# Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies

arXiv: https://arxiv.org/abs/2609.27008 · HTML: https://mlsanity.com/paper/2609.27008/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 52 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Emergence of meta-stable clustering in mean-field transformer models](https://mlsanity.com/paper/2410.23228/) (2024)
- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [Quantitative Clustering in Mean-Field Transformer Models](https://mlsanity.com/paper/2504.14697/) (2025)
- [Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel](https://mlsanity.com/paper/2310.03054/) (2023)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Deep MMD Gradient Flow without adversarial training](https://mlsanity.com/paper/2405.06780/) (2024)
- [Quantitative Convergence of Wasserstein Gradient Flows of Kernel Mean Discrepancies](https://mlsanity.com/paper/2603.01977/) (2026)
- [A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces](https://mlsanity.com/paper/2310.17610/) (2023)
- [Convergence of drift-diffusion PDEs arising as Wasserstein gradient flows of convex functions](https://mlsanity.com/paper/2507.12385/) (2025)
- [Propagation of Chaos in Contextual Flow Maps](https://mlsanity.com/paper/2605.16747/) (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)

## Abstract (arXiv)

We study the long-time behavior of Wasserstein gradient flows for interaction energies [ mathsf E[μ] = frac12iintM× MK(x,y),mathrm dμ(x),mathrm dμ(y) ] on a closed manifold M. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density u0∈ Lp(M), p>1, the energy gap is integrable in time and satisfies [ mathsf E[μt]-mathsf Emin=o(t-1). ] If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the o(t-1) rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any δ>0, we construct a solution of the linearized flow whose energy is comparable to t-1-δ along a sequence of times tending to infinity. Moreover, for any δ>0, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to t-1-δ. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.