segmenting the image into many disjointed regions. The
essence is the clustering process of pixels, and the $k$-means method is one
of the simplest clustering methods [51]. Image segmentation of the present
study dataset is performed using the classical machine learning method
described above. (2) Markov random field (MRF) is a powerful stochastic tool
that models the joint probability distribution of an image based on its local
spatial action [52]. It can extract the texture features of the image and
model the image segmentation problem. (3) OTSU algorithm is a global adaptive
binarized threshold segmentation algorithm that uses the maximum inter-class
variance between the image background and the target image as the selection
criterion [53]. The image is grouped into foreground and background parts
based on its grayscale characteristics independent of the brightness and
contrast. (4) Watershed algorithm is a region-based segmentation method, that
takes the similarity between neighboring pixels as a reference and connects
those pixels with similar spatial locations and grayscale values into a closed
contour to achieve the segmentation effect [54]. (5) Sobel algorithm has two
operators, where one detects horizontal edges and the other detects vertical
flat edges. An image is the final result of its operation. Sobel edge
detection operator is a set of directional operators that can be used to
perform edge detection from different directions [55]. The segmentation
results are shown in Figure 2.
Figure 2: Five types of data segmentation results obtained by different
classical machine learning methods.
The performance of EBHI-Seg for different machine learning methods is observed
by comparing the images segmented using classical machine learning methods
with the corresponding ground truth. The segmentation evaluation metrics
results are shown in Table 3. The Dice ratio algorithm is a similarity
measure, usually used to compare the similarity of two samples. The value of
one for this metric is c onsidered to indicate the best effect, while the
value of the worst impact is zero. The Table 3 shows that $k$-means has a good
Dice ratio algorithm value of up to 0.650 in each category. The MRF and Sobel
segmentation results also achieved a good Dice ratio algorithm value of around
0.6. In terms of image precision and recall segmentation coefficients,
$k$-means is maintained at approximately 0.650 in each category. In the
classical machine learning methods, $k$-means has the best segmentation
results, followed by MRF and Sobel. OTSU has a general effect, while the
watershed algorithm has various coefficients that are much lower than those in
the above methods. Moreover,there are apparent differences in the segmentation
results when using the above methods.
Table 3: Evaluation metrics for five different segmentation methods based on classical machine learning. | | DiceRatio | JaccardIndex | ConformityCoefficient | Precision | Recall
---|---|---|---|---|---|---
| $k$-means | 0.648 | 0.488 | -0.184 | 0.646 | 0.663
| MRF | 0.636 | 0.473 | -0.230 | 0.637 | 0.658
Normal | OTSU | 0.410 | 0.265 | -2.871 | 0.515 | 0.351
| Watershed | 0.461 | 0.300 | -1.375 | 0.668 | 0.356
| Sobel | 0.652 | 0.487 | -0.102 | 0.763 | 0.579
| $k$-means | 0.592 | 0.430 | -0.528 | 0.546 | 0.663
| MRF | 0.511 | 0.362 | -2.133 | 0.540 | 0.502
Polyp | OTSU | 0.400 | 0.259 | -3.108 | 0.413 | 0.399
| Watershed | 0.433 | 0.277 | -1.675 | 0.551 | 0.362
| Sobel | 0.583 | 0.416 | -0.499 | 0.626 | 0.562
| $k$-means | 0.626 | 0.478 | -0.467 | 0.650 | 0.620
| MRF | 0.550 | 0.441 | -30.85 | 0.614 | 0.526
High-grade IN | OTSU | 0.249 | 0.150 | -12.06 | 0.373 | 0.191
| Watershed | 0.472 | 0.309 | -1.258 | 0.738 | 0.350
| Sobel | 0.634 | 0.469 | -0.200 | 0.728 | 0.577
| $k$-means | 0.650 | 0.492 | -0.172 | 0.651 | 0.663
| MRF | 0.554 | 0.404 | -1.808 | 0.643 | 0.504
Low-grade IN | OTSU | 0.886 | 0.811 | 0.6998 | 0.832 | 0.979
| Watershed | 0.464 | 0.303 | -1.345 | 0.676 | 0.357
| Sobel | 0.656 | 0.492 | -0.079 | 0.771 | 0.582
| $k$-means | 0.633 | 0.481 | -0.414 | 0.655 | 0.645
| MRF | 0.554 | 0.404 | -1.808 | 0.643 | 0.504
Adenocarcinoma | OTSU | 0.336 | 0.215 | -5.211 | 0.454 | 0.282
| Watershed | 0.458 | 0.298 | -1.437 | 0.700 | 0.349
| Sobel | 0.553 | 0.388 | -0.733 | 0.692 | 0.484
| $k$-means | 0.636 | 0.473 | -0.230 | 0.637 | 0.658
| MRF | 0.571 | 0.419 | -0.898 | 0.656 | 0.547
Serrated adenoma | OTSU | 0.393 | 0.248 | -2.444 | 0.565 | 0.315
| Watershed | 0.449 | 0.290 | -1.494 | 0.656 | 0.345
| Sobel | 0.698 | 0.541 | 0.7484 | 0.662 | 0.572
In summary, EBHI-Seg has significantly different results when using different
classical machine learning segmentation methods. Different classical machine
learning methods have an obvious differentiation according to the image
segmentation evaluation metrics. Therefore, EBHI-Seg can effectively evaluate
the segmentation performance of different segmentation methods.
### 4.3 Deep Learning Methods
Besides the classical macine learning metheds tested above, some popular deep
learning methods are also tested. (1) Seg-Net is an open source project for
image segmentation [56]. The network is identical to the convolutional layer
of VGG-16, with the removal of the fully-connected hierarchy and the addition
of max-pooling indices resulting in improved boundary delineation. Seg-Net
performs better in large datasets. (2) U-Net network structure was first
proposed in 2015 [57] for medical imaging. U-Net is lightweight, and its
simultaneous detection of local and global information is helpful for both
information extraction and diagnostic results from clinical medical images.
(3) MedT is a network published in 2021, which is a transformer structure that
applies an attention mechanism based on medical image segmentation [58]. The
segmentation results are shown in Figure 3.
Figure 3: Three types of data segmentation results obtained by different deep
learning methods.
The segmentation effect is test on the present dataset using three deep
learning models. In the experiments, each model is trained using the ratio of
the training set, validation set, and test set of $4:4:2$. The model learning
rate is set to $3e-6$, epochs are set to 100, and batch-size is set to 1. The
dataset segmentation results of using three different models are shown in
Figure 3. The experimental segmentation evaluation metrics are shown in Table
4. Overall, deep learning performs much better than classical machine learning
methods. Among them, the evaluation indexes of the training results using the
U-Net and Seg-Net models can reach 0.90 on average. The evaluation results of
the MedT model are slightly worse at a level, between 0.70 and 0.80. The
training time is longer for MedT and similar for U-Net and Seg-Net.
Table 4: Evaluation metrics for three different segmentation methods based on deep learning. | | DiceRatio | JaccardIndex | ConformityCoefficient | Precision | Recall
---|---|---|---|---|---|---
| U-Net | 0.411 | 0.263 | -2.199 | 0.586 | 0.328
Normal | Seg-Net | 0.777 | 0.684 | -0.607 | 0.895 | 0.758
| MedT | 0.676 | 0.562 | -0.615 | 0.874 | 0.610
| U-Net | 0.965 | 0.308 | -1.514 | 0.496 | 0.470
Polyp | Seg-Net | 0.937 | 0.886 | 0.858 | 0.916 | 0.965
| MedT | 0.771 | 0.643 | 0.336 | 0.687 | 0.920
| U-Net | 0.895 | 0.816 | 0.747 | 0.847 | 0.961
High-grade IN | Seg-Net | 0.894 | 0.812 | 0.757 | 0.881 | 0.913
| MedT | 0.824 | 0.707 | 0.556 | 0.740 | 0.958
| U-Net | 0.911 | 0.849 | 0.773 | 0.879 | 0.953
Low-grade IN | Seg-Net | 0.924 | 0.864 | 0.826 | 0.883 | 0.977
| MedT | 0.889 | 0.808 | 0.730 | 0.876 | 0.916
| U-Net | 0.887 | 0.808 | 0.718 | 0.850 | 0.950
Adenocarcinoma | Seg-Net | 0.865 | 0.775 | 0.646 | 0.792 | 0.977
| MedT | 0.735 | 0.595 | 0.197 | 0.662 | 0.864
| U-Net | 0.938 | 0.886 | 0.865 | 0.899 | 0.983
Serrated adenoma | Seg-Net | 0.907 | 0.832 | 0.794 | 0.859 | 0.963
| MedT | 0.670 | 0.509 | -0.043 | 0.896 | 0.544
Based on the above results, EBHI-Seg achieved a clear differentiation using
deep learning image segmentation methods. Image segmentation metrics for
different deep learning methods are significantly different so that EBHI-Seg
can evaluate their segmentation performance.
## 5 Discussion
### 5.1 Discussion of Image Segmentation Results Using Classical Machine
Learning Methods
Six types of tumor differentiation stage data in EBHI-Seg were analyzed using
classical machine learning methods to obtain the results in Table 3. Base on
the Dice ratio metrics, $k$-means, MRF and Sobel show no significant
differences among the three methods around 0.55. In contrast, Watershed
metrics are $\sim$0.45 on average, which is lower than the above three
metrics. OTSU index is around $\sim$0.40 because the foreground-background is
blurred in some experimental samples and OTSU had a difficulty extracting a
suitable segmentation threshold, which resulted in undifferentiated test
results. Precision and Recall evaluation indexes for k-means, MRF, and Sobel
are also around 0.60, which is higher than those for OTSU and Watershed
methods by about 0.20. In these three methods, $k$-means and MRF are higher
than Sobel in the visual performance of the images. Although Sobel is the same
as these two methods in terms of metrics, it is difficult to distinguish
foreground and background images in real images.The segmentation results for
MRF are obvious but the running time for MRF is too long in comparison with
other classical learning methods. Since classical machine learning methods
have a rigorous theoretical foundation and simple ideas, they have been shown
to perform well when used for specific problems. However, the performance of
different methods varied in the present study.
### 5.2 Discussion of Image Segmentation Results Using Deep Learning Methods
In general, deep learning models are considerably superior to classical
machine learning methods, and even the lowest MedT performance is still higher
than the highest accuracy of classical machine learning methods. In EBHI-Seg,
the Dice ratio evaluation index of MedT reaches $\sim$0.75. However, the MedT
model size was larger and as a result the training time was too long. U-Net
and Seg-Net have higher evaluation indexes than MedT, both of about 0.88.
Among them, Seg-Net has the least training time and the lowest training model
size. Because the normal category has fewer sample images than other
categories, the evaluation metrics of the three deep learning methods in this
category are significantly lower than those in other categories. The
evaluation metrics of the three segmentation methods are significantly higher
in the other categories, with Seg-Net averaging above 0.90 and MedT exceeding
0.80.
## 6 Conclusion and Future Work
The present stduy introduced a publicly available colorectal pathology image
dataset containing 5,710 magnified $400\times$ pathology images of six types
of tumor differentiation stages. EBHI-Seg has high segmentation accuracy as
well as good robustness. In the classical machine learning approach,
segmentation experiments were performed using different methods and evaluation
metrics analysis was carried out utilizing segmentation results. The highest
and lowest Dice ratios are 0.65 and 0.30, respectively. The highest Precision
and Recall values are 0.70 and 0.90, respectively, while the lowest values are
0.50 and 0.35, respectively. All three models performed well when using the
deep learning method, with the highest Dice ratio reaching above 0.95 and both
Precision and Recall values reaching above 0.90. The segmentation experiments
using EBHI-Seg show that this dataset effectively perform the segmentation
task in each of the segmentation methods. Furthermore, there are significant
differences among the segmentation evaluation metrics. Therefore, EBHI-Seg is
practical and effective in performing image segmentation tasks.
## Data Availability Statement
EBHI-Seg is proved by “Research Project Ethics Certification” (No. 202229)
from Cancer Hospital of China Medical University, Shenyang, China. EBHI-Seg is
publicly available at: https://doi.org/10.6084/m9.figshare.21540159.v1
## Author Contributions
L. Shi: Data preparation, experiment, result analysis, paper writing; X. Li:
Corresponding author, data collection, medical knowledge; W. Hu: Data
collection, data preparation, paper writing; H. Chen: Data preparation, paper
writing; J. Chen, Z. Fan, M. Gao, Y. Jing, G. Lu, D. Ma, Z. Ma, Q. Meng, D.
Tang: Data preparation; H. Sun: Medical knowledge; M. Grzegorzek: Result
analysis; S. Qi: Method; Y. Teng; Result analysis; C. Li; Corresponding
author, data collection, method, experiment, result analysis, paper writing,
proofreading.
## Funding
This work is supported by the “National Natural Science Foundation of China”
(No. 82220108007) and “Beijing Xisike Clinical Oncology Research Foundation”
(No. Y-tongshu2021/1n-0379).
## Acknowledgements
We thank Miss. Zixian Li and Mr. Guoxian Li for their important discussion in
this work.
## Conflict of Interest
The authors declare that they have no conflict of interest in this paper.
## References
* [1] H. Sung, J. Ferlay, R. L. Siegel, M. Laversanne, I. Soerjomataram, A. Jemal, F. Bray, Global cancer statistics 2020: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA: a cancer journal for clinicians 71 (3) (2021) 209–249.
* [2] V. Pamudurthy, N. Lodhia, V. J. Konda, Advances in endoscopy for colorectal polyp detection and classification, in: Baylor University Medical Center Proceedings, Vol. 33, Taylor & Francis, 2020, pp. 28–35.
* [3] J. Thijs, A. Van Zwet, W. Thijs, H. Oey, A. Karrenbeld, F. Stellaard, D. Luijt, B. Meyer, J. Kleibeuker, Diagnostic tests for helicobacter pylori: a prospective evaluation of their accuracy, without selecting a single test as the gold standard., American Journal of Gastroenterology (Springer Nature) 91 (10) (1996).
* [4] R. Labianca, B. Nordlinger, G. Beretta, S. Mosconi, M. Mandalà, A. Cervantes, D. Arnold, Early colon cancer: Esmo clinical practice guidelines for diagnosis, treatment and follow-up, Annals of oncology 24 (2013) vi64–vi72.
* [5] A. H. Fischer, K. A. Jacobson, J. Rose, R. Zeller, Hematoxylin and eosin staining of tissue and cell sections, Cold spring harbor protocols 2008 (5) (2008) pdb–prot4986.
* [6] J. K. Chan, The wonderful colors of the hematoxylin–eosin stain in diagnostic surgical pathology, International journal of surgical pathology 22 (1) (2014) 12–32.
* [7] V. Gupta, M. Vasudev, A. Doegar, N. Sambyal, Breast cancer detection from histopathology images using modified residual neural networks, Biocybernetics and Biomedical Engineering 41 (4) (2021) 1272–1287.
* [8] T. Mathew, J. R. Kini, J. Rajan, Computational methods for automated mitosis detection in histopathology images: A review, Biocybernetics and Biomedical Engineering 41 (1) (2021) 64–82.
* [9] C. Li, K. Wang, N. Xu, A survey for the applications of content-based microscopic image analysis in microorganism classification domains, Artificial Intelligence Review 51 (4) (2019) 577–646.
* [10] J. Zhang, C. Li, Y. Yin, J. Zhang, M. Grzegorzek, Applications of artificial neural networks in microorganism image analysis: a comprehensive review from conventional multilayer perceptron to popular convolutional neural network and potential visual transformer, Artificial Intelligence Review (2022) 1–58.
* [11] J. Zhang, C. Li, S. Kosov, M. Grzegorzek, K. Shirahama, T. Jiang, C. Sun, Z. Li, H. Li, Lcu-net: A novel low-cost u-net for environmental microorganism image segmentation, Pattern Recognition 115 (2021) 107885.
* [12] P. Zhao, C. Li, M. M. Rahaman, H. Xu, H. Yang, H. Sun, T. Jiang, M. Grzegorzek, A comparative study of deep learning classification methods on a small environmental microorganism image dataset (emds-6): from convolutional neural networks to visual transformers, Frontiers in Microbiology 13 (2022).
* [13] F. Kulwa, C. Li, J. Zhang, K. Shirahama, S. Kosov, X. Zhao, T. Jiang, M. Grzegorzek, A new pairwise deep learning feature for environmental microorganism image analysis, Environmental Science and Pollution Research (2022) 1–18.
* [14] P. Ma, C. Li, M. M. Rahaman, Y. Yao, J. Zhang, S. Zou, X. Zhao, M. Grzegorzek, A state-of-the-art survey of object detection techniques in microorganism image analysis: from classical methods to deep learning approaches, Artificial Intelligence Review (2022) 1–72.
* [15] F. Kulwa, C. Li, M. Grzegorzek, M. M. Rahaman, K. Shirahama, S. Kosov, Segmentation of weakly visible environmental microorganism images using pair-wise deep learning features, Biomedical Signal Processing and Control 79 (2023) 104168.
* [16] J. Zhang, C. Li, M. M. Rahaman, Y. Yao, P. Ma, J. Zhang, X. Zhao, T. Jiang, M. Grzegorzek, A comprehensive survey with quantitative comparison of image analysis methods for microorganism biovolume measurements, Archives of Computational Methods in Engineering (2022).
* [17] J. Zhang, C. Li, M. M. Rahaman, Y. Yao, P. Ma, J. Zhang, X. Zhao, T. Jiang, M. Grzegorzek, A comprehensive review of image analysis methods for microorganism counting: from classical image processing to deep learning approaches, Artificial Intelligence Review (2021) 1–70.
* [18] M. M. Rahaman, C. Li, Y. Yao, F. Kulwa, M. A. Rahman, Q. Wang, S. Qi, F. Kong, X. Zhu, X. Zhao, Identification of covid-19 samples from chest x-ray images using deep learning: A comparison of transfer learning approaches, Journal of X-ray Science and Technology 28 (5) (2020) 821–839.
* [19] H. Chen, C. Li, G. Wang, X. Li, M. M. Rahaman, H. Sun, W. Hu, Y. Li, W. Liu, C. Sun, et al., Gashis-transformer: A multi-scale visual transformer approach for gastric histopathological image detection, Pattern Recognition 130 (2022) 108827\.
* [20] Y. Li, X. Wu, C. Li, X. Li, H. Chen, C. Sun, M. M. Rahaman, Y. Yao, Y. Zhang, T. Jiang, A hierarchical conditional random field-based attention mechanism approach for gastric histopathology image classification, Applied Intelligence (2022) 1–22.
* [21] W. Hu, C. Li, X. Li, M. M. Rahaman, J. Ma, Y. Zhang, H. Chen, W. Liu, C. Sun, Y. Yao, et al., Gashissdb: A new gastric histopathology image dataset for computer aided diagnosis of gastric cancer, Computers in biology and medicine (2022) 105207.
* [22] H. Chen, C. Li, X. Li, M. M. Rahaman, W. Hu, Y. Li, W. Liu, C. Sun, H. Sun, X. Huang, et al., Il-mcam: An interactive learning and multi-channel attention mechanism-based weakly supervised colorectal histopathology image classification approach, Computers in Biology and Medicine 143 (2022) 105265.
* [23] W. Hu, H. Chen, W. Liu, X. Li, H. Sun, X. Huang, M. Grzegorzek, C. Li, A comparative study of gastric histopathology sub-size image classification: from linear regression to visual transformer, Frontiers in Medicine (2022).
* [24] Y. Li, C. Li, X. Li, K. Wang, M. M. Rahaman, C. Sun, H. Chen, X. Wu, H. Zhang, Q. Wang, A comprehensive review of markov random field and conditional random field approaches in pathology image analysis, Archives of Computational Methods in Engineering 29 (1) (2022) 609–639.
* [25] C. Sun, C. Li, J. Zhang, M. M. Rahaman, S. Ai, H. Chen, F. Kulwa, Y. Li, X. Li, T. Jiang, Gastric histopathology image segmentation using a hierarchical conditional random field, Biocybernetics and Biomedical Engineering 40 (4) (2020) 1535–1555.
* [26] X. Li, C. Li, M. M. Rahaman, H. Sun, X. Li, J. Wu, Y. Yao, M. Grzegorzek, A comprehensive review of computer-aided whole-slide image analysis: from datasets to feature extraction, segmentation, classification and detection approaches, Artificial Intelligence Review (2022) 1–70.
* [27] M. M. Rahaman, C. Li, X. Wu, Y. Yao, Z. Hu, T. Jiang, X. Li, S. Qi, A survey for cervical cytopathology image analysis using deep learning, IEEE Access 8 (2020) 61687–61710.
* [28] M. Mamunur Rahaman, C. Li, Y. Yao, F. Kulwa, X. Wu, X. Li, Q. Wang, Deepcervix: A deep learning-based framework for the classification of cervical cells using hybrid deep feature fusion techniques, Computers in Biology and Medicine 136 (2021) 104649.
* [29] W. Liu, C. Li, M. M. Rahaman, T. Jiang, H. Sun, X. Wu, W. Hu, H. Chen, C. Sun, Y. Yao, et al., Is the aspect ratio of cells important in deep learning? a robust comparison of deep learning methods for multi-scale cytopathology cell image classification: From convolutional neural networks to visual transformers, Computers in biology and medicine (2021) 105026.
* [30] W. Liu, C. Li, N. Xu, T. Jiang, M. M. Rahaman, H. Sun, X. Wu, W. Hu, H. Chen, C. Sun, et al., Cvm-cervix: A hybrid cervical pap-smear image classification framework using cnn, visual transformer and multilayer perceptron, Pattern Recognition (2022) 108829.
* [31] A. Chen, C. Li, S. Zou, M. M. Rahaman, Y. Yao, H. Chen, H. Yang, P. Zhao, W. Hu, W. Liu, et al., Svia dataset: A new dataset of microscopic videos and images for computer-aided sperm analysis, Biocybernetics and Biomedical Engineering (2022).
* [32] S. Zou, C. Li, H. Sun, P. Xu, J. Zhang, P. Ma, Y. Yao, X. Huang, M. Grzegorzek, Tod-cnn: An effective convolutional neural network for tiny object detection in sperm videos, Computers in Biology and Medicine 146 (2022) 105543.
* [33] I. Pacal, D. Karaboga, A. Basturk, B. Akay, U. Nalbantoglu, A comprehensive review of deep learning in colon cancer, Computers in Biology and Medicine 126 (2020) 104003.
* [34] E. Miranda, M. Aryuni, E. Irwansyah, A survey of medical image classification techniques, in: 2016 International Conference on Information Management and Technology (ICIMTech), IEEE, 2016, pp. 56–61.
* [35] H. Kotadiya, D. Patel, Review of medical image classification techniques, in: Third International Congress on Information and Communication Technology, Springer, 2019, pp. 361–369.
* [36] E. Trivizakis, G. S. Ioannidis, I. Souglakos, A. H. Karantanas, M. Tzardi, K. Marias, A neural pathomics framework for classifying colorectal cancer histopathology images based on wavelet multi-scale texture analysis, Scientific Reports 11 (1) (2021) 1–10.
* [37] S. P. Oliveira, P. C. Neto, J. Fraga, D. Montezuma, A. Monteiro, J. Monteiro, L. Ribeiro, S. Gonçalves, I. M. Pinto, J. S. Cardoso, Cad systems for colorectal cancer from wsi are still not ready for clinical acceptance, Scientific Reports 11 (1) (2021) 1–15.
* [38] T. Kausar, A. Kausar, M. A. Ashraf, M. F. Siddique, M. Wang, M. Sajid, M. Z. Siddique, A. U. Haq, I. Riaz, Sa-gan: Stain acclimation generative adversarial network for histopathology image analysis, Applied Sciences 12 (1) (2021) 288.
* [39] J. Wei, A. Suriawinata, B. Ren, X. Liu, M. Lisovsky, L. Vaickus, C. Brown, M. Baker, M. Nasir-Moin, N. Tomita, et al., Learn like a pathologist: curriculum learning by annotator agreement for histopathology image classification, in: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, 2021, pp. 2473–2483.
* [40] M. Bilal, Y. W. Tsang, M. Ali, S. Graham, E. Hero, N. Wahab, K. Dodd, H. Sahota, W. Lu, M. Jahanifar, et al., Ai based pre-screening of large bowel cancer via weakly supervised learning of colorectal biopsy histology images, medRxiv (2022).
* [41] M. P. De Leon, C. Di Gregorio, Pathology of colorectal cancer, Digestive and Liver Disease 33 (4) (2001) 372–388.
* [42] H. S. Cooper, L. M. Deppisch, E. I. Kahn, R. Lev, P. N. Manley, R. R. Pascal, A. H. Qizilbash, R. R. Rickert, J. F. Silverman, J. A. Wirman, Pathology of the malignant colorectal polyp, Human pathology 29 (1) (1998) 15–26.
* [43] W. Ren, J. Yu, Z.-M. Zhang, Y.-K. Song, Y.-H. Li, L. Wang, Missed diagnosis of early gastric cancer or high-grade intraepithelial neoplasia, World Journal of Gastroenterology: WJG 19 (13) (2013) 2092.
* [44] J. R. Jass, L. H. Sobin, Histological typing of intestinal tumours, Springer Science & Business Media, 2012.
* [45] K. J. Spring, Z. Z. Zhao, R. Karamatic, M. D. Walsh, V. L. Whitehall, T. Pike, L. A. Simms, J. Young, M. James, G. W. Montgomery, et al., High prevalence of sessile serrated adenomas with braf mutations: a prospective study of patients undergoing colonoscopy, Gastroenterology 131 (5) (2006) 1400–1407.
* [46] S. C. Li, L. Burgart, Histopathology of serrated adenoma, its variants, and differentiation from conventional adenomatous and hyperplastic polyps, Archives of pathology & laboratory medicine 131 (3) (2007) 440–445.
* [47] K. H. Zou, S. K. Warfield, A. Bharatha, C. M. Tempany, M. R. Kaus, S. J. Haker, W. M. Wells III, F. A. Jolesz, R. Kikinis, Statistical validation of image segmentation quality based on a spatial overlap index1: scientific reports, Academic radiology 11 (2) (2004) 178–189.
* [48] P. Jaccard, Étude comparative de la distribution florale dans une portion des alpes et des jura, Bull Soc Vaudoise Sci Nat 37 (1901) 547–579.
* [49] S. Naz, H. Majeed, H. Irshad, Image segmentation using fuzzy clustering: A survey, in: 2010 6th international conference on emerging technologies (ICET), IEEE, 2010, pp. 181–186.
* [50] N. M. Zaitoun, M. J. Aqel, Survey on image segmentation techniques, Procedia Computer Science 65 (2015) 797–806.
* [51] N. Dhanachandra, K. Manglem, Y. J. Chanu, Image segmentation using k-means clustering algorithm and subtractive clustering algorithm, Procedia Computer Science 54 (2015) 764–771.
* [52] H. Deng, D. A. Clausi, Unsupervised image segmentation using a simple mrf model with a new implementation scheme, Pattern recognition 37 (12) (2004) 2323–2335.
* [53] C. Huang, X. Li, Y. Wen, An otsu image segmentation based on fruitfly optimization algorithm, Alexandria Engineering Journal 60 (1) (2021) 183–188.
* [54] M. S. H. Khiyal, A. Khan, A. Bibi, Modified watershed algorithm for segmentation of 2d images., Issues in Informing Science & Information Technology 6 (2009).
* [55] H. Zhang, Q. Zhu, X.-f. Guan, Probe into image segmentation based on sobel operator and maximum entropy algorithm, in: 2012 International Conference on Computer Science and Service System, IEEE, 2012, pp. 238–241.
* [56] V. Badrinarayanan, A. Kendall, R. Cipolla, Segnet: A deep convolutional encoder-decoder architecture for image segmentation, IEEE transactions on pattern analysis and machine intelligence 39 (12) (2017) 2481–2495.
* [57] O. Ronneberger, P. Fischer, T. Brox, U-net: Convolutional networks for biomedical image segmentation, in: International Conference on Medical image computing and computer-assisted intervention, Springer, 2015, pp. 234–241.
* [58] J. M. J. Valanarasu, P. Oza, I. Hacihaliloglu, V. M. Patel, Medical transformer: Gated axial-attention for medical image segmentation, in: International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2021, pp. 36–46.
|
Thermalization of long range Ising model in different dynamical regimes: a
full counting statistics approach.
Nishan Ranabhat1,2$\star$, Mario Collura1
1 SISSA - International School for Advanced Studies, via Bonomea 265, 34136
Trieste, Italy
2 The Abdus Salam International Centre for Theoretical Physics, Strada
Costiera 11, 34151 Trieste, Italy
⋆<EMAIL_ADDRESS>
## Abstract
We study the thermalization of the transverse field Ising chain with a power
law decaying interaction $\sim 1/r^{\alpha}$ following a global quantum quench
of the transverse field in two different dynamical regimes. The thermalization
behavior is quantified by comparing the full probability distribution function
(PDF) of the evolving states with the corresponding thermal state given by the
canonical Gibbs ensemble (CGE). To this end, we used the matrix product state
(MPS)-based Time Dependent Variational Principle (TDVP) algorithm to simulate
both real time evolution following a global quantum quench and the finite
temperature density operator. We observe that thermalization is strongly
suppressed in the region with strong confinement for all interaction strengths
$\alpha$, whereas thermalization occurs in the region with weak confinement.
###### Contents
1. 1 Introduction
2. 2 Model and Methods
3. 3 Numerical details
1. 3.1 Real and imaginary time evolution
2. 3.2 Extraction of effective temperature of a global quench
4. 4 Results
1. 4.1 Quench to dynamical ferromagnetic regime
2. 4.2 Quench to dynamical paramagnetic regime
5. 5 Conclusion
6. A Exact results for smaller systems
7. B Simulations details
1. B.1 Simulation of finite temperature density operator
2. B.2 Calculating full counting statistics with MPS
3. B.3 Errors and data convergence
8. C Thermal phase transition in long range Ising model
9. D Confinement dynamics in different regimes
## 1 Introduction
The investigation of non-equilibrium dynamics in isolated many-body systems
has garnered significant attention in recent decades, owing to advancements in
the manipulation of synthetic quantum systems in laboratory settings [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11] and the development of analytical and numerical
techniques [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. An
enduring question in quantum many-body dynamics pertains to the potential
thermalization of a closed system that has been perturbed from equilibrium.
Thermalization implies that the long-term behavior of a dynamic system can be
anticipated using the principles of statistical mechanics. Generally, in the
case of a non-integrable closed system, one would expect thermalization in
accordance with the Eigenstate Thermalization Hypothesis (ETH) [26, 27, 28,
29, 30]. Nonetheless, certain studies have presented contradictory evidence,
at least within their specific regime and time scales of investigation [31,
32, 33, 34]. In a scenario where a closed system is initially prepared in a
generic state, denoted as $\ket{\psi_{i}}$ (which is not an eigenstate of the
Hamiltonian), and subsequently evolved using unitary dynamics with a non-
integrable Hamiltonian, $\hat{H}$, thermalization is said to occur if the
local observables eventually relax to an equilibrium state that corresponds to
the predictions of the thermal ensemble,
$\langle\hat{O}\rangle_{t|t\rightarrow\infty}\rightarrow\langle\hat{O}\rangle_{\text{eq}}=\langle\hat{O}\rangle_{\text{MCE}}=\frac{1}{N_{E_{i},\delta\mathcal{E}}}\sum_{\absolutevalue{E_{n}-E_{i}}<\delta\mathcal{E}}\matrixelement{\psi_{n}}{\hat{O}}{\psi_{n}},$
(1)
$\langle\hat{O}\rangle_{t|t\rightarrow\infty}\rightarrow\langle\hat{O}\rangle_{\text{eq}}=\langle\hat{O}\rangle_{\text{CGE}}=\frac{\text{Tr}(\rho_{\beta}\hat{O})}{\text{Tr}(\rho_{\beta})}.$
(2)
Equation (1) indicates thermalization according to the microcanonical ensemble
(MCE). The summation encompasses all the eigenstates of the Hamiltonian within
a narrow energy range $\delta\mathcal{E}$ centered around the initial energy
$E_{i}=\matrixelement{\psi_{i}}{\hat{H}}{\psi_{i}}$. The normalization factor
$N_{E_{i},\delta\mathcal{E}}$ tallies the energy eigenstates within this
range, spanning $2\delta\mathcal{E}$. This approach requires either full
diagonalization [28, 35, 36] or partial diagonalization centered on the
initial energy density [37] of the Hamiltonian. Consequently, computational
limitations arise as a function of the system size. Equation (2) implies
thermalization in accordance with the canonical Gibbs ensemble (CGE). The
trace is performed over the density operator $\hat{\rho}_{\beta}$, defined as
the inverse temperature $\beta=1/T$, which is determined by the system’s
initial energy
$E_{i}=\frac{\text{Tr}(\hat{\rho}_{\beta}\hat{H})}{\text{Tr}(\hat{\rho}_{\beta})}$.
Further elaboration on how to extract $\beta$ and $\hat{\rho}_{\beta}$ is
provided in section 3.2. Equations (1) and (2) are recognized as the
conditions for strong thermalization. An alternative weak thermalization
condition occurs when the time-averaged local order parameter converges to the
thermal prediction [33, 34].
Disordered systems demonstrating many-body localization impede thermalization
[38, 39, 40, 41]. In clean systems, dynamical confinement hinders information
propagation and the thermalization process [42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53]. The Long Range Ising model (LRIM) exhibits confinement [44,
45] as outlined in Appendix D. A recent study [4] observed the suppression of
thermalization in the confined regime of LRIM simulated with trapped ions.
Furthermore, employing high-scale exact diagonalization, the validity of ETH
has been examined for various interaction strength parameters, denoted as
$\alpha$ (see section 2), in LRIM. The results indicate that a strong ETH
typically holds at least within the range $\alpha\geq 0.6$ [35]. In our
previous work [54], we explored the relaxation of order parameter statistics
following a global quench, revealing two distinct dynamical regimes based on
the gaussification of the full counting statistics (FCS) of subsystem
magnetization. Building on this foundation, the present study investigates the
thermalization of LRIM under the CGE framework following a global quench into
different dynamical regimes. We evaluate thermalization using the most
rigorous criteria by comparing the FCS of the time-evolving state post-global
quench with that of the corresponding thermal state.
The remainder of this paper is organized as follows: In Section 2, we
introduce the model, the order parameter, its distribution, and a metric for
quantifying the proximity of the time-evolved state to the thermal state.
Section 3 provides comprehensive details of the numerical methods, quench
protocol, and extraction of effective temperatures associated with the global
quench. Specific details on simulating finite temperature density operators
and calculating the full counting statistics of the order parameter is
provided in Appendix B.1 and Appendix B.2 respectively. Section 4 presents the
outcomes of our study. Appendix B.3 details the error analysis of the
numerical results and Appendices C and D provide supplementary information on
thermal phase transitions and correlation propagation in the Long Range Ising
model. Finally, we conclude by summarizing our findings and suggesting
potential avenues for future research in Section 5.
## 2 Model and Methods
We investigate the ferromagnetic long-range Ising model (LRIM) described by
the Hamiltonian in Equation (3),
$\hat{H}(J,\alpha,h)=-\frac{1}{\mathcal{K}(\alpha)}\sum_{i<j}^{N}\frac{\absolutevalue{J}}{|i-j|^{\alpha}}\hat{s}^{x}_{i}\hat{s}^{x}_{j}-h\sum_{i=1}^{N}\hat{s}^{z}_{i}$
(3)
where $\hat{s}^{\mu}_{i},\mu={x,y,z}$ is the spin one-half matrices at site
$i$. We consider open boundary condition, that is relevant to existing
experimental setups. The ferromagnetic interaction between two spins falls as
the inverse power of the distance between them and is parameterized by the
interaction strength parameter $\alpha$. For $\alpha\leq 1$, the inverse
power-law interaction series diverges with the lattice size and is normalized
using the Kac normalization constant, as defined in Equation (4).
$\mathcal{K}(\alpha)=\frac{1}{N-1}\sum_{i<j}^{N}\frac{1}{|i-j|^{\alpha}}=\frac{1}{N-1}\sum_{n=1}^{N}\frac{N-n}{n^{\alpha}}.$
(4)
This normalization ensures the intensivity of the energy density in the regime
$\alpha\leq 1$. The static and dynamic behaviors of this model are strongly
influenced by the interaction strength parameter $\alpha$. At $\alpha=\infty$,
the model simplifies to the transverse field Ising model (TFIM), which can be
solved exactly by mapping it to a system of spinless fermions through Jordan-
Wigner transformations [55]. TFIM exhibits a quantum phase transition from the
ferromagnetic phase to the paramagnetic phase at $h=J/2$. This quantum phase
transition persists as $\alpha$ decreases, with the transition point shifting
towards higher values of the magnetic field $h$ [56, 57, 5]. At the opposite
extreme of $\alpha=0$, we have a fully connected regime that is amenable to
analytical treatment for both the static and dynamic properties [58, 59, 54].
For $\alpha<2$, this model displays a long-range ferromagnetic order at low
finite temperatures [60, 61]. Given the absence of spontaneous symmetry
breaking in finite systems and the $\mathbb{Z}_{2}$ symmetry of the model in
Equation (3), the finite-temperature states with ferromagnetic order also
exhibit $\mathbb{Z}_{2}$ symmetry (see appendix C). The regime with $\alpha<2$
is particularly intriguing and features a wealth of exotic phenomena such as
prethermalization [11], nonlinear propagation of light cones[62, 63],
dynamical phase transitions [64, 65, 66, 2, 3, 54, 67, 68, 69], and dynamical
confinement [44, 45, 4]. Furthermore, this model has garnered significant
attention owing to its experimental relevance, particularly in systems
involving trapped ions with adjustable transverse field strengths and
interaction ranges [1, 2, 3, 4, 11].
The complete information of a generic time evolving quantum state, expanded in
the computational basis
$\ket{\psi_{t}}=\sum_{\\{\sigma_{i}\\}}C_{\\{\sigma_{i}\\}}(t)\ket{\sigma_{1},\sigma_{2},\ldots,\sigma_{i},\ldots,\sigma_{N}}$,
is encapsulated within the set of time dependent coefficients
$\\{C_{\\{\sigma_{i}\\}}(t)\\}$. In many-body systems, these coefficients
scales exponentially with the number of spins, rendering their study
exceedingly challenging. A common approach for investigating dynamics in such
systems is to monitor the evolution of the expectation value of a local
observable $\matrixelement{\psi_{t}}{\hat{O}}{\psi_{t}}$, such as the order
parameter in systems exhibiting order-disorder transitions. A more robust
strategy involves tracking the full probability distribution function (PDF) of
this observable, which provides comprehensive information on quantum
fluctuations in the system, including all moments and cumulants. Specifically,
when the operator $\hat{O}$ is diagonal in computational basis, the
corresponding PDF is defined as
$P(O,t)=\sum_{\\{\sigma_{i}\\}}\absolutevalue{C_{\\{\sigma_{i}\\}}(t)}^{2}\hskip
14.22636pt\text{for all
}\\{\sigma_{i}\\}:\matrixelement{\sigma_{1},\sigma_{2},\ldots,\sigma_{i},\ldots,\sigma_{N}}{\hat{O}}{\sigma_{1},\sigma_{2},\ldots,\sigma_{i},\ldots,\sigma_{N}}=O,$
(5)
which represents the histogram of the squared coefficients of the many-body
wave function within the range of possible outcomes of the measurements of
$\hat{O}$. The PDF allows for straightforward calculation of moments (or
cumulants) of any order. In this study, we employ the eigenvectors of the
total spin operator in the longitudinal direction, that is,
$\hat{S}^{x}=\sum_{i=1}^{N}\hat{s}_{i}^{x}$, as our computational basis. In
this context, the order parameter of interest is the longitudinal
magnetization defined for a subsystem of size $l$ within a system of $N$
spins:
$\hat{M}_{l}=\sum_{i=1}^{l}\hat{s}^{x}_{i}.$ (6)
The observable $\hat{M}_{l}$ is suitable because it typically relaxes to a
stationary state [70], ultimately approaching a stationary statistical
distribution in a subsystem of dimension $l$. In addition, because
$\hat{M}_{l}$ is diagonal in computational basis, the definition in Equation
(5) applies. The probability distribution function of the subsystem
magnetization $\hat{M}_{l}$ within a generic state $\hat{\rho}_{t}$ (whether
pure or mixed) is given by
$P_{l}(m,t)=\text{Tr}(\hat{\rho}_{t}\delta(\hat{M}_{l}-m)),$ (7)
which can be Fourier-transformed into an integral form:
$P_{l}(m,t)=\int_{-\pi}^{\pi}\frac{d\theta}{2\pi}e^{-i\theta
m}\text{Tr}\big{(}\hat{\rho}_{t}e^{i\theta\hat{M}_{l}}\big{)},$ (8)
where
$G_{l}(\theta,t)=\text{Tr}\big{(}\hat{\rho}_{t}e^{i\theta\hat{M}_{l}}\big{)}$
denotes the moment-generating function. Given that the Hamiltonian (3)
involves a system of spin one-half particles, the values of $m$ span either
integers or half-integers within the range
$m\in\Big{\\{}-\frac{l}{2},-\frac{l}{2}+1,\ldots,\frac{l}{2}-1,\frac{l}{2}\Big{\\}}$
depending on whether $l$ is even or odd. In Appendix B.2 we illustrate the
detailed calculation of $G_{l}(\theta,t)$ with matrix product state (MPS)
representation. Historically, the PDF has been studied as the full counting
statistic (FCS) of electron fluctuations in mesoscopic systems [71, 72, 73].
More recently, FCS has been explored in quantum many-body systems in both
equilibrium and non-equilibrium scenarios [18, 74, 75, 19, 76, 43, 54, 77, 78,
79].
To assess thermalization, we introduce a metric called "Distance to
Thermalization", $\text{DT}(t)$, initially introduced in [80]. This metric
quantifies the Euclidean distance between the probability distribution
function (PDF) of the order parameter at time $t$ following a quantum quench,
denoted as $P_{l}(m,t)$, and the corresponding thermal PDF, represented as
$P_{l}^{\text{TH}}(m)$. Mathematically, it is defined as
$\text{DT}(t)=\sqrt{\sum_{m}\big{[}P_{l}(m,t)-P_{l}^{\text{TH}}(m)\big{]}^{2}}.$
(9)
It is noteworthy that the convergence of the PDF provides a more rigorous
criterion for thermalization than the convergence of the expectation value.
This is because the former implies the latter, whereas the reverse is not
necessarily true. A similar approach has been employed in previous studies to
investigate thermalization dynamics [80, 37, 81, 79]. Comprehensive details of
how to extract the thermal state corresponding to a global quantum quench are
discussed in Section 3.2. In cases where the system undergoes thermalization,
$\text{DT}(t)$ is expected to converge to zero in the long time limit.
Figure 1: (a) Global Quench protocol: System is initialized as the ground
state of a trivial hamiltonian $\hat{H}_{i}$ (in our case the initial state is
a Greenberger-Horne-Zeilinger (GHZ) state), at time $t=0$ the system is
suddenly quenched to a final Hamiltonian $\hat{H}_{f}$ and the initial state
is unitarily evolved with the final Hamiltonian. (b) The nonequilibrium state
following a global quantum quench can exhibit different relaxation behavior;
Path 1: a direct relaxation to thermal equilibrium with a single time scale,
Path 2: a quick relaxation to a long lived prethermal state eventually
followed a relaxation to thermal equilibrium, Path 3: a strong retention of
initial memory and suppression of relaxation to thermal equilibrium.
## 3 Numerical details
### 3.1 Real and imaginary time evolution
The numerical simulations in this study are classified into two distinct
categories:
* •
real time evolution of pure state following a global quench.
* •
simulation of finite temperature density operators.
For both of these simulation tasks, we employ the MPS-based Time Dependent
Variational Principle (TDVP) algorithm [23, 24] with second order integration
scheme. This choice affords us a significant advantage over exact
diagonalization methods, allowing us to simulate systems of much larger sizes
than can be accommodated by the current exact diagonalization techniques.
The quench protocol implemented in this study, with energy rescaled to
$\absolutevalue{J}=1$, is as follows: At time $t=0$, the system is prepared in
the ground state of the Hamiltonian $\hat{H}_{i}(\alpha,0)$, which takes the
form of a $\mathbb{Z}_{2}$ symmetric Greenberger-Horne-Zeilinger (GHZ) state
oriented along the longitudinal direction:
$\ket{\psi_{i}}=\frac{1}{\sqrt{2}}(\ket{\rightarrow,\ldots\rightarrow,\rightarrow,\rightarrow\ldots,\rightarrow}_{x}+\ket{\leftarrow,\ldots\leftarrow,\leftarrow,\leftarrow\ldots,\leftarrow}_{x}),$
(10)
GHZ state is characterized by the PDF
$P^{\text{GHZ}}_{l}(m)=\frac{\delta_{m,\absolutevalue{l}/2}}{2}$ which is
sharply bimodal with two peaks at $m=l/2$ and $m=-l/2$ respectively. Equation
(10) can be explicitly represented as an exact Matrix Product State (MPS) with
bond dimension $\chi=2$. Subsequently, a global quench is initiated along the
transverse field $h$ to a final Hamiltonian $\hat{H}_{f}(\alpha,h_{f})$ and
the system is evolved unitarily using the expression
$\ket{\psi_{t+dt}}=e^{-idt\hat{H}_{f}}\ket{\psi_{t}}$. The evolution is
monitored by calculating the Full Counting Statistics (FCS) of the order
parameter at each time step. The details of the quench protocol is pictorially
represented in Figure 1 (a). The finite temperature density operator is
simulated by an imaginary time evolution starting from a maximally mixed state
at infinite temperature. Additional details pertaining to the calculation of
the thermal density operator are provided in Appendix B.1. For both sets of
simulations, we maintain a fixed maximum bond dimension of the MPS at
$\chi_{max}=128$. Furthermore, Trotter time steps of $dt=0.05$ and
$d\beta=0.001$ are used for real and imaginary time evolution, respectively.
There is a finite time-step error of $O(dt^{3})$ per time step and $O(dt^{2})$
per unit time [82]. In Appendix B.3 we access the accuracy of the TDVP data by
comparing the TDVP results with the exact results obtained by the full
diagonalization of a system of size $N=14$. Furthermore, we test the
convergence of the data by calculating the relative error for three increasing
bond dimensions.
### 3.2 Extraction of effective temperature of a global quench
A global quantum quench
$\hat{H}_{i}(\alpha,0)\xrightarrow{}\hat{H}_{f}(\alpha,h)$ in an isolated
system adds an extensive amount of energy to the system. Consequently, the
system relaxes to a state at a higher energy level than the ground state of
the post-quench Hamiltonian [70],
$\lim_{N\to\infty}\frac{1}{N}\frac{\matrixelement{\psi_{t}}{\hat{H}_{f}}{\psi_{t}}}{\innerproduct{\psi_{t}}{\psi_{t}}}>\lim_{N\to\infty}\frac{1}{N}\frac{\matrixelement{\psi_{0}}{\hat{H}_{f}}{\psi_{0}}}{\innerproduct{\psi_{0}}{\psi_{0}}}$
(11)
where $\ket{\psi_{0}}$ is the ground state of the post-quench Hamiltonian
$\hat{H}_{f}(\alpha,h)$. The left hand side of Equation (11) is a conserved
quantity because the real time evolution of $\ket{\psi_{t}}$ is unitary. For
every global quantum quench we can attribute an effective temperature
$\beta_{\text{eff}}$ which is the temperature at which the thermal energy
density above the ground state of the post-quench Hamiltonian matches the
conserved energy density of the system,
$\frac{1}{N}\frac{\matrixelement{\psi_{t}}{\hat{H}_{f}}{\psi_{t}}}{\innerproduct{\psi_{t}}{\psi_{t}}}=\frac{1}{N}\frac{\text{Tr}(\hat{\rho}_{\beta}\hat{H}_{f})}{\text{Tr}(\hat{\rho}_{\beta})}.$
(12)
Figure 2: Numerical extraction of $\beta_{\text{eff}}$ corresponding to a
global quantum quench. The horizontal black dashed line represent the energy
density attributed to the quench. The colored lines represents the energy
density as the function of inverse temperature $\beta$ for the corresponding
post-quench parameter (in legend). The point at which the colored lines
intersects the black dashed lines represents $\beta_{\text{eff}}$ for the
corresponding post-quench parameters (represented by vertical colored lines).
The effective temperature is extracted by solving Equation (12). The left hand
side of the equation is trivially calculated as
$\matrixelement{\psi_{t}}{\hat{H}_{f}}{\psi_{t}}=\matrixelement{\psi_{i}}{e^{it\hat{H}_{f}}\hat{H}_{f}e^{-it\hat{H}_{f}}}{\psi_{i}}=\matrixelement{\psi_{i}}{\hat{H}_{f}}{\psi_{i}}$,
and the right hand side can be calculated for a series of $\beta$ by
numerically solving Equation (23) and calculating the energy density at each
instance. The precision of $\beta_{\text{eff}}$ depends on the trotter steps
$d\beta$ in the solution to equation (23). In Fig. 2 we plot the numerical
solution of equation (12). The energy density attributed to quench
(represented by the black dashed line) in our setup is independent of the
post-quench parameters because the spin-spin interaction term in the
Hamiltonian (3) is normalized with the Kac normalization (4), whereas the
expectation value
$h\matrixelement{\psi_{i}}{\sum_{j}\hat{s}^{z}_{j}}{\psi_{i}}$ taken over the
transverse field term is trivially zero. If we extend the simulation to a
larger $\beta$ (i.e., lower temperature), all curves will converge to the
ground state energy density of $\hat{H}_{f}$ at the corresponding post-quench
parameters. Once $\beta_{\text{eff}}$ is extracted, we can calculate the
corresponding thermal PDF, $P^{\text{TH}}(m)=P^{\beta_{\text{eff}}}(m)$, using
equation (8).
## 4 Results
The global quantum quench, as discussed in Section 3.1, induces a dynamical
quantum phase transition (DQPT) [83, 84] in LRIM, which has garnered extensive
attention in recent years. This transition falls into two distinct categories:
the first, known as DQPT-I, is characterized by distinctive behaviors in the
time-averaged local order parameter following a global quench across the
dynamical critical point [54, 64, 2], and the second, DQPT-II, is marked by
non-analytic cusps in the Loschmidt echo rate [67, 68, 69, 85, 3]. In LRIM,
the dynamical critical points for DQPT-I and DQPT-II coincide at approximately
$h^{\text{dyn}}_{c}\approx 0.5$ for $\alpha\leq 2$ [64, 85]. When
$h_{f}<h^{\text{dyn}}_{c}$, the system is in the dynamical ferromagnetic
phase, which strongly retains the ferromagnetic order of the initial GHZ state
following a global quantum quench. This is evident from the persistent
oscillation of $P_{l}(m,t)$ around $P^{\text{GHZ}}_{l}(m)$. Conversely, when
$h_{f}>h^{\text{dyn}}_{c}$, the system transits to the dynamical paramagnetic
phase, characterized by the rapid dissolution of the initial ferromagnetic
order of the initial GHZ state following a global quantum quench. This is
signified by the Gaussification of $P_{l}(m,t)$ [54, 64]. The comprehensive
dynamical phase diagram and universality behavior related to the dynamical
phase transition in LRIM remain active areas of investigation [86, 87, 88,
66].
Figure 3: First row: Time evolution of the metric $\text{DT}(t)$ following a
global quantum quench to three interaction strength values
$\alpha\in\\{0.0,1.5,1.9\\}$ and transverse field $h_{f}=0.3$ at three
different subsystem sizes $l=\\{20,60,100\\}$. All three points are in
dynamical ferromagnetic phases[64, 54]. Second row: $P_{l}(m)$ versus $m$ for
$m\in[0,l/2]$ with $l=100$ at four time different slices $t=\\{2,6,20,50\\}$.
$P_{l}(m)$ versus $m$ for $m\in[-l/2,0)$ is its mirror image. The black dashed
curve represents the thermal PDF, $P^{\text{TH}}(m)$ attributed to the
corresponding global quantum quenches.
Our primary objective is to examine the convergence behavior of the metric
$\text{DT}(t)$ in two distinct dynamical phases of the LRIM. The convergence
of $\text{DT}(t)$ towards zero is an indicator of thermalization within a
particular phase under consideration. We initialize the system as
$\mathbb{Z}_{2}$ symmetric GHZ state presented in Equation (10), which
represents the ground state of the Hamiltonian given in Equation (3). This
choice is made because the model (3) undergoes a thermal transition from a
paramagnetic phase, characterized by a Gaussian probability density function
(PDF) at high temperatures, to a ferromagnetic phase with a $\mathbb{Z}_{2}$
symmetric bimodal PDF, as comprehensively detailed in Appendix C. We maintain
that $\alpha<2$ is crucial as this region exhibits an interesting landscape
encompassing both finite temperature phase transitions [89, 90] and dynamic
phase transitions [64, 65, 66, 2, 3, 54, 67, 68, 69]. Specifically, we
consider three distinct values for interaction strength, namely
$\alpha\in{0.0,1.5,1.9}$. At $\alpha=0.0$, the system exhibits integrability
because of its full connectivity and complete permutation symmetry, thereby
leading to a lack of thermalization[17]. On the other hand, the choices of
$\alpha=1.5$ and $\alpha=1.9$ are motivated by the relatively faster
equilibration and Gaussification of the PDF following a quench in the
dynamical paramagnetic phase, as previously observed [54].
Figure 4: First row: Time evolution of the metric $\text{DT}(t)$ following a
global quantum quench to three interaction strength values
$\alpha\in\\{0.0,1.5,1.9\\}$ and transverse field $h_{f}=0.6$ at three
different subsystem sizes $l=\\{20,60,100\\}$. All three points are in
dynamical paramagnetic phases[64, 54]. Second row: $P_{l}(m)$ versus $m$ for
$m\in[0,l/2]$ with $l=100$ at four time different slices $t=\\{2,6,20,50\\}$.
$P_{l}(m)$ versus $m$ for $m\in[-l/2,0)$ is its mirror image. The black dashed
curve represents the thermal PDF, $P^{\text{TH}}(m)$ attributed to the
corresponding global quantum quenches.
### 4.1 Quench to dynamical ferromagnetic regime
Figure 3 shows the temporal evolution of $\text{DT}(t)$ following a global
quantum quench of the transverse field to $h_{f}=0.3$ with
$\alpha=0.0,1.5,1.9$ for subsystem sizes $l=20,60,100$. Notably, all these
points belong to the dynamical ferromagnetic phase [64, 54]. For all three
quenches, a persistent oscillation in $\text{DT}(t)$ is evident, indicating
that the initial ferromagnetic order is strongly retained and thermalization
is suppressed. This behavior aligns with the relaxation mode represented by
Path 3 in figure 1(b). Specifically, $\alpha=0.0$ is in the integrable regime;
therefore, thermalization is expected to be absent [26] whereas we anticipate
thermalization for quenches with $\alpha=1.5$ and $\alpha=1.9$. The apparent
suppression of thermalization can be attributed to the confinement behavior.
The long-range interaction of the model effectively confines low-energy domain
wall kinks into heavier quasiparticles that typically travel slower than free
quasiparticles, thereby suppressing the spread of correlations in the system
[44, 45]. Consequently, thermalization is still expected but only at
significantly longer time scales [47]. Appendix D details the confinement
behavior in LRIM where we observe the spreading of connected correlation
function $\expectationvalue{\hat{s}^{x}_{k}\hat{s}^{x}_{k+\Delta}}_{c}$ for
$\alpha=1.9$ and $h_{f}=0.3$ shows a strong temporal suppression. In figure
3(d), (e), and (f), the colored scattered plots depict $P_{l}(m)$ as a
function of $m$ at four distinct time intervals post-quench. The black dashed
curve represents the Probability Density Function (PDF) of the expected
thermal state, $P_{l}^{\text{TH}}(m)$. We observe that the time-evolving
$P_{l}(m)$ oscillates persistently around $P_{l}^{\text{TH}}(m)$. Of
particular importance is the observation that, in all three cases, the thermal
PDFs are bimodal, indicating the presence of long-range ferromagnetic order.
This observation suggests that if the system eventually thermalizes for these
post-quench parameters at extended time scales, it would exhibit a long-range
ferromagnetic order. This finding further strengthens the argument that this
is indeed a dynamical ferromagnetic phase.
### 4.2 Quench to dynamical paramagnetic regime
Figure 4 illustrates the temporal evolution of $\text{DT}(t)$ following a
global quantum quench of the transverse field to $h_{f}=0.6$, with
$\alpha=0.0,1.5,1.9$ for subsystem sizes $l=20,60,100$. These points are
located within the dynamical paramagnetic phase [64, 54]. Notably, these
quenches exhibit a distinct relaxation behavior of $\text{DT}(t)$ compared
with the previous cases. In Figure 4(a), we observe rapid equilibration for
all values of $l$. However, it is essential to highlight that $\text{DT}(t)$
remains at or above the order of $O(10^{-1})$ following equilibration, which
suggests a lack of thermalization. This behavior aligns with expectations,
because $\alpha=0$ represents an integrable point. For $\alpha=1.5$,
$\text{DT}(t)$ does not exhibit stable equilibration (see figure 4 (b));
Finally, when $\alpha=1.9$, we observe equilibration for $l=60,100$ (see
figure 4 (c)). $\text{DT}(t)$ exhibits a stable oscillation around a constant
value of approximately $O(10^{-3})$. A more comprehensive picture is shown in
Fig. 4(f), where the late-time PDF perfectly overlaps with the corresponding
thermal PDF represented by a black dashed curve. This is indicative of
thermalization of the corresponding quench. Although we observed signatures of
thermalization, the system is not in a de-confined phase [4, 91]. In Appendix
D, the connected correlation function
$\expectationvalue{\hat{s}^{x}_{k}\hat{s}^{x}_{k+\Delta}}_{c}$ for
$\alpha=1.9$ and $h_{f}=0.6$ still exhibits weaker temporal suppression. A
recent study observed a de-confinement transition for a system of up to 31
spins for a much higher value of the transverse field [4]. This suggests that,
although strong confinement suppresses thermalization, signatures of
thermalization can be observed in the presence of weak confinement. This
suggests that while strong confinement suppresses thermalization, signatures
of thermalization can still be detected in the presence of weak confinement.
Figure 5: Time evolution of domain wall kinks, $\langle\hat{k}\rangle$,
following a global quantum quench to three interaction strength values
$\alpha\in\\{0.0,1.5,1.9\\}$ (Panels (a), (b), and (c) respectively) and
$h_{f}=0.3$ and $h_{f}=0.6$. The dashed horizontal lines represent the
expected thermal value of domain wall kinks,
$\langle\hat{k}\rangle^{\text{TH}}$ corresponding to the quenches.
To further support this observation we study the post quench temporal
evolution of domain wall kinks defined as,
$\hat{k}=\sum_{j=1}^{l-1}\frac{1-\hat{s}^{x}_{i}\hat{s}^{x}_{i+1}}{2}.$ (13)
$\hat{k}$ counts the number of nearest neighbor kinks in the $\hat{x}$
direction within subsystem $l$. Because confinement bounds the domain walls
kinks into heavier quasiparticles, it is a relevant parameter to study. In
Figure 5, we illustrate the temporal evolution of the average domain-wall
kinks, $\langle\hat{k}\rangle$, following a global quantum quench. As
anticipated, quenches to the dynamical ferromagnetic phase with $h_{f}=0.3$
display persistent oscillations around the thermal value, indicating a lack of
thermalization. Conversely, for quenches to the dynamical paramagnetic phase
with $h_{f}=0.6$, we observe distinctly different post-quench behavior. In the
case of $\alpha=0$, the domain wall kinks equilibrate to a stable value that
differs from the expected thermal value, as expected because it is an
integrable point. This observation complements the post-quench behavior of DT,
as depicted in Figure 4 (a). Although $\alpha=1.5$ is a non-integrable point,
thermalization is not observed within the simulation time. At later times, a
stable prethermal plateau, close but distinct from the expected thermal value,
becomes apparent. Conversely, for a quench corresponding to $\alpha=1.9$, the
average domain wall kinks converge to the expected thermal value. Notably,
before reaching the thermal value, the kink density exhibits a relatively
stable prethermal plateau until time $t\simeq 35$. This relaxation mode, which
is characterized by two time scales, is represented by Path 2 in Figure 1(b).
This discovery provides another robust indicator of thermalization in weakly
confined regimes.
## 5 Conclusion
We investigate the relaxation dynamics of the long-range Ising model
subsequent to a global quantum quench of the transverse field, assessing the
thermalization on a computationally viable time scale according to the
canonical Gibbs ensemble (CGE). The model is non-integrable for all values of
$\alpha$ except at the extremes ($\alpha={0.0,\infty}$), where we anticipate
thermalization following a global quantum quench. However, the long-range
Ising model exhibits confinement, which suppresses correlation spreading and
eventually impedes thermalization. Starting from the Greenberger-Horne-
Zeilinger (GHZ) state, we subject the system to two distinct dynamical
regimes. As anticipated, robust confinement hinders thermalization for smaller
quenches, specifically in the dynamical ferromagnetic region, where the metric
DT exhibits persistent oscillations characteristic of the masses of bound
mesons. Conversely, for quenches to the dynamical paramagnetic region, a
notably different behavior emerges. The persistent oscillation diminishes and
the DT relaxes more rapidly. Although conclusive evidence of thermalization
for $\alpha=1.5$ is not observed within the simulation time, compelling
indications of the thermalization surface for $\alpha=1.9$ are based on the
relaxation of DT. This observation gains additional support from the
convergence of the domain wall kinks to the expected thermal value.
## Acknowledgements
Nishan Ranabhat thanks Alvise Bastianello for fruitful discussion and
suggesting the future extension of the work. The numerical simulation of this
project was performed at the Ulysses v2 cluster at SISSA.
## Appendix A Exact results for smaller systems
For small systems we can calculate the time evolution of FCS and other
relevant order parameters by exact diagonalization of the post-quench
Hamiltonian. We begin from our initial state, $\mathbb{Z}_{2}$ symmetric GHZ
state,
$\ket{\psi_{0}}=\frac{1}{\sqrt{2}}(\ket{\rightarrow,\ldots\rightarrow,\rightarrow,\rightarrow\ldots,\rightarrow}+\ket{\leftarrow,\ldots\leftarrow,\leftarrow,\leftarrow\ldots,\leftarrow}).$
(14)
The time evolved state is given by
$\ket{\psi_{t}}=e^{-i\hat{H}t}\ket{\psi_{0}}$, where $\hat{H}$ is the post-
quench Hamiltonian 3. We proceed by expanding $\ket{\psi_{0}}$ in the
eigenbasis, of the post-quench Hamiltonian,
$\ket{\psi_{0}}=\sum_{j=0}^{2^{N}-1}q_{j}\ket{E_{j}},$ (15)
where $q_{j}=\innerproduct{E_{j}}{\psi_{0}}$. Further expanding $\ket{E_{j}}$
in the computational basis, $\ket{E_{j}}=\sum_{n}c^{j}_{n}\ket{n}$, we can
derive the expression for $q_{j}$ as,
$q_{j}=\innerproduct{E_{j}}{\psi_{0}}=\frac{\Big{(}c_{\ket{\rightarrow,\ldots,\rightarrow}}^{j}\Big{)}^{*}+\Big{(}c_{\ket{\leftarrow,\ldots,\leftarrow}}^{j}\Big{)}^{*}}{\sqrt{2}}.$
(16)
The post-quench state is
$\ket{\psi_{t}}=\sum_{n}X_{n}(t)\ket{n}$ (17)
where $X_{n}(t)=\sum_{j=0}^{N}q_{j}c_{n}^{j}e^{-iE_{j}t}$. We can now
calculate the time evolution of the expectation value of a generic parameter
$\hat{O}$ as,
$\displaystyle
O(t)=\bra{\psi_{t}}\hat{O}\ket{\psi_{t}}=\sum_{n,\tilde{n}}X^{\dagger}_{\tilde{n}}(t)X_{n}(t)\bra{\tilde{n}}\hat{O}\ket{n}.$
(18)
If $\ket{n}$ is the simultaneous eigenket of the order parameter $\hat{O}$
then 18 becomes,
$O(t)=\sum_{n}|X_{n}(t)|^{2}O_{n}.$ (19)
Finally, with the full eigenvalues of hamiltonian at hand we can also
calculate the energy density corresponding to a thermal density matrix
$\hat{\rho}_{\beta}$,
$\epsilon_{\beta}=\frac{\sum_{j}E_{j}e^{\beta E_{j}}}{\sum_{j}e^{\beta
E_{j}}}.$ (20)
## Appendix B Simulations details
In this section we present the details of numerical simulation complementary
to the results in the main text.
### B.1 Simulation of finite temperature density operator
The finite temperature states can be simulated by casting the density operator
as locally purified tensors [92, 93]. The thermal density operator is defined
by Gibbs distribution
$\hat{\rho}_{\beta}=\frac{e^{-\beta\hat{H}}}{Tr[e^{-\beta\hat{H}}]}$ where
$\beta=\frac{1}{T}$ is the inverse temperature. At $\beta=0$ (infinite
temperature) the state is maximally mixed and is given as the tensor product
of local identities
$\hat{\rho}_{0}=\bigotimes_{i=1}^{N}\mathbf{1}^{\sigma^{\prime}_{i},\sigma_{i}}=\mathbb{1}$,
where each $\mathbf{1}^{\sigma^{\prime}_{i},\sigma_{i}}$ is a unit matrix of
size $(d,d)$, i.e.
$\mathbf{1}^{\sigma^{\prime}_{i},\sigma_{i}}=[\delta_{\sigma^{\prime}_{i},\sigma_{i}}]_{d\times
d}$ and $d$ is the dimension of the physical space (for spin $\frac{1}{2}$,
$d=2$). The density operator for any finite temperature (non-zero $\beta$) is
$\displaystyle\hat{\rho}_{\beta}\propto e^{-\beta\hat{H}}$
$\displaystyle=e^{-\frac{\beta}{2}\hat{H}}\mathbb{1}e^{-\frac{\beta}{2}\hat{H}}$
(21a) $\displaystyle\propto
e^{-\frac{\beta}{2}\hat{H}}\hat{\rho}_{0}e^{-\frac{\beta}{2}\hat{H}}$ (21b)
We keep the density operator operator in locally purified form
$\hat{\rho}=\mathbb{X}\mathbb{X}^{\dagger}$ at each stage where $\mathbb{X}$
is represented as tensor
$\mathbb{X}^{\sigma_{1},\sigma_{2},\ldots\sigma_{i},\ldots,\sigma_{N}}_{k_{1},k_{2},\ldots,k_{i},\ldots,k_{N}}=\mathbf{X}^{\sigma_{1},k_{1}}_{c_{0},c_{1}}\mathbf{X}^{\sigma_{2},k_{2}}_{c_{1},c_{2}}\ldots\mathbf{X}^{\sigma_{i},k_{i}}_{c_{i-1},c_{i}}\ldots\mathbf{X}^{\sigma_{N},k_{N}}_{c_{N-1},c_{N}}$
(22)
where $\sigma_{i}=d$, $k_{i}=d$ are the physical index and the Kraus index are
are fixed through out and $1\leq c_{i}\leq\chi_{max}$ is the bond index and
$\chi_{max}$ is the maximum value of bond dimension. The density operator
initialized at infinite temperature can now be purified to a finite
temperature in trotterized steps
$\displaystyle\hat{\rho}_{\beta+d\beta}$
$\displaystyle=e^{-\frac{d\beta}{2}\hat{H}}\hat{\rho}_{\beta}e^{-\frac{d\beta}{2}\hat{H}}$
(23a)
$\displaystyle=e^{-\frac{d\beta}{2}\hat{H}}\mathbb{X}\mathbb{X}^{\dagger}e^{-\frac{d\beta}{2}\hat{H}}$
(23b)
$\displaystyle=e^{-\frac{d\beta}{2}\hat{H}}\mathbb{X}[e^{-\frac{d\beta}{2}\hat{H}}\mathbb{X}]^{\dagger}$
(23c)
Equation (23) can be simulated using imaginary time TDVP
($-idt\rightarrow-d\beta$) in only the half section of the density operator
operator and never contracting the $X$ and $X^{\dagger}$ layer during the
evolution, thus strictly preserving the locally purified form.
Figure 6: Maximally mixed density operator at $\beta=0$ as the tensor product
of local identities.
Figure 6 shows the tensor notation of the infinite temperature density
operator $\hat{\rho}_{\beta=0}$ which is a tensor product of identity matrices
of size $(d,d)$, where $d$ is the physical dimension. Rather than working with
the density operator as an MPO we represent the density operator in the
locally purified form [93, 94] which is positive semi-definite by construction
and keep it in locally purified form at every stage of the thermal
purification process. In Fig. 7 we represent $\hat{\rho}_{\beta=0}$ in the
locally purified form $\mathbb{X}_{\beta=0}\mathbb{X}_{\beta=0}^{\dagger}$,
where the index in purple is an auxiliary index called the Krauss index.
Figure 7: Representing $\hat{\rho}_{\beta=0}$ in the locally purified form.
we can now evolve one of the halves ($\mathbb{X}$ or $\mathbb{X}^{\dagger}$)
as shown in equation (23) and the evolution on the other half is its trivial
conjugate. This approach is computationally efficient as we can work with
cheaper MPS instead of more expensive MPDO. In Fig. 8 one half of the
$\hat{\rho}_{\beta=0}$ in locally purified form is shown, form here on we will
only work with this half.
Figure 8: One half of the $\hat{\rho}_{\beta=0}$ in the locally purified form.
Algebraically, $\mathbb{X}_{\beta=0}$ can be written as
$\mathbb{X}^{\sigma_{1},k_{1},\ldots\sigma_{i},k_{i},\ldots,\sigma_{N},k_{N}}=\mathbf{X}^{\sigma_{1},k_{1}}\otimes\ldots\mathbf{X}^{\sigma_{i},k_{i}}\ldots\otimes\mathbf{X}^{\sigma_{N},k_{N}}$
(24)
For the system of spin one-half particles we choose $A$ as
$\mathbf{X}^{\sigma_{i},k_{i}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\hskip 28.45274pt\forall\hskip
2.84544pti\in\\{1,2,\ldots,N\\}$ (25)
Figure 9: Choice of $\mathbf{X}^{\sigma_{i},k_{i}}$ to preserve the trace of
$\hat{\rho}$.
as shown in Fig. 9. This particular choice is taken to preserve the trace of
the density operator,
$\sum_{k}\mathbf{X}^{\sigma,k}[\mathbf{X}^{\sigma^{\prime},k}]^{*}=\frac{1}{2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$ (26)
Finally, we reshape $\mathbb{X}_{\beta=0}$ from a string of $2\times 2$
matrices to a string of four legged tensors of shape $(1,2,2,1)$ as shown in
Fig. 10, which is a MPS of bond dimension $1$.
Figure 10: $\mathbb{X}_{\beta=0}$ in MPS form.
Now that we have our initial state as an MPS, we can simulate a finite
temperature density operator by solving the equation (23),
Figure 11: Expectation of the local operator $\hat{O}_{i}$ in thermal density
operator $\hat{\rho}_{\beta}$
Numerically, equation (23) can be solved for long-range spin systems through
imaginary time evolution, (where $idt$ is transformed into $d\beta$) using the
Time-Dependent Variational Principle (TDVP). The TDVP algorithm employed for
simulating the thermal state remains fundamentally identical to that used for
the real-time evolution of the pure state, with the distinction of an
additional auxiliary Krauss index. However, in the thermal purification of a
closed system, the Krauss index becomes obsolete because all physical
operators act solely on the physical index, and the Krauss indices are
contracted among themselves [93]. Figure 11 illustrates the tensor network
diagram for computing the expectation value of a two point operator
$\hat{O}_{i}\hat{O_{j}}$ acting on site $i$ and $j$ within the thermal state
$\hat{\rho}_{\beta}$.
### B.2 Calculating full counting statistics with MPS
The Central object in the calculating the full probability distribution
function of an order parameter is the moment generating function
$G_{l}(\theta,t)=\text{Tr}(\hat{\rho}_{t}e^{i\theta\hat{M}_{l}})$. For pure
state, the density matrix can be written as
$\hat{\rho}_{t}=|\phi_{t}\rangle\langle\phi_{t}|$ such that
$G_{l}(\theta,t)=\langle\phi_{t}|e^{i\theta\hat{M}_{l}}|\phi_{t}\rangle=\langle\phi_{t}|\prod_{j=i}^{i+l-1}e^{i\theta\hat{s}^{x}_{j}}|\phi_{t}\rangle$.
The state $|\phi_{t}\rangle$ can be represented as a matrix product state
(MPS) [95], and the single-site operator $e^{i\theta\hat{s}^{x}_{j}}$ can be
expressed as a two-by-two matrix. Utilizing this representation, the moment
generating function can be computed by sandwiching the operators between the
matrix product states, as depicted in Figure 12. By obtaining
$G_{l}(\theta,t)$, the complete probability distribution $P_{l}(m,t)$ is
computed numerically by discretizing the Fourier integral in equation 8.
Figure 12: Computing the generating function $G_{l}(\theta,t)$ in matrix
product state representation. The site $i$ is chosen such that the subsystem
of size $l$ is in the center of the full system.
### B.3 Errors and data convergence
Figure 13: Absolute error in the energy
density,$|\epsilon_{\beta}^{\text{ED}}-\epsilon_{\beta}^{\text{TDVP}}|$, of
thermal states - (a). Absolute errors in the evolution domain wall kinks
$|\langle\hat{k}\rangle^{\text{ED}}-\langle\hat{k}\rangle^{\text{TDVP}}|$
following a quantum quench - (b),(c),(d). The numerically exact results are
calculated using equations 19 and 20 as detailed in A. TDVP results are
obtained with bond dimension $\chi=128$. The system size considered is N=14.
Figure 14: Convergence of the TDVP data for $\text{DT}(t)$ with increasing
bond dimensions, $\chi=60,90,128$, for six different post-quench parameters
considered in the main text. The black dashed line is for visual guidance.
We conducted two types of error analysis to assess the accuracy of the
numerical results. In figure 13, we assess the absolute error of the TDVP
algorithm in comparison with the numerically exact full diagonalization
results for a system with size $N=14$ and various post-quench parameters.
Figure 13, panel (a), shows the absolute error in the energy density of the
thermal states, defined as
$|\epsilon_{\beta}^{\text{ED}}-\epsilon_{\beta}^{\text{TDVP}}|$. The absolute
error remains of the order $O(10^{-5})$ or smaller across the entire
temperature range under consideration. Figures 13, (b), (c), and (d) show the
absolute error in domain wall kinks, defined as
$|\langle\hat{k}\rangle^{\text{ED}}-\langle\hat{k}\rangle^{\text{TDVP}}|$,
following a quantum quench to various post-quench parameters.
$\langle\hat{k}\rangle$ is defined in equation 13 and the computational basis
$\\{\ket{n}\\}$ is its simultaneous eigenbasis. Notably, the error rapidly
converge and is of order $O(10^{-6})$ or smaller for all the cases studied.
In Figure 14, we investigate the convergence of the TDVP data for
$\text{DT}(t)$ by computing the relative errors
$|\text{DT}^{\chi_{1}}(t)-\text{DT}^{\chi_{2}}(t)|$ for three increasing bond
dimensions. Our observations reveal that the relative error eventually
converges and consistently remains in the order $O(10^{-3})$ or smaller for
all cases. It is noteworthy that the error for $\alpha=0.0$ is several orders
of magnitude smaller than that for the other values of $\alpha$. This is
attributed to $\alpha=0.0$ being an integrable point with an extensive number
of conserved quantities, and therefore has a smaller Hilbert space to be
explored compared to non-integrable points. Furthermore, for
$\alpha=\\{1.5,1.9\\}$, the error for $h_{f}=0.3$ is approximately two orders
of magnitude smaller than that for $h_{f}=0.6$. This discrepancy arises
because the former case exhibits dynamical confinement, which effectively
suppresses the spread of correlations and constrains the total Hilbert space
that can be explored during time evolution.
## Appendix C Thermal phase transition in long range Ising model
Figure 15: Thermal phase transition of long range Ising model at four
different points in parameter space. The initial state in all cases is the
maximally mixed state at infinite temperature represented by $\rho_{\beta=0}$,
refer to 6. The color coding from red to blue signifies decreasing temperature
.
For values of $\alpha>2$, the long-range Ising model falls within the regime
of short-range interactions and does not exhibit any finite-temperature phase
transitions [89]. Extensive investigations into the critical properties of the
thermal phase transition in the quantum long-range Ising model have been
conducted using numerically exact path integral Monte Carlo methods [90]. The
thermal phase transition is qualitatively depicted in Figures 15 for specific
parameter values: $\alpha=1.5,1.9$ and $h=0.3,0.6$. As described in Section
B.1, the simulation begins with a maximally mixed state at $\beta=0$. This
initial state is characterized by a sharply peaked Gaussian distribution of
$P(m)$ centered around $m=0$, which signifies a strongly paramagnetic phase.
As the system is gradually cooled by increasing $\beta$, the distribution
gradually widens, eventually becoming nearly flat around the critical
temperature. A further reduction in temperature leads to the emergence of a
bimodal distribution of $P(m)$, which is indicative of the ferromagnetic
phase. Notably, this transition from a unimodal Gaussian distribution to a
bimodal distribution highlights the $\mathbb{Z}_{2}$ symmetry that is inherent
in the long-range Ising Hamiltonian.
## Appendix D Confinement dynamics in different regimes
Figure 16: Real time dynamics of half chain connected correlation function
$\expectationvalue{\hat{s}^{x}_{k}\hat{s}^{x}_{k+\Delta}}_{c}$after a global
quantum quench of the transverse field starting from a fully polarized initial
state. The dashed black lines is $v_{max}=2h$ line for nearest neighbor
transverse field Ising model[42].
Confinement phenomena in the long-range Ising model result from ferromagnetic
interactions extending over long distances between the interacting spins.
However, the strength of confinement varies within different regions of the
phase space [44, 45]. In this section, we present comprehensive numerical
results pertaining to the temporal spreading of correlations in the long-range
Ising chain following a sudden quench to various post-quench Hamiltonians
starting from a fully polarized initial state denoted as
$\ket{\psi_{i}}=\ket{\leftarrow,\leftarrow,\ldots,\leftarrow,\ldots,\leftarrow,\leftarrow}_{x}$.
Figure 15 illustrates the time evolution of the half chain connected
correlation function
$\expectationvalue{\hat{s}^{x}_{k}\hat{s}^{x}_{k+\Delta}}_{c}=\expectationvalue{\hat{s}^{x}_{k}\hat{s}^{x}_{k+\Delta}}-\expectationvalue{\hat{s}^{x}_{k}}\expectationvalue{\hat{s}^{x}_{k+\Delta}}$
in a chain of 200 spins, where $k$ is kept fixed at the center of the chain.
In panels (a), (b), and (c), we examine a fixed value of $\alpha=1.9$ while
varying the transverse field $h={0.3,0.6,0.8}$. Notably, panel (a) shows a
pronounced signature of confinement, which gradually diminishes as the value
of $h$ increases, as shown in panels (b) and (c). This behavior is expected
because the transverse field competes with long-range interactions and weakens
the confinement effect. In panel (d), we observe a linear light cone spreading
of the correlation with the maximum possible velocity, $v_{\text{max}}=2h$
[42].
## References
* [1] R. Islam, E. Edwards, K. Kim, S. Korenblit, C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. Joseph Wang, J. Freericks and C. Monroe, _Onset of a quantum phase transition with a trapped ion quantum simulator_ , Nature Communications 2 (2011), https://doi.org/10.1038/ncomms1374.
* [2] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong and C. Monroe, _Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator_ , Nature 551, 601 (2017), https://doi.org/10.1038/nature24654.
* [3] P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt and C. F. Roos, _Direct observation of dynamical quantum phase transitions in an interacting many-body system_ , Phys. Rev. Lett. 119, 080501 (2017), 10.1103/PhysRevLett.119.080501.
* [4] W. L. Tan, P. Becker, F. Liu, G. Pagano, K. S. Collins, A. De, L. Feng, H. B. Kaplan, A. Kyprianidis, R. Lundgren, W. Morong, S. Whitsitt _et al._ , _Domain-wall confinement and dynamics in a quantum simulator_ , Nature Physics 17, 742747 (2021), 10.1038/s41567-021-01194-3.
* [5] S. Fey and K. P. Schmidt, _Critical behavior of quantum magnets with long-range interactions in the thermodynamic limit_ , Phys. Rev. B 94, 075156 (2016), 10.1103/PhysRevB.94.075156.
* [6] J. Bohnet, B. Sawyer, J. Britton, M. Wall, A. Rey, M. Foss-Feig and J. Bollinger, _Quantum spin dynamics and entanglement generation with hundreds of trapped ions_ , Science 352, 1297 (2016), 10.1126/science.aad9958.
* [7] T. Kinoshita, T. Wenger and D. Weiss, _A quantum newton’s cradle_ , Nature 440, 900 (2006), https://doi.org/10.1038/nature04693.
* [8] T. Langen, R. Geiger and J. Schmiedmayer, _Ultracold atoms out of equilibrium_ , Annual Review of Condensed Matter Physics 6(1), 201 (2015), 10.1146/annurev-conmatphys-031214-014548.
* [9] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, A. J. Daley and H.-C. Nägerl, _Quantum quench in an atomic one-dimensional ising chain_ , Phys. Rev. Lett. 111, 053003 (2013), 10.1103/PhysRevLett.111.053003.
* [10] C. Gross and I. Bloch, _Quantum simulations with ultracold atoms in optical lattices_ , Science 357, 995 (2017), 10.1126/science.aal3837.
* [11] B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X. Gong, A. V. Gorshkov and C. Monroe, _Observation of prethermalization in long-range interacting spin chains_ , Science Advances 3 (2017), 10.1126/sciadv.1700672.
* [12] P. Calabrese, F. H. L. Essler and M. Fagotti, _Quantum quench in the transverse field ising chain: I. time evolution of order parameter correlators_ , Journal of Statistical Mechanics: Theory and Experiment 2012(07), P07016 (2012), 10.1088/1742-5468/2012/07/P07016.
* [13] P. Calabrese, F. H. L. Essler and M. Fagotti, _Quantum quench in the transverse-field ising chain_ , Phys. Rev. Lett. 106, 227203 (2011), 10.1103/PhysRevLett.106.227203.
* [14] P. Calabrese and J. Cardy, _Quantum quenches in extended systems_ , Journal of Statistical Mechanics: Theory and Experiment 2007(06), P06008 (2007), 10.1088/1742-5468/2007/06/P06008.
* [15] P. Calabrese and J. Cardy, _Evolution of entanglement entropy in one-dimensional systems_ , Journal of Statistical Mechanics: Theory and Experiment 2005(04), P04010 (2005), 10.1088/1742-5468/2005/04/P04010.
* [16] P. Calabrese and J. Cardy, _Time dependence of correlation functions following a quantum quench_ , Phys. Rev. Lett. 96, 136801 (2006), 10.1103/PhysRevLett.96.136801.
* [17] J.-S. Caux and F. H. L. Essler, _Time evolution of local observables after quenching to an integrable model_ , Phys. Rev. Lett. 110, 257203 (2013), 10.1103/PhysRevLett.110.257203.
* [18] S. Groha, F. H. L. Essler and P. Calabrese, _Full counting statistics in the transverse field Ising chain_ , SciPost Phys. 4, 043 (2018), 10.21468/SciPostPhys.4.6.043.
* [19] M. Collura, _Relaxation of the order-parameter statistics in the Ising quantum chain_ , SciPost Phys. 7, 072 (2019), 10.21468/SciPostPhys.7.6.072.
* [20] A. J. Daley, C. Kollath, U. Schollwöck and G. Vidal, _Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces_ , Journal of Statistical Mechanics: Theory and Experiment 2004(04), P04005 (2004), 10.1088/1742-5468/2004/04/P04005.
* [21] S. R. White and A. E. Feiguin, _Real-time evolution using the density matrix renormalization group_ , Phys. Rev. Lett. 93, 076401 (2004), 10.1103/PhysRevLett.93.076401.
* [22] G. Vidal, _Efficient simulation of one-dimensional quantum many-body systems_ , Phys. Rev. Lett. 93, 040502 (2004), 10.1103/PhysRevLett.93.040502.
* [23] J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde and F. Verstraete, _Time-dependent variational principle for quantum lattices_ , Phys. Rev. Lett. 107, 070601 (2011), 10.1103/PhysRevLett.107.070601.
* [24] J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken and F. Verstraete, _Unifying time evolution and optimization with matrix product states_ , Phys. Rev. B 94, 165116 (2016), 10.1103/PhysRevB.94.165116.
* [25] G. Lami, G. Carleo and M. Collura, _Matrix product states with backflow correlations_ , Phys. Rev. B 106, L081111 (2022), 10.1103/PhysRevB.106.L081111.
* [26] M. Srednicki, _Chaos and quantum thermalization_ , Phys. Rev. E 50, 888 (1994), 10.1103/PhysRevE.50.888.
* [27] J. M. Deutsch, _Quantum statistical mechanics in a closed system_ , Phys. Rev. A 43, 2046 (1991), 10.1103/PhysRevA.43.2046.
* [28] H. Kim, T. N. Ikeda and D. A. Huse, _Testing whether all eigenstates obey the eigenstate thermalization hypothesis_ , Phys. Rev. E 90, 052105 (2014), 10.1103/PhysRevE.90.052105.
* [29] R. Steinigeweg, J. Herbrych and P. Prelovšek, _Eigenstate thermalization within isolated spin-chain systems_ , Phys. Rev. E 87, 012118 (2013), 10.1103/PhysRevE.87.012118.
* [30] M. Rigol, V. Dunjko and M. Olshanii, _Thermalization and its mechanism for generic isolated quantum systems_ , Nature 452, 854858 (2008), 10.1038/nature06838.
* [31] C. Kollath, A. M. Läuchli and E. Altman, _Quench dynamics and nonequilibrium phase diagram of the bose-hubbard model_ , Phys. Rev. Lett. 98, 180601 (2007), 10.1103/PhysRevLett.98.180601.
* [32] S. R. Manmana, S. Wessel, R. M. Noack and A. Muramatsu, _Strongly correlated fermions after a quantum quench_ , Phys. Rev. Lett. 98, 210405 (2007), 10.1103/PhysRevLett.98.210405.
* [33] F. Chen, Z.-H. Sun, M. Gong, Q. Zhu, Y.-R. Zhang, Y. Wu, Y. Ye, C. Zha, S. Li, S. Guo, H. Qian, H.-L. Huang _et al._ , _Observation of strong and weak thermalization in a superconducting quantum processor_ , Phys. Rev. Lett. 127, 020602 (2021), 10.1103/PhysRevLett.127.020602.
* [34] M. C. Bañuls, J. I. Cirac and M. B. Hastings, _Strong and weak thermalization of infinite nonintegrable quantum systems_ , Phys. Rev. Lett. 106, 050405 (2011), 10.1103/PhysRevLett.106.050405.
* [35] S. Sugimoto, R. Hamazaki and M. Ueda, _Eigenstate thermalization in long-range interacting systems_ , Phys. Rev. Lett. 129, 030602 (2022), 10.1103/PhysRevLett.129.030602.
* [36] A. Russomanno, M. Fava and M. Heyl, _Quantum chaos and ensemble inequivalence of quantum long-range ising chains_ , Phys. Rev. B 104, 094309 (2021), 10.1103/PhysRevB.104.094309.
* [37] K. R. Fratus and M. Srednicki, _Eigenstate thermalization and spontaneous symmetry breaking in the one-dimensional transverse-field ising model with power-law interactions_ , 10.48550/ARXIV.1611.03992 (2016).
* [38] D. A. Abanin, E. Altman, I. Bloch and M. Serbyn, _Colloquium: Many-body localization, thermalization, and entanglement_ , Rev. Mod. Phys. 91, 021001 (2019), 10.1103/RevModPhys.91.021001.
* [39] C. Gogolin, M. P. Müller and J. Eisert, _Absence of thermalization in nonintegrable systems_ , Phys. Rev. Lett. 106, 040401 (2011), 10.1103/PhysRevLett.106.040401.
* [40] R. Nandkishore and D. A. Huse, _Many-body localization and thermalization in quantum statistical mechanics_ , Annual Review of Condensed Matter Physics 6(1), 15 (2015), 10.1146/annurev-conmatphys-031214-014726, https://doi.org/10.1146/annurev-conmatphys-031214-014726.
* [41] M. Fava, R. Fazio and A. Russomanno, _Many-body dynamical localization in the kicked bose-hubbard chain_ , Phys. Rev. B 101, 064302 (2020), 10.1103/PhysRevB.101.064302.
* [42] M. Kormos, M. Collura, G. Takács and P. Calabrese, _Real-time confinement following a quantum quench to a non-integrable model_ , Nature Physics 13, 249246 (2017), 10.1038/nphys3934.
* [43] R. J. V. Tortora, P. Calabrese and M. Collura, _Relaxation of the order-parameter statistics and dynamical inement_ , EPL (Europhysics Letters) 132(5), 50001 (2020), http://dx.doi.org/10.1209/0295-5075/132/50001.
* [44] A. Lerose, B. Žunkovič, A. Silva and A. Gambassi, _Quasilocalized excitations induced by long-range interactions in translationally invariant quantum spin chains_ , Phys. Rev. B 99, 121112 (2019), 10.1103/PhysRevB.99.121112.
* [45] F. Liu, R. Lundgren, P. Titum, G. Pagano, J. Zhang, C. Monroe and A. V. Gorshkov, _Confined quasiparticle dynamics in long-range interacting quantum spin chains_ , Phys. Rev. Lett. 122, 150601 (2019), 10.1103/PhysRevLett.122.150601.
* [46] S. Scopa, P. Calabrese and A. Bastianello, _Entanglement dynamics in confining spin chains_ , Phys. Rev. B 105, 125413 (2022), 10.1103/PhysRevB.105.125413.
* [47] S. Birnkammer, A. Bastianello and M. Knap, _Prethermalization in one-dimensional quantum many-body systems with confinement_ , Nature Communications 13 (2022), 10.1038/s41467-022-35301-6.
* [48] M. Rigobello, S. Notarnicola, G. Magnifico and S. Montangero, _Entanglement generation in $(1+1)\mathrm{D}$ qed scattering processes_, Phys. Rev. D 104, 114501 (2021), 10.1103/PhysRevD.104.114501.
* [49] R. Verdel, F. Liu, S. Whitsitt, A. V. Gorshkov and M. Heyl, _Real-time dynamics of string breaking in quantum spin chains_ , Phys. Rev. B 102, 014308 (2020), 10.1103/PhysRevB.102.014308.
* [50] R. C. Myers, M. Rozali and B. Way, _Holographic quenches in a confined phase_ , Journal of Physics A: Mathematical and Theoretical 50(49), 494002 (2017), 10.1088/1751-8121/aa927c.
* [51] A. C. Cubero and N. J. Robinson, _Lack of thermalization in (1+1)-d quantum chromodynamics at large nc_ , Journal of Statistical Mechanics: Theory and Experiment 2019(12), 123101 (2019), 10.1088/1742-5468/ab4e8d.
* [52] J. Vovrosh and J. Knolle, _Confinement and entanglement dynamics on a digital quantum computer_ , Scientific Reports 11 (2021), 10.1038/s41467-022-35301-6.
* [53] G. Lagnese, F. M. Surace, M. Kormos and P. Calabrese, _False vacuum decay in quantum spin chains_ , Phys. Rev. B 104, L201106 (2021), 10.1103/PhysRevB.104.L201106.
* [54] N. Ranabhat and M. Collura, _Dynamics of the order parameter statistics in the long range Ising model_ , SciPost Phys. 12, 126 (2022), 10.21468/SciPostPhys.12.4.126.
* [55] E. Lieb, T. Schultz and D. Mattis, _Two soluble models of an antiferromagnetic chain_ , Annals of Physics 16(3), 407 (1961), https://doi.org/10.1016/0003-4916(61)90115-4.
* [56] T. Koffel, M. Lewenstein and L. Tagliacozzo, _Entanglement entropy for the long-range ising chain in a transverse field_ , Phys. Rev. Lett. 109, 267203 (2012), 10.1103/PhysRevLett.109.267203.
* [57] M. Gabbrielli, L. Lepori and L. Pezzè, _Multipartite-entanglement tomography of a quantum simulator_ , New Journal of Physics 21(3), 033039 (2019), 10.1088/1367-2630/aafb8c.
* [58] B. Žunkovič, A. Silva and M. Fabrizio, _Dynamical phase transitions and Loschmidt echo in the infinite-range XY model_ , Phil. Trans. R. Soc. A. 374 (2016), 10.1098/rsta.2015.0160.
* [59] A. Das, K. Sengupta, D. Sen and B. K. Chakrabarti, _Infinite-range ising ferromagnet in a time-dependent transverse magnetic field: Quench and ac dynamics near the quantum critical point_ , Phys. Rev. B 74, 144423 (2006), 10.1103/PhysRevB.74.144423.
* [60] A. Dutta and J. K. Bhattacharjee, _Phase transitions in the quantum ising and rotor models with a long-range interaction_ , Phys. Rev. B 64, 184106 (2001), 10.1103/PhysRevB.64.184106.
* [61] E. Gonzalez-Lazo, M. Heyl, M. Dalmonte and A. Angelone, _Finite-temperature critical behavior of long-range quantum Ising models_ , SciPost Phys. 11, 76 (2021), 10.21468/SciPostPhys.11.4.076.
* [62] M. Foss-Feig, Z.-X. Gong, C. W. Clark and A. V. Gorshkov, _Nearly linear light cones in long-range interacting quantum systems_ , Phys. Rev. Lett. 114, 157201 (2015), 10.1103/PhysRevLett.114.157201.
* [63] A. S. Buyskikh, M. Fagotti, J. Schachenmayer, F. Essler and A. J. Daley, _Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models_ , Phys. Rev. A 93, 053620 (2016), 10.1103/PhysRevA.93.053620.
* [64] B. Žunkovič, M. Heyl, M. Knap and A. Silva, _Dynamical quantum phase transitions in spin chains with long-range interactions: Merging different concepts of nonequilibrium criticality_ , Phys. Rev. Lett. 120, 130601 (2018), 10.1103/PhysRevLett.120.130601.
* [65] J. C. Halimeh and V. Zauner-Stauber, _Dynamical phase diagram of quantum spin chains with long-range interactions_ , Phys. Rev. B 96, 134427 (2017), 10.1103/PhysRevB.96.134427.
* [66] R. Khasseh, A. Russomanno, M. Schmitt, M. Heyl and R. Fazio, _Discrete truncated wigner approach to dynamical phase transitions in ising models after a quantum quench_ , Phys. Rev. B 102, 014303 (2020), 10.1103/PhysRevB.102.014303.
* [67] J. C. Halimeh, V. Zauner-Stauber, I. P. McCulloch, I. de Vega, U. Schollwöck and M. Kastner, _Prethermalization and persistent order in the absence of a thermal phase transition_ , Phys. Rev. B 95, 024302 (2017), 10.1103/PhysRevB.95.024302.
* [68] I. Homrighausen, N. O. Abeling, V. Zauner-Stauber and J. C. Halimeh, _Anomalous dynamical phase in quantum spin chains with long-range interactions_ , Phys. Rev. B 96, 104436 (2017), 10.1103/PhysRevB.96.104436.
* [69] J. Lang, B. Frank and J. C. Halimeh, _Concurrence of dynamical phase transitions at finite temperature in the fully connected transverse-field ising model_ , Phys. Rev. B 97, 174401 (2018), 10.1103/PhysRevB.97.174401.
* [70] F. H. L. Essler and M. Fagotti, _Quench dynamics and relaxation in isolated integrable quantum spin chains_ , Journal of Statistical Mechanics: Theory and Experiment 2016(6), 064002 (2016).
* [71] L. S. Levitov, H. Lee and G. B. Lesovik, _Electron counting statistics and coherent states of electric current_ , Journal of Mathematical Physics 37(4845) (1996), https://doi.org/10.1063/1.531672.
* [72] M. Esposito, U. Harbola and S. Mukamel, _Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems_ , Rev. Mod. Phys. 81, 1665 (2009), 10.1103/RevModPhys.81.1665.
* [73] M. Campisi, P. Hänggi and P. Talkner, _Colloquium: Quantum fluctuation relations: Foundations and applications_ , Rev. Mod. Phys. 83, 771 (2011), 10.1103/RevModPhys.83.771.
* [74] P. Calabrese, M. Collura, G. D. Giulio and S. Murciano, _Full counting statistics in the gapped xxz spin chain_ , Europhysics Letters 129(6), 60007 (2020), 10.1209/0295-5075/129/60007.
* [75] M. Collura, F. H. L. Essler and S. Groha, _Full counting statistics in the spin-1/2 heisenberg xxz chain_ , Journal of Physics A: Mathematical and Theoretical 50(41), 414002 (2017), 10.1088/1751-8121/aa87dd.
* [76] M. Collura and F. H. L. Essler, _How order melts after quantum quenches_ , Phys. Rev. B 101, 041110 (2020), 10.1103/PhysRevB.101.041110.
* [77] D. J. Luitz, N. Laflorencie and F. Alet, _Many-body localization edge in the random-field heisenberg chain_ , Phys. Rev. B 91, 081103 (2015), 10.1103/PhysRevB.91.081103.
* [78] R. Singh, J. H. Bardarson and F. Pollmann, _Signatures of the many-body localization transition in the dynamics of entanglement and bipartite fluctuations_ , New Journal of Physics 18(2), 023046 (2016), 10.1088/1367-2630/18/2/023046.
* [79] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler and J. Schmiedmayer, _Relaxation and prethermalization in an isolated quantum system_ , Science 337(6100), 1318 (2012), 10.1126/science.1224953.
* [80] Y. Tang, W. Kao, K.-Y. Li, S. Seo, K. Mallayya, M. Rigol, S. Gopalakrishnan and B. L. Lev, _Thermalization near integrability in a dipolar quantum newton’s cradle_ , Phys. Rev. X 8, 021030 (2018), 10.1103/PhysRevX.8.021030.
* [81] B. Blaß and H. Rieger, _Test of quantum thermalization in the two-dimensional transverse-field ising model_ , Scientific Reports 6 (2016), 10.1038/srep38185.
* [82] S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck and C. Hubig, _Time-evolution methods for matrix-product states_ , Annals of Physics 411, 167998 (2019), 10.1016/j.aop.2019.167998.
* [83] A. A. Zvyagin, _Dynamical quantum phase transitions_ (2017), 1701.08851.
* [84] T. Mori, T. N. Ikeda, E. Kaminishi and M. Ueda, _Thermalization and prethermalization in isolated quantum systems: a theoretical overview_ , Journal of Physics B: Atomic, Molecular and Optical Physics 51(11), 112001 (2018), 10.1088/1361-6455/aabcdf.
* [85] N. Defenu, T. Enss and J. C. Halimeh, _Dynamical criticality and domain-wall coupling in long-range hamiltonians_ , Phys. Rev. B 100, 014434 (2019), 10.1103/PhysRevB.100.014434.
* [86] P. Titum, J. T. Iosue, J. R. Garrison, A. V. Gorshkov and Z.-X. Gong, _Probing ground-state phase transitions through quench dynamics_ , Phys. Rev. Lett. 123, 115701 (2019), 10.1103/PhysRevLett.123.115701.
* [87] P. Titum and M. F. Maghrebi, _Nonequilibrium criticality in quench dynamics of long-range spin models_ , Phys. Rev. Lett. 125, 040602 (2020), 10.1103/PhysRevLett.125.040602.
* [88] J. C. Halimeh, D. Trapin, M. Van Damme and M. Heyl, _Local measures of dynamical quantum phase transitions_ , Phys. Rev. B 104, 075130 (2021), 10.1103/PhysRevB.104.075130.
* [89] A. Dutta and J. K. Bhattacharjee, _Phase transitions in the quantum ising and rotor models with a long-range interaction_ , Phys. Rev. B 64, 184106 (2001), 10.1103/PhysRevB.64.184106.
* [90] E. G. Lazo, M. Heyl, M. Dalmonte and A. Angelone, _Finite-temperature critical behavior of long-range quantum ising models_ , SciPost Phys. 11, 076 (2021), 10.21468/SciPostPhys.11.4.076.
* [91] N. Ranabhat, A. Santini, E. Tirrito and M. Collura, _Dynamical deconfinement transition driven by density of excitations_ (2023), 2310.02320.
* [92] F. Verstraete, J. J. García-Ripoll and J. I. Cirac, _Matrix product density operators: Simulation of finite-temperature and dissipative systems_ , Phys. Rev. Lett. 93, 207204 (2004), 10.1103/PhysRevLett.93.207204.
* [93] A. H. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert and S. Montangero, _Positive tensor network approach for simulating open quantum many-body systems_ , Phys. Rev. Lett. 116, 237201 (2016), 10.1103/PhysRevLett.116.237201.
* [94] D. Jaschke, S. Montangero and L. D. Carr, _One-dimensional many-body entangled open quantum systems with tensor network methods_ , Quantum Science and Technology 4(1), 013001 (2018), 10.1088/2058-9565/aae724.
* [95] U. Schollwöck, _The density-matrix renormalization group in the age of matrix product states_ , Annals of Physics 326(1), 96 (2011), https://doi.org/10.1016/j.aop.2010.09.012, January 2011 Special Issue.
|
# On the geometry of uniform meandric systems
Jacopo Borga Stanford University Ewain Gwynne University of Chicago Minjae
Park University of Chicago
###### Abstract
A meandric system of size $n$ is the set of loops formed from two arc diagrams
(non-crossing perfect matchings) on $\\{1,\dots,2n\\}$, one drawn above the
real line and the other below the real line. A uniform random meandric system
can be viewed as a random planar map decorated by a Hamiltonian path
(corresponding to the real line) and a collection of loops (formed by the
arcs). Based on physics heuristics and numerical evidence, we conjecture that
the scaling limit of this decorated random planar map is given by an
independent triple consisting of a Liouville quantum gravity (LQG) sphere with
parameter $\gamma=\sqrt{2}$, a Schramm-Loewner evolution (SLE) curve with
parameter $\kappa=8$, and a conformal loop ensemble (CLE) with parameter
$\kappa=6$.
We prove several rigorous results which are consistent with this conjecture.
In particular, a uniform meandric system admits loops of nearly macroscopic
graph-distance diameter with high probability. Furthermore, a.s., the uniform
infinite meandric system with boundary has no infinite path. But, a.s., its
boundary-modified version has a unique infinite path whose scaling limit is
conjectured to be chordal SLE6.
###### Contents
1. 1 Introduction
1. 1.1 Meandric systems
2. 1.2 Conjectures for scaling limit and largest loop exponent
3. 1.3 Macroscopic loops in finite meandric systems
4. 1.4 The uniform infinite meandric system (UIMS)
5. 1.5 The uniform infinite half-plane meandric system (UIHPMS)
2. 2 Percolation interpretation
1. 2.1 Non-crossing perfect matchings and non-crossing integer partitions
2. 2.2 Meandric systems as boundaries of clusters of open edges
3. 2.3 Meandric systems and critical percolation on random planar maps
4. 2.4 Box crossings
5. 2.5 The lack of positive association and two open problems
3. 3 Existence of macroscopic loops in the UIMS
4. 4 Bounding distances via the mated-CRT map
1. 4.1 The mated-CRT map
2. 4.2 Proof of lower bounds for graph distances
3. 4.3 Proof of Proposition 1.4, Theorem 1.5, and Theorem 1.9
5. 5 Estimate for the mated-CRT map via SLE and LQG
1. 5.1 SLE/LQG description of the mated-CRT map
2. 5.2 Proof of lower bounds for mated-CRT graph distances
6. 6 Proofs for the UIHPMS
1. 6.1 Equivalence of the definitions of the UIHPMS
2. 6.2 Aperiodicity via good boundary points
3. 6.3 No semi-infinite paths started from the boundary
4. 6.4 Infinitely many paths separating the origin from $\infty$
5. 6.5 Proofs of Theorem 1.12 and Proposition 1.14
7. 7 Justification for Conjectures 1.2 and 1.3
1. 7.1 SLE8 on $\sqrt{2}$-LQG via mating of trees
2. 7.2 Physics argument
3. 7.3 Predictions for exponents via KPZ
4. 7.4 Simulations
Acknowledgments. We thank two anonymous referees for helpful comments on an
earlier version of this article. We thank Ahmed Bou-Rabee, Valentin Féray,
Gady Kozma, Ron Peled, and Xin Sun for helpful discussions. E.G. was partially
supported by a Clay research fellowship. M.P. was partially supported by an
NSF grant DMS 2153742.
## 1 Introduction
### 1.1 Meandric systems
Throughout this paper, we identify $\mathbbm{R}$ (resp. $\mathbbm{Z}$) with
the set $\mathbbm{R}\times\\{0\\}$ (resp. $\mathbbm{Z}\times\\{0\\}$).
###### Definition 1.1.
A meandric system of size $n\in\mathbbm{N}$ is a configuration
${\mathfrak{S}}_{n}$ consisting of a finite collection of simple loops in
$\mathbbm{R}^{2}$ with the following properties:
* •
No two loops of ${\mathfrak{S}}_{n}$ intersect each other.
* •
Each loop of ${\mathfrak{S}}_{n}$ intersects the real line $\mathbbm{R}$ at
least twice, and does not intersect $\mathbbm{R}$ without crossing it.
* •
The total number of intersection points between the loops of
${\mathfrak{S}}_{n}$ and $\mathbbm{R}$ is equal to $2n$.
We view such configurations as being defined modulo orientation-preserving
homeomorphisms from $\mathbbm{R}^{2}$ to $\mathbbm{R}^{2}$ which take
$\mathbbm{R}$ to $\mathbbm{R}$.
Figure 1: A meandric system of size $n=6$. The two corresponding arc diagrams
are shown in red and blue. The graphs of the simple walk excursions
corresponding to these arc diagrams are shown in black. The walk excursions
are translated (in time) by a $1/2$-factor to the right so that each step of
the walks is centered with the corresponding integer point in the real line.
We made this choice (here and in all our graphical representations of meandric
systems) to highlight better the correspondence between walks and arcs. See
(1.1) and the surrounding text for further explanation.
See Figure 1 for an illustration of a meandric system. If ${\mathfrak{S}}_{n}$
is a meandric system of size $n$, then by applying a homeomorphism, we can
always arrange the set of intersection points of arcs in ${\mathfrak{S}}_{n}$
with $\mathbbm{R}$ so that it is equal to $[1,2n]\cap\mathbbm{Z}$. We will
make this assumption throughout the paper. There have been several recent
works in probability and combinatorics which studied meandric systems (see,
e.g., [CKST19, FN19, GNP20, Kar20, FT22]). These works were in part motivated
by the study of decorated planar maps and by the connection between meandric
systems, non-crossing partitions, and meanders, as we discuss just below.
Additional motivations come from the fact that random meandric systems are
equivalent to a certain percolation-type model on a random planar map (Section
2) and to a version of the fully packed $O(0\times 1)$ loop model on a random
planar map (Section 7.2).
A meander of size $n$ is a meandric system of size $n$ with a single loop.
Each of the loops in a meandric system can be viewed as a meander by
forgetting the other loops. However, a typical loop in a uniformly sampled
meandric system of size $n$ is not the same as a uniformly sampled meander
[FT22, Section 4]. The study of meanders dates back to at least the work of
Poincaré in 1912 [Poi12] and is connected to a huge number of different areas
of math and physics. See [La 03, Zvo21] for surveys of results on meanders.
Most features of meanders are notoriously difficult to analyze mathematically.
For example, determining the $n\rightarrow\infty$ asymptotics of the total
number of meanders of size $n$ is a long-standing open problem (but see
[DFGG00] for a conjecture).
Meandric systems are significantly more tractable than meanders. The main
reason for this is that meandric systems are in bijection with pairs of arc
diagrams (non-crossing perfect matchings). An arc diagram of size
$n\in\mathbbm{N}$ is a collection of arcs in the upper half-plane
$\mathbbm{R}\times[0,\infty)$, each of which joint two points in
$[1,2n]\cap\mathbbm{Z}$, subject to the condition that no two of the arcs
cross. If ${\mathfrak{S}}_{n}$ is a meandric system of size $n$, then the
segments of loops in ${\mathfrak{S}}_{n}$ above (resp. below) the real line
form an arc diagram. Conversely, any two arc diagrams of size $n$ give rise to
a meandric system by drawing one above and one below the real line, and
considering the set of loops that they form. It is well known that arc
diagrams of size $n$ are counted by the Catalan number
$\operatorname{Cat}_{n}=\frac{1}{n+1}\binom{2n}{n}$. Consequently, the number
of meandric systems of size $n$ is $\operatorname{Cat}_{n}^{2}$.
Furthermore, arc diagrams of size $n$ are in bijection with $2n$-step simple
walks on $\mathbbm{Z}_{\geq 0}$ from $0$ to $0$, often called (non-negative)
simple walk excursions or Dyck paths. If
$\mathcal{X}:[0,2n]\cap\mathbbm{Z}\rightarrow\mathbbm{Z}$ is a $2n$-step
simple walk excursion, then the corresponding arc diagram is defined as
follows. Two points $x_{1},x_{2}\in[1,2n]\cap\mathbbm{Z}$ with $x_{1}<x_{2}$
are joined by an arc if and only if
$\mathcal{X}_{x_{1}-1}=\mathcal{X}_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]\cap\mathbbm{Z}}\mathcal{X}_{y};$
(1.1)
see Figure 1. So, one can sample a uniform random meandric system of size $n$
by sampling two independent simple random walk excursions with $2n$ steps,
drawing one of the two corresponding arc diagrams above the real line and the
other below the real line, then looking at the loops formed by the union of
the two arc diagrams.
Let $\mathfrak{S}_{n}$ be a uniform meandric system of size $n$. There are a
number of natural questions about the large-scale geometry of
$\mathfrak{S}_{n}$, e.g., the following:
1. 1.
How many loops does $\mathfrak{S}_{n}$ typically have?
2. 2.
What is the size of the largest loop of $\mathfrak{S}_{n}$, in terms of the
number of intersection points with $\mathbbm{R}$? What about for other notions
of size, e.g., graph-distance diameter in the 4-regular graph whose vertices
are the intersection points of loops with $\mathbbm{R}$, and whose edges are
the segments of loops and the segments of $\mathbbm{R}$ between these
vertices?
3. 3.
Is there typically a single loop of $\mathfrak{S}_{n}$ which is much larger
(in some sense) than the other loops, or are there multiple large loops of
comparable size?
4. 4.
Is there some sort of scaling limit of $\mathfrak{S}_{n}$ as
$n\rightarrow\infty$?
Due to the bijection between meandric systems and pairs of $2n$-step simple
walk excursions, questions of the above type, in principle, can be reduced to
questions about simple random walks on $\mathbbm{Z}$. However, the encoding of
the meandric system loops in terms of the pair of walk excursions is
complicated, so the answers to the above questions are far from trivial.
Question 1 was largely solved by Féray and Thévenin [FT22], who showed that
there is a constant $c>0$ (expressed in terms of a sum over meanders) such
that the number of loops in $\mathfrak{S}_{n}$ is asymptotic to $cn$ as
$n\rightarrow\infty$. Regarding Question 2, Kargin [Kar20] showed that the
number of intersection points with $\mathbbm{R}$ of the largest loop is at
least constant times $\log n$, and presented some numerical simulations which
suggested that this quantity in fact behaves like $n^{\alpha}$ for
$\alpha\approx 4/5$. Question 3 is closely related to the question of whether
there exists a so-called “infinite noodle”, i.e., an infinite path in the
infinite-volume limit of a uniform meandric system. It was shown in [CKST19]
that there is at most one such path. The existence is still open, but it is
conjectured in [CKST19] that such a path does not exist. See Section 1.4 for
further discussion.
In this paper, we present conjectures for the answers to each of Questions 2,
3, and 4 (see Conjectures 1.2 and 1.3). In particular, if $k\in\mathbbm{N}$ is
fixed, then as $n\rightarrow\infty$ the number of intersection points with
$\mathbbm{R}$ of the $k$th largest loop should grow like $n^{\alpha}$ where
$\alpha=\frac{1}{2}(3-\sqrt{2})\approx 0.7929$. Moreover, the scaling limit of
$\mathfrak{S}_{n}$ should be described by a $\sqrt{2}$-Liouville quantum
gravity sphere, a Schramm-Loewner evolution curve with parameter $\kappa=8$,
and a conformal loop ensemble with parameter $\kappa=6$.
We also prove several rigorous results in the direction of the above
questions. We show that a uniform meandric system admits loops of nearly
macroscopic graph-distance diameter (Theorem 1.5). This leads to an explicit
power-law lower bound for the size of the largest loop in a uniform meandric
system (Corollary 1.6). We also construct the _uniform infinite half-plane
meandric system (UIHPMS)_ and show that it does not admit any infinite paths
of arcs (Theorem 1.12). But, a minor modification of the UIHPMS admits a
unique infinite path of arcs which should converge to SLE6 (Proposition 1.14).
Most of our proofs use only elementary discrete arguments, but we need to use
the theory of Liouville quantum gravity at one step in the proof, namely in
Section 5.
### 1.2 Conjectures for scaling limit and largest loop exponent
We want to state a conjecture for the scaling limit of the uniformly sampled
meandric system $\mathfrak{S}_{n}$ as $n\rightarrow\infty$. To formulate this
conjecture, we let $\mathcal{M}_{n}$ be the planar map whose vertices are the
$2n$ intersection points of the loops $\ell$ in $\mathfrak{S}_{n}$ with the
real line, whose edges are the segments of the loops or the line $\mathbbm{R}$
between these intersection points (we consider the two infinite rays of
$\mathbbm{R}$ as being an edge from the leftmost to the rightmost intersection
point), and whose faces are the connected components of
$\mathbbm{R}^{2}\setminus\mathopen{}\mathclose{{}\left(\bigcup_{\ell\in\mathfrak{S}_{n}}\ell\cup\mathbbm{R}}\right)$.
The planar map $\mathcal{M}_{n}$ is equipped with a Hamiltonian path
$P_{n}:[1,2n]\cap\mathbbm{Z}\rightarrow\\{\text{vertices of
$\mathcal{M}_{n}$}\\}$ which traverses the vertices of $\mathcal{M}_{n}$ and
the segments of $\mathbbm{R}$ between these vertices in left-right numerical
order. The planar map is also equipped with a collection of loops $\Gamma_{n}$
(simple cycles in $\mathcal{M}_{n}$) corresponding to the loops in
$\mathfrak{S}_{n}$. See Figure 2 for a simulation of
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$.
To talk about convergence, we can, e.g., view
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$ as a metric measure space (equipped with
the graph metric and the counting measure on vertices) decorated by a path and
a collection of loops. We can then ask whether this decorated metric measure
space has a scaling limit with respect to the generalization of the Gromov-
Hausdorff topology for metric measure spaces decorated by curves and/or loops
[GM17, GHS21]. Alternatively, we could embed
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$ into $\mathbbm{C}$ in some manner (e.g.,
Tutte embedding [GMS21] as in Figure 2 or circle packing [Ste03]) and ask
whether the resulting metric, measure, curve, and collection of loops in
$\mathbbm{C}$ have a joint scaling limit in law.
We now state a conjecture for the scaling limit of
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$. The limiting object is described in
terms of several random objects whose definitions we will not write down
explicitly.
* •
The Liouville quantum gravity (LQG) sphere with parameter $\gamma\in(0,2]$ is
a random fractal surface with the topology of the sphere first introduced in
[DMS21, DKRV16]. A $\gamma$-LQG sphere can be described by a random metric and
a random measure on the Riemann sphere $\mathbbm{C}\cup\\{\infty\\}$. LQG
spheres (and other types of LQG surfaces) describe the scaling limits of
various types of random planar maps. See [Gwy20, She22, BP] for expository
articles on LQG.
* •
Schramm-Loewner evolution (SLEκ) with parameter $\kappa>0$ is a random fractal
curve introduced in [Sch00]. The curve is simple for $\kappa\in(0,4]$, has
self-intersections but not self-crossings for $\kappa\in(4,8)$, and is space-
filling for $\kappa\geq 8$.
* •
The conformal loop ensemble (CLEκ) with parameter $\kappa\in(8/3,8)$ is a
random countable collection of loops which do not cross themselves or each
other and which locally look like SLEκ curves [She09]. We allow our CLE loops
to be nested (i.e., we do not restrict attention to the outermost loops). CLE
on the whole plane was first defined in [MWW16] for $\kappa\in(4,8)$ and in
[KW16] for $\kappa\in(8/3,4]$.
Figure 2: Simulation of a uniform meandric system with boundary of size
$n=10^{6}$ (see Section 7.4 for a precise definition and for the details of
simulations). The left picture shows the corresponding arc diagrams. The right
picture shows the associated planar map $\mathcal{M}_{n}$, embedded in the
disk via the Tutte embedding [GMS21], together with some of the loops in
$\Gamma_{n}$. The largest 300 loops in $\Gamma_{n}$ (in terms of number of
vertices) are each shown in color, as indicated by the color bar. Smaller
loops and edges between consecutive vertices of $\mathbbm{R}$ are shown in
gray. Note that the distribution of colors in the arc diagram picture is
rather chaotic – this is consistent with the fact that the meandric system
loops are a complicated functional of the arc diagrams. According to
Conjecture 1.2, the embedded planar map $\mathcal{M}_{n}$ together with the
path $P_{n}$ and the loops in $\Gamma_{n}$ should converge to $\sqrt{2}$-LQG
decorated by SLE8 and CLE6.
###### Conjecture 1.2.
Let $(\mathcal{M}_{n},P_{n},\Gamma_{n})$ be the random planar map decorated by
a Hamiltonian path and a collection of loops associated to a uniform meandric
system of size $n$, as described just above. Then
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$ converges under an appropriate scaling
limit to an independent triple consisting of a $\sqrt{2}$-LQG sphere, a whole-
plane SLE8 from $\infty$ to $\infty$, and a whole-plane CLE6.
In the setting of Conjecture 1.2, the metric and measure on $\mathbbm{C}$
corresponding to the $\sqrt{2}$-LQG sphere, the SLE8 curve (viewed modulo time
parametrization), and the CLE6 are independent. At first glance, this may be
surprising since $(\mathcal{M}_{n},P_{n})$ and $(\mathcal{M}_{n},\Gamma_{n})$
each determine each other. However, we expect that the function which goes
from $(\mathcal{M}_{n},P_{n})$ to $(\mathcal{M}_{n},\Gamma_{n})$ depends only
on microscopic features of $(\mathcal{M}_{n},P_{n})$ which are not seen in the
scaling limit, and the same is true for the function which goes in the
opposite direction. This independence is numerically justified by Figure 18
(Right), using the discussion in Section 7.3.
An equivalent formulation of the conjecture is that
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$ should be in the same universality class
as a uniform triple consisting of a planar map decorated by a spanning tree
(represented by its associated discrete Peano curve) and a critical Bernoulli
percolation configuration (represented by the loops which describe the
interfaces between open and closed clusters).
Conjecture 1.2 is based on a combination of rigorous results, physics
heuristics, and numerical simulations. We will explain the reasoning leading
to the conjecture in Section 7. A similar scaling limit conjecture for
meanders, rather than meandric systems, is stated as [BGS22, Conjecture 1.3]
(building on [DFGG00]). In the meander case, one has
$\gamma=\sqrt{\frac{1}{3}\mathopen{}\mathclose{{}\left(17-\sqrt{145}}\right)}$
instead of $\gamma=\sqrt{2}$ and there are two SLE8 curves instead of an SLE8
and a CLE6. The heuristic justification for Conjecture 1.2 is similar to the
argument leading to this meander conjecture. In fact, as we will explain in
Section 7.2, Conjecture 1.2 may be viewed as a special case of a conjecture
for the $O(n\times m)$ loop model on a random planar map from [DFGG00]. See
also [DDGG22] for additional related predictions.
As we will explain in Section 7.3, Conjecture 1.2 together with the KPZ
formula [KPZ88] leads to the following conjectural answer to Questions 2 and 3
above.
###### Conjecture 1.3.
Let $\mathfrak{S}_{n}$ be a uniform meandric system of size $n$. For each
fixed $k\in\mathbbm{N}$, it holds with probability tending to 1 as
$n\rightarrow\infty$ that
$\\#\\{\text{vertices of $k$th largest loop of
$\mathfrak{S}_{n}$}\\}=n^{\alpha+o(1)},\quad\text{where}\quad\alpha=\frac{1}{2}\mathopen{}\mathclose{{}\left(3-\sqrt{2}}\right)\approx
0.7929.$ (1.2)
Conjecture 1.3 for $k=1$ is consistent with the numerical study of [Kar20,
Section 3], which suggested that the size of the largest loop in
$\mathfrak{S}_{n}$ is of order $n^{\alpha}$ for $\alpha$ close to $4/5$. We
have also run some numerical simulations of our own which are consistent with
Conjecture 1.3 for $k=1,2,3,4,5$. See Section 7.4 for more details.
### 1.3 Macroscopic loops in finite meandric systems
Let $d=d_{\sqrt{2}}$ be the Hausdorff dimension of the $\sqrt{2}$-Liouville
quantum gravity metric space (this quantity is well-defined thanks to [GP22,
Corollary 1.7]). A reader not familiar with Liouville quantum gravity can
simply think of $d$ as a certain constant. The number $d$ is not known
explicitly, but fairly good rigorous upper and lower bounds are available. In
particular, it was shown in [GP19, Corollary 2.5], building on [DG18, Theorem
1.2], that
$3.5504\approx 2(9+3\sqrt{5}-\sqrt{3})(4-\sqrt{15})\leq
d\leq\frac{2}{3}(3+\sqrt{6})\approx 3.6330.$ (1.3)
The following proposition can be proven using previously known techniques for
bounding distances in random planar maps [GHS20, GP21].
###### Proposition 1.4.
Let $\mathcal{M}_{n}$ be the planar map associated with a uniform meandric
system of size $n$. For each $\zeta\in(0,1)$, it holds except on an event of
probability decaying faster than any negative power of $n$ that the graph-
distance diameter of $\mathcal{M}_{n}$ is between $n^{1/d-\zeta}$ and
$n^{1/d+\zeta}$.
Proposition 1.4 is proven via a coupling with a so-called mated-CRT map, a
certain type of random planar map which is directly connected to Liouville
quantum gravity. See Section 4.3 for details. It is possible to prove
Proposition 1.4 via exactly the same argument as in [GP21, Theorem 1.9], which
gives an analogous statement for spanning-tree decorated random planar maps.
But, we will give a more self-contained proof in Section 4.3.
Our first main result tells us that a uniform meandric system admits loops
whose graph-distance diameter is nearly of the same order as the graph-
distance diameter of $\mathcal{M}_{n}$, in the following sense, c.f.
Proposition 1.4.
###### Theorem 1.5.
Let $\mathcal{M}_{n}$ be the planar map associated with a uniform meandric
system of size $n$ and let $\Gamma_{n}$ be the associated collection of loops
on $\mathcal{M}_{n}$. For each $\zeta\in(0,1)$, it holds except on an event of
probability decaying faster than any negative power of $n$ that the following
is true. There is a loop in $\Gamma_{n}$ which has $\mathcal{M}_{n}$-graph-
distance diameter at least $n^{1/d-\zeta}$.
The proof of Theorem 1.5 is based on a combination of two results. The first
input is a purely discrete argument, based on a parity trick, which shows that
the infinite-volume analog of $(\mathcal{M}_{n},\Gamma_{n})$ admits with
positive probability loops which are “macroscopic” in a certain sense (Theorem
3.2). The second input is a lower-bound for certain graph distances in
$\mathcal{M}_{n}$ which is proven via a combination of discrete arguments and
SLE/LQG techniques (Proposition 4.1). The continuum part of the argument,
given in Section 5, is short and simple, but very far from elementary since it
relies on both the mating of trees theorem [DMS21] and the existence of the
LQG metric [DDDF20, GM21b].
The reason why we have an error of order $n^{\zeta}$ in Proposition 1.4 and
Theorem 1.5 is that we are only able to estimate graph distances in
$\mathcal{M}_{n}$ up to $o(1)$ errors in the exponent. If we had up-to-
constants bounds for graph distances in $\mathcal{M}_{n}$, then our arguments
would show that with high probability, there exist loops in $\Gamma_{n}$ whose
$\mathcal{M}_{n}$-graph-distance diameter is comparable, up to constants, to
the graph-distance diameter of $\mathcal{M}_{n}$. Hence, Theorem 1.5 suggests
that the scaling limit of $(\mathcal{M}_{n},\Gamma_{n})$ should be non-
degenerate, in the sense that the loops of $\Gamma_{n}$ do not collapse to
points. This is consistent with Conjecture 1.2.
Theorem 1.5 is similar in spirit to the recent work [DCGPS21], which proves
the existence of macroscopic loops for the critical $O(n)$ loop model on the
hexagonal lattice when $1\leq n\leq 2$ (see also [CGHP20] for a similar result
for a different range of parameter values). However, our proof of Theorem 1.5
is very different from the arguments in [DCGPS21, CGHP20]. Part of the reason
for this is that we are not aware of any positive association (FKG) inequality
in our setting (see Question 2.2), in addition to the fundamental difference
that we are working on a random lattice.
Loops in $\Gamma_{n}$ are connected subsets of $\mathcal{M}_{n}$, so a loop of
graph-distance diameter at least $n^{1/d-\zeta}$ must hit at least
$n^{1/d-\zeta}$ vertices of $\mathcal{M}_{n}$. We therefore have the following
corollary of Theorem 1.5.
###### Corollary 1.6.
Let $\mathfrak{S}_{n}$ be a uniform meandric system of size $n$. For each
$\zeta\in(0,1)$, it holds except on an event of probability decaying faster
than any negative power of $n$ that there is a loop in $\mathfrak{S}_{n}$
which crosses the real line at least $n^{1/d-\zeta}$ times.
We note that the bounds for $d$ from (1.3) show that
$0.2753\approx\frac{1}{2}(3-\sqrt{6})\leq\frac{1}{d}\leq\frac{1}{72-38\sqrt{3}+30\sqrt{5}-18\sqrt{15}}\approx
0.2817.$ (1.4)
Corollary 1.6 gives a power-law lower bound for the number of vertices of the
largest loop in a typical meandric system. To our knowledge, the best lower
bound for this quantity prior to our work is [Kar20, Theorem 3.4], which shows
that the number of vertices in the largest loop is typically at least a
constant times $\log n$.
If Conjecture 1.3 is correct, then the lower bound of Corollary 1.6 is far
from optimal. Nevertheless, it would require substantial new ideas to get any
exponent larger than $1/d$ for the number of vertices in the largest loop of
$\mathfrak{S}_{n}$. Indeed, to do this one would need to show that the largest
loop in $\mathfrak{S}_{n}$ is much longer than an $\mathcal{M}_{n}$-graph
distance geodesic.
### 1.4 The uniform infinite meandric system (UIMS)
The uniform infinite meandric system (UIMS) is the local limit (in the sense
of Benjamini-Schramm [BS01]) of a uniform meandric system of size $n$ based at
a uniform vertex. It is shown in [FT22, Proposition 5] that this local limit
exists (in a quenched sense; see [FT22, Section 1.2] for further explanations)
and is the same as the object studied in [CKST19].
Figure 3: A subset of the uniform infinite meandric system (UIMS) together
with the graphs of the corresponding increments of the encoding walks
$\mathcal{L}$ and $\mathcal{R}$. Each horizontal line below the graph of
$\mathcal{L}$ (resp. $\mathcal{R}$) corresponds to an arc above (resp. below)
the real line. See (1.5) and the surrounding text for further explanation.
We now define the UIMS, following [CKST19]. Let111 The reason why we write
$\mathcal{L}$ and $\mathcal{R}$ for the walks is that $\mathcal{L}$ (resp.
$\mathcal{R}$) describes the arcs which lie to the left (resp. right) of the
real line when we traverse the real line from left to right. This notation is
chosen to be consistent with the mating of trees literature [DMS21, GHS23,
GHS20]. $\mathcal{L}$ and $\mathcal{R}$ be independent two-sided simple random
walks on $\mathbbm{Z}$ with $\mathcal{L}_{0}=\mathcal{R}_{0}=0$. See Figure 3
for an illustration. We define two infinite arc diagrams (non-crossing perfect
matchings of $\mathbbm{Z}$), one above $\mathbbm{R}$ and one below
$\mathbbm{R}$, as follows. For $x_{1},x_{2}\in\mathbbm{Z}$ with $x_{1}<x_{2}$,
we draw an arc above (resp. below) the real line joining $x_{1}$ and $x_{2}$
if and only if
$\mathcal{L}_{x_{1}-1}=\mathcal{L}_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]\cap\mathbbm{Z}}\mathcal{L}_{y}\qquad\text{(resp.}\>\mathcal{R}_{x_{1}-1}=\mathcal{R}_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]\cap\mathbbm{Z}}\mathcal{R}_{y}\>\text{)};$
(1.5)
c.f. (1.1). It is easy to see that there is exactly one arc above (resp.
below) the real line incident to each $x\in\mathbbm{Z}$, and that the arcs
above (resp. below) the real line can be taken to be non-intersecting. We then
define the UIMS to be the set of the loops and bi-infinite paths formed by the
union of the arcs in the above two arc diagrams.
It is also possible to recover the walks $\mathcal{L}$ and $\mathcal{R}$ from
the arc diagrams. Indeed, for each $x\in\mathbbm{Z}$, the increment
$\mathcal{L}_{x}-\mathcal{L}_{x-1}$ is $+1$ (resp. $-1$) if and only if the
arc above $\mathbbm{R}$ which is incident to $x$ has its other endpoint
greater than (resp. less than) $x$. A similar statement holds for
$\mathcal{R}$.
The UIMS is easier to work with than a finite uniform meandric system since
there is no conditioning on the walks. Most of our proofs will be in the
setting of the UIMS.
It is shown in [CKST19, Theorem 1] that either a.s. the collection of loops
(and possibly bi-infinite paths) in the UIMS has a unique infinite path, or
a.s. it has no infinite paths. This infinite path, if it exists, is called the
infinite noodle in [CKST19]. It is not known rigorously which of these two
possibilities holds. But, the following is conjectured in [CKST19].
###### Conjecture 1.7 ([CKST19]).
Almost surely, the UIMS has no infinite path.
Exactly as in the case of a finite meandric system, we can associate to the
UIMS an infinite planar map $\mathcal{M}$ decorated by a bi-infinite
Hamiltonian path $P$ and a collection of loops (and possibly bi-infinite
paths) $\Gamma$. Namely, the vertex set of $\mathcal{M}$ is $\mathbbm{Z}$; the
edges of $\mathcal{M}$ are the arcs above and below the real line together
with the segments $[x-1,x]$ for $x\in\mathbbm{Z}$; the path $P$ traverses the
vertex set $\mathbbm{Z}$ in left-right numerical order; and $\Gamma$ is the
set of loops (and possibly bi-infinite paths) formed by the two arc diagrams
as above.
We will now state the infinite-volume analog of Conjecture 1.2. The
$\sqrt{2}$-quantum cone is the most natural $\sqrt{2}$-LQG surface with the
topology of the plane. It arises as the local limit of the $\sqrt{2}$-quantum
sphere based at a point sampled from its associated area measure [DMS21,
Proposition 4.13(ii)].
###### Conjecture 1.8.
Let $(\mathcal{M},P,\Gamma)$ be the infinite random planar map decorated by a
bi-infinite Hamiltonian path and a collection of loops (and possibly bi-
infinite paths) associated to the UIMS, as described just above. Then
$(\mathcal{M},P,\Gamma)$ converges under an appropriate scaling limit to an
independent triple consisting of a $\sqrt{2}$-LQG cone, a whole-plane SLE8
from $\infty$ to $\infty$, and a whole-plane CLE6.
Conjecture 1.8 is consistent with Conjecture 1.7, since all of the loops in a
whole-plane CLE6 are compact sets [MWW16]. Our main result concerning the UIMS
gives a polynomial lower tail bound for the graph-distance diameter (and hence
also the number of vertices) of the loop containing the origin. It turns out
to be an easy consequence of one of the intermediate results in the proof of
Theorem 1.5 (see, in particular, Proposition 4.4).
###### Theorem 1.9.
Let $(\mathcal{M},\Gamma)$ be the planar map and collection of loops
associated with the UIMS and let $\ell_{0}$ be the loop (or infinite path) in
$\Gamma$ which passes through $0\in\mathbbm{Z}$. Then with $d$ as in (1.3),
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\text{$\mathcal{M}$-graph-
distance diameter of $\ell_{0}$}}\right)\geq k}\right]\geq
k^{-(d-1)-o(1)},\quad\text{as $k\rightarrow\infty$}.$ (1.6)
Unlike in Theorem 1.5, we expect that the exponent in Theorem 1.9 is not
optimal. Rather, we have the following conjecture.
###### Conjecture 1.10.
Let $\ell_{0}$ be as in Theorem 1.9. Then with
$\alpha=\frac{1}{2}(3-\sqrt{2})$ as in Conjecture 1.3,
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\text{$\mathcal{M}$-graph-
distance diameter of $\ell_{0}$}}\right)\geq k}\right]\geq
k^{-d(1-\alpha)-o(1)},\quad\text{as $k\rightarrow\infty$},$ (1.7)
and
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\text{number
of vertices in $\ell_{0}$}}\right)\geq k}\right]\geq
k^{-(1/\alpha-1)-o(1)},\quad\text{as $k\rightarrow\infty$}.$ (1.8)
If Conjecture 1.3 were true, then, at least heuristically, the same arguments
as in the proof of Theorem 1.9 would lead to a proof of Conjecture 1.10.
### 1.5 The uniform infinite half-plane meandric system (UIHPMS)
The uniform infinite half-plane meandric system (UIHPMS) is a natural variant
of the UIMS which is half-plane like in the sense that it has a bi-infinite
sequence of distinguished “boundary vertices” (points in $\mathbbm{Z}$ which
are not disconnected from $\infty$ below the real line by any path in the
UIHPMS), but “most” vertices are not boundary vertices. We will now give two
equivalent definitions of this object, see Figure 4.
_By cutting:_ Start with a sample of the UIMS on $\mathbb{Z}$, represented by
a pair of arc diagrams as in Section 1.4. We assume that the arcs of the two
arc diagrams are drawn in $\mathbb{R}^{2}$ in such a way that the arcs do not
cross each other and each arc with endpoints $x<y$ is contained in the
vertical strip $[x,y]\times\mathbbm{R}$ (the arc diagrams in all of the
figures in this paper have this property). We cut all of the arcs below the
real line which intersect the vertical ray $\\{1/2\\}\times(-\infty,0]$,
leaving two unmatched ends corresponding to each arc. Then, we rewire
successive pairs of unmatched ends to form new arcs. That is, we enumerate the
unmatched ends from left to right as $\\{e_{j}\\}_{j\in\mathbbm{Z}}$, with
positive indices corresponding to ends to the right of
$\\{1/2\\}\times(-\infty,0]$ and non-positive indices corresponding to ends to
the left. Then, we link up $e_{2j-1}$ and $e_{2j}$ for each $j\in\mathbbm{Z}$.
See Figure 4 (Left). The resulting infinite collection of the loops (and
possibly bi-infinite paths) is the UIHPMS.
A point $x\in\mathbbm{Z}$ is called a boundary point if it can be connected to
the ray $\\{1/2\\}\times(-\infty,0]$ by some continuous path without crossing
any arc or the real line. Each endpoint of arcs which were cut is a boundary
point, but there are also other boundary points. See Figure 4 (Left) for an
illustration. It is clear from the random walk description of the meandric
system (1.5) that each time $x\in\mathbbm{Z}$ at which the walk $\mathcal{R}$
attains a running infimum when run forward or backward from time 0 gives rise
to a boundary point of the UIHPMS (but not every boundary point arises in this
way). Hence, there are infinitely many boundary points. Denote by
$J_{k}\in\mathbbm{Z}$ (resp. $J_{-k+1}\in\mathbbm{Z}$) the $k$th positive
(resp. non-positive) boundary point. Note that in the UIHPMS we always have
that $J_{0}=0$ and $J_{1}=1$ and that $J_{0}$ and $J_{1}$ are not joined by an
arc.
_By random walks:_ We also have a random walk description for the UIHPMS
similar to that of the UIMS. The only difference is that we use a _reflected_
simple random walk instead of an ordinary simple random walk for the arc
diagram below the real line. Let $\mathcal{L}$ and $\mathcal{R}$ be
independent two-sided simple random walks on $\mathbb{Z}$ with
$\mathcal{L}_{0}=\mathcal{R}_{0}=0$. For $x_{1},x_{2}\in\mathbbm{Z}$ with
$x_{1}<x_{2}$, we draw an arc above (resp. below) the real line joining
$x_{1}$ and $x_{2}$ if and only if
$\mathcal{L}_{x_{1}-1}=\mathcal{L}_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]\cap\mathbbm{Z}}\mathcal{L}_{y}\quad\text{(resp.}\>|\mathcal{R}|_{x_{1}-1}=|\mathcal{R}|_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]\cap\mathbbm{Z}}|\mathcal{R}|_{y}\>\text{)};$
(1.9)
c.f. (1.5).
A point $x\in\mathbbm{Z}$ is called a boundary point if $|\mathcal{R}|$
crosses $1/2$ in between time $x-1$ and $x$, i.e.,
$\mathcal{R}_{x-1},\mathcal{R}_{x}\in\\{-1,0,1\\}$. As before, denote by
$J_{k}\in\mathbbm{Z}$ (resp. $J_{-k+1}\in\mathbbm{Z}$) the $k$th positive
(resp. non-positive) boundary point. See Figure 4 (Right) for an illustration.
Since $\mathcal{R}_{0}=0$, we always have $J_{0}=0$, $J_{1}=1$, and $J_{0}$
not matched with $J_{1}$ as in the previous definition.
Figure 4: Left: The UIHPMS constructed by cutting. The dash-dotted arcs
straddling $0$ below the real line in the UIMS (defined in Section 1.4) are
cut by the gray ray $\\{1/2\\}\times(-\infty,0]$, and these arcs are rewired
successively to result in the UIHPMS. Boundary points are labeled as
$\\{J_{k}\\}_{k\in\mathbbm{Z}}$ and are colored green together with their
connecting arcs. Note that the points $J_{-2}$ and $J_{-3}$ are boundary
points but are not endpoints of any cut arcs. Right: The UIHPMS constructed by
random walks. We show a subset of the UIHPMS together with the graphs of the
corresponding increments of the encoding walks $\mathcal{L}$ and
$|\mathcal{R}|$. Each horizontal line below the graph of $\mathcal{L}$ (resp.
|$\mathcal{R}$|) corresponds to an arc above (resp. below) the real line. The
boundary points and arcs incident to them are colored green.
The following lemma will be proven in Section 6.1 using a discrete version of
Lévy’s theorem [Sim83] which relates (roughly speaking) the zero set and the
running minimum times of a random walk.
###### Lemma 1.11.
The law of the UIHPMS is the same under the above two definitions.
Just as in the case of other types of meandric systems, we can view the UIHPMS
as a planar map $\mathcal{M}^{\prime}$ decorated by a collection of loops
(plus possibly bi-infinite paths) $\Gamma^{\prime}$ and a Hamiltonian path
$P^{\prime}$. The law of the UIHPMS is invariant under even translations along
the boundary. That is, if $\\{J_{k}\\}_{k\in\mathbbm{Z}}$ is the set of
boundary points as above, then
$(\mathcal{M}^{\prime}-J_{2k},P^{\prime}-J_{2k},\Gamma^{\prime}-J_{2k})\overset{d}{=}(\mathcal{M}^{\prime},P^{\prime},\Gamma^{\prime}),\quad\forall
k\in\mathbbm{Z}.$ (1.10)
This is clear from the random walk description.
As in Conjecture 1.7, we can also ask if there exists an infinite path in the
UIHPMS. The answer should be no if the scaling limit conjecture is true.
Unlike in the whole-plane setting, we are able to prove this in the half-plane
setting.
###### Theorem 1.12.
Almost surely, the set $\Gamma^{\prime}$ for the UIHPMS has no infinite path.
Theorem 1.12 will be proven in Section 6 via a purely discrete argument. See
the beginning of Section 6 for an outline. The proof is completely independent
of the proofs of our theorems for finite meandric systems and for the UIMS
(stated in Sections 1.3 and 1.4). So, the proofs can be read in any order.
In Conjecture 1.8, we discussed how the UIMS is conjecturally related to the
$\sqrt{2}$-quantum cone. Another natural $\sqrt{2}$-LQG surface – with the
half-plane topology instead of whole-plane topology – is the
$\sqrt{2}$-quantum wedge. See [DMS21, Section 1.4] or [GHS23, Section 3.4] for
a rigorous definition. In the continuum setting, cutting a $\sqrt{2}$-quantum
cone by an independent whole-plane SLE2 from 0 to $\infty$ yields the
$\sqrt{2}$-quantum wedge [DMS21, Theorem 1.5]. This is a continuum analog of
the above cutting description of the UIHPMS. That is, in the scaling limit,
the gray ray in Figure 4 should converge to some simple fractal curve from 0
to $\infty$. Cutting the previous loop-decorated whole-plane along this curve
yields the half-plane (or whole-plane with a slit) topology. The points
located immediately on the left or right side of the curve become boundary
points. Thus, Conjecture 1.8 has a natural half-plane version as follows.
###### Conjecture 1.13.
Let $(\mathcal{M}^{\prime},P^{\prime},\Gamma^{\prime})$ be the infinite random
planar map decorated by a bi-infinite Hamiltonian path and a collection of
loops associated to the UIHPMS, as described just above. Then
$(\mathcal{M}^{\prime},P^{\prime},\Gamma^{\prime})$ converges under an
appropriate scaling limit to an independent triple consisting of a
$\sqrt{2}$-LQG wedge, a space-filling SLE8 from $\infty$ to $\infty$ in the
half-plane, and a CLE6 in the half-plane.
In the setting of Conjecture 1.13, the boundary vertices of
$\mathcal{M}^{\prime}$ should correspond to boundary points for the
$\sqrt{2}$-LQG wedge. For example, if
$(\mathcal{M}^{\prime},P^{\prime},\Gamma^{\prime})$ is embedded into the half-
plane appropriately (e.g., via some version of Tutte embedding or circle
packing), in such a way that the boundary vertices of $\mathcal{M}^{\prime}$
are mapped to points on the real line, then one should have the convergence of
the embedded objects toward the $\sqrt{2}$-LQG wedge together with SLE8 and
CLE6.
Starting from a CLE6 in the half-plane, one can construct a chordal SLE6 curve
from 0 to $\infty$ by concatenating certain arcs of boundary-touching CLE6
loops (see the proof of [She09, Theorem 5.4]). The analogous path can be also
constructed in the UIHPMS by leaving exactly one boundary point unmatched as
follows.
We define the pointed infinite half-plane meandric system (PIHPMS) by rewiring
lower arcs between non-positive boundary points in the UIHPMS as follows.
Start with the UIHPMS and recall that $J_{0}=0$ is not matched with $J_{1}=1$.
Remove the arcs below the real line which join $J_{2k-1}$ and $J_{2k}$ for
each $k\leq 0$. Then, add new arcs joining $J_{2k-2}$ and $J_{2k-1}$ for each
$k\leq 0$, in such a way that the new arcs do not cross any other arcs. Note
that now $J_{0}=0$ is unmatched. This new configuration is defined to be the
PIHPMS and we think of it as pointed at $J_{0}=0$. One can also describe the
PIHPMS in terms of random walks by replacing $|\mathcal{R}|_{\cdot}$ in (1.9)
with the modified walk
$\mathcal{R}^{\circ}_{\cdot}:=|\mathcal{R}|_{\cdot}-\textbf{1}_{\\{\cdot\geq
0\\}}.$ (1.11)
As there is no $x<0$ such that
$\mathcal{R}^{\circ}_{x-1}=\mathcal{R}^{\circ}_{0}=-1$, we have a special
boundary point $J_{0}=0$ which is not incident to any arc below the real line
as desired. See Figure 5 (Top Left) for an illustration.
As per usual, we view the PIHPMS as a planar map $\mathcal{M}^{\circ}$
decorated by a Hamiltonian path $P^{\circ}$ and a collection of loops (plus
possibly infinite paths) $\Gamma^{\circ}$. Note that the path started from
$J_{0}=0$ must be an infinite path because no arc below the real line is
incident to the origin, so the path cannot come back to its starting point to
form a loop. We prove that this is the unique infinite path.
###### Proposition 1.14.
Consider the PIHPMS as defined above. Almost surely, there is a unique
infinite path $\gamma^{\circ}\in\Gamma^{\circ}$. This path starts at $J_{0}=0$
and hits infinitely many points in each of $\\{J_{k}\\}_{k<0}$ and
$\\{J_{k}\\}_{k>0}$.
The proof of Proposition 1.14 is given in Section 6, and uses many of the same
ideas as in the proof of Theorem 1.12. Another conjecture follows accordingly.
Figure 5: Top left: The random walks encoding a sample of the PIHPMS obtained
from the previous example in Figure 4 (Right). The yellow strip in the PIHPMS
highlights the special boundary point $J_{0}=0$ which is not incident to any
arc below the real line as a consequence of the fact that there is no $x<0$
such that $\mathcal{R}^{\circ}_{x-1}=\mathcal{R}^{\circ}_{0}=-1$; see the
yellow strip in the walk. Bottom left/right: A simulation of a finite-volume
version (i.e., with the topology of the disk) of the PIHPMS with two marked
points, of size $n=9\cdot 10^{6}$. The bottom left picture shows the arc
diagrams and the right picture shows the associated planar map, drawn in the
disk via the Tutte embedding [GMS21]. The red curve is conjectured to converge
to a chordal SLE6 between the two marked boundary points. See Section 7.4 for
the details of simulations.
###### Conjecture 1.15.
Let $(\mathcal{M}^{\circ},P^{\circ},\Gamma^{\circ})$ be the infinite random
planar map decorated by a bi-infinite Hamiltonian path and a collection of
loops (plus possibly infinite paths) associated to the PIHPMS, as described
just above. Also, let $\gamma^{\circ}$ be the path started from 0 in
$\Gamma^{\circ}$ (which is a.s. the unique infinite path by Proposition 1.14).
Then
$(\mathcal{M}^{\circ},P^{\circ},\gamma^{\circ},\Gamma^{\circ}\setminus\gamma^{\circ})$
converges under an appropriate scaling limit to a $\sqrt{2}$-LQG wedge, a
space-filling SLE8 from $\infty$ to $\infty$ in the half-plane, a chordal SLE6
from $0$ to $\infty$ in the half-plane, and a CLE6 in the complement of the
SLE6 curve (i.e., a union of conditionally independent CLE6s in the
complementary connected components).
###### Remark 1.16.
One can also consider finite or infinite meandric systems constructed from a
pair of random walks which are correlated rather than independent, i.e., the
step distribution of the pair of walks assigns different weights to steps in
$\\{(-1,-1),(1,1)\\}$ and steps in $\\{(-1,1),(1,-1)\\}$. We do not have a
simple combinatorial description for the random meandric systems obtained in
this way, so we do not emphasize them in this paper. Based on mating of trees
theory [DMS21, GHS23], it is natural to conjecture that the associated
decorated planar map converges to $\gamma$-LQG decorated by a space-filling
SLE${}_{16/\gamma^{2}}$ and an independent CLE6, where $\gamma\in(0,2)$ is
chosen so that the correlation of the encoding walks is
$-\cos(4\pi/\gamma^{2})$. We have run some numerical simulations (similar to
Section 7.4) which are consistent with this. Our proofs of Theorems 1.5 and
1.9 work verbatim in the case of correlated walks, with $d$ replaced by the
dimension of $\gamma$-LQG. However, the upper bound for the diameter of
$\mathcal{M}_{n}$ in Proposition 1.4 uses estimates from [GP21] which are only
proven for $\gamma=\sqrt{2}$. Most of our proofs for the UIHPMS do not work in
the case of correlated walks since we cannot apply the discrete Lévy theorem
(6.2) to one coordinate of the walk independently from the other.
## 2 Percolation interpretation
The main goal of this section is to explain how meandric systems can be viewed
as a model of (critical) percolation on planar maps. Throughout the section we
assume that the reader is familiar with the basic terminology and results in
percolation theory. This section is included only for intuition and
motivation: it is not needed for the proofs of our main results.
### 2.1 Non-crossing perfect matchings and non-crossing integer partitions
We start by recalling a classical bijection between non-crossing perfect
matchings and non-crossing integer partitions, see for instance [GNP20,
Section 3].
We write a partition $\pi$ of a finite set $A\subset\mathbbm{R}$ as
$\pi=\\{V_{1},\dots,V_{k}\\}$, where $V_{1},\dots,V_{k}$ are called the blocks
of $\pi$ and are non-empty, pairwise disjoint sets, with $\cup_{i}V_{i}=A$. We
say that a partition $\pi$ is non-crossing when it is not possible to find two
distinct blocks $V,W\in\pi$ and numbers $a<b<c<d$ in $A$ such that $a,c\in V$
and $b,d\in W$. Equivalently, $\pi$ is non-crossing if it can be plotted as a
planar graph with the vertices in $A$ arranged on the real line, so that the
blocks of $\pi$ are the connected components of the planar graph drawn in the
upper-half plane; see the orange planar graph in Figure 6 for an example.
Figure 6: We plot in orange the planar graph associated with the non-crossing
partition
$\pi=\\{\\{2-1/2,10-1/2,14-1/2,16-1/2\\},\\{4-1/2,6-1/2,8-1/2\\},\\{12-1/2\\}\\}$
of the points $2\mathbbm{Z}\cap[1,16]-1/2$ and in black the corresponding
planar graph of the non-crossing perfect matching $\psi(\pi)$ of the points
$[1,16]\cap\mathbbm{Z}$. Note that the black arcs are chosen so that they
“block” new potential orange edges.
We can now describe the aforementioned bijection. Given a non-crossing
partition $\pi=\\{V_{1},\dots,V_{k}\\}$ of the points
$2\mathbbm{Z}\cap[1,2n]-1/2$, we consider the unique non-crossing perfect
matching $\psi(\pi)$ of $[1,2n]\cap\mathbbm{Z}$ such that for every pair of
points $z<w\in 2\mathbbm{Z}\cap[1,2n]-1/2$ not contained in the same block
$V_{i}$, there exists a pair of matched points $x,y\in[1,2n]\cap\mathbbm{Z}$
in $\psi(\pi)$ such that either $x<z<y<w$ or $z<x<w<y$. See Figure 6 for a
graphical interpretation.
Conversely, given a non-crossing perfect matching $m$ of
$[1,2n]\cap\mathbbm{Z}$, we construct a non-crossing partition $\psi^{-1}(m)$
of the points $2\mathbbm{Z}\cap[1,2n]-1/2$ as follows: for every pair of
points $z,w\in 2\mathbbm{Z}\cap[1,2n]-1/2$ we say that $z\sim_{m}w$ if there
is no pair of matched points $x,y$ in $m$ with either $x<z<y<w$ or $z<x<w<y$.
We then set the equivalent classes of $(2\mathbbm{Z}\cap[1,2n]-1/2,\sim_{m})$
to be the blocks of $\psi^{-1}(m)$.
Note that the fact that $\psi$ and $\psi^{-1}$ are inverse maps is immediate.
### 2.2 Meandric systems as boundaries of clusters of open edges
Given a non-crossing perfect matching $m$ of $[1,2n]\cap\mathbbm{Z}$, one can
also consider (in the previous description of the inverse map $\psi^{-1}$) the
points $2\mathbbm{Z}\cap[0,2n]+1/2$ (see the green points in Figure 7). Then,
as before, one determines a second non-crossing integer partition
$\widetilde{\psi}^{-1}(m)$ of the points $2\mathbbm{Z}\cap[0,2n]+1/2$ (see for
instance the green planar graph on the upper-half plane in Figure 7).
We further notice that any element of the triple
$(m,\psi^{-1}(m),\widetilde{\psi}^{-1}(m))$ determines the two other elements
of the triple.
Figure 7: A meandric system $m=(m_{1},m_{2})$ plotted in black together with
its two pairs of non-crossing integer partitions
$(\psi^{-1}(m_{1}),\psi^{-1}(m_{2}))$ and
$(\widetilde{\psi}^{-1}(m_{1}),\widetilde{\psi}^{-1}(m_{2}))$ in orange and
green, respectively.
Consider now a meandric system of size $n$, which is a pair $(m_{1},m_{2})$ of
non-crossing perfect matchings of $[1,2n]\cap\mathbbm{Z}$; the first above and
the second below the real line. Using the bijections $\psi$ and
$\widetilde{\psi}$ described above, one can associate to the meandric system
$(m_{1},m_{2})$ two pairs of non-crossing integer partitions of
$2\mathbbm{Z}\cap[1,2n]-1/2$ and $2\mathbbm{Z}\cap[0,2n]+1/2$: the first pair
(resp. the second pair) is obtained applying the map $\psi^{-1}$ (resp. the
map $\widetilde{\psi}^{-1}$) to $m_{1}$ and $m_{2}$, obtaining the orange
(resp. the green) planar graph in Figure 7. From now on we will refer to these
two pairs of non-crossing integer partitions as the orange and the green
planar graphs (associated to a meandric system).
We now present a new way of thinking about a meandric system of size $n$
closer to percolation models (in what follows we explain what are the
classical quantities in percolation models in our setting; c.f. Figure 7).
_Main lattice:_ This is the planar map whose vertices are the union of the
green and orange vertices, and whose edges are the union of the green and
orange edges and the edges corresponding to the real line (between consecutive
vertices of different colors).
_Dual lattice:_ This is the planar map that we previously associated with a
meandric system, i.e. the map whose vertices are the black vertices, and whose
edges are the black edges and the edges corresponding to the real line
(between consecutive black vertices).
_Open edges in the main lattice:_ These are the orange edges.
_Closed edges in the main lattice:_ These are the green edges.
_Boundary of clusters of open edges:_ These are the black loops in the
meandric system.
Note that if one considers a uniform meandric system (or equivalently a
uniform pair of non-crossing integer partitions), then one obtains (through
the interpretation above) a model of percolation on random planar maps.
Analogously, if one considers the UIMS or the UIHPMS (Sections 1.4 and 1.5),
then one obtains (through the obvious adaptations of the constructions
above222We do not give the details of these constructions since they are not
needed to continue our discussion.) some natural model of percolation on
infinite random planar maps.
One of the main difficulties in the study of these models is that the
randomness for the environment (i.e. the planar map) and the randomness for
the percolation are strongly coupled. See also the beginning of Section 2.5
for a second major difficulty.
### 2.3 Meandric systems and critical percolation on random planar maps
Our interpretation of meandric systems as a percolation model described in the
previous section, gives also some natural new interpretations of our results.
In particular, Theorem 1.5, which states that a uniform meandric system admits
loops whose graph distance diameter is nearly of the same order as the graph
distance diameter of the associated planar map, implies that our percolation
model admits macroscopic clusters. Hence, it does not behave like subcritical
Bernoulli percolation on a fixed lattice.
Due to Corollary 3.3 below, a.s. the UIMS has a bi-infinite path if and only
if our above percolation model has an infinite open cluster. Theorem 1.12
implies that there is no infinite cluster for the percolation model on the
UIHPMS. Due to the construction of the UIHPMS by cutting the UIMS along an
infinite ray (Section 1.5), this is roughly analogous to the statement that
there is no infinite cluster for, say, Bernoulli percolation on
$\mathbbm{Z}^{2}\setminus(\\{0\\}\times\mathbbm{Z}_{+})$. In particular, our
percolation model does not behave like supercritical Bernoulli percolation on
a fixed lattice.
By combining the preceding two paragraphs, we see that our percolation model
is in some sense “critical” (see also Proposition 2.1 for further evidence).
Determining whether there is a bi-infinite path in the UIMS (Conjecture 1.7)
is therefore analogous to determining whether there is percolation at
criticality.
We note that the interpretation of meandric systems as a critical percolation
model is also consistent with Conjectures 1.2, 1.8 and 1.13, which state that
the scaling limit of the loops of a meandric system should be the conformal
loop ensemble with parameter $\kappa=6$. Indeed, the latter is also
conjectured to be the scaling limit of the cluster interfaces for critical
percolation models on various deterministic discrete lattices. This conjecture
has been proved in the case of critical site percolation on the triangular
lattice [Smi01, CN06, CN08].
### 2.4 Box crossings
A classical result for critical Bernoulli ($p=1/2$) bond percolation on
$\mathbb{Z}^{2}$ is that given a box
$B_{n}=\mathbbm{Z}^{2}\cap\\{[0,n]\times[0,n+1]\\}$, then the probability that
there exists a path of open edges connecting the top side of $B_{n}$ to the
bottom side of $B_{n}$ is always equal to $1/2$, independently of the size $n$
of the box.
We prove the analogous result in the context of the UIMS. We start by defining
a natural notion of box for the UIMS. Given the UIMS together with its orange
and green planar graphs as in the left-hand side of Figure 8, we define the
box of size $n$ rooted at $x\in\mathbb{Z}+\frac{1}{2}$ to be the collection of
(black/orange/green) vertices in $[x,x+2n-1]$ and (black/orange/green) edges
with at least one endpoint in $(x,x+2n-1)$. We denote such box by $B_{n}(x)$.
See Figure 8 for an example. Note that with this definition any box of size
$n$ contains the same number of green and orange vertices.
Figure 8: Left: A portion of the UIMS together with its orange and green
planar graphs. A box of size $2$ rooted at $1/2$ is highlighted in yellow. The
bottom-green edge between $-3.5$ and $2.5$ and the top-green edge between
$2.5$ and $4.5$ form a bottom-to-top green crossing of the box $B_{2}(1/2)$.
Note that $B_{2}(1/2)$ contains also a top-to-bottom green crossing of the
box. Right: An example of another box. Note that both the top and bottom green
edges from $0.5$ to $2.5$ crosses both the top-left boundary and the bottom-
left boundary of the box $B_{2}(1/2)$, according to our definition below.
Given a box $B_{n}(x)$, we say that an edge crosses the top-left (resp.
bottom-left, top-right, bottom-right) boundary of the box, if that edge
touches the vertical line $\\{x\\}\times[0,\infty)$ (resp.
$\\{x\\}\times(-\infty,0]$, $\\{x+2n-1\\}\times[0,\infty)$,
$\\{x+2n-1\\}\times(-\infty,0]$), where we highlight that the point $(x,0)$
(resp. $(x+2n-1,0)$) is included in the line. See Figure 8 for an example. We
then say that there exists a top-to-bottom orange crossing (resp. bottom-to-
top orange crossing) of $B_{n}(x)$ if there exists a path of all orange edges
such that the first edge in the path crosses the top-left (resp. bottom-left)
boundary of $B_{n}(x)$, the last edge in the path crosses the bottom-right
(resp. top-right) boundary of $B_{n}(x)$, and all the other edges have both
extremities in $B_{n}(x)$. We similarly define top-to-bottom and bottom-to-top
green crossings.
We can now state our analogous result for “the probability that there exists a
top-to-bottom crossing of open edges in a box is 1/2”.
###### Proposition 2.1.
Consider the UIMS together with its green and orange planar graphs. For all
$x\in\mathbb{Z}+\frac{1}{2}$ and all $n>0$, the probability that there exists
a top-to-bottom orange crossing of the box $B_{n}(x)$ is 1/2.
###### Proof.
Fix $x\in\mathbb{Z}+\frac{1}{2}$ and $n>0$. Note that by the construction of
the orange and green planar graphs (recall from Section 2.2 how the orange
arcs determine the green arcs), exactly one of the following two events holds:
either
$O=\\{\text{There exists a top-to-bottom orange crossing of the box
}B_{n}(x)\\}$
or
$G=\\{\text{There exists a bottom-to-top green crossing of the box
}B_{n}(x)\\}.$
Moreover, since $B_{n}(x)$ contains the same number of green and orange
vertices and the events $O$ and $G$ are symmetric, then they must must have
the same probability. Hence $\mathbb{P}(O)=\mathbb{P}(G)=1/2$. ∎
### 2.5 The lack of positive association and two open problems
Proposition 2.1 is a further evidence that meandric systems behave like a
critical percolation model on a planar map and also suggests that it might be
possible to prove analogs of other standard results in the theory of
percolation models. Unfortunately, as already mentioned in Section 1.3, we are
not aware of any positive association (FKG) inequality in our setting and for
the moment all the notions of monotonicity that we tried do not satisfy FKG.
This discussion naturally leads to the following open problem.
###### Question 2.2.
Is there a natural notion of monotonicity in infinite meandric configurations
such that some type of positive association (FKG) inequality holds (for
instance) for our notion of top-to-bottom orange crossings of proper boxes?
Another interesting question, which might be relevant to Conjecture 1.2 (or
one of its variants), is to compute the following loop crossing probability.
Recall the definition of the box $B_{n}(x)$ given at the beginning of Section
2.4. We define a top-to-bottom loop crossing of $B_{n}(x)$ in the same way as
we defined a top-to-bottom orange crossing of $B_{n}(x)$, where in the
definition orange edges are replaced by black edges.
###### Question 2.3.
Consider the UIMS. What is the asymptotic probability as $n\to\infty$ that
there exists a top-to-bottom loop crossing of $B_{n}(0)$?
Conjecture 1.8 gives a natural candidate for the answer to the question above,
which we now explain (assuming a certain familiarity with SLEs). Building on
Conjecture 1.8, the points in $B_{n}(0)$ are expected to converge under an
appropriate scaling limit to the points visited by the whole-plane SLE8 $\eta$
between time $0$ and $1$.333Here we assume that $\eta$ is parametrized by
$\sqrt{2}$-LQG area with respect to an independent unit area quantum cone, so
that $\eta:\mathbbm{R}\rightarrow\mathbbm{R}$.
The set $\eta((-\infty,0])\cap\eta([0,\infty))$ is the union of two disjoint
SLE2-type curves [DMS21, Footnote 4], one called the left-boundary of
$\eta((-\infty,0])$ and the other one called the right-boundary of
$\eta((-\infty,0])$. The same holds for
$\eta((-\infty,1])\cap\eta([1,\infty))$. Moreover, the left-boundaries (resp.
right-boundaries) of $\eta((-\infty,0])$ and $\eta((-\infty,1])$ a.s. merge
into each other. As a consequence, the set $\eta([0,1])$ forms a topological
rectangle. We refer to the piece of the left boundary of $\eta((-\infty,1])$
not in common with the left boundary of $\eta((-\infty,0])$ (resp. the piece
of the right boundary of $\eta((-\infty,0])$ not in common with the left
boundary of $\eta((-\infty,1])$) as the top-left boundary (resp. bottom-right
boundary) of $\eta([0,1])$.
We expect that the probability in Question 2.3 converges to the probability
that there exists a continuous portion of a loop in the whole-plane CLE6 of
Conjecture 1.8 crossing $\eta([0,1])$ from its top-left boundary to its
bottom-right boundary, without leaving $\eta([0,1])$.
We highlight that the latter crossing probability is not known explicitly, and
that computing it is an interesting problem in its own right.
## 3 Existence of macroscopic loops in the UIMS
Throughout this section, we let $(\mathcal{M},\Gamma)$ be the infinite planar
map (with vertex set $\mathbbm{Z}$) and collection of loops (and possibly bi-
infinite paths) associated with the UIMS, as defined in Section 1.4.
###### Definition 3.1.
Let $A,B\subset\mathbbm{R}^{2}$ and let $\ell$ be a loop or a path
$\mathbbm{R}^{2}\cup\\{\infty\\}$. We say that $\ell$ disconnects $A$ from $B$
if every path from $A$ to $B$ hits $\ell$.
The loops (and possibly bi-infinite paths) of the UIMS can be viewed as loops
in $\mathbbm{R}^{2}$ which hit $\mathbbm{R}$ only at integer points and which
are defined modulo orientation-preserving homeomorphisms from
$\mathbbm{R}^{2}$ to $\mathbbm{R}^{2}$ which fix $\mathbbm{R}$. If
$A,B\subset\mathbbm{R}\cup\\{\infty\\}$, such a homeomorphism does not alter
whether a loop disconnects $A$ from $B$. Hence it makes sense to talk about
loops in $\Gamma$ disconnecting subsets of $\mathbbm{R}\cup\\{\infty\\}$.
The goal of this section is to prove the following theorem, which can roughly
speaking be thought of as saying that $\Gamma$ has a positive chance to admit
macroscopic loops or paths at all scales. This theorem will eventually be
combined with estimates for distances in the infinite planar map $\mathcal{M}$
(which we prove in Sections 4 and 5.1) to prove Theorems 1.5 and 1.9, see
Section 4.3.
###### Theorem 3.2.
For each sufficiently large $n\in\mathbbm{N}$, it holds for each $C>1$ that
with probability at least $5-2\sqrt{6}\approx 0.1010$, at least one of the
following two conditions is satisfied:
1. A.
There is a loop in $\Gamma$ which disconnects $[-n,n]$ from $\infty$ and which
hits a point of $[n,(2C+1)n+3]\cap\mathbbm{Z}$.
2. B.
There is a loop or an infinite path in $\Gamma$ which hits a point in each of
$[-n,n]\cap\mathbbm{Z}$ and $\mathbbm{Z}\setminus[-Cn,Cn]$.
When we apply Theorem 3.2, we will take $C$ large but fixed independently of
$n$. For such a choice of $C$, a loop satisfying either of the two conditions
of Theorem 3.2 should be thought of as being “macroscopic”, in the sense that
it should give rise to a non-trivial loop when we send $n\rightarrow\infty$
and pass to an appropriate scaling limit (c.f. Conjecture 1.8).
Theorem 3.2 implies the following independently interesting corollary. We note
that a similar statement for the $O(n)$ loop model on the hexagonal lattice,
for a certain range of parameter values, is proven in [CGHP20, Theorem 1].
###### Corollary 3.3.
Exactly one of the following two conditions occurs with probability one, and
the other occurs with probability zero:
1. A′.
There is an infinite path in $\Gamma$.
2. B′.
For each $x\in\mathbbm{Z}$, there are infinitely many loops in $\Gamma$ which
disconnect $x$ from $\infty$.
It is possible to prove Corollary 3.3 via a more direct argument which does
not use Theorem 3.2, but we will deduce it from Theorem 3.2 for convenience.
###### Proof of Corollary 3.3.
By the zero-one law for translation invariant events, the event that $\Gamma$
has an infinite path has probability zero or one (see also [CKST19, Theorem
1]). Loops and infinite paths in $\Gamma$ cannot cross each other, so for any
$x\in\mathbbm{Z}$ which is hit by an infinite path in $\Gamma$, there is no
loop in $\Gamma$ which disconnects $x$ from $\infty$. Therefore, to prove the
corollary it suffices to assume that $\Gamma$ a.s. has no infinite paths and
show that this implies that for each $x\in\mathbbm{Z}$, a.s. there are
infinitely many loops in $\Gamma$ which disconnect $x$ from $\infty$. The
definition (1.5) of $(\mathcal{M},\Gamma)$ implies that translating by a fixed
$x\in\mathbbm{Z}$ preserves the law of $(\mathcal{M},\Gamma)$. So, we can
restrict attention to the case $x=0$.
Let $n\in\mathbbm{N}$ be large enough so that the conclusion of Theorem 3.2 is
satisfied. Since we are assuming that $\Gamma$ has no infinite paths, as
$C\rightarrow\infty$ ($n$ fixed) the probability that condition B of Theorem
3.2 tends to zero. Hence, the theorem implies that with probability at least
$5-2\sqrt{6}$, there is a loop in $\Gamma$ which disconnects $[-n,n]$ from
$\infty$. Let $G_{n}$ be the event that this is the case, so that
$\mathbbm{P}[G_{n}]\geq 5-2\sqrt{6}$ for each large enough $n\in\mathbbm{N}$.
Then
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}G_{n}}\right]\geq
5-2\sqrt{6},$
i.e., with probability at least $5-2\sqrt{6}$, there are infinitely many loops
in $\Gamma$ which disconnect 0 from $\infty$. By the zero-one law for
translation invariant events, this in fact holds with probability one. ∎
We now give an overview of the proof of Theorem 3.2. If $\Gamma$ admits an
infinite path, it is straightforward to check that condition B in the
proposition statement holds with high probability when $n$ is large. So, we
can assume without loss of generality that there is no infinite path.
The key idea of the proof is that a meandric system satisfies rather rigid
parity properties. In particular, any distinct $x,y\in\mathbbm{Z}$ such that
$x-1/2$ and $y-1/2$ are not separated by a loop or an infinite path have to
have the same parity (Lemma 3.4). Under the assumption that there is no
infinite path in $\Gamma$, this allows us to force the existence of a
macroscopic loop as follows.
Fix $C>1$ and let $E_{n}$ be the event that there is a loop in $\Gamma$ which
hits only vertices of $[-Cn,Cn]\cap\mathbbm{Z}$ and which disconnects $[-n,n]$
from $\infty$ (see Figure 10). If $\mathbbm{P}[E_{n}]$ is bounded below by an
$n$-independent constant, then condition A in the theorem statement holds with
uniformly positive probability. So, we can assume that $\mathbbm{P}[E_{n}]$ is
small, i.e., $\mathbbm{P}[E_{n}^{c}]$ is large.
The event $E_{n}$ depends only on the restriction of the encoding walk
$(\mathcal{L},\mathcal{R})$ to $[-Cn,Cn]\cap\mathbbm{Z}$ (Lemma 3.5).
Therefore, if $x\in\mathbbm{Z}$ with $x>2Cn$, then the probability that
$E_{n}^{c}$ occurs, and also $E_{n}^{c}$ occurs with the translated map
$\mathcal{M}-x$ in place of $\mathcal{M}$, is $\mathbbm{P}[E_{n}^{c}]^{2}$. If
we choose $x$ to be odd, then, using the definition of $E_{n}$, we can show
that with probability at least $\mathbbm{P}[E_{n}^{c}]^{2}/8$, there are pairs
of points $(y,y_{x})$ with $y\in[-n,n]$ and $y_{x}\in[x-n,x+n]$ which have
opposite parity and which are not disconnected from $\infty$ by loops whose
vertex sets are contained in $[-Cn,Cn]$ and $[x-Cn,x+Cn]$, respectively. Since
$y$ and $y_{x}$ have opposite parity, there has to be a loop which disconnects
$y$ from $y_{x}$ (Lemma 3.7 and Figure 11). It is then straightforward to
check that this loop has to satisfy one of the two conditions of Theorem 3.2.
The probability $5-2\sqrt{6}$ in the proposition statement comes from
considering the “worst case” possibility for $\mathbbm{P}[E_{n}]$.
We now commence with the proof, starting with the requisite parity lemma.
Figure 9: Illustration of the proof of Lemma 3.4. We prove the following
statement: suppose $x-1/2$ and $y-1/2$ are not disconnected by any loop or
infinite path in $\Gamma$, and each point $z-1/2$ with
$z\in[x+1,y-1]\cap\mathbbm{Z}$ is disconnected from $x$ and $y$ by a loop or
an infinite path in $\Gamma$. Then any loop or infinite path in $\Gamma$ must
hit $[x,y-1]\cap\mathbbm{Z}$ an even number of times (examples of a loop and
an infinite path are shown in purple and brown, respectively). This implies
that $y-x$ is even, which gives the contrapositive of the lemma statement.
###### Lemma 3.4.
Let $x\in\mathbbm{Z}$ be even and let $y\in\mathbbm{Z}$ be odd. There is a
loop or an infinite path in $\Gamma$ which disconnects $x-1/2$ from $y-1/2$.
###### Proof.
See Figure 9 for an illustration. Define an equivalence relation on
$\mathbbm{Z}$ by $x\sim y$ if there is no loop or infinite path in $\Gamma$
which disconnects $x-1/2$ from $y-1/2$. Let $X$ be any equivalence class. If
suffices to show that every element of $X$ has the same parity. By considering
the elements of $X$ in left-right numerical order, it suffices to show the
following. If $x$ and $y$ are two consecutive elements of $X$ (i.e., $x,y\in
X$, $x<y$, and there is no element of $X$ in $[x,y]$) then $x$ and $y$ have
the same parity.
Since $x\sim y$, there exists a path $P$ in $\mathbbm{R}^{2}$ from $x-1/2$ to
$y-1/2$ which does not hit any loop or infinite path in $\Gamma$. By erasing
loops made by $P$, we can take $P$ to be a simple path. Since loops in
$\Gamma$ do not intersect $\mathbbm{R}$ except at integer points, we can also
arrange that $P$ hits $(x-1/2,x)$ and $(y-1/2,y)$ only at their endpoints.
Since $x$ and $y$ are consecutive elements of $X$, the path $P$ does not hit
$[x,y-1]$ (otherwise, there would be an element of $X$ between $x$ and $y$).
Therefore, $[x-1/2,y-1/2]\cup P$ is a simple closed loop in $\mathbbm{R}^{2}$.
By the Jordan curve theorem, there are exactly two connected components of
$\mathbbm{R}^{2}\setminus([x-1/2,y-1/2]\cup P)$ whose common boundary is
$[x-1/2,y-1/2]\cup P$. Let $U$ and $V$ be these two connected components.
Consider a loop $\ell\in\Gamma$ which hits a point of
$[x,y-1]\cap\mathbbm{Z}$. We traverse $\ell$ counterclockwise, say, started
from a point of $\ell\setminus\mathbbm{R}$. Since $\ell$ cannot intersect
$\mathbbm{R}$ without crossing it (by the definition of a meandric system) and
$\ell$ cannot hit $P$ (by our choice of $P$), the number of times that $\ell$
intersects $[x,y-1]\cap\mathbbm{Z}$ is equal to the number of times that
$\ell$ crosses from $U$ to $V$ or from $V$ to $U$. Since $\ell$ starts and
ends at the same point, the number of times that $\ell$ crosses from $U$ to
$V$ is equal to the number of times that $\ell$ crosses from $V$ to $U$. Hence
the number of times that $\ell$ intersects $[x,y-1]\cap\mathbbm{Z}$ is even.
Similarly, if $\Gamma$ has an infinite path, then the number of times that
this infinite path intersects $[x,y-1]\cap\mathbbm{Z}$ is even. Since every
point of $[x,y-1]\cap\mathbbm{Z}$ is hit by either a loop or an infinite path
of $\Gamma$, we get that $[x,y-1]\cap\mathbbm{Z}$ is even. Hence, $y-x$ is
even. ∎
Figure 10: Illustration of the event $E_{n}$, which depends on the constant
$C>1$.
Fix $C>1$ and for $n\in\mathbbm{N}$, let $E_{n}$ be the event that the
following is true:
1. $(\dagger)$
There exists a loop in $\Gamma$ which hits only vertices in
$[-Cn,Cn]\cap\mathbbm{Z}$ and which disconnects $[-n,n]$ from $\infty$.
Note that $E_{n}$ depends on $C$. See Figure 10 for an illustration.
###### Lemma 3.5.
The event $E_{n}$ is determined by the encoding walk increment
$(\mathcal{L},\mathcal{R})|_{[-Cn,Cn]\cap\mathbbm{Z}}$.
###### Proof.
The event $E_{n}$ depends only on the arcs of the upper and lower arc diagrams
for $\mathcal{M}$ which join points of $[-Cn,Cn]\cap\mathbbm{Z}$. These arcs
are determined by $(\mathcal{L},\mathcal{R})|_{[-Cn,Cn]\cap\mathbbm{Z}}$ by
the relationship between the arc diagrams and the walks (1.5). ∎
It is clear that the conclusion of Theorem 3.2 is satisfied if
$\mathbbm{P}[E_{n}]\geq 5-2\sqrt{6}$. So, we need to show that the conclusion
of the theorem is also true if $\mathbbm{P}[E_{n}^{c}]\geq 1-(5-2\sqrt{6})$.
The following elementary topological lemma, in conjunction with Lemma 3.4,
will help us do so.
###### Lemma 3.6.
If $E_{n}^{c}$ occurs, then there exists $y\in[-n+1,n]\cap\mathbbm{Z}$ such
that $y-1/2$ is not disconnected from $\infty$ by any loop in $\Gamma$ which
hits only vertices in $[-Cn,Cn]\cap\mathbbm{Z}$.
###### Proof.
Let $\Gamma_{Cn}$ be the set of loops in $\Gamma$ which hit only vertices in
$[-Cn,Cn]\cap\mathbbm{Z}$. Then $\Gamma_{Cn}$ is a finite collection of simple
loops in $\mathbbm{R}^{2}$ which do not intersect each other. Let
$\Gamma_{Cn}^{\prime}$ be the set of outermost loops in $\Gamma_{Cn}$ (i.e.,
those which are not disconnected from $\infty$ by any other loop in
$\Gamma_{Cn}$). For $\ell\in\Gamma_{Cn}^{\prime}$, let $U_{\ell}$ be the open
region disconnected from $\infty$ by $\ell$. Then the closures
$\overline{U}_{\ell}$ of the sets $U_{\ell}$ for $\ell\in\Gamma_{Cn}^{\prime}$
are disjoint (since the loops $\ell\in\Gamma_{Cn}^{\prime}$ are disjoint and
non-nested) and their union is the same as the set of points which are
disconnected from $\infty$ by the union of the loops in $\Gamma_{Cn}$.
By the definition ($\dagger$) of $E_{n}$, if $E_{n}^{c}$ occurs then $[-n,n]$
is not contained in $\overline{U}_{\ell}$ for any
$\ell\in\Gamma_{Cn}^{\prime}$. Since $[-n,n]$ is connected and the sets
$\overline{U}_{\ell}$ for $\ell\in\Gamma_{Cn}^{\prime}$ are closed and
disjoint, it follows that $[-n,n]$ is not contained in the union of the sets
$\overline{U}_{\ell}$ for $\ell\in\Gamma_{Cn}^{\prime}$. Hence, there must be
a point $z\in[-n,n]$ which is not contained in $\overline{U}_{\ell}$ for any
$\ell\in\Gamma_{Cn}^{\prime}$. The set of such $z$ is an open subset of
$[-n,n]$, so we can take $z\in[-n,n]\setminus\mathbbm{Z}$. The point $z$ is
not disconnected from $\infty$ by any loop in $\Gamma_{Cn}$. Since loops in
$\Gamma$ hit $\mathbbm{R}$ only at integer points, if
$y\in[-n+1,n]\cap\mathbbm{Z}$ is chosen so that $z\in(y-1,y)$, then also
$y-1/2$ is not disconnected from $\infty$ by any loop in $\Gamma_{Cn}$. ∎
The following lemma is the main step in the proof of Theorem 3.2.
Figure 11: Illustration of the proof of Lemma 3.7. Let $x\in[2Cn+1,2Cn+3]$,
so that $[-Cn,Cn]\cap[x-Cn,x+Cn]=\emptyset$. Note that we are considering two
symmetric intervals of size $2Cn$ centered at $0$ and $x$ (recall Figure 10).
If (i) $E_{n}^{c}$ occurs, (ii) the analogous event for $[x-Cn,x+Cn]$ occurs,
and (iii) the “exposed” points $y$ and $y_{x}$ given by Lemma 3.6 have
opposite parity, then Lemma 3.4 (and our assumption that there is no infinite
path) tells us that there is a loop in $\Gamma$ which disconnects $y-1/2$ from
$y_{x}-1/2$ (purple). By symmetry, we can take this loop to disconnect $y-1/2$
from $\infty$. By our choice of $y$ (from Lemma 3.6), this loop must hit a
vertex outside of $[-Cn,Cn]\cap\mathbbm{Z}$. Since the loop disconnects $y$
from $y_{x}$, it must also hit a vertex in
$[y,y_{x}]\cap\mathbbm{Z}\subset[y,(2C+1)n+3]\cap\mathbbm{Z}$. We note that
the loop given by Lemma 3.7 can have many possible behaviors besides what is
shown in the figure. For example, it could disconnect $[-n,n]$ from $\infty$
and/or it could hit $(-\infty,-Cn]$.
###### Lemma 3.7.
Assume that there is no infinite path in $\Gamma$. With probability at least
$\mathbbm{P}[E_{n}^{c}]^{2}/8$, there is a loop $\ell\in\Gamma$ with the
following properties:
1. $(i)$
$\ell$ disconnects $y-1/2$ from $\infty$ for some
$y\in[-n+1,n]\cap\mathbbm{Z}$.
2. $(ii)$
$\ell$ hits a point of $\mathbbm{Z}\setminus[-Cn,Cn]$.
3. $(iii)$
$\ell$ hits a point of $[y,(2C+1)n+3]\cap\mathbbm{Z}$.
###### Proof.
See Figure 11 for an illustration. Write $p:=\mathbbm{P}[E_{n}^{c}]$. Recall
from Lemma 3.5 that $E_{n}$ is determined by
$(\mathcal{L},\mathcal{R})|_{[-Cn,Cn]\cap\mathbbm{Z}}$. On $E_{n}^{c}$, let
$y\in[-n+1,n]\cap\mathbbm{Z}$ be a point as in Lemma 3.6, chosen in some
manner which depends only on
$(\mathcal{L},\mathcal{R})|_{[-Cn,Cn]\cap\mathbbm{Z}}$ (on $E_{n}$, we
arbitrarily set $y=0$). Then one of the events
$E_{n}^{c}\cap\mathopen{}\mathclose{{}\left\\{\text{$y$ is
even}}\right\\}\quad\text{or}\quad
E_{n}^{c}\cap\mathopen{}\mathclose{{}\left\\{\text{$y$ is odd}}\right\\}$
(3.1)
has probability at least $p/2$. We will assume that
$\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{n}^{c}\cap\mathopen{}\mathclose{{}\left\\{\text{$y$
is even}}\right\\}}\right]\geq\frac{p}{2}.$ (3.2)
The other case is treated in an identical manner.
Let $x\in[2Cn+1,2Cn+3]\cap\mathbbm{Z}$ be an odd integer (so that
$[-Cn,Cn]\cap[x-Cn,x+Cn]=\emptyset$). Define the event $E_{n}(x)$ in the same
manner as the event $E_{n}$ from just above Lemma 3.5, but with the translated
meandric system $\mathcal{M}-x$ in place of $\mathcal{M}$. Also let
$y_{x}\in[x-n+1,x-n]\cap\mathbbm{Z}$ be defined in the same manner as the
point $y$ above but with $\mathcal{M}-x$ in place of $\mathcal{M}$. By (3.2)
and since $x$ is odd,
$\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{n}^{c}(x)\cap\mathopen{}\mathclose{{}\left\\{\text{$y_{x}$
is odd}}\right\\}}\right]\geq\frac{p}{2}.$ (3.3)
The translated meandric system $\mathcal{M}-x$ is encoded by the translated
pair of walks
$j\mapsto(\mathcal{L}_{j+x}-\mathcal{L}_{x},\mathcal{R}_{j+x}-\mathcal{R}_{x})$
in the same manner that $\mathcal{M}$ is encoded by
$(\mathcal{L},\mathcal{R})$. By Lemma 3.5 and the definition of $y_{x}$, the
event $E_{n}(x)$ and the point $y_{x}$ are determined by the walk increment
$(\mathcal{L}-\mathcal{L}_{x},\mathcal{R}-\mathcal{R}_{x})|_{[x-Cn,x+Cn]\cap\mathbbm{Z}}$.
Since $[-Cn,Cn]\cap[x-Cn,x+Cn]=\emptyset$, the pairs $(E_{n},y)$ and
$(E_{n}(x),y_{x})$ are independent. From this together with (3.2) and (3.3),
we obtain
$\mathbbm{P}\mathopen{}\mathclose{{}\left[E_{n}^{c}\cap\mathopen{}\mathclose{{}\left\\{\text{$y$
is even}}\right\\}\cap
E_{n}^{c}(x)\cap\mathopen{}\mathclose{{}\left\\{\text{$y_{x}$ is
odd}}\right\\}}\right]\geq\frac{p^{2}}{4}.$ (3.4)
Recall that we are assuming that there is no infinite path in $\Gamma$. By
Lemma 3.4, if the event in (3.4) occurs, then there is a loop $\ell\in\Gamma$
which disconnects $y-1/2$ from $y_{x}-1/2$. The loop $\ell$ has precisely two
complementary connected components (by the Jordan curve theorem), so $\ell$
must disconnect exactly one of $y-1/2$ or $y_{x}-1/2$ from $\infty$. Since the
law of $\mathcal{M}$ is invariant under integer translation and reflection
about the origin, symmetry considerations show that the probability that
$\ell$ disconnects $y-1/2$ from $\infty$ is at least $p^{2}/8$.
By our choice of $y$ (recall Lemma 3.6), the loop $\ell$ must hit a point of
$\mathbbm{Z}\setminus[-Cn,Cn]$. Since $\ell$ disconnects $y$ from $y_{x}$,
$\ell$ must hit an integer point in the interval
$[y,y_{x}]\subset[y,x+n]\subset[y,(2C+1)n+3].$
Therefore, $\ell$ satisfies the conditions in the lemma statement. ∎
###### Proof of Theorem 3.2.
First assume that there is an infinite path $\mathfrak{P}$ in $\Gamma$. It
holds with probability tending to 1 as $n\rightarrow\infty$ that
$\mathfrak{P}$ intersects $[-n,n]\cap\mathbbm{Z}$, so since $\mathfrak{P}$ is
infinite we get that condition B is satisfied with probability444Here and
throughout this paper, if $f,g:\mathbbm{N}\to(0,\infty)$ we write
$g(n)=o_{n}(f(n))$ (resp. $g(n)=O_{n}(f(n))$) if $g(n)/f(n)\to 0$ (resp.
$g(n)/f(n)$ remains bounded above) as $n\to\infty$. $1-o_{n}(1)$. We choose
$n$ large enough so that this probability is at least $5-2\sqrt{6}$.
Now assume that there is no infinite path in $\Gamma$. Since
$\mathbbm{P}[E_{n}^{c}]=1-\mathbbm{P}[E_{n}]\in[0,1]$,
$\max\mathopen{}\mathclose{{}\left\\{\mathbbm{P}[E_{n}],\mathbbm{P}[E_{n}^{c}]^{2}/8}\right\\}\geq
5-2\sqrt{6}.$ (3.5)
By the definition ($\dagger$) of $E_{n}$, if $E_{n}$ occurs then condition A
is satisfied, so condition A is satisfied with probability at least
$\mathbbm{P}[E_{n}]$. In light of (3.5), it remains to show that the
probability that either A or B is satisfied is at least
$\mathbbm{P}[E_{n}^{c}]^{2}/8$.
By Lemma 3.7, it holds with probability at least
$\mathbbm{P}[E_{n}^{c}]^{2}/8$ that there is a loop $\ell\in\Gamma$ satisfying
the three properties in that lemma. If $\ell$ disconnects $[-n,n]$ from
$\infty$, then property (iii) of Lemma 3.7 shows that condition A in the
theorem statement is satisfied. If $\ell$ does not disconnect $[-n,n]$ from
$\infty$, then since $\ell$ disconnects some point of $[-n,n]\cap\mathbbm{Z}$
from $\infty$ by property (i), we get that $\ell$ must hit a point of
$[-n,n]\cap\mathbbm{Z}$. By property (ii), this implies that condition B in
the theorem statement is satisfied. ∎
## 4 Bounding distances via the mated-CRT map
Recall that $\mathcal{M}$ denotes the planar map associated with the UIMS. In
this section, we will prove a lower bound for certain graph distances in
$\mathcal{M}$ (Proposition 4.1 just below), conditional on an estimate
(Proposition 4.3) which will be proven in Section 5 using Liouville quantum
gravity techniques. Then, in Section 4.3, we will combine Proposition 4.1 and
Theorem 3.2 to deduce our main results on the sizes of loops in meandric
systems (Proposition 1.4, Theorem 1.5, and Theorem 1.9).
To state our lower bound for distances in $\mathcal{M}$, we introduce some
notation. We define the submaps
$\mathcal{M}[a,b]:=\mathopen{}\mathclose{{}\left(\text{submap of $\mathcal{M}$
induced by $[a,b]\cap\mathbbm{Z}$}}\right),\quad\forall-\infty<a<b<\infty.$
(4.1)
For a graph $G$, we also define
$\operatorname{dist}_{G}(x,y):=\mathopen{}\mathclose{{}\left(\text{$G$-graph
distance from $x$ to $y$}}\right),\quad\text{$\forall$ vertices $x,y\in G$}.$
(4.2)
If $H$ is a subgraph of $G$ and $x,y\in H$, then
$\operatorname{dist}_{G}(x,y)\leq\operatorname{dist}_{H}(x,y)$. The inequality
can be strict since the minimal-length path in $G$ between $x$ and $y$ may not
be contained in $H$. We will frequently use this fact without comment, often
with $H=M[a,b]$ as in (4.1).
###### Proposition 4.1.
For each $\zeta\in(0,1)$ and each $p\in(0,1)$, there exists $C=C(p,\zeta)>3$
such that for each large enough $n\in\mathbbm{N}$, it holds with probability
at least $p$ that the following is true:
1. $(i)$
$\operatorname{dist}_{\mathcal{M}}\mathopen{}\mathclose{{}\left([-n,n]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-Cn,Cn]}\right)\geq
n^{1/d-\zeta}$.
2. $(ii)$
$\operatorname{dist}_{\mathcal{M}}\mathopen{}\mathclose{{}\left([-3Cn,3Cn]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]}\right)\geq
n^{1/d-\zeta}$.
3. $(iii)$
$\operatorname{dist}_{\mathcal{M}}\mathopen{}\mathclose{{}\left([-C^{2}n,C^{2}n]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-C^{3}n,C^{3}n]}\right)\geq
n^{1/d-\zeta}$.
4. $(iv)$
There are two paths $\Pi_{1},\Pi_{2}$ in $\mathcal{M}$ each going from a
vertex of $[-n,n]\cap\mathbbm{Z}$ to a vertex of
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$ which lie at
$\mathcal{M}[-C^{3}n,C^{3}n]$-graph distance at least $n^{1/d-\zeta}$ from
each other.
For non-uniform random planar maps, it appears to be quite difficult to
estimate graph distances directly. So, to prove Proposition 4.1, we will use
an indirect approach which was introduced in [GHS20]. The idea is as follows.
In Section 4.1, we will define the mated-CRT map $\mathcal{G}$, a random
planar map constructed from a pair of independent two-sided Brownian motions
via a semicontinuous analog of the construction of $\mathcal{M}$ from a pair
of independent two-sided random walks. We will also state a comparison result
for distances in $\mathcal{M}$ and distances in $\mathcal{G}$ (Theorem 4.2)
which follows from a more general result in [GHS20].
This comparison result allows us to reduce Proposition 4.1 to a similar
estimate for the mated-CRT map (Proposition 4.3). The proof of this latter
estimate is given in Section 5. Due to the results of [DMS21], the mated-CRT
map admits an alternative description in terms of $\sqrt{2}$-LQG decorated by
SLE8 (see Section 5.1). Using this description, the needed estimate for the
mated-CRT map turns out to be an easy consequence of known results for SLEs
and LQG.
### 4.1 The mated-CRT map
Let $(L,R)$ be a pair of standard linear two-sided Brownian motions, with
$L_{0}=R_{0}=0$. The mated-CRT map associated with $(L,R)$ is the graph
$\mathcal{G}$ with vertex set $\mathbbm{Z}$ and edge set defined as follows.
Two integers $x_{1},x_{2}\in\mathbbm{Z}$ with $x_{1}<x_{2}$ are joined by an
edge if and only if either
$\displaystyle\max\mathopen{}\mathclose{{}\left\\{\inf_{t\in[x_{1}-1,x_{1}]}L_{t},\,\inf_{t\in[x_{2}-1,x_{2}]}L_{t}}\right\\}\leq\inf_{t\in[x_{1},x_{2}-1]}L_{t}\quad\operatorname{or}\quad$
$\displaystyle\max\mathopen{}\mathclose{{}\left\\{\inf_{t\in[x_{1}-1,x_{1}]}R_{t},\,\inf_{t\in[x_{2}-1,x_{2}]}R_{t}}\right\\}\leq\inf_{t\in[x_{1},x_{2}-1]}R_{t}.$
(4.3)
We note that a.s. both conditions in (4.1) hold whenever $x_{2}=x_{1}+1$. If
both conditions in (4.1) hold and $x_{2}\geq x_{1}+2$, we declare that $x_{1}$
and $x_{2}$ are joined by two edges.
The edge set of $\mathcal{G}$ naturally splits into three subsets:
* •
Trivial edges, which join $x$ and $x+1$ for $x\in\mathbbm{Z}$.
* •
Upper edges, which join $x_{1},x_{2}\in\mathbbm{Z}$ with $x_{2}\geq x_{1}+2$
and arise from the first condition (the one involving $L$) in (4.1).
* •
Lower edges, which join $x_{1},x_{2}\in\mathbbm{Z}$ with $x_{2}\geq x_{1}+2$
and arise from the second condition (the one involving $R$) in (4.1).
We can assign a planar map structure to $\mathcal{G}$ by associating each
trivial edge with the line segment from $x$ to $x+1$ in $\mathbbm{R}^{2}$,
each upper edge $\\{x_{1},x_{2}\\}$ with an arc from $x_{1}$ to $x_{2}$ in the
upper half-plane, and each lower edge $\\{x_{1},x_{2}\\}$ with an arc from
$x_{1}$ to $x_{2}$ in the lower half-plane. In fact, $\mathcal{G}$ is a
triangulation when equipped with this planar map structure. See Figure 12.
Figure 12: Left: To construct the mated-CRT map $\mathcal{G}$ geometrically,
one can draw the graph of $C-L$ (red) and the graph of $R$ (blue) for some
large constant $C>0$ chosen so that the parts of the graphs over some time
interval of interest do not intersect. Here, this time interval is $[0,12]$.
One then divides the region between the graphs into vertical strips
(boundaries shown in orange). Each vertical strip corresponds to the vertex
$x\in\mathbbm{Z}$ which is the horizontal coordinate of its rightmost points.
Vertices $x_{1},x_{2}\in\mathbbm{Z}$ are connected by an edge if and only if
the corresponding vertical strips are connected by a horizontal line segment
which lies above the graph of $C-L$ or below the graph of $R$. For each pair
of vertices for which the condition holds for $C-L$ (resp. $R$), we have drawn
the highest (resp. lowest) segment which joins the corresponding vertical
strips in red (resp. blue). Equivalently, for each $x\in\mathbbm{Z}$, we let
$t_{x}$ be the time in $[x-1,x]$ at which $L$ attains its minimum value and we
draw in red the longest horizontal segment above the graph of $C-L$ which
contains $(t_{x},C-L_{t_{x}})$; and we perform a similar procedure for $R$.
Note that consecutive vertices are always joined by an edge. Right: One can
draw the graph $\mathcal{G}$ in the plane by connecting two non-consecutive
vertices $x_{1},x_{2}\in\mathbbm{Z}$ by an arc above (resp. below) the real
line if the corresponding vertical strips are connected by a horizontal
segment above (resp. below) the graph of $C-L$ (resp. $R$); and connecting
each pair of consecutive vertices of $\mathbbm{Z}$ by an edge. This gives
$\mathcal{G}$ a planar map structure under which it is a triangulation. A
similar figure and caption appeared in [GMS19].
Analogously to (4.1), we define
$\mathcal{G}[a,b]:=\mathopen{}\mathclose{{}\left(\text{submap of $\mathcal{G}$
induced by $[a,b]\cap\mathbbm{Z}$}}\right),\quad\forall-\infty<a<b<\infty.$
(4.4)
The definition of $\mathcal{G}$ is similar to the construction of the UIMS
$\mathcal{M}$ from the pair of bi-infinite simple random walks
$(\mathcal{L},\mathcal{R})$. In particular, by (1.5), two vertices
$x_{1},x_{2}\in\mathbbm{Z}$ are joined by an edge of $\mathcal{M}$ if and only
if
$\displaystyle
x_{2}=x_{1}+1\quad\text{or}\quad\mathcal{L}_{x_{1}-1}=\mathcal{L}_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]}\mathcal{L}_{y}\quad\operatorname{or}\quad\mathcal{R}_{x_{1}-1}=\mathcal{R}_{x_{2}}<\min_{y\in[x_{1},x_{2}-1]}\mathcal{R}_{y}$
(4.5)
which is a continuous time analog of (4.1). We note, however, that for
$x\in\mathbbm{Z}$ the number of arcs of $\mathcal{G}$ in the upper (resp.
lower) half-plane incident to $x$ can be any non-negative integer, whereas for
$\mathcal{M}$ the number of such arcs is always one.
Using the KMT strong coupling theorem for random walk and Brownian motion
[KMT76, Zai98] and an elementary geometric argument, it was shown in [GHS20,
Theorem 2.1] that one can couple $(L,R)$ and $(\mathcal{L},\mathcal{R})$ so
that the graph distances in their corresponding planar maps $\mathcal{M}$ and
$\mathcal{G}$ are close (actually, [GHS20] considers a more general class of
pairs of random walks).
###### Theorem 4.2 ([GHS20]).
For each $\alpha>0$, there exists $A=A(\alpha)>0$ and a coupling of $(L,R)$
with $(\mathcal{L},\mathcal{R})$ such that the following is true with
probability $1-O_{n}(n^{-\alpha})$. Let $\mathcal{G}$ be the mated-CRT map
constructed from $(L,R)$ as in (4.1) and let $\mathcal{M}$ be the infinite
planar map associated with the UIMS, constructed from
$(\mathcal{L},\mathcal{R})$ as in Section 1.4. For each
$x,y\in[-n,n]\cap\mathbbm{Z}$,
$A^{-1}(\log
n)^{-3}\operatorname{dist}_{\mathcal{G}[-n,n]}(x,y)\leq\operatorname{dist}_{\mathcal{M}[-n,n]}(x,y)\leq
A(\log n)^{3}\operatorname{dist}_{\mathcal{G}[-n,n]}(x,y),$ (4.6)
where here we use the notations (4.1) and (4.4).
### 4.2 Proof of lower bounds for graph distances
In this section we prove Proposition 4.1. We will deduce it from the
combination of Theorem 4.2 and the following analogous estimate for the mated-
CRT map.
###### Proposition 4.3.
For each $\zeta\in(0,1)$ and each $p\in(0,1)$, there exists $C=C(p,\zeta)>3$
such that for each large enough $n\in\mathbbm{N}$, it holds with probability
at least $p$ that the following is true:
1. $(a)$
$\operatorname{dist}_{\mathcal{G}}\mathopen{}\mathclose{{}\left([-n,n]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-Cn,Cn]}\right)\geq
n^{1/d-\zeta}$.
2. $(b)$
$\operatorname{dist}_{\mathcal{G}}\mathopen{}\mathclose{{}\left([-3Cn,3Cn]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]}\right)\geq
n^{1/d-\zeta}$.
3. $(c)$
$\operatorname{dist}_{\mathcal{G}}\mathopen{}\mathclose{{}\left([-C^{2}n,C^{2}n]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-C^{3}n,C^{3}n]}\right)\geq
n^{1/d-\zeta}$.
4. $(d)$
There are two paths $\Pi_{1},\Pi_{2}$ in $\mathcal{G}$ each going from a
vertex of $[-n,n]\cap\mathbbm{Z}$ to a vertex of
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$ which lie at $\mathcal{G}$-graph
distance at least $n^{1/d-\zeta}$ from each other.
The proof of Proposition 4.3 is given in Section 5, using the relationship
between the mated-CRT map and SLE-decorated LQG. We now deduce Proposition 4.1
from Proposition 4.3.
###### Proof of Proposition 4.1.
We divide the proof in three main steps.
Step 1: Regularity event. We first define a high-probability event which we
will work on throughout the rest of the proof. Let $A_{0}>0$ be as in Theorem
4.2 with $\alpha=1$, say, and let $A:=8A_{0}$. By Theorem 4.2 with $n^{2}$ in
place of $n$, we can couple $\mathcal{M}$ and $\mathcal{G}$ so that with
probability tending to 1 as $n\rightarrow\infty$, it holds for each
$x,y\in[-n^{2},n^{2}]\cap\mathbbm{Z}$ that
$A^{-1}(\log
n)^{-3}\operatorname{dist}_{\mathcal{G}[-n^{2},n^{2}]}(x,y)\leq\operatorname{dist}_{M[-n^{2},n^{2}]}(x,y)\leq
A(\log n)^{3}\operatorname{dist}_{\mathcal{G}[-n^{2},n^{2}]}(x,y).$ (4.7)
In order to make sure that we can compare distances in $\mathcal{M}$ and
$\mathcal{G}$ from points in $[-C^{2}n,C^{2}n]\cap\mathbbm{Z}$ to the set
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$, we also need to impose a further
regularity condition. By the adjacency conditions (4.5) and (4.1) for
$\mathcal{M}$ and $\mathcal{G}$ in terms of $(\mathcal{L},\mathcal{R})$ and
$(L,R)$, respectively, together with basic estimates for random walk and
Brownian motion, for any fixed $C>1$ it holds with probability tending to 1 as
$n\rightarrow\infty$ that
$\text{$\forall$ edges $\\{x,y\\}\in\mathcal{M}$ such that
$x\in[-C^{3}n,C^{3}n]\cap\mathbbm{Z}$, we have $y\in[-n^{2},n^{2}]$};$ (4.8)
and the same is true with $\mathcal{G}$ in place of $\mathcal{M}$. Note that
the quantity $n^{2}$ is unimportant in (4.8): the same would be true with
$n^{2}$ replaced by any function of $n$ which goes to $\infty$ as
$n\rightarrow\infty$.
We now take $C=C(1-(1-p)/2,\zeta/2)>0$ as in Proposition 4.3 with $1-(1-p)/2$
instead of $p$ and $\zeta/2$ instead of $\zeta$. By Proposition 4.3 and our
above estimates, for each large $n\in\mathbbm{N}$ it holds with probability at
least $p$ that (4.7) and (4.8) both hold and the numbered conditions in
Proposition 4.3 hold with $\zeta/2$ in place of $\zeta$. Henceforth assume
that this is the case. The rest of the argument is deterministic.
Step 2: Proofs of (i), (ii), and (iii). Consider a path $P$ in $\mathcal{M}$
from a point $x\in[-n,n]\cap\mathbbm{Z}$ to a point of
$\mathbbm{Z}\setminus[-Cn,Cn]$. By (4.8), $P$ hits a vertex in
$\mathbbm{Z}\cap([-n^{2},n^{2}]\setminus[-Cn,Cn])$. Let $y$ be the first such
vertex. Then the segment $P^{\prime}$ of $P$ between its starting point and
the first time it hits $y$ is a path in $\mathcal{M}[-n^{2},n^{2}]$. Using our
above estimates, we now get that the length of $P^{\prime}$ satisfies
$\displaystyle|P^{\prime}|$ $\displaystyle\geq A^{-1}(\log
n)^{-3}\operatorname{dist}_{\mathcal{G}[-n^{2},n^{2}]}(x,y)\quad\text{(by~{}\eqref{eqn-
use-ghs-coupling})}$ $\displaystyle\geq A^{-1}(\log
n)^{-3}\operatorname{dist}_{\mathcal{G}}\mathopen{}\mathclose{{}\left([-n,n]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-Cn,Cn]}\right)\quad\text{(choice
of $x$ and $y$)}$ $\displaystyle\geq A^{-1}(\log
n)^{-3}n^{1/d-\zeta/2}\quad\text{(\eqref{item-mcrt-in} of
Proposition~{}\ref{prop-mcrt-estimate} with $\zeta/2$ instead of $\zeta$)}.$
(4.9)
Since $P^{\prime}$ is a sub-path of $P$, we have $|P|\geq|P^{\prime}|$. Taking
the infimum over all $P$ now shows that
$\operatorname{dist}_{\mathcal{M}}\mathopen{}\mathclose{{}\left([-n,n]\cap\mathbbm{Z},\mathbbm{Z}\setminus[-Cn,Cn]}\right)$
is at least the right side of (4.2), which is at least $n^{1/d-\zeta}$ if $n$
is large enough. Thus (i) in the proposition statement holds.
The proofs of (ii) and (iii) are identical to the proof of (i), except that we
use (b) and (c) from Proposition 4.3 instead of (a).
Step 3: Proof of (iv). Let $\widetilde{\Pi}_{1}$ and $\widetilde{\Pi}_{2}$ be
the paths in $\mathcal{G}$ as in (d) of Proposition 4.3 with $\zeta/2$ instead
of $\zeta$. By possibly replacing $\widetilde{\Pi}_{1}$ and
$\widetilde{\Pi}_{2}$ by sub-paths (which can only increase the graph distance
between them), we can assume that each of these paths intersects
$[-n,n]\cap\mathbbm{Z}$ and $\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$ only at its
endpoints.
We view $\widetilde{\Pi}_{1}$ as a function
$\widetilde{\Pi}_{1}:[0,N]\cap\mathbbm{Z}\rightarrow\mathbbm{Z}$. For
$i=1,\dots,N$, we apply (4.7) with
$(\widetilde{\Pi}_{1}(i-1),\widetilde{\Pi}_{1}(i))$ in place of $(x,y)$ to get
$N$ new paths in $\mathcal{M}[-n^{2},n^{2}]$ (the ones realizing
$\operatorname{dist}_{\mathcal{M}[-n^{2},n^{2}]}(\widetilde{\Pi}_{1}(i-1),\widetilde{\Pi}_{1}(i))$),
each of length at most $A(\log n)^{3}$. Then we concatenate the $N$ new paths
in $\mathcal{M}[-n^{2},n^{2}]$. This results in a path $\Pi_{1}$ in
$\mathcal{M}[-n^{2},n^{2}]$ with the same endpoints as $\widetilde{\Pi}_{1}$
with the property that each point of $\Pi_{1}$ lies at
$\mathcal{M}[-n^{2},n^{2}]$-graph distance at most $A(\log n)^{3}$ from a
point of $\widetilde{\Pi}_{1}$. We similarly construct a path $\Pi_{2}$ in
$\mathcal{M}[-n^{2},n^{2}]$ but started from $\widetilde{\Pi}_{2}$ instead of
$\widetilde{\Pi}_{1}$.
Since $\Pi_{1},\Pi_{2}$ have the same endpoints as
$\widetilde{\Pi}_{1},\widetilde{\Pi}_{2}$, these paths each go from a vertex
of $[-n,n]\cap\mathbbm{Z}$ to a vertex of
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$. Furthermore, by the distance estimate
in the preceding paragraph and the triangle inequality, the
$\mathcal{M}[-n^{2},n^{2}]$-graph distance from any point of $\Pi_{1}$ to any
point of $\Pi_{2}$ is at least
$\operatorname{dist}_{\mathcal{M}[-n^{2},n^{2}]}(\widetilde{\Pi}_{1},\widetilde{\Pi}_{2})-2A(\log
n)^{3}$. By this and (4.7),
$\displaystyle\operatorname{dist}_{\mathcal{M}[-C^{3}n,C^{3}n]}(\Pi_{1},\Pi_{2})$
$\displaystyle\geq\operatorname{dist}_{\mathcal{M}[-n^{2},n^{2}]}(\Pi_{1},\Pi_{2})$
$\displaystyle\geq\operatorname{dist}_{\mathcal{M}[-n^{2},n^{2}]}(\widetilde{\Pi}_{1},\widetilde{\Pi}_{2})-2A(\log
n)^{3}$ $\displaystyle\geq A^{-1}(\log
n)^{-3}\operatorname{dist}_{\mathcal{G}}(\widetilde{\Pi}_{1},\widetilde{\Pi}_{2})-2A(\log
n)^{3}.$ (4.10)
By our initial choice of $\widetilde{\Pi}_{1}$ and $\widetilde{\Pi}_{2}$ the
right side of (4.2) is at least $A^{-1}(\log n)^{-3}n^{1/d-\zeta/2}-2A(\log
n)^{3}$, which is at least $n^{1/d-\zeta}$ for each large enough
$n\in\mathbbm{N}$. ∎
### 4.3 Proof of Proposition 1.4, Theorem 1.5, and Theorem 1.9
Combining Theorem 3.2 and Proposition 4.1 leads to the following result.
###### Proposition 4.4.
For each $\zeta\in(0,1)$ and each sufficiently large $n\in\mathbbm{N}$
(depending on $\zeta$), it holds with probability at least $1/10$ that the
following is true. There is a segment $\overline{\ell}$ of a loop or infinite
path in $\Gamma$ which is contained in $[-n,n]\cap\mathbbm{Z}$ such that the
$\mathcal{M}[-n,n]$-graph-distance diameter of $\overline{\ell}$ and the
$\mathcal{M}$-graph distance from $\overline{\ell}$ to
$\mathbbm{Z}\setminus[-n,n]$ are each at least $n^{1/d-\zeta}$.
###### Proof.
Fix $\zeta\in(0,1)$. By Theorem 3.2, for any choice of $C>1$, for each large
enough $n\in\mathbbm{N}$ it holds with probability at least $5-2\sqrt{6}>1/10$
that at least one of A or B in the statement of Theorem 3.2 is satisfied. By
Proposition 4.1, there is a universal constant $C>1$ so that for each large
enough $n$, it holds with probability at least $1-(5-2\sqrt{6}-1/10)$ that all
four of the numbered conditions in the statement of Proposition 4.1 are
satisfied. Henceforth assume that the events of Theorem 3.2 and Proposition
4.1 occur (both with this same choice of $C$), which happens with probability
at least $1/10$ if $n$ is large enough.
We will show that
1. $(*)$
there is a segment $\overline{\ell}$ of a loop or infinite path in $\Gamma$
which is contained in $[-C^{3}n,C^{3}n]\cap\mathbbm{Z}$ such that the
$\mathcal{M}[-C^{3}n,C^{3}n]$-graph-distance diameter of $\overline{\ell}$ and
the $\mathcal{M}$-graph distance from $\overline{\ell}$ to
$\mathbbm{Z}\setminus[-C^{3}n,C^{3}n]\cap\mathbbm{Z}$ are each at least
$n^{1/d-\zeta}$.
To extract the proposition statement from ($*$), we can apply ($*$) with
$\lfloor n/C^{3}\rfloor$ in place of $n$, then slightly shrink $\zeta$ in
order to absorb a factor of $1/C^{3\zeta}$ into a power of $n$.
To prove ($*$), we will treat the two possible scenarios in Theorem 3.2
separately. First suppose that
1. A.
there is a loop $\ell$ in $\Gamma$ which disconnects $[-n,n]$ from $\infty$
and which hits a point of $[n,(2C+1)n+3]\cap\mathbbm{Z}$.
If $\ell$ is contained in $[-C^{2}n,C^{2}n]\cap\mathbbm{Z}$, then by planarity
and since $\ell$ disconnects $[-n,n]$ from $\infty$, the loop $\ell$ must
intersect every path in $\mathcal{M}$ from $[-n,n]\cap\mathbbm{Z}$ to
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$. In particular, $\ell$ must intersect
the paths $\Pi_{1}$ and $\Pi_{2}$ from (iv) of Proposition 4.1. Hence, the
$\mathcal{M}[-C^{3}n,C^{3}n]$-graph-distance diameter of $\ell$ is at least
$n^{1/d-\zeta}$. Furthermore, by (iii) of Proposition 4.1, the
$\mathcal{M}$-graph distance from $\ell$ to
$\mathbbm{Z}\setminus[-C^{3}n,C^{3}n]\cap\mathbbm{Z}$ is at least
$n^{1/d-\zeta}$. So, we can take $\overline{\ell}=\ell$.
If $\ell$ is not contained in $[-C^{2}n,C^{2}n]\cap\mathbbm{Z}$, then since
$\ell$ hits $[n,(2C+1)n+3]\cap\mathbbm{Z}$, there is a segment
$\overline{\ell}$ of $\ell$ which is a path in $\mathcal{M}$ from a point of
$[n,(2C+1)n+3]\cap\mathbbm{Z}\subset[-3Cn,3Cn]\cap\mathbbm{Z}$ to a point of
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}n]$. By possibly replacing
$\overline{\ell}$ with a sub-path, we can assume that $\overline{\ell}$ is
contained in $[-C^{2}n,C^{2}n]\cap\mathbbm{Z}$ except for its terminal
endpoint.
By (ii) of Proposition 4.1, the $\mathcal{M}[-C^{3}n,C^{3}n]$-graph-distance
diameter of $\overline{\ell}$ is at least $n^{1/d-\zeta}$. By (iii) of
Proposition 4.1, the $\mathcal{M}$-graph distance from $\overline{\ell}$ to
$[-C^{3}n,C^{3}n]\cap\mathbbm{Z}$ is at least $n^{1/d-\zeta}$.
Next suppose that
1. B.
there is a loop or an infinite path in $\Gamma$ which hits a point of each of
$[-n,n]\cap\mathbbm{Z}$ and $\mathbbm{Z}\setminus[-Cn,Cn]$.
Let $\overline{\ell}$ be a segment of this loop or infinite path which is a
path from a point of $[-n,n]\cap\mathbbm{Z}$ to a point of
$\mathbbm{Z}\setminus[-Cn,Cn]$. By possibly replacing $\overline{\ell}$ with a
sub-path, we can assume that $\overline{\ell}$ is contained in
$[-Cn,Cn]\cap\mathbbm{Z}$ except for its terminal endpoint. Then (i) of
Proposition 4.1 shows that the $\mathcal{M}[-C^{3}n,C^{3}n]$-graph-distance
diameter of $\overline{\ell}$ is at least $n^{1/d-\zeta}$ and (iii) of
Proposition 4.1 shows that the $\mathcal{M}$-graph distance from
$\overline{\ell}$ to $[-C^{3}n,C^{3}n]\cap\mathbbm{Z}$ is at least
$n^{1/d-\zeta}$. ∎
Using Proposition 4.4, we obtain our lower tail bound for the diameter of the
origin-containing loop in the UIMS.
###### Proof of Theorem 1.9.
Let $\zeta\in(0,1)$, which we will eventually send to zero. For
$k\in\mathbbm{N}$, let $X=X_{k}$ be sampled uniformly from
$[-k^{d+\zeta},k^{d+\zeta}]\cap\mathbbm{Z}$, independently from everything
else, and let $\ell_{X}$ be the loop in $\Gamma$ which hits $X$. By the
translation invariance of the law of $(\mathcal{M},\Gamma)$, the translated
planar map / loop pair $(\mathcal{M}-X,\Gamma-X)$ has the same law as
$(\mathcal{M},\Gamma)$. Hence,
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\text{$\mathcal{M}$-graph-
distance diameter of $\ell_{0}$}}\right)\geq
k}\right]=\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\text{$\mathcal{M}$-graph-
distance diameter of $\ell_{X}$}}\right)\geq k}\right].$ (4.11)
We will now lower-bound the second quantity in (4.11).
By Proposition 4.4 (with $\lfloor k^{d+\zeta}\rfloor$ in place of $n$ and with
$\zeta$ possibly replaced by a smaller positive number), if $k$ is large
enough then it holds with probability at least $1/10$ that there is a segment
$\overline{\ell}$ of a loop or infinite path in $\Gamma$ which is contained in
$[-k^{d+\zeta},k^{d+\zeta}]\cap\mathbbm{Z}$ such that the
$\mathcal{M}[-k^{d+\zeta},k^{d+\zeta}]$-graph-distance diameter of
$\overline{\ell}$ and the $\mathcal{M}$-graph distance from $\overline{\ell}$
to $\mathbbm{Z}\setminus[-k^{d+\zeta},k^{d+\zeta}]$ are each at least $k$. Let
$E_{k}$ be the event that such an $\overline{\ell}$ exists. On $E_{k}$ let
$\overline{\ell}$ be a path as in the definition of $E_{k}$, chosen in some
manner which is measurable with respect to $\sigma(\mathcal{M},\Gamma)$.
We claim that if $E_{k}$ occurs, then the $\mathcal{M}$-graph-distance
diameter of $\overline{\ell}$, and hence also the number of vertices of
$\mathcal{M}$ hit by $\overline{\ell}$, are each at least $k$. Indeed, by
definition, there are two vertices
$x,y\in[-k^{d+\zeta},k^{d+\zeta}]\cap\mathbbm{Z}$ hit by $\overline{\ell}$
which lie at $\mathcal{M}[-k^{d+\zeta},k^{d+\zeta}]$-graph distance at least
$k$ from each other. Any path in $\mathcal{M}$ from $x$ to $y$ must either
stay in $[-k^{d+\zeta},k^{d+\zeta}]\cap\mathbbm{Z}$, in which case its length
is at least $k$ by our choice of $x,y$; or must cross from $x$ to
$\mathbbm{Z}\setminus[-k^{d+\zeta},k^{d+\zeta}]$, in which case its length is
also at least $k$ since the $\mathcal{M}$-graph distance from
$\overline{\ell}$ to $\mathbbm{Z}\setminus[-k^{d+\zeta},k^{d+\zeta}]$ is at
least $k$. Taking the infimum over all such paths gives our claim.
Since $X$ is sampled uniformly from
$[-k^{d+\zeta},k^{d+\zeta}]\cap\mathbbm{Z}$, independently from
$(\mathcal{M},\Gamma)$, the preceding paragraph implies that
$\mathbbm{P}\mathopen{}\mathclose{{}\left[X\in\overline{\ell}\,\middle|\,(\mathcal{M},\Gamma)}\right]\mathbbm{1}_{E_{k}}\geq\frac{1}{2}k^{-(d-1)-\zeta}\mathbbm{1}_{E_{k}}.$
(4.12)
Since $\mathbbm{P}[E_{k}]\geq 1/10$ and the $\mathcal{M}$-graph-distance
diameter of $\overline{\ell}$ is at least $k$ on $E_{k}$, taking the
expectations on both sides of (4.12) gives
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\text{$\mathcal{M}$-graph-
distance diameter of $\ell_{X}$}}\right)\geq
k}\right]\geq\frac{1}{20}k^{-(d-1)-\zeta}.$ (4.13)
By (4.11), (4.13) implies (1.6) upon sending $\zeta\rightarrow 0$. ∎
The following proposition is the analog of Theorem 1.5 for the UIMS. It is the
main input in the proof of Theorem 1.5.
###### Proposition 4.5.
For each $\zeta\in(0,1)$, there exists $\beta>0$ and $a_{0},a_{1}>0$,
depending on $\zeta$, such that for each $n\in\mathbbm{N}$, it holds with
probability at least $1-a_{0}e^{-a_{1}n^{\beta}}$ that there is a segment of a
loop or an infinite path in $\Gamma$ which hits only vertices of
$[1,2n]\cap\mathbbm{Z}$ and has $\mathcal{M}$-graph-distance diameter at least
$n^{1/d-\zeta}$.
###### Proof.
Let $\beta\in(0,1)$ to be chosen later, depending on $\zeta$. The idea of the
proof is as follows. We will consider a collection of
$\operatorname{const}\times n^{\beta}$ disjoint sub-intervals of $[1,2n]$ of
length $2\lfloor n^{1-\beta}\rfloor$. We will then use independence to show
that with high probability, the event of Proposition 4.4 (with $\lfloor
n^{1-\beta}\rfloor$ instead of $n$) occurs for at least one of these
intervals.
For $n\in\mathbbm{N}$ and $x\in\mathbbm{Z}$, let $G_{n}(x)$ be the event that
the following is true:
1. $(\ddagger)$
There is a segment $\overline{\ell}$ of a loop or infinite path in $\Gamma$
which is contained in $[x-n,x+n]\cap\mathbbm{Z}$ such that the
$\mathcal{M}[x-n,x+n]$-graph-distance diameter of $\overline{\ell}$ and the
$\mathcal{M}$-graph distance from $\overline{\ell}$ to
$\mathbbm{Z}\setminus[x-n,x+n]$ are each at least $n^{1/d-\beta}$.
By the translation invariance of the law of $\mathcal{M}$, Proposition 4.4
implies that for each large enough $n\in\mathbbm{N}$,
$\mathbbm{P}\mathopen{}\mathclose{{}\left[G_{n}(x)}\right]\geq\frac{1}{10},\quad\forall
x\in\mathbbm{Z}.$ (4.14)
Furthermore, the event $G_{n}(x)$ depends only on the set of edges of
$\mathcal{M}$ between vertices in $[x-n,x+n]\cap\mathbbm{Z}$ and the set of
vertices in $[x-n,x+n]\cap\mathbbm{Z}$ which are joined by edges of
$\mathcal{M}$ to vertices not in $[x-n,x+n]\cap\mathbbm{Z}$. This information
is determined by the restricted, shifted walk
$(\mathcal{L}-\mathcal{L}_{x},\mathcal{R}-\mathcal{R}_{x})|_{[x-n,x+n]\cap\mathbbm{Z}}$.
Consequently,
$G_{n}(x)$ and $G_{n}(y)$ are independent if $|x-y|\geq 2n$. (4.15)
There is a constant $c=c(\beta)>0$ such that for each $n\in\mathbbm{N}$, there
is a deterministic set $X_{n}\subset[1,2n]\cap\mathbbm{Z}$ of cardinality at
least $cn^{\beta}$ such that
$\text{the intervals $\mathopen{}\mathclose{{}\left[x-\lfloor
n^{1-\beta}\rfloor,x+\lfloor n^{1-\beta}\rfloor}\right]$ for $x\in X_{n}$ are
disjoint and contained in $[1,2n]$}.$ (4.16)
By (4.14) and (4.15),
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\text{$G_{\lfloor
n^{1-\beta}\rfloor}(x)$ occurs for at least one $x\in X_{n}$}}\right]\geq
1-\mathopen{}\mathclose{{}\left(\frac{9}{10}}\right)^{cn^{\beta}}.$ (4.17)
On the other hand, if $G_{\lfloor n^{1-\beta}\rfloor}(x)$ occurs, then the
segment $\overline{\ell}$ as in the definition of $G_{\lfloor
n^{1-\beta}\rfloor}$ has $\mathcal{M}[x-\lfloor n^{1-\beta}\rfloor,x+\lfloor
n^{1-\beta}\rfloor]$-graph-distance diameter at least $\lfloor
n^{1-\beta}\rfloor^{1/d-\beta}$ and the $\mathcal{M}$-graph distance from
$\overline{\ell}$ to $\mathbbm{Z}\setminus[x-\lfloor
n^{1-\beta}\rfloor,x+\lfloor n^{1-\beta}\rfloor]$ is at least $\lfloor
n^{1-\beta}\rfloor^{1/d-\beta}$. Hence, the $\mathcal{M}$-graph-distance
diameter of $\overline{\ell}$ is at least $\lfloor
n^{1-\beta}\rfloor^{1/d-\beta}$.
We now choose $\beta$ to be small enough, depending on $\zeta$, so that
$(1-\beta)(1/d-\beta)>1/d-\zeta.$
Then (4.17) and the preceding paragraph give that if $n$ is large enough, then
with probability at least $1-(9/10)^{cn^{\beta}}$, there is a segment of a
loop in $\Gamma$ which intersects $[1,2n]\cap\mathbbm{Z}$ and has
$\mathcal{M}$-graph-distance diameter at least $n^{1/d-\zeta}$. This gives the
proposition statement for an appropriate choice of $a_{0},a_{1}>0$. ∎
###### Proof of Theorem 1.5.
Let $(\mathcal{M},\Gamma)$ be the loop-decorated planar map associated with an
infinite meandric system. For $n\in\mathbbm{N}$, let $F_{n}$ be the event that
there is no arc of the upper or lower arc diagram associated with
$\mathcal{M}$ which has one endpoint in $[1,2n]\cap\mathbbm{Z}$ and one
endpoint not in $[1,2n]\cap\mathbbm{Z}$. Equivalently, by (1.5), $F_{n}$ is
the event that the encoding walks satisfy
$\mathcal{L}_{2n}=\mathcal{R}_{2n}=0$ and $\mathcal{L}_{x},\mathcal{R}_{x}\geq
0$ for each $x\in[0,2n]\cap\mathbbm{Z}$. By a standard random walk estimate,
there is a universal constant $c>0$ such that
$\mathbbm{P}[F_{n}]\sim cn^{-3}\quad\text{as $n\rightarrow\infty$}.$ (4.18)
By the definition of $F_{n}$, if $F_{n}$ occurs, then no infinite path in
$\Gamma$ can hit $[1,2n]\cap\mathbbm{Z}$ and the set $\Gamma_{n}$ of loops of
$\Gamma$ which hit $[1,2n]\cap\mathbbm{Z}$ is the same as the set of loops in
$\Gamma$ which do not hit any vertices in $\mathbbm{Z}\setminus[0,2n]$.
Moreover, the conditional law of
$(\mathcal{L},\mathcal{R})|_{[0,2n]\cap\mathbbm{Z}}$ given $F_{n}$ is that of
a pair of independent uniform $2n$-step simple random walk excursions. By the
discussion surrounding (1.1), this implies that the conditional law of the
loop-decorated planar map $(M[1,2n],\Gamma_{n})$ given $F_{n}$ is that of the
planar map associated with a uniform meandric system of size $n$. Hence, it
suffices to show that if we condition on $F_{n}$, then except on an event of
conditional probability decaying faster than any negative power of $n$, there
is a loop in $\Gamma_{n}$ which has $\mathcal{M}[1,2n]$-graph-distance
diameter at least $n^{1/d-\zeta}$.
By Proposition 4.5 and (4.18), if $\beta,a_{0},a_{1}$ are as in Proposition
4.5, then it holds with conditional probability at least
$1-a_{0}c^{-1}n^{3}e^{-a_{1}n^{\beta}}$ given $F_{n}$ that there is a segment
of a loop or an infinite path in $\Gamma$ which hits only vertices of
$[1,2n]\cap\mathbbm{Z}$ and has $\mathcal{M}$-graph-distance diameter (and
hence also $\mathcal{M}[1,2n]$-graph-distance diameter) at least
$n^{1/d-\zeta}$. By the first sentence of the preceding paragraph, on $F_{n}$
this segment is in fact a segment of a loop $\ell\in\Gamma_{n}$. This loop
$\ell$ has $\mathcal{M}[1,2n]$-graph-distance diameter at least
$n^{1/d-\zeta}$. Since $a_{0}c^{-1}n^{3}e^{-a_{1}n^{\beta}}=O(n^{-p})$ for
every $p>0$, this concludes the proof. ∎
###### Proof of Proposition 1.4.
Theorem 1.5 immediately implies that except on an event of probability
decaying faster than any negative power of $n$, the graph-distance diameter of
$\mathcal{M}_{n}$ is at least $n^{1/d-\zeta}$.
To prove an upper bound for the graph-distance diameter of $\mathcal{M}_{n}$,
we first apply [GHS19, Theorem 1.15], which tells us that there exists an
exponent $\chi>0$ such that for each $\zeta\in(0,1)$ and each
$n\in\mathbbm{N}$, it holds except on an event of probability decaying faster
than any negative power of $n$ that the graph-distance diameter of
$\mathcal{G}[1,2n]$ is at most $n^{\chi+\zeta/2}$. It was shown in [GP21,
Theorem 3.1] that $\chi=1/d$.
By combining the preceding paragraph with Theorem 4.2, we get that for each
$\alpha>0$, there exists $A=A(\alpha)>0$ such that with probability at least
$1-O_{n}(n^{-\alpha})$, the graph-distance diameter of $\mathcal{M}[1,2n]$ is
at most $A(\log n)^{3}n^{1/d+\zeta/2}$, which is at most $n^{1/d+\zeta}$ if
$n$ is large enough. Since $\alpha$ can be made arbitrarily large, we get that
except on an event of probability decaying faster than any negative power of
$n$, the graph-distance diameter of $\mathcal{M}[1,2n]$ is at most
$n^{1/d+\zeta}$.
To transfer this from $\mathcal{M}[1,2n]$ to $\mathcal{M}_{n}$, we define the
event $F_{n}$ as in the proof of Theorem 1.5, condition on $F_{n}$, and apply
(4.18), exactly as in the proof of Theorem 1.5. ∎
## 5 Estimate for the mated-CRT map via SLE and LQG
To complete the proofs of our main results, it remains to prove Proposition
4.3. We will do this using SLE and LQG.
### 5.1 SLE/LQG description of the mated-CRT map
Recall that we previously defined the mated-CRT map using Brownian motion in
Section 4.1. In this subsection we will give the SLE/LQG description of the
mated-CRT map, which comes from the results of [DMS21]. We will not need many
properties of the SLE/LQG objects involved, so we will not give detailed
definitions. Instead, we give precise references.
Let $h$ be the random generalized function on $\mathbbm{C}$ associated with
the $\sqrt{2}$-quantum cone. The generalized function $h$ is a minor variant
of the whole-plane Gaussian free field; see [DMS21, Definition 4.10] for a
precise definition. One can associate with $h$ a random locally finite measure
$\mu_{h}$ on $\mathbbm{C}$, the $\sqrt{2}$-LQG measure, which is a limit of
regularized versions of $e^{\sqrt{2}h}\,dx\,dy$, where $dx\,dy$ denotes
Lebesgue measure on $\mathbbm{C}$ [Kah85, DS11]. The measure $\mu_{h}$ assigns
positive mass to every open subset of $\mathbbm{C}$ and zero mass to every
point but is mutually singular with respect to Lebesgue measure. See [BP,
Chapter 2] for a detailed account of the construction and properties of
$\mu_{h}$.
One can similarly associate with $h$ a random metric (distance function)
$D_{h}$ on $\mathbbm{C}$, the $\sqrt{2}$-LQG metric [DDDF20, GM21b]. The
metric $D_{h}$ induces the same topology on $\mathbbm{C}$ as the Euclidean
metric, but the Hausdorff dimension $d$ of the metric space
$(\mathbbm{C},D_{h})$ is strictly larger than 2 (this is the same $d$
appearing in Proposition 1.4). See [DDG21] for a survey of results about
$D_{h}$.
The metric measure space $(\mathbbm{C},D_{h},\mu_{h})$ possesses a scale
invariance property which will be important for our purposes:
$\text{$(\mathbbm{C},D_{h},\mu_{h})\overset{d}{=}(\mathbbm{C},r^{1/d}D_{h},r\mu_{h})$
as metric measure spaces $\forall r>0$}.$ (5.1)
In fact, one has the following slightly stronger property: for each $r>0$,
there is a random $\rho_{r}>0$ such that
$\mathopen{}\mathclose{{}\left(D_{h},\mu_{h}}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(r^{1/d}D_{h}(\rho_{r}\cdot,\rho_{r}\cdot),r\mu_{h}(\rho_{r}\cdot)}\right),\quad\forall
r>0.$ (5.2)
The scaling property (5.2) follows from the scaling property of $h$ [DMS21,
Proposition 4.13(i)] together with the fact that adding the constant
$\frac{1}{\sqrt{2}}\log r$ to $h$ results in scaling $\mu_{h}$ by $r$ and
$D_{h}$ by $r^{-1/d}$, both of which are immediate from the constructions of
$\mu_{h}$ and $D_{h}$ (see the proof of [GS22, Proposition 2.17] for a more
detailed explanation).
Whole-plane SLE8 from $\infty$ to $\infty$ is a random space-filling curve
$\eta$ which travels from $\infty$ to $\infty$ in $\mathbbm{C}$. It can be
thought of as a two-sided version of chordal SLE8 (see [DMS21, Footnote 4] for
a precise version of this statement). For each $z\in\mathbbm{C}$, a.s. $z$ is
hit exactly once by $\eta$, but there exist zero-Lebesgue measure sets of
points which are hit twice or three times.
Now suppose that we sample $\eta$ independently from the random generalized
function $h$ above, then re-parametrize $\eta$ so that
$\eta(0)=0\quad\text{and}\quad\mu_{h}(\eta([a,b]))=b-a,\quad\forall
a,b\in\mathbbm{R}\>\text{with}\>a<b.$ (5.3)
The law of $\eta$ is invariant under spatial scaling (this is immediate from
the definition in [DMS21, Footnote 4]), so it follows from (5.2) that
$\mathopen{}\mathclose{{}\left(D_{h},\mu_{h},\eta}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(r^{1/d}D_{h}(\rho_{r}\cdot,\rho_{r}\cdot),r\mu_{h}(\rho_{r}\cdot),\rho_{r}^{-1}\eta(\cdot/r)}\right),\quad\forall
r>0.$ (5.4)
The connection between the pair $(h,\eta)$ and the mated-CRT map $\mathcal{G}$
comes by way of the following theorem, which is a consequence of [DMS21,
Theorems 1.9 and 8.18].
###### Theorem 5.1.
With $h$ and $\eta$ as above, let $\mathcal{G}$ be the graph with vertex set
$\mathbbm{Z}$, with two distinct vertices $x,y\in\mathbbm{Z}$ joined by an
edge if and only if
$\eta([x-1,x])\cap\eta([y-1,y])\not=\emptyset.$ (5.5)
Then $\mathcal{G}$ has the same law (as a graph) as the mated-CRT map as
defined in (4.1).
The mated-CRT map has some double edges, but we do not worry about such edge
multiplicity in Theorem 5.1 since in this section we are only interested in
graph distances.
###### Remark 5.2.
The results and proofs in this section all carry over verbatim if we replace
$\sqrt{2}$-LQG by $\gamma$-LQG for $\gamma\in(0,2)$ and SLE8 by space-filling
SLEκ for $\kappa=16/\gamma^{2}$. In this setting, the corresponding mated-CRT
map is constructed from a pair of correlated Brownian motions with correlation
$-\cos(\pi\gamma^{2}/4)$, instead of a pair of independent Brownian motions as
in Section 4.1; and the value of $d$ depends on $\gamma$.
### 5.2 Proof of lower bounds for mated-CRT graph distances
Henceforth assume that we are in the setting of Theorem 5.1. Our goal is to
prove Proposition 4.3. To this end, we first prove a lemma that allows us to
compare $D_{h}$-distances and graph distances in $\mathcal{G}$.
For $n\in\mathbbm{N}$ and $z,w\in\eta([-n,n])\subset\mathbb{C}$, we slightly
abuse notation by writing
$\operatorname{dist}_{\mathcal{G}[-n,n]}(z,w):=\min\mathopen{}\mathclose{{}\left\\{\operatorname{dist}_{\mathcal{G}[-n,n]}(x,y):x,y\in[-n,n]\cap\mathbbm{Z},z\in\eta([x-1,x]),w\in\eta([y-1,y])}\right\\}.$
(5.6)
Note that a.s. for Lebesgue-a.e. point $z$, the cell $\eta([x-1,x])$
containing $z$ is unique (since a.s. $z$ is hit exactly once by $\eta$), and
that $\operatorname{dist}_{\mathcal{G}[-n,n]}(z,w)=0$ if $z$ and $w$ belong to
the same cell.
###### Lemma 5.3.
Fix $\zeta\in(0,1)$. It holds with polynomially high probability as
$n\rightarrow\infty$ that
$\operatorname{dist}_{\mathcal{G}[-n,n]}(z,w)\geq
n^{-\zeta}D_{h}\mathopen{}\mathclose{{}\left(z,w}\right)-1,\quad\forall
z,w\in\eta([-n,n]).$ (5.7)
###### Proof.
By [GS22, Proposition 3.13], for each $p>0$ and each $t\in\mathbbm{R}$, the
$p$th moment of the random variable $\sup_{u,v\in\eta([t-1,t])}D_{h}(u,v)$ is
bounded above by a finite constant which depends only on $p$ (not on $t$).
Therefore, we can apply Chebyshev’s inequality (for large positive moments)
followed by a union bound over all $x\in[-n,n]\cap\mathbbm{Z}$ to get that
with superpolynomially high probability as $n\rightarrow\infty$,
$\max_{x\in[-n,n]\cap\mathbbm{Z}}\sup_{u,v\in\eta([x-1,x])}D_{h}(u,v)\leq
n^{\zeta}.$ (5.8)
Henceforth assume that (5.8) holds.
Let $z,w\in\eta([-n,n])$ and let
$N:=\operatorname{dist}_{\mathcal{G}[-n,n]}(z,w)$. By the description of
$\mathcal{G}$ in Theorem 5.1, there exists
$x_{0},x_{1},\dots,x_{N}\in[-n,n]\cap\mathbbm{Z}$ such that the union of the
SLE8 segments $\eta([x_{j}-1,x_{j}])$ for $j=0,\dots,N$ contains a path from
$z$ to $w$. By (5.8), the $D_{h}$-diameter of each of these cells is at most
$n^{\zeta}$. By the triangle inequality,
$D_{h}(z,w)\leq n^{\zeta}(N+1)$
which is (5.7). ∎
The following lemma is the main LQG estimate needed for the proof of
Proposition 4.3.
###### Lemma 5.4.
For each $t>0$, it holds with probability tending to 1 as
$C\rightarrow\infty$, uniformly over the choice of $t$, that
$D_{h}\mathopen{}\mathclose{{}\left(\eta([-t,t]),\partial\eta([-Ct,Ct])}\right)\geq
t^{1/d}.$ (5.9)
Moreover, for each fixed $C>1$ it holds with probability tending to 1 as
$\delta\rightarrow 0$, uniformly over the choice of $t$, that
$D_{h}\mathopen{}\mathclose{{}\left((-\infty,0]\cap\overline{\eta\mathopen{}\mathclose{{}\left([-Ct,Ct]\setminus[-t,t]}\right)},[0,\infty)\cap\overline{\eta\mathopen{}\mathclose{{}\left([-Ct,Ct]\setminus[-t,t]}\right)}}\right)\geq\delta
t^{1/d}$ (5.10)
where here we identify $\mathbbm{R}$ with
$\mathbbm{R}\times\\{0\\}\subset\mathbbm{C}$.
###### Proof.
By the scale invariance property (5.4), it suffices to prove both (5.9) and
(5.10) in the case when $t=1$. We start with (5.9). Since $\eta$ a.s. fills
all of $\mathbbm{C}$ and LQG metric balls of finite radius are a.s. compact,
a.s. there exists some $C>1$ such that
$\mathopen{}\mathclose{{}\left\\{z\in\mathbbm{C}\,:\,D_{h}(z,\eta([-1,1]))\leq
1}\right\\}\subset\eta([-C,C]).$ (5.11)
Hence, (5.11) holds with probability tending to 1 as $C\rightarrow\infty$.
This shows that (5.9) with $t=1$ holds with probability tending to 1 as
$C\rightarrow\infty$.
Since a.s. 0 is contained in the interior of $\eta([-1,1])$, for any fixed
$C>1$ the compact sets
$(-\infty,0]\cap\overline{\eta\mathopen{}\mathclose{{}\left([-C,C]\setminus[-1,1]}\right)}\quad\text{and}\quad[0,\infty)\cap\overline{\eta\mathopen{}\mathclose{{}\left([-C,C]\setminus[-1,1]}\right)}$
lie at positive Euclidean distance from each other. Moreover, since $D_{h}$
induces the Euclidean topology on $\mathbbm{C}$, the two compact sets above
lie at positive $D_{h}$ distance from each other. Hence (5.10) with $t=1$
holds with probability tending to 1 as $\delta\rightarrow 0$. ∎
Figure 13: Illustration of the SLE/LQG event used in the proof of Proposition
4.3. On the event, the LQG distance between the inner and outer boundaries of
the green, light-blue, and gray annular regions, as well as the LQG distance
between the blue and red sets, are all bounded below. We note that the blue
and red sets are not connected, but each of these sets contains a connected
set whose closure intersects each of $\eta([-n,n])$ and
$\overline{\mathbbm{C}\setminus\eta([-C^{2}n,C^{2}n])}$.
###### Proof of Proposition 4.3.
See Figure 13 for an illustration. By Lemma 5.4, we can find $C>1$ such that
for each $t>0$, (5.9) holds with probability at least $1-(1-p)/5$. Applying
this with $t=n$, $t=Cn$, and $t=C^{2}n$ shows that with probability at least
$1-3(1-p)/5$,
$\displaystyle
D_{h}\mathopen{}\mathclose{{}\left(\eta([-n,n]),\partial\eta([-Cn,Cn])}\right)\geq
n^{1/d},\quad
D_{h}\mathopen{}\mathclose{{}\left(\eta([-Cn,Cn]),\partial\eta([-C^{2}n,C^{2}n])}\right)\geq(Cn)^{1/d},\quad\text{and}$
$\displaystyle
D_{h}\mathopen{}\mathclose{{}\left(\eta([-C^{2}n,C^{2}n]),\partial\eta([-C^{3}n,C^{3}n])}\right)\geq(C^{2}n)^{1/d}.$
(5.12)
By (5.10) of Lemma 5.4 (applied with $C^{2}$ instead of $C$ and $t=n$), we can
find $\delta\in(0,1)$ (depending on $C$) such that with probability at least
$1-(1-p)/5$,
$D_{h}\mathopen{}\mathclose{{}\left((-\infty,0]\cap\overline{\eta\mathopen{}\mathclose{{}\left([-C^{2}n,C^{2}n]\setminus[-n,n]}\right)},[0,\infty)\cap\overline{\eta\mathopen{}\mathclose{{}\left([-C^{2}n,C^{2}n]\setminus[-n,n]}\right)}}\right)\geq\delta
n^{1/d}.$ (5.13)
By Lemma 5.3 (applied with $C^{3}n$ in place of $n$ and $\zeta/2$ in place of
$\zeta$), for each large enough $n\in\mathbbm{N}$ it holds with probability at
least $1-(1-p)/5$ that
$\operatorname{dist}_{\mathcal{G}[-C^{3}n,C^{3}n]}(z,w)\geq
n^{-\zeta/2}D_{h}\mathopen{}\mathclose{{}\left(z,w}\right)-1,\quad\forall
z,w\in\eta([-C^{3}n,C^{3}n]).$ (5.14)
Henceforth assume that (5.2), (5.13), and (5.14) all occur, which happens with
probability at least $p$ if $n$ is large enough. Recall that two vertices
$x,y\in\mathbbm{Z}$ are joined by an edge of $\mathcal{G}$ if and only if
$\eta([x-1,x])\cap\eta([y-1,y])\not=\emptyset$. Hence each path in
$\mathcal{G}$ from $[-n,n]\cap\mathbbm{Z}$ to $\mathbbm{Z}\setminus[-Cn,Cn]$
has a sub-path which is contained in $[-Cn,Cn]\cap\mathbbm{Z}$ and which goes
from $[-n,n]\cap\mathbbm{Z}$ to a vertex $x\in\mathbbm{Z}$ whose corresponding
cell $\eta([x-1,x])$ intersects $\partial\eta([-Cn,Cn])$. Therefore, the first
inequality in (5.2) together with (5.14) implies (a) in the lemma statement
(provided $n$ is large enough that $n^{1/d-\zeta/2}-1\geq n^{1/d-\zeta}$). We
similarly obtain (b) and (c) from the second and third inequalities in (5.2).
To get (d), let $\Pi_{1}^{0}$ (resp. $\Pi_{2}^{0}$) be the set of
$x\in\mathbbm{Z}$ such that the cell $\eta([x-1,x])$ intersects
$(-\infty,0]\cap\overline{\eta\mathopen{}\mathclose{{}\left([-C^{2}n,C^{2}n]\setminus[-n,n]}\right)}$
(resp.
$[0,\infty)\cap\overline{\eta\mathopen{}\mathclose{{}\left([-C^{2}n,C^{2}n]\setminus[-n,n]}\right)}$).
Then each of $\Pi_{1}^{0}$ and $\Pi_{2}^{0}$ contains a connected subset of
$\mathbbm{Z}$ which intersects both $[-n,n]\cap\mathbbm{Z}$ and
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}]$. So, we can find a path $\Pi_{1}$ (resp.
$\Pi_{2}$) in $\mathcal{G}$ from $[-n,n]\cap\mathbbm{Z}$ to
$\mathbbm{Z}\setminus[-C^{2}n,C^{2}]$ which is contained in $\Pi_{1}^{0}$
(resp. $\Pi_{2}^{0}$) and which is contained in
$[-C^{2}n,C^{2}n]\cap\mathbbm{Z}$ except for its terminal endpoint.
By (5.13) and (5.14), the $\mathcal{G}[-C^{3}n,C^{3}n]$-graph distance between
$\Pi_{1}$ and $\Pi_{2}$ is at least $\delta n^{1/d-\zeta/2}-1$, which is at
least $n^{1/d-\zeta}$ if $n$ is large enough. Furthermore, by (c) from the
proposition statement (which we have already proven), the $\mathcal{G}$-graph
distance from each of $\Pi_{1}$ and $\Pi_{2}$ to
$\mathbbm{Z}\setminus[-Cn^{3},Cn^{3}]$ is at least $n^{1/d-\zeta}$. Hence, any
path in $\mathcal{G}$ from $\Pi_{1}$ to $\Pi_{2}$ has to have length at least
$n^{1/d-\zeta}$. This gives (d). ∎
## 6 Proofs for the UIHPMS
This section is organized as follows. In Section 6.1, we prove the equivalence
of our two definitions of the UIHPMS (Lemma 1.11). The rest of the section is
devoted to the proofs of Theorem 1.12 and Proposition 1.14. We start out in
Section 6.2 by introducing a special sequence of boundary points for the
UIHPMS, $\\{H_{m}\\}_{m\in\mathbbm{Z}}\subset\\{J_{k}\\}_{k\in\mathbbm{Z}}$.
The points $\\{H_{m}\\}_{m\in\mathbbm{Z}}$ may be viewed as marked points of a
minor variant of the UIHPMS that satisfies an invariance property under
translation by $H_{m}$ as in (1.10), but (crucially) without the constraint
that $m$ is even (Lemma 6.1). This property will be important for our purposes
since we will eventually need to use some parity arguments.
Sections 6.3 and 6.4 constitute the core part of the argument. In Section 6.3,
we prove that a.s. there are no semi-infinite paths in the UIHPMS that start
at one of the special boundary points $H_{m}$ and never hit any other special
boundary point (Lemma 6.3). This is done using a Burton-Keane style argument,
combined with the existence of certain special times for the pair of encoding
walks $(\mathcal{L},|\mathcal{R}|)$ (Lemma 6.5).
The non-existence of such semi-infinite paths allows us to define a perfect
matching on $\mathbbm{Z}$ by saying that $m\in\mathbbm{Z}$ is matched to
$m^{\prime}\in\mathbbm{Z}$ if and only if the path $P_{m}$ of arcs started
from $H_{m}$ hits $H_{m^{\prime}}$ before hitting any other special boundary
point (see Definition 6.2 for a precise definition of this path). In Section
6.4, we prove a general lemma for ergodic perfect matchings on $\mathbbm{Z}$
(Lemma 6.9) which in our setting implies that there are infinitely many paths
between good boundary points which disconnect 0 from $\infty$ in the UIHPMS
(Lemma 6.6). As explained in Section 6.5, these paths act as “shields” which
cannot be crossed by any infinite path in the UIHPMS. This allows us to prove
Theorem 1.12 and Proposition 1.14.
Throughout this section, we assume that the arcs of the UIMS and the UIHPMS
are drawn in such a way that each arc joining $x<y$ is contained in
$[x,y]\times\mathbbm{R}$. With this convention, it is easy to see from either
of the two definitions of the UIHPMS from Section 1.5 that for each boundary
point $J_{k}$ there is no arc which crosses the downward ray
$\\{J_{k}\\}\times(-\infty,0]$.
### 6.1 Equivalence of the definitions of the UIHPMS
Recall the two definitions of the UIHPMS from Section 1.5, one by cutting the
UIMS and one in terms of a simple random walk and an independent reflected
random walk. We now prove that the two definitions are equivalent.
We use a discrete version of celebrated Lévy’s theorem [Sim83], which we now
recall. Let $\mathcal{X}=(\mathcal{X}_{n})_{n\geq 0}$ be a (one-sided) simple
random walk with $\mathcal{X}_{0}=0$ and $M^{\mathcal{X}}$ be the running
minimum process of $\mathcal{X}$ from time $0$, that is
$M^{\mathcal{X}}_{n}=\min_{k\in[0,n]}\mathcal{X}_{k},\qquad\text{for }n\geq
0.$ (6.1)
Then
$(\mathcal{X}-M^{\mathcal{X}},-M^{\mathcal{X}})\stackrel{{\scriptstyle
d}}{{=}}(|\mathcal{X}|-\textbf{1}_{\\{\mathcal{X}>0\\}},\ell^{\mathcal{X}})$
(6.2)
where $\ell^{\mathcal{X}}_{n}$ denotes the number of times that $\mathcal{X}$
crosses $1/2$ during the time interval $[0,n]$, for all $n\geq 0$.
###### Proof of Lemma 1.11.
Let $(\mathcal{L},\mathcal{R})$ be the pair of independent two-sided simple
random walks on $\mathbbm{Z}$ used to construct the UIMS as in Section 1.4. We
assume that $\mathcal{L}_{0}=\mathcal{R}_{0}=0$. We first analyze the cutting
description of the UIHPMS. For each integer $k\in\mathbbm{Z}_{>0}$, let
$T_{k}$ be the smallest time $x\in\mathbbm{Z}_{>0}$ such that
$\mathcal{R}_{x}=-k$. Also, let $T_{-k}$ be the largest time
$x\in\mathbbm{Z}_{<0}$ such that $\mathcal{R}_{x}=-k$. By (1.5), for each
$k\in\mathbbm{Z}_{>0}$ the point $T_{k}$ is joined to $T_{-k}+1$ by a lower
arc of the UIMS . Furthermore, arcs of this type are the only ones which cross
$\\{1/2\\}\times(-\infty,0]$. Therefore, the UIHPMS under the cutting
definition is obtained from the UIMS by first removing the lower arcs joining
$T_{k}$ and $T_{-k}+1$ for $k\in\mathbbm{Z}_{>0}$, then adding lower arcs
joining $T_{2k-1}$ and $T_{2k}$ for each $k\in\mathbbm{Z}_{>0}$; and lastly
joining $T_{2k+1}+1$ and $T_{2k}+1$ for each $k\in\mathbbm{Z}_{<0}$.
Let $\widetilde{\mathcal{R}}$ be the random walk on $\mathbbm{Z}$ obtained
from $\mathcal{R}$ by replacing each of the downward steps
$\mathcal{R}_{T_{2k-1}}-\mathcal{R}_{T_{2k-1}-1}$ for $k\in\mathbbm{Z}_{>0}$
by an upward step, and replacing each of the upward steps
$\mathcal{R}_{T_{2k+1}+1}-\mathcal{R}_{T_{2k+1}}$ for $k\in\mathbbm{Z}_{<0}$
by a downward step (and otherwise leaving the steps of $\mathcal{R}$
unchanged). Then the cutting description of the UIHPMS is equivalent to the
random walk description (1.5) with $\widetilde{\mathcal{R}}$ in place of
$\mathcal{R}$. Therefore, it suffices to show that
$\widetilde{\mathcal{R}}\overset{d}{=}|\mathcal{R}|$.
Let $M^{\mathcal{R}}$ and $\ell^{\mathcal{R}}$ be as in (6.1) and (6.2), with
$\mathcal{X}=\mathcal{R}\mid_{[0,\infty)\cap\mathbbm{Z}}$. Then for
$x\in\mathbbm{Z}_{\geq 0}$, we have
$\widetilde{\mathcal{R}}_{x}=\mathcal{R}_{x}-M_{x}^{\mathcal{R}}+\textbf{1}_{\\{\text{$M_{x}^{\mathcal{R}}$
is odd}\\}}.$ (6.3)
On the other hand, the number $\ell_{x}^{\mathcal{R}}$ of crossings of $1/2$
by $\mathcal{R}$ during the time interval $[0,x]$ is odd if and only if
$\mathcal{R}_{x}>0$, so
$|\mathcal{R}|_{x}=|\mathcal{R}|_{x}-\textbf{1}_{\\{\mathcal{R}_{x}>0\\}}+\textbf{1}_{\\{\text{$\ell_{x}^{\mathcal{R}}$
is odd}\\}}.$ (6.4)
By combining the preceding two identities with (6.2), we get that
$\widetilde{\mathcal{R}}\mid_{[0,\infty)\cap\mathbbm{Z}}\overset{d}{=}|\mathcal{R}|\mid_{[0,\infty)\cap\mathbbm{Z}}$.
We similarly get the desired equality in law for negative times. ∎
### 6.2 Aperiodicity via good boundary points
In this subsection we address the following technical point. The law of the
UIHPMS is only invariant under _even_ translations along the boundary, i.e.,
translations by $J_{2k}$ for $k\in\mathbbm{Z}$; see (1.10). In one step of our
proof of Theorem 1.12 (Section 6.4) we will use an ergodicity argument which
requires us to know that a certain event for boundary points of odd index has
the same probability as the corresponding event for boundary points of even
index. For this reason, we need to introduce a variant of the UIHPMS where we
have translation invariance for _all_ boundary points, not just even boundary
points. The idea is to replace the reflected random walk $|\mathcal{R}|$ by a
walk which has some constant steps at height 0. This is similar to how one can
make a Markov chain aperiodic by introducing constant steps.
Figure 14: Illustration of the variant $\mathcal{M}^{\operatorname{cut}}$ of
the UIHPMS $\mathcal{M}^{\prime}$ constructed using the walk $\mathcal{Y}$ of
(6.5). Dashed arcs are the ones which are part of the original UIHPMS
$\mathcal{M}^{\prime}$ but not part of $\mathcal{M}^{\operatorname{cut}}$. The
graph of the reflected random walk $|\mathcal{R}|$ used in the construction of
the UIHPMS is shown at the bottom of the figure. Below each excursion of
$|\mathcal{R}|$ we have also shown the outcomes of the i.i.d. Bernoulli$(1/2)$
random variables $\\{\xi_{k}\\}_{k\in\mathbbm{Z}}$.
Let $\mathcal{R}$ be the two-sided simple random walk on $\mathbbm{Z}$ such
that $|\mathcal{R}|$ is used to construct the UIHPMS. We define the modified
walk
$\mathcal{Y}_{x}:=|\mathcal{R}|_{x}-\textbf{1}_{\\{\mathcal{R}_{x}>0\\}}.$
(6.5)
For $m\in\mathbbm{Z}$, let
$H_{m}:=\mathopen{}\mathclose{{}\left(\text{$m$th smallest $x\in\mathbbm{Z}$
such that $\mathcal{Y}_{x}=\mathcal{Y}_{x-1}=0$}}\right),$ (6.6)
with the numbering chosen so that $0\in[H_{0},H_{1}-1]\cap\mathbbm{Z}$.
Recall that we denote by $J_{k}$ (resp. $J_{-k+1}$) the $k$th positive (resp.
non-positive) boundary point of the UIHPMS. In particular, for each
$k\in\mathbbm{Z}$, $J_{2k}=J_{2k+1}-1$ and
$\mathcal{R}_{J_{2k}}=\mathcal{R}_{J_{2k+1}-1}=0$; and these are all the zeros
of $\mathcal{R}$. Conditional on $|\mathcal{R}|$, the signs of the excursions
$\\{\mathcal{R}|_{[J_{2k-2},J_{2k}]}\\}_{k\in\mathbbm{Z}}$ of $|\mathcal{R}|$
away from 0 are i.i.d. Bernoulli$(1/2)$ random variables. Therefore,
$\mathcal{Y}$ can equivalently be constructed from $|\mathcal{R}|$ as follows.
Start with a collection $\\{\xi_{k}\\}_{k\in\mathbbm{Z}}$ of i.i.d.
Bernoulli$(1/2)$ random variables independent from $|\mathcal{R}|$. Then, for
each $k\in\mathbbm{Z}$ such that $\xi_{k}=1$, replace the steps
$|\mathcal{R}|_{J_{2k}}-|\mathcal{R}|_{J_{2k}-1}$ and
$|\mathcal{R}|_{J_{2k-1}}-|\mathcal{R}|_{J_{2k-1}-1}$ by constant steps. In
particular,
$\\{H_{m}\\}_{m\in\mathbbm{Z}}\subset\\{J_{k}\\}_{k\in\mathbbm{Z}}\quad\text{and}\quad\mathbbm{P}\mathopen{}\mathclose{{}\left[J_{k}\in\\{H_{m}\\}_{m\in\mathbbm{Z}}\,|\,|\mathcal{R}|}\right]=\frac{1}{2},\quad\forall
k\in\mathbbm{Z}.$ (6.7)
We call $H_{m}$ the $m$th good boundary point of the UIHPMS. See Figure 14 for
an illustration.
Let $(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$ be the
infinite planar map decorated by a collection of loops and paths which is
defined exactly as in (1.5) but with $\mathcal{Y}$ in place of $\mathcal{R}$,
with the convention that if $\mathcal{Y}_{x}=\mathcal{Y}_{x-1}=0$, then there
is no lower arc incident to $x\in\mathbbm{Z}$. The discussion just above
implies that $(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$
can be obtained from the UIHPMS by removing each lower arc joining boundary
points $J_{2k-1}$ and $J_{2k}$ for $k\in\mathbbm{Z}$ such that $\xi_{k}=1$.
The good boundary points are exactly these points $J_{2k-1}$ and $J_{2k}$ for
$k\in\mathbbm{Z}$ such that $\xi_{k}=1$. In particular, they are not
determined by the UIHPMS.
One can also see from the proof of the equivalence of the two definitions of
the UIHPMS (Lemma 1.11) that
$(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$ has the same
law as the decorated planar map obtained from the UIMS by removing all of the
lower arcs which intersect $\\{1/2\\}\times(-\infty,0]$. Moreover, in this
interpretation, the good boundary points are just the endpoints of such
removed arcs.
The following lemma is our main reason for introducing the objects in this
subsection. It will be used in the proof of Lemma 6.6 below.
###### Lemma 6.1.
Let $\widehat{\mathcal{Y}}$ be sampled from the conditional law of
$\mathcal{Y}$ given the event
$\\{\mathcal{Y}_{0}=\mathcal{Y}_{-1}=0\\}=\\{H_{0}=0\\}.$ (6.8)
Let
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
be defined in the same manner as
$(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$ with
$\widehat{\mathcal{Y}}$ in place of $\mathcal{Y}$ and let
$\\{\widehat{H}_{m}\\}_{m\in\mathbbm{Z}}$ be as in (6.6) with
$\widehat{\mathcal{Y}}$ in place of $\mathcal{Y}$. Then
$\mathopen{}\mathclose{{}\left(\widehat{\mathcal{M}}^{\operatorname{cut}}-\widehat{H}_{m},\widehat{\Gamma}^{\operatorname{cut}}-\widehat{H}_{m}}\right)\overset{d}{=}\mathopen{}\mathclose{{}\left(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}}}\right),\quad\forall
m\in\mathbbm{Z}.$ (6.9)
Furthermore, for each positive integer $r\in\mathbbm{Z}_{>0}$, each event
which is invariant under translations of the form (6.9) with $m$ restricted to
lie in $r\mathbbm{Z}$ has probability zero or one.
###### Proof.
Since
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
is constructed from $\widehat{\mathcal{Y}}$ and an independent two-sided
simple random walk $\mathcal{L}$ in the manner of (1.5), it suffices to show
that
$\widehat{\mathcal{Y}}_{\widehat{H}_{m}+\,\cdot}\overset{d}{=}\widehat{\mathcal{Y}}_{\cdot}\,,\quad\forall
m\in\mathbbm{Z},$ (6.10)
and that any event which is invariant under translations of the form (6.10)
with $m$ restricted to lie in $r\mathbbm{Z}$ has probability zero or one.
To this end, we first consider the unconditioned random walk
$\mathcal{Y}=|\mathcal{R}|-\textbf{1}_{\\{\mathcal{R}>0\\}}$ from (6.5). Let
$\\{\tau_{j}\\}_{j\in\mathbbm{Z}}$ be the times such that $\mathcal{Y}_{x}=0$,
numbered from left to right in such a way that $\tau_{0}=0$.
Let $(\mathcal{X},M^{\mathcal{X}})$ be a one-sided simple random walk on
$\mathbbm{Z}$ started from $0$ at time $0$ and its running minimum process, as
in (6.1). Also let $\mu$ be the law of $\mathcal{X}-M^{\mathcal{X}}$ stopped
at the first positive time when it reaches zero. A sample from $\mu$ can be
produced as follows: Flip a fair coin. If the coin comes up heads, we run a
simple random walk started from 0, conditioned on the event that its first
step is upward, until the first positive time when it reaches zero. If the
coin comes up tails, we instead take the path which takes one step from 0 to
0. From this description, we get that the law of a path sampled from $\mu$ is
invariant under time reversal.
By the discrete version of Lévy’s theorem (6.2), the process
$\mathcal{Y}|_{[0,\infty)\cap\mathbbm{Z}}$ has the same law as
$\mathcal{X}-M^{\mathcal{X}}$. By the strong Markov property of $\mathcal{X}$,
we get that the excursions
$\mathcal{Y}|_{[\tau_{j-1},\tau_{j}]\cap\mathbbm{Z}}$ for $j\geq 1$ are i.i.d.
samples from $\mu$. The law of $\mathcal{Y}$ is invariant under time reversal,
so the increments $\mathcal{Y}|_{[\tau_{j-1},\tau_{j}]\cap\mathbbm{Z}}$ for
$j\leq 0$ are also i.i.d., and have the same law as the time reversal of a
sample from $\mu$. From the time reversal symmetry of $\mu$, we get that the
law of the whole collection of increments
$\mathcal{Y}|_{[\tau_{j-1},\tau_{j}]\cap\mathbbm{Z}}$ for $j\in\mathbbm{Z}$
are i.i.d. samples from $\mu$.
The points $H_{m}$ for $m\in\mathbbm{Z}$ coincide precisely with the times
$\tau_{j}$ for which $\tau_{j}=\tau_{j-1}+1$. From this and the above
description of the increments
$\mathcal{Y}|_{[\tau_{j-1},\tau_{j}]\cap\mathbbm{Z}}$, we infer that the
increments
$\widehat{\mathcal{Y}}|_{[\widehat{H}_{m-1},\widehat{H}_{m}]\cap\mathbbm{Z}}$
for $m\in\mathbbm{Z}$ are i.i.d. samples from the law of
$\mathcal{Y}|_{[0,\infty)\cap\mathbbm{Z}}$ stopped at the first positive time
$x$ such that $\mathcal{Y}_{x}=\mathcal{Y}_{x-1}=0$. From this and the fact
that $\widehat{\mathcal{Y}}$ is sampled from the conditional law of
$\mathcal{Y}$ given the event $\\{\mathcal{Y}_{0}=\mathcal{Y}_{-1}=0\\}$, the
translation invariance property (6.10) is immediate. Furthermore, the desired
ergodicity property for $\widehat{\mathcal{Y}}$ follows from the zero-one law
for translation invariant events depending on a sequence of i.i.d. random
variables, applied to the i.i.d. random variables
$\widehat{\mathcal{Y}}|_{[\widehat{H}_{rn},\widehat{H}_{r(n+1)}]\cap\mathbbm{Z}}$
for $n\in\mathbbm{Z}$.
We remark that one could also prove (6.10) by showing that if $U_{m}$ is
sampled uniformly from $[-m,m]\cap\mathbbm{Z}$, then the law of
$\mathcal{Y}_{H_{U_{m}}+\cdot}$ converges as $m\to\infty$ to the law of
$\widehat{\mathcal{Y}}$. We will not give the details. ∎
### 6.3 No semi-infinite paths started from the boundary
Define the UIHPMS and its left-to-right ordered sequence of boundary points
$\\{J_{k}\\}_{k\in\mathbbm{Z}}\subset\mathbbm{Z}$ as in Section 1.5. Also
define the left-to-right ordered sequence of good boundary points
$\\{H_{m}\\}_{m\in\mathbbm{Z}}$ as in (6.6).
###### Definition 6.2.
For each good boundary point $H_{m}$ with $m\in\mathbbm{Z}$, let $P_{m}$ be
the unique directed path of arcs in the UIHPMS starting from $H_{m}$,
following the arc in the _upper_ half-plane incident to $H_{m}$, and ending at
the first good boundary point other than $H_{m}$ which is hit by the path, if
it exists; otherwise let $P_{m}$ be the whole semi-infinite path started from
$H_{m}$. We call $P_{m}$ the boundary path started from $H_{m}$.
By definition, the boundary path $P_{m}$ hits good boundary points of the
UIHPMS only at its endpoints. However, it is in principle possible that for
some values of $m$, the path $P_{m}$ never hits another good boundary point
other than $H_{m}$, in which case it is semi-infinite. The first step in the
proof of Theorem 1.12 is to rule this event out.
###### Lemma 6.3.
Almost surely, the boundary path $P_{m}$ is finite for all $m\in\mathbbm{Z}$.
The proof of Lemma 6.3 proceeds as follows. By even translation invariance
(1.10), the probability that $J_{k}$ is a good boundary point (i.e.,
$J_{k}=H_{m}$ for some $m\in\mathbbm{Z}$) and $P_{m}$ is semi-infinite depends
only on the parity of $k$. If this probability is positive for even values of
$k$, say, then by the ergodic theorem a.s. $P_{m}$ is semi-infinite for a
positive fraction of the indices $m\in\mathbbm{Z}$. We will show that this
cannot be the case by a Burton-Keane style argument. Roughly speaking, we will
use certain special times for the encoding walks (Lemma 6.5) to argue that
there is not enough “room” for there to be a positive density of values of $m$
for which $P_{m}$ is semi-infinite. We start with some bounds on the
probability that a bridge has at least some number of crossings of $1/2$.
###### Lemma 6.4.
For a simple random walk $\mathcal{X}$ on $\mathbbm{Z}$ with
$\mathcal{X}_{0}=0$, recall that $\ell^{\mathcal{X}}_{n}$ denotes the number
of times that $\mathcal{X}$ crosses $1/2$ during the time interval $[0,n]$.
For $n,k>0$, we have
$\frac{2^{2k-1}\binom{2n-2k}{n}}{\binom{2n}{n}}\leq\mathbbm{P}\mathopen{}\mathclose{{}\left[\ell^{\mathcal{X}}_{2n}\geq
2k\;\middle|\;\mathcal{X}_{2n}=0}\right]\leq\frac{2^{k}\binom{2n-k}{n}}{\binom{2n}{n}}.$
(6.11)
As a particular consequence, for each $\varepsilon>0$ and each integer $n>0$,
$C_{1}\leq\mathbbm{P}\mathopen{}\mathclose{{}\left[\ell^{\mathcal{X}}_{2n}\geq\varepsilon\sqrt{n}\;\middle|\;\mathcal{X}_{2n}=0}\right]\leq
C_{2}\quad\text{and}\quad
C_{1}\leq\mathbbm{P}\mathopen{}\mathclose{{}\left[\ell^{\mathcal{X}}_{2n}\geq\varepsilon\sqrt{n}\;\middle|\;(\mathcal{X}_{2n-1},\mathcal{X}_{2n})=(1,0)}\right]\leq
C_{2},$ (6.12)
for constants $C_{1},C_{2}\in(0,1)$ only depending on $\varepsilon$.
###### Proof.
We call a simple random walk path starting and ending at $0$ an _excursion_ if
it does not hit $0$ except at its two endpoints (we allow excursions to be
positive or negative). If $\mathcal{X}_{2n}=0$, we can decompose the path of
$\mathcal{X}\mid_{[0,2n]\cap\mathbbm{Z}}$ into several excursions. The number
$\ell^{\mathcal{X}}_{2n}$ is exactly twice the number of positive excursions.
Hence, if $\mathcal{X}_{2n}=0$ and $\ell^{\mathcal{X}}_{2n}\geq 2k$, then the
path of $\mathcal{X}\mid_{[0,2n]\cap\mathbbm{Z}}$ can be decomposed into at
least $k$ excursions. Therefore, we compute
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathcal{X}\mid_{[0,2n]\cap\mathbbm{Z}}\text{has
at least $k$ excursions}\;\middle|\;\mathcal{X}_{2n}=0}\right]$ in the next
paragraph to obtain an upper bound for
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\ell^{\mathcal{X}}_{2n}\geq
2k\;\middle|\;\mathcal{X}_{2n}=0}\right]$.
Assume that the path of $\mathcal{X}\mid_{[0,2n]\cap\mathbbm{Z}}$ with
$\mathcal{X}_{2n}=0$ can be decomposed into at least $k$ excursions. By making
the last $k$ excursions all positive (by flipping if necessary) and removing
the last downward step of each of these $k$ excursions, we obtain a simple
walk from $0$ to $k$ with $2n-k$ steps, i.e. with $n$ upward steps and $n-k$
downward steps. The total number of such walks is $\binom{2n-k}{n}$. In fact,
this defines a $2^{k}$-to-$1$ map, as the modified positive excursions can be
recovered from the resulting walk from $0$ to $k$ (the last visit of
$j\in[1,k]\cap\mathbbm{Z}$ corresponds to where a downward step was removed),
and there are $2^{k}$ ways to assign a sign for each excursion to obtain
$\mathcal{X}\mid_{[0,2n]}$. As there are $\binom{2n}{n}$ possible walks from
$0$ to $0$ with $2n$ steps, the upper bound in (6.11) follows.
The lower bound of (6.11) is obtained via a similar argument. It suffices to
notice that a simple walk from $0$ to $2k$ with $2n-2k$ steps can be turned
(as done above) in a walk from 0 to 0 with $2n$ steps and at least $k$
positive excursions among the last $2k$ ones in at least $2^{2k-1}$ ways (by
symmetry).
The first bound in (6.12) is obtained by setting
$k=\lfloor\varepsilon\sqrt{n}/2\rfloor$ in (6.11) and applying Stirling’s
formula. To add the extra condition $\mathcal{X}_{2n-1}=1$ in the second bound
in (6.12), we just fix the last excursion to be positive in the excursion
decomposition defined above, which gives a similar bound. ∎
Let $\mathcal{L}$ and $\mathcal{R}$ be the independent two-sided simple random
walks on $\mathbb{Z}$ with $\mathcal{L}_{0}=\mathcal{R}_{0}=0$ used in the
definition of the UIHPMS from (1.9). We call $x\in\mathbbm{Z}_{>0}$ an upper
block if $x$ is linked to some point in $(-\infty,0]\cap\mathbbm{Z}$ by an arc
above the real line. If $x$ is an upper block, no arc above the real line
connects $(0,x)\cap\mathbbm{Z}$ and $(x,\infty)\cap\mathbbm{Z}$; otherwise
such an arc would have to cross the upper arc incident to $x$. In terms of the
random walk description, $x\in\mathbbm{Z}_{>0}$ is an upper block if and only
if $\mathcal{L}\mid_{[0,x]\cap\mathbbm{Z}}$ attains its _unique_ minimum value
at time $x$.
Recall that for $k>0$ we denoted by $J_{k}$ the $k$th positive boundary point
of the UIHPMS. We call $x\in 2\mathbbm{Z}_{>0}$ a block if $x$ is an upper
block and $x=J_{2k}$ for some $k>0$. That is, $x\in 2\mathbbm{Z}_{>0}$ is a
block if and only if $|\mathcal{R}|_{x}=0$ and
$\mathcal{L}\mid_{[0,x]\cap\mathbbm{Z}}$ attains its unique minimum value at
time $x$. See Figure 15 for an example. Our motivation for the definition of a
block is that if $x$ is a block, then no path of arcs started at a point in
$(0,x)\cap\mathbbm{Z}$ can cross the vertical line $\\{x\\}\times\mathbbm{R}$
without first hitting $(-\infty,0]\cap\mathbbm{Z}$.
Figure 15: An illustration of a block $x=J_{2k}$ with some upper blocks and
boundary points in a subset of the UIHPMS. The right endpoints of the purple
arcs are upper blocks, while the red arcs are not incident to any upper
blocks. Boundary points are the endpoints of the lower green arcs. Lemma 6.5
describes a special type of block such that the number of purple arcs is much
fewer than the number of green arcs.
###### Lemma 6.5.
Fix $\varepsilon>0$. Almost surely, there are infinitely many $k>0$ such that
$J_{2k}$ is a block and there are at most $\varepsilon k$ upper blocks in
$(0,J_{2k}]\cap\mathbbm{Z}$.
###### Proof.
In light of Lévy’s theorem (6.2), consider another two-sided simple random
walk $\widetilde{\mathcal{L}}\overset{d}{=}\mathcal{L}$ such that $n>0$ is an
upper block of $\mathcal{L}$ if and only if $\widetilde{\mathcal{L}}$ crosses
$1/2$ in between times $n-1$ and $n$. Hence, the number of upper blocks in
$(0,2n]\cap\mathbbm{Z}$ is equal to $\ell_{2n}^{\widetilde{\mathcal{L}}}$,
where $\ell$ is defined in (6.2). Also, the number of zeros of $\mathcal{R}$
in $(0,2n]\cap\mathbbm{Z}$ is at least $\ell^{\mathcal{R}}_{2n}/2$ because at
most two crossings of $1/2$ can correspond to the same zero of $\mathcal{R}$.
Therefore, it is enough to prove that a.s. there are infinitely many integers
$n>0$ such that
$(\widetilde{\mathcal{L}}_{2n-1},\widetilde{\mathcal{L}}_{2n})=(1,0),\quad\mathcal{R}_{2n}=0,\quad\ell^{\mathcal{R}}_{2n}\geq
2\sqrt{n}\quad\text{ and
}\quad\ell^{\widetilde{\mathcal{L}}}_{2n}<\varepsilon\sqrt{n}.$ (6.13)
Indeed, the first two conditions guarantees that $2n$ is a block and so
$2n=J_{2k}$ for some $k>0$, the third condition guarantees that there are at
least $\sqrt{n}$ boundary points in $(0,J_{2k}]$ and so $2k\geq\sqrt{n}$, and
the fourth condition guarantees that there are at most $\varepsilon\sqrt{n}$
upper blocks in $(0,J_{2k}]$.
Let $A_{n}$ be the event that (6.13) holds. Using that
$\mathbbm{P}[(\widetilde{\mathcal{L}}_{2n-1},\widetilde{\mathcal{L}}_{2n},\mathcal{R}_{2n})=(1,0,0)]\sim\frac{1}{2\pi
n}$ for large $n$, the estimates in (6.12) implies that
$\mathbbm{P}\mathopen{}\mathclose{{}\left[A_{n}}\right]\geq\frac{C_{3}}{n}$
for some constant $C_{3}>0$ only depending on $\varepsilon$. In particular,
$\sum_{n=1}^{\infty}\mathbbm{P}\mathopen{}\mathclose{{}\left[A_{n}}\right]=\infty$.
We now want to apply the Kochen-Stone theorem [KS64], which asserts that if
$\sum_{n=1}^{\infty}\mathbbm{P}\mathopen{}\mathclose{{}\left[A_{n}}\right]=\infty$
and
$\liminf_{N\to\infty}\frac{\sum_{n_{1},n_{2}=1}^{N}\mathbbm{P}[A_{n_{1}}\cap
A_{n_{2}}]}{(\sum_{n=1}^{N}\mathbbm{P}[A_{n}])^{2}}<\infty,$ (6.14)
then $\mathbbm{P}[A_{n}\text{ infinitely often}]>0$. For $n_{2}>n_{1}>0$, we
have
$\mathbbm{P}[A_{n_{1}}\cap
A_{n_{2}}]\leq\mathbbm{P}[(\widetilde{\mathcal{L}}_{2n_{1}},\widetilde{\mathcal{L}}_{2n_{2}},\mathcal{R}_{2n_{1}},\mathcal{R}_{2n_{2}})=(0,0,0,0)]\leq\frac{C_{4}}{n_{1}(n_{2}-n_{1})},$
and similarly $\mathbbm{P}[A_{n}]\leq\frac{C_{4}}{n}$ for $n>0$, with some
constant $C_{4}$. It follows that
$\sum_{n_{1},n_{2}=1}^{N}\mathbbm{P}[A_{n_{1}}\cap A_{n_{2}}]\leq 2\sum_{1\leq
n_{1}\leq n_{2}\leq N}\mathbbm{P}[A_{n_{1}}\cap A_{n_{2}}]\leq
2C_{4}\mathopen{}\mathclose{{}\left(\sum_{n_{1}=1}^{N}\frac{1}{n_{1}}+\sum_{n_{1},n_{2}=1}^{N}\frac{1}{n_{1}n_{2}}}\right)\leq
C_{5}\log^{2}(N),$
for another constant $C_{5}$. Combined with our above lower bound for
$\mathbbm{P}[A_{n}]$, this implies (6.14). Therefore, $\mathbbm{P}[A_{n}\text{
infinitely often}]>0$, and Kolmogorov’s zero-one law assures that $A_{n}$
happens infinitely often, almost surely. ∎
###### Proof of Lemma 6.3.
We prove the lemma via a Burton-Keane type argument. Since the UIHPMS is
invariant under even translations along the boundary (1.10), we first look at
the case of good boundary points of the form $J_{2k}$ for $k\in\mathbbm{Z}$.
The odd case will be treated analogously at the end of the proof. For
$k\in\mathbbm{Z}$, let $E_{k}$ be the event that $J_{2k}$ is a good boundary
point, i.e., $J_{2k}=H_{m}$ for some $m\in\mathbbm{Z}$, and the boundary path
$P_{m}$ is semi-infinite, that is $P_{m}$ does not hit another good boundary
point other than $H_{m}$. By the translation invariance of the UIHPMS (1.10)
and the fact that the set of $k\in\mathbbm{Z}$ such that $J_{2k}$ is good is
independent from the UIHPMS (see the discussion just below (6.7)),
$p:=\mathbbm{P}[E_{k}]$ does not depend on $k$. We need to show that $p=0$.
Assume for contradiction that $p>0$. By the Birkhoff ergodic theorem,
$\lim_{k\to\infty}\frac{1}{k}\\#\mathopen{}\mathclose{{}\left\\{k^{\prime}\in[1,k]\cap\mathbbm{Z}:E_{k^{\prime}}\text{
occurs}}\right\\}=p.$ (6.15)
By Lemma 6.5 with $\varepsilon=p/4$ and (6.15), almost surely, there exist
arbitrarily large values of $k\in\mathbbm{Z}_{>0}$ such that:
1. (i)
$J_{2k}$ is a block;
2. (ii)
there are at most $pk/4$ upper blocks in $(0,J_{2k})\cap\mathbbm{Z}$;
3. (iii)
$\\#\\{k^{\prime}\in[1,k]\cap\mathbbm{Z}:E_{k^{\prime}}\text{
occurs}\\}>pk/2$.
Fix $k\in\mathbbm{Z}_{>0}$ with the above three properties. For each
$k^{\prime}\in(0,k)\cap\mathbbm{Z}$ such that $E_{k^{\prime}}$ occurs, let
$m(k^{\prime})\in\mathbbm{Z}$ such that $H_{m(k^{\prime})}=J_{2k^{\prime}}$.
Each semi-infinite path $P_{m(k^{\prime})}$ for
$k^{\prime}\in(0,k)\cap\mathbbm{Z}$ such that $E_{k^{\prime}}$ occurs does not
intersect any other such semi-infinite path. Since $J_{2k}$ is a block, there
is no arc connecting $(0,J_{2k})\cap\mathbbm{Z}$ and
$(J_{2k},\infty)\cap\mathbbm{Z}$. Therefore, each semi-infinite path
$P_{m(k^{\prime})}$ for $k^{\prime}\in(0,k)\cap\mathbbm{Z}$ must exit the
interval $(0,J_{2k})\cap\mathbbm{Z}$ via some arc joining
$(0,J_{2k})\cap\mathbbm{Z}$ to $(-\infty,0]\cap\mathbbm{Z}$. This arc must be
above the real line, since by the construction of the UIHPMS there are no arcs
below the real line that cross the ray $\\{1/2\\}\times(-\infty,0]$. In other
words, the right endpoint of such an arc is an upper block. It follows that
the number of upper blocks in $(0,J_{2k})\cap\mathbbm{Z}$ is at least the
number of semi-infinite paths started from points in
$(0,J_{2k})\cap\mathbbm{Z}$, which is at least $pk/2$. But, we have chosen $k$
so that the number of upper blocks in $(0,J_{2k})\cap\mathbbm{Z}$ is at most
$pk/4$. This yields a contradiction, and thus $p=0$.
Exactly the same argument shows that a.s. the event $E_{k}^{\prime}$ that
$J_{2k+1}=H_{m}$ for some $m\in\mathbbm{Z}$ and $P_{m}$ is semi-infinite does
not occur for any $k\in\mathbbm{Z}$. This finishes the proof. ∎
### 6.4 Infinitely many paths separating the origin from $\infty$
Lemma 6.3 implies that a.s. there are no semi-infinite paths in the UIHPMS
which start at a good boundary point and never hit another good boundary
point. The following lemma will allow us in Section 6.5 to rule out more
general types of infinite paths.
###### Lemma 6.6.
Almost surely, there exist infinitely many $m\in\mathbbm{Z}_{>0}$ such that
the boundary path $P_{m}$ ends at $H_{m^{\prime}}$ for some
$m^{\prime}\in\mathbbm{Z}_{\leq 0}$.
We will deduce Lemma 6.6 from a general result (Lemma 6.9) for random perfect
matchings.
###### Definition 6.7.
A perfect matching of $\mathbbm{Z}$ is a function
$\phi:\mathbbm{Z}\rightarrow\mathbbm{Z}$ such that $\phi(k)\not=k$ and
$\phi(\phi(k))=k$ for each $k\in\mathbbm{Z}$. We say that $j,k\in\mathbbm{Z}$
are matched if $\phi(j)=k$, equivalently, $\phi(k)=j$. We say that $\phi$ is
non-crossing if there do not exist integers $j_{1},j_{2}$ such that
$j_{1}<j_{2}<\phi(j_{1})<\phi(j_{2})$.
###### Definition 6.8.
A random perfect matching $\phi$ is stationary if
$\phi(\cdot+m)\overset{d}{=}\phi(\cdot)$ for each $m\in\mathbbm{Z}$. A
stationary perfect matching is strongly ergodic if any event which is
invariant under shifts of the form $\phi(\cdot)\mapsto\phi(\cdot+2m)$ for all
$m\in\mathbbm{Z}$ has probability zero or one.
###### Lemma 6.9.
Let $\phi$ be a strongly ergodic random perfect matching of $\mathbbm{Z}$.
Almost surely, for each $j\in\mathbbm{Z}$ there exists $k\geq j$ such that
$\phi(k)\leq j-1$. If $\phi$ is non-crossing, then a.s. for each
$j\in\mathbbm{Z}$ there are infinitely many integers $k\geq j$ such that
$\phi(k)\leq j-1$.
###### Proof of Lemma 6.9.
We say that $j\in\mathbbm{Z}$ is exposed if there does not exist $k\geq j$
such that $\phi(k)\leq j-1$. If we represent the matching by an arc diagram
(with each arc matching $x$ and $\phi(x)$ plotted in the region
$[x\wedge\phi(x),x\vee\phi(x)]\times[0,\infty)$) then $j$ being exposed is
equivalent to the condition that there are no arcs crossing
$\\{j-1/2\\}\times[0,\infty)$.
We claim that a.s. there are no exposed integers. By the stationarity of
$\phi$, the probability that $j\in\mathbbm{Z}$ is exposed does not depend on
$j$. By the Birkhoff ergodic theorem, if any integer has a positive
probability to be exposed, then a.s. there are infinitely many exposed
integers. Hence we just need to show that the probability that there are
infinitely many exposed integers is zero.
Suppose that $j_{1}\leq j_{2}$ are both exposed. By the definition of exposed,
every integer in the set $\\{j_{1},\dots,j_{2}-1\\}$ must have its match in
$\\{j_{1},\dots,j_{2}-1\\}$. Hence $\\#\\{j_{1},\dots,j_{2}-1\\}$ must be
even, so $j_{1}$ and $j_{2}$ must have the same parity. Consequently, a.s.
either every exposed integer is even or every exposed integer is odd. By
stationarity, the probabilities of the events
$\\{\text{$\exists$ infinitely many even exposed
integers}\\}\quad\text{and}\quad\\{\text{$\exists$ infinitely many odd exposed
integers}\\}$ (6.16)
are the same. The two events in (6.16) are invariant under translations of the
form $\phi\mapsto\phi(\cdot+2m)$ for $m\in\mathbbm{Z}$, so by strong
ergodicity each of these events has probability zero or one. Since we have
just seen that these two events are disjoint, they must each have probability
zero. Hence a.s. there are no exposed integers.
Now assume that $\phi$ is non-crossing and consider a $j\in\mathbbm{Z}$ with
the property that there are only finitely many integers $k\geq j$ such that
$\phi(k)\leq j-1$. We will show that there exists an exposed integer. This,
combined with our earlier result, will then imply that no such $j$ exists.
To construct an exposed integer, let $k_{*}$ be the largest $k\geq j$ such
that $\phi(k)\leq j-1$. We claim that $k_{*}+1$ is exposed. Indeed, suppose
for contradiction that there exists $k\geq k_{*}+1$ with $\phi(k)\leq k_{*}$.
By the maximality of $k_{*}$, we must have $\phi(k)\geq j$. Since
$\phi(k_{*})\leq j-1$, we have $\phi(k)\not=k_{*}$ and so $\phi(k)\leq
k_{*}-1$. Therefore,
$\phi(k_{*})\leq j-1<j\leq\phi(k)<k_{*}<k,$
which contradicts the non-crossing condition. ∎
###### Proof of Lemma 6.6.
We divide the proof in three main steps.
Step 1: Reducing to the setting of Lemma 6.1. Let
$(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$ be the
variant of the UIHPMS where we remove the lower arcs joining the good boundary
points $\\{H_{m}\\}_{m\in\mathbbm{Z}}$, as in Section 6.2. By Definition 6.2,
the boundary path $P_{m}$ for $m\in\mathbbm{Z}$ does not traverse any lower
arcs in the UIHPMS joining pairs of good boundary points. Hence, the
definition of $P_{m}$ and the statement of the lemma are unaffected if we
replace the UIHPMS by
$(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$.
Let
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
be sampled from the law of
$(\mathcal{M}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$ conditioned
on the event that $H_{0}=0$, as in Lemma 6.1. Define the good boundary points
$\\{\widehat{H}_{m}\\}_{m\in\mathbbm{Z}}$ and the boundary paths
$\\{\widehat{P}_{m}\\}_{m\in\mathbbm{Z}}$ with
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
in place of the original UIHPMS.
We claim that it suffices to show that a.s.
1. $(\boxdot)$
$\forall m_{0}\in\mathbbm{Z}$, $\exists$ infinitely many $m>m_{0}$ s.t. the
path $\widehat{P}_{m}$ in $\widehat{\mathcal{M}}^{\operatorname{cut}}$ ends at
$\widehat{H}_{m^{\prime}}$ for some $m^{\prime}\leq m_{0}$.
Indeed, the law of
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
is absolutely continuous with respect to the law of
$({\mathcal{M}}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$. So
($\boxdot$) implies that with positive probability, the event ($\boxdot$)
holds with $\mathcal{M}^{\operatorname{cut}}$, $P_{m}$ and $H_{m^{\prime}}$ in
place of $\widehat{\mathcal{M}}^{\operatorname{cut}}$, $\widehat{P}_{m}$ and
$\widehat{H}_{m^{\prime}}$. This event depends on the UIHPMS in a manner which
is invariant by translations of the form (1.10). By the zero-one law for
translation invariant events, we get that this event in fact has probability
one.
Step 2: Constructing a random perfect matching. Recall that $\widehat{P}_{m}$
is defined as in Definition 6.2 with
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
in place of the UIHPMS. Define $\phi:\mathbbm{Z}\to\mathbbm{Z}$ by the
condition that $\phi(m)=m^{\prime}$ if and only if $\widehat{P}_{m}$ ends at
$\widehat{H}_{m^{\prime}}$. By Lemma 6.3 and the absolutely continuity of
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$
with respect to
$({\mathcal{M}}^{\operatorname{cut}},\Gamma^{\operatorname{cut}})$, a.s. each
$\widehat{P}_{m}$ is a finite path ending at a good boundary point. Hence a.s.
$\phi$ is well-defined. We will prove ($\boxdot$) by applying Lemma 6.9 to
$\phi$.
We first check that $\phi$ is a perfect matching. By the definition of
$(\widehat{\mathcal{M}}^{\operatorname{cut}},\widehat{\Gamma}^{\operatorname{cut}})$,
none of the good boundary points $\widehat{H}_{m}$ for $m\in\mathbbm{Z}$ is
incident to a lower arc of $\widehat{\mathcal{M}}^{\operatorname{cut}}$.
Hence, $\widehat{P}_{m}$ traverses an upper arc immediately before hitting
$\widehat{H}_{\phi(m)}$. Therefore, $\widehat{P}_{\phi(m)}$ is the time
reversal of $\widehat{P}_{m}$ and so $\phi(\phi(m))=m$ and $\phi(m)\neq m$.
Thus $\phi$ is a perfect matching of $\mathbbm{Z}$.
Figure 16: The sets $A$ and $U$ used in Step 3 of the proof of Lemma 6.6.
Good boundary points are shown in green, and arcs not traversed by
$\widehat{P}_{m}\subset A$ are not shown. The key property of $A$ is that it
is not crossed by any arc of $\widehat{\mathcal{M}}^{\operatorname{cut}}$.
Step 3: Non-crossing, stationarity, and ergodicity. We next show that $\phi$
is non-crossing. Fix $m\in\mathbbm{Z}$. We are going to show that each point
in $(m\wedge\phi(m),m\vee\phi(m))\cap\mathbbm{Z}$ is matched with a point in
$(m\wedge\phi(m),m\vee\phi(m))\cap\mathbbm{Z}$. See Figure 16 for an
illustration. Let
$A:=\widehat{P}_{m}\cup\mathopen{}\mathclose{{}\left(\\{\widehat{H}_{m}\\}\times(-\infty,0]}\right)\cup\mathopen{}\mathclose{{}\left(\\{\widehat{H}_{\phi(m)}\\}\times(-\infty,0]}\right).$
(6.17)
By the definition of boundary points, for each $m^{\prime}\in\mathbbm{Z}$ no
arc of $\widehat{\mathcal{M}}^{\operatorname{cut}}$ can cross the infinite ray
$\\{\widehat{H}_{m^{\prime}}\\}\times(-\infty,0]$. This implies in particular
that $A$ is the trace of a bi-infinite simple path in $\mathbbm{C}$ from
$\infty$ to $\infty$. By the Jordan curve theorem $A$ separates $\mathbbm{C}$
into exactly two open connected components. Let $U$ be the open connected
component which contains an unbounded subset of the semi-infinite box
$(\widehat{H}_{\phi(m)}\wedge\widehat{H}_{m},\widehat{H}_{\phi(m)}\vee\widehat{H}_{m})\times(-\infty,0)$.
If $m^{\prime}\in\mathbbm{Z}\setminus\\{m,\phi(m)\\}$, then the infinite ray
$\\{\widehat{H}_{m^{\prime}}\\}\times(-\infty,0]$ intersects $U$ (resp.
$\mathbbm{C}\setminus\overline{U}$) provided
$m^{\prime}\in(m\wedge\phi(m),m\vee\phi(m))\cap\mathbbm{Z}$ (resp.
$m^{\prime}\in\mathbbm{Z}\setminus[m\wedge\phi(m),m\vee\phi(m)]$). This ray
cannot cross $A$ (by definition of boundary points), so must be entirely
contained in either $U$ or $\mathbbm{C}\setminus\overline{U}$. Hence
$U\cap\\{\widehat{H}_{m^{\prime}}\\}_{m^{\prime}\in\mathbbm{Z}}=\\{\widehat{H}_{m^{\prime}}\\}_{m^{\prime}\in(m\wedge\phi(m),m\vee\phi(m))\cap\mathbbm{Z}}.$
Since the paths $\widehat{P}_{m^{\prime}}$ for
$m^{\prime}\in\mathbbm{Z}\setminus\\{m,\phi(m)\\}$ cannot cross $A$, this
implies the desired property.
By Lemma 6.1, $\phi$ is stationary and strongly ergodic in the sense of
Definition 6.8. Therefore, we can apply Lemma 6.9 to get that a.s. ($\boxdot$)
holds. ∎
### 6.5 Proofs of Theorem 1.12 and Proposition 1.14
We are now ready to prove our main theorem (Theorem 1.12) for the UIHPMS. The
key idea is that the paths $P_{m}$ as in Lemma 6.6 act as “shields” which an
infinite path in the UIHPMS cannot cross.
###### Proof of Theorem 1.12.
Fix $n\in\mathbbm{N}$. We will show that a.s. there is no bi-infinite path of
arcs in the UIHPMS which intersects $[-n,n]$. Sending $n\to\infty$ shows that
a.s. there is no bi-infinite path of arcs in the UIHPMS. See Figure 17 for an
illustration.
Figure 17: Illustration of the proof of Theorem 1.12. The numbers
$m\in\mathbbm{Z}_{>0}$ and $m^{\prime}\in\mathbbm{Z}_{<0}$ are chosen so that
the boundary path $P_{m}$ ends at $H_{m^{\prime}}$ and is disjoint from
$[-n,n]\times(-\infty,0]$. This implies that any bi-infinite path of arcs
which hits a point of $[-n,n]\cap\mathbbm{Z}$ must enter the domain $U$. This
is possible only through the lower arcs $l_{m^{\prime}}$ and $l_{m}$ of the
UIHPMS incident to $H_{m^{\prime}}$ and $H_{m}$. We show that, regardless of
the configuration of these arcs, the bi-infinite path cannot enter $U$ (the
two possible configurations are shown in the upper and lower panels of the
figure) concluding that there is no bi-infinite path.
By, e.g., the definition of the UIHPMS by cutting, there are only finitely
many arcs of the UIHPMS which intersect the semi-infinite box
$[-n,n]\times(-\infty,0]$ (recall that we are assuming that each arc joining
$x<y$ is draw in such a way that it is contained in $[x,y]\times\mathbbm{R}$).
If $m,m^{\prime}\in\mathbbm{Z}$ are chosen so that $P_{m}$ does not end at
$H_{m^{\prime}}$, then the paths $P_{m}$ and $P_{m^{\prime}}$ are disjoint.
Therefore, there are only finitely many $m\in\mathbbm{Z}$ such that $P_{m}$
traverses an arc of the UIHPMS which intersects $[-n,n]\times(-\infty,0]$.
From this and Lemma 6.6, we get that a.s. there exists a large enough
$m\in\mathbbm{Z}_{>0}$ such that $P_{m}$ ends at $H_{m^{\prime}}$ for some
$m^{\prime}\in\mathbbm{Z}_{<0}$ and $P_{m}$ is disjoint from
$[-n,n]\times(-\infty,0]$.
Define $A$ and $U$ exactly as in the proof of Lemma 6.6 with $P_{m}$ instead
of $\widehat{P}_{m}$, see the paragraph including (6.17). The condition that
$P_{m}$ is disjoint from $[-n,n]\times(-\infty,0]$ implies that $[-n,n]\subset
U$. There are only finitely many arcs of the UIHPMS which intersect $U$, so a
bi-infinite path of arcs $\gamma$, if it exists, can spend only a finite
amount of time in $\overline{U}$. Since $\gamma$ is bi-infinite, it follows
that if $\gamma$ hits a point of $[-n,n]\cap\mathbbm{Z}\subset U$, then there
are at least two times when $\gamma$ traverses an arc which joins a point of
$\overline{U}\cap\mathbbm{Z}$ and a point of
$\mathbbm{Z}\setminus\overline{U}$.
By the definition of $A$ and the fact that no lower arcs of the UIHPMS cross
the infinite rays $\\{H_{m}\\}\times(-\infty,0]$ and
$\\{H_{m^{\prime}}\\}\times(-\infty,0]$, the only arcs of the UIHPMS which
intersect $A$ are the arcs traversed by $P_{m}$ and the lower arcs incident to
$H_{m^{\prime}}$ and $H_{m}$. Call these two lower arcs $l_{m^{\prime}}$ and
$l_{m}$, respectively. Since $P_{m}\subset A=\partial U$, these two lower arcs
$l_{m^{\prime}}$ and $l_{m}$ are the only arcs of the UIHPMS which can
possibly join a point of $\overline{U}\cap\mathbbm{Z}$ and a point of
$\mathbbm{Z}\setminus\overline{U}$. Hence, if $\gamma$ intersects
$[-n,n]\cap\mathbbm{Z}$ then each of $l_{m^{\prime}}$ and $l_{m}$ joins
$\overline{U}\cap\mathbbm{Z}$ and a point of
$\mathbbm{Z}\setminus\overline{U}$, i.e., $l_{m}$ joins $H_{m}$ to a point of
$\mathbbm{Z}\setminus\overline{U}$ and similarly for $l_{m^{\prime}}$ (see the
first panel in Figure 17).
But, $l_{m^{\prime}}$ and $l_{m}$ are also the only two arcs of the UIHPMS
which can possibly join a point of $U\cap\mathbbm{Z}$ to a point of
$\mathbbm{Z}\setminus U$ (see the second panel in Figure 17). But we have just
seen that if $\gamma$ intersects $[-n,n]\cap\mathbbm{Z}$, then neither
$l_{m^{\prime}}$ nor $l_{m}$ has an endpoint in $U\cap\mathbbm{Z}$. Hence, in
this case there are no arcs joining a point of $U\cap\mathbbm{Z}$ to a point
of $\mathbbm{Z}\setminus U$, so a.s. there is no bi-infinite path of arcs in
the UIHPMS which intersects $[-n,n]\cap\mathbbm{Z}\subset U$. ∎
Using the same idea, we prove that there is a unique infinite path in the
PIHPMS.
###### Proof of Proposition 1.14.
By the definition of the PIHPMS given in (1.11), we can couple the PIHPMS with
the UIHPMS in such a way that the upper arcs for both infinite meandric
systems are the same, the lower arcs joining non-boundary points for both
meandric systems are the same, and the sequence of boundary points
$\\{J_{k}\\}_{k\in\mathbbm{Z}}$ for both meandric systems is the same. Hence
we can identify the good boundary points $\\{H_{m}\\}_{m\in\mathbbm{Z}}$ with
boundary points of the PIHPMS. This implies that the boundary paths $P_{m}$
for $m\in\mathbbm{Z}$ (Definition 6.2) are unaffected if we replace the UIHPMS
by the PIHPMS.
Recall that the path of arcs $\gamma^{\circ}$ started from $0$ in the PIHPMS
is always semi-infinite because $\gamma^{\circ}$ cannot finish forming a loop
as no arc in the lower half plane is incident to $0$ by construction (note
that $\gamma^{\circ}$ is different from $P_{0}$ since it does not stop when it
hits a boundary point).
By Lemma 6.6, a.s. there exist infinitely many $m\in\mathbbm{Z}_{>0}$ such
that $P_{m}$ ends at $H_{m^{\prime}}$ for some $m^{\prime}\in\mathbbm{Z}_{\leq
0}$. Since $\gamma^{\circ}$ is semi-infinite, the same argument as in the
proof of Theorem 1.12 shows that $\gamma^{\circ}$ must hit either $H_{m}$ or
$H_{m^{\prime}}$ for each such $m$, and hence $\gamma^{\circ}$ must traverse
$P_{m}$ or $P_{m^{\prime}}$ (the time reversal of $P_{m}$) for each such $m$.
This shows that $\gamma^{\circ}$ hits infinitely many points in each of
$\\{H_{m}\\}_{m<0}\subset\\{J_{k}\\}_{k<0}$ and
$\\{H_{m}\\}_{m>0}\subset\\{J_{k}\\}_{k>0}$, almost surely. The same argument
as in the proof of Theorem 1.12 also shows that any other infinite path of
arcs in the PIHPMS must also traverse infinitely many of the paths $P_{m}$ for
$m\in\mathbbm{Z}_{>0}$ as above. Therefore, any other infinite path must share
a portion of arcs with $\gamma^{\circ}$. But this is possible only if such
infinite path is a subpath of $\gamma^{\circ}$. So $\gamma^{\circ}$ is the
unique maximal infinite path of arcs in the PIHMPS, i.e, the unique infinite
path in $\Gamma^{\circ}$. ∎
## 7 Justification for Conjectures 1.2 and 1.3
### 7.1 SLE8 on $\sqrt{2}$-LQG via mating of trees
Let $\mathfrak{S}_{n}$ be a uniform meandric system of size $n$ and let
$(\mathcal{M}_{n},P_{n},\Gamma_{n})$ be the associated planar map decorated by
a Hamiltonian path (corresponding to the real line) and a collection of loops,
as in the discussion at the beginning of Section 1.2. We have already seen in
Sections 4 and 5 that the infinite-volume analog $(\mathcal{M},P)$ of
$(\mathcal{M}_{n},P_{n})$ is closely connected to $\sqrt{2}$-LQG decorated by
SLE8. More precisely, due to the encoding of the UIMS by random walks (Section
1.4) and the convergence of random walk to Brownian motion, $(\mathcal{M},P)$
is connected to the mated-CRT map $\mathcal{G}$, equipped with the left-right
ordering of its vertices (Section 4). Furthermore, the mated-CRT map
$\mathcal{G}$ is closely connected to $\sqrt{2}$-LQG decorated by SLE8 due to
the results of [DMS21] (Theorem 5.1). This strongly suggests that the scaling
limit of $(\mathcal{M}_{n},P_{n})$ should be given by $\sqrt{2}$-LQG decorated
by SLE8.
In fact, the random walk excursions which encode the upper and lower arc
diagrams of $(\mathcal{M}_{n},P_{n})$ converge under appropriate scaling to
two independent Brownian excursions. This can already be interpreted as a
convergence statement for $(\mathcal{M}_{n},P_{n})$ toward $\sqrt{2}$-LQG
decorated by SLE8 in the so-called peanosphere sense. See [GHS23] for a survey
of this type of convergence and the results that can be proven using it.
To justify Conjecture 1.2, we still need to explain why $\Gamma_{n}$ should
converge to CLE6. To do this, we will give a physics argument which also
provides an alternative heuristic for the scaling limit of
$(\mathcal{M}_{n},P_{n})$.
### 7.2 Physics argument
We will now explain why Conjecture 1.2 can be viewed as a special case of
Conjecture [BGS22, Conjecture 6.2], which in turn is derived from physics
heuristics in [DFGG00], building on [JK98, JK99, DCN04, JZJ04].
Let $n,m>0$ and let $G$ be a finite 4-regular graph. The fully packed
$O(n\times m)$ loop model on $G$ is the probability measure on pairs of
collections of loops (simple cycles) $\Gamma_{1},\Gamma_{2}$ in $G$ with the
following properties:
* •
Each vertex of $G$ is visited by exactly one loop in each of $\Gamma_{1}$ and
$\Gamma_{2}$.
* •
Each edge of $G$ is visited by either exactly one loop in $\Gamma_{1}$ or
exactly one loop in $\Gamma_{2}$ (but not both).
The probability of each possible configuration $(\Gamma_{1},\Gamma_{2})$ is
proportional to $n^{\\#\Gamma_{1}}m^{\\#\Gamma_{2}}$.
Suppose now that $G$ is a planar map, instead of just a graph. This in
particular means that we have a canonical cyclic ordering of the four edges
incident to each vertex of $G$. The crossing fully packed $O(n\times m)$ loop
model on $G$ is the variant of the $O(n\times m)$ loop model where we impose
the additional constraint that a loop of $\Gamma_{1}$ and a loop of
$\Gamma_{2}$ cross at each vertex of $G$. Equivalently, the edges incident to
each vertex, in cyclic order, must alternate between edges of $\Gamma_{1}$ and
edges of $\Gamma_{2}$.
One can also define the (crossing) fully packed $O(n\times m)$ loop model when
one or both of $n$ or $m$ is equal to zero. In the case when, e.g., $n=0$, one
considers pairs $(\Gamma_{1},\Gamma_{2})$ as above subject to the constraint
that $\Gamma_{1}$ consists of a single loop (necessarily a Hamiltonian cycle),
and the probability of each configuration is proportional to
$m^{\\#\Gamma_{2}}$.
The relevance of the (crossing) fully packed $O(n\times m)$ loop model to the
present paper is that meandric systems are equivalent to the crossing fully
packed $O(0\times 1)$ loop model on a planar map, as we now explain. Let
$\mathfrak{S}_{n}$ be a meandric system of size $n$ and consider the
corresponding decorated planar map $(\mathcal{M}_{n},P_{n},\Gamma_{n})$. Then
$\mathcal{M}_{n}$ is a 4-regular planar map with $2n$ vertices, $P_{n}$ is a
Hamiltonian cycle on $\mathcal{M}_{n}$, $\Gamma_{n}$ is a collection of loops
on $\mathcal{M}_{n}$ such that each vertex is visited by exactly one loop, and
each edge is visited by either the Hamiltonian cycle or one of the loops (but
not both). Moreover, at each vertex of $\mathcal{M}_{n}$, the Hamiltonian
cycle $P_{n}$ is crossed by a loop of $\Gamma_{n}$ (this corresponds to the
condition that loops of $\mathfrak{S}_{n}$ do not touch $\mathbbm{R}$ without
crossing it in Definition 1.1).
Conversely, every 4-regular planar map decorated by a Hamiltonian path and a
collection of loops which satisfy the above conditions gives rise to a
meandric system: just choose a planar embedding of the map into
$\mathbbm{R}^{2}\cup\\{\infty\\}$ under which the Hamiltonian cycle is mapped
to the real line.
Hence, the set of possibilities for $(\mathcal{M}_{n},P_{n},\Gamma_{n})$ is
exactly the same as the set of possibilities for a 4-regular planar map with
$2n$ edges decorated by a realization of the crossing fully packed $O(0\times
1)$ loop model. In particular, if $\mathfrak{S}_{n}$ is sampled uniformly,
then $\mathcal{M}_{n}$ is a sample from the set of 4-regular planar maps with
$2n$ vertices, weighted by the number of possible crossing fully packed
$O(0\times 1)$ loop model configurations on the map. Moreover, the conditional
law of $(P_{n},\Gamma_{n})$ given $\mathcal{M}_{n}$ is that of the fully
packed $O(0\times 1)$ loop model on $\mathcal{M}_{n}$.
Jacobsen and Kondev [JK98, JK99] gave a prediction for the scaling limit of
the fully packed $O(n\times m)$ loop model on $\mathbbm{Z}^{2}$ in terms of a
conformal field theory whose central charge is an explicit function of $n$ and
$m$, which was later verified (at a physics level of rigor) in [DCN04, JZJ04].
Building on this, Di Franceso, Golinelli, and Guitter [DFGG00, Section 3] gave
a prediction for the scaling limit of the crossing fully packed $O(n\times m)$
loop model on a random planar map. Section 6 of [BGS22] reviews the arguments
leading to these predictions and translates the predictions into the language
of SLE and LQG. In particular, from [BGS22, Conjecture 6.2] for $n=0$ and
$m=1$, we get that the scaling limit of random planar maps decorated by the
crossing fully packed $O(0\times 1)$ loop model should be given by
$\sqrt{2}$-LQG decorated by SLE8 and CLE6. Since this decorated random planar
map model is equivalent to a uniform meandric system (as discussed just
above), this leads to Conjecture 1.2.
### 7.3 Predictions for exponents via KPZ
We now explain how Conjecture 1.2 leads to predictions for various exponents
associated with meandric systems (including Conjecture 1.3). In this and the
next section, we assume that the reader has some familiarity with SLE and LQG.
Consider a $\gamma$-LQG sphere, represented by a metric and a measure on
$\mathbbm{C}$. If $X\subset\mathbbm{C}$ is a random set sampled independently
from this metric and measure, then we can define the Hausdorff dimensions
$\Delta_{0}$ and $\Delta_{\gamma}$ of $X$ with respect to the Euclidean and
$\gamma$-LQG metrics, respectively. We re-scale $\Delta_{\gamma}$ by the
reciprocal of the dimension of the whole space, so that
$\Delta_{\gamma}\in[0,1]$. The Knizhnik-Polyakov-Zamolodchikov (KPZ) formula
[KPZ88] states that a.s.
$\Delta_{0}=\mathopen{}\mathclose{{}\left(2+\frac{\gamma^{2}}{2}}\right)\Delta_{\gamma}-\frac{\gamma^{2}}{2}\Delta_{\gamma}^{2}.$
(7.1)
See [BS09, DS11, RV11, BJRV13, DRSV14, Aru15, BGRV16, GHS19, GHM20, DMS21,
GM21a, GP22] for various rigorous versions of the KPZ formula.
Now consider $(\mathcal{M}_{n},P_{n},\Gamma_{n})$ as above and for
$k\in\mathbbm{N}$ let $\ell_{n}^{k}$ be the $k$th largest loop in
$\Gamma_{n}$, i.e., the one with the $k$th largest number of vertices.
Conjecture 1.2 tells us that if $k$ is fixed, then as $n\rightarrow\infty$,
the loop $\ell_{n}^{k}$ should converge to the $k$th largest loop in a CLE6 on
an independent $\sqrt{2}$-LQG sphere, i.e., the one with the $k$th largest
$\sqrt{2}$-LQG length. The Euclidean dimension of the $k$th largest loop in a
CLE6 is $\Delta_{0}=7/4$, the same as the dimension of an SLE6 curve [Bef08].
By (7.1), the (re-scaled) dimension of the $k$th largest loop in a CLE6 with
respect to the $\sqrt{2}$-LQG metric is
$\alpha:=\Delta_{\sqrt{2}}=\frac{1}{2}\mathopen{}\mathclose{{}\left(3-\sqrt{2}}\right)\approx
0.7929.$ (7.2)
This means that the number of $\sqrt{2}$-LQG balls of $\sqrt{2}$-LQG mass
$\varepsilon$ needed to cover the $k$th largest loop in a CLE6 is about
$\varepsilon^{-\alpha}$. The analog of this in the discrete setting says that
$\mathbbm{E}\mathopen{}\mathclose{{}\left[\\#\\{\text{vertices in
$\ell_{n}^{k}$}\\}}\right]\approx n^{\alpha},\quad\text{with $\alpha$ as
in~{}\eqref{eqn-loop-exponent-kpz}}.$ (7.3)
This gives Conjecture 1.3.
Using similar techniques to the ones above, we can also derive predictions for
other exponents associated with uniform meandric systems. For example, let
$\ell_{n}^{1}$ be the largest loop in $\Gamma_{n}$ and let
$\operatorname{Cross}_{n}$ be the number of times that $\ell_{n}^{1}$ crosses
the vertical line $\\{n\\}\times\mathbbm{R}$, i.e., the number of arcs of
$\ell_{n}^{1}$ which disconnect $n$ from $\infty$ on either of the two sides
of the horizontal line. To estimate the growth rate of
$\operatorname{Cross}_{n}$ as $n\rightarrow\infty$, we look at the continuum
analog of the set of crossing points. Consider an SLE8 curve $\eta$ from
$\infty$ to $\infty$ and an independent whole-plane CLE6. Assume that $\eta$
is parametrized by $\sqrt{2}$-LQG area with respect to an independent unit
area quantum sphere, so that $\eta:[0,1]\rightarrow\mathbbm{R}$. According to
Conjecture 1.2, the continuum analog of the set of crossing points in the
definition of $\operatorname{Cross}_{n}$ is the intersection of
$\eta([0,1/2])\cap\eta([1/2,1])$ with the largest (in the sense of
$\sqrt{2}$-LQG length) loop in the CLE6.
The set $\eta([0,1/2])\cap\eta([1/2,1])$ is the union of two SLE2-type curves
[DMS21, Footnote 4]. Therefore, the Euclidean Hausdorff dimension of the
intersection of this set with the largest loop in the CLE6 should be equal to
the Euclidean Hausdorff dimension of an SLE2 curve intersected with an
independent SLE6 curve. Using [Bef08], on the event that the intersection is
non-empty, its Euclidean Hausdorff dimension should be
$\Delta_{0}=2-(2-5/4)-(2-7/4)=1.$ (7.4)
Plugging this into (7.1) (with $\gamma=\sqrt{2}$) as above, we arrive at the
prediction
$\mathbbm{E}\mathopen{}\mathclose{{}\left[\operatorname{Cross}_{n}}\right]\approx
n^{\nu},\quad\text{where}\quad\nu=\frac{1}{2}(3-\sqrt{5})\approx 0.3820.$
(7.5)
### 7.4 Simulations
We give details on our numerical simulations in this section.
A sample of a (finite) uniform meandric system of size $n$ can be obtained
from a sample of two independent simple random walks of size $2n$ started from
$0$, conditioned to stay non-negative and to end at $0$ using (1.1). These
conditioned walks $\mathcal{X}_{i}$ can be sampled quickly from the
conditional probability
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathcal{X}_{i}-\mathcal{X}_{i-1}=1\,|\,\mathcal{X}_{1},\dots,\mathcal{X}_{i-1}}\right]$,
which boils down to counting the number of simple walk bridges conditioned to
be non-negative. Those numbers can be explicitly computed by the reflection
property of simple random walks in any case. The corresponding meandric system
of size $n$ is then constructed by (1.1) as a combinatorial graph and analyzed
by JuliaGraphs[FBS+21].
Our first simulation computes the median sizes (numbers of vertices) of the 5
largest loops from 100 uniformly sampled meandric systems of size $n$, for
increasing values of $n$. The result in Figure 18 (Left) suggests that the
constant $\alpha\approx 0.7929$ in Conjecture 1.3 is correct. This is
distinguishable from the value $0.8$ guessed in [Kar20] because the 95%
confidence interval for the largest loop exponent is $0.7929\pm 0.0022$. The
size of $k$th largest loop can be written as $C_{k}n^{\alpha}$ where $C_{k}$
is some random variable. In Figure 18 (Left), the $y$-intercept is related to
$\log(C_{k})$ for each $k$. One implication of Conjecture 1.2 is that the law
of $C_{k+1}/C_{k}$ is tight for each $k$.
Our second simulation computes the medians of $\operatorname{Cross}_{n}$
(defined in Section 7.3) from 100 uniformly sampled meandric systems of size
$n$, for increasing values of $n$. The result in Figure 18 (Right) suggests
that the KPZ prediction (7.5) is correct. This is strong evidence that our
conjectured limiting objects SLE8 and CLE6 are independent, which is not
trivial from the definition of meandric systems. See the discussion after
Conjecture 1.2.
Figure 18: Left: The plot of $\log(|\ell^{k}_{n}|)$ versus $\log(2n)$, where
$|\ell^{k}_{n}|$ is the number of vertices of the $k$th largest loop of
$\mathfrak{S}_{n}$. The 95% confidence intervals for the slopes are $0.7929\pm
0.0022,0.7932\pm 0.0021,0.7939\pm 0.0021,0.7938\pm 0.0020,0.7948\pm 0.0019$,
respectively, which all include our conjectured value $(3-\sqrt{2})/2\approx
0.7929.$ Right: The plot of $\log(|\operatorname{Cross}_{n}|)$ versus
$\log(2n)$, where $\operatorname{Cross}_{n}$ is defined in Section 7.3. The
95% confidence interval for the slope is $0.3815\pm 0.0015$, which includes
our conjectured value $\frac{1}{2}(3-\sqrt{5})\approx 0.3820$.
We now give the details of the simulations in Figure 2. There, we also
visualize how the arcs of arc diagrams (which are usually embedded as smooth
circular arcs in $\mathbbm{R}^{2}$ throughout the paper) associated with a
uniformly sampled meandric system may look like fractal loops of a CLE in a
proper embedding. For this purpose, we consider a uniform meandric system
_with boundary_ (as defined below; its infinite-volume version is defined in
Section 1.5) and embed the underlying planar map via the Tutte embedding
[GMS21], a.k.a. harmonic embedding, so that it conjecturally approximates the
$\sqrt{2}$-LQG wedge on the disk. We use a finite meandric system with
boundary because Tutte embeddings are easier to define for planar maps with
boundary.
The uniform meandric system with boundary can be constructed as in (1.1) from
two independent conditioned simple random walks with $2n$ steps started from
$0$, both conditioned to stay non-negative, but one ends at $0$ as before
while the other ends at $2\lfloor n^{1/2}\rfloor$. These conditioned random
walks $\mathcal{X}_{i}$ can be also sampled from the corresponding conditional
probability
$\mathbbm{P}\mathopen{}\mathclose{{}\left[\mathcal{X}_{i}-\mathcal{X}_{i-1}=1\,|\,\mathcal{X}_{1},\dots,\mathcal{X}_{i-1}}\right]$
exactly as before. Then we may use (1.1) to construct arc diagrams, but for
the second sampled walk, there are $2\lfloor n^{1/2}\rfloor$ unmatched points
called _(good) boundary points_ , c.f. Section 1.5 and Section 6.2. With these
boundary points, we may apply the Tutte embedding.
A path of arcs started from any of the boundary points ends at another
boundary point because boundary points have degree 1, whereas non-boundary
points have degree 2. Thus it defines a _boundary path_ , c.f. Definition 6.2.
We also link each pair of boundary points successively to form loops, c.f.
Figure 4 (Left) for the corresponding procedure in the UIHPMS. After coloring
loops according to their sizes, we finally obtain the pictures shown in Figure
2. This construction is analogous to the random walk construction of the
UIHPMS defined in Section 1.5.555Simple walks with $2n$ steps started from $0$
conditioned to stay positive and end at $2\lfloor n^{1/2}\rfloor$ are in
bijection with reflected simple walks with $2n+2\lfloor n^{1/2}\rfloor$ steps
started from $0$ conditioned to have exactly $2\lfloor n^{1/2}\rfloor$ zeros
(after time zero) and to end at $0$, from the bijection in the proof of Lemma
6.4. With the latter random walk, an arc diagram can be constructed from
(1.9), but no extra linking is necessary as there are no unmatched points. See
also Figure 4 (Right).
Instead of linking every successive pair of boundary points as above, we may
exclude two marked points (specifically, we marked $0$ and another point close
to $n$ in our simulations). As each point other than these two marked points
is incident to two arcs, there is always a path from one marked point to the
other. We expect this path approximates a chordal SLE6 curve between two
marked points. See the discussion prior to Conjecture 1.15. The simulation in
Figure 5 does not contradict any known qualitative behaviors of SLE6.
We also note that there are other embeddings, e.g., circle packing, Riemann
uniformization, or Smith embedding (square tiling) under which planar maps
without boundary should converge to LQG. However, the representation of paths
and loops on these embeddings may not be as straightforward as in the Tutte
embedding.
Every simulation in this section can also be done for random meandric systems
encoded by correlated random walks. For example, for $\gamma=1$ (correlation
$-\cos(\pi\gamma^{2}/4)=-1/\sqrt{2}$), the simulation suggests the size of the
largest loop grows like $n^{\alpha}$ where $\alpha$ is in the 95% confidence
interval $0.8379\pm 0.0100$. This is consistent with the prediction
$\frac{1}{2}(5-\sqrt{11})\approx 0.8417$ coming from KPZ if we assume that the
loops converge to CLE6. See Remark 1.16 for further discussion.
## References
* [Aru15] J. Aru. KPZ relation does not hold for the level lines and SLEκ flow lines of the Gaussian free field. Probab. Theory Related Fields, 163(3-4):465–526, 2015, 1312.1324. MR3418748
* [Bef08] V. Beffara. The dimension of the SLE curves. Ann. Probab., 36(4):1421–1452, 2008, math/0211322. MR2435854 (2009e:60026)
* [BGRV16] N. Berestycki, C. Garban, R. Rhodes, and V. Vargas. KPZ formula derived from Liouville heat kernel. J. Lond. Math. Soc. (2), 94(1):186–208, 2016, 1406.7280. MR3532169
* [BGS22] J. Borga, E. Gwynne, and X. Sun. Permutons, meanders, and SLE-decorated Liouville quantum gravity. ArXiv e-prints, July 2022, 2207.02319.
* [BJRV13] J. Barral, X. Jin, R. Rhodes, and V. Vargas. Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys., 323(2):451–485, 2013, 1202.5296. MR3096527
* [BP] N. Berestycki and E. Powell. Gaussian free field, Liouville quantum gravity, and Gaussian multiplicative chaos. Available at https://homepage.univie.ac.at/nathanael.berestycki/Articles/master.pdf.
* [BS01] I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001, 0011019. MR1873300 (2002m:82025)
* [BS09] I. Benjamini and O. Schramm. KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys., 289(2):653–662, 2009, 0806.1347. MR2506765 (2010c:60151)
* [CGHP20] N. Crawford, A. Glazman, M. Harel, and R. Peled. Macroscopic loops in the loop $O(n)$ model via the XOR trick. ArXiv e-prints, January 2020, 2001.11977.
* [CKST19] N. Curien, G. Kozma, V. Sidoravicius, and L. Tournier. Uniqueness of the infinite noodle. Ann. Inst. Henri Poincaré D, 6(2):221–238, 2019, 1701.01083. MR3950654
* [CN06] F. Camia and C. M. Newman. Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys., 268(1):1–38, 2006, math/0605035. MR2249794
* [CN08] F. Camia and C. M. Newman. ${\mathrm{S}LE}_{6}$ and ${\mathrm{C}LE}_{6}$ from critical percolation. In Probability, geometry and integrable systems, volume 55 of Math. Sci. Res. Inst. Publ., pages 103–130. Cambridge Univ. Press, Cambridge, 2008, math/0611116. MR2407594 (2009h:82032)
* [DCGPS21] H. Duminil-Copin, A. Glazman, R. Peled, and Y. Spinka. Macroscopic loops in the loop $O(n)$ model at Nienhuis’ critical point. J. Eur. Math. Soc. (JEMS), 23(1):315–347, 2021, 1707.09335. MR4186469
* [DCN04] D. Dei Cont and B. Nienhuis. The packing of two species of polygons on the square lattice. Journal of Physics A: Mathematical and General, 37(9):3085, 2004, cond-mat/0311244.
* [DDDF20] J. Ding, J. Dubédat, A. Dunlap, and H. Falconet. Tightness of Liouville first passage percolation for $\gamma\in(0,2)$. Publ. Math. Inst. Hautes Études Sci., 132:353–403, 2020, 1904.08021. MR4179836
* [DDG21] J. Ding, J. Dubedat, and E. Gwynne. Introduction to the Liouville quantum gravity metric. ArXiv e-prints, September 2021, 2109.01252.
* [DDGG22] P. Di Francesco, B. Duplantier, O. Golinelli, and E. Guitter. Exponents for Hamiltonian paths on random bicubic maps and KPZ. ArXiv e-prints, October 2022, 2210.08887.
* [DFGG00] P. Di Francesco, O. Golinelli, and E. Guitter. Meanders: exact asymptotics. Nuclear Phys. B, 570(3):699–712, 2000, cond-mat/9910453. MR1749396
* [DG18] J. Ding and E. Gwynne. The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds. Communications in Mathematical Physics, 374:1877–1934, 2018, 1807.01072.
* [DKRV16] F. David, A. Kupiainen, R. Rhodes, and V. Vargas. Liouville quantum gravity on the Riemann sphere. Comm. Math. Phys., 342(3):869–907, 2016, 1410.7318. MR3465434
* [DMS21] B. Duplantier, J. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees. Astérisque, (427):viii+257, 2021, 1409.7055. MR4340069
* [DRSV14] B. Duplantier, R. Rhodes, S. Sheffield, and V. Vargas. Renormalization of critical Gaussian multiplicative chaos and KPZ relation. Comm. Math. Phys., 330(1):283–330, 2014, 1212.0529. MR3215583
* [DS11] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math., 185(2):333–393, 2011, 1206.0212. MR2819163 (2012f:81251)
* [FBS+21] J. Fairbanks, M. Besançon, S. Simon, J. Hoffiman, N. Eubank, and S. Karpinski. Juliagraphs/graphs.jl: an optimized graphs package for the julia programming language, 2021.
* [FN19] M. Fukuda and I. Nechita. Enumerating meandric systems with large number of loops. Ann. Inst. Henri Poincaré D, 6(4):607–640, 2019, 1609.02756. MR4033682
* [FT22] V. Féray and P. Thévenin. Components in Meandric Systems and the Infinite Noodle. International Mathematics Research Notices, 07 2022, 2201.11572. rnac156.
* [GHM20] E. Gwynne, N. Holden, and J. Miller. An almost sure KPZ relation for SLE and Brownian motion. Ann. Probab., 48(2):527–573, 2020, 1512.01223. MR4089487
* [GHS19] E. Gwynne, N. Holden, and X. Sun. A distance exponent for Liouville quantum gravity. Probability Theory and Related Fields, 173(3):931–997, 2019, 1606.01214.
* [GHS20] E. Gwynne, N. Holden, and X. Sun. A mating-of-trees approach for graph distances in random planar maps. Probab. Theory Related Fields, 177(3-4):1043–1102, 2020, 1711.00723. MR4126936
* [GHS21] E. Gwynne, N. Holden, and X. Sun. Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense. Electron. J. Probab., 26:Paper No. 94, 58, 2021, 1905.06757. MR4278605
* [GHS23] E. Gwynne, N. Holden, and X. Sun. Mating of trees for random planar maps and Liouville quantum gravity: a survey. In Topics in statistical mechanics, volume 59 of Panor. Synthèses, pages 41–120. Soc. Math. France, Paris, 2023, 1910.04713. MR4619311
* [GM17] E. Gwynne and J. Miller. Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology. Electron. J. Probab., 22:1–47, 2017, 1608.00954.
* [GM21a] E. Gwynne and J. Miller. Characterizations of SLEκ for $\kappa\in(4,8)$ on Liouville quantum gravity. Astérisque, (429):vii+242, 2021, 1701.05174.
* [GM21b] E. Gwynne and J. Miller. Existence and uniqueness of the Liouville quantum gravity metric for $\gamma\in(0,2)$. Invent. Math., 223(1):213–333, 2021, 1905.00383. MR4199443
* [GMS19] E. Gwynne, J. Miller, and S. Sheffield. Harmonic functions on mated-CRT maps. Electron. J. Probab., 24:no. 58, 55, 2019, 1807.07511.
* [GMS21] E. Gwynne, J. Miller, and S. Sheffield. The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. Ann. Probab., 49(4):1677–1717, 2021, 1705.11161.
* [GNP20] I. P. Goulden, A. Nica, and D. Puder. Asymptotics for a class of meandric systems, via the Hasse diagram of ${\rm NC}(n)$. Int. Math. Res. Not. IMRN, (4):983–1034, 2020, 1708.05188. MR4073935
* [GP19] E. Gwynne and J. Pfeffer. Bounds for distances and geodesic dimension in Liouville first passage percolation. Electronic Communications in Probability, 24:no. 56, 12, 2019, 1903.09561.
* [GP21] E. Gwynne and J. Pfeffer. External diffusion-limited aggregation on a spanning-tree-weighted random planar map. Ann. Probab., 49(4):1633–1676, 2021, 1901.06860. MR4260464
* [GP22] E. Gwynne and J. Pfeffer. KPZ formulas for the Liouville quantum gravity metric. Trans. Amer. Math. Soc., 375(12):8297–8324, 2022, 1905.11790. MR4504639
* [GS22] E. Gwynne and J. Sung. The Minkowski content measure for the Liouville quantum gravity metric. ArXiv e-prints, November 2022, 2211.04701.
* [Gwy20] E. Gwynne. Random surfaces and Liouville quantum gravity. Notices Amer. Math. Soc., 67(4):484–491, 2020, 1908.05573. MR4186266
* [JK98] J. L. Jacobsen and J. Kondev. Field theory of compact polymers on the square lattice. Nuclear Physics B, 532(3):635–688, 1998, cond-mat/9804048.
* [JK99] J. L. Jacobsen and J. Kondev. Transition from the compact to the dense phase of two-dimensional polymers. Journal of statistical physics, 96(1):21–48, 1999, cond-mat/9811085.
* [JZJ04] J. Jacobsen and P. Zinn-Justin. Algebraic Bethe ansatz for the FPL2 model. Journal of Physics A: Mathematical and General, 37(29):7213, 2004, math-ph/0402008.
* [Kah85] J.-P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Québec, 9(2):105–150, 1985. MR829798 (88h:60099a)
* [Kar20] V. Kargin. Cycles in random meander systems. J. Stat. Phys., 181(6):2322–2345, 2020, 2011.13449. MR4179808
* [KMT76] J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent RV’s, and the sample DF. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34(1):33–58, 1976\. MR0402883
* [KPZ88] V. Knizhnik, A. Polyakov, and A. Zamolodchikov. Fractal structure of 2D-quantum gravity. Modern Phys. Lett A, 3(8):819–826, 1988.
* [KS64] S. Kochen and C. Stone. A note on the Borel-Cantelli lemma. Illinois Journal of Mathematics, 8(2):248–251, 1964.
* [KW16] A. Kemppainen and W. Werner. The nested simple conformal loop ensembles in the Riemann sphere. Probab. Theory Related Fields, 165(3-4):835–866, 2016, 1402.2433. MR3520020
* [La 03] M. La Croix. Approaches to the enumerative theory of meanders. Available at https://math.mit.edu/~malacroi/Latex/Meanders.pdf, 2003.
* [MWW16] J. Miller, S. S. Watson, and D. B. Wilson. Extreme nesting in the conformal loop ensemble. Ann. Probab., 44(2):1013–1052, 2016, 1401.0218. MR3474466
* [Poi12] H. Poincaré. Sur un téorème de géométrie. Rend. del Circ. Mat. Palermo, 33:375–407, 1912.
* [RV11] R. Rhodes and V. Vargas. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat., 15:358–371, 2011, 0807.1036. MR2870520
* [Sch00] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000, math/9904022. MR1776084 (2001m:60227)
* [She09] S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J., 147(1):79–129, 2009, math/0609167. MR2494457 (2010g:60184)
* [She22] S. Sheffield. What is a random surface? ArXiv e-prints, March 2022, 2203.02470.
* [Sim83] G. Simons. A discrete analogue and elementary derivation of ‘Lévy’s equivalence’ for Brownian motion. Statistics & Probability Letters, 1(4):203–206, 1983.
* [Smi01] S. Smirnov. Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math., 333(3):239–244, 2001, 0909.4499. MR1851632 (2002f:60193)
* [Ste03] K. Stephenson. Circle packing: a mathematical tale. Notices of the AMS, 50(11):1376–1388, 2003.
* [Zai98] A. Y. Zaitsev. Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. ESAIM Probab. Statist., 2:41–108, 1998. MR1616527
* [Zvo21] A. Zvonkin. Meanders: a Personal Perspective. Available at https://www.labri.fr/perso/zvonkin/Research/Meanders-for-Rosenstiehl.pdf, 2021\.
Data availability statement: Data sharing is not applicable to this article as
no new data were created or analyzed in this study.
Conflict of interest: The authors have no competing interests to declare that
are relevant to the content of this article.
|
# Super-resolution of positive near-colliding point sources ††thanks: This
work was supported in part by the Swiss National Science Foundation grant
number 200021–200307.
Ping Liu Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092
Zürich, Switzerland<EMAIL_ADDRESS>[email protected]).
Habib Ammari22footnotemark: 2
###### Abstract
In this paper, we analyze the capacity of super-resolution of one-dimensional
positive sources. In particular, we consider the same setting as in [2] and
generalize the results there to the case of super-resolving positive sources.
To be more specific, we consider resolving $d$ positive point sources with
$p\leqslant d$ nodes closely-spaced and forming a cluster, while the rest of
the nodes are well separated. Similarly to [2], our results show that when the
noise level $\epsilon\lesssim\mathrm{SRF}^{-2p+1}$, where
$\mathrm{SRF}=(\Omega\Delta)^{-1}$ with $\Omega$ being the cutoff frequency
and $\Delta$ the minimal separation between the nodes, the minimax error rate
for reconstructing the cluster nodes is of order
$\frac{1}{\Omega}\mathrm{SRF}^{2p-2}\epsilon$, while for recovering the
corresponding amplitudes $\left\\{a_{j}\right\\}$ the rate is of order
$\mathrm{SRF}^{2p-1}\epsilon$. For the non-cluster nodes, the corresponding
minimax rates for the recovery of nodes and amplitudes are of order
$\frac{\epsilon}{\Omega}$ and $\epsilon$, respectively. Our numerical
experiments show that the Matrix Pencil method achieves the above optimal
bounds when resolving the positive sources.
## 1 Introduction
In recent years, the problem of super-resolution (SR), which seeks to extract
fine details of a signal from its noisy Fourier data in a bounded frequency
domain, draws increasing interest in the field of applied mathematics. In
particular, considerable progress has been made in the study of super-
resolution of sparse signals, e.g. a host of algorithms [3, 9, 23, 26, 25, 22,
21, 7] were devised for resolving signals with a sparse prior. The sparse
signals are frequently modeled as discrete measures
$F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),\quad x_{j}\in\mathbb{R},$
where $\delta$ is the Dirac’s $\delta$-distribution. Let $\mathcal{F}[F]$
denote the Fourier transform of $F$ :
$\mathcal{F}[F](s)=\int_{-\infty}^{\infty}F(x)e^{-2\pi
isx}\mathrm{\leavevmode\nobreak\ d}x.$
The noisy spectral data of the signal $F$ is modeled as a function $\Phi$
satisfying,
$|\Phi(s)-\mathcal{F}[F](s)|\leqslant\epsilon,\quad s\in[-\Omega,\Omega],$
(1.1)
where $\epsilon>0$ represents the noise level and $\Omega>0$ is the cutoff
frequency. The sparse SR problem considered in this paper reads: given $\Phi$
as above, estimate the unknown parameters of $F$, namely the amplitudes
$\left\\{a_{j}\right\\}$ and the nodes $\left\\{x_{j}\right\\}$. The minimax
error rate for recovering the nodes and amplitudes from the spectral data
$\Phi(s)$ has been established in [2]. In the present paper, we aim at
exploring the corresponding minimax error rate for resolving positive signals.
Since our result is a generalization of the result in [2] which deals with
complex signals, we utilize the same notation, concepts, and configurations as
those in [2] for the sake of consistency of the two papers and the convenience
of reading.
### 1.1 Main contribution
The main contribution of this paper is the generalization of the estimates in
[2] for the minimax error rate for complex signals in the off-the-grid setting
to the case of resolving positive signals. We consider the case where the
nodes $\left\\{x_{j}\right\\}$ can take arbitrary real values and the
amplitudes $\left\\{a_{j}\right\\}$ are known to be positive. We consider the
same distribution of nodes as in [2] where it is assumed that $p$ nodes
(approximately uniformly distributed), $x_{\kappa},\ldots,x_{\kappa+p-1}$,
form a small cluster and the rest of the nodes are away from all the other
nodes (see Definition 2.4 below). We show in Theorem 2.3 that for
$\epsilon\lesssim(\Omega\Delta)^{2p-1}$ with $\Delta$ being the minimum
separation of the clustered nodes, in the worst-case scenario, the errors of
recovered nodes $x_{j}^{\prime}$ and amplitudes $a_{j}^{\prime}>0$ by any
minimax algorithm (see Definition 2.2 below), satisfy
* •
For the non-cluster nodes:
$\displaystyle\max_{j\notin\\{\kappa,\ldots,\kappa+p-1\\}}\left|x_{j}-x_{j}^{\prime}\right|\asymp\frac{\epsilon}{\Omega},$
$\displaystyle\max_{j\notin\\{\kappa,\ldots,\kappa+p-1\\}}\left|a_{j}-a_{j}^{\prime}\right|\asymp\epsilon;$
* •
For the cluster nodes:
$\displaystyle\max_{j\in\\{\kappa,\ldots,\kappa+p-1\\}}\left|x_{j}-x_{j}^{\prime}\right|\asymp\frac{\epsilon}{\Omega}(\Omega\Delta)^{-2p+2},$
$\displaystyle\max_{j\in\\{\kappa,\ldots,\kappa+p-1\\}}\left|a_{j}-a_{j}^{\prime}\right|\asymp\epsilon(\Omega\Delta)^{-2p+1}.$
Our results reveal that the minimax error rates for recovering the nodes and
amplitudes of positive signals in the super-resolution problem are the same as
those for resolving general complex signals [2]. To be more specific, for
$\epsilon\lesssim\mathrm{SRF}^{-2p+1}$ with
$\mathrm{SRF}:=(\Omega\Delta)^{-1}$, the minimax error rate for reconstructing
the cluster nodes of positive signals is of the order
$(\mathrm{SRF})^{2p-2}\frac{\epsilon}{\Omega}$, while for recovering the
corresponding amplitudes the rate is of the order
$(\mathrm{SRF})^{2p-1}\epsilon$. On the other hand, the corresponding minimax
rates for the recovery of the non-cluster nodes and amplitudes are of the
order $\frac{\epsilon}{\Omega}$ and $\epsilon$, respectively. These also
indicate that the non-cluster nodes
$\left\\{x_{j}\right\\}_{j\notin\\{\kappa,\ldots,\kappa+p-1\\}}$ can be
recovered with much better stability than the cluster nodes.
The main novelty we rely in analyzing the case of positive signals lies in a
crucial observation in estimating the lower bound of diameter of the error set
(Definition 2.3). In particular, in Theorem 4.1, we observe and demonstrate
that the recovered and the underlying signals in the example constructed in
[2] can actually be positive signals at the same time.
We also examine the performance limit of the Matrix Pencil method in resolving
positive signals by numerical experiments. The oberved error amplification in
the experiments exactly verifies our theory for the minimax error rate. This
also indicates that the Matrix Pencil method has the optimal performance in
super-resolving positive sources.
### 1.2 Related work and discussion
In 1992, Donoho first studied the possibility and difficulties of super-
resolving multiple sources from noisy measurements. In particular, he
considered measures supported on a lattice
$\\{k\Delta\\}_{k=-\infty}^{\infty}$ and regularized by a so-called “Rayleigh
index”. The measurement is then the noisy Fourier transform of the discrete
measure with cutoff frequency $\Omega$. He derived both the lower and upper
bounds for the minimax error of the amplitude recovery in terms of the noise
level, grid spacing, cutoff frequency, and Rayleigh index. His results
emphasize the importance of the sparsity in the super-resolution. The results
were improved in recent years for the case when resolving $n$-sparse on-the-
grid sources [6]. Concretely, the authors of [6] showed that the minimax error
rate for amplitudes recovery scales like $\mathrm{SRF}^{2n-1}\epsilon$, where
$\epsilon$ is the noise level and $\mathrm{SRF}:=\frac{1}{\Delta\Omega}$ is
the super-resolution factor. Similar results for multi-clumps cases were also
derived in [1, 10].
A closely related work to the present paper is [2], in which the authors
derived sharp minimax errors for the location and the amplitude recovery of
off-the-grid sources. They showed that for complex sources satisfying the
$(p,h,T,\tau,\eta)$-clustered configuration (Definition 2.4) and
$\epsilon\lesssim\mathrm{SRF}^{-2p+1}$ with $p$ being the number of the
cluster nodes, the minimax error rate for reconstructing of the cluster nodes
is of the order $(\mathrm{SRF})^{2p-2}\frac{\epsilon}{\Omega}$, while for
recovering the corresponding amplitudes the rate is of the order
$(\mathrm{SRF})^{2p-1}\epsilon$. Moreover, the corresponding minimax rates for
the recovery of the non-cluster nodes and amplitudes are of the order
$\frac{\epsilon}{\Omega}$ and $\epsilon$, respectively. As mentioned above, in
the present paper we have generalized these results to the case when resolving
positive sources. Thus, the minimax error estimations for super-resolving both
one-dimensional complex and positive sources are well established now.
On the other hand, in order to characterize the exact resolution in the number
and location recovery, in the earlier works [16, 15, 14, 12, 13, 11] the
authors have defined the so-called "computational resolution limits", which
characterize the minimum required distance between point sources so that their
number and locations can be stably resolved under certain noise level. It was
shown that the computational resolution limits for the number and location
recoveries in the $k$-dimensional super-resolution problem should be bounded
above by respectively
$\frac{C_{num}(k,n)}{\Omega}\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2n-2}}$
and
$\frac{C_{supp}(k,n)}{\Omega}\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2n-1}}$,
where $C_{num}(k,n)$ and $C_{supp}(k,n)$ are certain constants depending only
on the source number $n$ and the space dimensionality $k$. In particular,
these results were generalized to the case when resolving positive sources in
[13]. In this paper, a similar idea is used to generalize the miminax error
estimate to the positive cases.
For other works related to the limit of super-resolution, we refer the readers
to [20, 4] for understanding the resolution limit from the perceptive of
sample complexity and to [24, 5] for the resolving limit of some algorithms.
For the super-resolution of positive sources, to the best of our knowledge,
the theoretical possibility for the super-resolution of positive sources was
first considered in [8], where the authors characterized the relation between
the sparsity of the on-the-grid signal $\mathbf{x}$ and the possibility of
super-resolution in certain sense. Their definition and results focused on the
possibility of overcoming Rayleigh limit in the presence of sufficiently small
noise, while our work analyzes the non-asymptotic behavior of the
reconstructions.
In recent years, some researchers analyzed the stability of some super-
resolution algorithms in a non-asymptotic regime [22, 21, 7] and derived
similar stability results to those proved in this paper, which exhibit the
optimal performance of these algorithms.
### 1.3 Organization of the paper
The paper is organized in the following way. Section 2 presents the main
results of the minimax error rate and Section 3 exhibits the performance of
Matrix Pencil method by numerical experiments. Section 4 proves the main
results stated in Section 2.
## 2 Minimax bound for the location and amplitude recoveries
In this section, we present minimax error estimates for the location and
amplitude recoveries in the super-resolution of positive signals.
### 2.1 Notation and preliminaries
We shall denote by $\mathcal{P}_{d},\mathcal{P}_{d}^{+}$ the parameter space
of respectively general and positive signals $F$ with amplitudes $a_{j}$’s and
real, distinct and ordered nodes $x_{j}$’s:
$\displaystyle\mathcal{P}_{d}=\left\\{(\mathbf{a},\mathbf{x}):\mathbf{a}=\left(a_{1},\ldots,a_{d}\right)\in\mathbb{C}^{d},\mathbf{x}=\left(x_{1},\ldots,x_{d}\right)\in\mathbb{R}^{d},x_{1}<x_{2}<\ldots<x_{d}\right\\},$
$\displaystyle\mathcal{P}_{d}^{+}=\left\\{(\mathbf{a},\mathbf{x}):\mathbf{a}=\left(a_{1},\ldots,a_{d}\right)\in(\mathbb{R}^{+})^{d},\mathbf{x}=\left(x_{1},\ldots,x_{d}\right)\in\mathbb{R}^{d},x_{1}<x_{2}<\ldots<x_{d}\right\\},$
and identify the $F$’s with their parameters
$(\mathbf{a},\mathbf{x})\in\mathcal{P}_{d}$ or $\mathcal{P}_{d}^{+}$. We
denote
$\|F\|_{\infty}=\max\left(\|\mathbf{a}\|_{\infty},\|\mathbf{x}\|_{\infty}\right).$
We shall also denote the orthogonal coordinate projections of a signal $F$ to
the $j$-th node and $j$-th amplitude, respectively, by
$P_{\mathbf{x},j}:\mathcal{P}_{d}(\mathcal{P}_{d}^{+})\rightarrow\mathbb{R}$
and
$P_{\mathbf{a},j}:\mathcal{P}_{d}(\mathcal{P}_{d}^{+})\rightarrow\mathbb{C}(\mathbb{R}^{+})$.
Let $L_{\infty}[-\Omega,\Omega]$ be the space of bounded complex-valued
functions defined on $[-\Omega,\Omega]$ with the norm
$\|e\|_{\infty}=\max_{|s|\leqslant\Omega}|e(s)|$.
###### Definition 2.1.
Given $\Omega>0$ and $U\subseteq\mathcal{P}_{d}^{+}$, we denote by
$\mathfrak{F}(\Omega,U)$ the class of all admissible reconstruction
algorithms, i.e.,
$\mathfrak{F}(\Omega,U)=\left\\{f:L_{\infty}[-\Omega,\Omega]\rightarrow
U\right\\}.$
###### Definition 2.2.
Let $U\subset\mathcal{P}_{d}^{+}$. We consider the minimax error rate in
estimating a signal $F\in U$ from $\Omega$-bandlimited data as in (1.1), with
a measurement error $\epsilon>0$:
$\mathcal{E}^{+}(\epsilon,U,\Omega)=\inf_{\mathfrak{f}\in\mathfrak{F}(\Omega,U)}\sup_{F\in
U}\sup_{\|e\|_{\infty}\leqslant\epsilon}\|F-\mathfrak{f}(\mathcal{F}[F]+e)\|_{\infty}.$
Note that in order to analyze how the minimax error rate relates to the
separation of the nodes and the magnitudes of the amplitudes, we will consider
$U\in\mathcal{P}_{d}^{+}$ with certain specific constraints in the following
discussions.
Similarly, the minimax errors of estimating the individual nodes and the
amplitudes of $F\in U$ are defined respectively by
$\displaystyle\mathcal{E}^{+,\mathbf{x},j}(\epsilon,U,\Omega)=\inf_{\mathfrak{f}\in\mathfrak{F}(\Omega,U)}\sup_{F\in
U}\sup_{\|e\|\leqslant\epsilon}\left|P_{\mathbf{x},j}(F)-P_{\mathbf{x},j}(\mathfrak{f}(\mathcal{F}(F)+e))\right|,$
$\displaystyle\mathcal{E}^{+,\mathbf{a},j}(\epsilon,U,\Omega)=\inf_{\mathfrak{f}\in\mathfrak{F}(\Omega,U)}\sup_{F\in
U}\sup_{\|e\|_{\infty}\leqslant\epsilon}\left|P_{\mathbf{a},j}(F)-P_{\mathbf{a},j}(\mathfrak{f}(\mathcal{F}(F)+e))\right|.$
For a fixed $F\in\mathcal{P}_{d}^{+}$, we define the positive and general
$\epsilon$-error set as follows.
###### Definition 2.3.
The error set of positive signals
$E_{\epsilon,\Omega}^{+}(F)\subset\mathcal{P}_{d}^{+}$ is the set consisting
of all the signals $\hat{F}\in\mathcal{P}_{d}^{+}$ with
$\left|\mathcal{F}[\hat{F}](\omega)-\mathcal{F}[F](\omega)\right|\leqslant\epsilon,\quad\omega\in[-\Omega,\Omega].$
(2.1)
Moreover, the error set of general signal
$E_{\epsilon,\Omega}(F)\subset\mathcal{P}_{d}$ is the set consisting of all
the $\hat{F}\in\mathcal{P}_{d}$ satisfying (2.1).
We will denote by
$E_{\epsilon}^{+,\mathbf{x},j}(F)=E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)$ and
$E_{\epsilon}^{+,\mathbf{a},j}(F)=E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)$ the
projections of the error set of positive signals onto the individual nodes and
the amplitudes components, respectively:
$\displaystyle
E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)=\left\\{\mathbf{x}_{j}^{\prime}\in\mathbb{R}:\left(\mathbf{a}^{\prime},\mathbf{x}^{\prime}\right)\in
E_{\epsilon,\Omega}^{+}(F)\right\\}\equiv
P_{\mathbf{x},j}E_{\epsilon,\Omega}^{+}(F),$ $\displaystyle
E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)=\left\\{\mathbf{a}_{j}^{\prime}\in\mathbb{R}^{+}:\left(\mathbf{a}^{\prime},\mathbf{x}^{\prime}\right)\in
E_{\epsilon,\Omega}^{+}(F)\right\\}\equiv
P_{\mathbf{a}_{,}j}E_{\epsilon,\Omega}^{+}(F).$
Furthermore, we denote by
$E_{\epsilon}^{\mathbf{x},j}(F)=E_{\epsilon,\Omega}^{\mathbf{x},j}(F)$ and
$E_{\epsilon}^{\mathbf{a},j}(F)=E_{\epsilon,\Omega}^{\mathbf{a},j}(F)$ the
projections of the error set of general signals onto the individual nodes and
the amplitudes components, respectively:
$\displaystyle
E_{\epsilon,\Omega}^{\mathbf{x},j}(F)=\left\\{\mathbf{x}_{j}^{\prime}\in\mathbb{R}:\left(\mathbf{a}^{\prime},\mathbf{x}^{\prime}\right)\in
E_{\epsilon,\Omega}(F)\right\\}\equiv P_{\mathbf{x},j}E_{\epsilon,\Omega}(F),$
$\displaystyle
E_{\epsilon,\Omega}^{\mathbf{a},j}(F)=\left\\{\mathbf{a}_{j}^{\prime}\in\mathbb{C}:\left(\mathbf{a}^{\prime},\mathbf{x}^{\prime}\right)\in
E_{\epsilon,\Omega}(F)\right\\}\equiv
P_{\mathbf{a}_{,}j}E_{\epsilon,\Omega}(F).$
For any subset $V$ of a normed vector space with norm $\|\cdot\|$, the
diameter of $V$ is given by
$\mathrm{diam}(V)=\sup_{\mathbf{v}^{\prime},\mathbf{v}^{\prime\prime}\in
V}\left\|\mathbf{v}^{\prime}-\mathbf{v}^{\prime\prime}\right\|_{\infty}.$
By the theory of optimal recovery [19, 17, 18], the minimax errors are
directly linked to the diameter of the corresponding projections of the error
set. More specifically, we have the following proposition that is similar to
Proposition 2.4 in [2].
###### Proposition 2.1.
For $U\subset\mathcal{P}_{d}^{+},\Omega>0,1\leqslant j\leqslant d$ and
$\epsilon>0$, we have
$\displaystyle\frac{1}{2}\sup_{F:E_{\frac{1}{2}\epsilon,\Omega}^{+}(F)\subseteq
U}\mathrm{diam}\left(E_{\frac{1}{2}\epsilon,\Omega}^{+}(F)\right)\leqslant\mathcal{E}^{+}(\epsilon,U,\Omega)\leqslant\sup_{F\in
U}\mathrm{diam}\left(E_{2\epsilon,\Omega}^{+}(F)\right),$ (2.2)
$\displaystyle\frac{1}{2}\sup_{F:E_{\frac{1}{2}\epsilon,\Omega}^{+}(F)\subseteq
U}\mathrm{diam}\left(E_{\frac{1}{2}\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right)\leqslant\mathcal{E}^{+,\mathbf{x},j}(\epsilon,U,\Omega)\leqslant\sup_{F\in
U}\mathrm{diam}\left(E_{2\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right),$
$\displaystyle\frac{1}{2}\sup_{F:E_{\frac{1}{2}\epsilon,\Omega}^{+}(F)\subseteq
U}\mathrm{diam}\left(E_{\frac{1}{2}\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right)\leqslant\mathcal{E}^{+,\mathbf{a},j}(\epsilon,U,\Omega)\leqslant\sup_{F\in
U}\mathrm{diam}\left(E_{2\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right).$
### 2.2 Uniform estimates of minimax error for clustered configurations
Similarly to [2], the main goal of this paper is to estimate
$\mathcal{E}^{+,\mathbf{x},j}(\epsilon,U,\Omega),\mathcal{E}^{+,\mathbf{a},j}(\epsilon,U,\Omega)$,
where $U\subset\mathcal{P}_{d}^{+}$ are certain compact subsets of
$\mathcal{P}_{d}^{+}$ consisting of signals with $p\leqslant d$ nodes that are
nearly uniformly distributed, forming a cluster. To be more specific, the set
$U$ is defined as follows; See also [2, Definition 2.5].
###### Definition 2.4.
(Uniform cluster configuration)
Given $0<\tau,\eta\leqslant 1$ and $0<h\leqslant T$, a node vector
$\mathbf{x}=\left(x_{1},\ldots,x_{d}\right)\in\mathbb{R}^{d}$ is said to form
a $(p,h,T,\tau,\eta)$-clustered configuration, if there exists a subset of $p$
nodes
$\mathbf{x}^{c}=\left\\{x_{\kappa},\ldots,x_{\kappa+p-1}\right\\}\subset\mathbf{x},p\geqslant
2$, which satisfies the following conditions:
* (i)
for each $x_{j},x_{k}\in\mathbf{x}^{c},j\neq k$,
$\tau h\leqslant\left|x_{j}-x_{k}\right|\leqslant h;$
* (ii)
for $x_{\ell}\in\mathbf{x}\backslash\mathbf{x}^{c}$ and
$x_{j}\in\mathbf{x},\ell\neq j$,
$\eta T\leqslant\left|x_{\ell}-x_{j}\right|\leqslant T.$
One of the main contributions of [2] is an upper bound on
$\operatorname{diam}\left(E_{\epsilon,\Omega}(F)\right)$, and its coordinate
projections, for any signal $F$ forming a clustered configuration as above.
Here, we generalize the result to the positive signal cases, which is a direct
consequence of [2, Theorem 2.6].
###### Theorem 2.1.
(Upper bound)
Let the positive signal $F=(\mathbf{a},\mathbf{x})\in\mathcal{P}_{d}^{+}$,
such that $\mathbf{x}$ forms a $(p,h,T,\tau,\eta)$-clustered configuration and
$0<m\leqslant\|\mathbf{a}\|$. Then, there exist positive constants
$C_{1},\ldots,C_{5}$, depending only on $d,p,m$, such that for each
$\frac{C_{4}}{\eta T}\leqslant\Omega\leqslant\frac{C_{5}}{h}$ and
$\epsilon\leqslant C_{3}(\Omega\tau h)^{2p-1}$, it holds that
$\begin{gathered}\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right)\leqslant\frac{C_{1}}{\Omega}\epsilon\times\begin{cases}(\Omega\tau
h)^{-2p+2},&x_{j}\in\mathbf{x}^{c},\\\
1,&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c},\end{cases}\\\
\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right)\leqslant
C_{2}\epsilon\times\begin{cases}(\Omega\tau
h)^{-2p+1},&x_{j}\in\mathbf{x}^{c},\\\
1,&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}.\end{cases}\end{gathered}$
###### Proof.
By Theorem 2.6 in [2], under the same condition we have
$\begin{gathered}\operatorname{diam}\left(E_{\epsilon,\Omega}^{\mathbf{x},j}(F)\right)\leqslant\frac{C_{1}}{\Omega}\epsilon\times\begin{cases}(\Omega\tau
h)^{-2p+2},&x_{j}\in\mathbf{x}^{c},\\\
1,&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c},\end{cases}\\\
\operatorname{diam}\left(E_{\epsilon,\Omega}^{\mathbf{a},j}(F)\right)\leqslant
C_{2}\epsilon\times\begin{cases}(\Omega\tau
h)^{-2p+1},&x_{j}\in\mathbf{x}^{c},\\\
1,&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}.\end{cases}\end{gathered}$
On the other hand, note that for $\epsilon>0$, according to the definition of
$\mathrm{diam}(\cdot)$ we have
$\mathrm{diam}\left(E_{\epsilon,\Omega}^{+,\boldsymbol{\mathbf{x}},j}(F)\right)\leq\mathrm{diam}\left(E_{\epsilon,\Omega}^{\boldsymbol{\mathbf{x}},j}(F)\right),\quad\mathrm{diam}\left(E_{\epsilon,\Omega}^{+,\boldsymbol{\mathbf{a}},j}(F)\right)\leq\mathrm{diam}\left(E_{\epsilon,\Omega}^{\boldsymbol{\mathbf{a}},j}(F)\right).$
This proves the theorem. ∎
The above estimates are optimal, as shown by our next main theorem. This is
the main contribution of our paper, by which we non-trivially generalize the
results in [2, Theorem 2.7]. For simplicity and without loss of generality, we
assume that the index $\kappa$ is fixed in the result below.
###### Theorem 2.2.
(Lower bound)
Let $m\leqslant M,2\leqslant p\leqslant
d,\tau\leqslant\frac{1}{p-1},\eta<\frac{1}{d},T>0$ be fixed. There exist
positive constants $C_{1}^{\prime}\ldots,C_{5}^{\prime}$, depending only on
$d,p,m,M$, such that for every $\Omega,h$ satisfying $h\leqslant
C_{4}^{\prime}T$ and $\Omega h\leqslant C_{5}^{\prime}$, there exists
$F=(\mathbf{a},\mathbf{x})\in\mathcal{P}_{d}^{+}$, with $\mathbf{x}$ forming a
$(p,h,T,\tau,\eta)$-clustered configuration, and with
$0<m\leqslant\|\mathbf{a}\|\leq M<\infty$, such that for certain indices
$j_{1},j_{2}\in\\{\kappa,\ldots,\kappa+p-1\\}$ and every $\epsilon\leqslant
C_{3}^{\prime}(\Omega\tau h)^{2p-1}$, it holds that
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right)\geqslant\frac{C_{1}^{\prime}}{\Omega}\epsilon\times\begin{cases}(\Omega\tau
h)^{-2p+2},&j=j_{1},\\\ 1,&\forall
j\notin\\{\kappa,\ldots,\kappa+p-1\\},\end{cases}$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right)\geqslant
C_{2}^{\prime}\epsilon\times\begin{cases}(\Omega\tau h)^{-2p+1},&j=j_{2},\\\
1,&\forall j\notin\\{\kappa,\ldots,\kappa+p-1\\}.\end{cases}$
###### Proof.
See the proof in Section 4. ∎
Thus combining Theorems 2.1 and 2.2 with Proposition 2.1 now, we can obtain
the following theorem for the optimal rates of the minimax errors
$\mathcal{E}^{+,\boldsymbol{\mathbf{x}},j},\mathcal{E}^{+,\boldsymbol{\mathbf{a}},j}$.
This generalizes Theorem 2.8 in [2].
###### Theorem 2.3.
Let $m>0,2\leqslant p\leqslant d,\tau<\frac{1}{2(p-1)},\eta<\frac{1}{2d},T>0$
be fixed. There exist constants $c_{1},c_{2},c_{3}$, depending only on $d,p,m$
such that for all $\frac{c_{1}}{\eta
T}\leqslant\Omega\leqslant\frac{c_{2}}{h}$ and $\epsilon\leqslant
c_{3}(\Omega\tau h)^{2p-1}$, the minimax error rates for the set
$\displaystyle U:=$ $\displaystyle U(p,d,h,\tau,\eta,T,m)$ $\displaystyle=$
$\displaystyle\left\\{(\mathbf{a},\mathbf{x})\in\mathcal{P}_{d}^{+}:0<m\leqslant\|\mathbf{a}\|\leq
M<\infty,\text{$\mathbf{x}$ forms a $(p,h,T,\tau,\eta)$-clustered
configuration}\right\\}$
satisfy the following:
* •
For the non-cluster nodes:
$\forall
j\notin\\{\kappa,\ldots,\kappa+p-1\\}:\quad\left\\{\begin{array}[]{l}\mathcal{E}^{+,\mathbf{x},j}(\epsilon,U,\Omega)\asymp\frac{\epsilon}{\Omega},\\\
\mathcal{E}^{+,\mathbf{a},j}(\epsilon,U,\Omega)\asymp\epsilon.\end{array}\right.$
* •
For the cluster nodes:
$\displaystyle\max_{j=\kappa,\ldots,\kappa+p-1}\mathcal{E}^{+,\mathbf{x},j}(\epsilon,U,\Omega)$
$\displaystyle\asymp\frac{\epsilon}{\Omega}(\Omega\tau h)^{-2p+2},$
$\displaystyle\max_{j=\kappa,\ldots,\kappa+p-1}\mathcal{E}^{+,\mathbf{a},j}(\epsilon,U,\Omega)$
$\displaystyle\asymp\epsilon(\Omega\tau h)^{-2p+1}.$
The proportionality constants in the above statements depend only on
$d,p,m,M$.
###### Proof.
The proof is the same as the one for Theorem 2.8 in [2]. Here, we present the
details for the convenience of reading and completeness. Let
$C_{3},C_{3}^{\prime},C_{4},C_{4}^{\prime},C_{5},C_{5}^{\prime}$ be the
constants from Theorems 2.1 and 2.2. Let $c_{1}=C_{4}$ and
$c_{2}=\min\left(C_{5},C_{5}^{\prime},C_{4}C_{4}^{\prime}\right)$. Let
$\frac{c_{1}}{\eta T}\leqslant\Omega\leqslant\frac{c_{2}}{h}$, and
$\epsilon\leqslant c_{3}(\Omega\tau h)^{2p-1}$, where
$c_{3}\leqslant\min\left(C_{3},C_{3}^{\prime}\right)$ will be determined
below. It can be verified that $\Omega,h$ and $\epsilon$ as above satisfy the
conditions of both Theorems 2.1 and 2.2.
Upper bound Directly follows from the upper bounds in Theorem 2.1 and
Proposition 2.1.
Lower bound Denote $U_{\epsilon}=\left\\{F\in
U:E_{\frac{1}{2}\epsilon,\Omega}^{+}(F)\subseteq U\right\\}$. In order to
prove the lower bounds on $\mathcal{E}^{+,\mathbf{x},j}$ and
$\mathcal{E}^{+,\mathbf{a},j}$, by Proposition 2.1 it suffices to show that
there exists an $F\in U_{\epsilon}\neq\emptyset$ such that the conclusions of
Theorem 2.2 are satisfied for this $F$. Note that the set $U$ has a non-empty
interior. Furthermore, one can choose $m^{\prime},M^{\prime}$ satisfying
$m<m^{\prime}<M^{\prime}<M$, and also $T^{\prime}=0.99T,\tau^{\prime}=2\tau$
and $\eta^{\prime}=2\eta$, such that
$U^{\prime}=U\left(p,d,h,\tau^{\prime},\eta^{\prime},T^{\prime},m^{\prime},M^{\prime}\right)\subset
U,\quad\partial U^{\prime}\cap\partial U=\emptyset.$
By the construction of $U^{\prime}$, there exist positive constants
$\widetilde{C}_{1},\widetilde{C}_{2}$, independent of $\Omega,h$ and
$\tau,\eta$, such that
$\displaystyle\inf_{u\in\partial U,u^{\prime}\in\partial
U^{\prime}}\left|P_{\mathbf{x},j}(u)-P_{\mathbf{x},j}\left(u^{\prime}\right)\right|\geqslant\tilde{C}_{1}\times\begin{cases}\tau
h,&x_{j}\in\mathbf{x}^{c},\\\ \eta
T,&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c};\end{cases}$ (2.3)
$\displaystyle\inf_{u\in\partial U,u^{\prime}\in\partial
U^{\prime}}\left|P_{\mathbf{a},j}(u)-P_{\mathbf{a},j}\left(u^{\prime}\right)\right|\geqslant\tilde{C}_{2}.$
Now, we use the fact that $\epsilon<c_{3}(\Omega\tau h)^{2p-1}$. Applying
Theorem 2.1 to an arbitrary positive signal $F^{\prime}\in U^{\prime}$ and
using the conditions $\frac{1}{\Omega}\leqslant\frac{\eta T}{c_{1}}$ and
$\Omega\tau h\leqslant\Omega h\leqslant c_{2}$, we obtain that
$\displaystyle\operatorname{diam}\left(E_{\frac{1}{2}\epsilon}^{+,\mathbf{x},j}\left(F^{\prime}\right)\right)$
$\displaystyle\leqslant\begin{cases}\frac{C_{1}c_{3}}{2}\tau
h,&x_{j}\in\mathbf{x}^{c},\\\ \frac{C_{1}c_{3}}{2\Omega}(\Omega\tau
h)^{2p-1}\leqslant\frac{C_{1}c_{3}}{2c_{1}}c_{2}^{2p-1}\eta
T,&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c};\end{cases}$ (2.4)
$\displaystyle\operatorname{diam}\left(E_{\frac{1}{2}\epsilon}^{+,\mathbf{a},j}\left(F^{\prime}\right)\right)$
$\displaystyle\leqslant\begin{cases}\frac{C_{2}c_{3}}{2},&x_{j}\in\mathbf{x}^{c},\\\
\frac{C_{2}c_{3}}{2}c_{2}^{2p-1},&x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}.\end{cases}$
Next, we set
$c_{3}=\min\left(C_{3},C_{3}^{\prime},C_{3}^{\prime\prime}\right)$, where
$C_{3}^{\prime\prime}=\min\left(1,c_{1}\right)\times\min\left(1,c_{2}^{-2p+1}\right)\times\min\left(\frac{2\tilde{C}_{1}}{C_{1}},\frac{2\tilde{C}_{2}}{C_{2}}\right).$
Combining (2.3) and (2.4), we obtain that
$E_{\frac{1}{2}\epsilon,\Omega}^{+}(F^{\prime})\subseteq U$, i.e.
$F^{\prime}\in U_{\epsilon}$. Since $F^{\prime}\in U^{\prime}$ is arbitrary,
we conclude that $U^{\prime}\subseteq U_{\epsilon}$. Since clearly
$U^{\prime}\neq\emptyset$, applying Proposition 2.1 and Theorem 2.2 finishes
the proof. ∎
## 3 Numerical optimality of Matrix Pencil method (MP method)
Theorem 2.3 establishes the optimal error rate for super-resolving the
locations and amplitudes of positive sources. In this section, we demonstrate
by numerical experiments the optimal performance of MP method in recovering
the locations of positive sources. Note that the numerical experiments in [2]
have already demonstrated the optimal performance of Matrix Pencil method for
resolving general sources. Here we conduct experiments similar to those in [2]
but focusing on the case of resolving positive sources.
### 3.1 Review of the MP method
In this section, we review the MP method . We assume that the noisy Fourier
data of the signal $F$ is given by
$\boldsymbol{\mathbf{Y}}(\omega)=\sum_{j=1}^{d}a_{j}e^{-2\pi
ix_{j}\omega}+\epsilon(\omega),\quad\omega\in[-\Omega,\Omega].$
The measurements are usually taken at $N$ evenly spaced points,
$\omega_{1}=-\Omega,\omega_{2}=-\Omega+h,\cdots,\omega_{N}=\Omega$ with $h$
being the spacing. From the measurement
$\mathbf{Y}=(\mathbf{Y}(\omega_{1}),\mathbf{Y}(\omega_{2}),\cdots,\mathbf{Y}(\omega_{N}))^{\top}$
(3.1)
and $\hat{N}=\lfloor\frac{N-1}{2}\rfloor$, we assemble the
$(\hat{N}+1)\times(\hat{N}+1)$ Hankel matrix
$\boldsymbol{\mathbf{H}}=\begin{pmatrix}\mathbf{Y}(\omega_{1})&\mathbf{Y}(\omega_{2})&\cdots&\mathbf{Y}(\omega_{\hat{N}})\\\
\mathbf{Y}(\omega_{2})&\mathbf{Y}(\omega_{3})&\cdots&\mathbf{Y}(\omega_{\hat{N}+1})\\\
\cdots&\cdots&\ddots&\cdots\\\
\mathbf{Y}(\omega_{\hat{N}})&\mathbf{Y}(\omega_{\hat{N}+1})&\cdots&\mathbf{Y}(\omega_{2\hat{N}+1})\end{pmatrix}.$
(3.2)
Let $\boldsymbol{\mathbf{H}}_{u}:=\boldsymbol{\mathbf{H}}[1:\hat{N},:]$ (and
$H_{l}:=H[2:\hat{N}+1,:]$) be the $\hat{N}\times(\hat{N}+1)$ matrix obtained
from the Hankel matrix $\boldsymbol{\mathbf{H}}$ given by (3.2) by selecting
the first $\hat{N}$ rows (respectively, the second to the $(\hat{N}+1)$-th
rows). It turns out that, in the noiseless case, $e^{-2\pi ix_{j}h},1\leq
j\leq n$, are exactly the nonzero generalized eigenvalues of the pencil
$\boldsymbol{\mathbf{H}}_{l}-z\boldsymbol{\mathbf{H}}_{u}$. In the noisy case,
when the sources are well-separated, each of the first $n$ nonzero generalized
eigenvalues of the pencil
$\boldsymbol{\mathbf{H}}_{l}-z\boldsymbol{\mathbf{H}}_{u}$ is close to
$e^{-2\pi ix_{j}h}$ for some $j$. We summarize the Matrix Pencil method in
Algorithm 1.
Input: Source number $d$, measurement: $\mathbf{Y}$ in (3.1);
1: Let $\hat{N}=\lfloor\frac{N-1}{2}\rfloor$. Formulate the
$(\hat{N}+1)\times(\hat{N}+1)$ Hankel matrix $\mathbf{H}$ given by (3.2) from
$\mathbf{Y}$, and the matrices
$\boldsymbol{\mathbf{H}}_{u},\boldsymbol{\mathbf{H}}_{l}$;
2: Compute the truncated Singular Value Decomposition (SVD) of
$\boldsymbol{\mathbf{H}}_{u}$, $\boldsymbol{\mathbf{H}}_{l}$ of order $d$:
$\boldsymbol{\mathbf{H}}_{u}=U_{1}\Sigma_{1}V_{1}^{*},\quad\boldsymbol{\mathbf{H}}_{l}=U_{2}\Sigma_{2}V_{2}^{*},$
where $U_{1},U_{2},V_{1},V_{2}$ are $\hat{N}\times d$ matrices and
$\Sigma_{1},\Sigma_{2}$ are $d\times d$ matrices;
3: Generate the reduced pencil
$\mathbf{\hat{H}}_{u}=U_{2}^{*}U_{1}\Sigma_{1}V_{1}^{*}V_{2},\quad\mathbf{\hat{H}}_{l}=\Sigma_{2},$
where $\mathbf{\hat{H}}_{u}$, $\mathbf{\hat{H}}_{l}$ are $d\times d$ matrices;
4: Compute the generalized eigenvalues $\\{\hat{z}_{j}\\}$ of the reduced
pencil $(\mathbf{\hat{H}}_{u},\mathbf{\hat{H}}_{l})$, and put
$\\{\hat{x}_{j}\\}=\\{\arg(\hat{z}_{j})\\},j=1,\cdots,n$, where the $\arg(z)$
is the argument of $z$;
5: Solving the linear least squares problem
$\mathbf{\hat{b}}=\arg\min_{\mathbf{b}\in\mathbb{C}^{d}}\left|\left|{\boldsymbol{\mathbf{Y}}-V\mathbf{b}}\right|\right|_{2},$
where $V$ is the Vandermonde matrix $V=\left(e^{-2\pi
i\hat{x}_{j}\omega_{k}}\right)_{k=1,\cdots,N}^{j=1,\cdots,d}$;
6: Compute $\hat{a}_{j}$ by $|\mathbf{\hat{b}}_{j}|$;
Return: The estimated $\hat{x}_{j}$’s and $\hat{a}_{j}$’s.
Algorithm 1 The Matrix Pencil algorithm
### 3.2 Numerical experiments
We conduct 1000 random experiments (the randomness was in the choice of
$a_{j},x_{j},\epsilon$) to examine the error amplification in the recovery of
the nodes and amplitudes. In particular, we consider recovering $p=2$ cluster
nodes and $1$ non-cluster node and their corresponding amplitudes. Each single
experiment is summarized in Algorithm 2. The results are shown in Figure 3.1
and we observe that the error amplification is consistent with what we have
predicted, i.e., the error amplification factors for resolving nodes and
amplitudes are respectively $\mathrm{SRF}^{2p-2}$ and $\mathrm{SRF}^{2p-1}$
for the cluster nodes with size $p$. Moreover, for resolving non-cluster
nodes, both the corresponding error amplification factors are bounded by a
small constant.
Input: $p,d,N,\epsilon$;
1: Construct the signal $F$ with $p$ closely-spaced sources and $d-p$ non-
clustered sources;
2: Generate the measurement $\boldsymbol{\mathbf{Y}}$ defined by (3.1) with
$\epsilon$ being the noise level;
3: Execute the MP method (Algorithm 1) and obtain
$F_{MP}=\left(\mathbf{a}^{MP},\mathbf{x}^{MP}\right)$. The nodes in
$\mathbf{x}^{MP}$ are ordered in an increasing manner;
4: for _each $j$_ do
Compute the error for node $j$ :
$e_{j}=\left|\mathbf{x}_{j}^{MP}-\mathbf{x}_{j}\right|.$ The success for node
$j$ is defined as $\operatorname{Succ}_{j}=\left(e_{j}<\frac{\min_{\ell\neq
j}\left|\mathbf{x}_{\ell}-\mathbf{x}_{j}\right|}{3}\right).$ if _Succ
${}_{j}==$ true_ then
Compute normalized node error amplification factor
$\mathcal{K}_{\mathbf{x},j}=\frac{\left|\mathbf{x}_{j}-\mathbf{x}_{j}^{MP}\right|\cdot\Omega}{\epsilon};$
Compute normalized amplitude error amplification factor
$\mathcal{K}_{\mathbf{a},j}=\frac{\left|\mathbf{a}_{j}-\mathbf{a}_{j}^{MP}\right|}{\epsilon};$
Return:
$\left(\mathcal{K}_{\mathbf{x},j},\mathcal{K}_{\mathbf{a},j},\operatorname{Succ}_{j}\right)$
for each node $j=1,\ldots,d$.
Algorithm 2 A single experiment
(a)
(b)
Figure 3.1: The error amplification factors. For the cluster nodes, the error
amplification factors
$\mathcal{K}_{\boldsymbol{\mathbf{x}},j},\mathcal{K}_{\boldsymbol{\mathbf{a}},j}$
scale like $\mathrm{SRF}^{2p-2},\mathrm{SRF}^{2p-1}$, respectively. For the
non-cluster nodes, both error amplification factors are bounded by a small
constant.
## 4 Proof of Theorem 2.2
### 4.1 Normalization
Similarly to [2], for ease of exposition, we should normalize the cluster
configuration in some of the following discussions. Let us first define the
scale transformation on $\mathcal{P}_{d}^{+}$.
###### Definition 4.1.
For $F=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right)\in\mathcal{P}_{d}^{+}$
and $T>0$, we define
$SC_{T}:\mathcal{P}_{d}^{+}\rightarrow\mathcal{P}_{d}^{+}$ as follows:
$SC_{T}(F)(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-\frac{x_{j}}{T}\right).$
By the scale property of the Fourier transform, we have that for any
$\epsilon>0$,
$SC_{T}\left(E_{\epsilon,\Omega}^{+}(F)\right)=E_{\epsilon,\Omega
T}^{+}\left(SC_{T}(F)\right).$
Thus the following proposition holds.
###### Proposition 4.1.
Let $F=(\mathbf{a},\mathbf{x})\in\mathcal{P}_{d}^{+}$ and $T>0$. Then, for any
$\epsilon>0$ and $1\leqslant j\leqslant d$, we have
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right)=T\operatorname{diam}\left(E_{\epsilon,\Omega
T}^{+,\mathbf{x},j}\left(SC_{T}\left(F\right)\right)\right),$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right)=\operatorname{diam}\left(E_{\epsilon,\Omega
T}^{+,\mathbf{a},j}\left(SC_{T}\left(F\right)\right)\right).$
### 4.2 Auxiliary lemmas
In this subsection, we introduce some notation and lemmas that are used in the
following proofs. Set
$\phi_{s}(t)=\left(1,t,\cdots,t^{s}\right)^{\top}.$ (4.1)
###### Lemma 4.1.
Let $t_{1},\cdots,t_{k}$ be $k$ different real numbers and let $t$ be a real
number. We have
$\left(D_{k}(k-1)^{-1}\phi_{k-1}(t)\right)_{j}=\Pi_{1\leq q\leq k,q\neq
j}\frac{t-t_{q}}{t_{j}-t_{q}},$
where $D_{k}(k-1):=\big{(}\phi_{k-1}(t_{1}),\cdots,\phi_{k-1}(t_{k})\big{)}$
with $\phi_{k-1}(\cdot)$ defined by (4.1).
###### Proof.
This is [15, Lemma 5]. ∎
The following proposition is the main result for proving Theorem 2.2 and we
present its detailed proof in Section 4.4.
###### Proposition 4.2.
Let $F=(\mathbf{a},\mathbf{x})\in\mathcal{P}_{d}^{+}$, such that $\mathbf{x}$
forms a $(p,h,1,\tau,\eta)$-clustered configuration, with cluster nodes
$\mathbf{x}^{c}=\left(x_{1},\ldots,x_{p}\right)$ (according to Definition
2.4), and with $\mathbf{a}\in(\mathbb{R}^{+})^{d}$ satisfying
$m\leqslant\|\mathbf{a}\|\leqslant M$. Then, there exist constants
$c_{1},k_{1},k_{2},k_{3},k_{4}$, depending only on $(d,p,\tau,m,M)$, such that
for all $\epsilon<c_{1}(\Omega h)^{2p-1}$ and $\Omega h\leqslant 2$, there
exists a signal $F_{\epsilon}\in\mathcal{P}_{d}^{+}$ satisfying, for some
$j_{1},j_{2}\in\\{1,\ldots,p\\},$
$\displaystyle\left|P_{\mathbf{x},j_{1}}\left(F_{\epsilon}\right)-P_{\mathbf{x},j_{1}}(F)\right|\geqslant\frac{k_{1}}{\Omega}(\Omega
h)^{-2p+2}\epsilon,$
$\displaystyle\left|P_{\mathbf{a},j_{2}}\left(F_{\epsilon}\right)-P_{\mathbf{a},j_{2}}(F)\right|\geqslant
k_{2}(\Omega h)^{-2p+1}\epsilon,$
$\displaystyle\left|P_{\mathbf{x},j}\left(F_{\epsilon}\right)-P_{\mathbf{x},j}(F)\right|\geqslant\frac{k_{3}}{\Omega}\epsilon,\quad
x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c},$
$\displaystyle\left|P_{\mathbf{a},j}\left(F_{\epsilon}\right)-P_{\mathbf{a},j}(F)\right|\geqslant
k_{4}\epsilon,\quad x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c},$
$\displaystyle\left|\mathcal{F}\left(F_{\epsilon}\right)(s)-\mathcal{F}(F)(s)\right|\leqslant\epsilon,\quad|s|\leqslant\Omega.$
### 4.3 Proof of Theorem 2.2
###### Proof.
After employing Proposition 4.2, the arguments for proving Theorem 2.2 is just
the same as those in [2]. We present the details as follows.
Let $\mathbf{a}\in(\mathbb{R}^{+})^{d}$ be any positive amplitude vector
satisfying $m\leqslant\|\mathbf{a}\|\leqslant M$. Let $\Omega$, $h$ satisfy
$\Omega h\leqslant 2$, and choose nodes $\mathbf{x}$ satisfying that
$\mathbf{x}^{c}=\left(x_{1}=0,x_{1}=\tau h,\ldots,x_{p}=(p-1)\tau h\right),$
and the rest of the non-cluster nodes are equally spaced in $((p-1)\tau h,1)$.
Now, let $h^{\prime}=(p-1)\tau h$ and $\tau^{\prime}=\frac{1}{p-1}$. Clearly,
$\mathbf{x}$ is a $\left(p,h^{\prime},1,\tau^{\prime},\eta\right)$-clustered
configuration for all sufficiently small $h$ (for instance
$h<\frac{1}{d}<1-\eta(d-p+1)$).We now can apply Proposition 4.2 to the signal
$F=(\mathbf{a},\mathbf{x})$. It then follows that for
$\epsilon<c_{1}(p-1)^{2p-1}(\Omega\tau h)^{2p-1}$ and $\Omega
h<\frac{2}{(p-1)\tau}$, there exist $j_{1},j_{2}\in\\{1,\ldots,p\\}$ such that
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j_{1}}(F)\right)$
$\displaystyle\geqslant\frac{k_{1}}{\Omega}(p-1)^{-2p+2}\epsilon(\Omega\tau
h)^{-2p+2},$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j_{2}}(F)\right)$
$\displaystyle\geqslant k_{2}\epsilon(p-1)^{-2p+1}(\Omega\tau h)^{-2p+1}.$
Moreover,
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right)$
$\displaystyle\geqslant\frac{k_{3}}{\Omega}\epsilon,\quad
x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c},$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right)$
$\displaystyle\geqslant k_{4}\epsilon,\quad
x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}.$
For the general case that
$F=(\boldsymbol{\mathbf{a}},\boldsymbol{\mathbf{x}})\in\mathcal{P}_{d}^{+}$
such that $\mathbf{x}$ forms a $(p,h,T,\tau,\eta)$-clustered configuration, we
consider $SC_{T}(F)=(\boldsymbol{\mathbf{a}},\mathbf{\tilde{x}})$,
$\mathbf{\tilde{x}}=(\tilde{x}_{1},\cdots,\tilde{x}_{d})$, where
$\tilde{x}_{j}=\frac{x_{j}}{T},j=1,\cdots,d$. Now the node vector
$\mathbf{\tilde{x}}$ forms a $(p,\frac{h}{T},1,\tau,\eta)$-clustered
configuration. Applying Proposition 4.1 and the above results, we obtain that
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j_{1}}(F)\right)=T\operatorname{diam}\left(E_{\epsilon,\Omega
T}^{+,\mathbf{x},j_{1}}(SC_{T}(F))\right)\geqslant\frac{k_{1}}{\Omega}(p-1)^{-2p+2}\epsilon(\Omega\tau
h)^{-2p+2},$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j_{2}}(F)\right)=\operatorname{diam}\left(E_{\epsilon,\Omega
T}^{+,\mathbf{a},j_{2}}(SC_{T}(F))\right)\geqslant
k_{2}\epsilon(p-1)^{-2p+1}(\Omega\tau h)^{-2p+1},$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{x},j}(F)\right)=T\operatorname{diam}\left(E_{\epsilon,\Omega
T}^{+,\mathbf{x},j}(SC_{T}(F))\right)\geqslant\frac{k_{3}}{\Omega}\epsilon,\quad
x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c},$
$\displaystyle\operatorname{diam}\left(E_{\epsilon,\Omega}^{+,\mathbf{a},j}(F)\right)=\operatorname{diam}\left(E_{\epsilon,\Omega
T}^{+,\mathbf{a},j}(SC_{T}(F))\right)\geqslant k_{4}\epsilon,\quad
x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}.$
This finishes the proof of Theorem $2.7$ with
$C_{1}^{\prime}=\max\left(\frac{k_{1}}{(p-1)^{2p-2}},k_{3}\right)$,
$C_{2}^{\prime}=\max\left(k_{4},\frac{k_{2}}{(p-1)^{2p-1}}\right),C_{3}^{\prime}=c_{1}(p-1)^{2p-1},C_{4}^{\prime}=\frac{1}{d}$
and $C_{5}^{\prime}=2$.
### 4.4 Proof of Proposition 4.2
We separate the proof into three steps.
Step 1.
In this step we prove the following theorem.
###### Theorem 4.1.
Given the parameters $0<h\leqslant 2,0<\tau\leqslant 1,0<m\leqslant M<\infty$,
let the signal $F=(\mathbf{a},\mathbf{x})\in\mathcal{P}_{p}^{+}$ with
$\mathbf{a}\in(\mathbb{R}^{+})^{p}$ form a single uniform cluster as follows:
* •
(centered) $x_{p}=-x_{1}$;
* •
(uniform) for $1\leqslant j<k\leqslant p$ we have
$\tau h\leqslant\left|x_{j}-x_{k}\right|\leqslant h;$
* •
$m\leqslant\left\|a_{j}\right\|\leqslant M$.
Then, there exist constants $K_{1},\ldots,K_{5}$ depending only on
$(d,\tau,m,M)$ such that for every $\epsilon<K_{5}h^{2d-1}$, there exists a
signal $F_{\epsilon}=(\mathbf{b},\mathbf{y})\in\mathcal{P}_{p}^{+}$ satisfying
the following conditions:
1. (i)
$m_{k}(F)=m_{k}\left(F_{\epsilon}\right)$ for $k=0,1,\ldots,2p-2$, where
$m_{k}(F):=\sum_{j=1}^{p}a_{j}x_{j}^{k};$ (4.2)
2. (ii)
$m_{2p-1}\left(F_{\epsilon}\right)=m_{2p-1}(F)+\epsilon$;
3. (iii)
$K_{1}h^{-2p+2}\epsilon\leqslant\|\mathbf{x}-\mathbf{y}\|\leqslant
K_{2}h^{-2p+2}\epsilon$;
4. (iv)
$K_{3}h^{-2p+1}\epsilon\leqslant\|\mathbf{a}-\mathbf{b}\|\leqslant
K_{4}h^{-2p+1}\epsilon$.
###### Proof.
The case for
$F=(\boldsymbol{\mathbf{a}},\boldsymbol{\mathbf{x}})\in\mathcal{P}_{p}$ and
$F_{\epsilon}=(\boldsymbol{\mathbf{b}},\boldsymbol{\mathbf{y}})\in\mathcal{P}_{p}$
is the Theorem 6.2 in [2]. Now we prove that for the case when
$\boldsymbol{\mathbf{a}}\in(\mathbb{R}^{+})^{p}$, from condition (i) in the
theorem we actually have $\boldsymbol{\mathbf{b}}\in(\mathbb{R}^{+})^{p}$.
This is enough to prove the theorem.
Let $b_{j}$’s be elements in $\boldsymbol{\mathbf{b}}$ and $y_{j}$’s be
elements in $\boldsymbol{\mathbf{Y}}$. We first consider the case when
$x_{j^{*}}=y_{q^{*}}$ for certain $j^{*},q^{*}$. If $b_{q^{*}}\neq a_{j^{*}}$,
then condition (i) in the theorem yields that
$B\beta=A\alpha,$ (4.3)
where $\alpha=(a_{1},\cdots,\ a_{j^{*}-1},\ a_{j^{*}}-b_{q^{*}},\
a_{j^{*}+1},\cdots,a_{p})^{\top}$,
$\beta=(b_{1},\cdots,b_{q^{*}-1},b_{q^{*}+1},\cdots,b_{p})$ and
$A=\big{(}\phi_{2p-2}(x_{1}),\cdots,\phi_{2p-2}(x_{p})\big{)},\
B=\big{(}\phi_{2p-2}(y_{1}),\cdots,\phi_{2p-2}(y_{q^{*}-1}),\phi_{2p-2}(y_{q^{*}+1}),\cdots,\phi_{2p-2}(y_{p})\big{)},$
with $\phi_{2p-2}(\cdot)$ being defined by (4.1). Since all the elements in
$\alpha$ are nonzero by $b_{q^{*}}\neq a_{j^{*}}$ and $a_{j}>0,1\leq j\leq p$,
$A$ contains $p$ different Vandermonde vectors ($\phi_{2p-2}(\cdot)$), and $B$
contains at most $p-1$ Vandermonde vectors, it is impossible to have (4.3) by
[16, Theorem 3.12]. Thus, we must have $b_{q^{*}}=a_{j^{*}}>0$ for
$x_{j^{*}}=y_{q^{*}}$. Next, we prove that $b_{q}>0$ for those $y_{q}$ that
are not equal to any of the $x_{j}$’s. Without loss of generality, we can
actually assume that all the $y_{q}$’s are not equal to any of the $x_{j}$’s.
Further, since
$(\boldsymbol{\mathbf{b}},\boldsymbol{\mathbf{y}})\in\mathcal{P}_{p}$ and
$(\boldsymbol{\mathbf{a}},\boldsymbol{\mathbf{x}})\in\mathcal{P}_{p}^{+}$, all
the $y_{q}$’s and $x_{j}$’s are distinct from each other. We now claim that
$x_{1}<y_{1}<x_{2}<y_{2}<\cdots<x_{p}<y_{p},\ \text{ or }\
y_{1}<x_{1}<y_{2}<x_{2}<\cdots<y_{p}<x_{p}.$ (4.4)
We denote the $x_{j},y_{j}$’s from left to right by
$t_{1}<t_{2}<\cdots<t_{2p}$ and the corresponding $a_{j},-b_{j}$’s by
$\alpha_{1},\cdots,\alpha_{2p}$. By condition (i), it follows that
$A\alpha=0,$
where $\alpha=(\alpha_{1},\cdots,\alpha_{2p})^{\top}$ and
$A=\big{(}\phi_{2p-2}(t_{1}),\cdots,\phi_{2p-2}(t_{2p})\big{)}$ with
$\phi_{2p-2}(\cdot)$ being defined by (4.1). Thus we can have
$-\alpha_{2p}\phi_{2p-2}(t_{2p})=\big{(}\phi_{2p-2}(t_{1}),\cdots,\phi_{2p-2}(t_{2p-1})\big{)}(\alpha_{1},\cdots,\alpha_{2p-1})^{\top},$
and hence
$-\alpha_{2p}\left(\phi_{2p-2}(t_{1}),\cdots,\phi_{2p-2}(t_{2p-1})\right)^{-1}\phi_{2p-2}(t_{2p})=(\alpha_{1},\cdots,\alpha_{2p-1})^{\top}.$
(4.5)
If claim (4.4) does not hold, we have $t_{q}=x_{j_{q}}$ and
$t_{q+1}=x_{j_{q}+1}$ for certain $q,j_{q}$. Applying Lemma 4.1 to (4.5) and
considering the $q$-th and $(q+1)$-th elements in the vectors, we have
$\displaystyle-\alpha_{2p}\Pi_{1\leq j\leq 2p-1,j\neq
q}\frac{t_{2p}-t_{j}}{t_{q}-t_{j}}=\alpha_{q},$ (4.6)
$\displaystyle-\alpha_{2p}\Pi_{1\leq j\leq 2p-1,j\neq
q+1}\frac{t_{2p}-t_{j}}{t_{q+1}-t_{j}}=\alpha_{q+1}.$ (4.7)
Observe first that, for $1\leq k\leq 2p-1$, $\Pi_{1\leq j\leq 2p-1,j\neq
k}(t_{2p}-t_{j})$ is always positive. Moreover, it is obvious that $\Pi_{1\leq
j\leq 2p-1,j\neq q}(t_{q}-t_{j})$ and $\Pi_{1\leq j\leq 2p-1,j\neq
q+1}(t_{q+1}-t_{j})$ have different signs in (4.6) and (4.7), respectively. It
follows that $\alpha_{q}$ and $\alpha_{q+1}$ are of different signs. But the
$\alpha_{q}$ and $\alpha_{q+1}$ are amplitudes of positive sources located at
respectively $x_{q}$ and $x_{q+1}$, which yields a contradiction. Thus the
case that $t_{q}=x_{j_{q}}$ and $t_{q+1}=x_{j_{q}+1}$ for certain $q,j_{q}$
will not happen and the claim (4.4) is thus proved.
Suppose
$y_{1}<x_{1}<y_{2}<x_{2}<\cdots<y_{p}<x_{p},$ (4.8)
we now prove that the $b_{j}$’s are all positive. The another case can be
demonstrated in the same way as below. By this setting,
$t_{2j-1}=y_{j},\quad t_{2j}=x_{j},\quad j=1,\cdots,p.$
Since for $j=1,\cdots,2p-1$, we have
$\displaystyle-\alpha_{2p}\Pi_{1\leq k\leq 2p-1,k\neq
j}\frac{t_{2p}-t_{k}}{t_{j}-t_{k}}=\alpha_{j}.$ (4.9)
For $1\leq j\leq 2p-1$, $\Pi_{1\leq k\leq 2p-1,k\neq j}(t_{2p}-t_{k})$ is
always positive. For $j=2p-1$, since $\alpha_{2p}=a_{p}>0$,
$-\alpha_{2p}\Pi_{1\leq k\leq 2p-1,k\neq 2p-1}(t_{2p-1}-t_{k})$ is negative in
(4.9). Thus we have $\alpha_{2p-1}<0$. In the same fashion, we see that
$\alpha_{j}>0$ for even $j$ and $\alpha_{j}<0$ for odd $j$. Note that by the
setting (4.8), $\alpha_{2j-1}=-b_{j},\ j=1,\cdots,p$. This proves that
$F_{\epsilon}$ is actually a positive signal and completes the proof of
Theorem 4.1. ∎
Step 2. Now we start to prove Proposition 4.2. It is similar to the proof in
[2]. Define $F^{c}$ and $F^{nc}$ to be the cluster and the non-cluster parts
of $F\in\mathcal{P}_{d}^{+}$, respectively, i.e.,
$\displaystyle F^{c}$
$\displaystyle=\sum_{x_{j}\in\mathbf{x}^{c}}a_{j}\delta\left(x-x_{j}\right),$
$\displaystyle F^{nc}$
$\displaystyle=\sum_{x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}}a_{j}\delta\left(x-x_{j}\right).$
We first analyze the non-cluster nodes. We construct that
$F_{\epsilon}^{nc}=\sum_{x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}}a_{j}^{\prime}\delta\left(x-x_{j}^{\prime}\right)$
where $a_{j}^{\prime}=a_{j}+\frac{\epsilon}{4(d-p)}$ and
$x_{j}^{\prime}=x_{j}+\frac{\epsilon}{8\pi\Omega M(d-p)}$. For
$|s|\leqslant\Omega$, the difference between the Fourier transforms of
$F_{\epsilon}^{nc}$ and $F^{nc}$ satisfies
$\displaystyle\left|\mathcal{F}[F_{\epsilon}^{nc}](s)-\mathcal{F}[F^{nc}](s)\right|$
$\displaystyle\leq\sum_{x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}}\left|a_{j}e^{-2\pi
ix_{j}s}-a_{j}^{\prime}e^{-2\pi ix_{j}^{\prime}s}\right|$ (4.10)
$\displaystyle\leqslant\sum_{x_{j}\in\mathbf{x}\backslash\mathbf{x}^{c}}\left(\left|a_{j}e^{-2\pi
ix_{j}s}\left(1-e^{-2\pi i\frac{\epsilon}{8\pi\Omega
M(d-p)}s}\right)\right|+\frac{\epsilon}{4(d-p)}\right)$
$\displaystyle\leqslant\frac{\epsilon}{4}+\frac{\epsilon}{4}=\frac{\epsilon}{2}.$
Note that $a_{j}^{\prime}>0$ since
$a_{j}^{\prime}=a_{j}+\frac{\epsilon}{4(d-p)}$ and $F_{\epsilon}^{nc}$ is a
positive signal.
We next analyze the cluster nodes. We suppose that $x_{1}+x_{p}=0$. For the
case when $x_{1}+x_{p}\neq 0$, utilizing decomposition
$\sum_{j=1}^{p}a_{j}e^{-2\pi ix_{j}s}=\sum_{j=1}^{p}a_{j}e^{-2\pi
i\frac{x_{1}+x_{p}}{2}s}e^{-2\pi i\left(x_{j}-\frac{x_{1}+x_{p}}{2}\right)s},$
we nearly transfer the problem to the case when $x_{1}+x_{p}=0$ and it is not
difficult to see that the following arguments hold as well with only a small
modification, which is enough to prove the proposition.
Next, we define a blowup of $F^{c}$ by $\Omega$ as
$F_{(\Omega)}^{c}=SC_{\frac{1}{\Omega}}\left(F^{c}\right)=\sum_{x_{j}\in\mathrm{x}^{c}}a_{j}\delta\left(x-\Omega
x_{j}\right)$
where $SC$ is defined by Definition 4.1. Let $\tilde{h}=\Omega h$ and
$c_{1}=K_{5}(p,\tau,m,M)$ as in Theorem 4.1. Let $\epsilon\leqslant
c_{1}(\Omega h)^{2p-1}$. Now, we apply Theorem 4.1 with parameters
$p,\tilde{h},\tau,m,M,\tilde{\epsilon}=c_{2}\epsilon$ and the signal
$F_{(\Omega)}^{c}$, where $c_{2}\leqslant 1$ will be determined below. We can
obtain a signal $F_{(\Omega),\epsilon}^{c}\in\mathcal{P}_{p}^{+}$ such that
the following hold for the difference of signals
$H=F_{(\Omega),\epsilon}^{c}-F_{(\Omega)}^{c}$ :
$m_{k}(H)=0,\quad k=0,1,\ldots,2p-2,\quad m_{2p-1}(H)=\tilde{\epsilon};$
(4.11)
while also, for some $j_{1},j_{2}\in\\{1,\ldots,p\\}$
$\displaystyle\left|P_{\mathbf{x},j_{1}}\left(F_{(\Omega),\epsilon}^{c}\right)-P_{\mathbf{x},j_{1}}\left(F_{(\Omega)}^{c}\right)\right|\geqslant
K_{1}(\Omega h)^{-2p+2}\tilde{\epsilon},$ (4.12)
$\displaystyle\left|P_{\mathbf{x},j}\left(F_{(\Omega),\epsilon}^{c}\right)-P_{\mathbf{x},j}\left(F_{(\Omega)}^{c}\right)\right|\leqslant
K_{2}(\Omega h)^{-2p+2}\tilde{\epsilon},\quad j=1,\ldots,p,$
$\displaystyle\left|P_{\mathbf{a}j_{2}}\left(F_{(\Omega),\epsilon}^{c}\right)-P_{\mathbf{a},j_{2}}\left(F_{(\Omega)}^{c}\right)\right|\geqslant
K_{3}(\Omega h)^{-2p+1}\tilde{\epsilon}.$
Now, considering
$F_{\epsilon}^{c}=SC_{\Omega}\left(F_{(\Omega),\epsilon}^{c}\right),$
we obtain that
$\displaystyle\left|P_{\mathbf{x},j_{1}}\left(F_{\epsilon}^{c}\right)-P_{\mathbf{x},j_{1}}\left(F^{c}\right)\right|\geqslant\frac{K_{1}}{\Omega}(\Omega
h)^{-2p+2}\tilde{\epsilon},$
$\displaystyle\left|P_{\mathbf{a},j_{2}}\left(F_{\epsilon}^{c}\right)-P_{\mathbf{a},j_{2}}\left(F^{c}\right)\right|\geqslant
K_{3}(\Omega h)^{-2p+1}\tilde{\epsilon}.$
From the above definitions, we have
$H_{\Omega}=SC_{\Omega}(H)=F_{\epsilon}^{c}-F^{c}$. We next show that there is
a choice of $c_{2}$ such that
$\left|\mathcal{F}[H_{\Omega}](s)\right|\leqslant\frac{\epsilon}{2},\quad|s|\leqslant\Omega.$
(4.13)
Put $\omega=s/\Omega$. Then, by
$\mathcal{F}\left[H_{\Omega}\right](s)=\mathcal{F}[H](\omega)$, the above
inequality is equivalent to
$|\mathcal{F}[H](\omega)|\leq\frac{\epsilon}{2},\quad|\omega|\leq 1.$ (4.14)
Step 3. In this step we prove that (4.14) holds for a choice of $c_{2}$. Note
that we have the following Taylor expansion of $\mathcal{F}[H](\omega)$:
$\mathcal{F}[H](\omega)=\sum_{k=0}^{\infty}\frac{1}{k!}m_{k}(H)(-2\pi
i\omega)^{k}.$ (4.15)
Next we apply the following Taylor domination property [2, Theorem 6.3].
###### Theorem 4.2.
Let $H=\sum_{j=1}^{2p}\beta_{j}\delta\left(x-t_{j}\right)$, and put
$R=\min_{j=1,\ldots,2p}\left|t_{j}\right|^{-1}>0$. Then, for all $k\geqslant
2p$, we have the so-called Taylor domination property
$\left|m_{k}(H)\right|R^{k}\leqslant\left(\frac{2ek}{2p}\right)^{2p}\max_{\ell=0,1,\ldots,2p-1}\left|m_{\ell}(H)\right|R^{\ell}.$
Recall that $H=F_{(\Omega),\epsilon}^{c}-F_{(\Omega)}^{c}$. By Definition 2.4,
the nodes of $F_{(\Omega)}^{c}$ is inside the interval $\left[-\frac{\Omega
h}{2},\frac{\Omega h}{2}\right]$. The nodes of $F_{(\Omega),\epsilon}^{c}$, by
(4.12), satisfy
$\displaystyle\left|P_{\mathbf{x},j}\left(F_{(\Omega),\epsilon}^{c}\right)\right|\leqslant\frac{\Omega
h}{2}+K_{2}(\Omega h)^{-2p+2}\tilde{\epsilon}\leqslant\frac{\Omega
h}{2}+K_{2}(\Omega h)^{-2p+2}c_{1}(\Omega h)^{2p-1}=(\Omega
h)\left(c_{1}K_{2}+\frac{1}{2}\right).$
Since $\Omega h\leqslant 2$ by assumption, we can conclude that the factor $R$
in Theorem 4.2 is greater than
$C_{4}=\frac{1}{2\left(c_{1}K_{2}+\frac{1}{2}\right)}$.
Now, we continue the proof of (4.14). By Theorem 4.2 and (4.11), we have for
$k\geqslant 2p$,
$\displaystyle\left|m_{k}(H)\right|\leqslant\left(\frac{e}{p}\right)^{2p}k^{2p}R^{2p-1-k}\tilde{\epsilon}\leqslant
C_{5}C_{4}^{2p-1-k}k^{2p}\tilde{\epsilon}.$
Plugging this into (4.15) we obtain that
$|\mathcal{F}(H)(\omega)|\leqslant\frac{\tilde{\epsilon}|2\pi\omega|^{2p-1}}{(2p-1)!}+C_{5}C_{4}^{2p-1}\tilde{\epsilon}\sum_{k\geqslant
2p}\left(\frac{2\pi|\omega|}{C_{4}}\right)^{k}\frac{k^{2p}}{k!}.$
Let $\zeta=\frac{2\pi|\omega|}{C_{4}}$ and by $|\omega|\leq 1$, we further
have
$\displaystyle|\mathcal{F}(H)(\omega)|\leqslant
C_{6}\tilde{\epsilon}\sum_{k\geqslant 2p-1}\zeta^{k}\frac{k^{2p}}{k!}\leqslant
C_{7}\tilde{\epsilon}.$
Therefore, we can choose $c_{2}=\min\left(1,\frac{1}{2C_{7}}\right)$ to ensure
that
$|\mathcal{F}(H)(\omega)|\leqslant\frac{\epsilon}{2},\quad|\omega|\leqslant
1,$
which shows (4.13).
Finally, we construct the positive signal
$F_{\epsilon}=F_{\epsilon}^{nc}+F_{\epsilon}^{c}$. Thus we have
$\left|F_{\epsilon}(s)-F(s)\right|\leq\left|F_{\epsilon}^{nc}(s)-F^{nc}(s)\right|+\left|F_{\epsilon}^{c}(s)-F^{c}(s)\right|\leq\epsilon,\
s\in[-\Omega,\Omega].$
This completes the proof of Proposition 4.2 with
$k_{1}=K_{1},k_{2}=K_{3},k_{3}=\frac{1}{8\pi\Omega
M(d-p)},k_{4}=\frac{1}{4(d-p)}$. ∎
## References
* [1] Dmitry Batenkov, Laurent Demanet, Gil Goldman, and Yosef Yomdin. Conditioning of partial nonuniform fourier matrices with clustered nodes. SIAM Journal on Matrix Analysis and Applications, 41(1):199–220, 2020.
* [2] Dmitry Batenkov, Gil Goldman, and Yosef Yomdin. Super-resolution of near-colliding point sources. Information and Inference: A Journal of the IMA, 05 2020. iaaa005.
* [3] Emmanuel J. Candès and Carlos Fernandez-Granda. Towards a mathematical theory of super-resolution. Communications on Pure and Applied Mathematics, 67(6):906–956, 2014\.
* [4] Sitan Chen and Ankur Moitra. Algorithmic foundations for the diffraction limit. pages 490–503, 2021.
* [5] Maxime Ferreira Da Costa and Yuejie Chi. On the stable resolution limit of total variation regularization for spike deconvolution. IEEE Transactions on Information Theory, 66(11):7237–7252, 2020\.
* [6] Laurent Demanet and Nam Nguyen. The recoverability limit for superresolution via sparsity. arXiv preprint arXiv:1502.01385, 2015.
* [7] Quentin Denoyelle, Vincent Duval, and Gabriel Peyré. Support recovery for sparse super-resolution of positive measures. Journal of Fourier Analysis and Applications, 23(5):1153–1194, 2017\.
* [8] David L Donoho, Iain M Johnstone, Jeffrey C Hoch, and Alan S Stern. Maximum entropy and the nearly black object. Journal of the Royal Statistical Society: Series B (Methodological), 54(1):41–67, 1992.
* [9] Vincent Duval and Gabriel Peyré. Exact support recovery for sparse spikes deconvolution. Foundations of Computational Mathematics, 15(5):1315–1355, 2015\.
* [10] Weilin Li and Wenjing Liao. Stable super-resolution limit and smallest singular value of restricted fourier matrices. Applied and Computational Harmonic Analysis, 51:118–156, 2021.
* [11] Ping Liu and Habib Ammari. A mathematical theory of super-resolution and diffraction limit. arXiv preprint arXiv:2211.15208, 2022.
* [12] Ping Liu and Habib Ammari. Nearly optimal resolution estimate for the two-dimensional super-resolution and a new algorithm for direction of arrival estimation with uniform rectangular array. arXiv preprint arXiv:2205.07115, 2022.
* [13] Ping Liu, Yanchen He, and Habib Ammari. A mathematical theory of resolution limits for super-resolution of positive sources. arXiv preprint arXiv:2211.13541, 2022.
* [14] Ping Liu and Hai Zhang. A mathematical theory of computational resolution limit in multi-dimensional spaces. Inverse Problems, 37(10):104001, 2021.
* [15] Ping Liu and Hai Zhang. A theory of computational resolution limit for line spectral estimation. IEEE Transactions on Information Theory, 67(7):4812–4827, 2021\.
* [16] Ping Liu and Hai Zhang. A mathematical theory of computational resolution limit in one dimension. Applied and Computational Harmonic Analysis, 56:402–446, 2022.
* [17] CA Micchelli and TJ Rivlin. Lectures on optimal recovery. In Numerical Analysis Lancaster 1984, pages 21–93. Springer, 1985\.
* [18] Charles A Micchelli, Th J Rivlin, and Shmuel Winograd. The optimal recovery of smooth functions. Numerische Mathematik, 26(2):191–200, 1976.
* [19] Charles A Micchelli and Theodore J Rivlin. A survey of optimal recovery. Optimal estimation in approximation theory, pages 1–54, 1977.
* [20] Ankur Moitra. Super-resolution, extremal functions and the condition number of vandermonde matrices. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC ’15, pages 821–830, 2015.
* [21] Veniamin I Morgenshtern. Super-resolution of positive sources on an arbitrarily fine grid. J. Fourier Anal. Appl., 28(1):Paper No. 4., 2021.
* [22] Veniamin I. Morgenshtern and Emmanuel J. Candès. Super-resolution of positive sources: The discrete setup. SIAM Journal on Imaging Sciences, 9(1):412–444, 2016.
* [23] Clarice Poon and Gabriel Peyré. Multidimensional sparse super-resolution. SIAM Journal on Mathematical Analysis, 51(1):1–44, 2019.
* [24] Gongguo Tang. Resolution limits for atomic decompositions via markov-bernstein type inequalities. In 2015 International Conference on Sampling Theory and Applications (SampTA), pages 548–552. IEEE, 2015.
* [25] Gongguo Tang, Badri Narayan Bhaskar, and Benjamin Recht. Near minimax line spectral estimation. IEEE Transactions on Information Theory, 61(1):499–512, 2014.
* [26] Gongguo Tang, Badri Narayan Bhaskar, Parikshit Shah, and Benjamin Recht. Compressed sensing off the grid. IEEE transactions on information theory, 59(11):7465–7490, 2013\.
|
# Feedback-controlled solute transport through chemo-responsive polymer
membranes
Sebastian Milster Applied Theoretical Physics – Computational Physics,
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder
Strasse 3, D-79104 Freiburg, Germany Won Kyu Kim Korea Institute for
Advanced Study, Seoul 02455, Republic of Korea Joachim Dzubiella
<EMAIL_ADDRESS>Applied Theoretical Physics –
Computational Physics, Physikalisches Institut, Albert-Ludwigs-Universität
Freiburg, Hermann-Herder Strasse 3, D-79104 Freiburg, Germany Research Group
for Simulations of Energy Materials, Helmholtz-Zentrum Berlin für Materialien
und Energie, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany
###### Abstract
Polymer membranes are typically assumed to be inert and nonresponsive to the
flux and density of the permeating particles in transport processes. Here, we
study theoretically the consequences of membrane responsiveness and feedback
on the steady-state force–flux relations and membrane permeability using a
nonlinear-feedback solution-diffusion model of transport through a slab-like
membrane. Therein, the solute concentration inside the membrane depends on the
bulk concentration, $c_{0}$, the driving force, $f$, and the polymer volume
fraction, $\phi$. In our model, solute accumulation in the membrane causes a
sigmoidal volume phase transition of the polymer, changing its permeability,
which, in return, affects the membrane’s solute uptake. This feedback leads to
nonlinear force–flux relations, $j(f)$, which we quantify in terms of the
system’s differential permeability, $\mathcal{P}_{\text{\tiny
sys}}^{\Delta}\propto{\mathrm{d}j}/{\mathrm{d}f}$. We find that the membrane
feedback can increase or decrease the solute flux by orders of magnitude,
triggered by a small change in the driving force, and largely tunable by
attractive versus repulsive solute–membrane interactions. Moreover,
controlling the input, $c_{0}$ and $f$, can lead to steady-state bistability
of $\phi$ and hysteresis in the force–flux relations. This work advocates that
the fine-tuning of the membrane’s chemo-responsiveness will enhance the
nonlinear transport control features, providing great potential for future
(self-)regulating membrane devices.
## I Introduction
The precise and selective control of molecular transport through membranes is
of fundamental importance for various applications in industry and medicine,
such as water purification,[1, 2, 3] food-processing,[4, 5] nano-catalysis,[6,
7, 8, 9] drug-delivery,[10, 11, 12, 13] and tissue engineering.[14, 15] Modern
membrane technology becomes increasingly inspired by responsive bio-membranes
with nonlinear potential-, pressure- or flux-gated permeabilities, bistable
behavior and memristive properties.[16, 17, 18, 19, 20, 21, 22, 23] Such
features allow the design of highly selective membrane devices that
efficiently control molecular transport, autonomously regulate the chemical
milieu, and may act as logical operators, artificial synapses, or analogous
filters for electrical or chemical signals. Moreover, the possible memristive
properties create the foundation for information storage, adaptive responses
to stimuli based upon past events, and neuromorphic systems. [24, 25, 26]
In general, such self-regulation premises a feedback mechanism controlling the
transport properties in a nonlinear fashion.[27, 28, 29, 30] In the scope of
membrane applications this may arise from various system-dependent effects,
such as autocatalysis, substrate or product inhibition,[31, 32] the interplay
of voltage and hydrodynamic pressure,[33, 34] or, as highlighted in this work,
the reciprocal impacts of molecular fluxes and membrane permeability.[35, 36,
37, 38, 39, 40, 41, 42] In this regard many polymeric compounds offer great
potential as they are versatile in their response to various physico-chemical
stimuli and environmental conditions, such as temperature, electric field,
solvent quality, etc.[43, 44, 45] For example, the polymer responds with a
volume phase transition, either from a swollen to a collapsed state, or vice
versa, in which the polymer volume fraction, $\phi$, may change by orders of
magnitude.[46, 47, 48, 49, 50, 51, 36] Such a drastic change of the polymer’s
physical features, in turn, has substantial, nonlinear effects on the solute
permeability of the membrane device.
Figure 1: Essential feedback loop of chemo-responsive polymer membranes
pointing out the nonlinear, reciprocal dependence of the polymer volume
fraction, $\phi(c_{\text{\tiny in}})$, and the solute concentration inside the
membrane, $c_{\text{\tiny in}}(\phi)$. A change in the solute bulk
concentration, $c_{0}$, or the external force, $f$, acting on the solutes has
nontrivial effects on $c_{\text{\tiny in}}$ and $\phi$, and thus on the
transport properties of the membrane.
Very illustrative examples are so-called _smart gating membranes_ ,[52, 53,
54, 55, 56, 57, 58] which are (rather solid) porous membranes with polymer-
coated channels that can reversibly open and close, triggered by external
stimuli or, through autonomous feedback, by molecular recognition. Moreover,
literature on the solution-diffusion model[59, 60, 61, 62] suggests that the
use of more flexible, responsive polymeric membranes enables feedback-
controlled solute transport with further valuable features, such as multiple
steady states and hysteresis transitions.[35, 63, 38, 37, 36] However, more
research is needed here to understand the role of the membrane feedback,
especially in the presence of external driving, and how hysteresis transitions
can occur.
For nonresponsive polymer membranes, we have previously shown that the
Smoluchowski equation[64] well describes solute flux and concentration
profiles under stationary nonequilibrium conditions.[65] Therein, we reported
that the membrane’s solute uptake, $c_{\text{\tiny in}}$, is not only a
function of the polymer volume fraction, $\phi$, the membrane permeability,
$\mathcal{P}_{\text{\tiny mem}}(\phi)$, and bulk concentration, $c_{0}$, but
also tuneable in nonequilibrium by a the external driving force, $f$. The
latter leads to a nonlinear flux, $j(f)$, with significant differences between
the low- and high-force regimes. The nonlinear intermediate crossover was
quantified using the newly introduced system’s differential permeability,
$\mathcal{P}_{\text{\tiny sys}}^{\Delta}\propto{\mathrm{d}j}/{\mathrm{d}f}$.
Motivated by the above features and open questions, in this work we turn our
attention to polymer membranes that are responsive to the penetrants, and
highlight the key differences compared to nonresponsive membranes.
Specifically, we include a mean-field model for the polymer response in the
Smoluchowski framework, i.e., $\phi\to\phi(c_{\text{\tiny in}})$ is a
sigmoidal function of the average solute uptake, which enters
$\mathcal{P}_{\text{\tiny mem}}(\phi)$ and, in turn, controls $c_{\text{\tiny
in}}$, leading to a membrane-intrinsic feedback mechanism [Fig. 1].
Eventually, we use empirical expressions for $\mathcal{P}_{\text{\tiny
mem}}(\phi)$ to study the feedback effect on $j$ and $\mathcal{P}_{\text{\tiny
sys}}^{\Delta}$ as function of $c_{0}$ and $f$. Compared to nonresponsive
membranes, we find substantial enhancement of the nonlinear characteristics,
such as an order of magnitude change in $j$ due to a very small change in $f$,
and report the emergence of multiple steady sates, bifurcations, and
hysteresis in the force–flux relations.
## II Theoretical framework
### II.1 Steady-state Smoluchowski equation and system setup
We consider the solute transport across a polymer membrane as a one-
dimensional drift-diffusion process (in $z$-direction) of ideal solutes [see
the system sketch in Fig. 2(a)]. The membrane has the width $d$, and is
located in the center of the system of length $L$, yielding interfaces at
$(L\pm d)/2$. It is in contact with two solute reservoirs of equal
concentration $c_{0}$ via boundary layers on the feed and permeate sides. [66]
The steady-state flux in the overdamped limit derived from the Smoluchowski
equation, reads [67]
$\displaystyle j=-D(z)\mathopen{}\mathclose{{}\left[\frac{\partial
c(z)}{\partial z}+\beta c(z)\mathopen{}\mathclose{{}\left(\frac{\partial
G(z)}{\partial z}-f}\right)}\right],$ (1)
with the inverse temperature, $\beta=1/(k_{\text{\tiny B}}T)$, and the
(external) driving force, $f$, which may result from various sources. We
assume that the diffusion and energy landscapes, $D(z)$ and $G(z)$, are
piecewise homogeneous, cf. Fig. 2(b), precisely
$\displaystyle D(z)=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{l
l}D_{\text{\tiny in}}&\frac{L-d}{2}\leq z\leq\frac{L+d}{2},\\\
D_{0}&\text{elsewhere},\\\ \end{array}}\right.$ (4)
and
$\displaystyle G(z)=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{l
l}G_{\text{\tiny in}}&\frac{L-d}{2}\leq z\leq\frac{L+d}{2},\\\
G_{0}&\text{elsewhere},\\\ \end{array}}\right.$ (7)
where the subscripts ‘$0$’ and ‘in’ refer to the regions outside and inside
the membrane, respectively.
Figure 2: (a): System setup showing a membrane (red) of width $d$ in
$z$-direction (periodic in $x$ and $y$) in the center of the system of size
$L$, and an example solute concentration profile $c(z)$ (blue) in a steady
state with external driving, $f>0$. The system is in contact with identical
solute bulk reservoirs with constant concentration, $c(0)=c(L)=c_{0}$. The
concentration in the boundary and membrane layers, described by the
Smoluchowski framework, is determined by $f$, and the energy and diffusion
landscapes, $G(z)$ and $D(z)$, depicted in (b).
### II.2 The membrane permeability
The quantities $D_{\text{\tiny in}}$ and $\Delta G=G_{\text{\tiny in}}-G_{0}$
define the membrane permeability in the solution–diffusion picture,[59, 60,
61, 62]
$\displaystyle\mathcal{P}_{\text{\tiny mem}}=D_{\text{\tiny in}}\mathcal{K},$
(8)
where $\mathcal{K}=\exp(-\beta\Delta G)$ is the equilibrium partitioning. The
membrane permeability is a function of the polymer volume fraction, $\phi$,
and depends on the solute–polymer interactions.
For not too attractive solute-polymer interactions, the solute diffusivity
inside the membrane is well described by Yasuda’s free-volume theory,[60]
$\displaystyle D_{\text{\tiny
in}}(\phi)=D_{0}\exp\mathopen{}\mathclose{{}\left(-A\frac{\phi}{1-\phi}}\right),$
(9)
with $A$ a positive parameter accounting for steric solute–polymer effects.
For dilute solute systems, the partitioning of the ideal solutes can be well
approximated by[68]
$\displaystyle\mathcal{K}=\exp\mathopen{}\mathclose{{}\left(-B\phi}\right),$
(10)
where $B=2B_{2}v_{0}^{-1}$ is related to the second virial coefficient,
$B_{2}$, rescaled by effective monomer volume $v_{0}$.
In fact, there exist many extended versions of scaling laws for
$D_{\text{\tiny in}}$ and $\mathcal{K}$ which take into account further
microscopic details, such as the chemistry, shape and size of the solutes, the
solvent and the membrane types, and the architecture of the polymer
network.[69, 70, 68, 71, 72, 61, 73, 74, 75] However, we use Eqs. 9 and 10 to
explain the feedback effects of responsive polymers at the simplest level of
detail.
Further, the presented framework assumes that the equilibrium findings for
$\mathcal{K}$ and $D_{\text{\tiny in}}$ are also valid under moderate
nonequilibrium conditions, i.e., they are independent of the flux or the
force. The validity of this assumption was demonstrated in our preceding work
with nonequilibrium coarse-grained simulations of membrane-bulk systems.[65]
### II.3 Solute flux and concentration inside membrane
With known $\mathcal{P}_{\text{\tiny mem}}$ and $c(0)=c_{0}$, we solve Eq. 1
to obtain the concentration profile, yielding
$\displaystyle
c(z)=\mathopen{}\mathclose{{}\left[c_{0}-j\mathcal{I}(0,z)}\right]e^{-\beta(G(z)-fz)},$
(11)
with
$\displaystyle\mathcal{I}(0,z)=\int\limits_{0}^{z}\mathrm{d}y\frac{e^{\beta(G(y)-fy)}}{D(y)}.$
(12)
Using $c(L)=c_{0}$, the flux can be expressed as[65]
$\displaystyle j=D_{0}c_{0}\beta
f\mathopen{}\mathclose{{}\left[1+\mathopen{}\mathclose{{}\left(\frac{D_{0}}{\mathcal{P}_{\text{\tiny
mem}}}-1}\right)S(f)}\right]^{-1},$ (13)
with $S(f)=\sinh(\beta fd/2)/\sinh(\beta fL/2)$, which determines the impact
of $\mathcal{P}_{\text{\tiny mem}}/D_{0}$ in the low- and high-force limits
(see Appendix A for further details).
By reinserting Eq. 13, one obtains the solute concentration profile throughout
the system (see Appendix B for full expressions). We are particularly
interested in the mean solute concentration inside the membrane,
$c_{\text{\tiny in}}:=\langle c(z)\rangle_{\text{\tiny
in}}=d^{-1}\int_{\text{\tiny in}}\mathrm{d}z\ c(z)$, with $z_{\text{\tiny
in}}\in\mathopen{}\mathclose{{}\left[(L-d)/2,(L+d)/2}\right]$, because it is
the key stimulus for our membrane response model. After integration, we obtain
$\displaystyle c_{\text{\tiny in}}(c_{0},f,\phi)=$ $\displaystyle
c_{0}\mathcal{K}\
\frac{2\mathopen{}\mathclose{{}\left({D_{0}}-\mathcal{P}_{\text{\tiny
mem}}}\right)S(f)\sinh\mathopen{}\mathclose{{}\left({\beta
f(d-L)}/{2}}\right)+\beta fd{D_{0}}}{\beta
fd\mathopen{}\mathclose{{}\left[({D_{0}}-\mathcal{P}_{\text{\tiny
mem}})S(f)+\mathcal{P}_{\text{\tiny mem}}}\right]}.$ (14)
The detailed behavior of Eq. 14 is described in the results section with an
appropriate choice of the model parameters.
### II.4 The polymer response to the solute concentration
The above theoretical framework is straightforward for membranes with constant
$\phi$. We now extend the model to membranes that are responsive (in $\phi$)
to $c_{\text{\tiny in}}$. Motivated by experimental, theoretical, and
computational findings, [46, 47, 48, 49, 50, 51, 36, 76, 77, 78] we consider
that the polymer networks undergo a sigmoidal volume phase transition in the
vicinity of a crossover concentration $c_{\text{\tiny c}}$. We assume the
following form
$\displaystyle\phi(c_{\text{\tiny in}})=\phi_{\text{\tiny
c}}\pm\frac{\Delta\phi}{2}\tanh\mathopen{}\mathclose{{}\left(\frac{c_{\text{\tiny
in}}-c_{\text{\tiny c}}}{\Delta c}}\right),$ (15)
where $\Delta\phi=(\phi_{\text{\tiny max}}-\phi_{\text{\tiny min}})$ is the
maximum change, and $\phi_{\text{\tiny c}}=\phi(c_{\text{\tiny
c}})=(\phi_{\text{\tiny min}}+\phi_{\text{\tiny max}})/{2}$ the polymer volume
fraction at $c_{\text{\tiny c}}$. Hence, we call ($c_{\text{\tiny
c}},\phi_{\text{\tiny c}}$) the crossover point. The transition may occur from
a swollen state ($\phi_{\text{\tiny min}}$) to a collapsed state
($\phi_{\text{\tiny max}}$) or vice versa as $c_{\text{\tiny in}}$ increases
(denoted by the ‘$\pm$’-symbol in Eq. 15), depending on the interactions
between the solutes, the solvent, and the polymer. Effectively attractive
solute-membrane interactions ($B<0$) are expected to cause a swollen-to-
collapsed transition ($+$), while a transition from the collapsed to the
swollen phase ($-$) is expected for repulsive interactions ($B>0$). Further,
the parameter $\Delta c$ determines the sharpness of the transition, ranging
from almost irresponsive ($\Delta c\gg c_{\text{\tiny c}}$) to very sharp
transitions ($\Delta c\ll c_{\text{\tiny c}}$).
Note that Eq. 15 assumes continuous transitions although hysteresis has been
reported in experimental studies.[49, 77, 79, 80] However, this work will
demonstrate that hysteresis and bistability can result from the mutual
dependencies of $\phi$ and $c_{\text{\tiny in}}$.
Furthermore, Eq. 15 is a mean-field approach as it does not resolve spatial
inhomogeneities of $\phi$ and $c_{\text{\tiny in}}$ [cf. the example
concentration profile in Fig. 2(a)]. We assume that the system is small and
that the thin membranes do not change the width in the direction of the solute
flux. Despite the multiple assumptions, our simplified model enables the
investigation of the effect of a responsive membrane permeability on the
transport.
Table 1: Summary of the model parameters and the corresponding values for
$\mathcal{K},D_{\text{\tiny in}}$, and $\mathcal{P}_{\text{\tiny mem}}$ at
$\phi_{\text{\tiny min}}$, $\phi_{\text{\tiny max}}$, and $\phi_{\text{\tiny
c}}$. Length scales are given in units of $\sigma$, the radius of one monomer.
The transition width is rescaled by the crossover concentration
$c_{\text{\tiny c}}$ [cf. Eq. 15]. Permeabilities and diffusivities are
expressed in units of $D_{0}$, the solute bulk diffusion. The arrow
($\Rightarrow$) indicates that the presented values are direct consequences of
the parameter choice. The approximate Lennard-Jones energy
$\varepsilon_{\text{\tiny LJ}}$ stems from a comparison with the second virial
coefficient, $B=2B_{2}v_{0}^{-1}$. The symbols ‘$+$’ and ‘$-$’ correspond the
swollen-to-collapsed and the collapsed-to-swollen transition, respectively
[see also Eq. 15].
polymer response [Eq. 15]
---
| $\phi_{\text{\tiny min}}$ | $0.05$ | swollen
| $\phi_{\text{\tiny max}}$ | $0.35$ | collapsed
$\Rightarrow$ | $\phi_{\text{\tiny c}}$ | $0.2$ | |
$\Rightarrow$ | $\Delta\phi$ | $0.3$ | |
| | sharp | gradual | weak
| $\Delta c/c_{\text{\tiny c}}$ | $0.1$ | $1.0$ | $10.0$
lengths (Fig. 2)
| $L/\sigma$ | $100$ | system size
| $d/\sigma$ | $90$ | membrane width
solute diffusion inside membrane [Eq. 9]
| $A$ | $5$ | |
$\Rightarrow$ | ${D_{\text{\tiny in}}(\phi_{\text{\tiny min}})}/{D_{0}}$ | $0.77$ | |
$\Rightarrow$ | ${D_{\text{\tiny in}}(\phi_{\text{\tiny c}})}/{D_{0}}$ | $0.27$ | |
$\Rightarrow$ | ${D_{\text{\tiny in}}(\phi_{\text{\tiny max}})}/{D_{0}}$ | $0.07$ | |
solute–membrane interactions and partitioning [Eq. 10]
| | repulsive | weakly attr. | attractive
| $B$ | $5.26$ | $-6.25$ | $-17.8$
$\Rightarrow$ | $\beta\varepsilon_{\text{\tiny LJ}}$ (approx.) | $0.03$ | $0.55$ | $0.9$
$\Rightarrow$ | ${\mathcal{K}(\phi_{\text{\tiny min}})}$ | $0.77$ | $1.37$ | $2.4$
$\Rightarrow$ | ${\mathcal{K}(\phi_{\text{\tiny c}})}$ | $0.35$ | $3.49$ | $34.9$
$\Rightarrow$ | ${\mathcal{K}(\phi_{\text{\tiny max}})}$ | $0.16$ | $8.91$ | $501.2$
$\Rightarrow$ | transition [Eq. 15] | $-$ | $+$ | $+$
membrane permeability [Eq. 8]
| | repulsive | neutral | attractive
$\Rightarrow$ | ${\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny min}})}/{D_{0}}$ | $0.59$ | $1.05$ | $1.9$
$\Rightarrow$ | ${\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny c}})}/{D_{0}}$ | $0.10$ | $1.00$ | $10.0$
$\Rightarrow$ | ${\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny max}})}/{D_{0}}$ | $0.01$ | $0.60$ | $33.9$
### II.5 Model parameters
All length scales are expressed in units of $\sigma$, the effective diameter
of one monomer. We set the system size to $L=100\sigma$ and fix the membrane
width to $d=90\sigma$, i.e., the boundary layers between the membrane and the
two bulk reservoirs with concentration $c_{0}$ have the width $5\sigma$. The
concentrations $c_{0}$, $c_{\text{\tiny in}}$ and the transition width $\Delta
c$ are rescaled by the crossover concentration $c_{\text{\tiny c}}$ of the
volume phase transition [Eq. 15]. We choose three different transition widths,
$\Delta c/c_{\text{\tiny
c}}\in\mathopen{}\mathclose{{}\left\\{0.1,1.0,10}\right\\}$. The force, $\beta
f\sigma$, is rescaled by the thermal energy and the solute size. The solute
diffusivity inside the membrane, $D_{\text{\tiny in}}$, and the permeability,
$\mathcal{P}$, are expressed in units of the solute bulk diffusivity $D_{0}$.
The parameters $A$ and $B$, which enter $D_{\text{\tiny in}}(\phi)$ [Eq. 9]
and $\mathcal{K}(\phi)$ [Eq. 10], respectively, as well as the limits of the
polymer volume fraction, $\phi_{\text{\tiny min}}=0.05$ and $\phi_{\text{\tiny
max}}=0.35$, are based on our group’s coarse-grained simulations.[68, 70] We
fix $A=5$, assuming that the diffusion is dominated by steric exclusion. The
interaction parameter $B$ is chosen in a way to yield three different values
for the equilibrium membrane permeability at $\phi_{\text{\tiny c}}=0.2$, and,
hence, we denote the membranes as repulsive ($\mathcal{P}_{\text{\tiny
mem}}(\phi_{\text{\tiny c}})/D_{0}=0.1$), neutral ($\mathcal{P}_{\text{\tiny
mem}}(\phi_{\text{\tiny c}})/D_{0}=1.0$), and attractive
($\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny c}})/D_{0}=10.0$). In fact,
due to typical cancelling effects of $\mathcal{K}(\phi)$ and $D_{\text{\tiny
in}}(\phi)$,[68, 70] we can safely assume that the permeability of our neutral
membrane does not significantly deviate from unity throughout the range of
$\phi$. In a system with Lennard-Jones (LJ) interactions between the solutes
and the membrane monomers of equal size, the characteristic LJ interactions
strengths would take the approximate values $\beta\varepsilon\approx 0.03$
(repulsive), $\beta\varepsilon\approx 0.55$ (weakly attractive), and
$\beta\varepsilon\approx 0.9$ (attractive), respectively. All parameter values
and relevant quantities are summarized in Table 1.
## III Results and Discussion
Figure 3: Mean solute concentration inside the membrane [Eq. 14] as a
function of $\phi$ for different values of the driving force, $f$ [color-
coded, see colorbar in panel (b)], and different interaction strengths,
$B\in\mathopen{}\mathclose{{}\left\\{5.26,-6.25,-17.8}\right\\}$, which
correspond to a repulsive [(a): $\mathcal{P}_{\text{\tiny
mem}}(\phi_{\text{\tiny c}})=0.1D_{0}$], neutral [(b):
$\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny c}})=1.0D_{0}$], and
attractive membrane [(c): $\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny
c}})=10D_{0}$], respectively, as indicated above the panels. The blue and the
red dotted line represent the zero and the infinite force limits, $\lim_{f\to
0}c_{\text{\tiny in}}/c_{0}=\mathcal{K}(\phi)$ [Eq. 10] and
$\lim_{f\to\infty}c_{\text{\tiny in}}/c_{0}=D_{0}/D_{\text{\tiny in}}(\phi)$
[Eq. 9], respectively. While for the repulsive membrane [panel (a)] the
$c_{\text{\tiny in}}$ increases with an increase in force, it decreases for
the case of the attractive membranes [panel (c)]. The solute concentration in
the neutral membrane [panel (b)] depends on $f$, yet is essentially a function
of $\phi$.
### III.1 Force-controlled solute uptake
From Eq. 14, the low- and high-force limits for the mean solute concentration
inside the membrane, $c_{\text{\tiny in}}$, can be deduced, which has been
discussed and substantiated with concentration profiles from theory and
coarse-grained simulations in our previous work.[65] Here, we recapture the
main findings and discuss the results for the parameters used in this work.
In Fig. 3, we depict $c_{\text{\tiny in}}$ [Eq. 14] as a function of $\phi$
for different values of $\beta
f\sigma\in\mathopen{}\mathclose{{}\left[0.01,10}\right]$ (color-coded) and for
three different values of $\mathcal{P}_{\text{\tiny
mem}}(\phi_{c})/D_{0}\in\mathopen{}\mathclose{{}\left\\{0.1,1.0,10.0}\right\\}$
(different panels). In the zero force limit, $c_{\text{\tiny in}}$ reduces to
the expected equilibrium value, $\lim_{f\to 0}c_{\text{\tiny
in}}=c_{0}\mathcal{K}(\phi)$, which monotonously decreases for the repulsive
membrane [panel (a)] and increases otherwise [panels (b) and (c)]. The same
limiting result is obtained, if $\mathcal{P}_{\text{\tiny mem}}(\phi)=D_{0}\
\forall\phi$, which applies approximately for the ‘neutral’ membrane [panel
(b)]. In the high-force limit, the concentration profiles become piecewise
constant with $\lim_{f\to\infty}c_{\text{\tiny in}}=c_{0}D_{0}D_{\text{\tiny
in}}^{-1}(\phi)$, and one further finds $\lim_{f\to\infty}j=c_{0}\beta
fD_{0}=c_{\text{\tiny in}}\beta fD_{\text{\tiny in}}$ for all membrane types,
since it is independent of $\mathcal{K}$.
The solute uptake of the repulsive membrane at fixed $\phi$ increases with
$f$, while it decreases in the attractive membrane [see Figs. 3(a) and 3(c) ].
For ‘neutral’ membrane [panel (b)], $c_{\text{\tiny in}}$ shows no significant
force dependence.
Figure 4: Phase plane showing $\phi(c_{\text{\tiny in}})$ [Eq. 15] and
$c_{\text{\tiny in}}(\phi,c_{0},f)$ [Eq. 14]. The color-coded lines [see
colorbar in panel (c)] depict $c_{\text{\tiny in}}(\phi,c_{0},f)$ in the
attractive membrane with $\mathcal{P}(\phi_{\text{\tiny c}})=10D_{0}$ (cf.
Fig. 3(c)), for three different bulk concentrations $c_{0}$ as indicated above
the panels. The black lines depict the (swollen-to-collapsed) transition
function $\phi(c_{\text{\tiny in}})$ [Eq. 15] for three different values of
the transition sharpness $\Delta c$ [see legend in panel (a)]. Each
interception point of one colored line and one black line refers to a steady-
state solution ($c_{\text{\tiny in}}^{*},\phi^{*}$) that depends on $c_{0}$,
$\Delta c$ and $f$. The solutions, $\phi^{*}(f)$, are summarized in Fig. 5. In
panel (a), we show $c_{0}=c_{\text{\tiny c}}{D_{\text{\tiny
in}}(\phi_{\text{\tiny c}})}/{D_{0}}$, i.e., the high-force limit (red dotted
line) intercepts with the crossover point. In panel (c), we show
$c_{0}=c_{\text{\tiny c}}/{\mathcal{K}(\phi_{\text{\tiny c}})}$, and the zero
force limit (blue dotted line) intercepts with the crossover point
$(c_{\text{\tiny c}},\phi_{\text{\tiny c}})$. In panel (b) the geometric mean
of the two limits is chosen, i.e., $c_{0}=c_{\text{\tiny
c}}\sqrt{{D_{\text{\tiny in}}(\phi_{\text{\tiny
c}})}/\mathopen{}\mathclose{{}\left({D_{0}\mathcal{K}(\phi_{\text{\tiny
c}})}}\right)}$. In this phase plane, $c_{0}$ performs a horizontal shift of
$c_{\text{\tiny in}}/c_{\text{\tiny c}}$.
Figure 5: Steady-state solutions of the polymer volume fraction, $\phi^{*}$,
for attractive membranes ($\mathcal{P}(\phi_{\text{\tiny c}})=10D_{0}$) as
function of the external driving force, $f$. The columns differ in terms of
the bulk concentration, i.e., $c_{0}=c_{\text{\tiny c}}{D_{\text{\tiny
in}}(\phi_{\text{\tiny c}})}/{D_{0}}$ (panels in the left column),
$c_{0}=c_{\text{\tiny c}}\sqrt{{D_{\text{\tiny in}}(\phi_{\text{\tiny
c}})}\mathopen{}\mathclose{{}\left({D_{0}\mathcal{K}(\phi_{\text{\tiny
c}})}}\right)}$ (central column), $c_{0}=c_{\text{\tiny
c}}/{\mathcal{K}(\phi_{\text{\tiny c}})}$ (right column). Each row refers to
one value of the transition sharpness, $\Delta c$ (see labels right of rows).
In general, the force $f$ tunes $\phi^{*}$ from high ($\phi_{\text{\tiny
max}}$) to low ($\phi_{\text{\tiny min}}$) values (since $c_{\text{\tiny
in}}(f)$ decreases for attractive membranes). One observes regions of multiple
steady states (with two stable branches and one unstable branch) which may
occur in the entire force range [e.g., panels (g) and (h)].
### III.2 Multiple steady-state solution
With Eqs. 15 and 14 the feedback loop depicted in Fig. 1 is closed. We obtain
numerically the steady-state solutions, $(c_{\text{\tiny in}}^{*},\phi^{*},)$,
by finding the intersection points of $c_{\text{\tiny in}}(\phi,f)$ and
$\phi(c_{\text{\tiny in}})$ in the phase plane. In this section we show the
results with the attractive membrane only and demonstrate the general
procedure. (For the repulsive membrane, we show a representative phase plane
in Fig. 7 in Appendix C.)
In Fig. 5, the black lines depict the polymer’s volume phase transition of
$\phi(c_{\text{\tiny in}})$, Eq. 15, for three different values of $\Delta
c/c_{\text{\tiny c}}\in\mathopen{}\mathclose{{}\left\\{0.1,1,10}\right\\}$.
The colored lines, $c_{\text{\tiny in}}(c_{0},f,\phi)$, Eq. 14, are the
inverted images of Fig. 3(c) and are plotted in Fig. 5 for three different
bulk concentrations, $c_{0}$, from high [panel (a)] to low values [panel (c)].
As obvious, changing $c_{0}$ performs a shift of $c_{\text{\tiny
in}}(c_{0},f,\phi)$ along the horizontal axis. In panel (a), we choose
$c_{0}=c_{\text{\tiny c}}D_{\text{\tiny in}}(\phi_{c})/D_{0}\approx 0.29$ such
that the high-force limit of $c_{\text{\tiny in}}$ intercepts with the
crossover point $(c_{\text{\tiny c}},\phi_{\text{\tiny c}})$. In panel (c), we
impose that the low-force limit intercepts with the crossover point, i.e.,
$c_{0}=c_{\text{\tiny c}}/(\mathcal{K}(\phi_{c})\approx 0.03$. In panel (b),
the geometric mean, $c_{0}=c_{\text{\tiny c}}\sqrt{D_{\text{\tiny
in}}(\phi_{c})/(D_{0}\mathcal{K}(\phi_{\text{\tiny c}}))}\approx 0.09$ is used
as intermediate probe concentration.
For fixed $f$, $c_{0}$ and $\Delta c$, we find one or three interception
points of $c_{\text{\tiny in}}(c_{0}f,\phi)$ and $\phi(c_{\text{\tiny in}})$,
yielding the steady-state solutions,
$\mathopen{}\mathclose{{}\left(c_{\text{\tiny in}}^{*},\phi^{*}}\right)$. In
Fig. 5(b), for instance, the low-force limit (blue dotted line) intercepts
with the black solid line ($\Delta c/c_{\text{\tiny c}}=1$) only once at
$\phi^{*}\approx\phi_{\infty}$, while it has three interceptions with the
black dotted line ($\Delta c/c_{\text{\tiny c}}=0.1$) at
$\phi_{1}^{*}\approx\phi_{\text{\tiny min}}$, $\phi_{2}^{*}\approx 0.15$, and
$\phi_{3}^{*}\approx\phi_{\text{\tiny max}}$. In the case of triple solutions,
the intermediate one is an unstable solution, while the other two are stable
solutions. Precisely, the latter correspond to asymptotically stable solutions
of the time-dependent Smoluchowski equation, $\dot{c}(z,t)=-\partial
j(z)/\partial z$, i.e., the steady-state solution, $c^{*}(z)$, is restored
after a small perturbation. A more detailed discussion on the stability of
multiple solutions and consequences for the bistable domains is provided in a
separate section below.
We summarize the steady-state solutions for the attractive membrane by
plotting $\phi^{*}(f)$ for different $c_{0}$, and $\Delta c$ in Fig. 5. We
observe a swelling (decrease in $\phi$) with hysteresis due to an increase in
$f$ [see panels (a), (b), and (e)]. In more detail, higher $c_{0}$ can shift
the force-induced $\phi$-transition to higher force values [e.g., compare
panels (a) and (b)] and whether transitions may occur at all. For instance, as
in the case of low transition sharpness, $\Delta c/c_{\text{\tiny c}}=10$, and
low solute concentration [panel (c)], there is no significant effect on
$\phi^{*}$. Similarly for the moderate sharpness, $\Delta c/c_{\text{\tiny
c}}=1$, no transition is induced if $c_{0}$ is too high [panel (d)].
Further, $\Delta c$, plays an important role as it tunes the width of the
bistable domains, e.g., while only small force ranges with bistability are
observed for weakly responsive membranes ($\Delta c/c_{\text{\tiny c}}=10$),
see Figs. 5(a) and 5(b), it can exist in the entire force range for
sufficiently sharp transitions ($\Delta c/c_{\text{\tiny c}}=0.1$), see Figs.
5(g) and 5(h).
We already conclude that the membrane’s feedback response can lead to large
bistable domains in $\phi$ tuned by $f$ and $c_{0}$, which is characterized by
drastic switching of the membrane properties, such as the permeability, due to
the bifurcations at the critical values.
attractive membrane
repulsive membrane
Figure 6: Force-dependent permeability and flux of responsive membranes
undergoing a very sharp volume phase transition with $\Delta
c=0.1c_{\text{\tiny c}}$. All panels on the left (right) hand side show the
results for the attractive (repulsive) membrane. Top panels (a) and (h) depict
the system’s differential permeability, $\mathcal{P}_{\text{\tiny
sys}}^{\Delta}/D_{0}$, as heatmaps in the $f$-$c_{0}$ plane. The heatmaps
share the same color-code ranging from $10^{-2}$ to $10^{1}$ (see colorbar).
The white lines labeled with roman numbers, I-VI, depict selected values of
$c_{0}$, for which $j$ and $\mathcal{P}_{\text{\tiny sys}}^{\Delta}$ are
presented in the panels below. The black dotted lines indicate the bifurcation
at which the system changes from mono- to bi-stable (or vice versa), while the
two solutions in the bistable domain are presented in a striped pattern. In
examples I-VI, the solutions are distinguished by the membrane’s volume phase,
i.e., blue corresponds to $\phi<\phi_{\text{\tiny min}}$ (swollen), and red to
$\phi>\phi_{\text{\tiny max}}$ (collapsed). In fact, we find $\phi$ is either
fully swollen or collapsed except for example IV, where a gradual crossover
from $\phi(f=0)=\phi_{\text{\tiny max}}$ to $\phi(f\to\infty)\approx 0.15$ is
observed [cf. panel (h), where the loosely dotted line indicates
$\phi=\phi_{\text{\tiny c}}$]. The pale red and blue dotted lines in I-VI are
the references for nonresponsive membranes in the fully collapsed and swollen
case, respectively. The gray dashed lines in I-VI correspond to the bulk
references, i.e., $j=D_{0}c_{0}\beta f$ and $\mathcal{P}_{\text{\tiny
sys}}=D_{0}$, respectively, which yield the asymptotic values for $f\to\infty$
and $\phi\to 0$. More details are provided in the main text.
### III.3 Consequences for the transport properties
The flux $j(f)$, given by Eq. 13, is a nonlinear function of $f$ determined by
two contributions: The change in the membrane permeability
$\mathcal{P}_{\text{\tiny mem}}(\phi)$ (due to the change in $\phi$) and the
spatial setup (see Appendix A and our previous work[65]). The nonlinear
characteristics of $j(f)$ are quantified by the _differential_ system
permeability,[65] defined as
$\displaystyle\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny
sys}}(f)=\frac{1}{\beta c_{0}}\frac{\mathrm{d}j}{\mathrm{d}f},$ (16)
which describes the change in the steady-state flux induced by a change in the
external driving force.
We make use of $\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}(f)$ to
highlight the novel nonlinear effects on $j(f)$ caused by the membrane’s
feedback response. We limit ourselves to the very sharp membrane response
($\Delta c/c_{\text{\tiny c}}=0.1$), and point out the significant difference
between the fluxes in attractive and repulsive membranes.
In Fig. 6, we present heatmaps of
$\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}$ in the $f$-$c_{0}$
plane for the attractive [panel (a)] and the repulsive membrane [panel (h)].
The white lines labeled with roman numerals depict selected values of $c_{0}$,
and correspond to the panels below, showing $j$ and
$\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}$. The heatmaps share
the same color-scale (see colorbars), allowing a direct comparison between the
results of attractive and repulsive membranes.
The attractive membrane exhibits, in general, larger
$\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}$ values than the
repulsive one, particularly in the low-force and collapsed regime
($\phi=\phi_{\text{\tiny max}}$), in which the influence of
$\mathcal{P}_{\text{\tiny mem}}$ is the greatest. For $f\to 0$, we find
$\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}\approx 7D_{0}$ and
$\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}\approx 0.01D_{0}$ for
the attractive and repulsive membrane in the collapsed state, respectively. In
the high-force limit, the system permeability converges to $D_{0}$
irrespective of the volume phase. If the membrane is swollen
($\phi=\phi_{\text{\tiny min}}$), the permeability of the repulsive and the
attractive membrane are of the same order of magnitude, i.e.,
$\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny min}})\approx 0.6D_{0}$ and
$\mathcal{P}_{\text{\tiny mem}}(\phi_{\text{\tiny min}})\approx 2D_{0}$,
respectively, and $\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}$ does
not deviate significantly from bulk diffusivity $D_{0}$, even for low forces
(compare blue lines in lower panels of Fig. 6).
Due to the sharp response with $\Delta c=0.1c_{c}$, the membrane is either
fully swollen ($\phi_{\text{\tiny min}}$), or fully collapsed
($\phi_{\text{\tiny max}}$). Moreover, this also leads to large bistable
domains in the $c_{0}$-$f$ plane, visualized as striped patterns in Figs. 6(a)
and 6(h). Crossing the boundary of these domains leads to a discontinuous
volume phase transitions accompanied with an order of magnitude change in the
solute flux (examples I, III, V, and VI). In the case of the repulsive
membrane, the flux can be switched even by two orders of magnitude,
particularly for small $f$.
In examples I-VI [Figs. 6(b)–6(g), and 6(i)–6(n)], we also depict the results
for nonresponsive membranes in the fully swollen and collapsed case for
comparison. In the nonresponsive case,
$\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}$ can also be tuned in
the same range by only controlling $f$. Nonetheless, the membrane’s
responsiveness brings about more dramatic effects, such as bistability (I,
III, V, VI) and hysteresis (V), yielding new control mechanisms to switch
between two flux states. While nonresponsive membranes require large forces to
exhibit bulk-like properties ($D_{0}$), a transition to this neutral state can
be achieved in responsive membranes in a much sharper fashion and for lower
force values, e.g., see Fig. 6 I, V, and VI. In V, for example, the crossover
from the low- to the high-permeability state occurs abruptly around $\beta
f\sigma\approx 0.2$ (red solid line), whereas for the collapsed, nonresponsive
membrane a gradual change is observed in the range $\beta f\sigma\approx
0.1-1.0$.
Further, even if the polymer volume phase transition occurs without
bifurcation, nonlinearities in the force–flux relations can be significantly
enhanced. For instance, in panel (l) (example IV), we find a tenfold
maximization of $\mathcal{P}^{\text{\tiny$\Delta$}}_{\text{\tiny sys}}\approx
12D_{0}$ at roughly $\beta f\sigma\approx 0.5$, which even exceeds the maximum
differential system permeability measured for the attractive membrane.
### III.4 Discussion: nonequilibrium steady-state stability
Our model results into well-defined force–flux relations in the domains with
unique steady-state solutions. In the bistable domains, however, the question
arises whether the states coexist or whether only one survives under real
conditions. This far, the solutions were simply deduced from a deterministic
interpretation of the macroscopic model equations, i.e., by evaluating the
self-consistency equation $c_{\text{\tiny in}}=R(c_{\text{\tiny in}})$, with
$R(c_{\text{\tiny in}})=d^{-1}\int_{\text{\tiny in}}c(z,\phi(c_{\text{\tiny
in}}))\mathrm{d}z$. The steady state is asymptotically stable, if
${\mathrm{d}R(c_{\text{\tiny in}})}/{\mathrm{d}c_{\text{\tiny
in}}}|_{c_{\text{\tiny in}}^{*}}<1$.[81] However, our approach neglects larger
fluctuations in $\phi$ and $c_{\text{\tiny in}}$, and does not analyze further
nonequilibrium extremum principles.[82] In the following, we first discuss the
consequences of the deterministic perspective, and then briefly review
alternative interpretations.
In the case of negligible fluctuations the (deterministic) transition between
states can be either _reversible_ or _irreversible_.[35] One _reversible_
transition is example V [Figs. 6(j) and 6(m)]. Here, the membrane is in the
collapsed state (red line) for small $f$. With increasing $f$, the membrane is
driven into the bistable regime, yet remains in the collapsed state. Only if
$f$ exceeds the second bifurcation line, the membrane swells. In the same
example V, if $f$ is decreased from high force values, the membrane stays
swollen in the bistable domain and returns to the collapsed state until the
first bifurcation line is passed. Hence, we find a _reversible_ transition
with hysteresis between the two state in V.
In contrast, consider example I or VI, and assume that the membrane is in the
collapsed state at $f=0$, an increase in $f$ leads to a swelling when the
bifurcation line is crossed. Decreasing the force again, however, does not
induce a collapse, and, hence, this transition can be termed _irreversible_ in
the deterministic interpretation. This is because only the swollen case
survives once the threshold is surpassed. Analogously, see example III, a
collapsed-to-swollen transition cannot be induced by increasing $f$.
Although two stable states may coexist in the deterministic model, one of them
could be metastable and practically unoccupied under experimental conditions.
In literature one finds nonequilibrium principles, e.g., based on the
maximization of entropy, the minimization of entropy production (least
dissipation), or the minimization of power, providing various possible
routes.[83, 84, 85, 86, 31, 87, 88, 89, 82, 90] Such extremum principles may
lead to unique solutions in the _bistable_ regime, and to different values for
$f$ and $c_{0}$, where the switching between the high and low flux states
occurs. For example, it should be the minimum flux, if the least-dissipation
principle applies. This has direct consequences on the flux–force relation and
the critical transition values of $f$ and $c_{0}$, where the phase transition
in V would occur always at the first bifurcation line, i.e., without
bistability and hysteresis.
Furthermore, the presented diffusion process can also be modeled with the
stochastic Smoluchowski equation,[91] and possibly further coarse-grained to a
stochastic differential equation for $\dot{c}_{\text{\tiny in}}$.[92, 93]
Hence, given the fluctuations are large enough, a stochastic switching between
the two steady states may be observed in the bistable domains, and the
effective force–flux relations are determined by the averaged values of $\phi$
and $c_{\text{\tiny in}}$. Consequently, changing $f$ results in a continuous
transition between the two states, implying that example V does not exhibit
hysteresis behavior, but is rather similar to the transition in example IV.
Ultimately, the appropriate stability interpretation remains to be verified,
and is likely specific to the membrane material and the experimental
nonequilibrium conditions. Nonetheless, a strong amplification of nonlinear
characteristics and a critical switching in the force–flux relations can be
expected due to the membrane’s responsiveness.
## IV Summary
We have investigated the driven steady-state solute transport through
polymeric membranes with a sigmoidal volume phase response to the penetrant
uptake. The change in the polymer volume fraction is decisive for the membrane
permeability, which we modeled with exponential functions. This, in turn,
impacts on the solute uptake, leading to novel feedback-induced effects in
force–flux relations that cannot be achieved by nonresponsive membranes. We
quantified our findings in terms of the system’s differential permeability.
The feedback effects of responsive membranes are most pronounced in the low-
force regime, where the bulk concentration largely tunes the membrane density
between the swollen and the collapsed state. Increasing the force can lead to
a membrane swelling accompanied with a strong amplification of nonlinear
characteristics and critical switching in the force–flux relations. For
instance, the swelling of membranes with repulsive polymer-solute interactions
can be caused by a small change in the driving force, for which we report an
increase in the flux by two orders of magnitude, and a pronounced maximization
of the differential permeability, i.e., a tenfold increase compared to the
case of nonresponsive membranes.
Moreover, of particular note is the feedback-induced coexistence of two stable
steady states, while the size of the bistable domains increases with the
sharpness of the sigmoidal polymer response. The bifurcations from mono- to
bistability occur at critical values of the driving force, and the solute bulk
concentration, leading to discontinuous changes in the flux of up to two
orders of magnitude.
Thus, the force-dependent switching between high and low flux states provides
a valuable control mechanism for molecular transport. It can be fine-tuned
also to control the appearance of hysteresis, enabling the presented feedback
membranes to function as memristive devices. Moreover, the coupling of the
permeability hysteresis to (non-oscillatory) chemical reactions may lead to
biomimetic features, such as membrane excitability and autonomous
oscillations, as first proposed by theory,[35, 63, 31, 32] and eventually
validated by experiments.[36, 37, 38]
Hysteresis transitions found in literature[36, 37, 38, 80, 49, 77, 79] are
usually rationalized by a bistability in the polymer’s conformational free
energy,[94, 95, 79] and attributed to the complex microscopic interactions or
the competition between entropic and energetic contributions.[94, 95, 79]
Nonetheless, many polymers exhibit a hysteresis-free response, for which the
presented feedback mechanism provides a novel explanation of how hysteresis
transitions can be generated and tuned in polymer membranes.
We disclosed in this work how nonlinear solute transport through chemo-
responsive polymer membranes is controlled by membrane feedback. It thus
provides the theoretical basis for the rational design of self-regulating
membranes with nonlinear control features for molecular transport. Adaptations
employing more complex functions for partitioning, diffusivity, and the
polymer response, to differing feed and permeate bulk concentrations as well
as extensions to different spatial arrangements and geometries could be
interesting for future studies.
## Acknowledgments
The authors thank Matej Kanduč for fruitful discussions. This work was
supported by the Deutsche Forschungsgemeinschaft (DFG) via the Research Unit
FOR 5099 “Reducing complexity of nonequilibrium systems”. W.K.K. acknowledges
the support by the KIAS Individual Grants (CG076001 and CG076002) at Korea
Institute for Advanced Study.
## Author declarations
### Conflict of Interest
The authors have no conflicts to disclose.
## Appendix A The flux in the low- and high-force limits
The flux, $j$ [Eq. 13], in the small-force regime reads[65]
$\displaystyle\lim_{f\to 0}j=D_{0}\beta
fc_{0}\mathopen{}\mathclose{{}\left[1+\mathopen{}\mathclose{{}\left(\frac{D_{0}}{\mathcal{P}_{\text{\tiny
mem}}}-1}\right)\frac{d}{L}}\right]^{-1},$ (17)
which converges to $\lim_{f\to 0}j\to\mathcal{P}_{\text{\tiny mem}}c_{0}\beta
f$ for $d\to L$. The membrane thickness, $d$, determines the crossover to the
high-force regime, for which $\lim_{f\to\infty}j=D_{0}c_{0}\beta f$ results.
For moderate to large forces, we find $S(f)\approx\exp(-\beta f(L-d)/2)$. With
increasing $f$, the denominator in Eq. 13 converges to unity, governed by
$(L-d)$. So, the larger $d$ with respect to $L$, the higher $f$ has to be in
order to reach the high-force limit. Obviously, the onset of the high-force
limit also depends on the membrane permeability, precisely, large values of
$\mathcal{P}_{\text{\tiny mem}}$ will shift the crossover to smaller force
values.
## Appendix B Concentration profiles
The solute concentration profile in a system depicted in described in Sec. II
reads[65]
$\displaystyle c(z)=c_{0}\mathopen{}\mathclose{{}\left[1-\dfrac{D_{0}\beta
f\mathcal{I}(0,z)}{1+\mathopen{}\mathclose{{}\left(\frac{D_{0}}{\mathcal{P}_{\text{\tiny
mem}}}-1}\right)S(f)}}\right]e^{-\beta\mathopen{}\mathclose{{}\left(G(z)-fz}\right)},$
(18)
We can split Eq. 18 into the piecewise homogeneous layers, precisely, the feed
boundary, membrane (‘in’), and permeate boundary layer, and can write the
respective full expressions as
$\displaystyle c(z)|_{\text{\tiny feed}}=$ $\displaystyle c_{0}\mathcal{K}\
\dfrac{e^{\beta
fz}\mathopen{}\mathclose{{}\left({D_{0}}-\mathcal{P}_{\text{\tiny
mem}}}\right)S(f)+{P_{\text{\tiny mem}}}}{({D_{0}}-\mathcal{P}_{\text{\tiny
mem}})S(f)+\mathcal{P}_{\text{\tiny mem}}},$ (19) $\displaystyle
c(z)|_{\text{\tiny in}}=$ $\displaystyle c_{0}\mathcal{K}\
\dfrac{\frac{e^{\beta
f\mathopen{}\mathclose{{}\left(z-L/2}\right)}}{\sinh(\beta
fL/2)}\mathopen{}\mathclose{{}\left({D_{0}}-\mathcal{P}_{\text{\tiny
mem}}}\right){\sinh\mathopen{}\mathclose{{}\left(\beta
f\frac{d-L}{2}}\right)}+{D_{0}}}{({D_{0}}-\mathcal{P}_{\text{\tiny
mem}})S(f)+\mathcal{P}_{\text{\tiny mem}}},$ (20) $\displaystyle
c(z)|_{\text{\tiny permeate}}=$ $\displaystyle c_{0}\mathcal{K}\
\dfrac{e^{\beta
f(z-L)}\mathopen{}\mathclose{{}\left({D_{0}}-\mathcal{P}_{\text{\tiny
mem}}}\right)S(f)+{P_{\text{\tiny mem}}}}{({D_{0}}-\mathcal{P}_{\text{\tiny
mem}})S(f)+\mathcal{P}_{\text{\tiny mem}}}.$ (21)
## Appendix C Phase plane for repulsive membranes
Figure 7: Phase plane showing $\phi(c_{\text{\tiny in}})$ [Eq. 15] and
$c_{\text{\tiny in}}(\phi,c_{0},f)$ [Eq. 14]. The color-coded lines (see
colorbar) depict $c_{\text{\tiny in}}(\phi,c_{0},f)$ for the repulsive
membrane ($\mathcal{P}(\phi_{\text{\tiny c}})=0.1D_{0}$, cf. Fig. 3(a)), with
selected probe concentration $c_{0}$. The black lines depict the (collapsed-
to-swollen) transition function $\phi(c_{\text{\tiny in}})$ [Eq. 15] for three
different values of the transition sharpness $\Delta c$ (see legend). Each
interception point of a colored line [Eq. 14] and a black line [Eq. 15] refers
to a steady-state solution ($c_{\text{\tiny in}}^{*},\phi^{*}$) that depends
on $c_{0}$, $\Delta c$ and $f$.
In Fig. 7, $c_{\text{\tiny in}}(c_{0},f,\phi)$ [Eq. 14] and
$\phi(c_{\text{\tiny in}})$ [Eq. 15] are presented in the $c_{0}$-$\phi$
plane. Interception points of $c_{\text{\tiny in}}$ and $\phi$ correspond to
the force-dependent steady-state solution. The results were used to calculate
$\phi^{*}(c_{0},f)$, which enter the flux and the differential permeability as
depicted in Figs. 6(h)–(n). The bulk concentration, $c_{0}$, and the external
force, $f$, yield, in general, an increase in the mean inside concentration,
$c_{\text{\tiny in}}$, resulting in a swelling of the membrane. For very sharp
transitions, e.g., $\Delta c=0.1c_{\text{\tiny c}}$, one can find multiple
solutions for fixed $c_{0}$ and $f$, giving rise to bistability and
hysteresis.
## References
## References
* Shannon _et al._ [2008] M. A. Shannon, P. W. Bohn, M. Elimelech, J. G. Georgiadis, B. J. Mariñas, and A. M. Mayes, Nature 452, 301 (2008).
* Geise _et al._ [2010] G. M. Geise, H.-S. Lee, D. J. Miller, B. D. Freeman, J. E. McGrath, and D. R. Paul, J. Polym. Sci. Part B Polym. Phys. 48, 1685 (2010).
* Pendergast and Hoek [2011] M. M. Pendergast and E. M. Hoek, Energy Environ. Sci. 4, 1946 (2011).
* Cuperus and Nijhuis [1993] F. P. Cuperus and H. H. Nijhuis, Trends Food Sci. Technol. 4, 277 (1993).
* Kumar _et al._ [2013] P. Kumar, N. Sharma, R. Ranjan, S. Kumar, Z. F. Bhat, and D. K. Jeong, Asian-Australasian J. Anim. Sci. 26, 1347 (2013).
* Renggli _et al._ [2011] K. Renggli, P. Baumann, K. Langowska, O. Onaca, N. Bruns, and W. Meier, Adv. Funct. Mater. 21, 1241 (2011).
* Roa _et al._ [2017] R. Roa, W. K. Kim, M. Kanduč, J. Dzubiella, and S. Angioletti-Uberti, ACS Catal. 7, 5604 (2017).
* Lu and Ballauff [2011] Y. Lu and M. Ballauff, Prog. Polym. Sci. 36, 767 (2011).
* Kanduč _et al._ [2020] M. Kanduč, W. K. Kim, R. Roa, and J. Dzubiella, Mol. Syst. Des. Eng. 5, 602 (2020).
* Li and Mooney [2016] J. Li and D. J. Mooney, Nat. Rev. Mater. 1, 1 (2016).
* Brudno and Mooney [2015] Y. Brudno and D. J. Mooney, J. Control. Release 219, 8 (2015).
* Moncho-Jordá _et al._ [2020] A. Moncho-Jordá, A. B. Jódar-Reyes, M. Kanduč, A. Germán-Bellod, J. M. López-Romero, R. Contreras-Cáceres, F. Sarabia, M. García-Castro, H. A. Pérez-Ramírez, and G. Odriozola, ACS Nano 14, 15227 (2020).
* Mitchell _et al._ [2021] M. J. Mitchell, M. M. Billingsley, R. M. Haley, M. E. Wechsler, N. A. Peppas, and R. Langer, Nat. Rev. Drug Discov. 20, 101 (2021).
* Stamatialis _et al._ [2008] D. F. Stamatialis, B. J. Papenburg, M. Girones, S. Saiful, S. N. M. Bettahalli, S. Schmitmeier, and M. Wessling, J. Memb. Sci. 308, 1 (2008).
* Lee [2018] J. H. Lee, Biomater. Res. 22, 1 (2018).
* Hodgkin and Huxley [1952] A. L. Hodgkin and A. F. Huxley, J. Physiol. 117, 500 (1952).
* Henisch and Smith [1974] H. K. Henisch and W. R. Smith, Appl. Phys. Lett. 24, 589 (1974).
* Yang _et al._ [2006] Y. Yang, J. Ouyang, L. Ma, R. J. H. Tseng, and C. W. Chu, Adv. Funct. Mater. 16, 1001 (2006).
* Keener and Sneyd [2009] J. Keener and J. Sneyd, _Mathematical Physiology I: Cellular Physiology_ (Springer-Verlag New York, 2009) p. 547.
* Lee Takhee and Chen Yong [2012] Lee Takhee and Chen Yong, Marerials Res. Soc. Bull. 37, 144 (2012).
* Lei _et al._ [2014] Y. Lei, Y. Liu, Y. Xia, X. Gao, B. Xu, S. Wang, J. Yin, and Z. Liu, AIP Adv. 4, 077105 (2014).
* Hasan _et al._ [2018a] M. S. Hasan, J. S. Najem, R. Weiss, C. D. Schuman, A. Belianinov, C. P. Collier, S. A. Sarles, and G. S. Rose, IEEE 13th Nanotechnol. Mater. Devices Conf. , 1 (2018a).
* Minoura [1998] N. Minoura, Langmuir 14, 2145 (1998).
* Hasan _et al._ [2018b] M. S. Hasan, C. D. Schuman, J. S. Najem, R. Weiss, N. D. Skuda, A. Belianinov, C. P. Collier, S. A. Sarles, and G. S. Rose, IEEE 13th Dallas Circuits Syst. Conf. , 1 (2018b).
* Jo _et al._ [2010] S. H. Jo, T. Chang, I. Ebong, B. B. Bhadviya, P. Mazumder, and W. Lu, Nano Lett. 10, 1297 (2010).
* Prodromakis and Toumazou [2010] T. Prodromakis and C. Toumazou, 2010 17th IEEE Int. Conf. Electron. Circuits Syst. , 934 (2010).
* Frank [2005] T. Frank, _Nonlinear Fokker-Planck equations: fundamentals and applications_ , 1st ed. (Springer Science & Business Media, Heidelberg, 2005).
* Giacomelli _et al._ [2012] G. Giacomelli, F. Marino, M. A. Zaks, and S. Yanchuk, Europhys. Lett. 99, 58005 (2012).
* Gernert and Klapp [2015] R. Gernert and S. H. L. Klapp, Phys. Rev. E 92, 22132 (2015).
* Holubec _et al._ [2022] V. Holubec, A. Ryabov, S. A. M. Loos, and K. Kroy, New J. Phys. 24, 023021 (2022).
* Katchalsky and Spangler [1968] A. Katchalsky and R. Spangler, Q. Rev. Biophys. 1, 127 (1968).
* Borckmans _et al._ [2007] P. Borckmans, P. De Kepper, A. R. Khokhlov, and S. Métens, _Chemomechanical Instabilities in Responsive Materials_ (Springer, 2007).
* Teorell [1951] T. Teorell, Zeitschrift für Elektrochemie und Angew. Phys. Chemie 55, 460 (1951).
* Teorell [1962] T. Teorell, Biophys. J. 2, 25 (1962).
* Hahn, Ortoleva, and Ross [1973] H. S. Hahn, P. J. Ortoleva, and J. Ross, J. Theor. Biol. 41, 503 (1973).
* Bell _et al._ [2021] D. J. Bell, D. Felder, W. G. von Westarp, and M. Wessling, Soft Matter 17, 592 (2021).
* Zou and Siegel [1999] X. Zou and R. A. Siegel, J. Chem. Phys. 110, 2267 (1999).
* Leroux and Siegel [1999] J. C. Leroux and R. A. Siegel, Chaos An Interdiscip. J. Nonlinear Sci. 9, 267 (1999).
* Sparr _et al._ [2009] E. Sparr, C. Åberg, P. Nilsson, and W. Håkan, Soft Matter 5, 3225 (2009).
* Costa-Balogh _et al._ [2005] F. O. Costa-Balogh, C. Åberg, J. J. Sousa, and E. Sparr, Langmuir 21, 10307 (2005).
* Åberg and Wennerström [2009] C. Åberg and H. Wennerström, Phys. Chem. Chem. Phys. 11, 9075 (2009).
* Tikhonov and Tokmachev [2011] N. A. Tikhonov and M. G. Tokmachev, J. Math. Chem. 49, 629 (2011).
* Stuart _et al._ [2010] M. A. C. Stuart, W. T. S. Huck, J. Genzer, M. Müller, C. Ober, M. Stamm, G. B. Sukhorukov, I. Szleifer, V. V. Tsukruk, M. Urban, F. Winnik, S. Zauscher, I. Luzinov, and S. Minko, Nat. Mater. 9, 101 (2010).
* Schattling, Jochum, and Theato [2014] P. Schattling, F. D. Jochum, and P. Theato, Polym. Chem. 5, 25 (2014).
* Wandera, Wickramasinghe, and Husson [2010] D. Wandera, S. R. Wickramasinghe, and S. M. Husson, J. Memb. Sci. 357, 6 (2010).
* Tanaka [1978] T. Tanaka, Phys. Rev. Lett. 40, 820 (1978).
* Tanaka _et al._ [1980] T. Tanaka, D. Fillmore, S.-T. Sun, I. Nishio, G. Swislow, and A. Shah, Phys. Rev. Lett. 45, 1636 (1980).
* Hirokawa and Tanaka [1984] Y. Hirokawa and T. Tanaka, J. Chem. Phys. 81, 6379 (1984).
* Koga, Sasaki, and Maeda [2001] S. Koga, S. Sasaki, and H. Maeda, J. Phys. Chem. B 105, 4105 (2001).
* Roiter and Minko [2005] Y. Roiter and S. Minko, J. Am. Chem. Soc. 127, 15688 (2005).
* Kim _et al._ [2017] W. K. Kim, A. Moncho-Jorda, R. Roa, M. Kanduč, and J. Dzubiella, Macromolecules 50, 6227 (2017).
* Liu _et al._ [2016] Z. Liu, W. Wang, R. Xie, X. J. Ju, and L. Y. Chu, Chem. Soc. Rev. 45, 460 (2016).
* Ito _et al._ [2002] T. Ito, T. Hioki, T. Yamaguchi, T. Shinbo, S. I. Nakao, and S. Kimura, J. Am. Chem. Soc. 124, 7840 (2002).
* Ito _et al._ [2004] T. Ito, Y. Sato, T. Yamaguchi, and S. I. Nakao, Macromolecules 37, 3407 (2004).
* Bhattacharyya _et al._ [2013] D. Bhattacharyya, T. Schäfer, S. R. Wickramasinghe, and S. Daunert, _Responsive Membranes and Materials_ (John Wiley & Sons, 2013).
* Li _et al._ [2020] Y. Li, C. Chen, E. R. Meshot, S. F. Buchsbaum, M. Herbert, R. Zhu, O. Kulikov, B. McDonald, N. T. Bui, M. L. Jue, S. J. Park, C. A. Valdez, S. Hok, Q. He, C. J. Doona, K. J. Wu, T. M. Swager, and F. Fornasiero, Adv. Funct. Mater. 30, 1 (2020).
* Xie, Li, and Chu [2007] R. Xie, Y. Li, and L. Y. Chu, J. Memb. Sci. 289, 76 (2007).
* Tokarev and Minko [2010] I. Tokarev and S. Minko, Adv. Mater. 22, 3446 (2010).
* Wijmans and Baker [1995] J. G. Wijmans and R. W. Baker, J. Memb. Sci. 107, 1 (1995).
* Yasuda _et al._ [1969] H. Yasuda, A. Peterlin, C. K. Colton, K. A. Smith, and E. W. Merrill, Die Makromol. Chemie 126, 177 (1969).
* Gehrke _et al._ [1997] S. H. Gehrke, J. P. Fisher, M. Palasis, and M. E. Lund, Ann. N. Y. Acad. Sci. 831, 179 (1997).
* Missner and Pohl [2009] A. Missner and P. Pohl, Chem. Phys. Chem 10, 1405 (2009).
* Siegel and Pitt [1995] R. A. Siegel and C. G. Pitt, J. Control. Release 33, 173 (1995).
* Marian von Smoluchowski [1916] Marian von Smoluchowski, Ann. Phys. 353, 1103 (1916).
* Kim _et al._ [2022] W. K. Kim, S. Milster, R. Roa, M. Kanduč, and J. Dzubiella, Macromolecules 55, 7327 (2022).
* Grassi and Colombo [1999] M. Grassi and I. Colombo, J. Control. Release 59, 343 (1999).
* Risken [1996] H. Risken, in _Fokker-Planck Equ._ (Springer, 1996).
* Kim _et al._ [2020] W. K. Kim, R. Chudoba, S. Milster, R. Roa, M. Kanduč, and J. Dzubiella, Soft Matter 16, 1 (2020).
* Milster _et al._ [2019] S. Milster, R. Chudoba, M. Kanduč, and J. Dzubiella, Phys. Chem. Chem. Phys. 21, 6588 (2019).
* Milster _et al._ [2021] S. Milster, W. K. Kim, M. Kanduč, and J. Dzubiella, J. Chem. Phys. 154, 154902 (2021).
* Kanduč _et al._ [2018] M. Kanduč, W. K. Kim, R. Roa, and J. Dzubiella, Macromolecules 51, 4853 (2018).
* Kanduč _et al._ [2021] M. Kanduč, W. K. Kim, R. Roa, and J. Dzubiella, ACS Nano 15, 614 (2021).
* Quesada-Pérez and Martín-Molina [2021] M. Quesada-Pérez and A. Martín-Molina, Adv. Colloid Interface Sci. 287, 102320 (2021).
* Amsden [1998] B. Amsden, Macromolecules 31, 8382 (1998).
* Hansing and Netz [2018] J. Hansing and R. R. Netz, Biophys. J. 114, 2653 (2018).
* Wolf and Willms [1978] B. A. Wolf and M. M. Willms, Die Makromol. Chemie 179, 2265 (1978).
* Kamemaru _et al._ [2018] K. Kamemaru, S. Usui, Y. Hirashima, and A. Suzuki, Gels 2018, Vol. 4, Page 45 4, 45 (2018).
* Mohammadyarloo and Sommer [2022] Z. Mohammadyarloo and J.-U. Sommer, Macromolecules 55, 8975 (2022).
* Annaka and Tanaka [1992] M. Annaka and T. Tanaka, Nat. 1992 3556359 355, 430 (1992).
* Katchalsky and Neumann [1972] A. Katchalsky and E. Neumann, Int. J. Neurosci. 3, 175 (1972).
* Frank [2004] T. D. Frank, Phys. Lett. A 327, 146 (2004).
* Donskoy [2022] I. Donskoy, Energies 15, 7152 (2022).
* Martyushev and Seleznev [2006] L. M. Martyushev and V. D. Seleznev, Phys. Rep. 426, 1 (2006).
* P. Glansdorff, G. Nicolis, and I. Prigogine [1974] P. Glansdorff, G. Nicolis, and I. Prigogine, Proc. Natl. Acad. Sci. 71, 197 (1974).
* Kondepudi and Prigogine [1998] D. Kondepudi and I. Prigogine, _Modern Thermodynamics_ (John Wiley & Sons, Chichester, 1998) p. 506.
* Attard [2009] P. Attard, Annu. Reports Prog. Chem. Sect. C Phys. Chem. 105, 63 (2009).
* Tomé [2006] T. Tomé, Brazilian J. Phys. 36, 1285 (2006).
* Graham and Tél [1985] R. Graham and T. Tél, Phys. Rev. A 31, 1109 (1985).
* Seifert [2012] U. Seifert, Reports Prog. Phys. 75, 58 (2012).
* Schmidt [2022] M. Schmidt, Rev. Mod. Phys. 94, 015007 (2022).
* Chavanis [2008] P.-H. Chavanis, Phys. A Stat. Mech. its Appl. 387, 5716 (2008).
* Archer and Rauscher [2004] A. J. Archer and M. Rauscher, J. Phys. A. Math. Gen. 37, 9325 (2004).
* Gardiner [2004] C. W. Gardiner, _Chemistry (Easton)._ , 3rd ed. (Springer-Verlag Berlin Heidelberg, 2004).
* Hiller and Rubner [2003] J. Hiller and M. F. Rubner, Macromolecules 36, 4078 (2003).
* Baker and Siegel [1996] J. P. Baker and R. A. Siegel, Macromol. Rapid Commun. 17, 409 (1996).
|
TTK-22-42, P3H-22-115 August 28, 2024
NLO QCD corrections and off-shell effects for $t\overline{t}H$ production in
the Higgs characterisation model
Jonathan Hermann111Work supported by German Research Foundation (DFG)
Collaborative Research Centre/Transregio project CRC/TRR 257:P3H - Particle
Physics Phenomenology after the Higgs Discovery, by the Research Training
Group GRK 2497: The physics of the heaviest particles at the Large Hadron
Collider and by a grant of the Bundesministerium für Bildung und Forschung
(BMBF).
Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen
University,
D-52056 Aachen, Germany
> In these proceedings we discuss $t\overline{t}H$ production at the Large
> Hadron Collider in the Higgs characterisation model. We demonstrate the
> importance of off-shell effects and higher-order QCD corrections for this
> process and highlight the importance of single-resonant contributions for
> the production of a $\mathcal{CP}$-odd Higgs boson.
> PRESENTED AT
>
>
>
>
> $15^{\mathrm{th}}$ International Workshop on Top Quark Physics
> Durham, UK, 4–9 September, 2022
## 1 Introduction
Despite its small cross-section compared to gluon-gluon and vector-boson
fusion, $t\overline{t}H$ production is one of the most important Higgs
production channels at the Large Hadron Collider (LHC). It is particularly
interesting for probing the top-Higgs Yukawa interaction as it already appears
at tree level in $t\overline{t}H$ production. The $\mathcal{CP}$-nature of
this interaction has been of great interest since the discovery of the Higgs
boson in 2012. Both CMS and ATLAS have performed measurements of the mixing
angle $\alpha_{CP}$ between a $\mathcal{CP}$-even and a $\mathcal{CP}$-odd
Higgs boson [1, 2]. In both cases the results are in agreement with the
Standard Model (SM) prediction of a $\mathcal{CP}$-even Higgs boson and the
measurements have allowed the exclusion of a pure $\mathcal{CP}$-odd state
with $3.7\,\sigma$ by CMS and $3.9\,\sigma$ by ATLAS. Nonetheless, there still
remains a lot of freedom in the mixing angle and CMS has actually found that
their fit results favour a $\mathcal{CP}$-mixed coupling.
With our work we aim to provide state-of-the-art predictions for
$t\overline{t}H$ production in the Higgs characterisation model [3] which
extends the SM Lagrangian by a $\mathcal{CP}$-odd top-Higgs Yukawa interaction
term. To this end, we present results for this process in the dilepton decay
channel at next-to-leading order (NLO) in QCD and including full off-shell
effects, i.e. $pp\to b\overline{b}e^{+}\mu^{-}\nu_{e}\overline{\nu}_{\mu}H+X$
production at order $\mathcal{O}\left(\alpha_{s}^{3}\alpha^{5}\right)$ for the
LHC at $\sqrt{s}=13$ TeV. Here, full off-shell effects mean that we describe
the unstable intermediate particles in the complex mass scheme and include all
double-, single- and non-resonant contributions. These results are based on
the ones presented in Ref. [4] and were computed using the HELAC-NLO framework
[5, 6]. Similar calculations have already been performed for this process in
the SM case in Refs. [7, 8, 9] but not for $\mathcal{CP}$-mixed top-Higgs
interactions. For the latter, the previous state-of-the-art predictions only
included NLO QCD corrections to the $t\overline{t}H$ production and not to the
top-quark decays. In addition, only double-resonant diagrams were considered
[10].
## 2 Phenomenological results
Table 1: Integrated fiducial cross-sections calculated in the NWA, NWA with LO top-quark decays and full off-shell approach for $\alpha_{CP}=0,\pi/4$ and $\pi/2$. Table was taken from [4]. $\alpha_{CP}$ | | Off-shell | NWA | Off-shell effects
---|---|---|---|---
$0$ (SM) | $\sigma_{\text{LO}}$ [fb] | $2.0313(2)^{+0.6275\,(31\%)}_{-0.4471\,(22\%)}$ | $2.0388(2)^{+0.6290\,(31\%)}_{-0.4483\,(22\%)}$ | $-0.37\%$
$\sigma_{\text{NLO}}$ [fb] | $2.466(2)^{+0.027\,(1.1\%)}_{-0.112\,(4.5\%)}$ | $2.475(1)^{+0.027\,(1.1\%)}_{-0.113\,(4.6\%)}$ | $-0.36\%$
$\sigma_{\text{NLO}_{\text{LOdec}}}$ [fb] | $-$ | $2.592(1)^{+0.161\,(6.2\%)}_{-0.242\,(9.3\%)}$ |
| ${\cal K}=\sigma_{\text{NLO}}/\sigma_{\text{LO}}$ | $1.21$ | $1.21$ (LOdec: $1.27$) |
$\pi/4$ | $\sigma_{\text{LO}}$ [fb] | $1.1930(2)^{+0.3742\,(31\%)}_{-0.2656\,(22\%)}$ | $1.1851(1)^{+0.3707\,(31\%)}_{-0.2633\,(22\%)}$ | $0.66\%$
$\sigma_{\text{NLO}}$ [fb] | $1.465(2)^{+0.016\,(1.1\%)}_{-0.071\,(4.8\%)}$ | $1.452(1)^{+0.015\,(1.0\%)}_{-0.069\,(4.8\%)}$ | $0.89\%$
$\sigma_{\text{NLO}_{\text{LOdec}}}$ [fb] | $-$ | $1.517(1)^{+0.097\,(6.4\%)}_{-0.144\,(9.5\%)}$ |
| ${\cal K}=\sigma_{\text{NLO}}/\sigma_{\text{LO}}$ | $1.23$ | $1.23$ (LOdec: $1.28$) |
$\pi/2$ | $\sigma_{\text{LO}}$ [fb] | $0.38277(6)^{+0.13123\,(34\%)}_{-0.09121\,(24\%)}$ | $0.33148(3)^{+0.11240\,(34\%)}_{-0.07835\,(24\%)}$ | $13.4\%$
$\sigma_{\text{NLO}}$ [fb] | $0.5018(3)^{+0.0083\,(1.2\%)}_{-0.0337\,(6.7\%)}$ | $0.4301(2)^{+0.0035\,(0.8\%)}_{-0.0264\,(6.1\%)}$ | $14.3\%$
$\sigma_{\text{NLO}_{\text{LOdec}}}$ [fb] | $-$ | $0.4433(2)^{+0.0323\,(7.3\%)}_{-0.0470\,(11\%)}$ |
| ${\cal K}=\sigma_{\text{NLO}}/\sigma_{\text{LO}}$ | $1.31$ | $1.30$ (LOdec: $1.34$) |
First we compare the integrated fiducial cross-sections for the
$\mathcal{CP}$-even, -mixed and -odd Higgs boson production in the narrow-
width approximation (NWA) and the full off-shell treatment at LO and NLO in
QCD. The results are listed in Table 1. We find that the cross-sections for
the $\mathcal{CP}$-mixed and $\mathcal{CP}$-even cases are about $3$ and $5$
times larger than for the $\mathcal{CP}$-odd one. NLO corrections are around
$20\%$ for the $\mathcal{CP}$-even and -mixed and about $30\%$ for the
$\mathcal{CP}$-odd Higgs boson. In all cases these corrections are consistent
between the NWA and the full off-shell treatment and within the LO
uncertainties. Concerning the off-shell effects, we find that these are of the
expected order of $\Gamma_{t}/m_{t}\sim 0.8\%$ for the $\mathcal{CP}$-even and
-mixed cases and thus negligible at the level of integrated fiducial cross-
sections when compared to the NLO scale uncertainties of around $5\%$.
However, in the pure $\mathcal{CP}$-odd case, the off-shell effects are
significantly larger than these uncertainties at $14\%$ for the NLO results
and should thus be taken into account even for the integrated fiducial cross-
section.
Figure 1: Differential distributions for the observables $p_{T,\,H}$ and
$\Delta\phi_{e^{+}\mu^{-}}$ at NLO in QCD. The lower panels show the
differential $\mathcal{K}\textrm{-factors}$. Figures were taken from [4].
Figure 2: Left: Differential distributions at NLO in QCD for $p_{T,H}$ for
the full off-shell case (solid lines) and the NWA (dashed lines). The ratio to
$\alpha_{CP}=0$ of the normalised differential distributions for the full off-
shell case is shown in the middle panel, the ratio NWA/off-shell is given in
the lower one.
Center: Differential distributions at NLO in QCD for $p_{T,H}$ for the full
off-shell case and the NWA as well as the double- (DR), single- (SR) and non-
resonant (NR) contributions for the SM case.
Right: Same as central plot but for the $\mathcal{CP}$-odd case. Figures were
taken from [4].
The general trend of larger corrections for the $\mathcal{CP}$-odd Higgs boson
can also be observed at the differential level. In Figure 1, where we show the
differential distributions for $p_{T,H}$ and $\Delta\phi_{e^{+},\mu^{-}}$, one
can for example see that NLO corrections are generally larger in the
$\mathcal{CP}$-odd case than for the others. For most observables the
$\mathcal{K}$-factors follow the same trend as for $p_{T,H}$, i.e. the
corrections increase towards the distribution tail but the behaviour of the
three $\mathcal{K}$-factors is roughly the same. However, for observables
which involve decay products of both top quarks, like
$\Delta\phi_{e^{+},\mu^{-}}$, we observe that the $\mathcal{K}$-factor for the
$\mathcal{CP}$-odd Higgs boson is flatter than for the other two cases. This
happens due to the harder Higgs boson radiation for the former case which
suppresses the real radiation corrections that are responsible for the large
$\mathcal{K}$-factors for small opening angles and large transverse momenta.
Off-shell effects are also enhanced at the differential level, as can be seen
from the lower panel of the left plot in Figure 2. For the depicted $p_{T,H}$
distribution, they exceed $30\%$ in the high-$p_{T}$ region for the
$\mathcal{CP}$-odd Higgs boson. On the other hand, the effects remain
negligible for the other two considered $\mathcal{CP}$-states. The different
size of the off-shell effects can be attributed to the single-resonant (SR)
contributions included in the full off-shell computation. In the central and
right plot in Figure 2 we show the splitting of the off-shell distribution
into its double-, single- and non-resonant parts as defined in Eq. (5.6) to
(5.9) Ref. [4]. In the SM case (central plot), the ratio between the three
contributions remains virtually constant throughout the entire spectrum which
mimics the behaviour of the off-shell effects in this case. In contrast, for
the $\mathcal{CP}$-odd state (right plot), the DR contribution decreases
significantly towards the distribution tail, in line with NWA distribution. At
the same time, the SR contribution increases to account for the difference.
Hence, the larger off-shell effects in the latter case can be explained by the
different behaviour of the SR contribution.
## 3 Summary
In these proceedings we have presented integrated and differential fiducial
cross-sections for $t\overline{t}H$ production with leptonic top-quarks decays
in the Higgs characterisation framework. We have underlined the importance of
both off-shell effects and higher-order QCD corrections for these predictions,
particularly for the production of a $\mathcal{CP}$-odd Higgs boson. In
addition, we have demonstrated that the large off-shell effects in the
$\mathcal{CP}$-odd case are a result of the missing single-resonant
contributions in the NWA.
## References
* [1] CMS Collaboration, arXiv:2208.02686.
* [2] ATLAS Collaboration, _Phys. Rev. Lett._ 125 (2020) 061802.
* [3] P. Artoisenet et al., _JHEP_ 11 (2013) 043.
* [4] J. Hermann, D. Stremmer, M. Worek, _JHEP_ 09 (2022) 138.
* [5] G. Bevilacqua et al., _Comput. Phys. Commun._ 184 (2013) 986.
* [6] G. Bevilacqua, H. B. Hartanto, M. Kraus, T. Weber, M. Worek, _JHEP_ 03 (2020) 154.
* [7] A. Denner and R. Feger, _JHEP_ 11 (2015) 209.
* [8] A. Denner, J.-N. Lang, M. Pellen and S. Uccirati, _JHEP_ 02 (2017) 053
* [9] D. Stremmer, M. Worek, _JHEP_ 02 (2022) 196.
* [10] F. Demartin, F. Maltoni, K. Mawatari, B. Page, M. Zaro, _Eur. Phys. J. C_ 74 (2014) 3065.
|
# Monitored Recurrence of a One-parameter Family of Three-state Quantum Walks
Martin Štefaňák Department of Physics, Faculty of Nuclear Sciences and
Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19
Praha 1 - Staré Město, Czech Republic<EMAIL_ADDRESS>
###### Abstract
Monitored recurrence of a one-parameter set of three-state quantum walks on a
line is investigated. The calculations are considerably simplified by choosing
a suitable basis of the coin space. We show that the Polya number (i.e. the
site recurrence probability) depends on the coin parameter and the probability
that the walker is initially in a particular coin state for which the walk
returns to the origin with certainty. Finally, we present a brief
investigation of the exact quantum state recurrence.
## 1 Introduction
Quantum walks [1, 2, 3] are generalizations of classical random walks to
evolution of a quantum particle on a graph or a lattice. Similar ideas
appeared already in 1960’s in the works of Feynman and Hibbs on discretization
of the Dirac equation [4] and in the 1980’s in the work of Gudder on quantum
graphic dynamics [5, 6]. Quantum walks became versatile tools in various field
of physics and computation, for a recent review see [7].
One of the interesting questions in stochastic processes is the problem of
recurrence. For classical random walks, this was studied in detail by Polya
[8] who proved that in 1 and 2 spatial dimensions the balanced walk returns
with certainty, i.e. is recurrent, while for $d\geq 3$ the recurrence
probability (also called Polya number) is strictly less than unity, implying
transience. Nevertheless, walks on infinite line and square lattice are null-
recurrent, since the expected recurrence time is infinite. On the other hand,
irreducible walks on finite graphs are positive-recurrent since the expected
recurrence time is finite. In fact, for finite state space Kac shown [9] that
it is an integer.
Studying recurrence of quantum walks requires a detailed description of the
measurement process since it has a nontrivial effect on the wavefunction of
the quantum walker. To minimize the influence of measurement we have proposed
a scheme [10, 11] where the quantum walk evolves freely for a certain number
of steps, then a measurement detecting the presence of the walker at the
origin is performed, after which the process is restarted. In this concept the
recurrence or transience of a quantum walk can be decided from the behaviour
of the probability of finding the walker at the origin after $t$ steps
(without prior measurement) in the same way as for the classical random walk.
In fact, for classical random walks this scheme is equivalent to the Polya
approach, where we monitor the origin after each step, in the sense that the
walk is recurrent in one approach if and only if it is recurrent in the other
scheme. This stems from the fact that in the classical setting, the
fundamental role is played by probabilities and there is a direct relation
between the return probabilities without prior measurement and the first
return probabilities of the monitored walk. However, this does not hold
anymore in the quantum case. Due to the effect of measurement on the
wavefunction the two probabilities cannot be accessed in the same experiment
and they are not related to each other. Nevertheless, there exists a relation
between the return amplitudes of the unitary quantum walk and the first return
amplitudes of the monitored walk, as shown in [12, 13] for a much broader
setting of iterated unitary evolutions. Note that some particular examples
were investigated before the general theory was developed, e.g. the monitored
recurrence of the Hadamard walk on a line [14] which was shown to be
equivalent to the absorption of the walk on the half-line [15]. In the
monitored recurrence approach, the expected return time to the exact initial
state (state recurrence) of a finite system is an integer [12] as in the
classical case. This holds even for iterated open quantum evolutions [16, 17].
Note that in the coined quantum walks return to the initial position can be
understood as a subspace recurrence, since we are not interested in the exact
state of the quantum coin. For finite systems the expected return time for a
subspace recurrence is a rational number [13]. We point out that in the recent
years the study of recurrence was considerably broadened to open quantum
systems, quantum Markov chains and open quantum walks, see eg. [18, 19, 20,
21, 22, 23, 24].
In contrast to the classical case the two schemes for detecting recurrence
(restart after measurement vs monitored after each step) are not equivalent
for quantum walks. As an example, the Hadamard walk on a line is recurrent in
the restart scheme [10] while transient in the monitored case [14]. This was
also demonstrated in an optical experiment [25] where the quantum walk was
simulated by a weak laser pulse cycling in a time-delay loop, utilizing time-
multiplexing to encode the position and number of steps of the walk into the
time of detection of a photon. It was a considerable experimental challenge to
perform the local measurement at the origin without disturbing the rest of the
pulse sequence. The deterministic out-coupling from the loop was achieved by a
programmable fast-switching electro-optical modulator, which allowed to
address specific time slots corresponding to the walker being at the origin.
In the present paper we take the monitored approach to the recurrence problem
and apply it to the three-state quantum walks on a line with a particular one-
parameter family of coins [26]. We show that there is a unique initial coin
state for which the walk stays at the origin in the first step, so in this
case the walk is recurrent. We then choose this initial state as one of the
basis vectors in the coin space and complement the basis with two vectors from
orthogonal complement. This leads to significant reduction of the complexity
of the follow-up analytical calculations. We show that the subspace recurrence
probability is determined by the coin parameter and the probability that the
walker is initially in the unique recurrent state. For the exact quantum state
recurrence we provide a numerical investigation. In particular, we identify
initial states for which the state recurrence is greater than the subspace
recurrence, a paradoxical feature of monitored quantum evolution which was
already discussed previously in the literature [13].
The paper is organized as follows. Section 2 reviews the basic concepts and
tools for investigation of monitored recurrence. Section 3 is dedicated to the
study of site recurrence for a one-parameter family of three-state quantum
walks. In Section 4 we present a numerical investigation of the state
recurrence. We conclude and present an outlook in Section 5.
## 2 Monitored site recurrence of a quantum walk
We begin by a brief overview of the methods for studying recurrence developed
in [12, 13], adopted for a coined quantum walks with constant coin. Consider a
discrete time quantum walk on a lattice starting from the origin. The Hilbert
space is a tensor product of the position and the coin spaces
$\mathcal{H}=\mathcal{H}_{p}\otimes\mathcal{H}_{c}.$
We denote the evolution operator by $U$, it has the usual decomposition
$U=S\cdot(I_{p}\otimes C),$ (1)
where $S$ is the shift operator and $C$ is the coin. Return of the quantum
walker to the original site can be understood as a subspace recurrence [13],
since we are not interested in the internal state of the coin, only the
position of the quantum walker. Let us denote the orthogonal projector onto
the origin subspace as
$\Pi_{0}=|0\rangle\langle 0|\otimes I_{c}.$ (2)
The evolution of the monitored quantum walk is given by the operator
$\tilde{U}=(I-\Pi_{0})U,$
i.e. the walk continues if we do not find the walker at the origin. Normalized
state after $n$ steps of the monitored walk reads
$|\psi_{n}\rangle=\frac{1}{\sqrt{s_{n}}}\tilde{U}^{n}|\psi\rangle,\quad|\psi\rangle=|0\rangle\otimes|\psi_{c}\rangle,$
where $s_{n}$ is the survival probability until the $n$-th step
$s_{n}=\lVert\tilde{U}^{n}\psi\rVert^{2}.$
$s_{n}$ is the probability that the walker has not returned to the origin in
the first $n$ steps. Hence, the complement of the limiting value of $s_{n}$
$P(\psi)=1-\lim\limits_{n\to\infty}s_{n},$
corresponds to the probability that the walker ever returns, i.e. the site
recurrence probability (or Polya number). Alternatively, we can derive the
Polya number from the first return probabilities. Let us denote the
probability of first return to the origin after $n$ steps as $q_{n}$. Since
these events are mutually exclusive, the overall site recurrence probability
is given by the sum
$P(\psi)=\sum_{n=1}^{\infty}q_{n}.$
In case of iterated unitary evolution the first return probability after $n$
steps is given by
$q_{n}=\lVert\Pi_{0}U\tilde{U}^{n-1}\psi\rVert^{2}=\lVert
a_{n}\psi\rVert^{2},$
where we have introduced the first return amplitude operator (note that
$\Pi_{0}\psi=\psi$)
$a_{n}=\Pi_{0}U\tilde{U}^{n-1}\Pi_{0}.$
The Polya number of a quantum walk for a given initial state is therefore
given by
$P(\psi)=\sum_{n=1}^{\infty}\lVert a_{n}\psi\rVert^{2}.$
Determining the operators $a_{n}$ directly is rather difficult, however, they
can be related to the $n$-th step return amplitude operators without prior
monitoring, which we denote as
$\mu_{n}=\Pi_{0}U^{n}\Pi_{0}.$
We define the operator valued generating functions (for complex variable $z$
with $|z|<1$)
$\mu(z)=\sum_{n=0}^{\infty}\mu_{n}z^{n},\quad
a(z)=\sum_{n=1}^{\infty}a_{n}z^{n}.$
Note that $\mu(z)$ is also called the Stieltjes operator and the first return
generating function $a(z)$ is related to the Schur function $f(z)$ by
$a(z)=zf^{\dagger}(z),$
which is more extensively used in the literature [12, 13, 20, 23]. Introducing
the resolvents
$G(z)=\sum_{n=0}^{\infty}U^{n}z^{n}=(I-zU)^{-1},\quad\tilde{G}(z)=\sum_{n=0}^{\infty}\tilde{U}^{n}z^{n}=(I-z\tilde{U})^{-1},$
we see that the generating functions can be written in the form
$\mu(z)=\Pi_{0}G(z)\Pi_{0},\quad a(z)=z\Pi_{0}U\tilde{G}(z)\Pi_{0}.$ (3)
Using the resolvent identities
$G(z)-\tilde{G}(z)=zG(z)\Pi_{0}U\tilde{G}(z)=z\tilde{G}(z)\Pi_{0}UG(z),$
we can derive the renewal equations [12, 13]
$\mu(z)a(z)=a(z)\mu(z)=\mu(z)-\Pi_{0}.$ (4)
In the above equation all operators act on the origin subspace (see (2) and
(3)), i.e. they are of the form
$\mu(z)=|0\rangle\langle 0|\otimes\mu_{c}(z),\quad a(z)=|0\rangle\langle
0|\otimes a_{c}(z),$
where $\mu_{c}(z)$ and $a_{c}(z)$ act on the coin space. Thus, we can rewrite
(4) into operator equation on the coin space
$a_{c}(z)=I_{c}-\mu_{c}(z)^{-1}.$ (5)
Hence, we can express the first return generating function $a_{c}(z)$ from the
return generating function $\mu_{c}(z)$, which is easier to obtain. All
functions can be extended to the unit circle in the complex plane by the
radial limit. The site recurrence probability is expressed in terms of the
boundary values [12, 13] by
$P(\psi_{c})=\int\limits_{0}^{2\pi}\lVert
a_{c}(e^{it})\psi_{c}\rVert^{2}\frac{dt}{2\pi}=\langle\psi_{c}|R|\psi_{c}\rangle,$
(6)
where we have denoted the recurrence probability operator
$R=\int\limits_{0}^{2\pi}a_{c}^{\dagger}(e^{it})a_{c}(e^{it})\frac{dt}{2\pi}.$
(7)
In summary, the recipe to obtain the Polya number is to find the Stieltjes
operator $\mu_{c}(z)$, from (5) we find the first return generating operator,
construct the recurrence probability operator (7) and find its average value
for a given initial state of the coin $\psi_{c}$ (6).
The site recurrence probability of a two-state quantum walk on a line, where
the walker can move to the right or left in each step, was evaluated
explicitly in [12]. The unitary coin operator was parameterized in the
following way
$C=\begin{pmatrix}\rho&-\gamma\\\
\overline{\gamma}&\rho\end{pmatrix},\quad\rho=\sqrt{1-|\gamma|^{2}}.$ (8)
Since the considered walk is translationally invariant, one can use Fourier
transformation to diagonalize the step operator and evaluate the Stieltjes
operator with the formula
$\mu_{c}(z)=\int\limits_{0}^{2\pi}\frac{dp}{2\pi}(I_{c}-zU(p))^{-1}.$ (9)
Here $U(p)$ is the evolution operator (1) in the Fourier picture
$U(p)=S(p)\cdot C=\begin{pmatrix}e^{ip}&0\\\
0&e^{-ip}\end{pmatrix}\cdot\begin{pmatrix}\rho&-\gamma\\\
\overline{\gamma}&\rho\end{pmatrix},$
which is a multiplication operator. Using the substitution $e^{ip}=x$,
$dp=\frac{dx}{ix}$ the RHS of (9) is turned into an integral over the unit
circle in the complex plane, which can be evaluated with the residue theorem.
After some calculations it is found that the recurrence operator (7) for a
two-state walk is a multiple of identity, i.e. the Polya number does not
depend on the initial coin state, only on the quantum coin. Explicitly, the
site recurrence probability for a two-state quantum walk on a line with the
coin (8) is given by [12, 27]
$P=\frac{2\left(|\gamma|\sqrt{1-|\gamma|^{2}}-(1-2|\gamma|^{2})\arcsin|\gamma|\right)}{\pi|\gamma|^{2}}.$
## 3 Site recurrence of a three-state walk on a line
Let us now consider a three-state walk on a line, where the walker can move to
the right, stay, or move to the left. This corresponds to the standard basis
of the coin space $|R\rangle$, $|S\rangle$ and $|L\rangle$. We choose the
following one-parameter set of coins [26, 28]
$C=\begin{pmatrix}-\rho^{2}&\rho\sqrt{2(1-\rho^{2})}&1-\rho^{2}\\\
\rho\sqrt{2(1-\rho^{2})}&2\rho^{2}-1&\rho\sqrt{2(1-\rho^{2})}\\\
1-\rho^{2}&\rho\sqrt{2(1-\rho^{2})}&-\rho^{2}\end{pmatrix},\quad 0<\rho<1,$
(10)
which reduces to the 3x3 Grover matrix for $\rho=\frac{1}{\sqrt{3}}$. Before
turning to the derivation of the Polya number, we make the following
observation. Since the coin operator (10) is equal to its inverse, the initial
coin state
$|\alpha_{1}\rangle=C|S\rangle=\rho\sqrt{2(1-\rho^{2})}(|R\rangle+|L\rangle)+(2\rho^{2}-1)|S\rangle=\begin{pmatrix}\rho\sqrt{2(1-\rho^{2})}\\\
2\rho^{2}-1\\\ \rho\sqrt{2(1-\rho^{2})}\end{pmatrix},$
is mapped to $|S\rangle$. Hence, if we take $|\alpha_{1}\rangle$ as the
initial coin state, the walker is absorbed with certainty after one step, i.e.
for this initial state the site recurrence probability is one. Numerical
simulations indicate that for all coin states in the orthogonal complement to
$|\alpha_{1}\rangle$ the Polya number has the same value, which is less than
unity. Let us complete the orthonormal basis in the coin space by
$\displaystyle|\alpha_{2}\rangle$
$\displaystyle=\frac{2\rho^{2}-1}{\sqrt{2}}(|R\rangle+|L\rangle)-2\rho\sqrt{1-\rho^{2}}|S\rangle=\begin{pmatrix}\frac{2\rho^{2}-1}{\sqrt{2}}\\\
-2\rho\sqrt{1-\rho^{2}}\\\ \frac{2\rho^{2}-1}{\sqrt{2}}\end{pmatrix},$
$\displaystyle|\alpha_{3}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|R\rangle-|L\rangle)=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
0\\\ -1\end{pmatrix}.$
The discussion above indicates that the recurrence probability operator should
be diagonal when expressed in the basis formed by
$\left\\{|\alpha_{1}\rangle,|\alpha_{2}\rangle,|\alpha_{3}\rangle\right\\}$.
From now on all matrices will be expressed in this basis. For this we will
utilize the transition matrices $T$ and $T^{\dagger}$, where
$T=\begin{pmatrix}\rho\sqrt{2(1-\rho^{2})}&2\rho^{2}-1&\rho\sqrt{2(1-\rho^{2})}\\\
\frac{2\rho^{2}-1}{\sqrt{2}}&-2\rho\sqrt{2(1-\rho^{2})}&\frac{2\rho^{2}-1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\end{pmatrix}.$
We denote the matrix $M$ in the $\alpha$ basis as ${}^{\alpha}M$.
Turning to the derivation of the Polya number, we begin by expressing the
Stieltjes operator through the Fourier transformation (9), where
${}^{\alpha}U(p)$ is given by
${}^{\alpha}U(p)$ $\displaystyle=T\cdot\begin{pmatrix}e^{ip}&0&0\\\ 0&1&0\\\
0&0&e^{-ip}\end{pmatrix}\cdot C\cdot T^{\dagger}$
$\displaystyle=\begin{pmatrix}2\rho^{2}-1&-2\rho\sqrt{1-\rho^{2}}\cos{p}&-2i\rho\sqrt{1-\rho^{2}}\sin{p}\\\
-2\rho\sqrt{1-\rho^{2}}&(1-2\rho^{2})\cos{p}&i(1-2\rho^{2})\sin{p}\\\
0&-i\sin{p}&-\cos{p}\end{pmatrix}.$
The resolvent in the Fourier space is equal to
${}^{\alpha}(I_{c}-zU(p))^{-1}$ $\displaystyle=\
\frac{1}{(z-1)(1+z(2-2\rho^{2}+z)+2\rho^{2}z\cos{p})}\times$
$\displaystyle\begin{pmatrix}u(z)-2\rho^{2}z\cos
p&2\rho\sqrt{1-\rho^{2}}z(\cos p+z)&2i\rho\sqrt{1-\rho^{2}}z\sin p\\\
2\rho\sqrt{1-\rho^{2}}z(z\cos p+1)&w(z)(z\cos p+1)&-izv(z)\sin p\\\
-2i\rho\sqrt{1-\rho^{2}}z^{2}\sin p&i(u(z)+z+1)\sin p&zv(z)\cos
p+w(z)\end{pmatrix},$
where to shorten the formula we have denoted
$u(z)=z^{2}(1-2\rho^{2})-1,\quad v(z)=1-2\rho^{2}+z,\quad
w(z)=z(2\rho^{2}-1)-1.$
The integral in (9) can be again evaluated with residues. We will utilize the
following integrals (with $n=1,0,-1$)
$\displaystyle{\cal
I}(n)=\int\limits_{0}^{2\pi}\frac{dp}{2\pi}\frac{e^{inp}}{(z-1)(1+z(2-2\rho^{2}+z)+2\rho^{2}z\cos{p})}$
$\displaystyle=\frac{1}{2\pi i}\oint\frac{x^{n}dx}{b(x)},$
where we have denoted
$b(x)=(z-1)(\rho^{2}z+(1+2(1-\rho^{2})z+z^{2})x+\rho^{2}zx^{2}).$
The roots of the equation $b(x)=0$ are
$x_{\pm}=\frac{2(\rho^{2}-1)z-z^{2}-1\pm\sqrt{\left(1+2(1-\rho^{2})z+z^{2}\right)^{2}-4\rho^{4}z^{2}}}{2\rho^{2}z},$
with $|x_{-}|>1$ for $|z|\leq 1$. Hence, for $n=0,1$ there is only the residue
at $x_{+}$ and we find
$\displaystyle{\cal I}(0)$ $\displaystyle={\rm
Res}\left(\frac{1}{b(x)},x_{+}\right)=\frac{1}{(z^{2}-1)g(z)},$
$\displaystyle{\cal I}(1)$ $\displaystyle={\rm
Res}\left(\frac{x}{b(x)},x_{+}\right)=\frac{(z+1)g(z)-1-2(1-\rho^{2})z-z^{2}}{2\rho^{2}z(z^{2}-1)g(z)},$
where we have used the notation
$g(z)=\sqrt{1+2(1-2\rho^{2})z+z^{2}}.$
For $n=-1$ there is an additional residue at $x=0$ and we obtain
$\displaystyle{\cal I}(-1)$ $\displaystyle={\rm
Res}\left(\frac{1}{xb(x)},x_{+}\right)+{\rm
Res}\left(\frac{1}{xb(x)},0\right)$
$\displaystyle=\frac{2\rho^{2}z}{(z^{2}-1)g(z)[(z+1)g(z)-1-2(1-\rho^{2})z-z^{2}]}+\frac{1}{\rho^{2}z(z-1)}.$
After some algebra, we express the Stieltjes operator in the form
$\displaystyle^{\alpha}\mu_{c}(z)$
$\displaystyle=\frac{1}{(z-1)g(z)}\begin{pmatrix}2(\rho^{2}-1)z-g(z)&\frac{\sqrt{1-\rho^{2}}(g(z)+w(z))}{\rho}&0\\\
\frac{\sqrt{1-\rho^{2}}z(g(z)-v(z))}{\rho}&\frac{w(z)(g(z)-v(z))}{2\rho^{2}}&0\\\
0&0&\frac{v(z)g(z)-1-z(z+2-4\rho^{2})}{2\rho^{2}}\end{pmatrix}.$ (11)
From (5) we find the first return generating operator
${}^{\alpha}a_{c}(z)=\begin{pmatrix}(2\rho^{2}-1)z&-2\rho\sqrt{1-\rho^{2}}-\frac{4\rho^{3}\sqrt{1-\rho^{2}}(z-1)}{g(z)-v(z)}&0\\\
-2\rho\sqrt{1-\rho^{2}}z&1-2\rho^{2}+\frac{2\rho^{2}(2\rho^{2}-1)(z-1)}{g(z)-v(z)}&0\\\
0&0&-\frac{j(z)}{2(\rho^{2}-1)}\end{pmatrix}.$
Here we have denoted
$j(z)=1-2\rho^{2}z+z^{2}+(z-1)g(z).$
For the product ${}^{\alpha}a^{\dagger}_{c}(z)\ ^{\alpha}a_{c}(z)$ we obtain a
diagonal matrix as expected
$^{\alpha}r(z)=^{\alpha}a^{\dagger}_{c}(z)\
^{\alpha}a_{c}(z)=\begin{pmatrix}|z|^{2}&0&0\\\
0&\frac{|j(z)|^{2}}{4(\rho^{2}-1)^{2}}&0\\\
0&0&\frac{|j(z)|^{2}}{4(\rho^{2}-1)^{2}}\end{pmatrix}.$ (12)
To determine the matrix of the recurrence probability operator in the $\alpha$
basis ${}^{\alpha}R$ we have to take $z=e^{it}$ and perform the integral (7).
The first diagonal entry in (12) is 1, and so is the result of the integral,
corresponding to the fact that for the $|\alpha_{1}\rangle$ state the walker
returns to the original site with certainty. Then we are left with evaluating
$Q=\int\limits_{0}^{2\pi}\frac{dt}{8\pi(\rho^{2}-1)^{2}}|j(e^{it})|^{2}.$
We have to carefully express $g(e^{it})$ and $j(e^{it})$. We find the
following
$g(e^{it})=\left\\{\begin{array}[]{cc}e^{\frac{it}{2}}\operatorname{sgn}{(\sin{t})}\sqrt{2(\cos{t}+1-2\rho^{2})},&\cos{t}\geq
2\rho^{2}-1\\\ \\\
-ie^{\frac{it}{2}}\sqrt{-2(\cos{t}+1-2\rho^{2})},&\cos{t}<2\rho^{2}-1\end{array}\right..$
Hence, for $t\in(0,\arccos(2\rho^{2}-1))\cup(2\pi-\arccos(2\rho^{2}-1),2\pi)$
, we obtain
$j(e^{it})=2e^{it}\left(\cos{t}-\rho^{2}+i\operatorname{sgn}{(\sin{t})}\sin(t/2)\sqrt{2(\cos{t}+1-2\rho^{2})}\right),$
which has a constant modulus square
$\displaystyle|j(e^{it})|^{2}=4(\rho^{2}-1)^{2}.$
The contribution to $Q$ is then proportional to the length of the interval and
is found to be
$Q_{1}=\frac{\arccos(2\rho^{2}-1)}{\pi}.$
Turning to the case $\cos{t}<-2\rho^{2}-1$, i.e. the interval
$t\in(\arccos(2\rho^{2}-1),2\pi-\arccos(2\rho^{2}-1))$, we find
$j(e^{it})=2e^{it}\left(\cos{t}-\rho^{2}+\sin(t/2)\sqrt{-2(\cos{t}+1-2\rho^{2})}\right).$
The resulting integral can be evaluated directly
$\displaystyle Q_{2}=$
$\displaystyle\int\limits_{\arccos{(2\rho^{2}-1)}}^{2\pi-\arccos(2\rho^{2}-1)}\frac{dt}{8\pi(\rho^{2}-1)}|j(e^{it})|^{2}$
$\displaystyle=$
$\displaystyle\int\limits_{\arccos{(2\rho^{2}-1)}}^{2\pi-\arccos(2\rho^{2}-1)}\frac{dt}{2\pi(\rho^{2}-1)}\left(\cos{t}-1+\sin(t/2)\sqrt{-2(\cos{t}+1-2\rho^{2})}\right)^{2}$
$\displaystyle=\rho\frac{2(1+2\rho^{2})\sqrt{1-\rho^{2}}-\rho(2+\rho^{2})\arccos(2\rho^{2}-1)}{\pi}.$
In summary, we find
$Q=Q_{1}+Q_{2}=\frac{2\rho\left(2\rho^{2}+1\right)\sqrt{1-\rho^{2}}+\left(1-4\rho^{2}\right)\arccos\left(2\rho^{2}-1\right)}{\pi\left(\rho^{2}-1\right)^{2}}.$
(13)
The matrix of the recurrence probability operator in the $\alpha$ basis is
then given by
${}^{\alpha}R=\begin{pmatrix}1&0&0\\\ 0&Q&0\\\ 0&0&Q\end{pmatrix}.$
This means that the Polya number for the three-state walk with the coin (10)
depends on the probability $p_{1}$ to be initially in the $|\alpha_{1}\rangle$
state
$p_{1}=|\langle\alpha_{1}|\psi_{c}\rangle|^{2},$
and the coin parameter $\rho$. The resulting formula is
$P(\psi_{c})=p_{1}+Q(1-p_{1}).$ (14)
The site recurrence probability thus ranges between 1 and $Q$ (13). In
particular, for the Grover walk corresponding to $\rho=\frac{1}{\sqrt{3}}$, we
find
$P(\psi_{c})=p_{1}+\frac{10\sqrt{2}-3\arccos{(-1/3)}}{4\pi}(1-p_{1})\stackrel{{\scriptstyle.}}{{=}}0.67+0.33p_{1}.$
We illustrate our results in Figures 1 and 2. Figure 1 shows the site
recurrence probability (14) in dependence of $\rho$ and $p_{1}$. In Figure 2
we display the value of $Q$ (13) as a function of the coin parameter $\rho$.
Figure 1: Density plot of the Polya number (14) of the three-state quantum
walk as a function of the probability $p_{1}$ and the coin parameter $\rho$.
Dashed lines show the contours of $P=k/10$ for $k=1,\ldots 9$. Figure 2: $Q$
(13) as a function of the coin parameter $\rho$. $Q$ represents the minimal
value of the recurrence probability for the walk with a given parameter
$\rho$.
## 4 State recurrence of a three-state walk on a line
Let us now briefly comment on the state recurrence [12]. In this case we are
interested in the return to the exact initial state, so the orthogonal
projector (2) has the form
$\Pi_{0}=|0\rangle\langle 0|\otimes|\psi_{c}\rangle\langle\psi_{c}|.$
The recipe to determine the state recurrence probability $S(\psi_{c})$ is
similar to the site recurrence, but instead of the operator valued generating
functions we will deal with scalars. The matrix element of the Stieltjes
operator (11) with the initial coin state $|\psi_{c}\rangle$ yields the
generating function for return amplitudes without prior monitoring
$\mu_{\psi_{c}}(z)=\langle\psi_{c}|\mu_{c}(z)|\psi_{c}\rangle.$
The generating function for the first arrival amplitudes is then given by [12]
$a_{\psi_{c}}(z)=1-\frac{1}{\mu_{\psi_{c}}(z)}.$
Finally, the state recurrence probability is obtained by a formula analogous
to (7)
$S(\psi_{c})=\int\limits_{0}^{2\pi}\lvert
a_{\psi_{c}}(e^{it})\rvert^{2}\frac{dt}{2\pi}.$ (15)
In contrast to the site recurrence, which was tractable, the integrand in (15)
is usually a rather complicated function and we rely on numerical integration.
