Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and
  Hardness   Understanding the singular value spectrum of a matrix $A \in \mathbb{R}^{n
\times n}$ is a fundamental task in countless applications. In matrix
multiplication time, it is possible to perform a full SVD and directly compute
the singular values $\sigma_1,...,\sigma_n$. However, little is known about
algorithms that break this runtime barrier.
  Using tools from stochastic trace estimation, polynomial approximation, and
fast system solvers, we show how to efficiently isolate different ranges of
$A$'s spectrum and approximate the number of singular values in these ranges.
We thus effectively compute a histogram of the spectrum, which can stand in for
the true singular values in many applications.
  We use this primitive to give the first algorithms for approximating a wide
class of symmetric matrix norms in faster than matrix multiplication time. For
example, we give a $(1 + \epsilon)$ approximation algorithm for the
Schatten-$1$ norm (the nuclear norm) running in just $\tilde O((nnz(A)n^{1/3} +
n^2)\epsilon^{-3})$ time for $A$ with uniform row sparsity or $\tilde
O(n^{2.18} \epsilon^{-3})$ time for dense matrices. The runtime scales smoothly
for general Schatten-$p$ norms, notably becoming $\tilde O (p \cdot nnz(A)
\epsilon^{-3})$ for any $p \ge 2$.
  At the same time, we show that the complexity of spectrum approximation is
inherently tied to fast matrix multiplication in the small $\epsilon$ regime.
We prove that achieving milder $\epsilon$ dependencies in our algorithms would
imply faster than matrix multiplication time triangle detection for general
graphs. This further implies that highly accurate algorithms running in
subcubic time yield subcubic time matrix multiplication. As an application of
our bounds, we show that precisely computing all effective resistances in a
graph in less than matrix multiplication time is likely difficult, barring a
major algorithmic breakthrough.

30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
9
The categories of this paper are: nlin