Sparse covariance matrices play crucial roles by encoding the interdependencies between variables in numerous fields such as genetics and neuroscience. Despite substantial studies on sparse covariance matrices, existing methods face several challenges such as the correlation among the elements in the sample covariance matrix, positive definiteness and unbiased estimation of the diagonal elements. To address these challenges, we formulate a linear covariance model for estimating sparse covariance matrices and propose a penalized regression. This method is general enough to encompass existing sparse covariance estimators and can additionally consider correlation among the elements in the sample covariance matrix while avoiding unnecessary bias in the diagonal elements and preserving positive definiteness. We develop a consensus ADMM algorithm for estimation and derive $\ell_2$ convergence rate of the proposed estimator. We apply our estimator to simulated data and real data from neuroscience and genetics to describe the efficacy of our proposed method.

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