메뉴 건너뛰기




Volumn 74, Issue 8, 2008, Pages 1289-1308

Towards a theoretical foundation for Laplacian-based manifold methods

Author keywords

Graph Laplacian; Laplace Beltrami operator; Manifold methods

Indexed keywords

GRAPH THEORY; LAPLACE TRANSFORMS; SEMI-SUPERVISED LEARNING;

EID: 55449104028     PISSN: 00220000     EISSN: 10902724     Source Type: Journal    
DOI: 10.1016/j.jcss.2007.08.006     Document Type: Article
Times cited : (329)

References (38)
  • 1
    • 55449128783 scopus 로고    scopus 로고
    • M. Belkin, Problems of learning on manifolds, PhD Dissertation, The University of Chicago, 2003
    • M. Belkin, Problems of learning on manifolds, PhD Dissertation, The University of Chicago, 2003
  • 2
    • 55449083300 scopus 로고    scopus 로고
    • M. Belkin, P. Niyogi, Using manifold structure for partially labeled classification, in: NIPS 2002
    • M. Belkin, P. Niyogi, Using manifold structure for partially labeled classification, in: NIPS 2002
  • 3
    • 0042378381 scopus 로고    scopus 로고
    • Laplacian eigenmaps for dimensionality reduction and data representation
    • Belkin M., and Niyogi P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15 6 (2003) 1373-1396
    • (2003) Neural Comput. , vol.15 , Issue.6 , pp. 1373-1396
    • Belkin, M.1    Niyogi, P.2
  • 4
    • 55449086486 scopus 로고    scopus 로고
    • M. Belkin, P. Niyogi, Towards a theoretical foundation for Laplacian-Based manifold methods, in: COLT 2005
    • M. Belkin, P. Niyogi, Towards a theoretical foundation for Laplacian-Based manifold methods, in: COLT 2005
  • 5
    • 33745465341 scopus 로고    scopus 로고
    • M. Belkin, P. Niyogi, V. Sindhwani, On manifold regularization, in: AI Stats 2005
    • M. Belkin, P. Niyogi, V. Sindhwani, On manifold regularization, in: AI Stats 2005
  • 6
    • 55449124420 scopus 로고    scopus 로고
    • 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, in: NIPS 2003
    • 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, in: NIPS 2003
  • 7
    • 55449092487 scopus 로고    scopus 로고
    • M. Bernstein, V. de Silva, J.C. Langford, J.B. Tenenbaum, Graph approximations to geodesics on embedded manifolds, Technical Report, 2000
    • M. Bernstein, V. de Silva, J.C. Langford, J.B. Tenenbaum, Graph approximations to geodesics on embedded manifolds, Technical Report, 2000
  • 8
    • 55449136301 scopus 로고    scopus 로고
    • O. Bousquet, O. Chapelle, M. Hein, Measure based regularization, in: NIPS 2003
    • O. Bousquet, O. Chapelle, M. Hein, Measure based regularization, in: NIPS 2003
  • 9
    • 55449112618 scopus 로고    scopus 로고
    • O. Chapelle, J. Weston, B. Schoelkopf, Cluster kernels for semi-supervised learning, in: NIPS 2002
    • O. Chapelle, J. Weston, B. Schoelkopf, Cluster kernels for semi-supervised learning, in: NIPS 2002
  • 11
    • 19644394100 scopus 로고    scopus 로고
    • Geometric diffusions as a tool for harmonic analysis and structure definition of data, Parts I and II
    • Coifman R.R., Lafon S., Lee A., Maggioni M., Nadler B., Warner F., and Zucker S. Geometric diffusions as a tool for harmonic analysis and structure definition of data, Parts I and II. Proc. Natl. Acad. Sci. 102 (May 2005) 7426-7438
    • (2005) Proc. Natl. Acad. Sci. , vol.102 , pp. 7426-7438
    • Coifman, R.R.1    Lafon, S.2    Lee, A.3    Maggioni, M.4    Nadler, B.5    Warner, F.6    Zucker, S.7
  • 13
    • 0037948870 scopus 로고    scopus 로고
    • Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data
    • Donoho D.L., and Grimes C.E. Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Arts Sci. 100 (2003) 5591-5596
    • (2003) Proc. Natl. Acad. Arts Sci. , vol.100 , pp. 5591-5596
    • Donoho, D.L.1    Grimes, C.E.2
  • 14
    • 55449116088 scopus 로고    scopus 로고
    • E. Gine, V. Kolchinski, Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results, preprint, 2006
    • E. Gine, V. Kolchinski, Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results, preprint, 2006
  • 15
    • 55449107945 scopus 로고    scopus 로고
    • M. Hein, J.-Y. Audibert, U. von Luxburg, From graphs to manifolds-Weak and strong pointwise consistency of graph Laplacians, in: Proceedings of the 18th Annual Conference on Learning Theory (COLT 2005)
    • M. Hein, J.-Y. Audibert, U. von Luxburg, From graphs to manifolds-Weak and strong pointwise consistency of graph Laplacians, in: Proceedings of the 18th Annual Conference on Learning Theory (COLT 2005)
  • 16
    • 55449116383 scopus 로고    scopus 로고
    • M. Meila, J. Shi, Learning segmentation by random walks, in: NIPS 2000
    • M. Meila, J. Shi, Learning segmentation by random walks, in: NIPS 2000
  • 17
    • 55449084777 scopus 로고    scopus 로고
    • F. Memoli, G. Sapiro, Comparing point clouds, IMA Technical Report, 2004
    • F. Memoli, G. Sapiro, Comparing point clouds, IMA Technical Report, 2004
  • 18
    • 55449115812 scopus 로고    scopus 로고
    • S. Lafon, Diffusion maps and geodesic harmonics, PhD thesis, Yale University, 2004
    • S. Lafon, Diffusion maps and geodesic harmonics, PhD thesis, Yale University, 2004
  • 19
    • 55449123251 scopus 로고    scopus 로고
    • U. von Luxburg, M. Belkin, O. Bousquet, Consistency of spectral clustering, Max Planck Institute for Biological Cybernetics Technical Report TR 134, 2004
    • U. von Luxburg, M. Belkin, O. Bousquet, Consistency of spectral clustering, Max Planck Institute for Biological Cybernetics Technical Report TR 134, 2004
  • 20
    • 55449111223 scopus 로고    scopus 로고
    • A. Ng, M. Jordan, Y. Weiss, On spectral clustering: Analysis and an algorithm, in: NIPS 2001
    • A. Ng, M. Jordan, Y. Weiss, On spectral clustering: Analysis and an algorithm, in: NIPS 2001
  • 21
    • 55449098962 scopus 로고    scopus 로고
    • presented at IPAM Workshop on Multiscale Structures in the Analysis of High-Dimensional Data, 2004
    • Niyogi P. 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 High-Dimensional Data, 2004
    • Estimating functional maps on Riemannian submanifolds from sampled data
    • Niyogi, P.1
  • 22
    • 33750684023 scopus 로고    scopus 로고
    • Heat Kernels on Weighted Manifolds and Applications
    • Grigoryan A. Heat Kernels on Weighted Manifolds and Applications. Contemp. Math. vol. 398 (2006)
    • (2006) Contemp. Math. , vol.398
    • Grigoryan, A.1
  • 23
    • 40349102105 scopus 로고    scopus 로고
    • Finding the homology of submanifolds with high confidence from random samples
    • Niyogi P., Smale S., and Weinberger S. Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39 1-3 (2008) 419-441
    • (2008) Discrete Comput. Geom. , vol.39 , Issue.1-3 , pp. 419-441
    • Niyogi, P.1    Smale, S.2    Weinberger, S.3
  • 25
    • 0034704222 scopus 로고    scopus 로고
    • Nonlinear dimensionality reduction by locally linear embedding
    • Roweis S.T., and Saul L.K. Nonlinear dimensionality reduction by locally linear embedding. Science 290 (2000)
    • (2000) Science , vol.290
    • Roweis, S.T.1    Saul, L.K.2
  • 26
    • 55449118818 scopus 로고    scopus 로고
    • A. Smola, R. Kondor, Kernels and regularization on graphs, in: COLT/KW 2003
    • A. Smola, R. Kondor, Kernels and regularization on graphs, in: COLT/KW 2003
  • 28
    • 33745378890 scopus 로고    scopus 로고
    • From graph to manifold Laplacian: The convergence rate
    • Singer A. From graph to manifold Laplacian: The convergence rate. Appl. Comput. Harmon. Anal. 21 1 (2006) 135-144
