Strengthening the Entropy Power Inequality

Abstract: We tighten the Entropy Power Inequality (EPI) when one of the random summands is Gaussian. Our strengthening is closely connected to the concept of strong data processing for Gaussian channels and generalizes the (vector extension of) Costa's EPI. This leads to a new reverse entropy power inequality and, as a corollary, sharpens Stam's inequality relating entropy power and Fisher information. Applications to network information theory are given, including a short self-contained proof of the rate region for the two-encoder quadratic Gaussian source coding problem. Our argument is based on weak convergence and a technique employed by Geng and Nair for establishing Gaussian optimality via rotational-invariance, which traces its roots to a `doubling trick' that has been successfully used in the study of functional inequalities.