Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency

TL;DR

This paper studies the estimation of high-dimensional Laplacian constrained precision matrices, providing conditions for existence and proving high-dimensional consistency under Stein's loss, with validation through numerical experiments.

Contribution

It establishes a necessary and sufficient condition for estimator existence and proves high-dimensional consistency for Laplacian constrained precision matrices.

Findings

Existence condition linked to graph connectivity

Estimator is consistent in high dimensions under Stein's loss

Error rate independent of graph sparsity

Abstract

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.