Algebraic Stability of Zigzag Persistence Modules

TL;DR

This paper extends the algebraic stability theorem to zigzag persistence modules by functorially embedding them into two-dimensional modules, strengthening stability results for Reeb graphs and level set persistent homology.

Contribution

It introduces a novel algebraic stability theorem for zigzag persistence modules via functorial extension to 2D modules, providing new stability insights.

Findings

Established an algebraic stability theorem for zigzag modules.

Extended zigzag modules to 2D persistence modules.

Strengthened stability results for Reeb graphs and level set homology.

Abstract

The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a more general algebraic form, in the language of \emph{persistence modules} and \emph{interleavings}. In this paper, we establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions. One part of our argument yields a stability result for free two-dimensional persistence modules. As an application of our main theorem, we strengthen a result of Bauer et al. on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al.