Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization

TL;DR

This paper introduces a randomized polynomial-time algorithm for column subset selection that leverages eigenvalue minimization and matrix factorization, connecting functional analysis results with practical spectral matrix approximation.

Contribution

It presents the first efficient algorithm to find well-conditioned column submatrices as guaranteed by Bourgain and Tzafriri, using eigenvalue minimization and Grothendieck factorization.

Findings

Algorithm produces large well-conditioned column submatrices.

New approximation method for the NP-hard (∞,1)-norm.

Reveals a connection between matrix factorization and Max-Cut SDP.

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.