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

Jingbo Liu,Sergio Verdú,Thomas Courtade,Paul Cuff

doi:10.3390/e20060418

Jingbo Liu, Sergio Verdú + Show 2 more

Open Access

https://doi.org/10.3390/e20060418

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Journal: Entropy	Publication Date: May 30, 2018
Citations: 17	License type: CC BY 4.0

Affiliation: Princeton University, University of California, Berkeley

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.

Highlights

The Brascamp-Lieb inequality and its reverse [1] concern the optimality of Gaussian functions in a certain type of integral inequality. (Not to be confused with the “variance Brascamp-Lieb inequality”(cf. [2,3,4]), which generalizes the Poincaré inequality)
We prove that the forward-reverse Brascamp-Lieb inequality (11) has an entropic formulation, which turns out to be very close to the rate region of certain multiuser information theory problems
In a similar vein as the proverbial result that “Gaussian functions are optimal” for the forward or the reverse Brascamp-Lieb inequality, we show in this paper that Gaussian functions are optimal for the forward-reverse Brascamp-Lieb inequality, particularized to the case of Gaussian reference measures and linear maps

Summary

Introduction

The Brascamp-Lieb inequality and its reverse [1] concern the optimality of Gaussian functions in a certain type of integral inequality. We prove that the forward-reverse Brascamp-Lieb inequality (11) has an entropic formulation, which turns out to be very close to the rate region of certain multiuser information theory problems (but we will clarify the difference in the text). Theorem 1 (Dual formulation of the forward-reverse Brascamp-Lieb inequality). In a similar vein as the proverbial result that “Gaussian functions are optimal” for the forward or the reverse Brascamp-Lieb inequality, we show in this paper that Gaussian functions are optimal for the forward-reverse Brascamp-Lieb inequality, particularized to the case of Gaussian reference measures and linear maps. In the literature on the forward or the reverse Brascamp-Lieb inequalities, it is known that a certain geometric condition (5) ensures that the best constant equals one. To completely prove Theorem 2, in Appendix F, we use a limiting argument to drop the non-degenerate assumption and apply the equivalence between the functional and entropic formulations

Review of the Legendre-Fenchel Duality Theory

Compact X

Noncompact X

Gaussian Optimality

Non-Degenerate Forward Channels

Analysis of Example 1 Using Gaussian Optimality

Hypercontractivity