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.

翻译：稀疏协方差矩阵在遗传学和神经科学等诸多领域中通过编码变量间的相互依赖关系发挥着关键作用。尽管对稀疏协方差矩阵已有大量研究，但现有方法仍面临若干挑战，例如样本协方差矩阵元素间的相关性、正定性以及对角线元素的无偏估计。为应对这些挑战，我们构建了一个用于估计稀疏协方差矩阵的线性协方差模型，并提出了一种惩罚回归方法。该方法具有足够的通用性，能够涵盖现有的稀疏协方差估计器，同时还能考虑样本协方差矩阵元素间的相关性，避免对角线元素产生不必要的偏差，并保持正定性。我们开发了用于估计的共识ADMM算法，并推导出所提出估计器的$\ell_2$收敛速率。我们将该估计器应用于模拟数据以及来自神经科学和遗传学的真实数据，以验证所提出方法的有效性。