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# A Fast and Scalable Joint Estimator for Learning Multiple Related Sparse Gaussian Graphical Models

; Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:1168-1177, 2017.

#### Abstract

Estimating multiple sparse Gaussian Graphical Models (sGGMs) jointly for many related tasks (large $K$) under a high-dimensional (large $p$) situation is an important task. Most previous studies for the joint estimation of multiple sGGMs rely on penalized log-likelihood estimators that involve expensive and difficult non-smooth optimizations. We propose a novel approach, FASJEM for \underlinefast and \underlinescalable \underlinejoint structure-\underlineestimation of \underlinemultiple sGGMs at a large scale. As the first study of joint sGGM using the M-estimator framework, our work has three major contributions: (1) We solve FASJEM through an entry-wise manner which is parallelizable. (2) We choose a proximal algorithm to optimize FASJEM. This improves the computational efficiency from $O(Kp^3)$ to $O(Kp^2)$ and reduces the memory requirement from $O(Kp^2)$ to $O(K)$. (3) We theoretically prove that FASJEM achieves a consistent estimation with a convergence rate of $O(\log(Kp)/n_tot)$. On several synthetic and four real-world datasets, FASJEM shows significant improvements over baselines on accuracy, computational complexity and memory costs.