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Persistent Intersection Homology

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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.

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References

  1. P.K. Agarwal, H. Edelsbrunner, J. Harer, Y. Wang, Extreme elevation on a 2-manifold, Discrete Comput. Geom. 36, 553–572 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  2. Y.-H.A. Ban, H. Edelsbrunner, J. Rudolph, Interface surfaces for protein-protein complexes, J. Assoc. Comput. Mach. 53, 361–378 (2006).

    MathSciNet  Google Scholar 

  3. M. Belkin, P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, in Neural Computation (2003), pp. 347–356.

    Google Scholar 

  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. 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. P. Bendich, J. Harer, Extreme elevation on a stratified surface. Manuscript, Duke University Mathematics Department (2010).

  7. P. Bendich, S. Mukherjee, B. Wang, Towards stratification learning via homology inference. Manuscript, Duke University Computer Science Department (2010).

  8. F. Cagliari, M. Ferri, P. Pozzi, Size functions from the categorical viewpoint, Acta Appl. Math. 67, 225–235 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  9. G. Carlsson, A. Collins, L. Guibas, A. Zomorodian, Persistence barcodes for shapes. Int. J. Shape Model. (2005).

  10. G. Carlsson, A. Zomorodian, Computing persistent homology, Discrete Comput. Geom. 33, 249–274 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Carlsson, A. Zomorodian, Localized homology, Comput. Geom., Theory Appl. 41, 126–148 (2008).

    MathSciNet  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  13. D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37, 103–120 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Extending persistence using Poincare and Lefschetz duality, Found. Comput. Math. 9, 79–103 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. H. Edelsbrunner, J. Harer, Computational Topology. An Introduction (AMS, Providence, 2009).

    Google Scholar 

  18. H. Edelsbrunner, J. Harer, A. Zomorodian, Hierarchical Morse–Smale complexes for piecewise linear 2-manifolds, Discrete Comput. Geom. 30, 87–107 (2003).

    MathSciNet  MATH  Google Scholar 

  19. H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28, 511–533 (2002).

    MathSciNet  MATH  Google Scholar 

  20. G. Friedman, Stratified fibrations and the intersection homology of regular neighborhoods of bottom strata, Topol. Appl. 134, 69–109 (2003).

    Article  MATH  Google Scholar 

  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. M. Goresky, R. MacPherson, Intersection homology I, Topology 19, 135–162 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Goresky, R. MacPherson, Intersection homology II, Invent. Math. 71, 77–129 (1983).

    Article  MathSciNet  Google Scholar 

  24. M. Goresky, R. Macpherson, Stratified Morse Theory (Springer, Berlin, 1987).

    Google Scholar 

  25. B. Hughes, S. Weinberger, Surgery and stratified spaces, Ann. Math. Stud. 2, 319–352 (2001).

    MathSciNet  Google Scholar 

  26. H. King, Topological invariance of intersection homology without sheaves, Topol. Appl. 20, 149–160 (1985).

    Article  MATH  Google Scholar 

  27. F. Kirwan, J. Woolf, An Introduction to Intersection Homology Theory (CRC Press, Boca Raton, 2006).

    MATH  Google Scholar 

  28. J. Milnor, Morse Theory (Princeton University Press, Princeton, 1963).

    MATH  Google Scholar 

  29. J.R. Munkres, Elements of Algebraic Topology (Addison-Wesley, Redwood City, 1984).

    MATH  Google Scholar 

  30. V. Robins, Toward computing homology from finite approximations, Topol. Proc. 24, 503–532 (1999).

    MathSciNet  MATH  Google Scholar 

  31. J. Tenenbaum, V. De Silva, J. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290, 2319–2323 (2000).

    Article  Google Scholar 

  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 

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Authors and Affiliations

  1. IST Austria (Institute for Science and Technology Austria), Klosterneuburg, Austria

    Paul Bendich

  2. Departments of Mathematics and of Computer Science, and the Center for Systems Biology, Duke University, Durham, NC, USA

    John Harer

Authors
  1. Paul Bendich
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  2. John Harer
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Corresponding author

Correspondence to Paul Bendich.

Additional information

Communicated by Herbert Edelsbrunner.

This research was partially supported by the Defense Advanced Research Projects Agency (DARPA) under grant HR0011-05-1-0007.

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Bendich, P., Harer, J. Persistent Intersection Homology. Found Comput Math 11, 305–336 (2011). https://doi.org/10.1007/s10208-010-9081-1

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  • DOI: https://doi.org/10.1007/s10208-010-9081-1

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