Advertisement

Abstract

Laplacian eigenmaps (Leigs) method is based on the idea of manifold unsupervised learning. Let H be the observed high-dimensional data, which reside on a low-dimentional manifold M. Let h be the coordinate mapping on M so that Y = h(H)is a DR of H. Each component of the coordinate mapping h is a linear function on M. Hence, all components of h nearly reside on the numerically null space of the Laplace-Beltrsmi operator on M. In Leigs method, a Laplace-Beltrami operator is constructed on the data graph. Then the DR data set is derived from the eigenvectors of the operator corresponding to several smallest eigenvalues. The chapter is organized as follows. In Section 12.1, we describe the Laplacian eigenmaps method and focus the discussion on the construction of the Laplace-Beltrami operator on the data set. In Section 12.2, the Leigs DR algorithm is developed. In Section 12.3, some experiments of the algorithm are presented.

Keywords

Weight Matrix Coordinate Mapping Graph Neighborhood Noise Standard Deviation Laplacian Eigenmaps 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Belkin, M.: Problems of Learning on Manifolds. Ph.D. thesis, The University of Chicago (2003).Google Scholar
  2. [2]
    Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15(6), 1373–1396 (2003).zbMATHCrossRefGoogle Scholar
  3. [3]
    Belkin, M., Niyogi, P.: Semi-supervised learning on Riemannian manifolds. Machine Learning 56(1), 209–239 (2004).zbMATHCrossRefGoogle Scholar
  4. [4]
    Law, M.H.C., Jain, A.K.: Incremental nonlinear dimensionality reduction by manifold learning. IEEE Trans. Pattern Analysis and Machine Intelligence 28(3), 377–391 (2006).CrossRefGoogle Scholar
  5. [5]
    Park, J.H., Zhang, Z.Y., Zha, H.Y., Kasturi, R.: Local smoothing for manifold learning. In: CVPR II:, pp. 452–459 (2004).Google Scholar
  6. [6]
    Yosida, K.: Functional Analysis. Springer-Verlag (1980).Google Scholar
  7. [7]
    Rosenberg, S.: The Laplacian on a Riemmannian Manifold. Cambridge University Press (1997).Google Scholar
  8. [8]
    Donoho, D.L., Grimes, C.: Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. USA 100, 5591–5596 (2003).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jianzhong Wang
    • 1
  1. 1.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA

Personalised recommendations