Article

Convergence of Laplacian eigenmaps

Authors:

Mikhail Belkin,

Partha NiyogiAuthors Info & Claims

NIPS'06: Proceedings of the 20th International Conference on Neural Information Processing Systems

Pages 129 - 136

Published: 04 December 2006 Publication History

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Abstract

Geometrically based methods for various tasks of machine learning have attracted considerable attention over the last few years. In this paper we show convergence of eigenvectors of the point cloud Laplacian to the eigenfunctions of the Laplace-Beltrami operator on the underlying manifold, thus establishing the first convergence results for a spectral dimensionality reduction algorithm in the manifold setting.

References

[1]

M. Belkin, Problems of Learning on Manifolds, Univ. of Chicago, Ph.D. Diss., 2003.

Digital Library

Google Scholar

[2]

M. Belkin, P. Niyogi, Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering, NIPS 2001.

Google Scholar

[3]

M. Belkin, P. Niyogi, Towards a Theoretical Foundation for Laplacian-Based Manifold Methods, COLT 2005.

Digital Library

Google Scholar

[4]

O. Bousquet, O. Chapelle, M. Hein, Measure Based Regularization, NIPS 2003.

Google Scholar

[5]

F. R. K. Chung. (1997). Spectral Graph Theory. Regional Conference Series in Mathematics, number 92.

Google Scholar

[6]

R.R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner and S. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data, submitted to the Proceedings of the National Academy of Sciences (2004).

Google Scholar

[7]

D. L. Donoho, C. E. Grimes, Hessian Eigenmaps: new locally linear embedding techniques for high-dimensional data, PNAS, vol. 100 pp. 5591-5596.

Google Scholar

[8]

E. Gine, V. Kolchinski, Empirical Graph Laplacian Approximation of Laplace-Beltrami Operators: Large Sample Results, preprint.

Google Scholar

[9]

M. Hein, J.-Y. Audibert, U. von Luxburg, From Graphs to Manifolds – Weak and Strong Pointwise Consistency of Graph Laplacians, COLT 2005.

Digital Library

Google Scholar

[10]

S. Lafon, Diffusion Maps and Geodesic Harmonics, Ph.D. Thesis, Yale University, 2004.

Google Scholar

[11]

U. von Luxburg, M. Belkin, O. Bousquet, Consistency of Spectral Clustering, Max Planck Institute for Biological Cybernetics Technical Report TR 134, 2004.

Google Scholar

[12]

S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge Univ. Press, 1997.

Crossref

Google Scholar

[13]

Sam T. Roweis, Lawrence K. Saul. (2000). Nonlinear Dimensionality Reduction by Locally Linear Embedding, Science, vol 290.

Crossref

Google Scholar

[14]

J.B. Tenenbaum, V. de Silva, J. C. Langford. (2000). A Global Geometric Framework for Nonlinear Dimensionality Reduction, Science, Vol 290.

Crossref

Google Scholar

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Convergence of Laplacian eigenmaps

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NIPS'06: Proceedings of the 20th International Conference on Neural Information Processing Systems

December 2006

1632 pages

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MIT Press

Cambridge, MA, United States

Publication History

Published: 04 December 2006

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Hermanns JSkitsas KTsitsulin AMunkhoeva MKyster ANielsen SBronstein AMottin DKarras P(2023)GRASP: Scalable Graph Alignment by Spectral Corresponding FunctionsACM Transactions on Knowledge Discovery from Data10.1145/356105817:4(1-26)Online publication date: 24-Feb-2023
https://dl.acm.org/doi/10.1145/3561058
Tsitsulin AMottin DKarras PBronstein AMüller EGuo YFarooq F(2018)NetLSDProceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining10.1145/3219819.3219991(2347-2356)Online publication date: 19-Jul-2018
https://dl.acm.org/doi/10.1145/3219819.3219991
Patané GMadeira JPatow G(2016)STARProceedings of the 37th Annual Conference of the European Association for Computer Graphics: State of the Art Reports10.5555/3059330.3059334(599-624)Online publication date: 9-May-2016
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Patané G(2016)STAR - Laplacian Spectral Kernels and Distances for Geometry Processing and Shape AnalysisComputer Graphics Forum10.5555/3028584.302863735:2(599-624)Online publication date: 1-May-2016
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