Abstract: We propose a class of closed-form estimators for sparsity-structured graphical models, expressed as exponential family distributions, under high-dimensional settings. Our approach builds on observing the precise manner in which the classical graphical model MLE ``breaks down'' under high-dimensional settings. Our estimator uses a carefully constructed, well-defined and closed-form backward map, and then performs thresholding operations to ensure the desired sparsity structure. We provide a rigorous statistical analysis that shows that surprisingly our simple class of estimators recovers the same asymptotic convergence rates as those of the $\ell_1$-regularized MLEs that are much more difficult to compute. We corroborate this statistical performance, as well as significant computational advantages via simulations of both discrete and Gaussian graphical models.