Randomized Nyström approximation of non-negative self-adjoint operators
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915054072365056 |
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| author | Persson, David Boullé, Nicolas Kressner, Daniel |
| author_facet | Persson, David Boullé, Nicolas Kressner, Daniel |
| contents | The randomized singular value decomposition (SVD) has become a popular approach to computing cheap, yet accurate, low-rank approximations to matrices due to its efficiency and strong theoretical guarantees. Recent work by Boullé and Townsend (FoCM, 2023) presents an infinite-dimensional analog of the randomized SVD to approximate Hilbert-Schmidt operators. However, many applications involve computing low-rank approximations to symmetric positive semi-definite matrices. In this setting, it is well-established that the randomized Nyström approximation is usually preferred over the randomized SVD. This paper explores an infinite-dimensional analog of the Nyström approximation to compute low-rank approximations to non-negative self-adjoint trace-class operators. We present an analysis of the method and, along the way, improve the existing infinite-dimensional bounds for the randomized SVD. Our analysis yields bounds on the expected value and tail bounds for the Nyström approximation error in the operator, trace, and Hilbert-Schmidt norms. Numerical experiments on integral operators arising from Gaussian process sampling and Bayesian inverse problems are used to validate the proposed infinite-dimensional Nyström algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00960 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Randomized Nyström approximation of non-negative self-adjoint operators Persson, David Boullé, Nicolas Kressner, Daniel Numerical Analysis 65F30, 68W20, 65R10 The randomized singular value decomposition (SVD) has become a popular approach to computing cheap, yet accurate, low-rank approximations to matrices due to its efficiency and strong theoretical guarantees. Recent work by Boullé and Townsend (FoCM, 2023) presents an infinite-dimensional analog of the randomized SVD to approximate Hilbert-Schmidt operators. However, many applications involve computing low-rank approximations to symmetric positive semi-definite matrices. In this setting, it is well-established that the randomized Nyström approximation is usually preferred over the randomized SVD. This paper explores an infinite-dimensional analog of the Nyström approximation to compute low-rank approximations to non-negative self-adjoint trace-class operators. We present an analysis of the method and, along the way, improve the existing infinite-dimensional bounds for the randomized SVD. Our analysis yields bounds on the expected value and tail bounds for the Nyström approximation error in the operator, trace, and Hilbert-Schmidt norms. Numerical experiments on integral operators arising from Gaussian process sampling and Bayesian inverse problems are used to validate the proposed infinite-dimensional Nyström algorithm. |
| title | Randomized Nyström approximation of non-negative self-adjoint operators |
| topic | Numerical Analysis 65F30, 68W20, 65R10 |
| url | https://arxiv.org/abs/2404.00960 |