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Consider a random vector with finite second moments. If its precision matrixis an M-matrix, then all partial correlations are non-negative. If that randomvector is additionally Gaussian, the corresponding Markov random field (GMRF)is called attractive. We study estimation of M-matrices taking the role ofinverse second moment or precision matrices using sign-constrainedlog-determinant divergence minimization. We also treat the high-dimensionalcase with the number of variables exceeding the sample size. The additionalsign-constraints turn out to greatly simplify the estimation problem: weprovide evidence that explicit regularization is no longer required. To solvethe resulting convex optimization problem, we propose an algorithm based onblock coordinate descent, in which each sub-problem can be recast asnon-negative least squares problem. Illustrations on both simulated and realworld data are provided.

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