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Sampling from large matrices: an approach through geometric functional analysis (2006)
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| Venue: | Journal of the ACM |
| Citations: | 131 - 5 self |
BibTeX
@ARTICLE{Rudelson06samplingfrom,
author = {Mark Rudelson and Roman Vershynin},
title = {Sampling from large matrices: an approach through geometric functional analysis},
journal = {Journal of the ACM},
year = {2006}
}
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Abstract
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�2 F /�A�22 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. 1.
Keyphrases
large matrix geometric functional analysis numerical rank spectral norm random submatrices possible size random submatrix slight improvement small error optimal guarantee optimal estimate large number known sample complexity operator-valued random variable approximation algorithm cut-norm yield banach space max-2csp problem low-rank approximation stable relaxation
