Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913958698418176 |
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| author | Draganov, Ondřej Rosenmeier, Sophie Zava, Nicolò |
| author_facet | Draganov, Ondřej Rosenmeier, Sophie Zava, Nicolò |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack Draganov, Ondřej Rosenmeier, Sophie Zava, Nicolò Metric Geometry Computational Geometry 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. |
| title | Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack |
| topic | Metric Geometry Computational Geometry |
| url | https://arxiv.org/abs/2507.17994 |