2017/02/21 by Jingbo Liu, Thomas A. Courtade, Liu, Jingbo +5 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #FOS: Computer and information sciences #Information Theory (cs.IT) #Point processes and geometric inequalities #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1702.06260
openalex publication_date 2017/02/21 · openalex created_date 2017/03/16 · openalex updated_date 2026/07/28
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.