Robust Resource Allocation for Over-the-Air Computation Networks with Fluid Antenna Array

Ghosheh Abed Hodtani; Jean-Yves Chouinard; Leslie A Rusch; Ming Zeng; Mohsen Ahmadzadeh; Saeid Pakravan

REVIEW 2 major objections 5 minor 23 references

Robust Resource Allocation for Over-the-Air Computation Networks with Fluid Antenna Array

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that jointly optimizing transmit coefficients, receive beamforming, and fluid antenna positions under angle-of-arrival uncertainty lowers the mean squared error of over-the-air computation compared with fixed-position…

desk verdict Combines fluid-antenna AirComp with robust design, but the MSE objective assumes diagonal channel-error covariance that its own scalar angle-error model contradicts. read the letter →

arxiv 2504.16221 v1 pith:ROMSX72E submitted 2025-04-22 eess.SP

Saeid Pakravan , Mohsen Ahmadzadeh , Ming Zeng , Ghosheh Abed Hodtani , Jean-Yves Chouinard , Leslie A Rusch This is my paper · ORCID

classification eess.SP

keywords over-the-aircomputationfluidantennaarrayrobustresourceallocationmeansquarederrorblockcoordinatedescentchanneluncertaintybeamformingdesignpositionoptimization

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

Over-the-air computation lets a base station recover the average of many users' signals from the superposition of simultaneous transmissions, but accuracy depends on channel knowledge. This paper asks whether a fluid antenna array—whose elements slide along a line to chosen positions—can keep the aggregation accurate when arrival angles are known only approximately. It claims that jointly choosing transmit gains, the receive beamformer, and the antenna positions lowers the mean squared error compared with fixed-position antennas or designs that ignore angle uncertainty. An iterative block-coordinate algorithm alternates closed-form transmit and receive updates with a quasi-Newton search over antenna locations; simulations show the gain widens as uncertainty grows.

What carries the argument

The engine is a first-order Taylor model of the line-of-sight channel: each user's channel is written as an estimated steering vector plus an error term proportional to the angle error and to the antenna position, so the robust MSE acquires a sum over antennas of $|b_k|^2 \psi_k \theta_{k,0}^2 |m_n x_n|^2$. That quadratic form makes the uncertainty penalty depend on where the fluid antennas sit. The optimization separates into three blocks: a closed-form soft-limiter for each transmit coefficient $b_k$, a closed-form MMSE beamforming vector $m = R^{-1} \sum_k \bar{h}_k b_k$, and a BFGS quasi-Newton step with a log-barrier for the spacing constraints on the antenna position vector $x$. Iterating the three blocks until the variables stop changing yields the proposed design.

What would settle it

Compute the exact MSE of the proposed design under the true uniform angle-error model—without the first-order Taylor expansion and without dropping cross-antenna terms—for a small system such as $N=2$, $K=1$, $\theta_0=1$ rad, and compare it with the diagonal approximation used in the paper. If the exact MSE differs substantially, or if the optimal antenna positions move away from the BFGS solution, the robustness conclusion depends on the approximation rather than on the physical setup.

Watch

Extended reading notes

Core claim

The central claim is that antenna position is a usable resource for robust over-the-air computation. Under bounded angle-of-arrival error, the mean squared error of the aggregated average is minimized by a transmit coefficient that balances signal alignment against an uncertainty penalty, a receive beamformer of regularized least-squares form, and an antenna placement found by the BFGS quasi-Newton method. The paper derives a robust MSE expression in which each antenna contributes an error term proportional to its squared position and to the angle-uncertainty variance, then minimizes it by block coordinate descent. In simulation, the proposed design keeps MSE below both a fixed-position-antenna benchmark and a benchmark optimized without including uncertainty, and the gap widens as the uncertainty level rises from 0 to 1 radian.

Load-bearing premise

The robustness result rests on treating the channel error at each antenna as independent, even though the underlying angle error is a single scalar that shifts every antenna at once, so the true errors are correlated across antennas.

Editorial extensions

If this is right

For the simulated geometry, the robust fluid-antenna design achieves lower AirComp MSE than fixed-position antennas at both 0 dB and 10 dB SNR across the tested uncertainty range.
The gap over the non-robust benchmark widens as angle uncertainty grows, so explicitly modeling CSI error is most valuable when the channel estimates are poor.
Increasing the number of fluid antennas from 8 to 12 lowers MSE for all schemes, but optimized positions keep an advantage over uniform fixed positions.
Enlarging the available aperture L improves MSE and widens the gap over fixed-position antennas, showing that positioning flexibility drives the gain.
With very small angle error, the non-robust design is nearly as good; the robust design matters most in the regime that motivates modeling error.

