Concentration of Measure under Diffeomorphism Groups: A Universal Framework with Optimal Coordinate Selection

TL;DR

This paper introduces a universal framework for concentration inequalities based on invariance under diffeomorphisms, enabling optimal coordinate choices that significantly tighten bounds and improve statistical efficiency across various data models.

Contribution

It develops a theory linking concentration inequalities to geometric invariance under diffeomorphisms, including optimal coordinate selection and strict improvement results for heavy-tailed data.

Findings

Optimal diffeomorphisms minimize concentration constants.

Exponential improvements for heavy-tailed and multiplicative data.

Applications show orders of magnitude efficiency gains in statistics.

Abstract

We establish a universal framework for concentration inequalities based on invariance under diffeomorphism groups. Given a probability measure $\mu$ on a space $E$ and a diffeomorphism $\psi: E \to F$, concentration properties transfer covariantly: if the pushforward $\psi_*\mu$ concentrates, so does $\mu$ 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 $\psi$ becomes a free parameter that can be optimized. We prove that for any distribution class $\Pc$, there exists an optimal diffeomorphism $\psi^*$ minimizing the concentration constant, and we characterize $\psi^*$ in terms of the Fisher-Rao geometry of $\Pc$. We establish \emph{strict improvement theorems}: for heavy-tailed or multiplicative data, the optimal $\psi$ 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 $\alpha$-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.