Semi-Riemannian Graph Convolutional Networks

Graph Convolutional Networks (GCNs) are typically studied through the lens of Euclidean geometry. Non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or spherical data, but cannot align well with data of mixed topologies. We consider a larger class of semi-Riemannian manifolds with indefinite metric that generalize hyperboloid and sphere as well as their submanifolds. We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected semi-Riemannian manifolds. As a consequence, we derive a principled Semi-Riemannian GCN that first models data in semi-Riemannian manifolds of constant nonzero curvature in the context of graph neural networks. Our method provides a geometric inductive bias that is sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles. Empirical results demonstrate that our method outperforms Riemannian counterparts when embedding graphs of complex topologies.