Advances in Applied Probability

The normalized graph cut and Cheeger constant: from discrete to continuous


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Let M be a bounded domain of ∝d 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.

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Adv. in Appl. Probab. Volume 44, Number 4 (2012), 907-937.

First available in Project Euclid: 5 December 2012

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Cheeger isoperimetric constant of a manifold conductance of a graph; neighborhood graph spectral clustering U-process empirical process


ARIAS-CASTRO, ERY; PELLETIER, BRUNO; PUDLO, PIERRE. The normalized graph cut and Cheeger constant: from discrete to continuous. Adv. in Appl. Probab. 44 (2012), no. 4, 907--937. doi:10.1239/aap/1354716583.

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  • Arora, S., Hazan, E. and Kale, S. (2004). $O(\sqrt{\log n})$-approximation to sparsest cut in $\tilde{O}(n^2)$ time. In Proc. 45th Ann. IEEE Symp. on Foundations of Computer Science, IEEE Computer Society, Washington, DC, pp. 238–247.
  • Avin, C. and Ercal, G. (2007). On the cover time and mixing time of random geometric graphs. Theoret. Comput. Sci. 380, 2–22.
  • 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.
  • Belkin, M. and Niyogi, P. (2008). Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. System Sci. 74, 1289–1308.
  • Biau, G., Cadre, B. and Pelletier, B. (2007). A graph-based estimator of the number of clusters. ESAIM Prob. Statist. 11, 272–280.
  • Biau, G., Cadre, B. and Pelletier, B. (2008). Exact rates in density support estimation. J. Multivariate Anal. 99, 2185–2207.
  • 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 C. Demetrescu, R. Sedgewick, and R. Tamassia, SIAM, pp. 240–249.
  • 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.
  • Buser, P. (1982). A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15, 213–230.
  • Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46, 255–308.
  • Carlsson, G. and Zomorodian, A. (2009). The theory of multidimensional persistence. Discrete Comput. Geom. 42, 71–93.
  • 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.
  • Chazal, F. and Lieutier, A. (2005). Weak feature size and persistent homology: computing homology of solids in $\mathbb{R}^n$ from noisy data samples. In Computational Geometry (SCG'05), ACM, New York, pp. 255–262.
  • 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.
  • 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.
  • Chung, F. R. K. (1997). Spectral Graph Theory. (COMS Regional Conf. Ser. Math. 92). American Mathematical Society, Providence, RI.
  • Cuevas, A., Fraiman, R. and Rodríguez-Casal, A. (2007). A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35, 1031–1051.
  • De la Peña, V. H. and Giné, E. (1999). Decoupling, Springer, New York.
  • Doob, J. L. (1994). Measure Theory (Graduate Texts Math. 143). Springer, New York.
  • Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL.
  • Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93, 418–491.
  • 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.
  • Giusti, E. (1984). Minimal Surfaces and Functions of Bounded Variations (Monogr. Math. 80). Birkhäuser, Basel.
  • Gray, A. (2004). Tubes (Progress Math. 221), 2nd edn. Birkhäuser, Basel.
  • Henrot, A. and Pierre, M. (2005). Variation et Optimisation de Formes (Math. Appl. (Berlin) 48), Springer, Berlin.
  • Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30.
  • Khmaladze, E. and Weil, W. (2008). Local empirical processes near boundaries of convex bodies. Ann. Inst. Statist. Math. 60, 813–842.
  • Kolmogorov, A. N. and Tikhomirov, V. M. (1961). $\varepsilon $-entropy and $\varepsilon $-capacity of sets in functional space. Amer. Math. Soc. Transl. (2) 17, 277–364.
  • 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.
  • 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.
  • 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).
  • 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.
  • 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.
  • 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.
  • Pelletier, B. and Pudlo, P. (2011). Operator norm convergence of spectral clustering on level sets. J. Mach. Learning Res. 12, 385–416.
  • Penrose, M. (2003). Random Geometric Graphs (Oxford Stud. Prob. 5). Oxford University Press.
  • 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.
  • Singer, A. (2006). From graph to manifold Laplacian: the convergence rate. Appl. Comput. Harmon. Anal. 21, 128–134.
  • Spielman, D. A. and Teng, S.-H. (2007). Spectral partitioning works: planar graphs and finite element meshes. Linear Algebra Appl. 421, 284–305.
  • von Luxburg, U. (2007). A tutorial on spectral clustering. Statist. Comput. 17, 395–416.
  • von Luxburg, U., Belkin, M. and Bousquet, O. (2008). Consistency of spectral clustering. Ann. Statist. 36, 555–586.
  • Walther, G. (1997). Granulometric smoothing. Ann. Statist. 25, 2273–2299.
  • Weyl, H. (1939). On the volume of tubes. Amer. J. Math. 61, 461–472.
  • Zomorodian, A. and Carlsson, G. (2005). Computing persistent homology. Discrete Comput. Geom. 33, 249–274.