A Set-to-Set Distance Measure in Hyperbolic Space

TL;DR

This paper introduces HS2SD, a novel hyperbolic set-to-set distance measure that combines global geodesic and local topological information, improving set comparison in hyperbolic space for various applications.

Contribution

The paper presents a new hyperbolic set-to-set distance measure that integrates global and local structures, with an efficient topological approximation using Thue-Morse sequences.

Findings

Outperforms existing methods in entity matching.

Effective in standard and few-shot image classification.

Captures hierarchical and complex relationships effectively.

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.