Diffusion Approximations for Online Principal Component Estimation and Global Convergence
This work provides theoretical insights into the convergence of an online PCA method, which is incremental as it builds on existing stochastic gradient descent techniques.
The paper tackled the problem of analyzing the dynamics of Oja's iteration for online principal component estimation by using diffusion approximation tools, resulting in a finite-sample error bound that matches the minimax lower bound under bounded sample assumptions.
In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our three-phase analysis further provides a finite-sample error bound for the running estimate, which matches the minimax information lower bound for principal component analysis under the additional assumption of bounded samples.