On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling

Carlo Fischione; Saeed Razavikia

REVIEW 4 major objections 6 minor 1 cited by

On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Pyramid sampling reduces digital over-the-air computation encoder design from checking all $q^K$ superimposed symbol combinations to $\mathcal{O}(q^{K-p+1})$ representative histograms, and at $p=K$ leaves only $q$ consensus points that…

desk verdict The advertised O(q^{K-p+1}) complexity reduction rests on a false cardinality count and a sampling set that is empty for many p, but the qualitative pyramid idea and the p=K majority scheme are salvageable. read the letter →

arxiv 2506.16208 v2 pith:ZNNY5HMQ submitted 2025-06-19 eess.SP

Saeed Razavikia , Carlo Fischione This is my paper · ORCID

classification eess.SP

keywords over-the-aircomputationpyramidsamplingmajority-basedconstellationdesigndigitalmodulationsymmetricaggregationquantizationwirelessedgecomputing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the main obstacle to scalable digital over-the-air computation—the need to design the modulation by comparing all $q^K$ possible superimposed symbol combinations—can be removed for symmetric aggregation functions. It introduces pyramid sampling: instead of every input histogram, the designer keeps only histograms whose bin counts are multiples of a sampling order $p$, cutting the number of constraints from $\mathcal{O}(q^K)$ to $\mathcal{O}(q^{K-p+1})$ at a controlled cost in function accuracy. At the extreme $p=K$, only the $q$ combinations where all nodes transmit the same symbol remain, so the received signals are $q$ scaled copies of one constellation point that cannot overlap; standard modulations such as QAM, PSK, and ASK then work without bespoke design. If the paper is right, large-scale wireless computation becomes computationally feasible, and earlier one-bit and balanced-vote aggregation schemes reappear as special cases of a single design rule.

What carries the argument

The load-bearing object is the level-$p$ pyramid-sampling set $\Omega_p$, the sub-lattice of input histograms whose bin counts are all multiples of $p$; histograms are the right domain because a symmetric function depends only on counts per quantization level. Sampling on this lattice is what turns an $\mathcal{O}(q^K)$ constraint set into an $\mathcal{O}(q^{K-p+1})$ one. Two supporting identities carry the counting: the balls-and-bins count $\binom{K+q-1-p}{q-1}$ for each transition set, summed by the hockey-stick binomial identity into $\binom{K+q-1}{q}$, and the structural fact that at $p=K$ each received sum is $K$ times a single constellation point, which makes overlap impossible.

What would settle it

Set $K=3$, $q=4$, and $p=2$. The claimed cardinality formula gives $\binom{3-2+4}{3}=10$ sampled histograms, but a valid histogram must have every bin count in $\{0,2,4,\ldots\}$ and those counts must sum to $3$; no such histogram exists, so the actual sampled set is empty. Running the proposed complexity analysis on this example—where the formula's prediction and the actual count disagree—settles whether the central claim holds as stated.

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Extended reading notes

Core claim

Pyramid sampling is defined on the histogram of the $K$ quantized inputs, which is legitimate because the aggregation function is symmetric and therefore depends only on how many nodes chose each of the $q$ quantization levels. The full histogram domain $\Omega$ has $\binom{K+q-1}{q-1}$ elements, growing as $\mathcal{O}(q^K)$; the level-$p$ sub-domain $\Omega_p$ allows only bin counts that are multiples of $p$ and has cardinality $\binom{K-p+q}{q-1}=\mathcal{O}(q^{K-p+1})$. Solving the Part I max–min constellation optimization over $\Omega_p$ instead of $\Omega$ yields a design whose retained points are separated according to the function values, and the paper proves that at $p=K$ the $q$ agreement histograms produce received sums $K x(i)$ that are distinct whenever the $q$ transmit points $x(i)$ are distinct, so destructive overlaps are impossible by construction. Its Proposition 1 formalizes the payoff: using modulation level $q^K$ at $p=K$ makes the normalized sampling error a factor $K/q^{K-1}$ of the exhaustive design's error with design-cost ratio $\mathcal{O}(K/((q-1)\log_q K))$, while using level $qK$ matches the exhaustive error exactly at cost ratio $\mathcal{O}(1/(q-1))$. Simulations over sum, product, and max functions confirm that moderate $p$ reaches the quantization error floor with orders-of-magnitude fewer constraints.

Load-bearing premise

The paper's complexity count silently assumes that the sampling order $p$ divides the number of nodes $K$; when $K$ is not a multiple of $p$, the sampled input set can be empty and the claimed speedup has no optimization to run on.

Editorial extensions

If this is right

Moderate sampling orders preserve the large-scale geometry of the designed constellation: in the paper's product-function experiment with $K=16$, $q=8$, increasing $p$ from 1 to 8 cuts the constraint count from $\mathcal{O}(q^{16})$ to $\mathcal{O}(q^2)$ while only one pair of constellation points collides.
At $p=K$, any standard modulation format can be used without overlap, and the design problem reduces from an intractable $\mathcal{O}(q^K)$-constraint optimization to a linear-in-$q$ problem; one-bit over-the-air aggregation and balanced-vote aggregation appear as special cases.
Proposition 1 implies that raising the modulation level at $p=K$ can either beat the accuracy of an exhaustive low-level design at comparable cost or match its accuracy at a fraction $\mathcal{O}(1/(q-1))$ of the cost.
For the sum function, the optimal constellation is PAM for every sampling order, so sampling does not change the optimum for that function.
The end-to-end accuracy at high SNR is dominated by the quantization floor of the sampler, while at low SNR the modulation's minimum distance governs robustness, with rectangular QAM outperforming hexagonal and PAM in the simulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference — Because the argument only uses histogram symmetry and Lipschitz dependence, the same complexity–accuracy trade-off should extend to other symmetric aggregates such as medians, variance, and order statistics, which the paper does not simulate.
Editorial inference — The exact cardinality formula needs $p$ to divide $K$; a practical version of the scheme for network sizes where $K$ is not a multiple of $p$ would have to round counts to the nearest valid lattice or randomize the sampling, which the paper does not spell out.
Editorial inference — The $p=K$ receiver is effectively a vote over constellation indices, so the problem becomes a conventional point-to-point codebook design; this suggests layering standard channel codes or shaping schemes on top of the consensus constellation, a direction the paper only gestures at in its conclusion.
Editorial inference — A natural stress test is to replace pyramid sampling with a random subset of constraints of the same size; if random subsets match pyramid performance, the exact lattice structure is not what is buying the accuracy.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Quick read of 2506.16208. The headline complexity reduction does not hold as stated. Definition 1's Ωp requires every histogram bin to be a multiple of p and the bins to sum to K. That set is empty whenever p does not divide K, and when p does divide K its size is C(K/p+q−1, q−1), not the printed C(K−p+q, q−1). For K=16, q=8, p=8, the true count is 36, not 11440, so the O(q^{K−p+1}) scaling, the δ(p) curves in Figure 4, and the general statement of Proposition 1 are unsupported. Lemma 1 has a separate counting problem: the set S^{(p)}_{i,j} as defined—p indices changed i→j—has size C(K,p)q^{K−p}, not C(K+q−1−p, q−1). The balls-and-bins argument counts histograms of K−p balls, a different object.

What is genuinely nice: the qualitative idea of sampling histograms at multiples of p is new, and it does unify OBDA, FSK voting, and balanced-numeral schemes as p=K special cases. The p=K case itself is correct and clean: the received sums are K·x_i, so they are distinct and overlap-free, and any standard modulation works. That part is solid. The simulations are thin but consistent for the p values tested (1,2,4,8 with K=16), all of which divide K, so the experiments do not expose the empty-set problem.

The fix is not deep: redefine Ωp so it is nonempty for all p (for instance, allow a remainder bin), correct the cardinality, recompute δ(p), and repair or drop Lemma 1. Until that is done, the central trade-off curves and complexity claims are not reliable. I would send this to a referee with combinatorics expertise as a major revision, not a desk reject, because the core idea is salvageable and the p=K majority scheme is useful. For my own work, I would not cite the complexity claim in its current form.

