Nonlinear modeling of scattered multivariate data and its application to shape change

We are given a set of points in a space of high dimension. For instance, this set may represent many visual appearances of an object, a face, or a hand. We address the problem of approximating this set by a manifold in order to have a compact representation of the object appearance. When the scattering of this set is approximately an ellipsoid, then the problem has a well-known solution given by principal components analysis (PCA). However, in some situations like object displacement learning or face learning, this linear technique may be ill-adapted and nonlinear approximation has to be introduced. The method we propose can be seen as a nonlinear PCA (NLPCA), the main difficulty being that the data are not ordered. We propose an index which favors the choice of axes preserving the closest point neighborhoods. These axes determine an order for visiting all the points when smoothing. Finally, a new criterion, called "generalization error", is introduced to determine the smoothing rate, that is, the knot number for the spline fitting. Experimental results conclude this paper: The method is tested on artificial data and on two databases used in visual learning.