Quasi-optimal hierarchically semi-separable matrix approximation
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909773480329216 |
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| author | Amsel, Noah Chen, Tyler Keles, Feyza Duman Halikias, Diana Musco, Cameron Musco, Christopher Persson, David |
| author_facet | Amsel, Noah Chen, Tyler Keles, Feyza Duman Halikias, Diana Musco, Cameron Musco, Christopher Persson, David |
| contents | We present a randomized algorithm for producing a quasi-optimal hierarchically semi-separable (HSS) approximation to an $N\times N$ matrix $A$ using only matrix-vector products with $A$ and $A^T$. We prove that, using $O(k \log(N/k))$ matrix-vector products and ${O}(N k^2 \log(N/k))$ additional runtime, the algorithm returns an HSS matrix $B$ with rank-$k$ blocks whose expected Frobenius norm error $\mathbb{E}[\|A - B\|_F^2]$ is at most $O(\log(N/k))$ times worse than the best possible approximation error by an HSS rank-$k$ matrix. In fact, the algorithm we analyze in a simple modification of an empirically effective method proposed by [Levitt & Martinsson, SISC 2024]. As a stepping stone towards our main result, we prove two results that are of independent interest: a similar guarantee for a variant of the algorithm which accesses $A$'s entries directly, and explicit error bounds for near-optimal subspace approximation using projection-cost-preserving sketches. To the best of our knowledge, our analysis constitutes the first polynomial-time quasi-optimality result for HSS matrix approximation, both in the explicit access model and the matrix-vector product query model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-optimal hierarchically semi-separable matrix approximation Amsel, Noah Chen, Tyler Keles, Feyza Duman Halikias, Diana Musco, Cameron Musco, Christopher Persson, David Numerical Analysis Data Structures and Algorithms 65F55, 68W20, 68W25 We present a randomized algorithm for producing a quasi-optimal hierarchically semi-separable (HSS) approximation to an $N\times N$ matrix $A$ using only matrix-vector products with $A$ and $A^T$. We prove that, using $O(k \log(N/k))$ matrix-vector products and ${O}(N k^2 \log(N/k))$ additional runtime, the algorithm returns an HSS matrix $B$ with rank-$k$ blocks whose expected Frobenius norm error $\mathbb{E}[\|A - B\|_F^2]$ is at most $O(\log(N/k))$ times worse than the best possible approximation error by an HSS rank-$k$ matrix. In fact, the algorithm we analyze in a simple modification of an empirically effective method proposed by [Levitt & Martinsson, SISC 2024]. As a stepping stone towards our main result, we prove two results that are of independent interest: a similar guarantee for a variant of the algorithm which accesses $A$'s entries directly, and explicit error bounds for near-optimal subspace approximation using projection-cost-preserving sketches. To the best of our knowledge, our analysis constitutes the first polynomial-time quasi-optimality result for HSS matrix approximation, both in the explicit access model and the matrix-vector product query model. |
| title | Quasi-optimal hierarchically semi-separable matrix approximation |
| topic | Numerical Analysis Data Structures and Algorithms 65F55, 68W20, 68W25 |
| url | https://arxiv.org/abs/2505.16937 |