Discrete Morse Homology and the Perseus Software Project. At its core, the computation of homology -- persistent or otherwise -- involves performing elementary row and column operations on possibly gigantic matrix representations of boundary operators defined on cell complexes. Discrete Morse theory provides a powerful and flexible framework for drastically reducing the sizes of these matrices in almost linear time while preserving all the underlying homological information. In this talk, we outline the scope and limitations of such Morse-theoretic methods. We also provide a brief overview of the Perseus software project which implements these methods in order to efficiently compute persistent homology of various types of filtered complexes.

References in zbMATH (referenced in 25 articles )

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  1. Calcina, Sabrina S.; Gameiro, Marcio: Parameter estimation in systems exhibiting spatially complex solutions via persistent homology and machine learning (2021)
  2. Tanaka, Kohei: Simple homotopy theory and nerve theorem for categories (2021)
  3. Bendich, Paul; Bubenik, Peter; Wagner, Alexander: Stabilizing the unstable output of persistent homology computations (2020)
  4. Gonzalez, Georgina; Ushakova, Arina; Sazdanovic, Radmila; Arsuaga, Javier: Prediction in cancer genomics using topological signatures and machine learning (2020)
  5. Som, Anirudh; Ramamurthy, Karthikeyan Natesan; Turaga, Pavan: Geometric metrics for topological representations (2020)
  6. Mémoli, Facundo; Singhal, Kritika: A primer on persistent homology of finite metric spaces (2019)
  7. Alan Hylton, Gregory Henselman-Petrusek, Janche Sang, Robert Short: Tuning the Performance of a Computational Persistent Homology Package (2018) arXiv
  8. Chung, Yu-Min; Day, Sarah: Topological fidelity and image thresholding: a persistent homology approach (2018)
  9. Dłotko, Paweł; Wanner, Thomas: Rigorous cubical approximation and persistent homology of continuous functions (2018)
  10. Port, Alexander; Gheorghita, Iulia; Guth, Daniel; Clark, John M.; Liang, Crystal; Dasu, Shival; Marcolli, Matilde: Persistent topology of syntax (2018)
  11. Xia, Kelin; Li, Zhiming; Mu, Lin: Multiscale persistent functions for biomolecular structure characterization (2018)
  12. Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H.: Extremal examples of collapsible complexes and random discrete Morse theory (2017)
  13. Alsing, Paul M.; Blair, Howard A.; Corne, Matthew; Jones, Gordon; Miller, Warner A.; Mischaikow, Konstantin; Nanda, Vidit: Topological signals of singularities in Ricci flow (2017)
  14. Bauer, Ulrich; Kerber, Michael; Reininghaus, Jan; Wagner, Hubert: \textscPhat-- persistent homology algorithms toolbox (2017)
  15. Bubenik, Peter; Dłotko, Paweł: A persistence landscapes toolbox for topological statistics (2017)
  16. de Floriani, Leila; Fugacci, Ulderico; Iuricich, Federico: Homological shape analysis through discrete Morse theory (2016)
  17. Dłotko, Paweł; Wanner, Thomas: Topological microstructure analysis using persistence landscapes (2016)
  18. Wang, Bao; Wei, Guo-Wei: Object-oriented persistent homology (2016)
  19. Bubenik, Peter: Statistical topological data analysis using persistence landscapes (2015)
  20. Cang, Zixuan; Mu, Lin; Wu, Kedi; Opron, Kristopher; Xia, Kelin; Wei, Guo-Wei: A topological approach for protein classification (2015)

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