ArnoldiGCL: Graph Contrastive Learning via Learnable Arnoldi-Based Guided Spectral Chebyshev Polynomial Filters

Graph Contrastive Learning (GCL) emerged as a powerful paradigm in self-supervised graph representation learning. While earlier applications of GCL rely on homophily assumptions, spectral graph neural networks (GNNs) enhance the effectiveness of GCL on heterophilic graphs by incorporating both low-pass and high-pass filters. However, due to numerical considerations, existing approaches oversimplify low-pass and high-pass filters by modeling them as basic linear operations, failing to capture complex topological relationships. Here, we propose ArnoldiGCL, a novel algorithm that enables the application of complex spectral filters for Graph Contrastive Learning (GCL). Using Arnoldi orthonormalization-based Chebyshev interpolation, ArnoldiGCL overcomes the difficulties posed by ill-conditioned Vandermonde systems that arise in the modeling of complex filters. By introducing learnable filters, our method generates diverse spectral views and effectively captures nuanced graph structures. Theoretical analysis demonstrates that ArnoldiGCL accurately interpolates complex filters, thus forming a solid foundation for contrastive learning on graphs with complex structures. Extensive experiments on real-world datasets confirm that ArnoldiGCL significantly outperforms state-of-the-art GCL algorithms on both homophilic and heterophilic graphs, showcasing its robustness and versatility.