Alignment and Outer Shell Isotropy for Hyperbolic Graph Contrastive
  Learning
Yifei Zhang Hao Zhu Jiahong Liu Piotr Koniusz Irwin King
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.