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On the Joint Beamforming Design for Large-scale Downlink RIS-assisted Multiuser MIMO Systems

T0 review · 2 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that joint precoder and RIS phase-shift optimization for weighted sum rate in large-scale downlink RIS-assisted multiuser MIMO systems can be solved with per-iteration cost that is linear in the number of BS antennas and…

desk verdict A solid, useful extension with a real complexity win, but the stationarity claim for Algorithm 2 is not actually proven—the SPG scaling breaks the standard convergence theorem. read the letter →

arxiv 2412.08320 v1 pith:GX6FILYY submitted 2024-12-11 eess.SP

Eduard E. Bahingayi , Nemanja Stefan Perović , Le-Nam Tran This is my paper · ORCID
classification eess.SP
keywords reconfigurableintelligentsurfaceweightedsumratemaximizationmultiuserMIMObeamformingdesignscaledprojectedgradientmethodphaseshiftoptimizationalternatinglinearcomplexity
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

This paper addresses the deployment bottleneck of reconfigurable intelligent surface (RIS) systems: jointly choosing the base-station precoders and the phase shifts of thousands of passive RIS elements. It claims that the weighted-sum-rate problem for the downlink of a multiuser MIMO system can be solved with per-iteration complexity that grows linearly with the number of base-station antennas and RIS elements, rather than cubically as in standard WMMSE-based designs. The argument splits the problem by formulation: the precoders are updated with a closed-form successive convex approximation rule on an equivalent low-dimensional problem, while the RIS phases are updated with one scaled projected gradient step on the original problem. A deliberately loose line-search rule, initialized at $\alpha = 1/R(\theta)$, usually accepts that step on the first try. If the claims hold, joint beamforming for RIS-assisted massive MIMO with hundreds or thousands of elements becomes computationally practical.

What carries the argument

The paper's workhorses are two different formulations of the weighted-sum-rate objective. The precoder update runs on the equivalent low-dimensional problem (P2), built from $\bar H = HH^H$, and rests on the determinant inequality (7), which provides a tight concave quadratic minorant and turns the SCA subproblem into the closed-form matrix formula $F = (\mu I + \tilde A \bar H)^{-1} \tilde B$. The phase update runs on the original problem and is carried by the scaled projected gradient step, with the diagonal normalization $\Xi = \operatorname{diag}(1/|\nabla_\theta R(\theta)|)$ acting as a per-element gain control, and by the line-search condition (18), which deliberately accepts small objective increases from the phase block so that the precoder block drives most of the WSR improvement. The initial step-size choice $\alpha = 1/R(\theta)$ makes the step size self-adjusting as the weighted sum rate grows.

What would settle it

Run Algorithm 2 on a random channel realization with, say, $N_t=64$, $N_s=400$, $K=4$, and record the phase-gradient norm $\lVert \nabla_\theta R(\theta^{(\ell)})\rVert$ across many outer iterations. The Section III-D claim predicts that every accumulation point has zero phase-block gradient; if the WSR stops increasing while $\lVert \nabla_\theta R(\theta^{(\ell)})\rVert$ stays clearly above zero, the convergence guarantee fails on that instance.

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Extended reading notes

Core claim

The central discovery is that the two subproblems of the joint design should not be solved in the same coordinate system. For the precoders, the paper exploits the equivalent lower-dimensional problem (P2), where the transmit power constraint is folded into the objective; the closed-form SCA update is $F = (\mu I + \tilde A \bar H)^{-1} \tilde B$, with $\bar H = HH^H$, and the physical precoder is recovered as $W = \sqrt{\xi}H^H F$, $\xi = P_{\mathrm{BS}}/\lVert H^H F\rVert^2$. For the RIS phases, the paper keeps the original problem (P1), because numerical estimates show the gradient of the equivalent objective with respect to $\theta$ has roughly twice the Lipschitz constant, so gradient methods on (P2) converge more slowly. It then proposes a scaled projected gradient step $\theta \leftarrow \Pi_Q(\theta + \alpha \Xi \nabla_\theta R(\theta))$ with $\Xi = \operatorname{diag}(1/|\nabla_\theta R(\theta)|)$, which normalizes the gradient entrywise and compensates for the weak indirect channel through the RIS. The line search accepts the step as soon as $R(\theta^{(\ell+1)}) \ge R(\theta^{(\ell)}) + \frac{\beta}{2N_s}\lVert \theta^{(\ell+1)} - \theta^{(\ell)}\rVert^2$, with $\alpha$ initialized to $1/R(\theta^{(\ell)})$, and the paper reports that this condition is typically met in one trial. The complexity analysis counts the dominant per-outer-iteration cost as $O(N_t N_r N_d K^2 + I_\theta N_s N_t N_r K + I_w N_r^3 K^3)$, which is linear in $N_t$ and $N_s$.

