Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T17:32:04.066Z Has data issue: false hasContentIssue false
  • English
  • Français

The Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous

Part of: Nonparametric inference

Published online by Cambridge University Press:  04 January 2016

Ery Arias-Castro ,
Bruno Pelletier  and
Pierre Pudlo
Ery Arias-Castro*
Affiliation:
University of California
Bruno Pelletier*
Affiliation:
Université Rennes II
Pierre Pudlo*
Affiliation:
Université Montpellier II
*
∗ Postal address: Department of Mathematics, University of California, San Diego, USA.
∗∗ Postal address: Département de Mathématiques, IRMAR – UMR CNRS 6625, Université Rennes II, France. Email address: [email protected]
∗∗∗ Postal address: Département de Mathématiques, I3M – UMR CNRS 5149, Université Montpellier II, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be a bounded domain of with a smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.

Keywords

Cheeger isoperimetric constant of a manifoldconductance of a graph; neighborhood graphspectral clusteringU-processempirical process

MSC classification

Primary: 62G05: Estimation 62G20: Asymptotic properties
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 907 - 937
Copyright
© Applied Probability Trust 

References

Arora, S., Hazan, E. and Kale, S. (2004). -approximation to sparsest cut in Õ(n 2) time. In Proc. 45th Ann. IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, Washington, DC, pp. 238–247.Google Scholar
Avin, C. and Ercal, G. (2007). On the cover time and mixing time of random geometric graphs. Theoret. Comput. Sci. 380, 2–22.Google Scholar
Belkin, M. and Niyogi, P. (2001). Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems, Vol. 1, MIT Press, Cambridge, pp. 585–592.Google Scholar
Belkin, M. and Niyogi, P. (2008). Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. System Sci. 74, 1289–1308.Google Scholar
Biau, G., Cadre, B. and Pelletier, B. (2007). A graph-based estimator of the number of clusters. ESAIM Prob. Statist. 11, 272–280.Google Scholar
Biau, G., Cadre, B. and Pelletier, B. (2008). Exact rates in density support estimation. J. Multivariate Anal. 99, 2185–2207.Google Scholar
Boyd, S. P., Ghosh, A., Prabhakar, B. and Shah, D. (2005). Mixing times for random walks on geometric random graphs. In SIAM Workshop on Analytic Algorithmics & Combinatorics (ANALCO). eds Demetrescu, C., Sedgewick, R., and Tamassia, R., SIAM, pp. 240–249.Google Scholar
Bräker, H. and Hsing, T. (1998). On the area and perimeter of a random convex hull in a bounded convex set. Prob. Theory Relat. Fields 111, 517–550.Google Scholar
Buser, P. (1982). A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15, 213–230.CrossRefGoogle Scholar
Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46, 255–308.Google Scholar
Carlsson, G. and Zomorodian, A. (2009). The theory of multidimensional persistence. Discrete Comput. Geom. 42, 71–93.CrossRefGoogle Scholar
Caselles, V., Chambolle, A. and Novaga, M. (2010). Some remarks on uniqueness and regularity of Cheeger sets. Rend. Sem. Mat. Univ. Padova 123, 191–201.Google Scholar
Chazal, F. and Lieutier, A. (2005). Weak feature size and persistent homology: computing homology of solids in R n from noisy data samples. In Computational Geometry (SCG'05), ACM, New York, pp. 255–262.Google Scholar
Chazal, F., Guibas, L. J., Oudot, S. Y. and Skraba, P. (2009). Analysis of scalar fields over point cloud data. In Proc. 20th Ann. ACM-SIAM Symp. on Discrete Algorithms, SIAM, Philadelphia, PA, pp. 1021–1030.Google Scholar
Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), Princeton University Press, Princeton, NJ, pp. 195–199.Google Scholar
Chung, F. R. K. (1997). Spectral Graph Theory. (COMS Regional Conf. Ser. Math. 92). American Mathematical Society, Providence, RI.Google Scholar
Cuevas, A., Fraiman, R. and Rodrı´guez-Casal, A. (2007). A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35, 1031–1051.Google Scholar
De la Peña, V. H. and Giné, E. (1999). Decoupling, Springer, New York.Google Scholar
Doob, J. L. (1994). Measure Theory (Graduate Texts Math. 143). Springer, New York.Google Scholar
Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL.Google Scholar
Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93, 418–491.Google Scholar
Giné, E. and Koltchinskii, V. (2006). Empirical graph Laplacian approximation of Laplace-Beltrami operators: large sample results. In High Dimensional Probability (IMS Lecture Notes Monogr. Ser. 51), Institute of Mathematical Statistics, Beachwood, OH, pp. 238–259.Google Scholar
Giusti, E. (1984). Minimal Surfaces and Functions of Bounded Variations (Monogr. Math. 80). Birkhäuser, Basel.CrossRefGoogle Scholar
Gray, A. (2004). Tubes (Progress Math. 221), 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Henrot, A. and Pierre, M. (2005). Variation et Optimisation de Formes (Math. Appl. (Berlin) 48), Springer, Berlin.Google Scholar
Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30.CrossRefGoogle Scholar
Khmaladze, E. and Weil, W. (2008). Local empirical processes near boundaries of convex bodies. Ann. Inst. Statist. Math. 60, 813–842.Google Scholar
Kolmogorov, A. N. and Tikhomirov, V. M. (1961). ε-entropy and ε-capacity of sets in functional space. Amer. Math. Soc. Transl. (2) 17, 277–364.Google Scholar
Levina, E. and Bickel, P. J. (2005). Maximum likelihood estimation of intrinsic dimension. In Advances in Neural Information Processing Systems, Vol. 17, MIT Press, Cambridge, pp. 777–784.Google Scholar
Maier, M., Von Luxburg, U. and Hein, M. (2009). Influence of graph construction on graph-based clustering measures. In Advances in Neural Information Processing Systems, Vol. 22, MIT Press, Cambridge, pp. 1025–1032.Google Scholar
Narayanan, H. and Niyogi, P. (2009). On the sample complexity of learning smooth cuts on a manifold. In 22nd Ann. Conf. on Learning Theory (COLT).Google Scholar
Narayanan, H., Belkin, M. and Niyogi, P. (2007). On the relation between low density separation, spectral clustering and graph cuts. In Advances in Neural Information Processing Systems, Vol. 19, MIT Press, Cambridge.Google Scholar
Ng, A. Y., Jordan, M. I. and Weiss, Y. (2002). On spectral clustering: Analysis and an algorithm. In Advances in Neural Information Processing Systems, Vol. 14, MIT Press, Cambridge, pp. 849–856.Google Scholar
Niyogi, P., Smale, S. and Weinberger, S. (2008). Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 419–441.CrossRefGoogle Scholar
Pelletier, B. and Pudlo, P. (2011). Operator norm convergence of spectral clustering on level sets. J. Mach. Learning Res. 12, 385–416.Google Scholar
Penrose, M. (2003). Random Geometric Graphs (Oxford Stud. Prob. 5). Oxford University Press.CrossRefGoogle Scholar
Robins, V. (1999). Towards computing homology from finite approximations. In Proc. 14th Summer Conf. on General Topology and Its Applications (Brookville, NY, 1999; Topology Proc. 24), pp. 503–532.Google Scholar
Singer, A. (2006). From graph to manifold Laplacian: the convergence rate. Appl. Comput. Harmon. Anal. 21, 128–134.Google Scholar
Spielman, D. A. and Teng, S.-H. (2007). Spectral partitioning works: planar graphs and finite element meshes. Linear Algebra Appl. 421, 284–305.CrossRefGoogle Scholar
von Luxburg, U. (2007). A tutorial on spectral clustering. Statist. Comput. 17, 395–416.Google Scholar
von Luxburg, U., Belkin, M. and Bousquet, O. (2008). Consistency of spectral clustering. Ann. Statist. 36, 555–586.Google Scholar
Walther, G. (1997). Granulometric smoothing. Ann. Statist. 25, 2273–2299.Google Scholar
Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61, 461–472.CrossRefGoogle Scholar
Zomorodian, A. and Carlsson, G. (2005). Computing persistent homology. Discrete Comput. Geom. 33, 249–274.Google Scholar