Generalized Wasserstein Barycenters
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912463743614976 |
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| author | Tornabene, Francesco Veneroni, Marco Savaré, Giuseppe |
| author_facet | Tornabene, Francesco Veneroni, Marco Savaré, Giuseppe |
| contents | We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is atomic, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the $L^2$-barycenter of the quantiles on the cone of nonincreasing functions in $L^2(0,1)$. Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in $\mathbb{R}^2$. Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06838 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Wasserstein Barycenters Tornabene, Francesco Veneroni, Marco Savaré, Giuseppe Probability Functional Analysis Optimization and Control 49J27, 49J45, 49Q22 We study the existence and uniqueness of the barycenter of a signed distribution of probability measures on a Hilbert space. The barycenter is found, as usual, as a minimum of a functional. In the case where the positive part of the signed measure is atomic, we can show also uniqueness. In the one-dimensional case, we characterize the quantile function of the unique minimum as the orthogonal projection of the $L^2$-barycenter of the quantiles on the cone of nonincreasing functions in $L^2(0,1)$. Further, we provide a stability estimate in dimension one and a counterexample to uniqueness in $\mathbb{R}^2$. Finally, we address the consistency of the barycenters and we prove that barycenters of a sequence of approximating measures converge (up to subsequences) to a barycenter of the limit measure. |
| title | Generalized Wasserstein Barycenters |
| topic | Probability Functional Analysis Optimization and Control 49J27, 49J45, 49Q22 |
| url | https://arxiv.org/abs/2411.06838 |