R\'enyi Divergence and Kullback-Leibler Divergence

TL;DR

This paper reviews and extends key properties of Re9nyi and Kullback-Leibler divergences, exploring their mathematical relations, generalizations, and applications in information theory.

Contribution

It introduces new generalizations of the Pythagorean inequality and extends minimax results to continuous channels for all divergence orders.

Findings

Re9nyi divergence of order 1 equals Kullback-Leibler divergence.

Extended the Pythagorean inequality to orders different from 1.

Generalized channel capacity and minimax redundancy results for continuous inputs.

Abstract

R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by R\'enyi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the R\'enyi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of R\'enyi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of $\sigma$-algebras and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results.