### Abstract

_{i}), we introduce a zigzag Morse filtration whose complexes (A

_{i}) are Morse reductions of the original complexes (K

_{i}), 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.

Language | English |
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Status | Submitted - 2019 |

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### Cite this

*Discrete Morse Theory for Computing Zigzag Persistence*.

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

Research output: Contribution to conference › Paper › Research › peer-review

}

TY - CONF

T1 - Discrete Morse Theory for Computing Zigzag Persistence

AU - Schreiber, Hannah

AU - Maria, Clément

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

M3 - Paper

ER -