  Oja's algorithm of principal component analysis (PCA) has been one of the
methods utilized in practice to reduce dimension. In this paper, we focus on
the convergence property of the discrete algorithm. To realize that, we view
the algorithm as a stochastic process on the parameter space and semi-group. We
approximate it by SDEs, and prove large time convergence of the SDEs to ensure
its performance. This process is completed in three steps. First, the discrete
algorithm can be viewed as a semigroup: $S^k\varphi=\mathbb{E}[\varphi(\mathbf
W(k))]$. Second, we construct stochastic differential equations (SDEs) on the
Stiefel manifold, i.e. the diffusion approximation, to approximate the
semigroup. By proving the weak convergence, we verify that the algorithm is
'close to' the SDEs. Finally, we use the reversibility of the SDEs to prove
long-time convergence.

Numerical Analysis;Numerical Analysis;Dynamical Systems;Probability