  We consider the maximum likelihood estimation of sparse inverse covariance
matrices. We demonstrate that current heuristic approaches primarily encourage
robustness, instead of the desired sparsity. We give a novel approach that
solves the cardinality constrained likelihood problem to certifiable
optimality. The approach uses techniques from mixed-integer optimization and
convex optimization, and provides a high-quality solution with a guarantee on
its suboptimality, even if the algorithm is terminated early. Using a variety
of synthetic and real datasets, we demonstrate that our approach can solve
problems where the dimension of the inverse covariance matrix is up to 1,000s.
We also demonstrate that our approach produces significantly sparser solutions
than Glasso and other popular learning procedures, makes less false
discoveries, while still maintaining state-of-the-art accuracy.
