Sublinear Randomized Algorithms for Skeleton Decompositions
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Chiu-2013-SUBLINEAR RANDOMIZED.pdf
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Author(s)
Chiu, Jiawei
Demanet, Laurent
Date Issued
September 2013
Journal
SIAM Journal on Matrix Analysis and Applications
Publisher
Society for Industrial and Applied Mathematics
Citation
Chiu, Jiawei, and Laurent Demanet. “Sublinear Randomized Algorithms for Skeleton Decompositions.” SIAM Journal on Matrix Analysis and Applications 34, no. 3 (July 9, 2013): 1361-1383. © 2013, Society for Industrial and Applied Mathematics
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Final published version
Abstract
A skeleton decomposition of a matrix A is any factorization of the form A[subscript :C]ZA[subscript R:], where A[subscript :C] comprises columns of A, and A[subscript R:] comprises rows of A. In this paper, we investigate the conditions under which random sampling of C and R results in accurate skeleton decompositions. When the singular vectors (or more generally the generating vectors) are incoherent, we show that a simple algorithm returns an accurate skeleton in sublinear O(ℓ[superscript 3]) time from ℓ ~ k log n rows and columns drawn uniformly at random, with an approximation error of the form O([n over ℓ]σ[subscript k]) whereσ[subscript k] is the kth singular value of A. We discuss the crucial role that regularization plays in forming the middle matrix U as a pseudoinverse of the restriction A[subscript RC] of A to rows in R and columns in C. The proof methods enable the analysis of two alternative sublinear-time algorithms, based on the rank-revealing QR decomposition, which allows us to tighten the number of rows and/or columns to k with error bound proportional to σ[subscript k].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1137/110852310
