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Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization

Tropp, Joel A. (2008) Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization. ACM Technical Reports, 2008-02. California Institute of Technology , Pasadena, CA. (Unpublished)

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Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing QR, which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri. The method involves random sampling of columns, followed by a matrix factorization that exposes the well-conditioned subset of columns. This factorization, which is due to Grothendieck, is regarded as a central tool in modern functional analysis. The primary novelty in this work is an algorithm, based on eigenvalue minimization, for constructing the Grothendieck factorization. These ideas also result in a novel approximation algorithm for the (∞, 1) norm of a matrix, which is generally NP-hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous MAXCUT semidefinite program.

Item Type:Report or Paper (Technical Report)
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URLURL TypeDescription ItemPublished Version
Tropp, Joel A.0000-0003-1024-1791
Additional Information:Date: 26 June 2008. Supported in part by ONR award no. N000140810883. The author thanks Ben Recht for helpful discussions about eigenvalue minimization.
Group:Applied & Computational Mathematics
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-08-1-0883
Series Name:ACM Technical Reports
Issue or Number:2008-02
Record Number:CaltechAUTHORS:20111011-161421093
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:27170
Deposited On:19 Oct 2011 18:09
Last Modified:03 Oct 2019 03:21

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