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

1. Problem Statement

The paper addresses the design of Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RIS) in Multiple-Input Multiple-Output (MIMO) systems.

• Context: Unlike traditional diagonal RIS (which only control phase/amplitude of individual elements), BD-RIS allows tunable interconnections between elements, offering greater flexibility for channel shaping.
• Challenge:
  • Computational Complexity: Existing methods to optimize BD-RIS for MIMO rate maximization rely on complex iterative algorithms (e.g., Manifold Optimization) or suboptimal approximations. These are computationally expensive, especially for large surfaces.
  • Hardware Complexity: Fully connected BD-RIS architectures require a massive number of tunable circuits, impacting energy efficiency and implementation feasibility.
  • Symmetry Constraint: Practical BD-RIS devices are passive and reciprocal, requiring the scattering matrix $\Theta$ to be symmetric ($\Theta = \Theta^T$). Finding closed-form solutions for symmetric BD-RIS in MIMO scenarios has been an open problem.
• Goal: To derive a closed-form, low-complexity solution for configuring a symmetric BD-RIS that maximizes the achievable rate in a MIMO system (specifically when the direct Tx-Rx link is blocked).

2. Methodology

The authors propose a novel approach that shifts the optimization objective from directly maximizing the achievable rate to maximizing the absolute value of the determinant of the equivalent MIMO channel.

• Determinant as a Proxy:
  • The achievable rate is given by $R = \log \det(I + \rho H H^H)$.
  • At high Signal-to-Noise Ratio (SNR) or with a large number of RIS elements ($M$), the term $\log \det(H H^H)$ dominates the rate expression.
  • The authors prove that maximizing $|\det(H(\Theta))|$ serves as an asymptotically optimal proxy for rate maximization.
• Closed-Form Derivation (Max-Det Solution):
  • The problem is formulated as finding a symmetric, unitary scattering matrix $\Theta = QQ^T$ that maximizes $|\det(F \Theta G^H)|$, where $F$ and $G$ are the backward and forward channel matrices.
  • Using Singular Value Decomposition (SVD) of the channels and properties of the Stiefel manifold, they derive an explicit solution:
    $$\Theta_{opt} = U \begin{bmatrix} I_r & 0 \\ 0 & -I_r \end{bmatrix} U^T$$
    where $U$ is derived from the SVD of a composite matrix formed by the right singular vectors of $F$ and the complex conjugate of the right singular vectors of $G$.
• Low-Rank Structure:
  • The optimal solution $\Theta_{opt}$ has a rank of exactly $2r$, where $r = \min(N_t, N_r)$ is the degrees of freedom (DoF) of the MIMO channel.
  • This is a "low-rank" solution because typically $M \gg 2r$.
• Rate Gap Analysis:
  • Using log-majorization theory, the authors bound the rate loss between their symmetric Max-Det solution and the theoretical optimal (non-symmetric) unitary solution.
  • They prove that this rate gap vanishes as SNR $\to \infty$ or as the number of RIS elements $M \to \infty$.
• Hardware Implementation (q-stem Topology):
  • The low-rank nature of the solution ($rank = 2r$) is mapped to the q-stem architecture, a reduced-complexity BD-RIS topology.
  • They prove that the optimal Max-Det solution can be perfectly implemented using a q-stem structure with only $q = 2r - 1$ stems, significantly reducing the number of required reconfigurable circuits compared to fully connected designs.

3. Key Contributions

1. First Closed-Form Symmetric Solution: The paper provides the first closed-form solution for a symmetric BD-RIS in a MIMO setting that maximizes the channel determinant (and asymptotically the rate).
2. Optimal Low-Rank Characterization: It is shown that the optimal scattering matrix has a rank of $2r$. This is a critical insight because it dictates the minimum hardware complexity required.
3. Theoretical Performance Bounds: Using log-majorization, the authors rigorously prove that the rate penalty of the symmetric Max-Det solution compared to the optimal non-symmetric solution is negligible in high-SNR or large-$M$ regimes.
4. Hardware-Efficient Implementation: The work bridges the gap between mathematical optimization and hardware design by demonstrating that the optimal solution fits perfectly into a q-stem topology with $q = 2r - 1$ stems, minimizing the number of impedance elements.
5. Computational Efficiency: The proposed algorithm (Algorithm 1) involves only SVD computations with complexity $O(Mr^2)$, which is orders of magnitude faster than iterative Manifold Optimization methods.

4. Simulation Results

The authors validate their theory through simulations in a 4x4 MIMO scenario with a BD-RIS:

• Blocked Direct Link: The Max-Det solution achieves rates nearly identical to the optimal iterative Manifold Optimization (MO) solution and the theoretical non-symmetric upper bound. The difference is negligible, especially at high SNR.
• Unblocked Direct Link: While the closed-form solution is derived for blocked links, the authors propose a phase-correction heuristic for unblocked links. This suboptimal approach performs competitively, particularly when the direct link is weak or moderate.
• q-stem Implementation: Simulations confirm that using a q-stem topology with $q = 2r - 1 = 7$ stems (for a 4x4 system where $r=4$) recovers the performance of a fully connected BD-RIS perfectly, validating the low-rank implementation theory.

5. Significance

• Feasibility for Large-Scale Deployment: By providing a non-iterative, closed-form solution, this work makes real-time BD-RIS configuration feasible even for surfaces with thousands of elements, where iterative algorithms would be too slow.
• Cost Reduction: The identification of the $2r$ rank and the corresponding $q = 2r - 1$ stem architecture drastically reduces the hardware cost (number of tunable circuits) and power consumption, addressing a major barrier to commercial BD-RIS adoption.
• Theoretical Foundation: It establishes a strong theoretical link between determinant maximization and rate maximization in symmetric RIS systems, offering a new design paradigm that avoids the complexity of non-convex rate optimization.

In summary, this paper transforms BD-RIS design from a computationally heavy, iterative problem into a tractable, closed-form engineering solution that is both theoretically near-optimal and practically implementable with minimal hardware resources.