  Online principal component analysis (PCA) has been an efficient tool in
practice to reduce dimension. However, convergence properties of the
corresponding ODE are still unknown, including global convergence, stable
manifolds, and convergence rate. In this paper, we focus on the stochastic
gradient ascent (SGA) method proposed by Oja. By regarding the corresponding
ODE as a Landau-Lifshitz-Gilbert (LLG) equation on the Stiefel manifold, we
proved global convergence of the ODE. Moreover, we developed a new technique to
determine stable manifolds. This technique analyzes the rank of the initial
datum. Using this technique, we derived the explicit expression of the stable
manifolds. As a consequence, exponential convergence to stable equilibrium
points was also proved. The success of this new technique should be attributed
to the semi-decoupling property of the SGA method: iteration of previous
components does not depend on that of later ones. As far as we know, our result
is the first complete one on the convergence of an online PCA flow, providing
global convergence, explicit characterization of stable manifolds, and closed
formula of exponential convergence depending on the spectrum gap.
