{"id":"3d4565e0-4170-464f-88f6-c5545214da78","arxiv_id":"2412.08320","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An alternating optimization scheme with a closed-form SCA precoder update and a scaled projected gradient phase-shift update achieves linear per-iteration complexity in antennas and RIS elements for RIS-assisted MU-MIMO.","lead":"This paper presents an algorithm that jointly designs the base station precoders and the RIS phase shifts in a large-scale multiuser MIMO downlink, maximizing the weighted sum rate. If the complexity claims hold, it makes RIS-assisted massive MIMO optimization practical at scale, which is relevant for beyond-5G system design.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence to a stationary point is not proven: the SPG scaling Ξ=diag(1/|∇θR|) is unbounded at stationarity, so the cited [45, Chap. 2.7] theorem does not apply to the alternating SCA/SPG updates.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the convergence proof in Section III-D is only sketched and does not verify the assumptions of the cited theorem. I agree because the specific scaling Ξ=diag(1/|∇θR|) is unbounded at points where the gradient vanishes, and the alternating update (SCA for W, single SPG step for θ) does not directly fit the gradient-projection framework of [45, Chap. 2.7]. This is the central theoretical pillar of the paper: without a valid stationarity guarantee, Algorithm 2 is a heuristic, even though the numerical results are favorable. I also considered the complexity claim; the per-outer-iteration complexity is convincingly shown to be linear in Nt and Ns, but the total complexity depends on Iθ and the number of outer iterations, which are not rigorously bounded. Still, the convergence gap is the more fundamental issue. The proposed concrete test is an analytical re-derivation: if one cannot prove gradient-relatedness with bounded scaling, the stationarity claim should be weakened. This does not change the reader's CONDITIONAL verdict; it reinforces it. The numerical experiments provide partial support, but the missing proof and the absence of the closest linear-complexity baseline [30] justify conditional acceptance rather than outright rejection.","tokens_in":22446,"tokens_out":30861,"duration_ms":297756,"concrete_test":"Independently re-derive the stationarity claim in Section III-D by checking the two conditions needed to apply [45, Chap. 2.7] to the alternating updates: (i) the direction d(ℓ)=Ξ(ℓ)∇θR(θ(ℓ)) is gradient-related and the eigenvalues of Ξ(ℓ) are bounded away from 0 and ∞ on the feasible set; (ii) the W-update produces a point satisfying the first-order optimality conditions of (P1) for θ fixed. If condition (i) fails because |∇θR_n| can be arbitrarily small (exactly 0 at a stationary point), then the theorem does not apply and Algorithm 2's convergence guarantee is unsupported; in that case, a bounded regularization of Ξ should be stated and analyzed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III-D asserts that any accumulation point of {W(ℓ),θ(ℓ)} is stationary for (P1), citing [45, Chap. 2.7]. That theorem requires a gradient-related direction with a bounded positive-definite scaling and a sufficient-decrease line search. The proposed SPG step sets Ξ=diag(1/|∇θR(θ(ℓ))|), which is unbounded whenever any component of the phase gradient vanishes; at a stationary point all components vanish, so Ξ→∞. Thus the standard gradient-projection convergence theorem is not directly applicable. Moreover, the W-update is not a gradient step but the output of the iterative SCA Algorithm 1, so the alternating sequence is not a gradient-projection sequence on the joint variable; invoking [45, Chap. 2.7] on the combined updates is unjustified. Lemma 1 only proves finite termination of the inner line search for a fixed Ξ; it does not establish that the outer limit satisfies the stationarity condition. Hence the convergence guarantee is unsupported, and the algorithm is only empirically convergent. Separately, the claimed 'linear in Ns' complexity in (20) counts Iθ line-search iterations; if Iθ grows with Ns (the Lipschitz constant of ∇θR generally grows with Ns), the strict linear-scaling claim is not established by the analysis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":22736,"tokens_out":32867,"duration_ms":315596,"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":[{"comment":"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).","section":"Section III-D; Lemma 1 (Appendix F); Theorem 2 (Appendix C)"},{"comment":"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).","section":"Section III-E, Eq. (20); abstract"}],"minor_comments":[{"comment":"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.","section":"Section III-A; Appendix B"},{"comment":"The sentence 'which is obviously not impractical for very large Nt and Ns' should read 'impractical'.","section":"Section II-B"},{"comment":"The word 'nontrival' should be 'nontrivial'.","section":"Section III-D"},{"comment":"The acronym 'BSL1' appears in the discussion of Fig. 6 where 'BLS1' is meant.","section":"Section IV-B (discussion of Fig. 6)"},{"comment":"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.","section":"Section III-C; Fig. 2"},{"comment":"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.","section":"Appendix B, Eq. (28)"},{"comment":"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.","section":"Section IV-B"},{"comment":"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.","section":"Section III-E"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's numerical core is solid and the proposed algorithm is clearly useful in practice; my recommendation of major_revision is driven by the theory gap in Section III-D and the overstrong complexity claim, both of which are fixable in revision. There are no citation-pattern concerns: the equivalence (P2) is properly credited to [30], and self-citations are contextual. I suggest asking the revision to include a KKT-residual experiment if the stationarity proof is not completed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: solid extension, not a breakthrough. The genuinely new parts are the SCA closed-form W-update for the reduced problem in Theorem 1 and the scaled projected gradient with the 1/R(θ) line-search initialization. The complexity analysis is careful, and the run-time plots support the linear-in-Ns story. The empirical Lipschitz comparison in Section III-C is a nice, honest justification for using the original WSR formulation for the θ-update.