Exact Exponents for Concentration and Isoperimetry in Product Polish Spaces

TL;DR

This work derives exact, variational characterizations of the asymptotic exponents governing concentration and isoperimetry in product Polish spaces, expressing them through relative entropy and optimal transport costs. The authors establish a dimension-free bound for the concentration exponent and prove single-letterized, computable forms via auxiliary-variable alphabets of bounded size, with a detailed treatment for general costs, complete metrics, and the Hamming metric. They also develop dual formulas and demonstrate applications to Strassen-type OT problems and the classic isoperimetric setting, yielding asymptotically sharp inequalities under suitable conditions. The approach unifies information theory and OT techniques to advance understanding of measure concentration and geometric-functional inequalities in broad spaces. These results provide computable, sharp tools for analyzing large-scale probabilistic systems and contribute to the theoretical foundation connecting transport, entropy, and concentration phenomena.

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.

Figures (1)

• Figure 1: Illustration of the function $\underline{\phi}$ which corresponds to the asymptotic concentration exponent of atomless measures $P_{X}=P_{Y}$ under the Hamming metric. In this graph, the bases of the logarithm and exponent are changed to $2$. Given each $\alpha$, the function $\underline{\phi}$ is zero when $\tau$ is smaller than the value on the black curve. (The curve looks not so smooth in the figure due to the precision of computation, but should be smooth in theory.)

Theorems & Definitions (40)

• Example 1: Countable Alphabet and Bounded Cost
• Example 2: Wasserstein Metric Induced by a Bounded Metric
• Theorem 1: Asymptotics of $E_{1}^{(n)}$ and Dimension-Free Bound
• Corollary 1: Improved Talagrand's Concentration Inequality
• Remark 1
• Remark 2
• Theorem 2: Asymptotics of $E_{1}^{(n)}$ for Complete Metrics
• Example 3: Gaussian Distribution and Euclidean Distance
• Theorem 3: Asymptotics of $E_{1}^{(n)}$ for Hamming Metric
• Remark 3