Householder QR Factorization with Randomization for Column Pivoting (HQRRP). FLAME Working Note #78

TL;DR

This paper introduces a randomized pivoting technique for Householder QR factorization that achieves classical flop counts, accelerates computation significantly, and maintains pivot quality, enabling efficient, high-performance matrix factorizations.

Contribution

It presents a novel randomized pivot selection method for Householder QR that matches classical flop counts and improves computational speed while preserving pivot quality.

Findings

Achieves asymptotic flop count identical to classical algorithms.

Provides near tenfold speedup on modern multi-core CPUs.

Maintains comparable pivot quality to traditional methods.

Abstract

A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix-matrix multiplications, which greatly limits the benefits that can be derived from so-called blocking of algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic flop count for the proposed method is $2mn^2 - (2/3)n^3$ for an $m\times n$ matrix, identical to that of the best classical unblocked Householder QR factorization algorithm (with or without pivoting). Experiments demonstrate acceleration in speed of close to an order of magnitude relative to the {\sc geqp3} function in LAPACK, when executed on a modern CPU with multiple cores. Further, experiments demonstrate that the quality of the randomized pivot selection strategy is roughly the same as that of classical column pivoting. The described algorithm is made available under Open Source license and can be used with LAPACK or libflame.