A Schur Method for Low-Rank Matrix Approximation

The usual way to compute a low-rank approximant of a matrix $H$ is to take its singular value decomposition (SVD) and truncate it by setting the small singular values equal to 0. However, the SVD is computationally expensive. This paper describes a much simpler generalized Schur-type algorithm to co...

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Bibliographic Details
Published inSIAM journal on matrix analysis and applications Vol. 17; no. 1; pp. 139 - 160
Main Author van der Veen, Alle-Jan
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.1996
Subjects
Algorithms
Approximation
Decomposition
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Summary:The usual way to compute a low-rank approximant of a matrix $H$ is to take its singular value decomposition (SVD) and truncate it by setting the small singular values equal to 0. However, the SVD is computationally expensive. This paper describes a much simpler generalized Schur-type algorithm to compute similar low-rank approximants. For a given matrix $H$ which has $d$ singular values larger than $\epsilon $, we find all rank $d$ approximants $\hat H$ such that $H - \hat H$ has 2-norm less than $\epsilon $. The set of approximants includes the truncated SVD approximation. The advantages of the Schur algorithm are that it has a much lower computational complexity (similar to a QR factorization), and directly produces a description of the column space of the approximants. This column space can be updated and downdated in an on-line scheme, amenable to implementation on a parallel array of processors.
ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479893261340