Liu, J.; Courtade, T.A.; Cuff, P.W.; Verdú, S. A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality. Entropy2018, 20, 418.
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.
Liu, J.; Courtade, T.A.; Cuff, P.W.; Verdú, S. A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality. Entropy2018, 20, 418.
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.
Abstract
Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality which 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.
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