Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase Retrieval From the Magnitudes of Affine Linear Measurements   In this paper, we consider the phase retrieval problem in which one aims to
recover a signal from the magnitudes of affine measurements. Let $\{{\mathbf
a}_j\}_{j=1}^m \subset {\mathbb H}^d$ and ${\mathbf b}=(b_1, \ldots,
b_m)^\top\in{\mathbb H}^m$, where ${\mathbb H}={\mathbb R}$ or ${\mathbb C}$.
We say $\{{\mathbf a}_j\}_{j=1}^m$ and $\mathbf b$ are affine phase retrievable
for ${\mathbb H}^d$ if any ${\mathbf x}\in{\mathbb H}^d$ can be recovered from
the magnitudes of the affine measurements $\{|<{\mathbf a}_j,{\mathbf
x}>+b_j|,\, 1\leq j\leq m\}$. We develop general framework for affine phase
retrieval and prove necessary and sufficient conditions for $\{{\mathbf
a}_j\}_{j=1}^m$ and $\mathbf b$ to be affine phase retrievable. We establish
results on minimal measurements and generic measurements for affine phase
retrieval as well as on sparse affine phase retrieval. In particular, we also
highlight some notable differences between affine phase retrieval and the
standard phase retrieval in which one aims to recover a signal $\mathbf x$ from
the magnitudes of its linear measurements. In standard phase retrieval, one can
only recover $\mathbf x$ up to a unimodular constant, while affine phase
retrieval removes this ambiguity. We prove that unlike standard phase
retrieval, the affine phase retrievable measurements $\{{\mathbf
a}_j\}_{j=1}^m$ and $\mathbf b$ do not form an open set in ${\mathbb
H}^{m\times d}\times {\mathbb H}^m$. Also in the complex setting, the standard
phase retrieval requires $4d-O(\log_2d)$ measurements, while the affine phase
retrieval only needs $m=3d$ measurements.

22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