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arXiv · 2507.17994

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

Ondřej Draganov·Sophie Rosenmeier·Nicolò Zava
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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.

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Ondřej Draganov, Sophie Rosenmeier, Nicolò Zava. 2025-07-24. Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack. https://arxiv.org/abs/2507.17994

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