Multidimensional persistent homology is stable

TL;DR

This paper proves that multidimensional persistence invariants are stable under small function perturbations, enabling reliable shape comparison using multidimensional persistent homology.

Contribution

It establishes the stability of multidimensional rank invariants with respect to function perturbations, a key theoretical advancement in topological data analysis.

Findings

Multidimensional rank invariants are stable under continuous function perturbations.

A new distance between rank invariants quantifies stability.

Results enable robust shape comparison methods.

Abstract

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.