Structure learning of antiferromagnetic Ising models
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In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. samples. We first observe that the notoriously difficult problem of learning parities with noise can be captured as a special case of learning graphical models. This leads to an unconditional computational lower bound of Ω (p^d/2) for learning general graphical models on p nodes of maximum degree d, for the class of so-called statistical algorithms recently introduced by Feldman et al (2013). The lower bound suggests that the O(p^d) runtime required to exhaustively search over neighborhoods cannot be significantly improved without restricting the class of models. Aside from structural assumptions on the graph such as it being a tree, hypertree, tree-like, etc., many recent papers on structure learning assume that the model has the correlation decay property. Indeed, focusing on ferromagnetic Ising models, Bento and Montanari (2009) showed that all known low-complexity algorithms fail to learn simple graphs when the interaction strength exceeds a number related to the correlation decay threshold. Our second set of results gives a class of repelling (antiferromagnetic) models that have the opposite behavior: very strong interaction allows efficient learning in time O(p^2). We provide an algorithm whose performance interpolates between O(p^2) and O(p^d+2) depending on the strength of the repulsion.
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