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

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Summary
This summary is machine-generated.

This study derives closed-form Rényi and Natural Rényi differential cross-entropy for exponential family distributions. These measures are crucial for improving deep learning generative adversarial networks and analyzing information rates.

Keywords:
Gaussian processesMarkov sourcesRényi information measurescross-entropydivergence measuresexponential family distributions

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Area of Science:

  • Information Theory
  • Machine Learning
  • Statistical Inference

Background:

  • Shannon cross-entropy is a fundamental concept in information theory.
  • Rényi-type cross-entropies offer generalizations with applications in deep learning.
  • Generative Adversarial Networks (GANs) benefit from improved loss functions.

Purpose of the Study:

  • To derive closed-form expressions for Rényi and Natural Rényi differential cross-entropy.
  • To analyze these measures for common continuous distributions within the exponential family.
  • To summarize cross-entropy rates for Gaussian processes and Markov sources.

Main Methods:

  • Derivation of analytical expressions for differential cross-entropy measures.
  • Tabulation of results for exponential family distributions.
  • Summarization of cross-entropy rates for specific stochastic processes.

Main Results:

  • Closed-form solutions for Rényi and Natural Rényi differential cross-entropy are presented.
  • Results are provided for a broad class of exponential family distributions.
  • Cross-entropy rates for stationary Gaussian processes and finite-alphabet Markov sources are summarized.

Conclusions:

  • The derived measures provide valuable tools for deep learning applications, particularly GANs.
  • Tabulated results facilitate practical application and comparison of different distributions.
  • The study enhances understanding of information-theoretic rates in complex systems.