A support vector machine formulation to PCA analysis and its kernel version

In this paper, we present a simple and straightforward primal-dual support vector machine formulation to the problem of principal component analysis (PCA) in dual variables. By considering a mapping to a high-dimensional feature space and application of the kernel trick (Mercer theorem), kernel PCA is obtained as introduced by Scholkopf et al. While least squares support vector machine classifiers have a natural link with the kernel Fisher discriminant analysis (minimizing the within class scatter around targets +1 and -1), for PCA analysis one can take the interpretation of a one-class modeling problem with zero target value around which one maximizes the variance. The score variables are interpreted as error variables within the problem formulation. In this way primaldual constrained optimization problem interpretations to the linear and kernel PCA analysis are obtained in a similar style as for least square-support vector machine (LS-SVM) classifiers.