One-dimensional approximation of measures in Wasserstein distances

Antonin Chambolle; Vincent Duval; Joao Miguel Machado

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arXiv · 2304.14781

One-dimensional approximation of measures in Wasserstein distances

Antonin Chambolle·Vincent Duval·Joao Miguel Machado

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Abstract

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.

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Antonin Chambolle, Vincent Duval, Joao Miguel Machado. 2023-04-28. One-dimensional approximation of measures in Wasserstein distances. https://arxiv.org/abs/2304.14781

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