Semi-proximal augmented Lagrangian method for sparse estimation of high-dimensional inverse covariance matrices

Abstract

. Estimating a large and sparse inverse covariance matrix is a fundamental problem in modern multivariate analysis. Recently, a generalized model for a sparse estimation was proposed in which an explicit eigenvalue bounded constraint is involved. It covers a large number of existing estimation approaches as special cases. It was shown that the dual of the generalized model contains five separable blocks, which cause more challenges for minimizing. In this paper, we use an augmented Lagrangian method to solve the dual problem, but we minimize the augmented Lagrangian function with respect to each variable in a Jacobian manner, and add a proximal point term to make each subproblem easy to solve. We show that this iterative scheme is equivalent to adding a proximal point term to the augmented Lagrangian function, and its convergence can be directly followed. Finally, we give numerical simulations by using the synthetic data which show that the proposed algorithm is very effective in estimating high-dimensional sparse inverse covariance matrices.