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 20 articles )

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  1. Mémoli, Facundo; Singhal, Kritika: A primer on persistent homology of finite metric spaces (2019)
  2. Alan Hylton, Gregory Henselman-Petrusek, Janche Sang, Robert Short: Tuning the Performance of a Computational Persistent Homology Package (2018) arXiv
  3. Chung, Yu-Min; Day, Sarah: Topological fidelity and image thresholding: a persistent homology approach (2018)
  4. Dłotko, Paweł; Wanner, Thomas: Rigorous cubical approximation and persistent homology of continuous functions (2018)
  5. Port, Alexander; Gheorghita, Iulia; Guth, Daniel; Clark, John M.; Liang, Crystal; Dasu, Shival; Marcolli, Matilde: Persistent topology of syntax (2018)
  6. Xia, Kelin; Li, Zhiming; Mu, Lin: Multiscale persistent functions for biomolecular structure characterization (2018)
  7. Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H.: Extremal examples of collapsible complexes and random discrete Morse theory (2017)
  8. 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)
  9. Bauer, Ulrich; Kerber, Michael; Reininghaus, Jan; Wagner, Hubert: \textscPhat-- persistent homology algorithms toolbox (2017)
  10. Bubenik, Peter; Dłotko, Paweł: A persistence landscapes toolbox for topological statistics (2017)
  11. de Floriani, Leila; Fugacci, Ulderico; Iuricich, Federico: Homological shape analysis through discrete Morse theory (2016)
  12. Dłotko, Paweł; Wanner, Thomas: Topological microstructure analysis using persistence landscapes (2016)
  13. Wang, Bao; Wei, Guo-Wei: Object-oriented persistent homology (2016)
  14. Bubenik, Peter: Statistical topological data analysis using persistence landscapes (2015)
  15. Cang, Zixuan; Mu, Lin; Wu, Kedi; Opron, Kristopher; Xia, Kelin; Wei, Guo-Wei: A topological approach for protein classification (2015)
  16. Giusti, Chad; Pastalkova, Eva; Curto, Carina; Itskov, Vladimir: Clique topology reveals intrinsic geometric structure in neural correlations (2015)
  17. Benedetti, Bruno; Lutz, Frank H.: Random discrete Morse theory and a new library of triangulations (2014)
  18. Berwald, Jesse; Gidea, Marian: Critical transitions in a model of a genetic regulatory system (2014)
  19. Máté, Gabriell; Heermann, Dieter W.: Persistence intervals of fractals (2014)
  20. Mischaikow, Konstantin; Nanda, Vidit: Morse theory for filtrations and efficient computation of persistent homology (2013)