    • (2006) Appl. Comput. Harmon. Anal. , vol.21 , Issue.1 , pp. 135-144
    • Singer, A.1
  • 29
    • 0030381960 scopus 로고    scopus 로고
    • D. Spielman, S. Teng, Spectral partitioning works: Planar graphs and finite element meshes, in: FOCS 1996
    • D. Spielman, S. Teng, Spectral partitioning works: Planar graphs and finite element meshes, in: FOCS 1996
  • 30
    • 55449085926 scopus 로고    scopus 로고
    • Martin Szummer, Tommi Jaakkola, Partially labeled classification with Markov random walks, in: NIPS 2001
    • Martin Szummer, Tommi Jaakkola, Partially labeled classification with Markov random walks, in: NIPS 2001
  • 31
    • 0034704229 scopus 로고    scopus 로고
    • A global geometric framework for nonlinear dimensionality reduction
    • Tenenbaum J.B., de Silva V., and Langford J.C. A global geometric framework for nonlinear dimensionality reduction. Science 290 (2000)
    • (2000) Science , vol.290
    • Tenenbaum, J.B.1    de Silva, V.2    Langford, J.C.3
  • 32
    • 55449106184 scopus 로고    scopus 로고
    • R. Kannan, S. Vempala, A. Vetta, On clusterings-Good, bad and spectral, Technical Report, Computer Science Department, Yale University, 2000
    • R. Kannan, S. Vempala, A. Vetta, On clusterings-Good, bad and spectral, Technical Report, Computer Science Department, Yale University, 2000
  • 33
    • 55449098963 scopus 로고    scopus 로고
    • I. Kondor, J. Lafferty, Diffusion kernels on graphs and other discrete input spaces, in: Proc. 19th Intl. Conf. on Machine Learning, ICML
    • I. Kondor, J. Lafferty, Diffusion kernels on graphs and other discrete input spaces, in: Proc. 19th Intl. Conf. on Machine Learning, ICML
  • 34
    • 0011326268 scopus 로고    scopus 로고
    • Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions
    • Stochastic Processes, Physics and Geometry: New Interplays. II: A Volume in Honour of S. Albeverio, Amer. Math. Soc.
    • Smolyanov O.G., von Weizsäcker H., and Wittich O. Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions. Stochastic Processes, Physics and Geometry: New Interplays. II: A Volume in Honour of S. Albeverio. Can. Math. Soc. Conf. Proc. vol. 29 (2000), Amer. Math. Soc. 589-602
    • (2000) Can. Math. Soc. Conf. Proc. , vol.29 , pp. 589-602
    • Smolyanov, O.G.1    von Weizsäcker, H.2    Wittich, O.3
  • 35
    • 33846013541 scopus 로고    scopus 로고
    • Chernoff's theorem and discrete time approximations of Brownian motion on manifolds
    • Smolyanov O.G., von Weizsacker H., and Wittich O. Chernoff's theorem and discrete time approximations of Brownian motion on manifolds. Potential Anal. 26 (2007) 1-29
    • (2007) Potential Anal. , vol.26 , pp. 1-29
    • Smolyanov, O.G.1    von Weizsacker, H.2    Wittich, O.3
  • 36
    • 55449114418 scopus 로고    scopus 로고
    • D. Zhou, O. Bousquet, T.N. Lal, J. Weston, B. Schoelkopf, Learning with local and global consistency, in: NIPS 2003
    • D. Zhou, O. Bousquet, T.N. Lal, J. Weston, B. Schoelkopf, Learning with local and global consistency, in: NIPS 2003
  • 37
    • 1942484430 scopus 로고    scopus 로고
    • X. Zhu, J. Lafferty, Z. Ghahramani, Semi-supervised learning using Gaussian fields and harmonic functions, in: ICML 2003
    • X. Zhu, J. Lafferty, Z. Ghahramani, Semi-supervised learning using Gaussian fields and harmonic functions, in: ICML 2003
  • 38
    • 4544362289 scopus 로고    scopus 로고
    • A. Zomorodian, G. Carlsson, Computing persistent homology, in: 20th ACM Symposium on Computational Geometry, 2004
    • A. Zomorodian, G. Carlsson, Computing persistent homology, in: 20th ACM Symposium on Computational Geometry, 2004


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.