A constrained l1 minimization approach for estimating multiple Sparse Gaussian or Nonparanormal Graphical Models
Authors: Beilun Wang, Ritambhara Singh, Yanjun Qi

Abstract: The flood of multi-context measurement data from many scientific domains has created an urgent need to reconstruct context-specific variable networks, that could significantly simplify network-driven studies. Computationally, this problem can be formulated  as jointly estimating multiple different, but related, sparse Undirected Graphical Models (UGM) from  samples aggregated across several contexts. Previous joint-UGM studies could not address this challenge since they mostly focus on Gaussian Graphical Models (GGM) and have used likelihood-based formulation to infer multiple graphs toward a common pattern. Differently, we propose a novel approach, SIMULE (learning Shared and Individual parts of MULtiple graphs Explicitly) to solve multi-task UGM using a $\ell$1 constrained optimization. SIMULE is cast as independent subproblems of linear programming that can be solved  efficiently. It automatically infers specific dependencies that are unique to each context as well as shared substructures preserved among all the contexts. SIMULE can handle both multivariate Gaussian and multivariate Nonparanormal data that greatly relax the normality assumption. Theoretically we prove that SIMULE achieves a consistent result at rate $O(\log(Kp)/n_{tot})$ (not been proved before). On four synthetic datasets, SIMULE shows significant improvements over state-of-the-art multi-sGGM and single-UGM baselines.