Stochastic Information Geometry: Characterization of Fréchet Means of Gaussian Fields in Poisson Networks

We develop a unified framework for distributed inference, semantic communication, and exploration in spatial networks by integrating stochastic geometry with information geometry - a direction that has not been explored in prior literature. Specifically, we study the problem of estimating and aggregating a field of Gaussian distributions indexed by a spatial Poisson point process (PPP), under both the Fisher--Rao and 2-Wasserstein geometries. We derive non-asymptotic concentration bounds and Palm deviations for the empirical Fr\'echet mean, thereby quantifying the geometric uncertainty induced by spatial randomness. Building on these results, we demonstrate applications to wireless sensor networks, where our framework provides geometry-aware aggregation methods that downweight unreliable sensors and rigorously characterize estimation error under random deployment. Further, we extend our theory to semantic communications, proposing compression protocols that guarantee semantic fidelity via distortion bounds on Fr\'echet means under PPP sampling. Finally, we introduce the \texttt{Fr\'echet-UCB} algorithm for multi-armed bandit problems with heteroscedastic Gaussian rewards. This algorithm combines upper confidence bounds with a geometry-aware penalty reflecting deviation from the evolving Fr\'echet mean, and we derive regret bounds that exploit geometric structure. Simulations validate the theoretical predictions across wireless sensor networks, semantic compression tasks, and bandit environments, highlighting scalability, robustness, and improved decision-making. Our results provide a principled mathematical foundation for geometry-aware inference, semantic communication, and exploration in distributed systems with statistical heterogeneity.