Brascamp-Lieb inequality and its reverse: An information theoretic view
Brascamp-Lieb inequality and its reverse: An information theoretic view
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布拉斯坎-利布不等式及其反面：信息论观点
DOI：
10.1109/isit.2016.7541459
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
S. Verdú
中科院分区：
文献类型：
--
作者：
Jingbo Liu;T. Courtade;P. Cuff;S. Verdú
我们通过允许随机变换（广播信道）来概括 Carlen 和 Cordero-Erausquin 关于 Brascamp-Lieb 不等式与相对熵的次可加性之间的等价性的结果。这导致了对在证明不可能结果的背景下越来越流行的几种函数不等式的统一观点。我们证明了布拉斯坎普-利布不等式的信息论对偶是证明数据处理、张量化、凸性和高斯最优性等性质的便捷设置。后者的后果包括允许高斯随机变换的 Brascamp-Lieb 不等式的扩展、高斯源的多元 Wyner 公共信息的确定以及纳尔逊超收缩定理的多元版本。最后，我们提出了涉及随机变换（多址信道）的逆 Brascamp-Lieb 不等式的信息论表征。
We generalize a result by Carlen and Cordero-Erausquin on the equivalence between the Brascamp-Lieb inequality and the subadditivity of relative entropy by allowing for random transformations (a broadcast channel). This leads to a unified perspective on several functional inequalities that have been gaining popularity in the context of proving impossibility results. We demonstrate that the information theoretic dual of the Brascamp-Lieb inequality is a convenient setting for proving properties such as data processing, tensorization, convexity and Gaussian optimality. Consequences of the latter include an extension of the Brascamp-Lieb inequality allowing for Gaussian random transformations, the determination of the multivariate Wyner common information for Gaussian sources, and a multivariate version of Nelson's hypercontractivity theorem. Finally we present an information theoretic characterization of a reverse Brascamp-Lieb inequality involving a random transformation (a multiple access channel).