Information-Theoretic Limits of Selecting Binary Graphical Models in High Dimensions

The problem of graphical model selection is to estimate the graph structure of a Markov random field given samples from it. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size and the number of edges k, and/or the maximal node degree d, are allowed to increase to infinity as a function of the sample size n. For pair-wise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class G_p,k of graphs on vertices with at most k edges, and over the class G_p,d of graphs on p vertices with maximum degree at most d. For the class G_p,k, we establish the existence of constants c and c´ such that if n <; ck log p, any method has error probability at least 1/2 uniformly over the family, and we demonstrate a graph decoder that succeeds with high probability uniformly over the family for sample sizes n >; c´ k² log p. Similarly, for the class G_p,d, we exhibit constants c and c´ such that for n <; cd² log p, any method fails with probability at least 1/2, and we demonstrate a graph decoder that succeeds with high probability for n >; c´ d³ log p.