Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 30 March 2018 / Approved: 2 April 2018 / Online: 2 April 2018 (06:02:33 CEST)

How to cite: Liu, J.; Courtade, T.A.; Cuff, P.W.; Verdú, S. A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality. Preprints 2018, 2018040005 (doi: 10.20944/preprints201804.0005.v1). Liu, J.; Courtade, T.A.; Cuff, P.W.; Verdú, S. A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality. Preprints 2018, 2018040005 (doi: 10.20944/preprints201804.0005.v1).

Abstract

Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality which 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.

Subject Areas

Brascamp-Lieb inequality; hypercontractivity; functional-entropic duality; Gaussian optimality; network information theory; image size characterization

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