Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912907336351744 |
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| author | Wild, Paul Schröder, Lutz Messing, Karla König, Barbara Forster, Jonas |
| author_facet | Wild, Paul Schröder, Lutz Messing, Karla König, Barbara Forster, Jonas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_23552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich Wild, Paul Schröder, Lutz Messing, Karla König, Barbara Forster, Jonas Logic in Computer Science Probability 03B45, 03B52, 68Q85 F.4.1 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. |
| title | Generalized Kantorovich-Rubinstein Duality beyond Hausdorff and Kantorovich |
| topic | Logic in Computer Science Probability 03B45, 03B52, 68Q85 F.4.1 |
| url | https://arxiv.org/abs/2510.23552 |