Optimization of Beyond Diagonal RIS: A Universal Framework Applicable to Arbitrary Architectures

Bruno Clerckx; Zheyu Wu

REVIEW 1 major objections 5 minor 1 cited by

Optimization of Beyond Diagonal RIS: A Universal Framework Applicable to Arbitrary Architectures

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proposes a single ADMM-based optimization framework that works for every beyond-diagonal RIS architecture, not just fully- or group-connected ones.

desk verdict A genuinely useful architecture-independent ADMM framework for BD-RIS, with solid derivations, but the power-minimization convergence theorem rests on an assumption that fails on feasible equal-channel instances. read the letter →

arxiv 2412.15965 v2 pith:WAIWIRLD submitted 2024-12-20 eess.SP cs.ITmath.ITmath.OC

Zheyu Wu , Bruno Clerckx This is my paper · ORCID

classification eess.SPcs.ITmath.ITmath.OC

keywords beyonddiagonalRISreconfigurableintelligentsurfacearchitecture-independentoptimizationadmittancematrixpartiallyproximalADMMsum-ratemaximizationtransmitpowerminimizationMU-MISO

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

The paper claims to close a gap in reconfigurable-intelligent-surface (RIS) research: every prior optimization algorithm for beyond-diagonal RIS (BD-RIS) was tied to specific circuit topologies, such as fully-connected or group-connected architectures. The authors propose a framework in which any architecture, specified by the sparsity pattern of its admittance matrix, is optimized by the same partially proximal ADMM (pp-ADMM) algorithm. They apply it to sum-rate maximization and transmit-power minimization in multiuser MISO systems, and each update step either admits a closed form or is easy to solve. If the claim holds, wireless system designers could test new BD-RIS circuit layouts without designing a new optimizer for each one. The paper reports that the approach matches state-of-the-art performance at substantially lower CPU time.

What carries the argument

The central object is the susceptance matrix $B$ (the imaginary part of the admittance), whose sparsity pattern is the circuit architecture. The load-bearing mechanism is the exact reformulation of the scattering equation into the low-dimensional bilinear constraint $(I - iZ_0 B)U = (I + iZ_0 B)H$, combined with fractional programming for the sum-rate objective and an auxiliary variable $Y = U^\dagger GW$ for the QoS constraints. These transformations turn a nonconvex problem with matrix-inversion constraints into a form ADMM can handle, and the proximal terms on $W$ and $B$ make those subproblems strongly convex, which is what the convergence proofs rely on.

What would settle it

Run the pp-ADMM updates on a wide range of random Rician or Rayleigh channel realizations with the paper's own setup and monitor whether $U_t$ and $\lambda_t$ stay bounded and whether $G^\dagger U_t$ keeps a uniformly positive smallest singular value; if any realization produces divergence or a iterates that fail to approach a stationary point, Theorems 1 and 2 are falsified as stated.

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Extended reading notes

Core claim

The central discovery is that the hard part of BD-RIS optimization, the non-diagonal scattering matrix $\Theta$ with unitary and symmetry constraints, disappears once the problem is re-expressed through the admittance matrix $B$. Writing $\Theta = (I + iZ_0 B)^{-1}(I - iZ_0 B)$ and constraining $B$ to be real symmetric with zeros wherever the circuit has no interconnection encodes every architecture as a sparsity pattern. The authors introduce $U = \Theta^\dagger H$ to replace the matrix-inversion constraint by the bilinear constraint $(I - iZ_0 B)U = (I + iZ_0 B)H$, cutting the constraint dimension from $M^2$ to $MK$ and making the problem tractable by ADMM. Their pp-ADMM algorithm, with proximal terms on $W$ and $B$, has each subproblem solvable in closed form or by a simple one-dimensional search, and Theorems 1 and 2 assert convergence to a stationary point under boundedness assumptions. This is claimed to be the first architecture-independent framework for BD-RIS optimization, and simulations show a better performance-versus-CPU-time trade-off than the PDD baseline [31] and the two-stage FP baseline [29].

Load-bearing premise

The convergence theorems assume that the algorithm's intermediate variables stay bounded (and, for power minimization, that $G^\dagger U_t$ keeps a uniformly positive smallest singular value), but the paper offers no proof that these conditions always hold, only that simulations consistently satisfy them.