Load-bearing premise

The proof that the alternating algorithm converges to a stationary point assumes, without proof, that the single normalized gradient step with the diagonal scaling $\Xi = \operatorname{diag}(1/|\nabla_\theta R|)$ and the new line-search rule always makes enough progress in the right direction, and that the alternating scheme satisfies the standard block-coordinate convergence assumptions.

Editorial extensions

If this is right

  • If the complexity claim is correct, joint beamforming for RIS-assisted massive MIMO with $N_s$ in the hundreds or thousands becomes tractable: the per-iteration cost grows linearly in $N_s$ and $N_t$, whereas the WMMSE-MM and BCD baselines scale cubically or quadratically.
  • The proposed algorithm's simulations show higher weighted sum rate than the WMMSE-MM and BCD baselines while using fewer complex multiplications and less run time, so the linear-scaling design does not sacrifice WSR.
  • The paper's empirical finding that the equivalent formulation has a larger gradient Lipschitz constant for the phase variables implies that reformulation choice can matter more than the optimizer; keeping the original problem for the phase block is itself a design principle.
  • Because the W-update is a low-dimensional closed form, the number of base-station antennas affects the per-iteration cost only linearly, through channel construction and the gradient evaluation, not through an $O(N_t^3)$ matrix inversion.
  • The line-search design, which accepts a modest increase from the phase update and lets the precoder update drive most of the rate gain, yields the reported one-step acceptance and shorter runtime.

Reading between the lines

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

  • A testable extension would be to replace the diagonal scaling $\Xi = \operatorname{diag}(1/|\nabla_\theta R|)$ with other positive scalings, such as per-user block scalings or Barzilai-Borwein-type step lengths, and compare the one-step acceptance rate and total runtime; the paper's argument does not pin down $\Xi$ uniquely.
  • The Lipschitz-constant comparison in Section III-C is computed for a MISO setup; applying the same measurement to MIMO with quantized RIS phases would show whether the 'keep the original formulation for phases' rule survives under finite-resolution constraints.
  • If the linear-complexity method is extended to the imperfect-CSI setting listed as future work, the equivalent reformulation would need a statistical version, since the closed-form precoder update relies on the perfect instantaneous channel $H$.
  • A direct check of whether the accepted step direction is descent on random channels would either close the gap between the practical algorithm and the invoked block-coordinate convergence theorem, or produce a counterexample.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 8 minor

Summary. This paper considers the weighted sum-rate (WSR) maximization problem for a downlink RIS-assisted multiuser MIMO system with Nt base-station antennas, K users with Nr antennas each, and Ns RIS elements. The authors propose an alternating optimization algorithm. For fixed RIS phases, the precoders are updated by applying successive convex approximation (SCA) to an equivalent lower-dimensional unconstrained reformulation (P2) borrowed from [30], yielding a closed-form update (Theorem 1, Eq. (10)). For fixed precoders, the RIS phases are updated by a single scaled projected gradient (SPG) step with scaling Xi = diag(1/|grad R|) and a new line-search rule (Eqs. (14)-(18)). The paper claims that the per-iteration complexity is linear in Nt and Ns (Eq. (20)), justifies the use of the original formulation rather than (P2) for the theta-update through an empirical Lipschitz-constant comparison (Fig. 2), and presents extensive simulations against several baselines (Figs. 3-7). The authors further claim that every accumulation point of the AO sequence is a stationary solution of (P1) (Section III-D).

