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Discrete & Computational Geometry

, Volume 62, Issue 4, pp 945–989 | Cite as

On the Structural Theorem of Persistent Homology

  • Killian Meehan
  • Andrei Pavlichenko
  • Jan SegertEmail author
Article
  • 74 Downloads

Abstract

We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. We show that the Matrix Structural Theorem is equivalent to the Krull–Schmidt property of the category of filtered chain complexes. We separately establish the Krull–Schmidt property by abstract categorical methods, yielding a novel nonconstructive proof of the Matrix Structural Theorem. These results provide the foundation for an alternate categorical framework for decomposition in persistent homology, bypassing the usual persistence vector spaces and quiver representations.

Keywords

Persistent homology Matrix reduction Krull–Schmidt category 

Mathematics Subject Classification

55U99 18G99 

References

  1. 1.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Online Edition (2004). http://katmat.math.uni-bremen.de/acc
  2. 2.
    Alperin, J.L., Bell, R.B.: Groups and Representations. Graduate Texts in Mathematics, vol. 162. Springer, New York (1995)Google Scholar
  3. 3.
    Atiyah, M.F.: On the Krull–Schmidt theorem with application to sheaves. Bull. Soc. Math. France 84, 307–317 (1956). http://www.numdam.org/item?id=BSMF_1956__84__307_0
  4. 4.
    Awodey, S.: Category Theory. Oxford Logic Guides, vol. 52, 2nd edn. Oxford University Press, Oxfrord (2010)zbMATHGoogle Scholar
  5. 5.
    Boissonnat, J.-D., Chazal, F., Yvinec, M.: Geometric and Topological Inference, online notes dated January 17, 2017. http://geometrica.saclay.inria.fr/team/Fred.Chazal/papers/CGLcourseNotes/main.pdf
  6. 6.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46, 255–308 (2009) http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/
  7. 7.
    Carlsson, G.: Topological pattern recognition for point cloud dat. Acta Numer. 23, 289–368 (2014). http://math.stanford.edu/~gunnar/actanumericathree.pdf
  8. 8.
    Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010). arXiv:0812.0197
  9. 9.
    Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings of the 25th Annual Symposium on Computational Geometry (SoCG’09). ACM, New York (2009). http://www.mrzv.org/publications/zigzags/socg09/
  10. 10.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Morozov, D.: Persistent homology for kernels, images, and cokernels. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’09), pp. 1011–1020. SIAM, Philadelphia (2009). http://www.mrzv.org/publications/kic/full-soda09/
  12. 12.
    Cohen-Steiner, D., Edelsbrunner, H., Morozov, D.: Vines and vineyards by updating persistence in linear time. In: Proceedings of the 22nd Annual Symposium on Computational Geometry (SCG’06), pp. 119–126. ACM, New York (2006). https://pdfs.semanticscholar.org/69c7/12bb113bd06b6c5d6d71f5f3a24f5b243336.pdf
  13. 13.
    de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Dualities in persistent (co)homology. Inverse Probl. 27(12), (2011). arXiv:1107.5665v1 [math.AT]
  14. 14.
    Edelsbrunner, H.: CPS296.1: Computational Topology, Duke University Course Notes (2006). https://www.cs.duke.edu/courses/fall06/cps296.1/
  15. 15.
    Edelsbrunner, H., Harer, J.L.: Computation Topology: An Introduction. American Mathematical Society, Providence (2009)CrossRefGoogle Scholar
  16. 16.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002). https://users.cs.duke.edu/~edels/Papers/2002-J-04-TopologicalPersistence.pdf
  17. 17.
    Freyd, P.J.: Abelian Categories. Harper’s Series in Modern Mathematics. Harper and Row, New York (1964). Reprints in Theory and Applications of Categories, vol. 3 (2003). ftp://ftp.sam.math.ethz.ch/EMIS/journals/TAC/reprints/articles/3/tr3.pdf
  18. 18.
    Geck, M.: An Introduction to Algebraic Geometry and Algebraic Groups. Oxford Graduate Texts in Mathematics, vol. 20. Oxford University Press, Oxford (2003)Google Scholar
  19. 19.
    Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45(1), 61–75 (2008). http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01191-3/
  20. 20.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001). https://www.math.cornell.edu/~hatcher/AT/ATpage.html
  21. 21.
    Krause, H.: Krull–Schmidt categories and projective covers. Expo. Math. 33(4), 535–549 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liu, S.: Auslander–Reiten theory in a Krull–Schmidt category. São Paulo J. Math. Sci. 4(3), 425–472 (2010). https://www.ime.usp.br/~spjm/articlepdf/432.pdf
  23. 23.
    Lusztig, G.: Bruhat decomposition and applications (2010). arXiv:1006.5004
  24. 24.
    MacLane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, New York (1971)Google Scholar
  25. 25.
    Maria, C., Oudot, S.Y.: Zigzag persistence via reflections and transpositions. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15). SIAM, Philadelphia (2015). https://hal.inria.fr/hal-01091949
  26. 26.
    Maria, C., Oudot, S.: Computing zigzag persistent cohomology (2016). arXiv:1608.06039
  27. 27.
    Miyachi, J.: Derived categories with applications to representations of algebras. http://www.u-gakugei.ac.jp/~miyachi/papers/ChibaSemi.pdf
  28. 28.
    Oudot, S.Y.: Persistence Theory: From Quiver Representations to Data Analysis. Mathematical Surveys and Monographs, vol. 209. American Mathematical Society, Providence (2015)CrossRefGoogle Scholar
  29. 29.
    Pavlichenko, A., Segert, J., Meehan, K.: On kernels and cokernels in persistent homology (in preparation)Google Scholar
  30. 30.
    Shiffler, R.: Quiver Representations, CMS Books in Mathematics. Springer, Berlin (2014)Google Scholar
  31. 31.
    Skraba, P., Vejdemo-Johansson, M.: Parallel & scalable zig-zag persistent homology (2012)Google Scholar
  32. 32.
    Stacks Project Authors: Homological Algebra, The Stacks Project. http://stacks.math.columbia.edu/download/homology.pdf
  33. 33.
    Weinberger, S.: What is...persistent homology? Notices Am. Math. Soc. 58(1), 36–39 (2011). http://www.ams.org/notices/201101/rtx110100036p.pdf
  34. 34.
    Zomorodian, A.J.: Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics, vol. 16. Cambridge University Press, Cambridge (2005)Google Scholar
  35. 35.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zomorodian, A., Carlsson, G.: Localized homology. Comput. Geom. 41(3), 126–148 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hiraoka LaboratoryKUIAS, Kyoto UniversityKyotoJapan
  2. 2.Mathematics DepartmentUniversity of MissouriColumbiaUSA

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