%0 Generic %1 halko2009finding %A Halko, Nathan %A Martinsson, Per-Gunnar %A Tropp, Joel A. %D 2009 %K 15b52-random-matrices-algebraic-aspects 60b20-random-matrices-probabilistic-aspects 62-07-data-analysis 65f20-overdetermined-systems-pseudoinverses 65f30-other-matrix-algorithms 65f99-numerical-linear-algebra-none-of-the-above 65y05-parallel-computation 68w20-randomized-algorithms 68w30-symbolic-computation-algebraic-computation %R 10.1137/090771806 %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 = {2021-01-22T01:28:58.000+0100}, author = {Halko, Nathan and Martinsson, Per-Gunnar and Tropp, Joel A.}, biburl = {https://www.bibsonomy.org/bibtex/2a6bb9689df557369a6c8a864c871e500/gdmcbain}, doi = {10.1137/090771806}, howpublished = {arXiv.org > math > arXiv:0909.4061}, interhash = {f06397caa74820f92e0edbaf01ece2d3}, intrahash = {a6bb9689df557369a6c8a864c871e500}, keywords = {15b52-random-matrices-algebraic-aspects 60b20-random-matrices-probabilistic-aspects 62-07-data-analysis 65f20-overdetermined-systems-pseudoinverses 65f30-other-matrix-algorithms 65f99-numerical-linear-algebra-none-of-the-above 65y05-parallel-computation 68w20-randomized-algorithms 68w30-symbolic-computation-algebraic-computation}, note = {cite arxiv:0909.4061}, timestamp = {2021-01-22T01:28:58.000+0100}, title = {Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions}, url = {http://arxiv.org/abs/0909.4061}, year = 2009 }