Referee Report

4 major / 6 minor

Summary. The paper proposes pyramid sampling for digital over-the-air computation under symmetric aggregation. The idea is to constrain the constellation design to histograms whose bin counts are multiples of a sampling order p, thereby reducing the number of constraints in the max-min constellation design problem. The paper claims that this reduces encoder-design complexity from O(q^K) to O(q^{K-p+1}), introduces majority-based sampling (p=K) as a special case that uses only q consensus points, and supports the approach with simulations for sum, product, geometric-mean, and max functions. The central combinatorial statement underlying the complexity reduction is, however, incorrect.

Significance. If the complexity claim were correct, the framework would be valuable: it unifies several existing digital OAC schemes as special cases, permits standard modulation formats without overlap constraints in the p=K regime, and offers a principled complexity-accuracy trade-off. The paper is not fitting parameters to data; the analysis is analytical and the simulations are used to illustrate qualitative behavior. The majority-based sampling idea is intuitive and likely useful. However, the stated quantitative contribution depends on a false cardinality formula for the sampling set, so the headline complexity reduction and the general trade-off curves are not established as written.

major comments (4)

[Definition 1, Eq. (7)] The claimed cardinality |Ωp| = C(K-p+q, q-1) is false for the set defined in (7). Since every h_j must be a multiple of p, writing h_j = p m_j gives Σ_j m_j = K/p, so the set is empty unless p divides K, and when it is nonempty the count is C(K/p + q - 1, q - 1). For example, K=3, q=4, p=2 gives an empty set, not C(5,3)=10; for K=16, q=8, p=8 the printed formula gives C(16,7)=11440, whereas the true count is C(9,7)=36. Because Eq. (14), the complexity discussion, Figure 4, and the headline O(q^{K-p+1}) all rely on this count, the central reduction claim is unsupported and the method is undefined for non-dividing p.
[Eq. (14) and Figure 4] The general complexity-accuracy trade-off is based on log|Ωp|. With the corrected cardinality, for p | K and fixed K while q grows, log|Ωp| ≈ (K/p) log q, not (K-p+1) log q. Thus δ(p)/δ(1) does not behave as suggested by the O(q^{K-p+1}) formula, and Figure 4 needs to be recomputed. The special p=K case used in Proposition 1 may survive, because |Ω_K| = q is correct, but the general pyramid-sampling complexity claim for intermediate p is not established.
[Proposition 2, Eq. (19)] The proof asserts that Ω_p ⊆ Ω_{p'} for every p' ≤ p, but this inclusion follows from the definition only when p' divides p. For instance, for p=3 and p'=2, the histogram (3,0,0) is in Ω_3 but not in Ω_2. Consequently, the statement that constraints satisfied for sampling order p automatically prevent overlaps for all lower orders p' ≤ p is not proved. The proposition needs a divisibility condition or a different nesting argument.
[Lemma 1 and Eq. (32)] The count |S^{(p)}_{i,j}| = C(K+q-1-p, q-1) does not match the definition in (20). Once p labelled coordinates are fixed to transition from i to j, each of the remaining K-p coordinates can take any of q values, giving q^{K-p} pairs, not a stars-and-bars count. For example, K=3, q=3, p=1 gives 9 pairs, not C(4,2)=6. The union over p therefore has size (q^K - 1)/(q - 1), not C(K+q-1, q), and the claimed 1/|S_{i,j}| constraint-reduction factor in Section III-A is incorrect.

minor comments (6)

