# Wasserstein gradient flows for Coulomb discrepancies

arXiv: https://arxiv.org/abs/2607.12579 · HTML: https://mlsanity.com/paper/2607.12579/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 24 references that are in the map.
- First posted on arXiv 14 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Wasserstein Steepest Descent Flows of Discrepancies with Riesz Kernels](https://mlsanity.com/paper/2211.01804/) (2022)
- [On the Wasserstein Gradient Flow Interpretation of Drifting Models](https://mlsanity.com/paper/2605.05118/) (2026)
- [Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies](https://mlsanity.com/paper/2603.01977/) (2026)
- [Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions](https://mlsanity.com/paper/2408.07498/) (2024)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Sharp Rates of MMD Empirical Estimation with Power Kernels](https://mlsanity.com/paper/2605.18497/) (2026)

## Built on by (most cited first, 2 of 2)

- [Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels](https://mlsanity.com/paper/2608.01182/) (2026)
- [Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies](https://mlsanity.com/paper/2609.27008/) (2026)

## Abstract (arXiv)

We study the long-time behavior of the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) between a probability measure ρ and a target measure μ, where the underlying kernel is given by a Coulomb potential. For L^∞ target densities μ, we establish the existence of global weak solutions starting from arbitrary Borel probability measures and prove that the density ρt belongs to L^∞ for any t>0. We also show that the H"older norm can grow exponentially in time. On the flat torus mathbbT^mathsfd, we prove a global metric PL inequality for every finite-Coulomb-energy source and nearly uniform target. For general bounded, uniformly positive targets, we prove exponential decay of the squared MMD without requiring a lower bound on the initial data, using a defective PL inequality. We also prove that the usual PL inequality may fail when the target vanishes only at one point and that, when mathsfd≥2, no PL constant can hold uniformly over all targets satisfying a prescribed lower bound. On ℝ^mathsfd, for mathsfd≥2, under radial symmetry, source-support inclusion, and target-positivity assumptions, we establish a PL inequality and exponential convergence. On the unrestricted whole-space class, neither a multiplicative squared-MMD decay modulus uniform over the initial datum nor a global PL inequality can hold. Finally, in every dimension and in both spatial settings, we prove that every Lagrangian critical point coincides with the target when (ρ-μ)^+ is absolutely continuous. In dimension two, the energy supplies uniform tightness. This implies that if our constructed solutions have finite energy at some positive time, then they converge to the target narrowly and strongly in negative-order Sobolev spaces.