A Refined Alternating Optimization for Sum Rate Maximization in SIM-Aided Multiuser MISO Systems

Eduard E. Bahingayi; Le-Nam Tran; Marco Di Renzo; Murat Uysal; Shuying Lin

REVIEW 4 major objections 5 minor 20 references

A Refined Alternating Optimization for Sum Rate Maximization in SIM-Aided Multiuser MISO Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Optimizing SIM phase shifts before the digital beamformer, with an iterated projected-gradient loop, raises the achievable sum rate by up to 115.53% over benchmark schemes and lets the rate keep growing with SIM layer count instead of satur

desk verdict Useful design-guideline letter on SIM AO ordering, but the headline gains rest on a single simulation trajectory without error bars. read the letter →

arxiv 2508.15257 v1 pith:NWBVUPZ7 submitted 2025-08-21 eess.SP

Eduard E. Bahingayi , Shuying Lin , Murat Uysal , Marco Di Renzo , Le-Nam Tran This is my paper · ORCID

classification eess.SP

keywords stackedintelligentmetasurfacealternatingoptimizationprojectedgradientsumratemaximizationmultiuserMISOwave-domainbeamformingachievabledigital

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

The paper studies how to maximize the total downlink data rate in a multiuser system whose base station transmits through a stacked intelligent metasurface (SIM)—several layers of tunable meta-atoms that shape the signal in the electromagnetic domain before it reaches single-antenna users. The standard solution method is alternating optimization (AO): update the digital beamformer, then update the SIM phase shifts, and repeat. This letter argues that the recipe has been applied badly in two small-looking ways, and that these details matter enormously. It shows numerically that updating the SIM phase shifts first, before the digital beamformer, and solving that update with an iterative projected-gradient loop instead of a single step, yields up to 115.53% higher achievable sum rate than the common baselines, and—unlike them—keeps growing with the number of SIM layers instead of saturating. If correct, the result means the saturation observed in prior SIM studies can be avoided by more careful algorithm design, at least within the simulated setting.

What carries the argument

The load-bearing mechanism is the ordering and refinement inside an alternating optimization loop. Algorithm 1 cycles between an iterative projected-gradient ascent on the SIM phase-shift vector θ—using the closed-form complex gradient of the sum rate from Theorem 1, with backtracking line search and a stopping tolerance—and an SCA-based update of the digital beamforming matrix W. The phase-shift gradient is derived from the Rayleigh–Sommerfeld diffraction model of inter-layer propagation, which makes the entire wave-domain channel a structured function of θ, and the paper's two design rules—optimize θ first, and iterate the PG loop to convergence before returning to W—are what convert that

What would settle it

Repeat the comparison of the four algorithm variants (Figs. 1–3) under a different inter-layer propagation law—for example, full-wave electromagnetic simulation of the metasurface stack or measurements from a SIM prototype—and across many random initial points and channel realizations. If the θ-first iterative-PG scheme no longer consistently outperforms the W-first single-PG scheme, or if its sum rate saturates as L grows under the alternative model, the paper's central claims are refuted.

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Extended reading notes

Core claim

The paper's discovery is that the performance of alternating optimization for SIM-aided systems hinges on two implementation details prior work treated as interchangeable. First, the update order is not neutral: optimizing the SIM phase-shift vector θ before the digital beamformer W converges to a significantly higher sum rate than the reverse order, because the SIM's wave-domain beamforming is the dominant source of interference suppression and benefits from early optimization. Second, the phase-shift subproblem, usually handled by a single projected-gradient step, is far better solved by an iterative projected-gradient loop that runs to near-convergence inside each AO cycle before switchin

Load-bearing premise

The ranking of the four schemes—including the 115.53% gain and the absence of saturation—is established in simulation under a single propagation model (Rayleigh–Sommerfeld diffraction between layers and sinc-correlated Rayleigh fading to users), started from one randomly generated initial point; if real SIM propagation or the user-channel distribution differs from that simulator, the ordering of the methods could change.