Significance. The paper addresses a timely problem and provides a practically efficient solution. The per-iteration complexity analysis is careful and honest: the W-update in (10) works with KNr x KNr matrices rather than Nt x Nt ones; the gradient computation in (16) is kept linear in Ns by extracting only the diagonal of the relevant product; and Theorem 3 gives an explicit, verifiable gradient. The empirical Lipschitz comparison in Fig. 2 is a concrete, falsifiable design rationale, and the numerical study (six baselines, MIMO and MISO settings, flop counts and run times, scaling with Ns and Nt) is extensive; the reported reductions in complexity and runtime are substantial and credible. These are genuine strengths. The paper's main weakness is theoretical: the stationarity claim of Section III-D is not proved, and the supporting Lemma 1 has an inconsistency in the printed proof. If the authors either supply a rigorous two-block convergence argument or honestly downgrade the claim and support it with a numerical stationarity check, the paper would be a solid contribution to the RIS beamforming literature.

major comments (2)
  1. [Section III-D; Lemma 1 (Appendix F); Theorem 2 (Appendix C)] The stationarity claim in Section III-D, that any accumulation point of the sequence {W^(l), theta^(l)} is a stationary solution to (P1), is not established by the arguments provided. The cited [45, Chap. 2.7] concerns the projected gradient method for a single block with bounded (typically fixed) positive-definite scaling and a sufficient-decrease line search; the theta-update in (14)-(15) instead uses the state-dependent scaling Xi = diag(1/|grad R(theta^(l))|), which is unbounded whenever a gradient component vanishes and undefined when a component is exactly zero, so the cited theorem does not apply directly. In addition, the W-block update is not a gradient step but the output of the inner SCA loop of Algorithm 1 (stopped by an unspecified convergence test), so the joint sequence is not a gradient-projection sequence on (W, theta); Theorem 2 (Appendix C) likewise asserts stationarity 'following the same arguments in [45, Chap. 2]' without supplying the SCA-specific argument (e.g., gradient consistency of the minorant (7) and closedness of the subproblem map). The proof of Lemma 1 in Appendix F also contains an exponent inconsistency: inequalities (61)-(63) bound the objective increase by terms linear in the step norm, whereas the algorithm's line-search condition (18) is quadratic in the step norm, so the printed proof does not establish finite termination for the condition actually used; the argument can be repaired with the standard quadratic descent lemma, but the required bound then depends on min_n |grad R_n| and is not uniform near stationarity. I recommend either supplying a proper two-block convergence proof (for instance, establishing that the scaled projected steps are gradient-related with a sufficient-decrease line search and invoking a suitable alternating-optimization theorem) or softening the claim to convergence of the objective value and validating stationarity numerically (e.g., with a KKT-residual experiment).
  2. [Section III-E, Eq. (20); abstract] Equation (20) is a correct accounting of one outer iteration, and the diagonal-only evaluation of the vecd terms that keeps the gradient computation linear in Ns is a genuine contribution. However, the abstract's claim that the complexity 'scales linearly with the number of BS antennas and RIS reflective elements' is stronger than what (20) establishes: the expression is linear in Ns and Nt only for fixed values of I_theta and I_w, and no bound on I_theta is given. Because each line-search trial recomputes H at cost O(Ns Nt Nr K), a growing I_theta would break the linear-scaling claim; the paper's own Fig. 2 shows that the Lipschitz constant of grad R grows with Ns, which in a Lipschitz-based line search typically implies more evaluations as Ns increases. The numerical results in Figs. 5-7 do support near-linear total complexity in the tested range (Ns up to 202, Nt up to 128), so the claim is empirically credible, but the analysis establishes only a per-iteration bound, and the statement should be qualified accordingly (e.g., by reporting the observed I_theta and I_w counts alongside the complexity figures).
minor comments (8)
  1. [Section III-A; Appendix B] The full-row-rank condition on H(theta) used in (30d), and underlying the (P1)-(P2) equivalence from [30], should be stated as an explicit assumption in the main text, since the validity of the closed-form update in Theorem 1 at every AO iterate depends on it.
  2. [Section II-B] The sentence 'which is obviously not impractical for very large Nt and Ns' should read 'impractical'.
  3. [Section III-D] The word 'nontrival' should be 'nontrivial'.
  4. [Section IV-B (discussion of Fig. 6)] The acronym 'BSL1' appears in the discussion of Fig. 6 where 'BLS1' is meant.
  5. [Section III-C; Fig. 2] The procedure for turning 10^6 random samples of theta into the reported 'estimated Lipschitz constant' should be described (e.g., the norm used and whether the maximum is taken over all sample pairs), so that the figure is reproducible.
  6. [Appendix B, Eq. (28)] The trace manipulations leading from (28) to the gradient expression (29) are written without explicit matrix dimensions; rewriting them with dimensions would make the derivation of Theorem 1 considerably easier to verify.
  7. [Section IV-B] Adding the reduced WMMSE algorithm of [30] as a baseline for the W-update would directly benchmark the claimed advantage of the SCA update over the three-step WMMSE loop.
  8. [Section III-E] The dominance assumption Ns >> Nt >> KNr >= KNd >= K used in the complexity analysis should be introduced together with the system model, since the dimensionality reduction in (P2) is the source of the claimed savings.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivations rest on external equivalence [30], standard SCA arguments, and explicit heuristic choices; the unsupported convergence guarantee is a rigor gap, not circularity.

