A Well-Conditioned and Sparse Estimate of Covariance and Inverse Covariance Matrix Using Joint Penalty

• Activity Number: 311
• Type: Contributed
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Ashwini Maurya*
• Companies: Michigan State University
• Keywords: Sparsity ; Eigenvalue Penalty ; Graphical Models ; Penalized Likelihood.

Abstract

We develop a method for estimating a well-conditioned and sparse covariance and inverse covariance matrix from a sample of vectors drawn from a sub-gaussian distri-bution in high dimensional setting. The proposed estimator is obtained by minimizing the squared loss function and joint penalty of L-1 norm and sum of squared deviation penalty on the eigenvalues. The joint penalty plays two important roles: i) L-1 penalty on each entry of covariance matrix reduces the effective number of parameters and consequently the estimate is sparse and ii) the sum of squared deviations penalty on the eigenvalues controls the over-dispersion in the eigenvalues of sample covariance matrix. In contrast to some of the existing methods of covariance and inverse covariance matrix estimation, where often the interest is to estimate a sparse matrix, the proposed method is flexible in estimating both a sparse and well-conditioned covariance matrix simultaneously. We establish the theoretical consistency of the proposed estimators in both Frobenius and Operator norm. The proposed algorithm is very fast and efficient. We compare the performance of the proposed estimator to some other existing methods on