Towards a theoretical foundation for Laplacian-based manifold methods

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 3559 LNAI, Issue , 2005, Pages 486-500

  Belkin, Mikhail ^a   Niyogi, Partha ^a  

^a THE UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; GRAPH THEORY; LEARNING SYSTEMS; MATHEMATICAL OPERATORS;

GRAPH LAPLACIAN; LAPLACE-BELTRAMI OPERATOR; MANIFOLD METHODS;

LAPLACE TRANSFORMS;

EID: 26944439046     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/11503415_33     Document Type: Conference Paper

References (31)

1. M. Belkin, Problems of Learning on Manifolds, The University of Chicago, Ph.D. Dissertation, 2003.
2. M. Belkin, P. Niyogi, Using Manifold Structure for Partially Labeled Classification, NIPS 2002.
3. M. Belkin, P. Niyogi. (2003). Laplacian Eigenmaps for Dimensionality Reduction and Data Representation, Neural Computation, Vol. 15, No. 6, 1373-1396.
4. M. Belkin, P. Niyogi, V. Sindhwani, On Manifold Regularization, AI Stats 2005.
5. Y. Bengio, J.-F. Paiement, P. Vincent, O. Delalleau, N. Le Roux, M. Ouimet, Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering, NIPS 2003.
6. M. Bernstein, V. de Silva, J.C. Langford, J.B. Tenenbaum, Graph approximations to geodesies on embedded manifolds, Technical Report, 2000.
7. O. Bousquet, O. Chapelle, M. Hein, Measure Based Regularization, NIPS 2003.
8. Y. Bengio, J-F. Paiement, and P. Vincent,Out-of-Sample Extensions for LLE, Isomap, MDS, Eigenmaps, and Spectral Clustering, NIPS 2003.
9. Chapelle, O., J. Weston and B. Schoelkopf, Cluster Kernels for Semi-Supervised Learning, NIPS 2002.
10. F. R. K. Chung. (1997). Spectral Graph Theory. Regional Conference Series in Mathematics, number 92.
11. M. do Carmo, Riemannian Geometry, Birkhauser, 1992.
12. D. L. Donoho, C. E. Grimes, Hessian Eigenmaps: new locally linear embedding techniques for high-dimensional data, Proceedings of the National Academy of Arts and Sciences vol. 100 pp. 5591-5596.
13. M. Hein, J.-Y. Audibert, U. von Luxburg, From Graphs to Manifolds - Weak and Strong Pointwise Consistency of Graph Laplacians, COLT 2005, (to appear).
14. M. Meila, J. Shi, Learning segmentation by random walks, NIPS 2000.
15. F. Memoli, G. Sapiro, Comparing Point Clouds, IMA Technical Report, 2004.
16. S. Lafon, Diffusion Maps and Geodesic Harmonies, Ph. D. Thesis, Yale University, 2004.
17. U. von Luxburg, M. Belkin, O. Bousquet, Consistency of Spectral Clustering, Max Planck Institute for Biological Cybernetics Technical Report TR 134, 2004.
18. A. Ng, M. Jordan, Y. Weiss, On Spectral Clustering: Analysis and an Algorithm, NIPS 2001.
19. P. Niyogi, Estimating Functional Maps on Riemannian Submanifolds from Sampled Data, http://www.ipam. ucla.edu/publications/mgaws3/mgaws3_5188.pdf, presented at IPAM Workshop on Multiscale structures in the analysis of HighDimensional Data, 2004.
20. P. Niyogi, S. Smale, S. Weinberger, Finding the Homology of Submanifolds with High Confidence from Random Samples, Univ. of Chicago Technical Report TR2004-08, 2004.
21. S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge University Press, 1997.
22. Sam T. Roweis, Lawrence K. Saul. (2000). Nonlinear Dimensionality Reduction by Locally Linear Embedding, Science, vol 290.
23. A. Smola and R. Kondor, Kernels and Regularization on Graphs, COLT/KW 2003.
24. J. Shi, J. Malik, Normalized Cuts and Image Segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 8, 2000.
25. D. Spielman, S. Teng, Spectral partitioning works: planar graphs and finite element meshes, FOCS 1996.
26. Martin Szummer, Tommi Jaakkola, Partially labeled classification with Markov random walks, NIPS 2001.
27. J.B.Tenenbaum, V. de Silva, J. C. Langford. (2000). A Global Geometric Framework for Nonlinear Dimensionality Reduction, Science, Vol 290.
28. R. Kannan, S. Vempala, A. Vetta, On Clusterings - Good, Bad and Spectral, Technical Report, Computer Science Department, Yale University, 2000.
29. D. Zhou, O. Bousquet, T.N. Lal, J. Weston and B. Schoelkopf, Learning with Local and Global Consistency, NIPS 2003.
30. X. Zhu, J. Lafferty and Z. Ghahramani, Semi-supervised learning using Gaussian fields and harmonic functions, ICML 2003.
31. A. Zomorodian, G. Carlsson, Computing persistent homology, 20th ACM Symposium on Computational Geometry, 2004.