Estimation of a sparse and spiked covariance matrix

Abstract

We suggest a method for estimating a covariance matrix that can be represented as a sum of a sparse low-rank matrix and a diagonal matrix. Our formulation is based on penalized quadratic loss, which is a convex problem that can be solved via incremental gradient and proximal method. In contrast to other spiked covariance matrix estimation approaches that are related to principal component analysis and factor analysis, our method has a simple formulation and does not constrain entire rows and columns of the matrix to be zero. We further discuss a penalized entropy loss method that is nevertheless nonconvex and necessitates a majorization-minimization algorithm in combination with the incremental gradient and proximal method. We carry out simulations to demonstrate the finite-sample properties focusing on high-dimensional covariance matrices. Finally, the proposed method is illustrated using a gene expression data set.

Keywords:

• cross-validation
• factor analysis
• nuclear norm
• principal component analysis
• sparsity norm

Acknowledgments

The authors sincerely thank Professor Irene Gijbels, the Associate Editor, and two reviewers for their insightful comments and suggestions that greatly improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The research of Heng Lian is supported by a Singapore Ministry of Education Tier 2 grant and University of New South Wales Start Up Grant.