Michael W. Berry, Shakhina A. Pulatova, G. W. Stewart
In many applications---latent semantic indexing, for example---it is required to obtain a reduced rank approximation to a sparse matrix A. Unfortunately, the approximations based on traditional decompositions, like the singular value and QR decompositions, are not in general sparse. Stewart [(1999), 313--323] has shown how to use a variant of the classical Gram--Schmidt algorithm, called the quasi--Gram-Schmidt--algorithm, to obtain two kinds of low-rank approximations. The first, the SPQR, approximation, is a pivoted, Q-less QR approximation of the form (XR11−1)(R11 R12), where X consists of columns of A. The second, the SCR approximation, is of the form the form A ≅ XTYT, where X and Y consist of columns and rows A and T, is small. In this article we treat the computational details of these algorithms and describe a MATLAB implementation.
ArticleIn many applications---latent semantic indexing, for example---it is required to obtain a reduced rank approximation to a sparse matrix A. Unfortunately, the approximations based on traditional decompositions, like the singular value and QR decompositions, are not in general sparse. Stewart [(1999), 313--323] has shown how to use a variant of the classical Gram--Schmidt algorithm, called the quasi--Gram-Schmidt--algorithm, to obtain two kinds of low-rank approximations. The first, the SPQR, approximation, is a pivoted, Q-less QR approximation of the form (XR11−1)(R11 R12), where X consists of columns of A. The second, the SCR approximation, is of the form the form A ≅ XTYT, where X and Y consist of columns and rows A and T, is small. In this article we treat the computational details of these algorithms and describe a MATLAB implementation.
Michael W. Berry, Shakhina A. Pulatova, G. W. Stewart, et al. "Algorithm 844: Computing sparse reduced-rank approximations to sparse matrices." Journal ACM Transactions on Mathematical Software (TOMS). doi:10.1145/1067967.1067972. Retrieved 07/31/2021 from researchcompendia.org/compendia/2013.6/
Compendium Type: Published Papers Primary Research Field: Computer and Information Sciences Secondary Research Field: Mathematics Content License: Public Domain Mark Code License: MIT License
created 12/12/2013
modified 01/16/2014
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