Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency
This work addresses the challenge of precision matrix estimation in high-dimensional settings for statistical modeling, though it appears incremental as it builds on existing Laplacian constraints and loss functions.
The paper tackles the problem of estimating high-dimensional Laplacian-constrained precision matrices by minimizing Stein's loss, establishing a necessary and sufficient condition for estimator existence based on graph connectivity and proving consistency with an error rate independent of graph sparsity.
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 consists on checking whether a certain data dependent graph is connected. We also prove consistency in the high dimensional setting under the symmetrized 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, the matrix tree theorem, and a characterization of the proposed estimator based on effective graph resistances. We validate our theoretical claims with numerical experiments.