Persistent Homology for Labeled Datasets: Stability and Generalized Landscapes

In many applications, point cloud data comes with a finite set of labels. Modeling this situation, we introduce variants of the Gromov–Hausdorff distance for labeled metric spaces. We consider variants with equal and unequal numbers of classes, and give a lower bound for the case of an equal number of classes. Next, we examine the case of generalized persistence, where the simplicial complexes are parametrized by a product of the real line and a partially ordered set induced by the inclusion structure of the classes in a dataset. We discuss the appropriate notion of the interleaving distance for such persistence modules and show stability under small perturbations of the points. To facilitate computations, we propose a notion of a landscape function suitable for this type of persistence, and discuss its theoretical properties and computational aspects.

This is a joint work with Yaoying Fu, Lander Ver Hoef, Shiying Li, Tom Needham, and Morgan Weiler.