Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 419–441 | Cite as

Finding the Homology of Submanifolds with High Confidence from Random Samples



Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.


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  1. 1.
    Amenta, N., Bern, M.: Surface reconstruction by Voronoi filtering. Discrete Comput. Geom. 22, 481–504 (1999) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amenta, N., Choi, S., Dey, T.K., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. Int. J. Comput. Geom. Appl. 12, 125–141 (2002) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Belkin, M., Niyogi, P.: Semisupervised learning on Riemannian manifolds. Mach. Learn. 56, 209–239 (2004) MATHCrossRefGoogle Scholar
  4. 4.
    Bjorner, A.: Topological methods. In: Graham, R., Grotschel, M., Lovasz, L. (eds.) Handbook of Combinatorics, pp. 1819–1872. North-Holland, Amsterdam (1995) Google Scholar
  5. 5.
    Chazal, F., Lieutier, A.: Weak feature size and persistent homology: computing homology of solids in ℝn from noisy data samples. Preprint Google Scholar
  6. 6.
    Cheng, S.W., Dey, T.K., Ramos, E.A.: Manifold reconstruction from point samples. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 1018–1027 (2005) Google Scholar
  7. 7.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proceedings of the 21st Symposium on Computational Geometry, pp. 263–271 (2005) Google Scholar
  8. 8.
    Dey, T.K., Edelsbrunner, H., Guha, S.: Computational topology. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry, Contemporary Mathematics, vol. 223, pp. 109–143. AMS, Providence (1999) Google Scholar
  9. 9.
    Do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1992) MATHGoogle Scholar
  10. 10.
    Donoho, D., Grimes, C.: Hessian eigenmaps: new locally-linear embedding techniques for high-dimensional data. Preprint. Department of Statistics, Stanford University (2003) Google Scholar
  11. 11.
    Edelsbrunner, H., Mucke, E.P.: Three-dimensional alpha shapes. ACM Trans. Graph. 13, 43–72 (1994) MATHCrossRefGoogle Scholar
  12. 12.
    Fischer, K., Gaertner, B., Kutz, M.: Fast smallest-enclosing-ball computation in high dimensions. In: Proceedings of the 11th Annual European Symposium on Algorithms (ESA), pp. 630–641 (2003) Google Scholar
  13. 13.
    Friedman, J.: Computing Betti numbers via combinatorial laplacians. Algorithmica 21, 331–346 (1998) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004) MATHGoogle Scholar
  15. 15.
    Munkres, J.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984) MATHGoogle Scholar
  16. 16.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000) CrossRefGoogle Scholar
  17. 17.
    Tenenbaum, J.B., De Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000) CrossRefGoogle Scholar
  18. 18.
    Valiant, L.G.: A theory of the learnable. Commun. ACM 27(11), 1134–1142 (1984) MATHCrossRefGoogle Scholar
  19. 19.
    Website for smallest enclosing ball algorithm. http://www2.inf.ethz.ch/personal/gaertner/miniball.html
  20. 20.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2005) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Partha Niyogi
    • 1
  • Stephen Smale
    • 2
  • Shmuel Weinberger
    • 3
  1. 1.Departments of Computer Science and StatisticsUniversity of ChicagoChicagoUSA
  2. 2.Toyota Technological InstituteChicagoUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

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