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.

We study the identifiability of the interaction kernels in mean-field equations for intreacting particle systems. The key is to identify function spaces on which a probabilistic loss functional has a unique minimizer. We prove that identifiability holds on any subspace of two reproducing kernel Hilbert spaces (RKHS), whose reproducing kernels are intrinsic to the system and are data-adaptive. Furthermore, identifiability holds on two ambient L2 spaces if and only if the integral operators associated with the reproducing kernels are strictly positive. Thus, the inverse problem is ill-posed in general. We also discuss the implications of identifiability in computational practice.

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