{"id":"5b4b196e-5532-4300-b01f-f8c1fceebed4","arxiv_id":"2509.17101","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors propose a functional WMMSE algorithm for multiuser downlink beamforming in continuous aperture array systems, showing higher sum rate and lower complexity than discretization baselines.","lead":"This paper develops an algorithm for designing continuous aperture arrays, smooth antenna surfaces that focus wireless signals to multiple users at once. The method aims to improve connection speed in next-generation 6G base stations by computing optimal current distributions over the antenna surface.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Results hinge on an unspecified quadrature order M_U and a complexity count that omits stream-dimension (d) costs; neither is tested, so the claimed advantages are not yet established.","rationale":"I read the central claim as a numerical-algorithmic claim: functional WMMSE, implemented through quadrature, achieves higher sum rate at lower complexity than discretization-based baselines. The continuous-domain mathematics is largely standard: Proposition 1 has the form of a per-user SINR-matrix rate expression (the proof is omitted, which is a weakness, but not the decisive one), and the first-order stationary-point updates in (26), (28), and (35) are plausible given the basis expansions. The propositions are not circular. The fragile step is the transition from the infinite-dimensional functional equations to the numbers in Figures 2-4 and Table II. That transition is described only as Gauss-Legendre quadrature with an unspecified M_U, with M_B tied to M_U by a self-cited area-ratio rule, and with no sensitivity or convergence analysis. The reader's weakest_assumption identified exactly this dependence, and I agree. I additionally note a second consistency issue: the stated d = min{d_B,d_U} is about 8649 for the simulated 0.5 m user aperture, while the claimed complexity omits d-dependent operations such as forming W_k in (28) and evaluating T_k in (35). This reinforces the conditional verdict rather than changing it. A reproducible run with explicit M_U, explicit d, and a sweep over M_U would settle whether the headline advantages are real.","tokens_in":9180,"tokens_out":17027,"duration_ms":159367,"concrete_test":"Reproduce the Figure 2 and Table II experiments with M_U varied over 16, 32, 64, and 128, scaling M_B by the paper's area rule, and record sum rate and per-iteration wall time; simultaneously instrument the implementation to log the actual number of streams d and the sizes of the matrices passed to the inverse routine for (28). If varying M_U changes the reported gains by more than about 10%, or if the actual d differs from min(d_B,d_U) as stated, then the central performance/complexity comparison is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV-C implements the functional WMMSE updates (26), (28), (35) by Gauss-Legendre quadrature, but the number of quadrature points M_U is never reported. The continuous kernel inverses J_k^{-1} in (26) and T_k^{-1} in (35) are replaced by sampled matrix inversions; no convergence test in M_U, no comparison against the exact continuous inverse, and no sensitivity analysis are provided. This matters physically: at the stated 28 GHz with a 2 m BS aperture, the channel kernel (2) varies over roughly 373 wavelengths across the aperture, so the quadrature must resolve rapid phase variation. A too-small M_U would bias every reported sum-rate gain, while a large M_U would erode the claimed complexity advantage. The complexity count O(M_B^3+M_U^3) also omits the cost of forming and inverting the d-by-d weight matrix W_k in (28) and of building T_k in (35) from K users' d-dimensional beamformers; with the paper's own parameter d = min{d_B,d_U} ≈ 8649 for L_U = 0.5 m, these d-dependent terms can dominate. The runtimes in Table II are therefore not reproducible unless M_U and d are specified, and the headline claim that functional WMMSE outperforms Fourier and SPDA baselines in both sum rate and complexity remains conditional on this unverified numerical approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a functional weighted minimum mean-squared error (WMMSE) algorithm for downlink beamforming in multiuser continuous aperture array (CAPA) systems where both the base station and the users are equipped with CAPAs. The authors first state a closed-form achievable rate expression for the multiuser multi-CAPA downlink (Proposition 1), then establish an equivalence between sum-rate maximization and weighted sum-MSE minimization (Proposition 2). Using orthonormal basis expansions, they derive first-order optimality conditions in the functional