Distributed Distortion-Aware Robust Optimization for Movable Antenna-aided Cell-Free ISAC Systems

Dusit Niyato; Guangyi Liu; Ning Wei; Ran Yang; Wanting Lyu; Yang Zhao; Yue Xiu; Zeyuan Zhang; Zheng Dong

REVIEW 2 major objections 5 minor 32 references

Distributed Distortion-Aware Robust Optimization for Movable Antenna-aided Cell-Free ISAC Systems

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proposes a worst-case robust beamforming and antenna-positioning design that treats power-amplifier distortion as bounded uncertainty, and claims it improves the communication-sensing trade-off in cell-free ISAC systems.

desk verdict A fresh problem setup, but the key robustness bound is mathematically wrong: Eq. (69) drops a positive cross term, so Theorem 1 does not certify worst-case performance. read the letter →

arxiv 2508.13839 v2 pith:ZSYHHRZU submitted 2025-08-19 eess.SP

Yue Xiu , Yang Zhao , Ran Yang , Zheng Dong , Wanting Lyu , Zeyuan Zhang , Dusit Niyato , Guangyi Liu

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Ning Wei

This is my paper · ORCID

classification eess.SP

keywords cell-freeISACmovableantennapoweramplifiernonlinearitythird-orderdistortioncoefficientworst-caserobustoptimizationgraphneuralnetworkCRLBconstraint

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

Integrated sensing and communication networks that share spectrum across many access points degrade when power amplifiers distort transmitted signals, and the distortion coefficients vary from one amplifier to the next. This paper asks whether movable antennas can make the system robust to that uncertainty, and answers by treating the third-order distortion coefficient $\beta_{3,a}$ of each access point as lying in a bounded set $|\beta_{3,a}|\le\epsilon$. It derives a conservative worst-case lower bound on the achievable sum rate and a conservative sensing-accuracy constraint, then solves the joint beamforming and antenna-positioning problem with successive convex approximation and a graph neural network. The reported simulations indicate that the resulting design improves the communication-sensing trade-off under distortion and outperforms fixed-position antenna baselines in both robustness and capacity. If correct, this would let cell-free ISAC systems operate under realistic amplifier impairments without knowing the exact distortion coefficients.

What carries the argument

The load-bearing machinery is the combination of a third-order memoryless polynomial PA model with the Bussgang decomposition, which expresses the distorted output as $\mathbf{c}_a=\boldsymbol{\Xi}_a\mathbf{s}_a+\mathbf{d}_a$ with an uncorrelated distortion vector. The uncertainty set $|\beta_{3,a}|\le\epsilon$ converts distortion into a bounded perturbation, and Theorem 1 supplies the conservative worst-case rate bound $L(\epsilon)$ and the sensing constraint $g(\epsilon,\epsilon_0)$. Successive convex approximation handles the nonconvexity of the resulting max-min problem, and the SACGNN, a heterogeneous graph network with Transformer-style self-attention and convolutional layers, coordinates beamforming and antenna-position decisions across access points. The graph structure lets each access point aggregate channel, position, and distortion features from its neighbors before decoding transmit beamformers and MA positions.

What would settle it

Compute $L(\epsilon)$ and the true minimum over $\beta_{3,a}$ of the objective in Eq. (17) for a fixed small network with two access points, two users, and one target; if the true worst-case sum rate falls below $L(\epsilon)$ for any feasible beamforming and MA positions, the claimed guaranteed robustness is not established. The same check applies to the sensing constraint by comparing the exact Fisher information matrix, including the derivative of the round-trip delay phase, with the conservative bound $g(\epsilon,\epsilon_0)$.

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Extended reading notes

Core claim

The paper's central claim is that PA-induced nonlinear distortion can be handled by a distributed worst-case design rather than by exact distortion compensation. Under a third-order memoryless polynomial model and the Bussgang decomposition, the amplified signal splits into a linear term and an uncorrelated distortion term whose covariance depends on the beamformers. The authors bound the unknown third-order coefficients as $|\beta_{3,a}|\le\epsilon$ and derive a conservative lower bound $L(\epsilon)$ for the worst-case sum rate together with a conservative CRLB constraint $g(\epsilon,\epsilon_0)$ on sensing accuracy. They then maximize this lower bound over beamforming matrices and movable-antenna positions. The claimed discovery is that this worst-case formulation, solved by the proposed SACGNN, substantially improves the communication-sensing trade-off under distortion and beats fixed-position antenna baselines in robustness and capacity.

Load-bearing premise

The argument rests on the validity of a triangle-inequality bound in Eq. (69), specifically that $|A+B|^2$ is no larger than $|A|^2+|B|^2$ for the linear and distortion terms; because the cross term $2\operatorname{Re}(AB^*)$ can be positive, the derived $L(\epsilon)$ may not be a guaranteed lower bound on the worst-case sum rate.

Editorial extensions

If this is right

Under the bounded-3RDC assumption, the proposed max-min beamforming and MA-position policy yields higher worst-case sum rates than designs that ignore distortion uncertainty.
Movable antenna repositioning adds spatial degrees of freedom that can be steered away from distortion-dominated directions, with sum-rate gains growing as the number of antennas per access point increases.
The SACGNN solution is decentralized and has per-layer complexity linear in the number of graph edges, making it scalable to large cell-free deployments.
The conservative CRLB constraint gives a tunable link between sensing accuracy and communication rate, since relaxing $\gamma_b$ frees beamforming degrees of freedom for communication.

Reading between the lines

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

Because the worst-case bound in Appendix B uses a triangle-inequality estimate that can be checked numerically, the design could be re-run with an exact worst-case evaluation over $\beta_{3,a}$; a comparison of the true max-min rate with $L(\epsilon)$ would show how much of the reported gain is real robustness versus conservatism of the bound.
The same bounded-uncertainty treatment could be applied to other amplifier imperfections, such as phase noise or I/Q imbalance, by adding their coefficients to the Bussgang distortion covariance.
In a deployment where each access point can estimate its own $\beta_{3,a}$ online, the global fixed bound $\epsilon$ could be replaced by per-AP adaptive uncertainty sets, potentially improving the trade-off further.

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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

Dear Colleague,

This paper is worth a look, but the central robustness guarantee does not hold. The combination of movable antennas, PA distortion uncertainty, and a GNN-based distributed solver for cell-free ISAC is genuinely new, and the problem is well motivated. The system model is carefully laid out, and the simulation study is extensive, even though it lacks error bars and in Fig. 7 compares different antenna counts.

The soft spot is the core theorem. In Appendix B, Eq. (69) bounds |A+B|^2 by |A|^2+|B|^2 when |B| <= eps|C|. That is not an upper bound because the cross term 2 Re(A^*B) can be positive. A scalar example with A=1, B=0.4 gives |A+B|^2 = 1.96, larger than 1 + 0.16 = 1.16. Consequently L(eps) in Theorem 1 is not a guaranteed lower bound, and the max-min problem is not a conservative reformulation of (23). This is not a small typo; the bound is load-bearing for the paper's claim of certified robustness. The CRLB derivation in Appendix A also omits derivatives of the round-trip delay phase with respect to the target position, so the sensing constraint has the same kind of problem.

Credit where it's due: the paper identifies a real hardware bottleneck, the problem formulation is plausible, and the SACGNN is a sensible way to reach distributed decisions. If the authors replace the invalid inequality with the correct triangle inequality and rework the analysis, the paper could be salvageable. The convergence claim should be proven or explicitly stated as empirical, and the simulations would be more convincing with multiple runs.

The paper should be read by researchers working on robust beamforming for ISAC or GNN-based physical-layer optimization; they will find the problem setup and architecture ideas useful, but they should not rely on the theoretical certificate. It deserves a serious referee because the topic is timely and the paper is substantial, but in current form the verdict should be reject. I recommend sending it to review with a note asking the authors to fix the inequality before any acceptance.

Referee Report

2 major / 5 minor

Summary. The manuscript studies a movable-antenna (MA)-aided cell-free integrated sensing and communication (CF-ISAC) system in which power amplifier nonlinearity is modeled by a third-order memoryless polynomial with bounded uncertain coefficients. It formulates a worst-case max-min problem that jointly optimizes beamforming and MA positions under transmit-power and sensing-accuracy constraints, and it proposes a graph-neural-network (SACGNN) solution trained in an unsupervised manner. The main theoretical contribution is Theorem 1, which claims a tractable conservative reformulation of the robust problem using a lower bound L(epsilon) on the communication objective and a conservative CRLB constraint g(epsilon, epsilon_0). The numerical section compares the proposed algorithm against DRL and fixed-position-antenna baselines.

Significance. If the robustness certificates were valid, the paper would address a timely practical problem—PA distortion uncertainty in distributed ISAC—and the combination of MAs with a distributed learning-based solver would be a useful engineering contribution. The paper also models hardware variability explicitly and compares against several baselines. However, the central theoretical guarantee is not established: the key inequality in Appendix B is mathematically incorrect, and the CRLB derivation omits a substantial dependence of the sensing channel on the target location. Consequently, the claimed conservative reformulation is not supported, and the numerical results remain a heuristic demonstration rather than a validated robust design. The paper provides no machine-checked proofs or reproducible code, so the burden falls entirely on the derivations, which fail at the load-bearing step.

major comments (2)

[Appendix B, Eq. (69)] The claimed bound |sum_b h_{b,k}^H Xi_b w_{b,j}|^2 <= |sum_b beta1 h_{b,k}^H w_{b,j}|^2 + epsilon^2 |sum_b 2 h_{b,k}^H diag{W_a W_a^H} w_{b,j}|^2 is not valid. Writing A = sum_b beta1 h^H w and B = sum_b 2 beta_{3,b} h^H diag{WW^H} w, the exact identity is |A+B|^2 = |A|^2 + |B|^2 + 2 Re(A^* B), and the cross term can be positive. Since this inequality is used to upper-bound the interference contribution in delta, the resulting L(epsilon) in Eq. (34) is not a guaranteed lower bound on the worst-case objective; Theorem 1 therefore does not provide the claimed conservative reformulation of problem (23). A concrete scalar counterexample with beta1 = 1, beta3 = 0.2, and h = w = 1 gives |1.4|^2 = 1.96 on the left-hand side, while the claimed bound evaluates to 1 + 0.16 = 1.16.
[Appendix A, Eqs. (55)-(57)] The FIM derivation omits the derivative of the round-trip delay phase e^{-j2 pi f_s tau_{a,b}} with respect to the target location e. The sensing channel in Eq. (9) depends on e through tau_{a,b}(e) = (l_{a,d}(e) + l_{b,d}(e))/c, where l_{a,d} and l_{b,d} are defined in Eqs. (11)-(12). This dependence contributes additional terms to dot H_{a,b} and ddot H_{a,b} that are not present in Eqs. (55)-(57). Without these terms, CRLB_b(e) in Eq. (22) and the robust sensing constraint in (23c)/(33b) are not correctly characterized, so the sensing-side robustness guarantee is also unsupported.

minor comments (5)

[Appendix A, Eq. (60)] Eq. (60) writes E{F^b_{n1,n2}} while Eq. (58) defines F^b_{n1,n2}; the expectation and the expansion are not clearly derived, and the indexed quantities dot Upsilon, ddot Upsilon, and tilde Upsilon are introduced without defining all variants explicitly.
[Section IV.A, Eq. (33)] The statement that problem (33) is convex is not justified: L(epsilon) contains absolute values and products involving the optimization variables, and g(epsilon, epsilon_0) is a first-order Taylor expansion of a non-convex function. A proof or a more precise convexity argument is needed.
[Section V, Eq. (39)] The loss function in Eq. (39) uses a parameter beta > 0 that is not defined, and C1 is not explicitly connected to the constraint set of problem (36). Please define all symbols in the training objective.
[Section IV.B, Eq. (35)] Eq. (35) repeats the formulation already given in Eq. (32); if a different sensing-oriented problem is intended, the text should distinguish the two formulations clearly.
[Section VI, Fig. 4] The axis labels in Fig. 4 appear inconsistent with the text description: the horizontal axis is labeled 'Sum rate [bits/s]' while the text discusses 'number of episodes with high sum rate.' Please redraw the figure or revise the caption and description.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the robust derivation is self-contained; Eq. (69) is a correctness concern, not a circular one.

full rationale

The paper's central claims are produced by a constructive robust-optimization pipeline, not by re-importing its own target as an input. The worst-case rate bound L(epsilon) in Eq. (34) is derived from explicit model assumptions (third-order polynomial PA model, Bussgang decomposition Eqs. (3)-(6), and the channel models in Section III), and the uncertainty radius epsilon is treated as a design/uncertainty parameter that is optimized or swept in Theorem 1, not fitted to the sum-rate or CRLB values used in the evaluation. Similarly, the sensing CRLB bound in Appendix A is a derivation from the stated likelihood (Eq. (21)) using the standard Slepian-Bangs FIM expression, with the derivative terms then expanded in Eqs. (54)-(60); the citation to the authors' prior work [32] supplies a standard formula rather than the paper's own conclusion, so it is not a load-bearing self-citation chain. The SACGNN is trained in an unsupervised manner to maximize the worst-case sum rate subject to constraints, and the numerical comparison is against explicit external baselines (DRL schemes and an FPA SCA baseline) under a common simulated channel model; evaluating an optimization algorithm on the same model used for training is standard self-contained benchmarking and does not make the measured performance a fitted prediction. I therefore find no step where a prediction reduces to an input by construction. I note for completeness that Eq. (69) in Appendix B asserts |A+B|^2 <= |A|^2 + |B|^2, which drops the nonnegative cross term and is not a valid upper bound; this is a genuine correctness risk for the claimed guarantee L(epsilon), but an erroneous inequality is not a circularity pattern and does not change the circularity score.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The central design rests on the Bussgang decomposition under Gaussian inputs, the third-order memoryless PA model, statistical independence of distortion across APs, known geometric channels, and the questionable premise that H_{a,b} depends on target location only through steering vectors. No new physical entities are introduced. Free parameters include the uncertainty bound epsilon, the SCA linearization point epsilon_0, the loss penalty lambda, and the GNN training configuration.

free parameters (4)

epsilon (3RDC uncertainty bound) = swept 0.05 to 0.25 in simulations; also appears as optimization variable in Eq. (33)
The robust solution and all bounds are functions of this manually set or swept quantity; it is not derived from measured hardware data.
epsilon_0 (SCA linearization point) = not specified
Theorem 1 and Eq. (34) require a Taylor expansion of g around epsilon_0; no value or update rule is given, so the approximation is unspecified.
lambda (constraint penalty in SACGNN loss) = not specified
The composite loss in Eq. (39) balances rate against constraint violations; the value is not reported and affects which feasible solution is learned.
GNN hyperparameters = L=4, 1600x800, lr=0.01, batch 500, 5000 epochs
Hand-chosen training configuration; performance of the learned policy depends on these choices.

assumptions (8)

domain assumption Third-order memoryless polynomial model of PA distortion, z = beta1*x + beta3*x*|x|^2 (Eq. 1)
Real PAs have memory, higher-order terms, and frequency dependence; this model is a simplification used throughout.
domain assumption Bussgang decomposition holds under Gaussian input assumption; distortion d_a is uncorrelated with input and has covariance Eq. (6)
The linearization and all SINDR/CRLB formulas use this decomposition; the Gaussian assumption may be violated by finite-alphabet modulations.
domain assumption Distortion components across tAPs are statistically independent
Stated after Eq. (6); needed to sum distortion powers in SINDR and CRLB.
domain assumption Geometric multipath channel model with known AoDs, delays, and gains for MA position-dependent channels (Eqs. 7-14)
The optimization relies on perfect channel state information and known geometry; no channel estimation error is modeled.
standard math CRLB is a valid performance bound for deterministic target location estimation (Sec. III.D)
Standard estimation theory, but the likelihood in Eq. (21) is misspecified with a transpose and positive exponent, so the bound is not correctly applied.
ad hoc to paper Sensing channel H_{a,b} depends on target location only through the array steering vectors; derivatives of the delay phase e^{-j2*pi*fs*tau} w.r.t. target location are neglected (Appendix A, Eqs. 55-57)
No justification is given; the round-trip delay changes with target position and carries ranging information, so omitting its derivative likely distorts the FIM and CRLB.
domain assumption MA positions can be adjusted in real time with negligible latency
Invoked in Sec. III.B via [21]; needed for the joint beamforming and position optimization to be meaningful in a time-varying system.
domain assumption 3RDC uncertainty set is a known bounded interval |beta3,a| <= epsilon
The robust framework requires this bound; in practice it must be estimated from hardware data, and the paper instead treats epsilon as a design variable in Eq. (33).

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Cite this review

Pith. "Pith review of Distributed Distortion-Aware Robust Optimization for Movable Antenna-aided Cell-Free ISAC Systems." pith.science (2026). https://pith.science/paper/ZSYHHRZU

@misc{pith2026250813839,
  author       = {Pith},
  title        = {Pith review of: Distributed Distortion-Aware Robust Optimization for Movable Antenna-aided Cell-Free ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSYHHRZU}},
  note         = {Machine review of arXiv:2508.13839}
}

read the original abstract

The cell-free integrated sensing and communication (CF-ISAC) architecture is a promising enabler for 6G, offering spectrum efficiency and ubiquitous coverage. However, real deployments suffer from hardware impairments, especially nonlinear distortion from power amplifiers (PAs), which degrades both communication and sensing. To address this, we propose a movable antenna (MA)-aided CF-ISAC system that mitigates distortion and enhances robustness. The PAs nonlinearities are modeled by a third-order memoryless polynomial, where the third-order distortion coefficients (3RDCs) vary across access points (APs) due to hardware differences, aging, and environmental conditions. We design a distributed distortion-aware worst-case robust optimization framework that explicitly incorporates uncertainty in 3RDCs. First, we analyze the worst-case impact of PA distortion on both the Cramer-Rao lower bound (CRLB) and communication rate. Then, to address the resulting non-convexity, we apply successive convex approximation (SCA) for estimating the 3RDCs. With these, we jointly optimize beamforming and MA positions under transmit power and sensing constraints. To efficiently solve this highly non-convex problem, we develop an MA-enabled self-attention convolutional graph neural network (SACGNN) algorithm. Simulations demonstrate that our method substantially enhances the communication-sensing trade-off under distortion and outperforms fixed-position antenna baselines in terms of robustness and capacity, thereby highlighting the advantages of MA-aided CF-ISAC systems.

Figures

Figures reproduced from arXiv: 2508.13839 by the authors.

**Figure 2.** The illustration of the architecture of the SACGNN, which comprises three components, i.e., The GNN maps the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Simulation setup of the MA-aided CF-ISAC system. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: depicts the robustness and adaptability of the proposed worst-case SACGNN optimization framework under both channel estimation uncertainty and hardware nonidealities. In particular, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 4.** Figure 4: (a) Episodes versus sum rate without PA distortion. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 6.** Figure 6: (a) Sum rate versus SINDR. (b) Sum rate versus CRLB [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Sum rate versus the number of MAs [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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