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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2512.03726 |
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| _version_ | 1866912746335895552 |
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| author | Vauthier, Christophe |
| author_facet | Vauthier, Christophe |
| contents | Let $M$ be a complete connected Riemannian manifold. For $n \geq 0$, we endow the Wasserstein space $P^{(n)}_2(M) = P_2(\ldots P_2(M)\ldots)$, equipped with the Wasserstein distance $W_2$, with a variational structure that generalizes the standard variational structure on $P_2(M)$ provided by optimal transport theory. Our approach makes use of tools from category theory to lift the geometric structure of the manifold $M$ to the spaces $P^{(n)}_2(M)$, in order to establish in a principled way a rigorous theoretical framework for variational analysis on the space $P^{(n)}_2(M)$. In particular, we obtain a precise characterization of the constant speed geodesics of the space $P^{(n)}_2(M)$ in terms of optimal velocity plans. Moreover, we introduce a notion of gradient for functionals defined on $P^{(n)}_2(M)$, which allows us to study the differentiability and the convexity of various types of such functionals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational Analysis in the Wasserstein Hierarchy Vauthier, Christophe Optimization and Control Let $M$ be a complete connected Riemannian manifold. For $n \geq 0$, we endow the Wasserstein space $P^{(n)}_2(M) = P_2(\ldots P_2(M)\ldots)$, equipped with the Wasserstein distance $W_2$, with a variational structure that generalizes the standard variational structure on $P_2(M)$ provided by optimal transport theory. Our approach makes use of tools from category theory to lift the geometric structure of the manifold $M$ to the spaces $P^{(n)}_2(M)$, in order to establish in a principled way a rigorous theoretical framework for variational analysis on the space $P^{(n)}_2(M)$. In particular, we obtain a precise characterization of the constant speed geodesics of the space $P^{(n)}_2(M)$ in terms of optimal velocity plans. Moreover, we introduce a notion of gradient for functionals defined on $P^{(n)}_2(M)$, which allows us to study the differentiability and the convexity of various types of such functionals. |
| title | Variational Analysis in the Wasserstein Hierarchy |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2512.03726 |