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Entropy 2019, 21(1), 89; https://doi.org/10.3390/e21010089

Poincaré and Log–Sobolev Inequalities for Mixtures

Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
Received: 30 November 2018 / Revised: 30 December 2018 / Accepted: 11 January 2019 / Published: 18 January 2019
(This article belongs to the Special Issue Entropy and Information Inequalities)
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Abstract

This work studies mixtures of probability measures on R n and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the χ 2 -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method. View Full-Text
Keywords: Poincaré inequality; log–Sobolev inequality; relative entropy; fisher information; Dirichlet form; mixture; finite Gaussian mixtures Poincaré inequality; log–Sobolev inequality; relative entropy; fisher information; Dirichlet form; mixture; finite Gaussian mixtures
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Schlichting, A. Poincaré and Log–Sobolev Inequalities for Mixtures. Entropy 2019, 21, 89.

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