Randomized Householder QR
Laura Grigori, Edouard Timsit

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 is equivalent to the standard Householder QR factorization of matrix , where 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…
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Taxonomy
TopicsBig Data and Business Intelligence
