A metric multidimensional scaling-based nonlinear manifold learning approach for unsupervised data reduction

Abstract

Manifold learning may be seen as a procedure aiming at capturing the degrees of freedom and structure characterizing a set of high-dimensional data, such as images or patterns. The usual goals are data understanding, visualization, classification, and the computation of means. In a linear framework, this problem is typically addressed by principal component analysis (PCA). We propose here a nonlinear extension to PCA. Firstly, the reduced variables are determined in the metric multidimensional scaling framework. Secondly, regression of the original variables with respect to the reduced variables is achieved considering a piecewise linear model. Both steps parameterize the (noisy) manifold holding the original data. Finally, we address the projection of data onto the manifold. The problem is cast in a Bayesian framework. Application of the proposed approach to standard data sets such as the COIL-20 database is presented.