R\'{e}nyi Cross-Entropy Measures for Common Distributions and Processes with Memory

TL;DR

This paper derives closed-form Rényi cross-entropy measures for common distributions and processes, facilitating their use in deep learning and information theory applications.

Contribution

It provides explicit formulas for Rényi cross-entropy for exponential family distributions and summarizes rates for Gaussian and Markov processes.

Findings

Closed-form Rényi cross-entropy formulas for exponential family distributions

Tabulated results for ease of reference

Summarized Rényi cross-entropy rates for Gaussian and Markov processes

Abstract

Two R\'{e}nyi-type generalizations of the Shannon cross-entropy, the R\'{e}nyi cross-entropy and the Natural R\'{e}nyi cross-entropy, were recently used as loss functions for the improved design of deep learning generative adversarial networks. In this work, we build upon our results in [1] by deriving the R\'{e}nyi and Natural R\'{e}nyi differential cross-entropy measures in closed form for a wide class of common continuous distributions belonging to the exponential family and tabulating the results for ease of reference. We also summarise the R\'{e}nyi-type cross-entropy rates between stationary Gaussian processes and between finite-alphabet time-invariant Markov sources.