The Strong Data Processing Inequality Under the Heat Flow

Abstract

Let $\nu $ and $\mu $ be probability distributions on $\mathbb {R}^{n}$ , and $\nu _{s},\mu _{s}$ be their evolution under the heat flow, that is, the probability distributions resulting from convolving their density with the density of an isotropic Gaussian random vector with variance s in each entry. This paper studies the rate of decay of $s\mapsto D(\nu _{s}\|\mu _{s})$ for various divergences, including the $\chi ^{2}$ and Kullback-Leibler (KL) divergences. We prove upper and lower bounds on the strong data-processing inequality (SDPI) coefficients corresponding to the source $\mu $ and the Gaussian channel. We also prove generalizations of de Bruijn’s identity, and Costa’s result on the concavity in s of the differential entropy of $\nu _{s}$ . As a byproduct of our analysis, we obtain new lower bounds on the mutual information between X and $Y=X+\sqrt {s} Z$ , where Z is a standard Gaussian vector in $\mathbb {R}^{n}$ , independent of X, and on the minimum mean-square error (MMSE) in estimating X from Y, in terms of the Poincaré constant of X.