Below we present plots with numerical results for several initial states.
For Figure 3 we consider the basis states $|\alpha_{i}\rangle$ and plot the
state recurrence probability as a function of the coin parameter $\rho$. For
$|\alpha_{2}\rangle$ and $|\alpha_{3}\rangle$ the integral in (15) can be
evaluated analytically and we find that the state recurrence is equal to the
site recurrence. In fact, numerical simulations reveal that for the initial
state $|\alpha_{3}\rangle$ the walker always returns to the origin in the
$|\alpha_{3}\rangle$ state. However, for $|\alpha_{2}\rangle$ the situation is
more complicated - the walker returns to the original site in a superposition
of $|\alpha_{1}\rangle$ and $|\alpha_{3}\rangle$ with different weights in
every time step. Nevertheless, the state recurrence probability equals (13).
For $|\alpha_{1}\rangle$ the state recurrence probability is smaller than site
recurrence probability (which equals 1), except for the boundary cases of
$\rho=0,1$.
Figure 3: State recurrence probability for the state $|\alpha_{1}\rangle$
(blue dots) and $|\alpha_{2}\rangle$ and $|\alpha_{3}\rangle$ (orange dots) in
dependence on $\rho$.
Figures 4 and 5 focus on the three-state Grover walk
($\rho=\frac{1}{\sqrt{3}}$). In Figure 4 we consider the initial state
$\frac{1}{\sqrt{2}}(|\alpha_{1}\rangle+e^{i\phi}|\alpha_{2}\rangle)$ and plot
$S$ as a function of the angle $\phi$. Note that for superpositions
$\frac{1}{\sqrt{2}}(|\alpha_{1}\rangle+e^{i\phi}|\alpha_{3}\rangle)$ and
$\frac{1}{\sqrt{2}}(|\alpha_{2}\rangle+e^{i\phi}|\alpha_{3}\rangle)$ the angle
$\phi$ does not affect the state recurrence probability due to the block-
diagonal form of the Stieltjes operator (11). Figure 5 shows state recurrence
probability for superposition states
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{2}\rangle$,
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ and
$a|\alpha_{2}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ as a function of $a$.
Figure 4: State recurrence probability of the Grover walk ($\rho=1/\sqrt{3}$)
for the state
$\frac{1}{\sqrt{2}}(|\alpha_{1}\rangle+e^{i\phi}|\alpha_{2}\rangle)$ as a
function of the angle $\phi$. Figure 5: State recurrence probabilities of the
Grover walk ($\rho=1/\sqrt{3}$) for the states
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{2}\rangle$ (blue dots),
$a|\alpha_{1}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ (orange dots) and
$a|\alpha_{2}\rangle+\sqrt{1-a^{2}}|\alpha_{3}\rangle$ (green dots) as a
function of $a$.
Finally, we point out that the three-state walk shows similar paradoxical
behavior as reported in [13] for a walk on a half-line or some 2D quantum
walks, where the state recurrence probability can be greater than the site
recurrence probability. In Figure 6 we consider the initial coin state
$|\psi_{c}\rangle=\sqrt{\frac{1-\rho^{2}}{2}}|R\rangle+\rho|S\rangle+\sqrt{\frac{1-\rho^{2}}{2}}|L\rangle=\rho|\alpha_{1}\rangle-\sqrt{1-\rho^{2}}|\alpha_{2}\rangle,$
(16)
which is one of the eigenvectors used in the construction of coin operator
[26]. The plot indicates that the site recurrence probability, depicted by the
orange curve, is smaller than the state recurrence probability up to
$\rho\approx 0.79$.
Figure 6: State recurrence (blue dots) versus site recurrence (orange curve)
for the initial coin state (16).
## 5 Conclusions
The presented results demonstrate that in contrast to the simple two-state
quantum walks the site recurrence of the three-state model depends on the
initial state. For the selected one-parameter family of coins we were able to
derive the site recurrence probability in closed form. Extension to an
arbitrary 3x3 unitary coin seems to be intractable due to the complexity and
large number of free parameters. Nevertheless, for all three-state walks there
will be an initial coin state with Polya number equal to one, namely the state
given by $C^{-1}|S\rangle$. Indeed, such state will remain on the initial
position after the first step, and thus is absorbed with certainty. It is an
open question if such behaviour of site recurrence is possible for models
without the staying put option in the shift operator, e.g. in quantum walks
driven by Wigner rotation matrices of order $2j+1$ studied in [29, 30] for the
case of half-integer $j$.
As discussed in Section 4, obtaining closed formulas for state recurrence
probability was possible only for particular initial coin states due to the
complexity of the involved integrals. The same applies to both site and state
recurrence for quantum walks on higher dimensional lattices, since in such
cases already the evaluation of the Stieltjes operator through
multidimensional Fourier transform cannot be easily reduced to calculation of
residues as in the one-dimensional case. Nevertheless, the formulas allow to
obtain approximations through numerical evaluation. It would be interesting to
find if there exists site recurrent initial conditions e.g. for 2D quantum
walks with Grover coin and its extensions. Additional twist can come from
considering coined quantum walks combined with some form of classical
randomness resulting in iterated open quantum evolution, e.g. quantum walks on
dynamically percolated lattices [31, 32, 33]. Recent extension of monitored
recurrence to arbitrary iterated closed operators on Banach spaces [20] and
quantum Markov chains [23] allows to investigate such scenarios.
MŠ is grateful for financial support from RVO14000 and ”Centre for Advanced
Applied Sciences”, Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778, supported
by the Operational Programme Research, Development and Education, co-financed
by the European Structural and Investment Funds. On a personal note, I want to
express my gratitude to prof. Igor Jex for years of stimulating discussions
and mediation of a myriad of personal contacts with excellent researchers
worldwide. Live long and prosper!
## References
## References
* [1] Y. Aharonov, L. Davidovich, and N. Zagury. Quantum random walks. Phys. Rev. A, 48:1687–1690, 1993.
* [2] D. A. Meyer. From quantum cellular automata to quantum lattice gases. J. Stat. Phys., 85:551–574, 1996.
* [3] E Farhi and S Gutmann. Quantum computation and decision trees. Phys. Rev. A, 58:915–928, 1998.
* [4] R. P. Feynman and A. R. Hibbs. Quantum mechanics and path integrals. International series in pure and applied physics. McGraw-Hill, New York, 1965.
* [5] S. P. Gudder. Quantum graphic dynamics. Found. Phys., 18:751–776, 1988.
* [6] S. P. Gudder. Quantum Probability. Academic Press, New York, 1988.
* [7] K. Kadian, S. Garhwal, and A. Kumar. Quantum walk and its application domains: A systematic review. Comput. Sci. Rev., 41, 2021.
* [8] G. Pólya. Uber eine aufgabe betreffend die irrfahrt im strassennetz. Math. Ann., 84:149–160, 1921.
* [9] M. Kac. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc., 53:1002, 1947.
* [10] M. Štefaňák, I. Jex, and T. Kiss. Recurrence and Pólya number of quantum walks. Phys. Rev. Lett., 100:020501, 2008.
* [11] M. Štefaňák, T. Kiss, and I. Jex. Recurrence properties of unbiased coined quantum walks on infinite d-dimensional lattices. Phys. Rev. A, 78:032306, 2008.
* [12] F. A. Grünbaum, L. Velázquez, A. H. Werner, and R. F. Werner. Recurrence for Discrete Time Unitary Evolutions. Communications in Mathematical Physics, 320:543, 2013.
* [13] J. Bourgain, F. A. Grünbaum, L. Velazquez, and J. Wilkening. Quantum recurrence of a subspace and operator-valued schur functions. Commun. Math. Phys., 329:1031–1067, 2014.
* [14] T. Kiss, L. Kecskés, M. Štefaňák, and I. Jex. Recurrence in coined quantum walks. Phys. Script., T135:014055, 2009.
* [15] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous. One-dimensional quantum walks. In STOC’01 Proceedings of the 33rd annual ACM symposium on Theory of computing, pages 37–49, 2001.
* [16] P. Sinkovicz, Z. Kurucz, T. Kiss, and J. K. Asbóth. Quantized recurrence time in unital iterated open quantum dynamics. Phys. Rev. A, 91:042108, 2015.
* [17] P. Sinkovicz, T. Kiss, and J. K. Asbóth. Generalized Kac lemma for recurrence time in iterated open quantum systems. Phys. Rev. A, 93:050101, 2016.
* [18] C. F. Lardizabal and R. R. Souza. On a Class of Quantum Channels, Open Random Walks and Recurrence. J. Stat. Phys., 159:772, 2015.
* [19] S. L. Carvalho, L. F. Guidi, and C. F. Lardizabal. Site recurrence of open and unitary quantum walks on the line. Quantum Inform. Process., 16:17, 2016.
* [20] F.A. Grünbaum and L. Velázquez. A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks. Adv. Math., 326:352, 2018.
* [21] I. Bardet, H. Bringuier, Y. Pautrat, and C. Pellegrini. Recurrence and Transience of Continuous-Time Open Quantum Walks. In C. DonatiMartin, A. Lejay, and A. Rouault, editors, Seminaire De Probabilites L, volume 2252, page 493. Springer International Publishing Ag, Cham, 2019.
* [22] A. Dhahri and F. Mukhamedov. Open quantum random walks, quantum Markov chains and recurrence. Rev. Math. Phys., 31:1950020, 2019.
* [23] F. A. Grunbaum, C. F. Lardizabal, and L. Velazquez. Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules. Ann. Henri Poincare, 21:189, 2020.
* [24] T. S. Jacq and C. F. Lardizabal. Homogeneous Open Quantum Walks on the Line: Criteria for Site Recurrence and Absorption. Quantum Inform. Comput., 21:37, 2021.
* [25] T. Nitsche, S. Barkhofen, R. Kruse, L. Sansoni, M. Štefaňák, A. Gábris, V. Potoček, T. Kiss, I. Jex, and Ch. Silberhorn. Probing measurement-induced effects in quantum walks via recurrence. Sci. Adv., 4:eaar6444, 2018.
* [26] M. Štefaňák, I. Bezděková, and I. Jex. Continuous deformations of the grover walk preserving localization. Eur. Phys. J. D, 66:142, 2012.
* [27] M. Sabri, E. Segawa, and M. Štefaňák. Conditional limit measure of a one-dimensional quantum walk with an absorbing sink. Phys. Rev. A, 98:012136, 2018.
* [28] M. Štefaňák, I. Bezděková, and I. Jex. Limit distributions of three-state quantum walks: The role of coin eigenstates. Phys. Rev. A, 90:012342, 2014.
* [29] T. Miyazaki, M. Katori, and N. Konno. Wigner formula of rotation matrices and quantum walks. Phys. Rev A, 76:012332, 2007.
* [30] I. Bezděková, M. Štefaňák, and I. Jex. Suitable bases for quantum walks with wigner coins. Phys. Rev. A, 92:022347, 2015.
* [31] B. Kollár, T. Kiss, J. Novotný, and I. Jex. Asymptotic dynamics of coined quantum walks on percolation graphs. Phys. Rev. Lett., 108:230505, 2012.
* [32] B. Kollár, J. Novotný, T. Kiss, and I. Jex. Percolation induced effects in two-dimensional coined quantum walks: analytic asymptotic solutions. New J. Phys., 16:023002, 2014.
* [33] B. Kollár, J. Novotný, T. Kiss, and I. Jex. Discrete time quantum walks on percolation graphs. Eur. Phys. J. Plus, 129:103, 2014.
|
001<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Predictive Sampling:
Real-time Behaviour Synthesis with MuJoCo
Taylor Howell Stanford University DeepMind Nimrod Gileadi DeepMind Saran
Tunyasuvunakool DeepMind Kevin Zakka DeepMind University of California
Berkeley Tom Erez DeepMind Yuval Tassa DeepMind
We introduce MuJoCo MPC (MJPC), an open-source, interactive application and
software framework for real-time predictive control, based on MuJoCo physics.
MJPC allows the user to easily author and solve complex robotics tasks, and
currently supports three shooting-based planners: derivative-based iLQG and
Gradient Descent, and a simple derivative-free method we call Predictive
Sampling. Predictive Sampling was designed as an elementary baseline, mostly
for its pedagogical value, but turned out to be surprisingly competitive with
the more established algorithms. This work does not present algorithmic
advances, and instead, prioritises performant algorithms, simple code, and
accessibility of model-based methods via intuitive and interactive software.
MJPC is available at github.com/deepmind/mujoco_mpc, a video summary can be
viewed at dpmd.ai/mjpc.
## 1 Introduction
Model-based approaches form the foundation of classical control and robotics.
Since Kalman’s seminal work [Kalman, 1960], the _state_ along with its
dynamics and observation models has played a central role.
The classical approach is being challenged by learning-based methods, which
forgo the explicit description of the state and associated models, letting
internal representations emerge from the learning process [Lillicrap et al.,
2015, Schulman et al., 2017, Salimans et al., 2017, Smith et al., 2022, Rudin
et al., 2022]. The flexibility afforded by learned representations makes these
methods powerful and general, but the requirement for large amounts of data
and computation makes them slow. In contrast, pure model-based methods, like
the ones described below, can synthesise behaviour in real time [Tassa et al.,
2014].
Since both approaches ultimately generate behaviour by optimising an
objective, there is reason to believe they can be effectively combined.
Indeed, well-known discrete-domain breakthroughs like AlphaGo [Silver et al.,
2016] are predicated on combining model-based search and learning-based value
and policy approximation. We believe the same could happen for robotics and
control, and describe our thinking on how this might happen in the Discussion
(Section 5). However, before the community can rise to this challenge, a core
problem must be overcome: Model-based optimisation is difficult to implement,
often depends on elaborate optimisation algorithms, and is generally
inaccessible.
To address this deficit, we present MJPC, an open-source interactive
application and software framework for predictive control, based on MuJoCo
physics [Todorov et al., 2012], which lets the user easily author and solve
complex tasks using predictive control algorithms in real time. The tool
offers implementations of standard derivative-based algorithms: iLQG (second-
order planner) and Gradient Descent (first-order planner). Additionally, it
introduces _Predictive Sampling_ , a simple zero-order, sampling-based
algorithm that works surprisingly well and is easy to understand.
Importantly, the interactive simulation can be slowed down asynchronously,
speeding up the planner with respect to simulation time. This means that
behaviours can be generated on older, slower machines, leading to a
democratisation of predictive control tooling.
MJPC and Predictive Sampling advance our central goal of lowering the barriers
to entry for predictive control in robotics research. An important secondary
goal is to accelerate research velocity. When tweaking a parameter, a
researcher should not need to wait hours or minutes, but should receive
instantaneous feedback – which will measurably enhance their own cognitive
performance [Lu and Dosher, 2022]. We believe that flexible, interactive
simulation with researcher-authored graphical user interface is not just a
“nice to have”, but a prerequisite for advanced robotics research.
## 2 Background
In this section we provide a general background on Optimal Control and
Trajectory Optimisation, then focus on Predictive Control and derivative-free
optimisation.
#### Optimal control.
Optimal Control means choosing actions in order to minimise future costs
(equivalently, to maximise future returns). A dynamical system with _state_
${x}\in\mathbf{R}^{n}$, which takes a user-chosen _control_ (or _action_)
${u}\in\mathbf{R}^{m}$, evolves according to the discrete-time111The
continuous-time formulation is generally equivalent, we choose discrete time
for notation simplicity. dynamics:
${y}=f({x},{u}),$ (1)
returning a new state, ${y}\in\mathbf{R}^{n}$. The behaviour of the system is
encoded via the running cost:
${c}({x},{u}),$ (2)
a function of state and control, where explicit time-dependence can be
realised by folding time into the state. Future costs (a.k.a _cost-to-go_ or
_value_), can be defined in several ways. The summation can continue to
infinity, leading to the _average-cost_ or _discounted-cost_ formulations,
favoured in temporal-difference learning, which we discuss in Section 5. Here
we focus on the _finite-horizon_ formulation, whereby the optimisation
objective ${J}$ is given by:
${J}({x}_{0:T},{u}_{0:T})=\sum_{t=0}^{T}{c}({x}_{t},{u}_{t}),$ (3)
where subscripts indicated discrete-time indices.
#### Trajectory optimisation.
Solving the finite-horizon optimal control problem (1, 3), i.e., optimising a
fixed-length trajectory, is commonly known as _planning_ or _trajectory
optimisation_. These algorithms [Von Stryk and Bulirsch, 1992, Betts, 1998]
have a rich history reaching back to the Apollo Program [NASA, 1971, Smith and
Yound, 1967]. An important distinction can be made between two classes of
algorithms:
* •
_Direct_ or _simultaneous_ methods have both states and controls as decision
variables and enforce the dynamics (1) as constraints. These methods (e.g.,
Von Stryk [1993]) specify a large, sparse optimisation problem, which is
usually solved with general-purpose software [Wächter and Biegler, 2006, Gill
et al., 2005]. They have the important benefit that non-physically-realisable
trajectories can be represented, for example in order to clamp a final state,
without initially knowing how to get to it.
* •
_Shooting_ methods like Differential Dynamic Programming [Jacobson and Mayne,
1970] use only the controls ${u}_{0:T}$ as decision variables and enforce the
dynamics via forward simulation. In the shooting approach only physically-
realisable trajectories can be considered, but they benefit from the reduced
search space and from the optimiser not having to enforce the dynamics. The
latter benefit is especially important for stiff systems like those with
contact, where the difference between a physical and non-physical trajectory
can be very small222For example, consider the physical scenario of a free
rigid box lying flat on a plane under gravity, and then consider the non-
physical scenario of the same box hovering above the plane or penetrating it
by a few microns.. Unlike _direct_ methods which require dynamics derivatives,
shooting methods can employ derivative-free optimisation, as discussed below.
#### Predictive control.
Algorithm 1 Predictive Control (asynchronous)
Agent: (repeat)
1:Read the current action ${u}$ from the _nominal_ plan ${\mathbf{\Pi}}$,
apply it to the controlled system.
Planner: (repeat)
1:Measure the current state ${x}$.
2:Using the nominal ${\mathbf{\Pi}}$ to warm-start, optimise the finite-
horizon objective ${J}$.
3:Update ${\mathbf{\Pi}}$.
The key idea of Predictive Control, invented in the late 60s and first
published in [Richalet et al., 1978], is to use trajectory optimisation in
_real-time_ as the system dynamics are evolving. This class of algorithm has
been successfully deployed in numerous real-world settings including: chemical
and nuclear process control [Na et al., 2003, Lopez-Negrete et al., 2013],
navigation for autonomous vehicles [Falcone et al., 2007], and whole-body
control of humanoid robots [Kuindersma et al., 2016].
In the real-time setting, the current state ${x}$ needs to be estimated or
measured, and the trajectory optimiser is required to return a set of optimal
or near-optimal controls for the finite-horizon (here often called the
_receding horizon_) problem, starting at ${x}$. We use ${\mathbf{\Pi}}$ to
denote the _plan_ , the finite-horizon policy. In the context of shooting
methods ${\mathbf{\Pi}}={u}_{0:T}$, though as we discuss later, in some cases
it can be re-parameterised, rather than using the discrete-time control
sequence directly. Predictive control is best thought of in terms of two
asynchronous processes, the _agent_ and the _planner_ , see Algorithm 1.
Predictive Control has the following notable properties:
* •
Faster computation improves performance. The reason for this is clear, the
more optimisation steps the planner can take in one unit of time, the better
the optimised control sequences will be.
* •
Warmstarting has a large beneficial effect. By reusing the plan from the
previous planning step, the optimiser only needs to make small modifications
in order to correct for the changes implied by the new state. Warm-starting
also leads to an amortisation of the optimisation process across multiple
planning steps.
* •
The optimisation is not required to converge, only to improve, see Section
5.1.
* •
Tasks with a behaviour timescale much longer than the planning horizon $T$ are
often still solvable, though this property is task dependent.
* •
Predictive controllers can easily get stuck in local minima, especially those
using derivatives. This is due to the myopic nature of the optimisation, and
can be addressed by terminating the rollout with a value function
approximation, see Section 5.3.
#### Derivative-free optimisation.
One of the main reasons for our initial focus on shooting methods is their
ability to make use of derivative-free optimisation, also known as _sampling-
based_ optimisation [Audet and Hare, 2017]. Despite not leveraging problem
structure or gradient information, this class of algorithms can discover
complex behaviours [Salimans et al., 2017, Mania et al., 2018].
Sampling-based methods maintain a search distribution over policy parameters
and evaluate the objective at sampled points in order to find an improved
solution. Popular algorithms include: random search [Matyas, 1965], genetic
algorithms [Holland, 1992], and evolutionary strategies [Rechenberg, 1973],
including CMA-ES [Hansen and Ostermeier, 2001].
This class of algorithms has a number of desirable properties. First, because
derivative information is not required, they are well-suited for tasks with
non-smooth and discontinuous dynamics. Second, these methods are trivially
parallelisable.
Sampling-based methods have been used in the predictive control context, but
not very widely. Notable exceptions are Hämäläinen’s work [Hämäläinen et al.,
2014, 2015], which is an early precursor to MJPC, and the oeuvre of Theodorou
including [Williams et al., 2017] and related papers. While these methods are
usually considered to be sample inefficient, we explain below why,
specifically in the predictive control context, they can be surprisingly
competitive.
## 3 MuJoCo MPC (MJPC)
We introduce MJPC, an open-source interactive application and software
framework for predictive control, that lets the user easily synthesise
behaviours for complex systems using predictive control algorithms in real
time. Behaviours are specified by simple, composable objectives that are risk-
aware. The planners, including: Gradient Descent, Iterative Linear Quadratic
Gaussian (iLQG), and Predictive Sampling are implemented in C++ and
extensively utilise multi-threading for parallel rollouts. The framework is
asynchronous, enabling simulation slow-down and emulation of a faster
controller, allowing this tool to run on slow machines. An intuitive graphical
user-interface enables real-time interactions with the environment and the
ability to modify task parameters, planner and model settings, and to
instantly see the effects of the modifications. The tool is available at:
https://github.com/deepmind/mujoco_mpc.
### 3.1 Physics Simulation
We build MJPC using the API of the open-source physics engine MuJoCo [Todorov
et al., 2012]. MuJoCo is a good infrastructure for an interactive framework
for robotics algorithms for two main reasons: first, MuJoCo supports
simulating multiple candidate future trajectories in parallel by offering a
thread-safe API, which maximises the utilisation of modern multi-core CPU
architectures; second, MuJoCo affords faster-than-realtime simulation of high-
dimensional systems with many contacts — for example, the humanoid (a 27-DoF
system) can be simulated 4000 times faster than realtime on a single CPU
thread.
### 3.2 Objective
MJPC provides convenient utilities to easily design and compose costs in order
to specify an objective (3); as well as automatically and efficiently compute
derivatives.
#### Costs.
We use a “base cost” of the form:
${l}({x},{u})=\sum_{i=0}^{M}w_{i}\cdot\text{n}_{i}\big{(}{r}_{i}({x},{u})\big{)}.$
(4)
This cost is a sum of $M$ terms, each comprising:
* •
A nonnegative weight $w\in\mathbf{R}_{+}$ determining the relative importance
of this term.
* •
A twice-differentiable norm
$\text{n}(\cdot):\mathbf{R}^{p}\rightarrow\mathbf{R}_{+}$, which takes its
minimum at $0^{p}$.
* •
The residual ${r}\in\mathbf{R}^{p}$ is a vector of elements that are “small
when the task is solved”.
#### Risk sensitivity.
Figure 1: Risk transformation $\rho(l;R)$. The function is evaluated between
$0$ and $1$ for different values of the risk parameter $R$.
We augment the base cost (4) with a risk-aware exponential scalar
transformation, $\rho:\mathbf{R}_{+}\times\mathbf{R}\rightarrow\mathbf{R}$,
corresponding to the classical risk-sensitive control framework [Jacobson,
1973, Whittle, 1981]. The final running cost ${c}$ is given by:
${c}({x},{u})=\rho({l}({x},{u});R)=\frac{e^{R\cdot{l}({x},{u})}-1}{R}$ (5)
The scalar parameter $R\in\mathbf{R}$ denotes risk-sensitivity. $R=0$ (the
default) is interpreted as risk-neutral, $R>0$ as risk-averse, and $R<0$ as
risk-seeking. The mapping $\rho$ (see Figure 1) has the following properties:
* •
Defined and smooth for any $R$.
* •
If $R=0$, is the identity $\rho({l};0)={l}$ (in the limit).
* •
Zero is a fixed point: $\rho(0;R)=0$.
* •
The derivative at 0 is 1: $\partial\rho(0;R)/\partial{l}=1$.
* •
Non-negative: If ${l}\geq 0$ then $\rho({l};R)\geq 0$.
* •
Monotonic: $\rho({l};R)>\rho(z;R)$ if ${l}>z$.
* •
If $R<0$ then $\rho$ is bounded: $\rho({l};R)<-\frac{1}{R}$.
* •
$\rho({l};R)$ carries the same units as ${l}$.
Note that for negative $R$, the transformation $\rho$ creates costs that are
similar to the bounded rewards commonly used in reinforcement learning. For
example, when using the quadratic norm $\text{n}(r)=r^{T}Wr$ for some SPD
matrix $W=\Sigma^{-1}$ and a risk parameter $R=-1$, we get an inverted-
Gaussian cost ${c}=1-e^{-r^{T}\Sigma^{-1}r}$, whose minimisation is equivalent
to maximum-likelihood maximisation of the Gaussian. This leads to the
interesting interpretation of the bounded rewards commonly used in RL as
_risk-seeking_. We do not investigate this relationship further in this paper.
#### Derivatives.
MuJoCo provides a utility for computing finite-difference (FD) Jacobians of
the dynamics, which is efficient in two ways. First, by avoiding re-
computation where possible, for example when differencing w.r.t. controls,
quantities that depend only on positions and velocities are not recomputed.
Second, because FD computational costs scale with the dimension of the _input_
, outputs can be added cheaply. MuJoCo’s step function ${y},{r}=f({x},{u})$,
computes both the next state ${y}$ and sensor values $r$, defined in the
model. Because the FD approximation of the Jacobians,
$\frac{\partial{y}}{\partial{x}},\frac{\partial{y}}{\partial{u}},\frac{\partial{r}}{\partial{x}},\frac{\partial{r}}{\partial{u}},$
(6)
scales like the combined dimension of ${x}$ and ${u}$, adding more sensors
${r}$ is effectively “free”. MJPC automatically and efficiently computes cost
derivatives as follows.
#### Gradients.
Cost gradients are computed with:
$\displaystyle\frac{\partial{c}}{\partial{x}}$
$\displaystyle=e^{R{l}}\frac{\partial{l}}{\partial{x}}=e^{R{l}}\sum\limits_{i=0}^{M}w_{i}\frac{\partial\text{n}_{i}}{\partial
r}\frac{\partial r_{i}}{\partial{x}},$ (7a)
$\displaystyle\frac{\partial{c}}{\partial{u}}$
$\displaystyle=e^{R{l}}\frac{\partial{l}}{\partial{u}}=e^{R{l}}\sum\limits_{i=0}^{M}w_{i}\frac{\partial\text{n}_{i}}{\partial
r}\frac{\partial r_{i}}{\partial{u}}.$ (7b)
The norm gradients, $\partial\text{n}/\partial r$, are computed analytically.
#### Hessians.
Second-order derivatives use the Gauss-Newton approximation, ignoring second
derivatives of ${r}$:
$\displaystyle\frac{\partial^{2}{c}}{\partial{x}^{2}}\approx$
$\displaystyle\,e^{R{l}}\left[\sum\limits_{i=0}^{M}w_{i}{\frac{\partial
r_{i}}{\partial{x}}}^{T}\frac{\partial^{2}\text{n}_{i}}{\partial
r^{2}}\frac{\partial
r_{i}}{\partial{x}}+R{\frac{\partial{l}}{\partial{x}}}^{T}\frac{\partial{l}}{\partial{x}}\right],$
(8a) $\displaystyle\frac{\partial^{2}{c}}{\partial{u}^{2}}\approx$
$\displaystyle\,e^{R{l}}\left[\sum\limits_{i=0}^{M}w_{i}{\frac{\partial
r_{i}}{\partial{u}}}^{T}\frac{\partial^{2}\text{n}_{i}}{\partial
r^{2}}\frac{\partial
r_{i}}{\partial{u}}+R{\frac{\partial{l}}{\partial{u}}}^{T}\frac{\partial{l}}{\partial{u}}\right],$
(8b) $\displaystyle\frac{\partial^{2}{c}}{\partial{x}\partial{u}}\approx$
$\displaystyle\,e^{R{l}}\left[\sum\limits_{i=0}^{M}w_{i}{\frac{\partial
r_{i}}{\partial{x}}}^{T}\frac{\partial^{2}\text{n}_{i}}{\partial
r^{2}}\frac{\partial
r_{i}}{\partial{u}}+R{\frac{\partial{l}}{\partial{x}}}^{T}\frac{\partial{l}}{\partial{u}}\right].$
(8c)
The norm Hessians, $\partial^{2}\text{n}/\partial r^{2}$, are computed
analytically.
### 3.3 Splines
Figure 2: Time-indexed spline representation of the controls. Parameter points
(black) utilised to construct: zero (magenta), linear (orange), and cubic
(blue) interpolants.
As we explain below, planners like iLQG require the _direct_ control-sequence
representation ${u}_{0:T}$ due to the requirements of the Bellman Principle.
Without this constraint, controls can be “compressed” into a lower-dimensional
object. There are many ways to do this, we picked the simplest: splines.
Action trajectories are represented as a time-indexed set of knots, or
control-points, parameterised by a sequence of monotonic time points
$\tau_{0:P}$ and parameter values $\theta_{0:P}$, where we use the shorthand
$\theta=\theta_{0:P}$. Given a query point $\tau$, the evaluation of the
spline is given by:
${u}=s\big{(}\tau;(\tau_{0:P},\theta)\big{)}.$ (9)
We provide three spline implementations: traditional cubic Hermite splines,
piecewise-linear interpolation, and zero-order hold. See (Fig. 2) for an
illustration.
The main benefit of compressed representations like splines is that they
reduce the search space. They also smooth the control trajectory, which is
often desirable. Spline functions belong to the class of linear bases, which
includes the Fourier basis and orthogonal polynomials. These are useful
because they allow easy propagation of gradients from the direct
representation $\partial{\mathbf{\Pi}}$ back to the parameter values
$\partial\theta$. In our case this amounts to computing:
$\frac{\partial s}{\partial\theta},$ (10)
which has a simple analytic formula (see code for details). Unlike other
linear bases, splines have the convenient property that bounding the values
$\theta$ also bounds the spline trajectory. This is exactly true for the zero
and linear interpolations, and mostly-true for cubic splines. Bounding is
important as most physical systems clamp controls to bounds, and there is no
point searching outside of them. Expressions for cubic, linear and zero
interpolations are provided in Appendix A.
### 3.4 Planners
MJPC includes two derivative-based planners.
#### iLQG.
Algorithm 2 iLQG
1:initial state ${x}_{0}$, nominal plan ${\mathbf{\Pi}}={u}_{0:T}$
2:Roll out nominal trajectory from ${x}_{0}$ using ${\mathbf{\Pi}}$
3:Compute action improvements and feedback policy using Dynamic Programming.
4:Roll out parallel line search with feedback policy (11).
5:Best actions are new nominal actions.
The iLQG333Equivalently, “iLQR”, since we don’t make use of the noise-
sensitive term for which iLQG was originally developed [Li and Todorov, 2004].
We keep the name “iLQG” due to its provenance. planner [Tassa et al., 2012], a
Gauss-Newton approximation of the DDP algorithm [Jacobson and Mayne, 1970],
utilises first- and second-order derivative information to take an approximate
Newton step over the open-loop control sequence ${u}_{0:T}$ via dynamic
programming [Kalman, 1964], producing a time-varying linear feedback policy:
${u}_{t}=\bar{{u}}_{t}+K_{t}({x}_{t}-\bar{{x}}_{t})+\alpha k_{t}.$ (11)
The _nominal_ , or current best trajectory, is denoted with overbars
($\bar{\phantom{x}}$), $K$ is a feedback gain matrix, and $k$ is an
improvement to the current action trajectory. A parallel line search over the
step size $\alpha\in[\alpha_{\textrm{min}},1]$ is performed to find the best
improvement. Additional enhancements include a constrained backward pass
[Tassa et al., 2014] that enforces action limits and adaptive regularisation.
The details of iLQG are too involved to restate here, we refer the reader to
the references above for details.
#### Gradient descent.
Algorithm 3 Gradient Descent
1:initial state ${x}_{0}$, nominal plan ${\mathbf{\Pi}}(\theta)$
2:Roll out nominal from ${x}_{0}$ using ${\mathbf{\Pi}}(\theta)$
3:Compute $\partial{J}/\partial{\mathbf{\Pi}}$ with (14)
4:Compute $\partial{J}/\partial\theta$ with (13)
5:Roll out parallel line-search with (12)
6:Pick the best one:
$\theta\leftarrow\mbox{argmin}\bigl{(}{J}(\theta^{(i)})\bigr{)}$
This first-order planner, known as Pontryagin’s Maximum Principle
[Mangasarian, 1966], utilises gradient information to improve action
sequences, here represented as splines. The gradient of the total return is
used to update the spline parameters, using a parallel line search over the
step size $\alpha\in[\alpha_{\textrm{min}},\alpha_{\textrm{max}}]$:
$\theta\leftarrow\theta-\alpha\frac{\partial J}{\partial\theta}.$ (12)
The total gradient is given by:
$\frac{\partial J}{\partial\theta}=\frac{\partial
J}{\partial{\mathbf{\Pi}}}\frac{\partial{\mathbf{\Pi}}}{\partial\theta},$ (13)
where the spline gradient $\partial{\mathbf{\Pi}}/\partial\theta$ is given by
(10), while $\partial J/\partial{\mathbf{\Pi}}$ is computed with the Maximum
Principle. Letting $\lambda$ denote the _co-state_ , the gradients with
respect to ${u}$ are given by:
$\displaystyle\lambda_{j}$ $\displaystyle=\frac{\partial
c}{\partial{x}_{j}}+\Bigl{(}\frac{\partial
f}{\partial{x}_{j}}\Bigr{)}^{T}\lambda_{j+1},$ (14a)
$\displaystyle\frac{\partial{J}}{\partial{u}_{j}}$
$\displaystyle=\frac{\partial c}{\partial{u}_{j}}+\Bigl{(}\frac{\partial
f}{\partial{u}_{j}}\Bigr{)}^{T}\lambda_{j+1}.$ (14b)
The primary advantage of this first-order method compared to a computationally
more expensive method like iLQG is that optimisation is performed over the
smaller space of spline parameters, instead of the entire (non-parametric)
sequences of actions.
#### Predictive Sampling.
Algorithm 4 Predictive Sampling
Parameters: $N$ rollouts, noise scale $\sigma$
1:initial state ${x}_{0}$, nominal plan ${\mathbf{\Pi}}(\theta)$
2:Get $N\\!-\\!1$ samples $\theta_{i}\sim\mathcal{N}(\theta,\sigma^{2})$
3:Including $\theta$, roll out all $N$ samples from ${x}_{0}$
4:Pick the best one:
$\theta\leftarrow\mbox{argmin}\bigl{(}{J}(\theta^{(i)})\bigr{)}$
This is a trivial, zero-order, sampling-based Predictive Control method that
works well and is easy to understand. Designed as an elementary baseline, this
algorithm turned out to be surprisingly competitive with the more elaborate
derivative-based algorithms.
#### Algorithm.
A nominal sequence of actions, represented with spline parameters, is
iteratively improved using random search [Matyas, 1965]. At each iteration,
$N$ candidate splines are evaluated: the nominal itself and $N-1$ noisy
samples from a Gaussian with the nominal as its mean and fixed standard
deviation $\sigma$. After sampling, the actions are clamped to the control
limits by clamping the spline parameters $\theta$. Each candidate’s total
return is evaluated and the nominal is updated with the best candidate. See
Algorithm 4 and pseudocode in Appendix C. Predictive Sampling is not
innovative or performant, but is presented as a simple baseline, see
Discussion below.
## 4 Results
We provide a short textual description of our graphical user interface (GUI)
for three example tasks. They are best understood by viewing the associated
video at dpmd.ai/mjpc or better yet, by downloading the software and
interacting with it.
### 4.1 Graphical User Interface
Figure 3: Graphical User Interface. The left tab includes modules for Tasks
and the Agent. In the Task module, models are selected from a drop-down menu
and the risk value is set; cost weights and residual parameters are specified
with interactive sliders. The Agent module provides access to a drop-down menu
for planners, settings, and switches that toggle the planner and controller.
The right tab includes simulation readings and predictions for the cost,
including individual terms, and actions. Live, planner-specific information
and various compute time results are shown below. Traces of the current policy
and sampled trajectories are visualised and the user can interactively pause
or slow down the simulation and apply external forces and moments to the
system.
The MJPC GUI, shown and described in Fig. 3, provides an interactive
simulation environment, as well as modules containing live plots and
parameters that can be set by the researcher. The intuitive interface makes
policy design easy by enabling the researcher to interactively change cost
parameters or planner settings and immediately see the results in both the
simulation environment and live plots, allowing fast debugging and an enhanced
understanding of the factors that influence behaviour.
### 4.2 Examples
(a) Humanoid standing up off the floor.
(b) Quadruped rolling off its back to stand up.
(c) Hand manipulating a cube to a goal orientation.
Figure 4: Behaviours generated with MuJoCo MPC. Time progresses left to right.
In the following examples, we demonstrate the ability to synthesise complex
locomotion and manipulation behaviours for a variety of high-dimensional
systems in simulation on a single CPU. Further, we demonstrate that the
behaviours are robust to disturbances and mismatch between the simulation and
planning model, and can adapt extremely quickly in new scenarios. For all of
the examples, the total planning time for a single update is between 1 and 20
milliseconds. We highlight three examples below and provide additional
examples with the software. Experimental details for objectives and planner
settings are found in the Appendix B.
#### Humanoid.
This 27-DOF human-like system, from DeepMind Control Suite [Tunyasuvunakool et
al., 2020], has 21 actions and is tasked with standing. The system can be
initialised on the floor and quickly stands in a manner that is robust to
large disturbances. If a sufficiently large disturbance knocks the humanoid
onto the floor, the system will stand back up (Fig. 4(a)).
#### Quadruped.
A Unitree A1 quadruped [Unitree, 2022], from MuJoCo Menagerie [MuJoCo
Menagerie Contributors, 2022], exhibits agile behaviour to traverse uneven
terrain which includes walking over a steep slope. On slower machines, the
quadruped often struggles to ascend. In this scenario, the simulation slow
down can be effectively utilised to provide the planner with addition
simulation time to plan a successful climb. The system is also capable of
rolling off its back and standing up (Fig. 4(b)). In order to perform long-
horizon tasks like continuously navigating the terrain, a series of target
poses are set. Once a goal is reached, an automatic transition occurs and the
next target is set.
#### Hand.
A Shadow Hand [Tuffield and Elias, 2003], also from MuJoCo Menagerie, performs
in-hand manipulation of a cube to a desired orientation (Fig. 4(c)), where
this goal can be set by the researcher in real-time by interactively setting
the target orientation. In-hand reorientation—a high-DoF system with complex
contact dynamics—is considered difficult to solve [Chen et al., 2022] and to
the best of our knowledge has not previously been solved from scratch, in real
time.
## 5 Discussion
The thrust of this paper is to make predictive control accessible via
customisable, interactive, open-source tooling. We believe that responsive,
GUI-based tools are a prerequisite for accelerated robotics research, and that
due to their importance, these tools should be modifiable and the inner
workings transparent to the researcher. We hope that our MJPC project will be
embraced by the community, and look forward to improving and extending it
together.
### 5.1 Predictive Sampling
The effectiveness of this simple method suggests that fast, approximate
optimisation can be competitive with more sophisticated methods which return
better solutions but at a lower rate. Does the higher planning rate completely
explain this surprising effectiveness? We believe there is another, subtler
reason. Predictive Control is not well-described by the tenets of traditional
optimisation. For example, it usually makes no sense to take more than one
step of optimisation. Once a single iteration is complete, it is more
important to measure a new value of the state and re-plan, than it is to
continue to converge to the minimum of an already-outdated problem. The
constant shifting of the optimisation landscape makes Predictive Control a
_qualitatively different problem_ , more like surfing than mountain climbing.
The goal is not to find the minimum, but to _remain in the basin-of-attraction
of the minimum_. This is a different, weaker criterion, at which simple
algorithms fair better than when measured by the traditional yardstick of
convergence.
To be clear, Predictive Sampling is not a novel algorithm; instead, it is a
baseline. A corner-case of many existing methods, it can variously be
described as “MPPI with infinite temperature”, “CEM with a non-adaptive
distribution” or just “trivial random search”. Better algorithms exist, but
none are so easy to describe or implement. We are introducing Predictive
Sampling not because it is good, but because it is _not good_. It is the
simplest possible sampling-based shooting method, and therefore establishes a
_lower bound_ for performance baselines.
### 5.2 Use cases
Before we discuss limitations and their possible resolutions, it is worth
asking how can MJPC be used _now_ , as it is described above?
1. 1.
Task design. MJPC makes it easy to add new tasks, expose task parameters to
the GUI, and quickly generate the desired behaviour. The task can then be re-
implemented in any other framework of choice. While we have not yet
implemented time-dependent tasks, it is possible and easy; we expect MJPC to
work especially well for motion-tracking tasks.
2. 2.
Data generation. MJPC can be used to generate data for learning-based
approaches, i.e., it can act like an “expert policy”. In this context, it is
often the case that the model and task from which the data is generated do not
have to exactly match the one used by the learner, and the data can likely be
useful for a wide range of setups.
3. 3.
Predictive Control research. For researchers interested in Predictive Control
itself, MJPC provides an ideal playground. MJPC can switch planners on-the-
fly, and its asynchronous design affords a fair comparison by correctly
accounting for and rewarding faster planners.
### 5.3 Limitation and Future Work
#### Can only control what MuJoCo can simulate.
This is a general limitation of Predictive Control and is in fact stronger
since one can only control what can be simulated _much faster than real-time_.
For example, it is difficult to imagine any simulation of a very-high-DoF
system, like fluid, cloth or soft bodies advancing so fast. One solution is
improved simulation using a combination of traditional physics modeling and
learning, e.g., [Ladicky et al., 2017]. Another possibility is to entirely
learn the dynamics from observations. This approach, often termed Model Based
Reinforcement Learning is showing great promise [Heess et al., 2015, Nagabandi
et al., 2020, Wu et al., 2022]. We would recommend that where possible, when
attempting Predictive Control using learned dynamics models, a traditional
simulator be employed as a fallback as in [Schrittwieser et al., 2020], to
disambiguate the effects of modeling errors.
#### Myopic.
The core limitation of Predictive Control is that it is _myopic_ and cannot
see past the fixed horizon. This can be ameliorated in three conceptually
straightforward ways:
1. 1.
Learned policies. By adding a learned policy, information from past episodes
can propagate to the present via policy generalisation [Byravan et al., 2021].
This approach is attractive since it can only _improve_ performance: when
rolling out samples, one also rolls out the proposal policy. If the rollout is
better, it becomes the new nominal. A learned policy is also expected to lead
to more stereotypical, periodic behaviours, which are important in locomotion.
2. 2.
Value functions. Terminating the rollout with a learned value function which
estimates the remaining cost-to-go is the obvious way by which to increase the
effective horizon. Combining learned policies and value functions with model-
based search would amount to an “AlphaGo for control” [Silver et al., 2016,
Springenberg et al., 2020].
3. 3.
High-level agent. A predictive controller could be used as the low-level
module in a hierarchical control setup. In this scenario, the actions of the
high-level agent have the semantics of setting the cost function of the
predictive controller. The predictive controller remains myopic while the
high-level agent contains the long-horizon “cognitive” aspects of the task. A
benefit of this scenario is that the high-level actions have a much lower
frequency than required for low-level control (e.g., torques).
#### Hardware.
MJPC is aimed at robotics research, which raises the question, can it be used
to control hardware?
1. 1.
Transfer learning. As mentioned in 5.2, using MJPC to generate data which can
then be transferred to a real robot is already possible.
2. 2.
Estimation. The most obvious yet difficult route to controlling hardware is to
follow in the footsteps of classic control and couple MJPC to an estimator
providing real-time state estimates. In the rare cases where estimation is
easy, for example with fixed-base manipulators and static objects, controlling
a robot directly with MJPC would be a straightforward exercise. The difficult
and interesting case involves free-moving bodies and contacts, as in
locomotion and manipulation. For certain specific cases, like locomotion on
flat, uniform terrain, reasonable estimates should not be difficult to obtain.
For the general case, we believe that contact-aware estimation is possible
following the approach of [Lowrey et al., 2014], but that remains to be seen.
Relatedly, we believe that high-quality estimators also require the same kind
of interactive, GUI-driven interface used by MJPC.
## References
* Audet and Hare [2017] Charles Audet and Warren Hare. _Derivative-Free and Blackbox Optimization_ , volume 2. Springer, 2017.
* Betts [1998] John T. Betts. Survey of numerical methods for trajectory optimization. _Journal of Guidance, Control, and Dynamics_ , 21(2):193–207, 1998.
* Byravan et al. [2021] Arunkumar Byravan, Leonard Hasenclever, Piotr Trochim, Mehdi Mirza, Alessandro Davide Ialongo, Yuval Tassa, Jost Tobias Springenberg, Abbas Abdolmaleki, Nicolas Heess, Josh Merel, et al. Evaluating model-based planning and planner amortization for continuous control. _arXiv:2110.03363_ , 2021.
* Chen et al. [2022] Tao Chen, Jie Xu, and Pulkit Agrawal. A system for general in-hand object re-orientation. In _Conference on Robot Learning_ , pages 297–307, 2022.
* Falcone et al. [2007] Paolo Falcone, Francesco Borrelli, Jahan Asgari, Hongtei Eric Tseng, and Davor Hrovat. Predictive active steering control for autonomous vehicle systems. _IEEE Transactions on Control Systems Technology_ , 15(3):566–580, 2007.
* Gill et al. [2005] Philip E. Gill, Walter Murray, and Michael A. Saunders. SNOPT: An SQP algorithm for large-scale constrained optimization. _SIAM Review_ , 47(1):99–131, 2005.
* Hämäläinen et al. [2014] Perttu Hämäläinen, Sebastian Eriksson, Esa Tanskanen, Ville Kyrki, and Jaakko Lehtinen. Online motion synthesis using sequential Monte Carlo. _ACM Transactions on Graphics_ , 2014.
* Hämäläinen et al. [2015] Perttu Hämäläinen, Joose Rajamäki, and C. Karen Liu. Online control of simulated humanoids using particle belief propagation. _ACM Transactions on Graphics_ , 2015.
* Hansen and Ostermeier [2001] Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self-adaptation in evolution strategies. _Evolutionary Computation_ , 9(2):159–195, 2001\.
* Heess et al. [2015] Nicolas Heess, Gregory Wayne, David Silver, Timothy Lillicrap, Tom Erez, and Yuval Tassa. Learning continuous control policies by stochastic value gradients. _Advances in Neural Information Processing Systems_ , 28, 2015.
* Holland [1992] John H. Holland. Genetic algorithms. _Scientific American_ , 267(1):66–73, 1992.
* Jacobson [1973] David Jacobson. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games. _IEEE Transactions on Automatic Control_ , 18(2):124–131, 1973.
* Jacobson and Mayne [1970] David H. Jacobson and David Q. Mayne. _Differential Dynamic Programming_. Number 24. Elsevier Publishing Company, 1970.
* Kalman [1960] Rudolf E. Kalman. On the general theory of control systems. In _International Conference on Automatic Control_ , pages 481–492, 1960.
* Kalman [1964] Rudolf E. Kalman. When Is a Linear Control System Optimal? _Journal of Basic Engineering_ , 86(1):51–60, 1964.
* Kuindersma et al. [2016] Scott Kuindersma, Robin Deits, Maurice Fallon, Andrés Valenzuela, Hongkai Dai, Frank Permenter, Twan Koolen, Pat Marion, and Russ Tedrake. Optimization-based locomotion planning, estimation, and control design for the Atlas humanoid robot. _Autonomous Robots_ , 40(3):429–455, 2016.
* Ladicky et al. [2017] Lubor Ladicky, SoHyeon Jeong, Nemanja Bartolovic, Marc Pollefeys, and Markus Gross. PhysicsForests: Real-time fluid simulation using machine learning. In _ACM SIGGRAPH Real Time Live!_ , page 22, 2017. ISBN 9781450350907.
* Li and Todorov [2004] Weiwei Li and Emanuel Todorov. Iterative linear quadratic regulator design for nonlinear biological movement systems. In _International Conference on Informatics in Control, Automation and Robotics_ , pages 222–229, 2004.
* Lillicrap et al. [2015] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. _arXiv:1509.02971_ , 2015.
* Lopez-Negrete et al. [2013] Rodrigo Lopez-Negrete, Fernando J. D’Amato, Lorenz T. Biegler, and Aditya Kumar. Fast nonlinear model predictive control: Formulation and industrial process applications. _Computers & Chemical Engineering_, 51:55–64, 2013.
* Lowrey et al. [2014] Kendall Lowrey, Svetoslav Kolev, Yuval Tassa, Tom Erez, and Emanuel Todorov. Physically-consistent sensor fusion in contact-rich behaviors. In _IEEE/RSJ International Conference on Intelligent Robots and Systems_ , pages 1656–1662, 2014.
* Lu and Dosher [2022] Zhong-Lin Lu and Barbara Anne Dosher. Current directions in visual perceptual learning. _Nature Reviews Psychology_ , 1(11):654–668, 2022\.
* Mangasarian [1966] O. L. Mangasarian. Sufficient conditions for the optimal control of nonlinear systems. _SIAM Journal on Control_ , 4(1):139–152, 1966\.
* Mania et al. [2018] Horia Mania, Aurelia Guy, and Benjamin Recht. Simple random search of static linear policies is competitive for reinforcement learning. In _Advances in Neural Information Processing Systems_ , pages 1800–1809, 2018.
* Matyas [1965] J. Matyas. Random optimization. _Automation and Remote Control_ , 26(2):246–253, 1965.
* MuJoCo Menagerie Contributors [2022] MuJoCo Menagerie Contributors. MuJoCo Menagerie: A collection of high-quality simulation models for MuJoCo, 2022. URL http://github.com/deepmind/mujoco_menagerie.
* Na et al. [2003] Man Gyun Na, Sun Ho Shin, and Whee Cheol Kim. A model predictive controller for nuclear reactor power. _Nuclear Engineering and Technology_ , 35(5):399–411, 2003.
* Nagabandi et al. [2020] Anusha Nagabandi, Kurt Konolige, Sergey Levine, and Vikash Kumar. Deep dynamics models for learning dexterous manipulation. In _Conference on Robot Learning_ , pages 1101–1112, 2020.
* NASA [1971] NASA. _Apollo 11 Mission Report_ , volume 238. Scientific and Technical Information Office, National Aeronautics and Space Administration, 1971.
* Rechenberg [1973] Ingo Rechenberg. Evolutionsstrategie. _Optimierung technischer Systeme nach Prinzipien derbiologischen Evolution_ , 1973.
* Richalet et al. [1978] Jacques Richalet, André Rault, J. L. Testud, and J. Papon. Model predictive heuristic control: Applications to industrial processes. _Automatica_ , 14(5):413–428, 1978.
* Rudin et al. [2022] Nikita Rudin, David Hoeller, Philipp Reist, and Marco Hutter. Learning to walk in minutes using massively parallel deep reinforcement learning. In _Conference on Robot Learning_ , pages 91–100, 2022.
* Salimans et al. [2017] Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever. Evolution strategies as a scalable alternative to reinforcement learning. _arXiv:1703.03864_ , 2017.
* Schrittwieser et al. [2020] Julian Schrittwieser, Ioannis Antonoglou, Thomas Hubert, Karen Simonyan, Laurent Sifre, Simon Schmitt, Arthur Guez, Edward Lockhart, Demis Hassabis, Thore Graepel, Timothy Lillicrap, and David Silver. Mastering Atari, Go, chess and shogi by planning with a learned model. _Nature_ , 588(7839):604–609, 2020.
* Schulman et al. [2017] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. _arXiv:1707.06347_ , 2017.
* Silver et al. [2016] David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. Mastering the game of Go with deep neural networks and tree search. _Nature_ , 529(7587):484–489, 2016.
* Smith et al. [2022] Laura Smith, Ilya Kostrikov, and Sergey Levine. A walk in the park: Learning to walk in 20 minutes with model-free reinforcement learning. _arXiv:2208.07860_ , 2022.
* Smith and Yound [1967] R. E. Smith and J. W. Yound. Trajectory optimization for an Apollo-type vehicle under entry conditions encountered during lunar return. Technical report, 1967.
* Springenberg et al. [2020] Jost Tobias Springenberg, Nicolas Heess, Daniel Mankowitz, Josh Merel, Arunkumar Byravan, Abbas Abdolmaleki, Jackie Kay, Jonas Degrave, Julian Schrittwieser, Yuval Tassa, et al. Local search for policy iteration in continuous control. _arXiv:2010.05545_ , 2020.
* Tassa et al. [2012] Yuval Tassa, Tom Erez, and Emanuel Todorov. Synthesis and stabilization of complex behaviors through online trajectory optimization. In _IEEE/RSJ International Conference on Intelligent Robots and Systems_ , pages 4906–4913, 2012.
* Tassa et al. [2014] Yuval Tassa, Nicolas Mansard, and Emo Todorov. Control-limited differential dynamic programming. In _IEEE International Conference on Robotics and Automation_ , pages 1168–1175, 2014.
* Todorov et al. [2012] Emanuel Todorov, Tom Erez, and Yuval Tassa. MuJoCo: A physics engine for model-based control. In _IEEE/RSJ International Conference on Intelligent Robots and Systems_ , pages 5026–5033, 2012.
* Tuffield and Elias [2003] Paul Tuffield and Hugo Elias. The Shadow robot mimics human actions. _Industrial Robot: An International Journal_ , 2003.
* Tunyasuvunakool et al. [2020] Saran Tunyasuvunakool, Alistair Muldal, Yotam Doron, Siqi Liu, Steven Bohez, Josh Merel, Tom Erez, Timothy Lillicrap, Nicolas Heess, and Yuval Tassa. dm_control: Software and tasks for continuous control. _Software Impacts_ , 6:100022, 2020.
* Unitree [2022] Unitree. Unitree A1 Quadruped. https://m.unitree.com/products/a1, 2022.
* Von Stryk [1993] Oskar Von Stryk. Numerical solution of optimal control problems by direct collocation. In _Optimal Control_ , pages 129–143. Springer, 1993.
* Von Stryk and Bulirsch [1992] Oskar Von Stryk and Roland Bulirsch. Direct and indirect methods for trajectory optimization. _Annals of Operations Research_ , 37(1):357–373, 1992.
* Wächter and Biegler [2006] Andreas Wächter and Lorenz T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. _Mathematical Programming_ , 106(1):25–57, 2006\.
* Whittle [1981] Peter Whittle. Risk-sensitive linear/quadratic/Gaussian control. _Advances in Applied Probability_ , 13(4):764–777, 1981.
* Williams et al. [2017] Grady Williams, Andrew Aldrich, and Evangelos A. Theodorou. Model predictive path integral control: From theory to parallel computation. _Journal of Guidance, Control, and Dynamics_ , 40(2):344–357, 2017.
* Wu et al. [2022] Philipp Wu, Alejandro Escontrela, Danijar Hafner, Ken Goldberg, and Pieter Abbeel. DayDreamer: World models for physical robot learning. _arXiv:2206.14176_ , 2022.
## Appendix A Interpolation
This section provides analytical expressions for computing interpolations. The
indices $j$ and $j+1$ correspond to the indices of the domain variables which
contain the query point $\tau$. These values are efficiently found using
binary search in $\mathbf{O}\left(\text{log}(n)\right)$ where $n$ is the
dimension of the domain variable set.
#### Zero.
The zero-order interpolation simply returns the parameter values at the lower
bound index:
$\theta_{j}\leftarrow s.$ (15)
#### Linear.
The linear interpolation returns:
$(1-q)\cdot\theta_{j}+q\cdot\theta_{j+1}\leftarrow s,$ (16)
where $\quad q=(\tau-\tau_{j})/(\tau_{j+1}-\tau_{j})$.
#### Cubic.
The cubic interpolation leverages finite-difference approximations of the
slope at the interval points:
$\displaystyle\phi_{j}$
$\displaystyle=\frac{1}{2}\left(\frac{\theta_{j+1}-\theta_{j}}{\tau_{j+1}-\tau_{j}}+\frac{\theta_{j}-\theta_{j-1}}{\tau_{j}-\tau_{j-1}}\right),$
(17) $\displaystyle\phi_{j+1}$
$\displaystyle=\frac{1}{2}\left(\frac{\theta_{j+2}-\theta_{j+1}}{\tau_{j+2}-\tau_{j+1}}+\frac{\theta_{j+1}-\theta_{j}}{\tau_{j+1}-\tau_{j}}\right),$
(18)
and returns:
$a\cdot\theta_{j}+b\cdot\phi_{j}+c\cdot\theta_{j+1}+d\cdot\phi_{j+1}\leftarrow
s,$ (19)
where,
$\displaystyle a$ $\displaystyle=2q^{3}-3q^{2}+1,$ (20) $\displaystyle b$
$\displaystyle=(q^{3}-2q^{2}+q)\cdot(\tau_{j+1}-\tau_{j}),$ (21)
$\displaystyle c$ $\displaystyle=-2q^{3}+3q^{2},$ (22) $\displaystyle d$
$\displaystyle=(q^{3}-q^{2})\cdot(\tau_{j+1}-\tau_{j}).$ (23)
## Appendix B Tasks
This section provides additional information about the examples provided in
Section 4, including objective formulations and planner settings.
#### Humanoid objective.
This task comprises $M=6$ cost terms:
* •
Term $0$:
* $\text{r}_{0}$: lateral center-of-mass position and average lateral feet position alignment
* $\text{n}_{0}$: hyperbolic cosine
* $w_{0}$: 100
* •
Term $1$:
* $\text{r}_{1}$: lateral torso position and lateral center-of-mass position alignment
* $\text{n}_{1}$: smooth absolute value
* $w_{1}$: 1
* •
Term $2$:
* –
$\text{r}_{2}$: head height and feet height difference minus target height
difference
* –
$\text{n}_{2}$: smooth absolute value
* –
$w_{2}$: 100
* •
Term $3$:
* $\text{r}_{3}$: lateral center-of-mass velocity
* $\text{n}_{3}$: smooth absolute value
* $w_{3}$: 10
* •
Term $4$:
* $\text{r}_{4}$: joint velocity
* $\text{n}_{4}$: quadratic
* $w_{4}$: 0.01
* •
Term $5$:
* $\text{r}_{5}$: control effort
* $\text{n}_{5}$: quadratic
* $w_{5}$: 0.025
#### Quadruped objective.
This task comprises $M=4$ cost terms:
* •
Term $0$:
* $\text{r}_{0}$: body height and average feet height difference minus target height
* $\text{n}_{0}$: quadratic
* $w_{0}$: 1
* •
Term $1$:
* $\text{r}_{1}$: body position minus goal position
* $\text{n}_{1}$: quadratic
* $w_{1}$: 5.0
* •
Term $2$:
* $\text{r}_{2}$: body orientation minus goal orientation
* $\text{n}_{2}$: quadratic
* $w_{2}$: 1.0
* •
Term $3$:
* $\text{r}_{3}$: control effort
* $\text{n}_{3}$: quadratic
* $w_{3}$: 0.25
#### Hand objective.
This task comprises $M=3$ cost terms:
* •
Term $0$:
* $\text{r}_{0}$: cube position minus hand palm position
* $\text{n}_{0}$: quadratic
* $w_{0}$: 20
* •
Term $1$:
* $\text{r}_{1}$: cube orientation minus goal orientation
* $\text{n}_{1}$: quadratic
* $w_{1}$: 3
* •
Term $2$:
* $\text{r}_{2}$: cube linear velocity
* $\text{n}_{2}$: quadratic
* $w_{2}$: 10
#### Planner settings.
We provide the settings used for Predictive Sampling in Table 1.
Table 1: Predictive Sampling settings Task | $P$ | $N$ | $T$ | $\sigma$
---|---|---|---|---
Humanoid | 3 | 10 | 23 | 0.125
Quadruped | 3 | 10 | 35 | 0.25
Hand | 6 | 10 | 25 | 0.1
## Appendix C Predictive Sampling Algorithm
Algorithm 5 PredictiveSampling
1:procedure OptimizePolicy
2: task: $f$, $c$, $R$
3: settings: $T$, $N$, $\sigma$, $s$
4: initialise: $(\tau,\theta)$
5: for $k=0,\dots,\infty$
6: $(x,\tau)\leftarrow$ get state
7: $(\bar{\tau},\theta)\leftarrow\text{resample}$
8: for $i=0,\dots,N$ (parallel)
9:
$\tilde{\theta}^{(i)}=\theta+\begin{cases}0,\phantom{\mathcal{N}(0,\sigma^{2}\cdot
I)}i=0,\\\ \mathcal{N}(0,\sigma^{2}\cdot I),\phantom{0}\text{else}\end{cases}$
10: $x^{(i)}_{0}=x,\,\tau_{0}^{(i)}=\tau$,
11: for $t=0,\dots,T$
12: $u^{(i)}_{t}=s(\tau^{(i)}_{t};(\bar{\tau},\tilde{\theta}^{(i)}))$
13: $c^{(i)}_{t}=c(x^{(i)}_{t},u^{(i)}_{t};R)$
14: $(x^{(i)}_{t+1},\tau^{(i)}_{t+1})=f(x^{(i)}_{t},u^{(i)}_{t})$
15: end for
16: end for
17: $\theta\leftarrow\text{argmin}\left(J(\tilde{\theta}^{(i)})\right)$
18: end for
19:end procedure
20:procedure ActionFromPolicy
21: $(x,\tau)\leftarrow$ get state
22: $u=s(\tau;(\bar{\tau},\theta))$
23: return $u$
24:end procedure
## Appendix D Compute Resources
Experiments were performed on a Lenovo ThinkStation P920 with 48GB of memory
and an Intel Xeon Gold 6154 72-core CPU. Additional experiments were performed
on an Apple MacBook Pro (2021) with 16 GB of memory and an M1 Pro CPU.
|
# Towards Explainability in Modular Autonomous Vehicle Software
Hongrui Zheng∗, Zirui Zang∗, Shuo Yang∗, Rahul Mangharam ∗Authors contributed
equally. All authors are with University of Pennsylvania, Department of
Electrical and Systems Engineering, 19104, Philadelphia, PA, USA. Emails:
{hongruiz, zzang, yangs1<EMAIL_ADDRESS>
###### Abstract
Safety-critical Autonomous Systems require trustworthy and transparent
decision-making process to be deployable in the real world. The advancement of
Machine Learning introduces high performance but largely through black-box
algorithms. We focus the discussion of explainability specifically with
Autonomous Vehicles (AVs). As a safety-critical system, AVs provide the unique
opportunity to utilize cutting-edge Machine Learning techniques while
requiring transparency in decision making. Interpretability in every action
the AV takes becomes crucial in post-hoc analysis where blame assignment might
be necessary. In this paper, we provide positioning on how researchers could
consider incorporating explainability and interpretability into design and
optimization of separate Autonomous Vehicle modules including Perception,
Planning, and Control.
## I Introduction
According to the Morning Consult and Politico poll [1], only 16% of
respondents are “very likely” to ride as a passenger in an autonomous vehicle,
while 28% of respondents state that they “not likely at all”. Moreover, only
22% of respondents believe self-driving cars are safer than the average human
driver, while 35% of them believing self-driving cars are less safe than the
average human driver. The public’s distrust in Autonomous Vehicles (AV) shows
that improving explainability in AV software is a necessity.
There exist many surveys on explainable AI (XAI) and robotics [2, 3, 4, 5].
Specifically, [6, 7, 8] surveys explainability in Autonomous Driving.
Atakishiyev et al. [7] believes that AVs need to provide regulatory compliant
operational safety and explainability in real-time decisions. It focuses on
providing discussion through the cause-effect-solution perspective. Zablocki
et al. [6] provides an in-depth overview of XAI methods in deep vision-based
methods, but is limited to the scope of perception only. Omeiza et al. [8]
also provides an overview of explanations in AVs in the full self-driving
pipeline. Gilpin et al. [9] proposes explainability as the trade-off between
interpretability and completeness. As described in [9], to be interpretable is
to describe the internals of a system in such a way that is understandable to
humans; to be complete is to describe the operation of a system in an accurate
way.
We position ourselves to provide insight in augmenting explanability in
Autonomous Vehicle’s sense-plan-act software modules as a task of balancing
interpretability and completeness. In this paper, we look at the
explainability in existing works and in our recent contributions in
localization, planning, and control. In each case, we want to be able to
quantify the uncertainty at each step of the decision making and interpret the
provenance of the outcome of the algorithm.
## II Explainability in Localization
Robot localization is a problem of finding a robot’s pose using a map and
sensor measurements, such as LiDAR scans. The map is pre-built and the
environment is assumed to not change significantly after the map is captured.
It is crucial for any moving robot to interact with the physical world
correctly. However, the problem of finding the mappings between measurements
and poses can be ambiguous, because sensor measurements from multiple distant
poses can be similar. Therefore, to tightly integrate the localization module
with other parts of the software stack and for the engineers implementing and
tuning the algorithm, the explainability of localization algorithms using
neural networks becomes important. We need to estimate the uncertainty of the
localization results, and in the worst case, to know when and why the robot
fails to localize on a certain map.
Monte Carlo Localization (MCL)[10], the widely adopted method, uses random
hypothesis sampling and sensor measurement updates to infer the pose. In MCL,
the proposed particles are explicit poses on the map and we can interpret the
distribution of the particles as the uncertainties. The random generation of
the particles can be tuned with parameters that have physical meaning,
providing an interface for humans to adjust the behavior of the algorithm.
Many developments in localization seek to improve within the framework of
MCL.[11, 12, 13] Although particle filter has been a popular localization
framework for its robustness and reliable performance, it introduces random
jitter into the localization results.
Other common approaches are to use Bayesian filtering[14] or to find more
distinguishable global descriptors on the map[15, 16]. In Bayesian filtering,
the explainability lies in the conditional probability attached with the
motion model and each measurement. The estimation of such probability is
challenging. For the global descriptor approach, oftentimes manual selection
of map features are needed, which increases the explainability of the system,
but also increases the human workload and reduces robustness. Developments in
localization research usually propose better measurement models or feature
extractors within these frameworks. [17, 18].
Recent research in localization has also focused on the use of learning-based
methods outside of the above frameworks [19]. Although learning-based methods
may provide better localization precision with lower latency, the
interpretability of the method decreases. While the traditional localization
methods can be manually tuned according to the specific user scenarios,
learning-based localization methods are usually not tunable once the network
is trained. Uncertainty estimations of the neural networks also become a
challenge for learning-based methods. There are efforts to approximate the
uncertainty[20, 21, 22], but it hasn’t been widely applied.
Our contribution: In our recent paper, Local_INN, we proposed a new approach
to frame the localization problem as an ambiguous inverse problem and solve it
with an invertible neural network (INN) [23]. It stores the map data
implicitly inside the neural network. With the assumption that the environment
doesn’t not change significantly from the map, by evaluating the reverse path
of the neural network, we can get robot poses from LiDAR scans. It also
provides uncertainty estimation from the neural network and is capable of
learning and providing localization for complex environments.
Localization is an inverse problem of finding a robot’s pose using a map and
sensor measurements. This reverse process of inferring the pose from sensor
measurements is ambiguous. Invertible neural networks such as normalizing
flows[24] have been used to solve ambiguous inverse problems in various
fields[25, 26, 27, 28]. The version of normalizing flows we used is called
RealNVP[29], which uses a mathematical structure called coupling layers to
ensure the invertibility while performing transformations with arbitrary
neural network layers, such as MLPs. This framework of solving inverse
problems with normalizing flows was introduced by Ardizonne et al.[25] and was
later extended by [26, 30] to include a conditional input that is concatenated
to the vectors inside the coupling layers. They proposed to use normalizing
flows to learn a bijective mapping between two distributions and use a normal-
distributed latent variable to encode the lost information in training due to
the ambiguity of the problem. The network can be evaluated in both forward and
reverse paths. During the evaluation, repeatedly sampling the latent variable
can give the full posterior distribution given the input.
In Local_INN, we use pose-scan data pairs to train such a bijective mapping.
As shown in Fig. 1, The forward path is from pose to scan and the reverse path
is from scan to pose. We use a conditional input calculated from the previous
pose of the robot to reduce the ambiguity of the problem. Because INNs require
the same input and output dimensions, we use a Variational Autoencoder[31] to
reduce the dimension of the LiDAR scans and use Positional Encoding[32] to
augment that of the poses. The network is trained with supervised loss
functions on both sides. The data used for training the Local_INN can be
simulated or real data recorded from LiDARs. In our experiments, we tested on
both real and simulated data with 2D and 3D LiDARs. To collect training data,
we uniformly sample $x,y$ position and heading $\theta$ on the drivable
surface of each map, and use a LiDAR simulator to find the corresponding LiDAR
ranges. This means the trained network will be able to localize everywhere on
the map. For each different map, we need to train a separate network. Map
files are compressed inside the neural network and are no longer needed during
evaluation.
TABLE I: Local_INN Experiments: Map Reconstruction and RMS Localization Errors with 2D LiDAR ($xy$[m], $\theta$[∘]) | Race Track (Simulation) | Outdoor (Real)
---|---|---
| Original Map
---
Reconstruction
Test Trajectory
|
Online PF | $0.168,2.107$ | $0.047,1.371$
Local_INN+EKF | $\mathbf{0.056},\mathbf{0.284}$ | $\mathbf{0.046},\mathbf{1.130}$
Figure 1: Top: Evaluation of Local_INN in forward direction gives compressed
map information, and in the reverse direction gives accurate localization with
fast runtime and uncertainty estimation. Bottom: Network structure of
Local_INN. Solid arrows are from pose to lidar scan. Dashed arrows are from
lidar scan to pose. Conditional input is calculated from the previous pose of
the robot.
We claim that INN is naturally suitable for the localization problem with
improved explainability compared to other learning-based methods. In
particular, uncertainty estimation and map representation are the two
advantages that Local_INN provides in the context of explainability.
Figure 2: Example of global localization finding the correct pose at the 2nd
iterations (green arrow). Top: Narrowing down of the candidate poses in the
first 3 iterations. We can see candidate poses on the map (orange dots),
correct pose (red dot), selected pose (green dot). Bottom: Examples of LiDAR
scan ranges at candidate poses at iteration 3 (orange boxes), and at the
selected pose (green box). In the range plots, the horizontal axis is the
angle of LiDAR scans and vertical axis is the measured distance. We can see a
comparison of network expected ranges (blue curve) at various poses, and the
actual LiDAR measurement (orange curve). The correct pose is selected where
the measurement best matches the expected shape. The network is trained on
simulated data and tested on real LiDAR data.
### II-A Explainability from Uncertainty Estimation
When we use Local_INN to localize, the input to the reverse path of the INN
consists of the LiDAR scans concatenated with a latent vector that is sampled
from normal distribution. With this sampling of latent vector, the network can
output not just a pose but a distribution of inferred poses. The covariance of
this distribution can be used as the confidence of the neural network when
fusing with other sensors. Uncertainty estimation improves explainability by
providing information on the measurement quality of the prediction. Compared
to learning methods that do not provide uncertainty estimates, it is much
easier to determine whether the prediction of the neural network is lower in
accuracy due to higher uncertainty, and improve the prediction results by
augmentation. In our experiments, we used an EKF to fuse the localization
result with the odometry information. The results show that this fusion
significantly improved localization accuracy where the map geometry is
ambiguous, which means this covariance is very effective in revealing the
confidence of the network.
As shown in Table I, The accuracy of Local_INN is at par with the current
localization efforts. See [23] for a comparative analysis of Local_INN
localization accuracy in 2D and 3D maps.
### II-B Explainability from Map Representation
Local_INN provides an implicit map representation and a localization method
within one neural network. The guaranteed invertibility of the neural network
provides the use a direct way to check the neural network’s ’understanding’ of
the map by reproducing part of the map with poses. That is, we can compare the
reconstructed map to the original map to see how much detail is used by the
neural network in localization. Again, this feature improves explainability in
failure scenarios. When the localization fails, this comparison can help us
explain the failure and guide us in improving the methods. For example, we can
train with more data from a particular location on the map that was difficult
to localize in.
As an example of how the stored map information in the forward path of the
neural network can help us explain the localization results, let us consider
the algorithm for global localization. Global localization is needed when a
robot starts with an unknown pose or when the robot encounters the ’kidnapping
problem’. In this case, it is challenging to find the correct position on the
map due to the ambiguity of problem. MCL algorithms usually do global
localization by spreading the covariance all around the map and using
iterations of control inputs and measurements to decrease the covariance,
which gives an explicit visualization to see the progress of the global
localization processes. For other learning-based method, this process is
usually hard to explain as we rely on the neural network to output poses as a
black box.
With Local_INN, we can randomly initialize a set of random poses on the map as
conditional inputs and use new lidar scans to narrow down the assumptions. In
other words, we initially have multiple random assumptions of the robot’s
location on the map and use them as the conditional inputs for the nerual
network. As shown in figure 2 iteration 1, when we input a LiDAR scan to the
network along with these assumptions, it will output multiple possible pose
distributions. In our algorithm, for each possible pose distribution, we
compare the sensor measurement with what the neural network expects at this
location, and use the reciprocal of the error term to weight the assumptions
differently. The weights for the assumptions is used to determine the amount
of latent variable we use. This process repeats with each new LiDAR scan we
get from the sensor. In our experiments, the convergence of candidate poses is
fast and accurate. As shown in iteration 2 and 3 in figure 2, even if we still
track multiple poses, in this example, the correct pose is determined at the
2nd iteration in a highly symmetrical map. The low part of figure 2 shows
plots of the expected lidar scan from the neural network and the current LiDAR
measurement. This reveals the black-box process of the neural network so that
we can see why multiple poses are possible and how we should decide which one
to pick.
To summarize, the explainability of localization methods generally lies in the
uncertainty estimation and the ability to explain and tune the methods when
localization fails. Traditional localization methods usually offer higher
interpretability than learning-based methods, whereas the learning-based
methods can provide better empirical performance. The new method, Local_INN,
we recently proposed uses an invertible network network architecture to solve
the localization problem. It offers interpretability by giving uncertainty
estimation through the covariance of the inferred pose distributions, and by
ensuring the invertibility of the network so that we can reveal the what
information the neural network is using during the localization. At the same
time, it does sacrifice completeness by using a Variational Autoencoder (VAE)
to model a latent space of the LiDAR scans.
## III Explainability in Planning
Planning is the task of finding a viable motion plan for the robot to reach
some predetermined goal given the current observation of the environment
through various sensors. The planning step is usually the next step after
perception, or localization. Traditionally, sampling-based and model-
predictive methods are the most popular choices in Autonomous Vehicle motion
planning. Planning algorithms provide explanability through human-designed
objectives: e.g. maximizing the distance of a planned trajectory to obstacles
and undrivable areas, maximizing velocity on the trajectories, minimizing the
lateral acceleration the vehicle experiences on a planned trajectory. We
propose a unique position in adding interpretability and completeness to
planning algorithms: explainability through abstraction. Next, we show our
approach based on our recent paper on Game-Theoretic objective space planning
[33].
Figure 3: Overview of Game-theoretic Objective Space Planning
In this case study, our primary context is two-player racing games in close
proximity. An overall depiction of the pipeline is shown in Figure 3. We
choose a game theoretic approach that models racing as a zero-sum extensive
game with imperfect information and perfect recall. Extensive games model
sequential decision-making of players and naturally form a game tree where
each node is a decision point for a player. However, the planning problem
presented in autonomous racing is continuous, and the state space of the
agents, in turn, the game tree in the extensive game, will also be infinitely
large if we model the game in the vehicle’s planning space. Since the decision
made by a game-theoretic algorithm in the planning space cannot be explained
in a way that a human can understand, we use a lower dimensional space for
planning. We define the notion of Objective Space $\mathcal{O}$. For each
short rollout in an adversarial environment, we can compute multiple metrics
regarding this agent’s performance, such as safety and aggressiveness. These
metrics also add to the interpretability of our planning algorithm while not
losing the completeness. $\mathcal{O}$ models the average outcome of each
agent against competent opponents. Using $\mathcal{O}$, our planner maps
complex agent behaviors to a lower dimension where only the episodic outcome
is recorded instead of the entire decision-making process in the planning
space. We define an action in our game as movements in a dimension of
$\mathcal{O}$. This action space is analogous to the planning space in a grid
world with actions that move the agent to a neighboring cell. This means the
planning problem is much simpler than the original problem.
In our case study, we choose aggressiveness and restraint as the two
dimensions of $\mathcal{O}$. Aggressiveness is scored on an agent’s lead over
the other at the end of the rollout, and restraint is scored on an agent’s
time to collision to the environment and the other agent. Two movements are
available for each dimension: increasing or decreasing for a fixed distance
along the axis. For example, four discrete actions are available at each turn
when $\mathcal{O}\in\mathbb{R}^{2}$. Even with the formulation of agent action
space, the possible objective values of an opponent agent or possible position
in $\mathcal{O}$ is still infinite. We propose a predictive model for regret
values within Counterfactual Regret Minimization (CFR) [34] to make the
problem tractable. Finally with head-to-head racing experiments, we
demonstrate that using the proposed planning pipeline above significantly
improves the win rate that generalizes to unseen opponents in an unseen
environment.
### III-A Explainability in Agent Actions
Figure 4: Trajectories of ego moves in $\mathcal{O}$. Figure 5: Effect of
making a move in $\mathcal{O}$ in motion planning space.
In this section, we examine specific cases of agents moving in the lower
dimension and investigate whether we achieve the task of instilling
explainability in our algorithm. We choose a 2-D space that encodes
aggressiveness and restraint. In Figure 4, we show four examples of races
between two agents. The Ego (orange) uses our proposed game-theoretic approach
in the Objective Space, and the Opponent (green) is static in the objective
space. In the first two cases, the opponent is in the lower right quadrant,
meaning that they’re more conservative than aggressive. Hence our planner
chooses to increase in aggressiveness continuously to win the races. In the
last two cases, the opponent is in the upper left quadrant, meaning that
they’re more aggressive than conservative. Hence our planner chooses to first
become more conservative, and once a chance presents itself, increase in
aggressiveness and win the races. In Figure 5, we inspect a specific move to
show the effect in the motion planning space. In the beginning of the rollout,
both agents are side by side. At a decision point, our planner locates the
opponent in the lower right quadrant as more conservative than aggressive.
Then the ego decides to increase in aggressiveness to finally overtake the
opponent. From these examples, it is clear that moving the planning problem in
a lower dimension that encodes interpretable metrics for humans doesn’t
decrease the capability of the planner, or the completeness in the algorithm.
## IV Explainability in Control
With robust localization, the autonomous vehicle can plan its trajectory but
requires a safe an interpretable controller to execute the plan. In this
section, we show our recent progress on learning-based safety-critical control
through control barrier functions (CBFs) and provide our positioning on
explainable safe control. Specifically, we show that our proposed
differentiable safety filter has more completeness than non-differentiable
safety filter without sacrificing interpretability.
### IV-A Safety-critical control
Learning-based control could provide high empirical performance thus it has
become popular for controlling complex dynamical systems. However, learning-
based controllers, such as neural network (NN), generally lack formal safety
guarantees because of their black-box nature. This limits their deployments
with complex safety-critical systems. To address safety risk, multiple methods
have been proposed, such as model predictive control (MPC) [35], Hamilton-
Jacobi reachability analysis [36], contraction theory [37], and control
barrier functions (CBF) [38].
Among the many safety-critical control techniques, CBF is becoming a popular
choice since it explicitly specifies a safe control set by using a Lyapunov-
like condition and guards the system inside a safe invariant set. When a
continuous-time control-affine system is considered, such projection reduces
to a convex quadratic program (QP) which is referred to as CBF-QP. Due to its
simplicity, flexibility, and formal safety guarantees, CBFs have been applied
in safe learning control with many successful applications [39, 40, 41, 42,
43].
Compared with MPC, which needs to handle a possibly nonconvex optimization
problem in the face of nonlinear dynamical systems, CBF-QP is computationally
efficient to solve online. However, unlike MPC, the QP-based safety filter
only operates in a minimally invasive manner, i.e., it generates the safe
control input closest to the reference control input (in the Euclidean norm),
as shown in Fig. 6, unaware of the long-term effects of its action. This
indicates that the effects of the safety filter on the performance of the
closed-loop system are hard to predict. Therefore, the application of the
safety filter may give rise to myopic controllers [44] that induce subpar
performance in the long term.
Figure 6: Illustration of QP: If the NN output is not in the safe control set
$Q$, it will be projected in minimally invasive way to $Q$; otherwise, the
control keeps the same. Figure 7: Illustration of the gauge map from the
$\ell_{\infty}$ ball $B_{\infty}$ to a polytopic set $Q$. The original point
in $B_{\infty}$ and mapped point in $Q$ are in the same level set and with the
same direction.
To address the issue of myopic CBF-based safety filters, in our recent work
[45], we propose to utilize CBF to construct safe-by-construction NN
controllers that allow end-to-end learning. Incorporating safety layers in the
NN controller allows the learning agent to take the effects of safety filters
into account during training in order to maximize long-term performance.
Figure 8: NN controller with CBF-QP safety filter.
(a) Gauge map-based safe NN controller architecture.
(b) CBF-QP-based safe NN controller architecture.
Figure 9: Safe-by-construction NN controllers that utilize CBFs to construct
differentiable safety layers (yellow blocks).
We design a differentiable safety layer using the gauge map [46] (as shown in
Fig. 7) which establishes a bijection mapping between the polytopic set of a
NN output (e.g., an $\ell_{\infty}$ norm ball) and the CBF-based safe control
set. The proposed architecture is denoted as NN-gauge (Fig. 9(a)). We compare
NN-gauge with an alternative differentiable safe-by-construction NN controller
called NN-diff-QP, which consists of a NN followed by a differentiable CBF-QP
layer (Fig. 9(b)). Specifically, NN-diff-QP (Fig. 9(b)) concatenates a
differentiable projection layer which can be implemented in a NN using
toolboxes such as cvxpylayers [47], qpth [48]. NN-gauge involves finding an
interior point of the safe control set since gauge map requires that $Q$ set
in Fig. 7 is convex and cover the origin. It can be reduced to implicitly
finding the Chebyshev center [49] of the safe control set.
###### Remark 1
In the online execution, NN-gauge requires closed-form evaluation or solving a
linear program (LP) while NN-diff-QP solves a quadratic program. Both methods
are significantly cheaper to run than MPC.
As an example, let’s consider the adaptive cruise control (ACC) in which the
ego car is expected to achieve the desired cruising speed while maintaining a
safe distance from the leading car. As shown in Fig. 10(b), applying the CBF-
QP safety filter directly to enforce safe control deteriorates the long-term
closed-loop performance of the NN controller. However, both NN-gauge and NN-
diff-QP achieve similar closed-loop performance, and they are comparable to
MPC.
(a) CBF values.
(b) Velocity of the ego car.
Figure 10: Results of adaptive cruise control. CBF values (left) and velocity
of the ego car (right) under different controllers are evaluated in closed-
loop for $20$s. A CBF value below zero indicates unsafety, and the optimal
behavior of the ego car is expected to have a steady state velocity of
$16m/s$, same as the leading car.
### IV-B Explainability of safety-critical control
#### IV-B1 Non-differentiable QP filter
Quadratic program (QP) is the classic minimally invasive safety filter to
modify the nominal unsafe controller. Due to its simplicity and the nature of
minimal modification of the original controller, it has always been equipped
to CBF as the safety filter. As shown in Fig. 6, it is very understandable to
humans, since one only needs to project the unsafe control input to the
“nearest” point in the safe control set, which is specified by CBF. Thus, QP
safety filter enjoys high interpretability.
However, on the other hand, the QP operation is not necessarily the best
choice from the perspective of optimizing a long-term system objective. For
example, in the long run, sometimes it might be better to choose a “farther”
point in the safe control set than the “nearest” point on the boundary when
the nominal control is not safe. These modified actions are unaware of the
long-term effects and may give rise to myopic controllers which deteriorate
their performance. From the ACC case study, we can also observe in Fig. 10(b)
that the modified control (i.e., NN-QP) is myopic with unpredictable long-term
effects. Thus, vanilla QP filter lacks the completeness in the sense of
controlling the system in the optimal way with the safety guarantee.
#### IV-B2 Differentiable safety filters
Our proposed two differentiable safety filters, i.e., differentiable QP and
gauge map, enjoy both a high level of interpretability and completeness.
* •
Differentiable QP: the QP structure is the same with vanilla CBF-QP, so there
is no interpretability loss in differentiable QP. On the other hand, since the
NN controller considers the gradient of QP structure while training to
minimize the long-term loss, so the trained NN adapts to QP much better than
non-differentiable QP. Thus, even if the QP is applied to ensure safety when
evaluating online, the modified controller could still have good performance.
Therefore, from the explainability perspective, completeness is improved with
no loss on interpretability.
* •
Gauge map: unlike QP, gauge map seeks to map the original control
“proportianally” to the safe control set with the same direction. This mapping
is also understandable to humans. Furthermore, since gauge map is sub-
differentiable itself, so the end-to-end training holds naturally, which
allows it to perform as good as differentiable QP after training. Thus,
explanation could find the high completeness in this method as well.
In the ACC example, it also has been shown that both NN-gauge and NN-diff-QP
have comparable performance with MPC, which implies the high completeness from
demonstration. Looking ahead, we are interested in exploring a more general
parameterized gauge map method, which even not necessarily maps to the same
level set. In that case, it will perhaps lose some interpretability but have
better completeness as it is not limited to the same level set mapping. This
way, there is more flexibility while choosing the safe control input.
## V Discussion and Conclusion
In any autonomous vehicle which operates a sense-plan-act processing loop, the
essential components are localization, planning and control. For humans to
trust the system to make decisions and actions on their behalf in safety-
critical situations, it is essential to have explainability across the stack.
We view explainability as a trade-off between _interpretability_ of the
machine’s actions and the _completeness_ in terms of describing the operation
of the system in an accurate way.
The explainability of localization methods generally lies in the uncertainty
estimation and the ability to explain and tune the methods when localization
fails. We introduce Local_INN, which utilizes an invertable neural network
architecture that provides explainability for uncertainty estimation and
explainability from map representation. Uncertainty estimation, through the
covariance of the inferred pose distributions, improves explainability by
providing information on the measurement quality of the prediction.
Furthermore, the guaranteed invertibility of the neural network provides the
use a direct way to check the neural network’s understanding of the map by
analyzing the reconstructed map. This feedback from the internals of the
localization engine allows the human to know where in the map the network has
lower confidence and that they should augment it with more data.
The expainability in planning is necessary to ensure the autonomous vehicle
maintains a safe trajectory at all times in terms of maximizing the distance
of a planned trajectory to obstacles and undrivable areas, maximizing velocity
on the trajectories, minimizing the lateral acceleration the vehicle
experiences on a planned trajectory, and so on. To achieve this, we introduced
a new approach on Game-Theoretic Objective Space Planning where we map these
complex planning objectives to a lower dimensional space of balancing
aggressiveness and restraint. In a racing context, we show how our planner
increases aggressiveness continuously to win the races. Similarly, it chooses
to be more conservative to maintain safety. By moving the planning problem in
a lower dimension that encodes interpretable metrics for humans we demonstrate
how it doesn’t decrease the capability of the planner, or reduce the
completeness in the algorithm.
Finally, to ensure the plan is executed in a safe manner, we describe our
efforts in explainable safe neural network controller design with
Differentiable CBF-QP filters. The Control Barrier Function structure of the
filter ensures the control output is always safe in an interpretable manner as
there is a direct projection of the NN controller output into the safe control
set. The performance of the proposed safe controllers is comparable to model-
predictive controllers, which implies the high completeness from
demonstrations.
Through these three learning-based architectures for localization, planning
and safe control, we have demonstrated the initial findings on explainability
of decisions in the sense-plan-act process of autonomous systems. Further
developments in explainability of autonomous systems will lead to more public
trust and safer systems.
## References
* [1] [Online]. Available: https://morningconsult.com/wp-content/uploads/2017/09/National-Tracking-Poll-170904.pdf
* [2] F. K. Došilović, M. Brčić, and N. Hlupić, “Explainable artificial intelligence: A survey,” in _2018 41st International convention on information and communication technology, electronics and microelectronics (MIPRO)_. IEEE, 2018, pp. 0210–0215.
* [3] S. R. Islam, W. Eberle, S. K. Ghafoor, and M. Ahmed, “Explainable artificial intelligence approaches: A survey,” _arXiv preprint arXiv:2101.09429_ , 2021\.
* [4] E. Tjoa and C. Guan, “A survey on explainable artificial intelligence (xai): Toward medical xai,” _IEEE transactions on neural networks and learning systems_ , vol. 32, no. 11, pp. 4793–4813, 2020.
* [5] T. Sakai and T. Nagai, “Explainable autonomous robots: a survey and perspective,” _Advanced Robotics_ , vol. 36, no. 5-6, pp. 219–238, 2022\.
* [6] É. Zablocki, H. Ben-Younes, P. Pérez, and M. Cord, “Explainability of deep vision-based autonomous driving systems: Review and challenges,” _International Journal of Computer Vision_ , pp. 1–28, 2022.
* [7] S. Atakishiyev, M. Salameh, H. Yao, and R. Goebel, “Explainable artificial intelligence for autonomous driving: A comprehensive overview and field guide for future research directions,” _arXiv preprint arXiv:2112.11561_ , 2021\.
* [8] D. Omeiza, H. Webb, M. Jirotka, and L. Kunze, “Explanations in autonomous driving: A survey,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
* [9] L. H. Gilpin, D. Bau, B. Z. Yuan, A. Bajwa, M. Specter, and L. Kagal, “Explaining explanations: An overview of interpretability of machine learning,” in _2018 IEEE 5th International Conference on data science and advanced analytics (DSAA)_. IEEE, 2018, pp. 80–89.
* [10] F. Dellaert, D. Fox, W. Burgard, and S. Thrun, “Monte carlo localization for mobile robots,” in _Proceedings 1999 IEEE international conference on robotics and automation (Cat. No. 99CH36288C)_ , vol. 2. IEEE, 1999, pp. 1322–1328.
* [11] C. Zhang, M. H. Ang, and D. Rus, “Robust lidar localization for autonomous driving in rain,” in _2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2018, pp. 3409–3415.
* [12] X. Chen, I. Vizzo, T. Läbe, J. Behley, and C. Stachniss, “Range Image-based LiDAR Localization for Autonomous Vehicles,” in _Proc. of the IEEE Intl. Conf. on Robotics & Automation (ICRA)_, 2021.
* [13] X. Chen, T. Läbe, L. Nardi, J. Behley, and C. Stachniss, “Learning an overlap-based observation model for 3d lidar localization,” in _2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2020, pp. 4602–4608.
* [14] V. Fox, J. Hightower, L. Liao, D. Schulz, and G. Borriello, “Bayesian filtering for location estimation,” _IEEE pervasive computing_ , vol. 2, no. 3, pp. 24–33, 2003.
* [15] R. Dube, A. Cramariuc, D. Dugas, H. Sommer, M. Dymczyk, J. Nieto, R. Siegwart, and C. Cadena, “Segmap: Segment-based mapping and localization using data-driven descriptors,” _The International Journal of Robotics Research_ , vol. 39, no. 2-3, pp. 339–355, 2020.
* [16] P.-E. Sarlin, C. Cadena, R. Siegwart, and M. Dymczyk, “From coarse to fine: Robust hierarchical localization at large scale,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 12 716–12 725.
* [17] M. A. Uy and G. H. Lee, “Pointnetvlad: Deep point cloud based retrieval for large-scale place recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 4470–4479.
* [18] L. Sun, D. Adolfsson, M. Magnusson, H. Andreasson, I. Posner, and T. Duckett, “Localising faster: Efficient and precise lidar-based robot localisation in large-scale environments,” in _2020 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2020, pp. 4386–4392.
* [19] W. Lu, Y. Zhou, G. Wan, S. Hou, and S. Song, “L3-net: Towards learning based lidar localization for autonomous driving,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 6389–6398.
* [20] M. Cai, C. Shen, and I. Reid, “A hybrid probabilistic model for camera relocalization,” 2019.
* [21] A. Kendall and R. Cipolla, “Modelling uncertainty in deep learning for camera relocalization,” in _2016 IEEE international conference on Robotics and Automation (ICRA)_. IEEE, 2016, pp. 4762–4769.
* [22] H. Deng, M. Bui, N. Navab, L. Guibas, S. Ilic, and T. Birdal, “Deep bingham networks: Dealing with uncertainty and ambiguity in pose estimation,” _International Journal of Computer Vision_ , pp. 1–28, 2022.
* [23] Z. Zang, H. Zheng, J. Betz, and R. Mangharam, “Local_inn: Implicit map representation and localization with invertible neural networks,” _arXiv preprint arXiv:2209.11925_ , 2022.
* [24] G. Papamakarios, E. T. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, “Normalizing flows for probabilistic modeling and inference.” _J. Mach. Learn. Res._ , vol. 22, no. 57, pp. 1–64, 2021.
* [25] L. Ardizzone, J. Kruse, S. Wirkert, D. Rahner, E. W. Pellegrini, R. S. Klessen, L. Maier-Hein, C. Rother, and U. Köthe, “Analyzing inverse problems with invertible neural networks,” _arXiv preprint arXiv:1808.04730_ , 2018.
* [26] L. Ardizzone, C. Lüth, J. Kruse, C. Rother, and U. Köthe, “Guided image generation with conditional invertible neural networks,” _arXiv preprint arXiv:1907.02392_ , 2019.
* [27] T. J. Adler, L. Ardizzone, A. Vemuri, L. Ayala, J. Gröhl, T. Kirchner, S. Wirkert, J. Kruse, C. Rother, U. Köthe _et al._ , “Uncertainty-aware performance assessment of optical imaging modalities with invertible neural networks,” _International journal of computer assisted radiology and surgery_ , vol. 14, no. 6, pp. 997–1007, 2019.
* [28] T. Wehrbein, M. Rudolph, B. Rosenhahn, and B. Wandt, “Probabilistic monocular 3d human pose estimation with normalizing flows,” in _International Conference on Computer Vision (ICCV)_ , Oct. 2021.
* [29] L. Dinh, J. Sohl-Dickstein, and S. Bengio, “Density estimation using real nvp,” _arXiv preprint arXiv:1605.08803_ , 2016.
* [30] C. Winkler, D. Worrall, E. Hoogeboom, and M. Welling, “Learning likelihoods with conditional normalizing flows,” _arXiv preprint arXiv:1912.00042_ , 2019\.
* [31] D. P. Kingma and M. Welling, “Auto-encoding variational bayes,” _arXiv preprint arXiv:1312.6114_ , 2013.
* [32] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” _Advances in neural information processing systems_ , vol. 30, 2017.
* [33] H. Zheng, Z. Zhuang, J. Betz, and R. Mangharam, “Game-theoretic objective space planning,” _arXiv preprint arXiv:2209.07758_ , 2022.
* [34] M. Zinkevich, M. Johanson, M. Bowling, and C. Piccione, “Regret Minimization in Games with Incomplete Information,” in _Advances in Neural Information Processing Systems_ , vol. 20. Curran Associates, Inc., 2007.
* [35] E. F. Camacho and C. B. Alba, _Model predictive control_. Springer science & business media, 2013.
* [36] S. Bansal, M. Chen, S. Herbert, and C. J. Tomlin, “Hamilton-jacobi reachability: A brief overview and recent advances,” in _2017 IEEE 56th Annual Conference on Decision and Control (CDC)_. IEEE, 2017, pp. 2242–2253.
* [37] G. Chou, N. Ozay, and D. Berenson, “Safe output feedback motion planning from images via learned perception modules and contraction theory,” _arXiv preprint arXiv:2206.06553_ , 2022.
* [38] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 8, pp. 3861–3876, 2016.
* [39] A. Anand, K. Seel, V. Gjærum, A. Håkansson, H. Robinson, and A. Saad, “Safe learning for control using control lyapunov functions and control barrier functions: A review,” _Procedia Computer Science_ , vol. 192, pp. 3987–3997, 2021.
* [40] C. Dawson, Z. Qin, S. Gao, and C. Fan, “Safe nonlinear control using robust neural lyapunov-barrier functions,” in _Conference on Robot Learning_. PMLR, 2022, pp. 1724–1735.
* [41] A. Taylor, A. Singletary, Y. Yue, and A. Ames, “Learning for safety-critical control with control barrier functions,” in _Learning for Dynamics and Control_. PMLR, 2020, pp. 708–717.
* [42] S. Dean, A. J. Taylor, R. K. Cosner, B. Recht, and A. D. Ames, “Guaranteeing safety of learned perception modules via measurement-robust control barrier functions,” _arXiv preprint arXiv:2010.16001_ , 2020.
* [43] R. Cheng, G. Orosz, R. M. Murray, and J. W. Burdick, “End-to-end safe reinforcement learning through barrier functions for safety-critical continuous control tasks,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 33, no. 01, 2019, pp. 3387–3395.
* [44] M. H. Cohen and C. Belta, “Approximate optimal control for safety-critical systems with control barrier functions,” in _2020 59th IEEE Conference on Decision and Control (CDC)_. IEEE, 2020, pp. 2062–2067.
* [45] S. Yang, S. Chen, V. M. Preciado, and R. Mangharam, “Differentiable safe controller design through control barrier functions,” _arXiv preprint arXiv:2209.10034_ , 2022.
* [46] D. Tabas and B. Zhang, “Computationally efficient safe reinforcement learning for power systems,” in _2022 American Control Conference (ACC)_. IEEE, 2022, pp. 3303–3310.
* [47] A. Agrawal, B. Amos, S. Barratt, S. Boyd, S. Diamond, and J. Z. Kolter, “Differentiable convex optimization layers,” _Advances in neural information processing systems_ , vol. 32, 2019.
* [48] B. Amos and J. Z. Kolter, “Optnet: Differentiable optimization as a layer in neural networks,” in _International Conference on Machine Learning_. PMLR, 2017, pp. 136–145.
* [49] S. Boyd, S. P. Boyd, and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
|
# Quantum walk based state transfer algorithms on the complete $M$-partite
graph
S. Skoupý and M. Štefaňák Department of Physics, Faculty of Nuclear Sciences
and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115
19 Praha 1 - Staré Město, Czech Republic
###### Abstract
We investigate coined quantum walk search and state transfer algorithms,
focusing on the complete $M$-partite graph with $N$ vertices in each
partition. First, it is shown that by adding a loop to each vertex the search
algorithm finds the marked vertex with unit probability in the limit of a
large graph. Next, we employ the evolution operator of the search with two
marked vertices to perform a state transfer between the sender and the
receiver. We show that when the sender and the receiver are in different
partitions the algorithm succeeds with fidelity approaching unity for a large
graph. However, when the sender and the receiver are in the same partition the
fidelity does not reach exactly one. To amend this problem we propose a state
transfer algorithm with an active switch, whose fidelity can be estimated
based on the single vertex search alone.
## I Introduction
Quantum walks [1] are quantum mechanical analogues of random walks. Their
dynamics can be formulated either in discrete time steps [2], as we consider
in the present paper, or in continuous time [3]. Both have found promising
applications in quantum information processing [4], notably in quantum spatial
search where an unsorted database is represented by a graph. The solution to
the search problem corresponds to a marked vertex, where the local dynamics is
different from the non-marked vertices. Usually, the initial state of the walk
is taken as the equal weight superposition of all basis states. The walk is
evolved coherently for $T$ steps, after which we perform a measurement which
collapses the state of superposition and the walker is found on a single
vertex. On various graphs quantum walk is capable of finding the marked vertex
with sufficiently high probability in a number of steps that grows with the
square root of the number of vertices $n$, i.e., the complexity is the same as
for the abstract Grover search algorithm [5] which is known to be optimal.
Initially, the investigation was mostly focused on graphs with some degree of
symmetry or regularity. Continuous time quantum walks were shown to be optimal
[6] for a complete graph, hypercube and lattices of dimensions greater than 4.
Discrete time quantum walks with coins [7] are also optimal on these graphs
[8, 9, 10, 11], in addition, they are optimal for lattices of dimensions
greater than 2 [9]. Scattering quantum walk [12, 13, 14], which represents an
alternative equivalent formulation of the coined quantum walk [15, 16], can
perform optimal search e.g., on a star graph or a complete $M$-partite graph
[17]. However, high symmetry is not required for the optimal performance of
the quantum walk search [18, 19, 20]. In fact, it was shown [21] that
continuous the time quantum walk is optimal on Erdös-Renyi random graphs as
long as the probability of an edge existing between any pair of vertices is
greater than $\left(\log^{\frac{3}{2}}n\right)/n$. However, on scale-free
networks [22] the application of quantum walk search appears to be limited
since the optimal run-time depends on the centrality of the marked node [23].
More recently [24] several sufficient and necessary conditions for continuous
time quantum walk search to be optimal were derived.
For the search algorithm (SA) it is not required that we find the marked
vertex with unit probability. As long as the success probability is constant,
we can repeat the SA several times depending on our error tolerance to find
the marked vertex with high probability without changing the overall
complexity of the algorithm. Even if the success probability is of the order
of $1/\log n$, as for the discrete time quantum walk on the 2D lattice [9], we
can use amplitude amplification [25] which increases the run-time of the SA by
a factor of $\sqrt{\log n}$. There are several graphs where the quantum walk
SA is exactly equivalent to the Grover search, e.g., the star graph [17],
which means that the success probability is unity. It is interesting that for
the discrete time quantum walks the success probability of the SA can be often
increased close to unity by adding loops of appropriate weights at each
vertex. This was found originally for the complete graph [9] and the hypercube
[10]. Later investigations [26, 27, 28, 29, 30] found that this result is much
more generic and the optimal weight of the loop depending on the size of the
graph and degree of the vertex was identified. Recently, it was proven that
adding loops improves the success probability of the SA on all regular locally
arc-transitive graphs [31].
Quantum walks were also applied to the task of state transfer [32] between two
vertices of a graph. In this context the initial state of the walk is
localized on the sender vertex and we want to transfer it with high
probability to the receiver vertex. Provided that the location of the sender
and the receiver vertices are known, we can globally design the dynamics such
that the walker is transferred from one to the other. This approach was
investigated on different graphs such as circle [33, 34], 2D lattice [35],
regular graphs [36] or more general networks [37]. When the sender and the
receiver don’t know each other’s position they can perform state transfer by
modifying the local coins at their own vertices, i.e., by implementing the
evolution operator of the SA for two marked vertices. This approach was
proposed for state transfer on lattices [38] and further analyzed on various
types of finite graphs, e. g. on cycles and their variants [39, 40], star and
complete graph with loops [41], complete bipartite graph [42], circulant
graphs [43] or butterfly network [44]. Similar approach can be also applied
for finding a path in a maze formed by star graphs [45] or trees [46].
We consider the state transfer algorithm (STA) following the second approach,
i.e., based on the evolution operator of the SA with two marked vertices. For
the STA it is desirable that we succeed in the first attempt, i.e., the
fidelity of state transfer should be ideally one. Natural candidates for
graphs where STA works with unit fidelity are those where also the SA succeeds
with certainty. Indeed, the graphs considered in [41, 42] were chosen exactly
based on this idea. However, in this paper we show an example of a graph where
the SA works with certainty yet in some instance the STA does not have unit
fidelity.
We investigate search and state transfer on the complete $M$-partite graph
with $N$ vertices in each partition, i.e., the graph has $n=NM$ vertices.
Search on the complete $M$-partite graph was already investigated in [17] in
the framework of the scattering quantum walk. In the coined walk the success
probability of SA reaches $\frac{1}{2}$. We show that by adding a loop to each
vertex the success probability tends to one for a large graph. Our approach is
based on dimensional reduction [47, 48, 49, 19]. First, we find an exact
invariant subspace $\cal I$ where the state of the algorithm evolves. Next, we
investigate the eigenvectors of the evolution operator in the limit of a large
graph $M\to\infty$ and $N\to\infty$ and determine those which have non-
vanishing overlap with the initial state. These eigenvectors form an
orthonormal basis of the relevant part of the invariant subspace. For the SA,
we find an exact invariant subspace with dimension 8, and in the asymptotic
limit the relevant part is three-dimensional. We then investigate the
evolution operator of the search with two marked vertices for the sake of
state transfer. There are two possible configurations - either sender and
receiver are in the same partition or not. In the first case, we find an exact
invariant subspace with dimension 11, while in the second case it has
dimension 22. To simplify the calculations, we employ the symmetry of the
graph which allows us to exchange the sender and the receiver vertex, or the
whole partitions containing them in the latter case. This symmetry splits the
invariant subspace $\cal I$ further into two closed subspaces - $\cal I_{+}$
in which the search with two marked vertices evolves, and the complementary
subspace $\cal I_{-}$ needed for the state transfer. In the configuration
where the sender and the receiver are in the same partition, the subspace
$\cal I_{+}$ has dimension 8 and the complementary subspace $\cal I_{-}$ is
three-dimensional. For the second configuration the subspaces have dimensions
12 and 10, respectively. Nevertheless, in the limit of a large graph only five
eigenvectors of the evolution operator remain relevant in both configurations
- three in the subspace $\cal I_{+}$ and two in $\cal I_{-}$. The
corresponding eigenvalues can be also determined analytically. We show that
when the sender and the receiver are in different partitions the phases of the
relevant eigenvalues are harmonic. Hence, state transfer is achieved with unit
fidelity. However, when the sender and the receiver are in the same partition,
the phases are not harmonic, and the fidelity of state transfer is less than
one. To fix this issue we propose an STA with an active switch, where
initially only the sender vertex is marked, and after some number of steps the
marking is switched to the receiver vertex. The fidelity reachable by this STA
can be estimated based on the properties of the search for a single vertex. We
show that STA with an active switch achieves perfect state transfer on the
complete $M$-partite graph in the limit of large $N$ and $M$ for both
configurations of the sender and the receiver vertex, and discuss its
applicability on other graphs.
The paper is organized as follows: In Section II we describe discrete time
quantum walks with coin on finite graphs and introduce the quantum walk search
and state transfer algorithms. In Section III search on the complete
$M$-partite graph with one marked vertex is investigated in detail. Section IV
is devoted to the STA. The cases of sender and receiver vertices being in the
same or different partitions are analyzed in Section IV.1 and Section IV.2,
respectively. In Section V we consider a STA with an active switch. We
conclude and provide an outlook in Section VI.
## II Preliminaries
In this section we overview the general design of the search and the state
transfer algorithms based on the discrete-time quantum walks with coins [8],
[9] and [38]. Before we turn to the algorithms, we describe the discrete time
quantum walks with coins. Let us start with Hilbert space of the walk. Having
a graph $G=\left(V,E\right)$ the corresponding Hilbert space ${\cal H}_{G}$
can be decomposed as a direct sum
${\cal H}_{G}=\bigoplus_{v}{\cal H}_{v},$
of local Hilbert spaces at each vertex $v\in V$. The orthonormal basis in
${\cal H}_{v}$ is given by vectors $\mathinner{|{v,w}\rangle}$ such that there
is an edge between vertex $v$ and $w$
${\cal H}_{v}={\rm Span}\left\\{|v,w\rangle|w\in V,\left\\{v,w\right\\}\in
E\right\\}.$
In the basis state $\mathinner{|{v,w}\rangle}$ the first index $v$ describes
the actual position of the walker, while the second index $w$ describes the
direction of propagation of the walker. Movement of the walker is achieved by
application of the flip-flop shift operator $\hat{S}$, which is defined in the
following way
$\displaystyle\hat{S}|v,w\rangle=|w,v\rangle.$ (1)
To generate a nontrivial evolution a coin operator $\hat{C}$ is applied at
every step before the shift takes place. The coin operator can be decomposed
into a direct sum
$\hat{C}=\bigoplus_{v}\hat{C}_{v}^{(l)},$
where $\hat{C}_{v}^{(l)}$ acts locally at a vertex $v$, i.e., it is a unitary
operator on ${\cal H}_{v}$. The evolution operator $\hat{U}$ of one step of
the walk is then given by
$\displaystyle\hat{U}=\hat{S}\hat{C}.$ (2)
The main idea of search and state transfer algorithms is to apply one local
coin operator at the marked vertices and different local coin operator at the
other vertices of the graph. Marked vertices are those that we want to find or
between which we want transfer the walker. Usually, the local coin that is
used at non-marked vertices is the Grover operator [5]
$\displaystyle\hat{G}_{v}^{(l)}=2|\Omega_{v}\rangle\langle\Omega_{v}|-\hat{I}_{v}^{(l)},$
(3)
where $\hat{I}_{v}^{(l)}$ is an identity operator at the subspace ${\cal
H}_{v}$ and $|\Omega_{v}\rangle$ is an equal weight superposition of all basis
states at the vertex $v$ given by
$\displaystyle|\Omega_{v}\rangle=\frac{1}{\sqrt{d\left(v\right)}}\sum_{\begin{subarray}{c}w\\\
\left\\{v,w\right\\}\in E\end{subarray}}|v,w\rangle.$ (4)
Here $d(v)$ denotes the degree of the vertex $v$, which is also the dimension
of the subspace ${\cal H}_{v}$. The coin operator of the SA with one marked
vertex $m$ reads
$\displaystyle\hat{C}_{m}=\bigoplus_{\begin{subarray}{c}v\in V\\\ v\neq
m\end{subarray}}\hat{G}_{v}^{(l)}\oplus\hat{C}_{m}^{(l)}$ (5)
where $\hat{C}_{m}^{(l)}$ is the local coin operator at the marked vertex. A
usual choice for the marked coin is either a simple phase shift by $\pi$
(i.e., $\hat{C}_{m}^{(l)}=-\hat{I}_{m}^{(l)}$), or the Grover operator
followed by a phase shift by $\pi$ (i.e.,
$\hat{C}_{m}^{(l)}=-\hat{G}_{m}^{(l)}$). In the present paper we consider the
latter case. Using the coin operator (5) we obtain the evolution operator of
the SA
$\hat{U}_{m}=\hat{S}\hat{C}_{m}.$
The steps of the SA are:
1. 1.
Initialize the walk in the equal weight superposition of all basis states
$\displaystyle|\Omega\rangle=\frac{1}{\sqrt{\sum\limits_{v\in
V}d(v)}}\sum_{v\in V}\sqrt{d(v)}|\Omega_{v}\rangle.$ (6)
2. 2.
Apply the evolution operator $\hat{U}_{m}$ $t$-times. The state of the walk
after $t$ steps is given by
$|\phi\left(t\right)\rangle=\hat{U}_{m}^{t}\mathinner{|{\Omega}\rangle}.$
3. 3.
Measure the walk.
The probability to find the walker at the marked vertex is given by the
summation over all basis states in the subspace ${\cal H}_{m}$
$\displaystyle P_{m}(t)=\sum_{\begin{subarray}{c}w\\\ \left\\{m,w\right\\}\in
E\end{subarray}}|\langle m,w|\phi\left(t\right)\rangle|^{2}.$ (7)
The optimal number of steps $T$ providing high success probability depends on
the structure of the graph.
In the case of STA we consider $2$ parties, sender and receiver sitting at
vertices $s$ and $r$, respectively, which want to establish communication
between each other. The sender and the receiver have access only to their
local Hilbert spaces ${\cal H}_{s}$ and ${\cal H}_{r}$, respectively. Typical
STA [38, 41, 42] uses the evolution operator of the search for two marked
vertices
$\hat{U}_{s,r}=S\hat{C}_{s,r},$
where we apply the same local coins at both marked vertices at the same time
$\displaystyle\hat{C}_{s,r}=\bigoplus_{\begin{subarray}{c}v\in V\\\ v\neq
s,r\end{subarray}}\hat{G}_{v}^{(l)}\oplus\hat{C}_{s}^{(l)}\oplus\hat{C}_{r}^{(l)}.$
(8)
However, the initial state of STA is different — it starts localized at the
sender vertex in some state $\mathinner{|{s}\rangle}$. Standard choice is the
equal weight superposition of all basis states at the vertex $s$, i.e.,
$\mathinner{|{s}\rangle}=\mathinner{|{\Omega_{s}}\rangle}$. The steps of the
STA are the following:
1. 1.
Sender initializes the walk at its vertex in the state
$\mathinner{|{s}\rangle}$.
2. 2.
The evolution operator $\hat{U}_{s,r}$ is applied $t$-times. The state of the
walk after $t$ steps is given by
$|\phi\left(t\right)\rangle=\hat{U}_{s,r}^{t}\mathinner{|{s}\rangle}.$
3. 3.
Receiver measures the walk at its vertex.
The fidelity of the STA, i.e., the probability that the receiver finds the
walker at its vertex, is given by
$\displaystyle{\cal F}(t)=\sum_{\begin{subarray}{c}w\\\
\left\\{r,w\right\\}\in E\end{subarray}}|\langle
r,w|\phi\left(t\right)\rangle|^{2}.$ (9)
The number of steps $T^{(st)}$ required to achieve state transfer with high
fidelity depends again on the size and the structure of the graph. In contrast
to the SA it is desirable that the STA performs with high fidelity in a single
run. In such a case we talk about perfect state transfer.
## III Search on the complete M-partite graph
Consider the complete $M$-partite graph (with $M>2$). Complete $M$-partite
graph is a graph that has the set of vertices $V$ divided into $M$ subsets,
where vertices have no edges between them, but are connected to all vertices
in other subsets. We label the vertices of the graph as $v_{\alpha}$, where
$\alpha=1,\ldots,M$ denotes the partition. The basis states of the quantum
walk are therefore given by $\mathinner{|{v_{\alpha},w_{\beta}}\rangle}$,
$\alpha\neq\beta$. We also limit ourselves to the case where all parts have
the same size $N$, thus the whole graph has $n=MN$ vertices. This choice
greatly simplifies the construction of the invariant subspace. The graph is
$d$-regular with the vertex degree
$d=N(M-1).$
Without loss of generality we assume that the marked vertex is in the first
partition.
Search on the complete $M$-partite graph was investigated in [17] in the
framework of the scattering quantum walk. The difference between the two
formulations is that in the coined walk the walker lives on the vertices,
while in the scattering walk it lives on the edges. The results of [17] can be
adopted for the coined quantum walk with a single modification. Namely, for
the evaluation of the success probability (7) we consider only the overlap
with the states where the walker is at the marked vertex, i.e., of the form
$\mathinner{|{m_{1},k_{\alpha}}\rangle}$, $k=1,\ldots,N$, $\alpha=2,\ldots,M$,
which form the basis of the local Hilbert space at marked vertex ${\cal
H}_{m}$. The equal weight superposition of such states corresponds to the
state $\mathinner{|{w_{2}}\rangle}$ in [17]. This gives us the success
probability of $\frac{1}{2}$ for a large graph. Note that with probability
close to $\frac{1}{2}$ we would find the walker in some state
$\mathinner{|{k_{\alpha},m_{1}}\rangle}$, $k=1,\ldots,N$, $\alpha=2,\ldots,M$,
i.e., where the walker is at the vertex $k_{\alpha}$ and would move to the
marked vertex after the application of the shift $\hat{S}$. In the scattering
walk framework these states are also considered in the evaluation of the
success probability. Nevertheless, we focus on the coined formulation. In the
end our goal is to investigate the state transfer between vertices where the
coined formulation is more natural, since sender and receiver are considered
to be restricted to their local Hilbert spaces ${\cal H}_{s}$ and ${\cal
H}_{r}$, respectively.
We show that one can improve the success probability of the coined walk search
by adding one loop at each vertex. This is done by adding basis states
$\mathinner{|{v_{\alpha},v_{\alpha}}\rangle}$ corresponding to the loop at
each vertex, i.e. the dimension of the local Hilbert spaces ${\cal
H}_{v_{\alpha}}$ increases by one. Moreover, we modify the local coin operator
(3) by replacing the state $\mathinner{|{\Omega_{v_{\alpha}}}\rangle}$ (4)
with the state $\mathinner{|{\Omega_{v_{\alpha}}(l)}\rangle}$ given by
$\mathinner{|{\Omega_{v_{\alpha}}(l)}\rangle}=\frac{1}{\sqrt{d+l}}\left(\sum_{\begin{subarray}{c}\beta=1\\\
\beta\neq\alpha\end{subarray}}^{M}\sum_{k=1}^{N}\mathinner{|{{v_{\alpha}},k_{\beta}}\rangle}+\sqrt{l}\mathinner{|{{v_{\alpha}},{v_{\alpha}}}\rangle}\right),$
where $l$ is the weight of the loop. Note, however, that the initial state of
the SA remains the same as before, i.e., we prepare the system in the equal
weight superposition (6) of all basis states excluding the states
corresponding to the loops. According to [29, 31] the optimal weight $l$ of
the loops is given by
$l=\frac{d}{NM}={1-\frac{1}{M}}.$ (10)
Since we focus on the limit of a large graph we put $l=1$. This choice
corresponds to the local coin operator being the Grover operator (3) of
dimensions $d+1$, except for the marked vertex where we include an additional
phase shift by $\pi$. First, we show that the SA evolves in an 8-dimensional
invariant subspace ${\cal I}$. We note that in principle the dimension of the
exact invariant subspace can be reduced further, since the evolution operator
restricted on $\cal I$ still has some degenerate eigenvalues. However, the
construction of the basis would not be as intuitive and the ensuing
calculations will not simplify. Second, we consider a limit of a large graph
which effectively reduces the dimension of the invariant subspace to three.
Let us begin with the construction of the basis of the invariant subspace
${\cal I}$. The numerical simulations indicate that the state of the walk
$\mathinner{|{\phi(t)}\rangle}$ evolves periodically close to a state
corresponding to the loop on the marked vertex
$\mathinner{|{m_{1},m_{1}}\rangle}$. Hence, we consider this desired target
state of the SA as the first basis vector of the invariant subspace
$\mathinner{|{\nu_{1}}\rangle}=\mathinner{|{m_{1},m_{1}}\rangle}.$ (11)
Next, we add an equal weight superposition of all edges leaving the marked
vertex
$|\nu_{2}\rangle=\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}|m_{1},k_{\alpha}\rangle.$
(12)
Concerning the non-marked vertices in the first partition, we add two states
corresponding to a superposition of all loops and a superposition of all edges
leaving the first partition
$\displaystyle|\nu_{3}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(N-1)}}\sum_{j\neq m}^{N}|j_{1},j_{1}\rangle,$
$\displaystyle|\nu_{4}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d(N-1)}}\sum_{j\neq
m}^{N}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}|j_{1},k_{\alpha}\rangle.$
Next, we consider the edges leading to the first partition from the outside
and ending either on the marked or non-marked vertex, and construct the
following two basis states
$\displaystyle|\nu_{5}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}|k_{\alpha},m_{1}\rangle,$
(13) $\displaystyle|\nu_{6}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d(N-1)}}\sum_{j\neq
m}^{N}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}|k_{\alpha},j_{1}\rangle.$
These states can be obtained by applying the shift operator on
$\mathinner{|{\nu_{2}}\rangle}$ and $\mathinner{|{\nu_{4}}\rangle}$. To
complete the basis we consider the states corresponding to the superposition
of all edges between the vertices outside of the first partition
$|\nu_{7}\rangle=\frac{1}{\sqrt{d(d-N)}}\sum_{\begin{subarray}{c}\alpha,\beta=2\\\
\beta\neq\alpha\end{subarray}}^{M}\sum_{j,k=1}^{N}|j_{\alpha},k_{\beta}\rangle$
and the superposition of all remaining loops
$|\nu_{8}\rangle=\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}|k_{\alpha},k_{\alpha}\rangle.$
Clearly, the initial state of the SA (6) lies in ${\cal I}$ and has the
following form
$\displaystyle\mathinner{|{\Omega}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{MN}}\left[|\nu_{2}\rangle+|\nu_{5}\rangle+\sqrt{d-N}|\nu_{7}\rangle+\right.$
(14)
$\displaystyle\left.+\sqrt{N-1}(\mathinner{|{\nu_{4}}\rangle}+\mathinner{|{\nu_{6}}\rangle})\right].$
By direct calculation one can show that the evolution operator $\hat{U}_{m}$
of the SA acts on the basis states according to
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left((d-1)\mathinner{|{\nu_{1}}\rangle}-2\sqrt{d}\mathinner{|{\nu_{5}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{2}}\rangle}$ $\displaystyle=$
$\displaystyle-\frac{1}{d+1}\left(2\sqrt{d}\mathinner{|{\nu_{1}}\rangle}+(d-1)\mathinner{|{\nu_{5}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left((1-d)\mathinner{|{\nu_{3}}\rangle}+2\sqrt{d}\mathinner{|{\nu_{6}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{d}\mathinner{|{\nu_{3}}\rangle}+(d-1)\mathinner{|{\nu_{6}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left((1-d)\mathinner{|{\nu_{2}}\rangle}+2\sqrt{N-1}\mathinner{|{\nu_{4}}\rangle}+\right.$
$\displaystyle\left.+2\sqrt{d-N}\mathinner{|{\nu_{7}}\rangle}+2\mathinner{|{\nu_{8}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{N-1}\mathinner{|{\nu_{2}}\rangle}-(d+3-2N)\mathinner{|{\nu_{4}}\rangle}+\right.$
$\displaystyle+2\sqrt{(d-N)(N-1)}\mathinner{|{\nu_{7}}\rangle}+$
$\displaystyle\left.+2\sqrt{N-1}\mathinner{|{\nu_{8}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{d-N}\mathinner{|{\nu_{2}}\rangle}+\right.$
$\displaystyle\left.+2\sqrt{(d-N)(N-1)}\mathinner{|{\nu_{4}}\rangle}+\right.$
$\displaystyle\left.+(d-1-2N)\mathinner{|{\nu_{7}}\rangle}+2\sqrt{d-N}\mathinner{|{\nu_{8}}\rangle}\right),$
$\displaystyle\hat{U}_{m}\mathinner{|{\nu_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\mathinner{|{\nu_{2}}\rangle}+2\sqrt{N-1}\mathinner{|{\nu_{4}}\rangle}+\right.$
(15)
$\displaystyle\left.+2\sqrt{d-N}\mathinner{|{\nu_{7}}\rangle}+(1-d)\mathinner{|{\nu_{8}}\rangle}\right).$
Hence, $\cal I$ is indeed an invariant subspace of the SA and the state of the
walk $\mathinner{|{\phi(t)}\rangle}$ remains in $\cal I$ for all $t$. Its
evolution is determined by the eigenvalues and eigenvectors of $\hat{U}_{m}$,
which is in the invariant subspace represented by an $8\times 8$ unitary
matrix with matrix elements given by (15). While it is possible to diagonalize
this matrix analytically, the procedure is rather onerous and the resulting
expressions are quite lengthy. Nevertheless, the analysis can be considerably
simplified in the limit of a large graph, i.e., when $N\to\infty$ and
$M\to\infty$. As we can see from the expansion (14) for large $N$ and $M$ the
initial state of the algorithm tends to $\mathinner{|{\nu_{7}}\rangle}$.
Hence, the only eigenvectors of the evolution operator $\hat{U}_{m}$, which
remain relevant in the asymptotic limit, are those which have non-vanishing
overlap with $\mathinner{|{\nu_{7}}\rangle}$. It turns out that there are only
three such states and their asymptotic form is given by
$\displaystyle\mathinner{|{\psi_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\mathinner{|{\nu_{7}}\rangle}-\mathinner{|{\nu_{1}}\rangle}),$
$\displaystyle\mathinner{|{\psi_{2}^{(\pm)}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathinner{|{\nu_{1}}\rangle}+\mathinner{|{\nu_{7}}\rangle}\pm
i(\mathinner{|{\nu_{5}}\rangle}-\mathinner{|{\nu_{2}}\rangle})).$ (16)
It can be shown that for the other eigenvectors of $\hat{U}_{m}$ the overlap
with the initial state decreases at least as $O(1/\sqrt{NM})$. Let us turn to
the eigenvalues. The eigenvector $\mathinner{|{\psi_{1}}\rangle}$ corresponds
to $\lambda_{1}=1$. For $\mathinner{|{\psi_{2}^{(\pm)}}\rangle}$ the
eigenvalues have the form
$\lambda_{2}^{(\pm)}=e^{\pm i\omega_{2}}.$
From the characteristic polynomial of $\hat{U}_{m}$ we find that
$\cos\omega_{2}$ is given by the largest root of the quadratic equation
$\displaystyle
x^{2}-\left(1-\frac{N}{d+1}\right)x-\frac{(d+1)(N-2)+N-3}{{(d+1)}^{2}}=0.$
This leads us to
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\arccos\left(1-\frac{1+NM-\sqrt{N^{2}M^{2}-6NM+4N+5}}{2{(d+1)}}\right)$
(17) $\displaystyle\approx$ $\displaystyle\frac{2}{\sqrt{NM}}.$
From the relations (16) we express the initial and the target state of the SA
in terms of the eigenvectors of the evolution operator as
$\displaystyle\mathinner{|{\Omega}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{2}\left(\mathinner{|{\psi_{2}^{(+)}}\rangle}+\mathinner{|{\psi_{2}^{(-)}}\rangle}\right),$
$\displaystyle\mathinner{|{\nu_{1}}\rangle}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{2}\left(\mathinner{|{\psi_{2}^{(+)}}\rangle}+\mathinner{|{\psi_{2}^{(-)}}\rangle}\right).$
(18)
The state after $t$ iterations of the SA reads
$\mathinner{|{\phi(t)}\rangle}=\frac{1}{\sqrt{2}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{2}\left(e^{i\omega_{2}t}\mathinner{|{\psi_{2}^{(+)}}\rangle}+e^{-i\omega_{2}t}\mathinner{|{\psi_{2}^{(-)}}\rangle}\right).$
(19)
The success probability (7) of the SA after $t$ steps can be expressed in the
form
$P_{m}(t)=|\langle\nu_{1}|\phi(t)\rangle|^{2}+|\langle\nu_{2}|\phi(t)\rangle|^{2}.$
(20)
From (19) and (18) we see that the probability to find the walker in the
target state $\mathinner{|{\nu_{1}}\rangle}$ is given by
$|\langle\nu_{1}|\phi(t)\rangle|^{2}=\sin^{4}\left(\frac{\omega_{2}t}{2}\right).$
(21)
The probability to find the walker in the state
$\mathinner{|{\nu_{2}}\rangle}$ reads
$|\langle\nu_{2}|\phi(t)\rangle|^{2}=\frac{1}{4}\sin^{2}\left(\omega_{2}t\right).$
(22)
Put together we obtain the overall success probability of the SA
$P_{m}(t)=\sin^{2}\left(\frac{\omega_{2}t}{2}\right).$ (23)
We see that for $t=\frac{\pi}{\omega_{2}}$ the state of the SA is very close
to the target state $\mathinner{|{\nu_{1}}\rangle}$. Hence, the number of
steps needed to find the marked vertex with probability close to one is given
by
$T=\frac{\pi}{\omega_{2}}\approx\frac{\pi\sqrt{NM}}{2}+O\left(\frac{1}{\sqrt{NM}}\right).$
(24)
For illustration we plot in Figure 1 the probability to find the marked vertex
(23) as a function of the number of steps for a graph with $N=40$ and $M=100$.
Figure 1: Overall success probability of SA (black dots) and the probability
to find the walker in the target state $\mathinner{|{\nu_{1}}\rangle}$ (blue
squares) as a function of the number of steps $t$ for $N=40$ and $M=100$. Red
curves correspond to the analytical results (dashed line to eq. (21) and full
line to eq. (23) ). Green diamonds denote the probability that the walker is
on the marked vertex but not in the loop, which follows the curve (22). The
success probability is close to one after $T\approx 100$ steps, in accordance
with (24).
We note that the result (23) holds in the limit of large $N$ and $M$. To
investigate how quickly does the success probability at the optimal time (24)
approaches unity we performed numerical simulations for various values of $N$
and $M$. The simulations indicate that the success probability is essentially
independent of $N$ and with $M$ it scales according to
$P_{m}(T)=1-O\left(\frac{1}{M}\right).$
The results are illustrated in Figure 2.
Figure 2: Overall success probability of SA as a function of the number of
partitions $M$ for $N=10$ (gray circles), $N=50$ (blue triangles) and $N=100$
(brown diamonds). For a given $N$ and $M$ we evaluate numerically the
evolution of SA for the optimal number of steps $T$ given by (24) and
determine $P_{m}(T)$ from the formula (20). To unravel the scaling of the
success probability we plot $1-P_{m}(T)$ on the log-log scale. Independent of
the value of $N$, the sets of data-points fit well onto the $1/M$ slope
indicated by the red line.
## IV State transfer
We now turn to the analysis of the STA. There are two possible configurations
— the sender and the receiver are in the same partition or not. Numerical
simulations indicate that for the graph without loops the STA does not work
well. When the sender and the receiver are in the same partition the fidelity
does not surpass 0.25. For the second configuration the first maximum of
fidelity tends to 0.8. Hence, we turn to the graph with loops. Choosing the
initial state of the sender as the equal weight superposition (4), i.e.,
$\mathinner{|{s}\rangle}=\mathinner{|{\Omega_{s}}\rangle}$, the numerical
simulation reveals that the fidelity tends to 1 for the second configuration.
However, when the sender and the receiver are in the same partition the
fidelity is still limited. As we show in Figure 3 it does not surpass 0.35. A
careful analysis would reveal that the culprit are two orthogonal eigenvectors
of the evolution operator of the STA corresponding to the eigenvalue -1, one
having a large overlap with $\mathinner{|{\Omega_{s}}\rangle}$ and the other
one with $\mathinner{|{\Omega_{r}}\rangle}$. In the second configuration this
does not happen, as both $\mathinner{|{\Omega_{s}}\rangle}$ and
$\mathinner{|{\Omega_{r}}\rangle}$ have overlaps with the same eigenvectors of
the evolution operator of the STA. Hence, the absence of an edge between the
sender and the receiver vertex in the first configuration significantly limits
the achievable fidelity when we use the equal weight superposition state
$\mathinner{|{\Omega_{s}}\rangle}$ as the initial state of STA.
Figure 3: The evolution of the fidelity ${\cal F}$ of the state transfer
during $1000$ steps for $N=40$ and $M=100$. Sender and receiver are in the
same part. The initial state $\mathinner{|{s}\rangle}$ is the equal weight
superposition on the sender vertex $\mathinner{|{\Omega_{s}}\rangle}$. We see
that the fidelity does not grow over $0.35$.
We show that the fidelity of STA in the first configuration can be improved
considerably by choosing the initial state $\mathinner{|{s}\rangle}$ as the
loop on the sender vertex. Moreover, we prove that this initial state works
well also in the second configuration. In both configurations the walker will
be with high probability transferred to the loop at the receiver vertex. We
denote this receiver state as $\mathinner{|{r}\rangle}$.
### IV.1 Sender and receiver in the same partition
Let us first consider the case when the sender and the receiver are in the
same partition of the graph , i.e., they are not connected directly by an
edge. Without loss of generality we consider that they are in the first one.
We begin by constructing the basis of the exact invariant subspace. The
procedure is similar to the one for the SA, but we have to consider two marked
vertices corresponding to the sender $s$ and the receiver $r$. Hence, the
basis states of the form (11), (12), (13) will appear twice - once for $m=s$
and once for $m=r$. In the end we find the following 11 basis vectors
$\displaystyle\mathinner{|{\nu_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{s_{1},s_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{2}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{j=1}^{N}\mathinner{|{s_{1},j_{\alpha}}\rangle},$
$\displaystyle\mathinner{|{\nu_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{r_{1},r_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{j=1}^{N}\mathinner{|{r_{1},j_{\alpha}}\rangle},$
$\displaystyle\mathinner{|{\nu_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-2}}\sum_{j\neq
s,r}^{N}\mathinner{|{j_{1},j_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d(N-2)}}\sum_{j\neq
s,r}^{N}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}\mathinner{|{j_{1},k_{\alpha}}\rangle},$
$\displaystyle\mathinner{|{\nu_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{j=1}^{N}\mathinner{|{j_{\alpha},s_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{j=1}^{N}\mathinner{|{j_{\alpha},r_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{9}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d(N-2)}}\sum_{j\neq
s,r}^{N}\sum_{\alpha=2}^{M}\sum_{k=1}^{N}\mathinner{|{k_{\alpha},j_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{10}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d(d-N)}}\sum_{\alpha=2}^{M}\sum_{\beta=2,\beta\neq\alpha}^{M}\sum_{j,k=1}^{N}\mathinner{|{j_{\alpha},k_{\beta}}\rangle},$
$\displaystyle\mathinner{|{\nu_{11}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{\alpha=2}^{M}\sum_{j}^{N}\mathinner{|{j_{\alpha},j_{\alpha}}\rangle}$
(25)
Let us denote the subspace spanned by these vectors as $\cal I$. Clearly, it
contains the initial and the desired target state of the STA
($\mathinner{|{s}\rangle}=\mathinner{|{\nu_{1}}\rangle}$ and
$\mathinner{|{r}\rangle}=\mathinner{|{\nu_{3}}\rangle}$). It can be shown by
direct calculation that it is closed under the action of $\hat{U}_{s,r}$.
However, we will not provide the expression of $\hat{U}_{s,r}$ in this basis
because $\cal I$ can be divided further into two invariant subspaces. This
comes from a fact that the evolution of the STA is invariant with respect to
the exchange of the sender and the receiver vertex. Let us denote the operator
corresponding to this symmetry as $\hat{P}$. Clearly, it holds that
$\hat{P}^{2}=\hat{I},\quad[\hat{P},\hat{U}_{s,r}]=0.$
$\hat{P}$ acts on the basis states (25) as
$\displaystyle\hat{P}\mathinner{|{\nu_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{3}}\rangle},\quad\hat{P}\mathinner{|{\nu_{2}}\rangle}=\mathinner{|{\nu_{4}}\rangle},\quad\hat{P}\mathinner{|{\nu_{7}}\rangle}=\mathinner{|{\nu_{8}}\rangle},$
$\displaystyle\hat{P}\mathinner{|{\nu_{j}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{j}}\rangle},\quad j=5,6,9,10,11.$
Since $\hat{P}$ commutes with $\hat{U}_{s,r}$, they have common eigenvectors.
From $\hat{P}^{2}=\hat{I}$ we see that the spectrum of $\hat{P}$ consists of
two eigenvalues $1$ and $-1$. Hence, the invariant subspace ${\cal I}$ can be
split into two subspaces ${\cal I}_{+}$ and ${\cal I}_{-}$ which correspond to
eigenvalues $1$ and $-1$ of the operator $\hat{P}$. Basis of the invariant
subspace ${\cal I}_{+}$ is spanned by eigenstates denoted as
$\mathinner{|{\sigma_{i}}\rangle}$, $i=1,\ldots,8$ and it has the following
form
$\displaystyle\mathinner{|{\sigma_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{1}}\rangle}+\mathinner{|{\nu_{3}}\rangle}\right),$
$\displaystyle\mathinner{|{\sigma_{2}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{2}}\rangle}+\mathinner{|{\nu_{4}}\rangle}\right),$
$\displaystyle\mathinner{|{\sigma_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{5}}\rangle},$
$\displaystyle\mathinner{|{\sigma_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{6}}\rangle},$
$\displaystyle\mathinner{|{\sigma_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{7}}\rangle}+\mathinner{|{\nu_{8}}\rangle}\right),$
$\displaystyle\mathinner{|{\sigma_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{9}}\rangle},$
$\displaystyle\mathinner{|{\sigma_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{10}}\rangle},$
$\displaystyle\mathinner{|{\sigma_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{11}}\rangle}$
Note that if we perform SA for two marked vertices instead of STA, the
subspace ${\cal I}_{+}$ would be invariant with respect to the search. This is
due to the fact that the initial state of SA algorithm lies within this
subspace. Basis of the invariant subspace ${\cal I}_{-}$ is spanned by
eigenstates denoted as $\mathinner{|{\tau_{i}}\rangle}$, $i=1,\ldots,3$ and it
has the following form
$\displaystyle\mathinner{|{\tau_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{1}}\rangle}-\mathinner{|{\nu_{3}}\rangle}\right),$
$\displaystyle\mathinner{|{\tau_{2}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{2}}\rangle}-\mathinner{|{\nu_{4}}\rangle}\right),$
$\displaystyle\mathinner{|{\tau_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{7}}\rangle}-\mathinner{|{\nu_{8}}\rangle}\right)$
This subspace is needed only in STA since the initial state of SA is
orthogonal to this subspace.
The sender and the receiver states in the new basis read
$\displaystyle\mathinner{|{s}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\sigma_{1}}\rangle}+\mathinner{|{\tau_{1}}\rangle}\right),$
$\displaystyle\mathinner{|{r}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\sigma_{1}}\rangle}-\mathinner{|{\tau_{1}}\rangle}\right).$
(26)
The evolution operator in the new basis is block diagonal, i.e., ${\cal
I}_{i}$ are the invariant subspaces of $\hat{U}_{s,r}$. We find the following
relations for the basis states of ${\cal I}_{+}$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left((d-1)\mathinner{|{\sigma_{1}}\rangle}-2\sqrt{d}\mathinner{|{\sigma_{5}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{2}}\rangle}$ $\displaystyle=$
$\displaystyle-\frac{1}{d+1}\left(2\sqrt{d}\mathinner{|{\sigma_{1}}\rangle}+(d-1)\mathinner{|{\sigma_{5}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left((1-d)\mathinner{|{\sigma_{3}}\rangle}+2\sqrt{d}\mathinner{|{\sigma_{6}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{d}\mathinner{|{\sigma_{3}}\rangle}+(d-1)\mathinner{|{\sigma_{6}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left((3-d)\mathinner{|{\sigma_{2}}\rangle}+2\sqrt{2(N-2)}\mathinner{|{\sigma_{4}}\rangle}+2\sqrt{2(d-N)}\mathinner{|{\sigma_{7}}\rangle}+2\sqrt{2}\mathinner{|{\sigma_{8}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{2(N-2)}\mathinner{|{\sigma_{2}}\rangle}-\left(d-2N+5\right)\mathinner{|{\sigma_{4}}\rangle}+2\sqrt{(d-N)(N-2)}\mathinner{|{\sigma_{7}}\rangle}+2\sqrt{N-2}\mathinner{|{\sigma_{8}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{2(d-N)}\mathinner{|{\sigma_{2}}\rangle}+2\sqrt{(d-N)(N-2)}\mathinner{|{\sigma_{4}}\rangle}+\left(d-2N-1\right)\mathinner{|{\sigma_{7}}\rangle}+2\sqrt{d-N}\mathinner{|{\sigma_{8}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{2}\mathinner{|{\sigma_{2}}\rangle}+2\sqrt{N-2}\mathinner{|{\sigma_{4}}\rangle}+2\sqrt{d-N}\mathinner{|{\sigma_{7}}\rangle}-\left(d-1\right)\mathinner{|{\sigma_{8}}\rangle}\right).$
(27)
In the second subspace ${\cal I}_{-}$ the evolution operator acts according to
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(\left(d-1\right)\mathinner{|{\tau_{1}}\rangle}-2\sqrt{d}\mathinner{|{\tau_{3}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{2}}\rangle}$ $\displaystyle=$
$\displaystyle-\frac{1}{d+1}\left(2\sqrt{d}\mathinner{|{\tau_{1}}\rangle}+\left(d-1\right)\mathinner{|{\tau_{3}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{3}}\rangle}$ $\displaystyle=$
$\displaystyle-\mathinner{|{\tau_{2}}\rangle}.$ (28)
Let us now investigate the dynamics of the STA in the limit of a large graph.
We denote by $\hat{U}_{\pm}$ the restriction of $\hat{U}_{s,r}$ on ${\cal
I}_{\pm}$, and determine the spectrum and eigenvectors of these operators. For
$\hat{U}_{+}$ the results are similar to those for the SA. In the subspace
${\cal I}_{+}$ three relevant eigenvectors remain, as for the others the
overlap with $\mathinner{|{\sigma_{1}}\rangle}$ tends to zero at least as
$O(1/\sqrt{NM})$. The limit form of the relevant eigenvectors is given by
$\displaystyle\mathinner{|{\psi_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\mathinner{|{\sigma_{1}}\rangle}-\frac{1}{\sqrt{3}}\mathinner{|{\sigma_{7}}\rangle},$
$\displaystyle\mathinner{|{\psi_{2}^{(\pm)}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\mathinner{|{\sigma_{1}}\rangle}+\frac{1}{\sqrt{3}}\mathinner{|{\sigma_{7}}\rangle}\pm\frac{i}{2}(\mathinner{|{\sigma_{5}}\rangle}-\mathinner{|{\sigma_{2}}\rangle}).$
(29)
The eigenvector $\mathinner{|{\psi_{1}}\rangle}$ corresponds to
$\lambda_{1}=1$. In the case of $\mathinner{|{\psi_{2}^{(\pm)}}\rangle}$ the
eigenvalues have the form
$\lambda_{2}^{(\pm)}=e^{\pm i\omega_{2}}.$
From the characteristic polynomial of $\hat{U}_{+}$ we find that
$\cos\omega_{2}$ is the largest root of the quadratic equation
$x^{2}-\left(1-\frac{N}{d+1}\right)x-\frac{{(d+1)}(N-3)+N-5}{{(d+1)}^{2}}=0,$
which leads us to
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\arccos\left(1-\frac{1+NM-\sqrt{N^{2}M^{2}-10NM+8N+9}}{2{(d+1)}}\right)$
(30) $\displaystyle\approx$ $\displaystyle\sqrt{\frac{6}{NM}}.$
In the subspace $\cal I_{-}$ there are two additional relevant eigenvectors in
the asymptotic limit
$\mathinner{|{\psi_{3}^{(\pm)}}\rangle}=\frac{1}{\sqrt{2}}\mathinner{|{\tau_{1}}\rangle}\pm\frac{i}{2}(\mathinner{|{\tau_{2}}\rangle}-\mathinner{|{\tau_{3}}\rangle}).$
(31)
For the last eigenvector of $\hat{U}_{-}$ the overlap with the state
$\mathinner{|{\tau_{1}}\rangle}$ behaves like $O(1/\sqrt{NM})$. The relevant
eigenvalues have the form
$\displaystyle\lambda_{3}^{(\pm)}$ $\displaystyle=$ $\displaystyle e^{\pm
i\omega_{3}},$
with
$\omega_{3}=\arccos\left(1-\frac{1}{{d+1}}\right)\approx\sqrt{\frac{2}{NM}}.$
(32)
Using the results (29), (31) we see that for a large graph the sender and the
receiver states (26) can be decomposed into the eigenvectors of the evolution
operator $\hat{U}_{s,r}$ according to
$\displaystyle\mathinner{|{s}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{\sqrt{12}}\left(\mathinner{|{\psi_{2}^{(+)}}\rangle}+\mathinner{|{\psi_{2}^{(-)}}\rangle}\right)+$
$\displaystyle+\frac{1}{2}\left(\mathinner{|{\psi_{3}^{(+)}}\rangle}+\mathinner{|{\psi_{3}^{(-)}}\rangle}\right),$
$\displaystyle\mathinner{|{r}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{\sqrt{12}}\left(\mathinner{|{\psi_{2}^{(+)}}\rangle}+\mathinner{|{\psi_{2}^{(-)}}\rangle}\right)-$
$\displaystyle-\frac{1}{2}\left(\mathinner{|{\psi_{3}^{(+)}}\rangle}+\mathinner{|{\psi_{3}^{(-)}}\rangle}\right).$
Hence, the evolution of STA takes place in a five dimensional subspace
$\displaystyle\mathinner{|{\phi(t)}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\mathinner{|{\psi_{1}}\rangle}+$ (33)
$\displaystyle+\frac{1}{\sqrt{12}}\left(e^{i\omega_{2}t}\mathinner{|{\psi_{2}^{(+)}}\rangle}+e^{-i\omega_{2}t}\mathinner{|{\psi_{2}^{(-)}}\rangle}\right)+$
$\displaystyle+\frac{1}{2}\left(e^{i\omega_{3}t}\mathinner{|{\psi_{3}^{(+)}}\rangle}+e^{-i\omega_{3}t}\mathinner{|{\psi_{3}^{(-)}}\rangle}\right).$
The fidelity of STA can be written as a sum
${\cal F}(t)=|\langle
r|\phi(t)\rangle|^{2}+|\langle\nu_{4}|\phi(t)\rangle|^{2},$ (34)
of probabilities that the walker is in the receiver state
$\mathinner{|{r}\rangle}=\mathinner{|{\nu_{3}}\rangle}$ corresponding to the
loop, or at the receiver vertex but not in the loop, i.e., the state
$\mathinner{|{\nu_{4}}\rangle}$. From the relations (29), (31) and (33) we
find that these probabilities are given by
$\displaystyle|\langle r|\phi(t)\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left(2+\cos\left(\omega_{2}t\right)-3\cos\left(\omega_{3}t\right)\right)^{2},$
(35) $\displaystyle|\langle\nu_{4}|\phi(t)\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{24}\left(\sin(\omega_{2}t)-\sqrt{3}\sin(\omega_{3}t)\right)^{2}.$
(36)
From the relations (30), (32) we see that the frequencies are not harmonic,
since for a large graph
$\omega_{2}=\sqrt{3}\omega_{3}.$
The overall fidelity of STA is then given by
$\displaystyle{\cal F}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{36}\left(2+\cos\left(\sqrt{3}\omega_{3}t\right)-3\cos\left(\omega_{3}t\right)\right)^{2}+$
(37)
$\displaystyle+\frac{1}{24}\left(\sin(\sqrt{3}\omega_{3}t)-\sqrt{3}\sin(\omega_{3}t)\right)^{2}.$
The fidelity of STA will not reach one exactly. However, for a large graph the
first maximum of fidelity reaches the value
${\cal F}_{1}\approx 0.94.$ (38)
The number of steps required to reach the first maximum is approximately given
by
$T^{(st)}\approx 2.39\sqrt{NM}.$ (39)
At this time the walker is with high probability in the receiver state
$\mathinner{|{r}\rangle}$.
For illustration we show in Figure 4 the evolution of fidelity for a graph
with $N=40$ and $M=100$.
Figure 4: Fidelity of the state transfer as a function of the number of steps
$t$ for $N=40$ and $M=100$. The sender and the receiver vertices are in the
same partition. Black dots are obtained from the numerical simulation, the
full red curve corresponds to (37). Since the frequencies (30), (32) are not
integer multiples the fidelity behaves an-harmonically. At the time of the
first maximum (39) the walker is with high probability in the receiver state
$\mathinner{|{r}\rangle}$, which is depicted by the blue squares. The
probability to be in the receiver state follows the curve (35) represented by
the red dashed curve. The green diamonds correspond to the probability that
the walker is at the marked vertex but not in the loop, which follows the
curve (36).
The fidelity (38) in the first maximum is reached in the limit of large $N$
and $M$. We have performed numerical simulations to investigate how quickly
does the fidelity at the optimal time (39) approaches the asymptotic value
(38). Similarly to the results for the SA, the simulations indicate that the
fidelity is essentially independent of $N$ and with $M$ it scales according to
${\cal F}(T^{(st)})={\cal F}_{1}-O\left(\frac{1}{M}\right).$
The results are illustrated in Figure 5.
Figure 5: Overall fidelity of STA as a function of the number of partitions
$M$ for $N=10$ (gray circles), $N=50$ (blue triangles) and $N=100$ (brown
diamonds). The sender and the receiver vertices are in the same partition. For
a given $N$ and $M$ we evaluate numerically the evolution of STA for the
number of steps $T^{(st)}$ needed to reach the first maximum (39) and
determine ${\cal F}(T^{(st)})$ according to (34). To unravel the scaling of
the fidelity we plot ${\cal F}_{1}-{\cal F}(T^{(st)})$ on the log-log scale.
The data-points follow the $1/M$ slope indicated by the red line, with almost
no dependence on $N$.
### IV.2 Sender and receiver in different partitions
Let us now turn to the case when the sender and the receiver are in different
parts of the complete M-partite graph. Without loss of generality we label the
partition containing the sender as 1 and the partition with the receiver as 2.
The construction of the basis of the invariant subspace $\cal I$ is more
involved, since we have to consider the second partition with the receiver
vertex separately from the rest of the graph. We begin with the states
starting at the sender vertex
$\displaystyle\mathinner{|{\nu_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{s_{1},s_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{2}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{s_{1},r_{2}}\rangle},$
$\displaystyle\mathinner{|{\nu_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-1}}\sum_{j\neq
r}^{N}\mathinner{|{s_{1},j_{2}}\rangle},$
$\displaystyle\mathinner{|{\nu_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d-N}}\sum_{\alpha=3}^{M}\sum_{j=1}^{N}\mathinner{|{s_{1},j_{\alpha}}\rangle},$
(40)
which correspond to the loop, edge from the sender to the receiver, equal
weight superposition of all edges from the sender to the remaining vertices in
the second partition, and edges to all remaining vertices. We repeat the same
for the receiver vertex
$\displaystyle\mathinner{|{\nu_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{r_{2},r_{2}}\rangle},$
$\displaystyle\mathinner{|{\nu_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{r_{2},s_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-1}}\sum_{j\neq
s}^{N}\mathinner{|{r_{2},j_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d-N}}\sum_{\alpha=3}^{M}\sum_{j=1}^{N}\mathinner{|{r_{2},j_{\alpha}}\rangle}$
(41)
Next, we consider the same edges but starting at the non-marked vertices in
the first partition and prepare the following superpositions
$\displaystyle\mathinner{|{\nu_{9}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-1}}\sum_{j\neq
s}^{N}\mathinner{|{j_{1},j_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{10}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-1}}\sum_{j\neq
s}^{N}\mathinner{|{j_{1},r_{2}}\rangle},$
$\displaystyle\mathinner{|{\nu_{11}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{N-1}\sum_{j\neq s}^{N}\sum_{k\neq
r}^{N}\mathinner{|{j_{1},k_{2}}\rangle},$
$\displaystyle\mathinner{|{\nu_{12}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(d-N)(N-1)}}\sum_{j\neq
s}^{N}\sum_{\alpha=3}^{M}\sum_{k=1}^{N}\mathinner{|{j_{1},k_{\alpha}}\rangle}.$
(42)
The same procedure is repeated in the second partition
$\displaystyle\mathinner{|{\nu_{13}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-1}}\sum_{j\neq
r}^{N}\mathinner{|{j_{2},j_{2}}\rangle},$
$\displaystyle\mathinner{|{\nu_{14}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N-1}}\sum_{j\neq
r}^{N}\mathinner{|{j_{2},s_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{15}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{N-1}\sum_{j\neq r}^{N}\sum_{k\neq
s}^{N}\mathinner{|{j_{2},k_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{16}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(d-N)(N-1)}}\sum_{j\neq
r}^{N}\sum_{\alpha=3}^{M}\sum_{k=1}^{N}\mathinner{|{j_{2},k_{\alpha}}\rangle}.$
(43)
Next, we consider the edges leading to the sender or the receiver vertex from
the outside of the first two partitions
$\displaystyle\mathinner{|{\nu_{17}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d-N}}\sum_{\alpha=3}^{M}\sum_{j=1}^{N}\mathinner{|{j_{\alpha},s_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{18}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d-N}}\sum_{\alpha=3}^{M}\sum_{j=1}^{N}\mathinner{|{j_{\alpha},r_{2}}\rangle}.$
(44)
Then we add states corresponding to edges leading to the non-marked vertices
in the first or the second partition from the outside
$\displaystyle\mathinner{|{\nu_{19}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(d-N)(N-1)}}\sum_{j\neq
s}^{N}\sum_{\alpha=3}^{M}\sum_{k=1}^{N}\mathinner{|{k_{\alpha},j_{1}}\rangle},$
$\displaystyle\mathinner{|{\nu_{20}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(d-N)(N-1)}}\sum_{j\neq
r}^{N}\sum_{\alpha=3}^{M}\sum_{k=1}^{N}\mathinner{|{k_{\alpha},j_{2}}\rangle}.$
(45)
Finally, we consider all edges between vertices in the rest of the graph, and
all remaining loops
$\displaystyle\mathinner{|{\nu_{21}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N(d-N)(M-3)}}\sum_{\alpha=3}^{M}\sum_{\beta=3,\beta\neq\alpha}^{M}\sum_{j,k=1}^{N}\mathinner{|{j_{\alpha},k_{\beta}}\rangle},$
$\displaystyle\mathinner{|{\nu_{22}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d-N}}\sum_{\alpha=3}^{M}\sum_{j=1}^{N}\mathinner{|{j_{\alpha},j_{\alpha}}\rangle}$
(46)
It can be shown by direct calculation that the 22 vectors (40)-(46) constitute
an invariant subspace of the STA. To proceed further we employ the symmetry
$\hat{P}$ which switches the sender and the receiver partitions. Its action on
the basis states $\mathinner{|{\nu_{j}}\rangle}$ is given by
$\displaystyle\hat{P}\mathinner{|{\nu_{j}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{j+4}}\rangle},\quad j=1,2,3,4,9,10,11,12,$
$\displaystyle\hat{P}\mathinner{|{\nu_{i}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{i+1}}\rangle},\quad i=17,19,$
$\displaystyle\hat{P}\mathinner{|{\nu_{k}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{k}}\rangle},\quad k=21,22.$
Since $\hat{P}$ commutes with the evolution operator of the STA we can split
$\cal I$ into subspaces ${\cal I}_{\pm}$ corresponding to eigenvalues $\pm 1$.
Subspace ${\cal I}_{+}$ has dimension $12$ and it is spanned by the following
eigenvectors of $\hat{P}$
$\displaystyle\mathinner{|{\sigma_{i}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{i}}\rangle}+\mathinner{|{\nu_{i+4}}\rangle}\right),\quad
i=1,2,3,4$ $\displaystyle\mathinner{|{\sigma_{j}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{j+4}}\rangle}+\mathinner{|{\nu_{j+8}}\rangle}\right),\quad
j=5,6,7,8$ $\displaystyle\mathinner{|{\sigma_{9}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{17}}\rangle}+\mathinner{|{\nu_{18}}\rangle}\right),$
$\displaystyle\mathinner{|{\sigma_{10}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{19}}\rangle}+\mathinner{|{\nu_{20}}\rangle}\right),$
$\displaystyle\mathinner{|{\sigma_{11}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{21}}\rangle},$
$\displaystyle\mathinner{|{\sigma_{12}}\rangle}$ $\displaystyle=$
$\displaystyle\mathinner{|{\nu_{22}}\rangle}$
Subspace ${\cal I}_{-}$ has dimension $10$ and it is spanned by the following
eigenvectors of $\hat{P}$ corresponding to the eigenvalue $-1$
$\displaystyle\mathinner{|{\tau_{i}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{i}}\rangle}-\mathinner{|{\nu_{i+4}}\rangle}\right),\quad
i=1,2,3,4$ $\displaystyle\mathinner{|{\tau_{j}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{j+4}}\rangle}-\mathinner{|{\nu_{j+8}}\rangle}\right),\quad
j=5,6,7,8$ $\displaystyle\mathinner{|{\tau_{9}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{17}}\rangle}-\mathinner{|{\nu_{18}}\rangle}\right),$
$\displaystyle\mathinner{|{\tau_{10}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\nu_{19}}\rangle}-\mathinner{|{\nu_{20}}\rangle}\right)$
In the new basis the sender and the receiver states have the following form
$\displaystyle\mathinner{|{s}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\sigma_{1}}\rangle}+\mathinner{|{\tau_{1}}\rangle}\right),$
$\displaystyle\mathinner{|{r}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\mathinner{|{\sigma_{1}}\rangle}-\mathinner{|{\tau_{1}}\rangle}\right).$
(47)
The evolution operator $\hat{U}_{s,r}$ is block diagonal. We find the
following relations for the basis vectors of ${\cal I}_{+}$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(\left(d-1\right)\mathinner{|{\sigma_{1}}\rangle}-2\mathinner{|{\sigma_{2}}\rangle}-2\sqrt{N-1}\mathinner{|{\sigma_{6}}\rangle}-2\sqrt{d-N}\mathinner{|{\sigma_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{2}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\mathinner{|{\sigma_{1}}\rangle}+\left(d-1\right)\mathinner{|{\sigma_{2}}\rangle}-2\sqrt{N-1}\mathinner{|{\sigma_{6}}\rangle}-2\sqrt{d-N}\mathinner{|{\sigma_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\sqrt{N-1}\mathinner{|{\sigma_{1}}\rangle}-2\sqrt{N-1}\mathinner{|{\sigma_{2}}\rangle}+\left(d-2N+3\right)\mathinner{|{\sigma_{6}}\rangle}-2\sqrt{(d-N)(N-1)}\mathinner{|{\sigma_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\sqrt{d-N}\mathinner{|{\sigma_{1}}\rangle}-2\sqrt{d-N}\mathinner{|{\sigma_{2}}\rangle}-2\sqrt{(d-N)(N-1)}\mathinner{|{\sigma_{6}}\rangle}-\left(d-2N-1\right)\mathinner{|{\sigma_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\mathinner{|{\sigma_{3}}\rangle}-\left(d-1\right)\mathinner{|{\sigma_{5}}\rangle}+2\sqrt{N-1}\mathinner{|{\sigma_{7}}\rangle}+2\sqrt{d-N}\mathinner{|{\sigma_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-\left(d-1\right)\mathinner{|{\sigma_{3}}\rangle}+2\mathinner{|{\sigma_{5}}\rangle}+2\sqrt{N-1}\mathinner{|{\sigma_{7}}\rangle}+2\sqrt{d-N}\mathinner{|{\sigma_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{N-1}\mathinner{|{\sigma_{3}}\rangle}+2\sqrt{N-1}\mathinner{|{\sigma_{5}}\rangle}-\left(d-2N+3\right)\mathinner{|{\sigma_{7}}\rangle}+2\sqrt{(d-N)(N-1)}\mathinner{|{\sigma_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{d-N}\mathinner{|{\sigma_{3}}\rangle}+2\sqrt{d-N}\mathinner{|{\sigma_{5}}\rangle}+2\sqrt{(d-N)(N-1)}\mathinner{|{\sigma_{7}}\rangle}+\left(d-2N-1\right)\mathinner{|{\sigma_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{9}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-\left(d-3\right)\mathinner{|{\sigma_{4}}\rangle}+4\sqrt{N-1}\mathinner{|{\sigma_{8}}\rangle}+2\sqrt{2N(M-3)}\mathinner{|{\sigma_{11}}\rangle}+2\sqrt{2}\mathinner{|{\sigma_{12}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{10}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(4\sqrt{N-1}\mathinner{|{\sigma_{4}}\rangle}-\left(d-4N+5\right)\mathinner{|{\sigma_{8}}\rangle}+2\sqrt{2N(N-1)(M-3)}\mathinner{|{\sigma_{11}}\rangle}+2\sqrt{2(N-1)}\mathinner{|{\sigma_{12}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{11}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{2N(M-3)}\mathinner{|{\sigma_{4}}\rangle}+2\sqrt{2N(N-1)(M-3)}\mathinner{|{\sigma_{8}}\rangle}+\left(d-4N-1\right)\mathinner{|{\sigma_{11}}\rangle}+2\sqrt{N(M-3)}\mathinner{|{\sigma_{12}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\sigma_{12}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(2\sqrt{2}\mathinner{|{\sigma_{4}}\rangle}+2\sqrt{2(N-1)}\mathinner{|{\sigma_{8}}\rangle}+2\sqrt{N(M-3)}\mathinner{|{\sigma_{11}}\rangle}-\left(d-1\right)\mathinner{|{\sigma_{12}}\rangle}\right).$
The action of the evolution operator on the basis vectors of ${\cal I}_{-}$
reads
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(\left(d-1\right)\mathinner{|{\tau_{1}}\rangle}+2\mathinner{|{\tau_{2}}\rangle}+2\sqrt{N-1}\mathinner{|{\tau_{6}}\rangle}-2\sqrt{d-N}\mathinner{|{\tau_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{2}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\mathinner{|{\tau_{1}}\rangle}-\left(d-1\right)\mathinner{|{\tau_{2}}\rangle}+2\sqrt{N-1}\mathinner{|{\tau_{6}}\rangle}-2\sqrt{d-N}\mathinner{|{\tau_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{3}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\sqrt{N-1}\mathinner{|{\tau_{1}}\rangle}+2\sqrt{N-1}\mathinner{|{\tau_{2}}\rangle}-\left(d-2N+3\right)\mathinner{|{\tau_{6}}\rangle}-2\sqrt{(d-N)(N-1)}\mathinner{|{\tau_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{4}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\sqrt{d-N}\mathinner{|{\tau_{1}}\rangle}+2\sqrt{d-N}\mathinner{|{\tau_{2}}\rangle}+2\sqrt{(d-N)(N-1)}\mathinner{|{\tau_{6}}\rangle}-\left(d-2N-1\right)\mathinner{|{\tau_{9}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{5}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\mathinner{|{\tau_{3}}\rangle}-\left(d-1\right)\mathinner{|{\tau_{5}}\rangle}-2\sqrt{N-1}\mathinner{|{\tau_{7}}\rangle}+2\sqrt{d-N}\mathinner{|{\tau_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{6}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(\left(d-1\right)\mathinner{|{\tau_{3}}\rangle}+2\mathinner{|{\tau_{5}}\rangle}-2\sqrt{N-1}\mathinner{|{\tau_{7}}\rangle}+2\sqrt{d-N}\mathinner{|{\tau_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{7}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\sqrt{N-1}\mathinner{|{\tau_{3}}\rangle}+2\sqrt{N-1}\mathinner{|{\tau_{5}}\rangle}+\left(d-2N+3\right)\mathinner{|{\tau_{7}}\rangle}+2\sqrt{(d-N)(N-1)}\mathinner{|{\tau_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{8}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{d+1}\left(-2\sqrt{d-N}\mathinner{|{\tau_{3}}\rangle}+2\sqrt{d-N}\mathinner{|{\tau_{5}}\rangle}-2\sqrt{(d-N)(N-1)}\mathinner{|{\tau_{7}}\rangle}+\left(d-2N-1\right)\mathinner{|{\tau_{10}}\rangle}\right),$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{9}}\rangle}$ $\displaystyle=$
$\displaystyle-\mathinner{|{\tau_{4}}\rangle},$
$\displaystyle\hat{U}_{s,r}\mathinner{|{\tau_{10}}\rangle}$ $\displaystyle=$
$\displaystyle-\mathinner{|{\tau_{8}}\rangle}.$ (48)
To investigate the dynamics of STA in more detail we again turn to the limit
of a large graph. We denote by $\hat{U}_{\pm}$ the restriction of
$\hat{U}_{s,r}$ on ${\cal I}_{\pm}$. For $\hat{U}_{+}$ there are three
eigenstates which have non-vanishing overlap with the state
$\mathinner{|{\sigma_{1}}\rangle}$, namely
$\displaystyle\mathinner{|{\psi_{1}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\mathinner{|{\sigma_{1}}\rangle}-\frac{1}{2\sqrt{3}}\mathinner{|{\sigma_{2}}\rangle}-\frac{1}{\sqrt{6}}\mathinner{|{\sigma_{11}}\rangle},$
(49) $\displaystyle\mathinner{|{\psi_{2}^{(\pm)}}\rangle}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{8}}(\mathinner{|{\sigma_{1}}\rangle}+\mathinner{|{\sigma_{2}}\rangle})\pm\frac{i}{2}(\mathinner{|{\sigma_{9}}\rangle}-\mathinner{|{\sigma_{4}}\rangle})+\frac{1}{2}\mathinner{|{\sigma_{11}}\rangle}.$
Note that for the other eigenstates the overlap with
$\mathinner{|{\sigma_{1}}\rangle}$ decreases at least as $O(1/\sqrt{NM})$.
Turning to the eigenvalues, we find that the eigenvector
$\mathinner{|{\psi_{1}}\rangle}$ has eigenvalue $\lambda_{1}=1$. From the
characteristic polynomial of $\hat{U}_{+}$ we find that the eigenvalues of
$\mathinner{|{\psi_{2}^{(\pm)}}\rangle}$ have the form
$\lambda_{2}^{(\pm)}=e^{\pm i\omega_{2}}$, where $\cos\omega_{2}$ is the
largest root of the cubic equation
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
x^{3}-\left(1-\frac{N+2}{d+1}\right)x^{2}-\frac{{(d+1)}(N-1)-N+5}{{(d+1)}^{2}}x+$
$\displaystyle+\frac{NM-4}{{(d+1)}^{2}}.$
We find that it has the following asymptotic form
$\omega_{2}\approx\arccos\left(1-\frac{4}{NM}\right)\approx
2\sqrt{\frac{2}{NM}}.$ (50)
Considering the subspace ${\cal I}_{-}$, there are two eigenvectors which
remain relevant in the asymptotic limit (for the others the overlap with
$\mathinner{|{\tau_{1}}\rangle}$ vanishes at least as $O(1/\sqrt{NM})$),
namely
$\mathinner{|{\psi_{3}^{(\pm)}}\rangle}=\frac{1}{\sqrt{2}}\mathinner{|{\tau_{1}}\rangle}\pm\frac{i}{2}(\mathinner{|{\tau_{9}}\rangle}-\mathinner{|{\tau_{4}}\rangle}).$
(51)
The eigenvalues are $\lambda_{3}^{(\pm)}=e^{\pm i\omega_{3}}$, where
$\cos\omega_{3}$ is the largest root of the quartic equation
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
x^{4}+\frac{N-2}{d+1}x^{3}-\left(1-\frac{NM}{{(d+1)}^{2}}\right)x^{2}-$
$\displaystyle-\frac{(N-2)(d-1)}{{(d+1)}^{2}}x+\frac{N(M-2)}{{(d+1)}^{2}}.$
Its asymptotic form is given by
$\omega_{3}\approx\arccos\left(1-\frac{1}{NM}\right)\approx\sqrt{\frac{2}{NM}}.$
(52)
From (49), (51) we see that for a large graph the sender and the receiver
states can be decomposed into the eigenvectors of the evolution operator
$\hat{U}_{s,r}$ according to
$\displaystyle\mathinner{|{s}\rangle}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{8}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{4}\left(\mathinner{|{\psi_{2}^{(+)}}\rangle}+\mathinner{|{\psi_{2}^{(-)}}\rangle}\right)+$
$\displaystyle+\frac{1}{2}\left(\mathinner{|{\psi_{3}^{(+)}}\rangle}+\mathinner{|{\psi_{3}^{(-)}}\rangle}\right),$
$\displaystyle\mathinner{|{r}\rangle}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{8}}\mathinner{|{\psi_{1}}\rangle}+\frac{1}{4}\left(\mathinner{|{\psi_{2}^{(+)}}\rangle}+\mathinner{|{\psi_{2}^{(-)}}\rangle}\right)-$
$\displaystyle-\frac{1}{2}\left(\mathinner{|{\psi_{3}^{(+)}}\rangle}+\mathinner{|{\psi_{3}^{(-)}}\rangle}\right).$
The evolution of STA takes place in a five dimensional subspace
$\displaystyle\mathinner{|{\phi(t)}\rangle}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{8}}\mathinner{|{\psi_{1}}\rangle}+$ (53)
$\displaystyle+\frac{1}{4}\left(e^{i\omega_{2}t}\mathinner{|{\psi_{2}^{(+)}}\rangle}+e^{-i\omega_{2}t}\mathinner{|{\psi_{2}^{(-)}}\rangle}\right)+$
$\displaystyle+\frac{1}{2}\left(e^{i\omega_{3}t}\mathinner{|{\psi_{3}^{(+)}}\rangle}+e^{-i\omega_{3}t}\mathinner{|{\psi_{3}^{(-)}}\rangle}\right).$
The fidelity of STA after $t$ steps (9) can be expressed as a sum
${\cal F}(t)=\sum_{j=5}^{8}|\langle\nu_{j}|\phi(t)\rangle|^{2}.$ (54)
From (49), (51) and (53) we find that the transfer probabilities to individual
states $\mathinner{|{\nu_{j}}\rangle}$ are given by
$\displaystyle|\langle\nu_{5}|\phi(t)\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{64}\left(3+\cos\left(\omega_{2}t\right)-4\cos\left(\omega_{3}t\right)\right)^{2},$
$\displaystyle|\langle\nu_{6}|\phi(t)\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\sin^{4}\left(\frac{\omega_{2}t}{2}\right),$
$\displaystyle|\langle\nu_{7}|\phi(t)\rangle|^{2}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle|\langle\nu_{8}|\phi(t)\rangle|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{32}\left(\sin(\omega_{2}t)-2\sin(\omega_{3}t)\right)^{2}.$
(55)
From the asymptotic expansions (50), (52) we see that for a large graph the
frequencies are harmonic
$\omega_{2}=2\omega_{3}.$
Hence, we find that with probability
$|\langle r|\phi(t)\rangle|^{2}=\sin^{8}{\left(\frac{\omega_{3}t}{2}\right)},$
(56)
the walker is in the receiver state
$\mathinner{|{r}\rangle}=\mathinner{|{\nu_{5}}\rangle}$, i.e., at the receiver
vertex in the loop, and with probability
$\displaystyle|\langle\nu_{6}|\phi(t)\rangle|^{2}+|\langle\nu_{8}|\phi(t)\rangle|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sin^{2}\left(\omega_{3}t\right)\sin^{4}\left(\frac{\omega_{3}t}{2}\right)+$
(57) $\displaystyle+\frac{1}{16}\sin^{4}\left(\omega_{3}t\right),$
it is at the receiver vertex but not in the loop. Overall, the fidelity of STA
for a large graph is given by
${\cal F}(t)=\sin^{4}\left(\frac{\omega_{3}t}{2}\right).$ (58)
We conclude that the state transfer is achieved after $T^{(st)}$ steps, where
$T^{(st)}\approx\pi\sqrt{\frac{NM}{2}}.$ (59)
At this time the walker is with high probability in the receiver state
$\mathinner{|{r}\rangle}$.
For illustration we show in Figure 6 the evolution of fidelity for a graph
with $N=40$ and $M=100$.
Figure 6: Overall fidelity of the STA as a function of the number of steps $t$
for $N=40$ and $M=100$. The sender and the receiver vertices are in different
partitions. Black dots are obtained from the numerical simulation, the full
red curve corresponds to (58). The walker is transferred to the receiver
vertex with fidelity close to one after $T^{(st)}\approx 140$ steps, in
accordance with (59). At this time the walker is found with high probability
in the loop (blue squares), as follows from the analytical prediction (56)
depicted by the red dashed curve. Green diamonds represent the probability
that the walker is at the marked vertex but not in the loop, which follows the
curve (57).
