Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich

TL;DR

This paper generalizes Kantorovich-Rubinstein duality beyond the classical Hausdorff and Kantorovich liftings by examining when Wasserstein and Kantorovich liftings coincide for various functor-based distance notions. It first presents a counterexample showing failure of duality for the $p$-Wasserstein distance with $p>1$, then proves duality for two substantial cases: the Lévy-Prokhorov distance on distributions and the standard distance on convex sets of distributions (the convex powerset), including their extension to Borel measures. The Lévy-Prokhorov result relies on a Wasserstein representation with a specialized predicate lifting and relational duality, while the convex powerset result leverages metric quotients and Lipschitz duality to establish a robust Kantorovich description and even gives a polynomial-time LP-based approach for computing distances. These dualities enable explicit, formula-based witnesses in quantitative coalgebraic logics and provide practical computational advantages for systems combining probabilistic and nondeterministic behavior.

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 coupling-based construction referred to as the Wasserstein lifting, and the price-function-based construction as the Kantorovich lifting, both based on a choice of quantitative modalities for the given functor. It is known that every Wasserstein lifting can be expressed as a Kantorovich 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 Wasserstein distances.

Figures (1)

• Figure 1: Illustration of \ref{['expl:convex-powerset']}. The thick line segments correspond to the sets $A = \mathsf{conv}(A_0)$ and $B = \mathsf{conv}(B_0)$, while the shaded region shows the radius-$\frac{1}{2}$ ball around $\mu_1$, which is part of a regular hexagon centered at $\mu_1$.

Theorems & Definitions (22)

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