Functional WMMSE Algorithm for Multiuser Continuous Aperture Array Systems

Jia Guo; Shengqian Han; Shiyong Chen

REVIEW 4 major objections 5 minor 13 references

Functional WMMSE Algorithm for Multiuser Continuous Aperture Array Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2509.17101 v3 pith:E4KZ2R2R submitted 2025-09-21 eess.SP eess.AS

Shiyong Chen , Shengqian Han , Jia Guo This is my paper · ORCID

classification eess.SPeess.AS

keywords continuousaperturearrayfunctionalbeamformingWMMSEsum-ratemaximizationorthonormalbasisexpansionkernelinversionmultiuserMIMOintegralapproximation

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

The 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.

None 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[Section III, Proposition 1] 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 IV-C, implementation] 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 IV-C and Table II] 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 IV, Table I] 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.

minor comments (5)

[Section IV-B, Eq. (30)] In Eq. (30), the term "µβ_k(s1)" appears to be a typo; it should be µβ(s1), consistent with the surrounding equations.
[Section IV-B, Eq. (34)] 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 V] 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 V, Fig. 2-4] 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 IV-C] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the WMMSE-to-rate equivalence and functional updates are derived from stated first-order conditions; the only self-citations are minor, non-load-bearing implementation/modeling choices.

full rationale

The central derivation is self-contained. Proposition 2 and Appendix A establish the equivalence between sum-rate maximization and weighted MSE minimization by substituting the optimal combining functions and weight matrices and applying the kernel-inverse identity in Lemma 1; the resulting expression E_opt^{-1} = I_d + Q_k is derived from the model, not assumed from the target rate formula. The functional updates (26), (28), and (35) follow from first-order optimality conditions of the basis-expanded problem (19) and are mapped back to the continuous domain using the completeness/orthonormality relations, so they are not fitted to or defined in terms of the quantity they predict. Proposition 1 states the rate expression without a proof, but Appendix A supplies the supporting calculation; this is an omitted-proof/support gap, not circularity. The self-citations that do appear are minor: reference [8] is used only for the LoS channel modeling assumption, and reference [11] is used only for the quadrature sampling ratio M_B = (area ratio) M_U. Neither carries the load of the paper's sum-rate or complexity claims. The unspecified quadrature order M_U and the complexity analysis that omits d-dimensional costs are legitimate reproducibility and correctness concerns, but they are not instances of a prediction reducing to its inputs by construction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on three structural assumptions: (1) completeness of the chosen orthonormal bases, a standard math tool; (2) invertibility of the continuous kernels that define the update equations, which is plausible for positive-definite kernels but not analyzed; (3) the LoS uni-polarized channel model, which is standard in this literature but limits generalization. The free parameters are implementation choices that the paper leaves unspecified, most importantly the quadrature sample size M_U and the initialization of the beamforming functions.

free parameters (2)

M_U (number of quadrature points for user CAPA integral approximation)
Chosen by hand but not specified in the paper; it determines the accuracy of the Gauss-Legendre approximation (Section IV-C) and the runtime reported in Table II. M_B is then set as M_B = (L_B^2/L_U^2) M_U, so M_U is the only free sampling parameter.
Initialization of v_k(s)
Paper only says 'Initialize v_k(s) such that the power constraint is satisfied' (Table I) but does not specify the initialization scheme, which can affect convergence and final sum rate.

assumptions (3)

standard math Completeness of the orthonormal basis functions {β(s)} and {α_k(r)} over S_B and S_k^U (Eqs. 11, 14, 17).
Used to derive the Dirac delta identities α_k(r1)α_k^H(r)=δ(r1-r) and β(s1)β^H(s)=δ(s1-s) in (24) and (33), which are essential for converting matrix optimality conditions into functional integral equations.
domain assumption Existence and invertibility of the continuous kernels J_k(r1,r) and T_k(s1,s) as defined in Definition 1 and used in (26) and (35).
The updates require computing inverses of continuous kernels; the paper assumes these inverses exist. For σ^2>0 and μ>0 the kernels are positive definite, but no explicit invertibility conditions or numerical stability analysis is provided.
domain assumption LoS propagation with uni-polarized CAPA aligned along the y-axis, modeled by (2).
The channel kernel h_k(r,s) in (2) is a near-field free-space Green's function with polarization vectors Λ_T and Λ_R. The algorithm computes a_kj(r) and c_k(s) from this model; the results may not transfer to other channel models.

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Cite this review

Pith. "Pith review of Functional WMMSE Algorithm for Multiuser Continuous Aperture Array Systems." pith.science (2026). https://pith.science/paper/E4KZ2R2R

@misc{pith2026250917101,
  author       = {Pith},
  title        = {Pith review of: Functional WMMSE Algorithm for Multiuser Continuous Aperture Array Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4KZ2R2R}},
  note         = {Machine review of arXiv:2509.17101}
}

read the original abstract

In this paper, we develop a functional weighted minimum mean-squared error (WMMSE) algorithm for downlink beamforming in multiuser continuous aperture array (CAPA) systems where both the base station (BS) and users are equipped with CAPAs. We first present a closed-form expression for the achievable rate in multiuser CAPA systems, based on which the equivalence between maximizing the sum rate and minimizing the sum of weighted mean-squared errors (MSE) is established. We then employ the orthonormal basis expansion to transform the formulated functional optimization problem into a parameter optimization problem. By deriving the first-order optimality conditions of the parameter optimization problem and mapping them back to the functional domain, we obtain the update equations of the proposed functional WMMSE algorithm. Simulation results show that the proposed method outperforms discretization-based baselines in both sum rate and computational complexity.

Figures

Figures reproduced from arXiv: 2509.17101 by the authors.

**Figure 2.** shows that the sum rate increases monotonically with the current budget for all schemes. The proposed method consistently outperforms the Fourier-based baseline, because the proposed method incurs only integral-approximation error, whereas the Fourier approach suffers from both discretization loss and approximation error. In addition, the proposed method yields a substantial gain over SPDA, which is attributed to th… view at source ↗

**Figure 3.** depicts the sum rate versus the number of users. The proposed approach outperforms both baselines across all user counts. This behavior reflects the stronger interference management of the functional WMMSE updates, which act directly on the continuous beamforming and combining functions to mitigate inter-user and intra-user (multi-stream) interference. In [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Sum rate versus CAPA size. Table II compares the computational time of the proposed method with that of baselines. The proposed method significantly reduces computational time compared to the baselines. TABLE II COMPARISON OF COMPUTATIONAL TIME Name Proposed Fourier S…

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Reference graph

Works this paper leans on

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