Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency

This paper considers the problem of estimating high dimensional Laplacian constrained precision matrices by minimizing Stein's loss. We obtain a necessary and sufficient condition for existence of this estimator, that boils down to checking whether a certain data dependent graph is connected. We also prove consistency in the high dimensional setting under the symmetryzed Stein loss. We show that the error rate does not depend on the graph sparsity, or other type of structure, and that Laplacian constraints are sufficient for high dimensional consistency. Our proofs exploit properties of graph Laplacians, and a characterization of the proposed estimator based on effective graph resistances. We validate our theoretical claims with numerical experiments.