Luiza H. F. de Andrade

arXiv2020

Conditions for the existence of a generalization of Rényi divergence

We give necessary and sufficient conditions for the existence of a generalization of Rényi divergence, which is defined in terms of a deformed exponential function. If the underlying measure $μ$ is non-atomic, we found that not all deformed exponential functions can be used in the generalization of Rényi divergence; a condition involving the deformed exponential function is provided. In the case $μ$ is purely atomic (the counting measure on the set of natural numbers), we show that any deformed exponential function can be used in the generalization.

cs.IT↗

arXiv2020

Properties of a Generalized Divergence Related to Tsallis Relative Entropy

In this paper, we investigate the partition inequality, joint convexity, and Pinsker's inequality, for a divergence that generalizes the Tsallis Relative Entropy and Kullback-Leibler divergence. The generalized divergence is defined in terms of a deformed exponential function, which replaces the Tsallis $q$-exponential. We also constructed a family of probability distributions related to the generalized divergence. We found necessary and sufficient conditions for the partition inequality to be satisfied. A sufficient condition for the joint convexity was established. We proved that the generalized divergence satisfies the partition inequality, and is jointly convex, if, and only if, it coincides with the Tsallis relative entropy. As an application of partition inequality, a criterion for the Pinsker's inequality was found.

cs.IT↗