{"id":"45302071-e5bc-4e7c-b78d-ae56cd7f6708","arxiv_id":"2502.03922","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A two-stage GNN jointly infers fluid antenna positions and beamforming vectors, trained unsupervisedly to maximize sum rate or energy efficiency in MU-MISO systems.","lead":"This paper uses a two-stage graph neural network to jointly choose fluid antenna positions and beamforming weights at a base station, maximizing user sum rate or energy efficiency. The method runs in milliseconds and adapts to different user counts, but the numerical comparison only pits it against a fixed-antenna convex baseline.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed feasibility mappings are internally inconsistent: Eq. (14) scales powers upward when over budget and Eq. (9) can yield negative spacings from unconstrained ξ_i, so the claimed feasible-solution guarantee fails as written.","rationale":"The reader's weakest assumption was the absence of small-scale fading, which is a valid external-validity concern. However, my reading found a more fundamental, internal problem: the two feasibility mappings that the entire method relies on are mathematically incorrect as stated. The power activation in Eq. (14) does not enforce the power budget; it does the opposite when the sum exceeds Pmax. The position parameterization in Eq. (9) does not guarantee nonnegative spacings for unconstrained real ξ_i, and the output layer in Eq. (23) provides no mechanism to make ξ_i positive. Because the loss function contains no penalty on constraint violations, the trained model may output solutions that are not feasible, making the claimed utility values unachievable under the stated constraints. This is not a matter of the authors disagreeing with the field; it is a formal flaw in the presented algorithm. The concerns are correctable in principle (replace the scaling by Pmax/Σp_i and use a softmax-like positive parameterization), and if those corrections are made and experiments are re-run, the work might be salvageable. But as the manuscript currently stands, the central feasibility guarantee is invalid, so I would reject the version under review and require the corrections and revised numerical evidence before reconsideration.","tokens_in":8636,"tokens_out":13121,"duration_ms":140087,"concrete_test":"Write a small script that implements Eq. (14) as printed and feed it p=(0.8, 0.8) with Pmax=1; the output total power is (Σp_i)^2/Pmax = 2.56 W > 1 W, so the power constraint (3b) is violated. Separately, evaluate Eq. (9) with ξ=(1, -1, 2) and any ξmax; the resulting δ for the negative ξ is negative, violating (3d). If both checks pass as written, the feasibility claim fails; then re-run Tables II-III with the corrected scaling and a positive-ξ parameterization to determine whether the reported sum-rate/EE gains persist.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim that the two-stage GNN outputs feasible antenna positions and beamforming vectors is not supported by the stated mathematics. First, the power-scaling activation in Eq. (14) is inverted: when Σp_i > Pmax, each pk is multiplied by (Σp_i)/Pmax, which is greater than 1, so the resulting total power becomes (Σp_i)^2/Pmax > Pmax, violating constraint (3b). For example, p=(0.8, 0.8) with Pmax=1 gives total output power 2.56 W. Second, the position parameterization in Eq. (9) is claimed to satisfy δ_n ≥ 0 and Σδ_n ≤ δmax 'regardless of the values of {ξ_i} and ξmax', but ξ_i are recovered in Eq. (23) as the real part of an unconstrained CFL output with no positivity guarantee. If the ξ_i have mixed signs, e.g., ξ=(1, -1, 2), the denominator is positive but δ for the negative ξ becomes negative, violating the minimum-spacing constraint (3d); a zero sum would divide by zero. Since the unsupervised loss in Eq. (28) is just the inverse utility with no penalty for infeasibility, the training procedure can accept and reward solutions that violate the power budget or the antenna spacing constraints. These are internal inconsistencies, not disagreements with external consensus, and they directly undermine every numerical comparison in Tables II-IV unless the printed equations are typos and the actual implementation uses the corrected forms (Pmax/Σp_i scaling and a positive-valued parameterization such as softmax).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a MU-MISO fluid antenna system (FAS) and formulates sum-rate and energy-efficiency maximization with constraints on transmit power and antenna spacing. It proposes a two-stage graph neural network (GNN) in which the first stage outputs antenna positions and the second stage outputs beamforming vectors, trained with an unsupervised loss equal to the inverse of the system utility. Numerical results report higher sum rate and EE than an equidistant-antenna CVX baseline and an MLP, with very low inference times and scalability to unseen numbers of users (Tables II–III), and the authors claim that the two stages can function separately (Table IV).","tokens_in":8929,"tokens_out":2957,"duration_ms":28582,"significance":"If the feasibility and performance claims are correct, the two-stage GNN is a useful contribution to real-time FAS