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In this paper, we propose to adopt the diffusion approximation tools to studythe dynamics of Oja's iteration which is an online stochastic gradient descentmethod for the principal component analysis. Oja's iteration maintains arunning estimate of the true principal component from streaming data and enjoysless temporal and spatial complexities. We show that the Oja's iteration forthe top eigenvector generates a continuous-state discrete-time Markov chainover the unit sphere. We characterize the Oja's iteration in three phases usingdiffusion approximation and weak convergence tools. Our three-phase analysisfurther provides a finite-sample error bound for the running estimate, whichmatches the minimax information lower bound for principal component analysisunder the additional assumption of bounded samples.

Diffusion Approximations for Online Principal Component Estimation and Global Convergence