URV Factorization with Random Orthogonal System Mixing

TL;DR

This paper introduces a randomized URV factorization method that uses orthogonal system mixing to reduce communication costs and improve the accuracy of unpivoted QR factorizations in numerical linear algebra.

Contribution

The paper proposes a novel randomized URV factorization technique using orthogonal system mixing to enhance efficiency and accuracy over traditional pivoted QR methods.

Findings

Reduces communication bottleneck in QR factorizations.

Achieves comparable accuracy to rank-revealing factorizations.

Potentially rank-revealing with high probability.

Abstract

The unpivoted and pivoted Householder QR factorizations are ubiquitous in numerical linear algebra. A difficulty with pivoted Householder QR is the communication bottleneck introduced by pivoting. In this paper we propose using random orthogonal systems to quickly mix together the columns of a matrix before computing an unpivoted QR factorization. This method computes a URV factorization which forgoes expensive pivoted QR steps in exchange for mixing in advance, followed by a cheaper, unpivoted QR factorization. The mixing step typically reduces the variability of the column norms, and in certain experiments, allows us to compute an accurate factorization where a plain, unpivoted QR performs poorly. We experiment with linear least-squares, rank-revealing factorizations, and the QLP approximation, and conclude that our randomized URV factorization behaves comparably to a similar randomized rank-revealing URV factorization, but at a fraction of the computational cost. Our experiments provide evidence that our proposed factorization might be rank-revealing with high probability.