{"id":"9cf1fdfb-0477-42be-96ea-223a4e0d744d","arxiv_id":"2506.15950","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A noise-aware max-min criterion for digital OAC constellation design yields noise-tailored distance metrics that generally reduce computation error, though the gains depend on the target function and noise regime.","lead":"This paper designs digital modulation constellations for over-the-air computation, tailoring the spacing between constellation points to the noise distribution of the wireless channel. It shows that Gaussian, Laplace, and heavy-tailed noise lead to different distance metrics, and that these noise-aware designs can lower computation error compared with the original Euclidean metric.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The noise-tailored distance is derived from a pairwise Euclidean upper bound that ignores ML decision-region geometry; the paper's own sum-function AWGN result shows this surrogate can misrank designs, so the central claim is not established as stated.","rationale":"I agree with the reader's weakest assumption. The single load-bearing claim is that replacing Euclidean distance by noise-tailored metrics in ChannelComp's P2 minimizes the maximum computation error over a noisy MAC. That claim requires the pairwise-distance bound to be a faithful proxy for the true ML error probability. The paper itself gives a concrete counterexample in Sec. V-B: for the sum function with low-variance Gaussian noise, the AWGN-derived exponential metric underperforms Euclidean ChannelComp, because the superimposed constellation's decision regions are nearly vertical and pairwise Euclidean separation is not the controlling quantity. The conclusion acknowledges this limitation. This is not merely a disagreement with prevailing consensus; it is an internal inconsistency between the broad claim and the paper's own reported results. Proposition 1 also proves equivalence only for a surrogate objective (Eq. (37)-(42)), not for the exact MSE. Because the reader's conditional verdict already captures the need to sharpen scope and repair the derivation, I do not move the verdict. The proposed check would settle whether the surrogate misranks designs by evaluating true ML error rather than the design-time pairwise bound.","tokens_in":20871,"tokens_out":5831,"duration_ms":68476,"concrete_test":"Re-run the sum-function AWGN experiment in Fig. 6(a) with K=12, q=16, sigma=0.2, and compare constellations designed under the AWGN exponential metric and the Euclidean ChannelComp metric by Monte Carlo simulation of the full ML decoder over the superimposed constellation (not the pairwise Q-function bound used to design them). If the exponential-metric constellation has higher measured MSE than the Euclidean one in this low-noise regime, the surrogate objective misranks designs and the central claim fails as stated. As a secondary check, independently re-derive Eq. (19) from Eq. (18); if the exponential metric does not actually follow, the design criterion lacks formal justification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central objective in (14) is the true maximum pairwise computation error, but the derivation in Sec. IV-B to IV-D replaces Pr(r_i -> r_j) by an upper bound depending only on |r_i - r_j| (Eq. (15), Eq. (21)). That replacement is only faithful if the ML decision boundary between r_i and r_j is their midpoint and pairwise errors dominate. In the full superimposed constellation the ML decoder's Voronoi cells are determined by all constellation points, so Euclidean pairwise separation does not control the actual error event. The paper's own Sec. V-B and Fig. 6(a) is a falsifying instance: for sum-function computation under low-variance Gaussian noise, the AWGN exponential metric is worse than Euclidean ChannelComp because the actual decision regions are nearly vertical, so the pairwise distance metric misdirects the design. The conclusion (Sec. VI) explicitly concedes that noise distribution alone is insufficient and that a rigorous treatment of misdetection probability is needed. Consequently the abstract's broad claim that the proposed design 'achieves a substantially lower MSE than leading digital OAC methods' is not established; at most, the metrics minimize a pairwise-distance surrogate. A secondary algebraic gap: Eq. (19) does not follow from Eq. (18) when lambda = ln(t)/(4sigma), since (18) gives d^2 >= 8 sigma ln|f_i - f_j| + e^{4 sigma lambda} rather than d^2 >= 4 sigma ln(lambda |f_i - f_j|).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a noise-aware constellation design for digital over-the-air computation (OAC), extending the authors' earlier ChannelComp framework. The encoder design is formulated as a max-min optimization problem (Problem P2) in which the distance metric between superimposed constellation points is tailored to the channel noise distribution: an exponential metric for Gaussian noise, an exponential metric for Laplace noise, and a power-law metric for heavy-tailed noise. The paper also claims that, under low-variance sub-exponential noise, minimizing the mean-square computation error is asymptotically equivalent to minimizing the maximum pairwise computation error (Proposition 1), and it reports simulations for sum, product, max, and mean/geometric functions over Gaussian and Cauchy noise. The central claim is that the noise-tailored distances yield substantially lower MSE than leading digital OAC baselines.","tokens_in":21105,"tokens_out":12066,"duration_ms":121464,"significance":"If the claims were fully established, this would be a useful contribution to digital OAC: it gives a concrete, optimization-based way to adapt constellation design to the noise tail, and it connects the max-min design criterion to the channel statistics in a manner that extends the ChannelComp framework. The intended distance metrics are simple and amenable to the existing SDP relaxation approach, and the numerical experiments cover several functions and noise models. However, the main theoretical derivation contains a concrete algebraic error, the heavy-tail section is internally inconsistent about the error exponent, and the paper's own Section V-B shows that the pairwise-distance surrogate can misrank designs. These issues bear directly on the paper's central claim, so the current version is not yet suitable for publication without substantial revision.","major_comments":[{"comment":"The change of variables does not transform (18) into (19). Taking the logarithm of the constraint e^{|r_i-r_j|^2/(4σ)} ≥ λ|f_i-f_j| gives |r_i-r_j|^2 ≥ 4σ ln λ + 4σ ln|f_i-f_j|, whereas (18) requires |r_i-r_j|^2 ≥ 8σ ln|f_i-f_j| + t with t = e^{4σλ}. These constraints are not equivalent, so the exponential distance metric in (20) does not follow from the Chernoff-bound derivation as written. Since this metric is the main product of Section IV-B, the derivation needs to be corrected or replaced by a direct argument from the Chernoff bound.","section":"Section IV-B, Eqs. (18)-(19)"},{"comment":"The derivation replaces the misdetection probability Pr(r_i → r_j) by a bound that depends only on the pairwise Euclidean distance |r_i - r_j|, with the ML decision boundary implicitly assumed to be the midpoint. In the full superimposed constellation, the ML decision regions are determined by all constellation points, so pairwise Euclidean separation does not control the true error event. The paper's own Figure 6(a) is a falsifying instance: for the sum function under low-variance Gaussian noise, the AWGN exponential metric is worse than the Euclidean ChannelComp metric because the decision regions are nearly vertical. Thus the abstract's broad claim that the proposed noise-aware design 'achieves a substantially lower MSE' is not established; at best the metrics minimize a pairwise-distance surrogate.","section":"Section IV-B, Remark 4, and Section V-B"},{"comment":"The heavy-tail derivation is internally inconsistent. Section IV-D defines the computation error with exponent η < 1 and states that MSE is not suitable for heavy-tailed noise, yet Table I uses the metric |r_1-r_2| and Section V-A says 'Cauchy noise with η = 2 (since the metric is MSE here)', and Figure 7 is labeled with η = 2 while the axes read 'MAE (dB)' despite the caption saying 'MSE performance'. Moreover, in the paper's discrete-decoder setting the computation error is bounded because the output of the tabular mapper is finite, so the claim that MSE is undefined is not justified. The reported heavy-tail gains therefore appear to be obtained under a different objective than the one derived in Section IV-D, and the heavy-tail numerical results need to be reinterpreted or rerun with a consistent error criterion.","section":"Sections IV-D and V-C"},{"comment":"The inequality arctan^{-1}(1/|x|) ≥ |x| is false in general; for example, x = 2 gives arctan(1/2) ≈ 0.46 < 2. Consequently, replacing the inverse-arctangent constraint in (28) by the linear constraint |r_i-r_j|/(2γ) ≥ t|f_i-f_j|^η is not a valid lower-bound relaxation, and the power-law metric (30) is not justified by this argument. A correct bound, or a direct derivation of the power-law metric, is needed.","section":"Section IV-D, Eq. (29)"},{"comment":"The proof of Proposition 1 is not sufficient for the stated claim. The proof minimizes an upper bound on Ξ_{i,j} obtained from the exponential tail bound in (36); without a matching lower bound, a minimizer of the upper bound need not be a minimizer of J_MSE. The log-sum-exp argument in (39)-(42) only shows that the surrogate objective converges to J_max, not that the MSE minimizers converge to J_max minimizers. In addition, the proof uses a specific exponential tail bound and does not establish the 'every accumulation point' statement for general sub-exponential noise as stated in the proposition.","section":"Proposition 1 and Appendix A"}],"minor_comments":[{"comment":"The stated integral inequality ∫_z^∞ e^{-u^β} du ≤ (1/β) z^{β-1} e^{-z^β} is incorrect; the correct exponent is z^{1-β}. The subsequent expression for the tail probability appears to use the correct exponent, so this is likely a typo, but it should be fixed for clarity.","section":"Section IV-C, Eq. (21)"},{"comment":"The caption of Figure 7 says 'MSE performance' while the axis labels read 'MAE (dB)', and the text refers to both MSE and MAE. These labels should be harmonized so the reader knows which quantity is plotted.","section":"Figure 7 and Section V-C"},{"comment":"The sentence 'Cauchy noise with η = 2 (since the metric is MSE here)' is confusing because Section IV-D explicitly argues that MSE is not suitable for heavy-tailed noise and defines η < 1. Please clarify which objective is actually used.","section":"Section V-A, paragraph after Figure 5"},{"comment":"The notation arctan^{-1}(·) is ambiguous: it could mean the inverse function or the reciprocal. Please define it explicitly at first use.","section":"Notation, Eq. (29)"}],"recommendation":"major_revision","confidential_remarks":"The most serious issue is the algebraic error in the derivation of Eq. (19), because the exponential distance metric is the central contribution of Section IV-B. The self-referential admission in Section VI that 'solely accounting for the channel noise distribution is insufficient' is consistent with the numerical contradiction in Figure 6(a), but the abstract and contributions section still make an unconditional claim. The manuscript would benefit from a complete rewriting of the distance-metric derivations and a clear statement of the regimes in which the proposed design is intended to outperform baselines."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a look, but read it as a proposal with a broken proof, not a finished result. The genuinely new thing is replacing ChannelComp's Euclidean distance with noise-tailored metrics—exponential for Gaussian/Laplace, power-law for heavy tails—and the simulations show real gains in some regimes, e.g., max function under AWGN and Cauchy-heavy-tail settings. The heavy-tail connection to Euclidean distance is a nice observation. Credit where due: the paper is honest about its limits, and the conclusion basically concedes the central weakness.\n\nThe soft spots are real. Eq. (19) does not follow from (18): substituting λ = ln(t)/(4σ) gives |ri−rj|^2 ≥ 8σ ln|fi−fj| + e^{4σλ}, which does not turn into exp(|ri−rj|^2/4σ) ≥ λ|fi−fj|. That's an algebraic gap in the derivation of the AWGN metric. You can recover the metric directly from the Chernoff bound, so the idea survives, but the paper's stated derivation is not valid.\n\nBigger issue: the entire framework replaces Pr(ri→rj) with a pairwise Euclidean upper bound. That surrogate ignores that ML decision regions in the superimposed constellation are determined by all points. Your own Figure 6(a) is a counterexample: for sum function under low-noise Gaussian, the AWGN exponential metric is worse than Euclidean ChannelComp because the decision regions are nearly vertical. The paper acknowledges this in the conclusion. So the abstract's 'substantially lower MSE' claim is too broad; at best the design minimizes a surrogate.\n\nHeavy-tail section also has an inconsistency: the derivation introduces η<1 because MSE is not finite for Cauchy, yet the simulations use η=2 and evaluate MSE. With Cauchy noise, sample MSE is erratic and not a meaningful objective; that experiment needs to be re-run with a metric like MAE or a truncated moment, or the parameter choice needs justification.\n\nWho is this for? People working on digital OAC constellation design. It's a within-subfield advance, not a breakthrough. The idea is plausible and the experiments, despite the flaws, suggest something is there. I'd send it to a serious referee, but the referee should demand a corrected derivation, a re-scoped abstract, and a repaired heavy-tail evaluation before publication. I wouldn't cite the central claim as it stands.","headline":"Plausible noise-aware extension of ChannelComp, but the central derivation has an algebraic gap and the paper's own sum-function result undercuts the broad MSE claim; worth a careful revise.","tokens_in":21678,"tokens_out":3641,"would_cite":false,"duration_ms":35640,"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":"Digital over-the-air computation can be made noise-aware by replacing Euclidean constellation distances with noise-tailored metrics, yielding a max-min design that reduces computation error.","keywords":["over-the-air computation","digital modulation","constellation design","noise-aware design","max-min optimization","distance metric","heavy-tailed noise","ChannelComp"],"falsifier":"Run the paper's sum-function experiment (K=12 nodes, q=16, AWGN, σ<1) and compute the empirical maximum pairwise error and MSE for the exponential-metric and Euclidean constellations; if the exponential-metric design does not beat Euclidean, the central claim fails in that regime.","tokens_in":20610,"feed_emoji":"📡","tokens_out":9637,"duration_ms":92500,"temperature":0.7,"pith_summary":"This paper claims that the constellation diagram used for digital over-the-air computation (OAC) should be designed with the channel noise distribution in mind, and