A Fast Algorithm for Graph Learning under Attractive Gaussian Markov Random Fields

We consider the problem of graph learning under Gaussian Markov random fields, where all partial correlations are nonnegative. Such model is called attractive Gaussian Markov random fields, and has received considerable attention in recent years. The graph learning problem under this model can be formulated as the ℓ
1
-norm regularized Gaussian maximum likelihood estimation of the precision matrix under sign constraints. In this paper, we propose a projected Newton-like algorithm, which is computationally efficient. By exploiting the structure of the Gaussian maximum likelihood estimation problem, the proposed algorithm significantly reduces the computational cost in computing the approximate Newton direction. Then we prove that the proposed method can recover the graph edges correctly under the irrepresentability condition. Numerical results on synthetic and financial time-series data sets demonstrate the effectiveness of the proposed method.