pith. sign in

arxiv: 2605.01389 · v1 · submitted 2026-05-02 · 💻 cs.IT · eess.SP· math.IT

RIS Optimization and Scaling Laws in Multi-Operator Systems: Is Quadratic Scaling Achievable?

Zheyu Wu , Matteo Nerini , Bruno Clerckx This is my paper

Pith reviewed 2026-05-09 17:57 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords reconfigurable intelligent surfacebeyond-diagonal RISmulti-operator systemsscaling lawsquadratic scalinginterference managementgroup-connected RIS
0
0 comments X

The pith

In multi-operator wireless systems, beyond-diagonal RIS achieves quadratic scaling of received power when group size meets or exceeds the number of operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that an RIS can be designed to boost the serving operator's signal strength while holding the effective channels seen by non-serving operators constant, which prevents inter-operator interference from rapid fluctuations. Closed-form solutions are derived for this optimization under the RIS matrix constraints. Conventional single-connected RIS yields only linear scaling of power with element count, but group-connected beyond-diagonal RIS delivers quadratic scaling once its group size reaches the total number of operators. This matters because it identifies the precise interconnection requirement for RIS to deliver its expected performance gains in shared-spectrum multi-operator deployments.

Core claim

For an L-operator system, the scaling law of the received signal power transitions at group size Gs = L: quadratic scaling is achieved when Gs >= L for group-connected BD-RIS, whereas linear scaling holds otherwise. The optimization maximizes the serving operator's received power subject to fixed RIS-reflected channels for the L-1 non-serving operators, with closed-form solutions obtained by handling the coupled unitary and linear equality constraints.

What carries the argument

Group-connected beyond-diagonal RIS reflection matrix that satisfies both the unitary constraint and linear equalities enforcing fixed non-serving reflected channels while maximizing serving power.

Load-bearing premise

The joint unitary and linear-equality constraints always admit a feasible closed-form solution that can be solved simultaneously for arbitrary numbers of operators and group sizes.

What would settle it

A direct measurement or simulation for a three-operator system with group size two that shows received power scaling linearly rather than quadratically with the number of RIS elements would disprove the claimed transition point at Gs = L.

Figures

Figures reproduced from arXiv: 2605.01389 by Bruno Clerckx, Matteo Nerini, Zheyu Wu.

Figure 1
Figure 1. Figure 1: Illustration of a multi-operator system model, wher view at source ↗
Figure 2
Figure 2. Figure 2: Average received signal power achieved by different view at source ↗
Figure 3
Figure 3. Figure 3: Average received signal power achieved by different view at source ↗
Figure 5
Figure 5. Figure 5: Average received signal power of different RIS archi view at source ↗
Figure 6
Figure 6. Figure 6: Sum rate of different RIS architectures versus the nu view at source ↗
read the original abstract

This paper studies multi-operator wireless communication systems aided by general reconfigurable intelligent surface (RIS), including both conventional single-connected RIS and beyond-diagonal RIS (BD-RIS). Specifically, we consider a system where multiple operators coexist in the same area over different frequency bands, each with a single-antenna base station, while one operator serves its single-antenna user with the aid of an RIS. In such a system, the RIS may unintentionally reflect signals from the non-serving operators, leading to inter-operator interference and rapid fluctuations of their effective channels. To address this issue, we propose a practical RIS design framework that maximizes the received signal power of the serving operator while enforcing fixed RIS-reflected channels of the non-serving operators. We derive closed-form solutions to the resulting optimization problem, based on a novel technique to deal with the coupled unitary and linear equality constraints. We further give scaling law analysis of the received signal power. For a two-operator system, the received signal power scales quadratically with the number of RIS elements for group-connected BD-RIS with group size Gs>=2, whereas for conventional single-connected RIS it scales only linearly. More generally, for an L-operator system with L-1 non-serving operators, the scaling-law transition occurs at Gs=L, where quadratic scaling is achieved when Gs>=L, and linear scaling otherwise. These results demonstrate that, in a multi-operator system, quadratic scaling is achievable only with BD-RIS architectures having enough interconnections. Simulation results validate the analysis and show the significant gain of BD-RIS over conventional RIS in multi-operator systems. In particular, group-connected BD-RIS with Gs=2 achieves a 13dB gain over conventional RIS in a two-operator system with a 128-element RIS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies multi-operator RIS systems (single-connected and BD-RIS) where one operator serves its user while non-serving operators experience fixed reflected channels. It proposes an optimization maximizing serving power subject to unitary (or block-unitary) constraints plus L-1 linear equalities, derives closed-form solutions via a novel technique for the coupled constraints, and obtains scaling laws: for L operators the received power scales quadratically with RIS size N when group size Gs >= L and linearly otherwise. Simulations with N=128 report a 13 dB gain for Gs=2 in the two-operator case.

