A randomized algorithm for the decomposition of matrices

Given an m × n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient when...

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Vydáno v:	Applied and computational harmonic analysis Ročník 30; číslo 1; s. 47 - 68
Hlavní autoři:	Martinsson, Per-Gunnar, Rokhlin, Vladimir, Tygert, Mark
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Elsevier Inc 2011
Témata:	Algorithm Algorithms Approximation Decomposition Estimates Integers Lanczos Low rank Mathematical analysis Matrix Randomized SVD Vectors (mathematics) SVD Lanczos Randomized Matrix Algorithm Low rank
ISSN:	1063-5203, 1096-603X
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Abstract	Given an m × n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and A T can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as l m times the ( k + 1 ) st greatest singular value σ k + 1 of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when l − k is a fixed small nonnegative integer. For example, according to one of our estimates for l − k = 20 , the probability that the spectral norm ‖ A − Z ‖ is greater than 10 ( k + 20 ) m σ k + 1 is less than 10 − 17 . The paper contains a number of estimates for ‖ A − Z ‖ , including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and A T can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.
AbstractList	Given an m × n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and A T can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as l m times the ( k + 1 ) st greatest singular value σ k + 1 of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when l − k is a fixed small nonnegative integer. For example, according to one of our estimates for l − k = 20 , the probability that the spectral norm ‖ A − Z ‖ is greater than 10 ( k + 20 ) m σ k + 1 is less than 10 − 17 . The paper contains a number of estimates for ‖ A − Z ‖ , including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and A T can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples. Given an mxn matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A super(T) to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and A super(T) can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as [inline image] times the (k+1)st greatest singular value [sigma] sub()k1of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when l-k is a fixed small nonnegative integer. For example, according to one of our estimates for l-k=20, the probability that the spectral norm [inline image]A-Z[inline image] is greater than [inline image] is less than 10 super(-17). The paper contains a number of estimates for [inline image]A-Z[inline image], including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and A super(T) can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.
Author	Rokhlin, Vladimir Tygert, Mark Martinsson, Per-Gunnar
Author_xml	– sequence: 1 givenname: Per-Gunnar surname: Martinsson fullname: Martinsson, Per-Gunnar email: per-gunnar.martinsson@colorado.edu organization: Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States – sequence: 2 givenname: Vladimir surname: Rokhlin fullname: Rokhlin, Vladimir organization: Departments of Computer Science, Mathematics, and Physics, Yale University, New Haven, CT 06511, United States – sequence: 3 givenname: Mark surname: Tygert fullname: Tygert, Mark email: tygert@aya.yale.edu organization: Courant Institute of Mathematical Sciences, NYU, New York, NY 10012, United States
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Snippet	Given an m × n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies... Given an mxn matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies...
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SubjectTerms	Algorithm Algorithms Approximation Decomposition Estimates Integers Lanczos Low rank Mathematical analysis Matrix Randomized SVD Vectors (mathematics)
Title	A randomized algorithm for the decomposition of matrices
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