On Renyi Entropy Power Inequalities

Abstract

This paper gives improved R\'{e}nyi entropy power inequalities (R-EPIs). Consider a sum $S_n = \sum_{k=1}^n X_k$ of $n$ independent continuous random vectors taking values on $\mathbb{R}^d$, and let $\alpha \in [1, \infty]$. An R-EPI provides a lower bound on the order-$\alpha$ R\'enyi entropy power of $S_n$ that, up to a multiplicative constant (which may depend in general on $n, \alpha, d$), is equal to the sum of the order-$\alpha$ R\'enyi entropy powers of the $n$ random vectors $\{X_k\}_{k=1}^n$. For $\alpha=1$, the R-EPI coincides with the well-known entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov which relies on the sharpened Young's inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix.