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Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes
Journal of the American Statistical Association ( IF 3.0 ) Pub Date : 2020-12-08 , DOI: 10.1080/01621459.2020.1840990
Jianqing Fan 1 , Yingying Fan 2 , Xiao Han 2 , Jinchi Lv 2
Affiliation  

Abstract

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. Supplementary materials for this article are available online.



中文翻译:

带发散尖峰的随机矩阵特征向量的渐近理论

摘要

表征大型随机矩阵的特征向量的渐近分布提出了重要挑战,但可以为一系列统计应用提供有用的见解。为此,在本文中,我们介绍了具有发散尖峰和异质方差的大尖峰随机矩阵的特征向量渐近理论的一般框架,并针对广义维格纳矩阵噪声的场景建立了尖峰特征向量和特征值的渐近性质。在一些温和的规律性条件下,我们提供了尖峰特征值的渐近展开,并表明它们在一些归一化后渐近正态。对于尖峰特征向量,我们建立了一般线性组合的渐近展开式,并进一步表明它在一些归一化后渐近正态,其中权重向量可以是任意的。我们还使用双线性形式为尖峰特征向量提供了更一般的渐近理论。仿真研究验证了我们新理论结果的有效性。我们的模型系列包括许多常用模型,例如用于网络分析的具有或不具有重叠社区的随机块模型和用于文本分析的主题模型,我们的一般理论可用于这些大规模应用中的统计推断。本文的补充材料可在线获取。我们的模型系列包括许多常用模型,例如用于网络分析的具有或不具有重叠社区的随机块模型和用于文本分析的主题模型,我们的一般理论可用于这些大规模应用中的统计推断。本文的补充材料可在线获取。我们的模型系列包括许多常用模型,例如用于网络分析的具有或不具有重叠社区的随机块模型和用于文本分析的主题模型,我们的一般理论可用于这些大规模应用中的统计推断。本文的补充材料可在线获取。

更新日期:2020-12-08
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