Stability of higher-dimensional interval decomposable persistence modules

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.

20 pages, 7 figures. Removed chapter about non-p.f.d. modules, added a chapter about Reeb graphs and zigzag modules and one about complexity. Other smaller changes

Cited by in corpus (8)

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• Universality of persistence diagrams and the bottleneck and Wasserstein distances
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• A family of metrics from the truncated smoothing of Reeb graphs
• Local Equivalence of Metrics for Multiparameter Persistence Modules
• Persistence and Stability of the ${\mathbb{A}}_n$ Quiver
• Reeb Graph Metrics from the Ground Up