Discrete Morse Theory for Computing Zigzag Persistence

Hannah Schreiber, Clément Maria

Research output: Contribution to conferencePaperResearchpeer-review

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
Publication statusSubmitted - 2019

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Discrete Morse Theory
Zigzag
Persistence
Filtration
Computing
Homology
Algorithm Design
Preprocessing
Deduce
Trivial
Inclusion
Update
Generalise

Cite this

Discrete Morse Theory for Computing Zigzag Persistence. / Schreiber, Hannah; Maria, Clément.

2019.

Research output: Contribution to conferencePaperResearchpeer-review

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