A Set-to-Set Distance Measure in Hyperbolic Space
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912444658483200 |
|---|---|
| author | Li, Pengxiang Wu, Wei Gao, Zhi Fan, Xiaomeng Yu, Peilin Wu, Yuwei Lu, Zhipeng Jia, Yunde Harandi, Mehrtash |
| author_facet | Li, Pengxiang Wu, Wei Gao, Zhi Fan, Xiaomeng Yu, Peilin Wu, Yuwei Lu, Zhipeng Jia, Yunde Harandi, Mehrtash |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Set-to-Set Distance Measure in Hyperbolic Space Li, Pengxiang Wu, Wei Gao, Zhi Fan, Xiaomeng Yu, Peilin Wu, Yuwei Lu, Zhipeng Jia, Yunde Harandi, Mehrtash Computer Vision and Pattern Recognition Machine Learning 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. |
| title | A Set-to-Set Distance Measure in Hyperbolic Space |
| topic | Computer Vision and Pattern Recognition Machine Learning |
| url | https://arxiv.org/abs/2506.18529 |