Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields

Consider a random vector with finite second moments. If its precision matrix\nis an M-matrix, then all partial correlations are non-negative. If that random\nvector is additionally Gaussian, the corresponding Markov random field (GMRF)\nis called attractive. We study estimation of M-matrices taking the role of\ninverse second moment or precision matrices using sign-constrained\nlog-determinant divergence minimization. We also treat the high-dimensional\ncase with the number of variables exceeding the sample size. The additional\nsign-constraints turn out to greatly simplify the estimation problem: we\nprovide evidence that explicit regularization is no longer required. To solve\nthe resulting convex optimization problem, we propose an algorithm based on\nblock coordinate descent, in which each sub-problem can be recast as\nnon-negative least squares problem. Illustrations on both simulated and real\nworld data are provided.\n