Single-Source Localization as an Eigenvalue Problem

TL;DR

The authors recast trilateration as a weighted least-squares problem on squared distances and transform it into a fixed-eigenvalue problem, proving that the global optimum aligns with the largest real eigenvalue of a constructed matrix. The approach accommodates diverse noise models through tailored residual normalizations and derives a robust, fast solver that gracefully handles degenerate configurations. Empirical results on synthetic and real data show competitive speed and numerical stability compared with state-of-the-art methods, with notable resilience in challenging degenerate scenarios. The work enables practical, high-precision single-source localization and motivates extensions to multilateration and measurement-type fusion.

Abstract

This paper introduces a novel method for solving the single-source localization problem, specifically addressing the case of trilateration. We formulate the problem as a weighted least-squares problem in the squared distances and demonstrate how suitable weights are chosen to accommodate different noise distributions. By transforming this formulation into an eigenvalue problem, we leverage existing eigensolvers to achieve a fast, numerically stable, and easily implemented solver. Furthermore, our theoretical analysis establishes that the globally optimal solution corresponds to the largest real eigenvalue, drawing parallels to the existing literature on the trust-region subproblem. Unlike previous works, we give special treatment to degenerate cases, where multiple and possibly infinitely many solutions exist. We provide a geometric interpretation of the solution sets and design the proposed method to handle these cases gracefully. Finally, we validate against a range of state-of-the-art methods using synthetic and real data, demonstrating how the proposed method is among the fastest and most numerically stable.

Figures (6)

• Figure 1: Three senders ${\mathbf{s}}_1, {\mathbf{s}}_2, {\mathbf{s}}_3$ and the maximum likelihood solution ${\mathbf{x}}$ to the trilateration problem. The circles indicate the corresponding distance measurements $d_1$, $d_2$, and $d_3$.
• Figure 2: Two degenerate sender configurations. (a) When the senders are collinear in 2D, there are two possible solutions, ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$. (b) When the senders are collinear in 3D, there are infinitely many solutions located on a circle.
• Figure 3: Degenerate case in the plane occurring when the senders are evenly distributed on the unit circle and the distance measurements are $d_j = 1.65$ for $j=1,\dotsc,4$. The blue circle with radius $0.85$ indicates the solutions to \ref{['eq:sls']}, while the four squares indicate the solutions to \ref{['eq:ls']}.
• Figure 4: Normalized errors in estimated receiver position for several trilateration methods over various amounts of noise. The whiskers in the boxplot indicate the 1.5 IQR value and outliers are not shown.
• Figure 5: (a) Median error in estimated receiver position and (b) success rate over 1000.0 trials for a range of scaling factors. Success is defined as an error smaller than $10^{-6}$. The scaling factor is multiplied with the x-coordinate of each sender causing them to become coplanar when the factor approaches zero.

Theorems & Definitions (10)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5: cf. adachi_solving_2017[Theorem 3.4]
• proof