Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich

TL;DR

This paper extends Kantorovich-Rubinstein duality beyond traditional settings to general set functors, establishing conditions under which coupling-based and price-function-based metrics coincide, with applications to Lévý-Prokhorov and convex set distances.

Contribution

It proves a generalized Kantorovich-Rubinstein duality for broader classes of metrics on probability distributions and convex sets, beyond Hausdorff and Kantorovich-Rubinstein distances.

Findings

Established duality for Lévý-Prokhorov distance.

Proved duality for combined Hausdorff and Kantorovich-Rubinstein metrics.

Demonstrated cases where modalities can be shared in duality.

Abstract

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 L\'evy-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.