Wasserstein空间中光滑势能相互作用能量的信赖域优化
Trust-Region Optimization for Smooth Potential-Interaction Energies in Wasserstein Space
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中文总结 AI 辅助
本文提出Wasserstein空间中光滑势能相互作用能量的信赖域优化方法，利用二阶信息保证收敛，达到$O(\varepsilon^{-2})$迭代复杂度，并适用于经验测度，数值实验验证了其有效性。
中文摘要 AI 辅助
寻找相互作用粒子的低能构型以及近似概率分布，可归结为在Wasserstein空间中最小化势能相互作用能量。这些能量可能是非凸的，因此利用二阶信息同时控制局部近似的可靠性显得尤为重要。我们研究了在具有有限二阶矩的概率测度的Wasserstein空间上，对光滑势能相互作用能量进行信赖域优化。该方法沿前推曲线使用二次模型，采用$L^2(\rho)$步长半径，并使用带有显式自伴二阶变分算子的Steihaug-Toint子求解器。通过比率检验决定是否接受迭代并指导半径更新。在能量下界以及势能和相互作用核的Hessian全局有界的假设下，我们证明了目标函数非增，Wasserstein梯度范数收敛到零，并且在$O(\varepsilon^{-2})$次总外部迭代（包括被拒绝的迭代）内达到$\varepsilon$-平稳迭代点。若势能是二次强制的，则每个弱聚点都是平稳点。该分析适用于具有有限二阶矩的任意初始测度。对于经验测度，该迭代是在$L^2(\rho_N)$内积下的有限维信赖域方法，当初始目标间隙一致有界时，其复杂度常数与粒子数和维度无关。数值实验包括光滑软粒子能量、非高斯目标的最大均值差异最小化、组件消融以及粒子数和维度上的扩展性研究。
英文摘要
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 $L^2(ρ)$ 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 $L^2(ρ_N)$ 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.
发表机构
- Nanjing University（南京大学）
- Great Bay University（大湾区大学）
- Huazhong University of Science and Technology（华中科技大学）
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