Wasserstein steepest descent flows of discrepancies with Riesz kernels (Q6542824)

In this paper, the interest in Wasserstein flows arises from the approximation of probability measures by empirical measures when halftoning images, i.e., the gray values of an image are considered as values of a probability density function of a measure.\N\NThe Wasserstein space \({\mathcal P}_2({\mathbb R}^d)\) is defined as metric space of all Borel measures with finite second moments equipped with the Wasserstein distance. First the authors introduce Wasserstein steepest descent flows which are locally absolutely continuous curves in \({\mathcal P}_2({\mathbb R}^d)\) whose tangent vectors point into a steepest descent direction of a given functional. This allows the use of Euler forward schemes instead of Jordan-Kinderlehrer-Otto schemes. For a \(\lambda\)-convex functional, the Wasserstein steepest descent flow coincides with the Wasserstein gradient flow.\N\NFurther, the authors study Wasserstein flows of the maximum mean discrepancy with respect to certain Riesz kernels. They present analytic expressions for Wasserstein steepest descent flows of the interaction energy starting at Dirac measures. Finally, for halftoning images they provide several numerical simulations of Wasserstein steepest descent flows of discrepancies.