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REVIEW 4 major objections 5 minor 21 references

Approximate Message Passing for Multi-Preamble Detection in OTFS Random Access

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A new approximate message passing algorithm called CSGL-AMP exploits both group and within-group sparsity to detect multiple OTFS preambles, reaching a 10^-2 misdetection probability in regimes where the one-step thresholding baseline is fl

desk verdict A plausible complex extension of sparse-group LASSO AMP for OTFS random access, with real simulation gains but an unproven AMP step for the structured dictionary. read the letter →

arxiv 2509.03980 v1 pith:AJNQJ26T submitted 2025-09-04 eess.SP

Alessandro Mirri , Vishnu Teja Kunde , Enrico Paolini , Jean-Francois Chamberland This is my paper · ORCID
classification eess.SP
keywords OTFSrandomaccesspreambledetectionapproximatemessagepassingsparsegroupLASSOcomplexAMPdelay-Dopplersparsity
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 tries to establish that multi-preamble detection in OTFS-based random access is best treated as a complex sparse-group-LASSO problem, and that an approximate message passing (AMP) algorithm with a denoiser enforcing both group-level and element-level sparsity can solve it far better than the existing one-step thresholding (OST) detector. In the underdetermined regime with a 0.3 measurement ratio and 20% active preambles, the proposed CSGL-AMP reaches a misdetection probability of 10^-2 at about 12.5 dB SNR, while OST exhibits a floor above 10^-2. The paper derives the algorithm's Onsager correction from the Wirtinger calculus of a two-stage complex denoiser, which is the novel step that makes double sparsity compatible with AMP. If the claim holds, OTFS receivers for grant-free massive access gain an iterative, scalable detector whose accuracy exceeds the current baseline.

What carries the argument

The key object is the complex sparse group LASSO (CSGL) denoiser η(r, λ1, λ2), the proximal operator of the regularization λ1||β||1 + λ2 Σ_g ||β_g||2. Per group, it is the composition of element-wise complex soft-thresholding η_st(r, λ1) = (1−λ1/|r|) r for |r|>λ1, followed by group shrinkage (1−λ2/||s_g||2) s_g. The second load-bearing piece is the Onsager correction: the average of the Wirtinger derivatives of η, derived in closed form as (1/N) Σ_g Σ_{j∈J_g} η′_{g,j}(r). That term keeps the residual approximately uncorrelated with the estimate across iterations, allowing AMP to converge at the low per-iteration cost of matrix-vector products. It reduces to the standard complex soft-threshol

What would settle it

Run CSGL-AMP on the exact Veh-A OTFS sensing matrix of Section IV at δ=0.3 and ρ_G=0.2, and compare the empirically averaged Wirtinger divergence of the denoiser with the closed-form Onsager term (26) at each iteration; if the two disagree beyond Monte Carlo fluctuation, or if removing the Onsager term leaves the misdetection curve unchanged, then the proposed correction is not the mechanism behind the reported gains.

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Extended reading notes

Core claim

The central claim is that CSGL-AMP, an AMP recursion whose denoiser is the proximal map of the complex sparse group LASSO regularizer, detects multiple OTFS preambles with substantially lower misdetection than the current one-step thresholding method. The recursion updates an effective observation r(t)=β(t)+X^H z(t), denoises it through the two-stage operator η, and corrects the residual with the averaged Wirtinger divergence of η. The denoiser first applies complex soft-thresholding with λ1 inside each group and then shrinks the entire group by λ2 based on the norm of the thresholded group, so a preamble is declared only when its delay-Doppler energy survives both tests. In Monte Carlo simu

Load-bearing premise

The AMP update rule uses an Onsager correction whose statistical justification assumes a sensing matrix that is i.i.d. or rotation-invariant; the OTFS matrix here is a fixed, deterministic matrix built from shifted Zadoff-Chu sequences, and the paper does not analyze whether the correction remains valid for that structure.

