Nonlinear Manifold Learning

• Activity Number: 276
• Type: Contributed
• Date/Time: Tuesday, August 10, 2004 : 2:00 PM to 3:50 PM
• Sponsor: General Methodology
• Author(s): Alan J. Izenman*+
• Companies: Temple University
• Keywords: multivariate ; high dimensionality ; PCA ; machine-learning ; nonlinearity ; kernels

Abstract

Classical multivariate analysis has focused mainly on linear methods for dimensionality reduction. Examples include principal components analysis, canonical variate analysis, and discriminant analysis. New techniques are now being introduced into the statistical literature and the machine-learning literature that attempt to generalize these linear methods for dimensionality reduction to nonlinear methods. One of the main aspects of such generalizations is that there are different strategies for viewing the presence of nonlinearity in high-dimensional space. Thus, we have recently seen a number of versions of "nonlinear" PCA, including polynomial PCA, principal curves and surfaces, auto-associative multilayer neural networks, smallest additive principal components, and kernel PCA. Other techniques for finding nonlinear structure in data are Isomap, local linear embedding, and Laplacian eigenmaps. We describe several of these methods and their connections to each other.