Convergence study of principal component analysis algorithms

We investigate the convergence properties of two different principal component analysis algorithms, and analytically explain some commonly observed experimental results. We use two different methodologies to analyze the two algorithms. The first methodology uses the fact that both algorithms are stochastic approximation procedures. We use the theory of stochastic approximation, in particular the results of Fabian (1968), to analyze the asymptotic mean square errors (AMSEs) of the algorithms. This analysis reveals the conditions under which the algorithms produce smaller AMSEs, and also the conditions under which one algorithm has a smaller AMSE than the other. We next analyze the asymptotic mean errors (AMEs) of the two algorithms in the neighborhood of the solution. This analysis establishes the conditions under which the AMEs of the minor eigenvectors go to zero faster. Furthermore, the analysis makes explicit that increasing the gain parameter up to an upper bound improves the convergence of all eigenvectors. We also show that the AME of one algorithm goes to zero faster than the other. Experiments with multi-dimensional Gaussian data corroborate the analytical findings presented here