交换群的熵界与Ruzsa发散
Entropy bounds on abelian groups and the Ruzsa divergence
摘要：
在过去的几年里，一个有趣的家庭新的不等式的熵的总和和差异的随机变量已开发的Ruzsa，陶和其他人，动机是类似的结果添加剂组合。目前的工作扩展这些早期的结果的情况下，随机变量取值在$\mathbb{R}^n$，或更一般地说，在任意的局部紧和波兰阿贝尔群。我们隔离和研究一个关键的数量，两个概率分布之间的Ruzsa分歧，我们表明，它的属性可以用来延长以前的不等式，目前的一般设置。建立的新结果包括几个变化的主题，两个独立的随机变量的和和差的熵严重相互约束。虽然设置是相当普遍的，结果已经感兴趣的（和新的）随机向量在$\mathbb{R}^n$。在这种特殊情况下，给出了熵幂不等式等式条件稳定性的定量界;证明了对数凹随机向量的逆熵幂不等式;建立了凸体Rogers-Shephard不等式的信息论模拟;并观察到其中一些结果导致了正定矩阵行列式的新不等式.此外，通过考虑复平面上的乘法子群，得到了非零复值随机变量的乘积和比的微分熵的新不等式。
英文摘要：
Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao and others, motivated by analogous results in additive combinatorics. The present work extends these earlier results to the case of random variables taking values in $\mathbb{R}^n$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the present general setting. The new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other. Although the setting is quite general, the result are already of interest (and new) for random vectors in $\mathbb{R}^n$. In that special case, quantitative bounds are provided for the stability of the equality conditions in the entropy power inequality; a reverse entropy power inequality for log-concave random vectors is proved; an information-theoretic analog of the Rogers-Shephard inequality for convex bodies is established; and it is observed that some of these results lead to new inequalities for the determinants of positive-definite matrices. Moreover, by considering the multiplicative subgroups of the complex plane, one obtains new inequalities for the differential entropies of products and ratios of nonzero, complex-valued random variables.
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