The Strong Data Processing Inequality Under the Heat Flow
Bo’az Klartag, Or Ordentlich · IEEE Transactions on Information Theory · 2025
Let$ u $and$\mu $be probability distributions on$\mathbb {R}^{n}$, and$ u _{s},\mu _{s}$be their evolution under the heat flow, that is, the probability distributions resulting from convolving their density with the density of an isotropic Gaussian random vector with variancesin each entry. This paper studies the rate of decay of$s\mapsto D( u _{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 $and the Gaussian channel. We also prove generalizations of de Bruijn’s identity, and Costa’s result on the concavity insof the differential entropy of$ u _{s}$. As a byproduct of our analysis, we obtain new lower bounds on the mutual information betweenXand$Y=X+\sqrt {s} Z$, whereZis a standard Gaussian vector in$\mathbb {R}^{n}$, independent ofX, and on the minimum mean-square error (MMSE) in estimatingXfromY, in terms of the Poincaré constant ofX.