Related papers: Entropy-variance inequalities for discrete log-con…

We develop the notion of discrete degrees of freedom of a log-concave sequence and use it to prove that geometric distribution minimises R\'enyi entropy of order infinity under fixed variance, among all discrete log-concave random variables…

We show that for any $\alpha>0$ the R\'enyi entropy of order $\alpha$ is minimized, among all symmetric log-concave random variables with fixed variance, either for a uniform distribution or for a two sided exponential distribution. The…

We establish a discrete analog of the R\'enyi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic…

We consider the entropy of sums of independent discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for…

We show that for log-concave real random variables with fixed variance the Shannon differential entropy is minimized for an exponential random variable. We apply this result to derive upper bounds on capacities of additive noise channels…

Using a sharp version of the reverse Young inequality, and a R\'enyi entropy comparison result due to Fradelizi, Madiman, and Wang, the authors are able to derive R\'enyi entropy power inequalities for log-concave random vectors when…

We investigate the role of convexity in R\'enyi entropy power inequalities. After proving that a general R\'enyi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the R\'enyi parameter $r\in(0,1)$, we show that…

We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…

The entropy power inequality for independent random vectors is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Several extensions of the entropy power inequality have been…

An important theme in recent work in asymptotic geometric analysis is that many classical implications between different types of geometric or functional inequalities can be reversed in the presence of convexity assumptions. In this note,…

We show that $h_\infty(X+Y)\leq h_\infty(Z+W)$, where $X, Y$ are independent log-concave random variables, and $Z, W$ are exponential random variables having the same respective $\infty$-R\'enyi entropies. Analogs for integer-valued…

Let $\mathsf{N}_{\rm d}\left[X\right]=\frac{1}{2\pi {\rm e}}{\rm e}^{2\mathsf{H}\left[X\right]}$ denote the entropy power of the discrete random variable $X$ where $\mathsf{H}\left[X\right]$ denotes the discrete entropy of $X$. In this…

We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information…

We prove a quantitative dimension-free bound in the Shannon-Stam Entropy inequality for the convolution of two log-concave distributions in dimension d interms of the spectral gap of the density. The method relies on the analysis of the…

We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n})…

Two-sided bounds are explored for concentration functions and R\'enyi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities.

We introduce a framework for obtaining tight mixing times for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities (MLSI) quantify the rate of relative entropy contraction for…

We establish a reversal of Lyapunov's inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is disproved through counter example. We also…

Entropy rate is a real valued functional on the space of discrete random sources which lacks a closed formula even for subclasses of sources which have intuitive parameterizations. A good way to overcome this problem is to examine its…

Shannon's Entropy Power Inequality can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies. The entropy power inequality has played a key…