  State statistics of linear systems satisfy certain structural constraints
that arise from the underlying dynamics and the directionality of input
disturbances. In the present paper we study the problem of completing partially
known state statistics. Our aim is to develop tools that can be used in the
context of control-oriented modeling of large-scale dynamical systems. For the
type of applications we have in mind, the dynamical interaction between state
variables is known while the directionality and dynamics of input excitation is
often uncertain. Thus, the goal of the mathematical problem that we formulate
is to identify the dynamics and directionality of input excitation in order to
explain and complete observed sample statistics. More specifically, we seek to
explain correlation data with the least number of possible input disturbance
channels. We formulate this inverse problem as rank minimization, and for its
solution, we employ a convex relaxation based on the nuclear norm. The
resulting optimization problem is cast as a semidefinite program and can be
solved using general-purpose solvers. For problem sizes that these solvers
cannot handle, we develop a customized alternating minimization algorithm
(AMA). We interpret AMA as a proximal gradient for the dual problem and prove
sub-linear convergence for the algorithm with fixed step-size. We conclude with
an example that illustrates the utility of our modeling and optimization
framework and draw contrast between AMA and the commonly used alternating
direction method of multipliers (ADMM) algorithm.
