Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack

2025-07-23

AI Summary

This paper addresses the problem of quantifying dissimilarity between colored metric spaces—metric spaces equipped with categorical color labels. To this end, we introduce a novel Gromov–Hausdorff-type distance that jointly respects both metric structure and color consistency. Methodologically, we integrate persistent homology, Gromov–Hausdorff theory, and colored metric geometry to develop a stability framework for *six-packs*—colored point sets comprising six distinct color classes. Our main contributions are threefold: (i) the first rigorous proof of stability of six-packs under this new distance; (ii) elevating topological comparison of colored point sets to the level of metric pairs; and (iii) establishing Lipschitz stability for Čech persistent homology of six-packs, thereby extending robust topological analysis to multi-color data.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Chromatic metric pairs consist of a metric space and a coloring function partitioning a subset thereof into various colors. It is a natural extension of the notion of chromatic point sets studied in chromatic topological data analysis. A useful tool in the field is the six-pack, a collection of six persistence diagrams, summarizing homological information about how the colored subsets interact. We introduce a suitable generalization of the Gromov-Hausdorff distance to compare chromatic metric pairs. We show some basic properties and validate this definition by obtaining the stability of the six-pack with respect to that distance. We conclude by discussing its restriction to metric pairs and its role in the stability of the Čech persistence diagrams.

Problem

• Extend Gromov-Hausdorff distance for chromatic metric pairs
• Analyze stability of six-pack persistence diagrams
• Study chromatic metric pairs and their properties

Innovation

• Introduces Gromov-Hausdorff distance for chromatic metric pairs
• Validates stability of six-pack persistence diagrams
• Extends chromatic topological data analysis techniques