that doing so lowers the computation error. The proposed method replaces the Euclidean distance in the ChannelComp encoder-design constraint with noise-tailored metrics: an exponential of squared distance for Gaussian noise, an exponential of distance for Laplace noise, and a power-law distance for heavy-tailed noise. Constellation design then becomes a max-min optimization that minimizes the worst-case pairwise computation error, and the paper proves that under sub-exponential noise with low variance this is asymptotically equivalent to minimizing the mean-square error. The paper also shows that the original Euclidean ChannelComp choice emerges naturally for heavy-tailed noise, and that stochastic fading can be handled by averaging the pairwise distances over the channel covariance. If correct, this gives a principled, distribution-aware way to tune digital OAC modulations for applications such as distributed learning, wireless control, and IoT.","feed_headline":"Noise-aware constellations cut over-the-air computation errors","feed_subtitle":"Matching constellation spacing to Gaussian, Laplace, or heavy-tailed noise lowers computation error.","key_machinery":"The central object is the distance metric $D_C$ inside the ChannelComp framework (a method for computing finite-valued functions over the MAC by designing digital modulations so that superimposed constellation points remain distinct). The optimization problem P2 turns the requirement that distinct function outputs have distinct superimposed points into a Lipschitz-style separation constraint. The paper's machinery is to replace the intractable true misdetection probability with an upper bound that depends only on the Euclidean separation $|r_i-r_j|$ and the noise tail: the Chernoff bound for sub-exponential noise, an arctangent tail bound for Cauchy noise, and a power-law tail for stable distributions. These bounds convert the min-max error problem into a quadratically constrained program in the modulation vector $x$, solvable by semidefinite relaxation. The load-bearing identity is the log-sum-exp bound used in Proposition 1: it shows that $\\min_x \\sum_{i,j}\\exp(-\\tilde A_{i,j}(x)/\\nu)$ and $\\min_x \\max_{i,j} \\tilde A_{i,j}(x)$ have the same minimizers as $\\nu\\to 0^+$, which is what justifies the max-min criterion as an MSE proxy.","core_discovery":"On its own terms, the paper's central discovery is that the distance function $D_C$ in the ChannelComp design problem—which enforces $D_C(r_i,r_j)\\ge \\lambda |f(i)-f(j)|^2$ for every pair of distinct function outputs $i,j$—is not a free choice but should be derived from the channel noise statistics. For AWGN, using the Chernoff bound on the ML misdetection probability $Pr(r_i\\to r_j)$ leads to $D_C(r_i,r_j)=\\exp(|r_i-r_j|^2/(4\\sigma))$; for generalized normal noise (which includes Laplace) it becomes $|r_i-r_j|^{\\beta-1}\\exp(|r_i-r_j|^\\beta/\\alpha^\\beta)$; and for heavy-tailed Cauchy noise it becomes a power law $|r_i-r_j|^{2/\\eta}$, recovering the Euclidean $|r_i-r_j|^2$ as $\\eta\\to 1$. The paper further proves (Proposition 1) that for sub-exponential noise with vanishing variance, the minimizer of the mean-square computation error converges to a minimizer of the maximum pairwise error, so the max-min formulation is asymptotically MSE-optimal. Under stochastic Rician or Rayleigh fading, the same framework applies by replacing each pairwise separation with its expectation under the channel covariance matrix.","pith_inferences":["A natural extension is to replace the pairwise-distance bound with a bound that accounts for the actual Voronoi (decision) geometry of the superimposed constellation; for sum-type functions this would likely favor axis-aligned or function-aware metrics, and could explain the low-noise sum-function reversal the paper reports.","The equivalence result suggests a more general principle: for any symmetric unimodal noise with a sub-exponential tail, the max-min and MSE objectives coincide at low variance, so one could derive noise-tailored metrics for other distributions (Gamma, Weibull, log-normal) and test them numerically in the same framework.","The power-law metric for heavy-tailed noise implies that as the tail becomes heavier (smaller stability index α), the required separation between constellation points grows without bound, so there is a fundamental limit to reliable OAC under very heavy-tailed noise; a testable prediction is that the achievable MSE degrades as a power of the scale parameter.","The fading extension via covariance expectation could be combined with non-Gaussian noise models to design constellations under correlated fading plus impulsive interference, a regime that current numerical experiments do not cover."],"forward_implications":["If the noise-aware metrics deliver what the paper claims, OAC constellations can be tuned to the channel's noise distribution, lowering mean-square computation error relative to the original Euclidean ChannelComp design in high-noise regimes.","Proposition 1 implies that in low-variance sub-exponential noise, an MSE-optimal constellation is also worst-case optimal, so practitioners may optimize either objective.","For heavy-tailed (Cauchy-like) noise, the design reduces to the Euclidean metric, explaining why ChannelComp's original choice is well suited to impulsive interference.","Under stochastic fading, the same max-min design can be optimized in expectation using only the channel covariance matrix, avoiding per-realization channel estimation.","The paper's own experiments show a boundary: for the sum function under low-noise Gaussian, the exponential AWGN metric underperforms Euclidean ChannelComp because the superimposed decision regions are nearly vertical; this indicates the metric should be aligned with the true decision-boundary orientation."],"supporting_citations":[{"why":"Introduces the ChannelComp framework and the original Euclidean distance choice that this paper generalizes; the optimization problem P2 is taken from here.","marker":"[26]"},{"why":"Establishes the over-the-air computation paradigm over multiple-access channels that the whole paper builds on.","marker":"[1]"},{"why":"Supplies the Chernoff bound used to upper-bound the Gaussian Q-function and derive the exponential distance metric for AWGN.","marker":"[50]"},{"why":"Provides the log-sum-exp bounds used in the proof of Proposition 1 to connect MSE minimization to the max-min criterion.","marker":"[54]"},{"why":"Defines the Gaussian Q-function used in the pairwise misdetection upper bound in Eq. (15).","marker":"[49]"},{"why":"The companion Part II paper whose sampling strategy is used in the numerical experiments to scale the optimization.","marker":"[42]"},{"why":"Prior digital OAC scheme used as a comparison baseline for the numerical results.","marker":"[27]"}],"fun_headline_variants":["Match modulation to noise for better over-the-air computation","Noise-derived metrics sharpen digital over-the-air computation","ChannelComp gets noise-aware constellation design","Optimal OAC modulation adapts to noise statistics","Heavy-tailed noise shapes digital OAC constellations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the probability of confusing two constellation points can be bounded using only the Euclidean distance between them, with the decision boundary halfway between, ignoring the full shape of the superimposed constellation.","fun_headline_variants_meta":{"raw":{"variants":["Match modulation to noise for better over-the-air computation","Noise-derived metrics sharpen digital over-the-air computation","ChannelComp gets noise-aware constellation design","Optimal OAC modulation adapts to noise statistics","Heavy-tailed noise shapes digital OAC constellations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2739,"prompt_tokens":1036,"completion_tokens":1703,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":1631}},"tokens_in":652,"tokens_out":1703,"duration_ms":13982,"temperature":1.0,"reasoning_tokens":1631,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:45:41.282051+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's sum-function experiment (K=12 nodes, q=16, AWGN, σ<1) and compute the empirical maximum pairwise error and MSE for the exponential-metric and Euclidean constellations; if the exponential-metric design does not beat Euclidean, the central claim fails in that regime.","supporting_citations":[{"cited_title":"ChannelComp: A general method for computation by communications,","cited_arxiv_id":null,"evidence_quote":"Introduces the ChannelComp framework and the original Euclidean distance choice that this paper generalizes; the optimization problem P2 is taken from here."},{"cited_title":"Computation over multiple-access channels,","cited_arxiv_id":null,"evidence_quote":"Establishes the over-the-air computation paradigm over multiple-access channels that the whole paper builds on."},{"cited_title":"Space-time codes for high data rate wireless communication: performance criterion and code construction,","cited_arxiv_id":null,"evidence_quote":"Supplies the Chernoff bound used to upper-bound the Gaussian Q-function and derive the exponential distance metric for AWGN."},{"cited_title":"A universal approximation result for difference of log-sum-exp neural networks,","cited_arxiv_id":null,"evidence_quote":"Provides the log-sum-exp bounds used in the proof of Proposition 1 to connect MSE minimization to the max-min criterion."},{"cited_title":"Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers,","cited_arxiv_id":null,"evidence_quote":"Defines the Gaussian Q-function used in the pairwise misdetection upper bound in Eq. (15)."},{"cited_title":"On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling","cited_arxiv_id":"2506.16208","evidence_quote":"The companion Part II paper whose sampling strategy is used in the numerical experiments to scale the optimization."},{"cited_title":"SumComp coding: Digital over-the-air computation via the ring of integers,","cited_arxiv_id":null,"evidence_quote":"Prior digital OAC scheme used as a comparison baseline for the numerical results."}],"review_version":1}