Structure-Oriented Symmetric Unitary Projection

• The paper introduces a projection method that maps candidate scattering matrices onto a feasibility set, ensuring symmetry, unitarity, and compliance with circuit constraints.
• It utilizes a constrained least-squares framework with Takagi decomposition to minimize projection error and reduce computational complexity compared to iterative methods.
• The approach generalizes across BD-RIS architectures, enabling efficient passive beamforming designs and hardware-friendly implementations for 6G wireless systems.

The structure-oriented symmetric unitary projection method is a general mathematical and algorithmic framework for designing optimal scattering matrices in advanced beyond-diagonal reconfigurable intelligent surfaces (BD-RIS). It enforces symmetry and unitarity, along with physically imposed circuit constraints, by “projecting” arbitrary candidate matrices onto a feasibility set tailored to the RIS architecture’s specific interconnection topology. This approach ensures compliance with reciprocity, energy conservation, and circuit implementability—serving as an enabling mechanism for highly efficient, optimal, and structurally consistent passive beamforming designs in 6G and future wireless communication systems (Zhou et al., 22 Sep 2025).

1. Role in BD-RIS Architecture Design

The structure-oriented symmetric unitary projection method targets the feasible set of scattering matrices for generalized BD-RIS, encompassing stem-connected and cluster-connected topologies. In this context, the scattering matrix Θ is required to be symmetric and unitary to satisfy both physical (reciprocity, losslessness) and hardware constraints arising from the underlying circuit topology. Mathematically, the feasible scattering matrix must have the form

$\Theta = (I + jZ_0 B)^{-1}(I - jZ_0 B)$

where $B$ is a real, symmetric susceptance matrix whose sparsity pattern is dictated by the RIS topology. The structure-oriented projection operates by mapping a candidate solution (e.g., from an unconstrained or “upper-bound” design) to the admissible set defined by these symmetry, unitarity, and structure constraints. This technique enables performance-complexity tradeoff exploration, allowing for a unified treatment of existing RIS architectures (fully-, group-, cluster-, tree-, and forest-connected) and new topological designs.

2. Mathematical Formulation and Algorithmic Structure

Formally, the projection is constructed as the following optimization problem: $$\min_{B \in \mathcal{B}_x} \|\mathsf{X} - \Theta\|_F^2 \quad \text{s.t.} \quad \Theta = (I + jZ_0 B)^{-1}(I - jZ_0 B),\ B = B^{\mathsf{T}},\ \text{with}\ B_{i,j}=0\ \forall (i,j) \notin \mathcal{P}$$ where $\mathsf{X}$ is any candidate (usually unconstrained or upper-bound) matrix, $\|\cdot\|_F$ is the Frobenius norm, and $\mathcal{P}$ identifies allowed nonzero entries of $B$ according to the architecture (e.g., stem- or cluster-connected).

A critical technical step is the Takagi decomposition: $$\frac{1}{2}(\mathsf{X} + \mathsf{X}^{\mathsf{T}}) = Q \Sigma Q^{\mathsf{T}}$$ yielding a block-diagonalization that reduces the core projection error to aligning parts of $\Theta$ with identity matrices within subspaces defined by active singular values. The projection condition is ultimately recast as a constrained linear system: $B \cdot C = D,$ where $C$ and $D$ depend on the Takagi factors and the implementation details of the circuit (including characteristic impedance $Z_0$). The system is vectorized via a transformation matrix $R$ that selects only independent entries (accounting for symmetry and sparsity), yielding: $A b = z$ with $b = \mathrm{vec}_i(B)$ and solution $b^* = (A^\mathsf{T}A)^{-1}A^\mathsf{T} z$, which is then reassembled into the full $B$.

