Random Laplacian Features for Learning with Hyperbolic Space

TL;DR

This paper introduces a simplified hyperbolic learning method that uses random Laplacian features to efficiently embed data, improving performance over existing hyperbolic neural networks while reducing complexity and computational costs.

Contribution

The authors propose a novel approach that maps hyperbolic embeddings to Euclidean space using Laplacian eigenfunctions, enabling easier and more scalable hyperbolic learning.

Findings

Significant performance improvements over hyperbolic baselines.

Enhanced efficiency and scalability in hyperbolic embedding.

Compatibility with various graph neural networks.

Abstract

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.