Editorial extensions

If this is right

Any existing SIM-aided AO design can adopt the two guidelines as a drop-in change—move the phase-shift update to the front of each AO cycle and iterate the PG loop to convergence—without changing the channel model or the objective.
Within the considered setting, the previously reported sum-rate saturation with increasing SIM layer count (at fixed thickness) is avoided, so a correctly tuned AO can keep extracting gains from added layers.
The reported gains widen with the number of meta-atoms per layer (roughly 55–78% at N=49 up to 115.53% at N=100), so the benefit of the refined AO matters most in the large-SIM regime.
The same algorithm and ordering apply to SIM without digital beamforming (power-allocation variant), with comparable gains (up to 73.53% at L=10), making the design principle architecture-independent within the SIM family.
The ordering advantage is attributed to SIM's interference-suppression capability, which implies the phase-shift subproblem should be solved more thoroughly than the beamforming subproblem in any AO-based SIM design.

Reading between the lines

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

An implicit corollary the paper does not develop: because the ordering advantage comes from θ being the higher-leverage variable, the θ-first principle plausibly transfers to other SIM optimization problems (energy efficiency, cell-free networks, joint sensing and communication), all of which currently use AO with single-step phase-shift updates.
The saturation-avoidance claim is demonstrated starting from a single randomly generated initial point; a natural extension is to test whether θ-first iterative PG preserves its advantage across many random initializations, since AO for non-convex problems is initial-point-sensitive and the paper explicitly identifies that sensitivity.
If the results generalize beyond the simulated Rayleigh–Sommerfeld/sinc-correlated model, comparative studies of SIM should include a well-tuned AO baseline; otherwise 'saturation' and other reported SIM limits may be algorithmic artifacts rather than physical ones.
A testable engineering prediction follows: the gap between θ-first iterative PG and the baselines should widen as the number of users K grows, because more users mean more interference for the SIM's wave-domain suppression to handle.

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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

Colleague,

This letter is a useful empirical follow-up in the SIM-aided MU-MISO line: the authors show that optimizing the SIM phase shifts before the digital beamformer, and running the projected-gradient inner loop to convergence, gives substantially higher sum rate than the usual W-first, single-PG AO, and avoids the saturation with layer count reported in earlier SIM papers. The contribution is a design guideline, not a new technique — the authors admit as much — and the four-way comparison is clear, with a standard simulation setup for the subfield.

What the paper does well: the ordering and inner-loop questions are explicitly isolated, the benchmarks are sensible, and the paper is honest that performance depends on the initial point. The closed-form gradient in Theorem 1 is a direct calculus derivation, and the appendix makes the chain of dependencies readable. The self-citations supply the SCA solver and prior SIM scaling results, which is fine.

The soft spots are mostly about statistical support. Fig. 1 shows a single random initial point; Figs. 2 and 3 have no error bars or trial counts. Given that the problem is non-convex and the authors themselves emphasize sensitivity to initialization, one trajectory does not establish that θ-first is generally better, nor that saturation is avoided. The 115.53% headline is measured at one configuration (N=100, L=10). The 'no saturation' conclusion is only checked up to L=10; phrasing it as 'no saturation in the considered range' would be more accurate. There's also a likely typo: the denominator in the second sum of eq. (9)/(11) runs over j≠1 instead of j≠k. If that's not a typo, the printed gradient is not the gradient of the stated objective; either way, it should be corrected.

These are addressable, not fatal. The paper would be much stronger with multi-initialization statistics, error bars, and a corrected formula. As it stands, it's a plausible engineering claim that merits a serious referee, though I wouldn't cite it in my own work until the empirical basis is firmer. If you work on SIM optimization or AO for phase-shift design, it's worth a look.

Referee Report

4 major / 5 minor

Summary. The paper considers sum-rate maximization for a stacked intelligent metasurface (SIM) aided multiuser MISO downlink, optimizing digital beamforming and SIM phase shifts alternately. The proposed method differs from prior AO schemes in two ways: it optimizes the SIM phase shifts before the digital beamformer, and it uses an iterative projected-gradient (PG) procedure for the phase-shift subproblem instead of a single PG step. The main theoretical contribution is a closed-form expression for the gradient of the sum rate with respect to the phase-shift vector (Theorem 1). The performance claims—that the theta-first ordering and iterative PG yield higher achievable sum rate and avoid saturation as the number of SIM layers grows—are supported only by simulation results for a Rayleigh-Sommerfeld propagation model and correlated Rayleigh user channels. The headline gain is stated as up to 115.53% over benchmark schemes.

