$h$-Wasserstein barycenters
Fuente:
arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910337622605824 |
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| author | Brizzi, Camilla Friesecke, Gero Ried, Tobias |
| author_facet | Brizzi, Camilla Friesecke, Gero Ried, Tobias |
| contents | We generalize the notion and theory of Wasserstein barycenters introduced by Agueh and Carlier (2011) from the quadratic cost to general smooth strictly convex costs $h$ with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. To establish the latter result we introduce a new approach which is not based on explicitly solving the optimality system, but instead deriving a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $h$-barycenter on the support of an optimal multi-marginal plan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $h$-Wasserstein barycenters Brizzi, Camilla Friesecke, Gero Ried, Tobias Analysis of PDEs Primary 49Q20, Secondary 49J40, 49K21 We generalize the notion and theory of Wasserstein barycenters introduced by Agueh and Carlier (2011) from the quadratic cost to general smooth strictly convex costs $h$ with non-degenerate Hessian. We show the equivalence between a coupled two-marginal and a multi-marginal formulation and establish that the multi-marginal optimal plan is unique and of Monge form. To establish the latter result we introduce a new approach which is not based on explicitly solving the optimality system, but instead deriving a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $h$-barycenter on the support of an optimal multi-marginal plan. |
| title | $h$-Wasserstein barycenters |
| topic | Analysis of PDEs Primary 49Q20, Secondary 49J40, 49K21 |
| url | https://arxiv.org/abs/2402.13176 |