Variable Selection for Gaussian Graphical Models

We present a variable-selection structure learning approach for Gaussian graphical models. Unlike standard sparseness promoting techniques, our method aims at selecting the most-important variables besides simply sparsifying the set of edges. Through simulations, we show that our method outperforms the state-of-the-art in recovering the ground truth model. Our method also exhibits better generalization performance in a wide range of complex real-world datasets: brain fMRI, gene expression, NASDAQ stock prices and world weather. We also show that our resulting networks are more interpretable in the context of brain fMRI analysis, while retaining discriminability. From an optimization perspective, we show that a block coordinate descent method generates a sequence of positive definite solutions. Thus, we reduce the original problem into a sequence of strictly convex (\ell_1,\ell_p) regularized quadratic minimization subproblems for p∈{2,∞}. Our algorithm is well founded since the optimal solution of the maximization problem is unique and bounded.