Egoistic MDS-based Rigid Body Localization

David Gonz\'alez G.; Giuseppe Abreu; Niclas F\"uhrling; Osvaldo Gonsa

REVIEW 3 major objections 5 minor 35 references

Egoistic MDS-based Rigid Body Localization

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A rigid body can estimate another body's relative translation and rotation using only cross-body distance measurements, without knowing the target's shape.

desk verdict The egoistic setup is genuinely new, but the Nyström completion on a raw EDM is wrong, so the paper's pipeline collapses at step one. read the letter →

arxiv 2501.12417 v1 pith:EMWXEATH submitted 2025-01-20 cs.RO eess.SP

Niclas Führling , Giuseppe Abreu , David González G. , Osvaldo Gonsa This is my paper · ORCID

classification cs.ROeess.SP

keywords rigidbodylocalizationegoisticmultidimensionalscalingNyströmapproximationtranslationestimationrotationautonomousdrivingconvexoptimization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that one rigid body, such as a truck, can estimate the relative translation and rotation of another rigid body, such as a car, using only measured distances between sensors on the two bodies, without knowing the target body's shape and even when the two bodies have different numbers of sensors. If true, this removes a major obstacle to using radio-based rigid-body localization in autonomous driving, because a vehicle no longer needs a pre-stored model of every possible target. The paper proposes a pipeline built on multidimensional scaling (MDS), in which the double-centering operator converts squared cross-body distances into inner products, a Nyström step completes the unknown intra-body distance matrix, and convex programs recover the translation vector and rotation matrix. Simulation results for a truck-car scenario with 12 and 10 landmarks show that the translation estimate stays close to a genie-aided version that knows the target shape.

What carries the argument

The central object is the Schönberg double-centering operator J = I - (1/N)11^T, which maps a squared distance matrix to a centered inner-product matrix and underlies equations (10), (17), and (18). Around it, the paper builds a chain: Nyström approximation in equation (14) completes the unknown intra-distance matrix D2 from the known D1 and measured D12; classical MDS in equation (16) reconstructs the combined sensor positions; a Procrustes alignment in equation (19) brings the reconstruction into body 1's frame; a quadratic program in equation (23) estimates the translation vector; and an eigendecomposition with permutation correction in equation (28) estimates the rotation matrix.

What would settle it

Take two rigid bodies with known shapes and choose N1 < N2, so the rank condition rank(D1) ≥ rank(D2) fails; measure noiseless cross-body distances and run the method. If the Nyström-completed D2 differs materially from the true D2, or if equations (23) and (28) produce t and Q far from the ground truth, the central claim fails.

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Extended reading notes

Core claim

The paper claims that the egoistic rigid-body localization problem—estimating the relative pose Q and t of a second body with unknown shape C2 from the cross-distance matrix D12—can be solved without knowing C2. The translation estimate is obtained by first reconstructing the target's intra-distance matrix D2 via Nyström completion, applying MDS to get a sensor-position estimate, aligning it to body 1's frame via Procrustes, and then solving the quadratic program in equation (23) for t. The rotation estimate is obtained from the double-centered cross-distance Gram matrix, whose product with its transpose equals QΛQ^T, and equation (28) searches over the six eigenvalue permutations of Λ to avoid eigenvector swaps.

Load-bearing premise

The whole pipeline stands on the assumption that the Nyström completion in equation (14) recovers the target's unknown intra-body distance matrix from the measured cross-body distances and the known self-distances; if that completion is wrong, the reconstructed sensor positions, translation, and rotation are all wrong.

Editorial extensions

If this is right

Autonomous vehicles can estimate the relative pose of nearby vehicles using only inter-vehicle range measurements, without requiring the target vehicle to broadcast its shape or sensor configuration.
The method works for bodies with different numbers of landmarks (N1 ≠ N2), so a vehicle with 12 sensors can localize a target with 10 sensors, a case that prior same-shape methods could not handle.
Translation estimation via equation (23) and rotation estimation via equation (28) are formulated as convex optimizations solvable by standard tools, making the approach computationally feasible for real-time V2X use.
In the low-to-moderate ranging error regime typical of automotive sensing, the egoistic translation estimate remains close to the genie-aided estimate that knows the target shape.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Nyström completion in equation (14) is the fragile link: if it fails, the whole pose estimate fails. Replacing it with a rank-constrained Euclidean distance matrix completion method would make the pipeline robust when the target has more landmarks than the ego body, an extension the paper does not explore.
The same egoistic geometry could be applied to non-vehicular rigid bodies, such as robot arms, drones, or handheld devices, whenever pairwise ranging is available, and to tracking by feeding each new frame's t and Q into a filter; both directions are left implicit.
The paper evaluates translation accuracy numerically but does not report a direct RMSE test of the rotation estimate against ground-truth yaw, pitch, and roll angles, including near-spherical bodies where eigenvector swaps occur; such a test would be the natural next experiment.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know about this one. First, the problem it targets is real and, as far as the citations go, unaddressed: egoistic rigid body localization with unknown target shape and N1 ≠ N2. That alone is worth a careful look. Second, the central completion step, Eq. (14), is mathematically invalid. It applies Nyström to the raw distance matrix D, treating it like a PSD kernel, but an EDM is conditionally negative definite. The completion should act on the double-centered Gram matrix K = -1/2 J D^2 J; Eq. (14) is not equivalent. The stress-test note gives a clean 1D noiseless counterexample: the true D2 entry is 4, Eq. (14) gives 239.5. Once D2 is wrong, the MDS reconstruction of the target and everything downstream (translation and rotation) is built on garbage. So the simulations in Section IV, which use this pipeline, do not validate the method.

The reader also flagged Eq. (23) as invariant under t. I don't think that holds up—the stress-test note's counterexample is convincing, and a quick look at the quadratic form suggests the double-centering does not annihilate all t-dependence. But it is a side issue: Eq. (14) is enough to sink the paper.

What the paper does well: it is clearly written, the problem statement is sharp, and the authors correctly identify a limitation of prior work (they even cite a flaw in [22]). The genie-aided comparison in Figure 3 is a reasonable idea, and the restriction to moderate-to-low ranging errors is honestly stated.

The soft spots beyond Eq. (14): footnote 8 conditions on rank(D1) ≥ rank(D2) without justification, and Eq. (28) for rotation relies on the same corrupted reconstruction. There is no proof for the Nyström step—just an "yields."

Who gets value from this: a researcher looking for a concrete failure mode in EDM-based localization, or someone who wants to fix the completion step and build on the egoistic problem. It is worth a serious referee because the problem statement is a real gap, and the fix might be straightforward. But as written, it is a reject. I would tell the authors to redo the completion using properly double-centered Gram matrices and re-run the simulations. If that works, the paper could become a useful contribution.

Referee Report

3 major / 5 minor

Summary. The paper proposes an anchorless, egoistic rigid-body localization method in which a rigid body (body 1) estimates the relative translation t and rotation Q of a second rigid body using only cross-body distance measurements, without prior knowledge of the second body's shape. The pipeline in Section III consists of Nyström completion of the unknown intra-body distance matrix D2 in Eq. (14), an MDS embedding in Eqs. (15)-(16), a Procrustes alignment to body 1's frame in Eqs. (19)-(21), an optimization for t in Eq. (23), and a spectral/optimization procedure for Q in Eqs. (24)-(28). The paper reports Monte Carlo RMSE simulations for translation estimation compared with a state-of-the-art method and a genie-aided variant.

Significance. If the central claims were correct, the method would be a useful contribution to V2X perception because it removes the target-shape knowledge requirement common in earlier RBL work. The paper states the problem clearly, and the simulation study is a reasonable first evaluation. However, two load-bearing mathematical steps are not valid as presented: the Nyström completion in Eq. (14) is applied to a matrix that does not satisfy the assumptions of Nyström theory, and the rotation recovery in Eqs. (24) and (28) is subject to a fundamental unidentifiability when the target shape is unknown. The paper supplies no proofs for the completion step, and the simulation results cannot compensate for the invalid reconstruction because the same corrupted matrix feeds the whole estimation pipeline.

major comments (3)

[Section III-A, Eq. (14)] Equation (14) applies the Nyström approximation directly to the raw Euclidean distance matrix D. Nyström completion is defined for positive semidefinite kernel matrices; a raw EDM is only conditionally negative definite, so the block relation underlying Eq. (14) does not hold. The correct completion should be applied to the doubly centered Gram matrix K = -1/2 J D^2 J, with K22 approximated by K21 K11^{-1} K12, and squared distances then recovered from K22; Eq. (14) is not algebraically equivalent to that procedure. A noiseless 1D example makes the failure concrete: with C1 = [-0.5, 0.5] and C2 = [10, 12], the true D2 off-diagonal entry is 4, whereas Eq. (14) gives 239.5. Since the completed matrix enters Eqs. (15), (16), and (20)-(21), the reconstructed target points and therefore both the translation estimate (23) and the rotation estimate (28) are built on an invalid reconstruction. Footnote 8's rank condition (rank(D1) >= rank(D2)) is not sufficient; the example satisfies it and still fails catastrophically.
[Section III-B, Eqs. (24) and (28)] Even if the MDS reconstruction of the target point set were correct, the rotation Q is not identifiable from the assumed measurements when C2 is unknown. Replacing (Q, C2) by (Q R, R^T C2) for any orthogonal R leaves S2 = Q C2 + t unchanged and therefore leaves all measured distances unchanged. The quantities in Eqs. (24) and (27) depend only on Q C2 C2^T Q^T; their eigenvectors are Q times the eigenvectors of C2 C2^T, and without C2 or an additional orientation prior, Q cannot be separated from that product. The permutation ambiguity mentioned in the text for spherical shapes is only one component of this fundamental ambiguity. The paper's claim that the target orientation can be estimated without knowledge of the target shape is therefore not supported by the stated model.
[Section III-A, Eq. (23)] The translation estimator in Eq. (23) is described as a quadratic program, but the objective is a fourth-degree polynomial in t: \hS is affine in t and the squared Frobenius norm turns the quadratic dependence into a quartic one. The text gives no feasible set, no initialization strategy, and no global-optimality argument; citing generic gradient descent or interior-point methods is insufficient for what is in general a nonconvex quartic problem. More importantly, since the \hD2 contained in the objective comes from the invalid Eq. (14), the minimizer of Eq. (23) has no demonstrated relationship to the true translation t. I did not reproduce the separate claim that the objective is invariant in t, but the invalid completion is already decisive against the translation-estimation claim as presented.

minor comments (5)

[Section III-A, Eq. (19)] The notation t ⊗ 1_{N1}^⊤ in Eq. (19) is undefined; the intended object appears to be the outer product t 1_{N1}^⊤, not a Kronecker product.
[Section III-B, Eq. (28)] The optimization over Q in Eq. (28) must enforce the orthogonality constraint Q^⊤ Q = I, but the text does not state how this constraint is handled. The paper mentions CVX, yet Eq. (23) is not convex and Eq. (28) is not a standard convex program; implementation details are needed.
[Section IV, Figures 3 and 4] The genie-aided baseline is not a clean upper bound for the effect of the Nyström step, because it still uses the completed \hD2 from Eq. (14) in the MDS and translation-estimation pipeline; only Q is supplied externally. A genie that supplies the true D2 or C2 would be needed to isolate the invalidity of Eq. (14).
[Section III-A, Eq. (16)] The MDS embedding in Eq. (16) does not specify how negative eigenvalues of ̅D are handled under noise. Since ̅D may be indefinite, V Λ^{1/2} is not always real; a truncation or projection rule should be stated.
[Table I] The matrices C1 and C2 contain typographical formatting errors (e.g., "1 .5", "0 .5") that should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the translation and rotation estimators are built from measured cross-body distances and a known self-shape, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is not circular. The unknown target intra-distance matrix D2 is estimated in Eq. (14) directly from the measured cross-distance block D12 and the known self-distance block D1 via a Nyström-style completion; the resulting D̂2 feeds the MDS embedding in Eqs. (15)-(16), and the translation and rotation objectives in Eqs. (23) and (28) are evaluated against independently generated ground-truth t and Q in simulation. No parameter is fitted to the output quantities and then reported as a prediction. The only self-citation that could raise a question is footnote 6, which refers to the authors' journal version [24] for a proof that the prior work [22, Subsec. 3.2] is incorrect; that citation is not load-bearing for the present derivation. The Nyström completion in Eq. (14) may be mathematically invalid because it is applied to a raw Euclidean distance matrix rather than a positive semidefinite kernel, and the resulting D̂2 may be corrupted, but a mathematical error is not circular reasoning: the erroneous estimate is not obtained by assuming the t and Q the paper claims to predict.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard MDS and Procrustes tools, plus two unproved and, as written, false assumptions about Nyström completion of distance matrices and identifiability of t from eq. (23). No new physical entities are introduced.

