{"id":"2ac9ad22-64b7-4009-a92e-5cb3fcc3db86","arxiv_id":"2603.16053","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"BeamINR, a WMMSE-structured GNN INR, nearly matches functional WMMSE sum rate for multiuser multi-CAPA beamforming with far lower inference latency and better scale/frequency generalization than prior INRs.","lead":"The paper derives a closed-form multiuser multi-CAPA sum rate, a functional WMMSE beamformer, and BeamINR, a GNN-based implicit neural representation that learns continuous beamforming functions. It matters because continuous-aperture arrays need fast online beamforming without heavy Fourier truncation or iterative integral solvers.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged perfect-CSI/quadrature assumptions.","rationale":"The paper's technical path is coherent: Prop. 1 gives an explicit multiuser multi-CAPA rate under both intra- and inter-user interference; the functional WMMSE is obtained by orthonormal expansion, first-order conditions, and mapping back to continuous kernels (with functional Woodbury); BeamINR then hard-wires the resulting aggregation/combination structure into a PE-respecting GNN. Simulations consistently place functional WMMSE highest and BeamINR closest among INRs, with clear latency and sample-complexity gains. The reader's CONDITIONAL verdict already correctly isolates the practical soft spots (perfect continuous CSI, pure LoS kernels, quadrature surrogates, missing artifacts). I find no additional internal inconsistency that would move the verdict toward REJECT or full ACCEPT. A denser-quadrature check is the most direct way to confirm that the reported BeamINR–WMMSE gap is not an artifact of the shared GL approximation used at train and test time.","tokens_in":23533,"tokens_out":526,"duration_ms":31833,"concrete_test":"Re-evaluate BeamINR and functional WMMSE on the same 10k test geometries using a denser GL grid (e.g., M_B,G^2 = 1600 instead of 400) and report the relative sum-rate gap; if the gap widens by more than ~3–5 percentage points or ranking vs. ConINR/VarINR reverses, the quadrature-surrogate claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that BeamINR approaches functional-WMMSE sum rate with lower latency and better generalization than INR baselines—is internally supported by the closed-form rate (Prop. 1), the functional WMMSE derivation (Sec. IV, App. A–E), the PE-aware GNN update (45) matched to the WMMSE recursion (Prop. 4), and consistent simulation trends (Figs. 3–6, Tables II–V). The reader's weakest assumption (perfect continuous LoS CSI + fixed-order GL/Sobol surrogates) is the genuine soft spot for operational significance, but it is not an internal contradiction: under the paper's stated model the math and experiments cohere. No stronger load-bearing flaw (e.g., PE misuse, rate-expression error, or baseline-invalidating training leakage) is evident from the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies multiuser multi-CAPA downlink beamforming where both the BS and users have continuous apertures. It derives a closed-form sum-rate expression (Prop. 1) that accounts for both intra- and inter-user interference, reformulates sum-rate maximization as an equivalent weighted MSE problem (Prop. 2) via a functional Woodbury identity, and obtains a functional WMMSE algorithm whose updates for the combining functions, weights, and beamforming functions are written in the continuous domain after orthonormal expansions (Sec. IV, Table I). Building on the PE property of the optimal policy (Prop. 3) and an explicit recursion of the functional WMMSE iterates (Prop. 4), the authors propose BeamINR, a GNN-based INR whose layer update (45) aggregates channel kernels weighted by previous-layer representations. Simulations under LoS uni-polarized channels show that functional WMMSE attains the highest sum rate, while BeamINR approaches it with much lower inference latency and better generalization to user count, CAPA size, and carrier frequency than ConINR/VarINR baselines (Figs. 3–6, Tables II–V).","tokens_in":23818,"tokens_out":1282,"duration_ms":11724,"significance":"If the results hold under the stated model, the paper makes two concrete contributions to CAPA beamforming: (i) an explicit multiuser multi-CAPA rate formula and a functional WMMSE algorithm that updates continuous beamforming functions without Fourier truncation, and (ii) a model-structured PE GNN INR that substantially reduces online latency and training sample/time cost relative to unstructured INRs while improving scale and frequency generalization. The appendices supply a coherent derivation path (KLE rate, functional Woodbury, optimality conditions mapped back to functions, and the WMMSE-to-GNN recursion), and the simulation suite is reasonably comprehensive (rate vs. power/users/size/frequency, generalization tables, and complexity). These are useful advances for continuous-aperture systems, even though operational significance remains limited by perfect continuous CSI and quadrature surrogates.","major_comments":[{"comment":"Sec. II (perfect-CSI paragraph) and channel model (3): all rate claims and both algorithms assume perfect continuous LoS uni-polarized kernels at the BS. The paper cites parametric estimators [35], [36] but never evaluates BeamINR or functional WMMSE under estimated or noisy kernels. Because the strongest claim is operational (near-WMMSE rate at low latency with better generalization), at least one imperfect-CSI or parametric-channel experiment is needed to show