Collect, Commit, Expand: Efficient CPQR-Based Column Selection for Extremely Wide Matrices

TL;DR

This paper introduces CCEQR, an efficient, deterministic modification of the CPQR algorithm for selecting columns from extremely wide matrices, improving performance in applications like spectral clustering and quantum chemistry.

Contribution

The paper presents CCEQR, a novel three-step algorithm that enhances efficiency and maintains accuracy in CPQR-based column selection for large, structured matrices.

Findings

CCEQR outperforms GEQP3 on structured problems

CCEQR reduces computational effort by limiting column manipulations

CCEQR is deterministic and provably equivalent to Golub-Businger CPQR

Abstract

Column-pivoted QR (CPQR) factorization is a computational primitive used in numerous applications that require selecting a small set of ``representative'' columns from a much larger matrix. These include applications in spectral clustering, model-order reduction, low-rank approximation, and computational quantum chemistry, where the matrix being factorized has a moderate number of rows but an extremely large number of columns. We describe a modification of the Golub-Businger algorithm which, for many matrices of this type, can perform CPQR-based column selection much more efficiently. This algorithm, which we call CCEQR, is based on a three-step ``collect, commit, expand'' strategy that limits the number of columns being manipulated, while also transferring more computational effort from level-2 BLAS to level-3. Unlike most CPQR algorithms that exploit level-3 BLAS, CCEQR is deterministic, and provably recovers a column permutation equivalent to the one computed by the Golub-Businger algorithm. Tests on spectral clustering and Wannier basis localization problems demonstrate that on appropriately structured problems, CCEQR can significantly outperform GEQP3.