Wasserstein gradient flows of MMD functionals with distance kernel and Cauchy problems on quantile functions

¹ Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
² Univesité Paris Dauphine-PSL and Inria Mokaplan, Paris, France

^* Corresponding author: richard This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 11 September 2024
Accepted: 3 December 2025

Abstract

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals ℱ_v: = MMD_K²(⋅,v) towards given target measures v on the real line, where we focus on the negative distance kernel K(x, y) := − |x − y|. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone C(0,1) ⊂ L₂(0, 1) of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on L₂(0,1). Based on the construction of an appropriate counterpart of ℱ_v on L₂(0, 1) and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures v, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of C(0,1). For certain ℱ_v-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme, which is easily computable by a bisection algorithm. For continuous targets v, also the explicit Euler scheme can be employed, although with limited convergence guarantees.