An ADMM algorithm for matrix completion of partially known state covariances

We study the inverse problem of reproducing partially known second-order statistics of a linear time invariant system by the least number of possible input disturbance channels. This can be formulated as a rank minimization problem, and for its solution, we employ a convex relaxation based on the nuclear norm. The resulting optimization problem can be cast as a semi-definite program and solved efficiently using general-purpose solvers for small- and medium-size problems. In this paper, we focus on issues and techniques that are pertinent to large-scale systems. We bring in a re-parameterization which transforms the problem into a form suitable for the alternating direction method of multipliers. Furthermore, we show that each iteration of this algorithm amounts to solving a system of linear equations, an eigenvalue decomposition, and a singular value thresholding. An illustrative example is provided to demonstrate the effectiveness of the developed approach.