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

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ 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 $\mathcal F_\nu$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\nu$, 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 $\mathcal F_\nu$-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 and demonstrate differences to the explicit Euler scheme, which is easier to compute, but comes with limited convergence guarantees.

Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces

Most commonly used $f$-divergences of measures, e.g., the Kullback-Leibler divergence, are subject to limitations regarding the support of the involved measures. A remedy consists of regularizing the $f$-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel $K$. In this paper, we use the so-called kernel mean embedding to show that the corresponding regularization can be rewritten as the Moreau envelope of some function in the reproducing kernel Hilbert space associated with $K$. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to prove properties of the MMD-regularized $f$-divergences and, in particular, their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized $f$-divergences. Finally, we consider Wasserstein gradient flows starting from empirical measures and provide proof-of-the-concept numerical examples with Tsallis-$\alpha$ divergences.