# Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies

arXiv: https://arxiv.org/abs/2603.01977 · HTML: https://mlsanity.com/paper/2603.01977/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2026.

## Built on by (most cited first, 10 of 10)

- [Gradient Flow Drifting: Generative Modeling via Wasserstein Gradient Flows of KDE-Approximated Divergences](https://mlsanity.com/paper/2603.10592/) (2026)
- [Sobolev Regularized MMD Gradient Flow](https://mlsanity.com/paper/2605.11884/) (2026)
- [Three-Body Scattering for Generative Modeling](https://mlsanity.com/paper/2607.18198/) (2026)
- [Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow](https://mlsanity.com/paper/2605.09456/) (2026)
- [Uniform-in-Time Weak Propagation-of-Chaos in Shallow Neural Networks](https://mlsanity.com/paper/2605.22010/) (2026)
- [Wasserstein gradient flows for Coulomb discrepancies](https://mlsanity.com/paper/2607.12579/) (2026)
- [Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels](https://mlsanity.com/paper/2608.01182/) (2026)
- [Wasserstein Saddle-Free Newton: Saddle Escape and Local Spectral Convergence](https://mlsanity.com/paper/2605.17963/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)
- [Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies](https://mlsanity.com/paper/2609.27008/) (2026)

## Abstract (arXiv)

We study the quantitative convergence of Wasserstein gradient flows of Kernel Mean Discrepancy (KMD) (also known as Maximum Mean Discrepancy (MMD)) functionals. Our setting covers in particular the training dynamics of shallow neural networks in the infinite-width and continuous time limit, as well as interacting particle systems with pairwise Riesz kernel interaction in the mean-field and overdamped limit. Our main analysis concerns the model case of KMD functionals given by the squared Sobolev distance  mathscrEnus(μ)= frac12lVert μ-nurVert_dot H-s2 for any s≥ 1  and nu a fixed probability measure on the d-dimensional torus. First, inspired by Yudovich theory for the 2d-Euler equation, we establish existence and uniqueness in natural weak regularity classes. Next, we show that for s=1 the flow converges globally at an exponential rate under minimal assumptions, while for s>1 we prove local convergence at polynomial rates that depend explicitly on s and on the Sobolev regularity of μ and nu. These rates hold both at the energy level and in higher regularity classes and are tight for nu uniform. We then consider the gradient flow of the population loss for shallow neural networks with ReLU activation, which can be cast as a Wasserstein--Fisher--Rao gradient flow on the space of nonnegative measures on the sphere mathbbSd. Exploiting a correspondence with the Sobolev energy case with s=(d+3)/2, we derive an explicit polynomial local convergence rate for this dynamics. Except for the special case s=1, even non-quantitative convergence was previously open in all these settings. We also include numerical experiments in dimension d=1 using both PDE and particle methods which illustrate our analysis.