RATE-OPTIMAL PERTURBATION BOUNDS FOR SINGULAR SUBSPACES WITH APPLICATIONS TO HIGH-DIMENSIONAL STATISTICS

• 成果类型: Article
• 署名作者: Cai, T. Tony; Zhang, Anru
• 署名单位: University of Pennsylvania; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1541
• 发表日期: 2018
• 页码: 60-89
• 关键词: canonical correlation-analysis; Low-rank Matrix; principal-components; gaussian-noise; completion; VALUES; Consistency; CONVERGENCE; vectors; PCA

摘要

Perturbation bounds for singular spaces, in particularWedin's sin Theta theorem, are a fundamental tool in many fields including high-dimensional statistics, machine learning and applied mathematics. In this paper, we establish separate perturbation bounds, measured in both spectral and Frobenius sin Theta distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given. The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering and canonical correlation analysis (CCA). In particular, separate matching upper and lower bounds are obtained for estimating the left and right singular spaces. To the best of our knowledge, this is the first result that gives different optimal rates for the left and right singular spaces under the same perturbation.