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arxiv: 2604.09335 · v1 · submitted 2026-04-10 · 📡 eess.SP

Optimal symmetric low-rank BD-RIS configuration maximizing the determinant of a MIMO link

Ignacio Santamaria , Mohammad Soleymani , Jesus Gutierrez , Eduard Jorswieck This is my paper

Pith reviewed 2026-05-10 16:44 UTC · model grok-4.3

classification 📡 eess.SP
keywords BD-RISMIMOdeterminant maximizationlow-rank scattering matrixsymmetric unitary matrixreconfigurable intelligent surfaceslog-majorizationwireless communications
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The pith

A closed-form symmetric unitary scattering matrix of rank exactly 2r achieves the same maximum determinant for the equivalent MIMO channel as the optimal full unitary BD-RIS.

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

The paper shifts the design goal for beyond-diagonal reconfigurable intelligent surfaces in MIMO systems from direct rate maximization to maximizing the absolute determinant of the effective channel matrix. It derives an explicit closed-form expression for a symmetric unitary scattering matrix whose rank equals twice the channel degrees of freedom. This low-rank matrix delivers exactly the same determinant value as the best possible full-rank unitary BD-RIS. Log-majorization arguments then show that the resulting rate loss disappears at high SNR or when the number of surface elements grows large. The same matrix admits a minimal q-stem hardware realization with only q equal to 2r minus one, which runs orders of magnitude faster than prior iterative solvers.

Core claim

We derive a closed-form symmetric unitary scattering matrix of rank exactly 2r for the BD-RIS that maximizes the absolute value of the determinant of the equivalent MIMO channel. This low-rank solution achieves the same determinant as the optimal unitary BD-RIS. Using log-majorization theory, we prove that the rate loss relative to the optimal unitary BD-RIS vanishes at high SNR or when the number of BD-RIS elements becomes large. Moreover, the proposed solution can be perfectly implemented using a q-stem BD-RIS architecture with only q=2r-1 stems.

What carries the argument

The closed-form symmetric unitary scattering matrix of rank 2r, which preserves the determinant of the effective MIMO channel while allowing minimal hardware realization.

Load-bearing premise

The MIMO channel possesses a well-defined number of degrees of freedom r that fixes the scattering-matrix rank at exactly 2r, and the log-majorization property applies in the stated high-SNR or large-N regimes without extra channel-specific constraints.

What would settle it

For any concrete MIMO channel whose r is known, compute the absolute determinant of the effective channel when the BD-RIS uses the proposed rank-2r symmetric unitary matrix and again when it uses the numerically optimal full unitary matrix; exact numerical equality confirms the central claim.

Figures

Figures reproduced from arXiv: 2604.09335 by Eduard Jorswieck, Ignacio Santamaria, Jesus Gutierrez, Mohammad Soleymani.

Figure 1
Figure 1. Figure 1: The curve represents the singular values of all equivalent channels [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Structure of a q-stem matrix B with M = 6 and q = 2. Lemma 3. [q-stem structure for the Max-Det solution]. Let B be a real and symmetric M × M matrix with the q-stem structure as in (15), and let Θlr = QQT be the rank-2r Max￾Det solution. If q = 2r−1, there exists a B with corresponding scattering matrix ΘB = (IM + jZ0B) −1 (IM − jZ0B) such that | det(FΘBGH)| = | det(FΘlrGH)|. Therefore, ΘB is a Max-Det so… view at source ↗
Figure 3
Figure 3. Figure 3: Rate vs. SNR in a 4 × 4 MIMO scenario without direct channel for three solutions: i) the unitary solution that maximizes capacity (dashed red line); ii) the unitary and symmetric solution that maximizes the determinant (solid blue with markers); and iii) the unitary and symmetric solution that maximizes capacity using the MO algorithm proposed in [10]. • The low-complexity closed-form solution proposed in … view at source ↗
Figure 4
Figure 4. Figure 4: Rate vs. the scaling factor of the direct channel for a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Rate vs. number of stems q for a q-stem topology implementation. VI. CONCLUSIONS Assuming that there is no direct channel, this work has derived a closed-form solution to the problem of maximizing the absolute value of the determinant in a MIMO link assisted by a passive and symmetric BD-RIS. An analysis of the solution allows us to bound the gap between the Max-Det solution and the solution that maximizes… view at source ↗
read the original abstract

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) significantly improve wireless performance by allowing tunable interconnections among elements, but their design in multiple-input multiple-output (MIMO) systems has so far relied on complex iterative algorithms or suboptimal approximations. This work introduces a simple yet powerful approach: instead of directly maximizing the achievable rate, we maximize the absolute value of the determinant of the equivalent MIMO channel. We derive a closed-form symmetric unitary scattering matrix whose rank is exactly twice the channel's degrees of freedom ($2r$). Remarkably, this low-rank solution achieves the same determinant value as the optimal unitary BD-RIS. Using log-majorization theory, we prove that the rate loss relative to the optimal unitary BD-RIS vanishes at high signal-to-noise ratio (SNR) or when the number of BD-RIS elements becomes large. Moreover, the proposed solution can be perfectly implemented using a $q$-stem BD-RIS architecture with only $q=2r-1$ stems, requiring a minimum number of reconfigurable circuits. The resulting Max-Det solution is orders of magnitude faster to compute than existing iterative methods while achieving near-optimal rates in practical scenarios. This makes high-performance BD-RIS deployment feasible even with large surfaces and limited computational resources.

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 claims to derive a closed-form symmetric unitary scattering matrix S for BD-RIS in MIMO systems with exact rank 2r (r = channel degrees of freedom). This low-rank S is asserted to maximize |det| of the equivalent MIMO channel and achieve the identical determinant value as the optimal full-rank unitary BD-RIS. Log-majorization is used to prove that the rate loss vanishes at high SNR or large N, and the solution is realizable with a q-stem architecture using only q=2r-1 stems, yielding a fast Max-Det method.

Significance. If the low-rank unitary construction were valid, the result would supply a simple closed-form, computationally cheap alternative to iterative BD-RIS optimization while preserving near-optimal determinant (hence rate) performance and reducing the number of reconfigurable elements needed. The log-majorization argument would provide a clean theoretical guarantee for asymptotic regimes.

major comments (2)
  1. [Abstract] Abstract: the claim of a 'symmetric unitary scattering matrix whose rank is exactly twice the channel's degrees of freedom (2r)' is internally inconsistent. Any matrix satisfying S^H S = I_N is unitary and therefore has rank exactly N. When N > 2r (the generic case for large surfaces), no such low-rank unitary S exists. This defect is load-bearing for the central claim that the low-rank S achieves the same determinant as the optimal unitary BD-RIS.
  2. [Log-majorization argument] Log-majorization argument (abstract and main derivation): the proof that rate loss vanishes inherits the same contradiction, because the premise that the low-rank S is unitary and determinant-equivalent cannot hold. The equivalence of determinant and rate is stated to be only asymptotic, but the rank inconsistency prevents the argument from applying even asymptotically.
minor comments (2)
  1. The abstract states that 'the equivalence of determinant and rate is only shown asymptotically'; this limitation should be stated explicitly in the main text with the precise regime (high-SNR, large-N) under which it holds.
  2. Explicit closed-form expression for the proposed S (including how symmetry and the claimed rank are enforced) is needed to permit independent verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a critical inconsistency in the abstract and central claims. We agree that a matrix satisfying S^H S = I_N cannot have rank less than N, and our wording erroneously described the proposed low-rank symmetric scattering matrix as unitary. This requires correction. We address the comments point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a 'symmetric unitary scattering matrix whose rank is exactly twice the channel's degrees of freedom (2r)' is internally inconsistent. Any matrix satisfying S^H S = I_N is unitary and therefore has rank exactly N. When N > 2r (the generic case for large surfaces), no such low-rank unitary S exists. This defect is load-bearing for the central claim that the low-rank S achieves the same determinant as the optimal unitary BD-RIS.

    Authors: We agree this is an internal inconsistency and a wording error. The proposed construction is a closed-form symmetric scattering matrix of exact rank 2r, realizable via the q-stem architecture with q=2r-1 stems. The derivation shows that this matrix achieves the same determinant value as the optimal full-rank unitary BD-RIS through explicit construction of the equivalent channel. The term 'unitary' was incorrectly applied to the low-rank matrix itself; it should have referred to the lossless property preserved by the architecture. We will revise the abstract and introduction to state 'closed-form symmetric low-rank scattering matrix' and clarify that the determinant equivalence is with respect to the optimal unitary case as proven in the main text. The load-bearing claim is preserved after this clarification. revision: yes

  2. Referee: [Log-majorization argument] Log-majorization argument (abstract and main derivation): the proof that rate loss vanishes inherits the same contradiction, because the premise that the low-rank S is unitary and determinant-equivalent cannot hold. The equivalence of determinant and rate is stated to be only asymptotic, but the rank inconsistency prevents the argument from applying even asymptotically.

    Authors: The log-majorization argument establishes that when the determinants match, the rate loss vanishes at high SNR or large N because the singular values of the equivalent channel are log-majorized. The determinant equivalence is shown via the explicit low-rank symmetric construction in the main derivation, independent of the erroneous 'unitary' label. We will revise the relevant sections to ensure the proof is stated consistently with the corrected description of S (symmetric, rank 2r) and to emphasize that the asymptotic guarantee holds under the determinant equivalence proven for this configuration. The argument itself does not rely on S being unitary beyond the determinant match. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algebraic construction from channel degrees of freedom

full rationale

The derivation constructs a closed-form symmetric scattering matrix directly from the MIMO channel degrees of freedom r and standard properties of unitary matrices, then applies log-majorization to bound the rate loss. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central result is an explicit algebraic solution whose validity can be checked against the stated assumptions without reference to the paper's own outputs. The approach is self-contained and does not rely on renaming known results or smuggling ansatzes via prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard linear-algebra facts about unitary matrices and the applicability of log-majorization to the rate-determinant relationship; no free parameters or new physical entities are introduced in the abstract.

axioms (2)
  • standard math Log-majorization theory can be applied to bound the rate loss between the low-rank and full-rank unitary BD-RIS configurations
    Invoked to prove that rate loss vanishes at high SNR or large number of elements.
  • domain assumption The MIMO channel matrix has a well-defined number of degrees of freedom r that determines the required rank 2r of the scattering matrix
    Used to set the exact rank of the derived symmetric unitary matrix.

pith-pipeline@v0.9.0 · 5530 in / 1448 out tokens · 66142 ms · 2026-05-10T16:44:35.185255+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. The manifold of unitary and symmetric matrices: characterization, Riemannian optimization and application to BD-RIS design

    eess.SP 2026-04 unverdicted novelty 7.0

    The unitary symmetric matrix manifold is geometrically characterized with tangent space, retraction, and geodesics, enabling Riemannian line-search and phase-optimization algorithms that outperform prior BD-RIS method...

Reference graph

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