  Identifying context-specific entity networks from aggregated data is an
important task, arising often in bioinformatics and neuroimaging.
Computationally, this task can be formulated as jointly estimating multiple
different, but related, sparse Undirected Graphical Models (UGM) from
aggregated samples across several contexts. Previous joint-UGM studies have
mostly focused on sparse Gaussian Graphical Models (sGGMs) and can't identify
context-specific edge patterns directly. We, therefore, propose a novel
approach, SIMULE (detecting Shared and Individual parts of MULtiple graphs
Explicitly) to learn multi-UGM via a constrained L1 minimization. SIMULE
automatically infers both specific edge patterns that are unique to each
context and shared interactions preserved among all the contexts. Through the
L1 constrained formulation, this problem is cast as multiple independent
subtasks of linear programming that can be solved efficiently in parallel. In
addition to Gaussian data, SIMULE can also handle multivariate Nonparanormal
data that greatly relaxes the normality assumption that many real-world
applications do not follow. We provide a novel theoretical proof showing that
SIMULE achieves a consistent result at the rate O(log(Kp)/n_{tot}). On multiple
synthetic datasets and two biomedical datasets, SIMULE shows significant
improvement over state-of-the-art multi-sGGM and single-UGM baselines.