full rationale

The derivation chain is self-contained conditional on external references, and the paper's central algorithmic claims do not reduce to fitted inputs or self-citations. The W-update uses the equivalent reformulation (P2) from the external reference [30], and the SCA surrogate in (7) is proved in Appendix A, with the closed-form solution (10) proved in Appendix B and the convergence of inner iterations argued in Appendix C by standard SCA monotonicity. The theta-update uses the closed-form gradient (16) proved in Appendix D, while the scaling matrix Xi=diag(1/|grad_theta R|) in (15) is explicitly described as an experience-based algorithmic heuristic, not a parameter fitted to the reported WSR or to any benchmark. The line-search termination in Lemma 1 is proved in Appendix F, and the complexity expression (20) is an independent operation count. Simulations compare WSR and run time against external baselines rather than being tuned to reproduce a target value. Self-citations [19], [20], [24], [25] are contextual literature references and are not load-bearing: the key external input [30] is not authored by the present authors. The skeptical concern that the Section III-D stationarity claim is not established, because Xi is unbounded at stationarity and the alternating SCA/SPG updates do not directly fit the theorem in [45, Chap. 2.7], is a correctness/rigor issue rather than circularity: the claim is asserted without proof, but it is not an input to the derivation disguised as an output. Accordingly, no specific circular step can be quoted or exhibited under the required standard.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central algorithm depends on the external reduced-problem equivalence of [30] and on algorithmic hyperparameters that are chosen by hand. No new physical entities are introduced. The main unproven load-bearing premise is the convergence behavior of the scaled projected gradient step within the alternating loop.

free parameters (5)
  • Line search reduction factor η = 0.5
    Chosen by hand in Algorithm 2; controls how quickly the step size is reduced if condition (18) is violated. The convergence proof only requires η<1, not this exact value.
  • Line search constant β = 1e-7
    Set small in Algorithm 2 for the sufficient increase condition (18); chosen by hand and not derived.
  • AO stopping tolerance ε = 1e-5
    Stopping criterion for the outer alternating optimization loop; a numerical tuning parameter.
  • SPG scaling matrix Ξ = diag(1/|∇θR(θ)|)
    The scaled projected gradient normalization is motivated by numerical experience and not derived from an optimality or convergence condition.
  • Rician factor κ and path-loss parameters = κ=10; 3GPP LoS/NLoS path-loss constants
    Used in the simulation scenario; standard choices, not fitted to target WSR, but they determine the reported numerical gains.
assumptions (5)
  • domain assumption Equivalence between (P1) and the lower-dimensional (P2) as stated in [30, Prop. 5]
    The W-update and closed-form Theorem 1 depend on this equivalence; the paper omits the proof and refers to [30].
  • domain assumption H has full row rank, i.e., N_t ≥ K N_r, so Hbar = H H^H is invertible
    Used in Appendix B, Eq. (30d), to cancel Hbar^{-1} Hbar in the derivation of Theorem 1. Likely true for large N_t but not proven in general.
  • domain assumption Perfect channel state information is available at the BS
    Stated in Section II-A; standard in the literature but idealized.
  • standard math The inequality (7) from [42] is a valid global concave minorant
    The SCA bound for W-update relies on this external inequality; the paper gives a proof in Appendix A.
  • domain assumption Interference is treated as Gaussian noise, making (3) an achievable rate
    Standard rate expression used in (3) and in all baselines.

