Brascamp-Lieb不等式及其逆的信息论观点
Information-Theoretic Perspectives on Brascamp-Lieb Inequality and Its Reverse
摘要：
本文引入了一个不等式，它可以看作是Brascamp-Lieb不等式及其逆不等式（Barthe不等式）的推广，并证明了它的信息论（即熵）公式。这一结果导致了一个统一的方法来函数不等式，如Rényi熵的变分公式，超压缩性及其逆，强数据处理不等式，和运输成本不等式，其效用在各种编码定理的证明最近得到了越来越多的普及。我们表明，我们的信息理论的设置是方便的证明属性，如数据处理，张量化，凸性（Riesz-Thorin插值）和高斯最优性。特别是，我们详细说明了一个“加倍技巧”使用的Lieb和Geng Nair证明高斯最优性的几个结果。几个应用进行了讨论，包括推广的Brascamp-Lieb不等式涉及高斯随机变换，确定Wyner的共同信息的矢量高斯源，和可实现的速率区域的情况下，某些关键的生成问题的矢量高斯源。
英文摘要：
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.
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