Convergence analysis of Xu's LMSER learning algorithm via deterministic discrete time system method

The convergence of Xu's LMSER algorithm with a constant learning rate, which is in the one unit case, is interpreted by analyzing an associated deterministic discrete time (DDT) system. Some convergent results relating to the Xu's DDT system are obtained. An invariant set and an ultimate bound are identified so that the non-divergence of the system can be guaranteed. It is rigorously proven that all trajectories of the system from points in this invariant set will converge exponentially to a unit eigenvector associated with the largest eigenvalue of the correlation matrix. By comparing Xu's algorithm with Oja's algorithm, it can be observed, on the whole, the Xu's algorithm evolves faster at a cost of larger computational complexity. Extensive simulations will be carried out to illustrate the theory.