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In this paper, we present $\ell_{1,p}$ multi-task structure learning forGaussian graphical models. We analyze the sufficient number of samples for thecorrect recovery of the support union and edge signs. We also analyze thenecessary number of samples for any conceivable method by providinginformation-theoretic lower bounds. We compare the statistical efficiency ofmulti-task learning versus that of single-task learning. For experiments, weuse a block coordinate descent method that is provably convergent and generatesa sequence of positive definite solutions. We provide experimental validationon synthetic data as well as on two publicly available real-world data sets,including functional magnetic resonance imaging and gene expression data.

On the Statistical Efficiency of $\ell_{1,p}$ Multi-Task Learning of Gaussian Graphical Models