Robust adaptive quasi-Newton algorithms for eigensubspace estimation

A novel quasi-Newton algorithm for adaptively estimating the principal eigensubspace of a covariance matrix by making use of an approximation of its Hessian matrix is derived. A rigorous analysis of the convergence properties of the algorithm by using the stochastic approximation theory is presented. It is shown that the recursive least squares (RLS) technique can be used to implement the quasi-Newton algorithm, which significantly reduces the computational requirements from O( pN2 ) to O( pN), where N is the data vector dimension and p is the number of desired eigenvectors. The algorithm is further generalised by introducing two adjustable parameters that efficiently accelerate the adaptation process. The proposed algorithm is applied to different applications such as eigenvector estimation and the Comon–Golub test in order to study the convergence behaviour of the algorithm when compared with others such as PASTd, NIC, and the Kang et al. quasi-Newton algorithm. Simulation results show that the new algorithm is robust against changes of the input scenarios and is thus well suited to parallel implementation with online deflation.