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

Robust Identification of "Sparse Plus Low-rank" Graphical Models: An Optimization Approach