The result (58) holds in the limit of large $N$ and $M$. To investigate how
quickly does the fidelity at the optimal time (59) approaches unity we
performed numerical simulations for various values of $N$ and $M$. The
simulations indicate that the fidelity can be again estimated by
${\cal F}(T^{(st)})=1-O\left(\frac{1}{M}\right),$
however, the dependence on $N$ is more complex than for search and STA with
the sender and the receiver in the same partition. The results are illustrated
in Figure 7.
Figure 7: Overall fidelity of STA as a function of the number of partitions
$M$ for $N=10$ (gray circles), $N=50$ (blue triangles) and $N=100$ (brown
diamonds). The sender and the receiver vertices are in different partitions.
For a given $N$ and $M$ we evaluate numerically the evolution of STA for the
optimal number of steps $T^{(st)}$ given by (59) and determine ${\cal
F}(T^{(st)})$ from the formula (54). To unravel the scaling of the fidelity we
plot $1-{\cal F}(T^{(st)})$ on the log-log scale. The full red line has the
$1/M$ slope, while the dashed red line follows $1/M^{2}$. The plot indicates
that $O(1/M)>1-{\cal F}(T^{(st)})>O(1/M^{2})$, and that the fluctuations
decrease with increasing $N$.
## V State transfer algorithm with an active switch
As we have shown in the previous section, the STA does not perform with unit
fidelity on the complete $M$-partite graph with loops when the sender and the
receiver are in the same partition. Note that if the receiver does not know
the position of the sender, the measurement should be made at the optimal time
(59) corresponding to the more likely configuration, i.e., when the sender and
the receiver are in different partition. This reduces the fidelity of STA
further to approximately 0.91.
To fix this issue we introduce an STA where the sender and the receiver will
actively switch the local coins at their vertices. We use that for
$M\to\infty$ and $N\to\infty$ the state of the SA (19) on the complete
$M$-partite graph with loops evolves periodically from the initial state
$\mathinner{|{\Omega}\rangle}$ to the target state
$\mathinner{|{\nu_{1}}\rangle}$ and back with a period of $2T$, where $T$ is
the run-time of the SA given by (24). Hence, we can perform the state transfer
in the following way. The sender initializes the walk on its vertex in the
target state of the search algorithm
$\mathinner{|{s}\rangle}=\mathinner{|{\nu_{1}}\rangle}$ which corresponds to
the loop at the sender vertex. For the first $T$ steps only the sender will
use the marked coin, i.e., the walk will evolve according to the operator
$\hat{U}_{s}$ of the SA with the marked vertex $s$. The sender state
$\mathinner{|{s}\rangle}$ will evolve close to the equal weight superposition
$\mathinner{|{\Omega}\rangle}$, i.e., the initial state of the SA (6).
Afterwards, the sender switches off the marked coin, and the receiver switches
it on, i.e., the walk evolves according to the operator $\hat{U}_{r}$ of the
SA with the marked vertex $r$. After another $T$ steps the walk will evolve
close to the state $\mathinner{|{r}\rangle}$ corresponding to the loop at the
receiver vertex, and the receiver will detect it with high probability. In
this way we can achieve state transfer with high fidelity on the complete
$M$-partite graph with loops irrespective of the relative position of the
sender and the receiver.
For illustration, we show in Figure 8 the comparison of the fidelities of
state transfer of the original STA and the STA with an active switch when the
sender and the receiver are in the same partition. We see that the STA with an
active switch takes more steps, however, the fidelity reaches one.
Figure 8: Comparison of fidelity of the original STA (37) (red diamonds) with
fidelity of the STA with an active switch (black dots) as a function of the
number of steps $t$ for $N=40$ and $M=100$. The sender and the receiver
vertices are in the same partition. The switch between $\hat{U}_{s}$ and
$\hat{U}_{r}$ is done after $T\approx 100$ steps, corresponding to the run-
time of the SA.
Let us now formalize the STA with an active switch on more general graphs.
Namely, we consider graphs where the optimal number of steps $T$ of the SA
does not depend on the position of the marked vertex $m$. The steps of the STA
with an active switch can be formulated as follows:
1. 1.
Sender initializes the walk at its vertex in the state
$\mathinner{|{s}\rangle}$ corresponding to the target state of the SA with the
marked vertex $s$.
2. 2.
Sender uses marked coin on his vertex for $T$ steps, i.e., the evolution
operator $\hat{U}_{s}$ is applied $T$-times.
3. 3.
Receiver uses marked coin on his vertex for $T$ steps, i.e., the evolution
operator $\hat{U}_{r}$ is applied $T$-times.
4. 4.
Receiver measures the walk at its vertex.
We show that the fidelity of the STA with an active switch can be lower
bounded using only the results from the SA with one marked vertex, which is
not true for the original STA. As we have seen in the case of the complete
$M$-partite graph with loops, the SA does not tell us anything about the
evolution of the STA in the subspace ${\cal I}_{-}$.
To derive the lower bound of fidelity of the STA with an active switch we
first introduce two conditions on the SA. The first condition is related to
target state of the SA. We suppose that after $T$ steps the state of the SA
can be expressed in the form
$\displaystyle\mathinner{|{\phi(T)}\rangle}=\hat{U}_{m}^{T}\mathinner{|{\Omega}\rangle}=\alpha_{m}\mathinner{|{m}\rangle}+\epsilon_{m}\mathinner{|{\eta_{m}}\rangle}$
(60)
for every marked vertex $m$ in the graph. Here
$\mathinner{|{m}\rangle}\in{\cal H}_{m}$ is the target state of the SA, i.e.,
if the walk is in this state the success probability of finding the marked
vertex $m$ is exactly $1$, and $\mathinner{|{\eta_{m}}\rangle}$ is a unit
vector orthogonal to $\mathinner{|{m}\rangle}$. Complex numbers $\alpha_{m}$
and $\epsilon_{m}$ are such that $|\alpha_{m}|$ is close to one and
$|\epsilon_{m}|\ll 1$. $|\alpha_{m}|^{2}$ is closely related to the success
probability of the SA, since
$\displaystyle P_{m}(T)$ $\displaystyle=$ $\displaystyle|\langle
m|\phi(T)\rangle|^{2}+\sum_{\begin{subarray}{c}\mathinner{|{j}\rangle}\in{\cal
H}_{m}\\\ \langle j|m\rangle=0\end{subarray}}|\langle j|\phi(T)\rangle|^{2}$
(61) $\displaystyle=$
$\displaystyle|\alpha_{m}|^{2}+|\epsilon_{m}|^{2}\sum_{\begin{subarray}{c}\mathinner{|{j}\rangle}\in{\cal
H}_{m}\\\ \langle j|m\rangle=0\end{subarray}}|\langle j|\eta_{m}\rangle|^{2}.$
Note that if the vector $\mathinner{|{\eta_{m}}\rangle}$ does not have a
support at the marked vertex then the success probability is exactly
$|\alpha_{m}|^{2}$. From the relation (60) for $m=s$ we express the initial
state of the STA with an active switch as
$\mathinner{|{s}\rangle}=\frac{1}{\alpha_{s}}\left(\hat{U}_{s}^{T}\mathinner{|{\Omega}\rangle}-\epsilon_{s}\mathinner{|{\eta_{s}}\rangle}\right).$
(62)
The second condition describes the periodicity of the SA. Namely, we suppose
that after $2T$ steps the state of the SA can be written in the form
$\displaystyle\hat{U}_{m}^{2T}\mathinner{|{\Omega}\rangle}=\beta_{m}\mathinner{|{\Omega}\rangle}+\delta_{m}\mathinner{|{\rho_{m}}\rangle},$
(63)
where $\mathinner{|{\rho_{m}}\rangle}$ is a unit vector orthogonal to the
initial state $\mathinner{|{\Omega}\rangle}$. $\beta_{m}$ and $\delta_{m}$ are
again complex numbers where $|\beta_{m}|^{2}$ is the return probability. We
assume that it is close to one and that $|\delta_{m}|\ll 1$. In other words,
this condition says that if we double the number of steps the SA returns close
to its initial state.
Assuming that the SA satisfies the conditions (60) and (63) we write the final
state of the STA with an active switch in the following manner
$\displaystyle\hat{U}_{r}^{T}\hat{U}_{s}^{T}|s\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha_{s}}\hat{U}_{r}^{T}\hat{U}_{s}^{T}\left(\hat{U}_{s}^{T}|\Omega\rangle-\epsilon_{s}|\eta_{s}\rangle\right)$
(64) $\displaystyle=$
$\displaystyle\frac{1}{\alpha_{s}}\hat{U}_{r}^{T}\left(\hat{U}_{s}^{2T}|\Omega\rangle-\epsilon_{s}\hat{U}_{s}^{T}|\eta_{s}\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha_{s}}\hat{U}_{r}^{T}\left(\beta_{s}|\Omega\rangle+\delta_{s}|\rho_{s}\rangle-\epsilon_{s}\hat{U}_{s}^{T}|\eta_{s}\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{\alpha_{r}}{\alpha_{s}}\beta_{s}|r\rangle+\frac{\beta_{s}}{\alpha_{s}}\epsilon_{r}|\eta_{r}\rangle+\frac{\delta_{s}}{\alpha_{s}}\hat{U}_{r}^{T}|\rho_{s}\rangle-$
$\displaystyle-\frac{\epsilon_{s}}{\alpha_{s}}\hat{U}_{r}^{T}\hat{U}_{s}^{T}|\eta_{s}\rangle$
where we have first used (62), then (63) and finally we again use (62) but for
the state $|r\rangle$. The fidelity of STA with an active switch can be
expressed as
${\cal F}=\left|\left\langle
r\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|s\right\rangle\right|^{2}+\sum_{\begin{subarray}{c}\mathinner{|{j}\rangle}\in{\cal
H}_{r}\\\ \langle j|r\rangle=0\end{subarray}}\left|\left\langle
j\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|s\right\rangle\right|^{2}.$
Hence, the square root of the fidelity can be bounded from below by
$\sqrt{\cal F}\geq\left|\left\langle
r\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|s\right\rangle\right|.$
To approximate $\left|\left\langle
r\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|s\right\rangle\right|$ we use the
following estimates
$\displaystyle\left|\left\langle
r\left|\hat{U}_{r}^{T}\right|\rho_{s}\right\rangle\right|$ $\displaystyle\leq$
$\displaystyle|||r\rangle||\left|\left|\hat{U}_{r}^{T}\left|\rho_{s}\right\rangle\right|\right|=|||r\rangle||\left|\left|\left|\rho_{s}\right\rangle\right|\right|=1,$
$\displaystyle\left|\left\langle
r\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|\eta_{s}\right\rangle\right|$
$\displaystyle\leq$
$\displaystyle|||r\rangle||\left|\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\left|\eta_{s}\right\rangle\right|\right|=1,$
(65)
which follow from the Cauchy–Schwarz inequality and the unitarity of evolution
operator $\hat{U}_{m}$. Combining (64) with (65) we find the lower bound for
the square root of the fidelity which reads
$\displaystyle\sqrt{\cal F}$ $\displaystyle\geq$
$\displaystyle\left|\frac{\alpha_{r}}{\alpha_{s}}\beta_{s}\langle
r|r\rangle+\frac{\beta_{s}}{\alpha_{s}}\epsilon_{r}\langle
r|\eta_{r}\rangle+\frac{\delta_{s}}{\alpha_{s}}\left\langle
r\left|\hat{U}_{r}^{T}\right|\rho_{s}\right\rangle-\frac{\epsilon_{s}}{\alpha_{s}}\left\langle
r\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|\eta_{s}\right\rangle\right|$ (66)
$\displaystyle\geq$
$\displaystyle\frac{|\alpha_{r}|}{|\alpha_{s}|}|\beta_{s}|-\frac{|\delta_{s}|}{|\alpha_{s}|}\left|\left\langle
r\left|\hat{U}_{r}^{T}\right|\rho_{s}\right\rangle\right|-\frac{|\epsilon_{s}|}{|\alpha_{s}|}\left|\left\langle
r\left|\hat{U}_{r}^{T}\hat{U}_{s}^{T}\right|\eta_{s}\right\rangle\right|$
$\displaystyle\geq$
$\displaystyle\frac{|\alpha_{r}|}{|\alpha_{s}|}|\beta_{s}|-\frac{|\delta_{s}|}{|\alpha_{s}|}-\frac{|\epsilon_{s}|}{|\alpha_{s}|}.$
It is easy to see from (66) that if $|\alpha_{s}|$, $|\alpha_{r}|$ and
$|\beta_{s}|$ are close to one and if $|\epsilon_{s}|\ll 1$ and
$|\delta_{s}|\ll 1$ then the fidelity of the state transfer is close to one.
The result derived above guarantees that if the SA succeeds with unit
probability in the limit of a large graph, then the STA with an active switch
achieves perfect state transfer. Moreover, even for small graphs the STA with
an active switch can actually achieve very good fidelity. For illustration we
have investigated numerically the STA with an active switch on the complete
$M$-partite graph for various values of $N$ and $M$. The results are similar
to those presented in Figure 7.
Figure 9: Overall fidelity of STA with an active switch as a function of the
number of partitions $M$ for $N=10$ (gray circles), $N=50$ (blue triangles)
and $N=100$ (brown diamonds). The walk is initialized at the sender vertex in
the loop. For a given $N$ and $M$ we evaluate numerically the evolution
operator $\hat{U}_{s}$ of SA with marked vertex $s$, apply it for the optimal
number of steps $T$ given by (24). Then we repeat the same with $\hat{U}_{r}$.
Finally, we make a measurement at the receiver vertex $r$ and determine the
fidelity ${\cal F}(2T)$. To unravel the scaling of the fidelity we plot
$1-{\cal F}(2T)$ on the log-log scale. The full red line has the $1/M$ slope,
while the dashed red line follows $1/M^{2}$. The plot indicates that
$O(1/M)>1-{\cal F}(2T)>O(1/M^{2})$.
## VI Conclusion
We have investigated search and state transfer algorithms based on the coined
quantum walks, focusing on the complete $M$-partite graph. It was shown that
adding loops to all vertices increases the success probability of the SA close
to one for a large graph. This is by now a standard method [9, 10, 26, 27, 28,
29, 30] which has a potential to significantly improve the success probability
on a much broader class of graphs [31]. However, the analysis of the SA does
not provide the necessary insight for the investigation of the STA, as the
latter requires larger invariant subspace. As we have seen on the example of
the complete $M$-partite graph with loops, success probability of the SA close
to one does not guarantee STA with unit fidelity. The reason is that the
phases of the relevant eigenvalues of the evolution operator are not harmonic
when the sender and the receiver are in the same partition. Although the
modification of the initial state has improved the fidelity considerably, the
absence of an edge between the sender and the receiver vertex on the complete
$M$-partite graph does not allow for perfect state transfer. It would be
interesting to find out if this occurs for different graphs as well.
In the present paper we have limited our investigations to the case where all
partitions have the same number of vertices $N$. This enabled us to find exact
invariant subspaces for SA and STA which have dimensions independent of $N$
and $M$. Allowing the partitions with different number of vertices appears to
break this feature, and the dimension of the invariant subspace is likely to
depend on $M$. We plan to investigate this behaviour in the near future for
small values of $M$.
To improve the STA on the complete $M$-partite graph we have introduced the
STA with an active switch. This approach allows for perfect state transfer in
the limit of a large graph in both configurations of the sender and the
receiver vertex. The trade-off is that the STA with an active switch requires
more steps than the original STA. Indeed, when the sender and the receiver are
in different partitions, the number of steps for the original STA to reach
unit fidelity is twice the number of steps of the search for two vertices. On
the other hand, STA with an active switch takes twice the number of steps of
the SA for one vertex. Since the search for two vertices is $\sqrt{2}$ faster
than the search for one vertex, the STA with an active switch is slower by the
same factor.
The main advantages of the STA with an active switch are that it can be
applied to other graphs, and that its fidelity can be estimated based on the
analysis of the SA for one marked vertex alone. In this way we can achieve
state transfer with high fidelity on graphs, where the SA for one marked
vertex has success probability close to one and evolves almost periodically.
For many symmetric graphs these conditions are well satisfied, at least in the
limit of a large graph. Moreover, exact periodicity of the Grover walk (i.e.,
without the marked vertex) was recently investigated for various graphs [50,
51, 52]. It would be of interest to determine if SA works on these graphs as
well.
The STA with an active switch also has some disadvantages. As we have already
mentioned, it will have a longer run-time in comparison with the original STA.
Moreover, the sender and the receiver have to actively switch off or on their
marked coins. However, this is only a local operations, and since we consider
that the run-time of the SA is independent of the location of the marked
vertex, the time of the switching is determined solely by the global
properties of the graph, e.g., the number of vertices. Hence, the sender and
the receiver still do not need to know each other’s position. Finally, we have
to determine the target state of SA, which serves as the initial state for the
STA with an active switch. Nevertheless, for highly symmetric graphs this
target state is usually either the equal weight superposition of all direction
or the state corresponding to a loop.
###### Acknowledgements.
Both authors received support from the Czech Grant Agency (GAČR) under grant
No. 17-00844S and from MSMT RVO 14000. This publication was funded by the
project “Centre for Advanced Applied Sciences”, Registry No.
CZ.$02.1.01/0.0/0.0/16\\_019/0000778$, supported by the Operational Programme
Research, Development and Education, co-financed by the European Structural
and Investment Funds and the state budget of the Czech Republic.
## References
* Aharonov _et al._ [1993] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48, 1687 (1993).
* Meyer [1996] D. A. Meyer, J. Stat. Phys. 85, 551 (1996).
* Farhi and Gutmann [1998] E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998).
* Aharonov _et al._ [2001] D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, in Proceedings of the 33rd STOC (ACM Press, New York, 2001) , 50 (2001).
* Grover [1997] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997).
* Childs and Goldstone [2004] A. M. Childs and J. Goldstone, Phy. Rev. A 70, 022314 (2004).
* Ambainis _et al._ [2001] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, in _STOC’01: Proc. of the 33rd annual ACM Symposium on Theory of Computing_ (2001) pp. 37–49.
* Shenvi _et al._ [2003] N. Shenvi, J. Kempe, and K. B. Whaley, Phys. Rev. A 67, 052307 (2003).
* Ambainis _et al._ [2005] A. Ambainis, J. Kempe, and A. Rivosh, in _Proc. 16th ACM-SIAM symposium on Discrete algorithms_ (Society for Industrial and Applied Mathematics, 2005) pp. 1099–1108.
* Potoček _et al._ [2009] V. Potoček, A. Gábris, T. Kiss, and I. Jex, Phys. Rev. A 79, 012325 (2009).
* Hein and Tanner [2009a] B. Hein and G. Tanner, J. Phys. A 42, 085303 (2009a).
* Hillery _et al._ [2003] M. Hillery, J. Bergou, and E. Feldman, Phys. Rev. A 68, 032314 (2003).
* Feldman and Hillery [2004] E. Feldman and M. Hillery, Phys. Lett. A 324, 277 (2004).
* Feldman and Hillery [2007] E. Feldman and M. Hillery, J. Phys. A: Math. Theor. 40, 11343 (2007).
* Andrade and da Luz [2009] F. M. Andrade and M. G. E. da Luz, Phys. Rev. A 80, 052301 (2009).
* Venancio _et al._ [2013] B. F. Venancio, F. M. Andrade, and M. G. E. da Luz, J. Phys. A 46, 165302 (2013).
* Reitzner _et al._ [2009] D. Reitzner, M. Hillery, E. Feldman, and V. Bužek, Phys. Rev. A 79, 012323 (2009).
* Janmark _et al._ [2014] J. Janmark, D. A. Meyer, and T. G. Wong, Phys. Rev. Lett. 112, 210502 (2014).
* Novo _et al._ [2015] L. Novo, S. Chakraborty, M. Mohseni, H. Neven, and Y. Omar, Sci. Rep. 5, 13304 (2015).
* Meyer and Wong [2015] D. A. Meyer and T. G. Wong, Phys. Rev. Lett. 114, 110503 (2015).
* Chakraborty _et al._ [2016] S. Chakraborty, L. Novo, A. Ambainis, and Y. Omar, Phys. Rev. Lett. 116, 100501 (2016).
* Albert and Barabási [2002] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
* Osada _et al._ [2020] T. Osada, B. Coutinho, Y. Omar, K. Sanaka, W. J. Munro, and K. Nemoto, Phys. Rev. A 101, 022310 (2020).
* Chakraborty _et al._ [2020] S. Chakraborty, L. Novo, and J. Roland, Phys. Rev. A 102, 032214 (2020).
* Brassard _et al._ [2002] G. Brassard, P. Høyer, M. Mosca, and A. Tapp, Contemp. Math. 350, 53 (2002).
* Wong [2015] T. G. Wong, J. Phys. A 48, 435304 (2015).
* Wong [2018] T. G. Wong, Quantum Inf. Process. 17, 68 (2018).
* Rhodes and Wong [2019] M. L. Rhodes and T. G. Wong, Phys. Rev. A 100, 042303 (2019).
* Rhodes and Wong [2020] M. L. Rhodes and T. G. Wong, Quantum Inf. Process. 19, 334 (2020).
* Chiang [2020] C. Chiang, Quantum Inf. Process. 19, 364 (2020).
* Høyer and Zhan [2020] P. Høyer and Y. Zhan, arXiv:2002.11234 (2020).
* Bose [2003] S. Bose, Phys. Rev. Lett. 91, 207901 (2003).
* Kurzynski and Wojcik [2011] P. Kurzynski and A. Wojcik, Phys. Rev. A 83, 062315 (2011).
* Yalcinkaya and Gedik [2015] I. Yalcinkaya and Z. Gedik, J. Phys. A 48, 225302 (2015).
* Zhan _et al._ [2014] X. Zhan, H. Qin, Z. H. Bian, J. Li, and P. Xue, Phys. Rev. A 90, 012331 (2014).
* Shang _et al._ [2018] Y. Shang, Y. Wang, M. Li, and R. Q. Lu, EPL 124, 60009 (2018).
* Chen _et al._ [2019] X. B. Chen, Y. L. Wang, G. Xu, and Y. X. Yang, IEEE ACCESS 7, 13634 (2019).
* Hein and Tanner [2009b] B. Hein and G. Tanner, Phys. Rev. Lett. 103, 260501 (2009b).
* Kendon and Tamon [2011] V. M. Kendon and C. Tamon, J. Comput. Theor. Nanosci. 8, 422 (2011).
* Barr _et al._ [2014] K. Barr, T. Proctor, D. Allen, and V. M. Kendon, Quantum Inf. Comput. 14, 417 (2014).
* Štefaňák and Skoupý [2016] M. Štefaňák and S. Skoupý, Phys. Rev. A 94, 022301 (2016).
* Štefaňák and Skoupý [2017] M. Štefaňák and S. Skoupý, Quantum Inf. Process. 16, 72 (2017).
* Zhan [2019] H. Zhan, Quantum Inf. Process. 18, 369 (2019).
* Cao _et al._ [2019] W. F. Cao, Y. G. Yang, D. Li, J. R. Dong, Y. H. Zhou, and W. M. Shi, Mod. Phys. Lett. A 34, 1950317 (2019).
* Reitzner _et al._ [2017] D. Reitzner, M. Hillery, and D. Koch, Phys. Rev. A 96, 032323 (2017).
* Koch and Hillery [2018] D. Koch and M. Hillery, Phys. Rev. A 97, 012308 (2018).
* Krovi and Brun [2007] H. Krovi and T. A. Brun, Phys. Rev. A 75, 062332 (2007).
* Feldman _et al._ [2010] E. Feldman, M. Hillery, H.-W. Lee, D. Reitzner, H. Zheng, and V. Bužek, Phys. Rev. A 82, 040301 (2010).
* Hillery _et al._ [2012] M. Hillery, H. Zheng, E. Feldman, D. Reitzner, and V. Bužek, Phys. Rev. A 85, 062325 (2012).
* Kubota _et al._ [2018] S. Kubota, E. Segawa, T. Taniguchi, and Y. Yoshie, Linear Algebra Appl. 554, 371 (2018).
* Kubota _et al._ [2019] S. Kubota, E. Segawa, T. Taniguchi, and Y. Yoshie, Linear Algebra Appl. 579, 217 (2019).
* Yoshie [2019] Y. Yoshie, Graphs Combin. 35, 1305 (2019).
|
# Reflections on reflections
Kuan-Nan Lin<EMAIL_ADDRESS>Department of Physics and Center for
Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan, R.O.C.
LeCosPA, National Taiwan University, Taipei 10617, Taiwan, R.O.C. Pisin Chen
<EMAIL_ADDRESS>Department of Physics and Center for Theoretical
Sciences, National Taiwan University, Taipei 10617, Taiwan, R.O.C. LeCosPA,
National Taiwan University, Taipei 10617, Taiwan, R.O.C. Kavli Institute for
Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory,
Stanford University, Stanford, California 94305, U.S.A.
###### Abstract
Analog Hawking radiation emitted by a perfectly reflecting mirror in
(1+3)-dimensional flat spacetime is investigated. This is accomplished by
studying the reflected frequency and momentum based on Einstein’s mirror,
instead of the canonical way of solving, if possible, wave equations subjected
to a dynamical Dirichlet boundary condition. In the case of a finite-size
mirror, diffraction pattern appears in the radiation spectrum. Based on the
relevant parameters in the proposed AnaBHEL experiment, where the Hawking
temperature $T_{H}=0.03$ eV and the mirror area $\mathcal{A}\sim 0.1$ mm2, the
Hawking photon yield is estimated to be $N\sim 16$ per laser shot (assuming a
high reflectivity mirror can be generated in the proposed AnaBHEL experiment).
## I Introduction
Since the discovery of Hawking radiation [1] in curved spacetime in the 1970s,
most of the works of mimicking Hawking radiation via mirror-induced radiation
(MIR) in flat spacetime have been devoted to (1+1) dimensions [2, 3, 4, 5, 6,
7, 8, 9, 10, 7, 8, 11]. This is because Hawking radiation only propagates
radially in the spacetime with a black hole. However, what is experienced in a
laboratory is a (1+3)-dimensional flat spacetime, and these additional degrees
of freedom complicate the situation of how the vacuum fluctuations interact
with a non-point-like mirror. The intent of this paper is to fill in this gap
in the literature.
The essence of Hawking radiation in curved spacetime is the gravitational red
shift of the wave mode that propagates to the future infinity. In the flying
mirror case, red shift of wave mode is accomplished by the Doppler effect,
which is induced upon reflection off the mirror, instead. Thus, it is
essential to study how wave modes are reflected by a flying mirror.
The canonical way of dealing with the moving mirror model is to impose, say, a
dynamical Dirichlet boundary condition on a scalar field, whose excitation
leads to mirror-induced radiation. This can be exactly solved in (1+1)
dimensions since everything is conformally flat in (1+1)D. However, to our
awareness, there is no general exact way to solve the problem in spacetime
dimension other than (1+1) when the mirror is relativistic. In (1+3)D,
however, the case of a non-relativistic perfectly reflecting plane mirror [12,
13, 14] or a relativistic semitransparent plane mirror [15, 16] have been
worked out.
The necessity of extensions beyond the standard mirror-black hole
correspondence is required by laboratory considerations. In particular, the
recent AnaBHEL (Analog Black Hole Evaporation via Lasers) Collaboration [17]
proposed to generate a flying mirror based on the Chen-Mourou [18, 19]
proposal, in which the mirror would have low reflectivity and finite area and
thickness, to investigate analog Hawking radiation and the quantum
entanglement with its partner.
In this paper, instead of working with the conventional approach, we begin by
dealing with the reflected frequency and momentum, which can be derived
according to Einstein’s special theory of relativity [20], and, from which, we
are able to obtain the corresponding phase of the reflected wave mode, and
finally obtain the quantum particle spectrum in higher spacetime dimensions by
taking the reflected wave mode’s Fourier component. In fact, this approach to
study particle creation is quite universal, i.e., it also applies to particle
creation in other contexts, e.g., typical QFT in flat/curved spacetime, Unruh
effect, MIR for perfect/semitransparent mirror, etc.
This paper is organized as follows. In Sec.II, we begin by reviewing
reflections from Einstein’s inertial mirror in (1+1)D and, from which,
generalization to an accelerated mirror is made. In Sec.III, we review several
crucial aspects, which do not appear in (1+1)D, of reflection by a mirror in
(1+3)D. In Sec.IV, we first make connection of a general wave mode’s Fourier
components to the Bogoliubov coefficients in various cases, e.g., cosmological
particle creation, Unruh effect, etc., and later focus on the mirror-induced
radiation. Particularly, the radiation spectrum emitted by a (in)finite-size
plane mirror, which shows diffraction patterns, in (1+3)D is worked out. In
addition, estimation of event yield based on proposed experiment is made.
Finally, conclusion is given in Sec.V.
Notation: throughout the paper, we use the mostly plus metric convention,
$G=\hbar=c=k_{B}=1,\;\mathbb{R}^{1}=(-\infty,+\infty)$, and
$\mathbb{R}^{-}=(-\infty,0)$.
## II Reflection in (1+1)D
Suppose, according to a static observer in the lab frame $(t,x)$, there is a
plane wave $\phi_{inc}=\text{exp}\left(-i\Omega t+iPx\right)$, where
$\Omega=|P|>0$ is frequency and $P$ is momentum, that incidents upon an
inertial moving mirror with trajectory $x_{r}(t_{r})=\beta t_{r}$, where
$\beta=\dot{x}_{r}(t_{r})=$ const. is its velocity. According to Einstein’s
theory of special relativity [20], upon reflection, the incident plane wave
will undergo Doppler effect and this reflected wave, as seen by the static
observer in the lab frame, can be described by
$\phi_{ref}=\text{exp}\left(-i\omega t+ipx\right)$, where
$\omega=\Omega\left(\frac{1\pm\beta}{1\mp\beta}\right),\quad
p=-P\left(\frac{1\pm\beta}{1\mp\beta}\right),$ (1)
are, respectively, the Doppler shifted frequency and momentum of the reflected
wave, and the upper sign is valid for $P<0$, while the lower sign is valid for
$P>0$.
A first attempt to generalize the above discussions to an accelerated mirror
can be made by giving the above mirror-related constant quantities time
dependencies. That is, we shall have
$\displaystyle\phi_{inc}$ $\displaystyle=\text{exp}\left(-i\Omega
t+iPx\right),$ $\displaystyle\phi_{ref}$ $\displaystyle=\text{exp}\left(-i\int
dt\;\omega(t,x)+i\int dx\;p(t,x)\right),$ (2)
where $\Omega$ and $P$ are constants, while $\omega(t,x)$ and $p(t,x)$ are,
respectively, the time-dependent reflected frequency and momentum given by
$\displaystyle\omega(t,x)$
$\displaystyle=\Omega\left(\frac{1\pm\beta(t,x)}{1\mp\beta(t,x)}\right),$ (3)
$\displaystyle p(t,x)$
$\displaystyle=-P\left(\frac{1\pm\beta(t,x)}{1\mp\beta(t,x)}\right).$
The observation point $(t,x)$ dependency in $\beta(t,x)$ is due to the fact
that the retarded time $t_{r}$ at which the reflection occurs is a function of
the observation point, i.e., $t_{r}=t_{r}(t,x)$.
For a right/left-moving reflected wave, $p=\pm\omega$, the reflected waves
become
$\phi_{ref}=\text{exp}\left(-i\int dx_{\mp}\;\omega(t,x)\right),$ (4)
where $x_{\mp}(t,x)=t\mp x$ are the light cone coordinates.
On the worldline of a given reflected light ray, $t\mp x=t_{r}\mp x_{r}=$
const., where $(t_{r},x_{r}(t_{r}))$ is the spacetime point where the
reflection occurs. From this, we also obtain
$dx_{\mp}(t,x)=\left(1\mp\beta(t_{r})\right)dt_{r}$, which leads to
$\phi_{ref}=\text{exp}\left(-i\Omega\int
dt_{r}\left(1\pm\beta(t_{r})\right)\right).$ (5)
In terms of the mirror’s proper time $\tau$, which is related to the retarded
time by $d\tau=\gamma^{-2}(t_{r})dt_{r}$, one obtains
$\phi_{ref}=\text{exp}\left(-i\Omega\int
d\tau\gamma^{2}(t_{r})\left(1\pm\beta(t_{r})\right)\right).$ (6)
These are the direct generalizations to an accelerated mirror in terms of
observation point $(t,x)$, light cone coordinates $(x_{+},x_{-})$, retarded
time $t_{r}$, and mirror’s proper time $\tau$ based on observations from the
results of an inertial mirror.
Alternatively, one may also begin by solving the (1+1)-dimensional wave
equation: $\Box\phi(t,x)=0$, where $\Box=\partial^{\mu}\partial_{\mu}$ is the
d’Alembertian operator, with the dynamical Dirichlet boundary condition:
$\phi(t_{r},x_{r}(t_{r}))=0$, where $x_{r}(t_{r})$ is the mirror’s arbitrary
trajectory [2, 3, 4, 5]. Suppose $\phi=\phi_{inc}-\phi_{ref}$, then for an
incident plane wave $\phi_{inc}=\text{exp}\left(-i\Omega t+iPx\right)$, the
reflected wave is solved to be
$\phi_{ref}=\text{exp}\left(-i\Omega\eta_{\mp}(x_{\mp})\right)$, where
$\eta_{\mp}(x_{\mp})=t_{r}\pm x_{r}(t_{r})$ is the ray tracing function, for
$P<0$ (upper sign) and $P>0$ (lower sign). Noting that
$\eta_{\mp}(x_{\mp})=\int dx_{\mp}\partial_{\mp}\eta_{\mp}(x_{\mp})$, the
reflected wave can also be expressed as
$\phi_{ref}=\text{exp}\left(-i\Omega\int
dx_{\mp}\partial_{\mp}\eta(x_{\mp})\right),$ (7)
which is identical to the previous generalization, Eq. (4), by identifying the
time derivative of the ray tracing function as the reflected frequency, i.e.,
$\omega(t,x)=\Omega\partial_{\mp}\eta(x_{\mp})$. This justifies the validity
of our previous generalizations to an accelerated mirror.
In the case of an accelerated mirror, the integrations physically mean summing
over distinct reflected waves with each wave labelled by the integration
variables. Then, for distinct reflected waves that are close to each other,
e.g., $x_{\mp}\approx\bar{x}_{\mp}+\varepsilon$, where $\bar{x}_{\mp}$ is a
fixed reference point and $\varepsilon\rightarrow 0$, the wave in the vicinity
of $\bar{x}_{\mp}$ is approximately, using Eq. (4),
$\phi_{ref}\approx
A(\bar{t},\bar{x})\cdot\text{exp}\left(-i\omega(\bar{t},\bar{x})\varepsilon\right),$
(8)
where $A(\bar{t},\bar{x})$ is some function of the reference point. If one
expands $\phi_{ref}=\text{exp}\left(-i\Omega\eta_{\mp}(x_{\mp})\right)$ around
$\bar{x}_{\mp}$ instead, one obtains
$\phi_{ref}\approx\text{exp}(-i\Omega\eta_{\mp}(\bar{x}_{\mp}))\cdot\text{exp}(-i\omega(\bar{t},\bar{x})\varepsilon).$
(9)
From which we may identify
$A(\bar{t},\bar{x})=\text{exp}(-i\Omega\eta_{\mp}(\bar{x}_{\mp}))$.
Since $\varepsilon$ is small, the mirror has approximately a constant velocity
in this small interval and thus the reflected wave has a similar expression,
aside from the factor of $A(\bar{t},\bar{x})$ due to the velocity variation,
as that reflected by an inertial mirror.
From the above discussions, Eq. (3), one may observe that (I) whether the
mirror is accelerated or not, the reflected frequencies are (I.a) only related
to the mirror’s velocity at the retarded time and (I.b) the relations are
exactly the same. (II) While the Doppler shifted frequencies are related to
the velocities in the same manner for both inertial and accelerated mirrors,
the expressions for the phase of the reflected waves are in fact
different111The treatment in Ref.[21] to extend the discussion to an
accelerated mirror is equivalent to treating the phase of the reflected wave
as $\omega(t,x)x_{\mp}$ instead of reserving $\omega(t,x)$ inside the
integral. However, in the case of constant four-acceleration, the resulting
expression for the reflected wave happens to coincide with the result based on
Eq. (4).. For waves reflected by an accelerated mirror, only those that are
close to each other do they share similar expressions of phase as that
reflected by an inertial mirror.
## III Reflection in (1+3)D
In higher dimensional spacetimes, it is known, if possible, to be difficult to
solve the wave equation: $\Box\phi(t,\mathbf{x})=0$ with the boundary
condition: $\phi(t_{r},x_{r}(t_{r}),\mathbf{x}_{\perp})=0$ since conformal
symmetry no longer holds. Nevertheless, from the previous section, we have
seen the alternative route to obtain the expression of the reflected wave from
its reflected frequency. We shall pursue this manner in this section.
Suppose a plane wave $\phi_{inc}=\text{exp}(-i\Omega
t+i\mathbf{P}\cdot\mathbf{x})$ incidents on a perfectly reflecting plane
mirror moving at a constant velocity $\beta$, the reflected wave can be
derived using Lorentz transformation222See, e.g., Ref.[22] for an alternative
derivation based on purely geometric considerations without using Lorentz
transformation. and the result is $\phi_{ref}=\text{exp}\left(-i\omega
t+i\mathbf{p}\cdot\mathbf{x}\right)$, where
$\displaystyle\omega$ $\displaystyle=\Omega\left(\frac{1\pm
2\beta\cos\theta_{i}+\beta^{2}}{1-\beta^{2}}\right),$ (10) $\displaystyle
p_{x}$ $\displaystyle=-P_{x}\left(\frac{1\pm
2\beta\sec\theta_{i}+\beta^{2}}{1-\beta^{2}}\right),\quad\mathbf{p}_{\perp}=\mathbf{P}_{\perp},$
(11)
where $\theta_{i}$ is the incident angle observed in the lab frame,
$P_{x}=\mp\Omega\cos\theta_{i}$, and the reflected angle observed in the lab
frame, $\theta_{r}=\cos^{-1}(\pm p_{x}/\omega)$, is given by
$\cos\theta_{r}=\frac{(1+\beta^{2})\cos\theta_{i}\pm 2\beta}{1\pm
2\beta\cos\theta_{i}+\beta^{2}}.$ (12)
The incident frequency $\omega^{\prime}$ and angle $\theta^{\prime}$ in the
mirror’s rest frame are related to the incident frequency $\Omega$ and angle
$\theta_{i}$ in the lab frame by
$\displaystyle\omega^{\prime}=\Omega\left(\frac{1\pm\beta\cos\theta_{i}}{\sqrt{1-\beta^{2}}}\right),\quad\cos\theta^{\prime}=\frac{\cos\theta_{i}\pm\beta}{1\pm\beta\cos\theta_{i}}.$
(13)
Since $\cos\theta^{\prime}$ should be non-negative, it leads to the
constraint: $\cos\theta_{i}>|\beta|$, which is also the condition for
frequency red shifting, for a receding mirror. Thus, $\theta_{i}$ can be no
greater than $\theta_{m}=\cos^{-1}|\beta|$. Physically, this means that waves
with incident angles larger than $\theta_{m}$ are not able to catch up the
receding mirror. In addition, when $\theta_{i}=\theta_{m}$, we have
$\theta_{r}=-\theta_{m}+\pi$. This shows that the wave can be reflected to the
other side of the receding mirror. This is known as the forward reflection.
The critical incident angle ($\theta_{i}=\theta_{c}$) beyond which the waves
are forwardly reflected can be found by setting $\theta_{r}=\pi/2$ and solve
for $\theta_{i}$ in Eq. (12). The result is
$\cos\theta_{i}\left.\right|_{\theta_{i}=\theta_{c}}=\cos\theta_{c}=\mp\left(\frac{2\beta}{1+\beta^{2}}\right),$
(14)
which corresponds to $\theta^{\prime}=\cos^{-1}|\beta|$ in the rest frame.
In summary, for a receding mirror and in the lab frame, incident waves with
$\theta_{i}\in[0,\theta_{c})$ remain on the same side of the mirror as the
incident ones after the reflection; waves with
$\theta_{i}\in(\theta_{c},\theta_{m})$ are reflected forwardly to the other
side of the mirror, and waves with $\theta_{i}\in(\theta_{m},\pi/2]$ are not
able to hit the mirror. On the other hand, in the mirror’s rest frame, waves
with $\theta_{i}\in[0,\theta_{m})$ are on the same side of the mirror after
the reflection.
For an ultra-relativistically receding mirror, $\beta\sim\mp 1\pm\delta\beta$,
$\delta\beta\rightarrow 0^{+}$, the reflected frequency and momentum are
$\displaystyle\frac{\omega}{\Omega}$
$\displaystyle\sim\frac{\delta\beta}{2}\ll
1,\quad\frac{p_{x}}{\Omega}\sim\pm\frac{\delta\beta}{2},\quad\text{for
}\theta_{i}\ll 1.$ (15)
In addition, we have $\theta_{c}\sim\delta\beta\ll 1$ and
$\theta_{m}\sim\sqrt{2}\sqrt{\delta\beta}\ll 1$. Thus, although waves with
$\theta_{i}\in(\theta_{c},\theta_{m})$ are reflected forwardly to the mirror’s
other side, they also experience Doppler red shift.
On the other hand, for an ultra-relativistically approaching mirror,
$\beta\sim\pm 1\mp\delta\beta$, $\delta\beta\rightarrow 0^{+}$, the reflected
frequency and momentum are
$\displaystyle\frac{\omega}{\Omega}$
$\displaystyle\sim\frac{1+\cos\theta_{i}}{\delta\beta}\gg
1,\quad\frac{p_{x}}{\Omega}\sim\pm\left(\frac{1+\cos\theta_{i}}{\delta\beta}\right),$
(16)
where $\theta_{i}\in[0,\pi/2]$. That is, the frequencies are always Doppler
blue shifted, and the reflected waves are mostly longitudinal, i.e.,
$\omega\sim|p_{x}|\gg|\mathbf{p}_{\perp}|$, and are always on the same side as
their incident counterparts.
In the case of an accelerated mirror, we expect to have
$\displaystyle\phi_{inc}$
$\displaystyle=e^{i\mathbf{P}_{\perp}\cdot\mathbf{x}_{\perp}}\text{exp}\left(-i\Omega
t+iP_{x}x\right),$ (17) $\displaystyle\phi_{ref}$
$\displaystyle=e^{i\mathbf{p}_{\perp}\cdot\mathbf{x}_{\perp}}\text{exp}\left(-i\int
dt\;\omega(t,\mathbf{x})+i\int dx\;p_{x}(t,\mathbf{x})\right),$
where
$\displaystyle\omega(t,\mathbf{x})=\Omega\left(\frac{1\pm
2\beta(t,\mathbf{x})\cos\theta_{i}+\beta^{2}(t,\mathbf{x})}{1-\beta^{2}(t,\mathbf{x})}\right),$
(18) $\displaystyle p_{x}(t,\mathbf{x})=-P_{x}\left(\frac{1\pm
2\beta(t,\mathbf{x})\sec\theta_{i}+\beta^{2}(t,\mathbf{x})}{1-\beta^{2}(t,\mathbf{x})}\right),$
(19)
$\displaystyle\cos\theta_{r}(t,\mathbf{x})=\frac{(1+\beta^{2}(t,\mathbf{x}))\cos\theta_{i}\pm
2\beta(t,\mathbf{x})}{1\pm
2\beta(t,\mathbf{x})\cos\theta_{i}+\beta^{2}(t,\mathbf{x})},$ (20)
where the additional dependence on $\mathbf{x}_{\perp}$ in
$\beta(t,\mathbf{x})$ is due to the fact that $(t,\mathbf{x})$ is related to
the retarded time by $t-t_{r}=\pm|\mathbf{x}-\mathbf{x}_{r}(t_{r})|$ for a
given reflected light ray.
## IV Quantum particle spectrum
In the usual context of quantum field theory in (1+$d$)-dimensional curved
spacetime or quantum field theory in (1+$d$)-dimensional flat spacetime in the
presence of a suitable external source, the quantum field, say, a massless
real scalar field $\phi(t,\mathbf{x})$, can be expanded in terms of the mode
$u_{\mathbf{P}}(t,\mathbf{x})$ or the mode $v_{\mathbf{p}}(t,\mathbf{x})$ by
$\displaystyle\hat{\phi}(t,\mathbf{x})$ $\displaystyle=\int
d^{d}P\left[\hat{a}_{\mathbf{P}}u_{\mathbf{P}}(t,\mathbf{x})+h.c.\right],$
(21) $\displaystyle=\int
d^{d}p\left[\hat{b}_{\mathbf{p}}v_{\mathbf{p}}(t,\mathbf{x})+h.c.\right],$
(22)
where $\Omega=|\mathbf{P}|$, $\omega=|\mathbf{p}|$,
$(\hat{a}_{\mathbf{P}},\hat{b}_{\mathbf{p}})$ are annihilation operators,
$h.c.$ denotes Hermitian conjugate, and the two mode bases are related by the
Bogoliubov transformations
$\displaystyle v_{\mathbf{p}}(t,\mathbf{x})$ $\displaystyle=\int
d^{d}P\left[\alpha_{\mathbf{pP}}u_{\mathbf{P}}(t,\mathbf{x})+\beta_{\mathbf{pP}}\bar{u}_{\mathbf{P}}(t,\mathbf{x})\right],$
(23) $\displaystyle u_{\mathbf{P}}(t,\mathbf{x})$ $\displaystyle=\int
d^{d}p\left[\alpha_{\mathbf{pP}}v_{\mathbf{p}}(t,\mathbf{x})-\bar{\beta}_{\mathbf{pP}}\bar{v}_{\mathbf{p}}(t,\mathbf{x})\right],$
(24)
where $\alpha$ and $\beta$ are the Bogoliubov coefficients and the overbar
refers to taking complex conjugation.
In the context of quantum field theory with asymptotic in and out regions
within a given coordinate system, whether in flat spacetime, e.g., particle
creation by a moving mirror, or in curved spacetime, e.g., cosmological
particle production, one often encounters
$\displaystyle\lim_{t\rightarrow-\infty}u_{\mathbf{P}}(t,\mathbf{x})$
$\displaystyle\sim\frac{1}{(2\pi)^{3/2}\sqrt{2\Omega}}\text{exp}\left(-i\Omega
t+i\mathbf{P}\cdot\mathbf{x}\right),$ (25)
$\displaystyle\lim_{t\rightarrow+\infty}v_{\mathbf{p}}(t,\mathbf{x})$
$\displaystyle\sim\frac{1}{(2\pi)^{3/2}\sqrt{2\omega}}\text{exp}\left(-i\omega
t+i\mathbf{p}\cdot\mathbf{x}\right),$ (26)
so by taking the Fourier transformation of the out mode at the infinite past,
i.e.,
$\displaystyle\lim_{t\rightarrow-\infty}v_{\mathbf{p}}(t,\mathbf{x})=\int$
$\displaystyle\frac{d^{3}P}{(2\pi)^{4}\sqrt{\Omega}}\biggl{[}e^{-i\Omega
t+i\mathbf{P}\cdot\mathbf{x}}\tilde{v}_{\mathbf{p}}(\Omega,\mathbf{P})$
$\displaystyle+e^{i\Omega
t-i\mathbf{P}\cdot\mathbf{x}}\tilde{v}_{\mathbf{p}}(-\Omega,-\mathbf{P})\biggr{]},$
(27)
allows us to identify the Bogoliubov coefficients as
$\alpha_{\mathbf{pP}}=\frac{\sqrt{2}}{2\pi}\frac{\tilde{v}_{\mathbf{p}}(\Omega,\mathbf{P})}{(2\pi)^{3/2}},\quad\beta_{\mathbf{pP}}=\frac{\sqrt{2}}{2\pi}\frac{\tilde{v}_{\mathbf{p}}(-\Omega,-\mathbf{P})}{(2\pi)^{3/2}}.$
(28)
On the other hand, in non-dynamical situations, e.g., Unruh effect in (1+1)
dimensions, which involves two different coordinate systems, one encounters
$\displaystyle v_{\omega}(t,x)$
$\displaystyle=\frac{1}{(2\pi)^{1/2}\sqrt{2\omega}}\text{exp}\left(-i\omega
t+i\omega x\right),$ (29) $\displaystyle u_{\Omega}(\tau,\xi)$
$\displaystyle=\frac{1}{(2\pi)^{1/2}\sqrt{2\Omega}}\text{exp}\left(-i\Omega\tau+i\Omega\xi\right),$
(30)
where $(t,x)$ are Minkowski coordinates and $(\tau,\xi)$ are Rindler
coordinates, which are related by the transformation:
$t=a^{-1}e^{a\xi}\sinh(a\tau)$, $x=a^{-1}e^{a\xi}\cosh(a\tau)$, $a>0$. On the
worldline of a Rindler observer at $\xi=0$, the Minkowski mode can be expanded
in terms of the Rindler mode by
$\displaystyle v_{\omega}(\tau)$
$\displaystyle=\frac{1}{\sqrt{4\pi\omega}}\cdot\text{exp}\left(\frac{i\omega}{a}e^{-a\tau}\right)$
$\displaystyle=\int_{0}^{\infty}\frac{d\Omega}{2\pi}\left[e^{-i\Omega\tau}\tilde{v}_{\omega}(\Omega)+e^{i\Omega\tau}\tilde{v}_{\omega}(-\Omega)\right],$
(31)
which allows the identifications
$\alpha_{\omega\Omega}=\frac{\sqrt{4\pi\Omega}}{2\pi}\;\tilde{v}_{\omega}(\Omega),\quad\beta_{\omega\Omega}=\frac{\sqrt{4\pi\Omega}}{2\pi}\;\tilde{v}_{\omega}(-\Omega),$
(32)
where
$\tilde{v}_{\omega}(\Omega)=a^{-1}(-i\omega/a)^{i\Omega/a}\Gamma(-i\Omega/a)/\sqrt{4\pi\omega}$.
In this paper, we are interested in particle creation by a relativistic
perfectly reflecting mirror, thus only the former dynamical situation will be
relevant to our following discussions.
### IV.1 Flying mirror in (1+1)D
To mimic Hawking radiation emitted by a black hole formed from gravitational
collapse, we shall consider a perfect mirror whose trajectory asymptotes the
Davies-Fulling one: $x_{r}(t_{r})\sim-t_{r}-A\text{exp}(-2\kappa
t_{r})-B,\left\\{A,B,\kappa\right\\}>0$ at late times [5] (see Fig.1).
Since the out mode $v_{\omega}(x_{+}(\mathcal{O}_{1}))\sim\text{exp}(-i\omega
x_{-})$ is a plane wave basis (up to a normalization factor) for an (out)
observer $\mathcal{O}_{1}$ at $x_{+}=x_{+}(\mathcal{O}_{1})$, the reflected
waves, which have experienced Doppler red shift, that reach
$x_{+}=x_{+}(\mathcal{O}_{1})$ can be expanded in terms of
$v_{\omega}(x_{+}(\mathcal{O}_{1}))$, which further leads to the
interpretation of Fourier modes as Bogoliubov coefficients. In this case, we
have $\omega(t_{r}(x_{-}))/\Omega\sim A\kappa\cdot\text{exp}(-2\kappa
t_{r}(x_{-}))\sim A\kappa\cdot\text{exp}(-\kappa x_{-})$, and the Fourier
transformation of in mode at $x_{+}(\mathcal{O}_{1})$:
$\displaystyle\tilde{u}_{\Omega}(\omega)=\int_{\mathbb{R}^{1}}dx_{-}e^{i\omega
x_{-}}u_{\Omega}(x_{+}(\mathcal{O}_{1}))$
$\displaystyle=-\frac{1}{\sqrt{4\pi\Omega}}\int_{\mathbb{R}^{1}}dx_{-}e^{i\omega
x_{-}}\text{exp}\left(-i\int dx_{-}\omega(t_{r}(x_{-}))\right)$
$\displaystyle\sim-\frac{1}{\sqrt{4\pi\Omega}}\int_{\mathbb{R}^{1}}dx_{-}e^{i\omega
x_{-}}\text{exp}\left(i\Omega Ae^{-\kappa x_{-}}\right)$
$\displaystyle=-\frac{1}{\sqrt{4\pi\Omega}}\frac{(-i\Omega
A)^{i\omega/\kappa}}{\kappa}\Gamma\left(\frac{-i\omega}{\kappa}\right),$ (33)
and the Bogoliubov coefficients are
$\alpha_{\omega\Omega}=\frac{\sqrt{4\pi\omega}}{2\pi}\tilde{u}_{\Omega}(\omega),\quad\bar{\beta}_{\omega\Omega}=-\frac{\sqrt{4\pi\omega}}{2\pi}\tilde{u}_{\Omega}(-\omega).$
(34)
On the other hand, one intuitively expects an observer $\mathcal{O}_{2}$ at
$x=x(\mathcal{O}_{2})$ should obtain the same spectrum as that obtained by
$\mathcal{O}_{1}$ at $x_{+}=x_{+}(\mathcal{O}_{1})$. Indeed, since
$x_{-}=t-x$, the integration over $x_{-}$ in the Fourier transform of
$u_{\Omega}(t,x)$ can also be regarded as an integration over the observation
time $t$ at a fixed spatial position $x=x(\mathcal{O}_{2})$. In this
perspective, we have
$\displaystyle\tilde{u}_{\Omega}(\omega)=\int_{\mathbb{R}^{1}}dt\;e^{i\omega
x_{-}}u_{\Omega}(x(\mathcal{O}_{2}))$
$\displaystyle=-\frac{1}{\sqrt{4\pi\Omega}}\int_{\mathbb{R}^{1}}dt\;e^{i\omega
x_{-}}\text{exp}\left(-i\int dt\;\omega(t_{r}(x_{-}))\right)$
$\displaystyle\sim-\frac{1}{\sqrt{4\pi\Omega}}e^{-i\omega
x}\int_{\mathbb{R}^{1}}dt\;e^{i\omega t}\text{exp}\left(i\Omega Ae^{\kappa
x}e^{-\kappa t}\right)$
$\displaystyle=-\frac{1}{\sqrt{4\pi\Omega}}\frac{(-i\Omega
A)^{i\omega/\kappa}}{\kappa}\Gamma\left(\frac{-i\omega}{\kappa}\right),$ (35)
which is identical to the result of $\mathcal{O}_{1}$.
Finally, it is obvious that identical result can also be obtained if one
integrates along the retarded time $t_{r}$.
Figure 1: Reflection of plane waves by a relativistically receding perfect
mirror in (1+1)-dimensions. Blue: mirror’s trajectory. Dashed line: observer
$\mathcal{O}_{1}$’s worldline $x_{+}=x_{+}(\mathcal{O}_{1})$. Dotted line:
observer $\mathcal{O}_{2}$’s worldline $x=x(\mathcal{O}_{2})$.
### IV.2 Flying mirror in (1+3)D
In a (1+3)-dimensional spacetime with an infinite-size plane mirror moving
along the $x$-axis, the concept of light cone coordinate $x_{\pm}=t\pm x$ is
not suitable for discussion. Nevertheless, discussions based on the
observation point $(t,\mathbf{x})$ remain valid and more convenient. Thus, we
shall consider an observer $\mathcal{O}_{1}$ with the worldline
$\mathbf{x}=\mathbf{x}(\mathcal{O}_{1})$, as shown in Fig.2. Since the plane
wave mode $v_{\mathbf{p}}(\mathbf{x}(\mathcal{O}_{1}))\sim\text{exp}(-i\omega
t+i\mathbf{p}\cdot\mathbf{x})$ can serve as a basis at
$\mathbf{x}=\mathbf{x}(\mathcal{O}_{1})$, this validates the action of taking
the Fourier transformation of the reflected wave mode and interpreting its
Fourier components as Bogoliubov coefficients.
Figure 2: Reflection of plane waves by a relativistically receding perfect
mirror in higher dimensions. Blue: mirror’s trajectory. Dashed line: observer
$\mathcal{O}_{1}$’s worldline $x=x(\mathcal{O}_{1})$.
We shall again consider a perfect mirror with trajectory that asymptotes:
$x_{r}(t_{r})\sim-t_{r}-A\text{exp}(-2\kappa
t_{r})-B,\left\\{A,B,\kappa\right\\}>0$ at late times. For incident in modes
catching up the receding mirror, the critical incident angle $\theta_{c}$ and
maximum incident angle $\theta_{m}$ are
$\displaystyle\theta_{c}(t_{r})$ $\displaystyle\sim 2A\kappa\;e^{-2\kappa
t_{r}}+\mathcal{O}(e^{-4\kappa t_{r}})\ll 1,$ (36)
$\displaystyle\theta_{m}(t_{r})$ $\displaystyle\sim\sqrt{4A\kappa}\;e^{-\kappa
t_{r}}+\mathcal{O}(e^{-3\kappa t_{r}})\ll 1.$ (37)
Since $\theta_{m}(t_{r})\ll 1$ at late times, only incident in modes with
$\theta_{i}\leq\theta_{m}\ll 1$ can catch up the receding mirror and get
reflected. Among the reflected in modes, those with $\theta_{i}<\theta_{c}\ll
1$ are reflected backwardly, while those with
$\theta_{c}<\theta_{i}\leq\theta_{m}\ll 1$ are reflected forwardly.
For an incident in mode with $\theta_{i}\leq\theta_{m}\ll 1$, its reflected
frequency and longitudinal momentum are
$\displaystyle\omega(t_{r})$ $\displaystyle\sim\Omega
A\kappa\cos^{2}\left(\theta_{i}/2\right)e^{-2\kappa
t_{r}}+\mathcal{O}(e^{-4\kappa t_{r}}),$ (38) $\displaystyle p_{x}(t_{r})$
$\displaystyle\sim\Omega A\kappa\cos^{2}\left(\theta_{i}/2\right)e^{-2\kappa
t_{r}}+\mathcal{O}(e^{-4\kappa t_{r}}).$ (39)
Thus, to an observer at $\mathbf{x}=\mathbf{x}(\mathcal{O}_{1})$, the only
non-trivial wave mode he/she will receive is the reflected in mode with
negligible transverse momentum which comes from the reflection of an incident
in mode with $\theta_{i}<\theta_{c}\ll 1$.
Having the above discussions in mind, we decompose the in annihilation
operator $\hat{a}_{\mathbf{P}}$ into longitudinal component
$\hat{a}_{\mathbf{P}}^{\parallel}\equiv
2\pi\delta^{(2)}(\mathbf{P}_{\perp})\hat{c}_{P_{x}}$ and its complement
$\hat{a}_{\mathbf{P}}^{c}$, where the commutation relation for
$\hat{c}_{P_{x}}$ is
$\left[\hat{c}_{P_{x}},\hat{c}_{P_{x}^{\prime}}^{\dagger}\right]=\frac{\delta(P_{x}-P_{x}^{\prime})}{\mathcal{A}_{\infty}},$
(40)
where $\mathcal{A}_{\infty}\rightarrow\infty$ denotes the mirror’s transverse
area.
Thus, the decomposition of a scalar field now becomes
$\displaystyle\hat{\phi}(t,\mathbf{x})$
$\displaystyle=\int_{P_{x}<0}d^{3}P\left[(\hat{a}_{\mathbf{P}}^{\parallel}+\hat{a}_{\mathbf{P}}^{c})u_{\mathbf{P}}(t,\mathbf{x})+h.c.\right]$
$\displaystyle=\int_{\mathbb{R}^{-}}dP_{x}\left[\hat{c}_{P_{x}}\cdot\left.2\pi
u_{\mathbf{P}}(t,\mathbf{x})\right|_{\mathbf{P}_{\perp}=0}+h.c.\right]$
$\displaystyle\quad+\int_{P_{x}<0}d^{3}P\left[\hat{a}_{\mathbf{P}}^{c}u_{\mathbf{P}}(t,\mathbf{x})+h.c.\right],$
(41)
$\displaystyle=\int_{p_{x}>0}d^{3}p\left[\hat{b}_{\mathbf{p}}v_{\mathbf{p}}(t,\mathbf{x})+h.c.\right],$
(42)
to the mirror’s right, where the in mode with $\mathbf{P}_{\perp}=0$ is
$\displaystyle\left.u_{\mathbf{P}}(t,\mathbf{x})\right|_{\mathbf{P}_{\perp}=0}=\frac{e^{-i\Omega
t+iP_{x}x}-e^{-i\int dx_{-}\omega(t_{r})}}{(2\pi)^{3/2}\sqrt{2\Omega}},$ (43)
where $\Omega=|P_{x}|$, and Eq. (38) will later be used as the explicit
expression for $\omega(t_{r})$.
For the out observer $\mathcal{O}_{1}$ at
$\mathbf{x}=\mathbf{x}(\mathcal{O}_{1})$, the (reflected) in mode can be
decomposed into Fourier components as
$\displaystyle u_{\mathbf{P}}(\mathbf{x}(\mathcal{O}_{1}))=-\frac{e^{-i\int
dx_{-}\omega(t_{r}(x_{-}))}}{(2\pi)^{3/2}\sqrt{2\Omega}}$ (44)
$\displaystyle=\int_{p_{x}>0}\frac{d^{3}p}{(2\pi)^{4}\sqrt{\omega}}\left[e^{ip\cdot
x}\tilde{u}_{\Omega}(\omega,\mathbf{p})+e^{-ip\cdot
x}\tilde{u}_{\Omega}(-\omega,-\mathbf{p})\right],$
where $\omega=p^{t}=-p_{t}=|\mathbf{p}|$, and $p\cdot x=-\omega
t+\mathbf{p}\cdot\mathbf{x}$, and the Bogoliubov coefficients are given by
$\displaystyle\alpha_{\mathbf{pP}}=\frac{\sqrt{2}}{2\pi}\frac{\tilde{u}_{\Omega}(\omega,\mathbf{p})}{(2\pi)^{3/2}},\quad\bar{\beta}_{\mathbf{pP}}=-\frac{\sqrt{2}}{2\pi}\frac{\tilde{u}_{\Omega}(-\omega,-\mathbf{p})}{(2\pi)^{3/2}}.$
(45)
Since the first line of Eq. (44) is independent of $\mathbf{x}_{\perp}$, the
Fourier components must have the form:
$\tilde{u}_{\Omega}(\pm\omega,\pm\mathbf{p})=\delta^{(2)}(\mathbf{p}_{\perp})\underline{u}_{\Omega}(\pm\omega,\pm
p_{x})$. Then, the Fourier components are related to the Fourier
transformation of the in mode by
$\displaystyle\underline{u}_{\Omega}(\omega,p_{x})$
$\displaystyle=-\frac{(2\pi)^{3/2}\sqrt{\Omega}}{\sqrt{2\omega}}e^{-i\Omega
x}\int_{\mathbb{R}^{1}}dt\;e^{i\Omega t}e^{-i\int dx_{-}\omega(t_{r})}$
$\displaystyle=-\frac{(2\pi)^{3/2}\sqrt{\Omega}}{\sqrt{2\omega}}\frac{(-i\Omega
A)^{i\omega/\kappa}}{\kappa}\Gamma\left(\frac{-i\omega}{\kappa}\right),$ (46)
where we have used Eq. (38) and the fact that $\theta_{i}\ll 1$. Thus, the
Bogoliubov coefficients of interest are
$\displaystyle\alpha_{\mathbf{pP}}$
$\displaystyle=-\frac{\delta^{(2)}(\mathbf{p}_{\perp})}{2\pi\kappa}\frac{\sqrt{\omega}}{\sqrt{\Omega}}(-i\Omega
A)^{i\omega/\kappa}\Gamma\left(-\frac{i\omega}{\kappa}\right),$ (47)
$\displaystyle\bar{\beta}_{\mathbf{pP}}$
$\displaystyle=\frac{\delta^{(2)}(\mathbf{p}_{\perp})}{2\pi\kappa}\frac{\sqrt{\omega}}{\sqrt{\Omega}}(-i\Omega
A)^{-i\omega/\kappa}\Gamma\left(\frac{i\omega}{\kappa}\right).$ (48)
The mean occupation number of out particles is thus
$\displaystyle\frac{d^{3}N}{d^{3}p}$
$\displaystyle=\left<0;\text{in}\right|\hat{b}_{\mathbf{p}}^{\dagger}\hat{b}_{\mathbf{p}}\left|0;\text{in}\right>$
$\displaystyle=\lim_{\mathbf{p}^{\prime}\rightarrow\mathbf{p}}\frac{(2\pi)^{2}}{\mathcal{A}_{\infty}}\int_{\mathbb{R}^{-}}dP_{x}\beta_{\mathbf{pP}}\bar{\beta}_{\mathbf{p^{\prime}P}}$
$\displaystyle=\frac{(2\pi)^{2}\delta^{(2)}(\mathbf{p}_{\perp})}{\mathcal{A}_{\infty}}\frac{\delta^{(2)}(\mathbf{p}_{\perp})\Delta
t}{2\pi}\left(\frac{1}{e^{\omega/T_{H}}-1}\right),$ (49)
where $p_{x}=\omega>0$, $\Delta
t=2\pi\delta(\omega-\omega^{\prime}=0)\rightarrow\infty$ is the infinite
accelerating time period measured by the out observer $\mathcal{O}_{1}$, the
first fraction in the last line equals to unity for
$\mathbf{p}_{\perp}=\mathbf{0}$, and $T_{H}=\kappa/2\pi$ is the Hawking
temperature. This is the analog Hawking radiation spectrum emitted by a
perfectly reflecting, infinite transverse area, plane mirror in
(1+3)-dimensional flat spacetime at late times and is to be observed by an out
observer $\mathcal{O}_{1}$ on the right-hand-side of the leftwardly receding
mirror.
By using the identity:
$\displaystyle\lim_{L\rightarrow\infty}\text{sinc}(pL/2)=2\pi\delta(p)/L,$
the Dirac delta functions can be replaced by sinc functions instead. This
alternative expression allows a direct extrapolation to the case of a finite-
size, say, square mirror with transverse area: $\mathcal{A}=L\times L$ as
$\displaystyle\frac{d^{3}N}{d^{3}p}\approx\frac{\mathcal{A}\Delta
t}{(2\pi)^{3}}\;\text{sinc}^{2}\left(\frac{p_{y}L}{2}\right)\text{sinc}^{2}\left(\frac{p_{z}L}{2}\right)\left(\frac{1}{e^{\omega/T_{H}}-1}\right)$
for $\omega=|\mathbf{p}|\sim p_{x}\gg|\mathbf{p}_{\perp}|$. If
$|\mathbf{p}_{\perp}|$ is not required to be negligible compared to $p_{x}$,
then we should additionally replace $T_{H}$ by
$T_{eff}=\kappa/[(1+\cos\theta)\pi]$ since the mirror is neither isotropic nor
translational invariant [15, 16]. The appearance of sinc functions is a
manifestation of diffraction due to the mirror’s finite transverse geometry.
In far field, using $p_{y}\sim\omega y/R$ and $p_{z}\sim\omega z/R$, where
$R\gg|\mathbf{x}_{\perp}|$ is the longitudinal distance between the observer
$\mathcal{O}_{1}$ and the mirror at the instant of emission, the arguments of
the sinc functions become $\omega Ly/(2R)\sim\omega L\theta\cos\phi/2$ and
$\omega Lz/(2R)\sim\omega L\theta\sin\phi/2$, which are also justified by
$p_{y}=\omega\sin\theta\cos\phi$ and $p_{z}=\omega\sin\theta\sin\phi$ when
$\theta\ll 1$.
Similarly, for a circular mirror of diameter $D$, the particle spectrum would
be
$\displaystyle\frac{d^{3}N}{d^{3}p}\approx\frac{\mathcal{A}\Delta
t}{(2\pi)^{3}}\left[2\text{jinc}\left(\frac{|\mathbf{p}_{\perp}|D}{2}\right)\right]^{2}\left(\frac{1}{e^{\omega/T_{H}}-1}\right),$
where $\mathcal{A}=D^{2}\pi/4$, $\text{jinc}(x)=J_{1}(x)/x$, $J_{1}(x)$ is the
Bessel function of the first kind of order 1, and
$|\mathbf{p}_{\perp}|\sim\omega\theta$ with $\theta\ll 1$. This leads to the
Airy pattern, and the first minimum of the jinc function occurs at
$\theta=\theta_{1}=7.66/(\omega D)=1.22\lambda/D$, where $\lambda=2\pi/\omega$
is the wavelength of the emitted out particle.
The total occupation number can be obtained by integrating over the momentum
$\mathbf{p}$. For $\mathcal{A}\rightarrow\infty$, only particles with
$|\mathbf{p}_{\perp}|=0$ can be emitted, and the above three types of mirrors
all lead to the same yield:
$\displaystyle N$ $\displaystyle=\frac{\Delta
t}{2\pi}\int_{\omega_{1}}^{\omega_{2}}\frac{d\omega}{e^{2\pi\omega/\kappa}-1}$
$\displaystyle=\frac{T_{H}\Delta
t}{2\pi}\log\left(\frac{1-e^{-\omega_{2}/T_{H}}}{1-e^{-\omega_{1}/T_{H}}}\right),$
(50)
where $(\omega_{1},\omega_{2})$ is the frequency range of interest.
On the other hand, for $\mathcal{A}=$ finite, the emitted particles can
acquire non-negligible transverse momenta. However, analytic expressions of
spectra can only be obtained under certain approximations. For simplicity, in
the case of a square mirror, the frequency spectrum in the low frequency
regime: $\omega\ll L^{-1}$ is
$\displaystyle\frac{dN}{d\omega}\approx\frac{\mathcal{A}\Delta
t}{(2\pi)^{2}}\frac{\kappa\omega}{\pi}\log\big{(}1+e^{-\pi\omega/\kappa}\big{)},$
(51)
while the angular spectrum with $\theta\ll 1$ is
$\displaystyle\frac{dN}{d\Omega}\approx\frac{\mathcal{A}\Delta
t}{(2\pi)^{3}}\frac{\kappa^{3}\zeta(3)}{4\pi^{3}}\Big{[}1-\frac{\zeta(5)(\kappa
L)^{2}-3\pi^{2}\zeta(3)}{4\pi^{2}\zeta(3)}\theta^{2}\Big{]},$ (52)
where $\zeta$’s are Riemann zeta functions. The full spectra are plotted in
Figs. 3 $\&$ 4, which are obtained from numerical integrations and they indeed
agree well with the above analytic formulae in the corresponding regimes.
Figure 3: Frequency spectrum of analog Hawking radiation. Figure 4: Angular
spectrum of analog Hawking radiation.
The frequency spectrum initially grows linearly in $\omega$ in the low
frequency regime. Since small frequency corresponds to long wavelength, the
spectrum is simply proportional to the mirror’s area. However, as the
frequency gets higher, the wavelength becomes comparable to the area and
diffraction emerges, leading to the wiggling in the figure. On the other hand,
as indicated in Eq. (52), for $\zeta(5)(\kappa L)^{2}>3\pi^{2}\zeta(3)$, which
is the case used in the figure, the angular spectrum has its maximum at
$\theta=0$ and decreases quadratically in $\theta$ as $\theta$ gets larger.
In the recently proposed AnaBHEL collaboration [17, 18, 19], the flying mirror
generated via plasma-laser interaction has a low reflectivity [23] and the
event yield per laser shot is estimated as $N\sim 0.3$ [16].
Assuming a perfectly reflecting plane mirror with the same trajectory and
parameter values, i.e., $L=254\;\text{eV}^{-1},\;\kappa=0.2\;\text{eV}$, can
be generated, which faces both physical and technical challenges, and noting
that the Davies-Fulling-like motion takes about $\Delta t\sim 1$ ps $\sim
1520$ eV-1 based on the design in Ref.[17], the yield of analog Hawking
particles per laser shot is then $N\sim 0.011\times 1520\sim 16$ events, where
$0.011$ eV is the area under the frequency spectrum in Fig. 3.
For a petawatt-class laser facility that provides 1 laser shot per minute and
8 hours of operation time per day, a 20-day experiment with an ideal detector
efficiency would then give the total yield as $N_{total}\sim 160,000$ events.
## V Conclusion
The conventional approach toward radiation induced by a relativistically
flying mirror in spacetimes other than (1+1) dimensions is, if possible,
difficult. If the mirror is non-relativistic and perfectly reflecting, then it
can be solved by perturbing a static mirror solution [12, 13, 14]; if the
mirror is relativistic but semitransparent, it can also be solved by
perturbing a free field solution [15, 16].
To circumvent the difficulty encountered by a perfectly reflecting mirror in
higher dimensional spacetimes, we begin by studying the frequency and momentum
of an incident plane wave after reflection from a flying mirror instead. From
this, we observe that, in general, the reflected frequency and momentum only
depend on the mirror’s velocity, while the phase of the reflected wave will
also depend on the acceleration. This viewpoint provides crucial physical
insights which allow us to generalize the discussion to higher dimensional
spacetimes, which is important for actual laboratory experiments.
It should be noted that by striking a flying mirror by a classical plane wave
and Fourier transforming the received reflected wave can indeed lead to a
frequency spectrum. However, this is only a classical signal in Fourier space.
It is for quantum fluctuations that the Fourier components of the reflected
wave modes have the interpretation as quantum particle creation from the
vacuum (although this quantum radiation spectrum has exactly the same form as
that of the previous classical spectrum).
Without going through rigorous derivation of the radiation spectrum, one may
also heuristically argue the expected expression from dimensional analysis.
Since the mean occupation number $d^{3}N/d^{3}p$ has a dimension of length3,
and, for a relativistically receding mirror, the reflected wave mode are
mostly longitudinal, one expects
$d^{3}N/d^{3}p\sim\delta^{(2)}(\mathbf{p}_{\perp})dN/dp_{x}$ up to a
dimensionless proportionality constant. By going through rigorous derivation,
the proportionality constant can be uniquely determined, which is important to
the estimation of event yield for a proposed experiment.
The results based on the viewpoint adopted in this paper also allow direct
generalization to mirrors with various geometries, e.g., square, circular,
etc., and diffraction in the radiation spectrum emerges.
Finally, having robust expressions for the analog Hawking radiation spectra,
we estimated, in the case of $T_{H}=0.03$ eV, the yield to be $N\sim 16$
events per laser shot for an ideal detector, and $N_{total}\sim 160,000$ in a
20-day experiment, which is roughly 50 times larger than the yield,
$N_{total}\sim 3,000$, for a square-root-Lorentzian (SRLD), semitransparent
mirror [16].
###### Acknowledgements.
This work is supported by ROC (Taiwan) Ministry of Science and Technology
(MOST), National Center for Theoretical Sciences (NCTS), and Leung Center for
Cosmology and Particle Astrophysics (LeCosPA) of National Taiwan University.
P.C. is in addition supported by U.S. Department of Energy under Contract No.
DE-AC03-76SF00515.
## References
* Hawking [1975] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
* Moore [1970] G. T. Moore, Quantum theory of the electromagnetic field in a variable‐length one‐dimensional cavity, J. Math. Phys. (N.Y.) 11, 2679 (1970).
* DeWitt [1975] B. S. DeWitt, Quantum field theory in curved spacetime, Phys. Rep. 19, 295 (1975).
* Fulling and Davies [1976] S. A. Fulling and P. C. W. Davies, Radiation from a moving mirror in two dimensional space-time: conformal anomaly, Proc. R. Soc. A 348, 393 (1976).
* Davies and Fulling [1977] P. C. W. Davies and S. A. Fulling, Radiation from moving mirrors and from black holes, Proc. R. Soc. A 356, 237 (1977).
* Carlitz and Willey [1987] R. D. Carlitz and R. S. Willey, Reflections on moving mirrors, Phys. Rev. D 36, 2327 (1987).
* Foo and Good [2021] J. Foo and M. R. R. Good, Hawking radiation particle spectrum of a Kerr-Newman black hole, JCAP 01, 019 (2021), arXiv:2006.09681 [gr-qc] .
* Good _et al._ [2020] M. R. R. Good, A. Zhakenuly, and E. V. Linder, Mirror at the edge of the universe: Reflections on an accelerated boundary correspondence with de Sitter cosmology, Phys. Rev. D 102, 045020 (2020), arXiv:2005.03850 [gr-qc] .
* Good [2020] M. R. R. Good, Extremal Hawking radiation, Phys. Rev. D 101, 104050 (2020), arXiv:2003.07016 [gr-qc] .
* Good and Ong [2020] M. R. R. Good and Y. C. Ong, Particle spectrum of the Reissner–Nordström black hole, Eur. Phys. J. C 80, 1169 (2020), arXiv:2004.03916 [gr-qc] .
* Good _et al._ [2017] M. R. R. Good, K. Yelshibekov, and Y. C. Ong, On horizonless temperature with an accelerating mirror, JHEP 03, 013 (2017), arXiv:1611.00809 [gr-qc] .
* Ford and Vilenkin [1982] L. H. Ford and A. Vilenkin, Quantum radiation by moving mirrors, Phys. Rev. D 25, 2569 (1982).
* Neto and Machado [1996] P. A. M. Neto and L. A. S. Machado, Quantum radiation generated by a moving mirror in free space, Phys. Rev. A 54, 3420 (1996).
* Rego _et al._ [2013] A. L. C. Rego, B. W. Mintz, C. Farina, and D. T. Alves, Inhibition of the dynamical Casimir effect with Robin boundary conditions, Phys. Rev. D 87, 045024 (2013), arXiv:1302.4709 [quant-ph] .
* Lin _et al._ [2021] K.-N. Lin, C.-E. Chou, and P. Chen, Particle production by a relativistic semitransparent mirror in (1+3)D Minkowski spacetime, Phys. Rev. D 103, 025014 (2021), arXiv:2008.12251 [gr-qc] .
* Lin and Chen [2021] K.-N. Lin and P. Chen, Particle production by a relativistic semitransparent mirror of finite transverse size (2021), arXiv:2107.09033 [gr-qc] .
* Chen _et al._ [2022] P. Chen _et al._ , [AnaBHEL Collaboration], AnaBHEL (Analog Black Hole Evaporation via Lasers) Experiment: Concept, Design, and Status (2022), arXiv:2205.12195 [gr-qc] .
* Chen and Mourou [2017] P. Chen and G. Mourou, Accelerating plasma mirrors to investigate black hole information loss paradox, Phys. Rev. Lett. 118, 045001 (2017), arXiv:1512.04064 [gr-qc] .
* Chen and Mourou [2020] P. Chen and G. Mourou, Trajectory of a flying plasma mirror traversing a target with density gradient, Phys. Plasmas 27, 123106 (2020), arXiv:2004.10615 [physics.plasm-ph] .
* Einstein [1905] A. Einstein, Zur elektrodynamik bewegter körper, Annalen der Physik 322, 891 (1905).
* Van Meter _et al._ [2001] J. R. Van Meter, S. Carlip, and F. V. Hartemann, Reflection of plane waves from a uniformly accelerating mirror, American Journal of Physics 69, 783 (2001).
* Gjurchinovski and Skeparovski [2008] A. Gjurchinovski and A. Skeparovski, Einstein’s mirror, The Physics Teacher 46, 416 (2008).
* Liu _et al._ [2021] Y.-K. Liu, P. Chen, and Y. Fang, Reflectivity and spectrum of relativistic flying plasma mirrors, Phys. Plasmas 28, 103301 (2021), arXiv:2012.05769 [physics.plasm-ph] .
|
# Classical-Quantum Combs, their Min-Entropy and their Measurement-Based
Applications
Isaac D. Smith<EMAIL_ADDRESS>Marius Krumm Lukas J. Fiderer Hendrik
Poulsen Nautrup Hans J. Briegel Institute for Theoretical Physics, UIBK,
6020 Innsbruck, Austria
###### Abstract
Learning a hidden property of a quantum system typically requires a series of
interactions. In this work, we consider a formalisation of such multi-round
learning processes that uses a generalisation of classical-quantum states,
called classical-quantum combs. Here, “classical” refers to a random variable
encoding the hidden property to be learnt, and “quantum” refers to the quantum
comb describing the behaviour of the system. By using the quantum combs
formalism, the optimal strategy for learning the hidden property can be
quantified via the comb min-entropy (Chiribella and Ebler, NJP, $2016$). With
such a tool on hand, we focus attention on an array of combs derived from
measurement-based quantum computation (MBQC) and related applications.
Specifically, we describe a known blind quantum computation (BQC) protocol
using the combs formalism and thereby leverage the min-entropy to provide a
proof of single-shot security for multiple rounds of the protocol, extending
the existing result in the literature. Furthermore, we introduce novel
connections between MBQC and quantum causal models and quantum causal
inference, which allows for the use of the min-entropy to quantify the optimal
strategy for causal discovery. We consider further operationally motivated
examples, including one associated to learning a quantum reference frame.
## 1 Introduction
With the rapid development of quantum technology, a plethora of increasingly
complex quantum devices has become available. Already, a range of noisy-
intermediate scale quantum computers (NISQ) [1] are accessible via the
internet. Year on year, these devices increase in size and quality, as
evidenced by the growing number of addressable qubits, the lengthening
coherence times and improving gate fidelity.
An important parallel development to that of the devices themselves, is in how
they interconnect. The current and future progress of quantum communication
networks [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] aims to create a global quantum
internet of networked quantum computers [12, 13], with the attendant benefits
in communication and cryptography. Current implementations of such networks
largely focus on protocols for quantum key distribution, however a variety of
other significant applications appear to be feasible in the near-term,
including distributed quantum computation [14, 15], quantum position
verification [16], and blind quantum computation [17].
With the increasing sophistication of quantum networks, a full description
using the usual quantum formalism becomes intractable. Consequently, other
formalisms are being developed that allow for the description of the relevant
parts of complex networks whilst remaining numerically and analytically
manageable. One particularly promising such formalism represents networks as
quantum causal models [18, 19, 20], in which the connectivity of the the
network is faithfully represented but the nodes are freely described by any
quantum device. This allows for investigating the statistics of the network
under varying quantum protocols at the nodes. The quantum causal modelling
formalism is a quantum generalisation of the sub-field of statistics and
machine learning known as (classical) causal modelling, which has been
successfully applied to modelling complex systems in various classical
domains, such as medicine, social science, and engineering [21, 22].
One key object of study in the quantum causal networks literature [23, 24] is
known as a quantum comb. Quantum combs naturally model multi-round quantum
communication protocols, and can be understood as a concatenation of quantum
channels where some information may be input and output from the comb at each
time-step and some may remain within the comb (and inaccessible) between time-
steps. These combs are represented by operators acting on the input and output
spaces only, which constitutes a distinct advantage of their use: any
forwarding of information between time-steps is modelled by an operator of the
same size. As such, they can model arbitrary sequential interactions with a
quantum environment, and have consequently become a fruitful tool to describe
e.g. non-Markovian noise [25].
A crucial discovery of the field of quantum causal networks is that the
optimisation of multi-round quantum communication protocols can be framed as a
semi-definite program [26], where the optimisation can be with respect to a
range of different performance measures. For a given measure, the resultant
optimal success probability leads to a notion of min-entropy for quantum
combs, which generalises the min-entropy of quantum states [27] in quantum
cryptography. In this work, we consider a performance measure based on
learning a hidden classical parameters of quantum networks, which are modelled
as classical-quantum combs: generalisations of classical-quantum states [28]
where a series of quantum combs are indexed by a classical parameter. We
analyse properties of combs of this form and investigate a range of examples
based on measurement-based quantum computation.
Measurement-based quantum computing (MBQC) [29, 30, 31, 32, 33, 34] is a well-
known paradigm for quantum computation distinct from the circuit model
formalism. It consists of a sequential interaction with a quantum state (via
time-ordered single qubit measurements) and forms the basis of a range of
cryptographic computation protocols within the sub-field of quantum
cryptography known as blind quantum computation (BQC) [35, 36, 37, 38, 17].
Due to existing characterisations of the components of MBQC, such as graph
states [39, 40, 41, 42] (see also [43]) and the sequence of measurements [32,
44, 45, 46, 47], as well as to the connection with stabiliser quantum
mechanics [48, 49], investigations into certain aspects of MBQC remain
tractable despite its multi-partite, high-dimensional nature. As such, MBQC
provides an exemplary test-bed for the application of the quantum causal
networks framework.
The first comb we consider models a specific BQC protocol, that of Mantri et
al. [36]. This protocol consists of entirely classical communication between a
client and a (quantum) server, allowing the client to obtain the results of a
quantum computation while maintaining no quantum capabilities. The classical
parameter in this context encodes the secret choice of computation by the
client which an untrustworthy server may attempt to discover. Due to the
restriction to classical communication, the quantum comb part of the
classical-quantum comb, the part that models a round of the protocol, is
actually classical (i.e. we consider a classical-classical comb). The security
of the protocol is analysed by calculating the min-entropy of this comb, which
has the interpretation of quantifying how much information the server has yet
to learn about the choice of computation when using the most informative
strategy. We provide a security analysis both for a single round and for
multiple rounds, thereby extending the analysis of the original paper [36].
We then consider a range of classical-quantum combs where the comb component
is truly quantum. In particular, we investigate a variety of related scenarios
where an unknown device performing a measurement-based computation is
interacted with in order to gain some piece of information regarding its
internal functioning. In all these scenarios, the classical-quantum combs are
built from a comb based upon the method of measurement adaptation required for
MBQC [44, 45]. We demonstrate that this base comb satisfies the conditions of
a quantum causal model [20] and moreover, we explicate an interpretation of
MBQC in the context of quantum causal inference [50], thereby establishing
novel connections between MBQC and quantum causality [20, 19, 18, 50, 51, 52,
53]. One classical-quantum comb we consider explicitly utilises this
connection: the classical parameter encodes the different possible causal
structures and the comb min-entropy quantifies the optimal causal inferential
strategy. Finally, two examples are considered, each concerned with learning
an aspect of the MBQC device required for its proper use: in one case,
learning the types of measurement that ensure deterministic computation, and
in the other, calibrating measurement devices to ensure the correct
computation is performed. The latter has an interesting interpretation as
learning a quantum reference frame [54].
The remainder of this paper is structured as follows. The next subsection
provides a brief and informal presentation of the key results in this work. In
Section 2, the required background knowledge regarding the formalisms of MBQC
and quantum combs is presented. We introduce and motivate the comb min-entropy
in Section 3, including some results regarding the min-entropy for classical-
classical combs in Section 3.1. Section 4 outlines the BQC protocol of [36]
(Section 4.1), defines the corresponding classical comb (Section 4.2), and
conducts an analysis of the security of the protocol (Section 4.3 and Section
4.4). Section 5 contains the definition of the comb describing the corrections
required for MBQC (Section 5.1), a discussion of the connection to quantum
causal models and inference (Section 5.2), and an analysis of a series of
examples (Section 5.3, Section 5.4 and Section 5.5). Section 6 discusses the
future applications and limitations of the methods presented in this work and
concludes.
### 1.1 Our Contributions
The following is a more specific, but informal, summary of the key
contributions of this paper.
* •
3.2 establishes upper and lower bounds for the min-entropy for classical-
quantum combs where the quantum combs are all diagonal in the same basis (i.e.
classical-classical combs). These bounds have an interpretation in terms of
maximal Bayesian updating, an interpretation which extends from the analogous
case for the state min-entropy (i.e. classical-classical states). These bounds
play a role in the security proof of the BQC protocol in Section 4.
* •
4.2 establishes that the BQC protocol of [36] is (partially) secure in a
single-round by proving a general lower bound on the min-entropy of the secret
choice of computation given the interactions via the protocol. This lower
bound is strictly positive, indicating that the secret computation cannot be
known with certainty after only a single round, even if the optimal strategy
is used. This result extends the existing result [[, Theorem
2,]]mantri2017flow via use of the min-entropy as opposed to the Shannon
entropy. Furthermore, we give a simple example that obtains the lower bound,
which gives some indication of its tightness.
* •
Similarly, 4.3 provides a positive lower bound for the min-entropy under any
number of rounds of the protocol, demonstrating that the server will remain
uncertain about at least some aspect of the computation even if the protocol
is repeated. We use the same example as for the single-round case to
demonstrate that the min-entropy can strictly decrease between rounds,
indicating that more information is leaked in the multi-round case in general.
* •
5.1 establishes the connection between MBQC and quantum causal models, by
demonstrating that the conditional adaptation channels defined by gflow [44]
satisfy the required commutation conditions of the channels defining a quantum
causal model.
* •
The above connection allows us, via the min-entropy, to frame quantum causal
discovery as a semi-definite program, thereby establishing a new tool in the
quantum causal inference toolbox. We use this approach to numerically quantify
the optimal causal inferential strategy for a given example.
* •
We demonstrate that in certain circumstances, not even the optimal strategy
can learn anything about the causal structure such as when the possible causal
structures are related by symmetries. 5.2 establishes this fact for when the
causal structures are given by gflows that are related by symmetries of the
underlying graph state.
## 2 Preliminaries
This section introduces the background information regarding quantum combs and
measurement-based quantum computation as required for the introduction of the
comb min-entropy in Section 3 and the applications presented in Section 4 and
Section 5. Some emphasis is placed on aspects of the discourse deemed
important for the sequel, otherwise the aim is for brevity without sacrificing
completeness.
### 2.1 Multi-round Quantum Protocols: Quantum Combs
The standard framework for quantum information theory typically consists of
density matrices, positive operator-valued measures (POVMs) and quantum
channels (completely positive trace-preserving maps) to describe the
processing of quantum information. Each of these can be defined for or act
upon multiple quantum systems, can be composed, and moreover, can be composed
on subsystems, such as when two quantum channels are composed over the domain
of the latter and a subsystem of the co-domain of the former. Quantum
networks, separated nodes with quantum processing capabilities and
interconnected via quantum communication, can be modelled via an array of such
compositions of quantum channels. In fact, it was realised [24, 23] that it is
always possible to represent a quantum network in a canonical form, namely as
a sequence of channels with memory, where some of the output of each channel
is passed directly as input to the next, and some is available for arbitrary
processing as depicted in Figure 1. Such a sequence comes with an explicit
ordering on inputs and outputs of the network, from which the interpretation
of interacting with a quantum system over multiple time steps can be naturally
understood. Throughout this work, we use standard notation: $\mathcal{H}_{A}$
denotes a Hilbert space for system $A$, $\mathcal{L}(\mathcal{H}_{A})$ the
space of linear operators on that space, $\rho_{A}$ a state of
$\mathcal{H}_{A}$,
$\mathcal{E}:\mathcal{L}(\mathcal{H}_{A})\rightarrow\mathcal{L}(\mathcal{H}_{B})$
for a CPTP map, and so on.
Instead of explicitly working with a quantum network as a sequence of
channels, it is convenient to work with the corresponding operator, a quantum
comb, which is a generalisation of the Choi operator [55, 56] to a sequence of
quantum channels. A quantum comb is an operator on a Hilbert space consisting
of the tensor product of all input spaces
$\mathcal{H}_{j}^{\operatorname{in}}$ and output spaces
$\mathcal{H}_{j}^{\operatorname{out}}$. Each interaction with the system is
associated to an input-output pair of Hilbert spaces, between which a quantum
instrument can be inserted specifying the interaction. The distinction between
input and output is indicated by superscripts in the notation, with time-steps
indicated by subscripts. We take the following as our definition of quantum
comb, which is sometimes presented as a theorem establishing the connection to
quantum networks instead (see e.g. [[, Theorem 1,]]chiribella2008quantum, [[,
Theorem 3,]]chiribella2009theoretical):
###### Definition 2.1.
A quantum comb is a positive semi-definite operator
$D\in\mathcal{L}\left(\bigotimes_{j=1}^{n}\mathcal{H}_{A_{j}^{\operatorname{in}}}\otimes\mathcal{H}_{A_{j}^{\operatorname{out}}}\right)$
for which there exists a sequence of positive semi-definite operators
$D_{k}\in\mathcal{L}\left(\bigotimes_{j=1}^{k}\mathcal{H}_{A_{j}^{\operatorname{in}}}\otimes\mathcal{H}_{A_{j}^{\operatorname{out}}}\right)$,
$k=0,...,n$ which satisfy
$\displaystyle\operatorname{Tr}_{A_{k}^{\operatorname{out}}}\left[D_{k}\right]=I_{A_{k}^{\operatorname{in}}}\otimes
D_{k-1}\quad\quad\forall k\in\\{1,...,n\\},$ (1)
with $D_{n}=D$ and $D_{0}=1$. A comb $D$ is called unnormalised if $D_{0}$
instead satisfies the relaxed constraint $D_{0}>0$.
The partial trace conditions in the above definition correspond to enforcing
trace preservation of the individual component channels and positive semi-
definiteness enforces complete positivity (for a more thorough explanation,
see Section A.1 or [24, 23]). Since we also consider combs in this work that
are classical, which is to say diagonal, it is useful to have an explicit
definition for later reference:
###### Definition 2.2.
A positive semi-definite operator
$C\in\mathcal{L}\left(\bigotimes_{j=1}^{n}\mathcal{H}_{A_{j}^{\operatorname{in}}}\otimes\mathcal{H}_{A_{j}^{\operatorname{out}}}\right)$
is a classical comb if it can be written as
$\displaystyle
C=\sum_{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}f(\boldsymbol{a}^{\operatorname{out}},\boldsymbol{a}^{\operatorname{in}})\ket{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\bra$
where
$\\{\ket{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\}$
is an orthonormal basis for the Hilbert space and where $f$ is a non-negative
function for which there exists a sequence of non-negative functions
$f^{(k)}$, $k=0,...,n$ that satisfy, for all $j=2,...,n$:
$\displaystyle\sum_{a_{j}^{\operatorname{out}}}f^{(j)}$
$\displaystyle(a_{1}^{\operatorname{out}},...,a_{j}^{\operatorname{out}},a_{1}^{\operatorname{in}},...,a_{j}^{\operatorname{in}})$
$\displaystyle=f^{(j-1)}(a_{1}^{\operatorname{out}},...,a_{j-1}^{\operatorname{out}},a_{1}^{\operatorname{in}},...,a_{j-1}^{\operatorname{out}})$
(2)
with $f^{(n)}=f$ and
$f^{(0)}:=\sum_{a_{1}^{\operatorname{out}}}f^{(1)}(a_{1}^{\operatorname{out}},a_{1}^{\operatorname{in}})>0$.
If $f^{(0)}=1$ then the comb is normalised, otherwise it is unnormalised.
The conditions described in Equation 2 are a simple rewriting of the trace
conditions in Equation 1. In the case where the classical comb is normalised,
it is most intuitive to write
$f(\boldsymbol{a}^{\operatorname{out}},\boldsymbol{a}^{\operatorname{in}})$ as
a conditional probability distribution
$P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})$
and the conditions of Equation 2 then detail independence conditions on this
distribution under sequential marginalisation of the
$a_{j}^{\operatorname{out}}$. Note that these distributions can be understood
as observable quantities by using $C$ simply as a channel, i.e. by inputting
all $\boldsymbol{a}^{\operatorname{in}}$ at once and collecting statistics of
the outputs.
Figure 1: A quantum comb is a representation of a quantum system evolving over
multiple time-steps. An example of a comb $D$ (grey outline) is shown,
corresponding to the sequence of quantum channels
$\mathcal{D}_{1},\mathcal{D}_{2},\mathcal{D}_{3}$, each representing the
evolution of the system between adjacent time-steps. Interaction with the
system at a given time-step can also be represented by a quantum channel, here
denoted $\mathcal{E}$.
We use the notation $\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$ to denote the set of all combs on
$\bigotimes_{j=1}^{n}\mathcal{H}_{A_{j}^{\operatorname{in}}}\otimes\mathcal{H}_{A_{j}^{\operatorname{out}}}$,
including unnormalised combs. It will be clear from context, whether we are
considering classical combs only for a specific example. If a given input or
output space is $1$-dimensional, the corresponding
$A_{j}^{\operatorname{in}/\operatorname{out}}$ is replaced with $\mathbb{C}$.
For example, we will often consider combs where the first input space is
$1$-dimensional, that is, elements of
$\operatorname{Comb}(\mathbb{C}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$.
For later sections, particularly Section 5, it useful to use the following
notation, which provides a convenient way of writing the contraction of a comb
with a compatible operator (for example, for contracting $\mathcal{E}$ with
the comb in Figure 1).
###### Definition 2.3 ([23]).
For two operators $M\in\mathcal{L}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ and
$N\in\mathcal{L}(\mathcal{H}_{B}\otimes\mathcal{H}_{C})$, the link product is
defined as:
$\displaystyle M*N:=\operatorname{Tr}_{B}\left[(M^{T_{B}}\otimes
I_{C})\cdot(I_{A}\otimes N)\right],$ (3)
where $T_{B}$ denotes the partial transpose over the system $B$.
(a) Classical-quantum comb $D$
(b) Multi-round classical-quantum comb $D^{(m)}$
Figure 2: In this work, we consider an extension to classical-quantum states,
called classical-quantum combs, where a classical random variable $X$ indexes
a set of quantum combs $\sigma_{x}$ (rather than quantum states). (a) The
variable $X$ is considered to be unknown and as such is represented as an
inaccessible system (upper part). The combs $\sigma_{x}$ are contingent on the
value of $X$ and describe the dynamics of a system which can be interacted
with (lower part) over a series of time-steps. By interacting with the
accessible system, updated knowledge of $X$ can be obtained. (b) Some
interactions are naturally modelled as a series of separate rounds, which are
independent apart from the dependence on $X$. This is represented by taking
each round to be a separate comb $\sigma_{x}^{(j)}$ which have no direct
influence on later rounds. An example of such a multi-round interaction is the
BQC protocol presented in Section 4.
In this work, we consider the comb extension of classical-quantum states,
called classical-quantum combs, which consist of a series of combs indexed by
a classical random variable. Explicitly, we consider operators of the form
$\displaystyle D=\sum_{x\in X}P(x)\ket{x}\\!\\!\bra{x}\otimes\sigma_{x}$ (4)
which is an element in
$\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}},\mathbb{C}\rightarrow X)$ where $X$ is a (finite,
discrete) random variable with outcomes defining an orthonormal basis
$\\{\ket{x}\\}_{x}$ of the $|X|$-dimensional Hilbert space $\mathcal{H}_{X}$,
and where the $\sigma_{x}$ are in combs in
$\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$. Note that, even though the state corresponding
to $\mathcal{H}_{X}$ is written on the left in $D$, $\mathcal{H}_{X}$ is
always taken to be the last output space of quantum comb, in order to be
compatible with the comb min-entropy defined below.
The appropriate perspective to take of $D$ for the remainder of this work,
which is elaborated upon below, is that we aim to learn something about the
classical state on $X$ by interacting only with the $\sigma_{x}$, i.e. we only
have access to the spaces
$\mathcal{H}_{A_{1}^{\operatorname{in}}},...,\mathcal{H}_{A_{n}^{\operatorname{out}}}$
and not $\mathcal{H}_{X}$. This is portrayed in the pictorial representation
of $D$ in Figure 2(a), where the lower part of the comb is accessible and the
upper part is not.
It is important to note that, due to the special form that $D$ takes, it is
both an element of $\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}},\mathbb{C}\rightarrow X)$ and an element of
$\operatorname{Comb}(\mathbb{C}\rightarrow
X,A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$, as demonstrated by the proof of the following
result (given in Section A.1).
###### Proposition 2.1.
The operator $D$ as above is an element of both
$\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}},\mathbb{C}\rightarrow X)$ and
$\operatorname{Comb}(\mathbb{C}\rightarrow
X,A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$.
This result is significant for our purposes since the latter inclusion
highlights that there is no back-in-time signalling from the inputs of the
$\sigma_{x}$ to the output at $X$ (cf. [[, Section
III.C,]]chiribella2009theoretical), a fact that is explicitly used in Section
3.1.
At times, we consider interactions with a system that occur in a series of
rounds, where the evolution of the system between rounds is independent except
for a common dependence on the parameter $X$ (this is represented in Figure
2(b)). For example, repeated instances of a multi-step cryptographic protocol
can be considered in this way (see Section 4). Despite being able to represent
these interactions using $D$ above, where independence of rounds is merely an
extra constraint on the $\sigma_{x}$, we find it more convenient to use
$\sigma_{x}$ to denote a single round, and write the multi-round operator as
follows (the round number is indicated by the superscript in brackets):
$\displaystyle D^{(m)}:=\sum_{x\in
X}P(x)\ket{x}\\!\\!\bra{x}\otimes\bigotimes_{j=1}^{m}\sigma_{x}^{(j)}$ (5)
which is an operator in
$\operatorname{Comb}\left((A_{1}^{\operatorname{in}})^{(1)}\\!\rightarrow\\!(A_{1}^{\operatorname{out}})^{(1)},...,(A_{n}^{\operatorname{in}})^{(m)}\\!\rightarrow\\!(A_{n}^{\operatorname{out}})^{(m)},\mathbb{C}\\!\rightarrow\\!X\right)$.
### 2.2 MBQC and Gflow
Measurement-based quantum computation [29, 30, 31, 32, 33] is a model for
quantum computation distinct from the circuit picture, where, as the name
indicates, the computation is driven by measurement operations rather than
unitary ones. One can consider MBQC as consisting of three components: a
highly entangled state, namely a graph state, which is the substrate that
supports the computation; a sequence of single qubit projective measurements
parametrised by an angle in a plane of the Bloch sphere, the positive outcomes
of which encode the desired computation; and a correction method which ensures
the desired computation is performed even when negative outcomes are obtained.
The latter refers to the notion of gflow [44, 45] which plays a key role in
the blind quantum computing protocol of [36] as well as in Section 4 and
Section 5 below. In this subsection, each of these three components is
introduced, with emphasis on the latter due to its importance in the sequel.
Graph states take their name from the connection to mathematical graphs. Let
$G$ be a (simple, connected) graph on vertex set $V=\\{1,...,n\\}$ and edge
set $E$. One can define a graph state, denoted $\ket{G}$, as
$\displaystyle\ket{G}:=\prod_{(i,j)\in E}CZ_{ij}\ket{+}^{\otimes n}$ (6)
where each vertex of the graph is assigned a qubit in the $\ket{+}$ state and
where each edge $(i,j)$ in the graph is associated to a controlled Pauli-$Z$
gate, denoted $CZ_{ij}$, between the corresponding qubits. Equivalently,
$\ket{G}$ can be specified as the unique stabiliser state of the set of
stabilisers [57] generated by
$\displaystyle\left\\{K_{v}:=X_{v}\bigotimes_{v^{\prime}\in
N_{v}^{G}}Z_{v^{\prime}}|v\in V\right\\}$ (7)
where $N_{v}^{G}$ denotes the set of neighbours of $v$ in $G$. The stabiliser
picture of graph states is a fruitful one since the correction of negative
measurement outcomes relies upon the associated symmetries (see below). We
will use the notation $\rho_{G}$ for $\ket{G}\\!\\!\bra{G}$.
The single qubit measurements are restricted to projective measurements
defined by states that lie in the intersection of the Bloch sphere with the
$XY$-, $XZ$\- and $YZ$-planes. When it is necessary to specify a particular
measurement, we denote the corresponding projectors as
$\ket{+_{\alpha}}\\!\\!\bra{+_{\alpha}}_{\text{mp}}$ and
$\ket{-_{\alpha}}\\!\\!\bra{-_{\alpha}}_{\text{mp}}$ where
$\text{mp}\in\\{XY,XZ,YZ\\}$ and $\alpha\in[0,2\pi)$ specifies the angle from
one of the axes in the plane. For example,
$\ket{+_{\alpha}}_{XY}=\frac{1}{\sqrt{2}}(\ket{0}+e^{-i\alpha}\ket{1})$. If no
measurement plane label is given, it should be understood as being an
$XY$-plane measurement, as this is the most commonly treated type of
measurement in this work. For a fixed graph state $\ket{G}$, the computation
is specified by stipulating an angle $\alpha_{v}$ and measurement plane for
each qubit/vertex $v$.
The key observation for understanding why the restriction to only these planes
is made, and also how the correction procedure given by gflow (outlined below)
works, is the following set of relations, which hold for all values of
$\alpha$:
$\displaystyle\begin{split}\ket{-_{\alpha}}\\!\\!\bra{-_{\alpha}}_{XY}&=Z^{\dagger}\ket{+_{\alpha}}\\!\\!\bra{+_{\alpha}}_{XY}Z;\\\
\ket{-_{\alpha}}\\!\\!\bra{-_{\alpha}}_{XZ}&=(XZ)^{\dagger}\ket{+_{\alpha}}\\!\\!\bra{+_{\alpha}}_{XZ}XZ;\\\
\ket{-_{\alpha}}\\!\\!\bra{-_{\alpha}}_{YZ}&=X^{\dagger}\ket{+_{\alpha}}\\!\\!\bra{+_{\alpha}}_{YZ}X.\\\
\end{split}$ (8)
Here, one should note that each of the associated unitaries ($X$, $Z$ or their
product) appears naturally in the $K_{v}$ stabilisers for $\ket{G}$ (or a
product thereof). One can thus interpret the occurrence of a negative outcome
as equivalent to first applying a $Z$ (or $X$ or $XZ$) to the graph state and
then receiving the correct (positive) measurement outcome. Applying such a
gate to the graph state alone changes the computation, however applying the
remainder of a stabiliser, of which the $Z$, $X$ or $XZ$ term is constituent,
completes a symmetry of the graph state and so the negative measurement
outcome is effectively transformed into a positive one. Since the evolution of
a state to be measured can equivalently be considered as the evolution of the
measurement observables themselves, we can in fact apply the remainder of the
stabiliser to the other measurement operators, leaving the graph state
untouched. This is significant, since, due to the form of the stabiliser
operators and the properties of the measurement operators, the application of
the stabiliser can be affected by an appropriate change to the measurement
angle. For example, the measurement projectors in the $XY$-plane (the other
measurement planes follow similar patterns), we have the following:
$\displaystyle\begin{split}X^{\dagger}\ket{\pm_{\alpha}}\\!\\!\bra{\pm_{\alpha}}_{XY}X&\equiv\ket{\pm_{-\alpha\bmod
2\pi}}\\!\\!\bra{\pm_{-\alpha\bmod 2\pi}}_{XY};\\\
Z^{\dagger}\ket{\pm_{\alpha}}\\!\\!\bra{\pm_{\alpha}}_{XY}Z&\equiv\ket{\pm_{\alpha+\pi\bmod
2\pi}}\\!\\!\bra{\pm_{\alpha+\pi\bmod 2\pi}}_{XY}.\end{split}$ (9)
So, applying the remainder of a stabiliser to such measurements amounts to
applying appropriate reflections or adding $\pi$ phases to the original
measurement angles. We thus have two equivalent perspectives for the
correction of measurements, both of which play a role in this paper:
corrections can either be implemented via applying (conditional) unitaries to
the graph state or via adjustments to the classical measurement parameters.
The latter perspective is most relevant for Section 4 where as the former is
used in Section 5. Due to the importance of this equivalence for this work, a
more thorough explanation is provided in Section A.2, which is particularly
useful for motivating the discourse in Section 5.
To summarise: computations on a graph state are specified by positive outcomes
of single qubit measurements; measurements are restricted to measurement
planes that exhibit useful symmetries; and negative outcomes are treated via
conditional applications of certain symmetries of the graph state. To ensure
deterministic computation, the appropriate symmetries are required for every
measurement, and the conditionality of their application induces an order in
which the measurements must be performed. However, for any given graph state,
not every choice of symmetry and measurements order are consistent (i.e. one
cannot adapt a measurement that has already been performed). The definition of
gflow below addresses this point: a gflow assigns to every measured qubit a
choice of corrective stabiliser and assigns a (partial) order of measurements
which ensures consistency.
###### Definition 2.4 ([44]).
Let $G=(V,E)$ be a graph, $I$ and $O$ be input and output subsets of $V$
respectively, and $\omega:O^{c}\rightarrow\\{XY,XZ,YZ\\}$ be a map assigning
measurement planes to qubits (the superscript $c$ denotes set complement). The
tuple $(G,I,O,\omega)$ has gflow if there exists a map
$g:O^{c}\rightarrow\mathcal{P}(I^{c})$, where $\mathcal{P}$ denotes the
powerset, and a partial order over $V$ such that the following hold for all
$v\in O^{c}$:
1. 1.
if $v^{\prime}\in g(v)$ and $v^{\prime}\neq v$, then $v<v^{\prime}$;
2. 2.
if $v^{\prime}\in\operatorname{Odd}(g(v))$ and $v^{\prime}\neq v$, then
$v<v^{\prime}$;
3. 3.
if $\omega(v)=XY$, then $v\notin g(v)$ and $v\in\operatorname{Odd}(g(v))$;
4. 4.
if $\omega(v)=XZ$, then $v\in g(v)$ and $v\in\operatorname{Odd}(g(v))$;
5. 5.
if $\omega(v)=YZ$, then $v\notin g(v)$ and $v\notin\operatorname{Odd}(g(v))$;
where $\operatorname{Odd}(K):=\\{\tilde{v}\in V:|N_{\tilde{v}}^{G}\cap
K|=1\bmod 2\\}$ for any $K\subseteq V$.
It is known that the presence of gflow is both necessary and sufficient for
deterministic MBQC [[, Theorems 2 and 3,]]browne2007generalized. Furthermore,
polynomial time algorithms exist for determining whether a given tuple
$(G,I,O,\omega)$ supports gflow [58, 59]. Many distinct gflows for a given
$(G,I,O,\omega)$ can exist, and characterising or counting all gflows for a
given graph (as the input and output sets vary) in general remains an open
problem.
We can understand this definition in light of the discussion preceding it as
follows. The map $g$ assigns a (product of) stabiliser(s) to each qubit being
measured: $g(v)$ is a subset of $V$ and identifies the stabiliser
$\displaystyle K_{g(v)}:=\prod_{w\in g(v)}K_{w}.$
Every element in the set $g(v)$ receives an $X$-correction from the above
product and every element of $\operatorname{Odd}(g(v))$ receives a
$Z$-correction; the vertices in their intersection will receive both. The
partial order and the first two conditions enforce that every correction
conditioned on the measurement at $v$ happens in the future of that
measurement. The remaining three conditions enforce that the component of the
product that acts on the qubit $v$ is precisely the required symmetry
associated to the measurement plane $\omega(v)$, as given in Equation 8.
Derived from the definition of gflow, it is useful to define, for each $v\in
V$, the set of vertices whose corrections induce an $X$-operation on $v$ and
the set whose correction induce a $Z$-operation:
$\displaystyle\begin{split}\mathcal{X}_{v}&:=\\{v^{\prime}\in V:v\in
g(v^{\prime})\setminus\\{v\\}\\};\\\ \mathcal{Z}_{v}&:=\\{v^{\prime}\in
V:v\in\operatorname{Odd}(g(v^{\prime}))\setminus\\{v\\}\\}.\end{split}$ (10)
We allow for $\mathcal{X}_{v}$ or $\mathcal{Z}_{v}$ to be empty (such as when
$v\in I$ for example).
## 3 Quantum Comb Min-Entropy
In this section, we introduce and motivate the primary tool of analysis used
in the remainder of this work: the comb min-entropy [26]. This entropic
quantity is an extension of the min-entropy for quantum states (see e.g. [60,
27, 61, 62]), a well-understood information-theoretic quantity with
interpretations relating to maximal guessing probability of a random variable
or the maximal singlet fraction achievable from a quantum state [27]. Below,
we consider the operational meaning of the comb min-entropy, with particular
emphasis on the cases relevant for subsequent sections, namely for combs of
the forms given in Equations 4 and 5.
We commence by stating the definition of the comb min-entropy via three
equivalent characterisations, and then contextualise these in the subsequent
discussion.
###### Definition 3.1 ([26]).
Let $D\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$. The min-entropy relative to $D$ is
$\displaystyle H_{\min}$ $\displaystyle(A_{n}|A_{1},...,A_{n-1})_{D}$
$\displaystyle:=-\log\left[\min_{\Gamma}\min\\{\lambda\in\mathbb{R}:I_{A_{n}^{\operatorname{in}}A_{n}^{\operatorname{out}}}\otimes\lambda\Gamma\geq
D\\}\right]$ (11)
$\displaystyle\equiv-\log\left[\min_{\begin{subarray}{c}\widehat{\Gamma},\\\
I_{A_{n}^{\operatorname{in}}A_{n}^{\operatorname{out}}}\otimes\widehat{\Gamma}\geq
D\end{subarray}}\frac{\operatorname{Tr}[\widehat{\Gamma}]}{\prod_{j=1}^{n-1}\dim
A_{j}^{\operatorname{in}}}\right]$ (12)
$\displaystyle\equiv-\log\left[\max_{E}\operatorname{Tr}\left[DE^{T}\right]\right]$
(13)
where $\Gamma\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n-1}^{\operatorname{in}}\rightarrow
A_{n-1}^{\operatorname{out}})$ is a normalised comb, $\widehat{\Gamma}$ is an
unnormalised comb over the same space, and $E$ ranges over elements of
$\operatorname{Comb}(\mathbb{C}\rightarrow
A_{1}^{\operatorname{in}},A_{1}^{\operatorname{out}}\rightarrow
A_{2}^{\operatorname{in}},...,A_{n-1}^{\operatorname{out}}\rightarrow
A_{n}^{\operatorname{in}}\otimes A_{n}^{\operatorname{out}})$.
Each of the three quantities above are equivalent and each can be calculated
via semi-definite programming methods. Despite their equivalence, we have
chosen to present all three due to the following reasons. The first
expression, Equation 11, is consistent with the definition given in [26] (cf.
Definition $4$). The third expression as written has a nice interpretation in
terms of the generalised Born rule: the term inside the logarithm is the
maximum probability given by the Born rule for any comb dual to $D$, that is,
a strategy for interacting with $D$. Equivalently, it can be interpreted as
maximising over dual combs on a slightly different space that generate
correlations with the output space of the comb $D$. This perspective is
particularly useful when $D$ is a classical-quantum comb where the output is
the variable we aim to learn about, and hence we elaborate upon it below (see
also Figure 3(b)). The third expression is equivalent to the first expression
via e.g., Theorem $2$ in [26], and reflects the strong duality in the
associated semi-definite programs. The second expression is the form of the
comb min-entropy that is most used throughout this paper for both analytical
and numerical results (such as in Section 3.1, Section 4 and Section 5), due
in part to its suitability for implementation in code. This expression is
equivalent to the first by combining $\lambda$ and $\Gamma$ into one operator.
In the present context, the expression for the min-entropy given in Equation
13 is particularly valuable conceptually . Above, $E$ is defined to be a dual
operator on the exact same Hilbert spaces as $D$, since this provides a closed
expression for the probability given by
$\operatorname{Tr}\left[DE^{T}\right]$. As dual operators, the $E$ can be
understood as multi-timestep, adaptive strategies for interacting with the
system. However, by considering operators
$\widehat{E}\in\operatorname{Comb}(\mathbb{C}\rightarrow
A_{1}^{\operatorname{in}},A_{1}^{\operatorname{out}}\rightarrow
A_{2}^{\operatorname{in}},...,A_{n-1}^{\operatorname{out}}\rightarrow
A_{n}^{\operatorname{in}}\otimes\widehat{A}_{n}^{\operatorname{out}})$
instead, where
$\mathcal{H}_{\widehat{A}_{n}^{\operatorname{out}}}\cong\mathcal{H}_{A_{n}^{\operatorname{out}}}$,
the min-entropy can equivalently be understood as optimising over strategies
$\widehat{E}$ that produce maximal correlations between two different spaces,
namely $\mathcal{H}_{\widehat{A}_{n}^{\operatorname{out}}}$ and
$\mathcal{H}_{A_{n}^{\operatorname{out}}}$. We find this perspective of
maximising correlations between separate spaces especially appropriate for our
purposes, since, for the classical-quantum combs considered here, the output
space of $D$ (the classical parameter space) is deemed to be inaccessible, and
we thus aim to learn the classical parameter via strategies with an output
space available to us.
To begin to formalise these equivalent perspectives, it can be shown (a
combination of [[, Proposition 6,]]chiribella2016optimal and [[, Theorem
2,]]konig2009operational), that
$\displaystyle\max_{E}$
$\displaystyle\operatorname{Tr}\left[DE^{T}\right]\equiv\max_{\widehat{E}}\dim
A_{n}^{\operatorname{out}}\operatorname{Tr}\left[D\widehat{E}^{T}\ket{\Phi^{+}}\\!\\!\bra{\Phi^{+}}\right]$
(14)
where $T$ is to be understood as denoting the transpose over spaces that $D$
and $E$ (respectively $D$ and $\widehat{E}$) share, and where
$\displaystyle\ket{\Phi^{+}}:=\frac{1}{\sqrt{\dim
A_{n}^{\operatorname{out}}}}\sum_{i}\ket{i}_{A_{n}^{\operatorname{out}}}\ket{i}_{\widehat{A}_{n}^{\operatorname{out}}}$
(15)
with $\\{\ket{i}\\}$ an orthonormal basis. The term
$\operatorname{Tr}_{A_{1}^{\operatorname{in}},A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}}[D\widehat{E}^{T}]$
is in particular a state on
$\mathcal{H}_{A_{n}^{\operatorname{out}}}\otimes\mathcal{H}_{\widehat{A}_{n}^{\operatorname{out}}}$
and thus the full trace term on the right-hand side of Equation 14 is a
measure of the overlap with the maximally-entangled state
$\ket{\Phi^{+}}\\!\\!\bra{\Phi^{+}}$. Maximising over $\widehat{E}$ amounts to
maximising this overlap. In the case where $D$ is a bipartite quantum state,
$\widehat{E}$ is the Choi representation of a CPTP map so Equation 14 is
directly related to the operational meaning of the state min-entropy (see the
proof of [[, Theorem 2,]]konig2009operational). For any comb with classical
output, $\ket{\Phi^{+}}\\!\\!\bra{\Phi^{+}}$ can be replaced by its diagonal
part $(\dim
A_{n}^{\operatorname{out}})^{-1}\sum_{i}\ket{ii}\\!\\!\bra{ii}_{A_{n}^{\operatorname{out}}\widehat{A}_{n}^{\operatorname{out}}}$.
Figure 3 depicts this perspective for both the state and comb cases.
Throughout this work, when discussing the maximum formulation of the min-
entropy, we will often identify $E$ with $\widehat{E}$ for simplicity (they
are in one-to-one correspondence) despite the abuse of notation regarding the
constituent Hilbert spaces.
(a) Min-entropy of a quantum state $\rho_{XA}$
(b) Comb min-entropy of a comb $D$
Figure 3: The min-entropy can be considered as a measure of the maximal amount
of correlation achievable from a given state of two systems by interacting
with one system alone. (a) The state min-entropy is calculated if the
interaction occurs at a single time step. For a quantum state $\rho_{XA}$, the
min-entropy is calculated via maximising over all quantum channels
$\mathcal{E}$ acting on the system $A$, and comparing the output to a
maximally-entangled state $\ket{\Phi}\\!\\!\bra{\Phi}_{XX^{\prime}}$. If
$\rho_{XA}$ is a classical-quantum state instead, one simply maximises over
POVMs and compares the output to a maximally classically-correlated state
(namely $(\dim X)^{-1}\sum_{x}\ket{xx}\\!\\!\bra{xx}_{X^{\prime}X}$). (b) If
interactions occur over a series of time steps, the comb min-entropy is
calculated instead. For a quantum (respectively, classical-quantum) comb $D$,
the min-entropy is calculated via maximising over strategies $\widehat{E}$
(dual combs to $D$) and again comparing the output to a maximally-entangled
(maximally classically-correlated) state. The comb min-entropy given for $D$ a
classical-quantum as depicted in Figure 2 are the primary concern of this
work.
Let us now consider more closely the classical-quantum combs as presented in
Equation 4 and Equation 5. For classical-quantum states, i.e. for
$D=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\rho_{x}^{A}$ where
$\rho_{x}^{A}\in\mathcal{L}(\mathcal{H}_{A})$, the term
$\max_{E}\operatorname{Tr}[DE^{T}]$ is called the guessing probability [27]
since it quantifies the best probability of guessing $X$ after interacting
with the state on $A$. We use the same terminology for the combs case:
$P_{\text{guess}}(X|A_{1},...,A_{n})_{D}$ is the probability of guessing $X$
after interacting with
$A_{1}^{\operatorname{in}},A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}},A_{n}^{\operatorname{out}}$
as given by
$\displaystyle
P_{\text{guess}}(X|A_{1},...,A_{n})_{D}:=\max_{E}\sum_{x}P(x)\operatorname{Tr}[\left(\ket{x}\\!\\!\bra{x}\otimes\sigma_{x}\right)E^{T}]$
where $E$ ranges over elements of $\operatorname{Comb}(\mathbb{C}\rightarrow
A_{1}^{\operatorname{in}},A_{1}^{\operatorname{out}}\rightarrow
A_{2}^{\operatorname{in}},...,A_{n}^{\operatorname{out}}\rightarrow X)$. Since
$P_{\text{guess}}(X|A_{1},...,A_{n})_{D}=2^{-H_{\min}(X|A_{1},...,A_{n})_{D}}$,
we will often use the terms ‘guessing probability’ and ‘min-entropy’
interchangeably.
Later sections will compare the comb min-entropy across different numbers of
rounds of interactions, i.e. comparing the min-entropy for $D^{(m)}$ for
different values of $m$. Intuitively, given the specific structure of
$D^{(m)}$, one would expect that as the number of rounds increases, the min-
entropy should not increase. This is indeed the case, as stated in the
following lemma whose proof is given in Appendix B.
###### Lemma 3.1.
For $D^{(m)}$ and $D^{(l)}$ of the form given in Equation 5, where $m\geq l$,
$\displaystyle H_{\min}(X|A_{1}^{(1)},...,A_{n}^{(l)})_{D^{(l)}}\geq
H_{\min}(X|A_{1}^{(1)},...,A_{n}^{(m)})_{D^{(m)}}.$
For emphasis, we reiterate comments made above: throughout this work, we take
the perspective that the min-entropy relates to the maximal guessing
probability under the optimal strategy of interacting with a comb, which we
find more intuitive, however calculations will largely be done via the
expressions involving the minimum (typically Equation 12), which is easier to
work with since the optimisation is over fewer Hilbert spaces.
### 3.1 Min-Entropy for Classical-Classical Combs
In the previous subsection, we considered the relation between the comb min-
entropy and guessing probability for classical-quantum combs $D$. Due to their
relevance for the analysis that will take place in Section 4, we present here
a general result regarding the guessing probability for a further restricted
class of combs: classical-classical combs. The result is 3.2 and in particular
provides upper and lower bounds for the guessing probability.
Classical-classical are combs of the form
$\displaystyle D=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\sigma_{x}$ (16)
where $D\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}},\mathbb{C}\rightarrow X)$ and where all the
$\sigma_{x}\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$ are normalised and diagonal in the same basis
(cf. 2.2). Due to normalisation, we can adopt the probability distribution
notation and write
$\displaystyle\sigma_{x}:=\sum_{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}P(\boldsymbol{a}^{\operatorname{out}}|x,\boldsymbol{a}^{\operatorname{in}})\ket{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\!\\!\bra{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}$
(17)
where
$P(\boldsymbol{a}^{\operatorname{out}}|x,\boldsymbol{a}^{\operatorname{in}})$
satisfies the required marginalisation conditions (Equation 2) for each $x$.
This allows $D$ to be rewritten as
$\displaystyle D$
$\displaystyle=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\sum_{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}P(\boldsymbol{a}^{\operatorname{out}}|x,\boldsymbol{a}^{\operatorname{in}})\ket{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\!\\!\bra{..}$
(18)
$\displaystyle=\sum_{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}P(x)P(\boldsymbol{a}^{\operatorname{out}}|x,\boldsymbol{a}^{\operatorname{in}})\ket{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\!\\!\bra{..}.$
(19)
In the case where all input spaces are trivial, that is, when $D$ is a
classical-classical state (all the $\sigma_{x}$ are states that are diagonal
in the same basis), the guessing probability is known to have a nice
interpretation in terms of a maximal Bayesian update (this interpretation is
explicated in Appendix B; see also [[, Section 6.1.4,]]tomamichel2015quantum).
In an attempt to arrive at similar interpretation for the case where the
inputs are non-trivial, let us consider the conditional version of Bayes’
rule, namely:
$\displaystyle
P(X|\boldsymbol{A}^{\operatorname{in}},\boldsymbol{A}^{\operatorname{out}})P(\boldsymbol{A}^{\operatorname{out}}|\boldsymbol{A}^{\operatorname{in}})=P(\boldsymbol{A}^{\operatorname{out}}|X,\boldsymbol{A}^{\operatorname{in}})P(X|\boldsymbol{A}^{\operatorname{in}})$
(20)
where the capitalised letters denote random variables (as opposed to their
values denoted by the lower case letters as above). One notes that the right-
hand side is almost what appears in the form of $D$ above, except instead of
$P(X)$ we have $P(X|\boldsymbol{A}^{\operatorname{in}})$. A consequence of 2.1
is that $X$ is independent of $\boldsymbol{A}^{\operatorname{in}}$, so we do
in fact have
$\displaystyle P(X|\boldsymbol{A}^{\operatorname{in}})=P(X).$ (21)
This allows $D$ to be written as
$\displaystyle
D=\sum_{\begin{subarray}{c}x,\boldsymbol{a}^{\operatorname{in}},\\\
\boldsymbol{a}^{\operatorname{out}}\end{subarray}}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})\ket{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\!\\!\bra{..}.$
(22)
Using this form for $D$, we can establish the following:
###### Proposition 3.2.
Let $D$ be as above. Then
$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{a}^{\operatorname{in}},\\\
\boldsymbol{a}^{\operatorname{out}}\end{subarray}}\max_{x}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})$
$\displaystyle
P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})\leq\min_{\begin{subarray}{c}\Gamma,\\\
I_{X}\otimes\Gamma\geq D\end{subarray}}\operatorname{Tr}\left[\Gamma\right]$
(23)
$\displaystyle\leq\sum_{\begin{subarray}{c}\boldsymbol{a}^{\operatorname{in}},\\\
\boldsymbol{a}^{\operatorname{out}}\end{subarray}}\max_{x,\boldsymbol{\tilde{a}}}P(x|\boldsymbol{\tilde{a}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{\tilde{a}})$
(24)
where the minimisation is over unnormalised combs $\Gamma$.
By normalising each term by the dimension of the input spaces, and in light of
the definition of the comb min-entropy(specifically Equation 12), the above
provides bounds for the guessing probability (this is discussed in more depth
below). The proof is given in Appendix B and demonstrates the left-hand
inequality by showing that every positive semi-definite operator $\Gamma$,
which clearly includes all unnormalised combs that satisfies
$I_{X}\otimes\Gamma\geq D$, must have trace greater than the left-hand
quantity above. The right-hand inequality is obtained by showing that
$\displaystyle\Gamma:=I_{\boldsymbol{A}^{\operatorname{in}}}\otimes\sum_{\boldsymbol{a}^{\operatorname{out}}}\max_{x,\boldsymbol{\tilde{a}}}P(x|\boldsymbol{\tilde{a}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{\tilde{a}})\ket{\boldsymbol{a}^{\operatorname{out}}}\bra{\boldsymbol{a}^{\operatorname{out}}}$
(25)
is indeed an unnormalised classical comb. It also is worth noting that, for
$D$ classical, the minimum over unnormalised combs can always be achieved by a
diagonal $\Gamma$ (see B.1 and B.2).
Explicitly, the consequences for the guessing probability are as follows:
$\displaystyle P_{\text{guess}}$
$\displaystyle(X|\boldsymbol{A}^{\operatorname{in}},\boldsymbol{A}^{\operatorname{out}})\geq\sum_{\begin{subarray}{c}\boldsymbol{a}^{\operatorname{in}},\\\
\boldsymbol{a}^{\operatorname{out}}\end{subarray}}\max_{x}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})\frac{P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})}{\dim\boldsymbol{A}^{\operatorname{in}}}$
(26)
$\displaystyle\equiv\sum_{\begin{subarray}{c}\boldsymbol{a}^{\operatorname{in}},\\\
\boldsymbol{a}^{\operatorname{out}}\end{subarray}}\max_{x}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})P_{\text{unif}}(\boldsymbol{a}^{\operatorname{in}})$
(27)
where $P_{\text{unif}}(\cdot)$ denotes the uniform probability distribution,
and
$\displaystyle
P_{\text{guess}}(X|\boldsymbol{A}^{\operatorname{in}},\boldsymbol{A}^{\operatorname{out}})$
$\displaystyle\leq\sum_{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\max_{x,\boldsymbol{\tilde{a}}}P(x|\boldsymbol{\tilde{a}},\boldsymbol{a}^{\operatorname{out}})\frac{P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{\tilde{a}})}{\dim\boldsymbol{A}^{\operatorname{in}}}$
(28)
$\displaystyle=\sum_{\boldsymbol{a}^{\operatorname{out}}}\max_{x,\boldsymbol{\tilde{a}}}P(x|\boldsymbol{\tilde{a}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{\tilde{a}}).$
(29)
If, for each $\boldsymbol{a}^{\operatorname{out}}$, the factor
$\max_{x}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})$
is the same for every $\boldsymbol{a}^{\operatorname{in}}$, then the upper and
lower bounds coincide, thus specifying the guessing probability exactly. As
discussed in Appendix B, this occurs trivially for the state case since the
input space is one-dimensional, but we will see in Section 4.3 a simple comb
example with non-trivial input spaces where this also holds.
The above proposition highlights that, even in the entirely classical case,
the generalisation of the maximum Bayesian updating interpretation of the
guessing probability for states to the classical comb case is more subtle.
This aligns with what one would expect: the fact that the comb case allows for
inputs to the system plays a non-trivial role in the interpretation of the
guessing probability. For the case of quantum combs, further subtleties likely
exist, not in the least since certain formulations of conditional quantum
Bayesian updating are known to be potentially problematic (see e.g. [[,
Section VII,]]leifer2013towards).
A final comment regarding the multi-round case. Since a classical multi-round
comb (recall Equation 5) can be viewed as a special case of Equation 19 above
with extra independence conditions on the distribution
$P(\boldsymbol{a}^{\operatorname{out}}|x,\boldsymbol{a}^{\operatorname{in}})$,
namely that
$\displaystyle
P(\boldsymbol{a}^{\operatorname{out}}|x,\boldsymbol{a}^{\operatorname{in}})=\prod_{j=1}^{m}P(\boldsymbol{a}^{\operatorname{out},(j)}|x,\boldsymbol{a}^{\operatorname{in},(j)})$
(30)
where $\boldsymbol{a}^{\operatorname{in},(j)}$ denotes the inputs for round
$j$ and similarly for $\boldsymbol{a}^{\operatorname{out},(j)}$, applying the
results of 3.2 to the multi-round case gives
$\displaystyle
P_{\text{guess}}(X|\boldsymbol{A}^{\operatorname{in},(1)},\boldsymbol{A}^{\operatorname{out},(1)},...,\boldsymbol{A}^{\operatorname{in},(m)},\boldsymbol{A}^{\operatorname{out},(m)})$
$\displaystyle\geq\sum_{\begin{subarray}{c}\boldsymbol{a}^{\operatorname{in},(1:m)},\\\
\boldsymbol{a}^{\operatorname{out},(1:m)}\end{subarray}}\max_{x}P(x|\boldsymbol{a}^{\operatorname{in},(1:m)},\boldsymbol{a}^{\operatorname{out},(1:m)})\prod_{j=1}^{m}\frac{P(\boldsymbol{a}^{\operatorname{out},(j)}|\boldsymbol{a}^{\operatorname{in},(j)})}{\dim\boldsymbol{A}^{\operatorname{in},(j)}}$
(31)
and
$\displaystyle
P_{\text{guess}}(X|\boldsymbol{A}^{\operatorname{in},(1)},\boldsymbol{A}^{\operatorname{out},(1)},...,\boldsymbol{A}^{\operatorname{in},(m)},\boldsymbol{A}^{\operatorname{out},(m)})$
$\displaystyle\leq\sum_{\boldsymbol{a}^{\operatorname{out},(1:m)}}\max_{x,\boldsymbol{\tilde{a}}^{(1:m)}}P(x|\boldsymbol{\tilde{a}}^{(1:m)},\boldsymbol{a}^{\operatorname{out},(1:m)})\prod_{j=1}^{m}P(\boldsymbol{a}^{\operatorname{out},(j)}|\boldsymbol{\tilde{a}}^{(j)})$
(32)
where the shorthand $\boldsymbol{a}^{\operatorname{in},(1:m)}$ has been used
for the inputs for all rounds and similarly for
$\boldsymbol{a}^{\operatorname{out},(j)}$.
## 4 Blind Quantum Computing Protocol Analysis
Blind quantum computing is a term that encompasses an array of cryptographic
quantum computational protocols (see e.g. [35, 36, 37, 38] and the review
[17]), many of which build upon, or are inspired by, aspects derived from
MBQC. In very broad terms, a BQC protocol consists of a client, who has
limited computational power, interacting with a server (multi-party variants
also exist), who has quantum computational capabilities, in order to carry out
a desired computation in such a way that the latter is “blind” to the details.
A range of theoretical tools for modelling quantum cryptographic protocols and
analysing their security exist (see e.g. [64, 65] and the review [66]).
In this work, we consider one specific BQC protocol, that of Mantri et al.
[36] (discussed in the next subsection), and our main method of security
analysis consists of calculating the comb min-entropy applied to the
corresponding classical comb, $D_{\operatorname{client}}$, which we define
below (Section 4.2). In Section 4.3, by establishing a strictly positive lower
bound for the min-entropy of $D_{\operatorname{client}}$, which means the
server must still learn more information in order to know the client’s choice
of computation with certainty, we prove the partial security of the protocol
for a single round. We further provide a simple example of the protocol run on
a three-qubit graph state that obtains the lower bound. In Section 4.4, we
extend the security analysis to multiple rounds by considering a comb
$D_{\operatorname{client}}^{(m)}$, where a positive lower bound is again shown
for all rounds $m$, and comment on the consequences for the blindness of the
protocol in this case.
### 4.1 A Classically Driven BQC Protocol
The protocol of [36] is distinct from many other BQC proposals in that it
considers purely classical communication between client and server. At its
core, the protocol leverages various properties of gflow and MBQC outlined in
Section 2.2, such as the non-uniqueness of gflows existing for a given graph
state and the symmetries of measurement projections as in Equation 9, along
with cryptographic primitives such as the use of random bits as one-time pads.
The protocol parameters consist of a choice of graph $G$ along with a total
order on vertices, and a discrete set of angles $\mathcal{A}$ which satisfies
the following property:
$\displaystyle\mathcal{A}=\\{(-1)^{x}\alpha+z\pi\bmod
2\pi:\alpha\in\mathcal{A};x,z\in\mathbb{Z}_{2}\\}.$ (33)
All measurements are made in the $XY$-plane (which is no restriction on the
universality of the resulting computations - see [67]) and so the above
property enforces that $\mathcal{A}$ is closed under the angle transformations
given in Equation 9. The specific graph, total order and angle set are agreed
upon collectively by both the client and server prior to the commencement of a
specific computation. In secret, the client also chooses their desired
computation, that is, a list of measurement angles
$\boldsymbol{\alpha}\in\mathcal{A}^{n}$ (where $n$ is the number of vertices
in $G$) and designated input and output sets $I$ and $O$, a bit-string one-
time pad $\boldsymbol{r}\in\mathbb{Z}_{2}^{n}$ uniformly at random, and a
gflow compatible with $(G,I,O)$ and the total order.
One round of computation proceeds as follows:
1. 1.
The server initialises the graph state $\rho_{G}$.
2. 2.
For $i=1,...,n$ according to the total order, the following sequence is
repeated:
1. (a)
The user reports a measurement angle $\alpha^{\prime}_{i}$ to the server,
where
$\displaystyle\alpha^{\prime}_{i}:=(-1)^{\bigoplus_{j\in\mathcal{X}_{i}}c_{j}}\alpha_{i}+\left(r_{i}\oplus\bigoplus_{j\in\mathcal{Z}_{i}}c_{j}\right)\pi\bmod
2\pi$ (34)
where $c_{j}$ denotes the measurement outcome for qubit $j<i$ recorded by the
user based on the outcome reported by the server and where $\mathcal{X}_{i}$
and $\mathcal{Z}_{i}$ are the corrections sets defined by the gflow (as in
Equation 10).
2. (b)
The server measures $\mathcal{M}_{\alpha^{\prime}_{i}}$ and reports
$c^{\prime}_{i}=0$ for a positive outcome and $c^{\prime}_{i}=1$ for a
negative outcome.
3. (c)
The user records $c_{i}=c^{\prime}_{i}\oplus r_{i}$.
3. 3.
The outcomes pertaining to the output qubits (which are known only to the
user) are processed to obtain the results of the computation.
There are two things worth noting about the one-time pads. Firstly, their
utility for obscuring angles is a direct consequence of the $Z$-relation given
in Equation 9 (i.e. a positive measurement outcome for $\alpha$ and negative
measurement outcome for $\alpha+\pi$ are equivalent). Secondly, the presence
of $r_{i}$ in both the equation for $\alpha^{\prime}_{i}$ as well as for any
$\alpha^{\prime}_{k}$ for which $i\in\mathcal{X}_{k}$ or $i\in\mathcal{Z}_{k}$
places constraints on the set of possible reported angles
$\boldsymbol{\alpha^{\prime}}$ for a given choice of true angles
$\boldsymbol{\alpha}$.
The protocol is known to be correct if both the client and server behave
accordingly [[, Theorem 1,]]mantri2017flow, however it also known that this
protocol is not verifiable: the client has no way of knowing whether the
server actually prepares a graph state, measures according to the reported
angles, and communicates the actual measurement outcomes. The server could in
fact do none of these and report random outcomes $c^{\prime}$ instead, with
the client only being aware of this fact if they could compute the measurement
statistics of the final state themselves anyway. We review the proof of
blindness for a single round of the protocol [[, Theorem 2,]]mantri2017flow
and provide our own analysis in Section 4.3 after introducing the classical
comb that models the protocol.
Some final remarks regarding minor differences between the analysis in [36]
and the one below. Mantri et al. place a further condition on the definition
of gflow, largely for the purpose of simplifying a counting argument (see [[,
Theorem 3,]]mantri2017flow), which effectively singles out one gflow for every
pair $(I,O)$ given $G$ and they thus identify a choice of computation with a
choice of angles and choice of gflow. Here, we work with the unconstrained
definition of gflow and so identify a choice of computation as a choice of
$\boldsymbol{\alpha}$ and $(I,O)$ (again for fixed $G$), for which any of the
compatible gflows can be chosen. Moreover, since the protocol is entirely
classical, the client can only prepare an input state on the qubits in $I$ via
measurement-based state preparation, which is thus indistinguishable from the
part of the measurement sequence implementing the unitary. As such,
computations are specified here purely by angles $\boldsymbol{\alpha}$ and a
choice of output set $O$ for which $(G,I,O)$ supports gflow for some $I$.
### 4.2 The Classical BQC Protocol as a Classical Comb
The protocol parameters outlined in the previous section also form the
starting point for defining the corresponding classical comb: we fix a choice
of graph $G$, say on $n$ vertices, a total order on the vertices $1<2<...<n$,
and a set of allowed angles $\mathcal{A}$ satisfying Equation 33. Since all
communication in the protocol is classical, we represent the reported angles
and measurement outcomes as basis states in corresponding Hilbert spaces:
$\ket{\alpha^{\prime}_{i}}\bra{\alpha^{\prime}_{i}}\in\mathcal{H}_{A^{\prime}_{i}}$
for the reported angle at step $i$ and
$\ket{c^{\prime}_{i}}\bra{c^{\prime}_{i}}\in\mathcal{H}_{C^{\prime}_{i}}$ for
the reported measurement outcome, where
$\dim\mathcal{H}_{A^{\prime}_{i}}=|\mathcal{A}|$ and
$\dim\mathcal{H}_{C^{\prime}_{i}}=2$.
Prior to the commencement of the protocol, the client selects the output space
$O$, the “true” angles for the computation
$\boldsymbol{\alpha}\in\mathcal{A}^{n}$, the one-time pads
$\boldsymbol{r}\in\mathbb{Z}_{2}^{n}$, and a gflow $g$ compatible with the
graph and total order and chosen output set. We denote compatibility with an
output set $O$ as $g\sim O$. We define the classical comb corresponding to the
protocol for the specific choice of $\boldsymbol{\alpha},\boldsymbol{r}$ and
$g$ as
$\displaystyle\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha},\boldsymbol{r},g}:=\sum_{\begin{subarray}{c}\boldsymbol{\alpha^{\prime}}\in\mathcal{A}^{n},\\\
\boldsymbol{c^{\prime}}\in\mathbb{Z}_{2}^{n}\end{subarray}}P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)\ket{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}\bra{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}$
(35)
where
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)$
is simply a deterministic distribution that encodes the angle adaptations:
$\displaystyle
P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)=\begin{cases}1,\text{
if \lx@cref{creftypecap~refnum}{eq:adapt_meas_angles} holds for all $i$}\\\
0,\text{ otherwise }\end{cases}.$ (36)
Note that Equation 34 contains the notation $c_{i}$, but since
$c_{i}:=c^{\prime}_{i}\oplus r_{i}$, we use only the $\boldsymbol{c^{\prime}}$
and $\boldsymbol{r}$ notation instead.
For a given computation, that is, for a given $\boldsymbol{\alpha}$ and $O$,
any choice of $g\sim O$ and $\boldsymbol{r}$ produce the desired result. Since
these choices are made in secret, the protocol for a given computation is
modelled as
$\displaystyle\sum_{g\sim
O,\boldsymbol{r}}P(g|O)P(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha},\boldsymbol{r},g}$
(37)
where $P(g|O)$ is the probability that the gflow $g$ is chosen and
$P(\boldsymbol{r})$ is the probability of the one-time pad $\boldsymbol{r}$ is
chosen. According to the above protocol, the latter distribution is taken to
be uniform, $P(\boldsymbol{r})=\frac{1}{2^{n}}$, and independent of all other
variables (see [[, Lemma 4,]]mantri2017flow). Since the specific choice of
gflow is irrelevant for the computation once $O$ is fixed, we take $P(g|O)$ to
have uniform weight for all $g\sim O$ and $0$ otherwise, and moreover we take
$P(g|O)=P(g|\boldsymbol{\alpha},O)$. We thus have the following:
$\displaystyle
P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)$
$\displaystyle=P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g,O)\quad\forall
g\sim O$ (38) $\displaystyle
P(\boldsymbol{r},g|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)$
$\displaystyle=P(\boldsymbol{r})P(g|O)$ (39)
where Equation 39 has used the independence of $g$ and $\boldsymbol{r}$ from
$\boldsymbol{c^{\prime}}$ which follows from similar reasoning as for Equation
21 in the previous section. We can thus expand Equation 37 as follows:
$\displaystyle\sum_{g\sim O,\boldsymbol{r}}$ $\displaystyle
P(g|O)P(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha},\boldsymbol{r},g}$
$\displaystyle=\sum_{\begin{subarray}{c}\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}},\\\
g\sim
O,\boldsymbol{r}\end{subarray}}P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g,O)P(\boldsymbol{r},g|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)\ket{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}\\!\\!\bra{..}$
(40)
$\displaystyle=\sum_{\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}}}P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)\ket{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}\\!\\!\bra{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}$
(41) $\displaystyle=:\sigma_{\boldsymbol{\alpha},O}$ (42)
where we have used $\bra{..}$ when its contents are the same as the
accompanying ket.
Having summed over $g\sim O$ and $\boldsymbol{r}$, the probability
distribution
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)$
is no longer a simple deterministic distribution, but in general a more
complicated one. In particular, it depends on the set of gflows compatible
with $O$ and the set of angles they can mutually report.
The final step to arrive at a comb of the form of Equation 4 is to include the
random variable. In this case, the random variable encodes the choice of
computation made by the client. Denoting the set of all possible output sets
for the fixed graph by $\mathcal{O}$, the values that the computation random
variable can take are given by
$\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\in\mathcal{H}_{\mathcal{A}^{n}}\otimes\mathcal{H}_{\mathcal{O}}$.
We can now write down the classical comb that fully represents the BQC
protocol for a fixed $G$ and $\mathcal{A}$ as:
$\displaystyle
D_{\operatorname{client}}=\sum_{\boldsymbol{\alpha}\in\mathcal{A}^{n},O\in\mathcal{O}}$
$\displaystyle
P(\boldsymbol{\alpha},O)\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\otimes\sigma_{\boldsymbol{\alpha},O}.$
(43)
This is a classical comb in $\operatorname{Comb}(\mathbb{C}\rightarrow
A^{\prime}_{1},C^{\prime}_{1}\rightarrow
A^{\prime}_{2},...,C^{\prime}_{n}\rightarrow\mathbb{C},\mathbb{C}\rightarrow\boldsymbol{A}\times\boldsymbol{O})$
and as such the results of Section 3.1 regarding the min-entropy apply, a fact
that will be used in the blindness proof in the next section.
### 4.3 Blindness in a Single Round
We begin this subsection with a discussion of the security proof given in [36]
for the protocol. The proof proceeds by showing that, for a single round, the
mutual information between the information received by the server and the
client’s choice of computation is bounded away from the required amount to
learn the computation with certainty. We state their blindness theorem below,
using notation as close to the original as possible, and then consider a
related form more suitable for comparison with our analysis using the min-
entropy.
Recalling the comments at the end of Section 4.1, since in [36] a restricted
version of gflow was used, a computation is stipulated by a choice of angles
and a choice of restricted gflow (instead of the choice of $O$ as above). We
let $\boldsymbol{A}$ denote the random variable for the angles, which takes
values in $\mathcal{A}^{n}$, and let $\boldsymbol{F}$ denote the random
variable which takes values in the set of restricted gflows. The random
variables for the reported angles and measurement outcomes are denoted
$\boldsymbol{A^{\prime}}$ and $\boldsymbol{C^{\prime}}$ respectively. With
this notation, the blindness theorem is:
###### Theorem 4.1 (Theorem 2, [36]).
In a single instance of the protocol, the mutual information between the
client’s secret input $\\{\boldsymbol{\alpha},\boldsymbol{f}\\}$ and the
information received by the server is bounded by
$\displaystyle
I(\boldsymbol{C^{\prime}},\boldsymbol{A^{\prime}};\boldsymbol{A},\boldsymbol{F})\leq
H(\boldsymbol{A^{\prime}})$ (44)
where $I(\cdot;\cdot)$ denotes the mutual information and $H(\cdot)$ denotes
the Shannon entropy.
Using part of the proof of the above theorem which in particular shows that
$H(\boldsymbol{A^{\prime}},\boldsymbol{C^{\prime}}|\boldsymbol{A},\boldsymbol{F})\geq
n$ (see [[, Lemma 4,]]mantri2017flow) along with the properties of the mutual
information and Shannon entropy, it is possible to arrive at a related
entropic inequality which indicates the at least partial security of the
protocol (‘partial’ because some information is leaked about the choice of
angles in a given round):
$\displaystyle
H(\boldsymbol{A},\boldsymbol{F}|\boldsymbol{C^{\prime}},\boldsymbol{A^{\prime}})\geq\log_{2}|\boldsymbol{F}|>0.$
(45)
One important aspect that we want to emphasise is that the Shannon entropy is
not a single-shot entropy and therefore only provides limited insight about
the security of the protocol in a single round. The min-entropy however, is a
single-shot quantity, and since it deals with optimal learning strategies
(optimal from the inquisitive server’s perspective), it provides a more
appropriate measure of security in this case: if it can be shown that the min-
entropy equivalent of the quantity
$H(\boldsymbol{A},\boldsymbol{F}|\boldsymbol{C^{\prime}},\boldsymbol{A^{\prime}})$
is bounded above zero, then the server is guaranteed to retain some
uncertainty about the true computation. This is the content of the following
theorem. It should first be noted that, in deriving Equation 45, the
assumption is made that the computation is chosen uniformly at random (i.e.
$H(\boldsymbol{A},\boldsymbol{F})=\log_{2}|\boldsymbol{A}|+\log_{2}|\boldsymbol{F}|$)
and we make the equivalent assumption here (however for the corresponding
change in variable from $\boldsymbol{F}$ to $\boldsymbol{O}$; explicitly, we
assume $P(\boldsymbol{\alpha},O)=\frac{1}{|\mathcal{A}|^{n}|\mathcal{O}|}$).
###### Theorem 4.2.
For $D_{\operatorname{client}}$ as above, and under the given assumptions, the
following holds for any choice of graph $G$ and angle set $\mathcal{A}$ that
satisfy the required conditions:
$\displaystyle
H_{\min}(\boldsymbol{A},\boldsymbol{O}|\boldsymbol{A^{\prime}},\boldsymbol{C^{\prime}})_{D_{\operatorname{client}}}\geq
n+\log_{2}(|\mathcal{O}|).$ (46)
The proof of this theorem is given in Section C.1. The key step of the proof
consists of providing an upper bound for the guessing probability (which
provides a corresponding lower bound for the min-entropy) via the upper bound
of 3.2 applied to the comb $D_{\operatorname{client}}$. Due to the specific
form of $D_{\operatorname{client}}$, this bound has an interpretation in terms
of the number of different ways each $\boldsymbol{\alpha^{\prime}}$ can be
reported for a given classical message (that is, for different gflows $g$ and
one-time pads $\boldsymbol{r}$) with a larger number corresponding to a lower
min-entropy bound (this is explained further after the proof of the theorem in
Section C.1).
One can rightfully ask how good this bound is in practice. To conclude this
subsection, we consider a simple example for which the gflows are
characterised and the min-entropy is in fact given exactly by the right-hand
side of Equation 46. Let $G$ be the three-vertex graph as shown in Figure 4
with total order given by the natural order on vertex labels. As demonstrated
in Section C.2, there is exactly one non-trivial output set ($O=\\{2,3\\}$)
that supports gflow for this example, and exactly two corresponding gflows
exist. It can further be shown that the upper and lower bounds of 3.2 for
$D_{\operatorname{client}}$ coincide, which establishes
$n+\log_{2}(|\mathcal{O}|)$ as the true min-entropy value. For this example,
$n=3$ and $|\mathcal{O}|=1$, so
$H_{\min}(\boldsymbol{A},\boldsymbol{O}|\boldsymbol{A^{\prime}},\boldsymbol{C^{\prime}})_{D_{\operatorname{client}}}=3$,
which is corroborated by the numerical results (for two choices of
$\mathcal{A}$, one such that $|\mathcal{A}|=4$ and the other such that
$|\mathcal{A}|=8$).
Figure 4: An example graph state for which the security for a single round of
the BQC protocol considered here is maximally bad (that is, the min-entropy
for the associated quantum comb obtains the lower bound given in 4.2). All
potential gflows for the protocol are required to be compatible with the order
$1<2<3$ on vertices. The only output set that supports such gflows is
$O=\\{2,3\\}$.
### 4.4 Regarding Multi-Round Blindness
In this subsection, we extend the blindness analysis to the situation where
the client and server engage in multiple rounds of the protocol, for example,
in order to collect measurement statistics of the output. It is not clear that
the protocol does indeed stay secure in this case (multi-round security was
not treated in the original work [36]). As an extension of 4.2 above, we
demonstrate that the min-entropy is bounded away from zero for all rounds.
We use similar assumptions to 4.2, namely that $P(g|O)$ and
$P(\boldsymbol{r})$ are uniform, however we make a slightly weaker assumption
on $P(\boldsymbol{\alpha},O)$: we take
$P(\boldsymbol{\alpha},O)=\frac{P(O)}{|\mathcal{A}|^{n}}$ for $P(O)$ an
arbitrary distribution over the output sets. Furthermore, we make the
additional assumptions that the gflow and one-time pads are chosen
independently for each round, allowing the multi-round protocol to be
represented as
$\displaystyle
D_{\operatorname{client}}^{(m)}=\sum_{\boldsymbol{\alpha},O}\frac{P(O)}{|\mathcal{A}|^{n}}\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}.$
(47)
We have the following theorem:
###### Theorem 4.3.
For any $m$, the following holds:
$\displaystyle
H_{\min}(\boldsymbol{A},\boldsymbol{O}|\boldsymbol{A^{\prime}}^{(1)},\boldsymbol{C^{\prime}}^{(1)},...,$
$\displaystyle\boldsymbol{A^{\prime}}^{(m)},\boldsymbol{C^{\prime}}^{(m)})_{D_{\operatorname{client}}^{(m)}}$
$\displaystyle\geq-\log\left(\sum_{O\in\mathcal{O}}\frac{P(O)}{2^{|O|}}\right).$
(48)
A key point of the proof, given in Section C.3, consists of the observation
that, for any choice of output set $O$, the one-time pads assigned to the
qubits in the output set cannot be learnt by the server since they do not
appear in the adaptation of any other angle (in contrast to the one-time pads
assigned to non-output qubits). The consequence is that the server can, at
best, only learn the output angles of $\boldsymbol{\alpha}$ up to a $\pi$
phase. However, the server may be able to learn all other measurement angles
with certainty, which correspond to the computation proper, in which case the
only remaining uncertainty consists of not knowing how to interpret the
measurement outcomes on the output state. This level of uncertainty is likely
insufficient to deem the protocol as secure in multiple rounds.
The above theorem does not say anything about whether the min-entropy does in
fact change from round to round, let alone whether the above bound can ever be
reached. To provide an answer to the former, we conclude by returning to the
minimal example presented above and consider the associated min-entropy for
two rounds. As outlined in Section C.2, for a specific choice of angle set
$\mathcal{A}$, the two round entropy is upper bounded by
$-\log_{2}(0.140625)\approx 2.830(0)$ and thus the min-entropy value does
indeed decrease from the single round to the two round case.
## 5 Grey Box MBQC
In this section, we consider a set of combs that are once again defined via
gflow, however there is now a mixture of quantum and classical interactions.
The inputs to the combs defined here are the same as for the BQC case
(measurement outcomes), but the outputs are now single qubit states of the
graph state, potentially acted upon by correction operators. We demonstrate
that these combs can be understood as quantum causal models in the style of
[20]. Moreover, the measurement channels inserted into the comb to perform a
measurement-based computation bear close resemblance to the notion of
observation put forward in [50] in the context of quantum causal inference. As
such, a key goal of this section is to explicate a novel connection between
measurement-based quantum computation and quantum causal models and causal
discovery.
In keeping with the rest of this work, we again apply the comb min-entropy to
quantify optimal learning of an unknown property, now of a “grey box MBQC
device” - a device which both prepares the graph state and conducts
measurement corrections (so that we simply need to insert measurement
channels) and for which we have only partial knowledge. Specifically, we
consider an example where the graph state and designated input and output
vertex sets are known but the particular gflow defining the corrections is
not. In light of the above connection, learning the gflow corresponds to
learning the causal structure of the device, i.e. a type of causal discovery.
The min-entropy for this example provides a benchmark for optimal causal
learning to which e.g. a learning strategy involving only observations
(defined below) can be compared.
In the final two subsections, we introduce two further pragmatic examples that
continue to demonstrate the utility of the quantum comb and min-entropic
approach to the type of partial-information analysis considered here. The
first considers the same specific graph as the causal discovery example,
however instead of learning the gflow itself, we aim now only to learn the
required measurement planes for which the gflow is compatible. For the given
level of knowledge about the device, i.e. knowing the graph state being
prepared and the input and output sets, this represents the minimal
information required to use the MBQC device for deterministic computation -
knowing the full details of the specific gflow is not required. However, to
ensure that any particular computation can be correctly performed, we also
need to calibrate the frame of reference of our measurement channels (outside
the device) with the frame of reference of the graph state preparation (inside
the device). Said another way, we want to ensure that measuring in, say, the
$XY$-plane at an angle $\alpha$ to the positive $X$-axis in our measuring
device does indeed correspond to an angle $\alpha$ with respect to the
positive $X$-axis implicit in the preparation of $\ket{G}$. The final example
is concerned with this scenario by aiming to learn the angle discrepancy
between the positive $X$-axes in the internal and external frame of references
under the assumption that the $Z$-axis is known. This can be considered as an
example of correlating reference frames in the quantum reference frames
literature [54].
### 5.1 Gflow Quantum Combs
In Section 2.2, and reinforced by the discourse in Section A.2, much emphasis
was placed on the two equivalent ways of correcting for a negative measurement
outcome at a given qubit in an MBQC: by classically adapting measurement
angles for measurement on other qubits or by applying a partial stabiliser on
the graph state. The latter perspective is the relevant one for our present
purposes. An MBQC on a graph state $\rho_{G}$ that prepares an output
(quantum) state on the output set $O$ can be written as
$\displaystyle\operatorname{Tr}\left[\left(\bigotimes_{v\in V\setminus
O}\ket{\pm_{\alpha_{v}}}\\!\\!\bra{\pm_{\alpha_{v}}}_{v}\right)\rho_{G}\right]$
(49)
where the trace is over all qubits in $V\setminus O$ and the measurement
planes can be considered as all $XY$ for simplicity. For a given vertex
$v_{0}\in V\setminus O$, there are two possibilities: either a positive
measurement outcome obtains in which case no corrections are required, or a
negative outcome obtains, for which corrections can be performed by applying a
partial stabiliser to $\rho_{G}$:
$\displaystyle\operatorname{Tr}$
$\displaystyle\left[\left(\ket{-_{\alpha_{v_{0}}}}\\!\\!\bra{-_{\alpha_{v_{0}}}}_{v_{0}}\bigotimes_{v\in
V\setminus(O\cup
v_{0})}\ket{\pm_{\alpha_{v}}}\\!\\!\bra{\pm_{\alpha_{v}}}_{v}\right)\rho_{G}\right]$
$\displaystyle\equiv\operatorname{Tr}\Bigg{[}\left(\ket{+_{\alpha_{v_{0}}}}\\!\\!\bra{+_{\alpha_{v_{0}}}}_{v_{0}}\bigotimes_{v\in
V\setminus(O\cup
v_{0})}\ket{\pm_{\alpha_{v}}}\\!\\!\bra{\pm_{\alpha_{v}}}_{v}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\cdot
K_{g(v_{0})}|_{\setminus v_{0}}\rho_{G}K_{g(v_{0})}|_{\setminus
v_{0}}^{\dagger}\Bigg{]}$ (50)
where $K_{g(v_{0})}$ is the (product of) stabiliser(s) assigned by a gflow $g$
to correct for measurements at $v_{0}$ and the notation
$K_{g(v_{0})}|_{\setminus v_{0}}$ denotes the operator derived from
$K_{g(v_{0})}$ with the $v_{0}$ tensor factor replaced with the identity.
So, for each measurement and conditional on the measurement outcome, we are
applying an operator on the graph state, either the identity (no correction
implemented) or the partial stabiliser (correction implemented). The ability
to correct for a negative outcome at every qubit via stabilisers only works if
each measurement is performed in the assigned plane for which the graph and
measurement plane assignment has gflow. If gflow exists and the restriction on
the measurement planes is followed, one can take the perspective that the
correction method defined from gflow is a mapping from measurement channels to
unitary channels, i.e. a higher order quantum map (a quantum comb).
To make this perspective precise, let us consider a tuple $(G,I,O,\omega)$ for
which a gflow $g$ exists. Our aim is then to write down an operator
$\sigma_{\operatorname{MBQC}}^{g}$ corresponding to the correction method
induced by $g$ such that the following quantity is the same for every set of
measurement outcomes (which in particular includes the all positive outcome
instance, and thus is equivalent to the desired computation):
$\displaystyle\left(\bigotimes_{v\in V\setminus
O}\mathcal{M}_{v,\alpha_{v},\omega(v)}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(51)
where the projections $\ket{\pm_{\alpha_{v}}}\bra{\pm_{\alpha_{v}}}$ in
measurement plane $\omega(v)$ are now described by appropriately defined
measurement channels $\mathcal{M}_{v,\alpha_{v},\omega(v)})$ (see below) and
where $\ast$ denotes the link product (2.3) which in particular includes the
trace over all spaces not related to the output space. The next few paragraphs
are devoted to defining the operator $\sigma_{\operatorname{MBQC}}^{g}$.
Let $g$ be a gflow for $(G,I,O,\omega)$, where $G$ is a graph on $n$ vertices
and as such we identify $V$ with $\\{1,...,n\\}$. The operator
$\sigma_{\operatorname{MBQC}}^{g}$ is defined on the Hilbert spaces
$\bigotimes_{i=1}^{n}\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{C_{i}}\otimes\mathcal{H}_{A^{\prime}_{i}}$,
where
$\mathcal{H}_{A_{i}}\cong\mathcal{H}_{C_{i}}\cong\mathcal{H}_{A^{\prime}_{i}}\cong\mathbb{C}^{2}$
for each $i$. Recalling from Equation 10 that $g$ defines the correction sets
$\mathcal{X}_{i}$ and $\mathcal{Z}_{i}$ for each $i\in V$, we define the
correction operator for $i$ as
$\displaystyle
U_{\text{corr}(\boldsymbol{c}),i}:=X_{A_{i}}^{\bigoplus_{j\in\mathcal{X}_{i}}c_{j}}Z_{A_{i}}^{\bigoplus_{j\in\mathcal{Z}_{i}}c_{j}}$
(52)
and for notational simplicity in the following, we take
$\displaystyle
U_{\text{corr}(\boldsymbol{c})}:=\bigotimes_{i=1}^{n}U_{\text{corr}(\boldsymbol{c}),i}.$
(53)
We can then define $\sigma_{\operatorname{MBQC}}^{g}$ as
$\displaystyle\sigma_{\operatorname{MBQC}}^{g}:=\sum_{\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}}U_{\text{corr}(\boldsymbol{c})}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}_{A}U_{\text{corr}(\boldsymbol{c})}^{\dagger}\otimes\ket{\boldsymbol{c},\boldsymbol{a}}\\!\\!\bra{\boldsymbol{c},\boldsymbol{b}}_{CA^{\prime}}$
(54)
where the subscripts $A$, $C$ and $A^{\prime}$ denote that the corresponding
state is in $\mathcal{H}_{A}:=\bigotimes_{i=1}^{n}\mathcal{H}_{A_{i}}$,
$\mathcal{H}_{C}:=\bigotimes_{i=1}^{n}\mathcal{H}_{C_{i}}$ or
$\mathcal{H}_{A^{\prime}}:=\bigotimes_{i=1}^{n}\mathcal{H}_{A^{\prime}_{i}}$
respectively, and where the sum is over the computational basis of these
spaces. The space $\mathcal{H}_{A^{\prime}}$ receives the graph state
$\rho_{G}$ via the link product as in Equation 51. To connect back to the
discussion above, $\sigma_{\operatorname{MBQC}}^{g}$ is the Choi operator of
the channel that conditionally applies the appropriate corrections via partial
stabiliser to the graph state. See Figure 5(a) for a depiction of
$\sigma_{\operatorname{MBQC}}^{g}$ for one of the gflows considered in the
example in Section 5.3.
It remains then only to define the measurement channels
$\mathcal{M}_{\alpha_{i},i,\omega(i)}$. Previously, we denoted measurements
simply via projection operators
$\ket{\pm_{\alpha}}\\!\\!\bra{\pm_{\alpha}}_{\text{mp}}$ for
$\text{mp}\in\\{XY,XZ,YZ\\}$, however for our present purposes we would like
to consider channels from $\mathcal{H}_{A_{i}}$ to $\mathcal{H}_{C_{i}}$ which
moreover prepare classical measurement outcomes ($\ket{0}\\!\\!\bra{0}$ or
$\ket{1}\\!\\!\bra{1}$) at $\mathcal{H}_{C_{i}}$. We define
$\mathcal{M}_{\alpha_{i},i,\omega(i)}:\mathcal{H}_{A_{i}}\rightarrow\mathcal{H}_{C_{i}}$
via its Choi representation:
$\displaystyle\mathcal{M}_{\alpha_{i},i,\omega(i)}:=$
$\displaystyle\ket{0}\\!\\!\bra{0}_{C_{i}}\otimes\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i),A_{i}}^{T}$
$\displaystyle+\ket{1}\\!\\!\bra{1}_{C_{i}}\otimes\ket{-_{\alpha_{i}}}\\!\\!\bra{-_{\alpha_{i}}}_{\omega(i),A_{i}}^{T}.$
(55)
(a) $\sigma_{\operatorname{MBQC}}^{g_{1}}$
(b) $D_{\text{gflow}}$
Figure 5: The classical-quantum combs defined here for the consideration of an
MBQC device have quantum comb component based on the flow of corrections
required for measurement-based computation (known as gflow). (a) An example of
such a quantum comb is $\sigma_{\operatorname{MBQC}}^{g_{1}}$ corresponding to
a gflow $g_{1}$ for a four-qubit graph state (see the example in Section 5.3
for more details). The graph state is input on the left and passed through the
conditional correction channels to each of the output spaces on the right.
Measurement channels performing single qubit measurements can be inserted into
the gaps (such as between $A_{1}$ AND $C_{1}$) (b) An example of the full MBQC
device (the grey outline) is modelled as a classical-quantum comb built from
the classical parameter that takes values $g$ denoting the gflow (equivalently
causal structure) and quantum comb $\sigma_{\operatorname{MBQC}}^{g}$ along
with the preparation of the graph state $\rho_{G}$. As with all classical-
quantum combs in this work, only the lower systems (those extending out of
$\sigma_{\operatorname{MBQC}}^{g}$) can be interacted with, and the upper
system is to be learnt about.
It should be noted that, since quantum combs are typically considered with a
total order, we endow $\sigma_{\operatorname{MBQC}}^{g}$ with a choice of
total order for which the partial order given by the gflow is compatible. Any
such total order is sufficient and we leave it implicit in the labelling of
the Hilbert spaces. Explicitly, we consider
$\sigma_{\operatorname{MBQC}}^{g}\in\operatorname{Comb}(\mathcal{H}_{A^{\prime}}\rightarrow\mathcal{H}_{A_{1}},H_{C_{1}}\rightarrow\mathcal{H}_{A_{2}},...,\mathcal{H}_{C_{|V\setminus
O|}}\rightarrow\bigotimes_{i\in O}\mathcal{H}_{A_{i}})$.
A proof that $\sigma_{\operatorname{MBQC}}^{g}$ is indeed correct in the sense
of ensuring that Equation 51 is equivalent for all measurement outcomes is
given in Section D.1. This proof is related to the existing theorem
establishing sufficiency of gflow for deterministic computation [[, Theorem
2,]]browne2007generalized and also provides justification for why the ordering
of operators in Equation 52 is valid.
### 5.2 Quantum Causal Models and Inference
Having defined $\sigma_{\operatorname{MBQC}}^{g}$ for an arbitrary $g$, we can
now make two connections to the literature of quantum causal models [20, 19,
68] and quantum causal inference [50, 52, 51]. Quantum causal models (QCMs)
are an extension to the classical causal modelling literature (see e.g. [21,
22, 69]) due, in part, to the realisation that the classical methodology is
insufficient for treating quantum correlations [70], and are typically
represented in the process matrix formalism [71]. In the first part of this
subsection, we show that each $\sigma_{\operatorname{MBQC}}^{g}$ satisfies the
conditions required for being a quantum causal model as per [20].
A key aspect of the classical causality literature consists of regarding the
differences between the probability distribution obtained under observations
and that obtained under an intervention on one or more of the causal
mechanisms generating the distribution. A full introduction and treatment of
these notions is beyond the scope of this work, however suffice it to say that
their counterparts for quantum causal inference are far more nuanced [50, 52,
51]. For our present purposes, we focus on the notion of observation as given
in [50] (discussed below) and showcase in the second part of this subsection
the close resemblance between this perspective of observation and the
measurement channels $\mathcal{M}_{\alpha_{i},i,\omega(i)}$ defined in the
previous subsection for MBQC.
As a first step towards the upcoming definition of quantum causal models, it
is worthwhile to briefly review the main components of their classical
counterparts. Classical causal models are specified by a directed acyclic
graph (DAG), with each node assigned a random variable, and functional
equations relating the value of a given node to the values of its parents. In
the quantum case, random variables are replaced by quantum nodes $B_{i}$,
which consist of two Hilbert spaces $\mathcal{H}_{B_{i}}$ and
$\mathcal{H}_{B_{i}}^{*}$ specifying incoming and outgoing state spaces of the
node (just as for combs), and with functional equations replaced by quantum
channels.
###### Definition 5.1 (Definition 3.3, [20]).
A quantum causal model is given by a directed acyclic graph over quantum nodes
$B_{1},...,B_{n}$ and for each node $B_{i}$, a quantum channel
$\rho_{B_{i}|\operatorname{Pa}(B_{i})}\in\mathcal{L}(\mathcal{H}_{B_{i}}\otimes\mathcal{H}_{\operatorname{Pa}(B_{i})}^{*})$
(where $\operatorname{Pa}(B_{i})$ denotes the set of parent nodes to $B_{i}$
with respect to the DAG) such that all channels mutually commute. This defines
a process operator
$\displaystyle\sigma_{B_{1},...,B_{n}}:=\prod_{i=1}^{n}\rho_{B_{i}|\operatorname{Pa}(B_{i})}.$
A note regarding notation: the asterisk on the $\mathcal{H}_{B_{i}}^{*}$ is
not (necessarily) to indicate that this is the dual space of
$\mathcal{H}_{B_{i}}$. It is merely to differentiate the outgoing space of the
node from the incoming one. We are avoiding using superscripts involving “in”
and “out” since the convention here would clash with that used in the combs
notation (ingoing to the node is outgoing from the comb and vice versa).
To show that $\sigma_{\operatorname{MBQC}}^{g}$ is a valid QCM, we first
realise that each gflow $g$ (including the partial order) induces a DAG on the
vertices of the corresponding graph $G$ (see D.2). In particular, for each
$i\in V$, the parents of $i$ are given by the union of the correction sets:
$\displaystyle\operatorname{Pa}(i):=\mathcal{X}_{i}\cup\mathcal{Z}_{i}.$ (56)
Using this notation, one can see how to write
$\sigma_{\operatorname{MBQC}}^{g}$ as given in Equation 54 as a product of
channels as in the definition above and what those channels are:
$\displaystyle\sigma_{\operatorname{MBQC}}^{g}=\prod_{i=1}^{n}\rho_{A_{i}|C_{j:j\in\operatorname{Pa}(i)},A^{\prime}_{i}}$
(57)
where
$\displaystyle\rho_{A_{i}|C_{j:j\in\operatorname{Pa}(i)},A^{\prime}_{i}}:=\sum_{\begin{subarray}{c}a_{i},b_{i}\\\
\boldsymbol{c}_{\operatorname{Pa}(i)}\end{subarray}}$ $\displaystyle
U_{\text{corr}(\boldsymbol{c}_{\operatorname{Pa}(i)}),i}\ket{a_{i}}\\!\\!\bra{b_{i}}U_{\text{corr}(\boldsymbol{c}_{\operatorname{Pa}(i)}),i}^{\dagger}$
$\displaystyle\quad\quad\otimes\ket{\boldsymbol{c}_{\operatorname{Pa}(i)}a_{i}}\\!\\!\bra{\boldsymbol{c}_{\operatorname{Pa}(i)}b_{i}}$
(58)
with $\boldsymbol{c}_{\operatorname{Pa}(i)}$ denoting the classical message
that are input into $\sigma_{\operatorname{MBQC}}^{g}$ for the vertices in
$\operatorname{Pa}(i)$. The channel
$\rho_{A_{i}|C_{j:j\in\operatorname{Pa}(i)},A^{\prime}_{i}}$ clearly has a
very similar structure to Equation 54, but contains only the relevant spaces
and corrections pertaining to $A_{i}$.
It is immediate to see that the remaining condition, that the
$\rho_{A_{i}|C_{j:j\in\operatorname{Pa}(i)},A^{\prime}_{i}}$ commute, is
satisfied: the only overlap between any two such channels occurs on the
classical message spaces, upon which the operators are diagonal. We have thus
shown:
###### Proposition 5.1.
For each gflow $g$, $\sigma_{\operatorname{MBQC}}^{g}$ is a quantum causal
model.
Having demonstrated the first connection between MBQC and quantum causality,
we return to the common theme running through this work: learning about a
random variable via interactions with a comb. Let the random variable $X$ take
values in the set of all gflows for a given graph $G$ (or perhaps a given
tuple $(G,I,O)$ or $(G,I,O,\omega)$) and consider
$\displaystyle D_{\text{gflow}}:=\sum_{g\in
X}P(g)\ket{g}\\!\\!\bra{g}\otimes\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}.$
(59)
In this case, we can understand learning about $X$ as a type of quantum causal
discovery: $X$ encodes information about $g$ and $g$ corresponds to a unique
QCM, so learning about $X$, via interactions with the
$\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$, is equivalent to learning the
causal structure. In the classical literature, a distinction is made between
learning the causal structure via observations and via interventions. For the
quantum case, Ried et al. [50] (see also [53]) define an observation as a
standard projective measurement channel (i.e. one the performs a projective
measurement and then prepares the post-measurement state according to the
standard update rule) which takes a maximally mixed state as input. The
requirement of a maximally mixed input is necessary and sufficient to enforce
an “informational symmetry” between the updated belief of the input state and
the knowledge of the output state of the channel (see Section III.A in the
supplementary information of [50]).
The measurement channels $\mathcal{M}_{\alpha_{i},i,\omega_{i}}$ defined in
Section 5.1 are not exactly the measurement channels as stipulated above, but
the difference is only cosmetic: the standard projective measurement channel
can be recovered simply by post-composing
$\mathcal{M}_{\alpha_{i},i,\omega_{i}}$ with a unitary channel (that depends
on $\alpha_{i}$ and $\omega_{i}$). In particular, the guiding principle of
informational symmetry between input and output is preserved. Regarding the
requirement of a maximally mixed input state, this is indeed ensured provided
that a graph state is input into $\sigma_{\operatorname{MBQC}}^{g}$ (each
single qubit reduced state of a graph state is maximally mixed). Thus, using
the present terminology, one could consider a measurement-based computation,
such as that described by Equation 51, as a specific set of observations of a
particular quantum causal model.
For the reader familiar with the classical causality literature, in particular
the work of Pearl (eg., [21]), the $A^{\prime}_{i}$ are analogous to the
exogenous variables for a classical causal model, and thus $\rho_{G}$ can be
considered as the equivalent of a distribution over the exogenous variables
and moreover one that does not factorise into a product distribution as is
often assumed. The last sentence of the previous paragraph can then be more
thoroughly reformulated as MBQC being a specific set of observations of a
particular quantum causal model for a certain state of the exogenous nodes.
See Section D.2 for more details of this correspondence.
### 5.3 Causal Discovery Example
Let us now consider a concrete example, which acts both as an explanatory aid
to the preceding subsection and as a vehicle to discuss the comparison between
the optimal strategy for causal discovery and an observational strategy. We
fix a graph and input and output sets for which all corresponding gflows are
characterised, and thereafter numerically calculate the min-entropy for
$D_{\text{gflow}}$ (Equation 59) as well as for a related comb with a
restriction on the included gflows. With this benchmark for the optimal causal
learning, we optimise over a restricted set of strategies constrained to
consist of observations only (as per the above notion of observation) thereby
quantifying the difference between the best such observational strategy and
the best overall strategy. We further demonstrate that, for cases where the
causal structures (gflows) differ only by symmetries of the input state (graph
state), even the optimal strategy is completely uninformative for causal
discovery.
Figure 6(a) depicts the four-vertex graph and choice of input and output sets
that forms the basis for our example. For this $(G,I,O)$ there are $15$
different possible gflows, which are catalogued in Appendix D by stipulating
the map $g$ and corresponding DAG for each (an example is depicted in Figure
6(b) for $g_{1}$; the labelling is given in the appendices).
(a) $(G,I,O)$
(b) DAG for $g_{1}$
Figure 6: To give a concrete example, we consider a specific graph state and
the set of all corresponding gflows. (a) The four qubit graph $G$ and choice
of input set $I=\\{1\\}$ and output set $O=\\{3,4\\}$. There are $15$ gflows
compatible with this choice of $(G,I,O)$ which are catalogued in Section D.4
including the corresponding directed acyclic graph and correction operators.
(b) The DAG for gflow $g_{1}$, which measures qubits $1$ and $2$ in the
$XY$-plane following the order $1<2$. The labels on the directed arrows depict
the conditional correction operators, with the head and tail of the arrow
denoting the target and control of the operation respectively.
We write the $D_{\text{gflow}}$ for this example as
$\displaystyle
D_{\text{gflow}}=\sum_{j=1}^{15}P(g_{j})\ket{j}\\!\\!\bra{j}\otimes\sigma_{\operatorname{MBQC}}^{g_{j}}\ast\rho_{G}$
(60)
which is a comb in $\operatorname{Comb}(\mathbb{C}\rightarrow
A_{1},C_{1}\rightarrow A_{2},C_{2}\rightarrow A_{3,4},\mathbb{C}\rightarrow
X)$ where $A_{1}$ and $A_{2}$ label the Hilbert spaces for graph state qubits
$1$ and $2$ respectively, $A_{3,4}$ labels the two-qubit Hilbert space for the
remaining graph state qubits, and $X$ labels a $15$-dimensional Hilbert space
for which the elements of a choice of orthonormal basis
$\\{\ket{j}\\}_{j=1}^{15}$ correspond to the different gflows. Note that no
measurement outcomes on the output qubits are input into $D_{\text{gflow}}$.
In this example, we once again assume a uniform prior:
$P(g_{j})=\frac{1}{15}$.
We will also consider a related comb to $D_{\text{gflow}}$, but instead of
including all $15$ gflows that exist for the choice of graph and input and
output sets, we consider only those for which the assigned measurement planes
for qubits $1$ and $2$ are both in the $XY$-plane and moreover only the gflows
which have partial order such that $1<2$. Using the labelling convention of
Appendix D, the gflows $g_{1},g_{2},g_{4}$ and $g_{5}$ satisfy these criteria.
We define
$\displaystyle
D_{XY,1<2}:=\sum_{j\in\\{1,2,4,5\\}}P(g_{j})\ket{j}\\!\\!\bra{j}\otimes\sigma_{\operatorname{MBQC}}^{g_{j}}\ast\rho_{G}$
(61)
with $P(g_{j})=\frac{1}{4}$. This is a comb in
$\operatorname{Comb}(\mathbb{C}\rightarrow A_{1},C_{1}\rightarrow
A_{2},C_{2}\rightarrow A_{3,4},\mathbb{C}\rightarrow X)$ where
$\mathcal{H}_{X}$ is $4$-dimensional in this case (we overload $X$ for
notational simplicity).
The guessing probability for each of these combs is given below, and, as for
all numerical calculations of guessing probabilities in this work, it is
calculated via the minimisation over unnormalised combs (recall Equation 12).
However, to conduct the comparison with an optimal observational strategy, it
is more convenient to use the maximisation form instead (c.f. Equation 13):
$\displaystyle P_{\text{guess}}=\max_{E}\operatorname{Tr}[DE^{T}]$ (62)
where $D$ is replaced by $D_{\text{gflow}}$ or $D_{XY,1<2}$ for the relevant
calculation, and $E$ is an element of $\operatorname{Comb}(A_{1}\rightarrow
C_{1},A_{2}\rightarrow C_{2},A_{3,4}\rightarrow X)$ with $X$ the appropriate
space for each case. The guessing probability under observational strategies
only is given by an analogous quantity to Equation 62 except that $E$ is now
restricted to having the form
$\displaystyle
E:=\sum_{c_{1},c_{2}}\mathcal{M}_{C_{1}|A_{1}}^{c_{1}}(\ket{\psi})\otimes\mathcal{M}_{C_{2}|A_{2}}^{c_{2}}(\phi)\otimes\mathcal{E}_{X|A_{3,4}}^{c_{1}c_{2}}$
(63)
where
$\displaystyle\mathcal{M}_{C_{i}|A_{i}}^{0}(\ket{\psi})$
$\displaystyle:=\ket{0}\\!\\!\bra{0}_{C_{i}}\otimes\ket{\psi}\\!\\!\bra{\psi}_{A_{i}}$
$\displaystyle\mathcal{M}_{C_{i}|A_{i}}^{1}(\ket{\psi})$
$\displaystyle:=\ket{1}\\!\\!\bra{1}_{C_{i}}\otimes(I-\ket{\psi}\\!\\!\bra{\psi})_{A_{i}}$
(64)
denote the observations, i.e. measurement channels
$\mathcal{M}_{C_{i}|A_{i}}(\psi))=\mathcal{M}_{C_{i}|A_{i}}^{0}(\ket{\psi})+\mathcal{M}_{C_{i}|A_{i}}^{1}(\ket{\psi})$,
and $\mathcal{E}_{X|A_{3,4}}^{c_{1}c_{2}}$ denotes the Choi state for a CPTP
map $\mathcal{E}:\mathcal{H}_{A_{3,4}}\rightarrow\mathcal{H}_{X}$ for the
relevant $\mathcal{H}_{X}$ for the corresponding $D$. We allow for the choice
of CPTP map to depend on the measurement outcomes, hence the $c_{1},c_{2}$ in
the superscript.
Numerically, the optimal guessing probability for $D_{\operatorname{gflow}}$,
which is to say, for selecting the correct causal structure after a single
round of interaction with the device, stands at $0.373(2)$ (see Appendix E for
details of the numerical implementation). This is a clear improvement over the
uniform distribution prior to interacting with the device
($\frac{1}{15}\approx 0.067(1)$). To calculate the guessing probability with
the restriction to observational strategies, a different approach is required
since the extra restriction on the set of operators that is maximised over
imposes non-linear constraints on the problem, so SDP methods no longer apply.
Namely, a brute force search was conducted over a mesh lattice of single qubit
projectors. Using this method, an approximate value for the guessing
probability under the optimal observational strategy is $0.199(1)$ which
indicates that there exist strategies for causal discovery that are more
informative than any observational strategy for this case.
For $D_{XY,1<2}$, the situation is different and rather interesting: the
guessing probability is $0.250(0)$. That is, in a single round even the
optimal learning strategy can not distinguish between the four different
causal structures. A trivial consequence of this fact is that, in this case,
any observational strategy is optimal (but still completely uninformative). In
fact, one can understand why the lack of distinguishability is to be expected
by considering the following proposition.
###### Proposition 5.2.
For any $c_{j}\in\\{0,1\\}$ for $j=1,...,|V\setminus O|$,
$\displaystyle\left(\bigotimes_{j=1}^{|V\setminus
O|}\ket{c_{j}}\\!\\!\bra{c_{j}}_{C_{j}}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(65)
is the same state for each gflow $g$ compatible with $(G,I,O,\omega)$ and have
mutually compatible partial orders.
The proof is given in Section D.3, but the key point is that for any given
measurement outcomes, the correction operators induced by each
$g\sim(G,I,O,\omega)$ are related to each other by some stabiliser and hence
are in fact equivalent.
There are a number of conclusions one can draw from the above analysis.
Firstly, as expected, an observational strategy for quantum causal discovery
is typically not optimal in general (at least in a single round). Secondly, in
certain cases, it is not guaranteed that anything about the causal structure
can be learnt in a single round by any strategy. This arises due to the fine-
tuned situation where the possible causal structures coincide precisely with
symmetries of the particular prepared state. This serves to highlight the
important role that noise likely plays in the ability to learn a causal
structure (a topic for future work). Finally, we must emphasise that the
numerical results here are for a single round of interaction only, largely due
to implementation constraints (see Appendix E).
### 5.4 Learning Measurement Planes
The example of $D_{XY,1<2}$ above demonstrates that barriers to learning the
exact flow of corrections exist, but fortunately, these details are not
required if our objective is to use the device to perform computations. In
this subsection, we consider the same example as given above, however we are
no longer concerned with learning the causal structure of the device but
rather learning enough information about the gflow being implemented in order
to ensure deterministic computation.
Specifically, we consider the case where the assignment of measurement planes
is unknown: that is, we know $(G,I,O)$ and need to learn $\omega$. We continue
with the same graph $(G,I,O)$ as in Figure 6 which has gflows catalogued in
Appendix D. Due to the properties of gflow, the input qubits are always
assigned the $XY$-plane by $\omega$, so for our example $\omega(1)\equiv XY$.
Since we allow for any measurements on the output qubits, it remains only to
learn the value of $\omega(2)$. It turns out that, for the current choice of
$(G,I,O)$, all three measurement planes on qubit $2$ are possible, with an
equal number of gflows ($5$) for each. Under the labelling as given in
Appendix D, gflows $g_{1},...,g_{5}$ correspond to $\omega(2)=XY$,
$g_{6},...,g_{10}$ to $\omega(2)=XZ$ and $g_{11},...,g_{15}$ to
$\omega(2)=YZ$. For this example, we let $X$ be the random variable that
encodes the measurement plane choice for the second qubit, i.e. takes values
in $\\{XY,XZ,YZ\\}$, and we denote an arbitrary measurement plane by mp. The
notation $g\sim\text{mp}$ indicates the gflow $g$ is defined for
$(G,I,O,\omega)$ where $\omega(2)=\text{mp}$.
We consider the comb
$\displaystyle D_{\text{mp}}:=\sum_{\text{mp}\in
X}P(\text{mp})\ket{\text{mp}}\\!\\!\bra{\text{mp}}\otimes\sum_{g\sim\text{mp}}P(g|\text{mp})\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(66)
in $\operatorname{Comb}(\mathbb{C}\rightarrow A_{1},C_{1}\rightarrow
A_{2},C_{2}\rightarrow A_{3,4},\mathbb{C}\rightarrow X)$ where
$\dim\mathcal{H}_{X}=3$. For simplicity, we once again make uniformity
assumptions on the distributions: $P(\text{mp})=\frac{1}{3}$ and
$P(g|\text{mp})=\frac{1}{5}$. Under these assumptions, the guessing
probability for $D_{\text{mp}}$ reaches $1.000(0)$ indicating that the
measurement planes can in fact be determined with certainty after just a
single use of the device. This is perhaps unsurprising since the correction
operators of gflows that differ in the measurement plane assignment for the
second qubit (e.g. for $g_{1}$ and $g_{6}$) are not related by a stabiliser
and so the type of indistinguishability problem that arose for $D_{XY,1<2}$
does not apply here.
### 5.5 Calibrating Measurements
In the previous example, the aim was to learn the correct measurement planes
for the proper functioning of the device, which in particular allows the user
to insert measurement channels $\mathcal{M}_{\alpha_{v},v,\text{mp}_{v}}$ for
the correct $\text{mp}_{v}$ for each $v$ (i.e. $\text{mp}_{v}=\omega(v)$),
thereby performing deterministic computation. However, this is not yet enough
to ensure the correctness of a specific computation: we need to ensure that
the $\alpha_{v}$ are also correct for each $v$. To explain this further,
consider a computation where all measurements are in the $XY$-plane, meaning
that the projection operators are given by
$\displaystyle\ket{\pm_{\alpha_{v}}}\\!\\!\bra{\pm_{\alpha_{v}}}=R_{Z}(\alpha_{v})\ket{\pm}\\!\\!\bra{\pm}R_{Z}(\alpha)^{\dagger}$
(67)
for each $v$. The $\ket{\pm}$ denote the eigenvectors of the $X$ operator and
moreover, there is an implicit assumption that the $\ket{+}$ state is the same
as that used in the graph state preparation (recall Equation 6). This
assumptions is typically a very reasonable one, however, since in this case
the state preparation occurs inside the device and the measurement channels
occur outside of it, there is no guarantee that this assumption is valid. If a
measurement channel $\mathcal{M}_{\alpha_{v},v,XY}$ , thought to consist of
projectors as in Equation 67, instead consists of projectors
$\displaystyle
R_{Z}(\alpha_{v})\ket{\pm_{\text{meas}}}\\!\\!\bra{\pm_{\text{meas}}}R_{Z}(\alpha_{v})^{\dagger}$
(68)
where
$\displaystyle\ket{\pm_{\text{meas}}}\\!\\!\bra{\pm_{\text{meas}}}=R_{Z}(\theta)\ket{\pm}\\!\\!\bra{\pm}R_{Z}^{\dagger}(\theta)$
(69)
for some $\theta$, then the effective measurement on the graph state qubit is
for an angle $\alpha_{v}+\theta$ rather than $\alpha_{v}$, meaning an
incorrect computation is performed. Even if the measurement planes for the
device are known, the specific measurement channels required to calibrate the
device in order to give the computations as expected (i.e. specifying the
computation using $\alpha_{v}-\theta$ recovers the correct computation). It is
interesting to note that this calibration can be considered as correlating two
quantum reference frames [54], the one inside the device with the one outside
of it.
Figure 7: The guessing probability for $D_{\text{calibr}}$ in a single round
as the size of possible angles set $\mathcal{A}$ varies (shown in blue). The
angles of $\mathcal{A}$ are taken to be evenly spaced. The probability of
guessing the correct value for the angle off-set $\theta$ prior to interacting
with the device is shown in black.
In this subsection, we consider a simple example along the lines of the
explanation above: we consider a scenario where the positive $Z$-axis of the
Bloch sphere is known, but the positive $X$-axis is not (we assume a right-
handed frame meaning that the positive $Y$-axis is determined if the positive
$X$\- and $Z$-axes are). In other words, we aim to learn $\theta$ in Equation
69 above. We assume $\theta$ can take values in a discrete set of angles which
are evenly spaced between $0$ and $2\pi$. We consider here a simple three-
vertex linear graph (i.e. with vertices $V=\\{1,2,3\\}$ and edges
$\\{(1,2),(2,3)\\}$), along with a single choice of gflow $g$ defined as
$1\mapsto\\{2\\}$ and $2\mapsto\\{3\\}$, which gives the correction sets (only
the non-empty such sets are shown):
$\displaystyle\mathcal{X}_{2}$ $\displaystyle=\\{1\\};$ (70)
$\displaystyle\mathcal{X}_{3}$ $\displaystyle=\\{2\\};$ (71)
$\displaystyle\mathcal{Z}_{3}$ $\displaystyle=\\{1\\}.$ (72)
For the purposes of writing down the comb $D_{\text{calibr}}$ for this
example, we take the basis
$\ket{\pm_{\text{meas}}}\\!\\!\bra{\pm_{\text{meas}}}$ as reference and write
the graph state $\rho_{G}$ and the correction operators
$U_{\text{corr}(\boldsymbol{c}),i}$ (recall Equation 52) with respect to this
basis:
$\displaystyle\rho_{G}^{\theta}$ $\displaystyle:=R_{Z}^{\otimes
n}(-\theta)\rho_{G}\left(R_{Z}^{\otimes n}(-\theta)\right)^{\dagger};$ (73)
$\displaystyle U_{\text{corr}(\boldsymbol{c}),i}^{\theta}$
$\displaystyle:=\left(R_{Z}(-\theta)X_{A_{i}}^{\bigoplus_{j\in\mathcal{X}_{i}}c_{j}}R_{Z}(-\theta)^{\dagger}\right)Z_{A_{i}}^{\bigoplus_{j\in\mathcal{Z}_{i}}c_{j}}.$
(74)
Denoting the comb defined from the
$U_{\text{corr}(\boldsymbol{c}),i}^{\theta}$ in analogy to Equation 54 as
$\sigma_{\operatorname{MBQC}}^{\theta}$ and the (discrete) set of possible
(regularly spaced) values for $\theta$ by $\mathcal{A}$, the comb of interest
for this example is
$\displaystyle
D_{\text{calibr}}:=\sum_{\theta\in\mathcal{A}}P(\theta)\ket{\theta}\\!\\!\bra{\theta}\otimes\sigma_{\operatorname{MBQC}}^{\theta}\ast\rho_{G}^{\theta}.$
(75)
As per usual, we take the prior distribution over $\theta$ to be uniform:
$P(\theta)=\frac{1}{|\mathcal{A}|}$. The guessing probability values for
$D_{\text{calibr}}$ for different sizes of $\mathcal{A}$, ranging from
$|\mathcal{A}|=2$ to $|\mathcal{A}|=32$, are shown in Figure 7. For
$|\mathcal{A}|=2$, the correct direction can be known with certainty in a
single round, and almost so for the case where $|\mathcal{A}|=3$
($P_{\text{guess}}\approx 0.992(5)$), but otherwise there is a steep decrease
in the guessing probability as the size of $\mathcal{A}$ increases.
## 6 Discussion
By interacting with a system, possibly via some complicated sequence of
actions, one can learn a certain property of the system that influences the
details of the interaction. By modelling such a situation within the quantum
combs formalism, it is possible to leverage the comb min-entropy [26], to
quantify how much one can learn about the unknown property in question. Due to
the general, and also natural, modelling of both quantum and classical
interactions as combs of the form considered above, the methodology showcased
here has broad applicability.
In this work, we restricted our attention to a novel set of combs defined by
the paradigm of measurement-based quantum computation. In this first instance,
we defined a classical comb which models a specific BQC protocol [36] and by
so doing, give a proof of partial security based on the comb min-entropy for
both a single round of the protocol as well as the multi-round case, extending
the security analysis in the existing literature.
We further defined a series of combs that model interactions with an MBQC
device under varying levels of knowledge regarding the inner working of the
device. This establishes a previously unmade connection between MBQC and
quantum causality. By observing that the gflow component from which the MBQC
combs are built satisfies the criteria of a quantum causal model, we were able
to utilise the comb min-entropy to quantify the optimal strategy for causal
discovery in specific cases, including cases where no information about the
causal structure could be determined by any strategy. We investigated further
examples related to learning the minimal extra information required in order
to ensure the correct functioning of the device as a computer.
### 6.1 Limitations of the Min-Entropy Approach
Despite the broad applicability and operational meaning of the methodology
used in this work, there are certain limitations of which one should be made
aware. Primarily, when using a numerical SDP solver, size issues quickly start
to play a role. For example, in the three-qubit graph state example for the
BQC protocol in Section 4.3, as a square matrix, the operator
$D_{\operatorname{client}}^{(m)}$ has dimension $|\mathcal{A}|^{3+3m}2^{3m}$
where $m$ is the number of rounds. For the minimal possible choice of angle
set, $|\mathcal{A}|=4$ and for a single round, this already equals $32768$.
For larger angle sets or greater number of rounds, the size problems became
prohibitive, disallowing any numerical calculations of the guessing
probability on the available hardware (see Appendix E).
### 6.2 Future Work
As mentioned above, the generality of the combs framework ensures that the
methodology considered in this work can be applied in a wide variety of
contexts (keeping in mind the limitations outlined above). We conclude by
outlining a couple of avenues for future work.
One feature of many SDP solvers that was not utilised in this work, regards
the possibility to return an optimal solution instance (and its dual) along
with the optimal value (e.g. the min-entropy) for the problem at hand.
Depending on the exact formulation used for the min-entropy (recall Equations
11, 12 and 13) either the primal solution or its dual will be a matrix
representing a strategy for interacting with the system in question. It would
be interesting to analyse these matrices in order to infer what the
corresponding strategy would be. For example, in Section 5.3, the min-entropy
quantified the best strategy for inferring the causal structure in a single
round of interaction. The solution that gives the optimal value could in
principle be deconstructed into components acting at each node of the quantum
causal model. This may help identify a type of quantum instrument or sequence
of instruments that are optimal for causal inference in general (rather than
just for the specific example).
A second interesting type of analysis that could be performed relates to the
effect of noise on the min-entropy. The examples considered in Section 5 all
included the preparation of a graph state $\rho_{G}$ and the operators
$\sigma_{\operatorname{MBQC}}$ which were specifically tailored to the graph
$G$. In realistic situations, these preparations and the implementations of
the corrections $U_{\text{corr}(\boldsymbol{c})}$ could be afflicted by noise.
Using the methodology here, one could, for example, investigate the effect
that different noise models could have on the guessing probability for
determining e.g. the correct measurement planes for the MBQC device. Since
adding noise to $\rho_{G}$ would not change the dimension of the comb to which
it is constituent, there is no added computational overhead and so the noise
analysis could be conducted for any case in which the noiseless case can be
calculated.
## 7 Acknowledgements
We would like to thank Atul Mantri for useful discussions about the security
of the BQC protocol of [36], Simon Milz for insightful comments regarding
quantum combs, Joshua Morris for invaluable suggestions regarding semi-
definite programming resources, and Sofiene Jerbi for useful conversations
regarding gflow and MBQC. We acknowledge support from the Austrian Science
Fund (FWF) through DK-ALM: W$1259$-N$27$ and SFB BeyondC F$7102$, and from the
University of Innsbruck. This work was also co-funded by the European Union
(ERC, QuantAI, Project No. $101055129$). Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union or the European Research Council. Neither the European
Union nor the granting authority can be held responsible for them.
## References
* [1] John Preskill “Quantum Computing in the NISQ era and beyond” In _Quantum_ 2 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2018, pp. 79 DOI: 10.22331/q-2018-08-06-79
* [2] H.-J. Briegel, W. Dür, J. I. Cirac and P. Zoller “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication” In _Phys. Rev. Lett._ 81 American Physical Society, 1998, pp. 5932–5935 DOI: 10.1103/PhysRevLett.81.5932
* [3] Rodney Van Meter “Quantum Networking” ISTE Ltd/John WileySons Inc, Hoboken, NJ, 2014 DOI: 10.1002/9781118648919
* [4] Davide Castelvecchi “The quantum internet has arrived (and it hasn’t).” In _Nature_ 554.7690 Nature Publishing Group, 2018, pp. 289–293 DOI: 10.1038/d41586-018-01835-3
* [5] Sheng-Kai Liao et al. “Satellite-relayed intercontinental quantum network” In _Physical review letters_ 120.3 APS, 2018, pp. 030501 DOI: 10.1103/PhysRevLett.120.030501
* [6] CT Nguyen et al. “Quantum network nodes based on diamond qubits with an efficient nanophotonic interface” In _Physical review letters_ 123.18 APS, 2019, pp. 183602 DOI: 10.1103/PhysRevLett.123.183602
* [7] Peter C Humphreys et al. “Deterministic delivery of remote entanglement on a quantum network” In _Nature_ 558.7709 Nature Publishing Group, 2018, pp. 268–273 DOI: 10.1038/s41586-018-0200-5
* [8] Boris Korzh et al. “Provably secure and practical quantum key distribution over 307 km of optical fibre” In _Nature Photonics_ 9.3 Nature Publishing Group, 2015, pp. 163–168 DOI: 10.1038/nphoton.2014.327
* [9] Rachel Courtland “China’s 2,000-km quantum link is almost complete” In _IEEE Spectrum_ 53.11 IEEE, 2016, pp. 11–12 DOI: 10.1109/MSPEC.2016.7607012
* [10] Mohamed Elboukhari, Mostafa Azizi and Abdelmalek Azizi “Quantum Key Distribution Protocols: A Survey.” In _International Journal of Universal Computer Science_ 1.2, 2010 DOI: 10.1109/ICWT.2018.8527822
* [11] Angela Sara Cacciapuoti et al. “Quantum internet: networking challenges in distributed quantum computing” In _IEEE Network_ 34.1 IEEE, 2019, pp. 137–143 DOI: 10.1109/MNET.001.1900092
* [12] Quantum Internet Alliance, https://quantum-internet.team, 2022
* [13] Antonio Acín et al. “The quantum technologies roadmap: a European community view” In _New Journal of Physics_ 20.8 IOP Publishing, 2018, pp. 080201 DOI: 10.1088/1367-2630/aad1ea
* [14] Robert Beals et al. “Efficient distributed quantum computing” In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 469.2153 The Royal Society Publishing, 2013, pp. 20120686 DOI: 10.1098/rspa.2012.0686
* [15] Vasil S Denchev and Gopal Pandurangan “Distributed quantum computing: A new frontier in distributed systems or science fiction?” In _ACM SIGACT News_ 39.3 ACM New York, NY, USA, 2008, pp. 77–95
* [16] Rene Allerstorfer, Harry Buhrman, Florian Speelman and Philip Verduyn Lunel “On the Role of Quantum Communication and Loss in Attacks on Quantum Position Verification” arXiv, arXiv:2208.04341, 2022 DOI: 10.48550/ARXIV.2208.04341
* [17] Joseph F Fitzsimons “Private quantum computation: an introduction to blind quantum computing and related protocols” In _npj Quantum Information_ 3.1 Nature Publishing Group, 2017, pp. 1–11 DOI: 10.1038/s41534-017-0025-3
* [18] John-Mark A Allen et al. “Quantum common causes and quantum causal models” In _Physical Review X_ 7.3 APS, 2017, pp. 031021 DOI: 10.1103/PhysRevX.7.031021
* [19] Fabio Costa and Sally Shrapnel “Quantum causal modelling” In _New Journal of Physics_ 18.6 IOP Publishing, 2016, pp. 063032 DOI: 10.1088/1367-2630/18/6/063032
* [20] Jonathan Barrett, Robin Lorenz and Ognyan Oreshkov “Quantum causal models” In _arXiv preprint arXiv:1906.10726_ , 2019 DOI: 10.48550/arXiv.1906.10726
* [21] Judea Pearl “Causality” Cambridge University Press, 2009 DOI: 10.1017/CBO9780511803161
* [22] Peter Spirtes, Clark N Glymour, Richard Scheines and David Heckerman “Causation, prediction, and search” MIT press, 2000 DOI: 10.7551/mitpress/1754.001.0001
* [23] Giulio Chiribella, Giacomo Mauro D’Ariano and Paolo Perinotti “Theoretical framework for quantum networks” In _Physical Review A_ 80.2 APS, 2009, pp. 022339 DOI: 10.1103/PhysRevA.80.022339
* [24] Giulio Chiribella, G Mauro D’Ariano and Paolo Perinotti “Quantum circuit architecture” In _Physical review letters_ 101.6 APS, 2008, pp. 060401 DOI: 10.1103/PhysRevLett.101.060401
* [25] Felix A Pollock et al. “Non-Markovian quantum processes: Complete framework and efficient characterization” In _Physical Review A_ 97.1 APS, 2018, pp. 012127 DOI: 10.1103/PhysRevA.97.012127
* [26] Giulio Chiribella and Daniel Ebler “Optimal quantum networks and one-shot entropies” In _New Journal of Physics_ 18.9 IOP Publishing, 2016, pp. 093053 DOI: 10.1088/1367-2630/18/9/093053
* [27] Robert Konig, Renato Renner and Christian Schaffner “The operational meaning of min-and max-entropy” In _IEEE Transactions on Information theory_ 55.9 IEEE, 2009, pp. 4337–4347 DOI: 10.1109/TIT.2009.2025545
* [28] Mark M Wilde “From classical to quantum Shannon theory” In _arXiv preprint arXiv:1106.1445_ , 2011 DOI: 10.48550/arXiv.1106.1445
* [29] Robert Raussendorf and Hans J Briegel “A one-way quantum computer” In _Physical review letters_ 86.22 APS, 2001, pp. 5188 DOI: 10.1103/PhysRevLett.86.5188
* [30] Hans J Briegel et al. “Measurement-based quantum computation” In _Nature Physics_ 5.1 Nature Publishing Group, 2009, pp. 19–26 DOI: 10.1038/nphys1157
* [31] Robert Raussendorf, Daniel E Browne and Hans J Briegel “Measurement-based quantum computation on cluster states” In _Physical review A_ 68.2 APS, 2003, pp. 022312 DOI: 10.1103/PhysRevA.68.022312
* [32] Robert Raussendorf and Hans Briegel “Computational model underlying the one-way quantum computer” In _arXiv preprint quant-ph/0108067_ , 2001 DOI: 10.48550/arXiv.quant-ph/0108067
* [33] Richard Jozsa “An introduction to measurement based quantum computation” In _NATO Science Series, III: Computer and Systems Sciences. Quantum Information Processing-From Theory to Experiment_ 199, 2006, pp. 137–158
* [34] D Gross and J Eisert “Novel schemes for measurement-based quantum computation” In _Physical review letters_ 98.22 APS, 2007, pp. 220503 DOI: 10.1103/PhysRevLett.98.220503
* [35] Anne Broadbent, Joseph Fitzsimons and Elham Kashefi “Universal blind quantum computation” In _2009 50th Annual IEEE Symposium on Foundations of Computer Science_ , 2009, pp. 517–526 IEEE DOI: 10.1109/FOCS.2009.36
* [36] Atul Mantri, Tommaso F Demarie, Nicolas C Menicucci and Joseph F Fitzsimons “Flow ambiguity: A path towards classically driven blind quantum computation” In _Physical Review X_ 7.3 APS, 2017, pp. 031004 DOI: 10.1103/PhysRevX.7.031004
* [37] Tomoyuki Morimae and Keisuke Fujii “Blind quantum computation protocol in which Alice only makes measurements” In _Physical Review A_ 87.5 APS, 2013, pp. 050301 DOI: 10.1103/PhysRevA.87.050301
* [38] Tomoyuki Morimae “Verification for measurement-only blind quantum computing” In _Physical Review A_ 89.6 APS, 2014, pp. 060302 DOI: 10.1103/PhysRevA.89.060302
* [39] Hans J Briegel and Robert Raussendorf “Persistent entanglement in arrays of interacting particles” In _Physical Review Letters_ 86.5 APS, 2001, pp. 910 DOI: 10.1103/PhysRevLett.86.910
* [40] Marc Hein et al. “Entanglement in graph states and its applications” In _Volume 162: Quantum Computers, Algorithms and Chaos_ , Proceedings of the International School of Physics ”Enrico Fermi” IOS Press Ebooks, 2006, pp. 115–218 DOI: 10.3254/978-1-61499-018-5-115
* [41] Marc Hein, Jens Eisert and Hans J Briegel “Multiparty entanglement in graph states” In _Physical Review A_ 69.6 APS, 2004, pp. 062311 DOI: 10.1103/PhysRevA.69.062311
* [42] Mehdi Mhalla et al. “Which graph states are useful for quantum information processing?” In _Conference on Quantum Computation, Communication, and Cryptography_ , 2011, pp. 174–187 Springer DOI: 10.1007/978-3-642-54429-3“–˙˝12
* [43] Marc Hein et al. “Entanglement in graph states and its applications” In _arXiv preprint quant-ph/0602096_ , 2006 DOI: 10.48550/arXiv.quant-ph/0602096
* [44] Daniel E Browne, Elham Kashefi, Mehdi Mhalla and Simon Perdrix “Generalized flow and determinism in measurement-based quantum computation” In _New Journal of Physics_ 9.8 IOP Publishing, 2007, pp. 250 DOI: 10.1088/1367-2630/9/8/250
* [45] Vincent Danos and Elham Kashefi “Determinism in the one-way model” In _Physical Review A_ 74.5 APS, 2006, pp. 052310 DOI: 10.1103/PhysRevA.74.052310
* [46] Damian Markham and Elham Kashefi “Entanglement, flow and classical simulatability in measurement based quantum computation” In _Horizons of the Mind. A Tribute to Prakash Panangaden_ Springer, 2014, pp. 427–453 DOI: 10.1007/978-3-319-06880-0“–˙˝22
* [47] Vincent Danos, Elham Kashefi and Prakash Panangaden “The measurement calculus” In _Journal of the ACM (JACM)_ 54.2 ACM New York, NY, USA, 2007, pp. 8–es DOI: 10.1145/1219092.1219096
* [48] Maarten Nest, Jeroen Dehaene and Bart De Moor “Graphical description of the action of local Clifford transformations on graph states” In _Physical Review A_ 69.2 APS, 2004, pp. 022316 DOI: 10.1103/PhysRevA.69.022316
* [49] Maarten Nest, Jeroen Dehaene and Bart De Moor “Local unitary versus local Clifford equivalence of stabilizer states” In _Physical Review A_ 71.6 APS, 2005, pp. 062323 DOI: 10.1103/PhysRevA.75.032325
* [50] Katja Ried et al. “A quantum advantage for inferring causal structure” In _Nature Physics_ 11.5 Nature Publishing Group, 2015, pp. 414–420 DOI: 10.1038/nphys3266
* [51] Joseph F Fitzsimons, Jonathan A Jones and Vlatko Vedral “Quantum correlations which imply causation” In _Scientific reports_ 5.1 Nature Publishing Group, 2015, pp. 1–7 DOI: 10.1038/srep18281
* [52] Giulio Chiribella and Daniel Ebler “Quantum speedup in the identification of cause–effect relations” In _Nature communications_ 10.1 Nature Publishing Group, 2019, pp. 1–8 DOI: 10.1038/s41467-019-09383-8
* [53] Jonas M Kübler and Daniel Braun “Two-qubit causal structures and the geometry of positive qubit-maps” In _New Journal of Physics_ 20.8 IOP Publishing, 2018, pp. 083015 DOI: 10.1088/1367-2630/aad612
* [54] Stephen D Bartlett, Terry Rudolph and Robert W Spekkens “Reference frames, superselection rules, and quantum information” In _Reviews of Modern Physics_ 79.2 APS, 2007, pp. 555 DOI: 10.1103/RevModPhys.79.555
* [55] Man-Duen Choi “Completely positive linear maps on complex matrices” In _Linear algebra and its applications_ 10.3 Elsevier, 1975, pp. 285–290 DOI: 10.1016/0024-3795(75)90075-0
* [56] Andrzej Jamiołkowski “Linear transformations which preserve trace and positive semidefiniteness of operators” In _Reports on Mathematical Physics_ 3.4 Elsevier, 1972, pp. 275–278 DOI: 10.1016/0034-4877(72)90011-0
* [57] Daniel Gottesman “Stabilizer codes and quantum error correction” California Institute of Technology, 1997
* [58] Mehdi Mhalla and Simon Perdrix “Finding optimal flows efficiently” In _International Colloquium on Automata, Languages, and Programming_ , 2008, pp. 857–868 Springer DOI: 10.1007/978-3-540-70575-8“–˙˝70
* [59] Niel De Beaudrap “Finding flows in the one-way measurement model” In _Physical Review A_ 77.2 APS, 2008, pp. 022328 DOI: 10.1103/PhysRevA.77.022328
* [60] Renato Renner “Security of quantum key distribution” In _International Journal of Quantum Information_ 6.01 World Scientific, 2008, pp. 1–127 DOI: 10.1142/S0219749908003256
* [61] Marco Tomamichel, Roger Colbeck and Renato Renner “A fully quantum asymptotic equipartition property” In _IEEE Transactions on information theory_ 55.12 IEEE, 2009, pp. 5840–5847 DOI: 10.1109/TIT.2009.2032797
* [62] Marco Tomamichel “Quantum information processing with finite resources: mathematical foundations” Springer, 2015 DOI: 10.1007/978-3-319-21891-5
* [63] Matthew S Leifer and Robert W Spekkens “Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference” In _Physical Review A_ 88.5 APS, 2013, pp. 052130 DOI: 10.1103/PhysRevA.88.052130
* [64] Christopher Portmann et al. “Causal boxes: quantum information-processing systems closed under composition” In _IEEE Transactions on Information Theory_ 63.5 IEEE, 2017, pp. 3277–3305 DOI: 10.1109/TIT.2017.2676805
* [65] Ueli Maurer “Abstract models of computation in cryptography” In _IMA International Conference on Cryptography and Coding_ , 2005, pp. 1–12 Springer DOI: 10.1007/11586821“–˙˝1
* [66] Christopher Portmann and Renato Renner “Security in quantum cryptography” In _Reviews of Modern Physics_ 94.2 APS, 2022, pp. 025008 DOI: 10.1103/RevModPhys.94.025008
* [67] Atul Mantri, Tommaso F Demarie and Joseph F Fitzsimons “Universality of quantum computation with cluster states and (X, Y)-plane measurements” In _Scientific reports_ 7.1 Nature Publishing Group, 2017, pp. 1–7 DOI: 10.1038/srep42861
* [68] Giacomo Mauro D’Ariano “Causality re-established” In _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 376.2123 The Royal Society Publishing, 2018, pp. 20170313 DOI: 10.1098/rsta.2017.0313
* [69] Bernhard Schölkopf et al. “On causal and anticausal learning” In _arXiv preprint arXiv:1206.6471_ , 2012 DOI: 10.48550/arXiv.1206.6471
* [70] Christopher J Wood and Robert W Spekkens “The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning” In _New Journal of Physics_ 17.3 IOP Publishing, 2015, pp. 033002 DOI: 10.1088/1367-2630/17/3/033002
* [71] Ognyan Oreshkov, Fabio Costa and Časlav Brukner “Quantum correlations with no causal order” In _Nature communications_ 3.1 Nature Publishing Group, 2012, pp. 1–8 DOI: 10.1038/ncomms2076
* [72] Isaac D. Smith and Marius Krumm “Min-Entropy and MBQC”, https://github.com/IsaacDSmith/Min-Entropy_and_MBQC
* [73] Steven Diamond and Stephen Boyd “CVXPY: A Python-embedded modeling language for convex optimization” In _Journal of Machine Learning Research_ 17.83, 2016, pp. 1–5
* [74] Akshay Agrawal, Robin Verschueren, Steven Diamond and Stephen Boyd “A rewriting system for convex optimization problems” In _Journal of Control and Decision_ 5.1, 2018, pp. 42–60
* [75] Brendan O’Donoghue, Eric Chu, Neal Parikh and Stephen Boyd “SCS: Splitting Conic Solver, version 3.2.2”, https://github.com/cvxgrp/scs, 2021
* [76] Brendan O’Donoghue, Eric Chu, Neal Parikh and Stephen Boyd “Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding” In _Journal of Optimization Theory and Applications_ 169.3, 2016, pp. 1042–1068 URL: http://stanford.edu/~boyd/papers/scs.html
## Appendix A Preliminaries Section Supplementary
This appendix houses some supporting discussion for Section 2.
### A.1 Motivating Quantum Combs
In this subsection, we briefly motivate the definition of quantum comb given
in the main text (2.1), including why this is an appropriate representation of
quantum networks and what the conditions of the definition correspond to. We
provide these explanations here for completeness; they are drawn from
references such as [24, 23] which should be referred to for further details.
The definition of quantum comb given earlier is as an operator on a tensor
product of Hilbert spaces, which represents a series of connected quantum
channels. This correspondence between operators on a tensor product of spaces
and maps between these spaces, that is, between elements of
$\mathcal{L}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ and
$\mathcal{L}(\mathcal{L}(\mathcal{H}_{A}),\mathcal{L}(\mathcal{H}_{B}))$
respectively, makes use of the Choi-Jamiolkowski isomorphism [55, 56]. Since
we are most interested in the operator form, we make the following definition.
###### Definition A.1.
For a linear map
$\mathcal{E}:\mathcal{L}(\mathcal{H}_{A})\rightarrow\mathcal{L}(\mathcal{H}_{B})$,
its Choi operator is defined as:
$E:=(\mathcal{E}\otimes
I_{\mathcal{H}_{A^{\prime}}})(\ket{\Phi^{+}}\\!\\!\bra{\Phi^{+}}).$ (76)
Here, $\mathcal{H}_{A^{\prime}}\simeq\mathcal{H}_{A}$ is a copy of the input
space, and $\ket{\Phi^{+}}:=\sum_{i=0}^{d_{A}-1}\ket{ii}_{A^{\prime}A}$ is an
unnormalised maximally entangled state on
$\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{A}$.
The defining properties of CPTP maps correspond to properties of the Choi
operator:
###### Lemma A.1.
The Choi operator $E$ of a linear map $\mathcal{E}$ satisfies the following
properties:
1. 1.
$E$ is positive semi-definite, denoted $E\geq 0$, iff $\mathcal{E}$ is
completely positive;
2. 2.
$E$ is Hermitian iff $\mathcal{E}$ is Hermitian-preserving;
3. 3.
$\operatorname{Tr}_{B}\left[E\right]\leq I_{A}$ iff $\mathcal{E}$ is trace
non-increasing;
4. 4.
$\operatorname{Tr}_{B}\left[E\right]=I_{A}$ iff $\mathcal{E}$ is trace-
preserving.
One notices that the first and last items of the above lemma feature in 2.1,
however iteratively for the latter item. To arrive at the iterative
constraints, consider a simple example of two CPTP maps
$\mathcal{E}_{1}:\mathcal{L}(\mathcal{H}_{A_{1}})\rightarrow\mathcal{L}(\mathcal{H}_{B_{1}}\otimes\mathcal{H}_{C})$
and
$\mathcal{E}_{2}:\mathcal{L}(\mathcal{H}_{A_{2}}\otimes\mathcal{H}_{C})\rightarrow\mathcal{H}_{B_{2}}$
with Choi operators $E_{1}$ and $E_{2}$ respectively. The composition of the
two maps over the space $H_{C}$, denoted
$\mathcal{E}_{2}\circ_{C}\mathcal{E}_{1}$ is a linear operator from
$\mathcal{L}(\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A_{2}})$ to
$\mathcal{L}(\mathcal{H}_{B_{1}}\otimes\mathcal{H}_{B_{2}})$, and the
corresponding Choi operator is given by
$\displaystyle E=E_{1}\ast_{C}E_{2}$ (77)
where $\ast_{C}$ denotes the link product over the space $C$, which is the
analogue of composition in the Choi operator picture. By tracing over $B_{2}$,
we get
$\displaystyle\operatorname{Tr}_{B_{2}}\left[E\right]$
$\displaystyle=\operatorname{Tr}_{B_{2}}\left[E_{1}\ast_{C}E_{2}\right]$ (78)
$\displaystyle=E_{1}\ast_{C}\operatorname{Tr}_{B_{2}}\left[E_{2}\right]$ (79)
$\displaystyle=E_{1}\ast_{C}(I_{C}\otimes I_{A_{2}})$ (80)
$\displaystyle=I_{A_{2}}\otimes\operatorname{Tr}_{C}\left[E_{1}\right]$ (81)
where we have used the trace-preservation criterion of A.1. Noting that
$\operatorname{Tr}_{C}\left[E_{1}\right]$ is a positive semi-definite
operator, we see that the last line above is indeed of the form of the
constraints in 2.1. By tracing also over $B_{1}$, we can use the trace-
preservation again to obtain the terminal constraint of normalisation to $1$.
This reasoning extends to any number of composed maps, which helps demonstrate
the conciseness of the comb notation.
The remainder of this appendix subsection provides the proof of 2.1 given in
Section 2.1.
###### Proof of 2.1.
The proof consists of showing that the sequence of partial trace constraints
(2.1) are satisfied for both the sequence where $X$ is traced over first then
$A_{n}^{\operatorname{out}},...,A_{1}^{\operatorname{out}}$ and the sequence
$A_{n}^{\operatorname{out}},...,A_{1}^{\operatorname{out}},X$. Starting with
the former, tracing over $X$ gives
$\displaystyle\operatorname{Tr}_{X}[D]=\sum_{x}P(x)\sigma_{x}\equiv
I_{\mathbb{C}}\otimes\sum_{x}P(x)\sigma_{x}.$ (82)
Since the $\sigma_{x}$ are normalised combs, and since the set of normalised
combs is convex, $\sum_{x}P(x)\sigma_{x}$ is also a normalised comb and so the
remaining trace conditions are satisfied, thus showing that
$D\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}},\mathbb{C}\rightarrow X)$.
For the other sequence, define a sequence of operators $D_{k}$, $k=0,...,n+1$
via
$\displaystyle D_{k}$ $\displaystyle:=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes
C_{x,k-1}\quad\quad\forall k=1,...,n+1$ (83) $\displaystyle D_{0}$
$\displaystyle:=\sum_{x}P(x)$ (84)
where $C_{x,0},...,C_{x,n}$ denote the positive semi-definite operators that
satisfy the comb conditions for $\sigma_{x}$, that is:
$\displaystyle\sigma_{x}$ $\displaystyle=C_{x,n};$ (85)
$\displaystyle\operatorname{Tr}_{A_{j}^{\operatorname{out}}}[C_{x,j}]$
$\displaystyle=I_{A_{j}^{\operatorname{in}}}\otimes C_{x,j-1}\quad\quad\forall
j=1,...,n;$ (86) $\displaystyle C_{x,0}$ $\displaystyle=1.$ (87)
It follows that $D_{n+1}=D$, that
$\displaystyle\operatorname{Tr}_{A_{k-1}^{\operatorname{out}}}\left[D_{k}\right]$
$\displaystyle=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\operatorname{Tr}_{A_{k-1}^{\operatorname{out}}}[C_{x,k-1}]$
(88) $\displaystyle=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes
I_{A_{k-1}^{\operatorname{in}}}\otimes C_{x,k-2}$ (89)
$\displaystyle=I_{A_{k-1}^{\operatorname{in}}}\otimes D_{k-1}$ (90)
for all $k=2,...,n+1$, that
$\displaystyle\operatorname{Tr}_{X}[D_{1}]$
$\displaystyle=\operatorname{Tr}_{X}\left[\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes
1\right]$ (91) $\displaystyle=\sum_{x}P(x)$ (92)
$\displaystyle=I_{\mathbb{C}}\otimes D_{0}$ (93)
and that $D_{0}=1$. Thus, $D$ is also in
$\operatorname{Comb}(\mathbb{C}\rightarrow
X,A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$. ∎
### A.2 Equivalent Correction Methods for Measurement Outcomes
Here, we elaborate on the discussion in Section 2.2 regarding the correction
of measurement outcomes in MBQC via products of stabilisers $K_{v}$ of graph
state $\ket{G}$. To recap what is written in the main text, the correction for
negative measurement outcomes starts by noticing that, due to the restriction
of the allowed measurement to the planes of the Bloch sphere, the negative
projection operators are related to the positive ones via conjugation by $X$,
$Z$ or their product, which each exist as a factor of a (product of)
stabiliser(s). By applying the remainder of an appropriate stabiliser in the
case when a negative outcome obtains, a symmetry of the graph state is
completed and in effect, a positive projection has been enacted instead.
We demonstrate this with the following example where a vertex $v\in V$ is
measured in the $XY$-plane (the same reasoning applies for the other
measurement planes). We let $v^{\prime}\in V$ be a distinguished neighbour of
$v$ and denote all pending measurements on $\ket{G}$ (also in the $XY$-plane
for simplicity) as
$\ket{\pm_{\alpha_{\tilde{v}}}}\bra{\pm_{\alpha_{\tilde{v}}}}$ for
$\tilde{v}\in V\setminus\\{v\\}$. The notation $K_{v^{\prime}}|_{\setminus v}$
denotes the stabiliser $K_{v^{\prime}}$ with the tensor factor corresponding
to $v$ replaced by $I_{v}$.
$\displaystyle\operatorname{Tr}$
$\displaystyle\left[\left(\ket{-_{\alpha_{v}}}\\!\\!\bra{-_{\alpha_{v}}}_{v}\bigotimes_{\tilde{v}\in
V\setminus\\{v\\}}\ket{\pm_{\alpha_{\tilde{v}}}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}}}_{\tilde{v}}\right)\rho_{G}\right]=\operatorname{Tr}\left[\left(Z_{v}^{\dagger}\ket{+_{\alpha_{v}}}\\!\\!\bra{+_{\alpha_{v}}}_{v}Z_{v}\bigotimes_{\tilde{v}\in
V\setminus\\{v\\}}\ket{\pm_{\alpha_{\tilde{v}}}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}}}_{\tilde{v}}\right)\rho_{G}\right]$
(94)
$\displaystyle=\operatorname{Tr}\left[\left(\ket{+_{\alpha_{v}}}\\!\\!\bra{+_{\alpha_{v}}}_{v}\bigotimes_{\tilde{v}\in
V\setminus\\{v\\}}\ket{\pm_{\alpha_{\tilde{v}}}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}}}_{\tilde{v}}\right)Z_{v}\rho_{G}Z_{v}^{\dagger}\right]$
(95)
$\displaystyle=\operatorname{Tr}\left[\left(\ket{+_{\alpha_{v}}}\\!\\!\bra{+_{\alpha_{v}}}_{v}\bigotimes_{\tilde{v}\in
V\setminus\\{v\\}}\ket{\pm_{\alpha_{\tilde{v}}}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}}}_{\tilde{v}}\right)K_{v^{\prime}}|_{\setminus
v}\rho_{G}K_{v^{\prime}}|_{\setminus v}^{\dagger}\right]$ (96)
$\displaystyle=\operatorname{Tr}\left[\left(\ket{+_{\alpha_{v}}}\\!\\!\bra{+_{\alpha_{v}}}_{v}\otimes
X_{v^{\prime}}^{\dagger}\ket{\pm_{\alpha_{v^{\prime}}}}\\!\\!\bra{\pm_{\alpha_{v^{\prime}}}}X_{v^{\prime}}^{\dagger}\bigotimes_{\tilde{v}\in
N_{v^{\prime}}^{G}\setminus\\{v\\}}Z_{\tilde{v}}^{\dagger}\ket{\pm_{\alpha_{\tilde{v}}}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}}}_{\tilde{v}}Z_{\tilde{v}}\bigotimes_{\hat{v}\in
V\setminus(N_{v^{\prime}}^{G}\cup\\{v^{\prime}\\})}\ket{\pm_{\alpha_{\hat{v}}}}\\!\\!\bra{\pm_{\alpha_{\hat{v}}}}_{\hat{v}}\right)\rho_{G}\right]$
(97)
$\displaystyle=\operatorname{Tr}\left[\left(\ket{+_{\alpha_{v}}}\\!\\!\bra{+_{\alpha_{v}}}_{v}\otimes
X_{v^{\prime}}^{\dagger}\ket{\pm_{\alpha_{v^{\prime}}}}\\!\\!\bra{\pm_{\alpha_{v^{\prime}}}}X_{v^{\prime}}^{\dagger}\bigotimes_{\tilde{v}\in
N_{v^{\prime}}^{G}\setminus\\{v\\}}Z_{\tilde{v}}^{\dagger}\ket{\pm_{\alpha_{\tilde{v}}}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}}}_{\tilde{v}}Z_{\tilde{v}}\bigotimes_{\hat{v}\in
V\setminus(N_{v^{\prime}}^{G}\cup\\{v^{\prime}\\})}\ket{\pm_{\alpha_{\hat{v}}}}\\!\\!\bra{\pm_{\alpha_{\hat{v}}}}_{\hat{v}}\right)\rho_{G}\right]$
(98)
$\displaystyle=\operatorname{Tr}\left[\left(\ket{+_{\alpha_{v}}}\\!\\!\bra{+_{\alpha_{v}}}_{v}\otimes\ket{\pm_{-\alpha_{v^{\prime}}}}\\!\\!\bra{\pm_{-\alpha_{v^{\prime}}}}\bigotimes_{\tilde{v}\in
N_{v^{\prime}}^{G}\setminus\\{v\\}}\ket{\pm_{\alpha_{\tilde{v}}+\pi}}\\!\\!\bra{\pm_{\alpha_{\tilde{v}}+\pi}}_{\tilde{v}}\bigotimes_{\hat{v}\in
V\setminus(N_{v^{\prime}}^{G}\cup\\{v^{\prime}\\})}\ket{\pm_{\alpha_{\hat{v}}}}\\!\\!\bra{\pm_{\alpha_{\hat{v}}}}_{\hat{v}}\right)\rho_{G}\right]$
(99)
where the last line uses the useful correspondence between applying $X$ or $Z$
and the update of measurement angles presented in Equation 9 (with the modulo
$2\pi$ notation suppressed). Thus, a positive outcome at $v$ is recovered at
the cost of applying the remainder of the stabiliser $K_{v^{\prime}}$ to
either the graph state $\rho_{G}$ (Equation 96) or the measurement operators
on $v^{\prime}\cup N_{v^{\prime}}^{G}\setminus\\{v\\}$ via a classical change
of measurement parameter (Equation 99). The former perspective is used in
Section 5 and the latter is key to the BQC protocol in Section 4. Furthermore,
we believe the above discussion can provide a more intuitive understanding of
corrections work in MBQC and motivate the definition of gflow (2.4).
## Appendix B Min-Entropy Section Supplementary
In Section 3.1, we considered combs of the form
$\displaystyle
D=\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\sigma_{x}\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}},\mathbb{C}\rightarrow X)$ (100)
where all the
$\sigma_{x}\in\operatorname{Comb}(A_{1}^{\operatorname{in}}\rightarrow
A_{1}^{\operatorname{out}},...,A_{n}^{\operatorname{in}}\rightarrow
A_{n}^{\operatorname{out}})$ are classical. The ultimate aim was to provide an
interpretation of the guessing probability,
$P_{\text{guess}}(X|\boldsymbol{A}^{\operatorname{in}},\boldsymbol{A}^{\operatorname{out}})$,
in terms of Bayesian updating since such an interpretation exists for the
state min-entropy (see eg., [[, Section 6.1.4,]]tomamichel2015quantum). For
completeness, we present this interpretation here and discuss the similarities
and differences with the combs case.
Let $D$ be a classical-quantum state on
$\mathcal{H}_{X}\otimes\mathcal{H}_{Y}$ (i.e.
$D\in\operatorname{Comb}(\mathbb{C}\rightarrow Y,\mathbb{C}\rightarrow X)$)
such that each $\sigma_{x}\in\mathcal{L}(\mathcal{H}_{Y})$ is classical, that
is, they can be written as
$\displaystyle\sigma_{x}=\sum_{y}P(y|x)\ket{y}\\!\\!\bra{y}$ (101)
for $\\{\ket{y}\\}_{y}$ a common choice of orthonormal basis for
$\mathcal{H}_{Y}$ and for $P(y|x)$ a conditional probability distribution. We
can thus write
$\displaystyle D=\sum_{x,y}P(x)P(y|x)\ket{xy}\\!\\!\bra{xy}.$ (102)
Applying Bayes’ rule, it follows that
$\displaystyle D=\sum_{x,y}P(x|y)P(y)\ket{xy}\\!\\!\bra{xy}.$ (103)
By maximising over $x$ for each $y$, we obtain the inequality:
$\displaystyle D$
$\displaystyle\leq\sum_{x,y}\left[\max_{\tilde{x}}P(\tilde{x}|y)P(y)\right]\ket{xy}\\!\\!\bra{xy}$
(104)
$\displaystyle=I_{X}\otimes\sum_{y}\max_{\tilde{x}}P(\tilde{x}|y)P(y)\ket{y}\\!\\!\bra{y}.$
(105)
The second tensor factor above is an (in general) unnormalised state on $Y$,
and moreover it can be shown that this unnormalised state is a minimal such
state $\rho_{Y}$ for which $D\leq I_{X}\otimes\rho_{Y}$ holds (this follows
from the proof of the left-hand inequality of 3.2). Thus, using the
unnormalised version of the min-entropy (recall Equation 12), we arrive at
$\displaystyle
P_{\text{guess}}(X|Y)_{D}=\sum_{y}\max_{\tilde{x}}P(\tilde{x}|y)P(y)$ (106)
and hence also at the desired interpretation of the guessing probability in
terms of Bayesian updating: the guessing probability is the maximal Bayesian
update for each interaction (denoted by $y$) averaged over all possible
interactions. Clearly, for $P_{\text{guess}}$ to take value $1$, we must have
perfect updates for every interaction (in the support of $P(y)$).
It is worthwhile emphasising here that, in the state case, the states
$\sigma_{x}$ can be considered as combs with trivial input spaces, ie.
$\sigma_{x}\in\operatorname{Comb}(\mathbb{C}\rightarrow Y)$, and
consequentially, we are guaranteed that
$\sum_{y}\max_{x}P(x|y)P(y)\ket{y}\\!\\!\bra{y}$ is an unnormalised state. For
the general case where the $\sigma_{x}$ are classical combs with non-trivial
input spaces, we have no analogous guarantee as we will now discuss.
In the main text, we used the conditional Bayes’ rule and an independence
condition to write the classical-classical comb $D$ as
$\displaystyle
D=\sum_{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})\ket{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\bra{x,\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}$
(107)
Similarly to above, we can obtain the inequality:
$\displaystyle D\leq
I_{X}\otimes\sum_{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\max_{x}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})\ket{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\!\\!\bra{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}$
(108)
However, unlike above, we are not guaranteed that the second tensor factor is
an unnormalised classical comb:
$\sum_{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\max_{x}P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})P(\boldsymbol{a}^{\operatorname{out}}|\boldsymbol{a}^{\operatorname{in}})\ket{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}\\!\\!\bra{\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}}}$
may fail the required marginalisation conditions due to the dependence on the
inputs $\boldsymbol{a}^{\operatorname{in}}$ that persists in the distribution
$P(x|\boldsymbol{a}^{\operatorname{in}},\boldsymbol{a}^{\operatorname{out}})$
in the maximum. The trace of this operator (appropriately normalised by the
dimension of the input spaces) still provides a lower bound for the guessing
probability, just as for the state case above, but the assurance that this
bound can be reached is lacking.
To obtain an upper bound, we can construct an operator that removes the
dependence on the inputs by also maximising over the
$\boldsymbol{a}^{\operatorname{in}}$, which then trivially satisfies the
required marginalisation conditions. The proof of these bounds are given more
formally in the following proof of 3.2. Note that, for notational simplicity,
we have replaced $\boldsymbol{a}^{\operatorname{in}}$ by $\boldsymbol{a}$ and
$\boldsymbol{a}^{\operatorname{out}}$ by $\boldsymbol{b}$ in the following.
Also note that the proof makes use of lemmas that are stated and proved after
the current proof.
###### Proof of 3.2.
The lower bound is established by showing that any positive semi-definite
operator $\Gamma$ on
$\bigotimes_{i=1}^{n}\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{B_{i}}$ that
satisfies $I_{X}\otimes\Gamma\geq D$ must have trace greater than or equal to
$\sum_{\boldsymbol{a},\boldsymbol{b}}\max_{x}P(x|\boldsymbol{a},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{a})$.
Let $\Gamma$ be any positive semi-definite operator on
$\bigotimes_{i=1}^{n}\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{B_{i}}$ such that
$I_{X}\otimes\Gamma\geq D$. By B.1, it follows that
$\operatorname{diag}(I_{X}\otimes\Gamma)=I_{X}\otimes\operatorname{diag}(\Gamma)\geq
D$. We write
$\displaystyle
I_{X}\otimes\operatorname{diag}(\Gamma)=\sum_{x,\boldsymbol{a},\boldsymbol{b}}\alpha_{\boldsymbol{a}\boldsymbol{b}}\ket{x\boldsymbol{a}\boldsymbol{b}}\\!\\!\bra{x\boldsymbol{a}\boldsymbol{b}}$
where the $\alpha_{\boldsymbol{a}\boldsymbol{b}}$ are all real and non-
negative by positive semi-definiteness of $\Gamma$. The condition
$I_{X}\otimes\operatorname{diag}(\Gamma)\geq D$ induces a further condition on
the $\alpha$ terms: we must have for all $\boldsymbol{a},\boldsymbol{b}$ that,
for all $x$,
$\displaystyle\alpha_{\boldsymbol{a}\boldsymbol{b}}\geq
P(x|\boldsymbol{a},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{a})$
which in particular enforces that
$\displaystyle\alpha_{\boldsymbol{a}\boldsymbol{b}}\geq\max_{x}P(x|\boldsymbol{a},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{a})$
and it thus follows that
$\displaystyle\operatorname{Tr}[\Gamma]\geq\sum_{\boldsymbol{a},\boldsymbol{b}}\max_{x}P(x|\boldsymbol{a},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{a}).$
The upper bound is established by showing that
$\displaystyle\Upsilon:=\sum_{\boldsymbol{a},\boldsymbol{b}}\max_{x,\boldsymbol{\tilde{a}}}P(x|\boldsymbol{\tilde{a}},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{\tilde{a}})\ket{\boldsymbol{a}\boldsymbol{b}}\\!\\!\bra{\boldsymbol{a}\boldsymbol{b}}$
(109)
is a valid unnormalised comb, since $I_{X}\otimes\Upsilon\geq D$ clearly
holds. Defining $f$ via
$f(\boldsymbol{b},\boldsymbol{a})=\max_{x,\boldsymbol{\tilde{a}}}P(x|\boldsymbol{\tilde{a}},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{\tilde{a}})$,
non-negativity is immediate. Moreover, due to the maximum over
$\boldsymbol{\tilde{a}}$, $f$ has no dependence on $\boldsymbol{a}$:
$f(\boldsymbol{b},\boldsymbol{a})=f(\boldsymbol{b},\boldsymbol{a^{\prime}})$
for all $\boldsymbol{a},\boldsymbol{a^{\prime}}$. By defining the functions
$f^{(k)}$ via
$\displaystyle
f^{(k)}(b_{1},...,b_{k},a_{1},...,a_{k}):=\sum_{b_{n},...,b_{k+1}}f(\boldsymbol{b},\boldsymbol{a})$
(110)
for all $k=1,...,n$, the required conditions (non-negativity and the
marginalisation conditions - recall Equation 2) are trivially satisfied due to
the non-negativity and independence from $\boldsymbol{a}$ exhibited by $f$.
The sum over $b_{1}$ of $f^{(1)}$ so defined is also clearly positive since
$\max_{x,\boldsymbol{a}}P(x|\boldsymbol{a},\boldsymbol{b})P(\boldsymbol{b}|\boldsymbol{a})$
must be non-zero for some $\boldsymbol{b}$, and so $\Upsilon$ is indeed an
unnormalised classical comb. ∎
The following two lemmas establish that, for any classical comb $D$, we need
only (un)normalised classical combs $\Gamma$ in the minimum formulations of
the min-entropy since if some non-classical comb achieves the minimum, then
there exists a related classical comb that does also.
###### Lemma B.1.
Let $A$ and $B$ be operators on $\mathcal{H}$, where $A$ is diagonal in a
specific basis and $B$ is an arbitrary positive semi-definite operator. If
$B-A\geq 0$ then $\operatorname{diag}(B)-A\geq 0$.
###### Proof.
Let $\mathcal{E}:\mathcal{L}(\mathcal{H})\rightarrow\mathcal{L}(\mathcal{H})$
be a decohering channel in the basis for which $A$ is diagonal. Since this is
in particular a positive map and $B-A$ is positive semi-definite by
assumption, it follows that $\mathcal{E}(B-A)$ is also positive semi-definite
Linearity of $\mathcal{E}$ ensures that
$\mathcal{E}(B-A)=\mathcal{E}(B)-\mathcal{E}(A)=\operatorname{diag}(B)-A$
giving the result. ∎
###### Lemma B.2.
If $\Gamma$ is a normalised (resp. unnormalised) quantum comb, then
$\operatorname{diag}(\Gamma)$ is a normalised (resp. unnormalised) classical
comb.
###### Proof.
The proof is essentially immediate by definition but the details are spelt out
nonetheless. Let $\Gamma$ be a normalised (resp. unnormalised) quantum comb in
$\operatorname{Comb}(A_{1}\rightarrow B_{1},...,A_{n}\rightarrow B_{n})$ and
let $C^{(k)}$, $k=0,...,n$ be positive semi-definite operators that satisfy
$C^{(n)}=\Gamma$,
$\displaystyle\operatorname{Tr}_{B_{k}}\left[C^{(k)}\right]=I_{A_{k}}\otimes
C^{(k-1)}$ (111)
for $k=1,...,n$ and $C^{(0)}=1$ (resp. $C^{(0)}>0$). Denoting the
$\boldsymbol{a}\boldsymbol{b}^{\text{th}}$ diagonal element of $\Gamma$ by
$\alpha_{\boldsymbol{a}\boldsymbol{b}}$, we define $f$ via
$f(\boldsymbol{b},\boldsymbol{a})=\alpha_{\boldsymbol{a}\boldsymbol{b}}$.
Thus, we can write
$\displaystyle\operatorname{diag}(\Gamma)=\sum_{\boldsymbol{a},\boldsymbol{b}}f(\boldsymbol{b},\boldsymbol{a})\ket{\boldsymbol{a}\boldsymbol{b}}\\!\\!\bra{\boldsymbol{a}\boldsymbol{b}}.$
(112)
Since $\Gamma$ is positive semi-definite every
$\alpha_{\boldsymbol{a}\boldsymbol{b}}$, and hence every
$f(\boldsymbol{b},\boldsymbol{a})$, is non-negative. By denoting the diagonal
elements of $C^{(k)}$ similarly as
$\alpha_{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k}}$, where
$\boldsymbol{a}_{1:k}:=a_{1}a_{2}...a_{k}$ and similarly for
$\boldsymbol{b}_{1:k}$, we define
$f^{(k)}(\boldsymbol{b}_{1:k},\boldsymbol{a}_{1:k}):=\alpha_{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k}}$
for all $k=1,...,n$ and take $f^{(0)}:=C^{(0)}$. Since taking the partial
trace and applying $\operatorname{diag}(\cdot)$ commute, that is,
$\displaystyle\operatorname{diag}\left(\operatorname{Tr}_{B_{k}}\left[C^{(k)}\right]\right)=\operatorname{Tr}_{B_{k}}\left[\operatorname{diag}(C^{(k)})\right]$
(113)
it follows that
$\displaystyle\operatorname{Tr}_{B_{k}}\left[\operatorname{diag}(C^{(k)})\right]=\operatorname{diag}(I_{A_{k}}\otimes
C^{(k-1)})=I_{A_{k}}\otimes\operatorname{diag}(C^{(k-1)}).$ (114)
The left-hand side can be written as
$\displaystyle\sum_{\boldsymbol{a}_{1:k},\boldsymbol{b}_{1:k-1}}\left(\sum_{b_{k}}\alpha_{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k}}\right)\ket{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}\\!\\!\bra{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}=\sum_{\boldsymbol{a}_{1:k},\boldsymbol{b}_{1:k-1}}\left(\sum_{b_{k}}f^{(k)}(\boldsymbol{b}_{1:k},\boldsymbol{a}_{1:k})\right)\ket{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}\\!\\!\bra{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}$
(115)
and the right-hand side as
$\displaystyle\sum_{\boldsymbol{a}_{1:k},\boldsymbol{b}_{1:k-1}}\alpha_{\boldsymbol{a}_{1:k-1}\boldsymbol{b}_{1:k-1}}\ket{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}\\!\\!\bra{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}=\sum_{\boldsymbol{a}_{1:k},\boldsymbol{b}_{1:k-1}}f^{(k-1)}(\boldsymbol{b}_{1:k-1},\boldsymbol{a}_{1:k-1})\ket{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}\\!\\!\bra{\boldsymbol{a}_{1:k}\boldsymbol{b}_{1:k-1}}$
(116)
which establishes the required marginalisation condition on the $f^{(k)}$:
$\sum_{b_{k}}f^{(k)}(\boldsymbol{b}_{1:k},\boldsymbol{a}_{1:k})=f^{(k-1)}(\boldsymbol{b}_{1:k-1},\boldsymbol{a}_{1:k-1})$
for $k=2,...,n$ as well as the edge case of
$\sum_{b_{1}}f^{(1)}(b_{1},a_{1})=f^{(0)}$. ∎
The following is the proof of 3.1 which states that the min-entropy for multi-
round combs, i.e. of the form $D^{(m)}$, is non-increasing as the number of
rounds $m$ increases.
###### Proof of 3.1.
Let $\lambda\in\mathbb{R}$ and
$\Gamma\in\operatorname{Comb}\left((A_{1}^{\operatorname{in}})^{(1)}\rightarrow(A_{1}^{\operatorname{out}})^{(1)},...,(A_{n}^{\operatorname{in}})^{(m)}\rightarrow(A_{n}^{\operatorname{out}})^{(m)}\right)$
be such that $I_{X}\otimes\lambda\Gamma\geq D^{(m)}$ and that $-\log(\lambda)$
gives the min-entropy for $D^{(m)}$. Let
$\Gamma^{\prime}\in\operatorname{Comb}\left((A_{1}^{\operatorname{in}})^{(1)}\rightarrow(A_{1}^{\operatorname{out}})^{(1)},...,(A_{n}^{\operatorname{in}})^{(l)}\rightarrow(A_{n}^{\operatorname{out}})^{(l)}\right)$
be given by
$\displaystyle\Gamma^{\prime}:=\frac{1}{\prod_{t=l+1}^{m}\dim(\boldsymbol{A}^{\operatorname{in}})^{(t)}}\operatorname{Tr}_{\boldsymbol{A}^{(l+1)}...\boldsymbol{A}^{(m)}}\left[\Gamma\right]$
(117)
where
$\dim(\boldsymbol{A}^{\operatorname{in}})^{(t)}:=\prod_{k=1}^{n}\dim(A_{k}^{\operatorname{in}})^{(t)}$
and the subscripts $\boldsymbol{A}^{(j)}$ in the trace indicate that the trace
is over all subspace related to $\sigma_{x}^{(j)}$, i.e.
$(\mathcal{H}_{A_{1}^{\operatorname{in}}})^{(j)},(\mathcal{H}_{A_{1}^{\operatorname{out}}})^{(j)},...,(\mathcal{H}_{A_{n}^{\operatorname{in}}})^{(j)},(\mathcal{H}_{A_{n}^{\operatorname{out}}})^{(j)}$.
It remains only to show that $I_{X}\otimes\lambda\Gamma^{\prime}\geq D^{(l)}$.
It suffices to show that
$\displaystyle\frac{1}{\prod_{t=l+1}^{m}\dim(\boldsymbol{A}^{\operatorname{in}})^{(t)}}\operatorname{Tr}_{\boldsymbol{A}^{(l+1)}...\boldsymbol{A}^{(m)}}\left[D^{(m)}\right]=D^{(l)}.$
(118)
Starting from the left-hand side:
$\displaystyle\frac{1}{\prod_{t=l+1}^{m}\dim(\boldsymbol{A}^{\operatorname{in}})^{(t)}}\operatorname{Tr}_{\boldsymbol{A}^{(l+1)}...\boldsymbol{A}^{(m)}}\left[D^{(m)}\right]$
$\displaystyle=\frac{1}{\prod_{t=l+1}^{m}\dim(\boldsymbol{A}^{\operatorname{in}})^{(t)}}\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\operatorname{Tr}_{\boldsymbol{A}^{(l+1)}...\boldsymbol{A}^{(m)}}\left[\bigotimes_{j=1}^{m}\sigma_{x}^{(j)}\right]$
(119)
$\displaystyle=\frac{1}{\prod_{t=l+1}^{m}\dim(\boldsymbol{A}^{\operatorname{in}})^{(t)}}\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\left(\prod_{k=l+1}^{m}\dim(\boldsymbol{A}^{\operatorname{in}})^{(k)}\right)\bigotimes_{j=1}^{l}\sigma_{x}^{(j)}$
(120) $\displaystyle=D^{(l)}$ (121)
where we have used the fact that each $\sigma_{x}^{(j)}$ is a comb, so
$\operatorname{Tr}[\sigma_{x}^{(j)}]$ is equal to the product of input space
dimensions by definition. So, it holds that
$I_{X}\otimes\lambda\Gamma^{\prime}\geq D^{(l)}$, which entails that
$\displaystyle
H_{\min}(X|\boldsymbol{A}^{(1)},...,\boldsymbol{A}^{(l)})_{D^{(l)}}\geq-\log(\lambda)=H_{\min}(X|\boldsymbol{A}^{(1)},...,\boldsymbol{A}^{(m)})$
(122)
thus proving that the min-entropy is non-increasing with increasing round
number. ∎
## Appendix C Blind Quantum Computing Protocol Supplementary
This appendix contains supporting results for the blindness theorems of
Section 4 and for the accompanying examples.
### C.1 Single Round Theorem Supporting Results
The proof of 4.2 makes use of the following lemma which demonstrates that for
$\boldsymbol{\alpha^{\prime}},\boldsymbol{\alpha},\boldsymbol{c^{\prime}}$ and
$g$ fixed, there is at most one $\boldsymbol{r}$ such that Equation 34 is
satisfied for each $i$.
###### Lemma C.1.
If there exists an $\boldsymbol{r}$ such that
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)=1$
for all other variables fixed, then it is unique, otherwise
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)=0$.
###### Proof.
Since
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)$
is a deterministic distribution, it takes values either $0$ or $1$. Let
$\boldsymbol{r}$ be such that
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)=1$.
Recalling Equation 34 and the fact that a total order is imposed on
communication, we have that for each $i$
$\displaystyle\alpha^{\prime}_{i}:=(-1)^{\bigoplus_{j\in\mathcal{X}_{i}}c^{\prime}_{j}\oplus
r_{j}}\alpha_{i}+\left(r_{i}\oplus\bigoplus_{j\in\mathcal{Z}_{i}}c^{\prime}_{j}\oplus
r_{j}\right)\pi\bmod 2\pi$ (123)
where $\mathcal{X}_{i}$ and $\mathcal{Z}_{i}$ are necessarily subsets of
$\\{1,...,i-1\\}$. In particular, this means that
$\displaystyle\alpha^{\prime}_{1}=\alpha_{1}+r_{1}\pi\bmod 2\pi$ (124)
and hence there is a unique value for $r_{1}$ given $\boldsymbol{\alpha}$ and
$\boldsymbol{\alpha^{\prime}}$ are fixed. The result then follows by
induction: since $\boldsymbol{c^{\prime}}$ and $g$ are fixed, for fixed values
$\boldsymbol{r}_{1:i-1}$ the quantities
$\bigoplus_{j\in\mathcal{X}_{i}}c^{\prime}_{j}\oplus r_{j}$ and
$\bigoplus_{j\in\mathcal{Z}_{i}}c^{\prime}_{j}\oplus r_{j}$ are determined for
each $i$, and thus there is a unique $r_{i}$ for which Equation 123 holds. ∎
We can now give the proof of the theorem.
###### Proof of 4.2.
From 3.2, we know that
$\displaystyle
H_{\min}(\boldsymbol{A},\boldsymbol{O}|\boldsymbol{A^{\prime}},\boldsymbol{C^{\prime}})_{D_{\operatorname{client}}}\geq-\log\left[\sum_{\boldsymbol{\alpha^{\prime}}}\max_{\boldsymbol{\alpha},O,\boldsymbol{c^{\prime}}}P(\boldsymbol{\alpha},O|\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}})P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}})\right].$
(125)
Instead of dealing directly with
$P(\boldsymbol{\alpha},O|\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}})P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}})$,
it is easier to revert back to the form before the Bayes’ rule was applied.
That is, we aim to find
$\displaystyle\sum_{\boldsymbol{\alpha^{\prime}}}\max_{\boldsymbol{\alpha},O,\boldsymbol{c^{\prime}}}P(\boldsymbol{\alpha},O)P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O).$
(126)
Using the uniformity assumptions for choosing the computation, gflow and one-
time pads, as well as the definition of
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)$
in the previous subsection, we have
$\displaystyle
P(\boldsymbol{\alpha},O)P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)=\frac{1}{|\mathcal{A}|^{n}|\mathcal{O}|}\sum_{g\sim
O,\boldsymbol{r}}\frac{P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)}{|g\sim
O|2^{n}}$ (127)
where $g\sim O$ indicates that the gflow is defined for the output set $O$ and
$|g\sim O|$ denotes the number of all such gflows. Recalling that
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)$
is a deterministic distribution, it can be shown (C.1) that for fixed $g$,
$\boldsymbol{\alpha}$ and $\boldsymbol{c^{\prime}}$ there is at most one
$\boldsymbol{r}$ for which
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)=1$.
Thus,
$\displaystyle\max_{\boldsymbol{\alpha},O,\boldsymbol{c^{\prime}}}\sum_{g\sim
O,\boldsymbol{r}}\frac{P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)}{|g\sim
O|2^{n}}$ (128)
can be interpreted as selecting the $\boldsymbol{\alpha}$, $O$ and
$\boldsymbol{c^{\prime}}$ for which $\boldsymbol{\alpha^{\prime}}$ is
reportable from $\boldsymbol{\alpha}$ for the greatest number of pairs
$(g,\boldsymbol{r})$ for gflows $g\sim O$. This quantity is clearly upper-
bounded by a situation where $\boldsymbol{\alpha^{\prime}}$ is reportable
under all gflows, hence
$\displaystyle\max_{\boldsymbol{\alpha},O,\boldsymbol{c^{\prime}}}\sum_{g\sim
O,\boldsymbol{r}}\frac{P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)}{|g\sim
O|2^{n}}$ $\displaystyle\leq\frac{1}{2^{n}}$ (129)
which holds for all $\boldsymbol{\alpha^{\prime}}$. Thus, returning to
Equation 126:
$\displaystyle\sum_{\boldsymbol{\alpha^{\prime}}}\max_{\boldsymbol{\alpha},O,\boldsymbol{c^{\prime}}}P(\boldsymbol{\alpha},O)P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},O)$
$\displaystyle\leq\sum_{\boldsymbol{\alpha^{\prime}}}\frac{1}{|\mathcal{A}|^{n}|\mathcal{O}|2^{n}}$
(130) $\displaystyle=\frac{1}{|\mathcal{O}|2^{n}}$ (131)
which proves the theorem. ∎
It is worthwhile making some further comments regarding the interpretation of
Equation 128 and the related inequality Equation 129 in the proof above.
Firstly, it is possible to find a simple example, namely that given in the
main text (see also below), for which Equation 129 is equality for every
$\boldsymbol{\alpha^{\prime}}$. That is, for every
$\boldsymbol{\alpha^{\prime}}$, there exists an $\boldsymbol{\alpha}$, $O$ and
$\boldsymbol{c^{\prime}}$ that $\boldsymbol{\alpha^{\prime}}$ is reportable
from $\boldsymbol{\alpha}$ for every gflow $g\sim O$. Said another way, the
pre-images of $\boldsymbol{\alpha^{\prime}}$ under the gflows $g\sim O$ (for
some fixed $\boldsymbol{c^{\prime}}$ and as $\boldsymbol{r}$ varies) have non-
empty mutual intersection. Since the pre-image of each gflow has a fixed size
(this follows from C.1), a larger mutual intersection corresponds to a smaller
total set of angles $\boldsymbol{\alpha}$ from which
$\boldsymbol{\alpha^{\prime}}$ can be reported. Since $\boldsymbol{\alpha}$ is
one part of the secret information, it is intuitive that a smaller set of
possible $\boldsymbol{\alpha}$ given the evidence
$\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}}$ corresponds to a lower
min-entropy. This also suggests that, to improve security, considering a graph
where the corresponding gflows have smaller mutual intersection is beneficial.
Obtaining equality in Equation 129 for all
$\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}}$ accordingly means that
equality is also achieved in Equation 46, however, it is unlikely that this
bound is obtained in general. The above discussion highlights how much is
dependent on the specific properties of gflows for the graph chosen for the
protocol. Since (to the best knowledge of the authors’) there is no
characterisation of gflows in terms of graph-theoretic properties, and since
the given example achieves the lower bound, it is unlikely that a better lower
bound that that given in the theorem can be given in general.
### C.2 BQC Minimum Example
This appendix provides further details regarding the minimal example given in
the main text (recall Figure 4) which obtains the lower bound in 4.2. We begin
by demonstrating that $O=\\{2,3\\}$ is indeed the only choice of output set
for which gflows compatible with the total order exist.
###### Lemma C.2.
For $G$ as in Figure 4 with partial order $1<2<3$ and the $XY$-plane the only
allowed measurement plane, $O=\\{2,3\\}$ is the only non-trivial output set
for which there exist $I\subset\\{1,2,3\\}$ such that a gflow compatible with
the total order exists. Moreover, there are just two possible gflows, $g_{1}$
and $g_{2}$, defined by
$\displaystyle g_{1}:1$ $\displaystyle\mapsto\\{2\\},$ (132) $\displaystyle
g_{2}:1$ $\displaystyle\mapsto\\{3\\},$ (133)
where $g_{1}$ is compatible with $(I,O)$ for $I$ equal to $\\{1\\}$, $\\{3\\}$
or $\\{1,3\\}$ and $g_{2}$ is compatible with $(I,O)$ for $I=\\{1\\}$,
$\\{2\\}$ and $\\{1,2\\}$, for $O$ as above.
###### Proof.
We begin by demonstrating $O=\\{2,3\\}$ is the only valid non-trivial output
set (note that we do not consider the trivial output set $O=\\{1,2,3\\}$ since
this does not allow for any computation). Due to the total order and
measurement plane restriction, $3$ must be in the output set, since if this
was not the case, then $3\in O^{c}$ meaning that any gflow would map $3$ to
$\\{1\\}$, $\\{2\\}$ or $\\{1,2\\}$. In any of these cases, we would require
that $3<1$ or $3<2$ which contradicts the compatibility with total order. Thus
$\\{1\\},\\{2\\}$ and $\\{1,2\\}$ cannot be output sets.
Suppose either $\\{3\\}$ or $\\{1,3\\}$ was a valid output set. Any gflow
compatible with either output must then map $2$ to some subset of
$\\{1,2,3\\}$, however no such set exists for which some contradiction does
not arise, as follows. If $2$ maps to $\\{1\\}$, $2<1$ which is a
contradiction to the total order. If $2$ maps to $\\{2\\}$, $\\{1,2\\}$ or
$\\{2,3\\}$, the gflow requirement $2\notin g(2)$ is contradicted. If $2$ maps
to $\\{3\\}$, $1\in\operatorname{Odd}(g(2))$ implying $2<1$, contradicting the
total order. If $2$ maps to $\\{1,3\\}$, then
$2\notin\operatorname{Odd}(g(2))$ which contradicts a gflow requirement.
This leaves $\\{2,3\\}$ as the only remaining possible output set. We now show
it does indeed support gflow and characterise them. For output set
$O=\\{2,3\\}$, the domain of any gflow map is $O^{c}=\\{1\\}$, so we can begin
to characterise valid gflows by where they map $1$. Due to the $XY$-plane
restriction, we can’t have $1\in g(1)$, so a valid gflow cannot map $1$ to
$\\{1\\},\\{1,2\\}$ or $\\{1,3\\}$. Since
$\operatorname{Odd}(\\{2,3\\})=\\{2,3\\}$ and since we require
$1\in\operatorname{Odd}(g(1))$, no valid gflow maps $1$ to $\\{2,3\\}$. This
leaves just the options $1\mapsto\\{2\\}$ and $1\mapsto\\{3\\}$. Both of these
are valid gflows since in both cases, $1\notin g(1)$ is satisfied and the
corresponding implications for the partial order, $1<2$ and $1<3$
respectively, are compatible with the total order. Similarly, for both maps
$1\in\operatorname{Odd}(g(1))$ is satisfied
($\operatorname{Odd}(\\{2\\})=\\{1,3\\}$ and
$\operatorname{Odd}(\\{3\\})=\\{1,2\\}$) and the again the implications for
the partial order are compatible with the total order.
Denote by $g_{1}$ the gflow that maps $1$ to $\\{2\\}$ and by $g_{2}$ the
gflow that maps $1$ to $\\{3\\}$. Since a gflow is a map from $O^{c}$ to
$\mathcal{P}(I^{c})$, $g_{1}$ is compatible with all sets $I$ for which
$\\{2\\}\in\mathcal{P}(I^{c})$, i.e. $I=\\{1\\},\\{3\\}$ and $\\{1,3\\}$, and
similarly, $g_{2}$ is compatible with $I=\\{1\\},\\{2\\}$ and $\\{1,2\\}$. ∎
Returning to the discussion of the BQC protocol for the example, it is useful
to explicitly write out the correction sets given by $g_{1}$ and $g_{2}$ (only
those that are non-empty are shown):
$\displaystyle\mathcal{X}_{2}^{g_{1}}$
$\displaystyle=\mathcal{Z}_{3}^{g_{1}}=\\{1\\};$
$\displaystyle\mathcal{Z}_{2}^{g_{2}}$
$\displaystyle=\mathcal{X}_{3}^{g_{2}}=\\{1\\}.$
Let $\mathcal{A}$ be any agreed upon set of angles that satisfies Equation 33.
For a single round of the protocol, if $g_{1}$ is chosen by the client, then
the reported angles are given by
$\displaystyle\alpha^{\prime}_{1}$ $\displaystyle=\alpha_{1}+r_{1}\pi\bmod
2\pi,$ (134) $\displaystyle\alpha^{\prime}_{2}$
$\displaystyle=(-1)^{r_{1}\oplus c^{\prime}_{1}}\alpha_{2}+r_{2}\pi\bmod
2\pi,$ (135) $\displaystyle\alpha^{\prime}_{3}$
$\displaystyle=\alpha_{3}+(r_{3}\oplus r_{1}\oplus c^{\prime}_{1})\pi\bmod
2\pi,$ (136)
where $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathcal{A}$ are the chosen (and
secret) angles for the computation, $c^{\prime}_{1}$ is the classical message
reported by the server after the first measurement and
$\boldsymbol{r}=r_{3}r_{2}r_{1}\in\\{0,1\\}^{3}$ is the one-time pad. If
$g_{2}$ is used instead, the angles are reported as
$\displaystyle\alpha^{\prime}_{1}$ $\displaystyle=\alpha_{1}+r_{1}\pi\bmod
2\pi,$ (137) $\displaystyle\alpha^{\prime}_{2}$
$\displaystyle=\alpha_{2}+(r_{2}\oplus r_{1}\oplus c^{\prime}_{1})\pi\bmod
2\pi,$ (138) $\displaystyle\alpha^{\prime}_{3}$
$\displaystyle=(-1)^{r_{1}\oplus c^{\prime}_{1}}\alpha_{3}+r_{3}\pi\bmod
2\pi.$ (139)
Note that the only classical message relevant to this example is
$c^{\prime}_{1}$. Let $\boldsymbol{\alpha^{\prime}}$ and
$\boldsymbol{c^{\prime}}$ be reported in a single round of the protocol. We
can write the pre-image of $\boldsymbol{\alpha^{\prime}}$ under $g_{1}$ given
$\boldsymbol{c^{\prime}}$ by inverting Equations 134, 135 and 136 (we drop the
$\bmod 2\pi$ notation and leave it implicit):
$\displaystyle\alpha_{1}$ $\displaystyle=\alpha^{\prime}_{1}+r_{1}\pi;$ (140)
$\displaystyle\alpha_{2}$ $\displaystyle=(-1)^{r_{1}\oplus
c^{\prime}_{1}}\alpha^{\prime}_{2}+r_{2}\pi;$ (141) $\displaystyle\alpha_{3}$
$\displaystyle=\alpha^{\prime}_{3}+(r_{3}\oplus r_{1}\oplus
c^{\prime}_{1})\pi.$ (142)
Restricting our focus to a subset of the pre-image defined by
$r_{1}=c^{\prime}_{1}$, that is, the angles
$\displaystyle\alpha_{1}$
$\displaystyle=\alpha^{\prime}_{1}+c^{\prime}_{1}\pi,$ (143)
$\displaystyle\alpha_{2}$ $\displaystyle=\alpha^{\prime}_{2}+r_{2}\pi,$ (144)
$\displaystyle\alpha_{3}$ $\displaystyle=\alpha^{\prime}_{3}+r_{3}\pi,$ (145)
one observes that $\boldsymbol{\alpha^{\prime}}$ can be reported from any one
of these angles under $g_{2}$ given $\boldsymbol{c^{\prime}}$, namely by the
one-time pads
$\boldsymbol{\hat{r}}=\hat{r}_{3}\hat{r}_{2}\hat{r}_{1}=r_{3}r_{2}c^{\prime}_{1}$
(as can be shown via simple substitution into Equations 137, 138 and 139).
Since this holds for any $\boldsymbol{\alpha^{\prime}}$ and
$\boldsymbol{c^{\prime}}$, we have thus shown both that the bound Equation 129
in the proof of 4.2 above is in fact equality for every
$\boldsymbol{\alpha^{\prime}}$ and moreover that the maximum is obtained for
every $\boldsymbol{c^{\prime}}$. Recalling the discussion after 3.2, this
means that we have found the guessing probability (equivalently min-entropy)
exactly for this example and that any strategy used by the server is equally
informative. These results are corroborated numerically (see Appendix E for
details of the implementation using an SDP solver): the min-entropy is given
as $-\log_{2}(0.125)=3$, which is consistent with $|\mathcal{O}|=1$ and $n=3$
for this example (this result was obtained for different choices of
$\mathcal{A}$: for $|\mathcal{A}|=4$ and for $|\mathcal{A}|=8$).
We now consider two rounds of the protocol in order to calculate the guessing
probability for $D_{\operatorname{client}}^{(2)}$, which, for this example,
can be written as
$\displaystyle D_{\operatorname{client}}^{(2)}$
$\displaystyle=\sum_{\boldsymbol{\alpha},O}\frac{1}{|\mathcal{A}|^{n}|\mathcal{O}|}\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\otimes\sigma_{\boldsymbol{\alpha},O}^{(1)}\otimes\sigma_{\boldsymbol{\alpha},O}^{(2)}$
(146)
$\displaystyle=\sum_{\boldsymbol{\alpha}}\frac{1}{|\mathcal{A}|^{3}}\ket{\boldsymbol{\alpha}}\\!\\!\bra{\boldsymbol{\alpha}}\otimes\sum_{\begin{subarray}{c}\boldsymbol{\alpha^{\prime}}^{(1)},\boldsymbol{\alpha^{\prime}}^{(2)},\\\
\boldsymbol{c^{\prime}}^{(1)},\boldsymbol{c^{\prime}}^{(2)}\end{subarray}}\sum_{\begin{subarray}{c}g^{(1)},g^{(2)},\\\
\boldsymbol{r}^{(1)},\boldsymbol{r}^{(2)}\end{subarray}}\frac{1}{2^{8}}P(\boldsymbol{\alpha^{\prime}}^{(1)}|\boldsymbol{c^{\prime}}^{(1)},\boldsymbol{\alpha},\boldsymbol{r}^{(1)},g^{(1)})P(\boldsymbol{\alpha^{\prime}}^{(2)}|\boldsymbol{c^{\prime}}^{(2)},\boldsymbol{\alpha},\boldsymbol{r}^{(2)},g^{(2)})\ket{\boldsymbol{\alpha^{\prime}}^{(1)}\boldsymbol{c^{\prime}}^{(1)}\boldsymbol{\alpha^{\prime}}^{(2)}\boldsymbol{c^{\prime}}^{(2)}}\\!\\!\bra{..}$
(147)
where $g^{(j)}$ take values in $\\{g_{1},g_{2}\\}$ where $g_{1}$ and $g_{2}$
are as above, and where we have used $P(g^{(j)})=\frac{1}{2}$ and
$P(\boldsymbol{r}^{(j)})=\frac{1}{2^{3}}$. For a choice of angle set
$\mathcal{A}=\\{\frac{\pi}{5},\frac{\pi}{3},\frac{-\pi}{3}+\pi,\frac{-\pi}{5}+\pi,\frac{\pi}{5}+\pi,\frac{\pi}{3}+\pi,\frac{-\pi}{3},\frac{-\pi}{5}\\}$,
the guessing probability for $D_{\operatorname{client}}^{(2)}$ takes value
between $0.140625$ and $0.28125$ as computed via numeric methods (see [72])
and as explained in the following. Note that various other choices of angle
set give the same or similar results.
These bounds are derived directly from 3.2 for the specific comb under
consideration. The lower bound (which includes the normalisation over the
input spaces) can be written as
$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{\alpha^{\prime}}^{(1)},\boldsymbol{\alpha^{\prime}}^{(2)},\\\
\boldsymbol{c^{\prime}}^{(1)},\boldsymbol{c^{\prime}}^{(2)}\end{subarray}}\frac{1}{|\mathcal{A}|^{3}2^{6}}\max_{\boldsymbol{\alpha}}\sum_{\begin{subarray}{c}g^{(1)},g^{(2)},\\\
\boldsymbol{r}^{(1)},\boldsymbol{r}^{(2)}\end{subarray}}\frac{1}{2^{8}}P(\boldsymbol{\alpha^{\prime}}^{(1)}|\boldsymbol{c^{\prime}}^{(1)},\boldsymbol{\alpha},\boldsymbol{r}^{(1)},g^{(1)})P(\boldsymbol{\alpha^{\prime}}^{(2)}|\boldsymbol{c^{\prime}}^{(2)},\boldsymbol{\alpha},\boldsymbol{r}^{(2)},g^{(2)}).$
(148)
Due to the redundancy of $(c^{\prime}_{2})^{(j)}$ and $(c^{\prime}_{3})^{(j)}$
(they don’t appear in Equations 134, 135, 136, 137, 138 and 139), this reduces
to
$\displaystyle\sum_{\begin{subarray}{c}\boldsymbol{\alpha^{\prime}}^{(1)},\boldsymbol{\alpha^{\prime}}^{(2)},\\\
(c^{\prime}_{1})^{(1)},(c^{\prime}_{1})^{(2)}\end{subarray}}\frac{1}{|\mathcal{A}|^{3}2^{2}}$
$\displaystyle\max_{\boldsymbol{\alpha}}\sum_{\begin{subarray}{c}g^{(1)},g^{(2)},\\\
\boldsymbol{r}^{(1)},\boldsymbol{r}^{(2)}\end{subarray}}\frac{1}{2^{8}}P(\boldsymbol{\alpha^{\prime}}^{(1)}|(c^{\prime}_{1})^{(1)},\boldsymbol{\alpha},\boldsymbol{r}^{(1)},g^{(1)})P(\boldsymbol{\alpha^{\prime}}^{(2)}|(c^{\prime}_{1})^{(2)},\boldsymbol{\alpha},\boldsymbol{r}^{(2)},g^{(2)}).$
(149)
The upper bound is given by
$\displaystyle\sum_{\boldsymbol{\alpha^{\prime}}^{(1)},\boldsymbol{\alpha^{\prime}}^{(2)}}\frac{1}{|\mathcal{A}|^{3}}$
$\displaystyle\max_{\boldsymbol{\alpha},(c^{\prime}_{1})^{(1)},(c^{\prime}_{1})^{(2)}}\sum_{\begin{subarray}{c}g^{(1)},g^{(2)},\\\
\boldsymbol{r}^{(1)},\boldsymbol{r}^{(2)}\end{subarray}}\frac{1}{2^{8}}P(\boldsymbol{\alpha^{\prime}}^{(1)}|(c^{\prime}_{1})^{(1)},\boldsymbol{\alpha},\boldsymbol{r}^{(1)},g^{(1)})P(\boldsymbol{\alpha^{\prime}}^{(2)}|(c^{\prime}_{1})^{(1)},\boldsymbol{\alpha},\boldsymbol{r}^{(2)},g^{(2)}).$
(150)
Computing these quantities for $\mathcal{A}$ above via the code in the
repository [72] gives the results as listed above.
### C.3 Multi-round Blindness Supplementary
For this section, it is useful to have the following notation. Let a graph $G$
on vertices $V$ be given and $O\subset V$ a choice of output set. For a given
$\boldsymbol{\alpha}\in\mathcal{A}^{n}$, define the set
$\mathcal{A}_{\boldsymbol{\alpha},O}:=\\{\boldsymbol{\alpha}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi\bmod
2\pi:b_{o_{i}}\in\\{0,1\\}\\}$ where the zero entries of
$(0,...,0,b_{o_{1}},...,b_{o_{|O|}})$ correspond to $V\setminus O$ and the
labels $o_{i}$ correspond to the elements of $O$. We define an equivalence
relation $\sim_{O}$ by
$\boldsymbol{\alpha}\sim_{O}\boldsymbol{\widetilde{\alpha}}$ for
$\widetilde{\alpha}\in\mathcal{A}_{\boldsymbol{\alpha},O}$. The set obtained
by quotienting $\mathcal{A}^{n}$ by the equivalence relation, i.e.
$\mathcal{A}^{n}/\sim_{O}$ has $\frac{|\mathcal{A}|^{n}}{2^{|O|}}$ elements.
###### Lemma C.3.
Consider $\sum_{g\sim
O,\boldsymbol{r}}P(g|O)p(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha},\boldsymbol{r},g}$
as in Equation 37 with $P(g|O)$ and $P(\boldsymbol{r})$ both uniform. Then,
for all $\boldsymbol{\widetilde{\alpha}}\sim_{O}\boldsymbol{\alpha}$:
$\displaystyle\sum_{g\sim
O,\boldsymbol{r}}P(g|O)P(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha},\boldsymbol{r},g}=\sum_{g\sim
O,\boldsymbol{r}}P(g|O)P(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\widetilde{\alpha}},\boldsymbol{r},g}$
(151)
###### Proof.
We begin by noting that, by definition of gflow, for any $g\sim O$, no element
of $O$ is in the domain of $g$. This means that, for all $i\in O$,
$i\notin\mathcal{X}_{k}$ and $i\notin\mathcal{Z}_{k}$ for all $k\in V$. The
consequence of this being that for any $r_{i}$ for $i\in O$, the only equation
for $\alpha^{\prime}_{k}$ (recall Equation 34) that contains $r_{i}$ is when
$k=i$. This results in the following symmetry:
$\displaystyle\alpha^{\prime}_{i}=(-1)^{\bigoplus_{j\in\mathcal{X}_{i}}c^{\prime}_{j}\oplus
r_{j}}\alpha_{i}+(r_{i}\oplus\bigoplus_{j\in\mathcal{Z}_{i}}c^{\prime}_{j}\oplus
r_{j})\pi\bmod 2\pi=(-1)^{\bigoplus_{j\in\mathcal{X}_{i}}c^{\prime}_{j}\oplus
r_{j}}(\alpha_{i}+\pi)+((r_{i}\oplus
1)\oplus\bigoplus_{j\in\mathcal{Z}_{i}}c^{\prime}_{j}\oplus r_{j})\pi\bmod
2\pi$ (152)
Thus, for fixed gflow $g$ and classical messages $\boldsymbol{c^{\prime}}$, if
$\boldsymbol{\alpha^{\prime}}$ is reportable from $\boldsymbol{\alpha}$ for
one-time pads $\boldsymbol{r}$, i.e.
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)=1$,
then $\boldsymbol{\alpha^{\prime}}$ is reportable from
$\boldsymbol{\alpha}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi$ for one-time pads
$\boldsymbol{r}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi$, i.e.
$P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi,\boldsymbol{r}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi,g)=1$.
In light of C.1, and with the assumption that $P(\boldsymbol{r})$ is uniform,
this in particular means that
$\displaystyle\sum_{\boldsymbol{r}}P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)P(\boldsymbol{r})=\sum_{\boldsymbol{r}}P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}}.\boldsymbol{\alpha}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi,\boldsymbol{r},g)P(\boldsymbol{r})$
(153)
and so
$\displaystyle\sum_{g\sim
O,\boldsymbol{r}}P(g|O)P(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha},\boldsymbol{r},g}$
$\displaystyle=\sum_{\begin{subarray}{c}g\sim O,\boldsymbol{r},\\\
\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}}\end{subarray}}P(g|O)P(\boldsymbol{r})P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha},\boldsymbol{r},g)\ket{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}\\!\\!\bra{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}$
(154) $\displaystyle=\sum_{\begin{subarray}{c}g\sim O,\boldsymbol{r},\\\
\boldsymbol{\alpha^{\prime}},\boldsymbol{c^{\prime}}\end{subarray}}P(g|O)P(\boldsymbol{r})P(\boldsymbol{\alpha^{\prime}}|\boldsymbol{c^{\prime}},\boldsymbol{\alpha}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi,\boldsymbol{r},g)\ket{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}\\!\\!\bra{\boldsymbol{\alpha^{\prime}}\boldsymbol{c^{\prime}}}$
(155) $\displaystyle=\sum_{g\sim
O,\boldsymbol{r}}P(g|O)P(\boldsymbol{r})\sigma_{\operatorname{BQC}}^{\boldsymbol{\alpha}+(0,...,0,b_{o_{1}},...,b_{o_{|O|}})\pi,\boldsymbol{r},g}$
(156)
∎
In the notation of Equation 42, the above lemma states that
$\sigma_{\boldsymbol{\alpha},O}=\sigma_{\boldsymbol{\widetilde{\alpha}},O}$
for all $\widetilde{\alpha}\sim_{O}\boldsymbol{\alpha}$ which is the notation
used in the following proof.
###### Proof of 4.3.
For this proof, we work with the maximisation formulation for the guessing
probability, and moreover in the form given by maximising over pairs of dual
combs and CPTP maps $(\widehat{E},\mathcal{F})$ as given in Equation 14. That
is, we consider
$\displaystyle\max_{\widehat{E},\mathcal{F}}\operatorname{Tr}_{\boldsymbol{A},\boldsymbol{O},\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\left[(I_{\boldsymbol{A},\boldsymbol{O}}\otimes\mathcal{F})\left(D_{\operatorname{client}}^{(m)}\widehat{E}^{T}\right)\left(\sum_{\boldsymbol{\alpha},O}\ket{\boldsymbol{\alpha},O,\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O,\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O},\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\right)\right]$
(157)
where
$\widehat{E}\in\operatorname{Comb}(\mathbb{C}\rightarrow\mathbb{C},(A^{\prime}_{1})^{(1)}\rightarrow(C^{\prime}_{1})^{(1)},...,(A^{\prime}_{n})^{(m)}\rightarrow(C^{\prime}_{n})^{(m)},\mathbb{C}\rightarrow
B)$ is a normalised dual comb and
$\mathcal{F}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{\boldsymbol{A^{\prime}}}\otimes\mathcal{H}_{\boldsymbol{O^{\prime}}}\cong\mathcal{H}_{\boldsymbol{A}}\otimes\mathcal{H}_{\boldsymbol{O}}$
is a CPTP map. The transpose $T$ is over all spaces
$\boldsymbol{A^{\prime}}_{1},\boldsymbol{C^{\prime}}_{1},...,\boldsymbol{A^{\prime}}_{n},\boldsymbol{C^{\prime}}_{n}$.
Let $(\widehat{E},\mathcal{F})$ be any such pair. Writing
$D_{\operatorname{client}}^{(m)}$ as
$\sum_{\boldsymbol{\alpha},O}P(\boldsymbol{\alpha},O)\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}$,
we get
$\displaystyle\operatorname{Tr}_{\boldsymbol{A},\boldsymbol{O},\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}$
$\displaystyle\left[(I_{\boldsymbol{A},\boldsymbol{O}}\otimes\mathcal{F})\left(D_{\operatorname{client}}^{(m)}\widehat{E}^{T}\right)\left(\sum_{\boldsymbol{\alpha},O}\ket{\boldsymbol{\alpha},O,\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O,\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O},\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\right)\right]$
$\displaystyle=\sum_{\boldsymbol{\alpha},O}\bra{\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O}}\bra{\boldsymbol{\alpha},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\left[\sum_{\boldsymbol{\alpha},O}P(\boldsymbol{\alpha},O)\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\otimes\mathcal{F}\left(\left(\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}\right)\widehat{E}^{T}\right)\right]\ket{\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O}}\ket{\boldsymbol{\alpha},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}$
(158)
By the fact that $\widehat{E}$ is a (normalised) dual comb,
$\left(\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}\right)\widehat{E}^{T}$
is a (normalised) state in $\mathcal{H}_{B}$. Moreover, since
$\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}=\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\widetilde{\alpha}},O}^{(j)}$
for all $\boldsymbol{\widetilde{\alpha}}\sim_{O}\boldsymbol{\alpha}$ by C.3,
$\left(\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}\right)\widehat{E}^{T}$
is the same state for each
$\widetilde{\alpha}\in\mathcal{A}_{\boldsymbol{\alpha},O}$, and hence so is
$\mathcal{F}\left(\left(\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}\right)\widehat{E}^{T}\right)$.
For simplicity, let us replace the notation
$\mathcal{F}\left(\left(\bigotimes_{j=1}^{m}\sigma_{\boldsymbol{\alpha},O}^{(j)}\right)\widehat{E}^{T}\right)$
by $\rho_{\sim_{O}\boldsymbol{\alpha}}$. With this notation and invoking the
assumption on $P(\boldsymbol{\alpha},O)$, Equation 158 becomes
$\displaystyle\sum_{\boldsymbol{\alpha},O}\bra{\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O}}\bra{\boldsymbol{\alpha},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\left(\sum_{\boldsymbol{\alpha},O}\frac{P(O)}{|\mathcal{A}|^{n}}\ket{\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O}\otimes\rho_{\sim_{O}\boldsymbol{\alpha}}\right)$
$\displaystyle\ket{\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O}}\ket{\boldsymbol{\alpha},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}$
$\displaystyle=\frac{1}{|\mathcal{A}|^{n}}\sum_{\boldsymbol{\alpha},O}P(O)\bra{\boldsymbol{\alpha},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\rho_{\sim_{O}\boldsymbol{\alpha}}\ket{\boldsymbol{\alpha},O}$
(159)
$\displaystyle=\frac{1}{|\mathcal{A}|^{n}}\sum_{O}P(O)\sum_{\boldsymbol{\alpha}\in\mathcal{A}^{n}/\sim_{O}}\sum_{\boldsymbol{\widetilde{\alpha}}\sim_{O}\boldsymbol{\alpha}}\bra{\boldsymbol{\widetilde{\alpha}},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\rho_{\sim_{O}\boldsymbol{\alpha}}\ket{\boldsymbol{\widetilde{\alpha}},O}$
(160)
where we are abusing notation slightly by denoting by $\boldsymbol{\alpha}$
the equivalence class $[\boldsymbol{\alpha}]\in\mathcal{A}^{n}/\sim_{O}$.
Since $\rho_{\sim_{O}\boldsymbol{\alpha}}$ is a normalised state for every
$\boldsymbol{\alpha},O$ and the term
$\sum_{\boldsymbol{\widetilde{\alpha}}\sim_{O}\boldsymbol{\alpha}}\bra{\boldsymbol{\widetilde{\alpha}},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\rho_{\sim_{O}\boldsymbol{\alpha},O}\ket{\boldsymbol{\widetilde{\alpha}},O}$
can be interpreted as part of the trace over
$\rho_{\sim_{O}\boldsymbol{\alpha}}$, we thus have
$\displaystyle\sum_{\boldsymbol{\widetilde{\alpha}}\sim_{O}\boldsymbol{\alpha}}\bra{\boldsymbol{\widetilde{\alpha}},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\rho_{\sim_{O}\boldsymbol{\alpha},O}\ket{\boldsymbol{\widetilde{\alpha}},O}\leq
1$ (161)
Thus,
$\displaystyle\frac{1}{|\mathcal{A}|^{n}}\sum_{O}P(O)\sum_{\boldsymbol{\alpha}\in\mathcal{A}^{n}/\sim_{O}}\sum_{\boldsymbol{\widetilde{\alpha}}\sim_{O}\boldsymbol{\alpha}}\bra{\boldsymbol{\widetilde{\alpha}},O}_{\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\rho_{\sim_{O}\boldsymbol{\alpha}}\ket{\boldsymbol{\widetilde{\alpha}},O}$
$\displaystyle\leq\frac{1}{|\mathcal{A}|^{n}}\sum_{O}P(O)\sum_{\boldsymbol{\alpha}\in\mathcal{A}^{n}/\sim_{O}}1$
(162)
$\displaystyle=\frac{1}{|\mathcal{A}|^{n}}\sum_{O}\frac{P(O)|\mathcal{A}|^{n}}{2^{|O|}}$
(163) $\displaystyle=\frac{1}{|\mathcal{O}|}\sum_{O}\frac{P(O)}{2^{|O|}}$
(164)
Since this holds for any pair $(\widehat{E},\mathcal{F})$, it in particular
also holds for the one that maximises the trace, so
$\displaystyle\max_{E}\operatorname{Tr}_{\boldsymbol{A},\boldsymbol{O},\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\left[D_{\operatorname{client}}^{(m)}E^{T}\left(\sum_{\boldsymbol{\alpha},O}\ket{\boldsymbol{\alpha},O,\boldsymbol{\alpha},O}\\!\\!\bra{\boldsymbol{\alpha},O,\boldsymbol{\alpha},O}_{\boldsymbol{A},\boldsymbol{O},\boldsymbol{A^{\prime}},\boldsymbol{O^{\prime}}}\right)\right]\leq\sum_{O}\frac{P(O)}{2^{|O|}}$
(165)
proving the theorem. ∎
## Appendix D Grey Box MBQC Supplementary
### D.1 Correctness of $\sigma_{\operatorname{MBQC}}^{g}$
###### Proposition D.1.
Let $g$ be a gflow for $(G,I,O,\omega)$ where $G$ is a graph on vertices $V$,
let $\sigma_{\operatorname{MBQC}}^{g}$ be defined as in Equation 54, and let
$\mathcal{M}_{\alpha_{i},i,\omega(i)}^{c_{i}}$ denote the $c_{i}$ measurement
outcome of the measurement channels defined in Equation 55 ($c_{i}=0$ for
positive outcome and $c_{i}=1$ for the negative outcome). Then
$\displaystyle\left(\bigotimes_{i\in V\setminus
O}\mathcal{M}_{\alpha_{i},i,\omega(i)}^{c_{i}}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(166)
is the same state for all $\boldsymbol{c}=c_{1}...c_{|V\setminus O|}$.
###### Proof.
The proposition is a consequence of the fact that
$\sigma_{\operatorname{MBQC}}^{g}$ is defined directly from gflow, and
essentially follows the same reasoning as that presented in Section A.2 but
written in the combs formalism and for general measurement planes. We write
out the details in part to highlight that no issues arise with the choice of
ordering of $X$ and $Z$ operators in Equation 52. We proceed by showing that
the state produced by any series of measurement outcomes is the same as that
produced by all positive measurement outcomes (which encodes the computation
for the MBQC):
$\displaystyle\left(\bigotimes_{i\in V\setminus
O}\mathcal{M}_{\alpha_{i},i,\omega(i)}^{c_{i}}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}=\left(\bigotimes_{i\in
V\setminus
O}\mathcal{M}_{\alpha_{i},i,\omega(i)}^{0}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(167)
Consider first only a single measurement channel for qubit $i\in V\setminus
O$, which obtains the negative outcome:
$\displaystyle\mathcal{M}_{\alpha_{i},i,\omega(i)}^{1}\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
$\displaystyle=\left(\ket{1}\\!\\!\bra{1}\otimes\ket{-_{\alpha_{i}}}\\!\\!\bra{-_{\alpha_{i}}}_{\omega(i)}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(168)
By the definition of gflow, we are guaranteed that
$\displaystyle\ket{-_{\alpha_{i}}}\\!\\!\bra{-_{\alpha_{i}}}_{\omega(i)}\equiv
K_{g(i)}|_{i}^{\dagger}\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}K_{g(i)}|_{i}$
(169)
where $K_{g(i)}|_{i}$ denotes the operator which appears as the tensor factor
of $i$ in the stabiliser $K_{g(i)}$. Thus, using the properties of the link
product and also that $K_{g(i)}|_{i}^{\dagger}\equiv K_{g(i)}|_{i}^{T}$ (in
fact $K_{v}^{\dagger}=K_{v}^{T}$ for all stabilisers $K_{v}$), we have
$\displaystyle\left(\ket{1}\\!\\!\bra{1}\otimes\ket{-_{\alpha_{i}}}\\!\\!\bra{-_{\alpha_{i}}}_{\omega(i)}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
$\displaystyle=\left(\ket{1}\\!\\!\bra{1}\otimes
K_{g(i)}|_{i}^{\dagger}\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}K_{g(i)}|_{i}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(170)
$\displaystyle=\left(\ket{1}\\!\\!\bra{1}\otimes\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}\right)\ast
K_{g(i)}|_{i}\sigma_{\operatorname{MBQC}}^{g}K_{g(i)}|_{i}^{\dagger}\ast\rho_{G}$
(171)
By contracting the link product over $C_{i}$ and writing
$\sigma_{\operatorname{MBQC}}^{g}$ in the form of Equation 54, we get
$\displaystyle\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}\ast\left(\sum_{\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}|_{\setminus
i}}K_{g(i)}|_{i}U_{\text{corr}(1\boldsymbol{c}|_{\setminus
i})}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}U_{\text{corr}(1\boldsymbol{c}|_{\setminus
i})}^{\dagger}K_{g(i)}|_{i}^{\dagger}\otimes\ket{\boldsymbol{c}|_{\setminus
i}\boldsymbol{a}}\\!\\!\bra{\boldsymbol{c}|_{\setminus
i}\boldsymbol{b}}\right)\ast\rho_{G}$ (172)
where the sum over $\boldsymbol{c}|_{\setminus i}$ indicates the sum over
basis element of all the $\mathcal{H}_{C_{j}}$ for $j\neq i$ and
$1\boldsymbol{c}|_{\setminus i}$ denotes the binary string $\boldsymbol{c}$
with the entry corresponding to $C_{i}$ a $1$ and the other entries given by
$\boldsymbol{c}|_{\setminus i}$. By the definition of the correction sets
$\mathcal{X}_{j}$ and $\mathcal{Z}_{j}$, as well as the definition of
$U_{\text{corr}(\boldsymbol{c})}$, we can write
$\displaystyle U_{\text{corr}(1\boldsymbol{c}|_{\setminus
i})}=(-1)^{f(\boldsymbol{c}|_{\setminus
i})}U_{\text{corr}(0\boldsymbol{c}|_{\setminus i})}K_{g(i)}|_{\setminus i}$
(173)
where the exponent $f(\boldsymbol{c}|_{\setminus i})$ encodes the coefficient
that arises from commuting the factors of $K_{g(i)}|_{\setminus i}$ through
the other factors of $U_{\text{corr}(1\boldsymbol{c}|_{\setminus i})}$ and
where $U_{\text{corr}(0\boldsymbol{c}|_{\setminus i})}$ denotes the correction
operator with $c_{i}=0$. In particular, this means that
$\displaystyle K_{g(i)}|_{i}U_{\text{corr}(1\boldsymbol{c}|_{\setminus i})}$
$\displaystyle=(-1)^{f(\boldsymbol{c}|_{\setminus
i})+t_{i}}U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}K_{g(i)}|_{i}K_{g(i)}|_{\setminus i}$ (174)
$\displaystyle=(-1)^{f(\boldsymbol{c}|_{\setminus
i})+t_{i}}U_{\text{corr}(0\boldsymbol{c}|_{\setminus i})}K_{g(i)}$ (175)
where $t_{i}$ encodes the coefficient arises from commuting $K_{g(i)}|_{i}$
through $U_{\text{corr}(0\boldsymbol{c}|_{\setminus i})}$. Since the same
coefficient arises from the conjugate term (i.e.
$U_{\text{corr}(1\boldsymbol{c}|_{\setminus
i})}^{\dagger}K_{g(i)}|_{i}^{\dagger}$), the net result is that no $-1$ factor
can appear. That is, we have
$\displaystyle K_{g(i)}|_{i}U_{\text{corr}(1\boldsymbol{c}|_{\setminus
i})}(\cdot)U_{\text{corr}(1\boldsymbol{c}|_{\setminus
i})}^{\dagger}K_{g(i)}|_{i}^{\dagger}=U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}K_{g(i)}(\cdot)K_{g(i)}^{\dagger}U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}^{\dagger}$ (176)
The cancelling of any phase factor arising from commuting $X$ and $Z$ factors
in $U_{\text{corr}(\boldsymbol{c})}$ is the reason why the ordering in
Equation 52 is justified. Continuing from Equation 172, we have
$\displaystyle\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}\ast$
$\displaystyle\left(\sum_{\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}|_{\setminus
i}}U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}K_{g(i)}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}K_{g(i)}^{\dagger}U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}^{\dagger}\otimes\ket{\boldsymbol{c}|_{\setminus
i}\boldsymbol{a}}\\!\\!\bra{\boldsymbol{c}|_{\setminus
i}\boldsymbol{b}}\right)\ast\rho_{G}$ (177)
$\displaystyle=\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}\ast\left(\sum_{\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}|_{\setminus
i}}U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}U_{\text{corr}(0\boldsymbol{c}|_{\setminus
i})}^{\dagger}\otimes\ket{\boldsymbol{c}|_{\setminus
i}}\\!\\!\bra{\boldsymbol{c}|_{\setminus i}}\otimes
K_{g(i)}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}K_{g(i)}^{\dagger}\right)\ast\rho_{G}$
(178)
$\displaystyle\equiv\left(\ket{0}\\!\\!\bra{0}\otimes\ket{+_{\alpha_{i}}}\\!\\!\bra{+_{\alpha_{i}}}_{\omega(i)}\right)\ast
K_{g(i),A^{\prime}}\sigma_{\operatorname{MBQC}}^{g}K_{g(i),A^{\prime}}^{\dagger}\ast\rho_{G}$
(179)
$\displaystyle=\mathcal{M}_{\alpha_{i},i,\omega(i)}^{0}\ast\sigma_{\operatorname{MBQC}}^{g}\ast
K_{g(i)}\rho_{G}K_{g(i)}^{\dagger}$ (180)
$\displaystyle=\mathcal{M}_{\alpha_{i},i,\omega(i)}^{0}\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(181)
where the extra subscript $A^{\prime}$ on the operators in Equation 179
indicate that they act on the appropriate spaces $A^{\prime}_{j}$ rather than
$A_{j}$ as before, and can thus be transferred to the $\rho_{G}$ via the
properties of the link product (which also uses the fact that
$K_{g(i)}^{\dagger}=K_{g(i)}^{T}$). We have also made use of properties of the
Choi operator in Equation 178 to transfer the $K_{g(i)}$ from the $A$-spaces
to the $A^{\prime}$-spaces in the first place.
Since the left-hand side of Equation 167 contains a tensor product of terms
$\mathcal{M}_{\alpha_{i},i,\omega(i)}^{c_{i}}$, we can apply the same
reasoning as above to each factor for which $c_{i}=1$ which thus establishes
that Equation 167 does indeed hold for any measurement outcomes
$\boldsymbol{c}$. This means that once all the link products are evaluated,
which in particular includes the trace over all spaces except for
$\bigotimes_{i\in O}\mathcal{H}_{A_{i}}$, the same state is produced on the
output space. ∎
### D.2 Gflow-Induced Quantum Causal Models
This subsection provides some further details regarding the the quantum causal
model (QCM) induced by gflow for MBQC and thus supports Section 5.2 in the
main text. Firstly, we confirm here that gflow does indeed define a DAG, which
is a requirement for showing 5.1. Thereafter we provide a table that
elucidates the comparisons between the components of the QCM defined for MBQC
and the components of classical causal models as presented eg., by Pearl in
[21].
###### Proposition D.2.
Let $(G,I,O,\omega)$ be such that a gflow exists and let $(g,<)$ be a choice
of such a gflow. The directed graph $\overline{G}$ on vertex set $V$ and edge
set defined from the gflow via
$\displaystyle E$ $\displaystyle:=\left\\{(i,j)\in V\times V|j\in
V,i\in\mathcal{X}_{j}\cup\mathcal{Z}_{j}\right\\}$
is acyclic.
###### Proof.
Suppose for a contradiction that $\overline{G}$ is not acyclic, that is, there
exists a directed path $v_{0}\rightarrow v_{1}\rightarrow...\rightarrow
v_{k}=v_{0}$ for some sequence of vertices $v_{0},...,v_{k}$. By the
definition of the edge set of $\overline{G}$, it follows that either
$v_{i+1}\in g(v_{i})\setminus\\{v_{i}\\}$ or $v_{i}\neq
v_{i+1}\in\operatorname{Odd}(g(v_{i}))$. In either case $v_{i}<v_{i+1}$ in the
ordering of the given gflow. Thus, $v_{0}<v_{k}=v_{0}$ by transitivity of the
order, which gives the desired contradiction as the order is strict by
definition. ∎
Classical causal models are typically presented as consisting of a set of
observed variables $V=\\{V_{1},...,V_{n}\\}$ (those within the model), a set
of unobserved variables $U=\\{U_{1},...,U_{n}\\}$ (those determined by factors
outside of the model), a DAG that specifies which variables (both unobserved
and observed) that have a causal influence on a given observed variable, and
set of functions $F=\\{f_{i}:\operatorname{Pa}(V_{i})\cup U_{i}\rightarrow
V_{i}\\}_{i=1}^{n}$ that map from the parents of a variable to the variable
itself. These functions are often called structural equations and uniquely
specify the values for the observed variables given values of the unobserved
variables. A probabilistic causal model further includes a distribution $P(U)$
over the unobserved variables. Using this terminology, the following table
outlines the correspondence between classical causal models and the QCM
$\sigma_{\operatorname{MBQC}}^{g}$ defined in Section 5.2.
| Comparing the QCM $\sigma_{\operatorname{MBQC}}^{g}$ to Classical Causal
Models
---|---
Classical Causal Model | Unobserved Variable $U_{i}$ | Observed Variable $V_{i}$ | $f_{i}:\operatorname{Pa}(V_{i})\cup U_{i}\rightarrow V_{i}$ | DAG
MBQC QCM $\sigma_{\operatorname{MBQC}}^{g}$ | Input Space $\mathcal{H}_{A^{\prime}_{i}}$ | Quantum Node $\mathcal{H}_{A_{i}}\otimes\mathcal{H}_{C_{i}}$ | $\rho_{A_{i}|C_{j:j\in\operatorname{Pa}(i)},A^{\prime}_{i}}$ | $\overline{G}$
### D.3 Causal Equivalence of Gflows
###### Proof of 5.2.
Consider
$\displaystyle\left(\bigotimes_{j=1}^{|V\setminus
O|}\ket{c_{j}}\\!\\!\bra{c_{j}}_{C_{j}}\right)\ast\sigma_{\operatorname{MBQC}}^{g}\ast\rho_{G}$
(182)
for some $g\sim(G,I,O,\omega)$. Evaluating the link product over the $C_{j}$
gives
$\displaystyle\left(\sum_{\boldsymbol{a},\boldsymbol{b}}U_{\text{corr}(\boldsymbol{c})}^{g}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}(U_{\text{corr}(\boldsymbol{c})}^{g})^{\dagger}\otimes\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}\right)\ast\rho_{G}=\left(\sum_{\boldsymbol{a},\boldsymbol{b}}\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}\otimes\ket{\boldsymbol{a}}\\!\\!\bra{\boldsymbol{b}}\right)\ast
U_{\text{corr}(\boldsymbol{c})}^{g}\rho_{G}(U_{\text{corr}(\boldsymbol{c})}^{g})^{\dagger}$
(183)
where $\boldsymbol{c}=c_{1}...c_{|V\setminus O|}$. To prove the proposition,
it suffices to show that all the $U_{\text{corr}(\boldsymbol{c})}^{g}$ as $g$
varies are mutually related by stabilisers of $G$ for each $\boldsymbol{c}$,
up to a phase (which is cancelled by the corresponding conjugate phase from
the adjoint $(U_{\text{corr}(\boldsymbol{c})}^{g})^{\dagger}$).
Let $g,g^{\prime}\sim(G,I,O,\omega)$ be arbitrary. By definition of
$U_{\text{corr}(\boldsymbol{c})}$, we have
$\displaystyle U_{\text{corr}(\boldsymbol{c})}^{g}$
$\displaystyle\propto\prod_{j=1}^{|V\setminus O|}K_{g(j)}^{c_{j}}|_{\setminus
j}$ (184) $\displaystyle U_{\text{corr}(\boldsymbol{c})}^{g^{\prime}}$
$\displaystyle\propto\prod_{j=1}^{|V\setminus
O|}K_{g^{\prime}(j)}^{c_{j}}|_{\setminus j}$ (185)
where $\propto$ denotes that a $-1$ phase may arise from commuting $X$ and $Z$
operators to arrive at the above form of the operator in terms of $K_{g(j)}$
from the canonical form of $U_{\text{corr}(\boldsymbol{c})}^{g}$ (respectively
$U_{\text{corr}(\boldsymbol{c})}^{g^{\prime}}$) as in Equation 53 and Equation
52. Despite that $K_{g(j)}$ and $K_{g^{\prime}(j)}$ may be different
stabilisers, by the fact that both $g$ and $g^{\prime}$ observe the same
measurement planes, the $j$th tensor factor of each is the same. Thus,
$\displaystyle U_{\text{corr}(\boldsymbol{c})}^{g}$
$\displaystyle\propto\left(\bigotimes_{l=1}^{|V\setminus
O|}K_{g(l)}^{c_{l}}|_{l}\right)\prod_{j=1}^{|V\setminus O|}K_{g(j)}$ (186)
$\displaystyle=\left(\bigotimes_{l=1}^{|V\setminus
O|}K_{g(l)}^{c_{l}}|_{l}\right)\prod_{j=1}^{|V\setminus
O|}K_{g^{\prime}(j)}K_{g^{\prime}(j)}K_{g(j)}$ (187)
$\displaystyle=\left(\bigotimes_{l=1}^{|V\setminus
O|}K_{g^{\prime}(l)}^{c_{l}}|_{l}\right)\left(\prod_{j=1}^{|V\setminus
O|}K_{g^{\prime}(j)}\right)\left(\prod_{i=1}^{|V\setminus
O|}K_{g^{\prime}(i)}K_{g(i)}\right)$ (188) $\displaystyle\propto
U_{\text{corr}(\boldsymbol{c})}^{g^{\prime}}\left(\prod_{i=1}^{|V\setminus
O|}K_{g(i)}K_{g^{\prime}(i)}\right)$ (189)
The restriction to only those gflows $g\sim(G,I,O,\omega)$ that have mutually
compatible partial orders that is made in the statement of the proposition is
required when considering $\sigma_{\operatorname{MBQC}}^{g}$ with a total
order on input and output spaces. ∎
### D.4 Gflow Catalogue
Figure 8, Figure 9, and Figure 10 depict the DAGs corresponding to the $15$
different gflows for the four-vertex graph considered in Section 5 and
depicted in Figure 6(a), grouped by the assigned measurement plane for the
second qubit (the first is always measured in the $XY$-plane). The details of
the gflow and corresponding $U_{\text{corr}(\boldsymbol{c})}$ for each DAG are
gives in the caption.
(a) $g_{1}:1\mapsto\\{2\\},2\mapsto\\{3,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{1}}=X_{2}^{c_{1}}\otimes
X_{3}^{c_{2}}Z_{3}^{c_{2}}\otimes X_{4}^{c_{2}}Z_{4}^{c_{1}c_{2}}$
(b) $g_{2}:1\mapsto\\{3\\},2\mapsto\\{3,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{2}}=X_{3}^{c_{1}c_{2}}Z_{3}^{c_{2}}\otimes
X_{4}^{c_{2}}Z_{4}^{c_{1}c_{2}}$
(c) $g_{3}:1\mapsto\\{3\\},2\mapsto\\{4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{2}}=Z_{1}^{c_{2}}\otimes
X_{3}^{c_{1}}Z_{3}^{c_{2}}\otimes X_{4}^{c_{2}}Z_{4}^{c_{1}}$
(d) $g_{4}:1\mapsto\\{4\\},2\mapsto\\{3,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{4}}=Z_{2}^{c_{1}}\otimes
X_{3}^{c_{2}}Z_{3}^{c_{1}c_{2}}\otimes X_{4}^{c_{1}c_{2}}Z_{4}^{c_{2}}$
(e) $g_{5}:1\mapsto\\{2,3,4\\},2\mapsto\\{3,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{5}}=X_{2}^{c_{1}}Z_{2}^{c_{1}}\otimes
X_{3}^{c_{1}c_{2}}Z_{3}^{c_{1}c_{2}}\otimes X_{4}^{c_{1}c_{2}}Z_{4}^{c_{2}}$
Figure 8: (a) - (e) display the directed acyclic graphs and correction
operators for the gflows $g_{1},...,g_{5}$ respectively which are compatible
with $(G,I,O,\omega)$ where $\omega(1)=\omega(2)=XY$. The corresponding
captions detail the gflows themselves and the associated
$U_{\text{corr}(\boldsymbol{c})}$. The partial order for $g_{1},g_{2},g_{4}$
and $g_{5}$ is given by $1<2$ , and that for $g_{3}$ is $2<1$.
(a) $g_{6}:1\mapsto\\{2\\},2\mapsto\\{2,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{6}}=X_{2}^{c_{1}}\otimes
Z_{3}^{c_{2}}\otimes X_{4}^{c_{2}}Z_{4}^{c_{1}c_{2}}$
(b) $g_{7}:1\mapsto\\{3\\},2\mapsto\\{2,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{7}}=X_{3}^{c_{1}}Z_{3}^{c_{2}}\otimes
X_{4}^{c_{2}}Z_{4}^{c_{1}c_{2}}$
(c) $g_{8}:1\mapsto\\{3\\},2\mapsto\\{2,3,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{8}}=Z_{1}^{c_{2}}\otimes
X_{3}^{c_{1}c_{2}}Z_{3}^{c_{2}}\otimes X_{4}^{c_{2}}Z_{4}^{c_{1}}$
(d) $g_{9}:1\mapsto\\{4\\},2\mapsto\\{2,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{9}}=Z_{2}^{c_{1}}\otimes
Z_{3}^{c_{1}c_{2}}\otimes X_{4}^{c_{1}c_{2}}Z_{4}^{c_{2}}$
(e) $g_{10}:1\mapsto\\{2,3,4\\},2\mapsto\\{2,4\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{10}}=X_{2}^{c_{1}}Z_{2}^{c_{1}}\otimes
X_{3}^{c_{1}}Z_{3}^{c_{1}c_{2}}\otimes X_{4}^{c_{1}c_{2}}Z_{4}^{c_{2}}$
Figure 9: (a) - (e) display the directed acyclic graphs and correction
operators for the gflows $g_{6},...,g_{10}$ respectively which are compatible
with $(G,I,O,\omega)$ where $\omega(1)=XY$ and $\omega(2)=XZ$. The
corresponding captions detail the gflows themselves and the associated
$U_{\text{corr}(\boldsymbol{c})}$. The partial order for $g_{6},g_{7},g_{9}$
and $g_{10}$ is given by $1<2$ , and that for $g_{8}$ is $2<1$.
(a) $g_{11}:1\mapsto\\{2\\},2\mapsto\\{2,3\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{11}}=X_{2}^{c_{1}}\otimes
X_{3}^{c_{2}}\otimes Z_{4}^{c_{1}}$
(b) $g_{12}:1\mapsto\\{3\\},2\mapsto\\{2\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{12}}=Z_{1}^{c_{2}}\otimes
X_{3}^{c_{1}}\otimes Z_{4}^{c_{1}c_{2}}$
(c) $g_{13}:1\mapsto\\{3\\},2\mapsto\\{2,3\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{13}}=X_{3}^{c_{1}c_{2}}\otimes
Z_{4}^{c_{1}}$
(d) $g_{14}:1\mapsto\\{4\\},2\mapsto\\{2,3\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{14}}=Z_{2}^{c_{1}}\otimes
X_{3}^{c_{2}}Z_{3}^{c_{1}}\otimes X_{4}^{c_{1}}$
(e) $g_{15}:1\mapsto\\{2,3,4\\},2\mapsto\\{2,3\\}$,
$U_{\text{corr}(\boldsymbol{c})}^{g_{15}}=X_{2}^{c_{1}}Z_{2}^{c_{1}}\otimes
X_{3}^{c_{1}c_{2}}Z_{3}^{c_{1}}\otimes X_{4}^{c_{1}}$
Figure 10: (a) - (e) display the directed acyclic graphs and correction
operators for the gflows $g_{11},...,g_{15}$ respectively which are compatible
with $(G,I,O,\omega)$ where $\omega(1)=XY$ and $\omega(2)=YZ$. The
corresponding captions detail the gflows themselves and the associated
$U_{\text{corr}(\boldsymbol{c})}$. The partial order for
$g_{11},g_{13},g_{14}$ and $g_{15}$ is given by $1<2$ , and that for $g_{12}$
is $2<1$.
## Appendix E Details of SDP Implementation
The numerical calculations of the guessing probabilities throughout this work
made use of the convex optimisation library CVXPY [73, 74]. We primarily used
the Splitting Conic Solver (SCS) [75, 76]. Our code is provided at [72].
Calculations were run on an HP $Z4$ $G4$ $9980XE$ workstation.
To provide some indication of the limitations of the numerical approach in its
current form, to complement those detailed in Section 6.1, we document here
some of hurdles we faced when calculating guessing probabilities. For the BQC
examples, we generated the classical combs as $1$-dimensional objects
representing the corresponding diagonals. Despite this, size issues played a
role when storing the combs even before calling the SDP solver:
$D_{\operatorname{client}}$ for a single round and for any $\mathcal{A}$ of
size greater than $12$ and $D_{\operatorname{client}}^{(2)}$ for $\mathcal{A}$
of size greater than $4$ caused problems. The guessing probability for
$D_{\operatorname{client}}^{(1)}$ for $|\mathcal{A}|=4$ was calculated within
a day, whereas for $|\mathcal{A}|=8$, it took on the order of a week and
required approximately $110GB$ of RAM. Attempting to calculate the guessing
probability for $D_{\operatorname{client}}^{(2)}$ with $|\mathcal{A}|=4$
exceeded the available RAM of the machine.
For the Grey Box MBQC examples, the guessing probabilities for all single
round combs could be calculated quickly (on the order of minutes) however all
multi-round combs again caused size problems. Typically, the primary
bottleneck occurred when enforcing the comb constraints (i.e. sequential
partial trace constraints) on the variable in the SDP solver, which tended to
dominate the runtime of the algorithm. Otherwise, some size error would occur
during the Cone Matrix Stuffing reduction phase of the solver (see the CVXPY
documentation for details).
## Appendix F Whose Prior?
Throughout this work, we have considered combs of the form
$\sum_{x}P(x)\ket{x}\\!\\!\bra{x}\otimes\sigma_{x}$. As made particularly
apparent in Section 3.1 (in conjunction with Appendix B), the distribution
$P(x)$ can be considered as a prior distribution and the min-entropy for such
a comb can be interpreted in terms of a maximum over posterior distributions.
One might then ask, whose prior is it? The natural answer seems to be that it
is the prior of the agent interacting with the system in each case: the server
in the BQC protocol or the user of the MBQC device. As such, the operators
$D_{\operatorname{client}}$, $D_{\text{gflow}}$, $D_{\text{mp}}$ or
$D_{\text{calibr}}$ can be considered as a model of the client internal to the
server in the former case, or as models of the MBQC device internal to the
user in the latter three cases. With this interpretation, it is interesting to
reconsider $D_{\text{gflow}}$ and $D_{\text{mp}}$: they essentially consist of
the same components but via a slight difference in their composition, which is
to say, a slight difference in the internal representation of the system by
the agent, the agent learns about a different property of the system.
From a practical point of view, the presence of the prior distribution, as in
any Bayesian updating, also allows for the inputting of specific knowledge of
the problem into the calculation of the min-entropy. Throughout, we have
typically taken prior distributions to be uniform in order to simplify the
numerics, however this need not be the case. For example, in the BQC example,
the assumption that the client chooses a computation uniformly at random is
not necessarily a reasonable one - most known quantum algorithms have specific
structure which would correspond to the requirement that certain entries of
$\boldsymbol{\alpha}$ to specific values. This does, however, seem to pose a
slight dilemma - since changing the prior changes the value for the min-
entropy, is it really acceptable to have a proof of security for a protocol
that depends on the (subjective) prior of the server?
|
11institutetext: Dept. of Computer & Systems Sciences, Stockholm University,
Stockholm, Sweden
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Early prediction of the risk of ICU mortality with Deep Federated Learning
Korbinian Randl 11 0000-0002-7938-2747 Núria Lladós Armengol 11
0000-0002-8584-5058 Lena Mondrejevski 11 0000-0002-1790-3842 Ioanna Miliou 11
0000-0002-1357-1967
###### Abstract
Intensive Care Units usually carry patients with a serious risk of mortality.
Recent research has shown the ability of Machine Learning to indicate the
patients’ mortality risk and point physicians toward individuals with a
heightened need for care. Nevertheless, healthcare data is often subject to
privacy regulations and can therefore not be easily shared in order to build
Centralized Machine Learning models that use the combined data of multiple
hospitals. Federated Learning is a Machine Learning framework designed for
data privacy that can be used to circumvent this problem. In this study, we
evaluate the ability of deep Federated Learning to predict the risk of
Intensive Care Unit mortality at an early stage. We compare the predictive
performance of Federated, Centralized, and Local Machine Learning in terms of
AUPRC, F1-score, and AUROC. Our results show that Federated Learning performs
equally well as the centralized approach and is substantially better than the
local approach, thus providing a viable solution for early Intensive Care Unit
mortality prediction. In addition, we show that the prediction performance is
higher when the patient history window is closer to discharge or death.
Finally, we show that using the F1-score as an early stopping metric can
stabilize and increase the performance of our approach for the task at hand.
###### Keywords:
Federated Learning Early Mortality Prediction Recurrent Neural Networks
Multivariate Time Series Intensive Care Unit
## 1 Introduction
Intensive Care Units (ICUs) usually treat patients with a heightened mortality
risk. A study conducted on patients admitted to 167 ICUs from 17 European
countries during four weeks in 2011, and 2012 registered that, out of $5,834$
patients, $1,113\,(19\,\%)$ died in the ICU alone and a total of $1,397$
patients $(24\%)$ died in the whole hospital [3]. Mortality increases even
further when considering patients admitted to an ICU during 2020 and 2021.
Among $1,686$ patients admitted to the ICU with COVID-19, the mortality rate
was $30\%$ [1].
The ICU is also one of the units where medical errors are most likely to
occur, given the complexity of care. ICU patients are severely ill and usually
subject to multiple complex interventions and treatments, thus leading to a
high risk of an adverse outcome. In order to enable clinicians to take action
and prevent such an outcome, the risk of patients’ mortality has to be
predicted not only as accurately as possible but also as early as possible; we
refer to this concept as early ICU mortality prediction. ICUs are therefore
equipped with advanced diagnostic and therapeutic resources in order to enable
quick response to changes in patients’ health. Traditionally, hospitals use
the collected Electronic Health Records (EHRs) to assess individual mortality
risks in the ICU with the help of scores like APACHE [13] and SAPS [5].
Recent research in the field of Machine Learning (ML) has shown improvements
in these scores in terms of predictive performance [2, 11]. The results of
Awad et al. [2] clearly showcase the performance improvement of classical ML
methods over clinical severity scores for early ICU mortality prediction.
Similarly, Johnson and Mark [11] show that classical ML models trained using
data from the first $24\,\mathrm{h}$ of the ICU stay can outperform clinical
scores.
These ML models rely on big amounts of data in order to learn correlations
between current patient data and their risk of mortality within a specific
time window in the future. Conventionally, ML approaches use all the available
data to train the model in a centralized manner. However, as patient data are
usually subject to privacy regulations, like the European GDPR, the data
cannot simply be shared between hospitals to train Centralized ML (CML)
models. Alternatively, locally available data could be used by each hospital
to set up its own independent Local ML (LML) early warning system. However,
this paradigm could suffer from a lack of sufficient training data and would
not consider the heterogeneity of patients across multiple medical centers.
A promising alternative to LML or CML is presented by Federated Learning (FL).
Instead of exchanging data, this ML framework relies on training many local
models of identical structure at the location of the data, which are then
combined into a global model. In our case, the data are stored at different
hospitals, which we refer to as clients. Apart from ensuring privacy by
design, FL enables parallel training using the clients as computational units
[14, 17].
FL can be perfectly integrated into hospital facilities that store patients’
information in the form of EHRs. An advantage of EHRs is that data are
collected in the form of Multivariate Time-Series (MTS), which means that each
feature consists of a stream of values changing over time. Deep Learning (DL)
architectures like Recurrent Neural Networks (RNNs) represent powerful tools
for dealing with this kind of data, as they also take into account the history
encoded by the time series data. One major challenge when training a DL model
is to find a trade-off between training the neural network enough to learn the
features of the training set and stopping before it overfits the training
data. One possible technique to find the optimal number of training epochs is
Early Stopping (ES). In case of overfitting, the validation performance of the
model begins to degrade, and the training process stops. Different metrics,
such as loss or F1-score, can be used to trigger ES. Their suitability depends
on the task.
Some approaches in literature use RNNs for binary classification on MTS from
the ICU; however, they do not tackle the problem in combination with FL.
Pattalung et al. [16] compare different RNN architectures to create benchmark
values for ICU mortality prediction tasks on different publicly accessible
databases. The authors take MTS data over a $48\,\mathrm{h}$ period in order
to predict whether a patient dies at the end of this period. Ge et al. [6]
combine Logistic Regression and LSTM for early ICU mortality prediction after
$48\,\mathrm{h}$ of ICU admission. Their results show that DL achieves higher
accuracy than Logistic Regression in identifying patients at high risk of
death.
Despite the benefits of using an FL setup, few papers address the problem of
mortality prediction with FL. In this study, we built upon the research of
Mondrejevski et al. [15] that propose FLICU, an FL workflow for mortality
prediction tasks in the ICU. However, their paper does not focus on predicting
ICU mortality at an early stage but it is rather a retrospective study.
This paper tackles the current limitations by combining FL with early
prediction of ICU mortality. The main contributions are summarized below:
1. 1.
We propose a workflow for predicting early ICU mortality using deep FL.
2. 2.
We analyze the predictive performance of our proposed solution on different
time windows and data cohorts.
3. 3.
We compare the predictive performance of the FL setup (with $2$, $4$, and
$8\,$clients) against CML and LML.
4. 4.
We compare two early-stopping criteria during training: loss and F1-score.
## 2 Method
In this paper, we propose a workflow for early ICU mortality prediction that
comprises three main phases: (i) Data Preparation, (ii) Window Selection, and
(iii) Modeling. A schematic view of our workflow is shown in Fig. 0a.
(a)
(b) Patient data is available from the admission to the ICU $t_{adm}$ to death
or discharge from the ICU $t_{end}$. Training data is sampled during $\Delta
t_{data}$ (from $t_{adm}$ to $t_{eval}$). This leaves a prediction window
$\Delta t_{pred}$, during which data is unknown to the model. (c) Data are
split into $F$ equally sized folds. Over $F$ iterations, each of the folds is
used $F-1$ times for training and validation and once for testing. For FL and
LML, we further split the combined training and validation data into the
number of clients. These partitions are, in turn, divided into training and
validation data.
Figure 1: (a) Schematic Workflow, (b) Time-window representation (c), Data
splits.
### 2.1 Problem Formulation
Our basic research problem can be formulated as a binary classification
problem for early ICU mortality prediction: given a patient cohort $D$ that
consists of $n$ patients, we aim to estimate the real class label
$y_{i}\in[0,1]$ for each patient $i\in[0,1,\dotsc,n]$ by predicting a label
$\hat{y}_{i}$. The label $y_{i}$ denotes whether patient $i$ died or survived
the ICU stay and $\hat{y}_{i}$ indicates the patient’s predicted mortality
risk. The label $\hat{y}_{i}$ is estimated based on an MTS feature stream
$X_{i}(t)=[x_{i0}(t),x_{i1}(t),\dotsc,x_{im}(t)]$, that consists of $m$
features (e.g. vital signs or laboratory values). Each
$x_{ij}(t),~{}j\in[0,1,\dotsc,m]$ represents the value of a univariate time
series at time $t$. Since the main focus of this study is early prediction,
the observations $X_{i}(t)$ that are used are collected only during the first
hours of the patient’s ICU stay.
Furthermore, we assume that the patients are randomly distributed over $K$
ICUs/hospitals. $D_{k}$ is the local cohort of patients (i.e., the set of
locally available $X_{i}(t)$) in each hospital $k$. Thus,
$D=D_{0}~{}\cup~{}D_{1}~{}\cup~{}\dotsc~{}\cup~{}D_{K}$ is the global dataset
of $n$ patients with $D_{a}\cap
D_{b}=\emptyset,\forall(a,b)\in[1,K]\times[1,K]$. In this paper, we compare
the efficiency of CML, LML, and FL on this problem.
### 2.2 Data Preparation
In the first phase of our workflow, we are preparing the data for the
predictive modeling, following five steps (based on the FLICU-workflow [15]):
(i) Patient selection: Initially, we select patients with an ICU stay and
filter the cohort according to the criteria below:
1. 1.
We dismiss all but the first ICU stay of each patient.
2. 2.
We dismiss patients with data recorded for less than $\Delta t_{min}$.
3. 3.
We dismiss patients staying longer than $\Delta t_{max}$.
The windows $\Delta t_{min}$ and $\Delta t_{max}$ allow us to define the
patient cohorts. $\Delta t_{min}$ denotes the minimum length of MTS data per
patient. Additionally, it guarantees that all patients will have at least the
required history length. The upper bound $\Delta t_{max}$ limits the
prediction window in order to dismiss patients that stay in the ICU for an
extensive period of time.
(ii) Feature selection: Consecutively, we extract vital signs and lab values
in the form of MTS. We extract vitals explicitly connected to ICU stays and
consider the time of the first vital of each patient to be the time of
admission to the ICU. If no vitals are recorded for an ICU stay, we use the
admission time logged by the hospital staff. Lab values are sampled during the
patient’s whole ICU stay, from ICU admission to ICU discharge or death.
(iii) Re-sampling: Vital signs and lab values are re-sampled to fixed length
intervals. The length of these intervals depends on the data collection
frequency in the hospitals and may differ for vitals and labs. If more than
one measurement falls in the same interval, the median is used for data
aggregation.
(iv) Imputation: Missing values are treated with forward, and then backward
imputation, starting from the beginning of the ICU stay. Non-observed features
for a patient’s ICU stay are replaced with $-1$.
(v) Labeling: Finally, if a patient dies in the ICU, we assign the class label
$y_{i}=1$ (death). Otherwise, we assume that the patient survived the ICU and
assign the label $y_{i}=0$ (discharge).
### 2.3 Window Selection
In the second phase of our workflow, we are interested in selecting the
patient history window that will be used as input to the predictive models,
defined as $\Delta t_{data}\leq\Delta t_{min}$. $\Delta t_{data}$ starts at
the time of ICU admission $t_{adm}$ and ends at the time of evaluation
$t_{eval}$ (see Fig. 0b). Since the main focus of this study is early ICU
mortality prediction, we aim to predict the label $\hat{y}_{i}$ ahead of the
patient’s time of ICU discharge or death, thus $\Delta t_{data}$ is considered
to end before the end of the ICU stay $t_{end}$. More precisely, there is a
prediction window $\Delta t_{pred}\geq 0$ that starts at $t_{eval}$ and ends
at $t_{end}$.
### 2.4 Modeling
To avoid bias induced by data partitioning, we use $F$-fold cross-validation
as the model evaluation technique. We split the data into $F$ partitions,
where each split is used as a testing dataset once, while the remaining folds
are split into training and validation sets for each client, respectively (see
Fig. 0c). To allocate the data points throughout the clients (in LML and FL),
we assume $K$ horizontal, stratified splits. This means that the data in each
hospital $k\in[0,1,\dotsc,K]$ has the same number of patients
$|D_{k}|=\frac{n}{K}$ with the same class distribution and that each patient’s
records are kept in only one hospital at the same time. Before being passed
into the model, all data streams are normalized according to the global minima
and maxima found in the available training and validation data.
To deal with class imbalance during training, class weights are added to the
training data. This method down-weighs the impact of classes with more
examples (in our case: the discharged patients) and increases the importance
of classes with fewer data examples during error back-propagation. Thus,
applying class weights enables us to achieve optimization-level class balance.
The weights $w_{c}$ for each class $c$ are calculated according to the
following formula: $w_{c}=\frac{\text{\\# of samples in dataset}}{\text{\\# of
samples in class }c}$.
This setup is aggregated to the three configurations of our predictive model,
FL, CML, and LML. As both main gated RNN architectures, LSTM [8], and GRU [4]
have shown to perform equally well on tasks comparable to ours [15, 16], we
focus on the less resource-intensive GRU in this study. Our basic DL model
architecture (similar to [15] and [16]) consists of two parallel input layers,
one for the vital signs and one for the laboratory variables, each followed by
three recurrent layers of 16 GRUs. Consecutively, we perform batch
normalization and combine the resulting outputs using two fully connected
layers. Finally, we add a sigmoid-layer for estimating the patient risk of ICU
mortality with a value between $0$ and $1$. Matching to this binary output of
the system, we use binary cross-entropy as the loss function $L(\cdot)$.
In CML, the model $f^{CML}(\cdot,\theta)$ is trained on the whole training
data $D$, where $\theta$ is a set of DL weights defining the function of the
model. The goal is to find the optimal set of weights $\theta^{CML}$ that
minimizes the error between $y_{i}$ and $\hat{y}_{i}=f^{CML}(X_{i}(t),\theta)$
for a given $X_{i}(t)$. More formally:
$\theta^{CML}~{}=~{}\underset{\theta^{*}\in\mathbb{R}}{\mathrm{argmin}}~{}L(D,\theta^{*})\vspace{-2mm}$
(1)
After each epoch of training, we evaluate the predictive performance of
$f^{CML}(\cdot,\theta)$ on the validation data using a previously selected
metric $M(\bf{\hat{y}},\bf{y})$, comparing a vector of predictions
$\bf{\hat{y}}$ with the vector of true class labels $\bf{y}$. In order to
avoid overfitting the training data, we stop the training and reset $\theta$
to the time of the best score $s^{*}$ if $s=M(\cdot)$ does not improve for a
predefined number of epochs $P$. We refer to this as early stopping (ES) with
patience $P$.
In LML, each hospital $k$ trains a local model $f^{LML}_{k}(\cdot,\theta)$,
using only the data in the local dataset $D_{k}$, where
$\hat{y}_{i}=f^{LML}_{k}(X_{i}(t),\theta_{k})$. As before, the goal is to
minimize the prediction error by producing an optimal set of weights
$\theta^{LML}_{k}$ for each client $k$:
$\theta^{LML}_{k}~{}=~{}\underset{\theta^{*}\in\mathbb{R}}{\mathrm{argmin}}~{}L(D_{k},\theta^{*})$
(2)
Similarly to CML, we calculate local validation scores
$s_{k}=M(\bf{\hat{y}}_{k},\bf{y}_{k})$ on each client $k$ in every epoch and
monitor them for ES.
For the FL approach, we use a slightly modified version of Federated Averaging
(FedAvg) [14]: (i) First, all local models of the $K$ hospitals are
initialized with the same set of starting weights $\theta^{0}$. (ii) By
performing local training for $E$ local epochs, each of the participating
hospitals derives an updated set of weight parameters $\theta_{k}$. The
fraction of participating hospitals per round is $C$. The local objective is
similar to Equation 2. (iii) After local training, a global model
$f^{FL}(\cdot,\theta)$ is created by averaging all $\theta_{k}$ to one set of
parameters $\theta$. (iv) These averaged weights are then sent back to the
hospitals, which overwrite their local weights with the new ones. (v)
Afterwards, an ES score $s$ is calculated by evaluating the local models on
the respective validation set of each client $k$ and then averaging the
results: $s=\sum^{K}_{k=1}\frac{n_{k}}{n}M(\bf{\hat{y}}_{k},\bf{y}_{k})$. The
above is repeated from step (ii) until the validation scores suggest an
optimal set of weights $\theta^{FL}$ and ES is activated. Here, the objective
is to optimize the global model $f^{FL}(\cdot,\theta)$ by calculating:
$\theta^{FL}~{}=~{}\underset{\theta^{*}\in\mathbb{R}}{\mathrm{argmin}}~{}\sum\limits^{K}_{k=1}\frac{n_{k}}{n}F_{k}(\theta^{*})\vspace{-2mm}$
(3)
where $F_{k}(\theta^{*})=\frac{1}{n_{k}}L(D_{k},\theta^{*})$, $n_{k}=|D_{k}|$.
## 3 Empirical Evaluation
### 3.1 Data Description
In this paper, we use the MIMIC-III (version 1.4) clinical dataset provided by
PhysioNet [7]. This dataset provides de-identified data collected from ICU
patients in Beth Israel Deaconess Medical Center in Boston, Massachusetts. The
data was collected from $46,476$ patients during the years 2001 to 2012 [9,
10]. The database includes patient information such as demographics, vital
sign measurements, and laboratory test results.
Initially, we select the patients based on the criteria described in Section
0, and in addition, we dismiss patients from the Neonatal Intensive Care Unit
(NICU) and Pediatric Intensive Care Unit (PICU). For this study, we use two
different cohorts of patients: (i) Cohort 1 is identified by $\Delta
t_{min}=24\,\mathrm{h}$ and $\Delta t_{max}=72\,\mathrm{h}$, and we compare
different $\Delta t_{data}\in[8\,\mathrm{h},16\,\mathrm{h},24\,\mathrm{h}]$;
and (ii) Cohort 2 is identified by $\Delta t_{min}=48\,\mathrm{h}$ and $\Delta
t_{max}=96\,\mathrm{h}$, and we compare $\Delta
t_{data}\in[8\,\mathrm{h},16\,\mathrm{h},24\,\mathrm{h},32\,\mathrm{h},40\,\mathrm{h},48\,\mathrm{h}$].
Both of these cohorts use $\Delta t_{max}=\Delta t_{min}+48\,\mathrm{h}$. For
simplicity, we therefore, refer to the cohorts by their $\Delta t_{min}$ only.
For the pre-processing and feature selection, we follow the approach from [15,
16]. Initially, we extract demographic information, such as gender and age,
that is used for describing the cohorts. We also extract $7$ vital signs and
$16$ lab values, shown in Tab. 0, in the form of MTS. In this paper, vital
signs are re-sampled in $1\,\mathrm{h}$ intervals, while we use a sampling
interval of $8\,\mathrm{h}$ for lab values.
Table 1: Vital and Laboratory Values Vital Signs: (1 per hour) | heart-rate, systolic blood-pressure, diastolic blood-pressure, mean blood-pressure, respiratory rate, core temperature, blood oxigen saturation (spo2)
---|---
Lab Values: (1 per 8 hours) | albumin, blood urea nitrogen (bun), bilirubin, lactate, bicarbonate, band neutrophils (bands), chloride, creatinine, glucose, hemoglobin, hematocrit, platelets, potassium, partial thromboplastin time (ptt), sodium, white blood-cells
Finally, we extract the label indicating whether a patient died or not based
on the column deathtime in table admissions of MIMIC-III. If a deathtime is
recorded for patient $i$ during the ICU stay, we assign the label $y_{i}=1$.
Otherwise, we assume that the patient survived the ICU and assign the label
$y_{i}=0$. As shown in Tab. 0, there is a heavily imbalanced class
distribution as there are more patients in the discharge class than in the
death class.
Table 2: Cohort Sizes and Class Distribution
| Cohort 1 [$\Delta t_{min}=24\,\mathrm{h}$] | Cohort 2 [$\Delta t_{min}=48\,\mathrm{h}$]
---|---|---
Deaths | Discharges | Total | Deaths | Discharges | Total
absolute | percent | absolute | percent | absolute | percent | absolute | percent
Patients | $804$ | $4.4\%$ | $17,477$ | $95.6\%$ | $18,281$ | $547$ | $5.4\%$ | $9,496$ | $94.6\%$ | $10,043$
Male | $420$ | $2.3\%$ | $10,075$ | $55.1\%$ | $10,495$ | $287$ | $2.9\%$ | $5,255$ | $52.3\%$ | $5,542$
Female | $384$ | $2.1\%$ | $7,402$ | $40.5\%$ | $7,786$ | $260$ | $2.6\%$ | $4,241$ | $42.2\%$ | $4,501$
Age 0 to 29 | $18$ | $0.1\%$ | $892$ | $4.9\%$ | $910$ | $19$ | $0.2\%$ | $421$ | $4.2\%$ | $440$
Age 30 to 59 | $177$ | $1.0\%$ | $5,777$ | $31.6\%$ | $5,954$ | $133$ | $1.3\%$ | $2,863$ | $28.5\%$ | $2,996$
Age 60 to 89 | $525$ | $2.9\%$ | $9,920$ | $54.3\%$ | $10,445$ | $352$ | $3.5\%$ | $5,692$ | $56.7\%$ | $6,044$
Age 90+ | $84$ | $0.5\%$ | $888$ | $4.9\%$ | $972$ | $43$ | $0.4\%$ | $520$ | $5.2\%$ | $563$
### 3.2 Model Training and Evaluation
In this study, we use $F=5$ folds for stratified cross-validation, where we
evaluate CML, LML, and FL on the same testing data splits within each cross-
validation round. In each round, the data of the remaining four folds are
again partitioned in LML and FL models, as those are trained with different
numbers of clients $K\in[2,4,8]$, whilst all remaining data are used in CML.
The available data in each scenario (either all remaining data in CML or each
client’s data in LML and FL) are split into $80\%$ training and $20\%$
validation (see Fig. 0c).
We use Adaptive Moment Estimation (ADAM) [12] as an optimizer for updating the
network weights in CML, FL, and LML. Additionally, we apply an initial
learning rate $\eta$ of $0.01$, which is reduced by $50\%$ every five epochs.
Lastly, we use ES via monitoring the loss or F1-score on the validation set
with patience $P=30$ and the maximum number of epochs set to $100$. In case
the F1-score is undefined, i.e., recall and precision are zero, we set it to
$-1$. The local minibatch size $B$ depends on the number of clients $K$
participating in each configuration: we use $B=512/K$ for performance reasons.
For FL, we use $E=1$ number of local epochs. The fraction of clients computing
in each FL round is $C=1$, meaning that all clients participate in each
iteration.
To compare the performance of the different settings, we use the evaluation
metrics AUPRC, AUROC, Precision, Recall, and F1-score. AUROC is chosen as it
is commonly used to assess the performance of ICU mortality prediction.
However, AUPRC and F1-score are more suitable for highly imbalanced classes,
which is the case for our problem.
The entire code produced in this paper is publicly available on GitHub
111https://github.com/randlbem/Early˙ICU˙mortality˙prediction˙with˙deep˙FL.git.
### 3.3 Results & Discussion
As previously described, we evaluate the ability of our proposed FL workflow
to predict the risk of ICU mortality at an early stage. We compare it with the
CML and LML approaches on two cohorts of patients ($\Delta
t_{min}=24\,\mathrm{h}$ and $48\,\mathrm{h}$), using two ES metrics (loss and
F1-score) and different time windows $\Delta t_{data}$.
The results for cohort $\Delta t_{min}=24\,\mathrm{h}$, using the minimum loss
for ES, are shown in Tab. 0. Overall, the results show that our model performs
well in the task of early ICU mortality prediction. For example, for FL with
$\Delta t_{data}=24\,\mathrm{h}$, we obtain an average AUROC of $0.90\pm 0.01$
and an AUPRC of $0.47\pm 0.04$. It’s important to highlight that due to the
class imbalance of $4.4\,\%$ towards the positive class, the baseline value
for the AUPRC is $0.044$.
In addition, we yield three conclusions from Tab. 0: (i) While an increasing
number of clients results in a decrease in predictive performance over all the
metrics in LML, the performance of FL remains close to that of CML, regardless
of the number of clients. (ii) With growing $\Delta t_{data}$, and
respectively shrinking $\Delta t_{pred}$, the model performance increases.
(iii) The relation between precision and recall is very unstable, which is
shown in the fluctuation of the averages and the high standard deviations.
While (i) clearly shows that FL has the potential to improve on LML for early
ICU mortality prediction, (ii) and (iii) are further explored in the following
experiments.
Table 3: ES with min. loss ($\Delta t_{min}=24\,\mathrm{h}$).
score | CML | FL | LML
---|---|---|---
| | $2~{}clients$ | $4~{}clients$ | $8~{}clients$ | $2~{}clients$ | $4~{}clients$ | $8~{}clients$
$\bf\Delta t_{data}=8\,\mathrm{h}$; avg. $\Delta t_{pred}=35.2\,\mathrm{h}$
AUROC | $0.87\pm 0.02$ | $0.86\pm 0.01$ | $0.86\pm 0.01$ | $0.87\pm 0.01$ | $0.85\pm 0.01$ | $0.82\pm 0.01$ | $0.81\pm 0.01$
AUPRC | $0.36\pm 0.03$ | $0.37\pm 0.02$ | $0.37\pm 0.02$ | $0.37\pm 0.03$ | $0.34\pm 0.04$ | $0.31\pm 0.03$ | $0.29\pm 0.02$
F1 | $0.29\pm 0.15$ | $0.38\pm 0.01$ | $0.36\pm 0.05$ | $0.38\pm 0.04$ | $0.34\pm 0.07$ | $0.34\pm 0.02$ | $0.25\pm 0.04$
precision | $0.62\pm 0.21$ | $0.38\pm 0.07$ | $0.39\pm 0.11$ | $0.41\pm 0.08$ | $0.53\pm 0.14$ | $0.41\pm 0.06$ | $0.48\pm 0.08$
recall | $0.24\pm 0.14$ | $0.41\pm 0.07$ | $0.42\pm 0.14$ | $0.39\pm 0.09$ | $0.29\pm 0.10$ | $0.30\pm 0.07$ | $0.17\pm 0.04$
$\bf\Delta t_{data}=16\,\mathrm{h}$; avg. $\Delta t_{pred}=27.2\,\mathrm{h}$
AUROC | $0.89\pm 0.01$ | $0.88\pm 0.01$ | $0.88\pm 0.01$ | $0.88\pm 0.01$ | $0.87\pm 0.01$ | $0.85\pm 0.01$ | $0.82\pm 0.01$
AUPRC | $0.44\pm 0.04$ | $0.41\pm 0.04$ | $0.41\pm 0.04$ | $0.42\pm 0.03$ | $0.40\pm 0.04$ | $0.35\pm 0.05$ | $0.32\pm 0.02$
F1 | $0.42\pm 0.01$ | $0.35\pm 0.07$ | $0.41\pm 0.02$ | $0.38\pm 0.08$ | $0.29\pm 0.05$ | $0.29\pm 0.12$ | $0.22\pm 0.02$
precision | $0.51\pm 0.10$ | $0.58\pm 0.13$ | $0.47\pm 0.05$ | $0.46\pm 0.07$ | $0.55\pm 0.18$ | $0.50\pm 0.14$ | $0.55\pm 0.09$
recall | $0.38\pm 0.07$ | $0.27\pm 0.09$ | $0.37\pm 0.04$ | $0.34\pm 0.10$ | $0.21\pm 0.03$ | $0.22\pm 0.10$ | $0.14\pm 0.02$
$\bf\Delta t_{data}=24\,\mathrm{h}$; avg. $\Delta t_{pred}=19.2\,\mathrm{h}$
AUROC | $0.89\pm 0.01$ | $0.90\pm 0.01$ | $0.90\pm 0.01$ | $0.89\pm 0.01$ | $0.89\pm 0.01$ | $0.87\pm 0.01$ | $0.83\pm 0.01$
AUPRC | $0.48\pm 0.03$ | $0.48\pm 0.03$ | $0.47\pm 0.04$ | $0.45\pm 0.04$ | $0.46\pm 0.03$ | $0.41\pm 0.05$ | $0.37\pm 0.03$
F1 | $0.10\pm 0.11$ | $0.33\pm 0.17$ | $0.34\pm 0.07$ | $0.30\pm 0.17$ | $0.22\pm 0.12$ | $0.26\pm 0.07$ | $0.32\pm 0.05$
precision | $0.66\pm 0.38$ | $0.70\pm 0.18$ | $0.75\pm 0.06$ | $0.72\pm 0.16$ | $0.51\pm 0.18$ | $0.47\pm 0.10$ | $0.59\pm 0.15$
recall | $0.05\pm 0.07$ | $0.27\pm 0.16$ | $0.22\pm 0.05$ | $0.23\pm 0.14$ | $0.15\pm 0.08$ | $0.19\pm 0.06$ | $0.23\pm 0.05$
#### 3.3.1 Studying the influence of the size of $\Delta t_{data}$ and
$\Delta t_{pred}$.
To assess whether the performance improves with an increasing $\Delta
t_{data}$, decreases with an increasing $\Delta t_{pred}$, or both, we compare
the test scores of the two cohorts ($\Delta t_{min}=24\,\mathrm{h}$ and
$\Delta t_{min}=48\,\mathrm{h}$). The comparison is shown in Fig. 0. The
figure shows that cohort $\Delta t_{min}=24\,\mathrm{h}$ achieves higher
performance than cohort $\Delta t_{min}=48\,\mathrm{h}$, at the same $\Delta
t_{data}$ (Fig. 0a). Nevertheless, both cohorts’ performance increases
alongside $\Delta t_{data}$. When comparing the performance over the length of
$\Delta t_{pred}$, we see that with a rising $\Delta t_{pred}$, all the curves
are decreasing in a similar manner (Fig. 0b). This behavior can be seen over
different models and metrics and is a strong indicator that the size of
$\Delta t_{pred}$ is more important than the size of $\Delta t_{data}$, since
$\Delta t_{pred}$ is bigger in cohort $\Delta t_{min}=48\,\mathrm{h}$ than
cohort $\Delta t_{min}=24\,\mathrm{h}$ for the same $\Delta t_{data}$. This
means that even for small $\Delta t_{data}$, $8\,\mathrm{h}$ or just
$6\,\mathrm{h}$ as Awad et al. [2] demonstrate, prediction should be possible,
if $\Delta t_{pred}$ is small enough.
Table 4: ES with max. F1 ($\Delta t_{min}=24\,\mathrm{h}$).
score | CML | FL | LML
---|---|---|---
| | $2~{}clients$ | $4~{}clients$ | $8~{}clients$ | $2~{}clients$ | $4~{}clients$ | $8~{}clients$
$\bf\Delta t_{data}=8\,\mathrm{h}$; avg. $\Delta t_{pred}=35.2\,\mathrm{h}$
AUROC | $0.87\pm 0.01$ | $0.87\pm 0.01$ | $0.87\pm 0.01$ | $0.87\pm 0.01$ | $0.86\pm 0.01$ | $0.84\pm 0.01$ | $0.80\pm 0.01$
AUPRC | $0.39\pm 0.04$ | $0.38\pm 0.04$ | $0.37\pm 0.04$ | $0.37\pm 0.03$ | $0.36\pm 0.05$ | $0.32\pm 0.04$ | $0.28\pm 0.02$
F1 | $0.40\pm 0.03$ | $0.38\pm 0.02$ | $0.39\pm 0.04$ | $0.39\pm 0.04$ | $0.39\pm 0.03$ | $0.36\pm 0.03$ | $0.35\pm 0.02$
precision | $0.42\pm 0.07$ | $0.38\pm 0.05$ | $0.35\pm 0.06$ | $0.35\pm 0.08$ | $0.37\pm 0.05$ | $0.37\pm 0.05$ | $0.35\pm 0.04$
recall | $0.41\pm 0.08$ | $0.41\pm 0.11$ | $0.46\pm 0.08$ | $0.47\pm 0.09$ | $0.43\pm 0.04$ | $0.36\pm 0.03$ | $0.34\pm 0.03$
$\bf\Delta t_{data}=16\,\mathrm{h}$; avg. $\Delta t_{pred}=27.2\,\mathrm{h}$
AUROC | $0.89\pm 0.01$ | $0.88\pm 0.01$ | $0.88\pm 0.01$ | $0.88\pm 0.01$ | $0.87\pm 0.01$ | $0.85\pm 0.01$ | $0.82\pm 0.01$
AUPRC | $0.44\pm 0.05$ | $0.41\pm 0.04$ | $0.41\pm 0.04$ | $0.42\pm 0.03$ | $0.40\pm 0.05$ | $0.37\pm 0.04$ | $0.33\pm 0.03$
F1 | $0.44\pm 0.02$ | $0.41\pm 0.01$ | $0.40\pm 0.05$ | $0.43\pm 0.03$ | $0.41\pm 0.02$ | $0.39\pm 0.03$ | $0.37\pm 0.02$
precision | $0.46\pm 0.05$ | $0.41\pm 0.04$ | $0.43\pm 0.06$ | $0.43\pm 0.04$ | $0.38\pm 0.08$ | $0.38\pm 0.05$ | $0.41\pm 0.03$
recall | $0.42\pm 0.05$ | $0.42\pm 0.07$ | $0.40\pm 0.11$ | $0.43\pm 0.06$ | $0.46\pm 0.08$ | $0.42\pm 0.06$ | $0.33\pm 0.04$
$\bf\Delta t_{data}=24\,\mathrm{h}$; avg. $\Delta t_{pred}=19.2\,\mathrm{h}$
AUROC | $0.90\pm 0.01$ | $0.90\pm 0.01$ | $0.90\pm 0.00$ | $0.89\pm 0.01$ | $0.89\pm 0.01$ | $0.87\pm 0.01$ | $0.83\pm 0.01$
AUPRC | $0.50\pm 0.04$ | $0.49\pm 0.04$ | $0.47\pm 0.04$ | $0.47\pm 0.03$ | $0.47\pm 0.04$ | $0.43\pm 0.04$ | $0.37\pm 0.03$
F1 | $0.49\pm 0.04$ | $0.48\pm 0.03$ | $0.46\pm 0.03$ | $0.44\pm 0.03$ | $0.45\pm 0.02$ | $0.42\pm 0.02$ | $0.40\pm 0.02$
precision | $0.55\pm 0.05$ | $0.52\pm 0.05$ | $0.49\pm 0.07$ | $0.45\pm 0.07$ | $0.48\pm 0.08$ | $0.39\pm 0.04$ | $0.46\pm 0.02$
recall | $0.46\pm 0.08$ | $0.46\pm 0.05$ | $0.44\pm 0.05$ | $0.46\pm 0.11$ | $0.46\pm 0.10$ | $0.48\pm 0.09$ | $0.35\pm 0.03$
(a)(b)
The dots represent the mean values over the 5-fold cross-validation
iterations. The vertical bars show the standard deviation. The scores were
calculated on the test sets, while the ES metric is the loss.
