Persistent Homology for Labeled Datasets: Gromov-Hausdorff Stability and Generalized Landscapes

TL;DR

This paper extends metric geometry tools like Gromov-Hausdorff distance and persistent homology to labeled datasets, ensuring stability and computability for real-world applications with categorical labels.

Contribution

It introduces a framework for labeled metric spaces and develops new stable Gromov-Hausdorff and persistent homology notions tailored for labeled data.

Findings

Persistent homology is stable under the new labeled Gromov-Hausdorff distance.

Labeled persistence landscapes are Lipschitz continuous.

Framework applies to datasets with categorical labels.

Abstract

Techniques from metric geometry have become fundamental tools in modern mathematical data science, providing principled methods for comparing datasets modeled as finite metric spaces. Two of the central tools in this area are the Gromov-Hausdorff distance and persistent homology, both of which yield isometry-invariant notions of distance between datasets. However, these frameworks do not account for categorical labels, which are intrinsic to many real-world datasets, such as labeled images, pre-clustered data, and semantically segmented shapes. In this paper, we introduce a general framework for labeled metric spaces and develop new notions of Gromov-Hausdorff distance and persistent homology which are adapted to this setting. Our main result shows that our persistent homology construction is stable with respect to our novel notion of Gromov-Hausdorff distance, extending a classic result in topological data analysis. To facilitate computation, we also introduce a labeled version of persistence landscapes and show that the landscape map is Lipschitz.