Stability of multidimensional persistent homology with respect to domain perturbations

TL;DR

This paper extends the stability analysis of persistent homology from function perturbations to domain perturbations, demonstrating bounds using various set distances and validating the approach with experiments.

Contribution

It introduces a method to bound the multidimensional matching distance of persistent homology by domain perturbation measures like Hausdorff and fuzzy set distances.

Findings

Multidimensional matching distance is bounded by Hausdorff distance.

The method preserves information about the original set.

Experimental results show practical potential of the approach.

Abstract

Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to results concerning the stability with respect to domain perturbations. Domain perturbations can be measured in a number of different ways. An important method to compare domains is the Hausdorff distance. We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance between sets. Moreover we prove that our construction maintains information about the original set. Other well known methods to compare sets are considered, such as the symmetric difference distance between classical sets and the sup-distance between fuzzy sets. Also in these cases we present results stating that the multidimensional matching distance between rank invariants of persistent homology groups is upperly bounded by these distances. An experiment showing the potential of our approach concludes the paper.