Learning Latent Variable Dynamic Graphical Models by Confidence Sets Selection

We consider the problem of learning dynamic latent variable graphical models. More precisely, given an estimate of the graphical model based on a finite data sample, we propose a new method, which accounts for the uncertainty in the estimation by computing a "confidence neighborhood" containing the true model with a prescribed probability. In this neighborhood, we search for a structured model in which we allow for the presence of a small number of latent variables in order to enforce sparsity of the identified graph. As a consequence, the resulting optimization problem involves just one regularization parameter balancing the tradeoff between the number of latent variables and the sparseness of the learned graph. Hence, the cross-validation procedure is performed on a one-dimensional grid significantly reducing the computational burden with respect to existing methods, which require a two-dimensional grid. A convenient matricial reformulation is proposed for which the variational analysis can be carried out. The properties of the dual problem are analyzed and existence of solutions is established. An alternating direction method of multiplier algorithm is finally presented to solve the dual problem numerically.