that the ranking in Figs. 3–6 and Tables II–V is not an artifact of oracle continuous CSI.","section":null},{"comment":"Sec. V-C and VI-A (GL/Sobol training and testing): continuous integrals in the rate objective, GNN layers (51), and evaluation are replaced by fixed-order quadrature (M_B,G^2=100 train / 400 test; M_B,S=100). There is no ablation of quadrature order, no comparison against denser or alternative integrators, and no quantification of the residual policy/rate error relative to the continuous functional WMMSE. A short sensitivity study is load-bearing for the claim that BeamINR “approaches” functional WMMSE rather than a shared discrete surrogate.","section":null},{"comment":"Table II (user generalizability): models trained at K=5 and tested for K=2…8 show BeamINR ratios as low as ~53% at K=2 and ~87% at K=8 versus functional WMMSE. The abstract and Sec. VII state improved generalization to the number of users without quantifying this degradation or discussing when retraining is still required. The claim should be tempered, or the table should be accompanied by absolute rates and a clear statement of the usable generalization range.","section":null}],"minor_comments":[{"comment":"Abstract vs. body: the abstract claims improved generalization to CAPA sizes and carrier frequencies; Table III shows strong size generalization for all INRs, while Table IV shows a clearer BeamINR/VarINR advantage on frequency. Align the abstract wording with the tables.","section":null},{"comment":"Notation: S_U is defined as the union of user apertures in Prop. 1, but several integrals (e.g., (6a) and later) mix S_U and S_k^U; a short clarifying sentence would help.","section":null},{"comment":"Fig. 1 caption is truncated (“Illustration of the downlink CAPA system L_x^B, L_y^B.”); complete the caption.","section":null},{"comment":"Table V reports inference time and training complexity to reach 95% of functional WMMSE; state hardware (CPU/GPU) and whether times include quadrature overhead so the latency comparison is reproducible.","section":null},{"comment":"Related-work placement: the conference precursor [34] is cited for functional WMMSE; a one-sentence delineation of what is new in the journal version (closed-form multiuser multi-CAPA rate, BeamINR, extended sims) already appears in footnote 1 and could be mirrored briefly in Sec. I-B.","section":null},{"comment":"Typos / wording: “It also provides additional simulation results…” (footnote 1); “Teb.” for Feb. in [20]; occasional missing spaces before citations. Light copy-edit would suffice.","section":null}],"recommendation":"major_revision","confidential_remarks":"The technical core (rate derivation, functional WMMSE, PE-structured INR) looks sound and is a natural journal extension of the authors’ conference WMMSE piece. I would not reject on novelty grounds. The revision bar is mainly experimental: imperfect CSI and quadrature sensitivity. If those are addressed cleanly, minor_revision or accept is realistic. Scope fits eess.SP / wireless-comms venues well."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a real methods contribution for multiuser multi-CAPA beamforming, not a rehash. The new pieces are the closed-form sum-rate for the case where both BS and users have continuous apertures (Prop. 1, App. A via KLE + functional Woodbury), the functional WMMSE that maps coefficient KKT conditions back to continuous updates (Sec. IV, App. B–E), and BeamINR: a PE GNN whose aggregation/combination is taken from the WMMSE recursion (Prop. 4 → eq. 45).\n\nWhat they do well is keep the math coherent and then use it. The rate and MSE equivalence are standard information-theoretic arguments under the LoS uni-polarized kernel; the appendices are careful. Simulations are consistent: functional WMMSE sits on top, BeamINR approaches it with ~0.05 s inference vs ~1.4 s, lower sample/time/space cost than ConINR/VarINR, and better generalization across K, aperture area, and carrier frequency (Figs. 3–6, Tables II–V). The hybrid GL + scrambled-Sobol training is a sensible fix for coordinate overfitting. Citation pattern is appropriate (Fourier/SPDA CAPA, PE GNNs, model-driven unfolding).\n\nSoft spots are operational, not internal contradictions. Perfect continuous CSI of pure LoS kernels and fixed-order quadrature as a surrogate for the continuous policy are strong assumptions; the paper states them and does not claim otherwise. No code/data and no error bars limit reproducibility. Hyperparameters (layer widths, α=0.1, sample counts) are free but ordinary for this literature. The stress-test note is right: nothing load-bearing breaks under the stated model.\n\nWho it is for: people working continuous-aperture / holographic MIMO beamforming and model-driven GNNs. They get a usable solver plus a faster learned policy that generalizes better than plain INRs. It deserves a serious referee; I would send it to peer review and would cite the rate expression and the functional-WMMSE → BeamINR construction if I am in this subfield. Engage.","headline":"Solid CAPA methods paper: closed-form multiuser multi-CAPA rate + functional WMMSE + structure-aware BeamINR that actually beats the INR baselines on rate, latency, and generalization under the stated model.","tokens_in":24493,"tokens_out":535,"would_cite":true,"duration_ms":5355,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"BeamINR embeds functional WMMSE iterations in a graph neural network so continuous multiuser CAPA beamformers approach optimal sum rate at far lower