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Approximating persistent homology in Euclidean space through collapses

  • Magnus Bakke Botnan
  • Gard Spreemann
Original Paper

Abstract

The Čech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the Čech filtration, the number of simplices grows exponentially in the number of input points. A practical consequence is that computations may have to terminate at smaller scales than what the application calls for. In this paper we propose two methods to approximate the Čech persistence module. Both are constructed on the level of spaces, i.e. as sequences of simplicial complexes induced by nerves. We also show how the bottleneck distance between such persistence modules can be understood by how tightly they are sandwiched on the level of spaces. In turn, this implies the correctness of our approximation methods. Finally, we implement our methods and apply them to some example point clouds in Euclidean space.

Keywords

Persistent homology Computational topology Approximation 

References

  1. 1.
    Arsuaga, J., Baas, N.A., DeWoskin, D., Mizuno, H., Pankov, A., Park, C.: Topological analysis of gene expression arrays identifies high risk molecular subtypes in breast cancer. Appl. Algebra Eng. Commun. Comput. 23(1–2), 3–15 (2012). doi: 10.1007/s00200-012-0166-8 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Attali, D., Lieutier, A., Salinas, D.: Efficient data structure for representing and simplifying simplicial complexes in high dimensions. Int. J. Comput. Geom. Appl. 22(04), 279–303 (2012)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Azumaya, G.: Corrections and supplementaries to my paper concerning Krull–Remak–Schmidt’s theorem. Nagoya Math. J. 1, 117–124 (1950)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Boissonnat, J.D., Dey, T., Maria, C.: A space and time efficient implementation for computing persistent homology. INRIA Research Report 8195 (2012)Google Scholar
  5. 5.
    Boissonnat, J.D., Maria, C.: The simplex tree: an efficient data structure for general simplicial complexes. In: Epstein, L., Ferragina, P. (eds.) Algorithms—ESA 2012, Lecture Notes in Computer Science, vol. 7501, pp. 731–742. Springer, Berlin (2012). doi: 10.1007/978-3-642-33090-2_63
  6. 6.
    Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vision 76(1), 1–12 (2008). doi: 10.1007/s11263-007-0056-x CrossRefGoogle Scholar
  7. 7.
    Chan, J.M., Carlsson, G., Rabadan, R.: Topology of viral evolution. Proc. Nat. Acad. Sci. 110(46), 18,566–18,571 (2013). doi: 10.1073/pnas.1313480110 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L., Oudot, S.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, SCG’09, pp. 237–246 (2009). doi: 10.1145/1542362.1542407
  9. 9.
    Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules. ArXiv e-prints (2012)Google Scholar
  10. 10.
    Chazal, F., Oudot, S.: Towards persistence-based reconstruction in Euclidean spaces. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG’08, pp. 232–241 (2008)Google Scholar
  11. 11.
    Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. ArXiv e-prints (2012)Google Scholar
  12. 12.
    de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proceedings of the First Eurographics Conference on Point-Based Graphics, SPBG’04, pp. 157–166. Eurographics Association (2004). doi: 10.2312/SPBG/SPBG04/157-166
  13. 13.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7, 339–358 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dey, T., Edelsbrunner, H., Guha, S., Nekhayev, D.: Topology preserving edge contraction. Publications de l’Institut Mathématique (Beograd) (NS) 66(80), 23–45 (1999)MathSciNetGoogle Scholar
  15. 15.
    Dey, T., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. ArXiv e-prints (2012)Google Scholar
  16. 16.
    Dey, T., Fan, F., Wang, Y.: Graph induced complex on point data. In: Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry, SoCG’13, pp. 107–116 (2013)Google Scholar
  17. 17.
    Dłotko, P., Wagner, H.: Simplification of complexes of persistent homology computations. Homol. Homotopy Appl. 16(1), 49–63 (2014)CrossRefzbMATHGoogle Scholar
  18. 18.
    Edelsbrunner, H.: The union of balls and its dual shape. In: Proceedings of the Ninth Annual Symposium on Computational Geometry, SCG’93, pp. 218–231 (1993)Google Scholar
  19. 19.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)Google Scholar
  20. 20.
  21. 21.
    Gärtner, B.: Fast and robust smallest enclosing balls. In: Nešetřil , J. (ed.) Algorithms - ESA ’99, Lecture Notes in Computer Science, vol. 1643, pp. 325–338. Springer, Berlin (1999). doi: 10.1007/3-540-48481-7_29. http://www.inf.ethz.ch/personal/gaertner/miniball.html
  22. 22.
    Harker, S., Mischaikow, K., Mrozek, M., Nanda, V.: Discrete Morse theoretic algorithms for computing homology of complexes and maps. Found. Comput. Math. 14(1), 151–184 (2014). doi: 10.1007/s10208-013-9145-0 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)Google Scholar
  24. 24.
    Kerber, M., Sharathkumar, R.: Approximate Čech complex in low and high dimensions. In: Cai, L., Cheng, S.W., Lam, T.W. (eds.) Algorithms and Computation, Lecture Notes in Computer Science, vol. 8283, pp. 666–676. Springer, Berlin (2013). doi: 10.1007/978-3-642-45030-3_62
  25. 25.
    Müllner, D.: fastcluster: fast hierarchical, agglomerative clustering routines for R and Python. J. Stat. Softw. 53(9), 1–8 (2013)Google Scholar
  26. 26.
    Perea, J., Carlsson, G.: A Klein-bottle-based dictionary for texture representation. Int. J. Comput. Vision 107(1), 75–97 (2014). doi: 10.1007/s11263-013-0676-2 CrossRefMathSciNetGoogle Scholar
  27. 27.
    Perea, J., Harer, J.: Sliding windows and persistence: an application of topological methods to signal analysis. Found. Comput. Math. pp. 1–40 (2014). doi: 10.1007/s10208-014-9206-z
  28. 28.
    Sheehy, D.: Linear-size approximations to the Vietoris–Rips filtration. Discrete Comput. Geom. 49(4), 778–796 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Takens, F.: Detecting strange attractors in turbulence. In: Rand, D., Young, L.S. (eds.) Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, Berlin (1981)Google Scholar
  30. 30.
    Zomorodian, A.: The tidy set: a minimal simplicial set for computing homology of clique complexes. In: Proceedings of the Twenty-sixth Annual Symposium on Computational Geometry, SoCG’10, pp. 257–266 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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