信息组合的Rényi界
Rényi Bounds on Information Combining
摘要：
信息组合的界限是熵不等式，它决定了一组随机变量的信息或熵在以某些规定的方式组合时如何变化。这样的界限在信息论，特别是在编码和香农理论中起着重要的作用。可以说，最基本的信息组合是两个二进制随机变量的相加，即CNOT门，并且在研究置信传播和极性编码时，所得到的量是基本的。在这项工作中，我们将推广到雷尼熵的概念。我们给出了条件Rényi熵组合后的最优界，基于一定的凸性或非凸性，并讨论了当这个属性确实成立。由于没有普遍同意的条件雷尼熵的定义，我们认为四个不同的版本从文献。最后，我们讨论了这些界的应用程序的极化码下的Rényi熵的极化。
英文摘要：
Bounds on information combining are entropic inequalities that determine how the information, or entropy, of a set of random variables can change when they are combined in certain prescribed ways. Such bounds play an important role in information theory, particularly in coding and Shannon theory. The arguably most elementary kind of information combining is the addition of two binary random variables, i.e. a CNOT gate, and the resulting quantities are fundamental when investigating belief propagation and polar coding. In this work we will generalize the concept to Rényi entropies. We give optimal bounds on the conditional Rényi entropy after combination, based on a certain convexity or concavity property and discuss when this property indeed holds. Since there is no generally agreed upon definition of the conditional Rényi entropy, we consider four different versions from the literature. Finally, we discuss the application of these bounds to the polarization of Rényi entropies under polar codes.
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