DOI: 10.1162/089976603321780317 ISSN:

Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

Mikhail Belkin, Partha Niyogi
  • Cognitive Neuroscience
  • Arts and Humanities (miscellaneous)

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.