High-Dimensional Graphical Model Selection: Tractable Graph Families and Necessary Conditions

We consider the problem of Ising and Gaussian graphical model selection given n i.i.d. samples from the model. We propose an efficient threshold-based algorithm for structure estimation based on conditional mutual information thresholding. This simple local algorithm requires only loworder statistics of the data and decides whether two nodes are neighbors in the unknown graph. We identify graph families for which the proposed algorithm has low sample and computational complexities. Under some transparent assumptions, we establish that the proposed algorithm is structurally consistent (or sparsistent) when the number of samples scales as n = Ω(J min log p), where p is the number of nodes and Jmin is the minimum edge potential. We also develop novel non-asymptotic techniques for obtaining necessary conditions for graphical model selection.