Foundations of Computational Mathematics

, Volume 10, Issue 4, pp 367–405 | Cite as

Zigzag Persistence

Open Access
Article

Abstract

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.

Keywords

Applied topology Persistent topology Quiver representations 

Mathematics Subject Classification (2000)

68W30 55N99 

References

  1. 1.
    M.F. Atiyah, On the Krull–Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84, 307–317 (1956). MATHMathSciNetGoogle Scholar
  2. 2.
    G. Carlsson, T. Ishkhanov, V. de Silva, A. Zomorodian, On the local behavior of spaces of natural images, Int. J. Comput. Vis. 76(1), 1–12 (2008). CrossRefGoogle Scholar
  3. 3.
    G. Carlsson, V. de Silva, D. Morozov, Zigzag persistent homology and real-valued functions, in Proceedings 25th ACM Symposium on Computational Geometry (SoCG), 2009, pp. 247–256. Google Scholar
  4. 4.
    F. Chazal, D. Cohen-Steiner, M. Glisse, L. Guibas, S. Oudot, Proximity of persistence modules and their diagrams, in Proceedings of the 25th Annual ACM Symposium on Computational Geometry (SoCG), 2009, pp. 237–246. Google Scholar
  5. 5.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37(1), 103–120 (2007). MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9(1), 79–103 (2009). MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    V. de Silva, G. Carlsson, Topological estimation using witness complexes, in Eurographics Symposium on Point-Based Graphics, ed. by M. Alexa, S. Rusinkiewicz (ETH, Zürich, 2004), pp. 157–166. Google Scholar
  8. 8.
    H. Derksen, J. Weyman, Quiver representations, Not. Am. Math. Soc. 52(2), 200–206 (2005). MATHMathSciNetGoogle Scholar
  9. 9.
    H. Edelsbrunner, E.P. Mücke, Three-dimensional alpha shapes, ACM Trans. Graph. 13(1), 43–72 (1994). MATHCrossRefGoogle Scholar
  10. 10.
    H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28, 511–533 (2002). MATHMathSciNetGoogle Scholar
  11. 11.
    P. Gabriel, Unzerlegbare Darstellungen I, Manuscr. Math. 6, 71–103 (1972). CrossRefMathSciNetGoogle Scholar
  12. 12.
    V.G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56(1), 57–92 (1980). MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    S. Lang, Algebra, 3rd edn. Graduate Texts in Mathematics (Springer, Berlin, 2005). MATHGoogle Scholar
  14. 14.
    A. Zomorodian, G. Carlsson, Computing persistent homology, Discrete Comput. Geom. 33(2), 249–274 (2005). MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsPomona CollegeClaremontUSA

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