Figure 2: Mean AUPRC of the two scenarios ($\Delta t_{min}=24\,\mathrm{h}$ and
$\Delta t_{min}=48\,\mathrm{h}$).
#### 3.3.2 Stabilizing precision & recall.
The curves represent the mean values of the validation scores over the 5-fold
cross-validation iterations using loss as the ES metric. The borders of the
shaded areas mark the standard deviation. The best loss and F1 values
recognized by ES fall within the red and purple regions.
Figure 3: Learning progress of CML ($\Delta t_{min}=24\,\mathrm{h}$).
To better understand the interplay between precision and recall, we examine
their fluctuations from the CML model, as it represents the most basic case.
Its learning curve in Fig. 0 shows that there is a trade-off between precision
and recall: precision increases during the first 12 epochs and then shows a
decline, while recall increases steadily. We can also see that the loss shows
a very flat minimum while precision is still stabilizing, and recall has just
begun to increase. This means that minimal changes in the loss’s progression
can greatly impact the models’ precision and recall. The F1-score, however,
shows a more defined maximum.
In order to stabilize precision and recall, we re-train the models whose
performance is shown in Tab. 0, using the F1-score as the ES metric. As the
F1-Score is the harmonic mean between precision and recall, using it for
stopping the training should create a model with an optimal balance between
precision and recall. The results are shown in Tab. 0. A comparison of tables
Tab. 0 and Tab. 0 shows that ES with the highest F1-score produces better and
more stable results for all models. For example, for FL with $\Delta
t_{data}=24\,\mathrm{h}$, we now obtain an average F1-score of $0.46\pm 0.03$,
instead of $0.32\pm 0.15$ (as shown in Tab. 0) in addition to marginally
higher AUROC and AUPRC values.
## 4 Conclusion
We present an FL workflow that allows for early ICU mortality prediction. Our
results show that FL performs equally well as the CML approach and
substantially better than the LML, especially as the number of clients
increases. While the performance remains stable in FL with 2, 4, and 8
clients, in LML, the performance decreases considerably with an increasing
number of clients. These findings are based on the AUROC score - widely used
in the literature but ill-suited for the heavily imbalanced data in this
problem - but also on the more meaningful AUPRC and F1-score.
Furthermore, our results indicate, in agreement with literature [2, 16], that
the size of the prediction window is much more important for the performance
in early prediction tasks than the length of patient history during which data
is collected. Thus, the results show better predictive performance when the
patient history window is closer to the end of the ICU stay. Lastly, we show
that using the F1-score as an ES metric can stabilize and increase the
predictive performance in tasks like ours.
Nevertheless, this study also creates the basis for future work. Since we
limit ourselves to horizontal and stratified client splits for comparability
reasons, it is necessary to re-evaluate our findings in more realistic
settings. In addition, our FL workflow performs considerably well in
predicting ICU mortality at an early stage using the MIMIC-III dataset.
However, the generalizability of the approach needs to be tested beyond this
specific dataset. Furthermore, it would be interesting to explore how far
ahead of death or discharge early ICU mortality can be reasonably predicted
and hereby expand on our findings.
## References
* [1] Auld, S.C., Harrington, K.R.V., Adelman, M.W., Robichaux, C.J., Overton, E.C., Caridi-Scheible, M., Coopersmith, C.M., Murphy, D.J.: Research collaborative trends in icu mortality from coronavirus disease 2019: A tale of three surges. the Emory COVID-19 Quality and Clinical 50(2), 245–255 (2022)
* [2] Awad, A., Bader-El-Den, M., Briggs, J., McNicholas, J., El-Sonbaty, Y.: Predicting hospital mortality for intensive care unit patients: Time-series analysis. Health Informatics Journal 26(2), 1043–1059 (2020)
* [3] Capuzzo, M., Volta, C., Tassinati, T., Moreno, R., Valentin, A., Guidet, B., Iapichino, G., Martin, C., Perneger, T., Combescure, C., Poncet, A., Rhodes, A., Oeyen, S., Matejovic, M., Toft, P., Wrigge, H., et al.: Hospital mortality of adults admitted to intensive care units in hospitals with and without intermediate care units: A multicentre european cohort study. Critical Care 18(5) (2014)
* [4] Cho, K., van Merrienboer, B., Bahdanau, D., Bengio, Y.: On the properties of neural machine translation: Encoder-decoder approaches (2014)
* [5] Gall, J.R., Lemeshow, S., Saulnier, F.: A new simplified acute physiology score (saps ii) based on a european/north american multicenter study. JAMA: The Journal of the American Medical Association 270(24), 2957–2963 (1993)
* [6] Ge, W., Huh, J.W., Rang, Y.P., Lee, J., Kim, Y.H., Turchin, A.: An interpretable icu mortality prediction model based on logistic regression and recurrent neural networks with lstm units. In: AMIA annual symposium. vol. 1, pp. 460 – 469. American Medical Informatics Association (2018)
* [7] Goldberger, A.L., Amaral, L.A., Glass, L., Hausdorff, J.M., Ivanov, P.C., Mark, R.G., Mietus, J.E., Moody, G.B., Peng, C.K., Stanley, H.E.: Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation 101(23), E215–220 (2000)
* [8] Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Computation 9(8), 1735 (1997)
* [9] Johnson, A.E.W., Pollard, T.J., Lehman, L.W.H., Feng, M., Ghassemi, M., Moody, B., Celi, L.A., Mark, R.G., Shen, L., Szolovits, P.: Mimic-iii, a freely accessible critical care database. Scientific Data 3 (2016)
* [10] Johnson, A.E.W., Pollard, T.J., Mark, R.G.: Mimic-iii clinical database (version 1.4). PhysioNet (2016)
* [11] Johnson, A.E., Mark, R.G.: Real-time mortality prediction in the intensive care unit. In: AMIA Annual Symposium Proceedings. vol. 2017, p. 994. American Medical Informatics Association (2017)
* [12] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
* [13] Knaus, W., Wagner, D., Draper, E., Zimmerman, J., Bergner, M., Bastos, P., Sirio, C., Murphy, D., Lotring, T., Damiano, A., Harrell Jr., F.: The apache iii prognostic system: Risk prediction of hospital mortality for critically iii hospitalized adults. Chest 100(6), 1619–1636 (1991)
* [14] McMahan, H.B., Moore, E., Ramage, D., Hampson, S., Arcas, B.A.y.: Communication-efficient learning of deep networks from decentralized data (2016)
* [15] Mondrejevski, L., Miliou, I., Montanino, A., Pitts, D., Hollmen, J., Papapetrou, P.: Flicu: A federated learning workflow for intensive care unit mortality prediction. In: 2022 IEEE 35th International Symposium on Computer-Based Medical Systems (CBMS). pp. 32–37. IEEE Computer Society, Los Alamitos, CA, USA (2022)
* [16] Na Pattalung, T., Ingviya, T., Chaichulee, S.: Feature explanations in recurrent neural networks for predicting risk of mortality in intensive care patients. Journal of Personalized Medicine 11(9) (2021)
* [17] Shokri, R., Shmatikov, V.: Privacy-preserving deep learning. In: Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security. p. 1310–1321. CCS ’15, Association for Computing Machinery, New York, NY, USA (2015)
|
# Quantum targeted energy transfer through machine learning tools
I. Andronis<EMAIL_ADDRESS>Department of Physics, University of
Crete, Heraklion 70013, Greece G. Arapantonis<EMAIL_ADDRESS>Department of
Physics, University of Crete, Heraklion 70013, Greece William H. Miller III
Department of Physics & Astronomy, Johns Hopkins University, Baltimore,
Maryland 21218, USA G. D. Barmparis Department of Physics, University of
Crete, Heraklion 70013, Greece G. P. Tsironis Department of Physics,
University of Crete, Heraklion 70013, Greece John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
02138, USA
###### Abstract
In quantum targeted energy transfer, bosons are transferred from a certain
crystal site to an alternative one, utilizing a nonlinear resonance
configuration similar to the classical targeted energy transfer. We use a
novel computational method based on machine learning algorithms in order to
investigate selectivity as well as efficiency of the quantum transfer in the
context of a dimer and a trimer system. We find that our method identifies
resonant quantum transfer paths that allow boson transfer in unison. The
method is readily extensible to larger lattice systems involving nonlinear
resonances.
## I Introduction
Nonlinear dynamical systems are notoriously difficult to study analytically
and in most cases, one needs to resort to numerical methods for their analysis
[1]. In the classical realm, they are typically described mathematically
through coupled nonlinear differential equations that very scarcely admit
exact solutions. In this case, one resorts to direct numerical integration of
the equations of motion. In the quantum domain, on the other hand, even though
the equations are linear, one needs to engage a very large part of the Hilbert
space in order to find good, yet approximate, solutions [2]. Direct numerical
methods, either in the classical or quantum domain, are relatively
straightforward, yet might fail in large or strongly coupled systems. The
recent widespread of Machine Learning (ML) techniques and their implementation
in the domain of dynamical systems aim to both facilitate and also improve the
discovery process in these systems [3]. The aim of the present work is to use
techniques motivated by ML and obtain results that would be otherwise very
complex to derive. The specific model we work with is that of Targeted Energy
Transfer (TET) that was inspired by energy transfer processes in chlorophyll
[4]. We have two targets here; the first one is to show how ML-motivated
techniques may be superior to standard numerical methods when applied in
quantum complex systems and the second is to find explicit results that would
be otherwise much more difficult to obtain.
In the semiclassical TET model, we focus on resonant exciton transfer between
non-identical molecules [4]. In each molecule a single energy state
participates in the process, thus we have non-identical energy states coupled
together via a non-zero transfer matrix element. In the simplest case of two
molecules, we deal with a non-degenerate linear dimer system. Due to the
energy mismatch, the exciton transfer from the first site to the second is
non-resonant and thus occurs only partially. In order to make the transfer
resonant we need to add local interaction with additional degrees of freedom,
such as phonons. In the anti-adiabatic approximation, this procedure
introduces effective qubic nonlinearities [5]. In TET, complete resonant
transfer is restored for specific nonlinearity parameter configurations
linking the local interaction with anti-adiabatic phonons and the actual
energy mismatch. The analysis is done within the context of the discrete
nonlinear Schrödinger (DNLS) equation, a ubiquitous model for a large class of
nonlinear phenomena [6],[7]. A number of analytical results known for DNLS
dimers are particularly useful for ML implementations in nonlinear systems [8,
9, 10].
While in the case of the semiclassical TET dimer system the resonant transfer
regime can be found analytically, similar analysis in larger systems is an
arduous task. In order to bypass this difficulty, ML-based approaches have
been introduced that enable the discovery of nonlinear resonances in a
straightforward way [11, 12]. The method was tested both in the TET dimer and
also in other analytically known DNLS equation results. Also, it was applied
to the case of a TET trimer model. This ML approach found readily the trimer
resonances that were very difficult to obtain differently. The implementation
of ML in this semiclassical regime shows that its application can be very
beneficial.
The next challenge is that of addressing the fully quantum TET regime; this is
the aim of the present article. When the TET dimer is quantized with bosonic
degrees of freedom a more general resonant condition, which involves the
number of quanta, arises [13]. When at resonance, these bosons may transfer
collectively from the first site to the second in a way similar to the
semiclassical TET, although with rates depending on the boson number and the
energy difference. As in the semiclassical case, in the fully quantum TET, the
resonant condition can be found analytically in the dimer model [13], but any
extension to larger systems is prohibitive analytically and involves a high
computational cost. We show in the present work that the implementation of ML
methods can help in overcoming these difficulties and be able to obtain
readily the required resonant transfer properties.
The structure of the present article is thus the following. In the next
section II, we introduce a general form of the DNLS equation that upon
quantization leads to fully quantum TET models. Once we define clearly the
problem in the quantum case, we discuss in the following section III the ML
technique we use. More specifically, we discuss the choice of the Loss
Function (LF) that enables the analysis of the resonant transfer both in the
dimer and also more generally to arbitrary chains. In section IV we detail the
optimization method, and how the quantum resonant paths are found for
different boson numbers. Section V is the central section of the article; we
not only recover the exact dimer results but also apply the method to a trimer
configuration. This gives not only the new result of the resonant paths but
also shows that the method is fully able to investigate in detail the
specifics of the resonant transfer. Finally, in section VI we conclude and
summarize the findings of the work and comment on possible extensions.
## II From the semiclassical to the fully quantum TET
It is known that the semiclassical DNLS equation can be derived from a
classical Hamiltonian through the use of Hamilton’s equations [5]. This
classical DNLS Hamiltonian is
${}H=\sum_{k=1}^{f}\omega_{k}|\psi_{k}|^{2}+\frac{1}{2}\chi_{k}|\psi_{k}|^{4}-\lambda\sum_{k=1}^{f-1}(\psi^{*}_{k}\psi_{k+1}+\psi^{*}_{k+1}\psi_{k}),$
(1)
where $\omega_{k}$ and $\chi_{k}$ denote the frequency and nonlinearity
parameter of the oscillator at site $k$ respectively. Also,
($\psi^{*}_{k},i\psi_{k})$ is a pair of conjugate variables, and the parameter
$\lambda$ is the coupling among neighboring sites. For simplicity, we assume
that it is the same between every pair of adjacent oscillators. In order to
move to the quantum mechanical case we need to focus on the Bose-Hubbard
operator [14]. The DNLS model for the quantum domain may be seen to arise also
from the Bose-Hubbard model by using the time-dependent variational principle
[15]. A simple way to quantize the Hamiltonian of Eq. (1) is by substituting
$\psi^{*}_{k},\psi_{k}$ with the creation and annihilation operators
$a^{\dagger}_{k},a_{k}$ respectively, as expressed in the second quantization
formalism [16]. These operators obey to the commutation relations
$[a_{k},a^{\dagger}_{m}]=\delta_{km}$, and $[a_{k},a_{m}]=0$, where
$\delta_{km}$ is the Kronecker delta. Therefore, the Hamiltonian operator in
the quantum case becomes
${}\hat{H}=\sum_{k=1}^{f}\omega_{k}\hat{N}_{k}+\frac{1}{2}\chi_{k}\hat{N}_{k}^{2}-\lambda\sum_{k=1}^{f-1}(\hat{a}^{\dagger}_{k}\hat{a}_{k+1}+\hat{a}^{\dagger}_{k+1}\hat{a}_{k}),$
(2)
where $\hat{N}_{k}=\hat{a}^{\dagger}_{k}\hat{a}_{k}$ is the boson number
operator for the site $k$. The dimension of the Hilbert space
$\mathcal{H}_{N}$ for this problem is finite. Each state corresponds to an
allowed configuration of $N$ indistinguishable bosons occupying $f$
distinguishable sites-nonlinear oscillators, with repetitions. Thus the
dimension of the Hilbert space is
${}\mathcal{D}=\frac{(N+f-1)!}{N!(f-1)!}.$
A basis associated with that problem is the one composed of the so-called Fock
states $\ket{n}\equiv\ket{n_{1},n_{2},\dots,n_{f}}$, where
$n_{1},n_{2},\dots,n_{f}$ is the number of bosons at each respective site
$1,2,\dots,f$ at the state indexed as n. We discuss the procedure of labeling
the Fock states in section III. The occupation numbers $\\{n_{i}\\}$ are
restricted to $\sum_{i}n_{i}=N$, while $n_{i}=0,1,\dots,N$. Additionally, the
Fock states are orthonormal, meaning
$\bra{n}\ket{m}=\delta_{n_{1}m_{1}}\dots\delta_{n_{f}m_{f}}.$
Continuing, the actions of the operators
$\hat{a}_{k},\hat{a}^{\dagger}_{k},\hat{N}_{k}$ on each component $\ket{n}$ of
the basis are described by
$\hat{a}_{k}\ket{\dots,n_{k},\dots}=\sqrt{n_{k}}\ket{\dots,n_{k}-1,\dots},$
(3a)
$\hat{a}^{\dagger}_{k}\ket{\dots,n_{k},\dots}=\sqrt{n_{k}+1}\ket{\dots,n_{k}+1,\dots},$
(3b) $\hat{N}_{k}\ket{\dots,n_{k},\dots}=n_{k}\ket{\dots,n_{k},\dots}.$ (3c)
## III Determination of an appropriate Loss Function
We now focus on the choice of an appropriate loss function. We assume that the
donor site has the lowest energy with nonlinearity parameter $\chi_{D}$ while
the highest energy site is the acceptor site with nonlinearity parameter
$\chi_{A}$. We further assume that all the bosons are placed initially (at
time $t=0$) to the donor, and investigate the TET configurations that allow
the complete transfer of these bosons to the acceptor site. To achieve this,
we employ an algorithm relying on the same principle as the ones presented in
[17, 11], where an optimization algorithm is used to minimize a quantity that
is defined as the LF. The LF is usually associated with some physical
parameters and thus, the problem becomes one where the algorithm, has to tune
the parameters.
The first step towards defining the LF is to construct a numerical scheme for
calculating the matrix elements of the Hamiltonian in Eq. (2). While this is
usually a simple process, there are special intricacies, with the main one
being the labeling of the Fock states. We manage to overcome this obstacle by
implementing a similar technique to [18], where the authors rank the states in
lexicographic order and assign indices $1,2,\dots,\mathcal{D}$ to each
configuration of bosons among the sites. For instance, assuming $N=2$ and
$f=3$, state 1 corresponds to $\ket{1}=\ket{2,0,0}$, state 2 corresponds to
$\ket{2}=\ket{1,1,0}$ and so on, assigning every state-configuration to a
distinct index.
Under this indexing policy, it is now straightforward to compute the elements
$\hat{H}_{ij}$ similarly to [13], considering that
${}\hat{H}_{ij}=\bra{i}\hat{H}\ket{j}\overset{(\ref{bosehubbard_hamiltonian})}{=}T_{1}+T_{2},$
(4a)
$T_{1}\equiv\bra{i}\sum_{k=1}^{f}\left[\omega_{k}\hat{N_{k}}+\frac{1}{2}\chi_{k}(\hat{N_{k}})^{2}\right]\ket{j},$
(4b)
$T_{2}\equiv-\lambda\bra{i}\sum_{k=1}^{f-1}\left[\hat{a}^{\dagger}_{k}\hat{a}_{k+1}+\hat{a}^{\dagger}_{k+1}\hat{a}_{k}\right]\ket{j}.$
(4c)
Calculating $T_{1}$ is simple since it involves the simple action of the boson
number operator on state $\ket{j}$, as shown in Eq. (3c). That said, Eq. (4b)
is equivalent to
$\displaystyle
T_{1}=\sum_{k=1}^{f}\left[\omega_{k}j_{k}+\frac{1}{2}\chi_{k}(j_{k})^{2}\right]\delta_{ij},$
where $j_{k}$ stands for the number of bosons on site $k$ for the state
$\ket{j}$. Evaluating the second term is nontrivial because it involves the
action of the operators $\hat{a}^{\dagger}_{k}\hat{a}_{k+1}$,
$\hat{a}^{\dagger}_{k+1}\hat{a}_{k}$. The consecutive action of these
operators can be explored by referring to Eq. (3a) and Eq. (3b). The operators
$\hat{a}^{\dagger}_{k}$, $\hat{a}_{k}$ create and annihilate a boson at a
given site $k$ respectively. However, when a pair of operators like
$\hat{a}^{\dagger}_{k}\hat{a}_{k+1}$ (or $\hat{a}^{\dagger}_{k+1}\hat{a}_{k}$)
acts on a Fock state $\ket{j}$, it creates a boson at the site $k$ (or $k+1$)
but also destroys a boson at the site $k+1$ (or $k$). Thus, their action on a
Fock state conserves the total number of bosons and the resulting state is
going to be, up to a constant, another state in $\mathcal{H}_{N}$.
Specifically,
$\bra{i}\hat{a}^{\dagger}_{k+1}\hat{a}_{k}\ket{j}=\sqrt{j_{k}(j_{k+1}+1)}\delta_{ip}\equiv
C_{k}^{(p)}\delta_{ip},$ (5)
$\bra{i}a_{k}^{\dagger}a_{k+1}\ket{j}=\sqrt{j_{k+1}(j_{k}+1)}\delta_{im}\equiv
D_{k}^{(m)}\delta_{im},$
where the two new states $\ket{p},\ket{m}$ are
${}\ket{p}=\ket{j_{1},\dots,j_{k}-1,j_{k+1}+1\dots,j_{f}},$ (6)
${}\ket{m}=\ket{j_{1},\dots,j_{k}+1,j_{k+1}-1\dots,j_{f}}.$
Combining the above yields
$\displaystyle{}T_{2}=-\lambda\sum_{k=1}^{f-1}\
\left[C_{k}^{(p)}\delta_{ip}+D_{k}^{(m)}\delta_{im}\ \right].$
Figure 1: Time evolution of the expectation value of the number of bosons for
the two sites of the dimer under the parameters ($\chi_{A}$, $\chi_{D}$,
$\omega_{A}$, $\omega_{D}$, $\lambda$, $N$, maxt) = (-2,2, 3, -3, 0.1, 3, 25).
The blue line denotes the donor’s expectation value, while the red one the
acceptor’s.
The matrix representation of the Hamiltonian can be produced and subsequently,
the eigenstates and eigenvalues can be calculated. Everything is developed in
Python, using the Tensorflow [19] library.
The initial distribution of bosons $\ket{\Psi(0)}$ can be expanded to the
basis of the eigenstates $\ket{\psi_{i}}$
$\ket{\Psi(0)}=\sum_{i=1}^{\mathcal{D}}C_{i}\ket{\psi_{i}},\ \
C_{i}=\bra{\psi_{i}}\ket{\Psi(0)}.$
Also, these eigenstates can be expanded to the basis of the Fock states
$\ket{\psi_{i}}=\sum_{j=1}^{\mathcal{D}}b_{j,i}\ket{j},\ \
b_{j,i}=\bra{j}\ket{\psi_{i}}.$
We can now express the time evolution of the initial distribution
$\ket{\Psi(0)}$ by applying the time evolution operator
$\hat{U}(t)=e^{-\textbf{i}\hat{H}t}:$
$\ket{\Psi(t)}=e^{-\textbf{i}\hat{H}t}\ket{\Psi(0)}=\sum_{i,j}^{\mathcal{D}}C_{i}b_{j,i}e^{-\textbf{i}E_{i}t}\ket{j},$
(7)
where $E_{i}$ is the $i$-th eigenvalue and i is the imaginary unit. Similarly,
the time evolution of the average number of bosons at the site k is given by
${}\langle{}\hat{N}_{k}(t)\rangle{}\ \ =\ \
\bra{\Psi(t)}\hat{N_{k}}\ket{\Psi(t)}.$ (8)
Combining Eq. (7) and Eq. (8), $\langle\hat{N}_{k}\rangle$ can be assessed in
the following way
$\displaystyle\langle\hat{N}_{k}(t)\rangle=\sum_{i,j,n}^{\mathcal{D}}j_{k}C_{n}^{*}C_{i}b^{*}_{j,n}b_{j,i}e^{\textbf{i}(E_{n}-E_{i})t}.$
(9)
We use Eq. (9) in order to compute the LF for the quantum TET problem.
Specifically, we time-evolve Eq. (9) for the acceptor energy level until a
pre-defined time `maxt`. During this period, the oscillator of the $f^{th}$
site (acceptor) has completed a few oscillations. The next step in computing
the LF is to extract the maximum value from that time evolution. Concluding,
the LF is defined as
${}LF=N-max\\{\langle\hat{N}_{f}(t)\rangle\\}=N-max\\{\langle\hat{N}_{A}(t)\rangle\\}.$
(10)
In Fig. 1 one can observe the characteristic oscillatory behavior, in the
dimer system, when the complete transfer occurs.
In the present work, we fix the frequency of the oscillators and optimize for
the nonlinearity parameters of the oscillators. While the LF might appear not
to have any explicit connection to the nonlinearity parameters we want to
optimize for, we can use Tensorflow’s $GradientTape$ to compute the
derivatives with respect to these parameters. In that way, we keep track of
gradient information in terms of the trainable variables throughout the whole
process described in this section; from creating the Hamiltonian to
calculating $\langle\hat{N}_{k}(t)\rangle$. Using this information, an
optimizer like $ADAM$ [20] can now update the parameters accordingly, so that
the LF is minimized, signifying complete transfer. It is important to note
that, while many other optimization algorithms (simulated annealing [21],
particle swarm [22], differential evolution [23, 24]) were tried on this
problem, we were not able to produce adequate results with none of them.
The parameter `maxt` is of major importance, because of the oscillation of the
bosons between the sites. If it is not set large enough, TET can be missed
since the system would not have time to complete an oscillation, while it also
has to be small enough, so that precious computational time is saved. Thus, it
has to be large enough to obtain at least one complete oscillation, producing
this way essential information. It is also observed that the period of each
oscillator is proportional to $\lambda^{-1}$, so as one would expect, changes
to these parameters should be made concurrently.
## IV Optimization of the Loss Function
Once we define the LF we may now proceed with its minimization through the
procedure outlined in Fig. 2. We emphasize the first step of this process
graph, i.e. a proper update method, which is essential for the present work.
We noticed by making several computational runs that optimizers did not work
very well when initialized with a random set of parameters; this is due to the
extreme selectivity of the TET resonant condition. Specifically, the parameter
space that the LF maps to, exhibits slowly varying gradients everywhere,
except for areas close to the optimal parameters for transfer, where
relatively large gradients are present.
Figure 2: Schematic representation of the parameter optimization procedure.
Our approach for bypassing this problem relies on exploring a wide range of
the parameter space at the same time. To be more specific, the optimization
procedure is comprised of two distinct phases. First, many test optimizers are
assigned some initial guesses and left to run simultaneously, as different
processes, until a stopping condition is reached. Then we gather the LF and
the parameters that each test optimizer yielded at the last iteration. For the
second part of the optimization procedure, a main optimizer is employed. Its
initial guesses are the parameters of the test optimizer with the best
performance (i.e smallest LF). The purpose of this additional step is to
further minimize the LF, if possible.
Two methods were developed for defining the initial guesses of the test
optimizers, given that we choose the range of the parameter space they will
investigate. In the first approach, which is represented by Fig. 3a, we define
a grid of points in the parameter space and use the lattice points as initial
guesses for the test optimizers. In the second method, we split the parameter
space into a number of regions and chose random initial combinations of
parameters from each region. On the one hand, the former method has a high
computational cost, since it requires a large number of optimizers running at
the same time, but has a greater potential to derive the optimal parameters.
On the other hand, the latter is fast but less robust. It is useful in cases
of stronger coupling, as the gradients of the parameter space smooth out. If
the first run of any method does not produce favorable results we have the
option of redefining the limits of the parameter space around the best
parameters provided by the test optimizers. Nevertheless, choosing one of the
above methodologies relies on the problem at hand. In our case, both of them
produce accurate results. Moreover, we need to disambiguate that the main
optimizer is an optional step, that aims to corroborate TET, by producing a LF
lower than the one deduced from the test optimizers.
Figure 3: Defining the initial guesses of the test optimizers. (a) An example
using the grid method, where we define 400 different initial guesses for the
same number of test optimizers. (b) Representative example of the second
method, where splitting of the parameter space into regions is employed. The
black lines represent the boundaries among the four different regions from
which 16 different guesses are sampled. In both graphs, the test optimizers
that were able to derive a $\text{LF}<1$ are displayed with normal opacity and
their trajectories, while the rest are faded. Both figures refer to a dimer
system with ($N$, $\omega_{A}$, $\omega_{D}$, maxt) = (3, 3, -3, 25). For the
purposes of explaining the difference between the two methods, the system
presented in the left figure has a coupling parameter of $\lambda=0.1$, while
for the system on the right figure $\lambda=1$. Figure 4: Optimization
Results. (a) The trajectories and initial guesses of our algorithm, using the
grid method, for $N=3$ bosons. The optimal parameters found -by the main
optimizer- in this example are ($\chi_{A}$, $\chi_{D}$) = (-1.99, 2). Only the
initial guesses of the test optimizers that resulted in a LF smaller than 0.5
are displayed. (b) Parameters for TET from the quantum limit to the
semiclassical limit, as deduced from the grid method and the main optimizer,
in the dimer system. (c) Comparison of the predicted nonlinearity parameters
$\chi_{D}=-\chi_{A}$, illustrated in the figure as red dots, with the
reference value derived from Eq. (11) represented in the figure as a dashed
line. The system’s constant parameters are ($\omega_{A}$, $\omega_{D}$,
$\lambda$, maxt) = (3, -3, 0.1, 25).
Regardless of whether we use the grid or the splitting into regions method,
the trainable parameters are updated in the same way for both the test
optimizers and the main optimizer. Given the initial guesses of
$\\{\chi_{k}\\}$, $k=1,2,\dots,f$, the nonlinearity parameters after the mth
iteration are updated as following:
$\chi_{j}^{(m+1)}=\chi_{j}^{(m)}-\alpha\vec{\nabla}_{j}(LF),$
where the gradient is computed with Tensorflow’s $GradientTape$, as discussed
in section III. The learning rate $\alpha$ is a positive real number that
defines the rate of change at each iteration while moving towards the minimum
of the LF. With the new set of nonlinearity parameters, the optimization
procedure moves to the next iteration and it will be interrupted either due to
slow convergence or because of reaching maximum iterations.
The threshold parameter of Fig. 2 has a dual role. On one hand, it checks
whether the nonlinearity parameters of the current iteration yield a LF close
to zero, signifying TET. On the other hand, it determines the slow convergence
of the algorithm, and its role is to pause the optimization procedure when
there is no significant improvement in minimizing the LF. Thus, its value
should be small enough (close to zero) to manifest TET, but still nonzero
because otherwise, the stochastic optimization procedure we introduce will
lead to an infinite loop. The latter problem is also resolved by terminating
the optimization procedure after reaching a predefined maximum number of
iterations.
The $Python$ code implementing this procedure is located in our GitHub
repository [25].
## V Results
### V.1 TET Quantum Dimer
In the preceding section, we describe the optimization technique, including
the possible alternatives. We may now apply this scheme to the dimer realm and
test our method from the semiclassical limit to the fully quantum one, using
the grid method described earlier. As we discussed in section I, the optimal
parameters for this case are already known for both the semiclassical [4] and
the quantum regime [13]:
${}\chi_{D}=-\chi_{A}=\frac{\omega_{A}-\omega_{D}}{N}.$ (11)
Our method is successful in obtaining the TET configurations. We fix the
frequencies of the donor and acceptor to $\omega_{D}=-3$ and $\omega_{A}=3$
respectively, the coupling parameter to $\lambda=0.1$, while the time
evolution of Eq. (9) for the acceptor is performed until `maxt` = 25. The Fig.
4a displays the outcome of the optimization procedure for $N=3$ bosons, with
the optimal nonlinearity parameters being:
$(\chi_{D},\chi_{A})=(2,-1.99).$
The illustrated test optimizers produce a LF lower than the threshold
mentioned in Fig. 2, which is set arbitrarily (in this case 0.5). The results
validate the sensitivity of each optimizer to the initial guesses.
We keep the same values for $\omega_{A},\omega_{D},\lambda$, and apply the
grid method for a variety total of bosons in the dimer system. The outcome of
the optimization scheme is shown in Fig. 4b, while we compare our results with
Eq. (11) in Fig. 4c. In every case, the proposed method succeeds in
identifying the TET paths. It is worth mentioning that we can deduce the same
results with the method of splitting the parameter space.
Moreover, our analysis proves that the fully quantum case of $N=1$ boson is of
special interest since TET occurs for a whole set of nonlinearity parameters
$\chi_{A},\chi_{D}$ instead of a single, very limited configuration.
Specifically, we identify this set as
${}\chi_{D}=\chi_{A}+2(\omega_{A}-\omega_{D}).$ (12)
This result is in agreement with previous analytical calculations. To be more
specific, Maniadis et al. in [13] prove that the condition for having TET is
for the detuning function to vanish. The latter is defined as the variation of
the energy of the oscillators during a transfer:
$\epsilon=[H_{D}(N)+H_{A}(0)]-[H_{D}(i)+H_{A}(N-i)].$ (13)
In Eq. (13) $i=0,1,\dots,N$ while $H_{D}$, $H_{A}$ are the donor and acceptor
parts of the Hamiltonian in Eq. (2). In this case, any nonzero $\lambda$ can
raise the degeneracy of the system and complete TET occurs in the limiting of
zero coupling. We can easily prove that the detuning function of Eq. (13) for
one boson vanishes under the parameters of Eq. (12).
For all the cases where the detuning function vanishes, the Hamiltonian
becomes quadratic of the bosons operators:
${}\hat{H}=\hat{H}_{D}(N)-\lambda(a^{\dagger}_{D}a_{A}+a^{\dagger}_{A}a_{D}).$
(14)
Figure 5: Time evolution of the average number of bosons for the three sites
of the trimer system. (a) The time evolution of the expectation values of the
boson number operators for nonlinearity parameters that produce near complete
TET where ($\chi_{A}$, $\chi_{M}$, $\chi_{D}$, $\omega_{D}$, $\omega_{M}$,
$\omega_{A}$, $\lambda$, $N$, maxt) = (-1.5, -38.39, 1.5, -3, -3, 3, 1, 4,
40). (b) The time evolution of the expectation values of the boson number
operators for a system with parameters that don’t produce complete TET, where
($\chi_{A}$, $\chi_{M}$, $\chi_{D}$, $\omega_{D}$, $\omega_{M}$, $\omega_{A}$,
$\lambda$, $N$, maxt) = (-1.5, 1.5, 1.5, -3, -3, 3, 1, 4, 40). For both
figures, the blue, yellow, and red lines refer to the expectation value of the
boson number operator for the donor, middle site, and acceptor respectively.
### V.2 TET Quantum Trimer
The trimer system has been investigated in [26, 11] in the context of a single
electron or boson. We aim to expand this investigation to the arbitrary case
of $N$ bosons. While the method is able to optimize the nonlinearity
parameters of all three sites, it proved computationally consuming. To
circumvent this, we set the acceptor and donor sites to their dimer values,
and optimize the parameter of the middle layer (labeled as “M”). Similarly to
the dimer case, we can set an arbitrary threshold for the LF (LF$<$ 0.2) and
begin the optimization process.
We observe that for TET to occur in the quantum realm, the middle layer had to
exhibit extremely high nonlinearity. One of the trimer systems examined in
this section has the following properties: It is strongly coupled,
$\lambda=1$, with frequencies $\omega_{D}=3,\;\omega_{M}=-3,\;\omega_{A}=-3$
and a maximum number of bosons $N=4$ initially at the donor site.
The nonlinearity parameter required to have TET, in this case, is
$\chi_{M}\approx 38.39$, much higher than that of Eq. (11), compared to the
parameters of the donor and acceptor sites $\chi_{D}=-\chi_{A}=1.5$. It is
important to mention that the opposite value $\chi_{M}\approx-38.39$ will
produce a similar LF value, but still small enough to exceed the threshold and
stop the iterative process. We carry out the same procedure for a variety of
system parameters, as seen in Fig. 7.
Figure 6: Lowest value of the LF in a trimer system with the parameters
displayed on the figure. The other system parameters are ($\omega_{A}$,
$\omega_{D}$, $\lambda$, maxt) = ($\omega_{M}$-1, -$\omega_{M}$+1, 1, 40).
Figure 7: Trimer Results. Absolute values of the nonlinearity parameter of the
middle layer with respect to (a) different values of the maximum number of
bosons in the system, (b) the frequency of the middle layer $\omega_{M}$. In
both cases we use the grid method while the system’s constant parameters for
both figures are ($\omega_{A}$, $\omega_{D}$, $\lambda$, maxt) =
($\omega_{M}$-1, -$\omega_{M}$+1, 1, 40).
.
We display the absolute value of $\chi_{M}$ as both positive and negative
values produce the desired result. We observe that the relation between
$\chi_{M}$ and the frequencies or the number of bosons of the system seems to
differ from the dimer case, as there appears to be a greater correlation with
the former rather than the latter. In the graph some points seem to differ a
lot from others, a fact that is attributed to an optimization process that was
not able to minimize the loss function enough before being terminated by the
rules we defined earlier in section IV. We also have plotted the minimum value
of the LF that we observed in Fig. 6. The time evolution of the boson number
operators for this system is shown in Fig. 5. As we can see in subplot (a) the
expectation value of the boson number operator for the middle layer, in the
optimal case, appears to be zero for all time steps which indicates that
bosons do not stay on this site for any significant amount of time, or at all.
However, this is not observed in cases where the nonlinearity parameters are
set to nonoptimal values, as seen in subplot (b), where the expectation value
is nonzero. It is apparent that the oscillation frequency of the number
operator is larger in the non-resonant system and that complete TET cannot be
achieved.
## VI Discussion
In this work, we introduced a ML method in the context of a quantum many-body
system and showed that its efficient implementation can produce results that
are very hard to obtain with more conventional methods. We focused on the
quantized version of the DNLS equation with arbitrary local energies and
nonlinearities and addressed the question of optimal transfer in between
different sites for the dimer and trimer cases. Since the fully quantum
transfer dimer case is known analytically we compared our method with these
results and showed perfect agreement. This successful comparison between
analytics and ML methods shows that the latter can be used confidently in more
complex cases where results are not known. Subsequently, we applied the method
to the trimer case that cannot be solved analytically. Our method enabled a
detailed search showing the specifics of the resonant transfer, the different
parameter regimes as well as the transfer efficiencies. In terms of physics,
we found in the trimer system that in the nonresonant transfer regime from
donor to acceptor sites the intermediate state retains some of the
probability. In the resonant case, on the other hand, the intermediate site is
essentially not populated. This shows that this site acts as some form of a
barrier between the donor and acceptor sites that can be completely bypassed
in the fully resonant regime. It is noteworthy to point out that the bosons
move in unison over to the acceptor site showing a very interesting collective
behavior in the transfer.
The collective boson transfer can be investigated also in more general chains
with a larger number of sites. The computational challenge is now larger since
the dimensionality of the system becomes large and the calculation of the
Hamiltonian and the evolution of $\langle\hat{N}_{A}(t)\rangle$ slows down. In
this regime, one needs to explore other loss functions that could improve
scalability and/or implementation of meta-learning methods described in [27].
In this work, we tested also alternative optimizers such as ones with
momentum. We found that they were more efficient in finding the resonant
transfer parameter regime but were highly dependent on hyper-parameters that
needed to be also optimized. Finally, it is possible that variants of the
gradient descent algorithm might help in reducing the computational cost[28].
The phenomenon of the collective transfer of bosons in the trimer case opens
up very interesting new questions on the interplay of nonlinearity and
disorder in the fully quantum regime for more extended systems. The ML method
provided in the present work can be readily generalized to this case and be
utilized to investigate this very exciting problem with applications in
condensed matter physics as well as quantum optics.
## ACKNOWLEDGMENTS
We acknowledge the cofinancing of this research by the European Union and
Greek national funds through the Operational Program Crete 2020-2024, under
the call ”Partnerships of Companies with Institutions for Research and
Transfer of Knowledge in the Thematic Priorities of RIS3Crete”, with project
title “Analyzing urban dynamics through monitoring the city magnetic
environment” (project KPHP1 - 0029067).
## AUTHOR CONTRIBUTIONS
I. Andronis and G. Arapantonis contributed equally to this work.
## References
* Ott [2002] E. Ott, _Chaos in Dynamical Systems_, 2nd ed. (Cambridge University Press, 2002).
* Akulin [2014] V. M. Akulin, _Dynamics of Complex Quantum Systems_, 2nd ed. (Springer Dordrecht, 2014).
* Neofotistos _et al._ [2019] G. Neofotistos, M. Mattheakis, G. D. Barmparis, J. Hizanidis, G. P. Tsironis, and E. Kaxiras, Frontiers in Physics 7, 10.3389/fphy.2019.00024 (2019).
* Kopidakis _et al._ [2001] G. Kopidakis, S. Aubry, and G. P. Tsironis, Phys. Rev. Lett. 87, 165501 (2001).
* Hennig and Tsironis [1999] D. Hennig and G. Tsironis, Physics Reports 307, 333 (1999).
* Eilbeck _et al._ [1985] J. Eilbeck, P. Lomdahl, and A. Scott, Physica D: Nonlinear Phenomena 16, 318 (1985).
* Kenkre [2022] V. M. N. Kenkre, _Interplay of Quantum Mechanics and Nonlinearity_, 1st ed. (Springer Cham, 2022).
* Kenkre and Campbell [1986] V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34, 4959 (1986).
* Tsironis and Kenkre [1988] G. Tsironis and V. Kenkre, Physics Letters A 127, 209 (1988).
* Tsironis [1993] G. Tsironis, Physics Letters A 173, 381 (1993).
* Barmparis and Tsironis [2021] G. D. Barmparis and G. P. Tsironis, JOSA B, Vol. 38, Issue 9, pp. C120-C126 38, C120 (2021).
* Tsironis _et al._ [2022] G. P. Tsironis, G. D. Barmparis, and D. K. Campbell, International Journal of Modern Physics B 36, 2240001 (2022), https://doi.org/10.1142/S021797922240001X .
* Maniadis _et al._ [2004] P. Maniadis, G. Kopidakis, and S. Aubry, Physica D: Nonlinear Phenomena 188, 153 (2004).
* Gersch and Knollman [1963] H. A. Gersch and G. C. Knollman, Physical Review 129, 959 (1963).
* Amico and Penna [1998] L. Amico and V. Penna, Phys. Rev. Lett. 80, 2189 (1998).
* Pearsall [2020] T. P. Pearsall, _Quantum Photonics_, 2nd ed. (Springer International Publishing, 2020).
* Karniadakis _et al._ [2021] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Nature Reviews Physics 3, 422 (2021).
* Zhang and Dong [2010] J. M. Zhang and R. X. Dong, European Journal of Physics 31, 591 (2010).
* Abadi _et al._ [2015] M. Abadi _et al._ , TensorFlow: Large-scale machine learning on heterogeneous systems (2015), software available from tensorflow.org.
* Kingma and Ba [2014] D. P. Kingma and J. L. Ba, 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings 10.48550/arxiv.1412.6980 (2014).
* Bertsimas and Tsitsiklis [1993] D. Bertsimas and J. Tsitsiklis, Statistical Science 8, 10 (1993).
* Kennedy and Eberhart [1995] J. Kennedy and R. Eberhart, in _Proceedings of ICNN’95 - International Conference on Neural Networks_, Vol. 4 (1995) pp. 1942–1948 vol.4.
* Brest _et al._ [2006] J. Brest, S. Greiner, B. Boskovic, M. Mernik, and V. Zumer, IEEE transactions on evolutionary computation 10, 646 (2006).
* Elsayed _et al._ [2011] S. M. Elsayed, R. A. Sarker, and D. L. Essam, in _2011 IEEE Congress of Evolutionary Computation (CEC)_ (IEEE, 2011) pp. 1041–1048.
* Andronis and Arapantonis [2022] J. Andronis and G. Arapantonis, Quantum-targeted-energy-transfer, https://github.com/JAndronis/Quantum-Targeted-Energy-Transfer (2022).
* S.Aubry and G.Kopidakis [2003] S.Aubry and G.Kopidakis, A non adiabatic theory for ultrafast and catalytic transfer of electrons at low temperature, in _Localization and Energy Transfer in Nonlinear Systems_ (WORLD SCIENTIFIC, 2003) pp. 1–27.
* Li and Malik [2016] K. Li and J. Malik, Learning to optimize (2016), arXiv:1606.01885v1 .
* Ruder [2016] S. Ruder, An overview of gradient descent optimization algorithms (2016), arXiv:1609.04747 .
|
# Deep Kernel Learning for Mortality Prediction in the Face of Temporal Shift
††thanks: Citation: Rios, M., Abu-Hanna, A. (2021). Deep Kernel Learning for
Mortality Prediction in the Face of Temporal Shift. In: Artificial
Intelligence in Medicine. AIME 2021. Lecture Notes in Computer Science, vol
12721. Springer, Cham. https://doi.org/10.1007/978-3-030-77211-6_22
Miguel Rios
Centre for Translation Studies
University of Vienna
<EMAIL_ADDRESS>
&Ameen Abu-Hanna
Department of Medical Informatics, Amsterdam UMC
University of Amsterdam
<EMAIL_ADDRESS>
###### Abstract
Neural models, with their ability to provide novel representations, have shown
promising results in prediction tasks in healthcare. However, patient
demographics, medical technology, and quality of care change over time. This
often leads to drop in the performance of neural models for prospective
patients, especially in terms of their calibration. The deep kernel learning
(DKL) framework may be robust to such changes as it combines neural models
with Gaussian processes, which are aware of prediction uncertainty. Our
hypothesis is that out-of-distribution test points will result in
probabilities closer to the global mean and hence prevent overconfident
predictions. This in turn, we hypothesise, will result in better calibration
on prospective data.
This paper investigates DKL’s behaviour when facing a temporal shift, which
was naturally introduced when an information system that feeds a cohort
database was changed. We compare DKL’s performance to that of a neural
baseline based on recurrent neural networks. We show that DKL indeed produced
superior calibrated predictions. We also confirm that the DKL’s predictions
were indeed less sharp. In addition, DKL’s discrimination ability was even
improved: its AUC was 0.746 $(\pm$0.014 std), compared to 0.739 ($\pm$ 0.028
std) for the baseline. The paper demonstrated the importance of including
uncertainty in neural computing, especially for their prospective use.
_K_ eywords Deep Kernel Learning, temporal shift, time series, calibration,
Gaussian process, mortality prediction.
## 1 Introduction
In the ICU, the prediction of in-hospital mortality is the task of providing
probabilities for Intensive Care patients to die in the hospital, either in
the ICU or after discharge to another ward. The (early) detection of such
patients is relevant for clinical decision making. Mortality prediction models
(MPMs) are often trained with large collections of electronic health records
(EHR) that contain structured patient information such as demographics and
physiological variables. MPMs based on deep learning are becoming prevalent in
medical applications [18]. One reason for this is that NNs automatically
derive representations for time series data, which may provide predictive
ability superior to that of standard regression models [21, 9]. Specifically,
neural models learn features from the input data by the incremental
composition of simpler layers, resulting in complex representations for non-
linear prediction models [1].
However, patient characteristics, medical technology, and clinical guidelines
change over time, thus forming a challenge for the validity of MPMs for
prospective patients, as these models were learned on historical data [14]. In
particular, due to their flexibility, NNs have the ability to leverage on
slight patterns in the data, but such patterns may not be stable over time and
hence NN models may be sensitive to such temporal shifts causing a change
(usually a drop) in performance [16]. For prediction models of a binary
outcome, not only the discriminatory capability of the model may suffer, but
especially its (mis)calibration. Calibration refers to the correspondence
between the predicted probabilities and the true probabilities. The true
probabilities are estimated on the test set by some measure of averaging the
number of events for a set of patients. Performance drift has consequences for
the task at hand, and the detrimental effects on benchmarking ICUs have been
demonstrated [14]. One way to tackle this problem is to augment NNs with the
notion of uncertainty: whenever the data distribution changes due to shift,
the predictions should be more uncertain [13].
In contrast to NNs, The Gaussian process (GP) is a probabilistic framework for
time series modelling that is able to increase model capacity with the amount
of available data, and to produce uncertainty estimates. A GP characterises a
distribution over possible functions that fit the input data. It is defined by
a Gaussian function with a certain mean and, more importantly, a kernel
function that captures the correlations between any two observations. The
kernel encompasses the notion of uncertainty by performing a pairwise
computation among all input data using some notion of similarity between the
observations. The kernel can be viewed as providing a probability distribution
over all possible models fitting the data.
The prediction models based on GPs successfully model time series data,
incorporate confidence regions to predictions, and offer interpretability of
the variables with the kernel function [20]. Moreover, the GP framework has
been used to develop clinical prediction models [5, 2]. In particular,
Ghassemi et al. [8] use a multitask GP to model time series with physiological
variables and clinical notes for mortality prediction. Directly relevant to
our paper is the proposition in [23] to combine both NNs and GPs on a common
framework of deep kernel learning (DKL). DKL leverages inductive biases from
the NNs and from the non-parametric GPs.
In this paper, we investigate the behaviour of mortality prediction models
based on DKL. In particular, we are interested in inspecting the robustness of
the DKL model to a temporal shift. We also compare it to a strong NN-based
baseline. Our hypothesis is that incorporation of uncertainty improves
predictions. More specifically, we expect the DKL, when faced with uncertainty
in the test set, to provide less extreme predictions that are closer to the
global mean rather than providing overconfident predictions. In turn, the
resultant prediction set would be less sharp than for the baseline model.
Sharpness, which is also referred to refinement in weather forecast [15]
measures the tendency of predictions to be close to 0 and 1. We therefore also
compare the sharpness of both models but check that this does not come at the
cost of discrimination. Finally, we also performed internal validation of the
DKL model with all the population (i.e. no temporal shift) to understand
whether the DKL’s behaviour is specific to temporal validation.
Our main contribution in this paper is the introduction of a DKL model for in-
hospital mortality prediction based on the first hours of an ICU stay in the
context of temporal validation. The GP component in the DKL is shown to be
robust to the shift in population and produces better calibrated predictions,
without sacrificing discrimination. Our feature extraction is based on an open
source benchmark [9] using the publicly available MIMIC-III [11] database.
This facilitates the reproducibility of our results111Code is available at:
https://github.com/mriosb08/dkl-temporal-shift.git.
## 2 Deep Kernel Learning
The Gaussian Process [19] is a Bayesian non-parametric framework based on
kernels for regression and classification. The set of functions that describes
a given input data is possibly infinite and the GP assigns a probability to
each one. For a dataset
$\mathcal{X}=\left\\{\left(\mathbf{x}_{1},y_{1}\right),\left(\mathbf{x}_{2},y_{2}\right),\ldots,\left(\mathbf{x}_{n},y_{n}\right)\right\\}$
where $\mathbf{x}$ is an input vector and $y$ a corresponding output, we want
to learn a function $f$ that is inferred from a GP prior:
$\displaystyle
f(\mathbf{x})\thicksim\operatorname{GP}(m(\mathbf{x}),k(\mathbf{x},\mathbf{x}^{\prime}))$
(1a)
where $m(\mathbf{x})$ defines a mean (often set to 0) and
$k(\mathbf{x},\mathbf{x}^{\prime})$ defines the covariance in the form of a
kernel function. The kernel function models the covariance between all
possible pairs $(\mathbf{x},\mathbf{x}^{\prime})$ and provides a measure of
uncertainty. The choice of kernel determines properties of the function that
we want to learn, usually this choice is based on background knowledge of the
problem.
Wilson et al. [22] propose kernels based on deep learning architectures for GP
regression. The DKL employs a GP with a base kernel as the last hidden layer
of a NN. In other words, the DKL is a pipeline for learning complex NN
features, and a distribution over functions that fit our input data. The base
kernel $k\left(\mathbf{x},\mathbf{x}^{\prime}\mid\theta\right)$ with
hyperparameters $\theta$ is parameterized by a non-linear function.
$\displaystyle
k\left(\mathbf{x},\mathbf{x}^{\prime}\mid\theta\right)\rightarrow
k\left(g\left(\mathbf{x},\omega\right),g\left(\mathbf{x}^{\prime},\omega\right)\mid\theta,\omega\right),$
(2a)
where $g(\mathbf{x},\omega)$ is a NN architecture with weights $\omega$. In
addition, the DKL jointly learns the NN weights and kernel hyperparameters
under the GP probabilistic framework. Learning a GP involves computing the
kernel function, and finding the best kernel hyperparameters. The DKL
optimises both the kernel hyperparameters and the NN weights, by maximising
the marginal likelihood.
In Figure 1, we define the architecture for extracting features
$g(\mathbf{x},\omega)$, $\mathbf{x}_{i}$ denotes the input vector in the ith
element of $\mathcal{X}$.
Figure 1: NN architecture $g(\mathbf{x},\omega)$ for extracting features
$\mathbf{f}_{i}$ for the GP prediction layer.
The input features are first projected with an affine layer
($\operatorname{linear}(.)$), then fed to a bidirectional LSTM
($\operatorname{birnn}(.)$) [10] for encoding time series. Next the result
goes through an affine layer with a non-linearity ($\operatorname{ReLU}(.)$)
that combines the hidden states of the bidirectional LSTM. Next the features
$\mathbf{f}_{i}$ are summarised by averaging ($\operatorname{avg}(.)$) and
then fed to the GP layer.
## 3 Experiments
The Medical Information Mart for Intensive Care (MIMIC-III) database includes
over 60,000 ICU stays across 40,000 critical care patients [11]. Harutyunyan
et al. [9] propose a public benchmark and baselines based on MIMIC-III for
modelling mortality, length of stay, physiologic decline, and phenotype
classification. We use the benchmark for predicting in-hospital mortality
based on the first 48 hours of an ICU stay. The cohort excludes all ICU stays
with unknown length-of-stay, patients under 18, multiple ICU stays, stays less
than 48 hours, and no observations during the first 48 hours. The in-hospital
mortality class is defined by comparing the date of death against hospital
admissions and discharge times with a resulting mortality rate of 13.23%.
We use the benchmark to extract $17$ input physiological variables (i.e.
features), that are a subset of the Physionet challenge
222https://physionet.org/content/challenge-2012/1.0.0/.
The benchmark [9] code processes the time series data with imputation of
missing values with the previous hour, and normalisation from MIMIC-III. The
normalisation of the features is performed by subtracting the mean and
dividing by the standard deviation. The features also provide a binary mask
for each variable indicating which time-step is imputed. All categorical
variables are encoded using one-hot vectors (e.g. Glasgow coma scales). The
final feature vector is formed by the concatenation of the clinical variables
and the one-hot vectors with a total of $76$ features. The clinical variables
are shown in Table1.
Variable
---
Capillary refill rate
Diastolic blood pressure
Fraction inspired oxygen
Glascow coma scale eye opening
Glascow coma scale motor response
Glascow coma scale total
Glascow coma scale verbal response
Glucose
Heart Rate
Height
Mean blood pressure
Oxygen saturation
Respiratory rate
Systolic blood pressure
Temperature
Weight
pH
Table 1: Clinical variables used in our experiments from MIMIC-III.
We use the architecture $g(.)$ as the baseline defined as: BiLSTM, which is
based on a bidirectional LSTM for feature representation, and a linear
prediction layer. We implement the DKL model with GPyTorch [7], with the
following components: the RBF kernel as the base kernel, feature extractor
$g(.)$, and grid size $100$ which is the number of inducing points used to
approximate the GP for faster computations. The computation of the posterior
distribution in the GP is expensive and several methods have been proposed to
accelerate it by approximating it with a function over a set of inducing
points [17, 24]. In addition, we perform a simple ablation on the architecture
by replacing the bidirectional LSTM with a LSTM for both models, baseline and
DKL defined as: LSTM, and DKL-LSTM.
We use the following hyperparameters: optimiser Adam [12], learning rate
$1\mathrm{e}{-3}$, epochs 30, encoder size 16, hidden size 16, batch size 100,
dropout 0.3 applied after the $\operatorname{linear}$ layer. We perform model
selection with the validation dataset based on AUC-ROC.
### 3.1 Temporal shift: strategy and results
The MIMIC-III dataset includes data using the CareVue electronic patient
record (EPR) system from 2001 to 2008. From 2008 to 2012 the MetaVision system
was used instead. In the first experiment for inspecting temporal shift, we
split the datataset into the CareVue period for training with $9,646$
instances and $1,763$ for validation (for tuning the hyper-parameters), and
the data in the MetaVision period with $7,689$ as the test set. We excluded
patients present in both registries. This constitutes a temporal validation
strategy in which the model is tested on data collected in the future relative
to the data on which it has learned. This means that the model faces possible
temporal shift due to changes that occur in time, and indeed possibly also due
to the change of the EPR system that collects the data that could have
affected the workflow and/or the way of registration. Performance was measured
in terms of: Discrimination, by the AUC-ROC; the balance between the positive
predicted value and sensitivity, by the AUC-PR; the accuracy of predictions by
the Brier score; and calibration by calibration graphs and the Cox
recalibration approach [3] in which the observed outcome in the test set is
regressed using logistic regression on the log odds of the predictions. If the
predictions were perfectly calibrated then the linear predictor of this model
would have an intercept of 0 and a slope of 1. We test deviations from these
ideal value of 0 and 1, respectively. To test our hypothesis whether the DKL
approach provides more conservative predictions due to uncertainty for areas
in the test set, we measure the (un)sharpness of the predictions. We use the
following measure of unsharpness: $\frac{\sum_{1}^{N}{p_{i}(1-p_{i})}}{N}$
where $p_{i}$ is the $i$th prediction and N is number of observations.
| Validation | Test
---|---|---
Model | AUC-ROC | AUC-PR | AUC-ROC | AUC-PR
LSTM | $0.838\pm 0.003$ | $0.532\pm 0.006$ | $0.693\pm 0.027$ | $0.317\pm 0.037$
BiLSTM | $0.857\pm 0.002$ | $0.572\pm 0.007$ | $0.739\pm 0.028$ | $\textbf{0.386}\pm 0.018$
DKL-LSTM | $0.854\pm 0.002$ | $0.562\pm 0.010$ | $0.701\pm 0.033$ | $0.327\pm 0.026$
DKL | $0.856\pm 0.002$ | $0.569\pm 0.004$ | $\textbf{0.746}\pm 0.014$ | $0.373\pm 0.018$
Table 2: In-hospital mortality results with a temporal population shift over
10 runs $\pm$ one standard deviation. The training and validation datasets are
on CareVue (2001-2008), and the test on MetaVision (2008-2012).
(a)
(b)
Figure 2: Receiver operating characteristic curve (a) and calibration curve
(b) for in-hospital mortality with temporal shift in population.
Table 2 shows the AUC-ROC and AUC-PR results for in-hospital mortality with a
temporal shift in population. The baseline outperforms the DKL model on the
validation (tuning) dataset for both metrics. On the test dataset, however,
the DKL shows competitive performance on the AUC-ROC. We use the best run from
the validation based on the AUC-ROC for reporting the ROC and calibration
curves. In addition, we select the best performing models from Table 2 based
on AUC-ROC, namely BiLSMT and DKL, for comparing the calibration and ROC
curves. The LSTM models consistently underperform compared to the
bidirectional ones. Figure 2 shows the ROC and calibration curves for in-
hospital mortality with a temporal shift. The Brier score for the DKL is 0.101
which is better that the 0.109 of the BiLSTM. The DKL outperforms the baseline
and it shows better calibration.
In the Cox re-calibration on both models the BiLSTM had a calibration
intercept of 1.965 (1.88, 2.049), and slope of 0.538 (0.5, 0.577) compared to
the DKL’s of 0.6615 (0.586, 0.734), 0.712 (0.652, 0.772). Although both models
deviated significantly from the ideal values (of 0 and 1), the DKL showed
significantly much better calibration. The DKL’s predictions were also much
less sharp: unsharpness of 0.061 for DKL versus 0.025 for BiLSTM.
### 3.2 Experiment 2: Internal validation
We report the results with all the sources (2001-2012) for in-hospital
mortality, with no shift in population. The training, validation and test
datasets consisted of respectively $14,681$, $3,222$, and $3,236$ instances.
| Validation | Test
---|---|---
Model | AUC-ROC | AUC-PR | AUC-ROC | AUC-PR
LSTM | $0.843\pm 0.003$ | $0.513\pm 0.006$ | $0.840\pm 0.005$ | $0.434\pm 0.008$
BiLSTM | $0.858\pm 0.004$ | $0.549\pm 0.010$ | $\textbf{0.851}\pm 0.004$ | $\textbf{0.478}\pm 0.016$
DKL-LSTM | $0.838\pm 0.002$ | $0.485\pm 0.014$ | $0.841\pm 0.003$ | $0.425\pm 0.013$
DKL | $0.854\pm 0.004$ | $0.536\pm 0.010$ | $0.847\pm 0.005$ | $0.454\pm 0.018$
Table 3: In-hospital mortality results over 10 runs $\pm$ one standard
deviation. Validation and test dataset from all sources (2001-2012).
(a)
(b)
Figure 3: Receiver operating characteristic curve (a) and calibration curve
(b) for in-hospital mortality with all sources.
Table 3 shows the AUC-ROC and AUC-PR results for in-hospital mortality with
all sources (2002-2012). The baseline outperforms the DKL model on the test
dataset for both metrics the AUC-ROC, and AUC-PR. Figure 3 shows the ROC and
calibration curves for in-hospital mortality with all sources. Both of our
models perform similarly on the ROC curve. The Brier score for the DKL is
$0.082$ slightly better than the $0.084$ of the BiLSTM.
In the Cox re-calibration the BiLSTM’s calibration intercept was -0.358
(-0.49, -0.229), and slope 0.802 (0.726, 0.88); compared to the DKL’s -0.066
(-0.185, 0.05), and 1.177 (1.062, 1.298). Unlike the BiLSTM the DKL showed no
significant deviations from the ideal values of 0 and 1. The DKL was slightly
more unsharp: 0.089 versus 0.081 for the BiLSTM.
## 4 Related Work
Dürichen et al. [5] propose a multi-task GP that jointly models physiological
variables for clinical time series. Cheng et al. [2] develop a real-time
clinical prediction model based on a GP model. Aside from producing confidence
regions in the predictions, the GP also scales to large patient databases, and
produces interpretable relations across (clinical) variables. The
interprtability is produced by inspecting the correlation across variables in
the kernel function. Futoma et al. [6] propose a sepsis prediction model based
on a pipeline with a GP that produces inputs for a NN classifier. The model
takes into account uncertainty estimates and outperforms strong sepsis
prediction baselines. On the other hand, our DKL model uses RNNs to model the
time series physiological variables and feed the resulting features into the
GP for prediction. Our work, however, is the first to investigate DKL in the
context of temporal shift.
## 5 Conclusions and Future Work
We investigated the DKL framework for the task of in-hospital mortality
prediction under a temporal shift in population. The DKL shows competitive
performance compared to a strong NN baseline, as well as a better calibration.
However, when the test dataset is in the same distribution as the training
both models show similar results. The GP component does not degrade the
overall performance, and in addition, it provides extra guarantees such as
uncertainty estimates. By contrasting the two experiments and inspecting the
sharpness of the predictions we can ascribe the improved performance on the
test set to the robustness of the GP when facing uncertainty.
For future work, we will analyse different base kernels, evaluate the
uncertainty estimate of the DKL, and use the framework described in [4] for
better understanding of discrepancies in performance over time.
## References
* Bengio [2009] Bengio, Y.: Learning deep architectures for AI. Foundations and Trends in Machine Learning 2(1), 1–127 (2009)
* Cheng et al. [2020] Cheng, L.F., Darnell, G., Chivers, C., Draugelis, M.E., Li, K., Engelhardt, B.E.: Sparse multi-output Gaussian processes for medical time series prediction. BMC Medical Informatics and Decision Making 20(152) (2020)
* Cox [1958] Cox, D.R.: Two further applications of a model for binary regression. Biometrika 45, 562–565 (1958)
* Debray et al. [2015] Debray, T.P.A., Vergouwe, Y., Koffijberg, H., Nieboer, D., Steyerberg, E.W., Moons, K.G.M.: A new framework to enhance the interpretation of external validation studies of clinical prediction models. Journal of clinical epidemiology 68 3, 279–89 (2015)
* Dürichen et al. [2014] Dürichen, R., Pimentel, M., Clifton, L., Schweikard, A., Clifton, D.: Multi-task gaussian processes for multivariate physiological time-series analysis. IEEE transactions on bio-medical engineering 62 (2014)
* Futoma et al. [2017] Futoma, J., Hariharan, S., Heller, K.: Learning to detect sepsis with a multitask gaussian process rnn classifier. In: International Conference on Machine Learning. JMLR.org (2017)
* Gardner et al. [2018] Gardner, J.R., Pleiss, G., Bindel, D., Weinberger, K.Q., Wilson, A.G.: Gpytorch: Blackbox matrix-matrix gaussian process inference with GPU acceleration. CoRR abs/1809.11165 (2018)
* Ghassemi et al. [2015] Ghassemi, M., Pimentel, M.A.F., Naumann, T., Brennan, T., Clifton, D.A., Szolovits, P., Feng, M.: A multivariate timeseries modeling approach to severity of illness assessment and forecasting in icu with sparse, heterogeneous clinical data. In: Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence. p. 446–453. AAAI’15, AAAI Press (2015)
* Harutyunyan et al. [2019] Harutyunyan, H., Khachatrian, H., Kale, D.C., Ver Steeg, G., Galstyan, A.: Multitask learning and benchmarking with clinical time series data. Scientific Data 6(1) (2019)
* Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Computation 9(8), 1735–1780 (1997)
* Johnson et al. [2016] Johnson, A.E., Pollard, T.J., Shen, L., Li-wei, H.L., Feng, M., Ghassemi, M., Moody, B., Szolovits, P., Celi, L.A., Mark, R.G.: Mimic-iii, a freely accessible critical care database. Scientific data 3, 160035 (2016)
* Kingma and Ba [2014] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization (2014), cite arxiv:1412.6980Comment: Published as a conference paper at the 3rd International Conference for Learning Representations, San Diego, 2015
* Mackay [1992] Mackay, D.J.C.: Bayesian Methods for Adaptive Models. Ph.D. thesis, USA (1992), uMI Order No. GAX92-32200
* Minne et al. [2012] Minne, L., Eslami, S., de Keizer, N., de Jonge, E., de Rooij, S.E., Abu-Hanna, A.: Effect of changes over time in the performance of a customized saps-ii model on the quality of care assessment. Intensive care medicine (2012)
* Murphy and Winkler [1987] Murphy, A., Winkler, R.: A general framework for forecast verification. Monthly Weather Review 115, 1330–1338 (1987)
* Nestor et al. [2019] Nestor, B., McDermott, M.B.A., Boag, W., Berner, G., Naumann, T., Hughes, M.C., Goldenberg, A., Ghassemi, M.: Feature robustness in non-stationary health records: Caveats to deployable model performance in common clinical machine learning tasks. In: Doshi-Velez, F., Fackler, J., Jung, K., Kale, D., Ranganath, R., Wallace, B., Wiens, J. (eds.) Proceedings of the 4th Machine Learning for Healthcare Conference. Proceedings of Machine Learning Research, vol. 106, pp. 381–405. PMLR, Ann Arbor, Michigan (2019)
* Quiñonero Candela et al. [2007] Quiñonero Candela, J., Ramussen, C.E., Williams, C.K.I.: Approximation methods for gaussian process regression. Tech. Rep. MSR-TR-2007-124 (2007)
* Rajkomar et al. [2018] Rajkomar, A., Oren, E., Chen, K., Dai, A.M., Hajaj, N., Liu, P.J., Liu, X., Sun, M., Sundberg, P., Yee, H., Zhang, K., Duggan, G.E., Flores, G., Hardt, M., Irvine, J., Le, Q.V., Litsch, K., Marcus, J., Mossin, A., Tansuwan, J., Wang, D., Wexler, J., Wilson, J., Ludwig, D., Volchenboum, S.L., Chou, K., Pearson, M., Madabushi, S., Shah, N.H., Butte, A.J., Howell, M., Cui, C., Corrado, G., Dean, J.: Scalable and accurate deep learning for electronic health records. CoRR abs/1801.07860 (2018)
* Rasmussen and Williams [2005] Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press (2005)
* Roberts et al. [2012] Roberts, S., Osborne, M., Ebden, M., Reece, S., Gibson, N., Aigrain, S.: Gaussian processes for timeseries modelling. Philosophical Transactions of the Royal Society (2012)
* Shickel et al. [2018] Shickel, B., Loftus, T.J., Ozrazgat-Baslanti, T., Ebadi, A., Bihorac, A., Rashidi, P.: Deepsofa: A real-time continuous acuity score framework using deep learning. CoRR abs/1802.10238 (2018)
* Wilson et al. [2016a] Wilson, A.G., Hu, Z., Salakhutdinov, R.R., Xing, E.P.: Stochastic variational deep kernel learning. In: Lee, D.D., Sugiyama, M., Luxburg, U.V., Guyon, I., Garnett, R. (eds.) Advances in Neural Information Processing Systems 29, pp. 2586–2594. Curran Associates, Inc. (2016a)
* Wilson et al. [2016b] Wilson, A.G., Hu, Z., Salakhutdinov, R., Xing, E.P.: Deep kernel learning. In: Gretton, A., Robert, C.C. (eds.) Proceedings of the 19th International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 51, pp. 370–378. PMLR, Cadiz, Spain (2016b)
* Wilson and Nickisch [2015] Wilson, A.G., Nickisch, H.: Kernel interpolation for scalable structured gaussian processes (kiss-gp). In: International Conference on Machine Learning. JMLR.org (2015)
|
Characterizations Of Weakly Conformally Flat
And Quasi Einstein Manifolds
Ramesh Sharma
University of New Haven, West Haven, CT 06516, USA, E-mail:
<EMAIL_ADDRESS>
###### Abstract
First, we show that a warped product of a line and a fiber manifold is weakly
conformally flat and quasi Einstein if and only if the fiber is Einstein.
Next, we characterize and classify contact (in particular, $K$-contact)
Riemannian manifold satisfying weakly (and doubly weakly) conformally flat and
quasi-Einstein ($\eta$-Einstein) conditions. Finally, we provide local
classification and characterization of a semi-Riemannian (including the
4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic
conformal (including closed) vector field, in terms of Petrov types and Bach
tensor.
## 1 Introduction
In [6] and [7], Chen and Yano introduced the idea of a Riemannian manifold
with quasi constant curvature and defined it as a Riemannian manifold ($M,g$)
which is conformally flat and whose Ricci tensor $Ric$ satisfies the condition
$Ric=ag+bu\otimes u$ (1.1)
where $a$ and $b$ are smooth functions and $u$ is a 1-form metrically
equivalent to a unit vector field $U$ on $M$. Examples of quasi constant
curvature manifolds are conformally flat hypersurfaces of a space of constant
curvature and canal hypersurfaces (without singularities) of a Euclidean
space. We let $g$ be semi-Riemannian and $g(U,U)=\epsilon$, such that $U$ is
spacelike if $\epsilon=1$ and timelike if $\epsilon=-1$. Another example of a
quasi constant curvature semi-Riemannian manifold is the warped product
$I\times_{f}F$ of a real open interval $I$ and a Riemannian manifold (fiber)
$F$ of constant curvature, with warping function $f>0$ on $I$. In case $I$ is
timelike with line-element $-dt^{2}$, $I\times_{f}F$ is the well known
Friedman - Robertson - Walker ($FRW$) cosmological model (see, for example,
Hawking and Ellis [13]).
Remark: The equation (1.1) defines a quasi Einstein manifold in the sense of
Chaki and Maity [5], existence and examples of these manifolds were provided
by De and Ghosh [9]. In the literature, there is another concept of a quasi
Einstein manifold in terms of Bakry-Emery tensor, studied by Case, Shu and Wei
[4].
In this paper, we follow the former definition [equation (1.1)]. Examples of a
quasi Einstein manifold are $\eta$-Einstein Sasakian manifolds (in particular,
Sasakian space forms and Heisenberg groups with left invariant metrics as
expanding non-gradient Ricci solitons). For details, we refer to Blair [2],
Chow et al. [8] and Ghosh and Sharma [12]. Another example with Lorentzian
metric $g$ and a unit timelike vector field $U$, is the perfect fluid solution
of Einstein’s field equations [13], described by
$Ric=4\pi(\rho-p)g+8\pi(\rho-3p)$
on a 4-dimensional spacetime with energy density $\rho$ and pressure $p$.
Henceforth we assume the dimension of a semi-Riemannian manifold ($M,g$) to be
$>3$ and denote arbitrary vector fields on $M$ by $X,Y,Z$. Let us weaken the
conformal flatness and say that a semi-Riemannian manifold ($M,g$) is weakly
conformally flat if and only if
$W(X,Y,V)=0,$ (1.2)
for a non-zero vector field $V$ on $M$, and where $W$ denotes the Weyl tensor
of type (1,3).
In dimension 4, a weakly conformally flat ($M,g$) with a non-null $V$ (i.e.
$g(V,V)\neq 0$) is conformally flat. This follows from the following result of
Eardley et al. [10]. “If a 4-dimensional semi-Riemannian manifold has its Weyl
tensor annihilating a non-null (spacelike or timelike) vector field, then the
Weyl tensor vanishes identically”. Though this result has been proved for a
spacetime, it holds in general, for any semi-Riemannian manifold. The proof is
based on symmetry properties, tracelessness and combinatorial computations.
In this paper we provide characterizations and classifications of manifolds
that are either weakly conformally flat or quasi Einstein, or both within the
frameworks of warped product, Lorentzian and contact geometries. First, we
establish the following result.
###### Theorem 1.1
The semi-Riemannian warped product $(M,g)=I\times_{f}F$ of dimension $n\geq
4$, with metric $g=\epsilon dt^{2}+f^{2}(t)g_{F}$ ($g_{F}$ being the metric on
a Riemannian manifold $F$) is weakly conformally flat and quasi-Einstein if
any one of the following conditions hold: (i) $g_{F}$ is Einstein, (ii)
$W(x,U,U,y)=0$ for arbitrary vector fields $x$ and $y$ tangent to $F$ and $U$
the lift of $\partial_{t}$ ($t$ being the standard coordinate on $I$
(spacelike or timelike)), (iii) Weyl tensor of $g$ is harmonic. Under any one
of these conditions, $g$ is Bach flat.
Remark: It follows from the above theorem that an example of a weakly
conformally flat and quasi-Einstein is $I\times_{f}F$ with Einstein fiber $F$.
In the 4-dimensional Lorentzian case ($\epsilon=-1$), it is the Friedman -
Robertson - Walker ($FRW$) cosmological model, as mentioned earlier.
Next, we recall from Blair [2] that a conformally flat contact Riemannian
manifold whose Reeb vector field $\xi$ is everywhere an eigen vector of its
Ricci tensor, is either of constant curvature 1, or flat (in which case the
dimension is 3). Prompted by this result, we consider a $K$-contact manifold
$M$ (for which we know that $\xi$ is Killing and is an eigen vector of the
Ricci tensor) and check the impact of the doubly weakly conformally flat
condition $W(X,\xi,\xi)=0$ and weakly conformally flat condition $W(X,Y)\xi=0$
. More precisely, we prove the following result.
###### Proposition 1.1
A $K$-contact manifold $M$ satisfies the doubly weakly conformally flat
condition $W(X,\xi,\xi)=0$ if and only if $M$ is $\eta$-Einstein. If a
$K$-contact manifold satisfies the weakly conformally flat condition
$W(X,Y)\xi=0$, then it is Sasakian and $\eta$-Einstein (i.e. quasi Einstein
with $U=\xi$).
Now we provide a classification of a contact Riemannian manifold that is
weakly conformally flat and quasi-Einstein with $U$ as the Reeb vector field,
by establishing the following result.
###### Theorem 1.2
If a contact Riemannian manifold is weakly conformally flat and quasi-Einstein
with $U$ as the Reeb vector field $\xi$, then it is Sasakian, flat
(3-dimensional), or locally isometric to $SU(2)$ or $SL(2,\mathbb{R})$ with a
left invariant metric.
## 2 Preliminaries
Let us briefly review the basic formulas of the $n$-dimensional semi-
Riemannian warped product $M=I\times_{f}F$ of an open real interval $I$ and an
($n-1$)-dimensional semi-Riemannian manifold $F$ (fiber), with warping
function $f>0$ on $I$ such that the metric $g$ on $M$ is given by
$g=\pi^{*}(g_{I})+(f\circ\pi)^{2}\sigma^{*}(g_{F})$
where $\pi$ and $\sigma$ are projections from $I\times_{f}F$ onto $I$ and $F$
respectively, $g_{I}$ and $g_{F}$ are the metrics on $I$ and $F$. Locally, we
express $g$ as
$\epsilon dt^{2}+f^{2}(t)(g_{F})_{ij}dx^{i}dx^{j}$
where $\epsilon=1$ or -1 according as $g(\partial_{t},\partial_{t})=1$ or -1,
$t$ is the standard coordinate on $I$ and $x^{i}$ are local coordinates on
$F$. If $U$ is the lift of $\partial_{t}$ and $x$, $y$, $z$, $w$ denote
arbitrary vector fields tangent to $F$, then we have the following formulas
$R(x,U)U=-\frac{\ddot{f}}{f}x,R(x,y)U=0$ (2.1)
$R(x,U)y=\frac{\epsilon\ddot{f}}{f}g(x,y)U$ (2.2)
$g(R(x,y,z),w)=g({{}^{F}}R(x,y,z),w)-\epsilon\bigg{(}\frac{\dot{f}}{f}\bigg{)}^{2}[g(y,z)g(x,w)-g(x,z)g(y,w)]$
(2.3) $Ric(U,U)=-\frac{n-1}{f}\ddot{f}$ (2.4) $Ric(U,x)=0$ (2.5)
$Ric(x,y)={{}^{F}}Ric(x,y)-\bigg{[}\frac{\epsilon\ddot{f}}{f}+(n-2)\bigg{(}\frac{\dot{f}}{f}\bigg{)}^{2}\bigg{]}g(x,y)$
(2.6)
$r=\frac{{{}^{F}}r}{f^{2}}-2\epsilon(n-1)\frac{\ddot{f}}{f}-(n-1)(n-2)\bigg{(}\frac{\dot{f}}{f}\bigg{)}^{2}$
(2.7)
where an over-dot means derivative with respect to $t$, and $R$, $Ric$ and $r$
denote the curvature tensor, Ricci tensor and scalar curvature of $M$, while
the symbols with a superscript $F$ denote the corresponding quantities of $F$.
For details we refer to O’Neill [15].
Secondly, we briefly review contact geometry. A ($2m+1$)-dimensional smooth
manifold $M$ is said to be a contact manifold if it carries a smooth 1-form
$\eta$ such that $\eta\wedge(d\eta)^{m}\neq 0$ (where $\wedge$ is the exterior
wedge product and $d$ is the exterior derivation). For a given contact form
$\eta$, there exists a vector field (Reeb vector field) $\xi$ such that
$\eta(\xi)=1$ and $(d\eta)(\xi,.)=0$. Polarizing $d\eta$ on the contact
subbundle ($\eta=0$) we obtain a Riemannian metric $g$ and a (1,1)-tensor
field $\varphi$ such that
$(d\eta)(X,Y)=g(X,\varphi Y),\eta(X)=g(\xi,X),\varphi^{2}=-I+\eta\otimes\xi.$
(2.8)
The Riemannian metric $g$ is an associated metric of $\eta$ and
($\varphi,\eta,\xi,g$) is a contact Riemannian structure on $M$. The
(1,1)-tensor field $h=\frac{1}{2}\pounds_{\xi}\varphi$ ($\pounds$ denoting the
Lie-derivative operator) on a contact Riemannian manifold is self-adjoint,
trace-free and anti-commutes with $\varphi$. The following formulas hold on a
contact Riemannian manifold:
$\nabla_{X}\xi=-\varphi X-\varphi hX,$ (2.9) $Ric(\xi,\xi)=2m-|h|^{2}.$ (2.10)
A contact Riemannian metric is said to be $K$-contact if $\xi$ is Killing,
equivalently, $h=0$, or
$R(X,\xi)\xi=X-\eta(X)\xi$ (2.11)
It follows from (2.10) that $K$-contact metrics are maxima for the Ricci
curvature along $\xi$. A contact Riemannian metric on $g$ on $M$ is called
Sasakian if the almost Kaehler structure induced on the cone $C(M)$ over $M$
with metric $dr^{2}+r^{2}g$, is integrable (i.e. Kaehler). This Sasakian
condition on a contact Riemannian manifold is equivalent to
$R(X,Y)\xi=\eta(Y)X-\eta(X)Y.$ (2.12)
A Sasakian metric is $K$-contact, however, the converse holds in general, only
in dimension 3. A generalization of a Sasakian manifold is the
($k,\mu$)-contact manifold defined as a contact Riemannian manifold $M$
satisfying the nullity condition:
$R(X,Y)\xi=k(\eta(Y)X-\eta(X)Y)+\mu(\eta(Y)hX-\eta(X)hY)$ (2.13)
for real constants $k$, $\mu$. We know that $k\leq 1$, with equality when $M$
is Sasakian. For $k<1$ we have
$\displaystyle Ric(X,Y)=$ $\displaystyle(2m-2-m\mu)g(X,Y)+(2m-2+\mu)g(hX,Y)$
(2.14) $\displaystyle+$ $\displaystyle(m(2k+\mu)-2m+2)\eta(X)\eta(Y),$
$r=[2m-2+k-m\mu].$ (2.15)
Finally, a contact Riemannian manifold $M$ is said to be $\eta$-Einstein
(Boyer, Galicki and Matzeu [3]) if there exist smooth functions $a$ and $b$ on
$M$ such that
$Ric=ag+b\eta\otimes\eta.$ (2.16)
For a $K$-contact manifold (in particular, Sasakian manifold) of dimension $>$
3, we know that $a$ and $b$ are constants. An example of an $\eta$-Einstein
manifold is the Sasakian space-form which is a Sasakian manifold of constant
$\varphi$-sectional curvature. Sasakian geometry has been extensively studied
since its recently perceived relevance in string theory. Sasakian Einstein
metrics have received a lot of attention in physics, for example, $p$-brane
solutions in superstring theory, Maldacena conjecture (AdS/CFS duality) [14].
## 3 Proofs of The Results
Proof of Theorem 1.1 The Weyl tensor $W$ of type (1,3) is defined by
$\displaystyle W(X,Y,Z)=$ $\displaystyle
R(X,Y)Z+\frac{1}{n-2}[Ric(X,Z)Y-Ric(Y,Z)X+g(X,Z)QY$ (3.1) $\displaystyle-$
$\displaystyle g(Y,Z)QX]-\frac{r}{(n-1)(n-2)}[g(X,Z)Y-g(Y,Z)X]$
on an $n$-dimensional semi-Riemannian manifold ($M,g$), where $Q$ is the Ricci
operator defined by $g(QX,Y)=Ric(X,Y)$. Using (3.1) and the formulas (2.1)
through (2.7) and after a lengthy computation we obtain the following
equations
$g(W(x,U,U),y)=-\frac{\epsilon}{n-2}{{}^{F}}Ric^{0}(x,y)$ (3.2)
$g(W(x,y,z),U)=0$ (3.3) $\displaystyle g(W(x,y,z),w)=$ $\displaystyle
g({{}^{F}}W(x,y,z),w)-\frac{1}{(n-2)(n-3)}[g(y,w){{}^{F}}Ric^{0}(x,z)$ (3.4)
$\displaystyle-$ $\displaystyle
g(x,w){{}^{F}}Ric^{0}(y,z)+g(x,z){{}^{F}}Ric^{0}(y,w)$ $\displaystyle-$
$\displaystyle g(y,z){{}^{F}}Ric^{0}(x,w)]$
where $x$, $y$, $z$, $w$ are tangent to $F$, ${{}^{F}}W$ the Weyl tensor of
$F$ and a superscript $0$ means the traceless part. Parts (i) and (ii) of the
theorem follows from (2.6), (3.2) and (3.3). Part (iii) follows from the
following result of Gebarowski [11], ”The Weyl tensor of the warped product
$I\times_{f}F$ is harmonic (divergence - free) if and only if $F$ is
Einstein”. Finally, under any one of the conditions (i), (ii) and (iii), the
components of $W$ along $U$ are zero. The Bach tensor $B$ introduced by Bach
[1] in the context of conformal relativity, is defined as a symmetric second
order tensor with components
$B_{ab}=\frac{1}{n-1}\nabla^{c}\nabla^{d}W_{cabd}+\frac{1}{n-2}R^{cd}W_{cabd}$
(3.5)
where $R^{cd}$ are the components of the Ricci tensor. Under any one of the
conditions $F$ is Einstein and hence by Gebarowski’s result, $W$ is harmonic,
also $g$ is quasi-Einstein (implied by $g$ being Weakly conformally flat and
quasi Einstein). Using all these facts in (3.5) shows that $B_{ab}=0$,
completing the proof.
Remark: Taking $x=\partial_{i}$, $y=\partial_{j}$ ($\partial_{i}$ is the
coordinate basis $\mathfrak{X}(F)$, we can write $g(W(x,U,U),y)$ as
$W_{i00j}$, because $U$ is the lift of $\partial_{t}$ and $t=x^{0}$ (the
coordinate on $I$). The components $W_{i00j}$ are known as the components of
the electric part of the Weyl tensor (see [17]) of the spacetime manifold (in
which case $\epsilon=-1$).
Proof of Proposition 1.1 Let ($M,\eta,\xi,g$) be a ($2m+1$)-dimensional
$K$-contact manifold. Using the definition (3.1) of the Weyl tensor and the
$K$-contact property (2.11) we find that
$W(X,\xi)\xi=\frac{r-2m}{2m(2m-1)}[X-\eta(X)\xi]-\frac{1}{2m-1}[QX-2m\eta(X)\xi]$
(3.6)
If $W(X,\xi)\xi=0$, then the above equation implies
$QX=\bigg{(}\frac{r}{2m}-1\bigg{)}X+\bigg{(}2m+1-\frac{r}{2m}\bigg{)}\eta(X)\xi$
and hence $g$ is $\eta$-Einstein. Conversely, let $g$ be $\eta$-Einstein, i.e.
(2.16) holds. Tracing it gives
$r=(2m+1)a+b$ (3.7)
Writing (2.16) as $QX=aX+b\eta(X)\xi$, substituting $\xi$ for $X$, and using
the $K$-contact property: $Q\xi=2m\xi$ [2], we get
$a+b=2m$ (3.8)
Solving (3.7) and (3.8) for $a$ and $b$ and substituting them in (2.16) we get
$QX=\bigg{(}\frac{r}{2m}-1\bigg{)}X+\bigg{(}2m+1-\frac{r}{2m}\bigg{)}\eta(X)\xi.$
(3.9)
Plugging this value of $QX$ in (3.6) provides $W(X,\xi)\xi=0$. This proves the
first part. For the second part, we have the stronger hypothesis $W(X,Y)\xi=0$
on a contact Riemannain manifold $M$. This implies $W(X,\xi,\xi)=0$, and hence
applying the first part, we conclude that $g$ is $\eta$-Einstein, and hence
weakly conformally flat and quasi Einstein. Now, a straight forward
computation using (3.9) gives
$W(X,Y)\xi=R(X,Y)\xi-\eta(Y)X+\eta(X)Y$
By our hypothesis, the left hand side is zero, and thus the above equation
implies equation (2.12) and hence $g$ is Sasakian. This completes the proof.
Proof of Theorem 1.2 By our hypothesis we have $W(X,Y,\xi)=0$ and equation
(2.16), i.e. $Ric=ag+b\eta\otimes\eta$. Hence $r=(2m+1)a+b$, by contraction.
Using these data we find, after a straight forward computation, that
$R(X,Y)\xi=\frac{a+b}{2m}[\eta(Y)X-\eta(X)Y]$ (3.10)
At this point, we recall the following Schur-type result (Sharma [16]), ”If
$R(X,Y)\xi=k(\eta(Y)X-\eta(X)Y)$ for a smooth function $f$ independent of the
choice of the vector fields $X$, $Y$ on a contact Riemannian manifold, then
$k$ is constant on $M$”. Applying this result, in view of equation (3.10), we
conclude that $\frac{a+b}{2m}$ is constant, say $k$. Therefore $M$ is a
($k,\mu$)-contact manifold with $\mu=0$. As $k\leq 1$, we have either $k=1$,
in which case $M$ is Sasakian, or $k<1$. For $k<1$, comparing (2.16) with
(2.14) and noting that $\mu=0$ and $a+b=2mk$, we have
$(a-2m+2)g(X,Y)+(2m-2-a)\eta(X)\eta(Y)=2(m-1)g(hX,Y)$ (3.11)
Contracting it at $X$ and $Y$, and noting that $tr.h=0$, we find $a=2(m-1)$.
Consequently, equation (3.11) reduces to $(m-1)h=0$. So, $m=1$, because in
this case $h\neq 0$ (non-Sasakian). Thus, (2.14) reduces to
$Ric=2k\eta\otimes\eta$, i.e. Ricci tensor has rank 1. For $k=0$, $M$ is flat,
and for $k\neq 0$, $M$ is locally isometric to $SU(2)$ ($k>0$) or
$SL(2,\mathbb{R}$) ($k<0$) with a left invariant metric (see Blair [2]). This
completes the proof.
## 4 Concluding Remark
We may extend the definition (1.1) of a quasi-Einstein manifold for a semi-
Riemannian metric by allowing the non-zero vector field $U$ to be null (light-
like, i.e $g(U,U)=0$). An example for the null case is the null dust solution
of Einstein’s field equations, in which the only mass-energy is due to some
kind of massless radiation and whose stress-energy tensor is $T=\Phi k\otimes
k$ where $k$ is a null vector field specifying the direction of motion of the
radiation and $\Phi$ the intensity. This includes plane fronted waves with
parallel rays ($pp$-waves) which, in turn include gravitational plane waves.
For details, we refer to Stephani et al. [17].
## References
* [1] Bach, R., Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung des Krummungstensorbegriffs, Math. Z. 9 (1921), 110-135.
* [2] Blair, D. Riemannian geometry of contact and symplectic manifolds, Springer, 2010.
* [3] Boyer, C., Galicki, K. and Matzeu, P. On eta-Einstein sasakian geometry. Commun. Math. Phys. 262 (2006), 177-208.
* [4] Case, J., Shu, Y. and Wei, G. Rigidity of quasi-Einstein metrics. Diff. Geom. Appls. 29 (2011), 93-100.
* [5] Chaki, M. and Maity, R.K. On quasi Einstein manifolds. Publ. Math. Debrecen. 57 (2000). 297-306.
* [6] Chen, B. Hypersurfaces of a conformally flat space. Tensor, NS 26(1972), 315-321.
* [7] Chen, B. and Yano, K. Special conformally flat spaces and canal hypersurfaces. Tohoku Math. J. 25 (1973), 177-184.
* [8] Chow, B., Chu, S., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F, and Ni, L., The Ricci flow: Techniques and Applications, Part I: Geometric Aspects, Mathematical Surveys and Monographs 135, American Math. Soc., 2004.
* [9] De, U. C. and Ghosh, G. On quasi Einstein manifolds. Periodica Math. Hungarica. 48 (2004), 223-231.
* [10] Eardley, D., Isenberg, J., Marsden, J. and Moncrief, V. Homothetic and conformal symmetries of solutions to Einstein’s equations. Commun. Math. Phys. 106 (1986), 137-158.
* [11] Gebarowski, A. Nearly conformally symmetric warped product manifolds. Bull. Inst. Math. Acad. Sinica. 20 (1992), 359-377.
* [12] Ghosh, A. and Sharma, R. Sasakian metric as a Ricci soliton and related results. J. Geom. Phys. 75 (2014), 1-6.
* [13] Hawking, S. and Ellis, G. G.R., The large scale structure of space-time, Cambridge University Press, 1973.
* [14] Maldacena, J., The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252.
* [15] O’Neill, B. Semi-Riemannian geometry with applications to relativity. Academic Press,1983.
* [16] Sharma, R. On the curvature of contact metric manifolds. J. Geom.. 53 (1995), 179-190.
* [17] Stephani, H., Kramer, D., MacCallum, M., Hoensselaers, C. and Herlt, E. Exact solutions of Einstein’s field equations, Cambridge Univ. Press, Cambridge, 2009.
* [18] Yano, K., Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970.
|
Collider testable low-scale seesaw models predict pseudo-Dirac heavy neutrinos, that can produce an oscillating pattern of and events.
We explore if such can be resolved at the $\HLLHC$.
To that end, we employ the first ever full simulation of the oscillations, for several example benchmark points, and show under which conditions the experiment is able to discover them.
The workflow builds on a FeynRules model file for the (pSPSS) and a patched version of MadGraph, able to simulate .
We use the fast detector simulation Delphes and present a statistical analysis capable of inferring the significance of oscillations in the simulated data.
Our results demonstrate that, for heavy neutrino mass splittings smaller than about [100]μeV, the discovery prospects for at the $\HLLHC$ are promising.
§ INTRODUCTION
The observed light neutrino flavour oscillations <cit.> can be explained by introducing at least two mass splittings between the light neutrinos.
It follows that at least two of the neutrinos have to be massive.
However, the of particle physics lacks the corresponding mass terms.
Therefore, it needs to be extended with a model able to generate these light neutrino masses, such as the type I seesaw mechanism <cit.>.
In order to generate the two light neutrino masses in this model at least two sterile neutrinos have to be added to the particle content of the .
In general, the type I seesaw mechanism is governed by two sets of parameters, that can be adjusted to obtain light neutrino masses at the right order of magnitude.
Besides their Yukawa couplings, the second set is given by their Majorana masses $m_M^{}$, which are only possible since the additional neutrinos are singlets.
After the Yukawa couplings result in Dirac mass terms $\vec m_D^{(i)}$ between each of the sterile neutrinos $N_i$ and the active neutrinos $\nu_\alpha$.
[We indicate quantities with a suppressed vector index by bold font.]
For the case of two heavy neutrinos, the generated light neutrino mass matrix then follows from the seesaw equation
\begin{equation} \label{eq:seesaw}
M_\nu = \frac{\vec m_D^{(1)} \otimes \vec m_D^{(1)}}{m_M^{(1)}} + \frac{\vec m_D^{(2)} \otimes \vec m_D^{(2)}}{m_M^{(2)}} \,.
\end{equation}
In order to obtain light neutrino masses, one could take the Yukawa couplings to be very small or choose the Majorana masses of the heavy neutrinos to be very large.
Those two limits of the seesaw mechanism are referred to as the small coupling and the high scale limit, respectively.
In both cases the parameters have to be taken to such extreme values that a direct detection of the sterile neutrinos at current collider experiments is not feasible.
A third possibility, which leads to collider testable low-scale seesaw models, consists in heavy neutrino pairs forming a pseudo-Dirac structure.
This leads to a cancellation between the two terms in (<ref>) and can be justified by an approximate symmetry.
The symmetry can be realised as an extension of the lepton number and we refer to it as a <cit.>.
It allows for the heavy neutrinos to be light enough, and at the same time have sufficiently large Yukawa couplings, to be directly observable at the <cit.>.
When the is exact, the light neutrinos are massless and the two heavy neutrinos are precisely mass-degenerate, forming a single Dirac neutrino.
However, if the is broken by small parameters, light neutrino masses are generated and typically a mass splitting between the heavy neutrinos is introduced.
The corresponding seesaw scenario is referred to as <cit.>.
The phenomenological effects of the slightly broken are captured by the introduced in <cit.>.
Similar to the case of other neutral particles, this leads to particle-antiparticle oscillations <cit.>, which are in this context called <cit.>.
The framework of external wave packets, which has been developed in <cit.> and applied to the case of in <cit.>, allows to describe this phenomenon including decoherence effects <cit.>.
Although the small breaking of the generates processes, one can argue on principle grounds that for prompt heavy neutrino decays it is not possible to observe at the <cit.>.
In contrast, generate observable amounts of when the heavy neutrinos are sufficiently long-lived compared to their oscillation period.
When the heavy neutrinos have large enough lifetimes, such that their time of decay can be reconstructed, it can be possible to resolve the oscillation pattern of the .
This allows to deduce the mass splitting of the heavy neutrinos, which might not be possible from measuring the amount of and processes alone.
The possibility of resolving the at colliders has been estimated to be feasible during the in <cit.>.
The goal of the present paper is to explore under which conditions it is possible to resolve at the at the reconstructed level, employing the process shown in <ref>.
To this end, the FeynRules implementation of the <cit.> is used to simulate in MadGraph.
Subsequently, Pythia is employed to simulate effects such as hadronisation, and the fast detector simulation Delphes is used to simulate the phase II detector.
A cut based analysis of the generated events is performed using custom code.
Finally, the prospects to resolve oscillation patterns for several scenarios of long-lived nearly mass degenerate heavy neutrinos at the are derived, using a detailed statistical analysis implemented in Mathematica.
This paper is structured as follows:
First, the and are briefly reviewed in <ref>.
Subsequently, in <ref> the simulation of signal events containing these oscillations and relevant background processes are discussed.
Afterwards, the statistical analysis is introduced in detail in <ref>.
Finally, the results are presented in <ref> and the paper is concluded in <ref>.
In <ref> we comment on the induction of residual oscillations through cuts on the transverse impact parameter.
When considering two heavy neutrinos under the assumption of an intact , the only allowed additions to the Lagrangian are
\begin{equation} \label{eq:symmetric Lagrangian}
\mathcal L_{\SPSS}^L =
\widebar{N_i^c} \ispd N_i^{}
- y_{1\alpha} \widebar{N_1^c} \widetilde H^\dagger \ell_\alpha
- \widebar{N_1^c} m_M^{} N_2^{}
+ \dots + \HC \,,
\end{equation}
where $N_1$ and $N_2$ are sterile neutrinos written here as left-chiral four-component spinor fields.
The Higgs and lepton doublets of the are denoted by $H$ and $\ell$, respectively, and $\vec y_1$ is the Yukawa coupling vector with components $y_{1\alpha}$.
The ellipses capture further contributions that can be generated by additional sterile neutrinos, but are assumed to be subdominant here.
In this case the two terms in the seesaw formula (<ref>) cancel precisely.
In addition, the following symmetry breaking terms can be introduced
\begin{equation} \label{eq:broken Lagrangian}
\mathcal L_{\SPSS}^{\cancel L} =
- y_{2\alpha} \widebar{N_2^c} \widetilde H^\dagger \ell_\alpha
- \mu_M^\prime \widebar{N_1^c} N_1^{}
- \mu_M^{} \widebar{N_2^c} N_2^{}
+ \dots + \HC \,.
\end{equation}
When the Yukawa coupling vector $\vec y_2$ times the as well as the Majorana mass parameters $\mu_M^{}$ and $\mu_M^\prime$ are small compared to $m_M^{}$, the light neutrino masses are guaranteed to be small as well.
Due to the approximate , it is possible for the heavy neutrinos to have a mass well below the $W$ boson mass, while at the same time their coupling can be large enough to obtain a sizable number of events at the , without violating constraints from light neutrino experiments such as searches for decay.
In the with exact (<ref>), the neutrino mass matrix for the interaction eigenstates $n = \row{\nu_e, \nu_\mu, \nu_\tau, N_1, N_2}^\trans$ is given by
\begin{equation}
M_n^L =
\begin{pmatrix}
0 & \vec m_D^{} & 0 \\
\vec m_D^\trans & 0 & m_M^{} \\
0 & m_M^{} & 0
\end{pmatrix}
\end{equation}
where $\vec m_D = \vec y_1 v$ with the Higgs $v\approx\unit[174]{GeV}$.
The mass term of the Lagrangian after is given by
\begin{equation}
\mathcal L_\text{mass} = - \frac12 \widebar{n^c} M_n n + \HC \,.
\end{equation}
This mass matrix does not generate neutrino masses.
Furthermore, the two heavy neutrinos are mass degenerate with a relative phase of $-i$ in their Yukawa couplings and can therefore be described as a single Dirac fermion.
However, in the presence of the small symmetry breaking terms (<ref>) the complete mass matrix reads
\begin{equation} \label{eq:broken mass matrix}
M_n^{\cancel L} =
\begin{pmatrix}
0 & \vec m_D^{} & \vec \mu_D^{} \\
\vec m_D^\trans & \mu_M^\prime & m_M^{} \\
\vec \mu_D^\trans & m_M^{} & \mu_M^{}
\end{pmatrix}
\,,
\end{equation}
where $\vec \mu_D^{} = \vec y_2 v$.
This mass matrix not only generates small masses for the light neutrinos, but additionally introduces a mass splitting between the two heavy neutrinos, that is also suppressed by the same small breaking parameters.
Thus, the pair of two Majorana fermions can no longer be described as a pure Dirac particle, but as a pseudo-Dirac particle.
The symmetry breaking terms can arise from specific low scale seesaw models, such as the linear or the inverse seesaw, which yield only the terms proportional to $\vec \mu_D^{}$ and $\mu_M^{}$, respectively.
However, the complete mass matrix (<ref>) can be generated from more complicated seesaw models.
Since the symmetry breaking terms are very small, their phenomenological effects can often be neglected in collider studies with the notable exception of , which can be phenomenologically significant as they are an interference effect.
At , the are fully described by the mass splitting $\Delta m$ between the heavy neutrinos together with an additional damping parameter $\lambda$, capturing the potential decoherence effects discussed in <ref>.
Therefore, instead of adding the terms (<ref>) to the Lagrangian (<ref>), in the , the mass splitting is directly introduced as a model parameter.
Consequently, the masses of the heavy neutrinos are given by
\begin{equation} \label{eq:Delta m mass splitting}
m_{\nicefrac45}^{} = m_M^{} \left(1 + \frac12 \abs{\vec \theta}^2 \right) \mp \frac12 \Delta m \,,
\end{equation}
where $\vec \theta = \flatfrac{\vec m_D^{}}{m_M^{}}$ is the active-sterile mixing parameter.
A detailed description of the can be found in <cit.>.
$\Delta m / \unit{\mu eV}$ $c \tau_\text{osc} / \unit{mm}$ $R_{ll}$
1 82.7 15 0.9729
2 207 6 0.9956
3 743 1.67 0.9997
[ points]
All points have a mass of [14]GeV and an active-sterile mixing parameter satisfying $\abs{\theta_\mu}^2 = 10^{-7}$, which results for all points in a decay width of $\Gamma = \unit[13.8]{\mu eV}$.
However, they vary in their mass splitting and consequently have different oscillation periods $\tau_\text{osc}$, which leads to different to ratios $R_{ll}$, defined in (<ref>).
[Number of expected events at the $\HLLHC$]
Contour lines for the number of expected displaced vertex events $N$ as well as bands for the over event ratio $R_{ll}$, definition (<ref>), for the three defined in <ref>.
The contour lines for $N$ apply to the detector at the $\HLLHC$ with $\mathscr L = \unit[3]{\inv{ab}}$ and cuts as defined in <ref>.
They have a nose-like shape and depend only on the heavy neutrino mass $m$ and the active-sterile mixing parameter $\theta_\mu$, with $\theta_e = \theta_\tau = 0$.
$N$ is therefore identical for all three , which differ only in their mass splittings $\Delta m$.
The position of the in the parameter plane is indicated by a purple cross.
The different $\Delta m$ of the result in different $R_{ll}$ bands.
The contour lines where $R_{ll} \in [0.1,0.9]$ are shown in yellow and orange for 1 and 3, respectively.
The relative position of the cross to the bands shows that the have an $R_{ll}$ close to one.
The grey area corresponds to the exclusion bounds from displaced vertex searches <cit.> and the shaded grey area to the bounds from searches for prompt processes <cit.>.
Since the prompt searches rely on signals, they apply only to models with an $R_{ll}$ close to one.
In order to obtain the significance with which could be observed at colliders, three that differ only in their mass splitting are used and given in <ref>.
The Majorana mass parameter and the active-sterile mixing parameters for all points are chosen to be $m_M^{} = \unit[14]{GeV}$ and $\abs{\theta_\mu}^2 = \expno{-7}$, leading to a decay width of $\Gamma = \unit[13.8]{\mu eV}$.
The active-sterile mixing parameter corresponds to a Yukawa coupling of $y_\mu = \expnumber{2.55}{-5}$, while the Yukawa couplings to the electron and $\tau$-lepton have been set to zero.
The number of expected events with muons as final state leptons in conjunction with the cuts presented in <ref>, are shown in <ref>.
It becomes clear that the chosen lie comfortably beyond the current bounds <cit.>.
In the minimal linear seesaw model, only one pseudo-Dirac pair of heavy neutrinos is added to the .
This results in the lightest neutrino being massless and the mass splitting of the heavy neutrinos $\Delta m$ being identical to the mass splitting of the light neutrinos, <cit.>.
The mass splitting of 3 is chosen to represent this possibility, where the light neutrino mass splitting is taken from a recent global fit assuming inverse light neutrino mass hierarchy <cit.>.
However, in top-down realisations of low scale seesaw models, the low scale linear seesaw models in <cit.>, it is common to have multiple pseudo-Dirac pairs of heavy neutrinos.
Since the light neutrino masses get contributions from all these heavy neutrinos, it is possible for the pseudo-Dirac pairs to have smaller mass splittings than in 3 without the need of cancellations in the mass matrix.
For simplicity it might be assumed that the collider phenomenology is dominated by only one of the pseudo-Dirac pairs.
We introduce two additional points reflecting this possibility.
§.§ Oscillations
Oscillations of neutral particles, and thus , can be described in the framework of external wave packets.
Compared to simple methods relying on plane waves, wave packets allow for a self-consistent description of oscillations.
This is due to the finite uncertainty in energy-momentum and spacetime, that allows to produce a coherent superposition of non-degenerate mass eigenstates, while simultaneously makes it possible to introduce the notion of a travelled time and distance.
The framework also allows to calculate the possible suppression of oscillations due to the loss of coherence between the propagating mass eigenstates.
For phenomenological studies these effects can be captured by a damping parameter $\lambda$, which is thus included in the .
At , the probability to obtain a or event is given by
\begin{equation}
P^{\nicefrac{\LNC}{\LNV}}_\text{osc}(\tau) = \frac{1 \pm \cos(\Delta m \tau) \exp(-\lambda)}2 \,.
\end{equation}
Therefore, the oscillation period, defined in the proper time frame, is
[We use time for quantities in the proper time frame and length for quantities in the lab frame, independent from the units of those quantities.]
\begin{equation} \label{eq:oscillation period}
\tau_\text{osc} = \frac{2 \pi}{\Delta m} \,.
\end{equation}
Since the probability density of the heavy neutrino to decay is
\begin{equation}
P_\text{decay}(\tau) = - \dv \tau \exp\left(- \Gamma \tau\right) = \Gamma \exp\left(- \Gamma \tau\right) \,,
\end{equation}
the probability for a heavy neutrino to decay between the proper times $\tau_{\min}$ and $\tau_{\max}$ and forming an or event is given by
\begin{equation} \label{eq:probability displacement}
P_{ll}^{\nicefrac{\LNC}{\LNV}}(\tau_{\min}, \tau_{\max}) = \int_{\tau_{\min}}^{\tau_{\max}} P^{\nicefrac{\LNC}{\LNV}}_\text{osc}(\tau) P_\text{decay}(\tau) \d \tau \,.
\end{equation}
By integrating the oscillations from the origin to infinity, one derives the total ratio between and events <cit.>
\begin{equation} \label{eq:Rll}
R_{ll} = \frac{P_{ll}^{\LNV}}{P_{ll}^{\LNC}} = \frac{\Delta m^2}{\Delta m^2 + 2 \Gamma^2} \,,
\end{equation}
that can be directly deduced from cut and count based analyses.
The expected number of events in a collider experiment is given by
\begin{equation} \label{eq:event numbers}
N^{\nicefrac{\LNC}{\LNV}} = \mathscr L \sigma \BR \int D(\vartheta, \gamma) P^{\nicefrac{\LNC}{\LNV}}_{ll}(\tau_{\min}(\vartheta, \gamma), \tau_{\max}(\vartheta, \gamma)) \d\vartheta \d\gamma\,,
\end{equation}
where the collider luminosity $\mathscr L$, the heavy neutrino production cross section $\sigma$, and the are used and $D(\vartheta, \gamma)$ is the probability density of the heavy neutrino to have a Lorentz factor $\gamma$ and an angle $\vartheta$ with respect to the beam axis.
The detector geometry is incorporated in the parameters $\tau_{\min}$ and $\tau_{\max}$ when boosting to the laboratory frame via $\tau(\vartheta,\gamma) = (\gamma^2-1)^{-\nicefrac12} L(\vartheta)$.
The in the have been discussed in detail in <cit.>.
We have checked that for the parameter points under consideration decoherence effects can be neglected and we thus take $\lambda = 0$ in the following <cit.>.
§ SIMULATION
[Diagram of heavy neutrino production, oscillation, and decay]
Diagram depicting the production, oscillation, and decay of a heavy neutrino.
The heavy neutrino interaction eigenstate $N$ is produced in association with a prompt antimuon.
Subsequently, the mass eigenstates oscillate such that finally a neutrino or antineutrino interaction eigenstate decays into a displaced muon or antimuon, respectively.
We indicate that the process is initiated by proton collisions and that, for our parameter points, the two final quarks, originating from the hadronic $W$ decay, immediately hadronise into a single jet.
In order to simulate events exhibiting , we employ the FeynRules <cit.> implementation of the <cit.>.
FeynRules is used to generate an output, passed to [2.9.10 (LTS)]MadGraph5_aMC@NLO <cit.> in order to generate events at parton level.
We use the patch introduced in <cit.>, that modifies the function calculating the particle's , to implement the oscillations.
When evaluating the matrix element, MadGraph treats the process as prompt, adding the information afterwards.
Therefore, almost no is obtained when interference between diagrams with different mass eigenstates are taken into account.
In order to circumvent this, the process is initially specified in such a way as to prevent interference between the heavy neutrino mass eigenstates.
This is achieved by writing the heavy neutrinos explicitly as intermediate states.
As a consequence the total cross sections for the and process are identical and the amount of generated and events is, except for statistical variations, the same.
The correct interference effects, including , are then included via the patch described in <cit.>.
The syntax to generate a process, as the one given in <ref>, is
where, in addition to the jet j and proton p multi particles, containing quarks and gluons, the multi particles
are used.
Here, the initial $W$ boson is additionally taken to be on-shell, since we focus on heavy neutrinos with masses far below the $W$ boson mass.
An additional hard initial jet is included by using
where matching was enabled with the standard parameter choices of MadGraph to prevent double counting.
MadGraph utilises [8.306]Pythia in order to hadronise and shower the parton level events <cit.>.
Finally [3.5.0]Delphes is used with the standard phase II card CMS_PhaseII_0PU to simulate the detector effects <cit.>.
Secondary vertex reconstruction and smearing
Since Delphes neither simulates displaced tracks properly, nor reconstructs secondary vertices, we implemented a smearing function affecting the displaced vertices, with the idea to capture experimental uncertainties.
To our knowledge, no results for the overall precision of the vertex reconstruction have been published by so far.
Therefore, we vary it in our simulations between zero and [4]mm, which we assume to be a conservative parametrisation of our ignorance, relying on private communication with members on the experimental collaborations.
We like to emphasise that it would be highly welcome if the experimental collaborations could provide such information and, ideally, if such uncertainties could be implemented in Delphes directly.
In detail, it is assumed that each reconstructed vertex is distributed with a Gaussian, with a standard deviation of $\unit[n]{mm}$, around its actual value.
The true value of the displaced vertex is obtained from the displaced muon.
The results are presented for different values of $n$, demonstrating the effects of the uncertainty in the vertex reconstruction.
§.§ Signal
In order to observe via using the process given by the diagram presented in <ref>, the two leptons have to be measured.
For the given in <ref> and indicated in <ref>, the final state quarks are soft and immediately hadronise into a multi pion state, which then forms a single jet that can be captured with a cone radius of $\Delta R = 0.4$.
Furthermore, the points are chosen such that the heavy neutrinos are long-lived.
Hence, the signal contains one prompt muon, one displaced muon, and one displaced jet.
A cut is used, which requires the muon to have an impact parameter of $\abs{d_0} \leq \unit[100]{\mu m}$, to ensure cut:one prompt muonexactly one prompt $\mu$.
Furthermore, this muon is required to have a transverse momentum $p_T^{}$ greater than [20]GeV, ensuring that the event can be triggered.
Events that do not contain cut:one disp muonat least one displaced $\mu$ are excluded by a cut that requires at least one muon with an impact parameter of $\abs{d_0} \geq \unit[2]{mm}$.
Furthermore, the displaced muon candidate is only valid if it emerges from the same vertex as the displaced jet.
A jet is considered displaced if it contains at least two displaced tracks that originate from the same vertex within a radius of [100]μm.
A track is considered displaced if it has an impact parameter of $\abs{d_0} \geq \unit[2]{mm}$.
Signal events are required to have cut:one disp jetexactly one displaced $j$.
In addition to those basic cuts that define the signal, we demand cut:muon isolation$\mu$ isolation by requiring that the displaced muon and displaced jet have a $\Delta R$ larger than $0.4$ in order to reject muons radiated from the jet.
Furthermore, restricting the reconstructed heavy neutrino mass to the cut:N mass window$N$ mass window of $\pm \unit[2]{GeV}$ around its theoretical value results in a cleaner signal.
Generally, displaced particles are only considered if their origin falls inside a cylinder with dimensions given by half the tracker size in each direction.
For the phase II detector, this means that displaced tracks must have an origin with a transverse distance smaller than [60]cm, and a longitudinal distance smaller than [150]cm, from the primary vertex.
This requirement ensures that the produced particles propagate through a volume of the tracker that should be large enough to facilitate detection.
At the moment this is an optimistic assumption about the performance of the displaced track reconstruction in the tracker, however, we think that pushing this capability is a worthwhile goal for the collaboration (see also <cit.> and references therein).
Although the reconstruction of discussed here mostly relies on displaced vertices appearing in the inner tracker, it is also possible to reconstruct muon tracks and vertices using the muon chambers <cit.>.
§.§ Background
Signal Background
Heavy hadrons
2*Produced events Physical $2854$ $\expno[8.882]{7}$
Simulated $\expno[5]{4}$ $\expno[1.1]{7}$
9*Cuts <ref> $-23196$ $-\expno[6.50]{6}$
<ref> $-21652$ $-\expno[4.48]{6}$
<ref> $-1396$ $-17411$
<ref> $-838$ $-45$
<ref> $0$ $0$
<ref> $0$ $0$
<ref> $-111$ $-3$
<ref> $-1211$ $0$
2*Remaining events Simulated $1596$ $0$
Physical $91$ $0$
[Cut flow of signal and background events]
Cut flow for the signal and the heavy hadron background events.
The physical events are given at an integrated luminosity of $\mathscr L = \unit[3]{ab^{-1}}$ with a signal cross section of $\sigma = \unit[951]{ab}$.
The cuts introduced above define the basic search strategy for the signal, which is based on reconstructing a displaced vertex.
The main sources of background for such a process are given by heavy flavour processes generating long-lived heavy hadrons, interaction of particles with dense detector material, and cosmic ray muons <cit.>.
The heavy hadrons can travel macroscopic distances before they decay, potentially forming displaced vertices.
However, this is only possible if those heavy flavour particles, and consequently their decay products, are highly boosted.
Since the selection rules that define the signal already require the displaced muon and the displaced jet to have a $\abs{d_0} > \unit[2]{mm}$, this background is strongly reduced <cit.>.
A cut:W mass window$W$ mass window cut around the reconstructed $W$ boson mass of $ \pm \unit[20]{GeV}$ is employed to reduce the background even further.
We simulated this background using the same programs as for the signal events.
The MadGraph syntax we used is
where the additional multi particle
is defined.
An additional hard initial jet was included by using
where matching was enabled with the standard parameter choices of MadGraph.
The whole process was simulated using the model sm-no_b_mass, which employs the five flavour scheme in the definitions of the protons and jets.
After the parton level events are passed to Pythia and Delphes, the above mentioned cuts and event selection rules were used, and it has been found that the background is eliminated entirely.
The cut flow for the simulated signal and background is given in <ref>.
Interactions of particles with detector material can also result in a displaced vertex signature and thus have to be considered as part of the background.
A map of regions containing dense detector material is required in order to accept only displaced vertices that are outside that region.
Simulating this background requires detailed knowledge about the detector structure and is beyond the scope of this work.
However, requiring a displaced vertex to be reconstructed from enough good tracks reduces this background.
In an experimental analysis, a sophisticated track reconstruction algorithm judges which tracks are good, in the sense that the track is likely to be produced from a charged particle rather than detector noise.
With no specific insight about this algorithm, we define the tracks as being good that pass our cut and selection rules, which ensure that the charged particles traverse at least half the tracker.
With the cuts introduced to define the signal, it is ensured that each event contains at least three displaced tracks from which the displaced vertex can be reconstructed.
In order to veto against cosmic ray muons, the cut:vertex directionvertex direction cut is implemented that requires the reconstructed heavy neutrino momentum to be in the same direction as the displaced vertex.
For this the distance in $\row{\eta, \phi}$-space between the reconstructed momentum $\vec p_N^{}$ and the displaced vertex direction $\vec d$ is limited to $\Delta R \leq 1.5$.
We assume that this background is eliminated when using this cut in combination with additional timing information, therefore we have not attempted to simulate it.
With the cuts described above, the signal region can be assumed to be background free.
Additionally we found that a cut:no prompt eprompt $e$ veto, where prompt is again defined as $\abs{d_0} < \unit[100]{\mu m}$, with a $p_T^{} \geq \unit[20]{GeV}$ does not affect the signal, such that it could be used to eliminate further background if that becomes necessary.
§ STATISTICAL ANALYSIS
The number of events entering the cut based analysis is given by the luminosity times the cross section for the considered process.
The number of expected events after the cut based analysis $N_\text{exp}$ is then given by
\begin{align} \label{eq:number of events}
N_\text{exp} &= \mathscr L \sigma f_\text{eff} \,, &
f_\text{eff} &= \frac{N_\text{after cuts}}{N_\text{all events}}\,,
\end{align}
where the efficiency factor captures how the cuts summarised in <ref> impact the number of signal events.
The set of events surviving all cuts is labeled $\mathcal D_\text{all}$, and can be divided into events $\mathcal D_{\LNC}$ and events $\mathcal D_{\LNV}$.
These datasets are then binned in the proper time $\tau$ space and the number of events in the $i$-th bin are given by
\begin{equation}\label{eq:bin counts}
N_i = \abs*{\set*{E \in \mathcal D \suchthat \tau_E^{} \text{ in $i$-th bin}}}\,,
\end{equation}
where $\abs{}$ is the cardinality such that $N_\text{exp} = \abs{\mathcal D_\text{all}}$.
The heavy neutrino $\tau_E^{}$ is defined as the proper time at which it decays in the event $E$.
§.§ Hypotheses
The shape of the histograms describing the heavy neutrino may be predicted by two hypotheses.
In contrast to the null hypothesis, the alternative hypothesis features oscillations.
Null hypothesis
[Examples for the null and alternative hypotheses]
Example histogram plot:M0 with [15]k events, demonstrating the proper time distribution after applying the cuts summarised in <ref>.
The inverse Gaussian distribution $\mu_i^\text{all}(\mathcal M_0)$ described in (<ref>) is depicted plot:M0 fit together with the prediction of the null hypothesis $\mu_i^{\nicefrac{\LNC}{\LNV}}(\mathcal M_0)$ plot:M0 fit half.
The inverse Gaussian is shown with the parameters given in <ref> on it:one, where the normalisation is approximated as $N_0 = 15500$.
The distribution of the alternative hypothesis (<ref>) is shown for plot:M1 LNC and plot:M1 LNV events of 1, with example value $\alpha = 0.05$ for the washout parameter.
The hypothesis without oscillations $\mathcal M_0$ is based on the assumption that, for each bin, the probability of the heavy neutrino superposition to decay in a process is equal to the probability to decay in a process.
Therefore, the mean number of predicted events in the $i$-th bin is given by
\begin{equation}
\mu_i^{\nicefrac{\LNC}{\LNV}}(\mathcal M_0) = \frac12 \mu_i^\text{all}(\mathcal M_0) \,.
\end{equation}
The probability of a given bin count can then be computed assuming a Poisson distribution of bin counts around this mean.
In principle, one would expect the predicted mean values of this hypothesis to follow an exponential due to the finite lifetime of the heavy neutrinos.
However, the employed cuts change this distribution into a non trivial one, which can be approximated by the of a generalized inverse Gaussian, described by four free parameters, yielding
\begin{equation} \label{eq:inverse Gaussian}
\mu_i^\text{all}(\mathcal M_0) = \frac{N_0}{2} \frac{\tau_i^{\param\theta - 1}}{\param\mu^{\param\theta} K_{\param\theta}(\nicefrac{\param\lambda}{\param\mu})} \exp\left(-\frac{\param\lambda}{\param\mu^2} \frac{\tau_i^2 + \param\mu^2}{2 \tau_i}\right)\,,
\end{equation}
where $N_0$ denotes the overall normalisation, $\param\theta$ is the index parameter, $\param\mu$ is the mean of the distribution, and $\param\lambda$ is a scale parameter.
Additionally, $\tau_i$ denotes the position of the middle point of the $i$-th bin, and $K_{\alpha}(z)$ is the modified Bessel function of the second kind.
The distribution is shown in <ref>, where the parameters correspond to the best fit point estimated in <ref> on it:one.
Alternative hypothesis
The second hypothesis $\mathcal M_1$ includes oscillations, with an oscillation period given by (<ref>), on top of the distribution described by the first hypothesis.
Due to the imperfect reconstruction of the Lorentz factor, an additional washout effect obscures the oscillation pattern for larger $\tau$.
This effect is included by an exponential factor $\alpha$, suppressing the oscillation.
The prediction for the mean number of expected events in a bin is then given by
\begin{equation} \label{eq:model 1}
\mu_i^{\nicefrac{\LNC}{\LNV}}(\mathcal M_1) = \left[1 \pm \cos(\Delta m \tau_i) e^{- \alpha \tau_i} \right] \mu_i^{\nicefrac{\LNC}{\LNV}}(\mathcal M_0) \,,
\end{equation}
where the two additional parameters $\Delta m$ and $\alpha$ are incorporated, resulting in a total of six free parameters.
Again, a Poisson distribution of bin counts, around this mean, is assumed.
One example of such oscillations is given in <ref>.
§.§ Likelihood ratio test
To test whether the hypothesis including oscillations is statistically preferred by the simulated data, we use a likelihood ratio test.
The main idea is to decide if the null hypothesis, given by $\mathcal M_0$, can be discarded in favour of the alternative hypothesis, given by $\mathcal M_1$.
The likelihood is denoted by $\lh(\set{N_i}, \mathcal M)$ and describes the probability of finding the measured bin counts $\set{N_i}$ given the hypothesis $\mathcal M$.
It is given by the product of likelihoods for all bins in both, the and , cases.
With the assumed Poisson distributed number of events around the mean value $\mu_i$ in the $i$-th bin, the likelihood of a single bin is given by
\begin{equation}
\lh(N_i, \mu_i) = \frac{{\mu_i}^{N_i} e^{-\mu_i}}{N_i !} \,,
\end{equation}
where, as introduced above, $N_i$ is the number of events in the $i$-th bin, which depends on the dataset and the binning.
The best fit point for each hypothesis, given the bin counts $\set{N_i}$, is evaluated by maximising the corresponding likelihood.
A hypothesis at its best fit point is denoted with a hat, $\widehat{\mathcal M}_0$.
Therefore, the likelihood ratio can be computed, given the bin counts, and yields
\begin{equation}
\lhr(\set{N_i})
= \frac{\lh(\set{N_i}, \widehat{\mathcal M}_0)}{\lh(\set{N_i}, \widehat{\mathcal M}_1)}
= \prod_i \frac{\lh(N_i^{\LNC}, \widehat{\mathcal M}_0) \lh(N_i^{\LNV}, \widehat{\mathcal M}_0)}{ \lh(N_i^{\LNC}, \widehat{\mathcal M}_1) \lh(N_i^{\LNV}, \widehat{\mathcal M}_1)} \,.
\end{equation}
Taking the washout parameter to infinity in the alternative hypothesis reproduces the null hypothesis.
The two hypotheses are therefore nested and as a consequence the inequality
This inequality can by violated if not a global but a local maximum of the likelihood is found, while searching for $\widehat{\mathcal M}_1$.
Such a local maximum might result in a smaller likelihood for the alternative hypothesis than the likelihood obtained by a fit of the null hypothesis.
\begin{equation}
\lh(\set{N_i}, \widehat{\mathcal M}_0) \leq \lh(\set{N_i}, \widehat{\mathcal M}_1) \,,
\end{equation}
holds and hence the likelihood ratio is restricted by
\begin{equation}
0 \leq \lhr(\set{N_i}) \leq 1 \,.
\end{equation}
A likelihood ratio close to zero means that the given binned data are much better fitted by the alternative hypothesis than by the null hypothesis.
In contrast, a ratio close to one shows that there is no clear distinction between the two hypotheses for the given binned data.
Since in practice the logarithm of the likelihood is better suited for numerical computations, we continue the discussion using the which is defined as
\begin{equation}
\llhr(\set{N_i}) = -2 \log(\lhr(\set{N_i})) \,.
\end{equation}
The ranges from zero to infinity, where now a value near zero states that both, the null hypothesis and the alternative hypothesis, produce an equally good fit of the binned data.
Contrary, a high value states that the alternative hypothesis produces a better fit.
$\llhr=5.70$, $p=\unit[21.5]{\%}$, $Z=\unit[0.79]{\sigma}$
Low significance
$\llhr=13.59$, $p=\unit[0.82]{\%}$, $Z=\unit[2.4]{\sigma}$
Intermediate significance
$\llhr=17.71$, $p=\unit[0.13]{\%}$, $Z=\unit[3.01]{\sigma}$
High significance
[Statistical fluctuation of the null hypothesis]
Three examples of statistical fluctuations in the null hypothesis, producing patterns that mimic .
From panel fig:low to panel fig:high the oscillatory pattern becomes more distinct while the probability that the depicted histogram is generated decreases.
Caused by statistical fluctuations of the bin counts around their predicted mean, the null hypothesis $\widehat{\mathcal M}_0$ can also feature an oscillation pattern, as demonstrated in <ref>.
Therefore, the value of a alone does not contain enough information to decide whether oscillations in a given dataset are significant or not.
To translate the into a significance, it is crucial to know the probability that the null hypothesis produces the same due to fluctuations.
Per construction, the most likely produced by $\widehat{\mathcal M}_0$ are small, and the larger the the less likely it is to be produced.
The distribution of , generated by statistical fluctuations, is the of the under the assumption that the null hypothesis is true.
We label it $\pdf$.
$\chi^2$ $\DOFs$ $k(\unit[5]{\sigma})$
1 2.25 30.94
2 3.28 34.05
3 3.92 35.82
[Null hypothesis fluctuations]
Statistical fluctuations in the bin counts, following the null hypothesis, lead to a $\chi^2$ distribution with given $\DOFs$.
The distributions are computed following the <ref> on it:one.
The corresponding to a $p$ value of [5]σ are found to be increasing from 1 to 3.
The challenge is then to find a value $k_p$, for which the probability of obtaining a $\llhr(\set{N_i}) \geq k_p$ through statistical fluctuations, is smaller than $p$.
This means that $p$ is the probability of rejecting the null hypothesis with respect to the alternative hypothesis, even though the null hypothesis is true.
Given $\pdf$, the value of $k_p$ can be obtained via
\begin{equation}
\int_{k_p}^{\infty} \pdf(k) \d k = p \,.
\end{equation}
While in the limit of large sample sizes the is typically assumed to approach a $\chi^2$ distribution, the form of the for finite sample sizes is generally unknown.
Therefore a simulation is performed, with the goal to sample the , as follows
* All simulated events that survive the cut based analysis are used to estimate the true distribution of the null hypothesis $\widehat{\mathcal M}_0$, which is depicted in <ref>.
For those events, the best fit parameters of $\widehat{\mathcal M}_0$ are given by $\param\mu = 10.81$, $\param\lambda = 17.28$, and $\param\theta = 0.71$.
The overall normalisation parameter $N_0$ is not relevant for the following steps as they only depend on the , which is normalised to unity.
To obtain a set of events that follows the null hypothesis, taking statistical fluctuations into account, are taken according to the from <ref>.
This is done until the set contains $N_\text{exp}$ signal events.
* Using these values, bin counts are computed using the binning parameters based on <ref>.
* The is computed based on the bin counts from <ref>.
By repeating <ref> the desired distribution can be obtained.
The obtained distributions are well approximated by $\chi^2$ distributions, where the are treated as a free parameter.
The values of the for the three are given in <ref>.
The simulated events are based on 1 with an oscillation period of [15]mm.
The best fit parameters are given by an oscillation period of $\unit[14.08^{+0.85}_{-0.71}]{mm}$ and a washout parameter of $\alpha = \expnumber{3.66^{+31.73}_{-3.66}}{-3}$.
The $\LLR$ was found to be $51.0$, which in this case yields a significance of [6.66]σ.
The simulated events are based on 3 with an oscillation period of [1.67]mm.
The best fit parameters are given by an oscillation period of $\unit[1.63^{+0.03}_{-0.04}]{mm}$ and a washout parameter of $\alpha = \expnumber{7.44^{+17.76}_{-5.45}}{-2}$.
The $\LLR$ was found to be $5.29$, which in this case yields a significance of [0.67]σ.
[Examples for the statistical fit of the oscillations to the $\MC$ data]
Examples for the best fit of the alternative hypothesis to the data.
For each parameter point a luminosity of [3]ab is used, resulting in a total of about $90$ events after cuts.
However, the number of events contributing to the fit is based on the range of $\tau$ values used as given in <ref> and therefore differs between the .
The fit has been performed based on the binning options in <ref>.
In these examples the secondary vertex smearing has been neglected.
The bands around the oscillations depict the errors of one standard deviation, assuming a Poisson distribution for the event count in each bin.
With the so obtained it is possible to compute the probability that a given produced from statistical fluctuations of the null hypothesis $\widehat{\mathcal M}_0$.
This probability $p$ can be translated to a significance $Z$ by comparing it to a standard normal distribution.
A smaller probability translates to a larger significance and a higher threshold $k_p$.
A significance of [5]σ corresponds to a probability of $p \approx \expno[2.87]{-7}$, which for 1 corresponds to a threshold of $k_p \approx 30.94$.
See <ref> for the values of the other .
Therefore, in the case of 1, a greater than $30.94$ can be interpreted as a discovery, where oscillations have been found with a significance larger than [5]σ.
For very small , it is possible that the obtained probability is greater than [50]%.
Since for the translation into a significance a standard Gaussian is used, <cit.>, the corresponding significance would be smaller than zero .
However, in cases where the are so small, the result should just be interpreted in the way that no oscillations could be proven in the given data.
The simulated events and the resulting statistical fit for two example oscillations are shown in <ref>.
§.§ Data pre-processing
$c \tau_\text{max}/\unit{mm}$ Bins Used events
1 60 30 87
2 60 30 85
3 15 60 42
[Binning parameters]
Binning parameters for the different points, as well as the number of events used in the computation of the $\LLR$.
Note that for small samples of event surviving the cut based analysis, as it is the case here, the number of events participating in the computation of the $\LLR$ undergoes fluctuations, which contribute to the fluctuations in the obtained significances.
Some pre-processing of the events is performed to stabilise the numerics.
In principle, it is expected that for sufficiently large proper times alternative hypotheses with a washout parameter greater zero produce the same bin counts as the corresponding null hypotheses.
Therefore, the are dominated by small proper times, for which the washout effect is also small.
We have found that the fitting algorithm used by us gives more reliable results when we consider only these dominant oscillations.
Therefore we do not use all events for the statistical analysis but consider only a window containing the first oscillations.
The details of this restriction for each point are given in <ref>.
To help reduce noise in the data a Gaussian filter of radius $r = 1$, which corresponds to a standard deviation of $\sigma = \flatfrac{r}{2}$ in bin space, is applied to the bin counts before the maximisation takes place.
§ RESULTS
In this paper, the first ever analysis of at reconstructed level is performed.
A detailed statistical analysis is employed to obtain a significance describing the feasibility to resolve the oscillations at the .
To that end, three points with different oscillation periods, given in <ref>, have been simulated.
All points feature a mean mass of the heavy neutrinos of [14]GeV as well as an active-sterile mixing parameter of $\abs{\theta_\mu}^2 = 10^{-7}$.
This leads to a decay width of $\Gamma = \unit[13.8]{\mu eV}$.
The parameters are chosen such that the bounds of current collider searches are well evaded.
While 3 captures the mass splitting of the minimal linear seesaw model, producing the measured light neutrino data with an inverted hierarchy, 1 and 2 feature a smaller mass splitting as discussed in <ref>.
Secondary vertex smearing
Secondary vertex smearing for $90$ signal events.
Lorentz factor reconstruction error in 3.
[Significance as function of vertex smearing and Lorentz factor reconstruction]
Panel fig:smearing:
Significance of the three points at a luminosity of $\mathscr L = \unit[3]{ab^{-1}}$ as a function of the secondary vertex smearing.
Panel fig:reconstruction error:
Significance of 3, as function of the number of events surviving the analysis, for three different relative errors of the reconstruction of the Lorentz factor (<ref>).
For this comparison no smearing has been taken into account.
Since Delphes does not provide the experimental uncertainty to reconstruct secondary vertices, we have introduced a free parameter governing Gaussian smearing of the otherwise perfectly reconstructed secondary vertices.
<Ref> shows the expected confidence with which oscillations could be observed in the data for a given smearing of the secondary vertex.
With our points, the number of signal events that survive the cuts, shown in <ref>, is roughly $90$.
This assumes a luminosity of $\unit[3]{ab^{-1}}$ which is the expected total integrated luminosity of the <cit.>.
A factor of two in the number of events can easily be achieved by choosing points closer to the excluded region.
Therefore we regard this analysis as conservative.
For each data point in the plot, $100$ have been computed to obtain a mean value and a standard deviation for the significance.
The figure shows that for an oscillation period of [15]mm in proper time space, corresponding to 1, a significance of [5.19]σ can be expected if no smearing is taken into account.
Moreover, the significance is above [5]σ up to a smearing of [2]mm, dropping to [4.46]σ for a smearing of [4]mm.
Parameter points with smaller oscillation periods are affected stronger by the smearing.
This is expected since a smaller oscillation period in proper time space is related to a smaller oscillation length in lab space.
A larger smearing therefore results in a stronger washout for smaller oscillation periods.
Lorentz factor reconstruction
Since it is crucial to reconstruct the Lorentz factor on an event by event basis, we show in <ref> how a better reconstruction of the Lorentz factor as well as a higher luminosity makes it possible to observe oscillations even for 3.
The reconstruction error directly effects the quality of the reconstructed oscillations in the proper time frame.
Therefore, if the error is too large for the mass splitting one wants to resolve oscillations for, a higher event numbers only yields a limited improvement.
This is shown by the lowest line in <ref>.
In contrast the higher lines, that represent a smaller reconstruction error of the Lorentz factor, benefit much more from a larger event number.
The quality of the reconstruction of the Lorentz factor is measured using the relative error of the Lorentz factor, which is defined as
\begin{equation} \label{eq:gamma error}
\Delta \gamma = \frac{\abs{\gamma_\text{gen}^2 - \gamma_\text{reco}^2}}{\gamma_\text{gen}^2 - 1} \,,
\end{equation}
where $\gamma_\text{gen}$ is the true Lorentz factor of the heavy neutrino and $\gamma_\text{reco}$ is the reconstructed one.
The set of events used for the analysis yields an exponential distribution of the relative errors of the Lorentz factor.
Most events have a small $\Delta \gamma$, while only a small fraction of events have a large $\Delta \gamma$.
The overall quality in the reconstruction can be measured by the standard deviation of that exponential distribution.
While a large standard deviation corresponds to many events with a large relative error, and therefore to a poor reconstruction, the opposite is true for a small standard deviation.
Without any improvement, with respect to the Lorentz factor achieved in this analysis, the events of 3 yield $\Delta \gamma$'s that result in an exponential distribution with a standard deviation of $0.16$.
By improving the reconstruction such that the standard deviation is reduced to $0.0096$, the significance is improved from around zero to $\unit[(3.37 \pm 1.10)]{\sigma}$ for $90$ events.
Additionally doubling the number of events yields a significance of $\unit[(5.13 \pm 2.28)]{\sigma}$.
This scaling is justified since we assume that it is possible to improve the analysis presented here, such that more signal events survive while the background is still eliminated.
Additionally, it is possible to choose a point closer to the experimentally excluded region as mentioned earlier.
Furthermore, future collider experiments, such as the ones at the <cit.>, might yield much higher luminosities as well as better reconstruction possibilities of the Lorentz factor of the heavy neutrino.
Mass splitting dependency
Oscillations can be used to resolve very small mass splittings.
However, from the discussion above it is clear that if the mass splitting becomes too large, the oscillation period becomes too small, the reconstruction of oscillation patterns will be challenging.
This is shown in <ref> where the oscillation period has been varied, using a fast simulation described below.
One can see that larger oscillation periods produce higher significances.
It is expected that the significance drops again if the oscillation period reaches the mean lifetime of the heavy neutrinos, since then oscillations can not develop before the mass eigenstates decay.
The fast simulation is based on the assumption that the kinematics of the events is independent of the oscillation period.
With this assumption, the oscillation period has no impact on the cut based analysis.
As a consequence, the sum of and events follows for all points the same distribution, given by the null hypothesis.
Thus it is possible to obtain a large sample of valid signal events by combining the events passing all cuts of the three simulated points.
It is then possible to give each event a new tag, describing if that events should be counted as or as .
For this the of the heavy neutrino is computed on a per event basis, using the relation
\begin{equation}
\tau = \frac{\abs{\vec d}}{\sqrt{\gamma^2 - 1}}\,,
\end{equation}
where $\vec d$ is the position of the displaced vertex with respect to the primary vertex.
The formula for the oscillation probability can then be used to tag the event based on the new oscillation period.
At this point one has generated a sample of valid signal events with the new oscillation period.
After that, one can pick a random subset of this sample containing the physical number of events, that can be computed using (<ref>).
Subsequently, the analysis to obtain the significance can be applied to this subset of events.
This strategy is orders of magnitude faster than performing the full simulation and cut based analysis for each oscillation period separately.
§ CONCLUSION
[Significance as function of the oscillation period]
Significance as function of the oscillation period for $90$ events.
Three independently simulated points as well as eight points simulated using a fast simulation are depicted.
Smearing of the secondary vertices is neglected.
In this paper, we have performed a first full analysis of at the reconstructed level.
The simulations are based on the FeynRules implementation of the introduced in <cit.>.
After the generation of events at parton level with a patched version of MadGraph, hadronisation and showers are simulated using Pythia.
Subsequently, a fast detector simulation of the detector has been preformed using Delphes.
The uncertainty in the reconstruction of the displaced vertex has been implemented with a smearing function, that randomly selects a value of the displaced vertex around its true value, based on a Gaussian.
The analysis of the events is performed using custom and Mathematica code.
In our analysis we have focused on three within the , defined in <ref>, with heavy neutrino parameters conservatively chosen inside the region allowed by current experimental constraints.
The differ by the heavy neutrino mass splitting, which is largest for 1 and smallest for 3.
Simulating events containing heavy hadrons, we have shown that with the event selection rules and cuts as defined in <ref> the corresponding background is completely evaded.
It has also been argued that with the given cuts other backgrounds that could not be simulated should also be evaded.
Thus, the surviving signal events can be treated as background-free.
The statistical method to obtain the significance with which oscillations can be found in the simulated data is described in <ref>.
Our analysis shows that for small enough heavy neutrino mass splittings, corresponding to large enough oscillation periods, it is possible to discover with the detector at the assuming [3]ab integrated luminosity.
The impact of the oscillation period, the displaced vertex smearing, the number of events, and the error in the reconstruction of the heavy neutrino Lorentz factor on the significance are shown in <ref>.
For resolving the , it is important that the smearing is smaller than the oscillation length in lab space.
Similarly, the variance of the , due to the error in reconstructing the Lorentz factor, should be smaller than the oscillation period.
This is the case for 1, for which a significance of $\unit[(5.01 \pm 0.9)]{\sigma}$ is obtained, assuming a smearing of [2]mm and around $90$ total events relevant for the analysis, <ref>.
For smaller oscillation periods, as in 2 and 3, the significance is below [3]σ even if smearing is not taken into account.
However, we like to stress that smaller mass splittings might also be resolved with higher significance if the reconstruction of the Lorentz factor is improved.
This would not only increase the significance itself but also improve the effect for larger event numbers as shown in <ref>.
The event number could, , be increased by choosing a parameter point closer to the experimentally excluded region, <ref>.
Additionally, it might be possible to increase the significance for smaller mass splittings by increasing the decay width of the heavy neutrinos, parameter points with increased Yukawa couplings or mass.
Then the heavy neutrinos would decay faster and there would be more events in the first oscillation cycles such that resolving the pattern becomes more feasible.
However, more events would be lost by the $d_0$ cut in this case.
In order to study the interdependence between such considerations, a scan over a larger sample of benchmark parameters is necessary.
In addition, the presented study might be improved by more sophisticated background reduction and by optimising the window of considered defined in <ref>.
In summary, we have shown that the offers the exciting possibility to not only discover the induced by of long-lived heavy neutrinos, but also to resolve the pattern.
Reconstructing the oscillation period would allow to measure this mass splitting and therefore discover the pseudo-Dirac nature of the heavy neutrino pair.
This would provide deep insight into the mechanism of neutrino mass generation and could help to shed light on whether leptogenesis is able to generate the baryon asymmetry of the universe (as discussed in <cit.>).
§.§ Acknowledgements
The work of J.H. as partially supported by the Portuguese through the projects CFTP-Unit UIDB/00777/2020, UIDP/00777/2020, CERN/FIS-PAR/0002/2021, and CERN/FIS-PAR/0019/2021, which are partially funded through POCTI (FEDER), COMPETE, QREN and the EU.
S.A. and J.R. acknowledge partial support from the Swiss National Science Foundation grant 200020/175502.
§ RESIDUAL OSCILLATIONS
Impact of a $d_T$ cut
oscillation pattern after a $d_0$ cut
[Residual oscillations induced by a $d_0$ cut]
Effects of the spin correlation sensitive cut $d_0$, in comparison to the non-sensitive cut $d_T$, on a set of $N_{\LNC} + N_{\LNV}$ events.
For each of the three datasets, represented by histograms, $\expno[5]{5}$ generator level events, based on the values of 3, have been simulated.
All histograms are normalised with the same factor, that ensures that the area under the uncut histogram in fig:dT sums to unity.
Panel fig:dT: Comparison of the data before and after a $d_T$ cut of [4]mm.
Panel fig:d0: Impact of a $d_0$ cut of [4]mm overlayed by the pattern of oscillations.
Due to the angular dependence appearing in (<ref>) the $d_0$ cut does not only generate a delayed onset but additionally imprints an oscillatory pattern originating in the oscillations.
The transverse impact parameter $d_0$, calculated for displaced tracks, is not only proportional to the transverse lifetime of the decaying particle, but additionally contains a component depending on the angle between the displaced vertex direction and the observed particle momentum.
If the magnetic field can be neglected, which we explicitly checked to be the case for the points discussed in this paper, the transverse impact parameter is given by <cit.>
\begin{equation} \label{eq:impact parameter}
d_0 = d_T \sin(\varphi(\vec p_T^N, \vec p_T^\mu))\,,
\end{equation}
where $d_T$ is the transverse distance of the displaced vertex and the sine measures the angle in the transverse plane between the momenta of the heavy neutrino and the muon it decays into.
This sine introduces an angular dependency, sensitive to spin correlations in the process under consideration.
Since the and processes expose dissimilar spin correlations, the $d_0$ cut effects them differently.
This leads to the observation of patterns in event samples that are a priory insensitive to the difference between and processes.
As an example, <ref> shows the residual oscillations appearing in a large sample of $N_{\LNC} + N_{\LNV}$ events after introducing a $d_0$ cut.
While the event sample with no cuts does not feature any oscillation pattern, it is shown that the $d_0$ cut results in residual oscillations, with peaks aligning with the ones of the oscillation pattern.
It can be concluded that the $d_0$ cut affect the events more severely than the ones.
By contrast, a cut on $d_T$ is independent of spin correlations and thus no residual oscillations appear.
We found that this effect is subdominant for smaller event samples, such as in this analysis, and therefore neglected it in the main part of the paper.
Missing 'biblatex' package
The bibliography requires the 'biblatex' package.
journaltitlePhys. Rev. Lett.
titleDirect evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory
journaltitlePhys. Lett. B
title$\mu \to e\gamma$ at a Rate of One Out of $10^9$ Muon Decays?
reportnumberPrint-77-0182 (BERN)
journaltitleConf. Proc. C
titleComplex Spinors and Unified Theories
journaltitlePhys. Rev. D
titleNeutrino Masses in $\mathrm{SU}(2)\times\mathrm U(1)$ Theories
journaltitlePhys. Rev. Lett.
titleNeutrino Mass and Spontaneous Parity Nonconservation
journaltitleProg. Theor. Phys.
titleHorizontal Symmetry and Masses of Neutrinos
journaltitlePhys. Rev. D
titleNeutrino Decay and Spontaneous Violation of Lepton Number
titleTesting sterile neutrino extensions of the Standard Model at future lepton colliders
journaltitleInt. J. Mod. Phys. A
titleSterile neutrino searches at future $e^-e^+$, $pp$, and $e^-p$ colliders
titleSimulating lepton number violation induced by heavy neutrino-antineutrino oscillations at colliders
titleSearch for heavy Majorana neutrinos with the ATLAS detector in $pp$ collisions at $ \sqrt{s} = 8$ TeV
journaltitlePhys. Rev. Lett.
titleSearch for heavy neutral leptons in events with three charged leptons in proton-proton collisions at $\sqrt{s} = 13$ TeV
titleHeavy Neutrinos in displaced vertex searches at the LHC and HL-LHC
journaltitlePhys. Rev. D
titleSearch for long-lived, massive particles in events with a displaced vertex and a muon with large impact parameter in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector
titleSearch for long-lived heavy neutral leptons with displaced vertices in proton-proton collisions at $ \sqrt s = 13$ TeV
journaltitlePhys. Rev.
titleObservation of Long-Lived Neutral $V$ Particles
journaltitlePhys. Lett. B
titleObservation of $B^0-\bar B^0$ Mixing
journaltitlePhys. Rev. Lett.
titleObservation of $B^0_s - \bar{B}^0_s$ Oscillations
journaltitlePhys. Rev. Lett.
titleObservation of $D^0 - \bar D^0$ oscillations
journaltitleMod. Phys. Lett. A
titleResolvable heavy neutrino–antineutrino oscillations at colliders
journaltitleAnnals of Physics
titleInterference phenomena of neutral $K$ mesons
journaltitlePhys. Rev. D
titleOn the Quantum Mechanics of Neutrino Oscillation
journaltitlePhys. Rev. D
titleOn the treatment of neutrino oscillations without resort to weak eigenstates
journaltitlePhys. Rept.
titleOscillations of neutrinos and mesons in quantum field theory
titleHeavy Neutrino-Antineutrino Oscillations in Quantum Field Theory
titleDecoherence effects on lepton number violation from heavy neutrino-antineutrino oscillations
journaltitlePhys. Rev. D
titleRight-Handed Neutrinos at CERN LHC and the Mechanism of Neutrino Mass Generation
subtitlePhenomenological symmetry protected seesaw scenario
titleaddonFeynRules model file
titleThe fate of hints: updated global analysis of three-flavor neutrino oscillations
journaltitlePhys. Rev. Lett.
titleNovel supersymmetric SO($10$) seesaw mechanism
journaltitleNucl. Phys. B
titleNon-universal $Z'$ from SO($10$) GUTs with vector-like family and the origin of neutrino masses
titleQuasi-Dirac neutrinos at the LHC
journaltitlePhys. Rev. D
titleSame Sign versus Opposite Sign Dileptons as a Probe of Low Scale Seesaw Mechanisms
journaltitleComput. Phys. Commun.
subtitleA complete toolbox for tree-level phenomenology
titleFeynRules 2.0
titleThe automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations
titleA comprehensive guide to the physics and usage of PYTHIA 8.3
reportnumberLU-TP 22-16,MCNET-22-04,FERMILAB-PUB-22-227-SCD
subtitleA modular framework for fast simulation of a generic collider experiment
titleDELPHES 3
titleQuasi-stable neutralinos at the LHC
titleLong-lived higgsinos as probes of gravitino dark matter at the LHC
titleSearch for long-lived particles that decay into final states containing two muons, reconstructed using only the CMS muon chambers
journaltitlePhys. Rev. Lett.
titleSearch for Displaced Supersymmetry in events with an electron and a muon with large impact parameters
collaborationParticle Data Group
titleReview of Particle Physics
titleHigh-Luminosity Large Hadron Collider (HL-LHC): Technical design report
journaltitleEur. Phys. J. ST
titleFCC-ee: The Lepton Collider: Future Circular Collider Conceptual Design Report Volume 2
journaltitleEur. Phys. J. ST
titleFCC-hh: The Hadron Collider: Future Circular Collider Conceptual Design Report Volume 3
titleProbing Leptogenesis at Future Colliders
subtitleThe Lund Monte Carlo for high-$p_T^{}$ physic
subtitleA Monte Carlo at next-to-leading order
subtitleA Mathematica package to calculate Feynman rules
subtitleA framework for fast simulation of a generic collider experiment
|
# The Characteristic Time Scale of Cultural Evolution
Tobias Wand1,2∗ Daniel Hoyer3,4
(1Westfälische Wilhelms-Universität Münster, Insitut für Theoretische Physik
2Center for Nonlinear Science, Münster
3 George Brown College, Toronto
4 Evolution Institute, San Antonio
$*$ Corresponding Author<EMAIL_ADDRESS>
December 2022, Last Revision July 2023 )
###### Abstract
Numerous researchers from various disciplines have explored commonalities and
divergences in the evolution of complex social formations. Here, we explore
whether there is a ’characteristic’ time-course for the evolution of social
complexity in a handful of different geographic areas. Data from the Seshat:
Global History Databank is shifted so that the overlapping time series can be
fitted to a single logistic regression model for all 23 geographic areas under
consideration. The resulting regression shows convincing out-of-sample
predictions and its period of extensive growth in social complexity can be
identified via bootstrapping as a time interval of roughly 2500 years. To
analyse the endogenous growth of social complexity, each time series is
restricted to a central time interval without major disruptions in cultural or
institutional continuity and both approaches result in a similar logistic
regression curve. Our results suggest that these different areas have indeed
experienced a similar course in the their evolution of social complexity, but
that this is a lengthy process involving both internal developments and
external influences.
##### Keywords:
Cliodynamics, Cultural Evolution, Time Scale
## 1 Introduction
### 1.1 Motivation to Find Characteristic Time Scales
Researchers from various disciplines have analysed commonalities and
divergences in the evolution of complex social systems [1, 2, 3, 4, 5, 6, 7].
The recent emergence of Cliodynamics as a discipline has started the analysis
of the dynamics of human societies and states with data-driven scrutiny and
modelling approaches from natural sciences [8, 9]. Previous work established
that a common set of factors associated with complex social formations
typically moved in tandem across a wide variety of regions and time-periods;
factors such as social scale, the use of informational media, administrative
hierarchies, monetary instruments, and others [6]. These were interpreted as
comprising the primary dimension of what could be called ’social complexity’
across cultures, though other dimensions can be adduced as well [3].
Various studies have already discussed or tried to identify the causal drivers
of cultural evolution and evaluated the evidence for different theories of why
cultures become more complex [7, 10, 11, 12, 13]. Beyond the causal
similarities behind cultural evolution across cultures, researchers have also
found evidence for temporal similarities and seemingly parallel time scales in
the dynamics of various social structures. For example, models for societal
collapse have been derived from demographic and fiscal data that show
characteristic oscillation periods of a few centuries and a fine structure
with a faster periodicity of approximately two human generations [4]. Other
theories suggest that cultural evolution leads to the emergence of similar
political institutions and schools of thought at roughly identical time
intervals across different geographic regions [14, 15, 16]. Another recent
study has evaluated the connection between the first emergence of complex
societies in different world regions and the age of widespread reliance on
agriculture in those areas [17], supporting the theory that agriculture is a
necessary condition for the evolution of complex societies. While the time lag
between the primary reliance on agriculture and the emergence of states was
found to decrease over time, an average time lag of roughly 3,400 years for
pristine states suggested the existence of a characteristic time scale, though
this was not the explicit focus of that study. Similarly, the study on causal
drivers in [7] also found that the time since the adoption of agriculture had
a statistically significant effect as a linear predictor variable (called
’AgriLag’) for sociopolitical complexity, providing additional evidence for
temporal regularities in the growth of complexity across different cultures
and civilisations. Finally, using the same data as our article (cf. section
1.2), it was possible to identify periods of cultural macroevolution with
either slow or rapid change in social complexity [5].
Nevertheless, as yet there is no consensus on whether there is a ’typical’
time scale for socio-political development cross-culturally, let alone what
that time-course might be. Such characteristic time scales of dynamic systems
are, however, well documented in different areas of the natural sciences such
as physics and chemistry [18, 19]. Differentiating between fast and slow time
scales in a dynamical system can lead to useful insights and can inform
modelling assumptions for data analysis [20, 21]. In particular, Haken’s
theory of the ’enslaving principle’ [22], according to which the dynamics of
fast-relaxing modes are dominated (enslaved) by the behaviour of slowly
relaxing modes in a dynamical system, pioneered the research on how dynamics
on different time scales influence each other in the same observed system. The
existence of temporal regularities among societal dynamics would suggest that
cultural evolution not only occurs in similar developmental stages across
geographic regions and time periods, but also in similar time intervals. This
would add an important dimension to our understanding of how complex social
formations evolve, and raise a number of critical questions about what drives
these cross-cultural patterns.
Here, we adapt some of the methods employed in the natural sciences in an
attempt to identify characteristic time scales in the evolution of complex
societies. We utilise data collected by the Seshat: Global History Databank
[23, 24, 25], a large repository of information about the dynamics of social
complexity across world regions from the Neolithic to the early modern period
[26]. We find that, despite significant differences in the timing and
intensity of major increases in social complexity reached by polities across
the Seshat sample, there is a typical, quantitatively identifiable time course
recognisable in the data. This result is robust to a variety of checks and
covers polities from all major world regions and across thousands of years of
history. Our findings offer a novel contribution to the study of cultural
evolution, indicating the existence of a general, cross-cultural pattern in
both the scale as well as the pace of social complexity development.
### 1.2 Seshat Databank
The Seshat: Global History Databank includes systematically coded information
on over 35 geographic areas and over 200 variables across up to 10,000 years
in time steps of 100 years ([23, 26]; see also publicly available data at
http://www.seshatdatabank.info/databrowser/). During the time interval
captured by the Seshat databank, these NGAs are occupied by over 370 different
identifiable polities, defined as an "independent political unit". This sample
is constructed by identifying all known polities that occupied part or all of
each NGA over time (see [23, 24, 25] for details). The recorded variables are
aggregated into nine complexity characteristics (CCs) and a principal
component analysis shows that 77% of the variation in the data can be
explained by the first principal component (SPC1), which has almost equal
contributions from all nine CCs [6]. In the case of missing data or expert
disagreement in [6], multiple imputation [27] was used to create several data
sets with the differently imputed values which were aggregated into the
principle component analysis. The NGAs in the Seshat data cover a wide
geographical range and different levels of social complexity, though it is
important to note that the Seshat sample is focused laregly on relatively
complex, sedentary societies (but not exclusively). Data on the CCs is sampled
at century intervals, giving a time series of each polity’s estimated social
complexity measure throughout its duration.
Seshat data has allowed researchers to quantitatively test hypotheses on
cultural evolution such as identifying drivers of social complexity and
predictors of change in military technology, for example gauging the effect of
moralising religions on cultural evolution or predicting historical grain
yields [7, 28, 29, 30]. Further analysis of the Seshat data includes a
discussion of ideas from biological evolutionary theory with respect to the
tempo of cultural macroevolution, defined as "rates of change, including their
acceleration and deceleration", concluding that "cultural macroevolution is
characterized by periods of apparent stasis interspersed by rapid change" [5].
These results strongly relate to the question of the present article, whether
there is some generality in the time scale of cultural evolution in the Seshat
data.
### 1.3 Data on Culture/Polity Boundaries and Duration
Each NGA’s time series can contain data about very different polities that
succeeded each other. Sometimes, a gradual and continuous change between the
polities justifies treating predecessor and successor polities as closely
related; for instance, in the Latium NGA (modern-day central Italy), Seshat
records three separate polities for the Roman Republic, indicating the Early,
Middle, and Late phases. These phases are culturally and (to a signfiicant
degree) institutionally continuous, so can be treated as a single polity-
sequence. In other cases, there may have been an invasion or mass migration as
a clear break-point between the two polity’s continuity; for instance, between
the Ptolemaic Kingdom and Roman Principate polities in the Upper Egypt NGA.
Data from [26] and other information recorded in the Seshat sample, notably
information on the relationship between polities, is here used to establish a
list of continuous polities. The continuity is evaluated either as cultural
continuity or as political-institutional continuity and our cutout data for
both approaches is published on [31].
### 1.4 Organisation of this Article
Section 2 explains how we transformed the time series data on each NGA in the
Seshat sample to establish a common reference point to investigate the time
course of changes in social complexity across NGAs. In short, we shift each
NGA’s time series with respect to a single anchor time such that the
transformed time variable RelTime shows major overlap between the RelTime-vs-
SPC1-curves of all NGAs. Exploratory data analysis for the whole dataset
reveals that there is a logistic relationship between RelTime and the SPC1
response variable. Section 3 identifies the time scale of growth from the
lower to the upper plateau of the logistic curve via bootstrapping. The
logistic curve is compared to a regression using only either the culturally or
institutionally continuous time series and moreover, the duration of those
continuous time series is compared to the estimated characteristic time scale.
Finally, the results of the analyses are summarised and discussed in section
4. The mathematical methods and technical details are discussed in the
appendices A and B and appendix C gives more details on the used data.
## 2 Approach: Data Transformation and Exploratory Analyses
First, all raw SPC1 time series are rescaled via a min-max scaling, i.e.
$SPC1=\frac{SPC1_{raw}-\min(SPC1_{raw})}{\max(SPC1_{raw})-\min(SPC1_{raw})}.$
(1)
This has the advantage of making the interpretation of high and low SPC1
values much easier as high/low correspond to close to 1 or close to 0,
respectively. It also makes the parametrisation of a logistic curve easier by
restricting the observed data to a range between 0 and 1.
### 2.1 Anchor Time
Considering that most NGAs have an SPC1 time series that starts at a low value
barely above $0$ and ends at a high value close to $1$, a logistic regression
model seems like a reasonable suggestion for the data. Although all NGAs
experience a growth in SPC1 over time, they start at very different calendar
years. Therefore, it is necessary to shift the time series via an anchor time
so that in the new "relative" time, the growth phase in each NGA’s time series
coincide. Then, one logistic regression can be used for all shifted time
series (cf. figure 1; also A.1). Hence, each NGA $i$ needs an anchor time
$T^{(i)}_{a}$ so that if all time series are shifted by $-T^{(i)}_{a}$, they
roughly overlap. The shifted time series of the NGAs and the logistic fit are
shown in the main part of figure 1.
Figure 1: Main figure: Time series of RelTime vs. SPC1 for all 23 NGAs that
cross $\textmd{SPC}1_{0}$ and the logistic regression. Marked in red is the
area of growth between the two plateaus of the curve as identified in section
3. The various time series are shown in appendix C in multiple plots to make
the identification easier. Insets: a) distribution of SPC1 for all 35 NGAs,
the associated KDE (red) and the threshold $\textmd{SPC}1_{0}$ (vertical); b)
residuals of the logistic regression.
The anchor time can be chosen as the year during which the NGA $i\textmd{'s}$
SPC1 value crosses a threshold value. It has already been shown that there is
a clear threshold $\textmd{SPC}1_{0}$ between high and low values of SPC1 in
the data, which was used to define the RelTime variable in [29]. A similar
methodology was also used in [32], but there, the authors used the emergence
of a moralising religious belief as the "year zero" to shift each NGA’s time
series. Copying the procedure from [29] to get the RelTime variable,
$\textmd{SPC}1_{0}$ is chosen as the minimum between the two maxima in the
kernel density estimation (KDE; explained in A.2) of the SPC1 values (figure
1, inset a). The anchor time $T^{(i)}_{a}$ is then selected as the first
recorded data point when the NGA $i$ exceeds $\textmd{SPC}1_{0}$. An
illustration of the anchor time shift is provided in the appendix B. Thus, the
12 NGAs that never exceed $\textmd{SPC}1_{0}$ are discarded from this
analysis. On the one hand, this is not too problematic because their limited
growth in SPC1 means that they would have only contributed little information
to the estimation of SPC1’s characteristic growth time, but on the other hand,
this discards all NGAs from the world region Oceania-Australia in the Seshat
sample, meaning that it might introduce a bias. We discuss this and other
possible limitations of the approach further in section 4 below.
### 2.2 Logistic Regression
The RelTime-vs-SPC1 data is fitted to a logistic regression curve (cf. A.1)
via the optimisation algorithm scipy.optimize.curve_fit from [33]. The quality
of the regression curve is evaluated with the methods from A.3. With the
exception of a few outlier observations occurring in several of the NGAs, all
time series qualitatively agree with the regression curve fairly well. Also,
the majority of the residuals shown in figure 1 (inset b) are distributed
roughly symmetrically in a neighbourhood of zero. The distribution of the
residuals and the rather low value of the root-mean-square-error $RMSE\approx
0.11$ both indicate that the logistic regression is a suitable model for the
shifted SPC1 data.
To further increase our trust in the quality of the regression, it is
evaluated via the coefficient of determination $\rho^{2}$ in an out-of-sample
prediction. The data is split randomly into equally sized training and testing
data sets and a logistic regression curve $f_{i}$ is estimated by only using
the training data. Then, $f_{i}$ is used to predict the values for the test
data and the prediction is evaluated via the $\rho^{2}$ metric in A.4. The
random training-test-split is repeated $i=1,\dots,100$ times, each time using
the estimated parameters from the full time series as initial values, and the
resulting $\rho^{2}$ values have an average of $\rho^{2}=0.81\pm 0.01$ far
above 0 and therefore further strengthens our trust in the logistic model.
## 3 Analysis of Time Scales
### 3.1 Finding a Characteristic Time Scale
Having established that the data can be accurately captured by a logistic
curve, we can investigate our research question; namely, how many years did it
typically take in these different regions to transition from a polity with low
SPC1 to one with high SPC1? Or to reformulate the question: when does the
curve leave the low plateau and when does it reach the high plateau? We
attempt to answer these questions by estimating the heights of the plateaus
and their respective uncertainties and by checking when the regression curve
crosses these thresholds.
We performed 1000 steps of bootstrapping by sampling from the list of NGAs and
by estimating the regression parameters
$(a_{i},b_{i},c_{i},d_{i})_{i=1,\dots,1000}$ for each sample (cf. A.5).
According to the asymptotic behaviour in A.1, the plateaus are given by
$b_{i}$ and $a_{i}+b_{i}$. In order to make conservative estimates instead of
being influenced by noise, an upper boundary for the lower plateau’s value
$Th_{1}$ and a lower boundary for the upper plateau’s value $Th_{2}$ are used
as the thresholds. $Th_{1}$ is chosen as $Th_{1}=\mu(b)+3\sigma(b)$ of the
bootstrapped distribution of $b$, $Th_{2}$ as $Th_{2}=\mu(a+b)-3\sigma(a+b)$.
For each bootstrapped logistic curve $f_{i}(t)$, it is then determined at
which RelTime values $t^{(i)}_{1}$ and $t^{(i)}_{2}$ it crosses the lower and
upper thresholds $Th_{1}$ and $Th_{2}$. We can then understand the mean value
$\mu\left(t^{(i)}_{2}-t^{(i)}_{1}\right)=\mu\left(t^{(i)}_{2}\right)-\mu\left(t^{(i)}_{1}\right)\approx$2500\text{\,}\mathrm{yr}$$
(2)
as the characteristic time scale for the period of rapid cultural evolution
between low and high plateaus of socio-political complexity, across geography
and not in reference to any specific time period. Note that one can also
choose less restrictive thresholds via $Th_{1}=\mu(b)+\sigma(b)$ and
$Th_{2}=\mu(a+b)-\sigma(a+b)$. With these thresholds, the regression curve
leaves the vicinity of the lower plateau rather quickly but needs much longer
until it is close enough to the upper plateau to be considered as having
reached the upper plateau. These $1\sigma$ thresholds would result in a longer
time scale of roughly
$\mu\left(t^{(i)}_{2}-t^{(i)}_{1}\middle|1\sigma\right)\approx$4000\text{\,}\mathrm{yr}$.$
(3)
We can check the general validity of these results by explicitly identifying
for each NGA $i$ the first time $\tau_{1}^{(i)}$ that their SPC1 value exceeds
$Th_{1}$ and the first time $\tau_{2}^{(i)}$ they exceed $Th_{2}$. With the
exception of the Ghanaian Coast, all NGAs cross $Th_{2}$ and therefore, this
procedure yields 22 time durations $d^{(i)}=\tau_{2}^{(i)}-\tau_{1}^{(i)}$.
Only for the two NGAs Kachi Plain and Middle Yellow River Valley (two
’pristine’ states, cf. the discussion in section 4.2) does the duration
$d^{(i)}$ exceed the 4000 years estimated as an upper boundary in (3). Both
the mean (approximately 2200 years) and median (2100 years) are in line with
the main estimation in (2).
### 3.2 Continuous Polities
There are two reasons why it makes sense to restrict the logistic regression
only to a central part of each NGA’s time series, during which the polities in
that NGA are not disrupted by external influence or major dislocations in
socio-political structures. First, the logistic regression starts at a plateau
of low values of SPC1 close to $0$ and ends at a plateau of high values close
to $1$. Therefore, even a bad interpolation for the central part can achieve a
good $RMSE$, if the plateaus of the high and low tails are sufficiently
accurate. However, this would not be a reliable estimation to make an
inference on the growth phase in the centre of the curve. Second, if the NGA’s
polity is e.g. annexed by another, more developed polity, then it inherits the
invading polity’s high SPC1 value and may make a sudden jump in the SPC1
curve. However, the logistic regression here is intended to model steady,
uninterrupted growth like in [34] and not major transitions driven by
developments experienced elsewhere, as through annexations by an external
invader. Therefore, it makes sense to divide each NGA’s time series into
intervals which are separated by sharp, discontinuous changes within each NGA
and to restrict the analysis of the NGA to its central interval, i.e. to the
time series from the polities that cross the $\textmd{SPC}1_{0}$ threshold.
As mentioned earlier, there are two ways of identifying such discontinuous
changes: either via cultural changes of via major institutional changes of the
polity’s governance. Both approaches are analysed separately. The central time
series for both methods and their resulting logistic regressions are shown in
figure 2.
Figure 2: Main figure: estimated logistic curves for the full data and the two
cutout methods. Insets: a) each NGA’s central time series and resulting
logistic curve for the culturally continuous time series; b) the same as
subfigure a) for the institutionally continuous time series.
#### 3.2.1 Cultural Continuity
One set of sequences was determined by the absence of a major cultural
dislocation; namely, the introduction of a new ideological and linguistic
system, major population displacement, or major technological advance (the
adoption of iron metallurgy, for instance). This is a very broad and lenient
definition of continuity, as it allows for very different social formations to
be part of a single sequence and can include significant developments. In
Egypt, for instance, we treat nearly the entire Pharaonic period (from the
Naqada period to the Achaemenid conquest) of over 3000 years, including the
so-called Intermediate periods when central rule was fragmented (though many
cultural and social features were retained), as a culturally continuous time
period.
For the 23 NGAs under consideration, the mean value of data points for the
culturally continuous central interval is approximately $11.7$, i.e. there is
on average a bit more than one millennium of data. While this is much shorter
than the characteristic time scale of roughly $2500$ years, the longest
continuous time series of the NGAs show a similar length to that of the
characteristic time scale (cf. the left half of table 1). Hence, the logistic
regression for these cutouts is rather close to the regression of the full
data (cf. main part of figure 2) and in particular, the regression curves’
steepness (i.e. their time of growth) is quite similar.
#### 3.2.2 Institutional Continuity
For institutionally continuous time periods, we follow a similar procedure as
above, though with different criteria for continuity leading to shorter
sequences. Namely, we break each sequence at any significant
political/institutional change, even if there was much continuity in socio-
cultural forms. In Egypt, for instance, the institutional sequence starts at
the 1st Dynasty period and ends a the end of the Old Kingdom period and the
First Intermediate Period, which we call the ’Period of the Regions’. The mean
value of data points for institutionally continuous central time series is
only $5.9$ and represents approximately 500 years of data. Even the longest
continuous sequences now do not last as long as the characteristic growth time
of $2500$ years (cf. right half of table 1). Moreover, the logistic regression
has only very little data for the parameter estimation (cf. inset b of figure
2) and hence, the logistic regression has a much lower SPC1 level for the
upper plateau than the regression to the full data (main part of figure 2),
because the cutout time series are too short to reach the high-SPC1 levels.
NGA | Cultural Continuity Length | NGA | Institutional Continuity Length
---|---|---|---
Yellow River | 38 | Susiana | 17
Upper Egypt | 33 | Crete | 17
Kachi Plain | 22 | Konya Plain | 15
Susiana | 21 | Upper Egypt | 10
Table 1: For both methods of identifying continuous time sequences, the four
longest continuous central time series are shown and the amount of data points
they contain (given as their length). The data points are sampled at intervals
of one century. The culturally continuous time series are much longer than the
longest institutionally continuous sequences.
