标题： 随机信息几何：泊松网络中高斯场的弗雷歇均值的表征 显示英文标题

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

摘要： 我们通过将随机几何与信息几何相结合，开发了一个统一的框架，用于空间网络中的分布式推断、语义通信和探索——这一方向在以往文献中尚未被探索。 具体而言，我们在Fisher-Rao和2-Wasserstein几何下，研究了通过空间泊松点过程（PPP）索引的高斯分布场的估计和聚合问题。 我们推导了经验Fréchet均值的非渐近集中界和Palm偏差，从而量化了由空间随机性引起的几何不确定性。 基于这些结果，我们展示了在无线传感器网络中的应用，我们的框架提供了感知几何的聚合方法，能够降低不可靠传感器的影响，并严格表征随机部署下的估计误差。 此外，我们将理论扩展到语义通信，提出了通过PPP采样下Fréchet均值的失真界限保证语义保真的压缩协议。 最后，我们引入了\texttt{弗雷歇-UCB}算法，用于具有异方差高斯奖励的多臂老虎机问题。 该算法结合了上限置信区间与反映从动态Fréchet均值偏离的几何感知惩罚，并推导了利用几何结构的遗憾界限。 仿真验证了理论预测在无线传感器网络、语义压缩任务和老虎机环境中的有效性，突显了可扩展性、鲁棒性和改进的决策能力。 我们的结果为具有统计异质性的分布式系统中的几何感知推断、语义通信和探索提供了原则性的数学基础。 显示英文摘要

摘要： 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.