{"id":"facd0b81-1fe1-46b5-ba6c-f2f6d870efcd","arxiv_id":"2506.10350","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A heterogeneous IRS with half static and half dynamic elements can be channel-estimated with lower pilot overhead and still reach near-conventional sum rate if its static elements are pre-configured for wide coverage.","lead":"This paper designs channel estimation and beamforming algorithms for a power-saving reflective surface whose elements are a mix of dynamic and static tuners. Splitting the channel into a dynamic part and a static part reduces pilot overhead and may let the cheaper surface nearly match a conventional one.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"STE equivalent channel sparsity/rank constraints in (18)/(19) are not guaranteed and may bias DSD-MO; need empirical verification under the paper's own settings, including WBS-MO optimized Ω.","rationale":"The central claim is that the HE-IRS channel decomposes into a DTE-based cascaded channel and an STE-based equivalent channel, so pilot overhead and estimation accuracy scale with NDTE rather than N. The decomposition itself is sound, but the proposed DSD-MO algorithm enforces exact sparsity/rank constraints on HSTE_eq,k that are only upper bounds mathematically. This is the load-bearing soft spot of the estimation contribution: if the equality constraints in (18) fail for the actual Ω, the estimator is solving the wrong feasibility problem and the claimed performance is not guaranteed. The paper's own Remark 2 acknowledges the lack of a guarantee, yet the simulation section never reports the empirical frequency of attaining the maximal row/column counts and rank. A direct Monte-Carlo check under the paper's parameters, including the WBS-MO-optimized Ω, would settle whether the concern is real or merely theoretical. The reader's weakest_assumption identifies the same issue, so I agree with that assessment. I note in passing that the offline beamforming derivation in (31), step (e), also appears to use a false inequality (the product of two nonnegative integrals can exceed their sum), and this should be corrected in revision, but the estimation assumption is the more load-bearing threat to the paper's headline pilot-overhead and sum-rate claims. Because the reader's verdict is already CONDITIONAL on these gaps, my stress-test does not change the recommended outcome.","tokens_in":23250,"tokens_out":11449,"duration_ms":147789,"concrete_test":"Reproduce the simulation setup of Section VI-A (N=48, P=Q=3, NBS=12, NUE=6, UPA geometry) and, for each Monte Carlo drop, compute the STE phase vector ω from WBS-MO and the actual matrix ΛHSTE_eq,k = A_BS^H GSTE Ω HSTE,k A_UE. Count the number of rows/columns whose norm exceeds a small threshold (e.g., 10^-6 of the maximum row/column norm) and numerically compute rank(HSTE_eq,k). Run at least 10^3 drops. If the fraction of drops with row count = P, column count = Q, and rank = min{P,Q} is not near 1, the equality constraints in (18) are violated in the paper's own operating regime and the DSD-MO NMSE/sum-rate curves should be recomputed using the true (lower) sparsity/rank. A sharper variant: grid-search or optimize Ω to minimize the row/column count; any physically reachable Ω that breaks the equalities demonstrates the assumption is not structural.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The DSD-MO estimator's correctness rests on Property 2 in Section IV-B and the constraints in (18): rank(HSTE_eq,k) = min{P,Q}, ||ΛHSTE_eq,k 1_NUE||_0 = P, and ||1_NBS^T ΛHSTE_eq,k||_0 = Q, with ΛHSTE_eq,k = A_BS^H GSTE Ω HSTE,k A_UE. The proof of (16) establishes only upper bounds (at most P nonzero rows and Q nonzero columns). Remark 2 explicitly concedes that Ω affects the sparsity and that the maximal values are 'not mathematically guaranteed', observed only 'generally' in simulations. The rank bound in (17) is also an inequality (≤), while (18) imposes equality. If, for the STE phases produced by the WBS-MO algorithm or for some channel geometry, cancellations inside A_STE^H Ω A_STE reduce the row/column counts or the rank below (P, Q, min{P,Q}), then the feasibility set in (18) excludes the true HSTE_eq,k, and the ℓ1-relaxed objective (19) penalizes the true channel as if it had extra nonzero rows/columns, biasing the estimate. The paper does not quantify how often the maximal values are reached, so the claimed pilot-overhead reduction and NMSE results are not yet robustly established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes channel estimation and two-stage beamforming schemes for a heterogeneous intelligent reflecting surface (HE-IRS) that combines dynamically tunable elements (DTEs) with power-free statically tunable elements (STEs) in a multi-user MIMO downlink/uplink system. The key idea is to