  The sparse inverse covariance estimation problem is commonly solved using an
$\ell_{1}$-regularized Gaussian maximum likelihood estimator known as
"graphical lasso", but its computational cost becomes prohibitive for large
data sets. A recent line of results showed--under mild assumptions--that the
graphical lasso estimator can be retrieved by soft-thresholding the sample
covariance matrix and solving a maximum determinant matrix completion (MDMC)
problem. This paper proves an extension of this result, and describes a
Newton-CG algorithm to efficiently solve the MDMC problem. Assuming that the
thresholded sample covariance matrix is sparse with a sparse Cholesky
factorization, we prove that the algorithm converges to an $\epsilon$-accurate
solution in $O(n\log(1/\epsilon))$ time and $O(n)$ memory. The algorithm is
highly efficient in practice: we solve the associated MDMC problems with as
many as 200,000 variables to 7-9 digits of accuracy in less than an hour on a
standard laptop computer running MATLAB.
