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Finding the homology of submanifolds with high confidence from random samples. (English) Zbl 1148.68048

Summary: 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.

MSC:

68T05 Learning and adaptive systems in artificial intelligence
60D05 Geometric probability and stochastic geometry
62-07 Data analysis (statistics) (MSC2010)
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[1] Amenta, N.; Bern, M., Surface reconstruction by Voronoi filtering, Discrete Comput. Geom., 22, 481-504, (1999) · Zbl 0939.68138
[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) · Zbl 1152.68653
[3] Belkin, M.; Niyogi, P., Semisupervised learning on Riemannian manifolds, Mach. Learn., 56, 209-239, (2004) · Zbl 1089.68086
[4] Bjorner, A.; Graham, R. (ed.); Grotschel, M. (ed.); Lovasz, L. (ed.), Topological methods, 1819-1872, (1995), Amsterdam · Zbl 0851.52016
[5] Chazal, F., Lieutier, A.: Weak feature size and persistent homology: computing homology of solids in ℝ \(n\) from noisy data samples. Preprint · Zbl 1380.68388
[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) · Zbl 1297.68235
[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) · Zbl 1387.68252
[8] Dey, T. K.; Edelsbrunner, H.; Guha, S.; Chazelle, B. (ed.); Goodman, J. E. (ed.); Pollack, R. (ed.), Computational topology, No. 223, 109-143, (1999), Providence
[9] Do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1992) · Zbl 0752.53001
[10] Donoho, D., Grimes, C.: Hessian eigenmaps: new locally-linear embedding techniques for high-dimensional data. Preprint. Department of Statistics, Stanford University (2003) · Zbl 1130.62337
[11] Edelsbrunner, H.; Mucke, E. P., Three-dimensional alpha shapes, ACM Trans. Graph., 13, 43-72, (1994) · Zbl 0806.68107
[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) · Zbl 1266.68190
[13] Friedman, J., Computing Betti numbers via combinatorial laplacians, Algorithmica, 21, 331-346, (1998) · Zbl 0911.57021
[14] Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004) · Zbl 1039.55001
[15] Munkres, J.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984) · Zbl 0673.55001
[16] Roweis, S. T.; Saul, L. K., Nonlinear dimensionality reduction by locally linear embedding, Science, 290, 2323-2326, (2000)
[17] Tenenbaum, J. B.; Silva, V.; Langford, J. C., A global geometric framework for nonlinear dimensionality reduction, Science, 290, 2319-2323, (2000)
[18] Valiant, L. G., A theory of the learnable, Commun. ACM, 27, 1134-1142, (1984) · Zbl 0587.68077
[19] Website for smallest enclosing ball algorithm. http://www2.inf.ethz.ch/personal/gaertner/miniball.html
[20] Zomorodian, A.; Carlsson, G., Computing persistent homology, Discrete Comput. Geom., 33, 249-274, (2005) · Zbl 1069.55003
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