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

Jocelyn Nembé

Paper

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

Abstract

We establish a universal framework for concentration inequalities based on invariance under diffeomorphism groups. Given a probability measure

on a space

and a diffeomorphism

, concentration properties transfer covariantly: if the pushforward

concentrates, so does

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

becomes a free parameter that can be optimized. We prove that for any distribution class

, there exists an optimal diffeomorphism

minimizing the concentration constant, and we characterize

in terms of the Fisher-Rao geometry of

. We establish \emph{strict improvement theorems}: for heavy-tailed or multiplicative data, the optimal

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

. Connections to information geometry (Amari's

-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.