# A Set-to-Set Distance Measure in Hyperbolic Space

Source: [A Set-to-Set Distance Measure in Hyperbolic Space](<https://www.sota2.com/research/paper/68fa79901e209131cea7e3f3>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-26T05:00:46.441Z

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

Published: 2025-06-23T11:31:40.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2506.18529>)

[PDF](<https://arxiv.org/pdf/2506.18529.pdf>)