Constant Curvature Graph Convolutional Networks
This work addresses the problem of modeling non-Euclidean data properties like hierarchy or cyclicity in graph neural networks, representing a novel method for a known bottleneck.
The authors tackled the limitation of graph neural networks to Euclidean geometry by proposing constant curvature graph convolutional networks that generalize to hyperbolic and spherical spaces, achieving better performance in node classification and distortion minimization for non-Euclidean data.
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.