On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

TL;DR

This paper investigates the subtle properties of mixtures of probability laws, especially Gaussian mixtures, focusing on concentration inequalities and Sobolev inequalities, revealing surprising stability and instability phenomena.

Contribution

It provides sharp bounds for concentration of measure and analyzes Sobolev inequalities for mixtures, highlighting novel stability and blow-up behaviors of functional constants.

Findings

Poincaré constant remains bounded as mixture proportions approach 0 or 1.

Logarithmic Sobolev constant can blow up even when Poincaré constant stays bounded.

Sub-Gaussian concentration is less stable than Poincaré inequalities for mixtures.

Abstract

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar\'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions.