Spectro-Riemannian Graph Neural Networks

TL;DR

This paper introduces Spectro-Riemannian Graph Neural Networks (CUSP), a novel framework that combines spectral filtering and curvature signals on Riemannian manifolds to improve graph representation learning.

Contribution

CUSP is the first model to unify spectral and geometric curvature insights in a mixed-curvature spectral GNN for enhanced node embeddings.

Findings

Outperforms state-of-the-art models by up to 5.3% in node classification.

Effective on both homophilic and heterophilic datasets.

Demonstrates the benefit of integrating spectral and curvature signals.

Abstract

Can integrating spectral and curvature signals unlock new potential in graph representation learning? Non-Euclidean geometries, particularly Riemannian manifolds such as hyperbolic (negative curvature) and spherical (positive curvature), offer powerful inductive biases for embedding complex graph structures like scale-free, hierarchical, and cyclic patterns. Meanwhile, spectral filtering excels at processing signal variations across graphs, making it effective in homophilic and heterophilic settings. Leveraging both can significantly enhance the learned representations. To this end, we propose Spectro-Riemannian Graph Neural Networks (CUSP) - the first graph representation learning paradigm that unifies both CUrvature (geometric) and SPectral insights. CUSP is a mixed-curvature spectral GNN that learns spectral filters to optimize node embeddings in products of constant-curvature manifolds (hyperbolic, spherical, and Euclidean). Specifically, CUSP introduces three novel components: (a) Cusp Laplacian, an extension of the traditional graph Laplacian based on Ollivier-Ricci curvature, designed to capture the curvature signals better; (b) Cusp Filtering, which employs multiple Riemannian graph filters to obtain cues from various bands in the eigenspectrum; and (c) Cusp Pooling, a hierarchical attention mechanism combined with a curvature-based positional encoding to assess the relative importance of differently curved substructures in our graph. Empirical evaluation across eight homophilic and heterophilic datasets demonstrates the superiority of CUSP in node classification and link prediction tasks, with a gain of up to 5.3% over state-of-the-art models. The code is available at: https://github.com/amazon-science/cusp.