domain and obtain update equations for the combining functions u_k(r), weight matrices W_k, and beamforming functions v_k(s) (Eqs. (26), (28), and (35)). The updates are implemented by Gauss-Legendre quadrature, with sampling points M_U and M_B. Simulations compare the proposed algorithm with Fourier-based and SPDA baselines in terms of sum rate versus current budget, number of users, and CAPA size, as well as runtime (Table II). The paper claims the proposed method outperforms both baselines in sum rate and computational complexity.","tokens_in":9410,"tokens_out":5036,"duration_ms":44091,"significance":"If the results hold, the paper makes a useful contribution by extending WMMSE-type optimization to the functional domain for multiuser multi-CAPA systems, an extension that is nontrivial because the optimization variables are continuous functions and the baselines rely on discretization or basis truncation. The derivations in Section IV are careful and self-contained in large part: the functional update equations follow from explicit first-order conditions, and Lemma 1 in Appendix A provides a functional analogue of the Woodbury identity that is used correctly. A notable strength is that the derivation does not assume the rate result in a circular way; the MSE formulation and its optimality conditions lead to the functional updates. However, the paper currently lacks a proof of its foundational rate expression, a convergence analysis of the proposed iterative algorithm, and a complete specification and validation of the quadrature-based numerical implementation. These gaps are load-bearing because the claimed performance and complexity advantages rest on them.","major_comments":[{"comment":"Proposition 1 (Eqs. (4)-(5)) is the foundation of the entire paper: it defines the objective in (6), is used in the equivalence proof of Proposition 2, and motivates the WMMSE algorithm. However, the proof is omitted entirely (\"The proof is omitted due to space constraints\"). In a journal submission this is a load-bearing gap. The authors should either provide a complete proof or give a precise reference that contains the result for the exact multiuser multi-CAPA scenario, and state the conditions under which (4) is achievable (e.g., Gaussian signalling, treat-interference-as-noise, and existence of the kernel inverses in (5a)-(5b)).","section":"Section III, Proposition 1"},{"comment":"The implementation replaces the continuous kernel inverses J_k^{-1} in (26) and T_k^{-1} in (35) with matrix inverses obtained by Gauss-Legendre quadrature, but the number of user-aperture quadrature points M_U is never reported. Because the channel kernel (2) at 28 GHz with L_B = 2 m and L_U = 0.5 m varies over hundreds of wavelengths across the aperture, M_U is a critical accuracy-versus-complexity parameter. Please specify M_U for every reported simulation, include a convergence test with respect to M_U (e.g., sum rate versus M_U), and quantify the quadrature error against a high-resolution reference. Without this, the sum-rate gains in Figs. 2-4 cannot be separated from discretization artifacts.","section":"Section IV-C, implementation"},{"comment":"The complexity statement O(M_B^3 + M_U^3) omits the cost of forming and inverting the d x d matrices W_k in (28) and T_k in (35), as well as the cost of computing c_j(s) for all K users. With d = min{d_B, d_U} and L_U = 0.5 m, d is on the order of 8649, so the d-dependent operations dominate the per-iteration cost. Please provide a complete complexity count in terms of M_U, M_B, K, and d, and report the exact M_U and d values used for the runtimes in Table II. As written, the claimed complexity advantage and the runtimes are not reproducible.","section":"Section IV-C and Table II"},{"comment":"Convergence of the proposed iterative algorithm is not established. The updates in Table I form a block-coordinate-type scheme, but the paper provides no proof of monotonicity, no stationary-point guarantee, and no numerical convergence study beyond stopping when the change in sum log det(W_k) is below a tolerance. Since Proposition 2 concerns the global optimum of (6), the authors should clarify what the algorithm provably converges to, or at least provide a rigorous monotonicity argument for the objective. This is especially important because the simulations report final sum-rate values that depend on the stopping criterion.","section":"Section IV, Table I"}],"minor_comments":[{"comment":"In Eq. (30), the term \"µβ_k(s1)\" appears to be a typo; it should be µβ(s1), consistent with the surrounding equations.","section":"Section IV-B, Eq. (30)"},{"comment":"The replacement β(s1)β^H(s) = δ(s1 - s) in (34) is a shorthand for the