We consider the problem of estimating high dimensional spatial graphical models with a total cardinality constraint (i.e., the ℓ-constraint). Though this problem is highly nonconvex, we show that its primal-dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this “blessing of massive scale” phenomenon. Motivated by this result, we propose an efficient algorithm to solve the dual problem (which is concave) and prove that the solution achieves optimal statistical properties. Extensive numerical results are also provided.
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