The classical Kantorovich-Rubinstein duality guarantees coincidence between metrics on the space of probability distributions defined on the one hand via transport plans (couplings) and on the other hand via price functions. Both constructions have been lifted to the level of generality of set functors, with the construction based on couplings referred to as the Wasserstein or simply the coupling-based lifting, and the price-function-based construction as the Kantorovich or codensity lifting, both based on a choice of quantitative modalities for the given functor. It is known that every coupling-based lifting can be expressed as a price-function-based lifting; however, the latter in general needs to use additional modalities. We give an example showing that this cannot be avoided in general. We refer to cases in which the same modalities can be used as satisfying the generalized Kantorovich-Rubinstein duality. We establish the generalized Kantorovich-Rubinstein duality in this sense for two important cases: The Lévy-Prokhorov distance on distributions, which finds wide-spread applications in machine learning due to its favourable stability properties, and the standard metric on convex sets of distributions that arises by combining the Hausdorff and Kantorovich-Rubinstein distances.

翻译：经典的康德罗维奇-鲁宾斯坦对偶性保证了概率分布空间上两种度量定义方式的一致性：一种通过传输方案（耦合）定义，另一种通过价格函数定义。这两种构造已被提升到集合函子的一般性层面，其中基于耦合的构造被称为Wasserstein提升或简称为基于耦合的提升，而基于价格函数的构造则被称为康德罗维奇提升或共密度提升，两者都基于对给定函子的量化模态的选择。已知每个基于耦合的提升都可以表示为基于价格函数的提升；然而，后者通常需要使用额外的模态。我们给出一个例子，表明在一般情况下这是不可避免的。我们将能够使用相同模态的情形称为满足广义康德罗维奇-鲁宾斯坦对偶性。我们针对两个重要情形建立了这种意义上的广义康德罗维奇-鲁宾斯坦对偶性：其一是分布上的Lévy-Prokhorov距离，由于其良好的稳定性性质在机器学习中得到广泛应用；其二是分布凸集上的标准度量，该度量通过结合豪斯多夫距离与康德罗维奇-鲁宾斯坦距离而产生。