completeness relation Σ_n β_n(s1)β_n^*(s) = δ(s1 - s). As written, the matrix dimensions do not match (β(s1)β^H(s) is N_s x N_s). Please clarify this step, since it is central to obtaining the functional form of the v_k update.","section":"Section IV-B, Eq. (34)"},{"comment":"The stopping tolerance ε in Table I and the initialization of v_k(s) are not reported in the simulations. Please specify these to make the results reproducible.","section":"Section V"},{"comment":"The simulation curves are described in text but no error bars or confidence intervals are given, and the number of random channel realizations is not stated. Please add this information, especially since the proposed method is claimed to outperform the baselines consistently.","section":"Section V, Fig. 2-4"},{"comment":"The baseline specifications are incomplete: the number of Fourier basis functions N_B and N_U and the SPDA discretization spacing are not reported, so the complexity and performance comparisons are not fully reproducible.","section":"Section IV-C"}],"recommendation":"major_revision","confidential_remarks":"The paper's central derivation is largely sound under the stated completeness assumptions, and the omitted proof of Proposition 1 seems likely to be available from the authors or related references. The main concerns are fixable: supply the proof, specify M_U and baseline parameters, give a complete complexity analysis including d, and add convergence support. I would not reject, but the current manuscript is not yet ready for publication in its present form. The heavy reliance on self-citations [8] and [11] for modeling and sampling rules is not circular for the main claim, but the authors should make those rules explicit or at least verify that the cited references are fully accessible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper extends WMMSE to the multiuser downlink where both the BS and users are equipped with continuous aperture arrays. That is a genuine gap in the CAPA literature: prior work covered single-antenna users, single-user multi-CAPA, or multicast, but not this combination. The functional update equations for the combining function, weight matrix, and beamforming function are derived from first-order optimality conditions under the orthonormal-basis completeness assumption, and the MSE equivalence is given a real proof in an appendix. The derivation is self-contained and, as far as I can tell, internally consistent. The core idea of avoiding a full basis expansion and working directly with kernels is sensible and likely to be useful.\n\nThe soft spots are significant but not fatal. Proposition 1, the closed-form achievable rate, is stated without proof. It is load-bearing: everything else depends on it. It is probably a straightforward extension of the kernel-inverse rate results in [6, 9], but 'probably' is not a proof. There is also no convergence analysis for the alternating updates; discrete WMMSE convergence does not automatically carry over.\n\nThe implementation section is where the paper gets weakest. The Gauss-Legendre quadrature order M_U is never reported. At 28 GHz and a 2 m aperture, the channel phase varies over hundreds of wavelengths, so the quadrature must be dense enough to resolve that; without a sensitivity test, the sum-rate gains could be partly quadrature artifacts. The complexity claim O(M_B^3+M_U^3) also omits the cost of forming and inverting the d-by-d weight matrix W_k in (28) and of constructing T_k in (35). For this setup d = min(d_B,d_U) ≈ 8,649, so those d-dependent terms can dominate the runtime. Table II is therefore not reproducible, and the claimed complexity advantage over Fourier/SPDA baselines is not yet established.\n\nNone of this undermines the central idea. The paper is a solid draft, not a finished result. For people working on CAPA beamforming and 6G, it will be directly relevant. I would send it to peer review with the clear expectation that the authors provide the missing proof (or a pointer to an extended version), report M_U and test its sensitivity, add a convergence check, and redo the complexity accounting with the d-dependent terms included. That is a serious referee's worth of work.","headline":"A functional-domain WMMSE for multiuser multi-CAPA systems, careful in derivation but with an unproven rate expression, no convergence check, and an unreported quadrature order that the complexity and gain claims depend on.","tokens_in":9987,"tokens_out":4912,"would_cite":false,"duration_ms":39970,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims a functional WMMSE algorithm that operates directly on continuous beamforming and combining functions achieves higher sum rate and lower