Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels

2026-08-02

AI Summary

This work addresses the failure of standard gradient flow theory in Wasserstein space for maximum mean discrepancy (MMD) induced by non-smooth energy kernels, stemming from the lack of displacement semiconvexity. To overcome this, the authors develop a novel framework for Wasserstein gradient flows tailored to non-displacement-semiconvex energies. By integrating the modulated energy method, Lagrangian critical point analysis, subcritical $L^p$ estimates, moment bounds, mean-field limits, and $\omega$-limit set theory, they establish the first global well-posedness result for MMD gradient flows. The study reveals a critical correspondence between particle systems and their continuum limit and constructs collision-free saddle-point equilibria. Furthermore, it proves compactness and convergence of solution trajectories, achieving convergence to the target measure under rigidity conditions, while simultaneously disproving the validity of a global Polyak–Łojasiewicz inequality and the existence of an initial-data-independent MMD decay modulus.

Abstract

We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $ω$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak--Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.

Problem

• Wasserstein gradient flow
• Maximum Mean Discrepancy
• energy kernels
• nonconvex energy
• particle systems

Innovation

• Wasserstein gradient flow
• Maximum Mean Discrepancy
• energy kernels
• modulated energy
• nonconvex interaction