Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

Mikhail Belkin; Partha Niyogi. 4 January, 2002.
Communicated by Partha Niyogi.


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 non-linear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.

Original Document

The original document is available in Postscript (uploaded 4 January, 2002 by Partha Niyogi).