A Set-to-Set Distance Measure in Hyperbolic Space

• URL: http://arxiv.org/abs/2506.18529v1
• Date: Mon, 23 Jun 2025 11:31:40 GMT
• Title: A Set-to-Set Distance Measure in Hyperbolic Space
• Authors: Pengxiang Li, Wei Wu, Zhi Gao, Xiaomeng Fan, Peilin Yu, Yuwei Wu, Zhipeng Lu, Yunde Jia, Mehrtash Harandi,
• Abstract summary: We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space.<n>By considering the topological differences, HS2SD provides a more nuanced understanding of the relationships between two hyperbolic sets.
• Score: 50.134086375286074
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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