Discrete Morse Theory for Computing Zigzag Persistence

Clément Maria, Hannah Schreiber

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes (Ki), we introduce a zigzag Morse filtration whose complexes (Ai) are Morse reductions of the original complexes (Ki), and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda, defined for standard persistence. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.
Original languageEnglish
Title of host publicationAlgorithms and Data Structures. WADS 2019
Place of PublicationCham
PublisherSpringer
Pages538-552|
ISBN (Print)978-3-030-24765-2
DOIs
Publication statusPublished - 2019
EventWADS 2019: 16th International Symposium - Edmonton, Canada
Duration: 5 Aug 20197 Aug 2019

Publication series

NameLecture Notes in Computer Science
Volume11646

Conference

ConferenceWADS 2019
CountryCanada
CityEdmonton
Period5/08/197/08/19

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  • Cite this

    Maria, C., & Schreiber, H. (2019). Discrete Morse Theory for Computing Zigzag Persistence. In Algorithms and Data Structures. WADS 2019 (pp. 538-552|). (Lecture Notes in Computer Science ; Vol. 11646). Cham: Springer. https://doi.org/10.1007/978-3-030-24766-9_39