This is an announcement for the paper "Column subset selection, matrix factorization, and eigenvalue optimization" by Joel A. Tropp.

Abstract: 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 {\sf 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 $(\infty, 1)$ norm of a matrix, which is generally {\sf NP}-hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous {\sc maxcut} semidefinite program.

Archive classification: math.NA math.FA

Mathematics Subject Classification: 15A60; 15A23; 65F30; 90C25

Remarks: Conference version

The source file(s), alg.sty: 7607 bytes macro-file.tex: 8456 bytes subset-selection-soda-v4.bbl: 4536 bytes subset-selection-soda-v4.tex: 88398 bytes, is(are) stored in gzipped %form as 0806.4404.tar.gz with size 30kb. The corresponding postcript file has gzipped size 109kb.

Submitted from: jtropp@acm.caltech.edu

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http://front.math.ucdavis.edu/0806.4404

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http://arXiv.org/abs/0806.4404

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