On the Identifiablility of Nonlocal Interaction Kernels in First-Order Systems of Interacting Particles on Riemannian Manifolds

TL;DR

This work tackles the identifiability of nonparametric interaction kernels for first‑order particle systems on Riemannian manifolds by recasting the learning task as a linear inverse problem $A\varphi=\mathbf{f}_{\varphi}$ between $\mathcal{H}$ and $\mathcal{F}$. The authors prove $A$ is bounded with $\|A\|\le \frac{N-1}{N}$ and establish well‑posedness through positivity of an associated integral operator, with refined results on spheres $\mathbb{S}^n$ and hyperbolic spaces $\mathbb{H}^n$. They show that, for i.i.d. data, the inverse problem admits a positive lower bound; on the sphere this bound is sharp, while on hyperbolic space a larger class of hypothesis spaces yields a strictly positive constant, improving stability in those settings. The paper also discusses mean‑field ill‑posedness, robustness under equivalent measures, and presents explicit singular cases where stability can fail, highlighting the role of absolute continuity. These results provide a rigorous foundation for stable, data‑driven recovery of interaction kernels on manifolds and guide regularization needs in the large‑population limit, with potential implications for physics, biology, and social science models on curved spaces.

Abstract

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.

Figures (2)

• Figure 1: The two examples of systems \ref{['system']} using code provided in Maggioni2021LearningIK. The left models the swarming behaviour of preys and predators on $\mathbb{S}^2$ and the right models the aggregation behaviours of opinions on Poincare disk. The color variations from orange to pink denotes the forward time evolution of particles.
• Figure 2: Two examples of $\rho$ from different observation regimes in the same opinion dynamics on $\mathbb{S}^2$ with a piecewise linear kernel $\phi$ that is compactly supported on $[0,\frac{5}{\sqrt{\pi}}]$, where there are 20 agents/opinions evolving as in Maggioni2021LearningIK and the injective radius is $\frac{5}{\sqrt{\pi}}$. To obtain empirical approximations of $\rho_1$ and $\rho_2$, we use 3000 trajectories. In the first example, denoted by $\rho_1$, we choose $\mu_1$ to be the product of uniform distribution on $\mathbb{S}^2$ and observe the position and velocity at $t=0$ using i.i.d samples from $\mu_1$. In the second example, denoted by $\rho_2$, we choose $\mu_2$ to be the distribution of positions by observing infinite i.i.d trajectories at time interval [0,5] using $\mu_1$ as the initial distribution. Note that while $\mu_1$ has i.i.d components, $\mu_2$ does not due to time evolution. Despite this difference, we observe numerical evidence that $\rho_2$ is equivalent to $\rho_1$ on $[0,\frac{5}{\sqrt{\pi}}]$. Hence, Corollary \ref{['cor']} suggests that the stability result also holds for $\rho_2$.

Theorems & Definitions (21)

• Remark 1
• Definition 1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 4.1
• proof
• Remark 2
• Theorem 4.2