Advanced computation of sparse precision matrices for big data

The precision matrix is the inverse of the covariance matrix. Estimating large sparse precision matrices is an interesting and a challenging problem in many fields of sciences, engineering, humanities and machine learning problems in general. Recent applications often encounter high dimensionality with a limited number of data points leading to a number of covariance parameters that greatly exceeds the number of observations, and hence the singularity of the covariance matrix. Several methods have been proposed to deal with this challenging problem, but there is no guarantee that the obtained estimator is positive definite. Furthermore, in many cases, one needs to capture some additional information on the setting of the problem. In this paper, we introduce a criterion that ensures the positive definiteness of the precision matrix and we propose the inner-outer alternating direction method of multipliers as an efficient method for estimating it. We show that the convergence of the algorithm is ensured with a sufficiently relaxed stopping criterion in the inner iteration. We also show that the proposed method converges, is robust, accurate and scalable as it lends itself to an efficient implementation on parallel computers.