Dual-Polarized Beyond Diagonal RIS

Bruno Clerckx; Matteo Nerini

REVIEW 3 major objections 3 minor 12 references

Dual-Polarized Beyond Diagonal RIS

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For dual-polarized line-of-sight links, a group-connected RIS whose groups pair one vertical and one horizontal element reaches the fully-connected performance bound while needing only 3N/2 tunable components.

desk verdict Solid scaling-law contributions, but the Pareto-frontier proof has a fixable algebraic error and an unstated import; publish after repair. read the letter →

arxiv 2412.16097 v2 pith:VN26EPU3 submitted 2024-12-20 cs.IT eess.SPmath.IT

Matteo Nerini , Bruno Clerckx This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords beyonddiagonalRISdual-polarizedMIMOreconfigurableintelligentsurfacecross-polardiscriminationline-of-sightchannelsperformance-complexitytrade-offParetofrontiergroup-connected

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 studies reconfigurable intelligent surfaces (RISs) whose scattering matrices go beyond the diagonal form (BD-RIS), in systems where the surface elements and the transmitter/receiver antennas are dual-polarized. It claims that when the transmitter and receiver have opposite polarizations, BD-RIS beats conventional diagonal RIS even in line-of-sight channels, with a gain that grows as the cross-polar discrimination worsens. The central result is that the fully-connected upper bound on received power is achieved by a much simpler group-connected architecture: split the $N$ elements into $N/2$ pairs, each containing one vertical and one horizontal element. That design needs $3N/2$ tunable impedance components instead of $N(N+1)/2$. The paper also gives the full Pareto frontier of the performance-complexity trade-off, claiming that the optimal BD-RIS with $N+n$ components has $n$ such pairs and $N-2n$ unpaired elements.

What carries the argument

The central object is the group-connected BD-RIS scattering matrix, whose tunable impedance network partitions the $N$ elements into groups. The load-bearing identity is that a group holding one vertical and one horizontal element, under opposite Tx/Rx polarization, gives $\lVert h_{R,g}\rVert=\lVert h_{T,g}\rVert=\sqrt{1+\chi}$ regardless of the line-of-sight phases, so each group contributes $1+\chi$ to the coherent sum and $N/2$ groups produce $((1+\chi)N/2)^2$, exactly the fully-connected bound $\lVert h_R\rVert^2\lVert h_T\rVert^2=((1+\chi)N/2)^2$. For intermediate complexity, the argument rewrites the received power as $(\sum_g \lVert h_{R,g}\rVert\lVert h_{T,g}\rVert)^2$ and maximizes the contribution of the $n$ pairs through the Cauchy–Schwarz upper bound $\lVert \hat h_R\rVert\lVert \hat h_T\rVert$, whose optimum at equal numbers of vertical and horizontal elements yields $n(1+\chi)$.

What would settle it

Enumerate all group partitions of a small surface (say $N=8$) into $n$ pairs and singletons under both polarization assignments, compute the maximum of $(\sum_g \lVert h_{R,g}\rVert\lVert h_{T,g}\rVert)^2$ over all orderings, and compare it with $(n(1+\chi)+(N-2n)\sqrt{\chi})^2$; any partition that beats the formula, or any corrected algebra showing a maximum away from $n_h=n$, would disprove Proposition 2.

Watch

Extended reading notes

Core claim

The paper establishes that in a dual-polarized RIS-aided link with opposite Tx/Rx polarization, the received-power upper bound for a lossless BD-RIS is $P_R^\mathrm{fully}=(1+\chi)^2 N^2/4$, where $\chi\in[0,1]$ is the inverse cross-polar discrimination; a diagonal RIS only reaches $\chi N^2$ in line-of-sight, giving a BD-RIS gain $G=(1+\chi)^2/(4\chi)$. Proposition 1 shows that any group-connected RIS with group size 2 whose groups pair opposite polarizations attains the upper bound exactly for every $\chi$, with circuit complexity $C=3N/2$. Proposition 2 extends this to intermediate complexities: for $C=N+n$, the optimal architecture is $n$ opposite-polarization pairs plus $N-2n$ singletons, delivering $(n(1+\chi)+(N-2n)\sqrt{\chi})^2$. In Rayleigh fading the paper derives separate scaling laws, with gain $4(1+\chi)^2/(\pi^2\chi)$ for opposite polarization and $16/\pi^2$ for same polarization.

Load-bearing premise

The proof of the performance-complexity frontier takes as given a previously established formula for how many groups an optimal surface has, even though that formula was derived for single-polarization systems, and one algebraic step in the same proof is written with the wrong coefficient, so the claimed optimum at equally many vertical and horizontal elements is not fully verified as printed.

Editorial extensions

If this is right

In dual-polarized line-of-sight links with opposite transmitter and receiver polarization, a group-connected RIS with group size 2 attains the fully-connected received power $(1+\chi)^2 N^2/4$ using $3N/2$ tunable impedance components instead of $N(N+1)/2$.
The BD-RIS gain over diagonal RIS in that scenario is $G=(1+\chi)^2/(4\chi)$, so the largest relative gains occur at small $\chi$ (for instance $G=3$ at $\chi=0.1$) and disappear at $\chi=1$.
For intermediate complexity $C=N+n$, the claimed Pareto frontier is $(n(1+\chi)+(N-2n)\sqrt{\chi})^2$, realized by $n$ opposite-polarization pairs and $N-2n$ unpaired single elements.
With Rayleigh fading and opposite polarization, the asymptotic gain is $4(1+\chi)^2/(\pi^2\chi)$, while with same polarization it is $16/\pi^2$ independent of $\chi$.
Because the group-of-2 design saturates the performance bound, the tree- and fully-connected architectures are not needed for maximum power in this scenario, which can simplify prototype hardware.

Reading between the lines

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

Because the group-of-2 gain in line-of-sight does not depend on the LoS phases, the same wiring pattern should remain optimal under phase drift or imperfect channel knowledge, so the architecture could be prototyped with fixed inter-element couplings; the paper does not explore this simplification.
The Pareto formula suggests a graceful deployment strategy: start with $N/2$ opposite-polarization pairs and remove pairs one at a time to lower complexity, with received power decreasing quadratically in the number of remaining pairs; the paper does not present this as a design procedure.
The paper's Rician simulations indicate the gain lies between the LoS and Rayleigh extremes; deriving the explicit Rician scaling law and checking whether group-of-2 remains Pareto-optimal for all Rician factors would be a natural next step that the paper leaves open.
The same polarization-pairing principle may carry over to multi-user BD-RIS architectures, but the paper's analysis is single-user and does not address that setting.

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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

Colleague,

The scaling-law work here is solid and worth citing: first dual-polarized BD-RIS analysis, a nice LoS gain result G=(1+chi)^2/(4chi), and Prop. 1 showing group size 2 hits the fully-connected bound at linear complexity. I verified the algebra in Sec. III and Table I; it checks out under the stated models.

The soft spot is Prop. 2, the Pareto-frontier characterization. The proof as printed has a real algebraic error in Eq. (27). The expansion should be 4n^2 chi + 2n(1-chi)^2 n_h - (1-chi)^2 n_h^2; the printed middle term lacks the square on (1-chi). With the printed expression the derivative vanishes at n_h = n/(1-chi), not n_h=n, so the claimed maximizer doesn't follow. With the corrected coefficient the derivative does vanish at n_h=n and Q_max = n(1+chi), so the result is likely salvageable, but the proof is invalid as-is.

There's also an unstated import: the first line uses [4, Prop. 1] to set G=N-n groups for complexity C=N+n. That result was for uni-polarized BD-RIS. It's probably a counting/forest-equivalence statement that transfers, but the paper doesn't say why. The same group authored [4], so it's not a citation-pattern problem, just a missing justification.

Minor: in Prop. 2's proof, the line after (25) writes both norms as ||hR_hat||; the second should be ||hT_hat||.

Net: the core claim—that a group-connected RIS with group size 2 is optimal in dual-polarized LoS—is plausible and likely correct, but it rests on a proof that needs repair. The scaling laws and Prop. 1 stand alone. I'd send this to a serious referee, with the expectation that the authors fix Eq. (27) and make the import of [4] explicit. I wouldn't cite Prop. 2 in its current form, but I would cite the scaling laws and the group-2 result.

Bring it to reading group if you want to walk through the proof repair; otherwise it's a solid IEEE Comm. Letters type contribution once cleaned up.

Best,

Referee Report

3 major / 3 minor

Summary. This paper studies dual-polarized beyond-diagonal RIS (BD-RIS) systems. It derives scaling laws for the received power and the BD-RIS gain over conventional diagonal RIS (D-RIS) in four channel scenarios: Rayleigh or LoS fading, with Tx and Rx having either the same or opposite polarization. It shows that in LoS with opposite polarization, BD-RIS can provide a gain over D-RIS, and that a group-connected RIS with group size 2, in which each group pairs one vertical and one horizontal element, achieves the fully-connected received power upper bound with a low circuit complexity of C = 3N/2 (Prop. 1). The paper further claims a complete Pareto frontier for the performance-complexity trade-off in this LoS scenario, with Prop. 2 interpolating between single-connected and group-2 architectures.

Significance. If the results hold, the paper makes a useful contribution: it identifies a practically relevant scenario in which BD-RIS provides gains over D-RIS under LoS, and it proposes a low-complexity architecture (group size 2 with mixed polarizations) that attains the fully-connected performance bound. The scaling laws in Table I and the gain expressions in Eqs. (5), (7), (10), (11), and (15) are clean, parameter-free, and check out under the stated model. Prop. 1 is a solid result with clear practical implications. However, the Pareto-frontier claim rests on Prop. 2, whose proof as written contains an algebraic error in Eq. (27) and relies on an imported result from [4] without re-derivation; additionally, the proof considers only architectures with exactly N-2n singleton groups, leaving other feasible architectures unaddressed. These are load-bearing gaps, though they appear repairable.

major comments (3)

[Section IV, Prop. 2 proof, Eq. (27)] The expansion of the radicand in Eq. (27) is algebraically incorrect: the middle term should be 2n(1-χ)^2 n_h rather than 2n(1-χ)n_h. With the printed expression, the derivative with respect to n_h vanishes at n_h = n/(1-χ), so the asserted maximizer n_h = n does not follow. With the corrected coefficient, the derivative does vanish at n_h = n and yields Q ≤ n(1+χ), so the step is repairable, but the proof as printed is invalid.
[Section IV, Prop. 2 proof, first paragraph] The proof assumes without re-derivation that an optimal architecture with C=N+n has G=N-n groups, citing [4, Prop. 1]. That result was established for uni-polarized BD-RIS and may depend on complexity-counting conventions that are not verified here for dual-polarized LoS channels. Unless the transfer is justified or the relation is derived directly in this setting, the Pareto-frontier characterization lacks a load-bearing foundation.
[Section IV, Prop. 2 proof, Eqs. (20)-(22)] The proof passes from "at least N-2n groups of size 1" to an expression (20) with exactly N-2n singleton groups and n non-singleton groups. This is not without loss of generality: when n≥3, architectures with fewer non-singleton groups are feasible under the complexity budget (e.g., one group of size 3 when the extra budget is 3). For such architectures, the constant term in (21) and the definition of Q over n groups in (22) are not valid. A complete proof must optimize over the number of singleton groups and the sizes of the remaining groups, not only over the polarization assignment of 2n elements partitioned into n pairs.

minor comments (3)

[Section IV, proof of Prop. 2, after Eq. (26)] The text reads "∥^h_R∥ = √(n_v + n_h χ)" but this is the norm of ^h_T, not ^h_R; the symbol is mislabeled and should be corrected to ∥^h_T∥.
[Section III-D, last paragraph] The statement that "additional numerical simulations show that the gain under Rician channels..." provides no simulation setup, parameter choices, or figure; either add details or remove the claim to keep the paper self-contained.
[Abstract and Section III-D] The abstract states that group-connected RIS with group size 2 provides gains in both Rayleigh and LoS channels; this is true for the opposite-polarization LoS scenario, but in the same-polarization LoS case the gain is G=1. Please add the qualifier "opposite polarization" to avoid overgeneralization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the group-2 optimality and scaling laws are derived from the stated dual-polarized model; the only imported prior result is a parameter-free complexity-counting relation, and the Eq. (27) defect is a correctness issue, not a circular reduction.

full rationale

The paper's derivation chain is self-contained. Section III obtains the scaling laws directly from the stated dual-polarized channel model (p_R = [1, sqrt(chi)]-type polarization vectors), the standard D-RIS formula (1), and the fully-connected BD-RIS formula (2); no parameter is fitted and no fitted quantity is relabeled as a prediction. Proposition 1 computes P_R^Group2 = (sum_g ||h_R,g|| ||h_T,g||)^2 = ((1+chi) N/2)^2 = (1+chi)^2 N^2 / 4, which is exactly the fully-connected bound (14), so the low-complexity group-2 claim is proven algebraically rather than imported. Proposition 2 invokes [4, Proposition 1] only to fix the number of groups via G = 2N - C; this is a parameter-free complexity-counting/graph-theoretic relation from the authors' prior work, not a restatement of the dual-polarized LoS result, and the remaining proof (at least N-2n singleton groups, Cauchy-Schwarz upper bound, and Proposition 1 for paired groups) is carried out in the paper. The algebraic error in Eq. (27) -- missing square on (1-chi) -- is a proof defect that affects correctness but is not a circular step: the proposition is not defined in terms of itself or of a fitted input. No known result is merely renamed, and no ansatz is smuggled in through citation. Hence no significant circularity.

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

No free parameters are fitted: chi and N are model inputs. The group-size-2 architecture with opposite polarization is an existing BD-RIS configuration with a new polarization assignment, not a new physical entity. The main load-bearing imports are [2]'s received-power formulas and [4, Prop. 1], both from the same research group.

assumptions (6)

domain assumption Wireless channel is h = h_R Theta h_T with direct Tx-Rx path neglected
Section II introduces this cascaded model as 'widely used'; it ignores the direct path, which matters only if the RIS is not the dominant link.
domain assumption Dual-polarization is modeled by p_R and p_T with a single scalar chi (inverse XPD) and equal N/2 vertical and N/2 horizontal elements
Section II, definitions of p_R and p_T; all gains and scaling laws are computed under this polarization imbalance model.
domain assumption Fading channels are i.i.d. Rayleigh or deterministic LoS with phases
Section II defines the two channel models; results are limited to these models.
standard math Received power formulas for single-, fully-, and group-connected RIS from [2] (Eqs. (1), (2), (16))
The paper imports these established BD-RIS results rather than re-deriving them from Maxwell equations.
domain assumption [4, Proposition 1] fixes the number of groups as G=2N-C for an optimal BD-RIS
Section IV, proof of Prop. 2: this prior same-group result is used without verifying its assumptions hold for dual-polarized LoS channels.
domain assumption The BD-RIS is lossless and has half-wavelength element spacing
Section II states this assumption; it is required for the scattering matrix model.

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Cite this review

Pith. "Pith review of Dual-Polarized Beyond Diagonal RIS." pith.science (2026). https://pith.science/paper/VN26EPU3

@misc{pith2026241216097,
  author       = {Pith},
  title        = {Pith review of: Dual-Polarized Beyond Diagonal RIS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VN26EPU3}},
  note         = {Machine review of arXiv:2412.16097}
}

read the original abstract

Beyond diagonal reconfigurable intelligent surface (BD-RIS) is a family of RIS architectures more flexible than conventional RIS. While BD-RIS has been primarily analyzed assuming uni-polarized systems, modern wireless deployments are dual-polarized. To address this gap, this paper investigates the fundamental limits of dual-polarized BD-RIS-aided systems. We derive the scaling laws governing the performance of BD-RIS and the Pareto frontier of the trade-off between performance and circuit complexity enabled by BD-RIS. Theoretical results show that the group-connected RIS with group size 2 provides remarkable gains over conventional RIS in both Rayleigh and line-of-sight (LoS) channels, while maintaining a reduced circuit complexity.

Figures

Figures reproduced from arXiv: 2412.16097 by the authors.

**Figure 2.** Gain of BD-RIS over D-RIS G as a function of χ, when the Tx and Rx have the same/opposite polarization, and with Rayleigh/LoS channels. impedance components C in the BD-RIS. We begin by introducing a low-complex BD-RIS architecture that can achieve the same performance as the fully-connected RIS. Proposition 1. Consider a dual-polarized RIS-aided system with Tx and Rx having opposite polarization and LoS channels. … view at source ↗

**Figure 3.** Pareto frontier of the performance-complexity trade-off achieved by [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages

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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, 2023

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Pith tools

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

[1] [4]

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, 2023

work page 2023

[2] [1]

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 Trans. Commun., vol. 69, no. 5, pp. 3313–3351, 2021

2021

[3] [2]

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, 2022

2022

[4] [3]

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, 2024

work page 2024

[5] [5]

A novel Q-stem connected architecture for beyond-diagonal reconfigurable intelligent surfaces,

X. Zhou, T. Fang, and Y . Mao, “A novel Q-stem connected architecture for beyond-diagonal reconfigurable intelligent surfaces,” arXiv preprint arXiv:2411.18480, 2024

arXiv 2024

[6] [6]

Limited feedback beamforming systems for dual-polarized MIMO channels,

T. Kim, B. Clerckx, D. J. Love, and S. J. Kim, “Limited feedback beamforming systems for dual-polarized MIMO channels,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3425–3439, 2010

work page 2010

[7] [7]

Dual-polarized RIS-assisted mobile communications,

Y . Han, X. Li, W. Tang, S. Jin, Q. Cheng, and T. J. Cui, “Dual-polarized RIS-assisted mobile communications,” IEEE Trans. Wireless Commun. , vol. 21, no. 1, pp. 591–606, 2022

work page 2022

[8] [8]

Dual-polarized IRS-assisted MIMO network,

M. Munawar and K. Lee, “Dual-polarized IRS-assisted MIMO network,” IEEE Trans. Wireless Commun. , vol. 23, no. 4, pp. 2519–2532, 2024

work page 2024