[Section IV, Figure 5] The legend in Figure 5 lists the scheme 'p=K, qK' twice, which makes the comparison difficult to read.
[Eq. (24)] Eq. (24c) contains 'gp0,2q = gp0,2q'; the intended relation is presumably gp0,2q = gp2,0q.
[After Eq. (7)] The sentence 'the cardinality of Ωp is p(K+q-1 q-1)' appears to have a formatting error, and the parenthetical 'where Ω1 indicates the whole input domain' should be phrased as 'for p=1, Ω1 = Ω'.
[Figure 4 caption] The caption says q ∈ {4,32,64} while the text says q ∈ {4,32,64,128}; these should be reconciled.
[Proposition 1 and Appendix B] The notation 'qK' is used both for the product q·K and for q^K in Proposition 1, Figure 5, and Appendix B; using an explicit superscript would remove the ambiguity.
[Section IV, Figure 5 discussion] The authors correctly note that the computation-error comparison assumes an ideal channel; this limitation should be stated at the start of the experiment rather than only after the results, since the figure could otherwise be misread as an end-to-end OAC comparison.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: complexity and overlap-avoidance claims are derived in-paper from Definition 1 and Proposition 3; self-citations to Part I and ChannelComp are published background, not load-bearing support.

full rationale

The paper's central claims—that pyramid sampling reduces encoder-design complexity from O(q^K) to O(q^{K-p+1}) and that majority-based sampling (p = K) confines aggregation to q consensus points while avoiding destructive overlaps—are derived in-paper, not imported from the authors' prior work: the complexity bound follows from counting the set Ω_p defined in Definition 1, and the overlap-avoidance follows from Proposition 3's direct proof that the received sum equals K·x(i) and that distinct constellation points remain distinct after scaling by K. No parameter is fitted to data and then renamed a prediction; the trade-off metrics δ(p) and ε(p) are computed from their definitions, and the MSE curves in Figures 8-10 are genuine Monte Carlo evaluations of the designed constellations. The self-citations are background rather than load-bearing: ChannelComp [10] is a published, peer-reviewed IEEE Trans. Commun. paper whose Lemma 1 (used in Proposition 2) qualifies as independent support, the communication model is re-reviewed in Section II, and Remark 1's PAM-invariance claim, delegated to 'similar steps as in [9]', is corroborated by the paper's own Figure 8(a) simulation. The unification of OBDA, FSK voting, and balanced-vote schemes as special cases of majority sampling is presented transparently as a unifying perspective, not as a derived prediction. The complexity reduction is definitional to the sampling construction (the optimization's constraint count is the size of the chosen subset), but the paper states this openly as the method's design, and the count is a combinatorial derivation from Definition 1 rather than a hidden fit. A caveat for correctness, not circularity: the printed cardinality |Ω_p| = C(K-p+q, q-1) is wrong (Ω_p is empty when p does not divide K, and the true count for p|K is C(K/p+q-1, q-1)), so the quantitative complexity claims and Figure 4 need correction, but the derivation chain is a definition-to-count-to-complexity argument, not an equivalence of inputs and outputs. Overall, the derivation is self-contained and no circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters or new physical entities. The load-bearing assumptions are the symmetry of g, the shared alphabet, the implicit divisibility condition K mod p = 0, and the asserted (and incorrect) cardinality formula for Ωp. These assumptions underpin the central complexity-reduction claim.

assumptions (5)

domain assumption The aggregation function g is symmetric in its K input arguments, so it depends only on the histogram of quantized values.
Invoked in Section III before Definition 1 to justify histogram-based sampling.
domain assumption All nodes share the same input alphabet X with |X|=q and use a common encoder.
Stated in Section III: 'assume that the input values come from the same set of alphabet, X1=...=XK=X.'
ad hoc to paper The sampling order p divides K, so each histogram bin count can be a multiple of p while still summing to K.
Not stated in the paper, but required for Ωp to be nonempty. For K=3, q=4, p=2 the set Ωp is empty, contradicting the paper's use of p up to K.
ad hoc to paper The cardinality |Ωp| = C(K-p+q, q-1) holds for the set defined in (7).
Asserted without proof and false for the given definition in general, e.g., q=2, K=4, p=2 gives 3 actual histograms but formula 4. The proof of Proposition 1 and the complexity metric depend on this identity.
domain assumption The function g is L-Lipschitz under the ℓ8 norm on histograms, and inputs lie in [0,1] with uniform quantization.
Used in Proposition 1 and Table I to upper-bound the normalized sampling error ε(p).

