SINGULAR VECTOR AND SINGULAR SUBSPACE DISTRIBUTION FOR THE MATRIX DENOISING MODEL

• 成果类型: Article
• 署名作者: Bao, Zhigang; Ding, Xiucai; Wang, Ke
• 署名单位: Hong Kong University of Science & Technology; University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1960
• 发表日期: 2021
• 页码: 370-392
• 关键词: sample covariance matrices; central limit-theorems; LARGEST EIGENVALUE; rank perturbations; spectral projectors; bounds; UNIVERSALITY; deformations; asymptotics

摘要

In this paper, we study the matrix denoising model Y = S + X, where S is a low rank deterministic signal matrix and X is a random noise matrix, and both are M x n. In the scenario that M and n are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of Y, under fully general assumptions on the structure of S and the distribution of X. More specifically, we derive the limiting distribution of angles between the principal singular vectors of Y and their deterministic counterparts, the singular vectors of S. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of Y and that spanned by the singular vectors of S. It turns out that the limiting distributions depend on the structure of the singular vectors of S and the distribution of X, and thus they are nonuniversal. Statistical applications of our results to singular vector and singular subspace inferences are also discussed.