Coulomb MMD flows: Exponential decay proven on torus, full-space obstructions identified

A new paper by Antonin Chodron de Courcel and Matthew Rosenzweig tackles the long-time behavior of Wasserstein gradient flows for the squared Maximum Mean Discrepancy (MMD) when the underlying kernel is a Coulomb potential (e.g., 1/r in 3D). The authors first prove that global weak solutions exist starting from arbitrary Borel probability measures, and that the density becomes instantly bounded in L^∞. They also show Hölder regularity can grow exponentially. On the flat torus 𝕋^d, they achieve a major result: exponential decay of the squared MMD toward any uniformly positive target measure μ, even when the initial density has zeros. This relies on a 'defective Polyak-Łojasiewicz (PL) inequality' whose defect handles vacuum regions, and they prove the standard PL inequality can fail when the target vanishes at even a single point.

On ℝ^d, the picture differs dramatically. For a compactly supported target, sources initially separated by distance D retain a fixed fraction of their initial squared MMD for times scaling with D. This implies no global PL inequality or uniform multiplicative decay modulus can exist on the unrestricted whole space. However, under radial symmetry, source-support inclusion (the initial measure's support contains the target's support), and target positivity, the authors recover a PL inequality and exponential convergence. These findings have direct implications for understanding convergence of MMD-based generative models and optimal transport algorithms in machine learning, highlighting when gradient flows are guaranteed to converge and when they are not.