In this paper, we tackle a critical issue in nonparametric inference for systems of interacting particles on Riemannian manifolds: the identifiability of the interaction functions. Specifically, we define the function spaces on which the interaction kernels can be identified given infinite i.i.d observational derivative data sampled from a distribution. Our methodology involves casting the learning problem as a linear statistical inverse problem using a operator theoretical framework. We prove the well-posedness of inverse problem by establishing the strict positivity of a related integral operator and our analysis allows us to refine the results on specific manifolds such as the sphere and Hyperbolic space. Our findings indicate that a numerically stable procedure exists to recover the interaction kernel from finite (noisy) data, and the estimator will be convergent to the ground truth. This also answers an open question in [MMQZ21] and demonstrate that least square estimators can be statistically optimal in certain scenarios. Finally, our theoretical analysis could be extended to the mean-field case, revealing that the corresponding nonparametric inverse problem is ill-posed in general and necessitates effective regularization techniques.

翻译：本文针对黎曼流形上相互作用粒子系统的非参数推断中的关键问题——相互作用函数的可辨识性展开研究。具体而言，我们定义了可基于无穷独立同分布观测导数数据（从某一分布中采样）识别相互作用核的函数空间。方法论上，我们利用算子理论框架将学习问题转化为线性统计反问题。通过建立相关积分算子的严格正定性，证明了该反问题的适定性，并在此基础上细化了球面、双曲空间等特定流形上的结论。研究表明：存在从有限（含噪）数据中恢复相互作用核的数值稳定算法，且该估计量将收敛于真实值。这不仅回答了文献[MMQZ21]中的开放问题，同时证明在某些场景下最小二乘估计量可达到统计最优性。最后，我们的理论分析可推广至平均场情形，揭示相应的非参数反问题通常具有不适定性，需采用有效的正则化技术。