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

TL;DR

This paper introduces a generalized Gromov-Hausdorff distance for chromatic metric pairs, demonstrating its stability and implications for the six-pack and ch persistence diagrams in topological data analysis.

Contribution

It defines a new distance measure for chromatic metric pairs and proves its stability, extending the analysis tools in chromatic topological data analysis.

Findings

The generalized Gromov-Hausdorff distance is well-defined and satisfies basic properties.

The six-pack is stable under this new distance measure.

The framework relates to the stability of ch persistence diagrams.

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 \v{C}ech persistence diagrams.