Randomized QR with Column Pivoting

TL;DR

This paper introduces a randomized QR with column pivoting method that reduces communication and computational costs, enabling faster low-rank matrix approximations with strong parallel scalability.

Contribution

It proposes a novel randomized sampling approach for QR with column pivoting that minimizes communication and extends to efficient low-rank truncated approximations.

Findings

Achieves comparable pivoting quality to QRCP with near unpivoted QR performance.

Reduces approximation time by nearly half for small truncation ranks.

Demonstrates strong parallel scalability on multi-core systems.

Abstract

The dominant contribution to communication complexity in factorizing a matrix using QR with column pivoting is due to column-norm updates that are required to process pivot decisions. We use randomized sampling to approximate this process which dramatically reduces communication in column selection. We also introduce a sample update formula to reduce the cost of sampling trailing matrices. Using our column selection mechanism we observe results that are comparable in quality to those obtained from the QRCP algorithm, but with performance near unpivoted QR. We also demonstrate strong parallel scalability on shared memory multiple core systems using an implementation in Fortran with OpenMP. This work immediately extends to produce low-rank truncated approximations of large matrices. We propose a truncated QR factorization with column pivoting that avoids trailing matrix updates which are used in current implementations of level-3 BLAS QR and QRCP. Provided the truncation rank is small, avoiding trailing matrix updates reduces approximation time by nearly half. By using these techniques and employing a variation on Stewart's QLP algorithm, we develop an approximate truncated SVD that runs nearly as fast as truncated QR.