On Renyi Entropy Power Inequalities

Abstract

This paper gives improved Rényi 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 $α\in [1, \infty]$. An R-EPI provides a lower bound on the order-$α$ Rényi entropy power of $S_n$ that, up to a multiplicative constant (which may depend in general on $n, α, d$), is equal to the sum of the order-$α$ Rényi entropy powers of the $n$ random vectors $\{X_k\}_{k=1}^n$. For $α=1$, 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.