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Cite this review

Pith. "Pith review of On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling." pith.science (2026). https://pith.science/paper/ZNNY5HMQ

@misc{pith2026250616208,
  author       = {Pith},
  title        = {Pith review of: On Designing Modulation for Over-the-Air Computation -- Part II: Pyramid Sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZNNY5HMQ}},
  note         = {Machine review of arXiv:2506.16208}
}

abstract

Over-the-air computation (OAC) harnesses the natural superposition of wireless signals to compute aggregate functions during transmission, thereby collapsing communication and computation into a single step and significantly reducing latency and resource usage. In Part I, digital OAC was formulated as a noise-aware constellation design problem by casting encoder design as a max-min optimization that aligns minimum Euclidean distances between superimposed constellation points with squared differences of their corresponding function outputs. In this paper, Part II, we address the prohibitive complexity and quantization challenges inherent in digital OAC constellation design for large-scale edge networks. More precisely, we introduce a pyramid sampling strategy that judiciously selects a subset of superimposed constellation points to reduce the encoder design complexity from $\mathcal{O}(q^K)$ to $\mathcal{O}(q^{K-p+1})$, where $p\in\{1,\dots, K\}$ denotes the sampling order, $q$ levels of modulation, and $K$ denotes the number nodes in the network. Under the assumption of symmetric aggregation, this approach enables a controlled trade-off between computational complexity and function computation accuracy. As a special case, we propose majority-based sampling ($p=K$), which confines aggregation to only $q$ consensus points, inherently avoiding destructive overlaps and permitting the use of standard digital modulations (e.g., QAM, PSK, ASK) without bespoke constellation designs. We also show via several simulations, across various aggregation functions, modulation levels, and noise levels, that moderate sampling orders attain acceptable performance with orders-of-magnitude fewer constraints than exhaustive designs.

Figures

Figures reproduced from arXiv: 2506.16208 by the authors.

**Figure 2.** This figure gives the main idea of Constellation points over [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Quantization for q “ 4 with a sampling order p “ 2. The figure illustrates the discretization of the constellation space into four levels by employing a sampling order p “ 2, highlighting the resulting grid structure derived from the Minkowski sum of the transmitted symbols. to Opq p q, where p denotes the pyramid level. More specifically, for a symmetric function g with quantized input values xk P X , we can defin… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Trade-off curves between the worst-case approximation error [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 6.** Figure 6: Illustration of the induced constellation diagram of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 5.** Figure 5: MSE versus input quantization level q P t2, 4, . . . , 32u for the sum (a) and product (b) functions over K “ 5 inputs, comparing three schemes: p “ 1 with order q, p “ K with order qK, and p “ K with order qK. The MSE is computed by Monte Carlo simulation (2 ˆ 103 sam…

**Figure 7.** Figure 7: Illustration of the sampling scheme for q “ 4 computing the product function fps1, s2q “ s1s2, with s1, s2 P t0, 1, 2, 3u. Only the corner points of the superimposed constellation are employed; overlaps at the centre between output symbols “2” and “0” are ignored, simp…

**Figure 8.** Figure 8: Constellation diagrams under full-enumeration (AWGN) and pyramid sampling orders [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: MSE versus per-node SNR for the p ś k skq 1{K, and maxx sk functions under extreme-point decoding with uniform quantization levels q P t16, 32, 64u. Each curve corresponds to one modulation format—rectangular QAM, one-dimensional PAM, or a hexagonal lattice—normalized …

**Figure 10.** Figure 10: MSE versus per-node SNR for the ř k sk, ś k sk, and maxk sk functions under two strategies. Scenario 1 employs an SDP-designed constellation of size q “ 4 over K “ 10 users, while Scenario 2 uses an SDP-designed constellation of size q “ 1024 (designed in the optimiza…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design
eess.SP 2025-06 conditional novelty 5.0 of 10

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.

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