How to reveal the rank of a matrix?

TL;DR

This paper analyzes algorithms for revealing the rank structure of matrices, establishing that local maximum volume pivoting in Gaussian elimination and QR is both necessary and sufficient, thus unifying and improving practical rank-revealing methods.

Contribution

It proves the importance of local maximum volume pivoting as a practical and optimal strategy for rank-revealers, unifying Gaussian elimination and QR approaches.

Findings

Local maximum volume pivoting is necessary and sufficient for rank-revealing.

A practical implementation of rank-revealing QR is at most twice as expensive as column-pivoted QR.

The success of pivoting strategies can be benchmarked against local maximum volume pivoting.

Abstract

We study algorithms called rank-revealers that reveal a matrix's rank structure. Such algorithms form a fundamental component in matrix compression, singular value estimation, and column subset selection problems. While column-pivoted QR has been widely adopted due to its practicality, it is not always a rank-revealer. Conversely, Gaussian elimination (GE) with a pivoting strategy known as global maximum volume pivoting is guaranteed to estimate a matrix's singular values but its exponential complexity limits its interest to theory. We show that the concept of local maximum volume pivoting is a crucial and practical pivoting strategy for rank-revealers based on GE and QR. In particular, we prove that it is both necessary and sufficient; highlighting that all local solutions are nearly as good as the global one. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and we implement a version that is observed to be at most $2\times$ more computationally expensive than CPQR. We unify the landscape of rank-revealers by considering GE and QR together and prove that the success of any pivoting strategy can be assessed by benchmarking it against a local maximum volume pivot.