Laplacian Eigenmaps for dimensionality reduction and data representation

  • Authors:
  • Mikhail Belkin;Partha Niyogi

  • Affiliations:
  • Department of Mathematics, University of Chicago, Chicago, IL;Department of Computer Science and Statistics, University of Chicago, Chicago, IL

  • Venue:
  • Neural Computation
  • Year:
  • 2003

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Abstract

One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.