optimization, extending graph-based learning to the new dimension of antenna positions. The paper is honest in comparing its second stage against CVX given identical antenna positions (Table IV), which isolates the beamforming loss, and the unsupervised loss is the natural utility objective rather than a fitted surrogate. However, the printed feasibility mappings contain internal errors that directly affect the reported numerical results, so the significance can only be assessed after those are corrected.","major_comments":[{"comment":"The power-normalization activation is inverted. When Σp_i > Pmax, the printed formula multiplies each p_k by (Σp_i)/Pmax, which is greater than 1, so the total output power becomes (Σp_i)^2/Pmax > Pmax. This violates constraint (3b) and contradicts the adjacent statement that the formula ensures Σ∥w_i∥² ≤ Pmax. The intended correction is p_k ← p_k · Pmax/Σp_i. As printed, every numerical result in Tables II–IV is obtained with a model that can exceed the power budget, making the reported rates/EE potentially unattainable.","section":"§III.A.2, Eq. (14)"},{"comment":"The claim that δ_n in Eq. (9) satisfies (8) “regardless of the values of {ξ_i} and ξmax” is false. The variables ξ_i are recovered in Eq. (23) as the real part of an unconstrained complex-layer output, so they can be negative or sum to zero. If any ξ_i is negative, the corresponding δ_n is negative, violating the minimum-spacing constraint (3d); if the sum is zero, Eq. (9) divides by zero. Even when all ξ_i are positive, there is no built-in guarantee of a positive sum. A positivity-preserving parameterization (e.g., exp or softplus followed by normalization) is needed. This error undermines the central claim that the two-stage GNN outputs feasible antenna positions.","section":"§III.A.1, Eqs. (9), (23), (24)"},{"comment":"The unsupervised loss in Eq. (28) is exactly the inverse of the system utility and contains no penalty for infeasibility. This is acceptable only if the architecture guarantees feasibility by construction. Given the errors in Eqs. (9) and (14), the training procedure can accept and reward solutions that violate the power budget or antenna-spacing constraints. The authors should either confirm the printed equations are typos and provide the actual implemented formulas, or add a projected/reparameterized model that provably satisfies (3b)–(3d) before the numerical comparisons can be trusted.","section":"§III.D, Eq. (28)"},{"comment":"The channel model is a deterministic line-of-sight steering vector with only large-scale path loss; no small-scale multipath fading is simulated. Since a primary motivation of FAS is to mitigate deep fading, the reported gains over equidistant antennas may not persist in the fading channels for which the system is designed. I recommend either simulating a fading channel (e.g., correlated Rayleigh or Rician over the fluid aperture) or explicitly limiting the conclusions to LoS scenarios.","section":"§II, Eq. (1)–(2) and §IV"}],"minor_comments":[{"comment":"“Sigmod” should be “Sigmoid” in the text.","section":"Eqs. (25)–(26)"},{"comment":"The use of the † symbol for KTe values is confusing because the column header already separates KTr and KTe; please clarify in the caption that † denotes a test-only (unseen) user count rather than a different performance metric.","section":"Tables II–III"},{"comment":"Even with positive ξ_i, the denominator Σ_{i∈N\\{1}} ξ_i should be explicitly normalized to avoid ambiguity; a softmax-style parameterization would make the feasibility claim cleaner and easier to verify.","section":"§III.A.1, Eq. (9)"},{"comment":"The statement that setting x1 = 0 causes no optimality loss should be justified in one sentence, e.g., by noting that a common shift of all antenna positions multiplies each steering vector by a common phase factor that cancels in the SINR expression.","section":"§III.A.1, after Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":"I do not see a novelty-disclosure problem: the self-cited GNN references provide background architecture components, not the specific FAS application. The main concern is strictly internal consistency of the feasibility parameterization; if the authors correct Eqs. (9) and (14) and rerun the experiments, the paper could become acceptable. I would also encourage the editor to ask for a clearer statement of the channel model limitations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline is: this paper has a plausible architecture for a real problem, but the printed feasibility math is wrong, and everything quantitative hangs off it. The error is not a reviewer nitpick—it is an internal contradiction.\n\nWhat is new: it is, as far as the cited literature goes, the first GNN applied to FAS joint antenna-position and beamforming design. The two-stage decomposition (position stage then beamforming stage) with a hybrid MRT/ZF beamformer is sensible, and the unsupervised loss (inverse utility) is the standard learning-to-optimize setup. The authors also deserve credit for isolating the beamforming stage’s optimality gap in Table IV: given identical antenna positions, their learned beamformer is only slightly behind CVX.\n\nThe soft spot is load-bearing. Equation (14) is supposed to enforce the power budget, but when the raw outputs exceed Pmax it multiplies each pk by (sum pi)/Pmax, which is larger than one. The total output power becomes (sum pi)^2/Pmax > Pmax—the opposite of enforcement. Equation (9) is also claimed to produce feasible spacings regardless of the ξi, but the ξi are read as the real part of an unconstrained complex layer, so they can be negative (making δn negative) or sum to zero (dividing by zero). So the “guaranteed feasible” solution is not guaranteed, and the unsupervised loss has no penalty that would push the network back into the feasible set. Unless the printed equations are typos and the implementation uses Pmax/sum and a softmax-type parameterization, every number in Tables II–IV is comparing against a method whose outputs may violate the power budget or spacing constraint.\n\nSmaller concerns: the channel model is deterministic LOS with only large-scale path loss, which sidesteps exactly the fading that motivates FAS; the CVX baseline uses equidistant antennas, so it never benchmarks the joint optimum; and no code, seeds, or error bars are provided. These are all addressable.\n\nWho this is for: anyone working on learning-to-optimize for FAS or movable antennas. The architectural idea is worth discussing, but the experiments need to be rerun with corrected feasibility layers and a fading channel before the results can be trusted.\n\nMy recommendation: send it to peer review, but expect a major revision. The approach is important enough and the error is likely fixable, yet the current manuscript is not sound as written.","headline":"Plausible two-stage GNN for FAS, but the printed feasibility equations invert the power scaling and allow negative spacings, so the central guarantee and all numerical tables are built on an internal inconsistency.","tokens_in":9484,"tokens_out":2952,"would_cite":false,"duration_ms":27100,"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":"A two-stage graph neural network, trained unsupervised on the system utility, jointly chooses fluid-antenna positions and beamforming to beat an equidistant-antenna convex baseline in sum rate and energy efficiency while inferring in…","keywords":["fluid antenna systems","graph neural networks","unsupervised learning","MU-MISO","sum-rate maximization","energy efficiency","beamforming","antenna position optimization"],"falsifier":"Run the same two-stage GNN under a standard Rayleigh or Rician fading model, or on measured channels, and compare sum rate and energy efficiency with the equidistant-antenna CVX baseline. If the GNN-FAS advantage shrinks or reverses, the claim that the learned positions deliver real-world gains is not supported.","tokens_in":8418,"feed_emoji":"📡","tokens_out":7889,"duration_ms":65964,"temperature":0.7,"pith_summary":"The paper tries to establish that the antenna positions of a fluid antenna system (FAS) are optimizable variables that can be learned together with the beamforming vectors. The authors formulate sum-rate and energy-efficiency maximization for a multi-user MISO downlink and solve both with a single two-stage graph neural network (GNN), trained without labels by taking the inverse of the system utility as the loss. The claim is that this approach beats a fixed-antenna convex-optimization baseline in both objectives, generalizes to user counts unseen in training, and cuts inference time from seconds to tens of microseconds. If correct, it would make FAS a practical real-time option and replace per-scenario convex solvers with one trained network.","feed_headline":"Two-stage GNN places fluid antennas and beams, beating CVX in 0.05 ms","feed_subtitle":"Unsupervised utility training gives real-time, scalable antenna placement and beamforming for fluid antenna systems.","key_machinery":"The load-bearing object is the two-stage complex GNN built from complex graph attention layers and complex fully-connected layers, with a virtual node in the first stage to produce graph-level antenna positions and residual connections to reduce over-smoothing. Feasibility is enforced inside the architecture: positions are derived from unconstrained auxiliary variables through a normalized readout that guarantees $\\delta_n \\ge 0$ and $\\sum_n \\delta_n \\le \\delta_{\\max}$, while beamformers are parameterized as $w_k = \\sqrt{p_k}\\,\\bar{w}_k(\\alpha_k)$ with $\\alpha_k \\in (0,1)$ blending MRT and ZF directions. The unsupervised loss $L(\\Theta) = \\frac{1}{M}\\sum_m 1/U^{(m)}$ back-propagates through the SINR expressions, coupling the two stages during training.","core_discovery":"The paper's central claim is that the joint antenna-position and beamforming problem for a MU-MISO FAS can be mapped onto a fully connected graph and solved end-to-end by a two-stage complex-valued GNN. The first stage outputs feasible antenna positions through an auxiliary-variable parameterization that enforces ordering, minimum spacing, and total aperture; the second outputs beamforming vectors through a hybrid maximum-ratio-transmission/zero-forcing decomposition of each beam into a power coefficient and a normalized direction. Both stages share a single unsupervised