Article
Version 1
Preserved in Portico This version is not peer-reviewed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)
A peer-reviewed article of this Preprint also exists.
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. 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.
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.
Keywords
Brascamp-Lieb inequality; hypercontractivity; functional-entropic duality; Gaussian optimality; network information theory; image size characterization
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment