Abstract-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 , and/or the maximal node degree , are allowed to increase to infinity as a function of the sample size . For pairwise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class of graphs on vertices with at most edges, and over the class of graphs on vertices with maximum degree at most . For the class , we establish the existence of constants and such that if , any method has error probability at least uniformly over the family, and we demonstrate a graph decoder that succeeds with high probability uniformly over the family for sample sizes . Similarly, for the class , we exhibit constants and such that for , any method fails with probability at least , and we demonstrate a graph decoder that succeeds with high probability for . Index Terms-High dimensional inference, KL divergence between Ising models, Markov random fields, sample complexity, structure of Ising models.
Santhanam and Wainwright REF obtained necessary and sufficient conditions for learning pairwise binary Markov Random Fields.
Information-Theoretic Limits of Selecting Binary Graphical Models in High Dimensions
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computer Science Mathematics