complexity than Fourier and SPDA baselines in a multiuser CAPA downlink.","keywords":["continuous aperture array","functional beamforming","WMMSE","sum-rate maximization","orthonormal basis expansion","kernel inversion","multiuser MIMO","integral approximation"],"falsifier":"Run the proposed algorithm with $M_U = 8, 16, 32, 64$ sampling points on the user surfaces while keeping $K=3$, $C_{\\max}=1000$, and the geometry of Fig. 2; if the reported sum-rate curve rises appreciably with $M_U$, or runtime grows as $M_U^3$, the claim that the method is both accurate and cheap at the paper's operating point would be falsified. Independently, evaluating the closed-form rate integral (4) against a Monte Carlo estimate of mutual information for random channel realizations would test Proposition 1 directly.","tokens_in":8934,"feed_emoji":"📡","tokens_out":9805,"duration_ms":81072,"temperature":0.7,"pith_summary":"Continuous aperture arrays replace discrete antennas with a continuous current distribution on a surface, turning beamforming design into a functional optimization. This paper develops a weighted MMSE algorithm that acts directly on those continuous functions instead of approximating the array by discrete elements or finite Fourier series. It derives a closed-form achievable sum rate for the downlink in which both the base station and each user have continuous apertures, proves that maximizing this rate is equivalent to minimizing a weighted sum of mean-squared errors, and obtains iterative update equations for the beamforming and combining functions. Simulations indicate that the functional updates outperform Fourier-based and spatially discretized baselines in both sum rate and runtime.","feed_headline":"Functional WMMSE raises sum rate and cuts runtime in CAPA downlinks","feed_subtitle":"Direct continuous-domain updates outperform Fourier and SPDA baselines in both sum rate and computation time.","key_machinery":"The machinery is the orthonormal basis expansion: $v_k(s)=\\beta(s)V_k$, $u_k(r)=\\alpha_k(r)U_k$, and $h_k(r,s)=\\alpha_k(r)H_k\\beta^{\\mathrm{H}}(s)$ for complete orthonormal sets $\\beta(s)$ and $\\alpha_k(r)$. Optimality conditions are first derived for the coefficient matrices $V_k$ and $U_k$, then multiplied by basis functions and mapped back to the functional domain using orthonormality, producing the continuous update equations (26) and (35) that require inverting the kernels $J_k$ and $T_k$. The paper defines kernel inversion through Definition 1 and uses Lemma 1, a continuous-kernel analogue of the Woodbury identity, to prove the WMMSE equivalence. For implementation, the kernel integrals are discretized with Gauss-Legendre quadrature.","core_discovery":"The paper's central claim is that the multiuser multi-CAPA downlink beamforming problem admits a functional WMMSE solution. After expanding the beamforming and combining functions in complete orthonormal bases, the first-order optimality conditions in coefficient space are pulled back to the continuous domain, yielding update rules in which the optimal combining function $u_k(r)$ is obtained by inverting the kernel $J_k(r_1,r)=\\sum_{j=1}^{K} a_{kj}(r_1)a_{kj}^{\\mathrm{H}}(r)+\\sigma^2\\delta(r_1-r)$, and the beamforming function $v_k(s)$ by inverting a companion kernel $T_k(s_1,s)$. The rate expression in Proposition 1 and the equivalence in Proposition 2 make these updates a legitimate way to pursue sum-rate maximization. The paper reports that in simulated 28 GHz line-of-sight CAPA links the method surpasses Fourier-based and SPDA baselines in achievable sum rate and reduces computational time substantially.","pith_inferences":["A testable extension is the uplink analog: reversing transmit and receive roles should produce the same functional WMMSE block structure, since only the channel kernel and the noise diagonal enter the derivation.","Because the quadrature order $M_U$ is never specified, any independent comparison should sweep $M_U$ before crediting the reported speedups; the paper gives no saturation or sensitivity evidence.","The closed-form rate of Proposition 1, whose proof is omitted, is the analytical foundation; an independent derivation or a Monte Carlo check of the integral in (4) against channel realizations would make that foundation directly testable.","The continuous-kernel Woodbury-like identity in Lemma 1 might be reusable for other functional CAPA problems, such as deriving rates under colored noise or optimizing with uncertain channel kernels."],"forward_implications":["If the closed-form rate in Proposition 1 is correct, it supplies an achievable rate expression for the multiuser multi-CAPA downlink that accounts for both inter-user and intra-stream