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Single-Source Localization as an Eigenvalue Problem

Larsson, Martin LU orcid ; Larsson, Viktor LU ; Astrom, Kalle LU orcid and Oskarsson, Magnus LU orcid (2025) In IEEE Transactions on Signal Processing 73. p.574-583
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... (More)

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.

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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
generalized trust-region subproblem, global optimization, source localization, trilateration
in
IEEE Transactions on Signal Processing
volume
73
pages
574 - 583
publisher
IEEE - Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:85216096074
ISSN
1053-587X
DOI
10.1109/TSP.2025.3532102
language
English
LU publication?
yes
additional info
Publisher Copyright: © 1991-2012 IEEE.
id
2bb6e63d-e62e-471e-8644-c4eea5cebe78
date added to LUP
2025-02-05 22:02:24
date last changed
2025-03-20 15:21:42
@article{2bb6e63d-e62e-471e-8644-c4eea5cebe78,
  abstract     = {{<p>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.</p>}},
  author       = {{Larsson, Martin and Larsson, Viktor and Astrom, Kalle and Oskarsson, Magnus}},
  issn         = {{1053-587X}},
  keywords     = {{generalized trust-region subproblem; global optimization; source localization; trilateration}},
  language     = {{eng}},
  pages        = {{574--583}},
  publisher    = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
  series       = {{IEEE Transactions on Signal Processing}},
  title        = {{Single-Source Localization as an Eigenvalue Problem}},
  url          = {{http://dx.doi.org/10.1109/TSP.2025.3532102}},
  doi          = {{10.1109/TSP.2025.3532102}},
  volume       = {{73}},
  year         = {{2025}},
}