Constant Curvature Graph Convolutional Networks

TL;DR

This paper introduces a unified framework for graph neural networks in constant curvature spaces, enabling better modeling of non-Euclidean data with demonstrated improvements over traditional Euclidean GCNs.

Contribution

It proposes a mathematically grounded generalization of GCNs to constant curvature spaces using gyro-barycentric coordinates and a smooth interpolation between geometries.

Findings

Outperforms Euclidean GCNs in node classification tasks.

Effective in modeling non-Euclidean symbolic data.

Demonstrates smooth transition to Euclidean models as curvature approaches zero.

Abstract

Interest has been rising lately towards methods representing data in non-Euclidean spaces, e.g. hyperbolic or spherical, that provide specific inductive biases useful for certain real-world data properties, e.g. scale-free, hierarchical or cyclical. However, the popular graph neural networks are currently limited in modeling data only via Euclidean geometry and associated vector space operations. Here, we bridge this gap by proposing mathematically grounded generalizations of graph convolutional networks (GCN) to (products of) constant curvature spaces. We do this by i) introducing a unified formalism that can interpolate smoothly between all geometries of constant curvature, ii) leveraging gyro-barycentric coordinates that generalize the classic Euclidean concept of the center of mass. Our class of models smoothly recover their Euclidean counterparts when the curvature goes to zero from either side. Empirically, we outperform Euclidean GCNs in the tasks of node classification and distortion minimization for symbolic data exhibiting non-Euclidean behavior, according to their discrete curvature.