Understanding two-scale criteria for Poincar\'{e} and log-Sobolev inequalities in the Euclidean case through \Phi-entropies

Abstract

We study settings in which mixture and joint distributions satisfy a Poincar\'{e} (or log-Sobolev) inequality induced by a marginal and a collection of conditional distributions that are assumed to satisfy Poincar\'{e} (or log-Sobolev, resp.) inequalities and supported over Euclidean spaces. In this note, we use the framework of $\Phi$-Sobolev inequalities (Chafa\"{i}, 2004) to provide a unified approach to arriving at these inequalities in the Euclidean case. This results in a simpler proof technique for establishing these functional inequalities under certain two-scale criteria. We also discuss applications of these results to certain sampling algorithms.