Singular Gaussian graphical models: Structure learning

ABSTRACTThe goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.