interference in one formula.","The equivalence in Proposition 2 means that sum-rate maximization in these systems can be solved by weighted-MSE minimization, so standard WMMSE-style block updates to $W_k$, $u_k(r)$, and $v_k(s)$ carry the functional optimization.","Using the derived update equations, the algorithm only incurs integral-approximation error rather than the additional representation error of Fourier truncation, which is the paper's stated reason for the observed sum-rate gains over the Fourier baseline.","The per-iteration complexity scales as $O(M_B^3+M_U^3)$ with quadrature points, versus Fourier methods' $O(N_B N_U M_B^2)$ plus $O(N_B^3+N_U^3)$, explaining the reported runtime reduction from hundreds of seconds to well under a second."],"supporting_citations":[{"why":"Supplies the inverse-kernel definition and the single-CAPA rate result that Proposition 1 extends to the multiuser multi-CAPA case.","marker":"[9]"},{"why":"Provides the orthonormal basis expansion of the channel kernel and the near-field line-of-sight channel model used throughout.","marker":"[6]"},{"why":"Provides the SPDA discretization baseline and the multiuser rate formulation that Proposition 1 generalizes to user apertures.","marker":"[4]"},{"why":"Provides the Fourier-series beamforming baseline whose finite coefficient optimization the proposed method is compared against.","marker":"[3]"},{"why":"Supplies the area-ratio rule for setting the quadrature sampling densities $M_B$ and $M_U$ in the implementation.","marker":"[11]"},{"why":"Justifies representing any beamforming or combining function as a linear combination of complete orthonormal basis functions.","marker":"[10]"}],"fun_headline_variants":["Functional WMMSE lifts CAPA sum rate, cuts compute time","Continuous-domain WMMSE outperforms discretized baselines in CAPA","CAPA beamforming: functional WMMSE wins on rate and runtime","New functional WMMSE for multiuser CAPA: faster, higher sum rate","Functional WMMSE beats Fourier and SPDA in CAPA downlinks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The closed-form rate of Proposition 1 is stated without proof in this version, and the performance results rest on approximating continuous kernel-inverse updates by Gauss-Legendre quadrature at sampling counts that the paper never specifies; if either the unproved rate formula or the approximate inversion is inaccurate, the claimed gains would not stand.","fun_headline_variants_meta":{"raw":{"variants":["Functional WMMSE lifts CAPA sum rate, cuts compute time","Continuous-domain WMMSE outperforms discretized baselines in CAPA","CAPA beamforming: functional WMMSE wins on rate and runtime","New functional WMMSE for multiuser CAPA: faster, higher sum rate","Functional WMMSE beats Fourier and SPDA in CAPA downlinks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1403,"prompt_tokens":899,"completion_tokens":504,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":406}},"tokens_in":515,"tokens_out":504,"duration_ms":4602,"temperature":1.0,"reasoning_tokens":406,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:48:50.650825+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the proposed algorithm with $M_U = 8, 16, 32, 64$ sampling points on the user surfaces while keeping $K=3$, $C_{\\max}=1000$, and the geometry of Fig. 2; if the reported sum-rate curve rises appreciably with $M_U$, or runtime grows as $M_U^3$, the claim that the method is both accurate and cheap at the paper's operating point would be falsified. Independently, evaluating the closed-form rate integral (4) against a Monte Carlo estimate of mutual information for random channel realizations would test Proposition 1 directly.","supporting_citations":[{"cited_title":"Wireless Commun., 2025, early access","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse-kernel definition and the single-CAPA rate result that Proposition 1 extends to the multiuser multi-CAPA case."},{"cited_title":"Zhang and L","cited_arxiv_id":null,"evidence_quote":"Provides the SPDA discretization baseline and the multiuser rate formulation that Proposition 1 generalizes to user apertures."},{"cited_title":"Sanguinetti, A","cited_arxiv_id":null,"evidence_quote":"Provides the Fourier-series beamforming baseline whose finite coefficient optimization the proposed method is compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the area-ratio rule for setting the quadrature sampling densities $M_B$ and $M_U$ in the implementation."},{"cited_title":"Commun., 2025, early access","cited_arxiv_id":null,"evidence_quote":"Justifies representing any beamforming or combining function as a linear combination of complete orthonormal basis functions."}],"review_version":2}