Editorial extensions

If this is right

If the framework is right, any BD-RIS circuit topology, including tree- and forest-connected layouts whose scattering matrices have no simple structure, can be optimized with exactly the same algorithm by feeding in a sparsity mask.
The per-iteration complexity is $O(M^3 K + \min\{\sum_i |S_i|, 2MK\}^3)$, which for fully-connected RIS avoids the $O(M^6)$ per-iteration cost of the prior PDD approach when $M \gg K$.
The paper's simulations show that tree-connected RIS, known to be optimal for single-user MISO, is no longer optimal in multiuser MISO, achieving performance comparable to group-connected RIS with group size 4 while using fewer impedances.
The same three-step recipe extends to max-min fairness, energy-efficiency maximization, and MU-MIMO systems.

Reading between the lines

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

Implicit in the paper but not pursued: because each architecture is just a binary mask on $B$, the framework could be used for architecture search, running the same pp-ADMM update over many masks and ranking topologies by the resulting sum-rate or transmit power.
A natural extension the authors leave open is adaptive or learned sparsity patterns, treating the interconnection mask as a discrete design variable that could be optimized jointly with the continuous susceptance values.
The CPU-time comparison against the PDD baseline could be sharpened by benchmarking against a dedicated solver for the PDD subproblems rather than a generic interior-point SOCP solver, which would test how much of the efficiency gain comes from the ADMM structure itself.

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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

The genuinely new thing here is the bilinear reformulation of the scattering-matrix constraint: instead of optimizing over Θ with its unitary and symmetric structure, the paper writes (I − iZ0B)U = (I + iZ0B)H and lets the admittance sparsity pattern define the architecture. That is clean, and it makes the algorithm template genuinely architecture-independent. The pp-ADMM updates are worked out carefully, each subproblem is either closed-form or a simple bisection, and the complexity comparison against PDD is honest. The paper also delivers something the subfield lacked: a direct performance comparison of tree-, group-, and fully-connected architectures in multiuser systems, which previously required separate bespoke algorithms. Credit where due: the appendices contain complete proofs, the FP and proximal tricks are standard but applied competently, and the simulation results support the efficiency claims for the sum-rate problem.

The soft spots are concentrated in the power-minimization half. Theorem 2 assumes a uniform lower bound σ_min(G†U_t) ≥ c > 0 to control the dual multiplier. The paper says this is technical and always satisfied in simulations. It is not always satisfied: take two users with identical RIS-user channels, h1 = h2 = h, with N = K = 4 and a low SINR threshold. The problem is feasible, and any feasible limit point has U = Θ†H with two identical columns, so G†U is singular and σ_min = 0. If the ADMM iterates converge to feasibility, the singular value must tend to zero, violating the theorem's hypothesis. The universal convergence guarantee for the transmit power problem is therefore not established as stated. The boundedness assumptions in both theorems are also unproven, though that is a more standard gap and less alarming. On the reproducibility side, there is no code or data, no error bars, and the simulation setup omits several implementation parameters; the comparisons are believable but not independently checkable. The extensions to MU-MIMO and general utilities are sketches, which is fine for a paper that already has two full algorithms, but readers should not treat those as validated.

Net assessment: this is a solid optimization paper with a real contribution and one load-bearing gap that is fixable. The convergence theory needs either a proof that the singular-value condition holds on the actual solution path, a restriction to generic channel realizations, or an explicit treatment of degenerate cases. With that repaired, the architecture-independent claim would be defensible. As it stands, the paper deserves peer review, not desk rejection, but a referee should push on Theorem 2.

Recommendation: send to review, and ask the authors to address the convergence assumptions and release code.

Referee Report

1 major / 5 minor

Summary. The paper proposes a universal, architecture-independent optimization framework for beyond-diagonal RIS (BD-RIS) in MU-MISO systems, covering sum-rate maximization and transmit power minimization. The key modeling step is to replace the scattering matrix Θ with the admittance matrix B through the Cayley transform, and to encode any BD-RIS architecture as a sparsity constraint on B. The resulting nonconvex problems are reformulated via auxiliary variables (U = Θ†H and Y = U†GW) and solved with custom partially proximal ADMM (pp-ADMM) algorithms, with each subproblem solved in closed form or via a simple one-dimensional search. The paper states convergence to stationary points under boundedness and rank assumptions, gives complexity estimates, reports simulations comparing with PDD and two-stage FP for fully- and group-connected RISs, and extends the framework to general utility functions and MU-MIMO.

