The Strong Data Processing Inequality Under the Heat Flow

Let ν and μ be probability distributions on ℝⁿ , and ν_s, μ_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 ⭲ D(ν_s||μ_s) for various divergences, including the χ 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 ν_s . As a byproduct of our analysis, we obtain new lower bounds on the mutual information between X and Y=X+√s Z , where Z is a standard Gaussian vector in ℝⁿ , independent of X, and on the minimum mean-square error (MMSE) in estimating X from Y, in terms of the Poincaré constant of X.