  Motivated by graphical models, we consider the "Sparse Plus Low-rank"
decomposition of a positive definite concentration matrix -- the inverse of the
covariance matrix. This is a classical problem for which a rich theory and
numerical algorithms have been developed. It appears, however, that the results
rapidly degrade when, as it happens in practice, the covariance matrix must be
estimated from the observed data and is therefore affected by a certain degree
of uncertainty. We discuss this problem and propose an alternative optimization
approach that appears to be suitable to deal with robustness issues in the
"Sparse Plus Low-rank" decomposition problem.The variational analysis of this
optimization problem is carried over and discussed.