Significance. If the convergence claims were fully established, the framework would be a valuable contribution: it unifies BD-RIS architectures under a single optimization algorithm, avoids expensive matrix inversions through low-dimensional bilinear constraints, provides closed-form subproblem updates, and demonstrates a superior performance-efficiency trade-off relative to existing algorithms in the simulated regimes. The appendices contain detailed proofs, and the numerical comparison isolates the algorithmic gain rather than fitting parameters to data. The main weakness is that the central convergence theorems rest on assumptions that are not proven and, for the power-minimization theorem, are actually incompatible with a feasible class of channel realizations; this limits the claimed universality of the theoretical guarantee.

major comments (1)

[Section III-C2 and IV-C2] The boundedness assumptions in Theorems 1 and 2 (boundedness of {U_t}, {λ_t}, and, in Theorem 2, of {W_t} and {μ_t}) are stated as "imposed for technical reasons" and supported only by simulation evidence. While such assumptions are standard in nonconvex ADMM analyses, they should be presented as explicit limitations and, if possible, replaced by verifiable sufficient conditions. More importantly, for the power-minimization algorithm the rank condition of Theorem 2 cannot be guaranteed a priori even in simple feasible scenarios, as detailed in the previous comment. The revision should either prove the required conditions under stated channel/parameter assumptions, or substantially weaken the convergence claim and explicitly describe the class of instances to which the theorem applies.

minor comments (5)

[Abstract and throughout] There are several typographical errors: "catogarized" (Section I), "trackable" (Section III-A), "vise versa" (Section III-A), "dependance" (Section III-B3), and "decomposing ... into an easy-to-projection constraint" (Section IV-A) which should read "easy-to-project constraint."
[Section III-C] The sentence "The complexity and convergence analysis of the proposed pp-ADMM algorithm is given in Section IV-C" should refer to Section III-C; Section IV-C contains the analysis for the power-minimization algorithm.
[Appendix B-B] The proof references "Eq. (50)" but the displayed equations in Appendix B-B are numbered (46); the cross-reference should be corrected.
[Section III-B4, Eq. (18)] The definition of S_i = {i, i+1, …, M} \ {j | j > i, B_{i,j}=0} depends on the current iterate B, making the construction of the vector x architecture-dependent only through the current values of B. It should be defined from the fixed sparsity pattern of the BD-RIS architecture, not from the instantiated B.
[Appendix A, Eq. (29)] In the expression for r*_{k,k} when a_{k,k}=0, the denominator is typeset ambiguously as "(1 + Γ_k)^2 + Γ_k σ^2"; the parentheses should be clarified so that it is clear whether Γ_kσ^2 is added inside or outside the fraction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pp-ADMM derivations are self-contained conditional arguments, and self-citations concern established modeling relations rather than the target algorithmic claims.

full rationale

The paper's central claims are algorithmic: a variable-substitution reformulation (U = Θ†H), custom pp-ADMM updates, and conditional convergence theorems. No parameters are fitted to data and then redeclared as predictions; the reported sum-rate/power-vs-CPU results are numerical comparisons against published benchmarks [29], [31]. The convergence results are stated as conditional theorems ('Assume ... bounded ... then any limit point ... is a stationary point'), and the boundedness/singular-value assumptions are explicitly flagged as technical ('imposed for technical reasons'), so the theorems are conditional statements rather than circular constructions. The heavy citation of prior work by the same group (especially [8], [32]) concerns the admittance/scattering model and graph-based topology characterization; those are independently established modeling results whose basic relation is standard microwave network theory [33]. The paper's own contribution, the pp-ADMM framework, does not reduce to those citations: it reformulates the problem with auxiliary variables and derives each subproblem update in the text. No step in the derivation chain equates an output to an input by construction, and no known result is merely renamed as a derivation. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The framework introduces no new physical entities. The central assumptions are the microwave-network relation between scattering and admittance, the representation of architectures via sparsity patterns, and the technical boundedness conditions needed for the convergence proofs. The algorithm parameters are free but not fitted to data.

free parameters (3)

