Fast, High-Accuracy, Randomized Nullspace Computations for Tall Matrices
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912912737566720 |
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| author | Epperly, Ethan N. Park, Taejun Nakatsukasa, Yuji |
| author_facet | Epperly, Ethan N. Park, Taejun Nakatsukasa, Yuji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast, High-Accuracy, Randomized Nullspace Computations for Tall Matrices Epperly, Ethan N. Park, Taejun Nakatsukasa, Yuji Numerical Analysis 65F15, 68W20, 65D15 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. |
| title | Fast, High-Accuracy, Randomized Nullspace Computations for Tall Matrices |
| topic | Numerical Analysis 65F15, 68W20, 65D15 |
| url | https://arxiv.org/abs/2602.16797 |