Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

Nathan Halko

doi:10.1137/090771806

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

2011-01-01

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Nathan Halko *

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Per‐Gunnar Martinsson

J

Joel A. Tropp

DOI：10.1137/090771806

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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, robustness, and/or speed. 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 of an m x n matrix. (i) For a dense input matrix, randomized algorithms require O(mnlog(k)) floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

Keywords:

dimension reduction

eigenvalue decomposition

interpolative decomposition

Johnson-Lindenstrauss lemma

matrix approximation

parallel algorithm

pass-efficient algorithm

principal component analysis

randomized algorithm

random matrix

rank-revealing QR factorization

singular value decomposition

streaming algorithm

Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions

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