Chandrasekaran, Parrilo and Willsky (2010) proposed a convex optimization
problem to characterize graphical model selection in the presence of unobserved
variables. This convex optimization problem aims to estimate an inverse
covariance matrix that can be decomposed into a sparse matrix minus a low-rank
matrix from sample data. Solving this convex optimization problem is very
challenging, especially for large problems. In this paper, we propose two
alternating direction methods for solving this problem. The first method is to
apply the classical alternating direction method of multipliers to solve the
problem as a consensus problem. The second method is a proximal gradient based
alternating direction method of multipliers. Our methods exploit and take
advantage of the special structure of the problem and thus can solve large
problems very efficiently. Global convergence result is established for the
proposed methods. Numerical results on both synthetic data and gene expression
data show that our methods usually solve problems with one million variables in
one to two minutes, and are usually five to thirty five times faster than a
state-of-the-art Newton-CG proximal point algorithm
