Computationally Tractable Riemannian Manifolds for Graph Embeddings

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum machine learning due to their desirable geometric inductive biases (e.g., hierarchical structures benefit from hyperbolic geometry). However, going beyond embedding spaces constant sectional curvature, while potentially more representationally powerful, proves be challenging one can easily lose the appeal computationally tractable tools such geodesic distances or gradients. Here, we explore two efficient matrix manifolds, showcasing how learn and optimize graph these spaces. Empirically, demonstrate consistent improvements over Euclidean geometry often outperforming elliptical based on various metrics that capture different properties. Our results serve new evidence for benefits non-Euclidean pipelines.