REVIEW 4 major objections 5 minor 11 references
Multi-beam Beamforming in RIS-aided MIMO Subject to Reradiation Mask Constraints -- Optimization and Machine Learning Design
T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read Joint precoder and RIS phase design can honor reradiation masks while shaping multiple beams and protecting the weaker user's rate.
desk verdict Legitimate MIMO extension of the authors' SISO reradiation-mask work, but the theta-subproblem is mislabeled convex and the feasibility claims do not match the plots; fixable, but the convergence proof needs a rewrite. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the alternating optimization that converts the original non-convex max-min problem into three convex QCQP subproblems. In the phase-shift update, the quadratic mask constraint $\boldsymbol{\theta}^H \mathbf{M} \boldsymbol{\theta} \le p$ is combined with the unit-amplitude constraints and a quadratic objective built from the Arimoto-Blahut rate expression; in the precoder updates, the vectorized precoders turn the objective into convex quadratic forms with a single power constraint. The neural network variant replaces the iterative solver by a feedforward network trained with the min-rate objective as loss, using a modified one-hot encoding that enlarges small angular differences and a projection step that scales the output precoders and phase vector to enforce the power and mask constraints.
What would settle it
Measure or full-wave simulate the radiated power of the optimized RIS configuration at the protected angles and compare with the mask threshold p; any violation above numerical tolerance would falsify the claim that the constraints are satisfied.
Extended reading notes
Core claim
The paper's central discovery is that the non-convex max-min problem of maximizing the minimum achievable rate in a two-user RIS-aided MIMO downlink, subject to transmit power and reradiation mask constraints, can be decomposed into three convex QCQP subproblems that are updated in alternation: the RIS phase vector (problem (22)), the first precoder (29), and the second precoder (34). Using the Arimoto-Blahut expression for mutual information, each update keeps the other variables fixed, and the sequence of objective values is monotonically non-decreasing and bounded, so the algorithm converges to a feasible point. The same formulation is extended to an arbitrary number of receivers in the appendix. The paper further claims that a model-based neural network whose loss function is the min-rate objective and whose inputs are one-hot encoded angles of incidence and reflection matches the alternating optimizer's beam patterns and rates while cutting execution time, and that projecting the phase solution onto four discrete levels loses only a small amount of beamforming gain.
Load-bearing premise
The whole scheme assumes that the simplified far-field channel model used for the mask constraint captures the true reradiated power at every protected angle; if it does not, the mask will not be respected in real deployment.
Editorial extensions
If this is right
- If the alternating optimization converges as claimed, the same QCQP decomposition applies to any number of users by the appendix construction, so the design scales beyond two receivers.
- The mask-aware designs keep main beams on target while suppressing sidelobes, meaning interference to other directions can be regulated at the RIS rather than only at the base station.
- The model-based neural network, once trained, produces near-identical beam patterns in less time, making online reconfiguration of RIS phase profiles feasible at the base station.
- Four discrete phase levels suffice to preserve most of the beamforming gain, so inexpensive RIS hardware with limited phase resolution remains viable under mask constraints.
- Because the mask constraint reduces the feasible set, the gap in minimum rate between mask-free and mask-aware operation quantifies the price of regulatory compliance.
Reading between the lines
- Inference: the approach's dependence on known angle-to-RIS channel matrices suggests the same optimizer could be paired with angle estimation and electromagnetic calibration modules to enforce masks under uncertainty, going beyond the small noise test in Fig. 10.
- Inference: the one-hot encoding idea could transfer to near-field or wideband RIS designs, where angular resolution requirements are even more stringent, but the loss function would need modification.
- Inference: the amplitude-reduction projection used when mask and unit-amplitude constraints conflict means the RIS is treated as nearly passive with controllable loss; a practical deployment could exploit this by trading a small gain loss for stronger mask compliance.
