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An Improved Approximation Algorithm for the Column Subset Selection Problem
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BibTeX
@MISC{Boutsidis_animproved,
author = {Christos Boutsidis and Michael W. Mahoney and Petros Drineas},
title = {An Improved Approximation Algorithm for the Column Subset Selection Problem},
year = {}
}
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Abstract
We consider the problem of selecting the “best ” subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn 2, m 2 n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-chosen probability distribution that depends on information in the topk right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let PC denote the projection matrix onto the span of those columns, and let Ak denote the “best ” rank-k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that ‖A − PCA‖2 ≤ O k 3 4 log 1
Keyphrases
column subset selection problem improved approximation algorithm first stage singular value decomposition rank-k approximation novel two-stage algorithm judiciously-chosen probability distribution deterministic column-selection procedure min mn deterministic stage singular subspace second stage projection matrix
