谁驱动系统？从轨迹数据对平均场粒子系统进行分类
Who Drives the System? Classifying Mean-Field Particle Systems from Trajectory Data
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- SAMM, Université Paris 1（SAMM，巴黎第一大学）
- LPSM, Sorbonne Université, Université Paris Cité, CNRS（LPSM，索邦大学，巴黎西岱大学，法国国家科学研究中心）
- CMAP, Ecole Polytechnique, IP Paris（CMAP，巴黎综合理工学院，巴黎理工学院）
- CEREMADE, Université Paris Dauphine 1（CEREMADE，巴黎第九大学）
机构由 AI 辅助整理，请以论文原文为准。
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中文总结 AI 辅助
本文针对相互作用粒子系统的轨迹分类问题，利用McKean-Vlasov极限提出基于漂移估计的插件分类方法，收敛速度为N^{-1/6+ε}，优于神经网络基线。
中文摘要 AI 辅助
我们研究了基于轨迹观测的相互作用粒子系统（IPS）的监督分类问题。IPS属于K个类别之一，每个类别由不同的相互作用漂移刻画。给定一个学习数据集，目标是基于单个粒子的轨迹预测新观测系统的类别标签。这一设定带来了若干统计挑战。首先，每个系统内的粒子相互交互，因此是依赖的。其次，尽管整个粒子系统是马尔可夫的，但单个粒子的轨迹并非如此。为解决这些困难，我们利用IPS的McKean-Vlasov极限，该极限描述了当粒子数趋于无穷时典型粒子的动力学。我们提出了一种基于从离散观测轨迹数据中估计每个类别相关相互作用漂移的插件分类程序。因此，对于每个类别，我们通过在B样条基上最小化岭正则化最小二乘对比度，提供了一种新的非参数漂移估计器。特别地，我们的理论发现表明，所得分类器的收敛速度对于任意ε>0为N^{-1/6+ε}阶，其中N表示每个系统中的粒子数。数值实验展示了所提方法的性能，并表明其优于不利用底层粒子系统结构的端到端神经网络基线。
英文摘要
We study the supervised classification problem for interacting particle systems (IPS) from trajectory observations. The IPS belongs to one of K classes, each characterized by a distinct interaction drift. Given a learning dataset, the goal is then to predict the class label of a newly observed system based on the trajectory of a single particle. This setting raises several statistical challenges. First, the particles within each system interact and are therefore dependent. Second, although the full particle system is Markovian, the trajectory of a single particle is not. To address these difficulties, we exploit the McKean-Vlasov limit of the IPS, which describes the dynamics of a typical particle as the number of particles tends to infinity. We propose a plug-in classification procedure based on estimating the interaction drift associated with each class from discretely observed trajectory data. Hence, for each class, we provide a new nonparametric estimator of the drifts by minimizing a ridge-regularized least-squares contrast over a B-spline basis. In particular, our theoretical findings reveal that the convergence rate of the resulting classifier is of order N -1/6+$ε$ for any $ε$ > 0, where N denotes the number of particles in each system. Numerical experiments illustrate the performance of the proposed method and show that it outperforms an end-to-end neural network baseline that does not exploit the underlying particle-system structure.
