# Algebraic Stability of Zigzag Persistence Modules

arXiv: https://arxiv.org/abs/1604.00655 · HTML: https://mlsanity.com/paper/1604.00655/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2016.

## Built on by (most cited first, 7 of 7)

- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [Kan Extensions in Data Science and Machine Learning](https://mlsanity.com/paper/2203.09018/) (2022)
- [Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data](https://mlsanity.com/paper/2502.16049/) (2025)
- [TopoFormer: Topology Meets Attention for Graph Learning](https://mlsanity.com/paper/2607.28259/) (2026)
- [Algebraic and Geometric Models for Space Networking](https://mlsanity.com/paper/2304.01150/) (2023)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)

## Abstract (arXiv)

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 ℝ-valued functions, the result was later cast in a more general algebraic form, in the language of persistence modules and 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.