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arxiv: 2510.13498 · v2 · submitted 2025-10-15 · 📡 eess.SP · math.OC

A Robust EDM Optimization Approach for 3D Single-Source Localization with Angle and Range Measurements

Mingyu Zhao , Qingna Li , Hou-Duo Qi This is my paper

Pith reviewed 2026-05-18 06:24 UTC · model grok-4.3

classification 📡 eess.SP math.OC
keywords 3D single-source localizationEuclidean distance matrixangle measurementsbox constraintsrobust optimizationmajorization penalty method
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The pith

Reformulating 3D angle measurements into box constraints on distances allows a robust EDM model for single-source localization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a method to convert 3D angle measurements into lower and upper bounds on distances to the source. It does this by reducing each angle measurement to a two-dimensional nonlinear optimization problem and finding its global minimum and maximum. These bounds serve as simple box constraints in an EDM optimization that also incorporates range measurements and uses the l1 norm to handle noise and outliers. A sympathetic reader would care because this integration of angle and range data in a single optimization framework promises more accurate localization in challenging environments like low signal-to-noise ratios.

Core claim

By reducing each of the 3D angle measurements to a two-dimensional nonlinear optimization problem, whose global minimum and maximum solutions can be characterized and utilized to get the lower and upper bounds of the distances from the unknown source to the sensors, the existing 3D angle measurements are rigorously reformulated into simple box constraints on the Euclidean distances. These constraints are then used in a rank-constrained EDM optimization model solved by the majorization penalty method.

What carries the argument

The reduction of 3D angle measurements to two-dimensional nonlinear optimization problems that yield box constraints on Euclidean distances.

If this is right

  • The new model integrates range and angle measurements simultaneously for improved 3D localization accuracy.
  • The majorization penalty method provides an efficient way to solve the rank-constrained problem.
  • Performance gains are particularly notable in low SNR scenarios compared to leading solvers.
  • The l1-norm criterion enhances robustness to measurement errors and outliers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This bounding approach could extend to other problems involving directional measurements in sensor networks.
  • Analytical solutions to the 2D optimization subproblems might further reduce computation time.
  • Testing the method with real-world radar data would validate its practical utility beyond simulations.

Load-bearing premise

That the global minimum and maximum of each reduced 2D nonlinear optimization problem for angle measurements can be reliably characterized to produce tight, useful box constraints on Euclidean distances without introducing significant looseness or bias into the overall EDM model.

What would settle it

Comparing the computed distance bounds against the true distances in simulated scenarios with known source positions to check if the bounds are tight enough to not degrade the localization solution.

Figures

Figures reproduced from arXiv: 2510.13498 by Hou-Duo Qi, Mingyu Zhao, Qingna Li.

Figure 1
Figure 1. Figure 1: Illustration of SSLAR including one transmitter and multiple receivers. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representation of the antenna beamwidth (adapted from [9]). [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometric configuration of radar localization system in target location [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between EDMARp and EDMRp, when (θ, ϕ) = (6.9 ◦, 4.9 ◦) and (θ, ϕ) = (7◦, 5 ◦). (a) θ = 0◦ , ϕ = 0◦ (b) θ = 4◦ , ϕ = 0◦ (c) θ = 6.9 ◦ , ϕ = 4.9 ◦ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: RMSE versus SNR0, when (θ, ϕ) = (7◦, 5 ◦). TABLE II AVERAGE TIME (ms) WHEN (θ, ϕ) = (7◦, 5 ◦). (θ, ϕ) SNR0 EDMAR1 EDMAR2 ARCE fmincon (0◦, 0 ◦) 0 dB 0.7 0.82 0.68 14.03 5 dB 0.65 0.73 0.74 14.23 10 dB 0.66 0.67 0.81 15.13 15dB 0.57 0.57 0.76 17.34 (4◦, 0 ◦) 0 dB 0.69 0.8 0.63 14.6 5 dB 0.66 0.68 0.71 14.32 10 dB 0.64 0.67 0.73 14.83 15 dB 0.57 0.57 0.72 15.83 (6.9 ◦, 4.9 ◦) 0 dB 0.67 0.77 0.6 14.61 5 dB 0.… view at source ↗
read the original abstract

Accurate source localization in Multi-Platform Radar Networks (MPRNs) benefits from exploiting both range and angle measurements under robust estimation. In this paper, we propose a robust Euclidean distance matrix (EDM) optimization model that simultaneously integrates range measurements, angle information, and the least absolute deviation ($\ell_1$-norm) criterion for the case of 3D single-source localization (3DSSL). A key theoretical contribution of this work is the rigorous reformulation of {existing} 3D angle measurements into simple box constraints on the Euclidean distances. Unlike previous approximations, we achieve this by reducing each of the 3D angle measurements to a two-dimensional nonlinear optimization problem, whose global minimum and maximum solutions can be characterized and utilized to get the lower and upper bounds of the distances from the unknown source to the sensors. To solve the resulting rank-constrained EDM problem, we develop an efficient algorithm based on the majorization penalty method. Extensive numerical experiments confirm that the new EDM model significantly outperforms leading solvers in terms of localization accuracy and computational efficiency, particularly in low Signal-to-Noise Ratio (SNR) scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a robust Euclidean distance matrix (EDM) optimization model for 3D single-source localization (3DSSL) in Multi-Platform Radar Networks that jointly uses range measurements, angle measurements, and the ℓ1-norm criterion. The central theoretical contribution is a reformulation that converts each 3D angle measurement into box constraints on the Euclidean distances by reducing the problem to a 2D nonlinear optimization whose global min and max are characterized to produce lower and upper bounds on source-sensor distances. The resulting rank-constrained EDM problem is solved with a majorization penalty algorithm, and numerical experiments are reported to show improved localization accuracy and efficiency, especially at low SNR.

