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Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix (2004)
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| Venue: | SIAM JOURNAL ON COMPUTING |
| Citations: | 213 - 20 self |
BibTeX
@ARTICLE{Drineas04fastmonte,
author = {Petros Drineas and Ravi Kannan and Michael W. Mahoney},
title = {Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix},
journal = {SIAM JOURNAL ON COMPUTING},
year = {2004},
volume = {36},
pages = {2006}
}
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Abstract
... matrix A. It is often of interest to find a low-rank approximation to A, i.e., an approximation D to the matrix A of rank not greater than a specified rank k, where k is much smaller than m and n. Methods such as the Singular Value Decomposition (SVD) may be used to find an approximation to A which is the best in a well defined sense. These methods require memory and time which are superlinear in m and n; for many applications in which the data sets are very large this is prohibitive. Two simple and intuitive algorithms are presented which, when given an m n matrix A, compute a description of a low-rank approximation D to A, and which are qualitatively faster than the SVD. Both algorithms have provable bounds for the error matrix A D . For any matrix X , let kXk and kXk 2 denote its Frobenius norm and its spectral norm, respectively. In the rst algorithm, c = O(1) columns of A are randomly chosen. If the m c matrix C consists of those c columns of A (after appropriate rescaling) then it is shown that from C C approximations to the top singular values and corresponding singular vectors may be computed. From the computed singular vectors a description D of the matrix A may be computed such that rank(D ) k and such that holds with high probability for both = 2; F . This algorithm may be implemented without storing the matrix A in Random Access Memory (RAM), provided it can make two passes over the matrix stored in external memory and use O(m + n) additional RAM memory. The second algorithm is similar except that it further approximates the matrix C by randomly sampling r = O(1) rows of C to form a r c matrix W . Thus, it has additional error, but it can be implemented in three passes over the matrix using only constant ...
Keyphrases
low-rank approximation fast monte carlo algorithm matrix ii singular vector provable bound external memory appropriate rescaling singular value decomposition frobenius norm second algorithm data set additional error spectral norm top singular value additional ram memory specified rank high probability random access memory intuitive algorithm many application rst algorithm
