Near-Optimal Stochastic Approximation for Online Principal Component Estimation
This provides a theoretical foundation for online PCA, which is important for streaming data analysis in fields like machine learning and data science, though it is incremental in nature.
The paper tackled the lack of theoretical convergence analysis for online PCA algorithms by casting it as a stochastic nonconvex optimization problem and proving a nearly optimal finite-sample error bound that closely matches the minimax lower bound under subgaussian assumptions.
Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests. Despite its rich applications, theoretical convergence analysis remains largely open. In this paper, we cast online PCA into a stochastic nonconvex optimization problem, and we analyze the online PCA algorithm as a stochastic approximation iteration. The stochastic approximation iteration processes data points incrementally and maintains a running estimate of the principal component. We prove for the first time a nearly optimal finite-sample error bound for the online PCA algorithm. Under the subgaussian assumption, we show that the finite-sample error bound closely matches the minimax information lower bound.