Reading between the lines

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

The paper's MSE model treats the angle error as producing independent errors per antenna, but a single shared angle actually correlates the errors across antennas. Re-deriving the updates with the full rank-one covariance could change the optimal antenna placement.
Because the position update is a local search, the reported gains reflect the chosen uniform initialization; a multi-start search would test whether the advantage is robust to starting positions.
The same block-coordinate template extends to other error statistics, such as Gaussian angle jitter or non-uniform bounded errors, by substituting the appropriate variance in the transmit and beamforming updates.
An experimental setup with movable antennas and controlled angle jitter could check the first-order Taylor model; if the model breaks near 1 radian of uncertainty, the simulated gain may not match reality.

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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

Read the paper over the weekend. The core idea is sensible: optimize FA positions, transmit coefficients, and receive beamforming jointly to minimize AirComp MSE under angle-of-arrival uncertainty. That combination isn't in the cited prior work, and the BCD decomposition into three subproblems is a reasonable way to attack the nonconvex problem. The paper is clearly written and the updates in (13) and (15) are algebraically consistent with the stated objective.

The problem is the objective itself. The channel model in (5) has a single scalar angle error for each user, so the error vector is a deterministic vector times a scalar random variable. That gives a rank-one error covariance. The paper instead asserts a diagonal covariance with entries ψ_k x_n^2 θ_{k,0}^2. That's not what the model implies. The correct covariance term would contain coherent cross-antenna phases, not independent per-antenna variances. The b and m updates, the FA position optimization, and the claimed robustness gains all rest on that incorrect covariance. On top of that, the Taylor approximation in (4) keeps only the linear term, yet simulations go to θ_0 = 1 rad with x_n up to 8λ; the phase argument changes by several radians, so the truncation isn't controlled.

The simulations don't rescue this. The curves evaluate the same approximate MSE that Algorithm 1 minimizes, so the observed improvement over the non-robust benchmark is partly expected and doesn't independently test the robust design against the actual channel model. There is no simulation that plugs the true rank-one covariance into the performance evaluation, which is the natural check.

So my verdict matches the reader's: reject. But it's not a hopeless paper. If the authors either explicitly assume independent per-antenna phase errors (a different physical model) or derive the correct rank-one covariance and redo the updates, the core idea could still work. A serious referee would be able to sort that out.

I'd bring it to reading group as a case study in how approximation choices can silently change the problem, but I wouldn't cite it in its current form.

Recommendation: send to peer review, because the flaw is substantive and the fix is nontrivial but feasible.

Referee Report

2 major / 5 minor

Summary. The paper considers an over-the-air computation (AirComp) system in which an access point with N fluid antennas receives from K single-antenna users under line-of-sight angle-of-arrival uncertainty. It formulates a robust design problem that jointly optimizes the user transmit coefficients b, the receive beamforming vector m, and the antenna position vector x to minimize a mean-squared-error objective that includes a CSI-error term. The proposed solution is block coordinate descent: closed-form updates for b and m, and a BFGS-based update for x. Simulation results compare the design with fixed-position antennas and a non-robust design, and the paper concludes that the proposed transceiver and antenna-position design improves MSE and resilience to channel uncertainty.

Significance. If the robust MSE model were correctly derived, the paper would be a useful contribution to the emerging FA-AirComp literature: the problem is clearly stated, the BCD decomposition is natural, and the closed-form updates would be computationally attractive. The paper also addresses a relevant and timely problem, namely antenna-position optimization under angle uncertainty. However, the central derivation is not sound as written: the channel-error covariance used to build the objective contradicts the paper's own channel model in Eq. (5), and the first-order Taylor approximation is used in a regime where it is not controlled. Because the numerical studies evaluate the same approximate objective that is optimized, they do not independently validate the robustness claims. For these reasons the contribution, in its current form, cannot support the conclusions.

major comments (2)

