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The problem of structure estimation in graphical models with latent variablesis considered. We characterize conditions for tractable graph estimation anddevelop efficient methods with provable guarantees. We consider models wherethe underlying Markov graph is locally tree-like, and the model is in theregime of correlation decay. For the special case of the Ising model, thenumber of samples $n$ required for structural consistency of our method scalesas $n=\Omega(\theta_{\min}^{-\delta\eta(\eta+1)-2}\log p)$, where p is thenumber of variables, $\theta_{\min}$ is the minimum edge potential, $\delta$ isthe depth (i.e., distance from a hidden node to the nearest observed nodes),and $\eta$ is a parameter which depends on the bounds on node and edgepotentials in the Ising model. Necessary conditions for structural consistencyunder any algorithm are derived and our method nearly matches the lower boundon sample requirements. Further, the proposed method is practical to implementand provides flexibility to control the number of latent variables and thecycle lengths in the output graph.

Learning loopy graphical models with latent variables: Efficient methods and guarantees