Wasserstein Gradient Flows of MMD Functionals with Distance Kernels under Sobolev Regularization

We consider Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals ${\text{MMD}}_K^2(\cdot ,\nu )$ for positive and negative distance kernels $K(x,y):=\pm \mathrm{\mid }x-y\mathrm{\mid }$ and given target measures $\nu$ on $\mathbb{R}$. Since in one dimension the Wasserstein space can be isometrically embedded into the cone $\mathcal{C}(0,1)\subset L_2(0,1)$ of quantile functions, Wasserstein gradient flows can be characterized by the solution of an associated Cauchy problem on $L_2(0,1)$. 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 $\mathrm{\mid }\cdot {\mathrm{\mid }}_{H^1(0,1)}^2$ 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 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.