Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

• 成果类型: Article
• 署名作者: Fan, Jianqing; Fan, Yingying; Han, Xiao; Lv, Jinchi
• 署名单位: Princeton University; University of Southern California; Chinese Academy of Sciences; University of Science & Technology of China, CAS
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2020.1840990
• 发表日期: 2022
• 页码: 996-1009
• 关键词: CENTRAL LIMIT-THEOREMS; LARGEST EIGENVALUE; community detection; eigenstructure; deformations; vectors; graphs; forms

摘要

Characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications. To this end, in this article we introduce a general framework of asymptotic theory of eigenvectors for large spiked random matrices with diverging spikes and heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise. Under some mild regularity conditions, we provide the asymptotic expansions for the spiked eigenvalues and show that they are asymptotically normal after some normalization. For the spiked eigenvectors, we establish asymptotic expansions for the general linear combination and further show that it is asymptotically normal after some normalization, where the weight vector can be arbitrary. We also provide a more general asymptotic theory for the spiked eigenvectors using the bilinear form. Simulation studies verify the validity of our new theoretical results. Our family of models encompasses many popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis, and our general theory can be exploited for statistical inference in these large-scale applications. for this article are available online.