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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2304.14781 |
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| _version_ | 1866929545235398656 |
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| author | Chambolle, Antonin Duval, Vincent Machado, Joao Miguel |
| author_facet | Chambolle, Antonin Duval, Vincent Machado, Joao Miguel |
| contents | We propose a variational approach to approximate measures with measures uniformly distributed over a 1 dimentional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of the support. As it is challenging to prove existence of solutions to this problem, we propose a relaxed formulation, which always admits a solution. In the sequel we show that if the ambient space is $\mathbb{R}^2$ , under techinical assumptions, any solution to the relaxed problem is a solution to the original one. Finally we manage to prove that any optimal solution to the relaxed problem, and hence also to the original, is Ahlfors regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14781 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | One-dimensional approximation of measures in Wasserstein distances Chambolle, Antonin Duval, Vincent Machado, Joao Miguel Analysis of PDEs We propose a variational approach to approximate measures with measures uniformly distributed over a 1 dimentional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of the support. As it is challenging to prove existence of solutions to this problem, we propose a relaxed formulation, which always admits a solution. In the sequel we show that if the ambient space is $\mathbb{R}^2$ , under techinical assumptions, any solution to the relaxed problem is a solution to the original one. Finally we manage to prove that any optimal solution to the relaxed problem, and hence also to the original, is Ahlfors regular. |
| title | One-dimensional approximation of measures in Wasserstein distances |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2304.14781 |