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Foundations of Computational Mathematics

, Volume 11, Issue 3, pp 305–336 | Cite as

Persistent Intersection Homology

  • Paul BendichEmail author
  • John Harer
Article

Abstract

The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper incorporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersection homology gives useful information about the relationship between an embedded stratified space and its singularities. We give an algorithm for the computation of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcomplexes, and we prove its correctness. We also derive, from Poincaré Duality, some structural results about persistent intersection homology.

Keywords

Persistence Intersection homology Algorithms Stratified spaces 

Mathematics Subject Classification (2000)

55N33 68 

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References

  1. 1.
    P.K. Agarwal, H. Edelsbrunner, J. Harer, Y. Wang, Extreme elevation on a 2-manifold, Discrete Comput. Geom. 36, 553–572 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Y.-H.A. Ban, H. Edelsbrunner, J. Rudolph, Interface surfaces for protein-protein complexes, J. Assoc. Comput. Mach. 53, 361–378 (2006). MathSciNetGoogle Scholar
  3. 3.
    M. Belkin, P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, in Neural Computation (2003), pp. 347–356. Google Scholar
  4. 4.
    P. Bendich, Analyzing stratified spaces using persistent versions of intersection and local homology. Dissertation, Duke University, Durham, North Carolina (2008). URL: http://hdl.handle.net/10161/680.
  5. 5.
    P. Bendich, D. Cohen-Steiner, H. Edelsbrunner, J. Harer, D. Morozov, Inferring local homology from sampled stratified spaces, in Proc. 48th Ann. Sympos. Found. Comp. Sci. (2007), pp. 536–546. Google Scholar
  6. 6.
    P. Bendich, J. Harer, Extreme elevation on a stratified surface. Manuscript, Duke University Mathematics Department (2010). Google Scholar
  7. 7.
    P. Bendich, S. Mukherjee, B. Wang, Towards stratification learning via homology inference. Manuscript, Duke University Computer Science Department (2010). Google Scholar
  8. 8.
    F. Cagliari, M. Ferri, P. Pozzi, Size functions from the categorical viewpoint, Acta Appl. Math. 67, 225–235 (2001). MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    G. Carlsson, A. Collins, L. Guibas, A. Zomorodian, Persistence barcodes for shapes. Int. J. Shape Model. (2005). Google Scholar
  10. 10.
    G. Carlsson, A. Zomorodian, Computing persistent homology, Discrete Comput. Geom. 33, 249–274 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    G. Carlsson, A. Zomorodian, Localized homology, Comput. Geom., Theory Appl. 41, 126–148 (2008). MathSciNetzbMATHGoogle Scholar
  12. 12.
    F. Chazal, D. Cohen-Steiner, M. Glisse, L. Guibas, S. Oudot, Proximity of persistence modules and their diagrams, in Proc. 25th Ann. Sympos. Comput. Geom. (2009), pp. 237–246. CrossRefGoogle Scholar
  13. 13.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37, 103–120 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Extending persistence using Poincare and Lefschetz duality, Found. Comput. Math. 9, 79–103 (2009). MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    D. Cohen-Steiner, H. Edelsbrunner, D. Morozov, Vines and vineyards by updating persistence in linear time, in Proc. 22nd Ann. Sympos. Comput. Geom. (2006), pp. 119–126. Google Scholar
  16. 16.
    H. Edelsbrunner, J. Harer, Persistent homology-a survey, in Twenty Years After, ed. by J.E. Goodman, J. Pach, R. Pollack (AMS, Providence, 2007). Google Scholar
  17. 17.
    H. Edelsbrunner, J. Harer, Computational Topology. An Introduction (AMS, Providence, 2009). Google Scholar
  18. 18.
    H. Edelsbrunner, J. Harer, A. Zomorodian, Hierarchical Morse–Smale complexes for piecewise linear 2-manifolds, Discrete Comput. Geom. 30, 87–107 (2003). MathSciNetzbMATHGoogle Scholar
  19. 19.
    H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28, 511–533 (2002). MathSciNetzbMATHGoogle Scholar
  20. 20.
    G. Friedman, Stratified fibrations and the intersection homology of regular neighborhoods of bottom strata, Topol. Appl. 134, 69–109 (2003). zbMATHCrossRefGoogle Scholar
  21. 21.
    P. Frosini, C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal. 9, 596–603 (1999). Google Scholar
  22. 22.
    M. Goresky, R. MacPherson, Intersection homology I, Topology 19, 135–162 (1980). MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M. Goresky, R. MacPherson, Intersection homology II, Invent. Math. 71, 77–129 (1983). MathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Goresky, R. Macpherson, Stratified Morse Theory (Springer, Berlin, 1987). Google Scholar
  25. 25.
    B. Hughes, S. Weinberger, Surgery and stratified spaces, Ann. Math. Stud. 2, 319–352 (2001). MathSciNetGoogle Scholar
  26. 26.
    H. King, Topological invariance of intersection homology without sheaves, Topol. Appl. 20, 149–160 (1985). zbMATHCrossRefGoogle Scholar
  27. 27.
    F. Kirwan, J. Woolf, An Introduction to Intersection Homology Theory (CRC Press, Boca Raton, 2006). zbMATHGoogle Scholar
  28. 28.
    J. Milnor, Morse Theory (Princeton University Press, Princeton, 1963). zbMATHGoogle Scholar
  29. 29.
    J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Redwood City, 1984). zbMATHGoogle Scholar
  30. 30.
    V. Robins, Toward computing homology from finite approximations, Topol. Proc. 24, 503–532 (1999). MathSciNetzbMATHGoogle Scholar
  31. 31.
    J. Tenenbaum, V. De Silva, J. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290, 2319–2323 (2000). CrossRefGoogle Scholar
  32. 32.
    Y. Wang, P.K. Agarwal, P. Brown, H. Edelsbrunner, J. Rudolph, Coarse and reliable geometric alignment for protein docking, in Proc. Pacific Sympos. Biocomput. (2005), pp. 65–75. Google Scholar

Copyright information

© SFoCM 2010

Authors and Affiliations

  1. 1.IST Austria (Institute for Science and Technology Austria)KlosterneuburgAustria
  2. 2.Departments of Mathematics and of Computer Science, and the Center for Systems BiologyDuke UniversityDurhamUSA

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