Editorial extensions

If this is right

  • If the claims hold, CSGL-AMP is a drop-in replacement for OST in OTFS random access receivers, with gains concentrated in the low-misdetection regime that matters for reliable grant-free access.
  • The same Wirtinger-derived Onsager machinery should apply to any complex-valued sparse-group problem, not just OTFS, since the denoiser and correction do not depend on OTFS specifics.
  • The validity-region curves give an explicit operating envelope: for a given SNR and target P_md=10^-2, the receiver knows the maximum active-preamble fraction it can support at a given measurement ratio.
  • Because each AMP step is just matrix-vector products, the performance gains come without an increase in per-iteration complexity relative to OST, making the approach scalable to larger OTFS grids and preamble pools.

Reading between the lines

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

  • Editorial inference: the denoiser derivation transfers directly to OTFS channel estimation and joint activity and data recovery, where the same delay-Doppler sparsity structure appears, so the method's practical reach is larger than the preamble-detection framing suggests.
  • Editorial inference: the paper leaves open whether the deterministic Zadoff-Chu sensing matrix satisfies the invariance conditions under which AMP state evolution is proven; a simulation-based or analytic state-evolution study for structured OTFS matrices would be needed to confirm that the Onsager correction remains exact in this setting.
  • Editorial inference: the adaptive threshold rules for λ1 and λ2 are described as online approximations; a natural testable extension is to benchmark these heuristics against oracle-tuned thresholds to see how much of the reported gain comes from threshold adaptation rather than from the double-sparsity structure.
  • Editorial inference: in a collision scenario where two active users select the same preamble, group sparsity alone cannot separate them; extending the framework to collision resolution would likely require coupling CSGL-AMP with successive interference cancellation or a different group code.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes an approximate message passing (AMP) algorithm, CSGL-AMP, for complex sparse group LASSO, applied to multi-preamble detection in OTFS random access. The received signal is modeled as y = Xβ + w with a dictionary of shifted OTFS preambles; β has group sparsity (few active preambles) and within-group sparsity (few multipath taps). The algorithm iterates matched-filter updates (17)–(19), uses a two-stage denoiser (complex soft-thresholding plus group shrinkage), and includes an Onsager correction computed via Wirtinger calculus. Simulations compare the misdetection probability with the one-step thresholding baseline [17] and with complex-LASSO/group-LASSO AMP variants, reporting gains at δ=0.3 and δ=0.8 and larger validity regions for Pmd≤1e−2. The contributions claimed are the first complex SGL-AMP, a structured denoiser with a derived Onsager term, and simulation gains.

Significance. If the algorithmic claims can be substantiated, this would be a useful contribution to grant-free random access in mMTC: the two-level sparsity model is natural for OTFS preambles over sparse multipath channels, and a working complex SGL-AMP would extend AMP tooling to an important structured problem. The reported gains over OST in Fig. 1 are substantial (CSGL-AMP reaches Pmd=1e−2 near 12.5 dB where OST floors above 1e−2). The paper is clearly organized, and the authors are right to attempt a denoiser-specific Onsager correction rather than treating the correction as a free parameter. However, the central claims are not yet supported: the compact denoiser formula is internally inconsistent, the AMP validity for the structured sensing matrix is unexamined, the Onsager derivation is opaque, and the adaptive threshold rule is undisclosed. These gaps make the numerical results non-reproducible and the 'rigorously derived' claim unverifiable.

