Cram$\acute{\text{e}}$r-Rao Bounds for Laplacian Matrix Estimation

In this paper, we analyze the performance of the estimation of Laplacian matrices under general observation models. Laplacian matrix estimation involves structural constraints, including symmetry and null-space properties, along with matrix sparsity. By exploiting a linear reparametrization that enforces the structural constraints, we derive closed-form matrix expressions for the Cramér-Rao bound (CRB) specifically tailored to Laplacian matrix estimation. We further extend the derivation to the sparsity-constrained case, introducing two oracle CRBs that incorporate prior information of the support set, i.e. the locations of the nonzero entries in the Laplacian matrix. We examine the properties and order relations between the bounds, and provide the associated Slepian-Bangs formula for the Gaussian case. We demonstrate the use of the new CRBs in three representative applications: (i) topology identification in power systems, (ii) graph filter identification in diffused models, and (iii) precision matrix estimation in Gaussian Markov random fields under Laplacian constraints. The CRBs are evaluated and compared with the mean-squared-errors (MSEs) of the constrained maximum likelihood estimator (CMLE), which integrates both equality and inequality constraints along with sparsity constraints, and of the oracle CMLE, which knows the locations of the nonzero entries of the Laplacian matrix. We perform this analysis for the applications of power system topology identification and graphical LASSO, and demonstrate that the MSEs of the estimators converge to the CRB and oracle CRB, given a sufficient number of measurements.