$p$-Wasserstein barycenters
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910658942992384 |
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| author | Brizzi, Camilla Friesecke, Gero Ried, Tobias |
| author_facet | Brizzi, Camilla Friesecke, Gero Ried, Tobias |
| contents | We study barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). We prove that
-- $p$-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely continuous
-- $p$-Wasserstein barycenters admit a multi-marginal formulation
-- the optimal multi-marginal plan is unique and of Monge form if the marginals are absolutely continuous, and its support has an explicit parametrization as a graph over any marginal space.
This extends the Agueh--Carlier theory of Wasserstein barycenters [SIAM J. Math. Anal. 43 (2011), no.2, 904--924] to exponents $p\neq 2$. A key ingredient is a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $p$-barycenter on the support of an optimal multi-marginal plan. We also discuss the statistical meaning of $p$-Wasserstein barycenters in one dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $p$-Wasserstein barycenters Brizzi, Camilla Friesecke, Gero Ried, Tobias Analysis of PDEs Probability Primary 49Q20, Secondary 49J40, 49K21 We study barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). We prove that -- $p$-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely continuous -- $p$-Wasserstein barycenters admit a multi-marginal formulation -- the optimal multi-marginal plan is unique and of Monge form if the marginals are absolutely continuous, and its support has an explicit parametrization as a graph over any marginal space. This extends the Agueh--Carlier theory of Wasserstein barycenters [SIAM J. Math. Anal. 43 (2011), no.2, 904--924] to exponents $p\neq 2$. A key ingredient is a quantitative injectivity estimate for the (highly non-injective) map from $N$-point configurations to their $p$-barycenter on the support of an optimal multi-marginal plan. We also discuss the statistical meaning of $p$-Wasserstein barycenters in one dimension. |
| title | $p$-Wasserstein barycenters |
| topic | Analysis of PDEs Probability Primary 49Q20, Secondary 49J40, 49K21 |
| url | https://arxiv.org/abs/2405.09381 |