## 4 Summary and Discussion
### 4.1 Summary
Exploratory data analysis shows in figure 1 that the logistic regression is a
suitable model for the RelTime-vs-SPC1 time series. Bootstrapping allows us to
narrow down the time interval of rapid SPC1 growth to approximately 2500
years, as highlighted in figure 1. Together, these results illustrate that
there is a uniform behaviour in growth of social complexity represented by the
time evolution of SPC1.
If the data is restricted to the central part of each NGA’s time series
without any discontinuous cultural or institutional transitions, the logistic
regression is still a reasonable model and shows a similar shape to the full
data as depicted in figure 2. In particular, the regression based on the
culturally continuous time series show a very similar steepness (i.e. growth
period) to the full regression curve.
### 4.2 Discussion of the Time Scale and Continuous Sequences
Figure 2 shows that the culturally continuous and institutionally continuous
time series result in a similar logistic regression to the full data. Notably,
table 1 shows that the culturally continuous time series have a much longer
duration than the institutionally continuous ones. In particular, in the
Yellow River Valley, Upper Egypt, Kachi Plain and Susiana, the culturally
continuous time series is approximately as long (or even longer) as the
characteristic time scale of SPC1 growth. This is expected for regions that
saw the emergence of large, complex states relatively early in history and
without any precedent from neighbouring societies – the so-called ’pristine’
or ’primary’ states [35, 36] – which these regions all experienced. However,
this is not the case for most other NGAs, indicating that in those NGAs, the
growth from the lower to the higher SPC1 plateau did not take place over the
course of just one culturally continuous era, but rather included developments
across cultural spheres and, in most cases, including developments being
’brought in from the outside’ in the form of direct conquest or more indirect
influence. The institutionally continuous time series are all significantly
shorter than the characteristic growth time, as expected from the criteria
used to generate those sequences. This is notable, as it suggests that in
order to transition from low to high social complexity, major shifts in the
NGA’s governing institutions are necessary to facilitate the increase in
social complexity. In other words, our findings suggest that major transitions
in social complexity are not feasible for a single polity to accomplish, but
require multiple social formations or ’phases’ of rule building successively
(but not monotonically, as the above figures illustrate) on prior
developments. Nevertheless, the general similarity of the three regression
curves in figure 2 shows that our analysis is stable with respect to the exact
selection of time periods and different cutout criteria used.
It is interesting to compare those NGAs that crossed the threshold
$\textmd{SPC}1_{0}$ to those that failed to do so and stayed at lower
complexity values. The latter group had a mean of only 6.4 recorded data
points, i.e. there were only complex social formations coded as part of the
Seshat sample for a period of roughly six centuries. On the other hand, the
NGAs that did reach a high complexity and exceeded the threshold
$\textmd{SPC}1_{0}$ had a mean of 57.3 recorded data points, corresponding to
almost 6 millennia of observed data. Partly this is explained by different
availability of historical and archaeological evidence in different regions,
but it suggests also that cultural developments in the low complexity NGAs
could have followed the same trajectory of logistic growth, if they had been
given enough time. Unfortunately, the necessity to identify an anchor time for
this analysis means that all NGAs from the Seshat world region Oceania-
Australia had to be discarded for this research. The bias introduced by this
has to be kept in mind while interpreting our results.
### 4.3 Interpretation and Comparison to Previous Work
With the shifted time index RelTime, the logistic regression model of the SPC1
time series achieves a high accuracy in capturing the evolution of socio-
political complexity measured by SPC1. Previous work has already demonstrated
a significant amount of cross-cultural generality in the factors contributing
to the evolution of socio-political complexity ([6], supplemented by findings
in [3, 37]). Notably, a previous study has already identified a characteristic
growth pattern of SPC1 and the second principal component SPC2 and found that
a rapid period of scale is first followed by a growth of information
processing and economic complexity and then by further growth in scale [3].
Here, we expand on this prior work by identifying that the time scales
involved in these developments also exhibit a general, characteristic shape.
Nevertheless, the evolution of social complexity is a lengthy and non-
monotonous process; this emerges clearly from our analyses distinguishing the
full regional time-series involved in the transition from low to high
thresholds of SPC1 from sequences of cultural or institutional continuity. We
see no examples of this evolution accomplished during a single
institutionally-continuous sequence. Further, in all NGAs there are noisy
periods during which SPC1 grows but also crises during which socio-political
complexity sharply declines, only to recover later and continue increasing.
These findings highlight both that different parts of the world experienced
similar processes of social complexity growth, involving multiple phases of
cultural and socio-political structures building off of (and occasionally
recovering from) prior developments in each region.
While the sample of past societies explored in this article is certainly not
exhaustive, they comprise a fairly representative sample of regions from
different parts of the world and include societies from different periods,
cultures and different developmental experiences. Our results thus lends novel
empirical support to the idea (from e.g. [14, 5]) that socio-cultural
evolution does indeed occur in similar time scales across different cultures
and geographies. Future research can expand these insights by including
additional societies and exploring alternate thresholds of complexity to
identify anchor times to include more NGAs from the original sample, because
the current thresholding procedure in particular excluded some NGAs from
modern-day Oceania-Australia from our analysis.
In terms of the underlying approach, our study tries to single out the
autocatalytic effect of social complexity growth. To this end, it not only
focused on one NGA at a time, but also compares our regression results to the
culturally and institutionally continuous periods for the respective NGA.
Thus, we uncover an empirical pattern in the temporal evolution of SPC1 that
has not yet been fully discussed by previous work, e.g in causal analyses of
the drivers of social complexity like in [7]. Our methodology differs from
e.g. the regression model in [7] by deliberately choosing a very simple model
to single out the temporal evolution whilst disregarding possible drivers of
the observed dynamics. We believe this approach can be utilised to answer
other questions about long-run cultural evolution, for instance the processes
by which key technologies (e.g. metallurgy, military technology,
communications media etc.) are invented in certain locations and then adopted
in others.
While the autocatalytic growth model provides an elegant interpretation of our
findings (the current level of complexity facilitates further growth until the
presence of an upper boundary of complexity is approached), it has to be
regarded with caution: We sought as far as possible to disentangle culturally
and institutionally endogenous developments from those driven by interactions
with other polities, though even the internal developments are not free from
external influence. Previous work, for instance, shows the strong effect of
military conflicts with other states on the growth of sociopolitical
complexity [7, 28]. Hence, the autocatalytic model might be a useful low-
dimensional description of the data, but not an exhaustive explanation. In
short, our findings exposes a cross-cultural temporal pattern whose causes
need to be fleshed out in future work.
Finally, the findings of the present article can be used as a benchmark for
future additions to the Seshat data: if a new NGA is added to the databank and
shows a clear divergence from the logistic curve, it may be prudent to either
check, if there are any mistakes in the data generation and interpolation, or
if the divergences can be explained by historical developments. Such a
benchmark may thus be useful for further expansion of the Seshat databank.
#### Author Contributions
TW performed all analyses and drafted the manuscript; DH assisted in
conceptual development and drafting the manuscript.
#### Acknowledgements
Initial ideas behind this paper were developed at a workshop held by the
Complexity & Collapse Research Group of the Complexity Science Hub, Vienna.
The authors thank all the participants at that event, particularly Mateusz
Iskrzyński for valuable contributions at early stages of this project.
Financial support for this work was provided by the “Complexity Science”
research initiative supported by the Austrian Research Promotion Agency FFG
under grant #873927 and by the German Academic Scholarship Foundation
(Studienstiftung des deutschen Volkes).
## References
* [1] David Carballo, Paul Roscoe and Gary Feinman “Cooperation and Collective Action in the Cultural Evolution of Complex Societies” In _Journal of Archaeological Method and Theory_ 21.1, 2014, pp. 98–133
* [2] “Cultural Evolution” Boston: MIT Press, 2013
* [3] Jaeweon Shin et al. “Scale and information-processing thresholds in Holocene social evolution” In _Nature Communications_ 11.1, 2020, pp. 2394
* [4] Peter Turchin “Historical Dynamics: Why States Rise and Fall” Princeton University Press, 2003
* [5] Peter Turchin and Sergey Gavrilets “Tempo and Mode in Cultural Macroevolution” In _Evolutionary Psychology_ 19.4, 2021
* [6] Peter Turchin et al. “Quantitative historical analysis uncovers a single dimension of complexity that structures global variation in human social organization” In _PNAS_ 115.2, 2018, pp. E144–E151
* [7] Peter Turchin et al. “Disentangling the evolutionary drivers of social complexity: A comprehensive test of hypotheses” In _Science Advances_ 8.25, 2022
* [8] Peter Turchin “Arise ’cliodynamics”’ In _Nature_ 454, 2008, pp. 34–35
* [9] Patrick Manning, Pieter François, Daniel Hoyer and Vladimir Zadorozhny “Collaborative Historical Information Analysis”, Chapter 3.09 in Reference Module in Earth Systems and Environmental Sciences, Elsevier, 2018
* [10] V.. Childe “Man Makes Himself” Watts & Company, 1936
* [11] L.. White “The Evolution of Culture” McGraw-Hill, 1959
* [12] Elman R. Service “Origins of the state and civilization” New York, NY: WW Norton, 1975
* [13] Patrick Vinton Kirch “From chiefdom to archaic state: Hawai’i in comparative and historical context” In _How Chiefs Became Kings_ University of California Press, 2010, pp. 1–28
* [14] Oswald Spengler “Der Untergang des Abendlandes” Verlag Braumüller, 1918
* [15] David Engels “Kulturmorphologie und Willensfreiheit” In _Der lange Schatten Oswald Spenglers_ Lüdinghausen: Manuscriptum, 2018, pp. 79–102
* [16] David Engels “Oswald Spengler - Werk, Deutung, Rezeption” Stuttgart: Kohlhammer Verlag, 2021
* [17] Oana Borcan, Ola Olsson and Louis Putterman “Transition to agriculture and first state presence” In _Explorations in Economic History_ 82, 2021, pp. 101404
* [18] Elliot J. Carr “Characteristic time scales for diffusion processes through layers and across interfaces” In _Physical Review E_ 97.4 American Physical Society (APS), 2018
* [19] Eva-Maria Wartha, Markus Bösenhofer and Michael Harasek “Characteristic Chemical Time Scales for Reactive Flow Modeling” In _Combustion Science and Technology_ 193.16 Taylor & Francis, 2021, pp. 2807–2832
* [20] Todd L. Parsons and Tim Rogers arXiv, Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: a constructive approach with examples from theoretical biology, 2015 DOI: 10.48550/ARXIV.1510.07031
* [21] Clemens Willers and Oliver Kamps “Non-parametric estimation of a Langevin model driven by correlated noise” In _The European Physical Journal B_ 94.7 Springer ScienceBusiness Media LLC, 2021
* [22] Hermann Haken “Synergetik” New York, NY: Springer, 1990
* [23] Peter Turchin et al. “Seshat: The Global History Databank” In _Cliodynamics_ 6.1 California Digital Library (CDL), 2015
* [24] Pieter François et al. “A Macroscope for Global History. Seshat Global History Databank: a methodological overview” In _Digital Humanities Quarterly_ 10.4, 2016
* [25] Peter Turchin et al. “An Introduction to Seshat” Number: 1-2 In _Journal of Cognitive Historiography_ 5.1, 2018, pp. 115–123
* [26] Peter Turchin et al. “seshatdb (Equinox Packaged Data)” Zenodo, 2022 DOI: 10.5281/zenodo.6642230
* [27] Donald B Rubin “Multiple imputation for nonresponse in surveys” John Wiley & Sons, 2004
* [28] Peter Turchin et al. “Rise of the war machines: Charting the evolution of military technologies from the Neolithic to the Industrial Revolution” In _PLOS ONE_ 16.10 Public Library of Science, 2021, pp. 1–23
* [29] Peter Turchin et al. “Explaining the rise of moralizing religions: a test of competing hypotheses using the Seshat Databank” In _Religion, Brain & Behavior_ Routledge, 2022, pp. 1–28
* [30] Peter Turchin et al. “An integrative approach to estimating productivity in past societies using Seshat: Global History Databank” In _The Holocene_ 31.6 SAGE Publications, 2021, pp. 1055–1065
* [31] Tobias Wand and Daniel Hoyer “Seshat: Equinox Release with Continuous Polities” Zenodo, 2023 DOI: 10.5281/zenodo.8120128
* [32] Harvey Whitehouse et al. “Testing the Big Gods hypothesis with global historical data: a review and “retake”” In _Religion, Brain & Behavior_, 2022, pp. 1–43
* [33] Pauli Virtanen et al. “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python” In _Nature Methods_ 17, 2020, pp. 261–272
* [34] P.-F. Verhulst “Notice sur la loi que la population poursuit dans son accroissement” In _Corresp. Math. Phys._ 10, 1838, pp. 113–121
* [35] Henri..M. Claessen “The emergence of pristine states” In _Social Evolution & History_ 15.1, 2016, pp. 3–57
* [36] Charles S. Spencer “Territorial expansion and primary state formation” In _PNAS_ 107.16, 2010, pp. 7119–7126
* [37] Timothy A Kohler, Darcy Bird and David H Wolpert “Social Scale and Collective Computation” In _Journal of Social Computing_ 3.1 TUP, 2022, pp. 1–17
* [38] David W. Hosmer and Stanley Lemeshow “Applied Logistic Regression” John Wiley & Sons, Ltd, 2000
* [39] Emanuel Parzen “On Estimation of a Probability Density Function and Mode” In _The Annals of Mathematical Statistics_ 33.3 Institute of Mathematical Statistics, 1962, pp. 1065–1076
* [40] Murray Rosenblatt “Remarks on Some Nonparametric Estimates of a Density Function” In _The Annals of Mathematical Statistics_ 27.3 Institute of Mathematical Statistics, 1956, pp. 832–837
* [41] Bradley Efron and Robert J. Tibshirani “An Introduction to the Bootstrap” Springer US, 1994
* [42] Peter Turchin et al. “The Equinox2020 Seshat Data Release” In _Cliodynamics_ 11.1, 2020
## Appendix A Methods and Technical Details
### A.1 Logistic Regression Curve
Logistic regression is used to model time series data which is mostly
distributed at two plateaus with a transitory area between them [38]. It is
based on the characteristic sigmoid curve of the logistic growth model
described in [34], which models an exponential growth process constrained by a
carrying capacity. The logistic curve has the functional form $f$ with an
asymptotic behaviour
$f(x)=\frac{a}{1+\exp(-c(x-d))}+b,\quad f(-\infty)=b\quad\textmd{ and }\quad
f(\infty)=a+b.$ (4)
Often, data is scaled such that $b=0$ and $a=1$, i.e. an asymptotic behaviour
between two binary plateaus at height $0$ and $1$.
#### A.1.1 Reversing the Direction
Estimating the coefficients $(a,b,c,d)$ can lead to numerical instabilities
because it is possible to transform a logistic curve with $c>0$ to an
equivalent equation $\hat{f}$ with $c<0$. Consider e.g. $a=1,b=0,c=1$ and
$d=0$, then
$f(x)=\frac{1}{1+\exp(-x)}=\frac{\exp(x)}{\exp(x)+1}=\frac{\exp(x)+1-1}{\exp(x)+1}=1+\frac{-1}{1+\exp(x)}.$
(5)
The last reformulation of $f$ can now be parametrised via
$\hat{a}=-1,\hat{b}=1,\hat{c}=-1$ and $\hat{d}=0$. This ambiguity can lead to
the regression algorithm yielding positive and negative results for $c$ during
multiple runs. This can be prevented by setting an initial parameter guess
with $c>0$, which locks the algorithm into positive values for $c$.
### A.2 Kernel Density Estimation (KDE)
A KDE tries to reconstruct a probability density function based on a sample
$x_{1},\dots,x_{n}$ of measurement data by smoothing the histogram of the data
[39, 40]. The estimated density $\hat{\rho}(x)$ is modelled as a weighted sum
of probability densities (kernels) centred around the measured $x_{i}$. In
this article, the Gaussian density is used as the kernel via
scipy.stats.gaussian_kde [33].
### A.3 Residuals and Root Mean Squared Error
For an algorithm $f$ which estimates values $\hat{y}$ from data $X$ with true
values $y$, there are several methods to evaluate the accuracy of $f$. One of
them is the root mean squared error $RMSE$. It is defined as
$RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}r_{i}^{2}}$ (6)
via the residuals $r_{i}=\hat{y}_{i}-y_{i}$. An $RMSE$ much smaller than the
range of measured values $y_{i}$ means that the model shows only little
deviation from the data. A roughly symmetric distribution of the residuals
around $0$ indicates that the model does not have a bias towards particular
values.
### A.4 Coefficient of Prediction $\rho^{2}$
Another method to evaluate the quality of an estimated function $f$ is the
coefficient of prediction $\rho^{2}$ used in [6]. It takes the value of
$\rho^{2}=1$, if the prediction is always exactly true, and $\rho^{2}=0$, if
the prediction is only as accurate as always using the mean $\bar{y}$. It is
defined by
$\rho^{2}=1-\frac{\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}{\sum_{i=1}^{n}(\bar{y}-y_{i})^{2}}.$
(7)
### A.5 Bootstrapping
Bootstrapping is used to estimate standard deviations and confidence intervals
in a model-free approach. A sample $z_{1},\dots,z_{n}$ is re-sampled with
replacement, i.e. a new sample $\tilde{Z}=z_{i_{1}},\dots,z_{i_{n}}$ is
created that for some $j\neq k$ fulfils $i_{j}=i_{k}$. This procedure is
repeated $N$ times so that there are $\tilde{Z}_{1},\dots,\tilde{Z}_{N}$
bootstrapped samples. If $N$ is large enough, then e.g. the mean
$\tilde{\mu}(z)$ of the re-sampled data will converge to the true mean of the
original sample, but the empirical distribution of the re-sampled means
$\tilde{\mu}_{1}(z),\dots,\tilde{\mu}_{N}(z)$ enables the calculation of the
confidence interval of the empirical mean [41]. This approach can be adapted
to make inference on the standard deviation and CIs of any statistical
property of the original sample.
## Appendix B Example of the Data Preprocessing
We illustrate the preprocessing of the raw $\textmd{SPC}1$ time series using
the NGA ’Latium’ (modern day central Italy) as an example. Figure 3 shows the
shift between the original time series to the RelTime time frame relative to
the anchor time $T^{(Latium)}_{a}=500\textmd{ BC}$. Additionally, this figure
depicts how the $\textmd{SPC}1$ time series of Latium is dissected into
culturally or institutionally continuous time series intervals of SPC values
for the NGA. Note that the time index is given in $RelTime$ for the shifted
data and real-world time for the original data.
Figure 3: Illustration of how the $\textmd{SPC}1$ time series of the Latium
NGA is shifted by its anchor time to the new relative time frame $RelTime$.
Hence, at $RelTime=0$, the shifted time series is equal to the threshold value
$\textmd{SPC}1_{0}$. The two recorded discontinuities for the Latium NGA (red)
are used to divide its $\textmd{SPC}1$ time series into different intervals.
The interval containing the threshold value $\textmd{SPC}1_{0}$ (the "Relevant
Snippet" with the solid line) is used for further analysis whereas the rest
(dotted) is discarded. Note that the times of the discontinuities do not line
up with the 100-year time intervals of the $\textmd{SPC}1$ measurement which
is why the transition between the black dashed/solid lines does not perfectly
align with the red vertical lines.
## Appendix C Detailed Data
The SPC1 data from figure 1 is shown in figure 4, but spread out onto several
subplots to help the reader identify different NGAs.
Figure 4: Spreading the various time series from figure 1 onto multiple plots
to make it easier to distinguish the NGAs from each other.
### C.1 Data Collection and Availability
As described in the main text, the data used in this paper is derived from
[26], supplemented with information provided by the authors. The original data
is described in [42]. It was gathered, cleaned, reviewed, and managed by
members of the Seshat Databank project following standard project methods, as
described in the references cited in the main text. For more details on the
social complexity data utilised here and development of the SPC1 values, see
especially [6] and [7].
The full dataset used in the analyses presented here is available on [31].
This shows:
* •
NGA: Name of the NGA
* •
PolID: Unique identifier for each polity in the sample
* •
AbsTime: The ’absolute’ or calendar time-point for each SPC1 value. Note that
we sample at 100-year intervals, for as many intervals as there are polities
in the sample occupying each NGA
* •
RelTime: The shifted time-series, such that RelTime = 0 in the century
interval in each NGA during which SPC1 values crossed the calculated
threshold, as described in the main text. Other rows in the NGA are express
their relation before or after this threshold, in century intervals. Note that
rows that fall outside of the central cultural or institutional sequence are
not expressed
* •
SPC1: Raw SPC1 values for each polity at each century interval, calculated as
described in the main text (but not yet min-max scaled)
* •
Culture.Sequence: Label indicating the central time-series identified as part
of the culturally-continuous sequence that surrounds each NGA’s RelTime=0
threshold (labelled ’cultural.continuity’); all other century intervals in the
NGA are labelled ’outside.central’ to indicate they fall outside of this
central interval sequence
* •
Institutions.Sequence: Same as above, but indicating the institutionally
continuous sequence.
|
# Reconstruction of a Hypersurface Singularity from its Moduli Algebra
João Hélder Olmedo Rodrigues João Hélder Olmedo Rodrigues
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Rua
Professor Marcos Waldemar de Freitas Reis, Campus Gragoatá, Bloco G - São
Domingos
24210-201, Niterói, Rio de Janeiro
Brazil<EMAIL_ADDRESS>
(Date: August 28, 2024)
###### Abstract.
In this paper we characterize ideals of the local ring
$\mathscr{O}_{\mathbb{C}^{n},0}$ of germs of holomorphic functions at
$0\in\mathbb{C}^{n}$ which arise as the moduli ideal $\langle
f,\mathfrak{m}\,j(f)\rangle$, for some
$f\in\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^{n},0}$. A consequence of our
characterization is an effective solution to the long standing Reconstruction
Problem of the hypersurface singularity from its moduli algebra.
###### Key words and phrases:
Hypersurface singularity, Mather–Yau Theorem, Gaffney-Hauser Theorem, Tjurina
algebras, Moduli algebras
###### 2010 Mathematics Subject Classification:
Primary 32S05, 32S15, 14B05, 14B07, 14H20
Research partially supported by FAPERJ, ARC 211.361/2019
###### Contents
1. 1 Introduction
2. 2 Preliminaries
3. 3 Computation of $\Delta_{1}$
4. 4 Two properties of ideals in $\mathscr{O}_{\mathbb{C}^{n},0}$
5. 5 Main Result
## 1\. Introduction
This work comes from our interest in the classification of germs of
hypersurface singularities in $(\mathbb{C}^{n},0)$. A germ of a complex
hypersurface $(X,0)\subset(\mathbb{C}^{n},0)$ at the origin
$0\in\mathbb{C}^{n}$ is defined as the zero set of some - non-trivial -
principal ideal $I_{X}$ of the local ring $\mathscr{O}_{\mathbb{C}^{n},0}$,
the ring of germs of holomorphic functions at $0\in\mathbb{C}^{n}$. A
generator of $I_{X}$ \- which is well-defined, modulo multiplication by an
invertible element in $\mathscr{O}_{\mathbb{C}^{n},0}$ \- is said to be an
_equation_ for $(X,0)$; if $I_{X}=\langle f\rangle$ we often say that the germ
_is defined by_ $f$ and we indicate this fact writing
$(X_{f},0)\subset(\mathbb{C}^{n},0)$.
We say that two germs of hypersurfaces $(X_{f},0)\subset(\mathbb{C}^{n},0)$
and $(X_{g},0)\subset(\mathbb{C}^{n},0)$ are _biholomorphically equivalent_ if
there exist small open neighbourhoods $U$ and $V$ of the origin
$0\in\mathbb{C}^{n}$, where $f,\,g$ converge and a (germ of) biholomorphism
$\phi:U\rightarrow V$ \- which sends the origin to itself - such that
$\phi(X_{f}\cap U)=X_{g}\cap V$. Put more algebraically, it is easy to verify
that this holds if and only if there exists an invertible element
$u\in\mathscr{O}_{\mathbb{C}^{n},0}$ such that $ug=\phi^{*}(f)$. It is often
said in this case that the functions $f,g$ are _contact equivalent_ , because
they lie in the same orbit of the action of the _contact group_ $\mathcal{K}$
on $\mathscr{O}_{\mathbb{C}^{n},0}$ (see the book [GLS] for definitions). The
totality of germs of hypersurfaces all biholomorphically equivalent to one
another is said to be a _biholomorphic class_. One of the most famous
numerical invariants of a biholomorphic class is the _multiplicity_ of its
elements. Let $\mathfrak{m}$ denote the unique maximal ideal of
$\mathscr{O}_{\mathbb{C}^{n},0}$. We recall that if a system
$\\{x_{1},\ldots,x_{n}\\}$ of generators of
$\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^{n},0}$ is chosen and if
$(X_{f},0)\subset(\mathbb{C}^{n},0)$ is a germ of hypersurface, its
multiplicity $\operatorname{mult}(X_{f},0)=\operatorname{mult}(f)$ is the
smallest degree $m$ of a non-zero homogeneous polynomial appearing in a series
expansion
$f(x_{1},\ldots,x_{n})=f_{m}(x_{1},\ldots,x_{n})+f_{m+1}(x_{1},\ldots,x_{n})+\ldots\,\,$
Clearly the multiplicity of a germ doesn’t depend on the choice of
$\\{x_{1},\ldots,x_{n}\\}$. So, two biholomorphically equivalent germs of
hypersurfaces in $(\mathbb{C}^{n},0)$ have the same multiplicity. Another
important numerical invariant of a biholomorphic class is the _Tjurina number_
of its elements. We recall that the Tjurina number of
$(X_{f},0)\subset(\mathbb{C}^{n},0)$ is defined to be the complex vector space
dimension - whenever it is finite - $\tau(X_{f})$ of the _Tjurina algebra_ of
$(X_{f},0)\subset(\mathbb{C}^{n},0)$, which is defined as the quotient algebra
$A(X_{f})=\mathscr{O}_{\mathbb{C}^{n},0}/\langle f,j(f)\rangle,$
where $j(f)$ is the ideal generated by the partial derivatives of $f$. It
turns out that $\tau(X_{f})$ is finite if and only if
$(X_{f},0)\subset(\mathbb{C}^{n},0)$ has an _isolated singularity_ at
$0\in\mathbb{C}^{n}$. Notice that having same multiplicity and same Tjurina
number are necessary conditions for $(X_{f},0)\subset(\mathbb{C}^{n},0)$ and
$(X_{g},0)\subset(\mathbb{C}^{n},0)$ to belong to the same biholomorphic
class, but certainly are not sufficient. In fact, the search for a set of
invariants separating biholomorphic classes of hypersurface singularities is a
problem which seems far from being solved.
Given the previous terminology, our research started with a famous result of
J. Mather and S. Yau [MY] in the early eighties, which relates certain
isomorphism classes of $\mathbb{C}$-algebras to biholomorphic classes of
_isolated_ hypersurface singularities. Few years later the main Theorem of
[MY] was generalized by T. Gaffney and H. Hauser [GH] to the case of non-
isolated hypersurface singularities. These two results, therefore, throw
important light into the problem of biholomorphic classification of
hypersurface singularities.
Let us introduce the notation necessary to explain Mather-Yau and Gaffney-
Hauser Theorems: let $(X_{f},0)\subset(\mathbb{C}^{n},0)$ be a germ of
hypersurface. We define the _moduli algebra_ of $(X_{f},0)$ \- or more
accurately of $f$ \- as the quotient ring
$B(X_{f})=B(f)=\mathscr{O}_{\mathbb{C}^{n},0}/\langle
f,\mathfrak{m}\,j(f)\rangle.$
The ideal $\langle f,\mathfrak{m}\,j(f)\rangle$ appearing as the denominator
will be called the _moduli ideal_ of $(X_{f},0)\subset(\mathbb{C}^{n},0)$ and
we denote it as $T_{\mathcal{K}}(f)$. If $g$ is another generator of
$I_{X_{f}}$ it is easy to check that $T_{\mathcal{K}}(f)=T_{\mathcal{K}}(g)$
and this shows that $B(X_{f})$ really doesn’t depend on the chosen generator
for $I_{X_{f}}$.
We observe, more generally, that when $(X_{f},0)$ and $(X_{g},0)$ are
biholomorphically equivalent germs of hypersurfaces, then from a relation of
type $ug=\phi^{*}(f)$ as above, it is straightforward to check that the moduli
algebras $B(X_{f})$ and $B(X_{g})$ are _isomorphic_ as $\mathbb{C}$-algebras.
The converse holds but it is much more subtle, being the essential part of the
aforementioned results, which are stated here in a simplified manner:
###### Theorem (Mather, Yau; Gaffney, Hauser).
Let $(X_{f},0)\subset(\mathbb{C}^{n},0)$ and
$(X_{g},0)\subset(\mathbb{C}^{n},0)$ denote two germs of complex
hypersurfaces. The statements are equivalent:
1. (1)
$(X_{f},0)\subset(\mathbb{C}^{n},0)$ and $(X_{g},0)\subset(\mathbb{C}^{n},0)$
are biholomorphically equivalent;
2. (2)
$B(X_{f})$ and $B(X_{g})$ are isomorphic as $\mathbb{C}$-algebras.
###### Remark 1.1.
A few comments are in order:
* •
The original statement of Mather-Yau theorem (cf. [MY]) in the case of
isolated hypersurface singularities says that (1) and (2) are also equivalent
to “ _$A(X_{f})$ and $A(X_{g})$ are isomorphic as $\mathbb{C}$-algebras_”. In
the general case (non-necessarily isolated hypersurface singularities) this is
not true, as shown in [GH].
* •
In [GH], the authors show, by means of the introduction of a quotient module
which plays the role of the moduli algebra $B(X_{f})$, that a similar
assertion holds beyond the case of hypersurface singularities.
* •
In [MY], the authors originally baptised $A(X_{f})$ and $B(X_{f})$ as the
_moduli algebras_ of the germ of hypersurface
$(X_{f},0)\subset(\mathbb{C}^{n},0)$, because the preceding result tells us
that the problem of the classification of germs of (isolated) hypersurfaces
singularities $(X_{f},0)\subset(\mathbb{C}^{n},0)$ up to biholomorphic
equivalence is equivalent to that of the classification of their moduli
algebras $A(X_{f})$ or $B(X_{f})$ up to $\mathbb{C}$-algebra isomorphism.
* •
The proofs presented in [MY] and [GH] are not constructive and until very
recently there was the open problem, called the Reconstruction Problem, of
reconstructing the (isolated) hypersurface singularity out of its Tjurina
algebra. For solutions in special cases we refer to [Y], [IK], [E] and the
very recent [ES]. In [OR] we solved the Reconstruction Problem, at least in
the case where the hypersurface can be characterized by its Tjurina algebra.
This is precisely the case where the hypersurface singularity is of _Isolated
Singularity Type_ (cf. [GH] for details). This case includes - strictly - the
case of isolated hypersurface singularities. The main purpose of this paper is
to show that we can push our techniques a little further and reconstruct the
hypersurface out of $B(X_{f})$, then closing the remaining cases.
* •
Connected to the Reconstruction problem is the _Recognition problem_ , which
is to decide whether a quotient algebra $\mathscr{O}_{\mathbb{C}^{n},0}/I$ is
isomorphic to the Tjurina algebra of some hypersurface singularity
$(X_{f},0)\subset(\mathbb{C}^{n},0)$. This is of course equivalent to
recognize whether the ideal $I$ is a Tjurina ideal $\langle f,j(f)\rangle$ for
some $f\in\mathfrak{m}$. This was the approach taken in [OR].
To deal with the problem of reconstruction described above for algebras of
type $B(X_{f})$ we introduce right away our guiding question through this
paper, namely
###### Problem 1.2.
Fixed $n\geqslant 1$, how to find necessary and sufficient conditions for a
(proper) ideal $I\,\subset\mathscr{O}_{\mathbb{C}^{n},0}$ to be the moduli
ideal of some $f\in\mathfrak{m}$?
Clearly, for all $n\geqslant 1$, the zero ideal and the maximal ideal
$\mathfrak{m}$ are the moduli ideals of $0$ and $x_{1}$, say, respectively.
###### Example 1.3.
If $n=1$ then any ideal
$I\subset\mathscr{O}_{\mathbb{C}^{1},0}=\mathbb{C}\\{x_{1}\\}$ is of the form
$I=\mathfrak{m}^{k}=\langle x_{1}^{k}\rangle$, $k\geqslant 1$; this is the
moduli ideal of $f(x_{1})=x_{1}^{k}$.
So, if $n=1$, Problem 1.2 has a trivial solution. However, beginning at $n=2$,
this is not true any more. Clearly a necessary condition for an ideal
$I\,\subset\mathscr{O}_{\mathbb{C}^{n},0}$ to be the moduli ideal of some
$f\in\mathscr{O}_{\mathbb{C}^{n},0}$ is that the minimal number of generators
of $I$ should be at most $n^{2}+1$, but this is by no means sufficient as we
see in the next
###### Example 1.4.
For any $n\geqslant 2$ the ideal $I_{n}=\langle
x_{1}^{2},x_{2},x_{3},\ldots,x_{n}\rangle\subset\mathscr{O}_{\mathbb{C}^{n},0}$
is not a moduli ideal. Indeed, since
$\dim_{\mathbb{C}}\,\Big{(}\frac{\mathscr{O}_{\mathbb{C}^{n},0}}{I_{n}}\Big{)}=2$
the Tjurina number of a possible $f$ satisfying $I_{n}=T_{\mathcal{K}}(f)$ is
at most $2$. Up to contact equivalence there are finitely many $f$ such that
$\tau(X_{f})\leqslant 2$, namely $f_{0}=x_{1}$,
$f_{1}=x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}$ and
$f_{2}=x_{1}^{3}+x_{2}^{2}+\ldots+x_{n}^{2}$. By direct inspection one checks
that the corresponding quotients
$\frac{\mathscr{O}_{\mathbb{C}^{n},0}}{T_{\mathcal{K}}(f_{i})}$ have
$\mathbb{C}$-vector space dimensions $1$, $n+1$ and $n+2$ respectively. Since
$n\geqslant 2$, none has $\mathbb{C}$-vector space dimension $2$.
We now briefly describe the contents of the paper. In Section 2, along with
some preliminaries on moduli ideals, we introduce and investigate some
properties of the set $\Delta_{1}(I)$ of anti-derivatives of an arbitrary
ideal $I$, which is a natural set of elements of
$\mathscr{O}_{\mathbb{C}^{n},0}$ to look at in the search for a solution $f$
for the equation $I=T_{\mathcal{K}}(f)$. In Section 3 we suggest an easily
applicable method for computation of $\Delta_{1}(I)$ in examples, with
routines already implemented in SINGULAR [DGPS]. In Section 4 we introduce two
easily checkable properties on arbitrary ideals of
$\mathscr{O}_{\mathbb{C}^{n},0}$. Our main result, to be presented in Section
5, is a characterization of the ideals $I$ for which the equation
$I=T_{\mathcal{K}}(f)$ admits a solution $f$. In other words, we characterize
moduli ideals giving an explicit solution for Problem 1.2. In the same
Section, we show by an example how to use SINGULAR to recognize a moduli ideal
and to reconstruct the hypersurface singularity from it.
### Acknowledgments
The author wishes to express his gratitude to T. Gaffney for patiently
answering questions related to Mather-Yau type results; and to G.-M. Greuel
for bringing to his attention an inaccuracy written in a previous preprint.
## 2\. Preliminaries
Let $\mathscr{O}_{\mathbb{C}^{n},0}$ the local ring of germs of holomorphic
functions at $0\in\mathbb{C}^{n}$ and let $\mathfrak{m}$ denote its maximal
ideal. For some germ of holomorphic function $f\in\mathfrak{m}$ we let $j(f)$
denote the _Jacobian ideal_ of $f$, that is, the ideal generated by the (first
order) partial derivatives of $f$ with respect to a chosen coordinate system
$\\{x_{1},\ldots,x_{n}\\}$ \- minimal set of generators - for $\mathfrak{m}$.
As indicated in the Introduction, we will be concerned with moduli ideals
$T_{\mathcal{K}}(f)=\langle f,\mathfrak{m}j(f)\rangle$
of functions $f\in\mathfrak{m}$. It is easy to verify that for any
biholomorphic change of coordinates
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ and any
$f\in\mathscr{O}_{\mathbb{C}^{n},0}$, we have
$T_{\mathcal{K}}(\phi^{*}(f))=\phi^{*}(T_{\mathcal{K}}(f))$. Hence we will fix
a coordinate system and we will always compute $T_{\mathcal{K}}(f)$ with
respect to this coordinate system.
The properties below of $T_{\mathcal{K}}$ are immediate to verify:
###### Remark 2.1.
For any $f,g\in\mathscr{O}_{\mathbb{C}^{n},0}$ we have:
(i) $T_{\mathcal{K}}(f+g)\subseteq T_{\mathcal{K}}(f)+T_{\mathcal{K}}(g)$;
(ii) $T_{\mathcal{K}}(fg)\subseteq T_{\mathcal{K}}(f)\,T_{\mathcal{K}}(g)$;
(iii) $T_{\mathcal{K}}(f)=T_{\mathcal{K}}(g)$ if $\langle f\rangle=\langle
g\rangle$.
We will find it convenient to introduce the definition below:
###### Definition 2.2.
Let $J\subset\mathscr{O}_{\mathbb{C}^{n},0}$ the ideal generated by
$g_{1},\ldots,g_{q}\in\mathscr{O}_{\mathbb{C}^{n},0}$. We define the ideal
$T_{\mathcal{K}}(J)=T_{\mathcal{K}}(g_{1})+\ldots+T_{\mathcal{K}}(g_{q})\subseteq\mathscr{O}_{\mathbb{C}^{n},0}.$
Notice that this is a well-posed definition since $T_{\mathcal{K}}(J)$ does
not depend on the generators $g_{i}$ chosen but only on the ideal they
generate, as the reader can check using $(i),\,(ii)$ in Remark 2.1. Although
Problem 1.2 mentions moduli ideals of functions (or rather of principal
ideals, cf.(iii) of Remark 2.1), it seems fair to refer to
$T_{\mathcal{K}}(J)$ as the _moduli ideal_ of $J$.
###### Remark 2.3.
The properties below are easy to verify.
* (i)
$J\subseteq T_{\mathcal{K}}(J)$;
* (ii)
If $J_{1}\subseteq J_{2}$ is an inclusion of ideals then
$T_{\mathcal{K}}(J_{1})\subseteq T_{\mathcal{K}}(J_{2})$;
* (iii)
If $\\{J_{\lambda}\\}_{\lambda\in\Lambda}$ is any family of ideals of
$\mathscr{O}_{\mathbb{C}^{n},0}$ then
$T_{\mathcal{K}}(\sum_{\lambda}J_{\lambda})=\sum_{\lambda}T_{\mathcal{K}}(J_{\lambda})$;
* (iv)
$T_{\mathcal{K}}(J_{1}\cap J_{2})\subseteq T_{\mathcal{K}}(J_{1})\cap
T_{\mathcal{K}}(J_{2})$.
Now we introduce the main object - which is an adapted version of the object
used in [OR] and - that will ultimately lead us to our solution to Problem
1.2.
###### Definition 2.4.
Let $I\subset\mathscr{O}_{\mathbb{C}^{n},0}$ be an ideal. We define the _ideal
of anti-derivatives_ of $I$ as
$\Delta_{1}(I)=\\{f\in\mathscr{O}_{\mathbb{C}^{n},0}\,|\,T_{\mathcal{K}}(f)\subseteq
I\\}.$
It is easy to show that $\Delta_{1}(I)$ does not depend on the choice of
parameters $\\{x_{1},\ldots,x_{n}\\}$ used to compute Jacobian ideals.
Moreover, the properties below are straightforward to check.
###### Remark 2.5.
$\Delta_{1}(I)$ is an ideal of $\mathscr{O}_{\mathbb{C}^{n},0}$;
$I^{2}\subseteq\Delta_{1}(I)\subseteq I$;
If $\\{I_{\lambda}\\}_{\lambda\in\Lambda}$ is any family of ideals of
$\mathscr{O}_{\mathbb{C}^{n},0}$ then
$\Delta_{1}(\bigcap_{\lambda}I_{\lambda})=\bigcap_{\lambda}\Delta_{1}(I_{\lambda})$;
If $I\subseteq J$ is an inclusion of ideals in
$\mathscr{O}_{\mathbb{C}^{n},0}$ then $\Delta_{1}(I)\subseteq\Delta_{1}(J)$;
###### Example 2.6.
Let $I=\langle f^{k}\rangle\subset\mathscr{O}_{\mathbb{C}^{n},0}$, $n\geqslant
2$, be a principal ideal with $k\geqslant 1$ and $f$ being an irreducible
element. Then $\Delta_{1}(I)=\langle f^{k+1}\rangle$. Indeed, one inclusion is
immediate. For the opposite inclusion, assume $g\in\Delta_{1}(I)$ and write
$g=af^{k}$, for some $a\in\mathscr{O}_{\mathbb{C}^{n},0}$. For all
$i,j=1,\ldots,n$, $x_{i}\frac{\partial g}{\partial
x_{j}}=x_{i}f^{k}.\frac{\partial a}{\partial
x_{j}}+akf^{k-1}x_{i}\frac{\partial f}{\partial x_{j}}$ is also a multiple of
$f^{k}$. Hence $f$ must divide $ax_{i}\frac{\partial f}{\partial x_{j}}$, for
all $i,j=1,\ldots,n$. Since $f$ is irreducible, for every $i,j=1,\ldots,n$,
$f$ must divide $x_{i}$ or $\frac{\partial f}{\partial x_{j}}$ or $a$. If $f$
divides $\frac{\partial f}{\partial x_{j}}$ for all $j=1,\ldots,n$ then
$\langle\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial
x_{n}}\rangle\subseteq\langle f\rangle$ and we would deduce that
$f=0\in\mathscr{O}_{\mathbb{C}^{n},0}$, which is not the case. Since
$n\geqslant 2$, $f$ cannot divide all the $x_{i}$. So, for some $i,j$, $f$
does not divide $x_{i}\frac{\partial f}{\partial x_{j}}$. Being an irreducible
element, $f$ must divide $a$. Hence $g\in\langle f^{k+1}\rangle$.
###### Example 2.7.
Let $I=\langle f\rangle\subset\mathscr{O}_{\mathbb{C}^{n},0}$, $n\geqslant 2$,
be a non-trivial principal ideal and $f=f_{1}^{k_{1}}\ldots f_{r}^{k_{r}}$ be
a factorization of $f$ into irreducible, non-associated, elements with
positive $k_{1},\ldots,k_{r}$. Then $I=\langle
f_{1}^{k_{1}}\rangle\cap\ldots\cap\langle f_{r}^{k_{r}}\rangle$. It follows
from Remark 2.5, (iii) and Example 2.6 that $\Delta_{1}(I)=\langle
f_{1}^{k_{1}+1}\rangle\cap\ldots\cap\langle f_{r}^{k_{r}+1}\rangle=\langle
f_{1}^{k_{1}+1}\ldots f_{r}^{k_{r}+1}\rangle$.
###### Example 2.8.
Let $\mathfrak{m}^{k}$, with $k\geqslant 1$, denote the $k$-th power of the
maximal ideal of $\mathscr{O}_{\mathbb{C}^{n},0}$. Let $g=x_{1}^{k_{1}}\ldots
x_{n}^{k_{n}}$ with $k_{j}\geqslant 0$ and $\sum k_{j}=k$ be a monomial
generator of $\mathfrak{m}^{k}$. For any pair $i,j=1,\ldots,n$ it is easy to
check that $x_{i}\frac{\partial g}{\partial x_{j}}\in\mathfrak{m}^{k}$. Hence
$T_{\mathcal{K}}(g)\subset\mathfrak{m}^{k}$, which shows
$\Delta_{1}(\mathfrak{m}^{k})=\mathfrak{m}^{k}$.
###### Example 2.9.
Let $I_{n}=\langle
x_{1}^{2},x_{2},\ldots,x_{n}\rangle\subset\mathscr{O}_{\mathbb{C}^{n},0}$,
$n\geqslant 2$, as in Example 1.4. Then $\Delta_{1}(I_{n})=\mathfrak{m}^{2}$.
Indeed, since $\mathfrak{m}^{2}\subset I_{n}$ then according to Remark 2.5,
item (iv) and the previous example we have
$\mathfrak{m}^{2}=\Delta_{1}(\mathfrak{m}^{2})\subseteq\Delta_{1}(I_{n})$. For
the other inclusion, if
$g=a_{1}x_{1}^{2}+a_{2}x_{2}+\ldots+a_{n}x_{n}\in\Delta_{1}(I_{n})$, computing
$\frac{\partial g}{\partial x_{j}}$, we see that
$a_{j}\in(I_{n}:\mathfrak{m})=\mathfrak{m}$, for all $j\geqslant 2$. Hence
$a_{2}x_{2}+\ldots+a_{n}x_{n}\in\mathfrak{m}^{2}$ and we conclude that
$g\in\mathfrak{m}^{2}$.
## 3\. Computation of $\Delta_{1}$
Up to now we have obtained the ideal of anti-derivatives $\Delta_{1}(I)$ in a
few specific cases (cf. Examples 2.7, 2.8, 2.9). In this section we suggest a
method to compute $\Delta_{1}(I)$ for any ideal
$I\subset\mathscr{O}_{\mathbb{C}^{n},0}$, then showing that it is accessible
in concrete computations. We will illustrate our procedure with examples
obtained using basic routines already implemented in the software SINGULAR,
[DGPS].
Fix a basis $\\{x_{1},\ldots,x_{n}\\}$ of the maximal ideal
$\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^{n},0}=\mathbb{C}\\{x_{1},\ldots,x_{n}\\}$.
A free $\mathscr{O}_{\mathbb{C}^{n},0}$-module of certain rank $\ell$ will be
denoted by $F_{\ell}$; for any $\underline{b}=(b_{1},\ldots,b_{\ell})^{t}$,
$\underline{c}=(c_{1},\ldots,c_{\ell})^{t}\in F_{\ell}$ we put
$\underline{b}\cdot\underline{c}$ to denote
$\sum_{k=1}^{\ell}b_{k}c_{k}\in\mathscr{O}_{\mathbb{C}^{n},0}$.
Assume the ideal $I$ given by generators $I=\langle
f_{1},\ldots,f_{\ell}\rangle$. Then some element
$g=\underline{a}\cdot\underline{f}=\sum_{k}a_{k}f_{k}\in I$ is an element of
$\Delta_{1}(I)$ if and only if, for all $i,j=1,\ldots,n$,
$x_{i}\frac{\partial g}{\partial x_{j}}=\sum_{k}a_{k}x_{i}\frac{\partial
f_{k}}{\partial x_{j}}+\sum_{k}\frac{\partial a_{k}}{\partial
x_{j}}x_{i}f_{k}=\underline{a}\cdot x_{i}\frac{\partial\underline{f}}{\partial
x_{j}}+\frac{\partial\underline{a}}{\partial x_{j}}\cdot x_{i}\underline{f}$
belongs to $I$. In other words,
$g=\underline{a}\cdot\underline{f}\in\Delta_{1}(I)$ if and only if
$\underline{a}\cdot x_{i}\frac{\partial\underline{f}}{\partial x_{j}}\in I,$
for all $i,j=1,\ldots,n$.
For each $i,j=1,\ldots,n$ we denote by $E^{i,j}$ the submodule of $F_{\ell}$
consisting of the elements $\underline{a}\in F_{\ell}$ such that
$\underline{a}\cdot x_{i}\frac{\partial\underline{f}}{\partial x_{j}}\in I$.
Then $E:=\bigcap_{i,j=1}^{n}\,E^{i,j}\subseteq F_{\ell}$ is a finitely
generated $\mathscr{O}_{\mathbb{C}^{n},0}$-submodule of $F_{\ell}$ consisting
of all elements $\underline{a}\in\,F_{\ell}$ such that
$g=\underline{a}\cdot\underline{f}\in\Delta_{1}(I)$.
Notation being as above we have
###### Proposition 3.1.
Let $\underline{e}_{\,1},\ldots,\underline{e}_{\,q}\in F_{\ell}$ be generators
of $E$. Then the ideal of anti-derivatives $\Delta_{1}(I)$ is generated by
$\underline{e}_{\,t}\cdot\underline{f}$, for $t=1,\ldots,q$.
###### Proof.
As already discussed, $g\in\Delta_{1}(I)$ if and only if
$g=\underline{a}\cdot\underline{f}$ for some $\underline{a}\in E$. Since $E$
is generated by all the $\underline{e}_{\,t}$, the result follows from
$\mathscr{O}_{\mathbb{C}^{n},0}$-linearity of the $(-)\cdot\underline{f}$
product. ∎
Here we show how we used the software SINGULAR, [DGPS] to compute the ideal of
anti-derivatives of a given ideal. We claim no originality, since only
routines already implemented by the software developers and collaborators were
applied.
###### Example 3.2.
Let $I=\langle
yz,z^{3},xw,w^{2}\rangle\subset\mathscr{O}_{\mathbb{C}^{4},0}=\mathbb{C}\\{x,y,z,w\\}$.
With the interface of SINGULAR open, type
> ring r=0,(x,y,z,w),ds;
This declares you are working over a field of characteristic zero, variables
$x,y,z,w$ and set the corresponding ring of power series. We now declare the
generators of the ideal $I$ by means of a matrix with one row and (in the
present case) four columns: type
> matrix B[1][4]=yz,z3,xw,w2;
Now to compute the submodule $E^{1,1}$, (same notation as above), we declare a
matrix with entries the partial derivatives of the given generators in terms
of $x$ multiplied by $x$. We compute $E^{1,1}$ as follows:
> matrix e11[1][4]=x*diff(B,x); def E11=modulo(e11,B);
Likewise, compute $E^{1,2}$, $E^{1,3}$, $E^{1,4}$, $E^{2,1}$, $E^{2,2}$,
$E^{2,3}$, $E^{2,4}$, $E^{3,1}$, $E^{3,2}$, $E^{3,3}$, $E^{3,4}$, $E^{4,1}$,
$E^{4,2}$, $E^{4,3}$, $E^{4,4}$:
> matrix e12[1][4]=x*diff(B,y); def E12=modulo(e12,B);
> matrix e13[1][4]=x*diff(B,z); def E13=modulo(e13,B);
⋮
> matrix e44[1][4]=w*diff(B,w); def E44=modulo(e44,B);
Now, we put
> def
> e=intersect(E11,E12,E13,E14,E21,E22,E23,E24,E31,E32,E33,E34,E41,E42,E43,E44);
> def E=std(e);
This defines $E$ and computes a standard basis, with respect to the given
monomial order. Now, to obtain $E$, type
> print(E);
SINGULAR gives
$\left(\begin{array}[]{ccccccccccccccc}0&0&0&0&0&0&0&0&yz&0&z^{2}&xw&zw&w^{2}&0\\\
0&y&z&0&0&w&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&w&0&0&yz&0&0&0&0&0&0&z^{3}\\\
x&0&0&w&0&0&yz&0&0&z^{2}&0&0&0&0&0\\\ \end{array}\right)$
Hence, as in Proposition 3.1, we obtain generators of $\Delta_{1}(I)$
computing
$(-)\cdot(yz,z^{3},xw,w^{2}),$
for all (transposed) columns of the previous matrix. We speed up calculations
typing
> ideal d=B*E;
> ideal D=std(d);
> D;
The output lists the generators of $\Delta_{1}(I)$. In the case under
consideration we obtain the ten generated monomial ideal below:
$\Delta_{1}(I)=\langle
xw^{2},w^{3},y^{2}z^{2},yz^{3},z^{4},xyzw,yz^{2}w,z^{3}w,yzw^{2},z^{2}w^{2}\rangle.$
###### Example 3.3.
Let $I=\langle
3xy^{2}+x^{6},y^{3},x^{5}y,x^{7}\rangle\subset\mathscr{O}_{\mathbb{C}^{2},0}=\mathbb{C}\\{x,y\\}$.
Declare the ring and the generators of the ideal $I$ as
> ring r=0,(x,y),ds;
> matrix B[1][4]=3xy2+x6,y3,x5y,x7;
Now we compute the submodules $E^{1,1}$, $E^{1,2}$, $E^{2,1}$, $E^{2,2}$:
> matrix e11[1][4]=x*diff(B,x); def E11=modulo(e11,B);
> matrix e12[1][4]=x*diff(B,y); def E12=modulo(e12,B);
> matrix e21[1][4]=y*diff(B,x); def E21=modulo(e21,B);
> matrix e22[1][4]=y*diff(B,y); def E22=modulo(e22,B);
As before we define $E$ and compute a standard basis:
> def e=intersect(E11,E12,E21,E22);
> def E=std(e);
> print(E);
In this case, SINGULAR gives
$\left(\begin{array}[]{cccccc}0&0&0&0&y&x^{3}\\\ 0&1&x&y&0&0\\\ 0&1&0&0&0&0\\\
1&0&0&0&0&0\\\ \end{array}\right)$
Again, as in Proposition 3.1, generators of $\Delta_{1}(I)$ are obtained
computing
$(-)\cdot(3xy^{2}+x^{6},y^{3},x^{5}y,x^{7}),$
for all (transposed) columns of the previous matrix. We do this as
> ideal d=B*E;
> ideal D=std(d);
> D;
We obtain $\Delta_{1}(I)=\langle y^{3}+x^{5}y,x^{4}y^{2},x^{7},x^{6}y\rangle.$
## 4\. Two properties of ideals in $\mathscr{O}_{\mathbb{C}^{n},0}$
In this section we introduce the two relevant conditions present in our
characterization (see Theorem 5.1) of moduli ideals of hypersurface
singularities in $\mathscr{O}_{\mathbb{C}^{n},0}$.
### $T_{\mathcal{K}}$-fullness
Here we introduce $T_{\mathcal{K}}$-fullness, a quite natural notion related
to our Problem 1.2. Recall the definition of the moduli ideal of an arbitrary
ideal of $\mathscr{O}_{\mathbb{C}^{n},0}$ (cf. Definition 2.2) and observe
that, in general, we have $T_{\mathcal{K}}(\Delta_{1}(I))\subseteq I$.
###### Definition 4.1.
Let $I$ be an ideal of $\mathscr{O}_{\mathbb{C}^{n},0}$. We say that $I$ is
_$T_{\mathcal{K}}$ -full_ if $T_{\mathcal{K}}(\Delta_{1}(I))=I$.
###### Example 4.2.
Let $I_{n}=\langle
x_{1}^{2},x_{2},\ldots,x_{n}\rangle\subset\mathscr{O}_{\mathbb{C}^{n},0}$,
$n\geqslant 2$ as in Example 1.4. We have seen in Example 2.9 that
$\Delta_{1}(I_{n})=\mathfrak{m}^{2}$. A routine calculation now shows that
$T_{\mathcal{K}}(\Delta_{1}(I_{n}))=T_{\mathcal{K}}(\mathfrak{m}^{2})=\mathfrak{m}^{2}\subsetneq
I_{n}$ so $I_{n}$ is not $T_{\mathcal{K}}$-full.
The general significance of $T_{\mathcal{K}}$-fullness in our investigation is
apparent in the next result, which is, in view of Example 4.2, an alternative
proof that $I_{n}\subset\mathscr{O}_{\mathbb{C}^{n},0}$, as in Example 1.4, is
not a moduli ideal.
###### Proposition 4.3.
Let $I$ be an ideal of $\mathscr{O}_{\mathbb{C}^{n},0}$. If $I$ is a moduli
ideal then $I$ is $T_{\mathcal{K}}$-full.
###### Proof.
Let $\Delta_{1}(I)=\langle g_{1},\ldots,g_{q}\rangle$. If $I$ is a moduli
ideal, then there exists $f\in\Delta_{1}(I)$ such that $I=T_{\mathcal{K}}(f)$.
We may write $f=\sum_{k=1}^{q}r_{k}g_{k}$, for some
$r_{k}\in\mathscr{O}_{\mathbb{C}^{n},0}$. Using Remark 2.1, we obtain
$I=T_{\mathcal{K}}(f)\subseteq\sum_{k=1}^{q}T_{\mathcal{K}}(r_{k}g_{k})\subseteq\sum_{k=1}^{q}T_{\mathcal{K}}(g_{k})=T_{\mathcal{K}}(\Delta_{1}(I))\subseteq
I$ and equality holds throughout. We conclude that $I$ is
$T_{\mathcal{K}}$-full. ∎
###### Example 4.4.
(Example 3.2, continued) We have computed the anti-derivatives ideal of
$I=\langle yz,z^{3},xw,w^{2}\rangle\subset\mathbb{C}\\{x,y,z,w\\}$ as
$\Delta_{1}(I)=\langle
xw^{2},w^{3},y^{2}z^{2},yz^{3},z^{4},xyzw,yz^{2}w,z^{3}w,yzw^{2},z^{2}w^{2}\rangle$.
We check easily that $T_{\mathcal{K}}(\Delta_{1}(I))\subsetneq I$. Hence $I$
is not a moduli ideal because it is not $T_{\mathcal{K}}$-full.
Being $T_{\mathcal{K}}$-full is, however, not sufficient for $I$ to be a
moduli ideal.
###### Example 4.5.
Let $k\geqslant 5$ and let
$I=\mathfrak{m}^{k}\subset\mathscr{O}_{\mathbb{C}^{2},0}=\mathbb{C}\\{x,y\\}$.
Certainly $I$ is not a moduli ideal because it is minimally generated by more
than $5=2^{2}+1$ elements. However, we have seen (cf. Example 2.8) that
$\Delta_{1}(I)=I$. It is equally easy to check that
$T_{\mathcal{K}}(\Delta_{1}(I))=I$, which implies that $I$ is
$T_{\mathcal{K}}$-full.
### $T_{\mathcal{K}}$-dependence
We have shown above that $T_{\mathcal{K}}$-fullness is not sufficient for an
ideal to be a moduli ideal. Here we explain the last ingredient which is
needed, in addition to $T_{\mathcal{K}}$-fullness (cf. Definition 4.1), to
characterize moduli ideals. We begin with some examples.
###### Example 4.6.
Consider the ideals
$\mathfrak{m}^{k}\subset\mathscr{O}_{\mathbb{C}^{2},0}=\mathbb{C}\\{x,y\\}$,
$k\geqslant 1$. All of them are $T_{\mathcal{K}}$-full and Example 4.5 has
shown that, if $k\geqslant 5$, they are not moduli ideals. For $k\leqslant 4$,
we check easily that $\mathfrak{m}$, $\mathfrak{m}^{2}$, $\mathfrak{m}^{3}$
are moduli ideals of $x,\,xy,\,x^{3}+y^{3}$, respectively. We claim that
$\mathfrak{m}^{4}$ is not a moduli ideal of any $f\in\mathbb{C}\\{x,y\\}$.
Indeed, since
$\dim_{\mathbb{C}}\,\Big{(}\frac{\mathbb{C}\\{x,y\\}}{\mathfrak{m}^{4}}\Big{)}=10$,
the Tjurina number $\tau=\tau(X_{f})$ of a possible $f\in\mathbb{C}\\{x,y\\}$
such that $T_{\mathcal{K}}(f)=\mathfrak{m}^{4}$ is at most $10$. After a check
to the Arnold’s lists in [Ar] we see that any $f$ with $\tau\leqslant 10$ is
quasi-homogeneous; it follows that
$\dim_{\mathbb{C}}\,\Big{(}\frac{\mathbb{C}\\{x,y\\}}{T_{\mathcal{K}}(f)}\Big{)}=\tau+2$.
We deduce that the possible $f$ must define $A_{8}$, $D_{8}$ or $E_{8}$
singularities. Up to contact equivalence we can compute with the respective
normal forms $x^{2}+y^{9}$, $x^{2}y+y^{7}$ and $x^{3}+y^{5}$ to obtain
$T_{\mathcal{K}}(f)$ as $\langle x^{2},xy,y^{9}\rangle$, $\langle
x^{3},x^{2}y,xy^{2},y^{7}\rangle$ and $\langle
x^{3},x^{2}y,xy^{4},y^{5}\rangle$ respectively. All of them have elements of
multiplicity smaller than $4$, opposite to $\mathfrak{m}^{4}$. We conclude
that $\mathfrak{m}^{4}$ is not a moduli ideal but the reason is not anymore
its minimal number of generators.
We close this Example observing that the function germs $x,\,xy,\,x^{3}+y^{3}$
of which $\mathfrak{m}$, $\mathfrak{m}^{2}$, $\mathfrak{m}^{3}$ are moduli
ideals are not the more general possible: indeed in these cases, _general_
$\mathbb{C}$-linear combinations of the generators of $\Delta_{1}(I)$ are
function germs $f$ satisfying $I=T_{\mathcal{K}}(f)$ in each case, while in
the case $I=\mathfrak{m}^{4}$ _every_ linear combination satisfies
$T_{\mathcal{K}}(f)\subsetneq I$.
We aim to give a precise meaning to the intuition, that of a general linear
combination $f$ of the given generators of $\Delta_{1}(I)$ has the largest
possible $T_{\mathcal{K}}(f)$ and could reveal whether a given ideal $I$ is a
moduli ideal. Our results and definitions in this subsection are expressed
geometrically, since it seemed to us more appropriate to explain these ideas.
To this end, we use basic concepts on schemes, consistent with [H], Chapter
II, to which we refer for terminology.
We regard ideals in $\mathscr{O}_{\mathbb{C}^{n},0}$ as ideal sheaves on the
affine scheme $\mbox{Spec}\,\mathscr{O}_{\mathbb{C}^{n},0}$. Let
$J\subset\mathscr{O}_{\mathbb{C}^{n},0}$ be any ideal and assume that $J$ is
given by generators: $J=\langle
g_{1},\ldots,g_{q}\rangle\subset\mathscr{O}_{\mathbb{C}^{n},0}$; then we
consider the projective $(q-1)$-space over $\mathscr{O}_{\mathbb{C}^{n},0}$,
namely, $\mathbb{P}^{q-1}=\mbox{Proj}(S)$, where
$S=\mathscr{O}_{\mathbb{C}^{n},0}[\alpha_{1},\ldots,\alpha_{q}]=\bigoplus_{d\geqslant
0}\,S_{d}$ is the standard polynomial ring over
$\mathscr{O}_{\mathbb{C}^{n},0}$, with variables $\alpha_{i}$, graded so that
$\deg\alpha_{i}=1$, for all $i$.
Let
$\pi:\mathbb{P}^{q-1}\rightarrow\mbox{Spec}\,\mathscr{O}_{\mathbb{C}^{n},0}$
be the natural morphism of
$\mbox{Spec}\,\mathscr{O}_{\mathbb{C}^{n},0}$-schemes and let $\sigma$ denote
the global section $\sum_{i=1}^{q}\,g_{i}\alpha_{i}$ of
$\pi^{*}(J)\otimes\mathcal{O}_{\mathbb{P}^{q-1}}(1)$. Then there is a _moduli
ideal sheaf_ of $\sigma$ on $\mathbb{P}^{q-1}$, namely
$\mathscr{T}_{\mathcal{K}}(\sigma)$, the sheaf associated to the homogeneous
ideal $\langle\sigma,\mathfrak{m}\frac{\partial\sigma}{\partial
x_{1}},\ldots,\mathfrak{m}\frac{\partial\sigma}{\partial x_{n}}\rangle$ of
$S$. Clearly $\mathscr{T}_{\mathcal{K}}(\sigma)$ is a subsheaf of
$\pi^{*}(T_{\mathcal{K}}(J))$, so we can take the quotient sheaf
$\mathscr{F}=\frac{\pi^{*}(T_{\mathcal{K}}(J))}{\mathscr{T}_{\mathcal{K}}(\sigma)}$
on $\mathbb{P}^{q-1}$. Since $\mathscr{F}$ is coherent and $\mathbb{P}^{q-1}$
is noetherian, the support $\mbox{Supp}\mathscr{F}$ of $\mathscr{F}$ is a
closed subscheme of $\mathbb{P}^{q-1}$ given by the vanishing of
$\Big{(}\mathscr{T}_{\mathcal{K}}(\sigma):\pi^{*}(T_{\mathcal{K}}(J))\Big{)}$.
###### Definition 4.7.
We say that an ideal $J\subset\mathscr{O}_{\mathbb{C}^{n},0}$ is
_$T_{\mathcal{K}}$ -dependent_ if
$\pi^{-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F}$.
We check that $T_{\mathcal{K}}$-dependence for $J$ is a well-defined concept,
being independent on the choice of generators for $J$. For, let the ideal $J$
be given by another system of generators, say $J=\langle
h_{1},\ldots,h_{u}\rangle$. Let $\mathbb{P}^{u-1}=\mbox{Proj}(S^{\prime})$,
being
$S^{\prime}=\mathscr{O}_{\mathbb{C}^{n},0}[\,\beta_{1},\ldots,\beta_{u}]$. The
above construction can be carried out with the obvious morphism
$\pi^{\prime}:\mathbb{P}^{u-1}\rightarrow\mbox{Spec}\,\mathscr{O}_{\mathbb{C}^{n},0}$
and section $\sigma^{\prime}=\sum_{j=1}^{u}\,h_{j}\beta_{j}$ instead of $\pi$
and $\sigma$, obtaining a sheaf $\mathscr{F^{\prime}}$ on $\mathbb{P}^{u-1}$.
###### Lemma 4.8.
Keeping notation as above, we have
$\pi^{-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F}$ if and only if
$\pi^{\prime-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F^{\prime}}$.
###### Proof.
We will only show that
$\pi^{-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F}$ implies
$\pi^{\prime-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F^{\prime}}$. The
proof of the other implication is analogous.
Since the $g_{i}$’s and the $h_{j}$’s generate the same ideal $J$, we can
write for all $i$, $g_{i}=\sum_{j}r_{ji}h_{j}$ for some
$r_{ji}\in\mathscr{O}_{\mathbb{C}^{n},0}$ and at least one $r_{ji}$ is
invertible in $\mathscr{O}_{\mathbb{C}^{n},0}$. We construct an homomorphism
of graded $\mathscr{O}_{\mathbb{C}^{n},0}$-algebras
$\Phi:S^{\prime}\rightarrow S$ (preserving degrees) given by
$\beta_{j}\mapsto\sum_{i}r_{ji}\alpha_{i}$. This induces a natural morphism
$\varphi:U\rightarrow\mathbb{P}^{u-1}$, of
$\mathscr{O}_{\mathbb{C}^{n},0}$-schemes, where $U\subset\mathbb{P}^{q-1}$ is
the complement of the indeterminacy locus of $\varphi$, given by the vanishing
of $\langle\Phi(\beta_{1}),\ldots,\Phi(\beta_{u})\rangle$ in
$\mathbb{P}^{q-1}$. Notice that $\pi^{-1}(\mathfrak{m})$ is not contained in
the locus of indeterminacy of $\varphi$ because
$\Phi(\beta_{j})\not\in\mathfrak{m}S$ for at least one $j$. Observe also that
the construction of $\varphi$ implies both
$\sigma|_{U}=\varphi^{*}(\sigma^{\prime})$ and
$\pi|_{U}=\pi^{\prime}\circ\varphi$. In particular,
$\mathscr{T}_{\mathcal{K}}(\sigma)|_{U}=\varphi^{*}(\mathscr{T}_{\mathcal{K}}(\sigma^{\prime}))$
and
$\pi^{*}(T_{\mathcal{K}}(J))|_{U}=\varphi^{*}\pi^{\prime*}(T_{\mathcal{K}}(J))$.
According to the definition of $\mathscr{F}$ and $\mathscr{F^{\prime}}$, we
deduce $\mathscr{F}|_{U}=\varphi^{*}\mathscr{F}^{\prime}$.
Assume $\pi^{-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F}$. As observed
above $\mbox{Supp}\mathscr{F}$ is closed in the irreducible
$\mathbb{P}^{q-1}$. It follows that there exists some
$P\in\pi^{-1}(\mathfrak{m})\cap U$, $P\not\in\mbox{Supp}\mathscr{F}$. Since
$U\cap\mbox{Supp}\mathscr{F}=\varphi^{-1}(\mbox{Supp}\mathscr{F}^{\prime})$,
we have $\varphi(P)\not\in\mbox{Supp}\mathscr{F^{\prime}}$ and
$\mathfrak{m}S^{\prime}\subset\Phi^{-1}(\mathfrak{m}S)\subset\Phi^{-1}(P)=\varphi(P)$.
Therefore $\varphi(P)\in\pi^{\prime-1}(\mathfrak{m})$, showing that
$\pi^{\prime-1}(\mathfrak{m})\not\subset\mbox{Supp}\mathscr{F^{\prime}}$.∎
Now that we have defined $T_{\mathcal{K}}$-fullness and
$T_{\mathcal{K}}$-dependence, we present some examples showing that the two
notions are independent of each other.
###### Example 4.9.
If $J\subseteq\mathscr{O}_{\mathbb{C}^{n},0}$ is a principal ideal then $J$ is
$T_{\mathcal{K}}$-dependent. Indeed, in this case we see that
$\mbox{Supp}\mathscr{F}\subset\mathbb{P}^{0}$ is empty, opposite to
$\pi^{-1}(\mathfrak{m})$ which is not. It is easy to give examples of
principal ideals which are not $T_{\mathcal{K}}$-full.
###### Example 4.10.
Let $J=\langle x^{2},y\rangle\subset\mathbb{C}\\{x,y\\}$. We have seen before
(cf. Example 4.2) that $J$ fails to be $T_{\mathcal{K}}$-full. Let us show
that $J$ is $T_{\mathcal{K}}$-dependent. Here $\pi^{*}(T_{\mathcal{K}}(J))$ is
the sheaf associated to $T_{\mathcal{K}}(J)S=\mathfrak{m}S.$ Let
$\sigma=x^{2}\alpha_{1}+y\alpha_{2}$. Then the sections
$\sigma,\,x\frac{\partial\sigma}{\partial x},x\frac{\partial\sigma}{\partial
y},y\frac{\partial\sigma}{\partial x},y\frac{\partial\sigma}{\partial y}$
generate $\mathscr{T}_{\mathcal{K}}(\sigma)$. We investigate the support of
$\mathscr{F}$ in $\mathbb{P}^{1}$. We can compute generators of
$\Big{(}\mathscr{T}_{\mathcal{K}}(\sigma):\pi^{*}(T_{\mathcal{K}}(J))\Big{)}$
as $\langle\alpha_{1}x,\alpha_{2}\rangle\subset S$. Hence the fiber
$\pi^{-1}(\mathfrak{m})$ is not contained in the support of $\mathscr{F}$,
showing that $J$ is $T_{\mathcal{K}}$-dependent.
###### Example 4.11.
Let $J=\mathfrak{m}^{4}\subset\mathbb{C}\\{x,y\\}$. Clearly $J$ is
$T_{\mathcal{K}}$-full (cf. Examples 2.8, 4.5). Let us show that $J$ fails to
be $T_{\mathcal{K}}$-dependent. Notations being as before,
$\pi^{*}(T_{\mathcal{K}}(J))$ is the sheaf in $\mathbb{P}^{4}$ associated to
$T_{\mathcal{K}}(J)S=JS.$ Let
$\sigma=x^{4}\alpha_{1}+x^{3}y\alpha_{2}+x^{2}y^{2}\alpha_{3}+xy^{3}\alpha_{4}+y^{4}\alpha_{5}.$
Then $\sigma,\,x\frac{\partial\sigma}{\partial
x},x\frac{\partial\sigma}{\partial y},y\frac{\partial\sigma}{\partial
x},y\frac{\partial\sigma}{\partial y}$ are generators of
$\mathscr{T}_{\mathcal{K}}(\sigma)$. We check that the support of
$\mathscr{F}$ contains $\pi^{-1}(\mathfrak{m})$. To do this we show, with help
of SINGULAR, that
$\Big{(}\mathscr{T}(\sigma):\pi^{*}(T_{\mathcal{K}}(J))\Big{)}\subset\mathfrak{m}S$.
We proceed as follows:
> ring r=0,(x,y,a(1..5)),ds;
> ideal m=x,y;
> ideal M=std(m);
> poly s=a(1)*x4+a(2)*x3y+a(3)*x2y2+a(4)*xy3+a(5)*y4;
> ideal t=s,m*diff(s,x),m*diff(s,y);
> ideal q=quotient(t,m4);
> ideal Q=std(q);
> reduce(Q,M);
## 5\. Main Result
The purpose of this Section is to state and prove our result characterizing
moduli ideals, then solving Problem 1.2. We also derive Corollary 5.2 which
has as a consequence an explicit solution to both the Recognition and the
Reconstruction problems mentioned in the Introduction, even for the case of
non-isolated hypersurface singularities.
###### Theorem 5.1.
Let $I\subset\mathscr{O}_{\mathbb{C}^{n},0}$ be an ideal. Then $I$ is a moduli
ideal if and only if $I$ is $T_{\mathcal{K}}$-full and $\Delta_{1}(I)$ is
$T_{\mathcal{K}}$-dependent.
###### Proof.
Let $\Delta_{1}(I)=\langle g_{1},\ldots,g_{q}\rangle$ and let $\mathscr{F}$
denote the sheaf
$\pi^{*}(T_{\mathcal{K}}(\Delta_{1}(I)))/\mathscr{T}_{\mathcal{K}}(\sigma)$ on
$\mathbb{P}^{q-1}$, as described before. For any
$\lambda=(\lambda_{1},\ldots,\lambda_{q})\in\mathbb{C}^{q}\setminus\\{0\\}$ we
will denote by $p_{\lambda}\subseteq\mathbb{C}[\alpha_{1},\ldots,\alpha_{q}]$
the homogeneous prime ideal generated by
$\\{\lambda_{k}\alpha_{\ell}-\lambda_{\ell}\alpha_{k}\\}_{k,{\ell}}$. If $P\in
D_{+}(\alpha_{\ell})\subset\mathbb{P}^{q-1}$ one checks easily that
$\mathscr{T}_{\mathcal{K}}(\sigma)_{P}=T_{\mathcal{K}}\Big{(}\sigma/\alpha_{\ell}\Big{)}S_{(P)}$,
where parenthetical notation $-\,_{(P)}$ (here and in what follows) indicates
the submodule of degree zero elements of the localization of a graded module
at $P\in\mathbb{P}^{q-1}$. Moreover, if $p_{\lambda}S\subseteq P\in
D_{+}(\alpha_{\ell})$ then $\lambda_{\ell}\in\mathbb{C}\setminus 0$ and we can
use Remark 2.1 to obtain the following Estimate on ideals in the stalk of
structural sheaf $\mathcal{O}_{\mathbb{P}^{q-1},P}=S_{(P)}$:
(5.1)
$T_{\mathcal{K}}\Big{(}\sum_{k}\Big{(}\lambda_{k}-\frac{\lambda_{\ell}\alpha_{k}}{\alpha_{\ell}}\Big{)}g_{k}\Big{)}S_{(P)}\subseteq\sum_{k}T_{\mathcal{K}}\Big{(}\Big{(}\lambda_{k}-\frac{\lambda_{\ell}\alpha_{k}}{\alpha_{\ell}}\Big{)}g_{k}\Big{)}S_{(P)}\subseteq\sum_{k}\Big{(}\lambda_{k}-\frac{\lambda_{\ell}\alpha_{k}}{\alpha_{\ell}}\Big{)}T_{\mathcal{K}}(g_{k})S_{(P)}\subseteq
IP_{(P)}$
To prove the “$\Rightarrow$” assertion in our statement, assume $I$ to be a
moduli ideal. We have seen (cf. Proposition 4.3) that $I$ is
$T_{\mathcal{K}}$-full; hence
$\mathscr{F}=\pi^{*}(I)/\mathscr{T}_{\mathcal{K}}(\sigma)$. Now we verify that
$\Delta_{1}(I)$ is $T_{\mathcal{K}}$-dependent. Since this is clear if
$I=\langle 0\rangle$, we assume $I=T_{\mathcal{K}}(f)\neq\langle 0\rangle$,
with $f=\sum_{k}r_{k}g_{k}$ for some $r_{k}\in\mathscr{O}_{\mathbb{C}^{n},0}$.
We write $r_{k}=\lambda_{k}+s_{k}$, for certain $\lambda_{k}\in\mathbb{C}$ and
$s_{k}\in\mathfrak{m}$, for all $k=1,\ldots,q$. Since $I\neq\langle 0\rangle$
we can use Remark 2.5, (v) and Nakayama’s Lemma to check
$f\not\in\mathfrak{m}\Delta_{1}(I)$. Hence, at least one of the $r_{k}$’s is
an invertible element of $\mathscr{O}_{\mathbb{C}^{n},0}$. Using Remark 2.1
again we may assume
$(r_{1},\ldots,r_{q})=(\lambda_{1},\ldots,\lambda_{q})\in\mathbb{C}^{q}\setminus\\{0\\}$.
We consider $P=p_{\lambda}S+\mathfrak{m}S\in\mathbb{P}^{q-1}$. Then
$P\in\pi^{-1}(\mathfrak{m})$, so that $\pi^{*}(I)_{P}=I_{\pi(P)}\otimes
S_{(P)}=I_{\mathfrak{m}}\otimes S_{(P)}=IS_{(P)}$. On the other hand (say if
$P\in D_{+}(\alpha_{\ell})$), our assumption $I=T_{\mathcal{K}}(f)$ together
with Estimate (5.1) above produces
$IS_{(P)}=T_{\mathcal{K}}\Big{(}\sum_{k}\lambda_{k}g_{k}\Big{)}S_{(P)}\subseteq
T_{\mathcal{K}}\Big{(}\sum_{k}\Big{(}\lambda_{k}-\frac{\lambda_{\ell}\alpha_{k}}{\alpha_{\ell}}\Big{)}g_{k}\Big{)}S_{(P)}+T_{\mathcal{K}}\Big{(}\lambda_{\ell}\sigma/\alpha_{\ell}\Big{)}S_{(P)}\subseteq
IP_{(P)}+\mathscr{T}_{\mathcal{K}}(\sigma)_{P}\subseteq IS_{(P)}.$
It follows that $IP_{(P)}+\mathscr{T}_{\mathcal{K}}(\sigma)_{P}=IS_{(P)}$. Now
we use Nakayama’s Lemma to obtain
$\pi^{*}(I)_{P}=IS_{(P)}=\mathscr{T}_{\mathcal{K}}(\sigma)_{P}$. Hence
$P\not\in\mbox{Supp}\mathscr{F}$, which completes the proof that
$\Delta_{1}(I)$ is $T_{\mathcal{K}}$-dependent.
To prove the “$\Leftarrow$” assertion in the statement $I$ is assumed to be
$T_{\mathcal{K}}$-full and $\Delta_{1}(I)$ is assumed to be
$T_{\mathcal{K}}$-dependent. Then $I=T_{\mathcal{K}}(\Delta_{1}(I))$ and there
exists some $P\in\pi^{-1}(\mathfrak{m})\setminus\mbox{Supp}\mathscr{F}$. It
follows that
$\pi^{*}(I)_{P}=\pi^{*}(T_{\mathcal{K}}(\Delta_{1}(I))_{P}=\mathscr{T}_{\mathcal{K}}(\sigma)_{P}.$
We make use of the decomposition
$S=\mathbb{C}[\alpha_{1},\ldots,\alpha_{q}]+\mathfrak{m}S$ to simplify our
argument. Indeed, we obtain
$P=P\cap\mathbb{C}[\alpha_{1},\ldots,\alpha_{q}]+P\cap\mathfrak{m}S=P\cap\mathbb{C}[\alpha_{1},\ldots,\alpha_{q}]+\mathfrak{m}S$.
Therefore, there exists some
$(\lambda_{1},\ldots,\lambda_{q})\in\mathbb{C}^{q}\setminus\\{0\\}$,
$p_{\lambda}\supseteq P\cap\mathbb{C}[\alpha_{1},\ldots,\alpha_{q}]$, such
that the homogeneous prime ideal $p_{\lambda}S+\mathfrak{m}S$ also belongs to
$\pi^{-1}(\mathfrak{m})\setminus\mbox{Supp}\mathscr{F}$.
With this simplification - i. e., assuming $P=p_{\lambda}S+\mathfrak{m}S\in
D_{+}(\alpha_{\ell})$, as before - the displayed equality of stalks above
reads
$IS_{(P)}=T_{\mathcal{K}}\Big{(}\lambda_{\ell}\sigma/\alpha_{\ell}\Big{)}S_{(P)}.$
We will prove $I=T_{\mathcal{K}}(f)$, for
$f=\sum_{k}\lambda_{k}g_{k}\in\mathscr{O}_{\mathbb{C}^{n},0}$. Estimate (5.1)
here gives
$IS_{(P)}\subseteq
T_{\mathcal{K}}\Big{(}\sum_{k}\Big{(}\lambda_{k}-\frac{\lambda_{\ell}\alpha_{k}}{\alpha_{\ell}}\Big{)}g_{k}\Big{)}S_{(P)}+T_{\mathcal{K}}\Big{(}\sum_{k}\lambda_{k}g_{k}\Big{)}S_{(P)}\subseteq
IP_{(P)}+T_{\mathcal{K}}\Big{(}\sum_{k}\lambda_{k}g_{k}\Big{)}S_{(P)}\subseteq
IS_{(P)}.$
We obtain $IP_{(P)}+T_{\mathcal{K}}(\sum_{k}\lambda_{k}g_{k})S_{(P)}=IS_{(P)}$
and by Nakayama’s Lemma again we deduce $T_{\mathcal{K}}(f)S_{(P)}=IS_{(P)}$.
Since $T_{\mathcal{K}}(f)\subseteq I$, we should verify $I\subseteq
T_{\mathcal{K}}(f)$. It suffices to check that if $h\in I$ and there are some
$g\in T_{\mathcal{K}}(f)$ and $G_{1},G_{2}\in S$, homogeneous of same degree
with $G_{2}\not\in P$ such that $G_{2}h=gG_{1}$, then $h\in
T_{\mathcal{K}}(f)$. With this simplification, an argument with divisibility
and degree in the unique factorization domain $S$ shows $\xi h=gH$, for some
homogeneous element $H\in S$ and some degree zero factor $\xi$ of $G_{2}$. In
particular, $\xi\in\mathscr{O}_{\mathbb{C}^{n},0}\setminus\mathfrak{m}$. Again
by degree reasons, $H$ has degree zero, i.e., it belongs to
$\mathscr{O}_{\mathbb{C}^{n},0}$. This shows that $h\in T_{\mathcal{K}}(f)$.
Hence $I\subseteq T_{\mathcal{K}}(f)$ and we have shown that $I$ is a moduli
ideal of $\mathscr{O}_{\mathbb{C}^{n},0}$.∎
We will keep the notation as in the proof of Theorem 5.1. Then we have the
corollary below.
###### Corollary 5.2.
Let $I\subset\mathscr{O}_{\mathbb{C}^{n},0}$ be an ideal and let
$\Delta_{1}(I)=\langle g_{1},\ldots,g_{q}\rangle$. Then $I$ is a moduli ideal
if and only if $I=T_{\mathcal{K}}(\sum_{k}\lambda_{k}g_{k})$ for $p_{\lambda}$
varying in a (non empty) Zariski open subset of
$\mathbb{P}_{\mathbb{C}}^{q-1}$.
###### Proof.
For the proof of the non-trivial part of the statement, assume that $I$ is a
moduli ideal. The canonical inclusion
$\mathbb{C}\rightarrow\mathscr{O}_{\mathbb{C}^{n},0}$ induces
$\mathbb{C}[\alpha_{1},\ldots,\alpha_{q}]\rightarrow S$, which is a graded
inclusion of $\mathbb{C}$-algebras; hence we obtain a proper and dominant
morphism of schemes
$\rho:\mathbb{P}^{q-1}\rightarrow\mathbb{P}_{\mathbb{C}}^{q-1}$. From the
Theorem, we know that $\Delta_{1}(I)$ is $T_{\mathcal{K}}$-dependent, so that
$\pi^{-1}(\mathfrak{m})\setminus\mbox{Supp}\mathscr{F}$ is a non empty Zariski
open subset of $\pi^{-1}(\mathfrak{m})$. It follows that its image under
$\rho$ is a non-empty open Zariski subset
$U\subset\mathbb{P}_{\mathbb{C}}^{q-1}$. Homogeneous prime ideals
$p_{\lambda}$ in $U$ correspond bijectively, via $\rho$, to homogeneous prime
ideals of type
$p_{\lambda}S+\mathfrak{m}S\in\pi^{-1}(\mathfrak{m})\setminus\mbox{Supp}\mathscr{F}$.
Since the Theorem also guarantees that $I$ is $T_{\mathcal{K}}$-full, we can
proceed as in its proof to show that
$I=T_{\mathcal{K}}(\sum_{k}\lambda_{k}g_{k})$, for all
$\lambda=(\lambda_{1},\ldots,\lambda_{q})$ such that $p_{\lambda}\in U$.∎
###### Remark 5.3.
The practical character and the usefulness of the above Corollary comes from
its consonance with the fact outlined in Example 4.6. Precisely, it says that
computing the moduli ideal of a sufficiently general $\mathbb{C}$-linear
combination of generators of $\Delta_{1}(I)$ \- accessible in practical
examples, as we have illustrated in Section 3 \- reveals whether
$I\subset\mathscr{O}_{\mathbb{C}^{n},0}$ is a moduli ideal.
We illustrate how one can use the software SINGULAR to check whether an ideal
$I\subset\mathscr{O}_{\mathbb{C}^{n},0}$ is a moduli ideal, without knowing a
priori the function germ which realizes it as such. We do this with the ideal
presented in Example 3.3.
###### Example 5.4.
(Example 3.3, continued): We have computed the anti-derivatives ideal of
$I=\langle 3xy^{2}+x^{6},y^{3},x^{5}y,x^{7}\rangle\subset\mathbb{C}\\{x,y\\}$
as $\Delta_{1}(I)=\langle y^{3}+x^{5}y,x^{4}y^{2},x^{7},x^{6}y\rangle\subset
I$. A SINGULAR check shows that $I$ is $T_{\mathcal{K}}$-full. We consider the
corresponding sheaf $\mathscr{F}$ in $\mathbb{P}^{3}$ and investigate its
support. In order to do so we use SINGULAR again:
> ring r=0,(x,y,a(1..4)),ds;
> ideal i=3xy2+x6,y3,x5y,x7;
> poly s=a(1)*(y3+x5y)+a(2)*x4y2+a(3)*x7+a(4)*x6y;
> ideal m=x,y;
> ideal t=s,m*diff(s,x),m*diff(s,y);
> ideal q=quotient(t,i);
> ideal Q=std(q);
> Q;
The output is a list with twenty homogeneous generators of
$\Big{(}\mathscr{T}_{\mathcal{K}}(\sigma):\pi^{*}(T_{\mathcal{K}}(J))\Big{)}$,
the sixth element of which being
$15\alpha_{1}^{3}\alpha_{3}+52x\alpha_{1}^{2}\alpha_{3}\alpha_{4}-121y\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}+36y\alpha_{1}^{2}\alpha_{4}^{2}+15x^{2}\alpha_{1}\alpha_{3}\alpha_{4}^{2}+30xy\alpha_{1}^{2}\alpha_{2}^{2}+28x^{2}\alpha_{1}\alpha_{3}\alpha_{4}^{2}+\\\
24xy\alpha_{1}\alpha_{4}^{3}+42x^{3}\alpha_{1}^{2}\alpha_{2}\alpha_{4}+24x^{2}y\alpha_{1}\alpha_{2}^{2}\alpha_{4}-54x^{4}\alpha_{2}^{2}\alpha_{3}\alpha_{4}+41x^{4}\alpha_{1}\alpha_{2}\alpha_{4}^{2}+4x^{3}y\alpha_{2}^{2}\alpha_{4}^{2}+14x^{5}\alpha_{2}\alpha_{4}^{3}.$
Clearly this element does not belong to
$\mathfrak{m}[\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}]$ due to the
presence of the term $15\alpha_{1}^{3}\alpha_{3}$. It follows that
$\Delta_{1}(I)$ is $T_{\mathcal{K}}$-dependent. According to the proof of
Theorem, any $\mathbb{C}$-linear combination
$\lambda_{1}(y^{3}+x^{5}y)+\lambda_{2}x^{4}y^{2}+\lambda_{3}x^{7}+\lambda_{4}x^{6}y$
with both $\lambda_{1}\neq 0$ and $\lambda_{3}\neq 0$ represents a function
germ $f$ of which $I$ is a moduli ideal.
## References
* [Ar] Arnold, V. I. - Local normal forms of functions. Inventiones mathematicae, 35(1), (1976): 87-109.
* [DGPS] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H. - Singular 4-3-1 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2022).
* [E] Epure, R.-P. - Explicit and effective Mather-Yau correspondence in view of analytic gradings. 2020.
* [ES] Epure, R. and Schulze, M. - Hypersurface singularities with monomial Jacobian ideal. Bulletin of the London Mathematical Society (2022).
* [GH] Gaffney, T., and Hauser, H. - Characterizing singularities of varieties and of mappings. Inventiones mathematicae 81.3 (1985): 427-447.
* [GLS] Greuel, G. M., Lossen, C., Shustin, E. I. - Introduction to singularities and deformations. Springer Science Business Media(2007).
* [H] Hartshorne, R. - Algebraic geometry. Graduate texts in mathematics 52 (1977).
* [IK] Isaev, A; Kruzhilin, N. - Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras. Proceedings of the American Mathematical Society, v. 142, n. 2, p. 581-590, 2014.
* [MY] Mather, John N., and Stephen S-T. Yau. - Classification of isolated hypersurface singularities by their moduli algebras. Inventiones mathematicae 69.2: 243-251 (1982).
* [OR] Olmedo Rodrigues, J. H. - On Tjurina Ideals of Hypersurface Singularities. arXiv:2205.03527. (To appear)
* [Y] Stephen, S-T. Yau. - A necessary and sufficient condition for a local commutative algebra to be a moduli algebra: weighted homogeneous case. Complex analytic singularities 8: 687-697. (1987)
|
Measurement of the inclusive $t\bar{t}$ production cross section in the
lepton+jets channel in $pp$ collisions at $\sqrt{s}=7$ TeV with the ATLAS
detector using support vector machines A measurement of the top quark pair-
production cross section in the lepton+jets decay channel is presented. It is
based on 4.6 fb-1 of $\sqrt{s}=7$ TeV $pp$ collision data collected during
2011 by the ATLAS experiment at the CERN Large Hadron Collider. A three-class,
multidimensional event classifier based on support vector machines is used to
differentiate $t\bar{t}$ events from backgrounds. The $t\bar{t}$ production
cross section is found to be $\sigma_{t\bar{t}}=168.5\pm 0.7$(stat.)
${}^{+6.2}_{-5.9}$(syst.) ${}^{+3.4}_{-3.2}$(lumi.) pb. The result is
consistent with the Standard Model prediction based on QCD calculations at
next-to-next-to-leading order. TOPQ-2017-08 CERN-EP-2022-191 Phys. Rev. D
###### Contents
1. 1 Introduction
2. 2 ATLAS detector
3. 3 Object definitions
4. 4 Event selection
5. 5 Data samples
6. 6 Signal and background modeling
1. 6.1 Multi-jet background / fake leptons
2. 6.2 Monte Carlo samples
3. 6.3 Signal and background classes
7. 7 Analysis method
1. 7.1 The SVM discriminant
2. 7.2 Physics observables
3. 7.3 SVM training
4. 7.4 Class templates
5. 7.5 Cross-section measurement
6. 7.6 Systematic uncertainties
8. 8 Results
1. 8.1 Top quark mass dependence
9. 9 Summary
### 1 Introduction
In the Standard Model of particle physics [1, 2, 3], the top quark ($t$) and
the bottom quark ($b$) belong to a doublet representation of the weak-isospin
SU(2). The top quark is the most massive of the known elementary particles.
Because its mass is close to the electroweak symmetry breaking scale, it may
play a fundamental role in the mechanism of breaking of the SU(2) symmetry of
the electroweak interaction. Top quark production is also the dominant
background in many analyses looking for physics beyond the Standard Model at
high mass scales at the Large Hadron Collider (LHC), and a good understanding
of top quark production is a necessary step in many “new physics” searches.
The cross section is one of the simplest observables that can be measured in
the $t\bar{t}$ system. It allows one to make important comparisons with
theoretical predictions available at next-to-next-to-leading order in
perturbative QCD, including the soft-gluon resummation at next-to-next-to-
leading-log order (NNLO + NNLL); see Ref. [4] and references therein. For $pp$
collisions at a center-of-mass energy of $\sqrt{s}=7$ TeV, the predicted
$t\bar{t}$ production cross section is
$\sigma_{t\bar{t}}^{\text{NNLO+NNLL}}=177^{+10}_{-11}$ pb. This theoretical
value was calculated with the Top++ 2.0 program [5], including soft-gluon
resummation, assuming a top quark mass value of 172.5 GeV, and using the
PDF4LHC [6] procedures.
According to the Standard Model, top quarks from $pp$ collisions at the LHC
are produced mostly via the strong interaction as $t\bar{t}$ pairs, with each
top quark decaying into a $W$ boson and a $b$-quark nearly $100\%$ of the
time. The $t\bar{t}$ events in which one of the $W$ bosons decays into a quark
pair and the other into a charged lepton and a neutrino, are classified as
“lepton+jets”, as such events contain an electron or muon or $\tau$-lepton, a
neutrino and typically four hadronic jets (two of which originate from the
$b$-quark and $\bar{b}$-quark).
In this paper, a measurement of the top quark pair-production cross section at
$\sqrt{s}=7$ TeV using events with a single charged lepton (electron or muon)
and jets in the final state is presented. The previously published result from
the ATLAS Collaboration for the lepton+jets channel [7] uses 35 pb-1 of data
and obtains a precision of 12%. The most precise CMS $t\bar{t}$ cross-section
measurement in the same channel [8] has a precision of 7%. In the dilepton
channel, the best ATLAS result [9] achieves 3.5% precision, while CMS [10]
reaches 3.6%. These ATLAS and CMS dilepton results have been combined [11],
resulting in an uncertainty of 2.6% in $\sigma_{t\bar{t}}$.
The analysis presented in this paper is based on the full dataset collected
with the ATLAS detector at the LHC in 2011, corresponding to an integrated
luminosity of 4.6 fb-1, and attains statistical and systematic uncertainties
that are significantly lower than in previous ATLAS measurements in this final
state. In an extension of the usual application of binary multivariate
classifiers, this analysis uses a large number of variables to train three
different support vector machines (SVMs). The three SVMs are used to define a
three-dimensional space in which a multi-class event discriminator is
constructed to identify the $t\bar{t}$ events through a simultaneous profile
likelihood fit in four independent regions of this space.
### 2 ATLAS detector
The ATLAS detector is described in Ref. [12]. It is a multipurpose particle
detector with forward-backward symmetry and a cylindrical geometry.111 ATLAS
uses a right-handed coordinate system with its origin at the nominal
interaction point (IP) in the center of the detector and the $z$-axis along
the beam pipe. The $x$-axis points from the IP to the center of the LHC ring,
and the $y$-axis points upwards. Cylindrical coordinates $(r,\phi)$ are used
in the transverse plane, $\phi$ being the azimuthal angle around the $z$-axis.
The pseudorapidity is defined in terms of the polar angle $\theta$ as
$\eta=-\ln\tan(\theta/2)$, and the distance $\Delta R$ in the $\eta$–$\phi$
space is defined as $\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$.
The inner tracking detectors are surrounded by a thin superconducting
solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer
with a magnetic field generated by three superconducting toroidal magnets of
eight coils each. The inner-detector system (ID), in combination with the 2 T
magnetic field from the solenoid, provides precision momentum measurements for
charged particles within the pseudorapidity range $|\eta|<2.5$. Moving
radially outwards, it consists of a silicon pixel detector, a silicon
microstrip detector, and a straw-tube tracker that also provides transition
radiation measurements for electron identification. The calorimeter system
covers the pseudorapidity range $|\eta|<4.9$. A high-granularity liquid-argon
(LAr) sampling calorimeter with lead absorber measures electromagnetic showers
within $|\eta|<3.2$. In the region matched to the ID, $|\eta|<2.5$, the
innermost layer has fine segmentation in $\eta$ to improve the resolution of
the shower position and direction measurements. Hadronic showers are measured
by an iron/plastic-scintillator tile calorimeter in the central region,
$|\eta|<1.7$, and by a LAr calorimeter in the endcap region, $1.5<|\eta|<3.2$.
In the forward region, measurements of both electromagnetic and hadronic
showers are provided by a LAr calorimeter covering the pseudorapidity range
$3.1<|\eta|<4.9$. The muon spectrometer is instrumented with separate trigger
and high-precision tracking chambers. It provides muon identification for
charged-particle tracks within $|\eta|<2.7$. The combination of all ATLAS
detector systems provides charged-particle measurement along with lepton and
photon measurement and identification in the pseudorapidity range
$|\eta|<2.5$. Jets are reconstructed over the full range covered by the
calorimeters, $|\eta|<4.9$.
A three-level trigger system [13] is used to select interesting events. The
first-level (L1) trigger is implemented in hardware and uses a subset of
detector information to reduce the event rate to a design value of at most
$75\text{\,}\mathrm{kHz}$. This is followed by two software-based trigger
levels which together reduce the event rate to about
$200\text{\,}\mathrm{Hz}$.
An extensive software suite [14] is used in data simulation, in the
reconstruction and analysis of real and simulated data, in detector
operations, and in the trigger and data acquisition systems of the experiment.
### 3 Object definitions
Electrons
Electron candidates are selected using the offline identification with tight
requirements [15] within a fiducial region with transverse momentum
$p_{\mathrm{T}}>25$ GeV and $|\eta|<2.47$, excluding the calorimeter
transition region $1.37<|\eta|<1.52$. They are subjected to several other
strict criteria including requirements on track quality, impact parameter,
calorimeter shower shape, and track–cluster matching. The electron candidates
are also required to be isolated. The transverse energy ($E_{\mathrm{T}}$)
deposited in the calorimeter in a cone of size $\Delta R=0.2$ around the
electron is calculated. Additionally, the scalar sum of the $p_{\mathrm{T}}$
of tracks in a cone of size $\Delta R=0.3$ is determined. Both of these
quantities have selection requirements that depend on the $\eta$ and
$E_{\mathrm{T}}$ of the electron candidate, and which ensure 90% efficiency
for electrons from $W$ boson or $Z$ boson decays [16]. To avoid double
counting of jets and electrons (which are reconstructed by independent
algorithms), jets within $\Delta R=0.2$ of a reconstructed electron are
removed. Finally, electrons lying within $\Delta R=0.4$ of a selected jet are
discarded to reject leptons from heavy-flavor decays.
Muons
Muon candidates reconstructed from tracks in both the muon spectrometer and ID
are selected with the MuID algorithm [17]. Only candidates satisfying
$p_{\mathrm{T}}>20$ GeV and $|\eta|<2.5$ are selected. Muon candidates are
required to have a sufficient number of hits in the ID. The impact parameter
with respect to the primary vertex in the longitudinal direction along the
beam axis is required to satisfy $|z_{0}|<2$ mm. The tight muon candidates
used in this analysis are required to be isolated. The sum of the calorimeter
transverse energy within $\Delta R=0.2$ of a muon is required to be below 4
GeV, and the sum of the $p_{\mathrm{T}}$ of all the tracks within $\Delta
R=0.3$ (excluding the muon track) must be below 2.5 GeV. The efficiency of
this combined isolation requirement varies between 95% and 97%, depending on
the data-taking period. In order to reduce the background from muons produced
by heavy-flavor decays inside jets, muons are required to be separated by
$\Delta R>0.4$ from the nearest jet.
Jets
Jets are reconstructed from topological clusters [18] formed from energy
deposits in the calorimeters using the anti-$k_{t}$ algorithm [19, 20] with a
radius parameter of 0.4. Clusters are calibrated using the local cluster
weighting (LCW), which differentiates between the energy deposits arising from
electromagnetic and hadronic showers [21]. The jet reconstruction is done at
the electromagnetic scale and then a scale factor is applied in order to
obtain the jet energy at the hadronic scale. The jet energy scale (JES)
corrections account for the calorimeter response to the true jet energy by
using “truth jets” from simulation. The “truth jets” are formed through the
application of the anti-$k_{t}$ algorithm to stable particles, with the
exception of final-state muons and neutrinos. Jet calibration includes both
the LCW and JES calibrations. In addition, the jets are corrected for
distortions due to multiple $pp$ collisions per bunch crossing (pileup) using
a method which estimates the pileup activity on an event-by-event basis, as
well as the sensitivity of a given jet to pileup. With this method [21], a
contribution to the jet transverse momentum equal to the product of the jet
area in the $\eta$–$\phi$ plane and the average transverse energy density in a
given event is subtracted. The effects of additional collisions in either the
same bunch crossing or those adjacent in time are taken into account using
corrections which depend on the average number of interactions per bunch
crossing and the number of primary vertices. For this analysis, only jets in
the central region of the detector, $|\eta|<2.5$, and with transverse momentum
$p_{\mathrm{T}}>25$ GeV are considered. All jets, including those that do not
pass the analysis selection requirements, are used in the procedure to remove
overlaps with leptons.
Identification of b-jets
The identification of “$b$-jets” (jets arising from the decay of $B$-hadrons)
is performed with the MV1 algorithm [22], which combines the outputs of three
different tagging algorithms into a multivariate discriminant. Jets are
defined to be “$b$-tagged” if the MV1 discriminant value is larger than a
threshold (operating point) corresponding to 70% efficiency to identify
$b$-quark jets in simulated $t\bar{t}$ events. Approximately 20% of jets
originating from charm quarks are identified as $b$-jets, while light-flavor
jets are mistagged as $b$-jets at the 1% level.
Missing transverse momentum
The missing transverse momentum is calculated [23] as the complement of the
vector sum, in the transverse plane, of calorimeter cell energies within
$|\eta|<4.9$, after all corrections are applied to the associated physics
objects (including jets, electrons, and muons). A correction for significant
energy deposits not associated with high-$p_{\mathrm{T}}$ physics objects is
also included. The magnitude of the missing transverse momentum vector is
denoted by $E_{\text{T}}^{\text{miss}}$, while its direction in the transverse
plane is either denoted by an azimuthal angle $\phi$ or inferred through its
vector components $E_{\text{T}}^{\text{miss}}$ x and
$E_{\text{T}}^{\text{miss}}$ y.
### 4 Event selection
This analysis considers the single-lepton decay channel for the $t\bar{t}$
pair. The selected events are required to have exactly one lepton (either an
electron or a muon), a large amount of $E_{\text{T}}^{\text{miss}}$, and three
or more hadronic jets. The number of $b$-tagged jets in an event must be two
or less. Events must have at least one primary vertex, with five or more
tracks with $p_{\textrm{T}}>150$ MeV. If there is more than one primary
vertex, the one with the largest $\sum^{\text{tracks}}{p_{\textrm{T}}^{2}}$ is
chosen. Events were collected using single-lepton triggers, and each lepton
candidate must be matched to an appropriate lepton trigger. In the muon
channel, events are selected with a single-muon trigger with a
$p_{\mathrm{T}}$ threshold of 18 GeV. For the electron channel, a single-
electron trigger is required with $p_{\mathrm{T}}>20$ GeV. This is increased
to 22 GeV during high instantaneous luminosity periods. Three or more jets
with $p_{\mathrm{T}}$ greater than 25 GeV are required in each event. A large
amount of $E_{\text{T}}^{\text{miss}}$ is required to select events containing
a neutrino. For electron events the $E_{\text{T}}^{\text{miss}}$ must be
greater than 30 GeV, while for muon events the $E_{\text{T}}^{\text{miss}}$ is
required to be greater than 20 GeV.
In order to reduce the background due to multi-jet production containing
misidentified or nonprompt leptons, an additional selection requirement is
imposed. Typically, in an event arising from this background, the missing
transverse momentum vector points in the same direction in the transverse
plane as the charged-lepton candidate. Therefore, electron candidate events
must pass a requirement that $m_{\mathrm{T}}(W)>30$ GeV, while muon candidate
events must have $m_{\mathrm{T}}(W)+E_{\text{T}}^{\text{miss}}>60$ GeV. Here,
$m_{\mathrm{T}}(W)$ is defined as
$m_{\mathrm{T}}(W)=\sqrt{2\leavevmode\nobreak\
p_{\mathrm{T}}^{\ell}\leavevmode\nobreak\
E_{\text{T}}^{\text{miss}}\leavevmode\nobreak\ (1-\cos\Delta\phi)}\,,$
where $\Delta\phi$ is the difference in $\phi$ between the direction of the
charged-lepton transverse momentum, $p_{\mathrm{T}}^{\ell}$, and the missing
transverse momentum vector.
### 5 Data samples
The data sample used in this analysis comes from the $\sqrt{s}=7$ TeV $pp$
collisions collected during LHC Run 1 in 2011, and was recorded during stable
beam conditions with all relevant ATLAS subdetector systems operational. It
corresponds to an integrated luminosity of 4.6 fb-1, with an uncertainty of
1.8% [24].
### 6 Signal and background modeling
Except for the background due to multi-jet production leading to misidentified
leptons (which is estimated from the data), all signal and background samples
are modeled using Monte Carlo (MC) simulated events in conjunction with
factors to correct the simulations to data where required.
#### 6.1 Multi-jet background / fake leptons
The background from multi-jet events, in which a jet is misidentified as a
muon or electron, or where a nonprompt lepton within a jet passes the tight
lepton selection requirements, is sizable because of the large multi-jet
production cross section. Events from these two sources of multi-jet
background are referred to as fake-lepton events. This background is estimated
using the so-called “matrix method”, which is based on the measurement of the
selection efficiencies of leptons using data event samples satisfying relaxed
identification criteria [25, 26]. Loose electrons are electrons satisfying the
baseline selection criteria where the requirements on particle identification
using transition radiation measurements and on the energy-to-momentum ratio
($E$/$p$) are eased, and no requirement on the isolation is imposed. For loose
muons, isolation is not required, but all other selection criteria are
applied. The “matrix method” is based on the measurement of the efficiencies
for real and fake leptons in the loose lepton selection to pass the tight
selection criteria. The real-lepton efficiencies are measured in
$Z\rightarrow\ell\ell$ data samples, while the fake-lepton efficiencies are
determined in data control regions with selection requirements designed to
enhance the multi-jet content (1 lepton, $\geq 1$ jet, and only a small amount
of $E_{\text{T}}^{\text{miss}}$). These efficiencies depend on both the lepton
kinematics and event characteristics. To account for this, event weights are
computed from the efficiencies parameterized as a function of a number of
observables, which are then used to reweight the sample of data events with
lepton candidates that satisfy the loose but not the tight selection criteria.
The sums of these weights provide estimates of the multi-jet background.
#### 6.2 Monte Carlo samples
The samples used in this analysis were obtained from a simulation chain
consisting of an event generator interfaced to a parton shower and
hadronization model, the outputs of which were passed through a simulation of
the ATLAS detector and trigger system [27], and then reconstructed with the
same algorithms as the data. The ATLAS detector response was modeled with the
ATLAS full simulation (FS) based on Geant4 [28]. For $t\bar{t}$ samples used
to evaluate the signal modeling systematic uncertainties, the ATLAS fast
simulation AtlasFast-II (AF) [27, 29] was used to model the response of the
detector.
The $t\bar{t}$ signal was simulated using the NLO Powheg-hvq (patch 4) matrix
element generator [30] interfaced with Pythia 6.425 [31], using parameter
values set according to the C variant of the Perugia 2011 tune [32] to model
the underlying event and parton shower. The NLO CT10 [33] parton distribution
function (PDF) set was used for the NLO matrix element part, and the LO PDF
set CTEQ6L1 [34] was used with Pythia. The top quark mass was fixed at 172.5
GeV. This sample is referred to as Powheg+Pythia 6\. In order to evaluate the
dependence on the choice of parton shower and fragmentation models, additional
samples of $t\bar{t}$ events were created. The
$\textsc{MC@NLO}{+}\textsc{Herwig}$ sample was created with the NLO MC@NLO
[35] generator interfaced with Herwig [36] using the LO AUET2 tune [37]. The
Powheg+Herwig sample was created with Powheg interfaced to Herwig using the LO
AUET2 tune.
The largest backgrounds to the $t\bar{t}$ events in the selected sample are
from $W$+jets and $Z$+jets production. These were simulated with the LO event
generator Alpgen 2.13 [38] with LO PDF set CTEQ6L1, and interfaced with Herwig
6.52. Alpgen calculates matrix elements (ME) for final states with up to six
partons. The MLM [39] matching procedure was used to remove the overlaps
between ME and parton shower products in samples with $N$ and $N+1$ final-
state partons. In addition to the inclusive parton-flavor processes, separate
ME samples of $W$+$b\bar{b}$+jets, $W$+$c\bar{c}$+jets, $W$+$c$+jets, and
$Z$+$b\bar{b}$+jets were generated. The double counting of $b$\- and
$c$-quarks in $W$/$Z$+jets that occurs between the shower of the inclusive
samples and the ME of heavy-flavor production was eliminated using an overlap-
removal algorithm based on parton-to-jet $\Delta R$ matching [40]. The
$W$+jets and $Z$+jets event samples are referred to as
$\textsc{Alpgen}{+}\textsc{Herwig}$.
The single-top backgrounds were simulated at NLO using the MC@NLO generator
with the NLO PDF set CTEQ6.6 [34], and interfaced with Herwig, except the
$t$-channel samples, which were modeled with the LO AcerMC 3.8 generator [41]
interfaced with Pythia. Dibosons ($WW$, $WZ$, and $ZZ$) were generated with
Herwig using the LO PDF set CTEQ6L1. All samples generated with Herwig for the
parton shower evolution and hadronization used Jimmy 4.31 [42] for the
underlying-event model.
The effects of pileup were modeled by overlaying simulated minimum-bias events
on the hard-scattering events. The Monte Carlo events were then reweighted
such that the distribution of the number of interactions per bunch crossing,
$\langle\mu\rangle$, matched the shape and observed average of
$\langle\mu\rangle=9.1$ in the 2011 data.
#### 6.3 Signal and background classes
The most challenging backgrounds (i.e., those which most resemble $t\bar{t}$ )
are single-top and $W$/$Z$+$b\bar{b}$+jets. Therefore, the $t\bar{t}$ cross-
section measurement is expected to be affected most by the modeling of these
backgrounds. To improve discrimination between the $t\bar{t}$ signal and the
different types of background, this analysis separates the background events
into two classes and treats them independently. The “Heavy” background class
includes the Monte Carlo samples for single-top, $W$+$b\bar{b}$+jets, and
$Z$+$b\bar{b}$+jets. All other types of background, including fake leptons,
are assigned to the group designated as the “Light” class. Table 1 summarizes
the composition of the classes and lists the datasets which are used to model
them.
Table 1: Class definitions and compositions are presented. In the Process column, “lf” is defined as any partons that are not $b$-quarks. The other columns show the source of the events and the fractional contribution to the given class. Class | Process | Source | Fraction
---|---|---|---
Signal | $t\bar{t}$ | Powheg+Pythia 6 | 100.0%
Light | $W$+lf+jets | $\textsc{Alpgen}{+}\textsc{Herwig}$ | 74.9%
Light | $Z$+lf+jets | $\textsc{Alpgen}{+}\textsc{Herwig}$ | 7.6%
Light | Dibosons | Herwig | 1.4%
Light | Fake $e$ | Data | 6.4%
Light | Fake $\mu$ | Data | 9.7%
Heavy | $W$+$b\bar{b}$+jets | $\textsc{Alpgen}{+}\textsc{Herwig}$ | 43.3%
Heavy | $Z$+$b\bar{b}$+jets | $\textsc{Alpgen}{+}\textsc{Herwig}$ | 6.3%
Heavy | Single-top | MC@NLO / AcerMC | 50.4%
The expected numbers of signal and background events in the selected sample
are presented in Table 2. The uncertainties shown include theoretical
uncertainties in the production cross sections of the processes [4, 43, 44,
45, 46, 47]. The $W$/$Z$+jets and diboson uncertainties include a contribution
derived from an event yield comparison with Sherpa [48] Monte Carlo samples.
The uncertainty in the number of events with fake leptons is estimated to be
20% for muons and 50% for electrons [49, 50]. The observed number of events in
data is in good agreement with the prediction.
Table 2: The observed and expected numbers of events in the selected sample is shown. The first two columns list the contributions by physics process, and the two rightmost columns present events by class. The Heavy class includes the $W$/$Z$+$b\bar{b}$+jets and single-top processes, while the Light class includes all other backgrounds. Process | Events | Class | Events
---|---|---|---
$t\bar{t}$ | $86\,400$ p m 5700 | Signal | $86\,400$ p m 5700
$W$+jets | $184\,000$ p m 44000 | |
$Z$+jets | $19\,000$ p m 4300 | |
Dibosons | $3\,200$ p m 1600 | |
Single-top | $11\,040$ p m 670 | |
Fake leptons | $37\,500$ p m 8700 | |
| | Light | $233\,000$ p m 44000
| | Heavy | $21\,900$ p m 1600
Total | $341\,000$ p m 45000 | |
Observed | $344\,520$ | |
### 7 Analysis method
The SVM is a binary learning algorithm [51]. For any two classes of events,
the signed distance from a hyperplane that separates the events is the SVM
discriminant. For the analysis presented in this paper, a system of three
support vector machines is used to create a three-dimensional multi-class
event classifier to distinguish signal events from two classes of background
(i.e., Light and Heavy). For events from any dataset, the distances from the
three hyperplanes, trained to distinguish between Signal vs. Light, Signal vs.
Heavy, and Light vs. Heavy, are treated as the coordinates of points in a 3D
decision space. The resulting templates of the prediction model are used in a
binned likelihood fit to the analogous 3D distribution of the data events.
The SVM was chosen as the binary classifier because it is linear, it has firm
mathematical foundations, and it offers a simple geometrical interpretation.
Because the SVM method provides the solution to a straightforward convex
optimization problem, the minimum it finds is a global one. The stopping point
at the training stage is well defined, which therefore makes the method robust
against overtraining. The method also works well in problems involving a large
number of observables.
#### 7.1 The SVM discriminant
Each event is described by $N$ observables (i.e., features), and can be
represented as a point, $\vv{z}$, in an $N$-dimensional feature space. A
linear binary classifier finds a hyperplane of dimension $N-1$ that separates
the two classes. Once the separating hyperplane is found, its reconstruction
only requires the vectors that lie closest to the plane. These are the support
vectors from whence the method derives its name. If the two classes to be
discriminated are not linearly separable in the original feature space, this
$N$-dimensional space can be mapped, $\vv{z}\rightarrow\vv{\varphi}(\vv{z})$,
into a higher-dimensional space in which the problem is linearly separable.
Detailed knowledge of this mapping is not required when it is known how to
calculate the inner product of the mapped vectors [52, 53]. The distribution
of classes in the mapped space can be probed directly by analyzing the
multidimensional space which takes as its mathematical basis the SVM solutions
for different class pairings.
The soft-margin SVMs [54] used in this analysis are constructed using a
variant of the sequential minimal optimization algorithm [55] that includes
the improvements suggested in Ref. [56].
In the case of a three-class problem like the one considered in this analysis,
three different SVM classifiers are trained. Each SVM has the form
$f(\vv{z})=\langle
w|z\rangle-b=\sum_{i}^{\text{SVs}}\lambda_{i}y_{i}K(\vv{v_{i}},\vv{z})-b\,,$
which is the generalized equation of a plane. The $j^{\text{th}}$ SVM (with
$j$ in {1, 2, 3}) has a normal vector given by $|w_{j}\rangle$, and a constant
offset $b_{j}$ from the origin. The vectors $\vv{v}$ are the support vectors
from training (all other training vectors find their $\lambda=0$), the $y$’s
are their “truth” values ($\pm 1$), and the $\lambda$’s are parameters found
in the training process along with $b_{j}$. Hence, $|w_{j}\rangle$ is a linear
combination of training vectors mapped by $\vv{\varphi}$ to an alternative
vector space. The bra-ket notation here serves as a reminder that these
vectors belong to this mapped space. The inner product of two vectors in the
mapped space given their non-mapped vectors $\vv{x_{1}}$ and $\vv{x_{2}}$ is
determined via the kernel function $K(\vv{x_{1}},\vv{x_{2}})$. The SVMs in
this analysis use the Gaussian kernel:
$K(\vv{x_{1}},\vv{x_{2}})=\vv{\varphi}(\vv{x_{1}})\cdot\vv{\varphi}(\vv{x_{2}})=\exp(-|\vv{x_{1}}-\vv{x_{2}}|^{2}/2\sigma^{2})\,.$
The width $\sigma$ is an input parameter of the training process, along with
an additional positive constant $C$ which limits the range of the $\lambda$’s
and is necessary for soft-margin SVMs.
In order to construct an orthonormal basis from the three trained SVMs, the
Gram–Schmidt procedure [57] is used with their $|w_{j}\rangle$ vectors:
$|{w^{\prime}_{1}}\rangle=|w_{1}\rangle\quad\text{,}\quad|{w^{\prime}_{2}}\rangle=|w_{2}\rangle-\frac{\langle
w_{1}|w_{2}\rangle}{\langle
w_{1}|w_{1}\rangle}|w_{1}\rangle\quad\text{,}\quad|{w^{\prime}_{3}}\rangle=|w_{3}\rangle-\frac{\langle
w_{1}|w_{3}\rangle}{\langle w_{1}|w_{1}\rangle}|w_{1}\rangle-\frac{\langle
w^{\prime}_{2}|w_{3}\rangle}{\langle
w^{\prime}_{2}|w^{\prime}_{2}\rangle}|w^{\prime}_{2}\rangle\,.$
Using this basis, 3-tuples (X,Y, Z) for a decision space are created:
$\textrm{X}(\vv{z})=\frac{\langle z|w^{\prime}_{1}\rangle}{\sqrt{\langle
w^{\prime}_{1}|w^{\prime}_{1}\rangle}}\quad\text{,}\quad\textrm{Y}(\vv{z})=\frac{\langle
z|w^{\prime}_{2}\rangle}{\sqrt{\langle
w^{\prime}_{2}|w^{\prime}_{2}\rangle}}\quad\text{,}\quad\textrm{Z}(\vv{z})=\frac{\langle
z|w^{\prime}_{3}\rangle}{\sqrt{\langle
w^{\prime}_{3}|w^{\prime}_{3}\rangle}}\,.$
In this way, an input vector $\vv{z}$ describing an event has coordinates in
the XYZ space given by calculating ( X$(\vv{z})$, Y$(\vv{z})$, Z$(\vv{z})$ ).
It is these new coordinates in the decision space which are used to describe
all events, and this is the space in which the 3D templates of the likelihood
function are created.
#### 7.2 Physics observables
In this analysis, $21$ physics observables are used to distinguish $t\bar{t}$
events from background events (see Table 3). Twenty are kinematic variables,
and one comprises the $b$-tagging information of the event. These include the
electron or muon momentum, the number of jets in the event, the magnitude and
direction of the missing transverse momentum vector, sums of the jet momenta
components, the first five Fox–Wolfram moments (FWM), $H_{\mathrm{T}}$, the
two largest eigenvalues of the normalized momentum tensor, and the mass of the
lepton+jets system ($m_{\ell\mathrm{j}}$). The $H_{\mathrm{T}}$ is the scalar
sum of $E_{\text{T}}^{\text{miss}}$, electron $p_{\mathrm{T}}$ or muon
$p_{\mathrm{T}}$, and the $p_{\mathrm{T}}$ of all jets passing the selection
requirements.
Fox–Wolfram moments [58] were originally introduced for $e^{+}e^{-}$
colliders. The FWMs correspond to a decomposition of the event’s phase space
into Fourier modes on the surface of a sphere. They were modified for use at
the Tevatron and the LHC to characterize the complex shapes of final states at
hadron colliders [59, 60]. They form a set of event shape variables, and the
$l^{\text{th}}$ FWM ($H_{l}$) is defined in the following way:
$H_{l}=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}{\left|\sum_{i}^{\text{jets}}\frac{E_{\mathrm{T}}(i)}{E_{\mathrm{T}}(\text{total})}Y_{l}^{m}(\theta_{i},\phi_{i})\right|^{2}}\,.$
The $Y_{l}^{m}$’s are the spherical harmonics, $i$ runs over all selected jets
in the event, and $E_{\mathrm{T}}{(\text{total})}$ represents the sum of the
transverse energy from selected jets. The angles $\theta_{i}$ and $\phi_{i}$
indicate the direction of the $i^{\text{th}}$ jet. This analysis makes use of
$H_{1}$ through $H_{5}$.
The normalized momentum tensor uses the $E_{\text{T}}^{\text{miss}}$ and the
momenta of the lepton and up to five jets, and has the following form:
$P_{ij}=\frac{{E_{\text{T}}^{\text{miss}}}_{i}\leavevmode\nobreak\
{E_{\text{T}}^{\text{miss}}}_{j}}{|E_{\text{T}}^{\text{miss}}|^{2}}+\sum^{\text{lep}+5\text{jets}}\frac{p_{i}p_{j}}{|p|^{2}}\,.$
Here $i$ and $j$ run over the $x$, $y$, and $z$ components of momentum (for
$E_{\text{T}}^{\text{miss}}$, only the $x$ and $y$ components are nonzero).
The two largest eigenvalues of this “$p$-tensor” are used as SVM inputs.
Because the lepton+jets decays are rotationally invariant in $\phi$, some
variables are calculated with respect to the lepton direction in the plane
transverse to the beam. Hence, for the momenta of jets, $p_{\parallel}$ and
$p_{\bot}$ denote the components which are parallel and perpendicular to the
direction of the lepton in the transverse plane. Similarly, the
$\phi(E_{\text{T}}^{\text{miss}})$ variable is then the angle between the
transverse momentum of the lepton and the missing transverse momentum vector,
and $p_{\parallel}$ for the lepton corresponds to its entire transverse
momentum.
The SVMs treat each variable as one of the coordinates of a point in a
$21$-dimensional space. The algorithm requires that each variable should fall
roughly in the same numeric range so that all features contribute a similar
weight when evaluating the distance from the separating hyperplane. The
variables which have values outside the range of ($-1$, $+1$) are transformed
such that they approximately meet this requirement. All input variables and
the values that were used to scale them are listed in Table 3.
Table 3: List of the 21 variables used as input to the SVMs. The variables were divided by the given values to make them all of similar magnitude. Number | Feature | Divided by
---|---|---
11 | $E_{\text{T}}^{\text{miss}}$ [GeV] | 250
12 | $\phi(E_{\text{T}}^{\text{miss}})$ [radians] | $2\pi$
13 | Lepton $E$ [GeV] | 400
14 | Lepton $p_{\parallel}$ [GeV] | 400
15 | Lepton $p_{\mathrm{z}}$ [GeV] | 400
16 | Mass(lepton+jets) [GeV] | 750
17 | Fox–Wolfram moment 1 | 1
18 | Fox–Wolfram moment 2 | 1
19 | Fox–Wolfram moment 3 | 1
10 | Fox–Wolfram moment 4 | 1
11 | Fox–Wolfram moment 5 | 1
12 | Sum all jets $E_{\mathrm{T}}$ [GeV] | 500
13 | Sum all jets $E$ [GeV] | 750
14 | Sum all jets $p_{\parallel}$ [GeV] | 750
15 | Sum all jets $p_{\bot}$ [GeV] | 750
16 | Sum all jets $p_{\mathrm{z}}$ [GeV] | 750
17 | $H_{\mathrm{T}}$ [GeV] | 500
18 | $p$-tensor eigenvalue 1 | 1
19 | $p$-tensor eigenvalue 2 | 1
20 | Number of jets | 10
21 | Number of $b$-tags | 10
#### 7.3 SVM training
The SVMs are trained to separate three classes of events: the Light and Heavy
backgrounds and the $t\bar{t}$ signal. In order to train the SVMs, the Monte
Carlo simulation samples and a data sample representing the multi-jet
background are split into two subsamples. For training purposes, events from
each class are randomly selected from those passing the selection
requirements. The remaining events are used to test how well the trained SVMs
perform. Also, it is only these remaining events that are utilized in the
subsequent analysis.
The training process aims to find the set of support vectors that forms the
optimal decision plane in the mapped space induced by the kernel function. As
described in Section 7.1, there are two free parameters that need to be
specified when training. These are the $\sigma$ parameter of the Gaussian
kernel and $C$, the positive constant which constrains the $\lambda$’s in the
solution. A search grid over the values of these parameters was implemented,
and the performance of a given training was then evaluated based upon the area
of the resulting receiver operating characteristic (ROC) curve created with
the events not used in the training. As a result of this study, the values of
1.2 for $\sigma$ and 2.0 for $C$ were settled upon. The $t\bar{t}$ Signal
class and the two background classes, Light and Heavy, each used 8,000 events
for training, which is a small fraction of the total available events.
Increasing the number of training events was not found to improve performance.
It was also verified that the trained SVMs were not overtrained (i.e., that
their discriminant distributions generalize well from the training set to the
full class dataset).
#### 7.4 Class templates
Different physics processes can be distinguished by their distinctly different
distributions in the XYZ decision space. These distributions are obtained by
applying the Gram–Schmidt procedure to each event’s SVM output values.
Histogramming the different physics processes in the resulting 3D decision
space creates probability distribution functions (i.e., templates) that can
then be used in the likelihood fit.
Figure 1(a) shows a contour at a fixed value of the template function from
each of the classes. This highlights the different regions in 3D decision
space where the different event types congregate. As an alternative way of
illustrating these distributions, Figure 1(b) shows a sampling of 10,000
events from each class. Also shown are the three trained decision planes,
which serve to demonstrate the nonorthogonal nature of the basis defined by
the SVMs. The SvL plane at $\text{X}=0$ is seen to separate Signal from Light
as it extends downwards in Z. Similarly, the SvH and LvH planes separate their
training classes. The multiple band structure in the templates arises because
of training with the number of $b$-tags, which is a strong discriminant and is
discrete. These template bands represent groupings of events with 0, 1, or 2
$b$-tags.
(a)
(b)
Figure 1: Distribution of the three classes of events shown in the 3D
orthonormalized decision space. 1(a) Contours from each class are shown
together. 1(b) A view of the SVM decision space is shown with a random
sampling of 10,000 events from each class. The three planes corresponding to
each of the three trained SVMs are also depicted. SvL labels the plane which
comes from the Signal vs. Light training. Similarly, SvH is the Signal vs.
Heavy plane and LvH is the Light vs. Heavy solution.
In order to minimize the potential effect of small fluctuations in the
modeling, a small set of wider bins was constructed. The full 3D XYZ decision
space was organized into four quadrants by dividing the space at $\text{Y}=0$
and $\text{Z}=-0.01$. Each quadrant was then further divided into bins along
the X axis. The quadrants are designated YZ, Yz, yZ, and yz, where the
capitalization of the letters indicates where the quadrant is located (e.g.,
Yz means $\text{Y}>0.00$ and $\text{Z}\leq-0.01$). The division points for X
were chosen to keep a minimum of approximately 1,000 events in each bin while
preserving the shapes of the distributions. It is these four binned
distributions from the quadrants that are used for the final profile
likelihood fit.
#### 7.5 Cross-section measurement
A binned profile likelihood function is used in a fit to determine the
$t\bar{t}$ cross section from the data. In the likelihood fit, four templates
are used: $t\bar{t}$, $W$/$Z$, single-top, and fake lepton. In evaluating the
systematic uncertainties, particularly with respect to the modeling of
$t\bar{t}$, it was observed that a large uncertainty occurs because of the
similarity between the final states of $t\bar{t}$ events and single-top
background events. To alleviate this effect the single-top backgrounds, which
arise from electroweak processes rather than the strong interactions
responsible for the production of $t\bar{t}$ pairs, are combined into a single
template normalized to their predicted cross sections. The
$W$/$Z$+$b\bar{b}$+jets, the light-flavor $W$/$Z$+jets, and the diboson
backgrounds are combined into a $W$/$Z$ template. The normalizations of the
$t\bar{t}$, $W$/$Z$, and fake-lepton templates are free parameters of the fit.
The grouping of physics processes used when constructing the templates for the
fit can differ from the class definitions used for training. At the training
stage, the events are arranged in order to create SVMs that can distinguish
between the $t\bar{t}$ signal, the Light backgrounds, and the Heavy
backgrounds. After training, each physics process can be reassigned to
templates. The chosen allocation of the physics processes to four templates
($t\bar{t}$, $W$/$Z$, single-top, and fake lepton) results in smaller expected
uncertainties in the $t\bar{t}$ cross section.
The likelihood function uses the templates (projections onto the X axis from
each of the quadrants) that have been built in the XYZ decision space. Each
template has an associated strength parameter $\theta$ in the likelihood. The
maximum value of the likelihood is obtained in determining the central values
of the $\theta$ parameters. The systematic uncertainties of the fit results
are also included in the likelihood as nuisance parameters (NPs, $\alpha$’s,
or collectively $\vv{\alpha}$) with Gaussian constraints. Each template is a
function of the nuisance parameters in the likelihood, which is then able to
capture the effects due to each source of systematic uncertainty.
The likelihood of an unknown sample for an $n_{\text{T}}$ template problem is
defined as:
$L=\prod_{i}^{\text{bins}}P(\mathscr{E}_{i},o_{i})\times\prod_{j}^{\text{NPs}}G(\alpha_{j},\sigma_{j})\,.$
Here $G(\alpha_{j},\sigma_{j})$ is the Gaussian constraint for the
$j^{\text{th}}$ NP, $\alpha_{j}$, with the corresponding uncertainty
$\sigma_{j}$; and $P(\mathscr{E}_{i},o_{i})$ is the Poisson probability mass
function for the $i^{\text{th}}$ bin given the observed number of events,
$o_{i}$, and the expected number of events, $\mathscr{E}_{i}$ :
$\mathscr{E}_{i}=\mathscr{E}(\vv{\theta},\vv{\alpha})_{i}=\sum_{j}^{n_{\text{T}}}{\theta_{j}N(\vv{\alpha})_{j}T_{j}(i,\vv{\alpha})}\,.$
The templates are constructed such that $T_{j}(i,\vv{\alpha})$ gives the
fractional number of events in template $T_{j}$’s $i^{\text{th}}$ bin.
Consequently, the sum over all bins of a given template is equal to $1$. The
$N(\vv{\alpha})_{j}$ are defined to be the total number of events expected
from the $j^{\text{th}}$ template assuming an integrated luminosity
$\mathscr{L}$. To calculate this value, the following sum over all modeled
processes belonging to the $j^{\text{th}}$ template is computed:
$N(\vv{\alpha})_{j}=\sum_{k}^{\text{processes}}{\sigma_{k}\epsilon(\vv{\alpha})_{k}\mathscr{L}}\,.$
The $\sigma_{k}$ and $\epsilon_{k}$ are the cross section and acceptance for
the $k^{\text{th}}$ physics process. They are derived from MC simulation. For
the multi-jet background, $N_{j}$ is taken from the fake-lepton estimate.
A maximum-likelihood fit is performed to extract the values of the $\theta$
and $\alpha$ parameters. The $1\sigma$ uncertainty for a given parameter is
taken to be the change in the value of that parameter which causes $\ln L$ to
decrease by $0.5$ away from $\ln L_{0}$, when $\ln L$ is maximized with
respect to all other free parameters and where $\ln L_{0}$ is the global
maximum. All $\theta$ and $\alpha$ parameters have both their $+1\sigma$ and
$-1\sigma$ uncertainties determined in this way.
The $\theta$ for the $t\bar{t}$ class, multiplied by the assumed cross
section, gives the measured value of the $t\bar{t}$ cross section. Similarly,
the uncertainty in $\theta_{t\bar{t}}$ from the fit, multiplied by the assumed
cross section, gives the uncertainty in the $\sigma_{t\bar{t}}$ measurement.
#### 7.6 Systematic uncertainties
All systematic uncertainties were evaluated using the profile likelihood fit.
The systematic effects are incorporated into the templates, and each template
is associated with appropriate nuisance parameters ($\alpha$’s) in the
likelihood. A nuisance parameter that takes a value of $0$ in the fit keeps
the nominal template, while a value of $+1$ or $-1$ changes the template to
look like the $+1\sigma$ / $-1\sigma$ effect. Templates at intermediate values
of the $\alpha$’s are linearly interpolated. A Gaussian constraint is also
applied to each $\alpha$ in order to propagate its controlled uncertainty when
the data have no preference for that systematic effect. The profile likelihood
fit then provides a simultaneous measurement of the $\theta$ and $\alpha$
parameters. In this way, the systematic effects are converted into a
statistical framework that properly takes into account correlations and which
can potentially lower the uncertainties in the measurement.
The individual effects of various sources of systematic uncertainty are
displayed in Table 4. They are obtained by leaving groupings of nuisance
parameters out of the fit, and calculating the square of each effect as the
difference of the squares of the total error and the residual error.
Object modeling
Systematic uncertainties in the lepton selection arise from uncertainties in
lepton identification, reconstruction, and triggering. These are evaluated by
applying tag-and-probe methods to $Z\rightarrow\ell\ell$ events [16].
Uncertainties due to the energy scale and resolution are also considered for
electrons and muons. These effects are evaluated by assigning each of them a
separate nuisance parameter in the likelihood so as to allow the error source
to be shifted both upwards and downwards by its uncertainty. The resulting
systematic effects are summarized in Table 4 as Leptons.
For jets, the main source of uncertainty is the jet energy scale (JES). The
JES and its uncertainty are evaluated using a combination of test-beam data,
LHC collision data, and simulation [21]. As a result of the in situ analyses
for the calibration of the full 2011 dataset, the correlations between various
JES uncertainty components are encoded in 21 subcomponents. These include
statistical and method uncertainties, detector uncertainties, modeling and
theory uncertainties, mixed detector and modeling uncertainties, and pileup.
The JES uncertainty is evaluated by assigning a separate NP to each of these
21 JES subcomponents. The jet energy resolution (JER) is separated by process
($t\bar{t}$, single-top, and $W$/$Z$+jets) and is assigned three corresponding
NPs. These extra degrees of freedom allow differences in the kinematics and
prevalence of $b$-quark, light-quark, and gluon jets in these processes to be
better represented in the profile likelihood. The resulting uncertainties in
$\sigma_{t\bar{t}}$ from these sources are indicated in Table 4 as Jets.
The jet-flavor-dependent efficiencies of the $b$-tagging algorithm are
calibrated using dijet events, and dilepton $t\bar{t}$ events from data.
Differences in the $b$-tagging efficiency as well as $c$-jet and light-jet
mistag rates between data and simulation are parameterized using correction
factors, which are functions of $p_{\mathrm{T}}$ and $\eta$ [22]. The $b$-tag
systematic uncertainties were evaluated by constructing nine NPs that
correspond to unique bins in jet $p_{\mathrm{T}}$, as the uncertainties at low
and high jet $p_{\mathrm{T}}$ should be largely uncorrelated. Single NPs were
used for the $c$-tag and Mistag systematic uncertainties. These systematic
effects appear in Table 4 as three uncertainties labeled $b$-tag, $c$-tag, and
Mistag.
During the variation of nuisance parameters related to jets and leptons, the
$E_{\text{T}}^{\text{miss}}$ is recalculated in accordance with the changes
caused by those systematic effects. In this way, the jet and lepton
uncertainties are propagated to the $E_{\text{T}}^{\text{miss}}$. However, the
$E_{\text{T}}^{\text{miss}}$ uncertainty due to calorimeter cells not assigned
to any other physics object is evaluated individually. Also, an additional
6.6% uncertainty due to pileup is applied to $E_{\text{T}}^{\text{miss}}$.
Both of these are given separate NPs in the profile likelihood, and they are
listed in Table 4 under Missing transverse momentum.
Modeling of $\boldsymbol{t\bar{t}}$ events
Systematic uncertainties due to the choice of $t\bar{t}$ MC generator are
evaluated by taking the full difference between Powheg+Herwig (AF) and
$\textsc{MC@NLO}{+}\textsc{Herwig}$ (AF). The systematic uncertainty due to
the choice of parton shower model is taken as the full difference between
Powheg+Herwig (AF) and Powheg+Pythia 6 (AF). These are listed as Generator and
Shower/hadronization in Table 4, respectively.
The systematic error due to uncertainties in the modeling of initial- and
final-state radiation (ISR/FSR) is evaluated using Alpgen interfaced to Pythia
6\. Monte Carlo samples were created in which the parameter that controls the
amount of ISR/FSR in Alpgen was either halved or doubled. Half of the spread
between the Alpgen samples with raised and lowered ISR/FSR parameter values is
taken as the systematic error.
The uncertainty due to renormalization and factorization scales is evaluated
with two modified samples, generated with MC@NLO interfaced with Herwig, in
which parameters controlling the renormalization and factorization scales,
introduced to cure the ultraviolet and infrared divergences in ME
calculations, are simultaneously either halved or doubled. The full difference
between the two samples is taken as the Renormalization/factorization error.
Each of the major $t\bar{t}$ modeling systematic uncertainties (Generator,
Shower/hadronization, ISR/FSR, and Renormalization/factorization) is given a
shape NP in each quadrant. The uncertainty in the normalization of events is
assigned two NPs. One of these is used to track the migration of events
between quadrants, and it mirrors the movement of events seen when comparing
the nominal and systematically shifted samples. The second NP is taken as an
overall normalization error which corresponds to the normalization difference
seen for the full event selection (where all four quadrants are combined).
Therefore, each of the $t\bar{t}$ modeling uncertainties mentioned above has
six NPs (four for shape, and two for normalization).
The underlying-event modeling error is evaluated by comparing two different
$t\bar{t}$ MC event samples produced with varied parameters in Powheg+Pythia
6\. One was generated with the Perugia 2011 central tune, and the other with
Perugia 2011 mpiHI [32]. Both of these samples use the P2011 CTEQ5L Pythia
tune, and not P2011C CTEQ6L1, which applies to the nominal $t\bar{t}$ MC
sample. Their full difference is used as the measurement uncertainty for the
underlying event, using a single NP in the profile likelihood.
All particles in the final state from the LHC $pp$ collisions must be color
singlets. Different schemes for the color reconnection (CR) of the beam
remnant and other outgoing hard collision objects are examined. The $t\bar{t}$
cross-section uncertainty due to this effect is estimated by comparing two
different $t\bar{t}$ MC samples produced with Powheg+Pythia 6\. A reference
sample was obtained using the Perugia 2011 central tune. The other sample was
generated with Perugia 2011 noCR [32], and has modified color reconnection
parameters. These samples use the P2011 CTEQ5L Pythia tune. The full
difference between these two samples is taken as the CR uncertainty, using a
single NP.
To estimate the uncertainty due to the choice of parton distribution function,
the CT10 PDF set parameterization is examined using its 26 upwards and
downwards systematic variations. Each of the 26 CT10 eigenvector components is
assigned a separate NP in the profile likelihood.
Background modeling
To estimate the error due to the shape of the $W$/$Z$+jets backgrounds and to
assess the effect of any mismodeling, background samples are reweighted to
match data for each of the following variables, taken one at a time, in a
signal-depleted control region: lepton $E$,
$\phi(E_{\text{T}}^{\text{miss}})$, $m_{\ell\mathrm{j}}$,
$\sum_{\text{jets}}{p_{\parallel}}$, and $\sum_{\text{jets}}{p_{\bot}}$. This
control region was defined as events matching the nominal selection, but
containing exactly three jets, none of which are $b$-tagged. The reweighting
functions were applied only to $W$/$Z$+jets samples (leaving $t\bar{t}$,
single-top, and fake lepton untouched). Five NPs in the profile likelihood
implement these functions such that the NPs turn the reweighting effects on
and off, each according to the differences seen for these five variables in
the data. In Table 4, these effects appear under the heading $W$/$Z$
reweighting. For the single-top shape, AcerMC samples with raised and lowered
ISR/FSR parameter values are compared, and a single NP is assigned to this
systematic uncertainty.
The effects due to the uncertainty of the single-top, $W$+jets, and $Z$+jets
cross sections are investigated by varying these cross sections within their
theoretical errors. For the $W$+jets background, a 4% uncertainty is applied
to the inclusive $W$ boson cross section, with an additional 24% uncertainty
for each jet [39, 61]. This method was also applied to the $Z$+jets cross
sections. To evaluate the systematic uncertainty in the $t\bar{t}$ cross
section due to the theoretical uncertainties in the single-top cross section,
the single-top cross section is varied in accordance with the theoretical
results, taken from Refs. [45, 46, 47]. The relative normalization within the
$W$/$Z$+jets MC sample is varied by raising and lowering the corresponding
nominal relative yields of each jet multiplicity by their respective errors.
Similarly, the relative normalization between fake-electron and fake-muon
events is varied by raising and lowering their nominal predictions by their
errors. The resulting effects are evaluated using appropriate NPs added in the
profile likelihood fit. This uncertainty is quoted as $W$/$Z$ & fakes relative
normalization in Table 4. Also, the uncertainty due to variations of the
$W$+jets heavy-flavor fraction is included via three NPs in the profile
likelihood. These NPs place an additional $25$% uncertainty on each of the
assumed $W$+$b\bar{b}$+jets, $W$+$c\bar{c}$+jets, and $W$+$c$+jets cross
sections. Table 4 summarizes these in the row labeled Heavy-flavor fraction.
Template statistics / luminosity
For the profile likelihood fit, an additional fit parameter is introduced for
each bin. These parameters are used to represent the Poisson fluctuation of
the predicted number of events in each bin as estimated from the size of the
Monte Carlo samples. The error propagated to $\theta_{t\bar{t}}$ from these
additional parameters is then an appropriate representation of the MC
statistical error.
The integrated luminosity measurement has an uncertainty of 1.8% [24], and
therefore each physics process is assigned an uncertainty of this magnitude.
This systematic error is controlled by a single nuisance parameter in the
likelihood.
The total measurement uncertainty, including individual groups of
contributions, is listed in Table 4. The largest uncertainties are due to the
lepton selection and luminosity, followed by the uncertainties due to JES,
$b$-tagging, ISR/FSR, and other $t\bar{t}$ modeling.
Beam energy
The LHC beam energy during the 2012 $\sqrt{s}=8$ TeV $pp$ run was measured to
be within $0.1$% of the nominal value of 4 TeV per beam, using the revolution
frequency difference of protons and lead ions during $p+Pb$ runs in early 2013
combined with the magnetic model errors [62]. A similar uncertainty in the
beam energy is applicable to the 2011 LHC run. The approach used in Ref. [63]
was therefore applied to the measurement using the $\sqrt{s}=7$ TeV dataset.
The uncertainty in the $t\bar{t}$ theoretical cross section due to this energy
difference was calculated to be 0.27%, using the Top++ 2.0 program [5] and
assuming that the relative change of the $t\bar{t}$ cross section for a $0.1$%
change in $\sqrt{s}$ is as predicted by the NNLO + NNLL calculation. It is
negligible compared to other sources of systematic uncertainty.
Table 4: Summary table of the measurement uncertainties. Because the profile likelihood fit accounts for correlations, the total error is not simply the components added in quadrature. Individual effects were obtained by leaving groupings of NPs out of the fit, and calculating the square of each effect as the difference of the squares of the total error and the residual error. Source | $-1\sigma$ [pb] | $+1\sigma$ [pb] | $-1\sigma$ [%] | $+1\sigma$ [%]
---|---|---|---|---
Object Modeling | | | |
Leptons | $-3.1$ | $+3.3$ | $-1.8$ | $+2.0$
Jets | $-2.9$ | $+3.0$ | $-1.7$ | $+1.8$
$b$-tag | $-1.9$ | $+2.0$ | $-1.1$ | $+1.2$
$c$-tag | $-0.4$ | $+0.4$ | $-0.3$ | $+0.3$
Mistag | $-0.3$ | $+0.3$ | $-0.2$ | $+0.2$
Missing transverse momentum | $-0.2$ | $+0.2$ | $-0.1$ | $+0.1$
$t\bar{t}$ Modeling | | | |
Generator | $-1.6$ | $+1.8$ | $-1.0$ | $+1.1$
Shower/hadronization | $-2.4$ | $+2.6$ | $-1.4$ | $+1.5$
Renormalization/factorization | $-1.4$ | $+1.4$ | $-0.8$ | $+0.9$
ISR/FSR | $-2.4$ | $+2.5$ | $-1.4$ | $+1.5$
Underlying event | $-0.7$ | $+0.8$ | $-0.4$ | $+0.5$
Color reconnection | $-0.5$ | $+0.5$ | $-0.3$ | $+0.3$
PDF | $-1.8$ | $+1.9$ | $-1.1$ | $+1.1$
Background Modeling | | | |
$W$/$Z$ reweighting | $-1.0$ | $+1.0$ | $-0.6$ | $+0.6$
$W$/$Z$ & fakes relative normalization | $-1.2$ | $+1.2$ | $-0.7$ | $+0.7$
Heavy-flavor fraction | $-1.1$ | $+1.2$ | $-0.7$ | $+0.7$
Single-top | $-1.0$ | $+1.0$ | $-0.6$ | $+0.6$
Other | | | |
Data statistics | $-0.7$ | $+0.7$ | $-0.4$ | $+0.4$
Template statistics | $-1.0$ | $+1.0$ | $-0.6$ | $+0.6$
Luminosity | $-3.2$ | $+3.4$ | $-1.9$ | $+2.0$
Total | $-6.8$ | $+7.1$ | $-4.0$ | $+4.2$
### 8 Results
The top quark pair-production cross section for $pp$ collisions at a center-
of-mass energy of $\sqrt{s}=7$ TeV in the lepton+jets channel is found to be
$\sigma_{t\bar{t}}=168.5\pm 0.7$(stat.) ${}^{+6.2}_{-5.9}$(syst.)
${}^{+3.4}_{-3.2}$(lumi.) pb.
This result includes all systematic uncertainties as evaluated with the
profile likelihood fit, with the statistical and luminosity errors listed
separately.
Figure 2 shows a comparison between the observed and fitted numbers of events
in each of the quadrants. A correlated $\chi^{2}$ test was used to check that
there is good agreement between the data and the fit results within the
combined statistical and systematic error bands.
Figure 2: The fitted yields of signal and background processes compared with
data, shown in four YZ quadrants divided along the X axis, as used in the fit.
They are labeled quad1YZ, quad2Yz, quad3yZ, and quad4yz (the boundary letters
are appended for easy reference). The lower panel shows the ratio of data to
fit prediction. The shaded regions correspond to the statistical and
systematic uncertainties. The first and last bins also contain any events
found outside the range of the horizontal axis.
Comparison plots between data and the fit prediction are shown for a few
selected input variables in Figure 3 for all events. Analogous comparisons in
signal-rich and background-rich regions of the XYZ space are shown in Figures
4 and 5, respectively. The signal-rich region is defined by $\text{X}>0$,
$\text{Y}>0$, and $\text{Z}<0$, while the background-rich region lies in the
opposite octant of XYZ space, and has $\text{X}<0$, $\text{Y}<0$, and
$\text{Z}>0$. The X dimension corresponds to the Signal vs. Light decision
hyperplane, while the Y and Z dimensions are linear combinations of the other
SVM hyperplanes and are the directions orthogonal to X. Based on a correlated
$\chi^{2}$ test, the data and the fit agree well within the combined
statistical and systematic error bands for all 21 variables.
The measured $t\bar{t}$ cross section is in good agreement with the
theoretical predictions based on the NNLO + NNLL calculations of
$\sigma_{t\bar{t}}^{\text{NNLO+NNLL}}=177^{+5}_{-6}\>\mathrm{(scale)}\,\pm
9\>\mathrm{(PDF+}\alpha_{\text{s}})$ pb $=177\leavevmode\nobreak\
^{+10}_{-11}$ pb for $pp$ collisions at a center-of-mass energy of
$\sqrt{s}=7$ TeV and a top quark mass of 172.5 GeV [4].
The ATLAS measurement of the $t\bar{t}$ cross section at 7 TeV in the dilepton
channel [9] is $\sigma_{t\bar{t}}=182.9\pm 3.1$(stat.) $\pm$ 4.2(syst.) $\pm$
3.6(lumi.) pb. Depending upon the assumptions made for the systematic
uncertainty correlations between these two measurements, the significance of
their discrepancy was found to be in the $1.9\sigma$ to $2.1\sigma$ range.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3: The data distributions of six selected input variables are shown
with their post-fit predictions in the selected sample. The predicted signal
fraction is 24.8%. Shown are 3(a) the number of jets, 3(b) the number of
$b$-tagged jets, 3(c) $H_{\mathrm{T}}$, 3(d) the $4^{\text{th}}$ Fox–Wolfram
moment, 3(e) $E_{\text{T}}^{\text{miss}}$, and 3(f) the mass of the lepton and
jets. Data are shown with the overlaid dots. The predicted events are shown
for each of the templates used in modeling the data. The statistical and
systematic error bands are given by the shaded regions. The first and last
bins contain events found outside the range of the horizontal axis.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: The data distributions of six selected input variables are shown
with their post-fit predictions in the selected sample for the signal-rich
region with $\text{X}>0$, $\text{Y}>0$, and $\text{Z}<0$. The predicted signal
fraction in this region is 79.3%. Shown are 4(a) the number of jets, 4(b) the
number of $b$-tagged jets, 4(c) $H_{\mathrm{T}}$, 4(d) the $4^{\text{th}}$
Fox–Wolfram moment, 4(e) $E_{\text{T}}^{\text{miss}}$, and 4(f) the mass of
the lepton and jets. Data are shown with the overlaid dots. The predicted
events are shown for each of the templates used in modeling the data. The
statistical and systematic error bands are given by the shaded regions. The
first and last bins contain events found outside the range of the horizontal
axis.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5: The data distributions of six selected input variables are shown
with their post-fit predictions in the selected sample for the background-rich
region with $\text{X}<0$, $\text{Y}<0$, and $\text{Z}>0$. The predicted signal
fraction in this region is 3.1%. Shown are 5(a) the number of jets, 5(b) the
number of $b$-tagged jets, 5(c) $H_{\mathrm{T}}$, 5(d) the $4^{\text{th}}$
Fox–Wolfram moment, 5(e) $E_{\text{T}}^{\text{miss}}$, and 5(f) the mass of
the lepton and jets. Data are shown with the overlaid dots. The predicted
events are shown for each of the templates used in modeling the data. The
statistical and systematic error bands are given by the shaded regions. The
first and last bins contain events found outside the range of the horizontal
axis.
#### 8.1 Top quark mass dependence
The result of the profile likelihood fit depends on the assumed mass of the
top quark through differences in $t\bar{t}$ acceptance owing to lepton
kinematics, and also from minor variations in the shape of the discriminant.
The analysis in this paper assumes a top quark mass of $m_{\text{ref}}=172.5$
GeV. The 2014 average of Tevatron and LHC Run 1 measurements of the top quark
mass [64] gives a value of $m_{t}=173.34\pm 0.27$(stat.)
$\pm\leavevmode\nobreak\ 0.71$(syst.) GeV. The current ATLAS average from 2019
[65] yields $m_{t}=172.69\pm 0.25$(stat.) $\pm\leavevmode\nobreak\
0.41$(syst.) GeV.
The dependence of the $t\bar{t}$ cross section on the mass of the top quark
was determined through alternative profile likelihood fits that assume
different top quark masses. Monte Carlo samples for both $t\bar{t}$ and
single-top with top quark mass values of 165.0, 170.0, 172.5, 175.0, 177.5,
and 180.0 GeV were employed to measure the $t\bar{t}$ cross section assuming
each of these masses. These measurements were then fitted to obtain the
$t\bar{t}$ cross section’s top quark mass dependence. When constrained to go
through this measurement’s central value at 172.5 GeV, the best-fit second-
order polynomial for the $t\bar{t}$ cross section as a function of $\Delta
m_{t}=m_{t}-m_{\text{ref}}$ is $\sigma_{t\bar{t}}(\Delta
m_{t})=0.016\cdot\Delta m_{t}^{2}-0.75\cdot\Delta m_{t}+168.5$ pb, with
$\Delta m_{t}$ in GeV.
### 9 Summary
A measurement of the top quark pair-production cross section in the
lepton+jets channel was performed with the ATLAS experiment at the LHC, using
a multivariate technique based on support vector machines. The measurement was
obtained with a three-class, multidimensional event classifier. It is based on
$4.6\text{\leavevmode\nobreak\ fb}^{-1}$ of data collected during 2011 in $pp$
collisions at $\sqrt{s}=7$ TeV. The $t\bar{t}$ cross section is found to be
$\sigma_{t\bar{t}}=168.5\pm 0.7$(stat.) ${}^{+6.2}_{-5.9}$(syst.)
${}^{+3.4}_{-3.2}$(lumi.) pb, which has a relative uncertainty of
${}^{+4.2}_{-4.0}$%. This measurement is consistent with the Standard Model
prediction based on QCD calculations at next-to-next-to-leading order.
## Appendix: Fit visualization
The fit to the data can be difficult to visualize because of the 3D nature of
the decision space. A series of 2D and 1D projections of the 3D space have
been created in order to better illustrate its characteristics.
Projections are provided in the XY, XZ, and YZ planes to give a qualitative
comparison of the fit results with the data in XYZ space, and also to help
visualize the definitions of the signal-rich and background-rich regions
employed in Figures 4 and 5. The 2D projections of the 3D decision space use
the standard fit results in XYZ space, but project the constant binning
obtained with cubes of edge length 0.008 in that space. On the right side of
Figure 6 the 2D projection plots show the fractional relative difference
between the number of events predicted by the fit and the observed data. The
2D plots on the left side of Figure 6 show the class composition of each bin
when projected onto the XY, XZ, and YZ planes. The rectangle representing each
2D bin is colored from top to bottom such that the colors that fill it are in
the same ratio as the predictions for each class in that bin. Contour lines,
which each represent the concentration of events in the plane, are drawn on
top of the colored bins. Contours exist for each of the following fractional
values of the maximal 2D bin height: (0.05, 0.10, 0.20, 0.40, 0.60, 0.80,
0.90, 0.95).
The 1D projections of the 3D decision space onto the X, Y, and Z axes are
shown in Figure 7 for the results of the fit to the data. For visualization
purposes, the full 3D decision space is projected onto each of these axes.
These projections have the systematic error bands superimposed, and also
include the ratio of data to fit prediction.
---
Figure 6: (left) These 2D projections show the composition of each bin
according to the template fit results. Bins are shaded in the same ratio as
the fit prediction. The contours drawn on top of the bins represent the
overall concentration of events. Contours are provided at 5%, 10%, 20%, 40%,
60%, 80%, 90%, and 95% of the maximal 2D bin height. (right) The 2D bins are
shaded in accordance with the fractional difference between the observed data
and the number of events predicted by the fit. The contours shown are the same
as in the left column and provide a reference.
(a)
(b)
(c)
Figure 7: The 1D projections of the data and expected events after the
likelihood fit. X, Y, and Z form an orthogonal basis in the decision space
defined by the three trained SVMs. X corresponds to the Signal vs. Light
plane. Y and Z are the directions orthogonal to X found with the remaining
Signal vs. Heavy and Light vs. Heavy SVMs. The projections use the fit
parameters obtained with the profile likelihood fit to four quadrants, but
project a constant binning obtained with cubes of edge length 0.008 in XYZ
space. In each of these plots, the prediction is broken down by the templates
used in the fit. The first and last bins contain events found outside the
range of the horizontal axis.
### References
* [1] S.L. Glashow “Partial-symmetries of weak interactions” In _Nucl. Phys._ 22, 1961, pp. 579–588 DOI: 10.1016/0029-5582(61)90469-2
* [2] S. Weinberg “A model of leptons” In _Phys. Rev. Lett._ 19, 1967, pp. 1264–1266 DOI: 10.1103/PhysRevLett.19.1264
* [3] A. Salam “Elementary Particle Physics: Relativistic Groups and Analyticity” In _Eighth Nobel Symposium_ Stockholm: AlmquistWiksell, 1968, pp. 367
* [4] Michal Czakon, Paul Fiedler and Alexander Mitov “Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through $O(\alpha_{S}^{4})$” In _Phys. Rev. Lett._ 110, 2013, pp. 252004 DOI: 10.1103/PhysRevLett.110.252004
* [5] Michal Czakon and Alexander Mitov “Top++: A program for the calculation of the top-pair cross-section at hadron colliders” In _Comput. Phys. Commun._ 185, 2014, pp. 2930 DOI: 10.1016/j.cpc.2014.06.021
* [6] Michiel Botje et al. “The PDF4LHC Working Group Interim Recommendations”, 2011 arXiv:1101.0538 [hep-ph]
* [7] ATLAS Collaboration “Measurement of the top quark pair production cross-section with ATLAS in the single lepton channel” In _Phys. Lett. B_ 711, 2012, pp. 244 DOI: 10.1016/j.physletb.2012.03.083
* [8] CMS Collaboration “Measurement of the $t\bar{t}$ production cross section in $pp$ collisions at $\sqrt{s}=7\,\text{TeV}$ with lepton+jets final states” In _Phys. Lett. B_ 720, 2013, pp. 83 DOI: 10.1016/j.physletb.2013.02.021
* [9] ATLAS Collaboration “Measurement of the $t\bar{t}$ production cross-section using $e\mu$ events with $b$-tagged jets in $pp$ collisions at $\sqrt{s}=7$ and $8\,\text{TeV}$ with the ATLAS detector” In _Eur. Phys. J. C_ 74, 2014, pp. 3109 DOI: 10.1140/epjc/s10052-014-3109-7
* [10] CMS Collaboration “Measurement of the $t\bar{t}$ production cross section in the $e\mu$ channel in proton–proton collisions at $\sqrt{s}=7$ and $8\,\text{TeV}$” In _JHEP_ 08, 2016, pp. 029 DOI: 10.1007/JHEP08(2016)029
* [11] ATLAS and CMS Collaborations “Combination of inclusive top-quark pair production cross-section measurements using ATLAS and CMS data at $\sqrt{s}=7$ and $8\,\text{TeV}$”, 2022 arXiv:2205.13830 [hep-ex]
* [12] ATLAS Collaboration “The ATLAS Experiment at the CERN Large Hadron Collider” In _JINST_ 3, 2008, pp. S08003 DOI: 10.1088/1748-0221/3/08/S08003
* [13] ATLAS Collaboration “Performance of the ATLAS Trigger System in 2010” In _Eur. Phys. J. C_ 72, 2012, pp. 1849 DOI: 10.1140/epjc/s10052-011-1849-1
* [14] ATLAS Collaboration “The ATLAS Collaboration Software and Firmware”, ATL-SOFT-PUB-2021-001, 2021 URL: https://cds.cern.ch/record/2767187
* [15] ATLAS Collaboration “Electron performance measurements with the ATLAS detector using the 2010 LHC proton–proton collision data” In _Eur. Phys. J. C_ 72, 2012, pp. 1909 DOI: 10.1140/epjc/s10052-012-1909-1
* [16] ATLAS Collaboration “Electron reconstruction and identification efficiency measurements with the ATLAS detector using the 2011 LHC proton–proton collision data” In _Eur. Phys. J. C_ 74, 2014, pp. 2941 DOI: 10.1140/epjc/s10052-014-2941-0
* [17] ATLAS Collaboration “Measurement of the muon reconstruction performance of the ATLAS detector using 2011 and 2012 LHC proton–proton collision data” In _Eur. Phys. J. C_ 74, 2014, pp. 3130 DOI: 10.1140/epjc/s10052-014-3130-x
* [18] ATLAS Collaboration “Topological cell clustering in the ATLAS calorimeters and its performance in LHC Run 1” In _Eur. Phys. J. C_ 77, 2017, pp. 490 DOI: 10.1140/epjc/s10052-017-5004-5
* [19] Matteo Cacciari, Gavin P. Salam and Gregory Soyez “The anti-$k_{t}$ jet clustering algorithm” In _JHEP_ 04, 2008, pp. 063 DOI: 10.1088/1126-6708/2008/04/063
* [20] Matteo Cacciari, Gavin P. Salam and Gregory Soyez “FastJet user manual” In _Eur. Phys. J. C_ 72, 2012, pp. 1896 DOI: 10.1140/epjc/s10052-012-1896-2
* [21] ATLAS Collaboration “Jet energy measurement and its systematic uncertainty in proton–proton collisions at $\sqrt{s}=7\,\text{TeV}$ with the ATLAS detector” In _Eur. Phys. J. C_ 75, 2015, pp. 17 DOI: 10.1140/epjc/s10052-014-3190-y
* [22] ATLAS Collaboration “Performance of $b$-jet identification in the ATLAS experiment” In _JINST_ 11, 2016, pp. P04008 DOI: 10.1088/1748-0221/11/04/P04008
* [23] ATLAS Collaboration “Performance of missing transverse momentum reconstruction in proton–proton collisions at $\sqrt{s}=7\,\text{TeV}$ with ATLAS” In _Eur. Phys. J. C_ 72, 2012, pp. 1844 DOI: 10.1140/epjc/s10052-011-1844-6
* [24] ATLAS Collaboration “Improved luminosity determination in $pp$ collisions at $\sqrt{s}=7\,\text{TeV}$ using the ATLAS detector at the LHC” In _Eur. Phys. J. C_ 73, 2013, pp. 2518 DOI: 10.1140/epjc/s10052-013-2518-3
* [25] ATLAS Collaboration “Measurement of the top quark-pair production cross section with ATLAS in $pp$ collisions at $\sqrt{s}=7\,\text{TeV}$” In _Eur. Phys. J. C_ 71, 2011, pp. 1577 DOI: 10.1140/epjc/s10052-011-1577-6
* [26] ATLAS Collaboration “Estimation of non-prompt and fake lepton backgrounds in final states with top quarks produced in proton–proton collisions at $\sqrt{s}=8\leavevmode\nobreak\ \text{TeV}$ with the ATLAS Detector”, ATLAS-CONF-2014-058, 2014 URL: https://cds.cern.ch/record/1951336
* [27] ATLAS Collaboration “The ATLAS Simulation Infrastructure” In _Eur. Phys. J. C_ 70, 2010, pp. 823 DOI: 10.1140/epjc/s10052-010-1429-9
* [28] GEANT4 Collaboration and S. Agostinelli “Geant4 – a simulation toolkit” In _Nucl. Instrum. Meth. A_ 506, 2003, pp. 250 DOI: 10.1016/S0168-9002(03)01368-8
* [29] Wolfgang Lukas “Fast Simulation for ATLAS: Atlfast-II and ISF” In _J. Phys.: Conf. Ser._ 396.2, 2012, pp. 022031 DOI: 10.1088/1742-6596/396/2/022031
* [30] Stefano Frixione, Giovanni Ridolfi and Paolo Nason “A positive-weight next-to-leading-order Monte Carlo for heavy flavour hadroproduction” In _JHEP_ 09, 2007, pp. 126 DOI: 10.1088/1126-6708/2007/09/126
* [31] Torbjorn Sjöstrand, Stephen Mrenna and Peter Z. Skands “PYTHIA 6.4 physics and manual” In _JHEP_ 05, 2006, pp. 026 DOI: 10.1088/1126-6708/2006/05/026
* [32] Peter Zeiler Skands “Tuning Monte Carlo generators: The Perugia tunes” In _Phys. Rev. D_ 82, 2010, pp. 074018 DOI: 10.1103/PhysRevD.82.074018
* [33] H.-L. Lai “New parton distributions for collider physics” In _Phys. Rev. D_ 82, 2010, pp. 074024 DOI: 10.1103/PhysRevD.82.074024
* [34] J. Pumplin “New Generation of Parton Distributions with Uncertainties from Global QCD Analysis” In _JHEP_ 07, 2002, pp. 012 DOI: 10.1088/1126-6708/2002/07/012
* [35] Stefano Frixione, Paolo Nason and Bryan R. Webber “Matching NLO QCD and parton showers in heavy flavour production” In _JHEP_ 08, 2003, pp. 007 DOI: 10.1088/1126-6708/2003/08/007
* [36] G. Corcella “HERWIG 6: an event generator for hadron emission reactions with interfering gluons (including supersymmetric processes)” In _JHEP_ 01, 2001, pp. 010 DOI: 10.1088/1126-6708/2001/01/010
* [37] ATLAS Collaboration “New ATLAS event generator tunes to 2010 data”, ATL-PHYS-PUB-2011-008, 2011 URL: https://cds.cern.ch/record/1345343
* [38] Michelangelo L. Mangano “ALPGEN, a generator for hard multiparton processes in hadronic collisions” In _JHEP_ 07, 2003, pp. 001 DOI: 10.1088/1126-6708/2003/07/001
* [39] J. Alwall “Comparative study of various algorithms for the merging of parton showers and matrix elements in hadronic collisions” In _Eur. Phys. J._ C53, 2008, pp. 473–500 DOI: 10.1140/epjc/s10052-007-0490-5
* [40] M.. Mangano, M. Moretti, F. Piccinini and M. Treccani “Matching matrix elements and shower evolution for top-quark production in hadronic collisions” In _JHEP_ 01, 2007, pp. 013 DOI: 10.1088/1126-6708/2007/01/013
* [41] B.P. Kersevan and E. Richter-Was “The Monte Carlo event generator AcerMC versions 2.0 to 3.8 with interfaces to PYTHIA 6.4, HERWIG 6.5 and ARIADNE 4.1” In _Comput. Phys. Commun._ 184.3, 2013, pp. 919–985 DOI: 10.1016/j.cpc.2012.10.032
* [42] J.. Butterworth, J.. Forshaw and M.. Seymour “Multiparton interactions in photoproduction at HERA” In _Z. Phys. C_ 72, 1996, pp. 637 DOI: 10.1007/s002880050286
* [43] Charalampos Anastasiou, Lance Dixon, Kirill Melnikov and Frank Petriello “High-precision QCD at hadron colliders: Electroweak gauge boson rapidity distributions at next-to-next-to leading order” In _Phys. Rev. D_ 69, 2004, pp. 094008 DOI: 10.1103/PhysRevD.69.094008
* [44] John M. Campbell and R. Ellis “Update on vector boson pair production at hadron colliders” In _Phys. Rev. D_ 60, 1999, pp. 113006 DOI: 10.1103/PhysRevD.60.113006
* [45] Nikolaos Kidonakis “Differential and total cross section for top pair and single top production”, DESY-PROC-2012-02/251 DOI: 10.3204/DESY-PROC-2012-02/251
* [46] M. Aliev et al. “HATHOR – HAdronic Top and Heavy quarks crOss section calculatoR” In _Comput. Phys. Commun._ 182, 2011, pp. 1034–1046 DOI: 10.1016/j.cpc.2010.12.040
* [47] P. Kant et al. “HatHor for single top-quark production: Updated predictions and uncertainty estimates for single top-quark production in hadronic collisions” In _Comput. Phys. Commun._ 191, 2015, pp. 74–89 DOI: 10.1016/j.cpc.2015.02.001
* [48] T. Gleisberg et al. “Event generation with SHERPA 1.1” In _JHEP_ 02, 2009, pp. 007 DOI: 10.1088/1126-6708/2009/02/007
* [49] ATLAS Collaboration “Simultaneous measurements of the $t\bar{t}$, $W^{+}W^{-}$, and $Z/\gamma^{*}\rightarrow\tau\tau$ production cross-sections in $pp$ collisions at $\sqrt{s}=7\,\text{TeV}$ with the ATLAS detector” In _Phys. Rev. D_ 91, 2015, pp. 052005 DOI: 10.1103/PhysRevD.91.052005
* [50] ATLAS Collaboration “Comprehensive measurements of $t$-channel single top-quark production cross sections at $\sqrt{s}=7\,\text{TeV}$ with the ATLAS detector” In _Phys. Rev. D_ 90, 2014, pp. 112006 DOI: 10.1103/PhysRevD.90.112006
* [51] B.. Boser, I.. Guyon and V.. Vapnik “A Training Algorithm for Optimal Margin Classifiers” In _Proceedings of the Fifth Annual Workshop on Computational Learning Theory (COLT’92)_ , 1992, pp. 144–152 DOI: 10.1145/130385.130401
* [52] J. Mercer “Functions of Positive and Negative Type, and their Connection with Theory of Integral Equations” In _Philos. Trans. Roy. Soc. London Ser. A_ 209, 1909, pp. 415–446 DOI: 10.1098/rsta.1909.0016
* [53] N. Aronszajn “Theory of Reproducing Kernels” In _Trans. Amer. Math. Soc._ 68, 1950, pp. 337–404 DOI: 10.1090/S0002-9947-1950-0051437-7
* [54] C. Cortes and V. Vapnik “Support-vector networks” In _Machine Learning_ 20.3, 1995, pp. 273 DOI: 10.1007/BF00994018
* [55] J.C. Platt “Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines”, MSR-TR-98-14, 1998 URL: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.4376
* [56] S.S. Keerthi “Improvements to Platt’s SMO Algorithm for SVM Classifier Design” In _Neural Comp._ 13.3, 2001, pp. 637 DOI: 10.1162/089976601300014493
* [57] Michael Reed and Barry Simon “Methods of Modern Mathematical Physics. I: Functional Analysis” New York: Academic Press, 1972 DOI: 10.1016/B978-0-12-585001-8.X5001-6
* [58] G.. Fox and S. Wolfram “Event Shapes in $e^{+}e^{-}$ Annihilation” In _Nucl. Phys. B_ 149, 1979, pp. 413 DOI: 10.1016/0550-3213(79)90003-8
* [59] C. Bernaciak, M… Buschmann, A. Butter and T. Plehn “Fox-Wolfram Moments in Higgs Physics” In _Phys. Rev. D_ 87, 2013, pp. 073014 DOI: 10.1103/PhysRevD.87.073014
* [60] L.. Spiller “Modification of Fox-Wolfram Moments for Hadron Colliders” In _JHEP_ 03, 2016, pp. 027 DOI: 10.1007/JHEP03(2016)027
* [61] ATLAS Collaboration “Measurement of the top quark pair production cross section in $pp$ collisions at $\sqrt{s}=7\,\text{TeV}$ in dilepton final states with ATLAS” In _Phys. Lett. B_ 707, 2012, pp. 459 DOI: 10.1016/j.physletb.2011.12.055
* [62] E. Todesco and J. Wenninger “Large Hadron Collider momentum calibration and accuracy” In _Phys. Rev. Accel. Beams_ 20, 2017, pp. 081003 DOI: 10.1103/PhysRevAccelBeams.20.081003
* [63] ATLAS Collaboration “Measurement of lepton differential distributions and the top quark mass in $t\bar{t}$ production in $pp$ collisions at $\sqrt{s}=8\,\text{TeV}$ with the ATLAS detector” In _Eur. Phys. J. C_ 77, 2017, pp. 804 DOI: 10.1140/epjc/s10052-017-5349-9
* [64] The ATLAS, CDF, CMS and D0 Collaborations “First combination of Tevatron and LHC measurements of the top-quark mass”, ATLAS-CONF-2014-008, 2014 arXiv:1403.4427 [hep-ph]
* [65] ATLAS Collaboration “Measurement of the top quark mass in the $t\bar{t}\to$ lepton+jets channel from $\sqrt{s}=8\,\text{TeV}$ ATLAS data and combination with previous results” In _Eur. Phys. J. C_ 79, 2019, pp. 290 DOI: 10.1140/epjc/s10052-019-6757-9
* [66] ATLAS Collaboration In _Eur. Phys. J. C_ 76, 2016, pp. 642 DOI: 10.1140/epjc/s10052-016-4501-2
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.