  Graph embeddings, wherein the nodes of the graph are represented by points in
a continuous space, are used in a broad range of Graph ML applications. The
quality of such embeddings crucially depends on whether the geometry of the
space matches that of the graph. Euclidean spaces are often a poor choice for
many types of real-world graphs, where hierarchical structure and a power-law
degree distribution are linked to negative curvature. In this regard, it has
recently been shown that hyperbolic spaces and more general manifolds, such as
products of constant-curvature spaces and matrix manifolds, are advantageous to
approximately match nodes pairwise distances. However, all these classes of
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. In this paper, we study graph embeddings
in a broader class of heterogeneous rotationally-symmetric manifolds. By adding
a single extra radial dimension to any given existing homogeneous model, we can
both account for heterogeneous curvature distributions on graphs and pairwise
distances. We evaluate our approach on reconstruction tasks on synthetic and
real datasets and show its potential in better preservation of high-order
structures and heterogeneous random graphs generation.
