Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality
Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality
复制标题
DOI：
10.1109/isit.2019.8849711
复制
发表时间：
2019-01
期刊：
影响因子：
--
通讯作者：
V. Anantharam;Varun Jog;Chandra Nair
中科院分区：
文献类型：
--
作者：
V. Anantharam;Varun Jog;Chandra Nair
熵幂不等式(EPI)和BrasCamp-Lieb不等式(BLI)是关于随机向量的线性变换的微分熵的基本不等式。EPI提供了具有独立分量的随机向量的线性变换的微分熵的下界。另一方面，BLI根据随机向量的一些线性变换的微分熵提供了随机向量的微分熵的上界。在这篇文章中，我们定义了一族熵泛函，我们证明了它们是次可加的。然后，我们通过模仿耿氏和奈尔(2014)中的想法，证明了对于这些泛函来说，高斯是极端的。因此，我们得到了一个新的熵不等式，它同时推广了BLI和EPI。通过考虑出现在这些泛函中的随机向量分量之间的各种独立关系，我们还得到了介于EPI和BLI之间的一族不等式族。
The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by mimicking the idea in Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI.