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一个关键问题：相互作用函数的可辨识性。具体而言，我们定义了函数空间，在该空间上，给定从某一分布中采样的无限独立同分布观测导数数据，相互作用核可被唯一识别。我们的方法是通过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，我们证明了该反问题的适定性，并且我们的分析允许我们在特定流形（如球面和双曲空间）上细化相关结果。研究结果表明，存在一种数值稳定的方法可从有限（含噪）数据中恢复相互作用核，且估计量将收敛于真实值。这也回答了[MMQZ21]中的一个开放性问题，并证明最小二乘估计器在某些场景下可以达到统计最优性。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是不适定的，需要有效的正则化技术。