Relative Optimal Transport
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914449450860544 |
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| author | Bubenik, Peter Elchesen, Alex |
| author_facet | Bubenik, Peter Elchesen, Alex |
| contents | We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, Kantorovich-Rubinstein duality and Monge-Kantorovich duality. We also prove relative versions of the Riesz-Markov-Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich-Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_05678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative Optimal Transport Bubenik, Peter Elchesen, Alex Metric Geometry Algebraic Topology Functional Analysis Optimization and Control Probability Primary: 49Q22, 28C05, Secondary: 51F99, 46A40, 55N31 We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich-Rubinstein norm, Wasserstein distance, Kantorovich-Rubinstein duality and Monge-Kantorovich duality. We also prove relative versions of the Riesz-Markov-Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich-Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest. |
| title | Relative Optimal Transport |
| topic | Metric Geometry Algebraic Topology Functional Analysis Optimization and Control Probability Primary: 49Q22, 28C05, Secondary: 51F99, 46A40, 55N31 |
| url | https://arxiv.org/abs/2411.05678 |