Introduction to Manifold Learning

• Activity Number: 673
• Type: Contributed
• Date/Time: Thursday, August 4, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Alan Julian Izenman*+
• Companies: Temple University
• Keywords: dimensionality reduction ; Isomap ; diffusion maps ; eigenmaps ; embedding ; nonlinear manifold learning

Abstract

One of the most popular research areas today in statistics and machine learning is that of manifold learning, which is related to the algorithmic techniques of dimensionality reduction. Manifold learning can be divided into linear and nonlinear methods. Linear methods, which have long been part of the statistician's toolbox for analyzing multivariate data, include principal component analysis (PCA) and multidimensional scaling (MDS). Recently, there has been a flurry of research activity in techniques for nonlinear manifold learning, which includes kernel PCA, Isomap, local linear embedding, Laplacian eigenmaps, Hessian eigenmaps, and diffusion maps. Some of these techniques are nonlinear generalizations of linear methods. The algorithmic process of most of these techniques consists of three steps, a nearest-neighbor search, a computation of distances between points, and an eigenproblem for embedding the r-dimensional points in a lower-dimensional space. In this talk, we give a brief survey of these new methods and indicate their strengths and weaknesses.