Cramer-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 Cramer-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.

Estimation of Complex-Valued Laplacian Matrices for Topology Identification in Power Systems

In this paper, we investigate the problem of estimating a complex-valued Laplacian matrix from a linear Gaussian model, with a focus on its application in the estimation of admittance matrices in power systems. The proposed approach is based on a constrained maximum likelihood estimator (CMLE) of the complex-valued Laplacian, which is formulated as an optimization problem with Laplacian and sparsity constraints. The complex-valued Laplacian is a symmetric, non-Hermitian matrix that exhibits a joint sparsity pattern between its real and imaginary parts. Leveraging the l1 relaxation and the joint sparsity, we develop two estimation algorithms for the implementation of the CMLE. The first algorithm is based on casting the optimization problem as a semi-definite programming (SDP) problem, while the second algorithm is based on developing an efficient augmented Lagrangian method (ALM) solution. Next, we apply the proposed SDP and ALM algorithms for the problem of estimating the admittance matrix under three commonly-used measurement models, that stem from Kirchhoff's and Ohm's laws, each with different assumptions and simplifications: 1) the nonlinear alternating current (AC) model; 2) the decoupled linear power flow (DLPF) model; and 3) the direct current (DC) model. The performance of the SDP and the ALM algorithms is evaluated using data from the IEEE 33-bus power system data under different settings. The numerical experiments demonstrate that the proposed algorithms outperform existing methods in terms of mean-squared-error (MSE) and F-score, thus, providing a more accurate recovery of the admittance matrix.