major comments (4)
  1. [III-B, Eq. (23)] The compact expression for the denoiser is not equivalent to the two-stage rule in Eqs. (20)–(22). For |r_{g,j}| ≤ λ1, the factor (1 − λ1/|r_{g,j}|) is non-positive; because the group factor in front can be positive when other elements in the group are large, the expression yields a nonzero output for a coefficient that Eq. (21) sets to zero. Example: r=(0.5,10), λ1=1, λ2=1 gives η_1 ≈ −0.444 instead of 0. The element factor needs a positive part, i.e. (1 − λ1/|r|)_+, or equivalently (|r|−λ1)_+ e^{i∠r}. Since the Onsager divergence in Eq. (26) is derived from Eq. (23), all subsequent formulas inherit this error.
  2. [III-A, Eqs. (17)–(19) and Section IV] The AMP recursion is used for the deterministic structured sensing matrix X defined in Eq. (5), which is neither i.i.d. nor right-orthogonally invariant; the standard AMP guarantees [14], [18] cited in Section II-D do not cover this case. No state evolution derivation, residual decorrelation test, or convergence diagnostic is provided for the actual X. Please add either a theoretical argument tailored to the OTFS/Zadoff–Chu structure or an empirical validation under the same matrix (e.g., compare AMP estimates against a convex sparse-group-LASSO solver, monitor residual correlations over iterations, and report failure rates over many channel realizations). Without this, the validity of the Onsager term in the regime of Figs. 1–3 is unsupported.
  3. [III-C, Eq. (26)] The 'after simplification' closed form for the Onsager term is the key novel claim, but the algebra is not shown. As typeset, Eq. (26) is also ambiguous: the placement of the λ2/2 factor and the square root makes the expression hard to parse. Please provide a complete derivation of the Wirtinger derivative starting from Eq. (23) — or from its corrected version — and give a clean, unambiguous formula. Without this, the central algorithmic mechanism cannot be checked.
  4. [III-A, adaptive thresholds] The update rule for λ1(t) and λ2(t) is never specified; the text only says 'suitable online approximations'. Since the numerical results in Section IV depend on these thresholds, the figures cannot be reproduced and it is unclear whether the reported gains rely on a particular tuning schedule. Provide the exact update equations, initialization, and any hyperparameters. If the thresholds are derived from noise variance or residual energy, give the formula.
minor comments (5)
  1. [III-B] Please define (x)_+ consistently and state explicitly that the element-wise factor in Eq. (23) should be (1 − λ1/|r|)_+.
  2. [III-C] The notation in Eq. (26) is hard to read; please rewrite using displayed fractions, define J_g, G_a, and the positive-part operator.
  3. [IV] The number of AMP iterations and the convergence criterion are not given. Report these in the table/caption or in Section III-A.
  4. [II-A] In Eq. (5), the index i in X_{j|S|+i} is not defined over the Cartesian product order; specify the ordering of S and the range of i.
  5. [Introduction/Related Work] The relationship to the real-valued SGL-AMP in [19] should be discussed; the authors claim a first complex extension but do not contrast with the real-valued analysis or explain what changes in the complex domain.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: CSGL-AMP derivation is self-contained; main risks are unvalidated AMP assumptions and unspecified threshold adaptation, not circularity.

full rationale

The derivation chain is self-contained. CSGL-AMP is obtained by instantiating the standard AMP recursion (17)-(19) with a proximal denoiser (20)-(23), and the Onsager term is computed from that denoiser via the Wirtinger divergence (24)-(26); no target quantity (e.g., Pmd, SNR thresholds) is fed into the algorithm derivation. The performance claims are Monte Carlo comparisons, not predictions derived from fitted parameters. The only same-author citation, [13], appears in Sec. I as background for 'multiple preamble detection in OTFS-based random access'; it is not load-bearing for the algorithm or its conclusions. The passage in Sec. III-A saying thresholds 'are updated adaptively at every iteration through suitable online approximations' is a genuine reproducibility gap, but it is not circular: no equation or description shows the thresholds being fitted to the reported Pmd values, so it does not reduce the claimed gains to inputs. AMP's applicability to the structured OTFS matrix is asserted without state-evolution validation, but that is a correctness risk, not circularity. Thus, no circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The algorithm relies on two free adaptive thresholds that are not specified, and on the untested applicability of AMP state evolution to the structured sensing matrix. No new physical entities are introduced.