Significance. If the empirical claims are robust, the paper offers a simple and practically relevant design guideline for SIM-aided systems: optimize phase shifts first and iterate PG to convergence. The derivation of the gradient is direct and appears correct apart from a likely typographical error in the denominators of Eqs. (9) and (11), and the simulation setup is clearly described, which aids reproducibility. However, the central claims are empirical ordering statements over non-convex problems, and the current evidence is not statistically grounded: Fig. 1 uses a single initial point, Figs. 2–3 show no error bars or trial counts, and the 'no saturation' conclusion is extrapolated from L up to 10. The comparison between iterative PG and single PG also conflates algorithmic refinement with additional computational effort. With additional Monte Carlo results and a corrected gradient expression, the contribution would be a valid and useful engineering insight.

major comments (4)

[Section IV, Figs. 1–3] The central claims are empirical: the theta-first ordering, the advantage of iterative PG, and the percentage gains in Fig. 3 all depend on random initializations and channel realizations. Fig. 1 is explicitly described as starting 'from the same randomly generated initial point' (singular), and Figs. 2–3 do not report the number of Monte Carlo runs, the number of initial points, or error bars. Since (P1) is non-convex and the paper itself notes in Remark 2 that AO performance is sensitive to the initial point, the observed ordering and the 115.53% gain could be specific to a favorable draw. Please provide statistics over many random channel realizations and initial points (e.g., mean ± standard deviation or box plots) and state the trial count.
[Eqs. (9) and (11)] The denominator of the second term in the gradient expression is written as Σ_{j≠1} |(θ^l)^T e^l_{k,j}|^2 + σ^2 in both Eq. (9) and Eq. (11). By the derivation in the Appendix, this term should be the interference-plus-noise power for user k, i.e., Σ_{j≠k} |(θ^l)^T e^l_{k,j}|^2 + σ^2. As printed, the PG update is not the gradient of the stated sum-rate objective. If this is a typographical error, please correct it and confirm that the simulations used the correct formula; if it is not a typo, the algorithm description is internally inconsistent and the simulation results may not correspond to the claimed objective.
[Section IV, Fig. 2 and Conclusion] The conclusion that 'saturation does not occur with the proposed method' is extrapolated from simulations with L only up to 10. A monotonically increasing curve over L=1,...,10 does not establish absence of saturation at larger L; many functions increase for small arguments and flatten later. To support this headline claim, either provide a scaling argument or simulate larger L (with computational cost reported) and show that the ASR continues to grow, for example by plotting the slope or relative increment versus L. As it stands, the evidence is too limited for a 'no saturation' claim.
[Section IV, Algorithm 1] The comparison between 'iterative PG' and 'single PG' does not equalize computational effort. In the proposed method, PG is repeated until convergence (Lines 12–13 of Algorithm 1) before updating W, whereas the single-PG benchmarks perform only one PG step per outer iteration. The observed gain may therefore be due to more inner iterations rather than to the algorithmic 'refinement' per se. To support the design guideline that iterative PG is inherently better, compare the variants under a matched computational budget (e.g., equal total number of PG iterations, equal runtime, or ASR versus per-iteration cost) and report the additional complexity.

minor comments (5)

