Distributed Sparse Covariance Matrix Estimation

Covariance matrix estimation is a crucial problem in many areas related to data analysis. While centralized sparse covariance matrix estimators have received extensive attention, practical considerations such as communication efficiency and privacy constraints often make centralizing data impractical in many real-world scenarios. This necessitates the development of distributed covariance matrix estimation methods. In this paper, we present a novel distributed estimator for a sparse covariance matrix over networks by minimizing the sum of all agents' losses based on
$\ell_{1}$
penalized Gaussian likelihood. To solve this constrained non-convex, non-Lipschitz-smooth optimization problem without relying on a central processor, we propose a straightforward network covariance iterative shrinkage-thresholding algorithm (network C-ISTA) with provable convergence. Numerical simulations demonstrate the convergence and impressive estimation performance of the network C-ISTA algorithm, confirming its effectiveness under decentralized settings.