Strengthening the Entropy Power Inequality

TL;DR

This paper enhances the Entropy Power Inequality for cases involving Gaussian summands, introduces a reverse inequality, and applies these results to network information theory, providing new bounds and proofs.

Contribution

It generalizes Costa's EPI and establishes a new reverse entropy power inequality using weak convergence and rotational invariance techniques.

Findings

Tightened EPI for Gaussian summands

Derived a new reverse entropy power inequality

Provided a simplified proof for the two-encoder Gaussian source coding rate region

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.