NONASYMPTOTIC UPPER BOUNDS FOR THE RECONSTRUCTION ERROR OF PCA

• 成果类型: Article
• 署名作者: Reiss, Markus; Wahl, Martin
• 署名单位: Humboldt University of Berlin
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1839
• 发表日期: 2020
• 页码: 1098-1123
• 关键词: Principal component analysis; spectral projectors; gram matrix; eigenspectrum; approximation; perturbation; inequalities

摘要

We analyse the reconstruction error of principal component analysis (PCA) and prove nonasymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. The bounds reveal that the excess risk differs significantly from usually considered subspace distances based on canonical angles. Our approach relies on the analysis of empirical spectral projectors combined with concentration inequalities for weighted empirical covariance operators and empirical eigenvalues.