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.