Exact Exponents for Concentration and Isoperimetry in Product Polish Spaces

TL;DR

This paper establishes variational formulas linking concentration and isoperimetric exponents in product Polish spaces with information theory and optimal transport, providing dimension-free bounds and computable bounds for finite alphabets.

Contribution

It introduces new variational formulas connecting concentration, isoperimetry, information theory, and optimal transport, with computable bounds and applications to classic inequalities.

Findings

Derived variational formulas for asymptotic exponents

Established dimension-free bounds in concentration regime

Provided a finite-alphabet computability bound for isoperimetric exponents

Abstract

In this paper, we derive variational formulas for the asymptotic exponents (i.e., convergence rates) of the concentration and isoperimetric functions in the product Polish probability space under certain mild assumptions. These formulas are expressed in terms of relative entropies (which are from information theory) and optimal transport cost functionals (which are from optimal transport theory). Hence, our results verify an intimate connection among information theory, optimal transport, and concentration of measure or isoperimetric inequalities. In the concentration regime, the corresponding variational formula is in fact a dimension-free bound in the sense that this bound is valid for any dimension. A cardinality bound for the alphabet of the auxiliary random variable in the expression of the asymptotic isoperimetric exponent is provided, which makes the expression computable by a finite-dimensional program for the finite alphabet case. We lastly apply our results to obtain an isoperimetric inequality in the classic isoperimetric setting, which is asymptotically sharp under certain conditions. The proofs in this paper are based on information-theoretic and optimal transport techniques.