The spectral norm of random inner-product kernel matrices

• 成果类型: Article
• 署名作者: Fan, Zhou; Montanari, Andrea
• 署名单位: Stanford University; Stanford University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0830-4
• 发表日期: 2019
• 页码: 27-85
• 关键词: Principal component analysis; sparse pca; semidefinite relaxations; Covariance matrices; LARGEST EIGENVALUE; local statistics; free convolution; Optimal Rates; UNIVERSALITY; limit

摘要

We study an inner-product kernel random matrix model, whose empirical spectral distribution was shown by Xiuyuan Cheng and Amit Singer to converge to a deterministic measure in the large n and p limit. We provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a Marchenko-Pastur law. By comparing the tracial moments of this matrix to an additive deformation of a Wigner matrix, we establish that for odd kernel functions, the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum. Our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components, and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting.