Title:Information-theoretic limits of selecting binary graphical models in high dimensions

Abstract: The problem of graphical model selection is to correctly estimate the graph structure of a Markov random field given samples from the underlying distribution. 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 p 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 pairwise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class Gp,k of graphs on p vertices with at most k edges, and over the class Gp,d of graphs on p vertices with maximum degree at most d. For the class Gp,k, we establish the existence of constants c and c′ such that if $\numobs < c k \log p$, 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 $\numobs > c' k^2 \log p$. Similarly, for the class Gp,d, we exhibit constants c and c′ such that for n<cd2logp, any method fails with probability at least 1/2, and we demonstrate a graph decoder that succeeds with high probability for n>c′d3logp.