In this paper we provide a new geometric characterization of the
Hirschfeld-Gebelein-R\'{e}nyi maximal correlation of a pair of random $(X,Y)$,
as well as of the chordal slope of the nontrivial boundary of the
hypercontractivity ribbon of $(X,Y)$ at infinity. The new characterizations
lead to simple proofs for some of the known facts about these quantities. We
also provide a counterexample to a data processing inequality claimed by Erkip
and Cover, and find the correct tight constant for this kind of inequality.

本文提供了随机变量对 $(X,Y)$ 的 Hirschfeld-Gebelein-R\'{e} nyi 最大相关性以及在无穷远处的超收缩带的非平凡边界的弦斜率的新的几何特征，这些新特征简化了这些量的一些已知事实的证明，还给出了 Erkip 和 Cover 声称的数据处理不等式的反例，以及找到了这种不等式的正确紧致常数。

关于最大相关性、超收缩性和数据处理不等式的研究（Erkip 和 Cover 的论文）

On Maximal Correlation, Hypercontractivity, and the Data Processing  Inequality studied by Erkip and Cover

Reshef et al. recently proposed a new statistical measure, the "maximal
information coefficient" (MIC), for quantifying arbitrary dependencies between
pairs of stochastic quantities. MIC is based on mutual information, a
fundamental quantity in information theory that is widely understood to serve
this need. MIC, however, is not an estimate of mutual information. Indeed, it
was claimed that MIC possesses a desirable mathematical property called
"equitability" that mutual information lacks. This was not proven; instead it
was argued solely through the analysis of simulated data. Here we show that
this claim, in fact, is incorrect. First we offer mathematical proof that no
(non-trivial) dependence measure satisfies the definition of equitability
proposed by Reshef et al.. We then propose a self-consistent and more general
definition of equitability that follows naturally from the Data Processing
Inequality. Mutual information satisfies this new definition of equitability
while MIC does not. Finally, we show that the simulation evidence offered by
Reshef et al. was artifactual. We conclude that estimating mutual information
is not only practical for many real-world applications, but also provides a
natural solution to the problem of quantifying associations in large data sets.

提出了一种新的统计量 —— 最大信息系数 (MIC)，用于量化随机数量对之间的任意依赖关系，但证明了 MIC 不满足平等性定义。因此，我们提出了自洽且更普遍的平等性定义，并表明互信息满足这一新的定义，而 MIC 则不满足。最后，证明了 Reshef 等人提供的模拟证据是人为的。因此，我们得出结论，估计互信息不仅在许多实际应用中是可行的，而且还提供了一个自然的解决方案来量化大数据集中的相关性问题。

公平性、互信息和最大信息系数

Equitability, mutual information, and the maximal information  coefficient

The data processing inequality (DPI) is a fundamental feature of information
theory. Informally it states that you cannot increase the information content
of a quantum system by acting on it with a local physical operation. When the
smooth min-entropy is used as the relevant information measure, then the DPI
follows immediately from the definition of the entropy. The DPI for the von
Neumann entropy is then obtained by specializing the DPI for the smooth
min-entropy by using the quantum asymptotic equipartition property (QAEP). We
provide a new, simplified proof of the QAEP and therefore obtain a
self-contained proof of the DPI for the von Neumann entropy.

该研究提供了一种基于量子渐近等分定理（QAEP）的新的、简化的证明方法，从而获得了约束 von Neumann 熵的数据处理不等式（DPI）的自包含证明。