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

Entropy

ThomasA. Courtade¹  Sergio Verdú²  Jingbo Liu²  PaulW. Cuff³ 

[1] Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720-1770, USA;Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA;Renaissance Technologies, LLC 600 Route 25A East Setauket, New York, NY 11733, USA;

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

DOI  :  10.3390/e20060418

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown