Wasserstein gradient flows of maximum mean discrepancy functionals with distance kernels under Sobolev regularization

Abstract

We consider Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals, [Formula: see text] for positive and negative distance kernels [Formula: see text] and given target measures [Formula: see text] on [Formula: see text]. Since in one dimension, the Wasserstein space can be isometrically embedded into the cone [Formula: see text] of quantile functions, Wasserstein gradient flows can be characterized by the solution of an associated Cauchy problem on [Formula: see text]. While for the negative kernel, the MMD functional is geodesically convex, this is not the case for the positive kernel, which needs to be handled to ensure the existence of the flow. We propose to add a regularizing Sobolev term [Formula: see text] corresponding to the Laplacian with Neumann boundary conditions to the Cauchy problem of quantile functions. Indeed, this ensures the existence of a generalized minimizing movement (GMM) for the positive kernel. Furthermore, for the negative kernel, we demonstrate by numerical examples how the Laplacian rectifies a 'dissipation-of-mass' defect of the MMD gradient flow.This article is part of the theme issue 'Partial differential equations in data science'.

Keywords: Sobolev regularization; Wasserstein gradient flow; distance kernels; maximum mean discrepancy; minimizing movement scheme.