Sparse Covariance Matrix Estimation With Eigenvalue Constraints

Abstract

We propose a new approach for estimating high-dimensional, positive-definite covariance matrices. Our method extends the generalized thresholding operator by adding an explicit eigenvalue constraint. The estimated covariance matrix simultaneously achieves sparsity and positive definiteness. The estimator is rate optimal in the minimax sense and we develop an efficient iterative soft-thresholding and projection algorithm based on the alternating direction method of multipliers. Empirically, we conduct thorough numerical experiments on simulated datasets as well as real data examples to illustrate the usefulness of our method. Supplementary materials for the article are available online.

Key Words:

• Explicit eigenvalue constraint
• High-dimensional data
• Positive-definiteness guarantee

ACKNOWLEDGMENTS

We thank the editor, associate editor, and two referees for their helpful comments. Han Liu and Tuo Zhao are supported by NSF grant III-1116730, and Lie Wang is supported by NSF grant DMS-1005539.