{"id":"76c9da4f-d6f5-4787-8dac-fa23366b0b11","arxiv_id":"2504.16221","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A joint optimization of transmit powers, receive beamforming, and fluid-antenna positions is proposed to minimize mean-squared aggregation error under angle-of-arrival uncertainty.","lead":"The paper proposes a method to position fluid antennas and set transmission and beamforming weights in a wireless aggregation network so that errors from uncertain channel angles are reduced. It matters because over-the-air computation networks, which compute averages of sensor data while transmitting, are sensitive to channel errors, and movable antennas add flexibility to compensate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Diagonal channel-error covariance in the MSE contradicts the paper's own scalar angle-error model, so the robust gains may stem from optimizing the wrong objective.","rationale":"The paper's stated goal is to show that jointly optimizing transmit coefficients, receive beamforming, and fluid antenna positions yields robust AirComp performance under channel uncertainty. The central mathematical step is the second-order statistics of the channel error. The paper's own model, Eq. (5), says the per-user perturbation is a single scalar Δθ_k multiplying a deterministic vector; hence the error covariance is rank-one. The paper instead uses a diagonal covariance in Eq. (9), Eq. (13), and Eq. (15). This is not a matter of an alternative modeling choice stated up front; it contradicts the preceding derivation. The numerical section evaluates the same approximate MSE that the optimizer minimizes, so the reported gains cannot reveal the mismatch. The first-order Taylor expansion is also used far outside its validity range when θ_0 reaches 1 rad and x_n reaches 8λ. I agree with the reader's weakest_assumption and with the REJECT verdict. The BCD splitting and BFGS update are standard, but the correctness problem sits in the objective itself, so the strongest claim is unsupported as submitted. No change to the reader's verdict is needed.","tokens_in":8066,"tokens_out":5113,"duration_ms":50422,"concrete_test":"Run a Monte Carlo simulation under the paper's own scalar angle-error model: for N=8, K=10, SNR at 0 and 10 dB, and θ_0 on a grid from 0 to 1 rad, generate many realizations of Δθ_k ~ U[−θ_0, θ_0] independently per user and compute each user's channel exactly as sqrt(l_k^{-α}) exp(j(2π/λ) x_n cos(ar θ_k + Δθ_k)). Take the transmit coefficients, receive beamformer, and APV produced by Algorithm 1 as well as the FPA and non-robust benchmark designs, and evaluate the true MSE of each design directly on the generated channels rather than using Eq. (9). If the proposed design does not consistently outperform both benchmarks across the θ_0 grid, the central robustness claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing flaw is the channel-error statistics used to build the MSE. Eq. (3) defines a single scalar angle uncertainty Δθ_k ~ U[−θ_{k,0}, θ_{k,0}] for user k. Equations (4)–(7) then give Δh_k = ar h_k ⊙ q(ar θ_k) Δθ_k, with q_n = j (2π/λ) x_n sin ar θ_k. This is a deterministic vector times a scalar random variable, so E[Δh_k Δh_k^H] is a rank-one matrix, not a diagonal matrix. The paper instead asserts a diagonal covariance with n-th entry ψ_k x_n^2 θ_{k,0}^2 and writes the error term in Eq. (9) as Σ_n |b_k|^2 ψ_k θ_{k,0}^2 |m_n x_n|^2. That equals |b_k|^2 m^H diag(ψ_k θ_{k,0}^2 x^2) m, whereas the actual expression under Eq. (5) is |b_k|^2 (θ_{k,0}^2/3) |Σ_n m_n^* ar h_{k,n} q_{k,n}|^2. The correct term contains coherent cross-antenna phases and cannot be reduced to independent per-antenna variances. The same incorrect covariance is propagated into the b_k update (13) and the m update (15). In addition, Eq. (4) keeps only the linear term of exp(j(2π/λ) x_n cos(ar θ_k + Δθ_k)) while simulations take θ_0 up to 1 rad; with x_n up to 8λ, the phase argument changes by many radians, so the first-order truncation is not a controlled approximation. Since the simulation curves evaluate the same approximate objective that Algorithm 1 minimizes, they do not independently validate robustness. If the rank-one covariance is used instead, the optimized antenna positions and beamformers change, and the claimed gains over FPA and non-robust designs may disappear. This is an internal inconsistency in the paper's own model, not merely a disagreement with prior work.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8538,"tokens_out":8177,"duration_ms":76015,"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":[{"comment":"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).","section":"III, Eqs. (13) and (15)"},{"comment":"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.","section":"IV, Figs. 2-4"}],"minor_comments":[{"comment":"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.","section":"II, Eq. (4)"},{"comment":"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.","section":"III, Eq. (13)"},{"comment":"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.","section":"III, Algorithm 1"},{"comment":"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).","section":"III, Eq. (15)"},{"comment":"Simulation parameters such as the BFGS barrier schedule, line-search tolerance, and initialization are not specified, so the numerical results are not fully reproducible.","section":"IV"}],"recommendation":"reject","confidential_remarks":"I agree with the reader's assessment that the paper is not sound in its current state. The inconsistency between Eq. (5) and Eq. (9) is internal and changes the problem being solved; replacing the covariance and re-deriving the updates would require new simulations and a re-examination of the Taylor approximation. Rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nSo 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.