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FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS
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@MISC{Halko_findingstructure,
author = {N. Halko and P. G. Martinsson and J. A. Tropp},
title = {FINDING STRUCTURE WITH RANDOMNESS: PROBABILISTIC ALGORITHMS FOR CONSTRUCTING APPROXIMATE MATRIX DECOMPOSITIONS},
year = {}
}
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Abstract
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition
Keyphrases
finding structure randomness low-rank matrix approximation singular value decomposition modern computational architecture modular framework powerful tool reduced matrix extensive numerical experiment central role desired low-rank factorization many case partial matrix decomposition input matrix scientific computing data analysis dominant component rank-revealing qr decomposition model problem specific benefit computational environment recent research classical competitor truncated singular value decomposition truly massive data set work survey classical method detailed error analysis
