Graph Laplacians and their convergence on random neighborhood graphs

Journal of Machine Learning Research

Volumn 8, Issue , 2007, Pages 1325-1370

  Hein, Matthias ^a   Audibert, Jean Yves ^b   Von Luxburg, Ulrike ^a  

^a MAX PLANCK INSTITUTE FOR BIOLOGICAL CYBERNETICS   (Germany)

^b UNIVERSITÉ PARIS EST   (France)

Author keywords

Dimensionality reduction; Graph Laplacians; Graphs; Semi supervised learning; Spectral clustering

Indexed keywords

CLUSTERING ALGORITHMS; MATHEMATICAL OPERATORS; PROBABILITY; SUPERVISED LEARNING;

DIMENSIONALITY REDUCTION; GRAPH LAPLACIANS; SEMI-SUPERVISED LEARNING; SPECTRAL CLUSTERING;

GRAPH THEORY;

EID: 34547313657     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Article

References (43)

1. R. Alexander and S. Alexander. Geodesies in Riemannian manifolds with boundary. Indiana Univ. Math. J., 30:481-488, 1981.
2. M. Belkin. Problems of Learning on Manifolds. PhD thesis, University of Chicago, 2003. http://www.people.cs.uchicago.edu /~misha /thesis.pdf.
3. M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comp., 15(6): 1373-1396, 2003.
4. M. Belkin and P. Niyogi. Semi-supervised learning on manifolds. Machine Learning, 56:209-239, 2004.
5. M. Belkin and P. Niyogi. Towards a theoretical foundation for Laplacian-based manifold methods. In P. Auer and R. Meir, editors, Proc. of the 18th Conf. on Learning Theory (COLT), pages 470-485, Berlin, 2005. Springer.
6. P. H. Bérard. Spectral Geometry: Direct and Inverse Problems. Springer, Berlin, 1986.
7. M. Bernstein, V. de Silva, J. C. Langford, and J.B. Tenenbaum. Graph approximations to geodesics on embedded manifolds. Technical report, Stanford University, 2001.
8. O. Bousquet, O. Chapelle, and M. Hein. Measure based regularization. In S. Thrun, L. Saul, and B. Schölkopf, editors, Adv. in Neur. Inf. Proc. Syst. (NIPS), volume 16. MIT Press, 2004.
9. S. Canu and A. Elisseeff. Regularization, kernels and sigmoid net. unpublished, 1999.
10. F. Chung. Spectral Graph Theory. AMS, Providence, RI, 1997.
11. F. Chung and R. P. Langlands. A combinatorial Laplacian with vertex weights. J. Combinatorial Th. Ser. A, 75:316-327, 1996.
12. S. Coifman and S. Lafon. Diffusion maps. Appl. Comput. Harmon. Anal., 21:5-30, 2006.
13. J. Dodziuk. Difference equation, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc., 284:787-794, 1984.
14. E. Giné and V. Koltchinskii. Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results. High Dimensional Probability, 51:238-259, 2006.
15. W. Greblicki, A. Krzyzak, and M. Pawlak. Distribution-free pointwise consistency of kernel regression estimate. Ann. Stat., 12:1570-1575, 1984.
16. A. Grigoryan. Heat kernels on weighted manifolds and applications. Cont. Math., 398:93-191, 2006.
17. L. Györfi, M. Kohler, A. Krzyżak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer, New York, 2004.
18. M. Hein. Geometrical aspects of statistical learning theory. PhD thesis, MPI für biologische Kybernetik/Technische Universität Darmstadt, 2005.
19. M. Hein. Uniform convergence of adaptive graph-based regularization. In G. Lugosi and H. Simon, editors, Proc. of the 19th Conf. on Learning Theory (COLT), pages 50-64, Berlin, 2006. Springer.
20. d. In L. De Raedt and S. Wrobel, editors, Proc. of the 22nd Int. Conf. on Machine Learning (ICML), pages 289-296, 2005.
21. M. Hein, J.-Y. Audibert, and U. von Luxburg. From graphs to manifolds - weak and strong pointwise consistency of graph Laplacians. In P. Auer and R. Meir, editors, Proc. of the 18th Conf. on Learning Theory (COLT), pages 486-500, Berlin, 2005. Springer.
22. J. Jost. Riemannian Geometry and Geometric Analysis. Springer-Verlag, Berlin, third edition, 2002. W. Klingenberg. Riemannian Geometry. De Gruyter, Berlin, 1982.
23. S. S. Lafon. Diffusion Maps and Geometric Harmonics. PhD thesis, Yale University, 2004. http://www.math.yale.edu /~sl349 /publications /dissertation.pdf.
24. J. M. Lee. Riemannian Manifolds. Springer, New York, 1997.
25. J. M. Lee. Introduction to Smooth Manifolds. Springer, New York, 2003.
26. P. McDonald and R. Meyers. Diffusions on graphs, Poisson problems and spectral geometry. Trans. Amer. Math. Soc., 354:5111-5136, 2002.
27. S. Rosenberg. The Laplacian on a Riemannian Manifold. Cambridge University Press, 1997.
28. 2-Index Theorem. PhD thesis, Universität Mainz, 1996.
29. T. Schick. Manifolds with boundary of bounded geometry. Math. Nachr., 223:103-120, 2001.
30. A. Singer. From graph to manifold Laplacian: The convergence rate. Appl. Comput. Harmon. Anal., 21:128-134, 2006.
31. O. G. Smolyanov, H. von Weizsäcker, and O. Wittich. Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions. In Stochastic Processes, Physics and Geometry: New Interplays, II, volume 29, pages 589-602, Providence, RI, 2000. AMS.
32. O. G. Smolyanov, H. von Weizsäcker, and O. Wittich. Chernoff's theorem and discrete time approximations of Brownian motion on manifolds. Potential Analysis, 26:1-29, 2007.
33. O. Sorkine. Differential representations for mesh processing. Computer Graphics Forum, 25(4): 789-807, 2006.
34. D. Spielman and S. Teng. Spectral partioning works: planar graphs and finite element meshes. In 37th Ann. Symp. on Found. of Comp. Science (FOCS), pages 96-105. IEEE Comp. Soc. Press, 1996.
35. N. T. Varopoulos. Brownian motion and random walks on manifolds. Ann. Inst. Fourier, 34:243-269, 1984.
36. U. von Luxburg. A tutorial on spectral clustering. Technical Report 149, Max Planck Institute for Biological Cybernetics, 2006.
37. U. von Luxburg, M. Belkin, and O. Bousquet. Consistency of spectral clustering, 2007. to appear in Ann. Stat.
38. W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge, 2000.
39. G. Xu. Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design, 21:767-784, 2004.
40. D. Zhou, O. Bousquet, T. N. Lal, J. Weston, and B. Schölkopf. Learning with local and global consistency. In S. Thrun, L. Saul, and B. Schölkopf, editors, Adv. in Neur. Inf. Proc. Syst. (NIPS), volume 16, pages 321-328. MIT Press, 2004.
41. D. Zhou, B. Schölkopf, and T. Hofmann. Semi-supervised learning on directed graphs. In S. Thrun, L. Saul, and B. Schölkopf, editors, Adv. in Neur. Inf. Proc. Syst. (NIPS), volume 17, pages 1633-1640. MIT Press, 2005.
42. X. Zhu and Z. Ghahramani. Learning from labeled and unlabeled data with label propagation. Technical Report CMU-CALD-02-107, 2002.
43. X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields and harmonic functions. In T. Fawcett and N. Mishra, editors, Proc. of the 20nd Int. Conf. on Machine Learning (ICML), 2003.