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Cramer-Rao Bounds for Laplacian Matrix Estimation

Morad Halihal, Tirza Routtenberg, H. Vincent Poor

TL;DR

The paper develops closed-form CRBs for Laplacian matrix estimation under symmetry, null-space, and sparsity constraints by a linear reparametrization that reduces the parameter dimension from $M^2$ to $ rac{M(M-1)}{2}$. It provides CRBs for complete and sparse graphs, including two oracle variants for sparsity, and specializes to Gaussian observations via the Slepian-Bangs formula. Theoretical insights are complemented by three applications—power-system topology identification, graph-filter identification in diffusion models, and Laplacian-based GMRFs—showing that (i) the constrained maximum likelihood estimator attains the oracle CRB asymptotically, and (ii) the proposed bounds serve as sharp performance benchmarks across varying noise and sample regimes. The results also illuminate how sparsity and graph structure influence identifiability and estimation accuracy, with practical implications for sensor placement and model-based graph learning.

Abstract

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.

Cramer-Rao Bounds for Laplacian Matrix Estimation

TL;DR

The paper develops closed-form CRBs for Laplacian matrix estimation under symmetry, null-space, and sparsity constraints by a linear reparametrization that reduces the parameter dimension from to . It provides CRBs for complete and sparse graphs, including two oracle variants for sparsity, and specializes to Gaussian observations via the Slepian-Bangs formula. Theoretical insights are complemented by three applications—power-system topology identification, graph-filter identification in diffusion models, and Laplacian-based GMRFs—showing that (i) the constrained maximum likelihood estimator attains the oracle CRB asymptotically, and (ii) the proposed bounds serve as sharp performance benchmarks across varying noise and sample regimes. The results also illuminate how sparsity and graph structure influence identifiability and estimation accuracy, with practical implications for sensor placement and model-based graph learning.

Abstract

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.

Paper Structure

This paper contains 27 sections, 3 theorems, 64 equations, 5 figures.

Table of Contents

  1. Introduction
  2. CRB for Laplacian Matrix Estimation
  3. Problem Formulation and Laplacian Matrix Properties
  4. Discussion on the parametric constraints
  5. Which Cram$\acute{\text{e}}$r-Rao-type bound should be used?
  6. Laplacian Reparametrization
  7. CRB for Complete Graphs
  8. CRB for Sparse Graphs
  9. Discussion and Special Cases
  10. Discussion
  11. Existence of the Bounds
  12. Relation with the Oracle CRB from ben2010cramer
  13. Order Relation
  14. Incidence-based Decomposition of Laplacian Matrices
  15. The Gaussian Case - Slepian-Bangs Formula
  16. ...and 12 more sections

Key Result

Theorem 1

Consider the estimation of the vector ${\hbox{\boldmath $\alpha$}}\in \mathbb{R}^{\frac{M(M-1)}{2}}$, defined in alpha_def, from the observation vector ${\bf{x}}$, assuming a complete graph structure. Then, under the regularity conditions of the crb (see, e.g.Kayestimation), the mse of any unbiased where in which and under the assumption that the fim are invertible. Equality in bound on mse is

Figures (5)

  • Figure 1: Illustration of the considered operators applied on the matrix ${\bf{A}} \in \mathbb{R}^{M\times M}$: $\mathrm{Vec}_{d}(\cdot)$ returns elements on the main diagonal; $\mathrm{Vec}_{\ell}(\cdot)$ returns elements below the main diagonal; and $\mathrm{Vec}_{u}(\cdot)$ returns elements above the main diagonal.
  • Figure 2: The susceptance matrix, ${\bf{L}}$, for IEEE 33-bus system.
  • Figure 3: The mse of the different estimators compared with the oracle crb for ${\bf{L}}$, for IEEE $33$-bus system with $N=600$.
  • Figure 4: Random planar graph with $100$ nodes used in the simulations of Subsection \ref{['simulation_GMRF_sec']}.
  • Figure 5: The mse of the different estimators compared with the oracle crb for planner graph versus $n/p$.

Theorems & Definitions (5)

  • Theorem 1: Laplacian-Constrained CRB for Complete Graphs
  • proof
  • Theorem 2: Oracle CRB for Sparse Graphs
  • proof
  • Theorem 3: FIM for General Matrix Estimation