Sheaf Graph Neural Networks via PAC-Bayes Spectral Optimization

TL;DR

This paper introduces SGPC, a novel sheaf GNN architecture that combines spectral regularization and optimal transport to improve semi-supervised node classification, especially on heterophilic graphs, with theoretical guarantees and state-of-the-art performance.

Contribution

The paper proposes SGPC, a unified sheaf GNN framework with PAC-Bayes spectral regularization, providing stability guarantees and scalable training for improved graph learning.

Findings

SGPC outperforms existing spectral and sheaf GNNs on multiple benchmarks.

Provides certified confidence intervals for node predictions.

Achieves performance bounds with linear computational complexity.

Abstract

Over-smoothing in Graph Neural Networks (GNNs) causes collapse in distinct node features, particularly on heterophilic graphs where adjacent nodes often have dissimilar labels. Although sheaf neural networks partially mitigate this problem, they typically rely on static or heavily parameterized sheaf structures that hinder generalization and scalability. Existing sheaf-based models either predefine restriction maps or introduce excessive complexity, yet fail to provide rigorous stability guarantees. In this paper, we introduce a novel scheme called SGPC (Sheaf GNNs with PAC-Bayes Calibration), a unified architecture that combines cellular-sheaf message passing with several mechanisms, including optimal transport-based lifting, variance-reduced diffusion, and PAC-Bayes spectral regularization for robust semi-supervised node classification. We establish performance bounds theoretically and demonstrate that end-to-end training in linear computational complexity can achieve the resulting bound-aware objective. Experiments on nine homophilic and heterophilic benchmarks show that SGPC outperforms state-of-the-art spectral and sheaf-based GNNs while providing certified confidence intervals on unseen nodes. The code and proofs are in https://github.com/ChoiYoonHyuk/SGPC.