Metric Entropy and Minimax Risk of Ellipsoids with an Application to Pinsker's Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915577158696960 |
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| author | Allard, Thomas |
| author_facet | Allard, Thomas |
| contents | We study how large an $\ell^2$ ellipsoid is by introducing type-$τ$ integrals that capture the average decay of its semi-axes. These integrals turn out to be closely related to standard complexity measures: we show that the metric entropy of the ellipsoid is asymptotically equivalent to the type-1 integral, and that the minimax risk in non-parametric estimation is asymptotically determined by the type-2 and type-3 integrals. This allows us to retrieve and sharpen classical results about metric entropy and minimax risk of ellipsoids through a systematic analysis of the type-$τ$ integrals, and yields an explicit formula linking the two. As an application, we improve on the best-known characterization of the metric entropy of the Sobolev ellipsoid, and extend Pinsker's Sobolev theorem in two ways: (i) to any bounded open domain in arbitrary finite dimension, and (ii) by providing the second-order term in the asymptotic expansion of the minimax risk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22441 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric Entropy and Minimax Risk of Ellipsoids with an Application to Pinsker's Theorem Allard, Thomas Statistics Theory Functional Analysis We study how large an $\ell^2$ ellipsoid is by introducing type-$τ$ integrals that capture the average decay of its semi-axes. These integrals turn out to be closely related to standard complexity measures: we show that the metric entropy of the ellipsoid is asymptotically equivalent to the type-1 integral, and that the minimax risk in non-parametric estimation is asymptotically determined by the type-2 and type-3 integrals. This allows us to retrieve and sharpen classical results about metric entropy and minimax risk of ellipsoids through a systematic analysis of the type-$τ$ integrals, and yields an explicit formula linking the two. As an application, we improve on the best-known characterization of the metric entropy of the Sobolev ellipsoid, and extend Pinsker's Sobolev theorem in two ways: (i) to any bounded open domain in arbitrary finite dimension, and (ii) by providing the second-order term in the asymptotic expansion of the minimax risk. |
| title | Metric Entropy and Minimax Risk of Ellipsoids with an Application to Pinsker's Theorem |
| topic | Statistics Theory Functional Analysis |
| url | https://arxiv.org/abs/2510.22441 |