Global regularity and sparsity for regularised optimal transport

Abstract

We study regularised optimal transport with subquadratic or entropic regularisation. Under $C^{1,\alpha }$-assumptions on the marginals and the domains of their support, we prove global gradient-Lipschitz bounds on the potentials, uniform in the regularisation parameter. Moreover, we show that, up to the boundary, the support of the conditional supports $\text{supp}{\pi }_{\epsilon }(\cdot |x)$ behaves like balls of radius ${\epsilon }^{\frac{4}{d(p-1)+2}}$. Our work was carried out concurrently and independently of [9]. We obtain the results of [9,Theorem 1.2] under weaker assumptions and with a different proof.

arXiv commentv2. Some small typos in text and bibliography corrected