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This paper considers the problem of estimating high dimensional Laplacianconstrained precision matrices by minimizing Stein's loss. We obtain anecessary and sufficient condition for existence of this estimator, thatconsists on checking whether a certain data dependent graph is connected. Wealso prove consistency in the high dimensional setting under the symmetrizedStein 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 forhigh dimensional consistency. Our proofs exploit properties of graphLaplacians, the matrix tree theorem, and a characterization of the proposedestimator based on effective graph resistances. We validate our theoreticalclaims with numerical experiments.

Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency