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

TL;DR

The work develops a comprehensive framework for Wasserstein gradient flows of MMD functionals with the non-smooth negative distance kernel in one dimension by embedding measures into the L2 cone of quantile functions. It constructs a continuous extension Fν on L2(0,1), computes its subdifferential, and proves that the Wasserstein gradient flow is governed by a Cauchy problem in L2(0,1) that remains in the quantile cone. The authors provide explicit pointwise solutions for the Cauchy problem, establish invariance and smoothing properties (including instantaneous regularization of point masses to absolutely continuous measures), and introduce practical implicit and explicit Euler schemes for numerically computing the flows, illustrating the dynamics with diverse target configurations. The results contribute precise, computable descriptions of 1D MMD flows with a non-smooth kernel, revealing how mass redistributes and how supports evolve under the influence of the target measure.

Abstract

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_ν:= \text{MMD}_K^2(\cdot, ν)$ towards given target measures $ν$ 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 $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\mathcal F_ν$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $ν$, 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 $\mathcal F_ν$-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 $ν$, also the explicit Euler scheme can be employed, although with limited convergence guarantees.

Figures (23)

• Figure 1: Wasserstein gradient flow of $\mathcal{F}_\nu$ from $\delta_{-1}$ towards $\nu = \delta_0$. The absolutely 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 flow changes immediately from a point measure to a uniform one with increasing support. It stays absolutely continuous until approaching the target support point $0$. Then it becomes the sum of an absolutely continuous measure and a discrete one, where the weight of the latter increases in time, see HBGS2023.
• Figure 2: Left: the CDF $R_{\mu}^+$ of a probability measure $\mu \in \mathop{\mathrm{\mathcal{P}}}\nolimits(\mathop{\mathrm{\mathbb R}}\nolimits)$. Right: the corresponding quantile function $Q_{\mu}$. Intervals of constancy translate to jumps and vice-versa.
• Figure 3: The map $[0, \infty) \to \mathop{\mathrm{\mathbb R}}\nolimits$, $t \mapsto [g(t)](s)$ for fixed $s \in (0, 1)$ with $Q_{\mu}(s) < Q_{\nu}(s)$(left) and $Q_{\mu}(s) > Q_{\nu}(s)$(right). The slopes of the affine linear pieces, determined by $s - R_{s, j}$, decrease for increasing $t$, that is, for increasing $j$.
• Figure 4: The Wasserstein gradient flow between the point measures in Example \ref{['example:TwoToThree']}. The gray regions illustrate the density of the absolute continuous part of the flow, whereas the red spikes illustrate the discrete part. In the lower row, the mass from $\delta_{-1}$ splits up, and a portion moves towards $\delta_{1}$. The movement speed of this portion, however, significantly decreases due to its attraction by $\delta_0$. Figuratively, the mass sticks to $\delta_0$ and can escape only slowly. For the corresponding quantile functions, see Figure \ref{['fig:pt-target_q']}.
• Figure 5: Left:$R_{\mu}^+$ is $\frac{1}{2}$-Lipschitz continuous, since there is a "double cone" (white) with extremal rays of slope $\frac{1}{2}$ such that if it is centered at any point of the graph of $R_{\mu}^+$, then the graph has no points within this double cone. Right:$Q_{\mu}$ admits a lower $2$-Lipschitz condition, since the white double cone with extremal rays of slope $2$ contains the whole graph of $Q_\mu$.

Theorems & Definitions (47)

• Theorem 2.1: Existence and uniqueness of Wasserstein gradient flows
• Remark 3.1
• Theorem 3.2
• Theorem 3.3
• Proposition 3.4
• Theorem 3.5
• proof
• Corollary 3.6
• Lemma 4.1
• proof