Proximal parameters τ and ξ = not specified
Used in the W- and B-subproblems (11c), (11d) to ensure strong convexity and stability. Their values affect the path and rate of convergence but are not reported or tuned systematically.
ADMM penalty parameter ρ (and ρλ, ρμ) = not specified
Appears in augmented Lagrangians (10), (25). Theorem 1 requires ρ > ρ0 for convergence, but ρ0 is not quantified; the paper does not report chosen values.
Simulation parameters (path loss exponents, Rician factor, noise power, group size) = α=2.2, κ=2 dB, σ²=-80 dBm, Gs=4
These are scenario parameters, not fitted to make the algorithm succeed, but they are degrees of freedom in the reported performance comparisons.

assumptions (5)

domain assumption The scattering matrix of the BD-RIS is related to the admittance matrix by Θ = (I + iZ0B)^{-1}(I - iZ0B) for a lossless, reciprocal network
Invoked in constraints (4b) and (5c). Based on microwave network theory [33] and BD-RIS modeling [8, 32]. The paper treats this as given.
domain assumption The BD-RIS architecture is fully characterized by the sparsity pattern of the susceptance matrix B (the set B in Section II-C)
This is the basis of the architecture-independent claim. It assumes that any interconnection topology can be represented by setting off-diagonal entries of B to zero or non-zero.
ad hoc to paper The sequences {U_t} and {λ_t} generated by the algorithm are bounded; and for the power-minimization algorithm, σ_min(G^†U_t) is uniformly bounded away from zero
Required for Theorems 1 and 2. The paper acknowledges these are not proven (Section III-C2) and claims simulations show they hold.
domain assumption The system is MU-MISO with no direct BS-user link, no mutual coupling, no structural scattering, and perfect matching, with a reciprocal reconfigurable impedance network
Stated in Section II-A, following prior BD-RIS literature [5], [32]. Simplifies the model and is used in the received signal model (1).
standard math The fractional programming equivalence (8) and the rotation-based SOC reformulation of the QoS constraints (23) are valid
These are established techniques from [38, 39] and [36]; the paper cites them. They are not derived in full but are standard.

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Pith. "Pith review of Optimization of Beyond Diagonal RIS: A Universal Framework Applicable to Arbitrary Architectures." pith.science (2026). https://pith.science/paper/WAIWIRLD

@misc{pith2026241215965,
  author       = {Pith},
  title        = {Pith review of: Optimization of Beyond Diagonal RIS: A Universal Framework Applicable to Arbitrary Architectures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WAIWIRLD}},
  note         = {Machine review of arXiv:2412.15965}
}

read the original abstract

Reconfigurable intelligent surfaces (RISs) are envisioned as a promising technology for future wireless communication systems due to their ability to control the propagation environment in a hardware- and energy-efficient way. Recently, the concept of RISs has been extended to beyond-diagonal RISs (BD-RISs), which unlock the full potential of RISs thanks to the presence of tunable interconnections between RIS elements. While various algorithms have been proposed for BD-RIS optimization, they mainly focus on specific architectures whose scattering matrices exhibit very special structures. A universal optimization framework that can accommodate different BD-RIS circuit topologies is still lacking. In this paper, we bridge this research gap by proposing an architecture-independent framework for BD-RIS optimization, with the main focus on sum-rate maximization and transmit power minimization in multiuser multi-input single-output (MU-MISO) systems. Specifically, we first incorporate BD-RIS architectures into the models by connecting the scattering matrix with the admittance matrix and introducing appropriate constraints to the admittance matrix. The formulated problems are then solved by our custom-designed partially proximal alternating direction method of multipliers (pp-ADMM) algorithms. The pp-ADMM algorithms are computationally efficient, with each subproblem either admitting a closed-form solution or being easily solvable. Simulation results demonstrate that the proposed approaches achieve a better trade-off between performance and computational efficiency compared to existing methods.

Figures

Figures reproduced from arXiv: 2412.15965 by the authors.

**Figure 1.** Different BD-RIS architectures, where (a) single-connected RIS, (b) fully-connected RIS, (c) group-connected RIS, (d) tree-connected RIS. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** An illustration of the BD-RIS aided MU-MISO system for simulation. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 5.** Transmit power versus number of RIS elements ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: CPU time versus number of RIS elements for transmit power [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: Sum-rate versus transmit power for different BD-RIS architectures [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Transmit power versus SINR threshold for different BD-RIS archi [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Fractional Programming and Manifold Optimization for Reciprocal BD-RIS Scattering Matrix Design
eess.SP 2025-11 conditional novelty 4.0 of 10

Applying standard fractional programming transforms to BD-RIS scattering-matrix optimization gives faster convergence and higher sum-rate than two prior methods in simulation, but the comparison lacks joint-design baselines.

