Tropp, Joel A.
Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization.
California Institute of Technology
, Pasadena, CA.
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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
|Item Type:||Report or Paper (Technical Report)|
|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 Agency||Grant Number|
|Other Numbering System:|
|Other Numbering System Name||Other Numbering System ID|
|Applied & Computational Mathematics Technical Report||2008-02|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited On:||19 Oct 2011 18:09|
|Last Modified:||06 Mar 2015 23:10|
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