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

In this letter we present a framework for estimating sparse covariance matrices, wherein we solve the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}-$</tex-math></inline-formula>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.