Reference graph

Works this paper leans on

47 extracted references · 38 canonical work pages · cited by 1 Pith paper

[1]

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,”IEEE Trans. Wireless Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019

2019
[2]

Weighted sum- rate maximization for reconfigurable intelligent surface aided wireless networks,

H. Guo, Y .-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum- rate maximization for reconfigurable intelligent surface aided wireless networks,” IEEE Trans. Wireless Commun. , vol. 19, no. 5, pp. 3064– 3076, May 2020

2020
[3]

Intelligent reflecting surface-aided wireless communications: A tutorial,

Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface-aided wireless communications: A tutorial,” IEEE Transactions Commun., vol. 69, no. 5, pp. 3313–3351, May 2021

work page 2021
[4]

Reconfigurable intelligent surfaces: Principles and opportunities,

Y . Liu, X. Liu, X. Mu, T. Hou, J. Xu, M. Di Renzo, and N. Al-Dhahir, “Reconfigurable intelligent surfaces: Principles and opportunities,” IEEE Commun. Surveys Tuts., vol. 23, no. 3, pp. 1546–1577, 3rd Quart. 2021

2021
[5]

Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,

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, Feb. 2022

work page 2022
[6]

Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,

H. Li, S. Shen, M. Nerini, and B. Clerckx, “Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,” IEEE Commun. Mag., vol. 62, no. 3, pp. 102–108, Mar. 2024

work page 2024
[7]

Beyond diagonal reconfigurable intelli- gent surfaces: From transmitting and reflecting modes to single-, group- , and fully-connected architectures,

H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelli- gent surfaces: From transmitting and reflecting modes to single-, group- , and fully-connected architectures,” IEEE Trans. Wireless Commun. , vol. 22, no. 4, pp. 2311–2324, Apr. 2023

work page 2023
[8]

Beyond diagonal reconfig- urable intelligent surfaces utilizing graph theory: Modeling, architecture design, and optimization,

M. Nerini, S. Shen, H. Li, and B. Clerckx, “Beyond diagonal reconfig- urable intelligent surfaces utilizing graph theory: Modeling, architecture design, and optimization,” IEEE Trans. Wireless Commun., vol. 23, no. 8, pp. 9972–9985, Aug. 2024

work page 2024

Show all 47 references

[9]

Pareto frontier for the performance- complexity trade-off in beyond diagonal reconfigurable intelligent sur- faces,

M. Nerini and B. Clerckx, “Pareto frontier for the performance- complexity trade-off in beyond diagonal reconfigurable intelligent sur- faces,” IEEE Commun. Lett., vol. 27, no. 10, pp. 2842–2846, Oct. 2023

2023
[10]

Beyond diagonal reconfigurable intelligent surfaces: A multi-sector mode enabling highly directional full-space wireless coverage,

H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces: A multi-sector mode enabling highly directional full-space wireless coverage,” IEEE J. Sel. Areas Commun. , vol. 41, no. 8, pp. 2446–2460, Aug. 2023

2023
[11]

Discrete-value group and fully connected architectures for beyond diagonal reconfigurable intelligent surfaces,

M. Nerini, S. Shen, and B. Clerckx, “Discrete-value group and fully connected architectures for beyond diagonal reconfigurable intelligent surfaces,” IEEE Trans. Vehicular Tech. , vol. 72, no. 12, pp. 16 354– 16 368, Dec. 2023

2023
[12]

Beyond diag- onal reconfigurable intelligent surfaces with mutual coupling: Modeling and optimization,

H. Li, S. Shen, M. Nerini, M. Di Renzo, and B. Clerckx, “Beyond diag- onal reconfigurable intelligent surfaces with mutual coupling: Modeling and optimization,” IEEE Commun. Lett. , vol. 28, no. 4, pp. 937–941, Apr. 2024

2024
[13]

Global optimal closed-form solutions for intelligent surfaces with mutual coupling: Is mutual coupling detrimental or beneficial?

M. Nerini, H. Li, and B. Clerckx, “Global optimal closed-form solutions for intelligent surfaces with mutual coupling: Is mutual coupling detrimental or beneficial?” 2024. [Online]. Available: https://arxiv.org/abs/2411.04949

2024
[14]

Channel reciprocity attacks using intelligent surfaces with non-diagonal phase shifts,

H. Wang, Z. Han, and A. L. Swindlehurst, “Channel reciprocity attacks using intelligent surfaces with non-diagonal phase shifts,” IEEE Open J. Commun. Soc. , vol. 5, pp. 1469–1485, 2024

2024
[15]

Beyond diagonal reconfigurable intelligent surfaces in wideband OFDM communications: Circuit-based modeling and optimization,

H. Li, M. Nerini, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces in wideband OFDM communications: Circuit-based modeling and optimization,” 2024. [Online]. Available: https://arxiv.org/abs/2405.07297

2024 arXiv
[16]

Maximizing spectral and energy efficiency in multi-user MIMO OFDM systems with RIS and hardware impairment,

M. Soleymani, I. Santamaria, A. Sezgin, and E. Jorswieck, “Maximizing spectral and energy efficiency in multi-user MIMO OFDM systems with RIS and hardware impairment,” 2024. [Online]. Available: https://arxiv.org/abs/2401.11921

2024 arXiv
[17]

Wideband channel capacity maximization with beyond diagonal RIS reflection matrices,

¨O. T. Demir and E. Bj¨ornson, “Wideband channel capacity maximization with beyond diagonal RIS reflection matrices,” 2024. [Online]. Available: https://arxiv.org/abs/2404.00982

2024 arXiv
[18]

Non-reciprocal beyond diagonal RIS: Multiport network models and performance benefits in full-duplex systems,

H. Li and B. Clerckx, “Non-reciprocal beyond diagonal RIS: Multiport network models and performance benefits in full-duplex systems,”
[19]

A dual-function radar-communication system empowered by beyond diagonal reconfig- urable intelligent surface,

B. Wang, H. Li, S. Shen, Z. Cheng, and B. Clerckx, “A dual-function radar-communication system empowered by beyond diagonal reconfig- urable intelligent surface,” IEEE Trans. Commun. (early access) , 2024

2024
[20]

Joint computation and communication resource optimization for beyond diagonal UA V-IRS empowered MEC networks,

A. Mahmood, T. X. Vu, W. U. Khan, S. Chatzinotas, and B. Ottersten, “Joint computation and communication resource optimization for beyond diagonal UA V-IRS empowered MEC networks,” 2024. [Online]. Available: https://arxiv.org/abs/2311.07199

2024 arXiv
[21]

Fully connected reconfigurable intelligent surface aided rate-splitting multiple access for multi-user multi-antenna transmission,

T. Fang, Y . Mao, S. Shen, Z. Zhu, and B. Clerckx, “Fully connected reconfigurable intelligent surface aided rate-splitting multiple access for multi-user multi-antenna transmission,” in IEEE Int. Conf. Commun. Workshop., Seoul, Korea, May 2022, pp. 675–680

2022
[22]

Synergizing beyond diagonal reconfig- urable intelligent surface and rate-splitting multiple access,

H. Li, S. Shen, and B. Clerckx, “Synergizing beyond diagonal reconfig- urable intelligent surface and rate-splitting multiple access,” IEEE Trans. Wireless Commun., vol. 23, no. 8, pp. 8717–8729, Aug. 2024

2024
[23]

A dynamic grouping strategy for beyond diagonal reconfigurable intelligent surfaces with hybrid transmitting and reflecting mode,

——, “A dynamic grouping strategy for beyond diagonal reconfigurable intelligent surfaces with hybrid transmitting and reflecting mode,” IEEE Trans. Vehicular Tech., vol. 72, no. 12, pp. 16 748–16 753, Dec. 2023

2023
[24]

Static grouping strategy design for beyond diagonal reconfigurable intelligent surfaces,

M. Nerini, S. Shen, and B. Clerckx, “Static grouping strategy design for beyond diagonal reconfigurable intelligent surfaces,” IEEE Commun. Lett., vol. 28, no. 7, pp. 1708–1712, Jul. 2024

2024
[25]

Localized and distributed beyond diagonal reconfigurable intelligent surfaces with lossy 16 interconnections: Modeling and optimization,

M. Nerini, G. Ghiaasi, and B. Clerckx, “Localized and distributed beyond diagonal reconfigurable intelligent surfaces with lossy 16 interconnections: Modeling and optimization,” 2024. [Online]. Available: https://arxiv.org/abs/2402.05881

