Rényi entropy power inequality and a reverse

Abstract

This paper is twofold. In the first part, we present a refinement of the Rényi Entropy Power Inequality (EPI) recently obtained in [11]. The proof largely follows the approach in [18] of employing Young's convolution inequalities with sharp constants. In the second part, we study the reversibility of the Rényi EPI, and confirm a conjecture in [5,24] in two cases. Connections with various p-th mean bodies in convex geometry are also explored.

Rényi Entropy Power Inequalities via Normal Transport and Rotation

“…As a perspective, the methods developed in this paper can perhaps be generalized to obtain reverse Rényi entropy power inequalities (see e.g., the discussion in [8]).…”

Rearrangement and Prékopa–Leindler Type Inequalities

“…For the converse, given x, y take λ 1 < f (x) and λ 2 < g(y), then z = (1 − t)x + ty ∈ (1 − t){f > λ 1 } + t{g > λ 2 }. By (26), h(z) > Ψ t (f (x), g(y)), and by the continuity assumption on Ψ t , Ψ t (f (x), g(y)) = sup λ Ψ t (λ 1 , λ 2 ) ≤ h(z). Thus we will…”

“…These results can also be motivated from an information theoretic perspective, where the BMI can be considered a Rényi entropy power inequality. There has been considerable recent work (see [6,8,9,26,27,29,39]) developing Rényi entropy [40] generalizations of the classical entropy power inequality (EPI) of Shannon-Stam [41,43]. One should compare the sharpening of PLI here to [45], where Madiman and Wang show that while spherically symmetric decreasing rearrangements of random variables preserve their Rényi entropy, they decrease the Rényi entropy of independent sums of random variables.…”

Majorization and Rényi entropy inequalities via Sperner theory

“…Theorem 2.6 uses the exponent c = 1 + α; Rényi entropy power inequalities with the same exponent in R were recently explored by Bobkov and Marsiglietti [53] (although it was shown soon after by Li [54] that this exponent can be improved). In fact, these authors proved similar inequalities in R d , with the exponent 1+α d , mimicking the 2/d exponent in the original Shannon-Stam entropy power inequality.…”