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We propose maximum likelihood estimation for learning Gaussian graphicalmodels with a Gaussian (ell_2^2) prior on the parameters. This is in contrastto the commonly used Laplace (ell_1) prior for encouraging sparseness. We showthat our optimization problem leads to a Riccati matrix equation, which has aclosed form solution. We propose an efficient algorithm that performs asingular value decomposition of the training data. Our algorithm isO(NT^2)-time and O(NT)-space for N variables and T samples. Our method istailored to high-dimensional problems (N gg T), in which sparseness promotingmethods become intractable. Furthermore, instead of obtaining a single solutionfor a specific regularization parameter, our algorithm finds the whole solutionpath. We show that the method has logarithmic sample complexity under thespiked covariance model. We also propose sparsification of the dense solutionwith provable performance guarantees. We provide techniques for using ourlearnt models, such as removing unimportant variables, computing likelihoodsand conditional distributions. Finally, we show promising results in severalgene expressions datasets.

Inverse Covariance Estimation for High-Dimensional Data in Linear Time and Space: Spectral Methods for Riccati and Sparse Models