A few online algorithms for extracting minor generalized eigenvectors

A few adaptive algorithms for generalized eigen-decomposition have been proposed, which are very useful in many applications such as digital mobile communications, blind signal separation, etc. These algorithms are all focusing on extracting principal generalized eigenvectors. However, in many practical applications such as dimension reduction and signal processing, extracting the minor generalized eigenvectors adaptively are needed. Because of little literatures in the community, we discuss several approaches that lead to a few novel algorithms for extracting minor generalized eigenvectors. First, we derive an adaptive algorithms by using a single-layer linear forward neural network from the viewpoint of linear discriminant analysis (LDA). And the algorithm to extract multiple minor generalized eigenvectors are also proposed by using orthogonality property. Second, by using gradient ascent approach of some objective functions, we can derive more algorithms and explain the first algorithm. Then, we extend these algorithms to minor generalized eigenvector problem. Theoretical analysis shows that these algorithms are stable and convergent to the minor generalized eigenvectors. Simulations have been conducted for illustration of the efficiency and effectiveness of our algorithms.