Information-Theoretic Perspectives on Brascamp-Lieb Inequality and Its Reverse

Abstract

We introduce an inequality which may be viewed as a generalization of both the Brascamp-Lieb inequality and its reverse (Barthe's inequality), and prove its information-theoretic (i.e.\ entropic) formulation. This result leads to a unified approach to functional inequalities such as the variational formula of Rényi entropy, hypercontractivity and its reverse, strong data processing inequalities, and transportation-cost inequalities, whose utility in the proofs of various coding theorems has gained growing popularity recently. We show that our information-theoretic setting is convenient for proving properties such as data processing, tensorization, convexity (Riesz-Thorin interpolation) and Gaussian optimality. In particular, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove several results on Gaussian optimality. Several applications are discussed, including a generalization of the Brascamp-Lieb inequality involving Gaussian random transformations, the determination of Wyner's common information of vector Gaussian sources, and the achievable rate region of certain key generation problems in the case of vector Gaussian sources.

Corrected some typos in the previous version

References in corpus (5)

• Wyner's Common Information: Generalizations and A New Lossy Source Coding Interpretation
• Brascamp-Lieb Inequality and Its Reverse: An Information Theoretic View
• An Extremal Inequality for Long Markov Chains
• Converses for distributed estimation via strong data processing inequalities
• Sylvester-Gallai for Arrangements of Subspaces

Cited by in corpus (5)

• Quantum Brascamp-Lieb Dualities
• Second-Order Converses via Reverse Hypercontractivity
• Brascamp-Lieb Inequality and Its Reverse: An Information Theoretic View
• The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems
• Dispersion Bound for the Wyner-Ahlswede-Körner Network via Reverse Hypercontractivity on Types