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.

翻译：由于双曲空间的几何特性，它能支持树状和图结构数据的高保真嵌入，并由此发展出多种双曲网络。现有双曲网络不仅对输入编码几何先验，而且在网络的每一层都如此。这种方法需要反复在双曲空间与欧氏空间之间映射，导致网络实现复杂、扩展计算昂贵且训练数值不稳定。本文提出一种更简单的方法：先学习输入数据的双曲嵌入，然后通过一种尊重双曲空间等距变换的映射将其一次性映射到欧氏空间，最后使用标准欧氏网络。关键思路是利用拉普拉斯算子特征函数构造随机特征映射，我们证明它能逼近双曲空间上的任意等距不变核。该方法可与任意图神经网络结合使用：即使在传递性和归纳性任务中，仅使用线性图模型也能在效率和性能上显著超越其他双曲基线方法。