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Pith. "Pith review of On the Joint Beamforming Design for Large-scale Downlink RIS-assisted Multiuser MIMO Systems." pith.science (2026). https://pith.science/paper/GX6FILYY

@misc{pith2026241208320,
  author       = {Pith},
  title        = {Pith review of: On the Joint Beamforming Design for Large-scale Downlink RIS-assisted Multiuser MIMO Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GX6FILYY}},
  note         = {Machine review of arXiv:2412.08320}
}
read the original abstract

Reconfigurable intelligent surfaces (RISs) have huge potential to improve spectral and energy efficiency in future wireless systems at a minimal cost. However, early prototype results indicate that deploying hundreds or thousands of reflective elements is necessary for significant performance gains. Motivated by this, our study focuses on \emph{large-scale } RIS-assisted multi-user (MU) multiple-input multiple-output (MIMO) systems. In this context, we propose an efficient algorithm to jointly design the precoders at the base station (BS) and the phase shifts at the RIS to maximize the weighted sum rate (WSR). In particular, leveraging an equivalent lower-dimensional reformulation of the WSR maximization problem, we derive a closed-form solution to optimize the precoders using the successive convex approximation (SCA) framework. While the equivalent reformulation proves to be efficient for the precoder optimization, we offer numerical insights into why the original formulation of the WSR optimization problem is better suited for the phase shift optimization. Subsequently, we develop a scaled projected gradient method (SPGM) and a novel line search procedure to optimize RIS phase shifts. Notably, we show that the complexity of the proposed method \emph{scales linearly with the number of BS antennas and RIS reflective elements}. Extensive numerical experiments demonstrate that the proposed algorithm significantly reduces both time and computational complexity while achieving higher WSR compared to baseline algorithms.

Figures

Figures reproduced from arXiv: 2412.08320 by the authors.

Figure 1
Figure 1. Illustration of a downlink RIS-aided MU-MIMO [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Estimated Lipschitz constants of ∇θR˜(θ) and ∇θR(θ) for a RIS-aided MISO system with Nt = 64, K = 4, and PBS = 30 dBm. us fix F and rewrite (6) as a function of θ, i.e. R˜ k(θ) = log det I + H¯ k(θ)FkF H k H¯ k(θ) H × XK j̸=k H¯ k(θ)FjF H j H¯ k(θ) H + σ 2 0 PBS XK i=1 Tr(F H i H¯ (θ)Fi)I −1  , (19) where the dependency of the equivalent channels on θ is explicitly shown.. For the sake of brevity, we restrict ou… view at source ↗
Figure 3
Figure 3. Algorithms convergence behavior results for (a). RIS [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 6
Figure 6. Figure 6: Number of complex multiplications versus [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: illustrates the effect of Ns on the average problem￾solving time across different algorithms in comparison. 16 32 64 128 107 108 109 1010 1011 Number of transmit antennas, Nt Number of complex multiplications Algorithm 2 BLS1 WMMSE-MM [28] (a) 16 32 64 128 107 108 109 …

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Cited by 1 Pith paper

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    Optimizing SIM phase shifts before the digital beamformer, with an inner iterative projected-gradient loop, achieves up to 115.53% higher achievable sum rate than benchmark AO schemes in simulation.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.