  Learning a Gaussian Mixture Model (GMM) is hard when the number of parameters
is too large given the amount of available data. As a remedy, we propose
restricting the GMM to a Gaussian Markov Random Field Mixture Model (GMRF-MM),
as well as a new method for estimating the latter's sparse precision (i.e.,
inverse covariance) matrices. When the sparsity pattern of each matrix is
known, we propose an efficient optimization method for the Maximum Likelihood
Estimate (MLE) of that matrix. When it is unknown, we utilize the popular
Graphical Least Absolute Shrinkage and Selection Operator (GLASSO) to estimate
that pattern. However, we show that even for a single Gaussian, when GLASSO is
tuned to successfully estimate the sparsity pattern, it does so at the price of
a substantial bias of the values of the nonzero entries of the matrix, and we
show that this problem only worsens in a mixture setting. To overcome this, we
discard the nonzero values estimated by GLASSO, keep only its pattern estimate
and use it within the proposed MLE method. This yields an effective two-step
procedure that removes the bias. We show that our "debiasing" approach
outperforms GLASSO in both the single-GMRF and the GMRF-MM cases. We also show
that when learning priors for image patches, our method outperforms GLASSO even
if we merely use an educated guess about the sparsity pattern, and that our
GMRF-MM outperforms the baseline GMM on real and synthetic high-dimensional
datasets.
