R\'enyi Entropy Power and Normal Transport

TL;DR

This paper introduces a unified framework for Re9nyi entropy-power inequalities using linearization and transportation methods, providing new inequalities, proofs, and insights into entropy and varentropy bounds for various densities.

Contribution

It develops a new framework for Re9nyi EPIs, unifies previous results, and offers novel proofs for key inequalities using transportation and entropy properties.

Findings

Unified Re9nyi EPI framework using linearization.

New proofs of Dembo-Cover-Thomas inequality via transportation.

Sharp varentropy bounds for log-concave densities.

Abstract

A framework for deriving R\'enyi entropy-power inequalities (EPIs) is presented that uses linearization and an inequality of Dembo, Cover, and Thomas. Simple arguments are given to recover the previously known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with constant c and a modification with exponent $\alpha$ of previous works. An information-theoretic proof of the Dembo-Cover-Thomas inequality---equivalent to Young's convolutional inequality with optimal constants---is provided, based on properties of R\'enyi conditional and relative entropies and using transportation arguments from Gaussian densities. For log-concave densities, a transportation proof of a sharp varentropy bound is presented.