Multiuser Beamforming for Pinching-Antenna Systems: An Element-wise Optimization Framework

Chongjun Ouyang; Mingjun Sun; Shaochuan Wu; Yuanwei Liu

REVIEW 2 major objections 5 minor 30 references

Multiuser Beamforming for Pinching-Antenna Systems: An Element-wise Optimization Framework

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Joint pinching-antenna beamforming reduces to a one-dimensional search per antenna.

desk verdict Useful element-wise PASS beamforming method with real but unproven local-optimality claims and no benchmark against existing PASS optimizers. read the letter →

arxiv 2506.03770 v2 pith:UPVALHKX submitted 2025-06-04 eess.SP

Mingjun Sun , Chongjun Ouyang , Shaochuan Wu , Yuanwei Liu This is my paper

classification eess.SP

keywords pinching-antennasystemselement-wiseoptimizationsum-ratemaximizationlinearbeamformingone-dimensionalsearchhybriddielectricwaveguidemultiuserMIMO

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 claims that the hard joint optimization in a pinching-antenna system—choosing both the baseband beamformer and the positions of antennas sliding on dielectric waveguides—can be reduced to a sequence of one-variable problems. For three standard linear precoders in the downlink (maximum-ratio transmission, zero-forcing, and minimum mean-square error) and the matching linear combiners in the uplink, the authors derive closed-form sum-rate expressions in which only one antenna position varies at a time. A sequential element-wise one-dimensional search then updates each pinching antenna in turn, removing the alternating inner-outer loops that earlier joint designs required. If the claim is correct, the spatial flexibility of PASS is obtained at low computational cost and, in the reported simulations, consistently beats a fixed-antenna hybrid MIMO baseline.

What carries the argument

The load-bearing mechanism is an element-wise coordinate-ascent loop over the discretized waveguide positions of the pinching antennas, sustained by algebraic rank-one updates. At each step only one position $p_{m,n}$ moves, so the effective channel changes only through its $m$-th column; the one-dimensional search maximizes a closed-form scalar objective over a grid of candidate positions, with a screening set that enforces the minimum-spacing constraint. The Sherman-Morrison, Woodbury, and Sylvester identities convert the costly matrix inversions and determinants of the ZF and MMSE objectives into scalar updates, which is what makes the sequential search competitive in complexity with fixed-antenna processing.

What would settle it

At a small configuration (e.g., $M=2$ waveguides, $N=2$ PAs, $K=2$ users) on a coarse position grid, compare the sum-rate of Algorithm 1 from many random initializations with the global optimum obtained by exhaustive enumeration of all valid position combinations. If exhaustive search finds a configuration with materially higher sum-rate than the element-wise output, or if the element-wise output falls below the fixed-antenna baseline for any beamformer, the paper's local-optimality and performance claims are falsified.

Watch

Extended reading notes

Core claim

The authors aim to establish that sum-rate maximization in a multiuser PASS is tractable without alternating between baseband and pinching optimization. They first prove by contradiction that the optimal downlink solution uses the full power budget, which lets them normalize MRT, ZF, and MMSE precoders. Substituting each precoder into the SINR expression, they obtain a compact closed-form objective in which, for a fixed environment, the only free variables are the pinching-antenna positions. Updating one position at a time reduces the objective to a scalar function of that position, and matrix inverses or determinants that would have to be recomputed at every candidate point are instead updated with rank-one identities such as the Sherman-Morrison formula and Sylvester's determinant identity. The same construction is repeated for uplink MRC, ZF, and MMSE combining. The paper's central assertion is that iterating these one-dimensional searches over all antennas yields a locally optimal solution, and the numerical experiments show PASS outperforming fixed-antenna hybrid MIMO systems under every beamformer considered.

Load-bearing premise

The load-bearing premise is that sequentially moving one antenna at a time on a discretized waveguide converges to a good locally optimal solution of a nonconvex coupled problem; the paper gives no convergence proof, so if the search stalls in a poor basin the reported PASS gains over fixed antennas may not reflect the true best achievable positions.

Editorial extensions

If this is right

For the simulated configurations, PASS achieves higher downlink and uplink sum-rate than the fixed-antenna hybrid MIMO baseline with every linear beamformer; at $N=6$ the reported downlink gains are about 137% for MMSE, 190% for ZF, and 124% for MRT, and the uplink gains are about 54% for MMSE, 74% for ZF, and 103% for MRC.
The performance gap between ZF and MMSE becomes negligible in PASS because repositioning the antennas lets the system operate in an interference-limited regime that suits zero-forcing.
The SNR range in which MRT or MRC beats ZF narrows substantially in PASS, so the choice among linear beamformers matters less than in fixed-antenna systems.
Sum-rate improves monotonically with the number of one-dimensional search samples and saturates around $10^5$ samples, indicating that near-optimal performance is reachable with a finite position grid.
The element-wise method avoids the inner-outer alternating iterations used by earlier PASS beamforming designs, lowering per-iteration computational complexity.

Reading between the lines

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

A natural stress test is update-order sensitivity: coordinate ascent on a nonconvex objective can stop at different local optima depending on how the antennas are swept, and the paper does not analyze this; if reversing the order changes the final sum-rate materially, the method would need an order-selection rule.
The rank-one structure exploited here is independent of the specific channel formula, so the same element-wise reduction could be re-derived for near-field spherical wavefronts, lossy waveguide models with different attenuation laws, or rate-energy multi-objective settings.
If the local optima found by the one-dimensional search are consistently good, the method also offers a fast initialization for more expensive refinement, such as local continuous optimization of PA positions or nonlinear precoding.

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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

Quick read on arXiv:2506.03770. The paper gives a low-complexity way to do joint baseband and pinching beamforming in PASS by turning sum-rate maximization into single-variable problems over each PA position, solved by a one-dimensional search. The closed-form reductions for MRT, ZF, and MMSE (downlink) and MRC, ZF, MMSE (uplink) are the genuinely new part; they avoid the alternating inner-outer loops of prior BCD and penalty methods. The Sherman-Morrison and determinant manipulations check out, and the complexity savings are real.

What the paper does well: the derivations are careful and the simulations are extensive across many parameters. The consistent PASS gains over fixed-antenna hMIMO are credible and align with the physical argument about path-loss and phase control.

The soft spots are two, both load-bearing. First, the claim that the sequential one-dimensional search yields a locally optimal solution is unproven. The search is over a finite grid (Ns=10^4 points), and the sum-rate as a function of a single PA position is oscillatory due to the e^{-jk0 d_k(p_m,n)} terms. A grid coordinate-wise maximum is not a local maximum of the continuous problem, and no convergence or grid-refinement analysis is given. This matters because the headline "no alternating updates needed" rests on this. Second, the numerical section never compares against the alternating-optimization methods [23]-[25] it cites. Without those baselines, the reported gains are not attributable to the framework; the PASS advantage over fixed antennas could come from the extra spatial DoFs alone. The abstract's "eliminates the need for alternating updates" is thus not empirically supported.

That said, the core algorithmic idea is sound and likely useful for system-level studies where low complexity matters. The gaps are fixable: a convergence proof for the coordinate ascent (or even a rigorous statement that it finds a stationary point of a suitably smoothed objective), and a comparison with at least one prior PASS optimizer on the same realizations. I'm not convinced the local-optimality claim can be proven as stated without additional assumptions, but the paper can be revised to soften it.

Who this is for: people working on PASS or flexible-antenna beamforming, and those doing complexity-aware 6G system design. It deserves a serious referee; the math is solid and the contribution is new within the subfield, but the missing baseline and unproven convergence claim need addressing before publication.

My recommendation: send it to peer review with the expectation of major revision. I'd want to see the missing comparison and an honest discussion of what "locally optimal" means under discretization.

Referee Report

2 major / 5 minor

Summary. The paper studies sum-rate maximization in pinching-antenna systems (PASS) for both downlink and uplink multiuser transmission. The authors propose an element-wise sequential optimization framework: for each of MRT, ZF, and MMSE baseband beamforming/combining, they derive a closed-form expression for the sum-rate as a function of a single pinching-antenna position, and then update the PA positions one by one using a one-dimensional search over a discrete grid. The manuscript claims that this procedure yields a locally optimal solution and eliminates the need for alternating optimization between the baseband and pinching beamformers. Numerical results are provided to show that the proposed PASS architecture consistently outperforms a fixed-antenna hybrid MIMO benchmark.

Significance. If the algorithmic claims were fully substantiated, the proposed framework would provide a genuinely low-complexity alternative to the alternating-optimization methods in [23]-[25] for PASS beamforming. The algebraic derivations are transparent, and the use of Sherman-Morrison, Woodbury, and Sylvester identities appears to be correct in the places I checked. The simulation protocol is clearly parameterized and reproducible in principle. However, the two central claims—local optimality of the coordinate-ascent solution and competitiveness against existing PASS joint-optimization methods—are not yet supported by the manuscript. These gaps are load-bearing because the abstract and conclusion advertise the method as achieving local optimality and as removing the need for alternating updates.

major comments (2)

The assertion that the sequential one-dimensional search over the discrete set X_m in (25) yields a locally optimal solution to the continuous sum-rate problem is not substantiated. No convergence proof is provided for Algorithm 1, and the single-variable objective functions contain rapidly oscillating phase terms of the form e^{-j k0 d_k(p_{m,n})}; a coordinate-wise maximum on a finite grid is not generally a local maximum of the continuous problem, and no grid-refinement or error-bound analysis is given. Furthermore, since the spacing constraints (15c) are coupled across n and the algorithm updates PAs in a fixed order without verifying that the previous value of p_{m,n} remains feasible after earlier updates in the same sweep, even monotonic improvement of the objective across sweeps is not guaranteed. This is load-bearing for the abstract's and Section III's claims of a locally optimal low-complexity solution.
The numerical study compares the proposed method only against a fixed-antenna hMIMO benchmark and does not include any of the existing PASS joint beamforming methods [23]-[25]. As a result, the reported gains over fixed antennas primarily demonstrate the additional spatial degrees of freedom of PASS rather than the effectiveness of the proposed element-wise optimizer. The paper's claim that the element-wise design achieves competitive sum-rate without alternating updates is therefore not directly tested. A comparison with at least one alternating-optimization baseline on the same realizations, with comparable initialization and computational budget, is needed to support the low-complexity claim.

minor comments (5)

The text refers to 'problem (39)' and 'problem (40)' in the MRT subsection, but these equation numbers correspond to later MMSE-related problems; the intended references are the downlink MRT problem (20a) and the update rule (26). Algorithm 1 line 6 similarly refers to 'problem (40)' but should refer to (26).
The set difference operation is written with the symbol '/', which is nonstandard and could be confused with division; the notation '\setminus' would be clearer.
The paper emphasizes low computational complexity but does not provide a formal complexity analysis of Algorithm 1, such as the number of objective evaluations or matrix operations per iteration. A brief complexity table or a comparison of runtime with [23]-[25] would strengthen the claim.
In Eq. (22a), a_{k,k'}(p_m) is defined as \bar{a}_k(p_{m,n}) \bar{a}_{k'}^H(p_{m,n}); since \bar{a}_k is a scalar, the superscript H is unnecessary and may confuse readers about the dimensions of the quantities involved.
The final paragraph contains a grammatical error: 'PASS significantly outperform' should be 'PASS significantly outperforms'. There are also several places where 'problem (39)' should refer to earlier equations, which should be fixed in revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation is present: the sum-rate objectives follow by exact algebraic transformation from the stated channel model, and the only structurally used self-citation ([24, Lemma 1]) is a minor constraint simplification, not a load-bearing premise.

full rationale

The central derivations in Sections III and IV take the stated PASS channel model from Section II and transform the SINR expressions into single-variable functions of each PA position using exact algebra, e.g., equations (19)-(24), (32)-(34), and (43)-(63). No fitted constants or inferred parameters enter these expressions, and no reported quantity is constructed from itself. The paper does cite prior work by overlapping authors, but these uses are not circular: [8] is a background architecture reference, [27] is a standard free-space channel formula, and [24, Lemma 1] is invoked only to relax the ordering constraints in (15c)-(16b) into a pairwise spacing constraint. That lemma, although from co-authored work, is a parameter-free supporting result and is not the source of the closed-form sum-rate expressions or the element-wise update rules. The main caveats are correctness and validation gaps rather than circularity: Section III-A after (26) and Algorithm 1 assert without proof that the discretized coordinate-ascent loop reaches a locally optimal solution, and the numerical section does not compare against the alternating-optimization baselines [23]-[25], so the claim that alternating updates are unnecessary is under-validated. Because the only self-citation with any structural role is minor and non-load-bearing, the circularity score is 2.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation introduces no curve-fitted constants. All numerical values, Dx, Dy, M, N, K, kappa, neff, and Ns, are simulation scenario settings, not parameters tuned to force the result. The derivation rests on the PASS channel and waveguide models taken from the prior literature, plus an unproved coordinate-ascent convergence assumption. No new physical entities are postulated.

assumptions (6)

domain assumption LoS-only channel model: h_{k,m,n} = lambda e^{-jk0 d}/(4 pi d), NLoS paths neglected.
Invoked in Section II-B to define all channel coefficients. If NLoS paths are significant, the sum-rate and optimal positions change.
domain assumption Waveguide propagation model: per-PA loss eta_{m,n} = 10^{-kappa |p-o|/10}/N and phase e^{-jkg(p-o)}; uplink uses zeta without the /N factor.
Adopted from [22] in Section II-A. The closed-form derivations depend on this loss and phase structure.
domain assumption Each PA acts as an isotropic antenna with no mutual coupling beyond the spacing constraint Delta.
Stated in Section II-A. Real dielectric particles may have directional patterns and additional coupling.
domain assumption Full-rank effective channel and M > K are assumed for ZF and for the Sherman-Morrison identities.
Remark 1 in Section III-B; needed for matrix inversion identities and ZF feasibility.
ad hoc to paper Coordinate-ascent one-dimensional search converges to a locally optimal solution of the original nonconvex problem.
Section III-A and Algorithm 1 assert local optimality without proof. Standard coordinate descent on nonconvex SINR objectives has no general convergence guarantee.
domain assumption PA position ordering is irrelevant, so spacing constraints can be relaxed to pairwise differences (Lemma 1 of [24]).
Used in Section II-C to simplify the constraints; accepted from cited work without proof in this paper.

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Cite this review

Pith. "Pith review of Multiuser Beamforming for Pinching-Antenna Systems: An Element-wise Optimization Framework." pith.science (2026). https://pith.science/paper/UPVALHKX

@misc{pith2026250603770,
  author       = {Pith},
  title        = {Pith review of: Multiuser Beamforming for Pinching-Antenna Systems: An Element-wise Optimization Framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPVALHKX}},
  note         = {Machine review of arXiv:2506.03770}
}

read the original abstract

The pinching-antenna system (PASS) reconstructs wireless channels through pinching beamforming, i.e., optimizing the activated locations of pinching antennas (PAs) along the waveguide. The aim of this article is to investigate the joint design of baseband beamforming and pinching beamforming. A low-complexity element-wise sequential optimization framework is proposed to address the sum-rate maximization problem in PASS-enabled downlink and uplink channels. i) For the downlink scenario, maximum ratio transmission (MRT), zero-forcing (ZF), and minimum mean square error (MMSE) beamforming schemes are employed as baseband beamformers. For each beamformer, a closed-form expression for the downlink sum-rate is derived as a single-variable function with respect to the pinching beamformer. Based on this, a sequential optimization method is proposed, where the positions of the PAs are updated element-wise using a low-complexity one-dimensional search. ii) For the uplink scenario, signal detection is performed using maximum ratio combining (MRC), ZF, and MMSE combiners. A closed-form sum-rate expression is derived for each linear combiner, and a similar element-wise design is applied to optimize the pinching beamforming. Numerical results are provided to validate the effectiveness of the proposed method and demonstrate that: (i) For all considered linear beamformers, the proposed PASS architecture outperforms conventional fixed-antenna systems in terms of sum-rate performance; (ii) in both downlink and uplink channels, ZF achieves performance close to that of MMSE and significantly outperforms MRT or MRC; and (iii) the proposed element-wise design eliminates the need for alternating updates between the baseband and pinching beamformers, thereby ensuring low computational complexity.

Figures

Figures reproduced from arXiv: 2506.03770 by the authors.

**Figure 2.** Achievable sum-rate performance under different one-dimensional search resolutions. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Achievable sum-rate performance under different power levels. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Achievable sum-rate performance under different side length levels. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Achievable sum-rate performance under different numbers of PAs. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Achievable sum-rate performance under different numbers of users. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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