ABSTRACT
This chapter discusses various methods for nonlinear dimensionality reduction, where the nonlinear aspect refers to the mapping between the high-dimensional space and the low-dimensional space. In general, multidimensional scaling (MDS) is a set of techniques for the analysis of proximity data measured on a set of objects in order to reveal hidden structure. The purpose of MDS is to find a configuration of the data points in a low-dimensional space such that the proximity between objects in the full-dimensional space is represented with some degree of fidelity by the distances between points in the low-dimensional space. Locally linear embedding (LLE) was developed by Roweis and Saul. The method is an eigenvector-based method, and its optimizations do not involve local minima or iterative algorithms. Some of the initial work in MDS was done by Shepard in 1962. The first paper in the series describes a computer program to reconstruct a configuration of points in Euclidean space.
