Diffusion Approximations for Online Principal Component Estimation and Global Convergence

TL;DR

This paper uses diffusion approximation to analyze the dynamics and convergence of Oja's online PCA algorithm, providing a detailed phase-wise characterization and finite-sample error bounds.

Contribution

It introduces a diffusion approximation framework for studying Oja's iteration, revealing its three-phase behavior and establishing optimal finite-sample error bounds.

Findings

Oja's iteration can be modeled as a Markov chain on the unit sphere.

The three-phase analysis accurately describes the convergence dynamics.

Finite-sample error bounds match the minimax lower bound under bounded samples.

Abstract

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.