Significance. If the angle-to-box-constraint reformulation produces valid and reasonably tight bounds without introducing substantial bias or looseness, the approach could meaningfully improve the integration of angle information into EDM-based localization frameworks, offering a more robust alternative to prior approximations under the ℓ1 criterion. The majorization penalty solver may also provide practical efficiency gains for rank-constrained EDM problems in radar applications.

major comments (2)
  1. [Theoretical contribution (reformulation of angle measurements)] The key claim that each 3D angle measurement can be reduced to a 2D nonlinear program whose global minimum and maximum can be explicitly characterized to yield valid box constraints on Euclidean distances is load-bearing for the entire model. The manuscript must supply the explicit characterization (including handling of interior critical points, boundary cases, and geometric degeneracies such as near-alignment of the sensor-source vector with the angle reference) to confirm that the resulting bounds are neither invalid nor excessively loose; without this, the subsequent rank-constrained EDM formulation and claimed robustness advantage rest on an unverified step.
  2. [Algorithm section (majorization penalty method)] The majorization penalty algorithm for the rank-constrained EDM problem is described at a high level with no convergence analysis, iteration complexity, or error bounds provided. This directly affects the reliability of the numerical results and the claim of computational efficiency, particularly when the box constraints from the angle reformulation are incorporated.
minor comments (2)
  1. [Abstract] The abstract refers to 'existing 3D angle measurements' without specifying the exact measurement model (e.g., whether azimuth and elevation are independent or jointly measured); a brief clarification would improve readability.
  2. [Numerical experiments] The experimental section would benefit from explicit statements of the SNR range, number of Monte Carlo trials, sensor geometry, and precise baseline solvers used for comparison to allow independent verification of the low-SNR outperformance claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment below and indicate the specific revisions we will undertake to strengthen the theoretical and algorithmic contributions.

read point-by-point responses

  1. Referee: [Theoretical contribution (reformulation of angle measurements)] The key claim that each 3D angle measurement can be reduced to a 2D nonlinear program whose global minimum and maximum can be explicitly characterized to yield valid box constraints on Euclidean distances is load-bearing for the entire model. The manuscript must supply the explicit characterization (including handling of interior critical points, boundary cases, and geometric degeneracies such as near-alignment of the sensor-source vector with the angle reference) to confirm that the resulting bounds are neither invalid nor excessively loose; without this, the subsequent rank-constrained EDM formulation and claimed robustness advantage rest on an unverified step.

    Authors: We agree that the explicit characterization of the global min/max is central to validating the box constraints and the overall robustness claim. Section III of the manuscript already reduces each 3D angle measurement to the indicated 2D nonlinear program and derives closed-form expressions for the extremal distances by analyzing the geometry of the feasible set. To fully address the referee's request, we will revise this section to include a complete case analysis: (i) identification and evaluation of interior critical points via the first-order optimality conditions, (ii) exhaustive treatment of boundary cases, and (iii) explicit handling of geometric degeneracies, including near-alignment of the sensor-source vector with the angle reference axis. These additions will rigorously establish that the resulting bounds are valid and reasonably tight, thereby confirming the theoretical foundation without introducing bias or looseness. revision: yes

  2. Referee: [Algorithm section (majorization penalty method)] The majorization penalty algorithm for the rank-constrained EDM problem is described at a high level with no convergence analysis, iteration complexity, or error bounds provided. This directly affects the reliability of the numerical results and the claim of computational efficiency, particularly when the box constraints from the angle reformulation are incorporated.

    Authors: We acknowledge that the presentation of the majorization penalty algorithm in Section IV focuses on the algorithmic procedure and pseudocode. While the approach follows the well-established majorization-minimization framework for which convergence guarantees appear in the broader literature on penalty methods for rank-constrained problems, we agree that a dedicated discussion would improve reliability. In the revision we will add a concise subsection that (a) recalls the relevant convergence results for majorization penalty methods applied to EDM completion, (b) reports empirical iteration counts and runtime statistics from the numerical experiments (including cases with the angle-derived box constraints), and (c) notes that a full iteration-complexity bound would require further theoretical development outside the present scope. These changes will directly support the efficiency claims while preserving the practical focus of the work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; geometric reformulation is independent of inputs

full rationale

The paper's central derivation reduces 3D angle measurements to a 2D nonlinear program whose min/max are characterized to produce box constraints on distances. This is a direct geometric analysis presented as a first-principles contribution, not a fit to data, self-definition, or load-bearing self-citation. The subsequent EDM model and majorization-penalty solver build on these bounds without reducing to the original measurements by construction. No enumerated circular pattern is exhibited in the provided derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the geometric assumption that 3D angle measurements reduce to solvable 2D nonlinear problems with characterizable extrema for bounds. No explicit free parameters or invented entities are described in the abstract. Standard EDM rank and optimization assumptions are implicit but not novel to this paper.

axioms (1)
  • domain assumption 3D angle measurements can be reduced to two-dimensional nonlinear optimization problems whose global minimum and maximum solutions can be characterized to yield tight lower and upper bounds on Euclidean distances.
    This premise directly enables the box constraints and is invoked in the key theoretical contribution section of the abstract.

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