发表机构
The University of Tokyo; Agency for Science, Technology and Research (A*STAR); Nanyang Technological University; RIKEN Center for Advanced Intelligence Project（东京大学; 新加坡科技研究局; 南洋理工大学; 理化学研究所先进智能研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析Hessian引导的扰动Wasserstein梯度流在有限粒子逼近下的跟踪准确性，证明在累积曲率条件下可实现增长时间范围的可靠逼近，并在多个模型中验证了相关条件。
AI 中文摘要
Wasserstein梯度流将梯度下降扩展到概率测度。其Hessian引导的扰动变体（PWGF）通过添加高斯扰动来逃离非凸问题中的鞍点。我们研究了当有限个相互作用粒子对其逼近在增长的时间范围内保持准确时的情况。我们的分析保留了沿总体驱动参考路径累积的曲率：负曲率可以放大逼近误差，而随后的正曲率可以抑制其影响。这捕捉了有利场景，其中暂时的不稳定性与在增长范围内准确跟踪兼容。在正则性假设和规定的共同扰动调度下，我们证明了在满足累积曲率显式条件的参考路径上，高概率事件下的粒子和目标值跟踪界限。为了处理状态依赖的高斯跳跃，我们构建了一种总体优先耦合，保留了参考粒子的条件独立性，并将跳跃误差简化为协方差比较。我们在方差加余弦模型中验证了这些条件，其中曲率恢复产生了增长范围的跟踪保证。我们还建立了正则化矩阵分解模型两个区域中的局部吸引、横向下降和正二阶变差，从而激发了正-负-正曲率模式。
英文摘要
Wasserstein gradient flow extends gradient descent to probability measures. Its Hessian-guided perturbed variant (PWGF) adds Gaussian perturbations to escape saddle points in nonconvex problems. We investigate when its approximation by finitely many interacting particles remains accurate over growing time horizons. Our analysis retains the curvature accumulated along the population-driven reference path: negative curvature can amplify approximation errors, while subsequent positive curvature can damp their influence. This captures favorable scenarios in which temporary instability is compatible with accurate tracking over growing horizons. Under regularity assumptions and a prescribed common perturbation schedule, we prove particle and objective-value tracking bounds on a high-probability event for reference paths satisfying explicit conditions on accumulated curvature. To handle state-dependent Gaussian jumps, we construct a population-first coupling that preserves the reference particles' conditional independence and reduces jump errors to covariance comparison. We verify the conditions in a variance-plus-cosine model, where curvature recovery yields a growing-horizon tracking guarantee. We also establish local attraction, transverse descent, and positive second variation in two regions of a regularized matrix-factorization model, motivating a positive-negative-positive curvature pattern.