\n\nThe soft spots are real but localized. First, the convergence claim in Section III-D is a sketch, not a proof. The scaling Ξ = diag(1/|∇θR|) is unbounded when any gradient component is zero, so the standard projected-gradient theorem from [45, Chap. 2.7] does not apply as stated. And the W-update is the output of SCA, not a gradient step, so the alternating sequence is not a projected-gradient sequence on the joint variable. Lemma 1 only shows finite termination of the inner line search for a fixed Ξ; it does not give outer stationarity. I would bet the algorithm converges in practice—the simulations are consistent with that—but the claim should be downgraded to 'empirically convergent' until a real proof appears.\n\nSecond, the 'linear in Ns' claim in (20) treats Iθ as a constant. That is the per-outer-iteration complexity; the total complexity is linear only if Iθ does not grow with Ns. They give no argument for that, and the Lipschitz constant of ∇θR generally grows with Ns. Their own plots suggest Iθ is small, so this is likely fine, but it should be stated as a per-iteration result with empirical support for the total.\n\nThird, since the reduced problem and the linear-in-Nt trick come from [30], and R-WMMSE from [30] is not included as a baseline, the superiority claim is not fully tested against the closest linear-complexity work. No code is posted either, so independent verification is limited. These are gaps, not fatal ones.\n\nBottom line: for someone in RIS-aided MIMO, this is a useful, credible algorithm paper. It deserves a serious referee. The convergence section needs major revision, and the scaling caveat should be said plainly. Accept for review.","headline":"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.","tokens_in":23297,"tokens_out":4469,"would_cite":true,"duration_ms":51205,"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":"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…","keywords":["reconfigurable intelligent surface","weighted sum rate maximization","multiuser MIMO","beamforming design","scaled projected gradient method","phase shift optimization","alternating optimization","linear complexity"],"falsifier":"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.","tokens_in":22234,"feed_emoji":"📡","tokens_out":10821,"duration_ms":106349,"temperature":0.7,"pith_summary":"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.","feed_headline":"Joint precoder and RIS phase design drops to linear cost","feed_subtitle":"A scaled projected gradient step makes weighted-sum-rate beamforming practical for thousand-element intelligent surfaces.","key_machinery":"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.","core_discovery":"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 $N_s$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It establishes the equivalence between the original WSR problem (P1) and the lower-dimensional unconstrained problem (P2), and the recovery formula $W = \\sqrt{\\xi}H^H F$ that the precoder update relies on.","marker":"[30]"},{"why":"It provides the BCD/FP baseline for RIS-aided MU-MISO that the paper compares against, and its $O(N_s^2)$ per-iteration complexity is the contrast for the linear-scaling claim.","marker":"[22]"},{"why":"It provides the WMMSE-MM baseline for RIS-aided MU-MIMO, the main benchmark for the MIMO simulations, with cubic complexities that motivate the new method.","marker":"[28]"},{"why":"The convergence arguments for both Algorithm 1 and Algorithm 2 invoke this reference for monotone convergence and accumulation-point stationarity.","marker":"[45, Chap. 2.7]"},{"why":"It supplies the determinant inequality (7) used to derive the concave quadratic minorant and the closed-form SCA precoder update.","marker":"[42]"},{"why":"It supplies the complex-valued matrix derivative rules and trace identities used to derive the closed-form gradient in (16) and to count multiplications in Section III-E.","marker":"[31]"},{"why":"It is cited for the NP-hardness of the precoder subproblem with fixed $\\theta$, which motivates the alternating approximate approach.","marker":"[35]"}],"fun_headline_variants":["Linear-complexity joint beamforming for large RIS-MIMO","Scaled gradient makes RIS phase tuning linear in size","Joint precoder and RIS phases at linear cost","RIS-assisted MIMO beamforming in linear time","Closed-form precoders plus scaled step for RIS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Linear-complexity joint beamforming for large RIS-MIMO","Scaled gradient makes RIS phase tuning linear in size","Joint precoder and RIS phases at linear cost","RIS-assisted MIMO beamforming in linear time","Closed-form precoders plus scaled step for RIS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000771,"raw_usage":{"total_tokens":3540,"prompt_tokens":1195,"completion_tokens":2345,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":811,"completion_tokens_details":{"reasoning_tokens":2270}},"tokens_in":811,"tokens_out":2345,"duration_ms":15588,"temperature":1.0,"reasoning_tokens":2270,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:59:00.446786+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Multicell MIMO communications relying on intelligent reflecting surfaces,","cited_arxiv_id":null,"evidence_quote":"It provides the WMMSE-MM baseline for RIS-aided MU-MIMO, the main benchmark for the MIMO simulations, with cubic complexities that motivate the new method."},{"cited_title":"Successive convex quadratic programming for quality-of-service management in full-duplex MU- MIMO multicell networks,","cited_arxiv_id":null,"evidence_quote":"It supplies the determinant inequality (7) used to derive the concave quadratic minorant and the closed-form SCA precoder update."},{"cited_title":"Hjørungnes, Complex-valued matrix derivatives: with applications in signal processing and communications","cited_arxiv_id":null,"evidence_quote":"It supplies the complex-valued matrix derivative rules and trace identities used to derive the closed-form gradient in (16) and to count multiplications in Section III-E."},{"cited_title":"Dynamic spectrum management: Complexity and duality,","cited_arxiv_id":null,"evidence_quote":"It is cited for the NP-hardness of the precoder subproblem with fixed $\\theta$, which motivates the alternating approximate approach."}],"review_version":1}