[III, Eqs. (13) and (15)] The closed-form updates inherit the covariance error. In Eq. (15), R is written as σ_z^2 I + Σ_k |b_k|^2 (h̄_k h̄_k^H + ψ_k x x^H θ_{k,0}^2 I), which is not even consistent with the paper's own diagonal covariance: the correct perturbation term would be diag(Σ_k |b_k|^2 ψ_k θ_{k,0}^2 x_n^2). More importantly, if the rank-one covariance implied by Eq. (5) were used, the m-subproblem would contain a rank-one correction, and Eq. (13) would involve |m^H(h̄_k ⊙ q_k)|^2 rather than Σ_n |m_n x_n|^2. The stated updates are therefore tied to the incorrect objective in Eq. (9).
[IV, Figs. 2-4] The numerical evaluation is circular with respect to the robustness claim. The figures plot the same MSE expression from Eq. (9) that Algorithm 1 minimizes, rather than an independent Monte Carlo evaluation of the MSE under the angle-error model in Eqs. (3)-(5). Consequently, the observed improvement over the non-robust and FPA benchmarks mainly reflects the fact that minimizing an objective lowers that objective; no evidence is provided that the optimized design is robust under the model that generated the channel errors.

minor comments (5)

[II, Eq. (4)] The derivative of exp(j(2π/λ)x_n cos θ) with respect to θ is -j(2π/λ)x_n sin θ times the exponential, so the plus sign in q(θ̄_k) appears to have the wrong sign; the sign should be corrected.
[III, Eq. (13)] In Eq. (13), b_k^* is written as a real magnitude, but b_k is complex and must align the phase of m^H h̄_k b_k with the target value 1; the solution should explicitly include the conjugate phase of m^H h̄_k.
[III, Algorithm 1] Algorithm 1 updates m before b, while Section III presents Subproblem 1 (b) before Subproblem 2 (m); the order should be aligned or the discrepancy should be explained.
[III, Eq. (15)] The definition of R in Eq. (15) is notationally ambiguous because ψ_k x x^H θ_{k,0}^2 I mixes a vector outer product with an identity-scaled term; under the paper's own diagonal covariance assumption the perturbation term should be diag(ψ_k θ_{k,0}^2 x_n^2).
[IV] Simulation parameters such as the BFGS barrier schedule, line-search tolerance, and initialization are not specified, so the numerical results are not fully reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in the derivation chain; the optimization and evaluation are aligned but not circularly defined.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs by construction. The MSE objective in Eq. (9) is built from the linearized channel model in Eqs. (4)-(7) and the stated diagonal covariance assumption; the transmit coefficients (13), receive beamformer (15), and antenna positions (BFGS update on Eq. (16)) are all derived by first-order optimality conditions or a standard quasi-Newton method on that objective. No parameter is fitted to a subset of data and then called a prediction, and no external benchmark is claimed that would require the optimized objective to be independent of the evaluation metric. The simulations evaluate the same MSE expression that Algorithm 1 minimizes, but this is a standard verification of an optimization procedure, not a circular reduction: minimizing an objective and then showing the minimized objective is lower than non-optimized benchmarks is the intended behavior. The only citations to co-authors' prior work ([15], and [21] as methodological background) are not load-bearing for the central derivation; no uniqueness theorem or fitted ansatz is imported from a self-citation. The internal inconsistency between the scalar angle-error model of Eq. (3) and the diagonal error covariance used in Eq. (9) is a technical correctness concern, not a circularity of the derivation chain, and is therefore outside the scope of this pass.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No empirical data or fitting is used; all numerical parameters are simulation inputs. The central model relies on LoS steering, uniform scalar angle uncertainty, a first-order Taylor expansion, and the diagonal-covariance approximation, the last of which contradicts Eq. (5).

assumptions (5)

domain assumption Line-of-sight channel model with far-field steering vector (Eq. 1).
The channel for each user is a pure LoS geometric vector; no multipath, no amplitude uncertainty, and the path loss l_k^{-α} is known exactly.
domain assumption Angle uncertainty ∆θ_k is independently uniform over [-θ_{k,0}, θ_{k,0}] for each user (Eq. 3).
This distribution is chosen for tractability and is not grounded in measured estimation-error statistics.
ad hoc to paper First-order Taylor expansion (Eq. 4) approximates exp(j 2π/λ x_n cos(θbar_k + ∆θ_k)) accurately for all simulated θ0 up to 1 rad.
The paper discards higher-order and periodic terms without bounding the truncation error or comparing with the exact exponential.
ad hoc to paper Channel error vector has diagonal covariance with entries ψ_k x_n^2 θ_{k,0}^2 (text after Eq. 7 and Eq. 9).
This contradicts Eq. 5, where a single scalar ∆θ_k multiplies a deterministic vector, giving a rank-one covariance. The entire MSE objective and both closed-form updates depend on this diagonal assumption.
standard math BFGS with a log-barrier finds an adequate local optimum of the nonconvex antenna-position subproblem, and the alternating loop converges to a stationary point (Algorithm 2).
The barrier weight is reduced each iteration and the position subproblem is only solved locally, so the global convergence claim and representativeness of the result are not established.

