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Laplacian Eigenmaps for Dimensionality Reduction and Data ...
LETTER Communicated by Joshua B. Tenenbaum Laplacian Eigenmaps for Dimensionality Reduction and Data Representation Mikhail Belkin Department of Mathematics, University of Chicago, Chicago, IL 60637, Partha Niyogi Department of Computer Science and Statistics, University of Chicago, Chicago, IL 60637 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.
dimensional representations when data arise from sampling a probabil-ity distribution on a manifold. In this letter, we present a geometrically Neural Computation 15, 1373–1396 (2003) c 2003 Massachusetts Institute of Technology. 1374 M. Belkin and P. Niyogi
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