  Due to its geometric properties, hyperbolic space can support high-fidelity
embeddings of tree- and graph-structured data, upon which various hyperbolic
networks have been developed. Existing hyperbolic networks encode geometric
priors not only for the input, but also at every layer of the network. This
approach involves repeatedly mapping to and from hyperbolic space, which makes
these networks complicated to implement, computationally expensive to scale,
and numerically unstable to train. In this paper, we propose a simpler
approach: learn a hyperbolic embedding of the input, then map once from it to
Euclidean space using a mapping that encodes geometric priors by respecting the
isometries of hyperbolic space, and finish with a standard Euclidean network.
The key insight is to use a random feature mapping via the eigenfunctions of
the Laplace operator, which we show can approximate any isometry-invariant
kernel on hyperbolic space. Our method can be used together with any graph
neural networks: using even a linear graph model yields significant
improvements in both efficiency and performance over other hyperbolic baselines
in both transductive and inductive tasks.