3. Application to Utility Maximization and Beamforming

The projection procedure is leveraged in various algorithmic schemes:

• UB-based SOSUP (Upper Bound-based Structure-Oriented Symmetric Unitary Projection): An unconstrained or upper-bound solution is first computed, then projected using the closed-form SOSUP approach. This can be used for sum channel gain or weighted sum rate maximization.
• SOSUP-based quasi-Newton: The projection method provides a high-quality initialization for a quasi-Newton iterative refinement within the admissible set, ensuring convergence to high-performance beamformers without violating circuit constraints.

These techniques efficiently solve core RIS optimization problems with a substantial reduction in computational cost compared to prior methods employing alternating projection or manifold optimization.

4. Generalization to Arbitrary Topologies and Structural Sets

A major advantage of the structure-oriented projection paradigm is its generality across BD-RIS architectures. By carefully constructing the indicator and cumulative counting matrices that underlie the transformation matrix $R$, the projection algorithm accommodates a wide range of interconnection topologies—including stem-, cluster-, group-, and tree-connected structures. This unification makes the method broadly applicable, simplifying the design pipeline and facilitating rapid exploration of architectural tradeoffs between complexity and performance.

For example, the cluster-connected RIS framework subsumes fully-, arrowhead-, forest-, and previously proposed group-connected topologies, providing a universal linear-algebraic mechanism for enforcing the required physical and topological constraints in the projection.

5. Computational Efficiency and Comparison with Previous Methods

The SOSUP method reduces the original nonconvex, combinatorial optimization to a tractable constrained least-squares problem. This leads to substantial gains in CPU efficiency: empirical results indicate that UB-based SOSUP consumes only ~13.7% of the CPU time required by advanced iterative projection primitives such as pp-ADMM for equivalent weighted sum rate performance. The closed-form or near-closed-form update and the ability to exploit matrix sparsity and symmetry are key to this efficiency.

Unlike methods based solely on iterative alternating projections or unconstrained optimization followed by aggressive thresholding, the structure-oriented approach ensures all physical requirements are met at every step, without post hoc repair or relaxation. It is also inherently more flexible, adaptively incorporating new hardware constraints with simple modification of the underlying matrix mask.

6. Hardware and 6G System Implications

By offering closed-form, structure-compatible projections, the method enables hardware-efficient implementation of advanced BD-RIS designs. For example, stem-connected RIS achieves the same maximal sum channel gain as the fully connected case but with significant circuit complexity reduction; cluster-connected architectures flexibly interpolate between full and minimal connectivity, balancing hardware cost with performance. These properties make the method directly relevant for large-scale, dense, and energy-efficient 6G wireless deployments, where scalability and rapid online (or real-time) reconfiguration are critical.

The reduction in control and switching complexity, coupled with the retention of optimal passive beamforming, is expected to foster the integration of RIS into practical 6G MIMO, cell-free, and integrated sensing/communication platforms, supporting full-space coverage and high spectral efficiency.

7. Numerical Results and Trade-off Analysis

Extensive simulations demonstrate that, when parameterized appropriately (e.g., tuning the number of stems or cluster sizes), the proposed structure-oriented projections achieve nearly identical performance to fully-connected designs—even under severe sparsity constraints—while dramatically reducing the number of tunable components. Trade-off curves and comparative tables (e.g., for sum channel gain, weighted sum rate, and CPU time) are reported for multiple system sizes and user numbers, substantiating that circuit complexity can be reduced without compromising communication performance—especially via stem- and cluster-connected configurations.

The method’s success across scenarios is attributed to the fact that structure-oriented projection—under correct symmetry and unitarity enforcement—preserves all critical physical invariants while efficiently navigating the combinatorial design space of BD-RIS interconnects.

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In summary, the structure-oriented symmetric unitary projection method provides the underpinning mathematical infrastructure for optimal, physically feasible beamforming design in modern BD-RIS architectures. It systematically enforces symmetry, unitary, and topology-induced constraints, enables high-performance and real-time optimization in resource-intensive wireless systems, and generalizes across arbitrary interconnect patterns—making it a foundational component for next-generation wireless metamaterials and 6G infrastructure (Zhou et al., 22 Sep 2025).