free parameters (2)
  • λ1 (within-group threshold) = adaptive, unspecified
    Controls elementwise sparsity. The paper states thresholds are updated adaptively via 'suitable online approximations' (Section III-A) but never gives the rule. Performance likely depends on this choice.
  • λ2 (group threshold) = adaptive, unspecified
    Controls group sparsity. Same issue as λ1; no update rule is provided. The simulations may rely on hand-tuned or silently adapted values.
assumptions (4)
  • domain assumption AMP state evolution assumptions hold for the structured OTFS sensing matrix X
    The algorithm uses standard AMP iterations with an Onsager correction (Section III-A). This is justified for i.i.d. Gaussian-like matrices, but X is built from Zadoff-Chu shifts and is deterministic and structured. The paper does not verify this assumption.
  • standard math The two-stage denoiser (elementwise soft-threshold then group soft-threshold) is the exact proximal operator of complex sparse group LASSO
    For disjoint groups, the composition of soft-thresholding and group soft-thresholding is known to be the proximal operator of the sum of ℓ1 and group ℓ2 penalties. The paper relies on this without proof, but it is a standard result.
  • domain assumption Collision-free random access with exactly K active users, each choosing a distinct preamble
    Used in the simulation setup (Section IV). It simplifies the group sparsity structure. Real systems may have collisions, which the paper defers to future work.
  • domain assumption The Veh-A channel model with L=6 dominant delay-Doppler components is representative
    Determines the sensing matrix and the within-group sparsity of β. The paper does not test other channel models, so the claimed robustness is limited to this assumption.

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Cite this review

Pith. "Pith review of Approximate Message Passing for Multi-Preamble Detection in OTFS Random Access." pith.science (2026). https://pith.science/paper/AJNQJ26T

@misc{pith2026250903980,
  author       = {Pith},
  title        = {Pith review of: Approximate Message Passing for Multi-Preamble Detection in OTFS Random Access},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJNQJ26T}},
  note         = {Machine review of arXiv:2509.03980}
}
read the original abstract

This article addresses the problem of multiple preamble detection in random access systems based on orthogonal time frequency space (OTFS) signaling. This challenge is formulated as a structured sparse recovery problem in the complex domain. To tackle it, the authors propose a new approximate message passing (AMP) algorithm that enforces double sparsity: the sparse selection of preambles and the inherent sparsity of OTFS signals in the delay-Doppler domain. From an algorithmic standpoint, the non-separable complex sparsity constraint necessitates a careful derivation and leads to the design of a novel AMP denoiser. Simulation results demonstrate that the proposed method achieves robust detection performance and delivers significant gains over state-of-the-art techniques.

Figures

Figures reproduced from arXiv: 2509.03980 by the authors.

Figure 1
Figure 1. Probability of misdetection Pmd versus SNR for the proposed CSGL￾AMP algorithm and the OST algorithm from [17]. The block sparsity is set to ρG = 0.2, and the measurement ratio is δ = 0.3. 0 2 4 6 8 10 12 14 10−4 10−3 10−2 10−1 1 SNR [dB] Pmd OST CL-AMP CGL-AMP CSGL-AMP [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Probability of misdetection Pmd versus SNR for the proposed CSGL￾AMP algorithm and the OST algorithm from [17]. The block sparsity is set to ρG = 0.3, and the measurement ratio is δ = 0.8. 12.5 dB. The weaker results of CL-AMP and CGL-AMP highlight the need to exploit both group and within-group sparsity. Fig.2 depicts a less sparse case with δ = 0.8, G = 72, and ρG = 0.3 (K = 21). The performance gap narrows, yet C… view at source ↗
Figure 3
Figure 3. Validity regions in the (ρG, δ) plane where the target misdetection probability P ⋆ md ≤ 10−2 is met, shown for two SNR levels. The comparison is carried out between the proposed CSGL-AMP algorithm and the OST algorithm from [17]. at high ρG. At 20 dB, CSGL-AMP remains superior, with higher SNR enlarging the region to support more active users. Although AMP requires multiple iterations, each step involves only simpl… view at source ↗

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