\n\nI'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.\n\nRecommendation: send to peer review, because the flaw is substantive and the fix is nontrivial but feasible.","headline":"Combines fluid-antenna AirComp with robust design, but the MSE objective assumes diagonal channel-error covariance that its own scalar angle-error model contradicts.","tokens_in":9013,"tokens_out":1907,"would_cite":false,"duration_ms":17222,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["over-the-air computation","fluid antenna array","robust resource allocation","mean squared error","block coordinate descent","channel uncertainty","beamforming design","antenna position optimization"],"falsifier":"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.","tokens_in":7886,"feed_emoji":"📡","tokens_out":9566,"duration_ms":81697,"temperature":0.7,"pith_summary":"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.","feed_headline":"Moving antennas cut AirComp error in uncertain channels","feed_subtitle":"Robust beamforming and antenna positions keep over-the-air averages accurate as angle estimates degrade.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines over-the-air computation and the superposition-based data aggregation paradigm the paper builds on.","marker":"[1]"},{"why":"Surveys AirComp designs and motivates the MSE aggregation criterion used as the objective.","marker":"[2]"},{"why":"Introduces fluid antenna systems and the movable-antenna degrees of freedom that the design exploits.","marker":"[10]"},{"why":"Prior work on fluid antenna array enhanced AirComp, the direct starting point this paper extends to channel uncertainty.","marker":"[14]"},{"why":"Supplies the line-of-sight steering-vector channel model that depends on the antenna position vector.","marker":"[16]"},{"why":"Provides the Taylor-expansion technique for modeling angle uncertainty that produces the robust channel error model.","marker":"[21]"},{"why":"The BFGS quasi-Newton method used to optimize antenna positions under spacing constraints.","marker":"[22]"}],"fun_headline_variants":["Fluid antenna arrays improve AirComp accuracy under uncertainty","Joint beamforming and antenna placement for robust AirComp","Optimizing fluid antenna positions to reduce AirComp error","Robust over-the-air computation with moving antenna arrays","Antenna location optimization counters channel uncertainty in AirComp"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fluid antenna arrays improve AirComp accuracy under uncertainty","Joint beamforming and antenna placement for robust AirComp","Optimizing fluid antenna positions to reduce AirComp error","Robust over-the-air computation with moving antenna arrays","Antenna location optimization counters channel uncertainty in AirComp"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1504,"prompt_tokens":794,"completion_tokens":710,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":634}},"tokens_in":410,"tokens_out":710,"duration_ms":6839,"temperature":1.0,"reasoning_tokens":634,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:10:02.056649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Fluid antenna systems,","cited_arxiv_id":null,"evidence_quote":"Introduces fluid antenna systems and the movable-antenna degrees of freedom that the design exploits."},{"cited_title":"Over-the-Air computing for wireless data aggregation in massive IoT,","cited_arxiv_id":null,"evidence_quote":"Defines over-the-air computation and the superposition-based data aggregation paradigm the paper builds on."},{"cited_title":"A survey on Over-the-Air computation,","cited_arxiv_id":null,"evidence_quote":"Surveys AirComp designs and motivates the MSE aggregation criterion used as the objective."},{"cited_title":"Fluid antenna array enhanced Over-the-Air computation,","cited_arxiv_id":null,"evidence_quote":"Prior work on fluid antenna array enhanced AirComp, the direct starting point this paper extends to channel uncertainty."},{"cited_title":"Multi-beam forming with movable-antenna array,","cited_arxiv_id":null,"evidence_quote":"Supplies the line-of-sight steering-vector channel model that depends on the antenna position vector."},{"cited_title":"Intelligent and robust UA V-aided multiuser RIS communication technique with jittering UA V and imperfect hardware constraints,","cited_arxiv_id":null,"evidence_quote":"Provides the Taylor-expansion technique for modeling angle uncertainty that produces the robust channel error model."},{"cited_title":"Broyden’s quasi-newton methods for a nonlinear system of equations and unconstrained optimization: a review and open problems,","cited_arxiv_id":null,"evidence_quote":"The BFGS quasi-Newton method used to optimize antenna positions under spacing constraints."}],"review_version":1}