Linear-Time Algorithm for Learning Large-Scale Sparse Graphical Models

The sparse inverse covariance estimation problem is commonly solved using an l1-regularizedGaussian 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 ǫ-accurate solution inO(n log(1/ǫ)) time andO(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.