2024 arXiv
[26]

Physically consistent modeling of stacked intelligent metasurfaces implemented with beyond diagonal RIS,

M. Nerini and B. Clerckx, “Physically consistent modeling of stacked intelligent metasurfaces implemented with beyond diagonal RIS,” IEEE Commn. Lett., vol. 28, no. 7, pp. 1693–1697, Jul. 2024

2024
[27]

Channel estimation and beamforming for beyond diagonal reconfigurable intelligent surfaces,

H. Li, S. Shen, Y . Zhang, and B. Clerckx, “Channel estimation and beamforming for beyond diagonal reconfigurable intelligent surfaces,” IEEE Trans. Signal Process. , vol. 72, pp. 3318–3332, 2024

2024
[28]

Closed-form global optimization of beyond diagonal reconfigurable intelligent surfaces,

M. Nerini, S. Shen, and B. Clerckx, “Closed-form global optimization of beyond diagonal reconfigurable intelligent surfaces,” IEEE Trans. Wireless Commun., vol. 23, no. 2, pp. 1037–1051, Feb. 2024

2024
[29]

A low-complexity beamforming design for beyond-diagonal RIS aided multi-user networks,

T. Fang and Y . Mao, “A low-complexity beamforming design for beyond-diagonal RIS aided multi-user networks,” IEEE Commun. Lett. , vol. 28, no. 1, pp. 203–207, 2024

2024
[30]

Channel shaping using beyond diagonal reconfigurable intelligent surface: Analysis, optimization, and enhanced flexibility,

Y . Zhao, H. Li, M. Franceschetti, and B. Clerckx, “Channel shaping using beyond diagonal reconfigurable intelligent surface: Analysis, optimization, and enhanced flexibility,” 2024. [Online]. Available: https://arxiv.org/abs/2407.15196

2024
[31]

Optimizing power consumption, energy efficiency, and sum-rate using beyond diag- onal RIS—a unified approach,

Y . Zhou, Y . Liu, H. Li, Q. Wu, S. Shen, and B. Clerckx, “Optimizing power consumption, energy efficiency, and sum-rate using beyond diag- onal RIS—a unified approach,” IEEE Trans. Wireless Commun., vol. 23, no. 7, pp. 7423–7438, Jul. 2024

2024
[32]

A universal framework for multiport network analysis of reconfigurable intelligent surfaces,

M. Nerini, S. Shen, H. Li, M. Di Renzo, and B. Clerckx, “A universal framework for multiport network analysis of reconfigurable intelligent surfaces,” IEEE Trans. Wireless Commun. , vol. 23, no. 10, pp. 14 575– 14 590, Oct. 2024

2024
[33]

Pozar, Microwave Engineering

D. Pozar, Microwave Engineering. John Wiley & Sons, 2011

2011
[34]

SCALE: A low-complexity dis- tributed protocol for spectrum balancing in multiuser DSL networks,

J. Papandriopoulos and J. S. Evans, “SCALE: A low-complexity dis- tributed protocol for spectrum balancing in multiuser DSL networks,” IEEE Trans. Inf. Theory , vol. 55, no. 8, pp. 3711–3724, Aug. 2009

2009
[35]

An iteratively weighted mmse approach to distributed sum-utility maximization for a MIMO interfering broadcast channel,

Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weighted mmse approach to distributed sum-utility maximization for a MIMO interfering broadcast channel,” IEEE Trans. Signal Process. , vol. 59, no. 9, pp. 4331–4340, Sept. 2011

2011
[36]

Linear precoding via conic optimization for fixed MIMO receivers,

A. Wiesel, Y . Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Process. , vol. 54, no. 1, pp. 161–176, Jan. 2006

2006
[37]

Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,

W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2646–2660, Jun. 2007

2007
[38]

Fractional programming for communication systems—part I: Power control and beamforming,

K. Shen and W. Yu, “Fractional programming for communication systems—part I: Power control and beamforming,” IEEE Trans. Signal Process., vol. 66, no. 10, pp. 2616–2630, May 2018

2018
[39]

Fractional programming for communication systems—part II: Uplink scheduling via matching,

——, “Fractional programming for communication systems—part II: Uplink scheduling via matching,” IEEE Transactions on Signal Pro- cessing, vol. 66, no. 10, pp. 2631–2644, May 2018

2018
[40]

Distributed optimization and statistical learning via the alternating direction method of multipliers,

S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein et al., “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. , vol. 3, no. 1, pp. 1–122, 2011

2011
[41]

A survey of recent advances in optimization methods for wireless communications,

Y .-F. Liu, T.-H. Chang, M. Hong, Z. Wu, A. Man-Cho So, E. A. Jorswieck, and W. Yu, “A survey of recent advances in optimization methods for wireless communications,” IEEE J. Sel. Areas Commun. , vol. 42, no. 11, pp. 2992–3031, Nov. 2024

2024
[42]

ADMM for multiaffine con- strained optimization,

W. Gao, D. Goldfarb, and F. E. Curtis, “ADMM for multiaffine con- strained optimization,” Optim. Methods Softw. , vol. 35, no. 2, pp. 257– 303, 2020

2020
[43]

Bilinear constraint based ADMM for mixed poisson-gaussian noise removal,

J. Zhang, Y . Duan, Y . Lu, M. K. Ng, and H. Chang, “Bilinear constraint based ADMM for mixed poisson-gaussian noise removal,” Inverse Problems and Imaging , vol. 15, no. 2, pp. 339–366, 2021

2021
[44]

Optimization of MIMO device-to-device networks via matrix fractional programming: A mi- norization–maximization approach,

K. Shen, W. Yu, L. Zhao, and D. P. Palomar, “Optimization of MIMO device-to-device networks via matrix fractional programming: A mi- norization–maximization approach,” IEEE/ACM Trans. Netw. , vol. 27, no. 5, pp. 2164–2177, Oct. 2019

2019
[45]

Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,

M. Hong, Z.-Q. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. Optim. , vol. 26, no. 1, pp. 337–364, 2016

2016
[46]

Global convergence of ADMM in nonconvex nonsmooth optimization,

Y . Wang, W. Yin, and J. Zeng, “Global convergence of ADMM in nonconvex nonsmooth optimization,” J. Sci. Comput. , vol. 78, no. 1, pp. 29–63, 2019

2019
[2024]

Available: https://arxiv.org/abs/2411.04370

[Online]. Available: https://arxiv.org/abs/2411.04370

arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

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,”IEEE Trans. Wireless Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019

2019

[2] [2]

Weighted sum- rate maximization for reconfigurable intelligent surface aided wireless networks,

H. Guo, Y .-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum- rate maximization for reconfigurable intelligent surface aided wireless networks,” IEEE Trans. Wireless Commun. , vol. 19, no. 5, pp. 3064– 3076, May 2020

2020

[3] [3]

Intelligent reflecting surface-aided wireless communications: A tutorial,

Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface-aided wireless communications: A tutorial,” IEEE Transactions Commun., vol. 69, no. 5, pp. 3313–3351, May 2021

work page 2021

[4] [4]

Reconfigurable intelligent surfaces: Principles and opportunities,

Y . Liu, X. Liu, X. Mu, T. Hou, J. Xu, M. Di Renzo, and N. Al-Dhahir, “Reconfigurable intelligent surfaces: Principles and opportunities,” IEEE Commun. Surveys Tuts., vol. 23, no. 3, pp. 1546–1577, 3rd Quart. 2021

2021

[5] [5]

Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,

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, Feb. 2022

work page 2022

[6] [6]

Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,

H. Li, S. Shen, M. Nerini, and B. Clerckx, “Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,” IEEE Commun. Mag., vol. 62, no. 3, pp. 102–108, Mar. 2024

work page 2024

[7] [7]

Beyond diagonal reconfigurable intelli- gent surfaces: From transmitting and reflecting modes to single-, group- , and fully-connected architectures,

H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelli- gent surfaces: From transmitting and reflecting modes to single-, group- , and fully-connected architectures,” IEEE Trans. Wireless Commun. , vol. 22, no. 4, pp. 2311–2324, Apr. 2023

work page 2023

[8] [8]

Beyond diagonal reconfig- urable intelligent surfaces utilizing graph theory: Modeling, architecture design, and optimization,

M. Nerini, S. Shen, H. Li, and B. Clerckx, “Beyond diagonal reconfig- urable intelligent surfaces utilizing graph theory: Modeling, architecture design, and optimization,” IEEE Trans. Wireless Commun., vol. 23, no. 8, pp. 9972–9985, Aug. 2024

work page 2024