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