{"id":"9f04b38f-b92c-47c2-ae22-8d9d6b0d2693","arxiv_id":"2602.05724","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An MMSE-based Bayesian calibration algorithm with von Mises denoisers estimates the up/downlink gain ratio of dual-antenna repeaters, beating NLS calibration by 4–14 dB RMSE at similar cost.","lead":"A Bayesian calibration method is proposed for estimating the gain-ratio mismatch of dual-antenna repeaters in MIMO systems, using MMSE estimation with circular von Mises denoisers. If it works, it improves calibration accuracy by 4–14 dB RMSE over existing nonlinear least squares at similar complexity, relevant to practical repeater-assisted 6G deployments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The MoM estimator of |γ|² in Eq. (48) is self-referential: s in Eq. (49c) requires V_D^γ from Eq. (44), which contains the unknown φ_γ=E|γ|², and Algorithm 3 never initializes φ_γ before using it. The γ-update is therefore undefined as written.","rationale":"The reader's weakest-assumption identification is exactly the load-bearing concern. The paper's contribution rests on Algorithm 3 outputting an MMSE estimate of γ; before that, Lemma 2 must produce a usable |ˇγ|². But Eq. (48) depends on s, which depends on V_D^γ, which depends on φ_γ — the unknown second moment that the MoM is supposed to estimate. The text acknowledges φ_γ is unavailable and proposes to substitute the MoM estimate, but no separate estimator of φ_γ is supplied; Algorithm 3 computes V_D^γ before any candidate |ˇγ|² exists. This is not merely a typographical issue: either the estimator is implicitly defined as a fixed point that must be solved and the paper omits that solve step, or it is circular and undefined. Both readings invalidate the claimed simulation results as reproducible from the printed algorithm. I credit the paper's solid derivations for the A/B updates, the von Mises denoiser, and the complexity analysis, but those do not rescue the final γ-step, which is the deliverable. The concern is not about disagreement with consensus or a stylistic weakness; it is an internal inconsistency in the central algorithm. A revised version that defines the fixed-point equation, initializes φ_γ explicitly (e.g., φ_γ=1 or a two-pass estimate), and updates the pseudocode could move the paper toward conditional acceptance, but as submitted the reader's REJECT verdict stands.","tokens_in":20991,"tokens_out":4693,"duration_ms":54600,"concrete_test":"Re-derive Lemma 2 under the deterministic-γ interpretation φ_γ=|γ|². Substitute (44) into (49c) and factor s = |γ|²·s′, where s′ is independent of |γ|²; solve Eq. (48) for x=|γ|² and check whether a nonnegative closed-form root exists. Then instrument Algorithm 3 with an uninitialized-variable check and run the Section V configuration, e.g., (M_A,M_B)=(64,32), exactly as written. If execution reaches line 21 with no value assigned to φ_γ, the algorithm is undefined; if the fixed-point root differs from the value implicitly used to generate Fig. 4, the missing initialization changes the numerical results.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central output of Algorithm 3 is the MMSE estimate of γ, but the MoM step that supplies the prior radius |ˇγ| is circular. Lemma 2 defines |ˇγ|² = (|q|²−u)/(u²+s), with s = ˆdᴴΣ_A⁻¹ V_D^γ Σ_A⁻¹ ˆd. Equation (44) defines V_D^γ = diag(v_{γ,ij}) where each v_{γ,ij} contains φ_γ = E|γ|². Since φ_γ is declared unavailable, the text proposes to replace it with the MoM estimate of |γ|² — the very quantity being estimated. Algorithm 3 lines 21–24 compute V_D^γ and s before any |ˇγ|² exists, with no initialization, fixed-point iteration, or clipping of negative estimates. Even if one treats γ as deterministic so φ_γ=|γ|², Eq. (48) becomes an implicit quadratic equation in x=|γ|²; the paper neither states nor solves this fixed-point equation. As printed, the algorithm is not executable, and the RMSE curves in Fig. 4 cannot be reproduced from Algorithm 3 without supplying an extra, unspecified value for φ_γ. This is a load-bearing gap because the claimed 4–14 dB gains depend on the complete γ-MMSE update being well defined.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Bayesian MMSE calibration method for dual-antenna repeaters in TDD MIMO systems. From four measurements (8), the repeater gain ratio γ=β/α and the antenna reciprocity matrices A, B are estimated. The proposed Algorithm 3 uses PDA-style alternating MMSE updates: point estimates for H and Z, von Mises denoisers for the diagonal entries of A and B with a uniform phase prior (β=0), and a MoM-based MMSE update for γ. Simulations claim 4–14 dB RMSE gains over NLS baselines at comparable complexity and convergence within about 4 iterations.","tokens_in":21279,"tokens_out":8981,"duration_ms":91834,"significance":"If correct, the paper would provide a practical Bayesian alternative to deterministic NLS calibration for repeater-assisted MIMO, with detailed derivations for the A/B updates (Eqs. 18–39), a self-contained von Mises denoiser derivation, and a complexity analysis distinguishing full covariance and diagonal-only implementations. However, the γ-estimation step is not executable as printed: the MoM estimator of |γ|² is self-referential, so the central output of Algorithm 3 is undefined. The claimed performance gains therefore are not supported by the printed algorithm until that step is repaired.","major_comments":[{"comment":"The MoM estimator of |γ|² is self-referential. Equation (49c) defines s = d̂ᴴΣ_A⁻¹ V_D^γ Σ_A⁻¹ d̂, and V_D^γ in (44) contains φ_γ = E|γ|². The text proposes to replace φ_γ with the MoM estimate |γ̂|², i.e. the very quantity being defined. Algorithm 3 computes V_D^γ (line 21) before any |γ̂|² exists (line 24) and gives no initialization or fixed-point update. If γ is treated as deterministic so φ_γ = |γ|², (48) becomes an implicit quadratic equation in x = |γ|²; the paper neither states nor solves this equation. Consequently Algorithm 3 is not executable, and the RMSE curves in Figs. 4 and 5 cannot be reproduced from the printed pseudocode without supplying an extra unspecified φ_γ. Please specify a well-defined procedure (e.g. initialize φ_γ and iterate the MoM update, or solve the implicit equation) and clip non-positive estimates.","section":"§IV.C.4 (Lemma 2, Eqs. (48)–(49c); Algorithm 3, lines 21–24)"},{"comment":"The estimator is called “consistent”, but the proof replaces the ensemble moment E|q|² by the single-sample quantity |q|². No averaging over repeated observations or concentration argument over the M_A M_B dimensions is provided. As written, the proof establishes only a moment-matching equation for one realization. Please state the asymptotic regime in which consistency is claimed and justify the approximation, or revise the wording.","section":"§IV.C.4, Lemma 2 proof (Eqs. (50)–(52))"}],"minor_comments":[{"comment":"The notation p_{vec(R4)|γ} denotes the conditional PDF of R4 given γ, but the surrounding text says “given R4”; please correct this notational inconsistency.","section":"§IV.C.4, Eq. (42)"},{"comment":"The symbols \\bar{Σ}_A and \\bar{Σ}_B are introduced without prior definition, while Section IV later uses Σ_A = Ψ_A ⊗ Ω_A. Please define these consistently.","section":"§IV.A.2, after Eq. (16)"},{"comment":"The γ update writes tr{(Â Ẑ ˆB)^H R4}; Eq. (14) and Algorithm 1 use tr{(Â Ẑ^T ˆB)^H R4}. The missing transpose appears to be a typo.","section":"Algorithm 2, line 12"},{"comment":"Please check the placement of the square in the posterior MSE formula. The standard expression for the von Mises denoiser at radius |γ̂| is |γ̂|²(1 − (I1/I0)²), and the typesetting in (53b) is ambiguous.","section":"Eq. (53b)"},{"comment":"Line 27 uses |γ̂| in the definition of ζ_γ, but the surrounding derivation and Eq. (54) use |γ̌|; please correct the symbol.","section":"Algorithm 3, line 27"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is real and it is the central blocker. The A/B half of the paper is solid, and the fix is plausible, so I am not recommending reject; however, the revised version must contain a corrected, well-defined γ estimator and new simulation results. If the authors cannot provide a well-defined γ update, the paper should ultimately be rejected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short take: this paper does something genuinely new—it casts dual-antenna repeater reciprocity calibration as Bayesian MMSE estimation with PDA bilinear inference and von Mises denoisers—but the step that estimates the gain ratio γ, the main output, is not actually well-defined as written. The A and B estimation is derived carefully (Eqs. 18–39) and the von Mises denoiser is a sensible application of standard results. The improvement over the NLS baseline [27] may well be real, but I can't fully trust the results until the γ step is fixed.\n\nThe problem is in Lemma 2. The MoM estimator of |γ|² in (48) is |γ̂|² = (|q|² − u)/(u² + s). But s in (49c) depends on V_D^γ in (44), which contains φ_γ = E|γ|². The text says to replace φ_γ by the MoM estimate—the very quantity being estimated. Algorithm 3 lines 21–24 compute V_D^γ and s before any |γ̂|² exists, with no initialization or fixed-point iteration. If you treat γ as deterministic so φ_γ = |γ|², (48) becomes a quadratic in |γ|²; the paper doesn't state or solve it. So as printed, the algorithm is not executable. This is load-bearing because the claimed 4–14 dB gains rest on this MMSE update.\n\nThere's a fallback mentioned in the text: set φ_γ = 1. That would make the step well-defined, but it isn't what Algorithm 3 does. A simple fix—initializing with the NLS estimate of γ, or using φ_γ = 1, or iterating the MoM equation—would likely resolve this, but it needs to be in the paper.\n\nSecondary issues: the simulation setup exactly reproduces the unit-modulus/negligible-amplitude-error assumption used in the derivation, so the gains are partly in-model. The abstract also overclaims \"complete statistical knowledge\" when the phase priors are non-informative and the γ prior is estimated from data.\n\nOverall: the core idea is credible, the A/B derivation is solid, and the problem is practically relevant. The flaw is real but localized and fixable. I'd send this to peer review—the authors need to fix the γ step before it can be accepted, but a good referee could help them get there.\n\nRecommendation: engage with it, but ask for a corrected Algorithm 3 and robustness experiments.","headline":"Credible Bayesian reformulation of repeater calibration, but the γ-prior MoM step is circular as written—Algorithm 3 is not executable without a fix.","tokens_in":21903,"tokens_out":3147,"would_cite":false,"duration_ms":32174,"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":"This paper claims that reciprocity calibration of a dual-antenna repeater—estimating the ratio γ = β/α between the repeater's forward and reverse gains—can be recast as a Bayesian MMSE problem whose estimator, built on von Mises denoisers a","keywords":["reciprocity calibration","dual-antenna repeater","repeater-assisted MIMO","MMSE estimation","von Mises denoiser","method of moments","Bayesian bilinear inference","TDD channel reciprocity"],"falsifier":"Run Algorithm 3 with φ_γ set to the true value versus the MoM estimate and compare the RMSE of γ̂; alternatively, monitor the quantity (|q|² − u)/(u² + s) from equation (48) across many trials. If negative values of |γ̂|² occur at a non-negligible rate, or if the two RMSE curves diverge, then the claimed MMSE gains do not follow from the algorithm as written.","tokens_in":20691,"feed_emoji":"📡","tokens_out":5376,"duration_ms":57127,"temperature":0.7,"pith_summary":"The paper is trying to establish that reciprocity calibration of a dual-antenna repeater can be solved as Bayesian bilinear inference rather than as a deterministic least-squares problem. The proposed MMSE algorithm incorporates statistical priors for the hardware reciprocity coefficients and noise, updates parameters with von Mises denoisers, and estimates the unknown prior statistic of |γ|² via the method of moments. Simulation results show RMSE gains of roughly 4 dB for small arrays and 14 dB for large arrays over the basic NLS baseline at RMSE = 10⁻¹, with convergence in about 4 iterations. If correct, this offers a practical, low-complexity way to maintain TDD channel reciprocity in repeater-assisted massive MIMO systems.","feed_headline":"Bayesian estimator cuts repeater calibration error by 14 dB","feed_subtitle":"MMSE calibration restores TDD downlink reciprocity for repeater-assisted MIMO at low SNR and large array sizes.","key_machinery":"The load-bearing component is the von Mises denoiser—a closed-form Bayes-optimal estimator for a point on a circle observed in AWGN. It converts each scalar observation into a posterior mean whose magnitude is the modified-Bessel ratio I₁(·)/I₀(·) and whose phase follows the observation's phase. The paper applies this denoiser to the diagonal entries of A and B and to γ, and couples it with a method-of-moments formula that estimates the unknown statistic φ_γ = E|γ|² from the same measurement, avoiding the need for long-term calibration statistics. The overall loop uses probabilistic data association with Gaussian approximation of the residual interference terms to propagate means and MSEs.","core_discovery":"The central claim is that the gain ratio γ = β/α can be estimated in the MMSE sense from the four effective measurements R1–R4 obtained by two bidirectional pilot transmissions with the repeater in its nominal and π-phase-shifted states. The paper's key argument is that the diagonal reciprocity matrices A and B have a phase-dominant structure captured by a von Mises prior, so their MMSE updates reduce to the closed-form denoiser η(y; v) = r·I₁(|ζ|)/I₀(|ζ|)·e^{j arg ζ}; the same denoiser is applied to γ after estimating |γ| by the method of moments. Under AWGN and standard Rayleigh-fading simulation settings, the paper reports consistently lower RMSE than NLS and, for large arrays, an error t","pith_inferences":["The phase-only prior assumption (β = 0, unit radius for A and B) is the main regularizer; a natural robustness check is to test the denoiser under non-uniform phase-error distributions or amplitude mismatches that violate approximation (26).","The self-referential MoM step in Lemma 2 suggests a testable fix: initialize φ_γ, iterate the estimate of |γ|², and clip negative values; comparing this fixed-point version with the paper's single-shot version would quantify how much of the reported gain depends on the unresolved initialization.","The same bilinear-inference-plus-circular-denoiser structure could extend to other cascaded hardware whose forward and reverse responses differ, such as RIS elements or relays, where only the ratio matters for reciprocity.","A direct experimental validation would be to run Algorithm 3 on measured repeater hardware with known α and β and check whether the achieved downlink beamforming gain follows the predicted 1/SNR scaling."],"forward_implications":["A dual-antenna repeater can be made effectively reciprocal by configuring its gain to compensate the estimated ratio γ̂, preserving downlink precoding in TDD repeater-assisted massive MIMO.","Calibration stays accurate in the low-SNR and large-array regimes where deterministic NLS saturates, because the prior acts as a regularizer on the estimation problem.","The algorithm converges in roughly 4 iterations, making it usable within a channel coherence interval with modest training overhead.","When only diagonal noise variances are used, the computational complexity is on the same order as basic NLS, so the accuracy gain does not require heavy matrix inversions.","The framework can exploit known colored noise through a separable Kronecker covariance structure at additional computational cost."],"fun_headline_variants":["MMSE calibration cuts repeater error by 14 dB","Von Mises denoiser sharpens dual-antenna calibration","Bayesian estimator boosts repeater reciprocity accuracy","Fast and accurate repeater calibration via closed-form MMSE"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the method-of-moments estimator in Lemma 2 can compute |γ|² without knowing |γ|², even though its formula for s in (49c) contains φ_γ = E|γ|²; if that circularity cannot be resolved by an initialization or fixed-point iteration, the final γ-MMSE output of Algorithm 3 is undefined.","fun_headline_variants_meta":{"raw":{"variants":["MMSE calibration cuts repeater error by 14 dB","Von Mises denoiser sharpens dual-antenna calibration","Bayesian estimator boosts repeater reciprocity accuracy","Fast and accurate repeater calibration via closed-form MMSE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000677,"raw_usage":{"total_tokens":2898,"prompt_tokens":710,"completion_tokens":2188,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":2134}},"tokens_in":454,"tokens_out":2188,"duration_ms":15466,"temperature":1.0,"reasoning_tokens":2134,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T04:10:03.223599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Algorithm 3 with φ_γ set to the true value versus the MoM estimate and compare the RMSE of γ̂; alternatively, monitor the quantity (|q|² − u)/(u² + s) from equation (48) across many trials. If negative values of |γ̂|² occur at a non-negligible rate, or if the two RMSE curves diverge, then the claimed MMSE gains do not follow from the algorithm as written.","supporting_citations":[],"review_version":1}