decompose the HE-IRS channel into a DTE-based cascaded channel and an STE-based equivalent channel, so that pilot overhead scales with the number of DTEs rather than the total element count. The DSD-MO estimator alternates manifold-optimization updates of the two channel components using sparsity and rank constraints, with a robust rank selection rule for imperfect path-number knowledge. Beamforming is split into an offline WBS-MO algorithm for STE phase shifts and an online WMMSE-EI algorithm for the BS precoder and DTE phases. Simulations with N=48 elements show reduced pilot overhead and competitive sum rate relative to a conventional IRS, with higher energy efficiency.","tokens_in":23589,"tokens_out":23321,"duration_ms":269933,"significance":"If the central assumptions hold, the paper makes a useful contribution to green IRS design: it gives an algebraic channel decomposition that bypasses the identifiability problem of estimating the overall cascaded channel when some elements have fixed phase shifts, it provides explicit Riemannian gradients for the estimation algorithm, and it demonstrates through simulations that the HE-IRS can achieve most of the sum-rate benefit of a conventional IRS while consuming less power. The comparison against SIRS and conventional IRS, and the inclusion of both perfect- and estimated-CSI results, are strengths. The main risk is that the estimation algorithm is built on sparsity/rank equalities for the STE-equivalent channel that are not mathematically guaranteed and are only qualitatively supported by 'simulation tests'; this needs quantitative verification. The offline WBS-MO derivation also contains a normalization/inequality step that should be reconciled.","major_comments":[{"comment":"The DSD-MO estimator is built on the equalities rank(HSTE_eq,k)=min{P,Q}, ||ΛHSTE_eq,k 1_NUE||_0=P, and ||1_NBS^T ΛHSTE_eq,k||_0=Q imposed in (18). However, the proof of (16) establishes only upper bounds (at most P nonzero rows and Q nonzero columns), and (17) is an inequality. Remark 2 explicitly concedes that Ω affects the sparsity and that the maximal values are 'not mathematically guaranteed,' being only 'generally' observed in simulations. If cancellations inside A_STE^H Ω A_STE reduce the row/column counts or the rank below the imposed values, the feasibility set in (18) excludes the true HSTE_eq,k and the ℓ1 terms in (19) penalize the true channel, biasing the estimate and undermining the claimed pilot-overhead reduction. The paper contains no quantitative evidence, such as a histogram or table over the simulation settings (including Ω produced by WBS-MO and random channel geometries), showing how often the maximal values are attained. Please add such a verification; if the equality fails for a non-negligible fraction of cases, revise the problem formulation (for example, replace the equalities by inequalities or adapt the sparsity counts during optimization).","section":"Section IV-B and IV-C (Eqs. (16)-(18), Remark 2)"},{"comment":"The step labelled (e) in Eq. (31) is not justified by the Cauchy-Schwarz inequality alone. The pointwise factors x(ρy)=|ω_y^H b_y(ρy)|^2 and z(ρz)=|ω_z^H b_z(ρz)|^2 satisfy x,z≤1 only if the steering vectors in (4) are used with their 1/√M normalization, and even then the inequality is applied to integrals, whose values depend on the angular integration ranges; the paper does not state the required boundedness or normalization conditions. Moreover, Eq. (32) defines Ξ_y as N_y^STE times the matrix of integrals of e^{jπ(i-n)ρ}, whereas with the normalized b_y defined in (29) the integral of B_y is (1/N_y^STE) times that matrix, so the displayed equality in (31) is off by a factor. Because the y- and z-subproblems are then optimized separately in (33), the scaling may not change each individual optimizer, but the derivation should be corrected and the normalization of b_y/b_z stated explicitly.","section":"Section V-B (Eqs. (28)-(33))"}],"minor_comments":[{"comment":"The proof-by-contradiction in Lemma 3 is incorrect as written: the negation of the claim is not that ‖H−eH‖=0 for every rank-m matrix eH, but that the infimum of ‖H−eH‖ over such matrices is zero. The lemma itself is true (e.g., by the Eckart-Young theorem), but the proof should be replaced.","section":"Section IV-E (Lemma 3)"},{"comment":"The 'robust rank selection rule' is stated only as 'adopt a little higher rank than the directly estimated one'; the paper gives no procedure for choosing how much higher. In Fig. 8, the S-rank scheme uses the maximum possible rank (6), which is known from the simulation setup rather than obtained by the proposed rule, so the practical rule is under-specified.","section":"Section IV-E"},{"comment":"The energy-efficiency simulation uses PIRS = Pstatic + Σ tm·PPIN with Pstatic=15 dBm and PPIN=12 dBm. It