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Entropy 2018, 20(6), 418; https://doi.org/10.3390/e20060418

A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality

1
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
2
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720-1770, USA
3
Renaissance Technologies, LLC 600 Route 25A East Setauket, New York, NY 11733, USA
*
Author to whom correspondence should be addressed.
Received: 30 March 2018 / Revised: 25 May 2018 / Accepted: 25 May 2018 / Published: 30 May 2018
(This article belongs to the Special Issue Entropy and Information Inequalities)
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Abstract

Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality that unifies both the Brascamp-Lieb inequality and Barthe’s inequality, which is a reverse form of the Brascamp-Lieb inequality. For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a “doubling trick” used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures. View Full-Text
Keywords: Brascamp-Lieb inequality; hypercontractivity; functional-entropic duality; Gaussian optimality; network information theory; image size characterization Brascamp-Lieb inequality; hypercontractivity; functional-entropic duality; Gaussian optimality; network information theory; image size characterization
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Liu, J.; Courtade, T.A.; Cuff, P.W.; Verdú, S. A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality. Entropy 2018, 20, 418.

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