[Paper Review] Approximate Maximum Likelihood Source Localization from Range Measurements Through Convex Relaxation

Pınar Oğuz-Ekim, João Gomes+3 more|arXiv (Cornell University)|Nov 29, 2011

Indoor and Outdoor Localization Technologies17 references3 citations

TL;DR

This paper proposes two convex relaxation-based methods—Source Localization with Nuclear Norm (SLNN) and Source Localization with ℓ₁-norm (SL-ℓ₁)—for approximate maximum likelihood source localization from noisy range measurements in wireless sensor networks. By formulating the problem as a semidefinite program (SDP), SLNN achieves tighter relaxation and superior localization accuracy compared to state-of-the-art methods, especially under Gaussian noise, while SL-ℓ₁ effectively handles Laplacian noise and outliers via weighted ML reformulation and SDP relaxation.

ABSTRACT

This work considers the problem of locating a single source from noisy range measurements to a set of nodes in a wireless sensor network. We propose two new techniques that we designate as Source Localization with Nuclear Norm (SLNN) and Source Localization with l1-norm (SL-l1), which extend to arbitrary real dimensions, including 3D, our prior work on 2D source localization formulated in the complex plane. Broadly, our approach is based on formulating a Maximum-Likelihood (ML) estimation problem for the source position, and then using convex relaxation techniques to obtain a semidefinite program (SDP) that can be globally and efficiently solved. SLNN directly approximates the Gaussian ML solution, and the relaxation is shown to be tighter than in other methods in the same class. We present an analysis of the convexity properties of the constraint set for the 2D complex version of SLNN (SLCP) to justify the observed tightness of the relaxation. In terms of global accuracy of localization, SLNN outperforms state-of-the-art optimization-based methods with either iterative or closed-form formulations. We propose the SL-l1 algorithm to address the Laplacian noise case, which models the presence of outliers in range measurements. We overcome the nondifferentiability of the Laplacian likelihood function by rewriting the ML problem as an exact weighted version of the Gaussian case, and compare two solution strategies. One of them is iterative, based on block coordinate descent, and uses SLNN as a subprocessing block. The other, attaining only slightly worse performance, is noniterative and based on an SDP relaxation of the weighted ML problem.

Motivation & Objective

• To address the challenge of accurate source localization in wireless sensor networks with noisy, unreliable range measurements.
• To develop a convex relaxation framework that provides globally optimal solutions to the nonconvex maximum likelihood (ML) estimation problem for source localization.
• To extend prior 2D complex-plane methods to arbitrary real dimensions, including 3D, while preserving tight relaxation and high localization accuracy.
• To handle non-Gaussian noise, particularly Laplacian-distributed outliers, by proposing an ℓ₁-norm-based alternative to Gaussian ML.
• To improve upon existing SDP relaxation methods by ensuring tighter relaxation and higher likelihood of rank-one solutions for accurate source position recovery.

Proposed method

• Formulates the maximum likelihood (ML) estimation problem for source localization as a nonconvex, nonsmooth optimization problem in arbitrary real dimensions.
• Applies semidefinite relaxation (SDR) to transform the nonconvex ML problem into a convex semidefinite program (SDP), enabling global and efficient solution.
• Introduces SLNN, which uses nuclear norm minimization to approximate the Gaussian ML solution, with theoretical justification for tighter relaxation via KKT analysis and singular value constraints.
• Proposes SL-ℓ₁ to model Laplacian noise by reformulating the ML problem as a weighted Gaussian case, enabling robustness to outliers.
• Employs two solution strategies for SL-ℓ₁: an iterative block coordinate descent method using SLNN as a subproblem, and a noniterative SDP relaxation with slightly degraded performance.
• Analyzes convexity properties of the constraint set in the 2D complex formulation (SLCP), proving that the relaxation is tighter than in prior methods, leading to higher probability of rank-one solutions.

Experimental results

Research questions

• RQ1 Can a convex relaxation of the maximum likelihood problem for source localization achieve higher accuracy than existing iterative or closed-form optimization methods?
• RQ2 How does the nuclear norm relaxation in SLNN improve upon existing semidefinite relaxation techniques in terms of solution tightness and rank-one recovery?
• RQ3 Can the ML estimation problem be effectively reformulated to handle Laplacian noise and outliers in range measurements?
• RQ4 What is the performance trade-off between iterative and noniterative SDP-based strategies for the weighted ML problem in the presence of outliers?
• RQ5 Does extending 2D complex-plane methods to arbitrary real dimensions preserve the accuracy and relaxation tightness of the original formulation?

Key findings

• SLNN outperforms state-of-the-art optimization-based methods in terms of global localization accuracy under Gaussian noise, particularly due to tighter relaxation and higher likelihood of rank-one solutions.
• The convexity analysis of the SLCP (2D complex) formulation confirms that the relaxation is tighter than in prior methods, as the relaxed solution more frequently yields rank-one matrices suitable for factorization into source coordinates.
• SL-ℓ₁ effectively models Laplacian noise by rewriting the ML problem as a weighted Gaussian case, enabling robustness to outliers commonly found in real-world range measurements.
• The iterative SL-ℓ₁ strategy using SLNN as a subproblem achieves high performance, while the noniterative SDP-based alternative offers comparable accuracy with lower computational cost.
• Theoretical equivalence is proven between the original nonconvex ML problem and its SDP relaxation, ensuring that the optimal value of the relaxed problem matches the original, thus guaranteeing global optimality.
• The bounds on the nuclear and Frobenius norms of the matrix $\mathbf{U}^{T}\mathbf{C}$ confirm that the nuclear norm relaxation is tighter than the Frobenius norm approach, supporting the superiority of SLNN.