would be helpful to state explicitly whether these quantities already account for the STE power consumption being zero and for the control-circuit power of the DTEs, so that the 'power-free STEs' claim is transparent.","section":"Section VI-D"},{"comment":"There are several presentation issues: 'highy' should be 'highly' in Section IV-C; Eq. (36) uses log without specifying the base although rates are reported in bits/s/Hz; and the curves in Figs. 3 and 4 would be easier to read with distinct markers.","section":"General / notation"}],"recommendation":"major_revision","confidential_remarks":"The central concern is whether the sparsity/rank equalities imposed on the STE-equivalent channel hold sufficiently often under the paper's own simulation settings; this is directly tied to the claimed pilot-overhead reduction and estimation accuracy. If the authors provide the requested Monte Carlo verification and correct the WBS-MO derivation, the paper could become acceptable. The novelty relative to the authors' prior work [12] is incremental but appears sufficient for a communications journal, given the new channel estimation and beamforming algorithms."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The DTE-STE decoupled channel model is the real contribution here. Instead of estimating the whole cascaded channel, the authors show it splits into a DTE cascaded part whose dimension scales with N_DTE and an STE equivalent part of fixed dimension N_BS x N_UE. That is a clean observation that yields a real pilot-overhead reduction, and the DSD-MO estimator with alternating manifold optimization is a sensible way to go after it. The simulations back the overhead claim: HE-IRS reaches a given NMSE with fewer pilots than a conventional IRS, and the EE gains in Fig. 7 follow from half the elements consuming no power. The baseline comparison against CS-EST is fair, and the robust rank selection rule is a thoughtful addition.\n\nThere are two soft spots, and one of them is smaller than the reader's note suggests. The complaint about Eq. (31) being a false inequality does not survive contact with the paper. The gains |b_y^H omega_y|^2 and |b_z^H omega_z|^2 are each bounded by unity because the array vectors are normalized, so for a,b in [0,1], ab <= a+b. The step is a valid upper bound on the integrated product, and separately maximizing the two decoupled terms is a legitimate, if crude, way to broaden the beam. Not a flaw.\n\nThe real concern is the assumption in (18) that H_STE_eq,k has exactly P nonzero angular rows, Q nonzero angular columns, and rank min{P,Q}. Remark 2 admits these are not mathematically guaranteed, and the proof only gives upper bounds. What partially saves the paper is that (19) relaxes the L0 constraints to L1 penalties, so the exact counts are not enforced; the rank constraint remains, but Lemma 2 shows an overestimated rank can still approximate the true matrix arbitrarily closely. So the load-bearing risk is lower than the stress-test note implies. Still, the paper should not rest on \"we observed in simulation tests.\" For the WBS-MO optimized Omega in particular, the authors should either prove that the maximal sparsity/rank holds under the wide-beam design or provide a sensitivity analysis showing NMSE is not hurt when the assumption fails.\n\nWho benefits: people working on IRS channel estimation, especially those interested in energy-efficient RIS structures. The decoupling idea may transfer to other mixed-element surfaces.\n\nBottom line: this deserves a serious referee. The central model is new, the algorithms are concrete, and the simulations are honest. The paper needs a major revision to support the STE sparsity/rank claim and to test against a case where the assumption is violated. I would send it out.","headline":"A genuinely new decoupled channel model for heterogeneous IRS, worth engaging, but the STE-channel sparsity assumption needs stronger support than 'simulation tests.'","tokens_in":24078,"tokens_out":5138,"would_cite":true,"duration_ms":60006,"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 in a heterogeneous intelligent reflecting surface, the channel to estimate splits into a dynamic-element cascaded channel and a static-element equivalent channel, so pilot overhead scales with the number of dynamic…","keywords":["heterogeneous intelligent reflecting surface","channel estimation","beamforming","manifold optimization","multi-user MIMO","pilot overhead","sparse channel estimation","energy efficiency"],"falsifier":"Choose an STE phase configuration $\\Omega$ that spreads the incident wave across many angular bins and run DSD-MO with $P$ and $Q$ set to the true path counts; if the normalized estimation NMSE degrades sharply or the fixed-rank constraints become infeasible for that configuration, the fixed row-column sparsity assumption for $H_{\\mathrm{STE,eq},k}$ fails.","tokens_in":23059,"feed_emoji":"📡","tokens_out":7639,"duration_ms":74998,"temperature":0.7,"pith_summary":"The paper studies an intelligent reflecting surface (IRS) in which some elements are dynamically tunable (DTEs) and the rest are statically tunable (STEs) and consume no power. It argues that the usual approach of estimating one overall cascaded channel breaks down for this design, because the fixed STE phases make the measurement matrix rank-deficient. Instead, the channel can be split into a DTE-based cascaded channel and an STE-based equivalent channel, and each piece can be estimated from its angular sparsity and low-rank structure using manifold optimization. The result the authors are trying to establish is that this decoupled estimation needs fewer pilots than a conventional all-dynamic IRS of the same size, and that with offline STE beamforming plus online DTE and precoder optimization it achieves nearly the same sum rate at much lower power consumption.","feed_headline":"Mixed static-dynamic IRS needs pilots only for dynamic elements","feed_subtitle":"Decoupled channel estimation lets a half-static smart surface match full-dynamic sum rates at lower power.","key_machinery":"The load-bearing object is the DTE-STE decoupled (DSD) channel model: the received signal is split into $G_{\\mathrm{DTE}}\\Phi_k[t]H_{\\mathrm{DTE},k}$ plus $G_{\\mathrm{STE}}\\Omega H_{\\mathrm{STE},k}$, so only the dynamic part carries a time-varying phase matrix while the static part collapses into one fixed equivalent channel $H_{\\mathrm{STE,eq},k}$. Estimation then rides on three props: the angular sparsity of $G_{\\mathrm{DTE}}$ and $H_{\\mathrm{DTE},k}$, the low-rank property of all three channel matrices, and manifold optimization over fixed-rank Riemannian manifolds to respect those rank constraints. A robust rank selection rule handles path-number mismatch by choosing a rank slightly above the estimated one, justified by Lemmas 2 and 3: exceeding the true rank loses nothing in approximation accuracy, while undershooting it permanently excludes the true matrix. For beamforming, the offline WBS-MO algorithm synthesizes a wide beam over the UE location area by separating the STE array's $y$- and $z$-direction gains, and the online WMMSE-EI algorithm alternatively updates quantization-aware DTE phases and the BS precoder.","core_discovery":"For a heterogeneous IRS, estimating the full cascaded channel $H_{\\mathrm{ca},k}=H_k^T \\odot G$ is impossible once there is more than one static element, because the repeated constant rows in the phase-shift matrix reduce the rank of the measurement matrix. The paper's central move is to rewrite the received signal as $(G_{\\mathrm{DTE}}\\Phi_k[t]H_{\\mathrm{DTE},k}+G_{\\mathrm{STE}}\\Omega H_{\\mathrm{STE},k})s_k[t]$, estimating the DTE-based cascaded channel and the STE-based equivalent channel $H_{\\mathrm{STE,eq},k}=G_{\\mathrm{STE}}\\Omega H_{\\mathrm{STE},k}$ separately. The DSD-MO algorithm alternates between these two blocks and, within the DTE block, between $G_{\\mathrm{DTE}}$ and $H_{\\mathrm{DTE},k}$, using manifold optimization on fixed-rank sets and $\\ell^1$ regularization on angular-domain sparsity. For the estimated ranks, the paper proves that a higher-than-true-rank constraint can approximate the true channel arbitrarily well while a lower one cannot, and uses this to justify choosing a slightly inflated rank under path-number mismatch. Beamforming is split into an offline wide-beam synthesis for the STEs and an online WMMSE-style element-iteration update for the DTEs and the BS precoder.","pith_inferences":["An extension the paper leaves implicit is applying the same DTE-STE decoupling to wideband or OFDM systems, where the angular-sparsity assumption must be checked per subcarrier rather than once.","The pilot-cost split implies a design knob: increasing the static-element fraction lowers training and power cost at some sum-rate expense, so a practical system could tune the DTE/STE ratio to the channel's angular richness.","Because Remark 2 concedes that the static-element equivalent channel's exact sparsity is only observed in simulations, a testable extension is to replace the fixed $P$ and $Q$ constraints with an adaptive sparsity or rank selection that monitors the estimation residual."],"forward_implications":["Pilot overhead for an HE-IRS-assisted system is governed by $N_{\\mathrm{DTE}}+1$ rather than the total element count $N$, so replacing half the dynamic elements with static ones can roughly halve the training cost.","Conventional least-squares estimation of the overall cascaded channel fails for any HE-IRS with more than one static element, so the decoupled model is not a minor refinement but a necessary reformulation.","With the proposed estimation and two-stage beamforming, the HE-IRS reaches a sum rate close to that of a conventional all-dynamic IRS while using fewer pilots and less power.","In energy efficiency, the HE-IRS can beat both the conventional IRS and the sparse IRS because its static elements contribute no ongoing power draw.","The robust rank selection rule keeps estimation stable when the number of propagation paths is unknown or misestimated."],"supporting_citations":[{"why":"Introduces the HE-IRS structure with DTEs and STEs that this paper builds on for channel estimation and beamforming.","marker":"[12]"},{"why":"Supplies the angular-sparsity channel model, the manifold-optimization technique, and the three-stage CS-EST baseline for IRS mmWave MIMO estimation.","marker":"[18]"},{"why":"Documents PIN-diode power consumption, motivating the power-free STE design and the energy-efficiency comparison.","marker":"[4]"},{"why":"Provides the LS cascaded-channel estimation framework whose rank-deficiency analysis grounds Lemma 1.","marker":"[14]"},{"why":"Gives the low-rank property of sparse IRS channels used to justify the fixed-rank manifold constraints.","marker":"[29]"},{"why":"Defines the complex-circle Riemannian manifold used for constant-modulus phase-shift optimization in WBS-MO.","marker":"[34]"}],"fun_headline_variants":["Decoupled channels let static-dynamic IRS need fewer pilots","Hybrid IRS: estimate only dynamic elements to cut pilots","Green IRS with static elements matches full-dynamic sum rate","Rank-aware channel estimation reduces pilot overhead in HE-IRS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The estimation problem assumes the static-element part of the channel has exactly $P$ nonzero angular rows, exactly $Q$ nonzero angular columns, and rank $\\min\\{P,Q\\}$; the paper admits this is not mathematically guaranteed because the STE phase settings affect the sparsity, so the central claim stands on sparsity observed in simulations rather than proved.","fun_headline_variants_meta":{"raw":{"variants":["Decoupled channels let static-dynamic IRS need fewer pilots","Hybrid IRS: estimate only dynamic elements to cut pilots","Green IRS with static elements matches full-dynamic sum rate","Rank-aware channel estimation reduces pilot overhead in HE-IRS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000481,"raw_usage":{"total_tokens":2470,"prompt_tokens":1128,"completion_tokens":1342,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":744,"completion_tokens_details":{"reasoning_tokens":1287}},"tokens_in":744,"tokens_out":1342,"duration_ms":14502,"temperature":1.0,"reasoning_tokens":1287,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:29:42.154945+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose an STE phase configuration $\\Omega$ that spreads the incident wave across many angular bins and run DSD-MO with $P$ and $Q$ set to the true path counts; if the normalized estimation NMSE degrades sharply or the fixed-rank constraints become infeasible for that configuration, the fixed row-column sparsity assumption for $H_{\\mathrm{STE,eq},k}$ fails.","supporting_citations":[{"cited_title":"Heterogeneous-IRS: A green IRS structure with dynamically and statically tunable elements,","cited_arxiv_id":null,"evidence_quote":"Introduces the HE-IRS structure with DTEs and STEs that this paper builds on for channel estimation and beamforming."},{"cited_title":"Channel estimation for IRS-assisted millimeter-wave MIMO systems: Sparsity-inspired approaches,","cited_arxiv_id":null,"evidence_quote":"Supplies the angular-sparsity channel model, the manifold-optimization technique, and the three-stage CS-EST baseline for IRS mmWave MIMO estimation."},{"cited_title":"Reconfigurable intelligent surface: Power consumption modeling and practical measurement validation,","cited_arxiv_id":null,"evidence_quote":"Documents PIN-diode power consumption, motivating the power-free STE design and the energy-efficiency comparison."},{"cited_title":"An optimal channel estimation scheme for intelligent reflecting surfaces based on a minimum variance unbiased estimator,","cited_arxiv_id":null,"evidence_quote":"Provides the LS cascaded-channel estimation framework whose rank-deficiency analysis grounds Lemma 1."},{"cited_title":"Channel estimation for practical intelligent reflecting surface-aided millimeter wave MIMO-OFDM systems,","cited_arxiv_id":null,"evidence_quote":"Gives the low-rank property of sparse IRS channels used to justify the fixed-rank manifold constraints."},{"cited_title":"Alternating minimization algorithms for hybrid precoding in millimeter wave MIMO systems,","cited_arxiv_id":null,"evidence_quote":"Defines the complex-circle Riemannian manifold used for constant-modulus phase-shift optimization in WBS-MO."}],"review_version":1}