Fast, High-Accuracy, Randomized Nullspace Computations for Tall Matrices

TL;DR

This paper introduces RLOBPCG, a fast and accurate randomized algorithm for computing small singular triplets of large tall matrices, combining sketch-based preconditioning with eigensolvers for improved efficiency.

Contribution

The paper presents RLOBPCG, a novel randomized method that accelerates singular value computations for large matrices by integrating sketching and preconditioning with eigensolvers.

Findings

RLOBPCG converges geometrically under standard assumptions.

Achieves near-optimal accuracy on matrices with up to 10^6 rows.

Outperforms classical methods with up to 12x speedup and better robustness.

Abstract

In this paper, we develop RLOBPCG, an efficient method for computing a small number of singular triplets corresponding to the smallest singular values of large, tall matrices. The algorithm combines randomized preconditioner from the sketch-and-precondition techniques with the LOBPCG eigensolver: a small sketch is used to construct a high-quality preconditioner, and LOBPCG is run on the Gram matrix to refine the singular vector. Under the standard subspace embedding assumption and a modest singular value gap between the two smallest singular values, we prove that RLOBPCG converges geometrically to the minimum singular vector. In numerical experiments, RLOBPCG achieves near-optimal accuracy on matrices with up to $10^6$ rows, outperforming classical LOBPCG and Lanczos methods by a speedup of up to $12\times$ and maintaining robustness when other iterative methods fail to converge.