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.
Description
Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions
%0 Generic
%1 halko2009finding
%A Halko, Nathan
%A Martinsson, Per-Gunnar
%A Tropp, Joel A.
%D 2009
%K linear-algebra multivariate statistics
%T Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions
%U http://arxiv.org/abs/0909.4061
%X 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.
@misc{halko2009finding,
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.},
added-at = {2016-05-21T23:32:30.000+0200},
author = {Halko, Nathan and Martinsson, Per-Gunnar and Tropp, Joel A.},
biburl = {https://www.bibsonomy.org/bibtex/2a6bb9689df557369a6c8a864c871e500/shabbychef},
description = {Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions},
interhash = {f06397caa74820f92e0edbaf01ece2d3},
intrahash = {a6bb9689df557369a6c8a864c871e500},
keywords = {linear-algebra multivariate statistics},
note = {cite arxiv:0909.4061},
timestamp = {2016-05-21T23:32:30.000+0200},
title = {Finding structure with randomness: Probabilistic algorithms for
constructing approximate matrix decompositions},
url = {http://arxiv.org/abs/0909.4061},
year = 2009
}