- Inference: the paper's own convergence argument shows only monotone convergence of the alternating objective, not global optimality; a direct comparison of the final min-rate against a global solver on small instances would reveal how far the suboptimality gap is.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies joint transmit precoding and RIS phase-shift design in a two-user RIS-aided MIMO downlink, maximizing the minimum achievable rate subject to transmit power and reradiation mask constraints. The authors reformulate the achievable rate via the Arimoto-Blahut variational bound, then propose an alternating optimization algorithm that splits the problem into three subproblems: a phase-shift update, and two precoder updates. They further propose a model-driven neural network that takes one-hot encoded incidence/reflection angles as inputs and is trained with a loss function derived from the rate objective, and a greedy search method for discrete phase shifts. Numerical results compare beam patterns, minimum rate, and execution time against an unconstrained baseline and a prior method [28].
Significance. If the algorithmic claims were fully correct, the paper would offer a practical approach to enforcing reradiation masks in RIS-aided MIMO systems, extending prior SISO work to multi-stream MIMO and providing a faster neural-network alternative. The problem is timely and the general AO framework is standard. However, the central mathematical claim—that subproblem (22) is a convex QCQP and the algorithm converges monotonically to a feasible point of (6)—is not correct as stated, and the implemented projection in (42) relaxes the unit-amplitude constraint, so the reported UACP results may solve a different problem. These issues are load-bearing and require a substantive revision.
major comments (4)
- [Section IV-B, Eq. (22)] The claim that subproblem (22) is a convex QCQP is incorrect. The constraints θ^H I(:,n) I(:,n)^H θ = 1 for every n are non-convex quadratic equalities, since the set {θ : |θ_n| = 1} is not convex. Moreover, the objective is the minimum of two concave quadratic functions, so maximizing it over any set is not a convex problem. Consequently, the assertion in Section IV-D that all three subproblems are convex, and the monotone convergence argument in Eq. (36) that relies on solving (22) to global optimality, are unsupported.
- [Section VI, Eq. (42) and Figs. 3-4] The projection P_θ(θ) defined in Eq. (42) scales θ to satisfy the mask constraint and is explicitly said to apply 'even if the unit amplitude condition is relaxed'. The text acknowledges that when the mask is imposed, some RIS elements have |θ_i| < 1, as visible in the amplitude color maps of Figs. 3 and 4. Therefore, the outputs labeled 'UACP with mask' do not necessarily satisfy constraint (6b), and the paper's convergence proof is not a convergence proof for the original unit-amplitude problem. The authors should either reformulate the problem to allow amplitude variations explicitly, or clearly characterize the reported results as an approximate solution.
- [Section IV-C, Eqs. (23)-(34)] The precoder subproblems are formulated with the mask constraint dropped, and the text justifies this by stating that the θ update in (22) enforces the mask. However, since the θ update itself may return a relaxed-amplitude θ via (42), it is not established that the final alternating iterate satisfies the mask constraint with the precoders that are ultimately used. The paper should specify conditions under which the final output (θ, F1, F2) is feasible for (6d), or provide a feasibility check with reported violation rates.
- [Section II, Eq. (10) and Section VII] The reradiation mask is evaluated with a simplified far-field model in which the RIS-to-observation channel G(θ^o) is a known matrix and each mask direction is treated independently. The paper motivates the problem with macroscopic reradiation models [45]-[49], but does not test whether masks enforced under the simplified model remain satisfied under those more realistic models. I recommend adding an explicit limitation statement and, if possible, a robustness experiment using one of the cited macroscopic models.
minor comments (5)
- [Throughout] The paper inconsistently uses 'QPQC' and 'QCQP'; the standard acronym for quadratically constrained quadratic program is QCQP.
- [Section VIII] The conclusion contains the typo 'optimizaion' instead of 'optimization'.
- [Tables I and II] In the version provided, Tables I and II are not legible; please ensure that all simulation parameters (e.g., the mask threshold p, noise power, path-loss constants, and the neural network hidden sizes n1 and n2) are fully specified in the final manuscript.
