Generalization Error Bound for Hyperbolic Ordinal Embedding

Atsushi Suzuki, Atsushi Nitanda, Jing Jing Wang, Linchuan Xu, Kenji Yamanishi, Marc Cavazza

Research output: Chapter in book / Conference proceedingConference article published in proceeding or bookAcademic researchpeer-review

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 are not applicable 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 comes at a reasonable cost considering its exponential representation ability.
Original languageEnglish
Title of host publicationProceedings of the 38th International Conference on Machine Learning
Chapter139
Pages10011-10021
Number of pages11
Publication statusPublished - 2021

Fingerprint

Dive into the research topics of 'Generalization Error Bound for Hyperbolic Ordinal Embedding'. Together they form a unique fingerprint.

Cite this