Stability of higher-dimensional interval decomposable persistence modules

TL;DR

This paper generalizes the algebraic stability theorem to higher-dimensional rectangle decomposable persistence modules, providing bounds that are proven to be optimal and applying the technique to related structures like zigzag modules and Reeb graphs.

Contribution

It introduces a stability theorem for n-dimensional rectangle decomposable persistence modules with optimal bounds, extending the algebraic stability theorem.

Findings

Stability bound for n-dimensional modules is (2n-1), proven to be optimal for n=2.

The proof reduces to a new proof of the algebraic stability theorem for n=1.

Applied technique to zigzag modules and Reeb graphs, establishing optimal stability bounds.

Abstract

The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant $(2n-1)$ that is a generalization of the algebraic stability theorem, and also has connections to the complexity of calculating the interleaving distance. The proof given reduces to a new proof of the algebraic stability theorem with $n=1$. We give an example to show that the bound cannot be improved for $n=2$. We apply the same technique to prove stability results for zigzag modules and Reeb graphs, reducing the previously known bounds to a constant that cannot be improved, settling these questions.