Probabilistic analysis of kernel principal components: mixture modeling, and classification

This paper presents a probabilistic approach to analyze kernel principal components by naturally combining in one treatment the theory of probabilistic principal component analysis and that of kernel principal component analysis. In this formulation, the kernel component enhances the nonlinear modeling power, while the probabilistic structure offers (i) a mixture model for nonlinear data structure containing nonlinear sub-structures, and (ii) an effective classification scheme. It also turns out that the original loading matrix is replaced by the newly defined empirical loading matrix. The expectation/maximization algorithm for learning parameters of interest is then developed. Computation of reconstruction error and Mahalanobis distance is also discussed. Finally, we apply this to a real application of face recognition.