Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line

Abstract: This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient flows of functionals on $\mathcal P_2(\mathbb R)$ can be characterized as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum mean discrepancy functional $\mathcal F_\nu := \mathcal D^2_K(\cdot, \nu)$ with the non-smooth negative distance kernel $K(x,y) = -|x-y|$, we deduce a formula for the associated functional. This functional appears to be convex, and we show that $\mathcal F_\nu$ is convex along (generalized) geodesics. For the Dirac measure $\nu = \delta_q$, $q \in \mathbb R$ as end point of the flow, this enables us to determine the Wasserstein gradient flows analytically. Various examples of Wasserstein gradient flows are given for illustration.