A Set-to-Set Distance Measure in Hyperbolic Space

2025-06-23

AI Summary

This work addresses the problem of measuring dissimilarity between sets in hyperbolic space. The proposed method introduces a novel distance metric that jointly captures global geometric and local topological structures: it models the global hierarchical organization of sets via geodesic distances between Einstein midpoints, while approximating local topological characteristics using finite Thue–Morse sequences. The framework end-to-end integrates hyperbolic geometry, graph-theoretic topology analysis, and sequence-based approximation—constituting the first approach to jointly model both geometric and topological attributes of sets within hyperbolic space. Experiments demonstrate that the method significantly outperforms state-of-the-art hyperbolic and Euclidean baselines on entity matching, standard image classification, and few-shot image classification tasks. These results validate its effectiveness and generalizability for modeling complex, hierarchically structured data.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Measure dissimilarity between sets in hyperbolic space
• Integrate global and local structural information in sets
• Improve hierarchical relationship modeling in hyperbolic data

Innovation

• Hyperbolic set-to-set distance measure HS2SD
• Global structure via geodesic Einstein midpoints
• Local topology with Thue-Morse sequence approximation