Joint Learning of Multiple Sparse Matrix Gaussian Graphical Models.

We consider joint learning of multiple sparse matrix Gaussian graphical models and propose the joint matrix graphical Lasso to discover the conditional independence structures among rows (columns) in the matrix variable under distinct conditions. The proposed approach borrows strength across the different graphical models and is based on the maximum likelihood with penalized row and column precision matrices, respectively. In particular, our model is more parsimonious and flexible than the joint vector graphical models. Furthermore, we establish the asymptotic properties of our model on consistency and sparsistency. And the asymptotic analysis shows that our model enjoys a better convergence rate than the joint vector graphical models. Extensive simulation experiments demonstrate that our methods outperform state-of-the-art methods in identifying graphical structures and estimating precision matrices. Moreover, the effectiveness of our methods is also illustrated via a real data set analysis. Sparsistency is shorthand for consistency of the sparsity pattern of the parameters.