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' k
²
log p. Similarly, for the class G
_p,d
, we exhibit constants c and c' such that for n <; cd
²
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' d
³
log p.