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 $\alpha \in [1,\infty ]$. An R-EPI provides a lower bound on the order-$\alpha$ Rényi 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-$\alpha$ Rényi 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.

arXiv commentRevised version of a submission to the IEEE Trans. on Information Theory. Presented in part at ISIT 2016, Barcelona, July 2016