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

Abstract

We propose maximum likelihood estimation for learning Gaussian graphical models with a Gaussian (ℓ[2 over 2]) prior on the parameters. This is in contrast to the commonly used Laplace (ℓ[subscript 1) prior for encouraging sparseness. We show that our optimization problem leads to a Riccati matrix equation, which has a closed form solution. We propose an efficient algorithm that performs a singular value decomposition of the training data. Our algorithm is O(NT[superscript 2])-time and O(NT)-space for N variables and T samples. Our method is tailored to high-dimensional problems (N >> T), in which sparseness promoting methods become intractable. Furthermore, instead of obtaining a single solution for a specific regularization parameter, our algorithm finds the whole solution path. We show that the method has logarithmic sample complexity under the spiked covariance model. We also propose sparsification of the dense solution with provable performance guarantees. We provide techniques for using our learnt models, such as removing unimportant variables, computing likelihoods and conditional distributions. Finally, we show promising results in several gene expressions datasets.