Next Article in Journal
Multivariate Matching Pursuit Decomposition and Normalized Gabor Entropy for Quantification of Preictal Trends in Epilepsy
Next Article in Special Issue
Entropy Inequalities for Lattices
Previous Article in Journal
Modeling the Comovement of Entropy between Financial Markets
Previous Article in Special Issue
On f-Divergences: Integral Representations, Local Behavior, and Inequalities
Open AccessArticle

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

Figure 1

MDPI and ACS Style

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.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map

1
Back to TopTop