熵的Wasserstein连续性与干扰信道的外界
Wasserstein continuity of entropy and outer bounds for interference channels
摘要：
证明了在适当的正则性条件下，微分熵是$n$维欧氏空间上关于二次Wasserstein距离的分布空间上的Lipschitz泛函.在类似的条件下，（离散）香农熵被证明是Lipschitz连续的产品空间上的分布相对于Ornstein的$\bar d$距离（Wasserstein距离对应的汉明距离）。这些结果与Talagrand的和马顿的运输信息不等式允许一个取代未知的多用户干扰与它的独立同分布。近似值作为应用，证明了两用户高斯干扰信道的一个新的外界，特别解决了Costa（1985）的“角点缺失”问题。
英文摘要：
It is shown that under suitable regularity conditions, differential entropy is a Lipschitz functional on the space of distributions on $n$-dimensional Euclidean space with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be Lipschitz continuous in distributions over the product space with respect to Ornstein's $\bar d$-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interference with its i.i.d. approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the "missing corner point" problem of Costa (1985).
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