Learning Propagation Geometry from Message-Passing Feedback

2026-09-28

AI Summary

This study addresses the limitations of conventional Graph Neural Network (GNN) aggregation, which often overlooks message discrepancies and inter-dimensional feature dependencies. To this end, we propose GeoF, a framework that jointly evolves node representations and propagation geometry through a feedback loop. Methodologically, GeoF parameterizes local geometry using block-logarithmic triangular coordinates and constructs a recursive shared controller by integrating symmetric positive definite manifolds, triangular frame transport, and residual analysis of second-order statistics, thereby achieving closed-loop optimization between geometric structure and message passing. Extensive experiments demonstrate that the proposed framework significantly outperforms existing state-of-the-art GNN baselines on multiple benchmark datasets across tasks such as node classification.

Abstract

Learning local geometry enables graph neural networks (GNNs) to adapt how they compare and integrate neighborhood information. However, estimating geometry from aggregated representations can overlook variation among individual messages and dependencies across feature dimensions. We propose GeoF, a recurrent framework that jointly evolves node features and propagation geometry through message-passing feedback. Each node maintains a local symmetric positive-definite geometry, initialized from a structure-aware prototype atlas and parameterized in block log-triangular coordinates. At each step, the geometry determines neighborhood weights, while triangular frame transport maps transformed source messages into the target node's local coordinates before aggregation. Weighted second-order statistics of residuals between aligned messages and the transformed target state capture directional variation and within-block dependencies, yielding a geometric update target. A shared controller learns complementary corrections through task supervision. A bounded log-triangular update combines these corrections, the target, and the previous geometric state while preserving positive definiteness. The geometry governs subsequent propagation, closing the feedback loop. With parameters shared across recurrent steps, task-specific readouts support node classification, link prediction, and graph classification. Experiments on benchmark datasets show that GeoF consistently outperforms state-of-the-art GNN baselines.

Problem

• Graph Neural Networks
• Local Geometry Learning
• Message Passing
• Feature Dependencies

Innovation

• Propagation Geometry
• Message-Passing Feedback
• Recurrent Framework
• Log-Triangular Parameterization
• Frame Transport