Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Randomized estimation of spectral densities of large matrices made
  accurate   For a large Hermitian matrix $A\in \mathbb{C}^{N\times N}$, it is often the
case that the only affordable operation is matrix-vector multiplication. In
such case, randomized method is a powerful way to estimate the spectral density
(or density of states) of $A$. However, randomized methods developed so far for
estimating spectral densities only extract information from different random
vectors independently, and the accuracy is therefore inherently limited to
$\mathcal{O}(1/\sqrt{N_{v}})$ where $N_{v}$ is the number of random vectors. In
this paper we demonstrate that the "$\mathcal{O}(1/\sqrt{N_{v}})$ barrier" can
be overcome by taking advantage of the correlated information of random vectors
when properly filtered by polynomials of $A$. Our method uses the fact that the
estimation of the spectral density essentially requires the computation of the
trace of a series of matrix functions that are numerically low rank. By
repeatedly applying $A$ to the same set of random vectors and taking different
linear combination of the results, we can sweep through the entire spectrum of
$A$ by building such low rank decomposition at different parts of the spectrum.
Under some assumptions, we demonstrate that a robust and efficient
implementation of such spectrum sweeping method can compute the spectral
density accurately with $\mathcal{O}(N^2)$ computational cost and
$\mathcal{O}(N)$ memory cost. Numerical results indicate that the new method
can significantly outperform existing randomized methods in terms of accuracy.
As an application, we demonstrate a way to accurately compute a trace of a
smooth matrix function, by carefully balancing the smoothness of the integrand
and the regularized density of states using a deconvolution procedure.

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The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs