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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.10075 |
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| _version_ | 1866908704842973184 |
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| author | Nembé, Jocelyn |
| author_facet | Nembé, Jocelyn |
| contents | We establish a universal framework for concentration inequalities based on invariance under diffeomorphism groups. Given a probability measure $μ$ on a space $E$ and a diffeomorphism $ψ: E \to F$, concentration properties transfer covariantly: if the pushforward $ψ_*μ$ concentrates, so does $μ$ in the pullback geometry. This reveals that classical concentration inequalities -- Hoeffding, Bernstein, Talagrand, Gaussian isoperimetry -- are manifestations of a single principle of \emph{geometric invariance}. The choice of coordinate system $ψ$ becomes a free parameter that can be optimized. We prove that for any distribution class $\Pc$, there exists an optimal diffeomorphism $ψ^*$ minimizing the concentration constant, and we characterize $ψ^*$ in terms of the Fisher-Rao geometry of $\Pc$. We establish \emph{strict improvement theorems}: for heavy-tailed or multiplicative data, the optimal $ψ$ yields exponentially tighter bounds than the identity. We develop the full theory including transportation-cost inequalities, isoperimetric profiles, and functional inequalities, all parametrized by the diffeomorphism group $\Diff(E)$. Connections to information geometry (Amari's $α$-connections), optimal transport with general costs, and Riemannian concentration are established. Applications to robust statistics, multiplicative models, and high-dimensional inference demonstrate that coordinate optimization can improve statistical efficiency by orders of magnitude. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10075 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration of Measure under Diffeomorphism Groups: A Universal Framework with Optimal Coordinate Selection Nembé, Jocelyn Statistics Theory 2020: Primary 60E15, 28C15, Secondary 53C21, 62B10, 49Q20 We establish a universal framework for concentration inequalities based on invariance under diffeomorphism groups. Given a probability measure $μ$ on a space $E$ and a diffeomorphism $ψ: E \to F$, concentration properties transfer covariantly: if the pushforward $ψ_*μ$ concentrates, so does $μ$ in the pullback geometry. This reveals that classical concentration inequalities -- Hoeffding, Bernstein, Talagrand, Gaussian isoperimetry -- are manifestations of a single principle of \emph{geometric invariance}. The choice of coordinate system $ψ$ becomes a free parameter that can be optimized. We prove that for any distribution class $\Pc$, there exists an optimal diffeomorphism $ψ^*$ minimizing the concentration constant, and we characterize $ψ^*$ in terms of the Fisher-Rao geometry of $\Pc$. We establish \emph{strict improvement theorems}: for heavy-tailed or multiplicative data, the optimal $ψ$ yields exponentially tighter bounds than the identity. We develop the full theory including transportation-cost inequalities, isoperimetric profiles, and functional inequalities, all parametrized by the diffeomorphism group $\Diff(E)$. Connections to information geometry (Amari's $α$-connections), optimal transport with general costs, and Riemannian concentration are established. Applications to robust statistics, multiplicative models, and high-dimensional inference demonstrate that coordinate optimization can improve statistical efficiency by orders of magnitude. |
| title | Concentration of Measure under Diffeomorphism Groups: A Universal Framework with Optimal Coordinate Selection |
| topic | Statistics Theory 2020: Primary 60E15, 28C15, Secondary 53C21, 62B10, 49Q20 |
| url | https://arxiv.org/abs/2512.10075 |