loss equal to the inverse of the chosen utility, so no labeled optimal solutions are needed. In simulation, the two-stage GNN achieves higher sum rate and energy efficiency than the equidistant-antenna CVX baseline, keeps working for user counts not seen during training, and runs in roughly $0.05$ ms per inference.","pith_inferences":["The line-of-sight-only channel model leaves open whether the learned positions help under multipath fading; testing the architecture on fading channels would directly extend the paper's claim.","Because the loss only needs a differentiable utility, the same two-stage scheme could be retargeted to max-min fairness, outage probability, or other objectives without changing the architecture.","The graph formulation suggests applicability to multi-cell or user-centric networks where users and access points are nodes, although the paper only simulates a single-cell MU-MISO setting."],"forward_implications":["The antenna-position stage carries most of the gain: given identical CVX beamforming, first-stage positions improve sum rate and EE over equidistant antennas in the (8,4) scenario.","The beamforming stage can run alone: for equidistant antennas it reaches near-CVX sum rate and EE with orders-of-magnitude faster inference, so it can serve as a standalone or warm-start beamformer.","Because the number of learnable parameters is independent of $K$, the same trained network applies to user counts it never saw during training.","The full system runs at about 0.05 ms per inference, which is fast enough for real-time fluid antenna control."],"supporting_citations":[{"why":"Defines fluid antenna systems and motivates optimizing antenna positions to combat deep fading, which is the problem this paper builds on.","marker":"[1]"},{"why":"Supplies the movable-antenna parameterization (setting the first antenna position to zero without optimality loss) and the MLP baseline that the GNN is compared against.","marker":"[4]"},{"why":"Motivates the graph conversion and parameter-sharing view that gives scalability to unseen user counts.","marker":"[5]"},{"why":"Provides the GNN sum-rate beamforming architecture that the two-stage design extends with a separate antenna-position stage.","marker":"[7]"},{"why":"Supplies the hybrid MRT/ZF beamforming decomposition used in the second stage to reduce the output space to power and hybrid coefficients.","marker":"[12]"}],"fun_headline_variants":["GNN optimizes fluid antenna positions and beams in 0.05 ms","Two-stage GNN beats CVX for fluid antenna placement and beamforming","Unsupervised GNN drives fluid antenna gains at 0.05 ms per inference","Fluid antenna system optimized by two-stage GNN: 0.05 ms real-time","Graph neural network places fluid antennas and beams, beating CVX"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical gains rest on a deterministic line-of-sight channel model with only large-scale path loss, so the learned antenna positions are not tested under the small-scale multipath fading that fluid antennas are meant to mitigate.","fun_headline_variants_meta":{"raw":{"variants":["GNN optimizes fluid antenna positions and beams in 0.05 ms","Two-stage GNN beats CVX for fluid antenna placement and beamforming","Unsupervised GNN drives fluid antenna gains at 0.05 ms per inference","Fluid antenna system optimized by two-stage GNN: 0.05 ms real-time","Graph neural network places fluid antennas and beams, beating CVX"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000264,"raw_usage":{"total_tokens":1563,"prompt_tokens":867,"completion_tokens":696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":594}},"tokens_in":483,"tokens_out":696,"duration_ms":5729,"temperature":1.0,"reasoning_tokens":594,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T00:12:35.769795+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same two-stage GNN under a standard Rayleigh or Rician fading model, or on measured channels, and compare sum rate and energy efficiency with the equidistant-antenna CVX baseline. If the GNN-FAS advantage shrinks or reverses, the claim that the learned positions deliver real-world gains is not supported.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines fluid antenna systems and motivates optimizing antenna positions to combat deep fading, which is the problem this paper builds on."},{"cited_title":"Deep learning enabled multicast beamforming with mov- able antenna array,","cited_arxiv_id":null,"evidence_quote":"Supplies the movable-antenna parameterization (setting the first antenna position to zero without optimality loss) and the MLP baseline that the GNN is compared against."},{"cited_title":"Graph neural networks for wireless networks: Graph representation, architecture and evaluation,","cited_arxiv_id":null,"evidence_quote":"Motivates the graph conversion and parameter-sharing view that gives scalability to unseen user counts."},{"cited_title":"Hybrid MRT and ZF learning for energy-efficient transmission in multi-RIS-assisted networks,","cited_arxiv_id":null,"evidence_quote":"Supplies the hybrid MRT/ZF beamforming decomposition used in the second stage to reduce the output space to power and hybrid coefficients."}],"review_version":1}