On the randomized SVD in infinite dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918324914356224 |
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| author | Kressner, Daniel Persson, David Uschmajew, André |
| author_facet | Kressner, Daniel Persson, David Uschmajew, André |
| contents | Randomized methods, such as the randomized SVD (singular value decomposition) and Nyström approximation, are an effective way to compute low-rank approximations of large matrices. Motivated by applications to operator learning, Boullé and Townsend (FoCM, 2023) recently proposed an infinite-dimensional extension of the randomized SVD for a Hilbert-Schmidt operator $A$ that invokes randomness through a Gaussian process with a covariance operator $K$. While the non-isotropy introduced by $K$ allows one to incorporate prior information on $A$, an unfortunate choice may lead to unfavorable performance and large constants in the error bounds. In this work, we introduce a novel infinite-dimensional extension of the randomized SVD that does not require such a choice and enjoys error bounds that match those for the finite-dimensional case. Our extension implicitly uses isotropic random vectors, reflecting a choice commonly made in the finite-dimensional case. In fact, the theoretical results of this work show how the usual randomized SVD applied to a discretization of $A$ approaches our infinite-dimensional extension as the discretization gets refined, both in terms of error bounds and the Wasserstein distance. We also present and analyze a novel extension of the Nyström approximation for self-adjoint positive semi-definite trace class operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_06882 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the randomized SVD in infinite dimensions Kressner, Daniel Persson, David Uschmajew, André Numerical Analysis 65F55, 65N80 Randomized methods, such as the randomized SVD (singular value decomposition) and Nyström approximation, are an effective way to compute low-rank approximations of large matrices. Motivated by applications to operator learning, Boullé and Townsend (FoCM, 2023) recently proposed an infinite-dimensional extension of the randomized SVD for a Hilbert-Schmidt operator $A$ that invokes randomness through a Gaussian process with a covariance operator $K$. While the non-isotropy introduced by $K$ allows one to incorporate prior information on $A$, an unfortunate choice may lead to unfavorable performance and large constants in the error bounds. In this work, we introduce a novel infinite-dimensional extension of the randomized SVD that does not require such a choice and enjoys error bounds that match those for the finite-dimensional case. Our extension implicitly uses isotropic random vectors, reflecting a choice commonly made in the finite-dimensional case. In fact, the theoretical results of this work show how the usual randomized SVD applied to a discretization of $A$ approaches our infinite-dimensional extension as the discretization gets refined, both in terms of error bounds and the Wasserstein distance. We also present and analyze a novel extension of the Nyström approximation for self-adjoint positive semi-definite trace class operators. |
| title | On the randomized SVD in infinite dimensions |
| topic | Numerical Analysis 65F55, 65N80 |
| url | https://arxiv.org/abs/2506.06882 |