Geometry-Aware Operator Families for Structured Representation Learning

2026-09-27

AI Summary

This study addresses the limitation of existing graph neural networks that rely on generic templates, making it difficult to uniformly derive adaptive propagation operators from fixed geometric structures. To this end, we propose the Geometry-Induced Operator Family (GIOF) framework, which transforms geometric information into reusable generating bases and dynamically constructs propagation operators via a context selector, thereby decoupling and adaptively fusing structure and content. Furthermore, continuous-time propagation, bottleneck residual layers, and adaptive scaling techniques are introduced. Theoretically, we establish guarantees for parameter compression, stability, and over-smoothing prevention. Experimental results demonstrate that under scenarios with 30% missing sensors on the PEMS-BAY and METR-LA datasets, GIOF reduces the Mean Absolute Error by 2.4%–8.8% compared to the strongest baselines.

Abstract

The geometry of latent representations governs which components should interact and how information should propagate, making geometry-aware operator design a fundamental ingredient of structured deep representation learning. However, existing neural architectures typically rely on generic operator templates or geometry-specific constructions, creating a need for a unified framework that can derive admissible operators directly from fixed structural information while remaining adaptive to changing contexts. We introduce \emph{Geometry-Induced Operator Families} (GIOF), a general framework that converts fixed geometry into a structured family of propagation operators and dynamically selects an appropriate member of this family according to the current context. GIOF first transforms geometry-derived interaction channels into reusable generator bases, then combines them through a context-dependent selector and adaptive propagation scale, and finally realizes the selected operator through stable continuous-time propagation and a bottleneck residual layer. We establish theoretical guarantees covering parameter compression, identifiability, stability, locality, compositional structure, and oversmoothing behavior, while controlled experiments validate these mechanisms and experiments on PEMS-BAY and METR-LA achieve the lowest mean MAE across all reported regional-outage settings, improving over the strongest retained baseline by 2.4\%--8.8\% at 30\% missing sensors.

Problem

• structured representation learning
• geometry-aware operators
• latent geometry
• operator families
• context adaptation

Innovation

• Geometry-Induced Operator Families
• Structured Representation Learning
• Continuous-Time Propagation
• Context-Dependent Selection
• Generator Bases