Structure learning of antiferromagnetic Ising models
In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i. 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 [3] 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 e cient learning in time Â O(p 2 ). We provide an algorithm whose performance interpolates between Â O(p 2 ) and Â O(p d+2 ) depending on the strength of the repulsion.
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 [3] 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 e cient learning in time Â O(p 2 ). We provide an algorithm whose performance interpolates between Â O(p 2 ) and Â O(p d+2 ) depending on the strength of the repulsion.
This paper presents some theoretical results for the complexity of learning pairwise graphical models. Arguably, the most important result is a lower bound of the complexity of the general case. I believe the initial part of the paper can be improved. For instance, a "graphical model" is never defined and the discussion of sect.1.1 and 1.2 are confusing because of that. While I understand what the authors want to say, they should formally define the problem of interest, including the graphical model and the objective function that they want to optimise as being the problem of "learning the structure of general graphical models". For instance (and if I understood correctly), the work considers graphical models with pairwise relations only, which could be emphasised in the initial description of the problem (currently missing, even though the paper later clarify most of my doubts, reading its initial part for the first time was not much informative). The goal is explained as "recovering a graph on p nodes", but this is too informal. The problem that is being solved only becomes clear in sect.2, and even there it is not really formally defined. Because of the presentation, theorem 1.1 is not so appealing in the first time one reads it. In a different note, the initial complexity discussion ignores amount of data to learn, although this is clarified later. All in all, the paper seems sound and with an interesting theoretical study. However, I admit not to be an expert on this exact topic, and it is not easy to me to assess well the relevance and novelty. For example, the main result (of theorem 1.1) does not seem surprising to me, but if it had not been proved before, its proof is of clear relevance.Minor issues:- "Thus algorithms... does" -> do- "turns out reconstruct" ?- Algorithm 1: S should start empty, algorithm 2 and 3, hat{E}, etc. Better at least to make clear that sets are assumed to start empty.- "Thus the ML estimate contains a subset of those edges e which have not..." Could you explain this with slightly more detail?