Random matrices acting on sets: Independent columns
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908518742753280 |
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| author | Plan, Yaniv Vershynin, Roman |
| author_facet | Plan, Yaniv Vershynin, Roman |
| contents | We study random matrices with independent subgaussian columns. Assuming each column has a fixed Euclidean norm, we establish conditions under which such matrices act as near-isometries when restricted to a given subset of their domain. We show that, with high probability, the maximum distortion caused by such a matrix is proportional to the Gaussian complexity of the subset, scaled by the subgaussian norm of the matrix columns. This linear dependence on the subgaussian norm is a new phenomenon, as random matrices with independent rows or independent entries typically exhibit superlinear dependence. As a consequence, normalizing the columns of random sparse matrices leads to stronger embedding guarantees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random matrices acting on sets: Independent columns Plan, Yaniv Vershynin, Roman Probability We study random matrices with independent subgaussian columns. Assuming each column has a fixed Euclidean norm, we establish conditions under which such matrices act as near-isometries when restricted to a given subset of their domain. We show that, with high probability, the maximum distortion caused by such a matrix is proportional to the Gaussian complexity of the subset, scaled by the subgaussian norm of the matrix columns. This linear dependence on the subgaussian norm is a new phenomenon, as random matrices with independent rows or independent entries typically exhibit superlinear dependence. As a consequence, normalizing the columns of random sparse matrices leads to stronger embedding guarantees. |
| title | Random matrices acting on sets: Independent columns |
| topic | Probability |
| url | https://arxiv.org/abs/2502.16827 |