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

2025-08-24

AI Summary

This paper addresses the distributed estimation and aggregation of Gaussian random fields indexed by spatial Poisson point processes (PPPs), within Fisher–Rao and 2-Wasserstein information-geometric frameworks. To overcome the limitation of conventional methods—neglecting the joint spatial-statistical structure—it pioneers an integrated approach unifying stochastic geometry and information geometry. The proposed framework includes a geometry-aware Fréchet mean estimator, a semantic compression protocol, and a Fréchet-UCB online learning algorithm. Theoretically, it establishes non-asymptotic concentration bounds and a Palm-deviation-driven error control mechanism. Empirically, the framework is validated across wireless sensor networks, semantic communication, and multi-armed bandit settings, demonstrating substantial improvements in scalability, robustness, and decision-making performance. Overall, it delivers a unified, geometrically grounded paradigm for distributed semantic inference.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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 Fréchet 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 Fréchet means under PPP sampling. Finally, we introduce the exttt{Fréchet-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 Fréchet 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.

Problem

• Estimating Gaussian field Fréchet means under spatial randomness
• Developing geometry-aware aggregation for unreliable sensor networks
• Designing semantic compression protocols with fidelity guarantees

Innovation

• Integrating stochastic and information geometry for distributed inference
• Deriving concentration bounds for empirical Fréchet mean estimation
• Developing geometry-aware algorithms for wireless and bandit applications