online cost.","keywords":["continuous aperture array","CAPA","beamforming","WMMSE","implicit neural representation","graph neural network","permutation equivariance","multiuser MIMO"],"falsifier":"Retrain and re-evaluate BeamINR versus functional WMMSE after replacing the ideal continuous LoS kernels with multipath or imperfectly estimated kernels, or after changing quadrature order; if BeamINR’s sum-rate ratio to WMMSE falls well below the reported high-nineties percentages, the central claim fails under realistic conditions.","tokens_in":24376,"feed_emoji":"📡","tokens_out":854,"duration_ms":17940,"temperature":0.7,"pith_summary":"When both the base station and users have continuous aperture arrays, beamforming becomes a functional design problem rather than a finite-vector one. This paper first derives a closed-form multiuser multi-CAPA sum-rate expression that accounts for both intra-user and inter-user interference, then converts sum-rate maximization into a functional WMMSE algorithm by expanding continuous kernels in orthonormal bases and mapping optimality conditions back to the continuous domain. From those iterations it builds BeamINR: a graph neural network that respects user permutation equivariance and whose layer update aggregates channel kernels exactly as the functional WMMSE steps do. Simulations show the functional algorithm attains the highest rates while BeamINR nearly matches them, with substantially lower inference latency, lower training cost, and better generalization to the number of users, aperture sizes, and carrier frequencies than prior INR baselines.","feed_headline":"Neural beamformer matches continuous-array rates at low latency","feed_subtitle":"A graph net built from functional WMMSE iterations generalizes across users, sizes and frequencies","key_machinery":"Functional WMMSE: orthonormal-basis conversion of the functional rate problem into coefficient-matrix MSE minimization, followed by first-order conditions that produce closed-form continuous updates for combining functions, weight matrices, and beamforming functions; BeamINR then uses those updates as its GNN aggregation/combination rule.","core_discovery":"A closed-form multiuser multi-CAPA sum rate plus a functional WMMSE algorithm whose continuous-domain updates can be turned into a GNN layer update yield BeamINR, an implicit neural representation that approaches the functional WMMSE sum rate while cutting inference latency and improving generalization over conventional INR beamformers.","pith_inferences":["The functional-iteration-to-GNN template may transfer to other continuous-domain wireless designs such as continuous RIS phase profiles or near-field focusing.","Hybrid fixed-quadrature plus scrambled-Sobol training is a reusable recipe for preventing coordinate overfitting whenever an INR must evaluate aperture integrals.","If continuous CSI must be estimated rather than given, the reported generalization edge may shrink unless the network is trained end-to-end with estimated kernels."],"forward_implications":["Continuous multiuser CAPA beamforming can be run online at near-WMMSE rates with inference times roughly an order of magnitude lower than iterative functional solvers.","Embedding permutation equivariance and WMMSE iteration structure reduces the samples and parameters needed to train CAPA beamforming INRs.","The same trained network generalizes across unseen user counts, CAPA areas, and carrier frequencies without retraining.","Fourier-truncated and discrete-array approximations leave measurable sum-rate gaps that pure functional and model-structured INR methods close."],"fun_headline_variants":["BeamINR GNN nears multi-CAPA sum rates at low latency","Functional WMMSE-driven GNN learns continuous CAPA beamformers","INR beamformer matches functional rates and generalizes users sizes freqs","Closed-form multiuser CAPA rate enables low-latency BeamINR","Graph net from WMMSE iterations cuts CAPA beamforming latency"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The base station is assumed to know the exact continuous line-of-sight channel kernels between every aperture point pair, and continuous integrals can be replaced by fixed-order quadrature without changing the learned policy.","fun_headline_variants_meta":{"raw":{"variants":["BeamINR GNN nears multi-CAPA sum rates at low latency","Functional WMMSE-driven GNN learns continuous CAPA beamformers","INR beamformer matches functional rates and generalizes users sizes freqs","Closed-form multiuser CAPA rate enables low-latency BeamINR","Graph net from WMMSE iterations cuts CAPA beamforming latency"]},"model":"grok-4.5","effort":"low","cost_usd":0.003866,"raw_usage":{"total_tokens":1197,"prompt_tokens":733,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":38660000,"prompt_tokens_details":{"text_tokens":733,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":386,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":733,"tokens_out":78,"duration_ms":3537,"temperature":1.0,"reasoning_tokens":386,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T20:25:15.395704+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Retrain and re-evaluate BeamINR versus functional WMMSE after replacing the ideal continuous LoS kernels with multipath or imperfectly estimated kernels, or after changing quadrature order; if BeamINR’s sum-rate ratio to WMMSE falls well below the reported high-nineties percentages, the central claim fails under realistic conditions.","supporting_citations":[],"review_version":1}