Computing Barycentres of Measures for Generic Transport Costs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912984506302464 |
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| author | Tanguy, Eloi Delon, Julie Gozlan, Nathaël |
| author_facet | Tanguy, Eloi Delon, Julie Gozlan, Nathaël |
| contents | Wasserstein barycentres represent average distributions between multiple probability measures for the Wasserstein distance. The numerical computation of Wasserstein barycentres is notoriously challenging. A common approach is to use Sinkhorn iterations, where an entropic regularisation term is introduced to make the problem more manageable. Another approach involves using fixed-point methods, akin to those employed for computing Fréchet means on manifolds. The convergence of such methods for 2-Wasserstein barycentres, specifically with a quadratic cost function and absolutely continuous measures, was studied by Alvarez-Esteban et al. (2016). In this paper, we delve into the main ideas behind this fixed-point method and explore how it can be generalised to accommodate more diverse transport costs and generic probability measures, thereby extending its applicability to a broader range of problems. We show convergence results for this approach and illustrate its numerical behaviour on several barycentre problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing Barycentres of Measures for Generic Transport Costs Tanguy, Eloi Delon, Julie Gozlan, Nathaël Numerical Analysis Probability Wasserstein barycentres represent average distributions between multiple probability measures for the Wasserstein distance. The numerical computation of Wasserstein barycentres is notoriously challenging. A common approach is to use Sinkhorn iterations, where an entropic regularisation term is introduced to make the problem more manageable. Another approach involves using fixed-point methods, akin to those employed for computing Fréchet means on manifolds. The convergence of such methods for 2-Wasserstein barycentres, specifically with a quadratic cost function and absolutely continuous measures, was studied by Alvarez-Esteban et al. (2016). In this paper, we delve into the main ideas behind this fixed-point method and explore how it can be generalised to accommodate more diverse transport costs and generic probability measures, thereby extending its applicability to a broader range of problems. We show convergence results for this approach and illustrate its numerical behaviour on several barycentre problems. |
| title | Computing Barycentres of Measures for Generic Transport Costs |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2501.04016 |