Generalization Error Bounds for Graph Embedding Using Negative Sampling: Linear vs Hyperbolic

Graph embedding, which represents real-world entities in a mathematical space, has enabled numerous applications such as analyzing natural languages, social networks, biochemical networks, and knowledge bases. It has been experimentally shown that graph embedding in hyperbolic space can represent hierarchical tree-like data more effectively than embedding in linear space, owing to hyperbolic space's exponential growth property. However, since the theoretical comparison has been limited to ideal noiseless settings, the potential for the hyperbolic space's property to worsen the generalization error for practical data has not been analyzed. In this paper, we provide a generalization error bound applicable for graph embedding both in linear and hyperbolic spaces under various negative sampling settings that appear in graph embedding. Our bound states that error is polynomial and exponential with respect to the embedding space's radius in linear and hyperbolic spaces, respectively, which implies that hyperbolic space's exponential growth property worsens the error. Using our bound, we clarify the data size condition on which graph embedding in hyperbolic space can represent a tree better than in Euclidean space by discussing the bias-variance trade-off. Our bound also shows that imbalanced data distribution, which often appears in graph embedding, can worsen the error.