arXiv：cs.LG· You Wan, Ting Gao, Jinqiao Duan·· 4 小时前AI 评分33
Wasserstein 空间中光滑势相互作用能量的信赖域优化
Trust-Region Optimization for Smooth Potential-Interaction Energies in Wasserstein Space
AI 导读
研究提出一种针对 Wasserstein 概率测度空间上光滑势相互作用能量的信赖域优化方法，沿 pushforward 曲线构造二次模型，采用 $L^2(\rho)$ 步长半径与 Steihaug-Toint 子求解器，并用比值检验决定接受与半径更新。
正文
Abstract:Finding low-energy configurations of interacting particles and approximating probability distributions lead to the minimization of potential-interaction energies in Wasserstein space. These energies can be nonconvex, making it important to exploit second-order information while controlling the reliability of local approximations. We study trust-region optimization of smooth potential-interaction energies on the Wasserstein space of probability measures with finite second moment. The method uses a quadratic model along pushforward curves, an $L^2(\rho)$ step radius, and a Steihaug-Toint subsolver with an explicit self-adjoint second-variation operator. A ratio test determines acceptance and guides the radius update. Under a lower energy bound and globally bounded Hessians of the potential and interaction kernel, we prove that the objective is nonincreasing, the Wasserstein-gradient norms converge to zero, and an $\varepsilon$-stationary iterate is reached within $O(\varepsilon^{-2})$ total outer trials, including rejected trials. If the potential is quadratically coercive, every weak accumulation point is stationary. The analysis applies to arbitrary initial measures with finite second moment. For empirical measures, the iteration is a finite-dimensional trust-region method in the $L^2(\rho_N)$ inner product, with complexity constants independent of particle number and dimension when the initial objective gaps are uniformly bounded. Numerical experiments include a smooth soft-particle energy, maximum-mean-discrepancy minimization for non-Gaussian targets, component ablations, and scaling studies in particle number and dimension.
| Comments: | 24 pages, 4 figures, 1 table; code and data in the ancillary files |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC); Probability (math.PR) |
| MSC classes: | 49Q22, 65K10, 60J60, 90C30 |
| Cite as: | arXiv:2610.08883 [stat.ML] |
| (or arXiv:2610.08883v1 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2610.08883 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: You Wan [view email]
[v1]
Tue, 6 Oct 2026 12:01:00 UTC (8,312 KB)
来源：arXiv：cs.LG · arxiv.org