Significance. If the closed-form solutions and scaling laws are valid, the work shows that quadratic scaling (previously limited to single-operator settings) remains achievable in multi-operator environments provided BD-RIS architectures have sufficient intra-group connections. The analytical scaling results and the explicit feasibility condition Gs >= L supply concrete design guidelines and performance predictions for shared-spectrum RIS deployments.

major comments (2)
  1. [closed-form derivation] Section deriving the closed-form solution (the novel technique handling unitary plus L-1 linear equality constraints): the construction of the block-unitary Theta for Gs = L assumes an isometry on the Gs-dimensional subspace that simultaneously satisfies the null constraints h_k^H Theta g_k = 0 (k != s) and preserves the Gram matrix of the g vectors. For generic channel realizations the required inner-product conditions on the image vectors conflict with the original <g_j, g_l>, so exact feasibility is not guaranteed; this directly undermines the claimed sharp transition at Gs = L and the quadratic-scaling regime.
  2. [scaling laws] Scaling-law analysis (the paragraph stating quadratic scaling for Gs >= L): the quadratic claim is obtained by substituting the closed-form solution into the power expression; because the closed-form is not shown to be feasible for all realizations when Gs = L, the scaling law holds only conditionally and the transition point requires additional qualification or a probabilistic feasibility statement.
minor comments (2)
  1. [abstract] Abstract and simulation section: the 13 dB gain figure lacks channel model details, number of Monte-Carlo trials, and error bars, making quantitative comparison difficult.
  2. [system model] Notation: the distinction between single-connected RIS and group-connected BD-RIS should be clarified with an explicit diagram or matrix structure in the system model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the closed-form derivation and scaling laws. We address each point below and will revise the manuscript to strengthen the feasibility analysis.

read point-by-point responses

  1. Referee: Section deriving the closed-form solution (the novel technique handling unitary plus L-1 linear equality constraints): the construction of the block-unitary Theta for Gs = L assumes an isometry on the Gs-dimensional subspace that simultaneously satisfies the null constraints h_k^H Theta g_k = 0 (k != s) and preserves the Gram matrix of the g vectors. For generic channel realizations the required inner-product conditions on the image vectors conflict with the original <g_j, g_l>, so exact feasibility is not guaranteed; this directly undermines the claimed sharp transition at Gs = L and the quadratic-scaling regime.

    Authors: We appreciate the referee highlighting the need for a rigorous feasibility proof. In the construction, the L-dimensional subspace (when Gs = L) provides exactly L-1 degrees of freedom after enforcing the null constraints on the non-serving operators. This leaves sufficient room to select an orthonormal frame that satisfies h_k^H v_k = 0 for k != s while preserving the Gram matrix of the original g vectors through an appropriate rotation within the allowable orthogonal complement. For generic channel realizations, where the h_k and g vectors are in general position, no conflict arises between the required inner products and the orthogonality conditions. We will add a detailed appendix proving existence of the isometry for Gs >= L. revision: yes

  2. Referee: Scaling-law analysis (the paragraph stating quadratic scaling for Gs >= L): the quadratic claim is obtained by substituting the closed-form solution into the power expression; because the closed-form is not shown to be feasible for all realizations when Gs = L, the scaling law holds only conditionally and the transition point requires additional qualification or a probabilistic feasibility statement.

    Authors: The quadratic scaling is obtained by substituting the closed-form solution, which exists with probability one over continuous channel distributions when Gs >= L due to the extra degrees of freedom from group connections. We agree that an explicit qualification improves clarity. In the revision we will state that the quadratic scaling holds almost surely for random channels when Gs >= L (and linearly otherwise), and we will reference the new feasibility appendix. revision: yes

Circularity Check

0 steps flagged

No significant circularity; closed-form derivation and scaling analysis are independent of inputs

full rationale

The paper derives closed-form solutions to the RIS optimization problem under unitary and linear equality constraints using a novel technique, then extracts scaling laws directly from those solutions for different group sizes Gs relative to the number of operators L. No parameter is fitted to data and then reused as a prediction, no quantity is defined in terms of the target scaling result, and the central claims do not reduce to self-citation chains or imported uniqueness theorems. The feasibility of the closed-form solution for arbitrary channel realizations is a separate correctness question rather than a circularity issue, as the derivation itself does not presuppose the scaling outcome by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard far-field channel models, the assumption that RIS reflection coefficients form a unitary matrix, and the existence of a feasible solution to the joint unitary-plus-linear-equality problem; no new physical entities are introduced and no parameters are fitted to data beyond the architectural choice of group size.

free parameters (1)
  • group size Gs
    Architectural parameter that controls interconnection density and determines the scaling regime; chosen by design rather than fitted.
axioms (2)
  • domain assumption RIS reflection matrix satisfies unitary constraint (phase-only passive reflection)
    Standard modeling assumption for passive RIS; invoked when formulating the optimization constraints.
  • domain assumption Perfect instantaneous knowledge of all operator channels is available at the RIS controller
    Required to enforce the fixed-channel constraints for non-serving operators.

pith-pipeline@v0.9.0 · 5625 in / 1542 out tokens · 41858 ms · 2026-05-09T17:57:29.498337+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Intelligent reflecting surface enhan ced wireless network via joint active and passive beamforming,

    Q. Wu and R. Zhang, “Intelligent reflecting surface enhan ced wireless network via joint active and passive beamforming,” IEEE Trans. Wireless Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019

    work page 2019

  2. [2]

    Intellige nt reflecting surface-aided wireless communications: A tutor ial,

    Q. Wu, S. Zhang, B. Zheng, C. Y ou, and R. Zhang, “Intellige nt reflecting surface-aided wireless communications: A tutor ial,” IEEE Trans. Commun., vol. 69, no. 5, pp. 3313–3351, May 2021

    work page 2021

  3. [3]

    Smart radio environments emp owered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,

    M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Y uen , J. de Rosny, and S. Tretyakov, “Smart radio environments emp owered 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, Nov. 2020

    work page 2020

  4. [4]

    An ove rview of signal processing techniques for RIS/IRS-aided wireles s systems,

    C. Pan, G. Zhou, K. Zhi, S. Hong, T. Wu, Y . Pan, H. Ren, M. D. Renzo, A. Lee Swindlehurst, R. Zhang, and A. Y . Zhang, “An ove rview of signal processing techniques for RIS/IRS-aided wireles s systems,” IEEE J. Sel. Topics Signal Process. , vol. 16, no. 5, pp. 883–917, Aug. 2022

    work page 2022

  5. [5]

    Modeling and architec ture design of reconfigurable intelligent surfaces using scatte ring parameter network analysis,

    S. Shen, B. Clerckx, and R. Murch, “Modeling and architec ture design of reconfigurable intelligent surfaces using scatte ring parameter network analysis,” IEEE Trans. Wireless. Commun. , vol. 21, no. 2, pp. 1229–1243, Feb. 2022

    work page 2022

  6. [6]

    Beyond diagonal reconfigu rable intelligent surfaces: From transmitting and reflecting mod es to single- , group-, and fully-connected architectures,

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

    work page 2023

  7. [7]

    Closed-form global o ptimization of beyond diagonal reconfigurable intelligent surfaces,

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

    work page 2024

  8. [8]

    A tutorial on be yond- diagonal reconfigurable intelligent surfaces: Modeling, a rchitectures, system design and optimization, and applications,

    H. Li, M. Nerini, S. Shen, and B. Clerckx, “A tutorial on be yond- diagonal reconfigurable intelligent surfaces: Modeling, a rchitectures, system design and optimization, and applications,” IEEE Commun. Surv. Tutor ., vol. 28, pp. 4086–4126, 2026

    work page 2026

  9. [9]

    End-to-end mutual coupling a ware communication model for reconfigurable intelligent surfac es: An electromagnetic-compliant approach based on mutual imped ances,

    G. Gradoni and M. Di Renzo, “End-to-end mutual coupling a ware communication model for reconfigurable intelligent surfac es: An electromagnetic-compliant approach based on mutual imped ances,” IEEE Wireless Commun. Lett. , vol. 10, no. 5, pp. 938–942, May 2021

    work page 2021

  10. [10]

    A u niversal framework for multiport network analysis of reconfigurable intelligent surfaces,

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

    work page 2024

  11. [11]

    Lossy beyond diago nal reconfigurable intelligent surfaces: Modeling and optimiz ation,

    Y . Peng, H. Li, Z. Wu, and B. Clerckx, “Lossy beyond diago nal reconfigurable intelligent surfaces: Modeling and optimiz ation,” IEEE Trans. Wireless Commun. , vol. 25, pp. 7365–7380, 2026

    work page 2026

  12. [12]

    Weighte d sum- rate maximization for reconfigurable intelligent surface a ided wireless networks,

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

    work page 2020

  13. [13]

    Channel estima tion and beamforming for beyond diagonal reconfigurable intelligen t surfaces,

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

    work page 2024

  14. [14]

    Optimization of beyond diagonal RIS: A universal framework applicable to arbitrary architectures,

    Z. Wu and B. Clerckx, “Optimization of beyond diagonal R IS: A universal framework applicable to arbitrary architectur es,” 2024. [Online]. Available: https://arxiv.org/abs/2412.15965

    work page arXiv 2024

  15. [15]

    Communication through a lar ge reflecting surface with phase errors,

    M.-A. Badiu and J. P . Coon, “Communication through a lar ge reflecting surface with phase errors,” IEEE Wireless Commun. Lett. , vol. 9, no. 2, pp. 184–188, Feb. 2020

    work page 2020

  16. [16]

    Secrecy analysis for IRS-aided wi retap MIMO communications: Fundamental limits and system design,

    X. Zhang and S. Song, “Secrecy analysis for IRS-aided wi retap MIMO communications: Fundamental limits and system design,” IEEE Trans. Inf. Theory , vol. 70, no. 6, pp. 4140–4159, Jun. 2024

    work page 2024

  17. [17]

    Global optimal closed -form solu- tions for intelligent surfaces with mutual coupling: Is mut ual coupling detrimental or beneficial?

    M. Nerini, H. Li, and B. Clerckx, “Global optimal closed -form solu- tions for intelligent surfaces with mutual coupling: Is mut ual coupling detrimental or beneficial?” IEEE Trans. Wireless Commun. , vol. 25, pp. 3201–3214, 2026

    work page 2026

  18. [18]

    Combating inter-operator pilot contamination in reconfigurable intelligent surface s assisted multi- operator networks,

    D. G¨ urg¨ unoglu, E. Bj¨ ornson, and G. Fodor, “Combating inter-operator pilot contamination in reconfigurable intelligent surface s assisted multi- operator networks,” IEEE Trans. Commun. , vol. 72, no. 9, pp. 5884– 5895, Sept. 2024

    work page 2024

  19. [19]

    Receive beamforming schemes to mitigate inter-operator pilot cont amination in RIS-aided MIMO networks,

    D. G¨ urg¨ unoglu, Z. G¨ ulg¨ un, E. Bj¨ ornson, and G. Fodo r, “Receive beamforming schemes to mitigate inter-operator pilot cont amination in RIS-aided MIMO networks,” IEEE Open J. Commun. Soc. , vol. 7, pp. 2130–2147, 2026

    work page 2026

  20. [20]

    On the impact of an RIS on t he out- of-band performance in sub-6 GHz and mmwave frequencies,

    L. Y ashvanth and C. R. Murthy, “On the impact of an RIS on t he out- of-band performance in sub-6 GHz and mmwave frequencies,” IEEE Trans. Commun., vol. 72, no. 12, pp. 7417–7434, Dec. 2024

    work page 2024

  21. [21]

    Distributed IRSs always benefit every mobile opera tor,

    ——, “Distributed IRSs always benefit every mobile opera tor,” IEEE Wireless Commun. Lett. , vol. 13, no. 11, pp. 2975–2979, Nov. 2024

    work page 2024

  22. [22]

    Performance a nalysis of multi-IRS aided multiple operator systems at mmwave freque ncies,

    S. Ghosh, L. Y ashvanth, and C. R. Murthy, “Performance a nalysis of multi-IRS aided multiple operator systems at mmwave freque ncies,” IEEE Trans. Commun. , vol. 73, no. 11, pp. 12 727–12 743, Nov. 2025

    work page 2025

  23. [23]

    Impact of inter-operator interference via re configurable intelligent surfaces,

    N. I. Miridakis, T. A. Tsiftsis, P . A. Karkazis, H. C. Lel igou, and P . Popovski, “Impact of inter-operator interference via re configurable intelligent surfaces,” IEEE Wireless Commun. Lett. , vol. 13, no. 9, pp. 2536–2540, Sept. 2024

    work page 2024

  24. [24]

    Spatially correlat ed multi-RIS communication: The effect of inter-operator interference ,

    N. I. Miridakis and P . A. Karkazis, “Spatially correlat ed multi-RIS communication: The effect of inter-operator interference ,” IEEE Trans. V eh. Tech., vol. 75, no. 3, pp. 5103–5108, Mar. 2026

    work page 2026

  25. [25]

    Irs-assis ted multicell multiband systems: Practical reflection model and joint bea mforming design,

    W. Cai, R. Liu, M. Li, Y . Liu, Q. Wu, and Q. Liu, “Irs-assis ted multicell multiband systems: Practical reflection model and joint bea mforming design,” IEEE Trans. Commun. , vol. 70, no. 6, pp. 3897–3911, Jun. 2022

    work page 2022

  26. [26]

    Resource allocation for RIS-aided mmwave s ystem with cooperative and non-cooperative base stations,

    X. Lin, Y . Fan, L. Zhang, K. Ma, Y . Sun, A. Tukmanov, Q. Abb asi, and M. A. Imran, “Resource allocation for RIS-aided mmwave s ystem with cooperative and non-cooperative base stations,” IEEE Trans. V eh. Tech., vol. 74, no. 3, pp. 4419–4431, Mar. 2025

    work page 2025

  27. [27]

    A hierarchic al DRL approach for resource optimization in multi-RIS multi- operator networks,

    H. Zhang, W. Wang, H. Zhou, Z. Lu, and M. Li, “A hierarchic al DRL approach for resource optimization in multi-RIS multi- operator networks,” IEEE Trans. Wireless Commun. , vol. 24, no. 6, pp. 4981– 4995, Jun. 2025

    work page 2025

  28. [28]

    Non-reciprocal beyond diagonal R IS: Multiport network models and performance benefits in full-duplex syst ems,

    H. Li and B. Clerckx, “Non-reciprocal beyond diagonal R IS: Multiport network models and performance benefits in full-duplex syst ems,” IEEE Trans. Commun. , vol. 73, no. 11, pp. 12 221–12 234, Nov. 2025

    work page 2025

  29. [29]

    Non-reciprocal beyond di agonal RIS: Sum-rate maximization in full-duplex communications,

    Z. Liu, H. Li, and B. Clerckx, “Non-reciprocal beyond di agonal RIS: Sum-rate maximization in full-duplex communications,” IEEE Trans. Commun., vol. 74, pp. 5169–5183, 2026

    work page 2026

  30. [30]

    E. M. Stein and R. Shakarchi, F ourier Analysis: An Introduction . Princeton, NJ: Princeton University Press, 2003

    work page 2003