Ng, “Learning factor graphs in polynomial time J. Bento and A. Montanari, “Which graphical models are difficult to learn?,”Ī. Sly and N. Sun, “The computational hardness of counting in two-spin models G. Bresler, D. Gamarnik, and D. Shah, “Hardness of parameter estimation inĪ. Montanari, “Computational Implications of Reducing Data to Sufficient This work was supported in part by NSF grants CMMI-1335155 and CNS-1161964, and by Army Research Office MURI Award W911NF-11-1-0036. ![]() We are grateful to Mina Karzand, Kuang Xu, and Luis Voloch for helpful comments on a draft of the paper, and to Vijay Subramanian for an interesting discussion on coordination games. Concretely, with θ representing either θ 0 or θ u v, (We only project the Gibbs measure to the clique, keeping node update indices over the entire original graph.) The initial configuration σ ( 1 ) is drawn according to the stationary measure \lx p a r a g r a p h s i g n θ for each model Q θ. Q θ represent the distribution of the observation X, which now consists of samples σ ( 1 ), …, σ ( n ) ∈ d 1 as well as node update indices I ( 1 ), …, I ( n ) ∈. Similarly, using θ as a placeholder for either θ 0 or θ u v, We therefore abuse notation slightly and write \lx p a r a g r a p h s i g n θ 0 and \lx p a r a g r a p h s i g n θ u v for the Gibbs distributions after projecting onto the relevant clique. It suffices to consider the projection (i.e., marginal) onto the size d 1 clique containing u and v, since the KL divergence between these projections is equal to the entire KL divergence. In this section we upper bound the KL divergence between the models parameterized by θ 0 and any θ u v (by symmetry of the construction this is the same for every θ u v). Require similar incoherence or restricted isometry-type conditions that are difficult to interpret in terms of model parameters, and likely also require the CDP. Other convex optimization-based algorithms such as While this algorithm is shown to work under certain incoherence conditions and does not explicitly require the CDP, Bento and Montanari showed through a careful analysis that the algorithm provably fails to learn ferromagnetic Ising models on simple families of graphs without the CDP. A variety of other papers including give alternative low-complexity algorithms, but also require the CDP.Ī number of structure learning algorithms are based on convex optimization, such as Ravikumar et al.’s Īpproach using regularized node-wise logistic regression. It was first observed in that it is possible to efficiently learn models with (exponential) decay of correlations, under the additional assumption that neighboring variables have correlation bounded away from zero. Maximum degree d can be learned in time f(d)p^2 p, for a functionį(d), using nearly the information-theoretic minimum number of samples. Specifically, we show that a binary pairwise graphical model on p nodes with Models is computationally tractable when we observe the Glauber dynamics. Work, we establish that the problem of reconstructing binary pairwise graphical Much of the research on graphical model learning has been directed towardsįinding algorithms with low computational cost. ![]() Literature, where one assumes access to i.i.d. ![]() Work deviates from the standard formulation of graphical model learning in the Glauber dynamics is a natural dynamical model in a variety of settings. Graphical model and it is frequently used to sample from the stationaryĭistribution (to which it converges given sufficient time). ![]() The Glauber dynamics isĪ Markov chain that sequentially updates individual nodes (variables) in a In this paper we consider the problem of learning undirected graphical modelsįrom data generated according to the Glauber dynamics.
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