title: 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.
authors.affiliation: Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 ( hanliu@princeton.edu ).
authors.forename: Han
authors.initials: H
authors.lastname: Liu
authors.affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 ( liewang@math.mit.edu ).
authors.forename: Lie
authors.initials: L
authors.lastname: Wang
authors.affiliation: Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218 ( tourzhao@jhu.edu ).
authors.forename: Tuo
authors.initials: T
authors.lastname: Zhao
keywords: Explicit eigenvalue constraint; High-dimensional data; Positive-definiteness guarantee