  We introduce a general framework for estimation of inverse covariance, or
precision, matrices from heterogeneous populations. The proposed framework uses
a Laplacian shrinkage penalty to encourage similarity among estimates from
disparate, but related, subpopulations, while allowing for differences among
matrices. We propose an efficient alternating direction method of multipliers
(ADMM) algorithm for parameter estimation, as well as its extension for faster
computation in high dimensions by thresholding the empirical covariance matrix
to identify the joint block diagonal structure in the estimated precision
matrices. We establish both variable selection and norm consistency of the
proposed estimator for distributions with exponential or polynomial tails.
Further, to extend the applicability of the method to the settings with unknown
populations structure, we propose a Laplacian penalty based on hierarchical
clustering, and discuss conditions under which this data-driven choice results
in consistent estimation of precision matrices in heterogenous populations.
Extensive numerical studies and applications to gene expression data from
subtypes of cancer with distinct clinical outcomes indicate the potential
advantages of the proposed method over existing approaches.
