Weighted Rayleigh quotients for minor and principal component extraction

New criteria are proposed for extracting in parallel multiple minor and principal components associated with the covariance matrix of an input process. The proposed minor and principal component analyzer (MCA/PCA) algorithms are based on optimizing a weighted inverse Rayleigh quotient so that the optimum equilibrium points are exactly the desired eigenvectors of a covariance matrix instead of an arbitrary orthonormal basis of the minor subspace. Variations of the derived MCA/PCA learning rules are obtained by imposing orthogonal and quadratic constraints and change of variables. Similar criteria are proposed for component analysis of the generalized eigenvalue problem. Some of the proposed MCA algorithms can also perform PCA by merely changing the sign of the step-size. These algorithms may be seen as generalization of Oja´s and Xu´s systems for computing multiple principal components.