R\'enyi Entropy Power Inequalities via Normal Transport and Rotation

TL;DR

This paper introduces a unified framework using normal transport and rotation to derive Rényi entropy power inequalities, recovering known results and establishing new bounds, especially for log-concave densities.

Contribution

It presents a comprehensive transport-based method to derive and unify various Rényi entropy power inequalities, including new bounds for log-concave densities.

Findings

Unified framework for Rényi EPIs using transport and rotation

Recovery of known Rényi EPIs through simple arguments

New sharp varentropy bounds for log-concave densities

Abstract

Following a recent proof of Shannon's entropy power inequality (EPI), a comprehensive framework for deriving various EPIs for the R\'enyi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation. 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. In particular, for log-concave densities, we obtain a simple transportation proof of a sharp varentropy bound.