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Rényi Entropy Power Inequalities via Normal Transport and Rotation.

Olivier Rioul1,2

  • 1LTCI, Télécom ParisTech, Université Paris-Saclay, 75013 Paris, France.

Entropy (Basel, Switzerland)
|December 3, 2020
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Summary
This summary is machine-generated.

This study presents a new framework for deriving entropy power inequalities (EPIs) for Rényi entropy using transport arguments. It unifies existing results and offers new bounds, particularly for log-concave densities.

Keywords:
Rényi entropyentropy power inequalitiesescort distributionslog-concave distributionsnormal distributionstransportation arguments

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Area of Science:

  • Information Theory
  • Probability Theory
  • Mathematical Physics

Background:

  • Shannon's entropy power inequality (EPI) is a fundamental result in information theory.
  • Recent proofs have spurred further research into generalized EPIs.
  • Rényi entropy offers a flexible generalization of Shannon entropy.

Purpose of the Study:

  • To develop a comprehensive framework for deriving entropy power inequalities (EPIs) for Rényi entropy.
  • To unify and extend existing results on Rényi EPIs.
  • To provide new EPIs and bounds, especially for log-concave densities.

Main Methods:

  • Utilizing transport arguments derived from normal densities.
  • Employing a change of variable via rotation.
  • Unifying multiplicative forms (constant c) and modified forms (exponent α).

Main Results:

  • A comprehensive framework for deriving Rényi entropy power inequalities is established.
  • Previously known Rényi EPIs are recovered through simpler arguments.
  • New Rényi EPIs are derived, unifying existing approaches.
  • A sharp varentropy bound for log-concave densities is proven using a transportation method.

Conclusions:

  • The presented framework offers a unified approach to Rényi entropy power inequalities.
  • The methods provide new insights and results in information theory.
  • The work simplifies proofs and extends the applicability of EPIs to broader classes of densities.