Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions

TL;DR

This paper characterizes Wasserstein gradient flows of MMD functionals with a distance kernel on the real line, providing explicit solutions, invariance, smoothing properties, and numerical schemes, especially for discrete and continuous target measures.

Contribution

It offers a novel characterization of Wasserstein gradient flows of MMD functionals via Cauchy problems on quantile functions, including explicit solutions and numerical methods.

Findings

Flow solutions become absolutely continuous instantly for certain targets.

Explicit piecewise linear solutions for discrete measures.

Numerical schemes with convergence guarantees for specific cases.

Abstract

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ 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 $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\mathcal F_\nu$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\nu$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $\mathcal C(0,1)$. For certain $\mathcal F_\nu$-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 $\nu$, also the explicit Euler scheme can be employed, although with limited convergence guarantees.