[Algorithm 1, line 4] The notation 'Compute ∇θ(m)R(θ(m))' is awkward; it should be 'Compute ∇θ R(θ)|_{θ=θ(m)}' or simply 'Compute ∇θR(θ(m))'.
[Fig. 2 caption] The caption '... (a). SIMwDB (b). SIMwoDB' is missing a separator verb; it should read '(a) SIMwDB, (b) SIMwoDB' for clarity.
[Section IV] When reporting percentage gains, please report the absolute ASR values and, if possible, confidence intervals; a percentage computed from a single point is sensitive to small variations.
[Section IV] The paper states that the SCA method of [20] achieves the same performance as WMMSE. Since the proposed algorithm relies on this SCA solver, a brief description of its computational complexity or a reference to a version with more details would improve reproducibility.
[General] Minor typos and notation inconsistencies exist (e.g., 'P A' in Section I, and the use of 'θ(m)' as both a subscript and an argument in Algorithm 1). A careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are empirical comparisons against benchmark schemes and a direct gradient derivation, not predictions that reduce to fitted inputs or self-citation chains.

full rationale

The paper's load-bearing claims are (i) the gradient formula in Theorem 1, (ii) the ordering recommendation (optimize θ first), (iii) the iterative-PG recommendation, and (iv) the numerical ASR gains up to 115.53%. The gradient is derived in Appendix A as a direct calculus derivative of the objective R(θ) with respect to θ*, using the definitions of e_{k,j}^l; it does not assume the result it claims to prove. The ordering and iterative-PG conclusions are obtained by simulating the same system model under four algorithm variants; no parameter is fitted to the target ASR and then reported as a prediction. The SCA method cited as [20] is a self-citation, but it is used symmetrically for the W-subproblem in all compared schemes and does not by itself determine the θ-first vs W-first ordering or the iterative-vs-single-PG comparison. The saturation-avoidance claim is an empirical observation under the stated Rayleigh-Sommerfeld and sinc-correlated Rayleigh fading model; even though the paper reports a single initial point and no error bars, that is a reproducibility/statistical-support concern, not circularity. The apparent denominator typo in Eq. (9)/(11) (j≠1 instead of j≠k) is a correctness issue, not a circularity issue. No step in the derivation reduces by construction to its own input, and no load-bearing result is imported solely from the authors' prior work.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. It assumes the standard SIM architecture with unit-modulus phase shifts, the Rayleigh-Sommerfeld propagation law, perfect CSI, and a specific correlated Rayleigh fading model. Free parameters are algorithm hyperparameters and standard path-loss constants, all fixed for the comparison.

free parameters (4)

PG step size α0 = 1
Initial step size for the backtracking line search in Algorithm 1. Same value for all compared schemes, so the comparison is fair, but the reported gains could in principle depend on this choice.
line search constants β, η = 0.5, 1e-7
Backtracking line-search parameters chosen by the authors. Not fitted to data, but they affect the projected-gradient trajectory.
convergence tolerances εθ, ε = 1e-6
Stopping thresholds for the inner PG loop and the outer AO loop, chosen by the authors.
path-loss constants c1, c2 = 2, 3.5
Standard free-space and urban path-loss exponents used in the channel model in Section IV. Domain assumptions, not fitted by this paper.

assumptions (4)

domain assumption Rayleigh-Sommerfeld diffraction model (eq. 1) describes inter-layer propagation in the SIM
Used to construct the wave-based beamforming matrix G in (2); all simulations rely on this physical model.
domain assumption Unit-modulus constraint on phase shifts (4b) with perfect CSI
The optimization constraint and perfect-CSI assumption are taken from prior SIM literature (e.g., [4], [6]) and are not justified in this paper.
domain assumption AO converges to a stationary point of (P1)
The paper assumes the alternating procedure in Algorithm 1 converges to a useful solution; no convergence proof is provided for the non-convex problem.
domain assumption sinc-correlated Rayleigh fading for user channels (Section IV)
R[n,n'] = sinc(2d/λ) is taken from [4]; the paper's conclusions about algorithm ordering are demonstrated only for this channel model.

