  We propose a penalized likelihood framework for estimating multiple precision
matrices from different classes. Most existing methods either incorporate no
information on relationships between the precision matrices, or require this
information be known a priori. The framework proposed in this article allows
for simultaneous estimation of the precision matrices and relationships between
the precision matrices, jointly. Sparse and non-sparse estimators are proposed,
both of which require solving a non-convex optimization problem. To compute our
proposed estimators, we use an iterative algorithm which alternates between a
convex optimization problem solved by blockwise coordinate descent and a
k-means clustering problem. Blockwise updates for computing the sparse
estimator require solving an elastic net penalized precision matrix estimation
problem, which we solve using a proximal gradient descent algorithm. We prove
that this subalgorithm has a linear rate of convergence. In simulation studies
and two real data applications, we show that our method can outperform
competitors that ignore relevant relationships between precision matrices and
performs similarly to methods which use prior information often uknown in
practice.
