Algebraic stability of zigzag persistence modules

Botnan, Magnus; Lesnick, Michael

doi:10.2140/agt.2018.18.3133

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Algebraic stability of zigzag persistence modules

Magnus Bakke Botnan and Michael Lesnick

Algebraic & Geometric Topology 18 (2018) 3133–3204

DOI: 10.2140/agt.2018.18.3133

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

Keywords

topological data analysis, persistent homology, interleavings

Mathematical Subject Classification 2010

Primary: 55N35

Secondary: 55U99

References

Publication

Received: 16 April 2017

Revised: 28 January 2018

Accepted: 11 March 2018

Published: 18 October 2018

Authors

Magnus Bakke Botnan
Zentrum Mathematik Technische Universität München Garching bei München Germany

Michael Lesnick
Princeton Neuroscience Institute Princeton University Princeton, NJ United States

Volume 18, issue 6 (2018)