Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

Jianqing Fan; Yingying Fan; Xiao Han; Jinchi Lv

doi:https://doi.org/10.1080/01621459.2020.1840990

Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

Jianqing Fan, Yingying Fan, Xiao Han, Jinchi Lv

Princeton University

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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.

Original language	American English
Pages (from-to)	996-1009
Number of pages	14
Journal	Journal of the American Statistical Association
Volume	117
Issue number	538
DOIs	https://doi.org/10.1080/01621459.2020.1840990
State	Published - 2022

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Keywords

Asymptotic distributions
Eigenvectors
Generalized Wigner matrix
High dimensionality
Low-rank matrix
Random matrix theory

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https://doi.org/10.1080/01621459.2020.1840990

Cite this

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title = "Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes",

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.",

keywords = "Asymptotic distributions, Eigenvectors, Generalized Wigner matrix, High dimensionality, Low-rank matrix, Random matrix theory",

author = "Jianqing Fan and Yingying Fan and Xiao Han and Jinchi Lv",

note = "Funding Information: This work was supported by NIH grants R01-GM072611-14 and 1R01GM131407-01, NSF grants DMS-1662139, DMS-1712591, and DMS-1953356, NSF CAREER Award DMS-1150318, a grant from the Simons Foundation, Adobe Data Science Research Award and NSF of China (no. 12001518). The authors sincerely thank the joint editor, associate editor, and referees for their valuable comments that helped improve the article substantially. Publisher Copyright: {\textcopyright} 2020 American Statistical Association.",

year = "2022",

doi = "https://doi.org/10.1080/01621459.2020.1840990",

language = "American English",

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TY - JOUR

T1 - Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

AU - Fan, Jianqing

AU - Fan, Yingying

AU - Han, Xiao

AU - Lv, Jinchi

N1 - Funding Information: This work was supported by NIH grants R01-GM072611-14 and 1R01GM131407-01, NSF grants DMS-1662139, DMS-1712591, and DMS-1953356, NSF CAREER Award DMS-1150318, a grant from the Simons Foundation, Adobe Data Science Research Award and NSF of China (no. 12001518). The authors sincerely thank the joint editor, associate editor, and referees for their valuable comments that helped improve the article substantially. Publisher Copyright: © 2020 American Statistical Association.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

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KW - Generalized Wigner matrix

KW - High dimensionality

KW - Low-rank matrix

KW - Random matrix theory

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Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

Abstract

ASJC Scopus subject areas

Keywords

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