Near-optimal stochastic approximation for online principal component estimation

• 成果类型: Article
• 署名作者: Li, Chris Junchi; Wang, Mengdi; Liu, Han; Zhang, Tong
• 署名单位: Princeton University; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1182-z
• 发表日期: 2018
• 页码: 75-97
• 关键词: eigenvalue; POWER; PCA

摘要

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.