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Cite this review

Pith. "Pith review of Robust Resource Allocation for Over-the-Air Computation Networks with Fluid Antenna Array." pith.science (2026). https://pith.science/paper/ROMSX72E

@misc{pith2026250416221,
  author       = {Pith},
  title        = {Pith review of: Robust Resource Allocation for Over-the-Air Computation Networks with Fluid Antenna Array},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ROMSX72E}},
  note         = {Machine review of arXiv:2504.16221}
}

read the original abstract

Fluid antenna (FA) array is envisioned as a promising technology for next-generation communication systems, owing to its ability to dynamically control the antenna locations. In this paper, we apply FA array to boost the performance of over-the-air computation networks. Given that channel uncertainty will impact negatively not only the beamforming design but also the antenna location optimization, robust resource allocation is performed to minimize the mean squared error of transmitted messages. Block coordinate descent is adopted to decompose the formulated non-convex problem into three subproblems, which are iteratively solved until convergence. Numerical results show the benefits of FA array and the necessity of robust resource allocation under channel uncertainty.

Figures

Figures reproduced from arXiv: 2504.16221 by the authors.

**Figure 1.** An illustration of the system model. A. Subproblem 1: Transmit power allocation First, we optimize b under given other parameters. The associated optimization problem with respect to b can be formulated as follows: min b X K k=1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Computation of MSE versus the uncertainty level under different SNR [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Computation of MSE versus the uncertainty level under different [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Computation of MSE versus the uncertainty level under different [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Over-the-Air computing for wireless data aggregation in massive IoT,

G. Zhu et al., “Over-the-Air computing for wireless data aggregation in massive IoT,” IEEE Wireless Commun , vol. 28, no. 4, pp. 57–65, Aug. 2021

work page 2021

[2] [2]

A survey on Over-the-Air computation,

A. Sahin and et al., “A survey on Over-the-Air computation,” IEEE Commun. Surveys Tuts , vol. 25, no. 3, pp. 1877–1908, Apr. 2023

work page 1908

[3] [3]

Integrated sensing, communication, and computation Over-the-Air: MIMO beamforming design,

X. Li and et al., “Integrated sensing, communication, and computation Over-the-Air: MIMO beamforming design,” IEEE Trans. Wireless Com- mun., vol. 22, no. 8, pp. 5383–5398, Jan. 2023

work page 2023

[4] [4]

Multi-level Over-the-Air aggregation of mobile edge computing over D2D wireless networks,

F. Wang and et al., “Multi-level Over-the-Air aggregation of mobile edge computing over D2D wireless networks,” IEEE Trans. Wireless Commun., vol. 21, no. 10, pp. 8337–8353, Apr. 2022

work page 2022

[5] [5]

Over-the-Air split machine learning in wireless MIMO networks,

Y . Yang and et al., “Over-the-Air split machine learning in wireless MIMO networks,” IEEE J. Sel. Areas Commun. , vol. 41, no. 4, pp. 1007–1022, Feb. 2023

work page 2023

[6] [6]

Federated learning via Over-the-Air computation,

K. Yang and et al., “Federated learning via Over-the-Air computation,” IEEE Trans. Wireless Commun., vol. 19, no. 3, pp. 2022–2035, Jan. 2020

work page 2022

[7] [7]

Beamforming design based on two-stage stochastic optimization for RIS-assisted Over-the-Air computation systems,

X. Zhai and et al., “Beamforming design based on two-stage stochastic optimization for RIS-assisted Over-the-Air computation systems,” IEEE Internet of Things Journal , vol. 9, no. 7, pp. 5474–5488, Aug. 2022

work page 2022

[8] [8]

Over-the-Air computation via reconfigurable intel- ligent surface,

W. Fang and et al., “Over-the-Air computation via reconfigurable intel- ligent surface,” IEEE Trans. Commun. , vol. 69, no. 12, pp. 8612– 8626, Sep. 2021

work page 2021