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

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals mathcal F_nu := MMDK2(·, 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 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 mathcal F_nu on L2(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.