A randomized preconditioned Cholesky-QR algorithm

TL;DR

This paper introduces rpCholesky-QR, a randomized preconditioned algorithm for efficient and accurate QR factorization of matrices, especially effective for highly singular matrices, with strong theoretical guarantees and empirical validation.

Contribution

The paper presents rpCholesky-QR, a novel randomized preconditioned method that achieves low orthogonalization error and robust performance for singular matrices, with rigorous perturbation bounds.

Findings

Achieves residuals on the order of machine precision.

Remains stable for highly singular matrices.

Performance improves with fewer preconditioner rows.

Abstract

We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of machine precision, and does not break down for highly singular matrices. We derive rigorous and interpretable two-norm perturbation bounds for rpCholesky-QR that require a minimum of assumptions. Numerical experiments corroborate the accuracy of rpCholesky-QR for preconditioners sampled from as few as 3n rows, and illustrate that the two-norm deviation from orthonormality increases with only the condition number of the preconditioned matrix, rather than its square -- even if the original matrix is numerically singular.