Laplacian Features for Learning with Hyperbolic Space

Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data. As a result, various hyperbolic networks have been developed which outperform Euclidean networks on many tasks: e.g. hyperbolic graph convolutional networks (GCN) can outperform vanilla GCN on some graph learning tasks. However, most existing hyperbolic networks are complicated, computationally expensive, and numerically unstable -- and they cannot scale to large graphs due to these shortcomings. With more and more hyperbolic networks proposed, it is becoming less and less clear what key component is necessary to make the model behave. In this paper, we propose HyLa, a simple and minimal approach to using hyperbolic space in networks: HyLa maps once from a hyperbolic-space embedding to Euclidean space via the eigenfunctions of the Laplacian operator in the hyperbolic space. We evaluate HyLa on graph learning tasks including node classification and text classification, where HyLa can be used together with any graph neural networks. When used with a linear model, HyLa shows significant improvements over hyperbolic networks and other baselines.