Learning good self-supervised graph representations that are beneficial to downstream tasks is challenging. Among a variety of methods, contrastive learning enjoys competitive performance. The embeddings of contrastive learning are arranged on a hypersphere that enables the Cosine distance measurement in the Euclidean space. However, the underlying structure of many domains such as graphs exhibits highly non-Euclidean latent geometry. To this end, we propose a novel contrastive learning framework to learn high-quality graph embedding. Specifically, we design the alignment metric that effectively captures the hierarchical data-invariant information, as well as we propose a substitute of uniformity metric to prevent the so-called dimensional collapse. We show that in the hyperbolic space one has to address the leaf- and height-level uniformity which are related to properties of trees, whereas in the ambient space of the hyperbolic manifold, these notions translate into imposing an isotropic ring density towards boundaries of Poincar\'e ball. This ring density can be easily imposed by promoting the isotropic feature distribution on the tangent space of manifold. In the experiments, we demonstrate the efficacy of our proposed method across different hyperbolic graph embedding techniques in both supervised and self-supervised learning settings.

翻译：学习有利于下游任务的良好自监督图表示具有挑战性。在众多方法中，对比学习表现出竞争性性能。对比学习的嵌入向量排布在超球面上，能够在欧氏空间中实现余弦距离度量。然而，图等许多领域的底层结构呈现出高度非欧几里得潜在几何特征。为此，我们提出一种新颖的对比学习框架，用于学习高质量图嵌入。具体而言，我们设计了有效捕捉层级数据不变信息的对齐度量，并提出了均匀度量的替代方案以防止所谓的维度坍缩。我们证明，在双曲空间中必须解决与树属性相关的叶级和高度级均匀性，而在双曲流形的环境空间中，这些概念转化为对庞加莱球边界施加各向同性环密度。这种环密度可通过促进流形切空间上的各向同性特征分布轻松实现。在实验中，我们证明了所提方法在监督和自监督学习设置下对不同双曲图嵌入技术的有效性。