Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices

TL;DR

This paper introduces a novel approach using symmetric positive definite matrices to embed complex graphs, enhancing the performance of graph neural networks beyond traditional Euclidean and hyperbolic spaces.

Contribution

The authors develop a new library leveraging SPD geometry to implement and improve graph neural networks for complex graph data.

Findings

SPD-based GNNs outperform Euclidean and hyperbolic GNNs on complex graph tasks

The library enables robust handling of diverse geometric features in graphs

Experimental results show significant improvements in node and graph classification

Abstract

Recent research has shown that alignment between the structure of graph data and the geometry of an embedding space is crucial for learning high-quality representations of the data. The uniform geometry of Euclidean and hyperbolic spaces allows for representing graphs with uniform geometric and topological features, such as grids and hierarchies, with minimal distortion. However, real-world graph data is characterized by multiple types of geometric and topological features, necessitating more sophisticated geometric embedding spaces. In this work, we utilize the Riemannian symmetric space of symmetric positive definite matrices (SPD) to construct graph neural networks that can robustly handle complex graphs. To do this, we develop an innovative library that leverages the SPD gyrocalculus tools \cite{lopez2021gyroSPD} to implement the building blocks of five popular graph neural networks in SPD. Experimental results demonstrate that our graph neural networks in SPD substantially outperform their counterparts in Euclidean and hyperbolic spaces, as well as the Cartesian product thereof, on complex graphs for node and graph classification tasks. We release the library and datasets at \url{https://github.com/andyweizhao/SPD4GNNs}.