  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 nonasymptotic necessary and sufficient conditions for
structure estimation.
