  Hyperbolic representations have shown remarkable efficacy in modeling
inherent hierarchies and complexities within data structures. Hyperbolic neural
networks have been commonly applied for learning such representations from
data, but they often fall short in preserving the geometric structures of the
original feature spaces. In response to this challenge, our work applies the
Gromov-Wasserstein (GW) distance as a novel regularization mechanism within
hyperbolic neural networks. The GW distance quantifies how well the original
data structure is maintained after embedding the data in a hyperbolic space.
Specifically, we explicitly treat the layers of the hyperbolic neural networks
as a transport map and calculate the GW distance accordingly. We validate that
the GW distance computed based on a training set well approximates the GW
distance of the underlying data distribution. Our approach demonstrates
consistent enhancements over current state-of-the-art methods across various
tasks, including few-shot image classification, as well as semi-supervised
graph link prediction and node classification.
