  Hyperbolic ordinal embedding (HOE) represents entities as points in
hyperbolic space so that they agree as well as possible with given constraints
in the form of entity i is more similar to entity j than to entity k. It has
been experimentally shown that HOE can obtain representations of hierarchical
data such as a knowledge base and a citation network effectively, owing to
hyperbolic space's exponential growth property. However, its theoretical
analysis has been limited to ideal noiseless settings, and its generalization
error in compensation for hyperbolic space's exponential representation ability
has not been guaranteed. The difficulty is that existing generalization error
bound derivations for ordinal embedding based on the Gramian matrix do not work
in HOE, since hyperbolic space is not inner-product space. In this paper,
through our novel characterization of HOE with decomposed Lorentz Gramian
matrices, we provide a generalization error bound of HOE for the first time,
which is at most exponential with respect to the embedding space's radius. Our
comparison between the bounds of HOE and Euclidean ordinal embedding shows that
HOE's generalization error is reasonable as a cost for its exponential
representation ability.
