Sampling from large matrices: an approach through geometric functional analysis
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.
Cite
@article{arxiv.math/0503442,
title = {Sampling from large matrices: an approach through geometric functional analysis},
author = {Mark Rudelson and Roman Vershynin},
journal= {arXiv preprint arXiv:math/0503442},
year = {2016}
}
Comments
Our initial claim about Max-2-CSP problems is corrected. We put an exponential failure probability for the algorithm for low-rank approximations. Proofs are a little more explained