Unbalanced Kantorovich-Rubinstein distance, plan, and barycenter on finite spaces: A statistical perspective
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916334424555520 |
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| author | Hundrieser, Shayan Heinemann, Florian Klatt, Marcel Struleva, Marina Munk, Axel |
| author_facet | Hundrieser, Shayan Heinemann, Florian Klatt, Marcel Struleva, Marina Munk, Axel |
| contents | We analyze statistical properties of plug-in estimators for unbalanced optimal transport quantities between finitely supported measures in different prototypical sampling models. Specifically, our main results provide non-asymptotic bounds on the expected error of empirical Kantorovich-Rubinstein (KR) distance, plans, and barycenters for mass penalty parameter $C>0$. The impact of the mass penalty parameter $C$ is studied in detail. Based on this analysis, we mathematically justify randomized computational schemes for KR quantities which can be used for fast approximate computations in combination with any exact solver. Using synthetic and real datasets, we empirically analyze the behavior of the expected errors in simulation studies and illustrate the validity of our theoretical bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08858 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unbalanced Kantorovich-Rubinstein distance, plan, and barycenter on finite spaces: A statistical perspective Hundrieser, Shayan Heinemann, Florian Klatt, Marcel Struleva, Marina Munk, Axel Methodology 65C60, 62R20, 05C05, 90C08, 62D99, 62G09 We analyze statistical properties of plug-in estimators for unbalanced optimal transport quantities between finitely supported measures in different prototypical sampling models. Specifically, our main results provide non-asymptotic bounds on the expected error of empirical Kantorovich-Rubinstein (KR) distance, plans, and barycenters for mass penalty parameter $C>0$. The impact of the mass penalty parameter $C$ is studied in detail. Based on this analysis, we mathematically justify randomized computational schemes for KR quantities which can be used for fast approximate computations in combination with any exact solver. Using synthetic and real datasets, we empirically analyze the behavior of the expected errors in simulation studies and illustrate the validity of our theoretical bounds. |
| title | Unbalanced Kantorovich-Rubinstein distance, plan, and barycenter on finite spaces: A statistical perspective |
| topic | Methodology 65C60, 62R20, 05C05, 90C08, 62D99, 62G09 |
| url | https://arxiv.org/abs/2211.08858 |