{"id":"e2cce6af-2719-4820-ad83-8f8742e6ec28","arxiv_id":"2502.08734","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Coded repetition of the same symbol over time slots, with jointly optimized constellation and slot patterns, reduces computation error in digital over-the-air computing by up to 7.5 dB in simulations.","lead":"This paper introduces a scheme called ReMAC that repeats coded transmissions over multiple time slots so a receiver can compute a function of many devices' digital signals directly over the air. Simulation results show up to 7.5 dB lower computation error than the previous ChannelComp method, especially for product functions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For many-to-one functions, the separability constraint (8a) is enforced on only one representative input tuple per output value, so the optimized ReMAC code is not guaranteed to distinguish all input tuples with different function outputs.","rationale":"The reader's weakest_assumption matches my own read: the most load-bearing point in the argument is the representative-only constraint. I considered two other candidates. (1) The baseline comparison: ChannelComp is transmitted once while ReMAC uses L slots, so part of the gain is time diversity; however, the paper's contribution is exactly the joint design of repetition and constellation, so this is not a fatal flaw. (2) The convergence proof in Proposition 2 relies on an alternating minimization over time-varying constraint sets, but this affects the optimality gap analysis, not the correctness of the construction. The representative constraint, by contrast, attacks the definition of valid computation itself: Eq. (3) is the only formal guarantee that the tabular map can compute f, and it is stated over the range of f rather than over input tuples. For sum, product, and max, there are many tuples per output; a single a_i per output value cannot encode their variety. The proposed concrete test is decisive because it enumerates all inputs for a configuration the paper itself simulates. If collisions appear, the 7.5 dB claim is at best average-case on non-representative inputs; if no collisions appear, the reader's concern is empirically void for that configuration, though a proof would still be needed. I therefore keep the reader's CONDITIONAL verdict: the scheme is promising and reproducible, but the correctness guarantee needs a revision or a proof.","tokens_in":23183,"tokens_out":8020,"duration_ms":74595,"concrete_test":"Run Algorithm 1 (code on GitHub) for a small configuration from the paper, e.g., K=4, Q=4, L=2, product function, obtaining the optimized (x,C). Enumerate all 4^4 = 256 input tuples and compute the noiseless received sequence v for each tuple. Check whether every pair of tuples with different product values satisfies v != v'. If any collision exists, constraint (8a) over representatives is insufficient and the error floor from such collisions should be measured by including the colliding tuples in the NMSE evaluation; if no collision exists, test larger K or the sum and max functions. This directly settles whether the many-to-one gap invalidates the reported gains.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"For the functions targeted in this paper (sum, product, max), the mapping from input tuple to function value is many-to-one. The correctness condition in Eq. (3) is stated over (i,j) in [M]^2, where M is the cardinality of the range of f, not the number of input tuples Q^K. In Eq. (6) and in constraint (8a), each output value i is represented by a single binary support vector a_i, i.e., a single representative input tuple. The optimization therefore enforces v(i) != v(j) only for those representatives. Two non-representative input tuples with different function values can produce the same noiseless received sequence v, and because the tabular decoder in Eq. (5) assigns one output per Voronoi cell, such a collision means the computation is not actually valid for those inputs. This is a correctness gap in the central claim that ReMAC guarantees valid function computation, not merely a performance issue; the NMSE reported in Figures 6-7 may be optimistic if random test inputs happen to avoid colliding tuples. No passage in the manuscript addresses or limits this many-to-one issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes ReMAC, a digital over-the-air computation scheme in which each node repeats its modulated symbol over L time slots according to a binary on/off pattern, and the access point applies a tabular decoder to the L received symbols. The authors formulate an optimization problem that jointly designs the constellation points and the repetition code under separability constraints, relax it into SDP and LP subproblems, solve them by alternating minimization, and report simulations showing up to about 7.5 dB NMSE improvement over ChannelComp, especially for product functions. The manuscript also contains convergence and optimality-gap analyses for the proposed algorithm.","tokens_in":23403,"tokens_out":11661,"duration_ms":125561,"significance":"The core idea is a natural and potentially useful extension of the ChannelComp framework: adding time diversity through coded repetition while keeping digital modulation is practically relevant and the numerical comparisons against ChannelComp, bit-slicing, and digital AirComp are informative. The paper is largely self-contained, the optimization formulation is developed from scratch, and the authors provide code. If the correctness conditions were fully established, the proposed scheme would be a credible step toward reliable digital AirComp. However, as detailed below, the formal guarantees in the current manuscript are not yet established for the many-to-one functions that the paper actually targets, and the feasibility of the proposed optimization problem is questionable for the reported product-function experiments.","major_comments":[{"comment":"The many-to-one issue raised in the stress-test note is confirmed by the manuscript text. Equation (3) is stated over output values (i,j) in [M]^2, where M is the size of the range of f, but for the functions considered (sum, product, max) each output value has many preimage input tuples. In Eq. (6), v(i) is generated from a single binary support vector a_i, i.e., a single representative input tuple per output value, and constraint (8a) enforces separation only for these representatives. This does not prevent two non-representative tuples u and u' with f(u) != f(u') from producing the same noiseless received sequence v. The tabular decoder in Eq. (5) then assigns one output value to the merged Voronoi cell, so the computation is incorrect for such tuples even without noise. The manuscript nowhere proves, or even states as an assumption, that separating one representative per output value implies separation of all input tuples with different outputs. This is a load-bearing gap in the central claim that ReMAC guarantees valid function computation, and it also means the NMSE results in Figures 6-7 may depend on whether the random test inputs happen to avoid colliding tuples.","section":"Section II-B (Eqs. (3)-(6)) and Section III (Eq. (8a))"},{"comment":"The threshold in constraint (8a), Delta f_{i,j} = sigma_z^2 |f(i)-f(j)|, appears incompatible with the power constraint ||x||_2^2 <= 1 for the product-function experiment in Fig. 4(b). For K=4 and Q=256, the product range is enormous, so |f(i)-f(j)| can be on the order of 10^9. With ||x||^2 <= 1, the maximum possible value of the left-hand side in (8a) is bounded by a constant that depends only on N=QK and L (roughly on the order of 2KL), while the right-hand side can be orders of magnitude larger even at the SNRs shown in Fig. 4. As written, Problem P0 is therefore likely infeasible for that experiment, and Algorithm 1 cannot produce a feasible code satisfying (8a). The authors should either normalize the function values, reformulate the constraint in terms of constellation spacing rather than raw output differences, or explain why the reported solutions are feasible despite this scaling issue.","section":"Eq. (8a)-(8b) and Fig. 4"},{"comment":"The proof of Proposition 2 does not justify the claimed Lipschitz constant L1=1 for the indicator function id_{S3}(W). An indicator function of a closed convex set is not Lipschitz continuous, so the invocation of Theorem 2 of [54] with L1=1 is not supported as written. Since the convergence-rate statement is one of the paper's stated theoretical contributions, the argument needs to be repaired or the proposition needs to be restricted to assumptions under which the alternating minimization objective satisfies the required regularity conditions.","section":"Appendix C (Eqs. (26)-(29))"},{"comment":"The remark assumes 1 < L < min{Q,K}, but the paper's own experiments use L=4 with K=4 in Fig. 4, which violates the stated strict inequality. The remark itself also discusses the case L=K as a legitimate operating point, so the assumption should be reformulated consistently with the experimental setup.","section":"Section II-C, Remark 3"}],"minor_comments":[{"comment":"The statement that ReMAC does not lose spectral efficiency compared to ChannelComp is imprecise: the scheme uses L time slots instead of one, so it consumes more time resources; what is preserved is the use of a single frequency channel.","section":"Section II-C"},{"comment":"The vector a_i is not formally defined. In particular, the manuscript should specify whether it has exactly one nonzero entry per node (a representative tuple) or whether it aggregates multiple preimage tuples in some way, and how its length and sparsity relate to K and Q.","section":"Eq. (6)"},{"comment":"When W^n has rank greater than one, the rank-one approximation via the leading eigenvector is not accompanied by a bound on how much the approximation degrades the constraints in (8a). Adding such a bound would make the projection step more informative.","section":"Eq. (14)"},{"comment":"The y-axis label 'Gap' is ambiguous. It would help to state explicitly in the caption that the numerical gap is the P2 objective difference at a fixed iteration count n and that the analytical gap is the bound from Proposition 2 evaluated at the same n.","section":"Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of a signal processing or communications journal and the underlying idea is worth pursuing. However, the formal correctness conditions for many-to-one functions and the feasibility of the optimization for the reported experiments need to be resolved before publication. The theoretical convergence proof also needs a careful check. I would support publication after these issues are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: the new idea is to repeat each digital symbol over L slots with a coded on/off pattern and to optimize the constellation and the pattern jointly. That is a real extension of ChannelComp, and the simulations show a convincing improvement on product functions. But the correctness guarantee in Eq. (3)/(8a) does not do what the paper says it does for many-to-one functions, so the main claim needs a fix before the results can be taken at face value.\n\nThe paper does several things well. The repetition-pattern example in Section II-C is clear: selective on/off transmission separates constellation points that would overlap under plain QPSK. The optimization framework is self-contained: lifting to SDP for the modulation, McCormick relaxation for the integer code, alternating minimization, and a convergence-rate bound. They ship code on GitHub, and the experiments cover sum, product, and max under noise and phase/magnitude fading, with bit-slicing and digital AirComp comparisons. The 7.5 dB gain for the product function is plausible as a time-diversity effect.\n\nThe soft spots:\n\n1. Major: the separability constraint is indexed by output values, with a single binary vector a_i representing 'the' input support for output i. For sum, product, and max, each output has many input tuples. Two non-representative tuples with different outputs can still produce the same noiseless received sequence, so the optimized ReMAC code is not guaranteed to compute the function correctly on all inputs. The paper never limits the claim or discusses this. This is not a cosmetic issue; it undermines the 'valid function computation' guarantee. It is fixable by enforcing distinguishability over all input tuples, or by restricting to function classes where the output value determines the support pattern.\n\n2. Moderate: the main comparison is against ChannelComp with L=1, so the gain from ReMAC is largely repetition diversity. A baseline that repeats the ChannelComp symbol and does maximum-ratio combining, or an equivalent repetition-code baseline, would make the 'joint design' contribution clearer. Also, Ns=100 with no error bars: the reported dB numbers are probably fine as trends, but not as tight claims.\n\n3. Minor: the optimality-gap result depends on quantities like R(x0,c0) that are not easy to compute, and the branch-and-bound step is worst-case exponential. The offline argument mitigates this, but the complexity section oversells practicality.\n\nWho is this for: people working on digital over-the-air computation for 6G aggregation or federated learning. If the many-to-one gap is fixed, this is a solid technique paper. I would send it to review, but with a request for major revision; a referee should check the many-to-one issue first.","headline":"Coded repetition with joint constellation and pattern design is a useful new trick, but the separability constraint only checks one representative input per output value, so the central correctness guarantee is not established for many-to-one functions.","tokens_in":23915,"tokens_out":4013,"would_cite":false,"duration_ms":39811,"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":"Repeated transmissions cut over-the-air computation error by 7.5 dB","keywords":["over-the-air computation","digital modulation","repetition coding","multiple access channel","function computation","constellation design","fading channels","alternating minimization"],"falsifier":"Enumerate all $Q^K$ input tuples for a small case, say $K=4$, $Q=4$ with the product function, run the ReMAC optimization, and compare the noiseless received sequences: if any two tuples with different function outputs produce the same sequence $\\mathbf{v}$, the tabular decoder cannot separate them and the claimed guarantee fails. A simpler proxy is to simulate at high SNR and look for an NMSE error floor that does not decrease as SNR grows.","tokens_in":22966,"feed_emoji":"📡","tokens_out":4022,"duration_ms":37501,"temperature":0.7,"pith_summary":"Many systems want a wireless receiver to compute a function—say a sum or product—of values held by several transmitters, without first decoding each message separately. ReMAC is a scheme for doing this with digital modulation: each transmitter repeats its symbol over several time slots in a coded on/off pattern, and the constellation and pattern are designed together so that different function outputs always produce different received sequences. The paper claims this reduces computation error by up to about 7.5 dB compared with the existing ChannelComp method in noisy and fading channels, with the largest gains for product functions. The payoff is a digital, repetition-coded alternative to analog over-the-air computation that handles arbitrary finite functions without orthogonal resource allocation.","feed_headline":"Repeated transmissions cut over-the-air computation error by 7.5 dB","feed_subtitle":"Joint constellation and repetition-code design makes digital function computation over the air far more reliable under noise and fading.","key_machinery":"The load-bearing object is the pair (constellation vector $\\mathbf{x}$, repetition matrix $\\mathbf{C}$), where each column $\\mathbf{c}_\\ell$ is a binary vector saying which constellation points are active in time slot $\\ell$. The computation constraint is $\\mathbf{v}^{(i)} = \\mathbf{a}_i^T (\\mathbf{x} \\otimes \\mathbf{1}_L) \\odot \\mathbf{C}$, and validity requires $\\mathbf{v}^{(i)} \\neq \\mathbf{v}^{(j)}$ whenever $f^{(i)} \\neq f^{(j)}$. The paper enforces the stronger squared-distance constraint $\\|\\mathbf{v}^{(i)} - \\mathbf{v}^{(j)}\\|_2^2 \\geq \\sigma_z^2 |f^{(i)} - f^{(j)}|$, then relaxes the NP-hard joint design into a semidefinite program for $\\mathbf{x}$ and a McCormick-relaxed linear program for $\\mathbf{C}$, alternates until a stationary point, and projects back with Cholesky decomposition and branch-and-bound.","core_discovery":"The central claim is that repetition resolves the destructive constellation-point overlaps that prevent a receiver from distinguishing function outputs. Whereas ChannelComp must redesign or enlarge the constellation when two input tuples with different outputs superimpose to the same point, ReMAC allocates each symbol to a subset of L time slots, so the aggregated points form an L-length sequence and equal points in a single slot can be separated by different patterns across slots. The paper formalizes this as an optimization problem that minimizes the total number of transmitted symbols subject to a lower bound on the squared distance between received sequences for distinct outputs, and solves it by alternating between a semidefinite relaxation for the constellation and a McCormick-based linear relaxation plus branch-and-bound for the binary repetition code. Numerical experiments report the 7.5 dB NMSE improvement for the product function under fading, with smaller but consistent gains for sum and maximum.","pith_inferences":["The representative-tuple constraint in Eq. (8a) could be augmented by an exhaustive verification step over all input tuples, since many-to-one functions may produce collisions among unrepresented tuples that the current constraint set misses.","The time-slot repetition pattern generalizes directly to other resource dimensions such as frequency subcarriers or MIMO spatial streams, turning ReMAC into a generic resource-allocation code for computation.","The 7.5 dB gain is demonstrated under perfect CSI and phase-aligned precoding; a natural test is to re-run the design with explicit precoding-error models, where the gap to ChannelComp may shrink and could motivate a co-designed pre-equalizer."],"forward_implications":["ReMAC with more than one time slot strictly reduces NMSE compared with its own single-slot case (ChannelComp) for sum, product, and maximum functions under both noise and fading.","For the product function under low fading variance, ReMAC reduces computation error by approximately 7.5 dB compared with ChannelComp.","The alternating minimization converges to a first-order stationary point of the relaxed problem, with an optimality gap that shrinks as $O(1/n)$ and grows with the number of constraints in the original problem.","ReMAC uses the same single frequency resource as ChannelComp, so it gains reliability through time diversity rather than extra bandwidth, at the cost of latency.","The optimization is performed offline and stored, so the NP-hard design cost is paid once during system setup and reused in real-time operation."],"supporting_citations":[{"why":"Supplies the ChannelComp digital constellation computation framework and the baseline whose single-slot case ReMAC extends.","marker":"[15]"},{"why":"Provides the distinguishability constraint v(i) != v(j) and the maximum-likelihood tabular decoding approach that ReMAC inherits.","marker":"[34]"},{"why":"Provides the bit-slicing digital AirComp baseline that ReMAC is compared against for the summation function.","marker":"[36]"},{"why":"Provides the digital AirComp baseline over which ReMAC and ChannelComp show computation-error gains.","marker":"[12]"},{"why":"Supplies the linear relaxation technique used to convert the mixed-integer quadratic subproblem P2 into the tractable LP P4.","marker":"[45]"},{"why":"Supplies the optimal power control policy that ReMAC assumes to invert channel coefficients before superposition.","marker":"[38]"}],"fun_headline_variants":["ReMAC repeats sends to cut AirComp error by 7.5 dB","Repeated transmissions improve over-the-air function computation","ReMAC hits 7.5 dB gain with repeated symbol codes","Digital AirComp: Repetition coding slashes error 7.5 dB","Repeating over the air: ReMAC reduces computation noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole design stands on the assumption that enforcing distinct received sequences for one representative input tuple per function output value is enough to guarantee correct decoding for every possible input tuple, which is not automatically true for many-to-one functions.","fun_headline_variants_meta":{"raw":{"variants":["ReMAC repeats sends to cut AirComp error by 7.5 dB","Repeated transmissions improve over-the-air function computation","ReMAC hits 7.5 dB gain with repeated symbol codes","Digital AirComp: Repetition coding slashes error 7.5 dB","Repeating over the air: ReMAC reduces computation noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1385,"prompt_tokens":918,"completion_tokens":467,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":377}},"tokens_in":534,"tokens_out":467,"duration_ms":4516,"temperature":1.0,"reasoning_tokens":377,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:52:22.169839+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all $Q^K$ input tuples for a small case, say $K=4$, $Q=4$ with the product function, run the ReMAC optimization, and compare the noiseless received sequences: if any two tuples with different function outputs produce the same sequence $\\mathbf{v}$, the tabular decoder cannot separate them and the claimed guarantee fails. A simpler proxy is to simulate at high SNR and look for an NMSE error floor that does not decrease as SNR grows.","supporting_citations":[{"cited_title":"ChannelComp: A general method for computation by communications,","cited_arxiv_id":null,"evidence_quote":"Supplies the ChannelComp digital constellation computation framework and the baseline whose single-slot case ReMAC extends."},{"cited_title":"Computing functions over-the-air using digital modulations,","cited_arxiv_id":null,"evidence_quote":"Provides the distinguishability constraint v(i) != v(j) and the maximum-likelihood tabular decoding approach that ReMAC inherits."},{"cited_title":"Broadband digital over- the-air computation for asynchronous federated edge learning,","cited_arxiv_id":null,"evidence_quote":"Provides the digital AirComp baseline over which ReMAC and ChannelComp show computation-error gains."},{"cited_title":"Global optimization algorithm for mixed integer quadratically constrained quadratic program,","cited_arxiv_id":null,"evidence_quote":"Supplies the linear relaxation technique used to convert the mixed-integer quadratic subproblem P2 into the tractable LP P4."}],"review_version":1}