A Set-to-Set Distance Measure in Hyperbolic Space

We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space. While point-to-point distances in hyperbolic space effectively capture hierarchical relationships between data points, many real-world applications require comparing sets of hyperbolic data points, where the local structure and the global structure of the sets carry crucial semantic information. The proposed the \underline{h}yperbolic \underline{s}et-\underline{to}-\underline{s}et \underline{d}istance measure (HS2SD) integrates both global and local structural information: global structure through geodesic distances between Einstein midpoints of hyperbolic sets, and local structure through topological characteristics of the two sets. To efficiently compute topological differences, we prove that using a finite Thue-Morse sequence of degree and adjacency matrices can serve as a robust approximation to capture the topological structure of a set. In this case, by considering the topological differences, HS2SD provides a more nuanced understanding of the relationships between two hyperbolic sets. Empirical evaluation on entity matching, standard image classification, and few-shot image classification demonstrates that our distance measure outperforms existing methods by effectively modeling the hierarchical and complex relationships inherent in hyperbolic sets.

Pengxiang Li, Wei Wu, Zhi Gao, Xiaomeng Fan, Peilin Yu, Yuwei Wu, Zhipeng Lu, Yunde Jia, Mehrtash Harandi · 2025