Positive-definite l1-penalized estimation of large covariance matrices

Lingzhou Xue; Shiqian Ma; Hui Zou

doi:10.1080/01621459.2012.725386

Positive-definite l₁-penalized estimation of large covariance matrices

Lingzhou Xue, Shiqian Ma, Hui Zou

Statistics (Twin Cities)

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

113 Scopus citations

Abstract

The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To fix this drawback of thresholding estimation, we develop a positive-definite l₁- penalized covariance estimator for estimating sparse large covariance matrices. We derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties. Under weak regularity conditions, nonasymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications.

Original language	English (US)
Pages (from-to)	1480-1491
Number of pages	12
Journal	Journal of the American Statistical Association
Volume	107
Issue number	500
DOIs	https://doi.org/10.1080/01621459.2012.725386
State	Published - 2012

Bibliographical note

Funding Information:
Lingzhou Xue is Postdoctoral Research Associate, Department of Operations Research & Financial Engineering, Princeton University, Princeton, NJ 08544. Shiqian Ma is Assistant Professor, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, Hong Kong. Hui Zou is Associate Professor, School of Statistics, University of Minnesota, Minneapolis, MN 55455 (E-mail: zouxx019@umn.edu). The article was completed when Lingzhou Xue was a Ph.D. student at the University of Minnesota and Shiqian Ma was a Postdoctoral Fellow in the Institute for Mathematics and Its Applications at the University of Minnesota. The authors thank Adam Rothman for sharing his code. We are grateful to the coeditor, the associate editor, and two referees for their helpful and constructive comments. Shiqian Ma was supported by the National Science Foundation postdoctoral fellowship through the Institute for Mathematics and Its Applications at the University of Minnesota. Lingzhou Xue and Hui Zou are supported in part by grants from the National Science Foundation and the Office of Naval Research.

Keywords

Alternating direction methods
Matrix norm
Positive-definite estimation
Soft-thresholding
Sparsity

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Cite this

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abstract = "The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To fix this drawback of thresholding estimation, we develop a positive-definite l1- penalized covariance estimator for estimating sparse large covariance matrices. We derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties. Under weak regularity conditions, nonasymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications.",

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