Low-complexity Modeling of Partially Available Second-order Statistics via Matrix Completion

Tuesday, October 6, 2015 - 1:25pm - 2:25pm
Lind 305
Mihailo Jovanovic (University of Minnesota, Twin Cities)
We study the problem of completing partially known state statistics of
complex dynamical systems using low-complexity linearized models. State
statistics of linear systems satisfy certain structural constraints that
arise from the underlying dynamics and the directionality of input
disturbances. The dynamical interaction between state variables is known
while the directionality of input excitation is uncertain. Thus, the goal
of the inverse problem that we formulate is to identify the dynamics and
directionality of input excitation so as to explain the observed sample
statistics. In particular, we seek to explain the data with 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 semidefinite program and solved efficiently using
general-purpose solvers for small- and medium-size problems. We develop a
customized alternating minimization algorithm (AMA) to solve the problem
for large-scale systems. We demonstrate that AMA works as a proximal
gradient for the dual problem and provide examples to illustrate that
identified colored-in-time stochastic disturbances represent an effective
means for explaining available second-order state statistics.
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