Heterogeneous manifolds for curvature-aware graph embedding

The quality of graph embeddings depends on whether the geometry of the space matches that of the graph. Euclidean spaces are often a poor choice and recently hyperbolic spaces and more general manifolds, such as products of constant-curvature spaces and matrix manifolds, have resulted advantageous to better matching nodes pairwise distances. However, all these manifolds are homogeneous, implying that the curvature distribution is the same at each point, making them unsuited to match the local curvature (and related structural properties) of the graph. We study embeddings in a broader class of heterogeneous rotationally-symmetric manifolds. By adding a single radial dimension to existing homogeneous models, we can both account for heterogeneous curvature distributions on graphs and pairwise distances. We evaluate our approach on reconstruction tasks.