- [Section IV-D, Eq. (37)] The complexity formula for the QCQP subproblem in (22) is garbled; please rewrite it with unambiguous exponents and definitions of K_qcqp, M_qcqp, and n_i,qcqp.
- [Equation (21) and notation] The notation I_{N_ris}(:,n) is nonstandard; please define e_n as the n-th standard basis vector and write the unit-modulus constraint as |θ_n|^2 = 1.
Circularity Check
No circular derivation found: the AO subproblems are algebraically derived from the rate and mask expressions, the neural network is trained on an unsupervised objective rather than fitted outputs, and the only questionable claim (convexity of the unit-modulus subproblem) is a correctness concern, not a circularity.
full rationale
The paper's derivation chain is self-contained. The three AO subproblems (22), (29), and (34) are obtained by explicit algebraic manipulation of the Arimoto-Blahut rate expression (15) and the mask power expression (20), not by assuming the result. The neural network in Section VI is model-driven: its loss (69) is the negative minimum achievable rate, and the network is trained by gradient descent on that unsupervised objective rather than fitted to output labels, so reporting rate and beam-pattern performance is an evaluation of the same objective, not a prediction forced by a fitted parameter. The mask projection (42) is an explicit constraint-enforcement step, so the statement that masks are satisfied is an algorithmic guarantee, not a fitted prediction masquerading as a result. The self-citations [1], [46], [49] motivate the problem and cite macroscopic reradiation models, but the actual optimization uses the path-loss formula [55, Eq. (8)] and the stated channel model; the cited prior work is not used to forbid alternatives or to supply the main claim. Comparison against the external method [28] provides an independent benchmark. The one serious technical weakness, that subproblem (22) is not a convex QCQP because of the non-convex unit-modulus equality constraints, affects the validity of the convergence argument, but it is a correctness gap rather than a circular reduction of a result to its own inputs. No equation is shown to be equivalent to another by construction, and no fitted input is renamed as a prediction.
Assumptions & free parameters
free parameters (2)
- Mask threshold p =
Values such as -105 dBm in Fig. 6
- Neural network hidden sizes n1 and n2 =
Not specified in text (Table II empty)
assumptions (3)
- standard math The rate expression can be reformulated using the Arimoto-Blahut structure with the optimal posterior being a complex Gaussian as in [23].
- domain assumption The channel model with U (3) and G_i (4) Rician plus the free-space path-loss formula [55, Eq. (8)] is accurate for the considered RIS geometry.
- ad hoc to paper Subproblem (22) is convex because the matrices E_1, E_2 and Q_theta0 are Hermitian positive semidefinite and the objective is concave in theta.
Cite this review
Pith. "Pith review of Multi-beam Beamforming in RIS-aided MIMO Subject to Reradiation Mask Constraints -- Optimization and Machine Learning Design." pith.science (2026). https://pith.science/paper/QWX45DRJ
@misc{pith2026250715367,
author = {Pith},
title = {Pith review of: Multi-beam Beamforming in RIS-aided MIMO Subject to Reradiation Mask Constraints -- Optimization and Machine Learning Design},
year = {2026},
howpublished = {\url{https://pith.science/paper/QWX45DRJ}},
note = {Machine review of arXiv:2507.15367}
}
read the original abstract
Reconfigurable intelligent surfaces (RISs) are an emerging technology for improving spectral efficiency and reducing power consumption in future wireless systems. This paper investigates the joint design of the transmit precoding matrices and the RIS phase shift vector in a multi-user RIS-aided multiple-input multiple-output (MIMO) communication system. We formulate a max-min optimization problem to maximize the minimum achievable rate while considering transmit power and reradiation mask constraints. The achievable rate is simplified using the Arimoto-Blahut algorithm, and the problem is broken into quadratic programs with quadratic constraints (QPQC) sub-problems using an alternating optimization approach. To improve efficiency, we develop a model-based neural network optimization that utilizes the one-hot encoding for the angles of incidence and reflection. We address practical RIS limitations by using a greedy search algorithm to solve the optimization problem for discrete phase shifts. Simulation results demonstrate that the proposed methods effectively shape the multi-beam radiation pattern towards desired directions while satisfying reradiation mask constraints. The neural network design reduces the execution time, and the discrete phase shift scheme performs well with a small reduction of the beamforming gain by using only four phase shift levels.
