Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields
This addresses structure learning in graphical models for researchers in statistics and machine learning, offering an incremental improvement by simplifying estimation under specific constraints.
The paper tackles the problem of estimating positive definite M-matrices for attractive Gaussian Markov Random Fields, showing that sign constraints simplify estimation by eliminating the need for explicit regularization, with evidence from high-dimensional scenarios where variables exceed sample size.
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.