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首页> 外文会议>IEEE International Symposium on Information Theory >Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality
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Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality

机译：统一Brascamp-Lieb不等式和熵幂不等式

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摘要

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.
机译：熵幂不等式（EPI）和Brascamp-Lieb不等式（BLI）是关于随机向量线性变换的微分熵的基本不等式。 EPI为具有独立分量的随机向量的线性变换的微分熵提供了下界。另一方面，就其线性变换的某些差分熵而言，BLI提供了随机矢量的差分熵的上限。在本文中，我们定义了一系列熵函数，这些函数表明是亚可加的。然后，我们通过模仿Geng和Nair（2014）中的想法，确定高斯人对于这些功能极端。结果，我们获得了一个新的熵不等式，它可以同时推广BLI和EPI。通过考虑出现在这些函数中的随机向量的组成部分之间的各种独立性关系，我们还获得了EPI和BLI之间的不等式族。

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