Randomized Householder QR

TL;DR

This paper presents a randomized Householder QR factorization method that efficiently produces well-conditioned bases with stability guarantees, suitable for large-scale applications and Krylov subspace methods.

Contribution

It introduces the RHQR method, combining randomized sketching with Householder QR, and analyzes its stability, efficiency, and applicability in Krylov subspace algorithms.

Findings

RHQR yields well-conditioned, numerically orthogonal bases.

The method is computationally efficient, requiring half the cost of traditional Householder QR.

Numerical experiments confirm stability and accuracy even in high-dimensional, low-precision settings.

Abstract

This paper introduces a randomized Householder QR factorization (RHQR). This factorization can be used to obtain a well conditioned basis of a vector space and thus can be employed in a variety of applications. The RHQR factorization of the input matrix $W$ is equivalent to the standard Householder QR factorization of matrix $\Psi W$, where $\Psi$ is a sketching matrix that can be obtained from any subspace embedding technique. For this reason, the RHQR factorization can also be reconstructed from the Householder QR factorization of the sketched problem, yielding a single-synchronization randomized QR factorization (recRHQR). In most contexts, left-looking RHQR requires a single synchronization per iteration, with half the computational cost of Householder QR, and a similar cost to Randomized Gram-Schmidt (RGS) overall. We discuss the usage of RHQR factorization in the Arnoldi process and then in GMRES, showing thus how it can be used in Krylov subspace methods to solve systems of linear equations. Based on Charles Sheffield's connection between Householder QR and Modified Gram-Schmidt (MGS), a BLAS2-RGS is also derived. A finite precision analysis shows that, under mild probabilistic assumptions, the RHQR factorization of the input matrix $W$ inherits the stability of the Householder QR factorization, producing a well-conditioned basis and a columnwise backward stable factorization, all independently of the condition number of the input $W$, and with the accuracy of the sketching step. We study the subsampled randomized Hadamard transform (SRHT) as a very stable sketching technique. Numerical experiments show that RHQR produces a well conditioned basis whose sketch is numerically orthogonal and an accurate factorization, even for the most difficult inputs and with high-dimensional operations made in half-precision.