  The problem of identifying geometric structure in heterogeneous,
high-dimensional data is a cornerstone of representation learning. While there
exists a large body of literature on the embeddability of canonical graphs,
such as lattices or trees, the heterogeneity of the relational data typically
encountered in practice limits the applicability of these classical methods. In
this paper, we propose a combinatorial approach to evaluating embeddability,
i.e., to decide whether a data set is best represented in Euclidean, Hyperbolic
or Spherical space. Our method analyzes nearest-neighbor structures and local
neighborhood growth rates to identify the geometric priors of suitable
embedding spaces. For canonical graphs, the algorithm's prediction provably
matches classical results. As for large, heterogeneous graphs, we introduce an
efficiently computable statistic that approximates the algorithm's decision
rule. We validate our method over a range of benchmark data sets and compare
with recently published optimization-based embeddability methods.
