Sampling from large matrices: an approach through geometric functional analysis

TL;DR

This paper presents a method for approximating large matrices using small random submatrices, achieving optimal guarantees in low-rank approximation and improving bounds for related computational problems.

Contribution

It introduces a novel approach leveraging geometric functional analysis to efficiently sample and reconstruct matrices with minimal spectral norm error, optimizing sample size.

Findings

Approximate matrix A from a submatrix of size O(r log r) with small spectral norm error.

Provides asymptotically optimal estimates for spectral and cut-norms of random submatrices.

Improves sample complexity bounds for MAX-2CSP approximation algorithms.

Abstract

We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the numerical rank of A. The numerical rank is always bounded by, and is a stable relaxation of, the rank of A. This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of A. We also prove asymptotically optimal estimates on the spectral norm and the cut-norm of random submatrices of A. The result for the cut-norm yields a slight improvement on the best known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables.