Erosion distance for generalized persistence modules

TL;DR

This paper extends the erosion distance to generalized persistence modules, providing a stable and lower-bounding metric for multidimensional persistent homology and sublevel set persistent homology.

Contribution

It introduces a new erosion distance for rank invariants of generalized persistence modules, unifying and extending previous metrics in persistent homology.

Findings

Erosion distance is stable with respect to the interleaving distance.

It provides a lower bound for the natural pseudo-distance in sublevel set persistent homology.

The extension applies to multidimensional persistent homology groups with torsion.

Abstract

The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with images. Patel also introduced a distance for persistence diagrams, the erosion distance. Motivated by this work, we extend the erosion distance to a distance of rank invariants of generalized persistence modules by using the generalization of the interleaving distance of Bubenik, de Silva, and Scott as a guideline. This extension of the erosion distance also gives, as a special case, a distance for multidimensional persistent homology groups with torsion introduced by Frosini. We show that the erosion distance is stable with respect to the interleaving distance, and that it gives a lower bound for the natural pseudo-distance in the case of sublevel set persistent homology of continuous functions.