Some theoretical results on nonlinear principal components analysis

Presents some results on the nonlinear principal components analysis (NLPCA) method of nonlinear feature extraction and discusses its relation to the principal curve/surface method. Both methods attempt to reduce the dimension of a set of multivariate observations by fitting a curve or surface through the middle of the observations and projecting the observations onto this curve/surface. The two methods fit their models under a similar objective function, with one important difference: NLPCA defines the function which maps observed variables to scores (projection index) to be continuous. The authors show that the effects of this constraint are (1) NLPCA is unable to model curves and surfaces which intersect themselves and (2) the NLPCA "projections" are suboptimal producing larger approximation error. The authors show how NLPCA score values can be interpreted and give the results of a small simulation study comparing the two methods.