{"ID":2850390,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.22441","arxiv_id":"2510.22441","title":"Metric Entropy and Minimax Risk of Ellipsoids with an Application to Pinsker's Theorem","abstract":"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.","short_abstract":"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 th...","url_abs":"https://arxiv.org/abs/2510.22441","url_pdf":"https://arxiv.org/pdf/2510.22441v1","authors":"[\"Thomas Allard\"]","published":"2025-10-25T21:52:06Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.FA\"]","methods":"[]","has_code":false}