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arXiv · 2602.16797

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

Ethan N. Epperly·Taejun Park·Yuji Nakatsukasa
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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.

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Ethan N. Epperly, Taejun Park, Yuji Nakatsukasa. 2026-02-18. Fast, High-Accuracy, Randomized Nullspace Computations for Tall Matrices. https://arxiv.org/abs/2602.16797

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