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Pith. "Pith review of A Refined Alternating Optimization for Sum Rate Maximization in SIM-Aided Multiuser MISO Systems." pith.science (2026). https://pith.science/paper/NWBVUPZ7

@misc{pith2026250815257,
  author       = {Pith},
  title        = {Pith review of: A Refined Alternating Optimization for Sum Rate Maximization in SIM-Aided Multiuser MISO Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWBVUPZ7}},
  note         = {Machine review of arXiv:2508.15257}
}

abstract

Stacked intelligent metasurfaces (SIMs) have emerged as a disruptive technology for future wireless networks. To investigate their capabilities, we study the sum rate maximization problem in an SIM-based multiuser (MU) multiple-input single-output (MISO) downlink system. A vast majority of pioneer studies, if not all, address this fundamental problem using the prevailing alternating optimization (AO) framework, where the digital beamforming (DB) and SIM phase shifts are optimized alternately. However, many of these approaches suffer from suboptimal performance, quickly leading to performance saturation, when the number of SIM layers increases assuming the \emph{fixed SIM thickness}. In this letter, we demonstrate that significant performance gains can still be achieved, and such saturation does not occur with the proposed method in the considered setting. To this end, we provide practical design guidelines to improve AO-based optimization of digital precoders and SIM phase shifts. Specifically, we show that (i) optimizing the SIM phase shifts first yields significant performance improvements, compared to optimizing the DB first; and (ii) when applying projected gradient (PG) methods, which are gradually becoming more popular to optimize the phase shifts thanks to their scalability, we find that using an iterative PG method achieves better performance than the single PG step, which is commonly used in existing solutions. Based on these customizations, the proposed method achieves a higher achievable sum rate (ASR) of up to $\ensuremath{115.53\%}$, compared to benchmark schemes for the scenarios under consideration.

Figures

Figures reproduced from arXiv: 2508.15257 by the authors.

**Figure 2.** ASR versus the number of SIM layers for an SIM [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** ASR versus the number of meta-atoms per layer for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Stacked intelligent metasurface-aided MIMO transceive r design,

J. An et al. , “Stacked intelligent metasurface-aided MIMO transceive r design,” IEEE Wireless Commun. Mag. , vol. 31, no. 4, pp. 123–131, 2024

work page 2024

[2] [2]

State of the art on stacked intelligent meta surfaces communication, sensing and computing in the wave domain,

M. Di Renzo, “State of the art on stacked intelligent meta surfaces communication, sensing and computing in the wave domain,” i n Eur . Conf. Antennas Propag. (EuCAP), Stockholm, Sweden , 2025, pp. 1–3

work page 2025

[3] [3]

Efﬁcient beamforming and radiation pattern control using stacked intelligent metasurfaces,

N. U. Hassan et al., “Efﬁcient beamforming and radiation pattern control using stacked intelligent metasurfaces,” IEEE Open J. Commun. Soc. , vol. 5, pp. 599–611, 2024

work page 2024

[4] [4]

Stacked intelligent metasurfaces for efﬁcient holograp hic MIMO communications in 6G,

J. An et al. , “Stacked intelligent metasurfaces for efﬁcient holograp hic MIMO communications in 6G,” IEEE J. Sel. Areas Commun. , vol. 41, no. 8, pp. 2380–2396, 2023

2023

[5] [5]

Stacked intell igent metasurfaces for multiuser beamforming in the wave domain,

J. An, M. Di Renzo, M. Debbah, and C. Y uen, “Stacked intell igent metasurfaces for multiuser beamforming in the wave domain, ” in ICC 2023-IEEE Int. Conf. Commun. IEEE, 2023, pp. 2834–2839

work page 2023

[6] [6]

Stacked intelligent metasurfaces for multiuser downlin k beamforming in the wave domain,

J. An et al. , “Stacked intelligent metasurfaces for multiuser downlin k beamforming in the wave domain,” IEEE Trans. Wireless Commun. , vol. 24, no. 7, pp. 5525–5538, 2025

work page 2025

[7] [7]

Stacked intelligent metasurface enabled LEO satellite communications relying on statistical CSI,

S. Lin et al. , “Stacked intelligent metasurface enabled LEO satellite communications relying on statistical CSI,” vol. 13, no. 5, pp. 1295– 1299, 2024

work page 2024

[8] [8]

Achievable rate optimization for large stacked intelligent metasurfaces based on statistical CSI ,

A. Papazafeiropoulos et al. , “Achievable rate optimization for large stacked intelligent metasurfaces based on statistical CSI ,” vol. 13, no. 9, pp. 2337–2341, 2024

work page 2024