Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering

Drawing on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivatedalgorithm for constructing a representation for data sampled from a low dimensional manifold embedded in a higher dimensional space. The algorithm provides a computationally e cient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Several applications are considered. In many areas of arti cial intelligence, information retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. For example, gray scale n n images of a xed object taken with a moving camera yield data points in Rn2 . However, the intrinsic dimensionality of the space of all images of the same object is the number of degrees of freedom of the camera { in fact the space has the natural structure of...

• Type: Conference
• Year: 2001
• Where: NIPS
• Authors: Mikhail Belkin, Partha Niyogi
• Keywords: Dimensional Space, Dimensionality Reduction, Low Dimensional Manifold, NIPS 2001, NIPS 2007