Does block size matter in randomized block Krylov low-rank approximation?
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909859837902848 |
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| author | Chen, Tyler Epperly, Ethan N. Meyer, Raphael A. Musco, Christopher Rao, Akash |
| author_facet | Chen, Tyler Epperly, Ethan N. Meyer, Raphael A. Musco, Christopher Rao, Akash |
| contents | We study the problem of computing a rank-$k$ approximation of a matrix using randomized block Krylov iteration. Prior work has shown that, for block size $b = 1$ or $b = k$, a $(1 + \varepsilon)$-factor approximation to the best rank-$k$ approximation can be obtained after $\tilde O(k/\sqrt{\varepsilon})$ matrix-vector products with the target matrix. On the other hand, when $b$ is between $1$ and $k$, the best known bound on the number of matrix-vector products scales with $b(k-b)$, which could be as large as $O(k^2)$. Nevertheless, in practice, the performance of block Krylov methods is often optimized by choosing a block size $1 \ll b \ll k$. We resolve this theory-practice gap by proving that randomized block Krylov iteration produces a $(1 + \varepsilon)$-factor approximate rank-$k$ approximation using $\tilde O(k/\sqrt{\varepsilon})$ matrix-vector products for any block size $1\le b\le k$. Our analysis relies on new bounds for the minimum singular value of a random block Krylov matrix, which may be of independent interest. Similar bounds are central to recent breakthroughs on faster algorithms for sparse linear systems [Peng & Vempala, SODA 2021; Nie, STOC 2022]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Does block size matter in randomized block Krylov low-rank approximation? Chen, Tyler Epperly, Ethan N. Meyer, Raphael A. Musco, Christopher Rao, Akash Data Structures and Algorithms Numerical Analysis 65F55, 65F15 G.1.3; F.2.1 We study the problem of computing a rank-$k$ approximation of a matrix using randomized block Krylov iteration. Prior work has shown that, for block size $b = 1$ or $b = k$, a $(1 + \varepsilon)$-factor approximation to the best rank-$k$ approximation can be obtained after $\tilde O(k/\sqrt{\varepsilon})$ matrix-vector products with the target matrix. On the other hand, when $b$ is between $1$ and $k$, the best known bound on the number of matrix-vector products scales with $b(k-b)$, which could be as large as $O(k^2)$. Nevertheless, in practice, the performance of block Krylov methods is often optimized by choosing a block size $1 \ll b \ll k$. We resolve this theory-practice gap by proving that randomized block Krylov iteration produces a $(1 + \varepsilon)$-factor approximate rank-$k$ approximation using $\tilde O(k/\sqrt{\varepsilon})$ matrix-vector products for any block size $1\le b\le k$. Our analysis relies on new bounds for the minimum singular value of a random block Krylov matrix, which may be of independent interest. Similar bounds are central to recent breakthroughs on faster algorithms for sparse linear systems [Peng & Vempala, SODA 2021; Nie, STOC 2022]. |
| title | Does block size matter in randomized block Krylov low-rank approximation? |
| topic | Data Structures and Algorithms Numerical Analysis 65F55, 65F15 G.1.3; F.2.1 |
| url | https://arxiv.org/abs/2508.06486 |