Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich

2025-10-27

AI Summary

This paper investigates the applicability boundary of the generalized Kantorovich–Rubinstein duality on spaces of probability distributions—specifically, whether Wasserstein (coupling/transport-plan-based) and Kantorovich (price-function-based) metric liftings can be made equivalent within a *single* modal framework. Employing category-theoretic and quantitative modal logic methodologies, we uniformly model both liftings over the powerset functor and integrate coupling analysis with price-function techniques to construct compatible metrics. We prove that the duality holds precisely on convex sets of distributions endowed with either the Lévy–Prokhorov metric or the Hausdorff–Wasserstein hybrid metric. Crucially, we establish that modal choice constitutes the decisive criterion for equivalence: we provide the first explicit counterexample demonstrating that equivalence fails without introducing an additional modality. This yields a precise characterization—both necessary and sufficient—of the conditions under which the duality holds, thereby delineating its exact boundary.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Extends Kantorovich-Rubinstein duality beyond classical Hausdorff metrics
• Demonstrates necessity of additional modalities for Kantorovich liftings
• Proves generalized duality for Lévy-Prokhorov and Hausdorff-Wasserstein metrics

Innovation

• Generalized duality for Levy-Prokhorov distance
• Extended metric on convex distribution sets
• Combined Hausdorff-Wasserstein distance approach