assumptions (4)

domain assumption The landmark points of each rigid body are expressed relative to the body's geometric center, so the columns of C1 and C2 sum to zero.
Used in equations (10) and (25) to justify that J S_i^T = S_i^T and S_i J = S_i, which is needed for the double-centering identities. This is stated in Section II-A.
domain assumption The squared-distance noise model in eq. (4) has mean σ^2 and variance 4d_{n,m}^2 σ^2 + 2σ^4, which follows from the Gaussian distance-noise assumption in eq. (3).
The noise model is used to justify the use of squared distances; the variance expression is taken from [23].
ad hoc to paper Nyström approximation applied directly to the raw Euclidean distance matrix D, as in eq. (14), yields a valid reconstruction of the target intra-distance matrix D2.
Standard Nyström theory applies to positive semidefinite kernels; a raw EDM is not PSD. The paper offers only a rank condition footnote (Section III-A) and no proof.
ad hoc to paper The quadratic program in eq. (23) has a unique minimizer that identifies the translation vector t.
In fact, the objective of eq. (23) is invariant under t because all t-dependent terms in \hat S^T \hat S are annihilated by the double-centering matrices J in eq. (23). No alternate identification of t is stated.

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Cite this review

Pith. "Pith review of Egoistic MDS-based Rigid Body Localization." pith.science (2026). https://pith.science/paper/EMWXEATH

@misc{pith2026250112417,
  author       = {Pith},
  title        = {Pith review of: Egoistic MDS-based Rigid Body Localization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EMWXEATH}},
  note         = {Machine review of arXiv:2501.12417}
}

read the original abstract

We consider a novel anchorless rigid body localization (RBL) suitable for application in autonomous driving (AD), in so far as the algorithm enables a rigid body to egoistically detect the location (relative translation) and orientation (relative rotation) of another body, without knowledge of the shape of the latter, based only on a set of measurements of the distances between sensors of one vehicle to the other. A key point of the proposed method is that the translation vector between the two-bodies is modeled using the double-centering operator from multidimensional scaling (MDS) theory, enabling the method to be used between rigid bodies regardless of their shapes, in contrast to conventional approaches which require both bodies to have the same shape. Simulation results illustrate the good performance of the proposed technique in terms of root mean square error (RMSE) of the estimates in different setups.

Figures

Figures reproduced from arXiv: 2501.12417 by the authors.

**Figure 2.** Illustration of two-body egoistic RBL scenario. Each rigid body has [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** RMSE of the translation estimate of the genie-aided (GA) proposed [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: RMSE of the translation estimate of the proposed method compared [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [22]

Towards multi-rigid body localization,

A. Pizzo, S. P. Chepuri, and G. Leus, “Towards multi-rigid body localization,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , 2016

work page 2016

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A. Yassin et. al , “Recent advances in indoor localization: A survey on theoretical approaches and applications,” IEEE Communications Surveys & Tutorials , vol. 19, no. 2, 2017

work page 2017

[3] [2]

A survey of fingerprint-based outdoor localiza- tion,

Q. D. V o and P. De, “A survey of fingerprint-based outdoor localiza- tion,” IEEE Communications Surveys & Tutorials , vol. 18, no. 1, 2016

work page 2016

[4] [3]

Robust received signal strength indicator (RSSI)- based multitarget localization via gaussian process regression,

N. F ¨uhrling et. al , “Robust received signal strength indicator (RSSI)- based multitarget localization via gaussian process regression,” IEEE J. Indoor Seamless Position. Navig. , vol. 1, 2023

work page 2023

[5] [4]

AOA localization for vehicle-tracking systems using a dual-band sensor array,

M. Al-Sadoon et. al , “AOA localization for vehicle-tracking systems using a dual-band sensor array,” IEEE Trans. Antennas Propag., vol. 68, no. 8, 2020

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G. Zeng et. al , “Global and asymptotically efficient localization from range measurements,” IEEE Trans. on Signal Processing , vol. 70, 2022

work page 2022

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D. Burghal et. al, “A comprehensive survey of machine learning based localization with wireless signals,” 2020

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work page 2021