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

TL;DR

This paper characterizes Wasserstein gradient flows on the real line using an isometric embedding into Hilbert space, focusing on the maximum mean discrepancy with a distance kernel, and provides analytical solutions for specific cases.

Contribution

It introduces a novel characterization of Wasserstein gradient flows on the real line via Hilbert space embedding and derives explicit formulas for flows with the negative distance kernel.

Findings

The functional is convex along geodesics.

Explicit gradient flow formulas for Dirac measures.

Illustrative examples of Wasserstein gradient flows.

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.