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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. The algorithm provides a computationally efficient ap- proach to nonlinear Dimensionality Reduction that has locality-preserving properties and a natural connection to clustering.
properties and a natural connection to clustering. Some potential appli-cations and illustrative examples are discussed. 1 Introduction In many areas of artificial intelligence, information retrieval, and data min-ing, one is often confronted with intrinsically low-dimensional data lying in a very high-dimensional space.
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