We consider the problem of high-dimensional Ising (graphical) model selection. We propose a simple algorithm for structure estimation based on the thresholding of the empirical conditional variation distances. We introduce a novel criterion for tractable graph families, where this method is efficient, based on the presence of sparse local separators between node pairs in the underlying graph. For such graphs, the proposed algorithm has a sample complexity of n =Omega(J_{min}^{-2} log p), where p is the number of variables and J_{min} is the minimum (absolute) edge potential in the model. We also establish non-asymptotic necessary and sufficient conditions for structure estimation.
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