$\ell_{0}$ Penalized Maximum Likelihood Estimation of Sparse Covariance Matrices

In this letter we present a framework for estimating sparse covariance matrices, wherein we solve the $\ell _{0}-$norm penalized maximum likelihood estimation problem using the extended Bayesian information criterion (EBIC), a high dimensional model selection rule. The framework combines choosing the sparsity pattern and estimating the covariance matrix in a single step, eliminating the need for any hyper-parameter tuning. Using the framework we propose a cyclic majorization-minimization based technique and apply it to synthetic data to evaluate its performance in terms of normalized root mean square error (NRMSE) and Kullback Leibler (KL) divergence.