Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory

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.