Single-Source Localization as an Eigenvalue Problem

TL;DR

This paper presents a fast, stable eigenvalue-based method for single-source localization via trilateration, effectively handling degenerate cases and outperforming existing methods in speed and stability.

Contribution

It formulates the localization problem as an eigenvalue problem, providing a novel, stable, and efficient solution that addresses degenerate cases and improves over prior approaches.

Findings

Method is among the fastest in the field.

Demonstrates high numerical stability.

Effective in handling degenerate cases.

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.