TY - JOUR

T1 - Matrix decompositions using sub-Gaussian random matrices

AU - Aizenbud, Yariv

AU - Averbuch, Amir

N1 - Publisher Copyright:
© The Author(s) 2018.

PY - 2019/9/19

Y1 - 2019/9/19

N2 - In recent years, several algorithms which approximate matrix decomposition have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We present a new algorithm, which achieves with high probability a rank-r singular value decomposition (SVD) approximation of an n × n matrix and derive an error bound that does not depend on the first r singular values. Although the algorithm has an asymptotic complexity similar to state-of-the-art algorithms and the proven error bound is not as tight as the state-of-the-art bound, experiments show that the proposed algorithm is faster in practice while providing the same error rates as those of the stateof- the-art algorithms. We also show that an i.i.d. sub-Gaussian matrix with large probability of having null entries is metric conserving. This result is used in the SVD approximation algorithm, as well as to improve the performance of a previously proposed approximated LU decomposition algorithm.

AB - In recent years, several algorithms which approximate matrix decomposition have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We present a new algorithm, which achieves with high probability a rank-r singular value decomposition (SVD) approximation of an n × n matrix and derive an error bound that does not depend on the first r singular values. Although the algorithm has an asymptotic complexity similar to state-of-the-art algorithms and the proven error bound is not as tight as the state-of-the-art bound, experiments show that the proposed algorithm is faster in practice while providing the same error rates as those of the stateof- the-art algorithms. We also show that an i.i.d. sub-Gaussian matrix with large probability of having null entries is metric conserving. This result is used in the SVD approximation algorithm, as well as to improve the performance of a previously proposed approximated LU decomposition algorithm.

KW - Johnson-Lindenstrauss Lemma

KW - LU decomposition

KW - Low-rank approximation

KW - Oblivious subspace embedding

KW - Random matrices

KW - SVD

KW - Sparse matrices

KW - Sub-Gaussian matrices

UR - http://www.scopus.com/inward/record.url?scp=85070872228&partnerID=8YFLogxK

U2 - 10.1093/imaiai/iay017

DO - 10.1093/imaiai/iay017

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AN - SCOPUS:85070872228

SN - 2049-8772

VL - 8

SP - 445

EP - 469

JO - Information and Inference

JF - Information and Inference

IS - 3

ER -