Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich
Wild P, Schröder L, Messing K, König B, Forster J (2026)
Publication Type: Conference contribution
Publication year: 2026
Journal
Publisher: Springer Science and Business Media Deutschland GmbH
Book Volume: 16503 LNCS
Pages Range: 636-657
Conference Proceedings Title: Lecture Notes in Computer Science
Event location: Turin
ISBN: 9783032227294
DOI: 10.1007/978-3-032-22730-0_30
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 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.
Authors with CRIS profile
Paul Wild
Lehrstuhl für Informatik 8 (Theoretische Informatik)
Lutz Schröder
Lehrstuhl für Informatik 8 (Theoretische Informatik)
Jonas Forster
Lehrstuhl für Informatik 8 (Theoretische Informatik)
Involved external institutions
Germany (DE)How to cite
APA:
Wild, P., Schröder, L., Messing, K., König, B., & Forster, J. (2026). Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich. In Nathalie Bertrand, Stefan Milius (Eds.), Lecture Notes in Computer Science (pp. 636-657). Turin, IT: Springer Science and Business Media Deutschland GmbH.
MLA:
Wild, Paul, et al. "Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich." Proceedings of the 29th International Conference on Foundations of Software Science and Computation Structures, FoSSaCS 2026, Held as Part of the International Joint Conferences on Theory and Practice of Software, ETAPS 2026, Turin Ed. Nathalie Bertrand, Stefan Milius, Springer Science and Business Media Deutschland GmbH, 2026. 636-657.
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