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Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies — arXiv:2609.27008

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.27008
License (as archived): CC0-1.0
Archived: 2026-10-08T10:57:52.849Z
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Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies — arXiv:2609.27008 — arXiv (en), https://arxiv.org/abs/2609.27008 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_911152e809dd (Metraipolis archive dc_911152e809dd, sha256 46dc28b5b7688c4294aa694057cef3f884cca971de4d68a0198e20cdf220e721)

BibTeX

@misc{dc911152e809dd,
  title = {Sharp Convergence of Wasserstein Gradient Flows for Spectrally Nonnegative Interaction Energies — arXiv:2609.27008},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.27008},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_911152e809dd; sha256 46dc28b5b7688c4294aa694057cef3f884cca971de4d68a0198e20cdf220e721},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.27008. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[\mu] = \frac12\iint_{M\times M}K(x,y)\,\mathrm d\mu(x)\,\mathrm d\mu(y) \] 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 $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies \[ \mathsf E[\mu_t]-\mathsf E_{\min}=o(t^{-1}). \] 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 $\delta>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-\delta}$ along a sequence of times tending to infinity. Moreover, for any $\delta>0$, 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 $t^{-1-\delta}$. 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.

Tema degli esploratori: agenti e intelligenza artificiale (retro-catalogo arXiv: annunci del 2026-09-24 in cs.LG).
Categorie arXiv: Analysis of PDEs (math.AP) ; Machine Learning (cs.LG); Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.27008 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T10:06:16.796Z dalla pagina https://arxiv.org/catchup/cs.LG/2026-09-24?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).