Information-theoretic bounds on model selection for Gaussian Markov random fields

The problem of graphical model selection is to estimate the graph structure of an unknown Markov random field based on observed samples from the graphical model. For Gaussian Markov random fields, this problem is closely related to the problem of estimating the inverse covariance matrix of the underlying Gaussian distribution. This paper focuses on the information-theoretic limitations of Gaussian graphical model selection and inverse covariance estimation in the high-dimensional setting, in which the graph size p and maximum node degree d are allowed to grow as a function of the sample size n. Our first result establishes a set of necessary conditions on n(p,d) for any recovery method to consistently estimate the underlying graph. Our second result provides necessary conditions for any decoder to produce an estimate Θ̂ of the true inverse covariance matrix Θ̂ satisfying ||Θ̂ - Θ|| <; δ in the elementwise ℓ_∞-norm (which implies analogous results in the Frobenius norm as well). Combined with previously known sufficient conditions for polynomial-time algorithms, these results yield sharp characterizations in several regimes of interest.