Abstract
Persistent homology is a popular data analysis technique that is used to capture the changing homology of an indexed sequence of simplicial complexes. These changes are summarized in persistence diagrams. A natural problem is to contract edges in complexes in the initial sequence to obtain a sequence of simplified complexes while controlling the perturbation between the original and simplified persistence diagrams. This paper is an extended version of Dey and Slechta (in: Discrete geometry for computer imagery, Springer, New York, 2019), where we developed two contraction operators for the case where the initial sequence is a filtration. In addition to the content in the original version, this paper presents proofs relevant to the filtration case and develops contraction operators for towers and multiparameter filtrations.
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Source code. https://github.com/rslechta/pers-contract. Accessed 15 Mar 2019
Acknowledgements
The authors would like to thank the National Elevation Dataset for their terrain data, the Aim@Shape repository for the models, and the Hera project for their bottleneck distance code [19]. In addition, the authors are grateful for the comments of the anonymous reviewers. This work was supported by NSF grants CCF-1740761, DMS-1547357, and CCF-1839252.
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Dey, T.K., Slechta, R. Filtration Simplification for Persistent Homology via Edge Contraction. J Math Imaging Vis 62, 704–717 (2020). https://doi.org/10.1007/s10851-020-00956-7
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DOI: https://doi.org/10.1007/s10851-020-00956-7