A characterization of dimension free concentration in terms of transportation inequalities

TL;DR

This paper establishes that dimension-free Gaussian concentration of probability measures is equivalent to satisfying Talagrand's  transportation inequality, providing new proofs and generalizations for related concentration phenomena using Large Deviations techniques.

Contribution

It proves the equivalence between dimension-free Gaussian concentration and Talagrand's  inequality, and extends the framework to other concentration types with simplified proofs.

Findings

Gaussian concentration is characterized by  transportation inequality

Poincare9 inequality relates to exponential concentration

New, concise proof of Otto and Villani's result

Abstract

The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincar\'e inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.