Generalization Error Bound for Hyperbolic Ordinal Embedding

TL;DR

This paper provides the first theoretical generalization error bound for hyperbolic ordinal embedding, demonstrating its effectiveness and limitations in representing hierarchical data with exponential growth properties.

Contribution

It introduces a novel characterization of HOE using decomposed Lorentz Gramian matrices and derives a generalization error bound that accounts for hyperbolic space's exponential capacity.

Findings

First generalization error bound for HOE

Bound is at most exponential in the space's radius

HOE's error is reasonable given its exponential representation ability

Abstract

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.