GLOBAL ANALYSIS OF OJAS FLOW FOR NEURAL NETWORKS

A detailed study of Oja's learning equation in neural networks is unde rtaken in this paper. Not only are such fundamental issues as existenc e, uniqueness, and representation of solutions completely resolved, bu t also the convergence issue is resolved. It is shown that the solutio n of Oja's equation is exponentially convergent to an equilibrium from any initial value. Moreover, the necessary and sufficient conditions are given on the initial value for the solution to converge to a domin ant eigenspace of the associated autocorrelation matrix. As a by-produ ct, this result confirms one of Oja's conjectures that the solution co nverges to the principal eigenspace from almost all initial values. So me other characteristics of the limiting solution are also revealed. T hese facilitate the determination of the limiting solution in advance using only the initial information. Two examples are analyzed demonstr ating the explicit dependence of the limiting solution on the initial value. In another respect, it is found that Oja's equation is the grad ient flow of generalized Rayleigh quotients on a Stiefel manifold.