Reference graph
Works this paper leans on
-
[1]
S. Wang, M. Di Renzo, and M. Poulakis, “Multibeam beamforming design in RIS-aided SISO communications subject to reradiation masks — optimization and machine learning methods,” in 19th ISWCS, 2024, pp. 1-6
work page 2024
-
[2]
M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” IEEE J. Sel. Areas Commun., vol. 38, no. 11, pp. 2450-2525, 2020
work page 2020
-
[3]
M. Di Renzo, F. H. Danufane, and S. Tretyakov, “Communication mod- els for reconfigurable intelligent surfaces: From surface electromagnetics to wireless networks optimization,’ Proceedings of the IEEE, vol. 110, no. 9, pp. 1164-1209, 2022
work page 2022
-
[4]
S. Shen, B. Clerckx, and R. Murch, “Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,” IEEE Trans. Wireless Commun., vol. 21, no. 2, pp. 1229-1243, 2022
work page 2022
-
[5]
Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,
Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” JEEE Trans. Wireless Commun., vol. 18, no. 11, pp. 5394-5409, 2019
work page 2019
-
[6]
Wireless networks design in the era of deep learning: Model-based, AI-based, or both?
A. Zappone, M. Di Renzo, and M. Debbah, “Wireless networks design in the era of deep learning: Model-based, AI-based, or both?” [EEE Trans. Commun., vol. 67, no. 10, pp. 7331-7376, 2019
work page 2019
-
[7]
Rate-splitting multiple access for RIS-aided cell-edge users with discrete phase-shifts,
D. Shambharkar, S. Dhok, A. Singh, and P. K. Sharma, “Rate-splitting multiple access for RIS-aided cell-edge users with discrete phase-shifts,” IEEE Commun. Lett., vol. 26, no. 11, pp. 2581-2585, 2022
work page 2022
-
[8]
Rate- splitting multiple access for intelligent reflecting surface aided multi- user communications,
A. Bansal, K. Singh, B. Clerckx, C.-P. Li, and M.-S. Alouini, “Rate- splitting multiple access for intelligent reflecting surface aided multi- user communications,” JEEE Trans. Veh. Technol., vol. 70, no. 9, pp. 9217-9229, 2021. [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] S. Ao, Y. Niu, Z. H...
arXiv 2021
Show all 11 references
-
[52]
Components separation algorithm for localization and classification of mixed near- field and far-field sources in multipath propagation,
A. M. Molaei, B. Zakeri, and S. M. Hosseini Andargoli, “Components separation algorithm for localization and classification of mixed near- field and far-field sources in multipath propagation,” [EEE Trans. Signal Process., vol. 68, pp. 404-419, 2020. A. Ertug Zorkun, M. A. Sal...
2020
-
[58]
He received his Ph.D
Cambridge [59] [60] Shumin Wang is a postdoctoral researcher at Uni- versité Paris-Saclay, France, and a visiting researcher at Ranplan Wireless in Cambridge, UK. He received his Ph.D. in Networks, Information and Commu- nication Sciences from Université Paris-Saclay in
-
[2024]
in Electronic Science and Technology from Central South Uni- versity, Changsha, China, in 2021, and his B.Eng
Prior to that, he earned his M.Sc. in Electronic Science and Technology from Central South Uni- versity, Changsha, China, in 2021, and his B.Eng. in Engineering from Guangxi University, Nanning, China, in 2018. His research interests include Re- configurable intelligent surfac...
2021
Reviewed August 6, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.