Precision matrix estimation using penalized Generalized Sylvester matrix equation

Estimating a precision matrix is an important problem in several research fields when dealing with large-scale data. Under high-dimensional settings, one of the most popular approaches is optimizing a Lasso or ℓ₁ norm penalized objective loss function. This penalization endorses sparsity in the estimated matrix and improves the accuracy under a proper calibration of the penalty parameter. In this paper, we demonstrate that the problem of minimizing Lasso penalized D-trace loss can be seen as solving a penalized Sylvester matrix equation. Motivated by this method, we propose estimating the precision matrix using penalized generalized Sylvester matrix equations. In our method, we develop a particular estimating equation and a new convex loss function constructed through this equation, which we call the generalized D-trace loss. We assess the performance of the proposed method using detailed numerical analysis, including simulated and real data. Extensive results show the advantage of the proposed method compared to other estimation approaches in the literature.