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

TL;DR

For the Dirac measure ν, it is shown that F ν is convex along (generalized) geodesics, and this functional appears to be convex, and the Wasserstein gradient is determined analytically.

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 $\mathcal F_ν:= \mathcal D^2_K(\cdot, ν)$ 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 $\mathcal F_ν$ is convex along (generalized) geodesics. For the Dirac measure $ν= δ_q$, $q \in \mathbb R$ 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.

Figures (4)

• Figure 1: Visualization of the Wasserstein gradient flow of $\mathcal{F}_{\delta_0}$ from $\delta_{-1}$ to $\delta_0$. At various times $t$, the absolute continuous part is visualized by its density in blue (area equals mass) and the atomic part by the red dotted vertical line (height equals mass). The atomic part at the end point $x=0$ starts to grow at time $t=\tfrac{1}{2}$, where the support of the density touches this point for the first time.
• Figure 2: Wasserstein gradient flow of $\mathcal{F}_{1,\nu}$ for $\nu = \delta_0$ (left) and $\nu=\frac{1}{2} \lambda_{[-1,1]}$ (right) from various initial points $\delta_{x}$, $x \in [-2,2]$. The support of the right measure $\nu$ is depicted by the blue region. The examples show that gradient flows may reach the optimal points in finite or infinite time.
• Figure 3: Wasserstein gradient flow $\mathcal{F}_{2,\delta_0}$ from $(m(0),\sigma(0))$ to $\delta_0$ (left) and from $\delta_{-1}$ to $\delta_0$ (right). In contrast Figure \ref{['fig:gradient_flow_dirac_dirac_P2R']} it is a uniform measure for all $t \in (0,1)$.
• Figure 4: Visualization of the energy landscapes of $\mathcal{F}_{2,\lambda_{[-1,1]}}$ for the convex negative distance kernel (left) and the non-convex, smooth kernel given in \ref{['eq:rbf_kernel']}. The red dot is the global minimizer $\lambda_{[-1,1]}$ (left and right) and the blue point (right) is the saddle point $\delta_0$. The black lines depict selected gradient flows.

Theorems & Definitions (8)

• theorem thmcountertheorem: Vil03
• theorem thmcountertheorem
• proof
• remark thmcounterremark
• lemma thmcounterlemma
• proof
• proposition thmcounterproposition
• proof