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The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shownto have strong statistical guarantees in recovering a sparse inverse covariancematrix, or alternatively the underlying graph structure of a Gaussian MarkovRandom Field, from very limited samples. We propose a novel algorithm forsolving the resulting optimization problem which is a regularizedlog-determinant program. In contrast to recent state-of-the-art methods thatlargely use first order gradient information, our algorithm is based onNewton's method and employs a quadratic approximation, but with somemodifications that leverage the structure of the sparse Gaussian MLE problem.We show that our method is superlinearly convergent, and present experimentalresults using synthetic and real-world application data that demonstrate theconsiderable improvements in performance of our method when compared to otherstate-of-the-art methods.

Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation