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

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.

• The paper derives a closed-form symmetric low-rank BD-RIS configuration that maximizes the MIMO channel determinant, offering significant computational efficiency.
• It employs compact SVD and log-majorization theory to ensure near-optimal rate performance at high SNR while drastically reducing hardware complexity.
• Simulation results confirm that the proposed design matches iterative methods' performance, paving the way for practical and efficient 6G implementations.

Optimal Symmetric Low-Rank BD-RIS Configuration for MIMO Channel Determinant Maximization

Introduction and Context

The deployment of beyond-diagonal reconfigurable intelligent surfaces (BD-RIS) in MIMO systems has attracted significant interest due to their enhanced capability for channel manipulation compared to diagonal RIS. BD-RIS models allow arbitrary amplitude and phase coupling among surface elements, substantially increasing spatial degrees of freedom (DoF) at the cost of greater algorithmic and hardware complexity. Existing rate-maximization approaches for BD-RIS-assisted MIMO either rely on iterative manifold optimization, which is computationally demanding for large surfaces, or use suboptimal approximations that cannot exploit the full potential of BD-RIS architectures.

This paper proposes a closed-form configuration for a symmetric, passive, low-rank BD-RIS aiming to maximize the absolute value of the determinant (Max-Det) of the equivalent MIMO channel. Maximizing the determinant is established as an asymptotically optimal proxy for rate maximization in high-SNR or large-surface regimes. The solution possesses a structured, symmetric, and unitary scattering matrix of rank $2r$ (where $r$ is the MIMO channel DoF), enabling dramatic reduction in computational effort and, via $q$-stem hardware architecture, considerable simplification of physical implementation.

System Model and Problem Formulation

Consider a MIMO system where transmission between transmit (Tx) and receive (Rx) arrays (with $N_t$ and $N_r$ antennas) is solely via a $M$-element BD-RIS (the direct path is blocked). The effective baseband channel is $$\mathbf{H}_\Theta = \mathbf{G} \mathbf{\Theta} \mathbf{F}^H$$, where $\mathbf{G}$ and $\mathbf{F}$ are the RIS-Rx and Tx-RIS channel matrices, respectively, and $\mathbf{\Theta}$ is the BD-RIS scattering matrix, constrained to be symmetric and passive ($r$0 and $r$1).

The paper departs from direct rate maximization (computationally unwieldy under symmetry) and instead considers maximizing the determinant of the equivalent MIMO channel:

$r$2

It is shown that, at high SNR or when $r$3, this objective yields equivalent solutions to explicit rate maximization, because the determinant term dominates with vanishing error.

Closed-Form Max-Det Solution

The core contribution is the derivation of a maximizer for $r$4 under the symmetry constraint. Given the compact SVDs:

• $r$5
• $r$6 with $r$7, the optimal configuration is:

$r$8

where $r$9 arises from the compact SVD of $q$0, and $q$1 is a block-diagonal matrix with $q$2. This symmetric solution has precisely rank $q$3.

This construction is orders of magnitude less complex than iterative manifold optimization: it requires only standard SVDs of size $q$4 or $q$5, entailing computational complexity $q$6. As $q$7 is typically much larger than $q$8, this is particularly significant for practical systems.

Rate Gap Analysis and Majorization Results

While both the unconstrained unitary and symmetric unitary solutions reach the same Max-Det value, their singular value distributions differ, impacting achievable rate due to Schur majorization. The paper leverages log-majorization theory to upper bound the worst-case gap in attainable rate between the optimal symmetric Max-Det BD-RIS and the unconstrained unitary solution. The derived bound

$q$9

vanishes as SNR or $N_t$0 increase, reflecting negligible rate loss in high-SNR or large-surface scenarios.

Hardware Implementation: $N_t$1-Stem Architecture

A crucial practical implication is the solution's rank-$N_t$2 structure, which translates directly into hardware savings when using $N_t$3-stem BD-RIS architectures. Unlike fully connected BD-RIS requiring $N_t$4 reconfigurable elements, the Max-Det solution admits perfect implementation with only $N_t$5 stems, i.e., $N_t$6 circuit elements. Lemma 4 formally connects the low-rank property to solvability within the $N_t$7-stem topology, indicating the wide practical applicability of the approach.

Numerical Results

Simulation studies confirm that, in representative $N_t$8 MIMO settings, the closed-form Max-Det symmetric solution achieves rates statistically indistinguishable from those provided by computationally intensive iterative algorithms, particularly for $N_t$9 or at high SNR. The Max-Det proxy is shown to be effective even with hardware-constrained $N_r$0-stem implementations. When the direct channel is present, a simple phase adjustment offers a close-to-optimal suboptimal strategy. Across all scenarios, significant reduction in computation and circuit complexity is observed.

Implications and Future Directions

This work bridges an urgent gap between the information-theoretic and implementation aspects of BD-RIS-assisted MIMO: it demonstrates that with proper structure exploitation, hardware-realizable, closed-form configurations can attain near-optimal rates under challenging symmetry and passivity constraints. This makes large-scale, high-performance BD-RIS deployment feasible for 6G and beyond, where computational and hardware efficiency are paramount.

Theoretically, the result suggests that for a class of MIMO transform design problems (under symmetric passivity), maximizing $N_r$1 not only admits elegant solutions but is operationally optimal in important regimes. The connection to log-majorization also motivates further investigation into more general cost functions and multi-user extensions.

Conclusion

The paper provides a rigorous, readily implementable, and computationally scalable solution for configuring symmetric BD-RIS in MIMO settings to maximize the determinant (and, asymptotically, the rate) of the equivalent channel. By establishing the minimal-rank structure and aligning it to minimal hardware complexity in $N_r$2-stem BD-RIS architectures, this work delivers both theoretical insight and practical guidance. Future research avenues include closed-form solutions for alternative objective functions, extension to multi-user and broadcast channels, and further optimization under practical constraints such as channel estimation uncertainty and non-ideal hardware.

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Reference: "Optimal symmetric low-rank BD-RIS configuration maximizing the determinant of a MIMO link" (2604.09335).