Generalization Error Bound for Hyperbolic Ordinal Embedding
This work addresses a theoretical gap for researchers using hyperbolic embeddings in hierarchical data applications, though it is incremental as it extends existing analysis to a new space.
The paper tackles the lack of theoretical guarantees for hyperbolic ordinal embedding (HOE) in noisy settings by deriving its first generalization error bound, which is shown to be at most exponential with respect to the embedding space's radius, and compares it favorably to Euclidean ordinal embedding as a reasonable cost for HOE's exponential representation ability.
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.