Extremely Large Beyond-Diagonal RIS: Low-Rank Modal Optimization for Near-Field Communications

Problem Statement

The paper addresses the scalability and performance challenges of Beyond-Diagonal Reconfigurable Intelligent Surfaces (BD-RIS) in the context of Extremely Large (XL) apertures operating in the radiative near-field.

1. Optimization and Hardware Burden: Fully connected BD-RIS architectures offer the highest performance by enabling non-diagonal, unitary scattering responses. However, optimizing an $M \times M$ scattering matrix (where $M$ is the number of elements) incurs a quadratic number of reconfigurable entries ($O(M^2)$) and cubic per-iteration computational complexity ($O(M^3)$). For XL surfaces with hundreds or thousands of elements, this burden is prohibitive.
2. Near-Field Modeling Breakdown: As aperture sizes increase at millimeter-wave and sub-THz frequencies, the Rayleigh distance extends, placing both the Base Station (BS) and users in the near-field. In this regime, wavefronts are spherical rather than planar. Traditional design tools relying on far-field assumptions (e.g., DFT beamspace, angle-only parameterization) degrade significantly because they fail to capture range information and the amplitude taper inherent in spherical waves.
3. The Gap: Existing dimensionality reduction techniques, such as fixed DFT beamspaces, reintroduce far-field assumptions that violate the near-field geometry, leading to significant performance penalties. There is a lack of a dimensionality reduction method that is matched to near-field geometry and provably equivalent to the fully connected optimum.

Methodology

The authors propose a Geometric Modal Architecture that decouples the optimization dimension from the physical panel size ($M$) and ties it instead to the propagation environment.

1. System Modeling:

   • The system is modeled as a continuous aperture using the free-space scalar Green function, which captures both spherical phase curvature and amplitude decay.
   • The cascaded channel (BS-RIS-User) is shown to live in a low-dimensional subspace spanned by spherical-wave atoms determined solely by the terminal positions (BS antennas and users).
   • The incident and radiated fields are expanded onto an orthonormal modal basis $\Phi$, reducing the infinite-dimensional scattering kernel to a compact $L \times L$ modal matrix $\Psi$.

2. Geometric Modal Basis Construction:

   • The authors construct a basis from the stacked atom matrix $[G, F^*]$, where $G$ represents the BS-to-RIS response and $F^*$ represents the conjugated RIS-to-user response.
   • A thin Singular Value Decomposition (SVD) of this matrix identifies the active subspace dimension $\rho$ (the numerical rank).
   • The basis $\Phi$ is formed by the dominant singular vectors plus an arbitrary orthonormal slack to ensure unitarity. The resulting dimension is $L = 2\rho \le 2(N+K)$, where $N$ is the number of BS antennas and $K$ is the number of users.

3. Theoretical Equivalence (Proposition 2):

   • Using a unitary dilation argument (Halmos dilation), the paper proves that an $L \times L$ unitary modal matrix $\Psi$ (embedded in the surface via a lossless completion) is provably equivalent to an unconstrained $M \times M$ unitary surface.
   • This equivalence holds for any Line-of-Sight (LoS) geometry, meaning the fully connected optimum can be attained with only $L^2$ reconfigurable entries, independent of the panel size $M$.

4. Scalable Optimization:

   • The authors develop a Weighted-MMSE (WMMSE) Riemannian alternating optimization algorithm.
   • Beamforming: Updated via closed-form solutions with a radial power projection to ensure monotonic convergence.
   • Modal Matrix Update: Optimized on the unitary group $U(L)$ using Riemannian gradient descent with Armijo backtracking.
   • Complexity: The per-iteration cost is $O(L^3)$, which is independent of $M$. The dependence on $M$ is limited to a one-time compression step ($O(M\rho^2)$).

Key Contributions

1. Geometric Modal Architecture: A novel design where the reconfigurable dimension is set by the number of coupled spherical-wave modes (determined by terminal positions) rather than the panel size.
2. Exact $M$-Free Equivalence: A proof that the fully connected optimum is attainable with $L = 2\rho$ degrees of freedom, provided the basis spans the active subspace of the channel. This eliminates the need for $O(M^2)$ entries.
3. Scalable Optimization Algorithm: A WMMSE-Riemannian algorithm that achieves monotone convergence with per-iteration complexity independent of the surface size.
4. Near-Field Analysis: A comprehensive analysis quantifying the penalties of mismatched designs (DFT beamspace) and the benefits of near-field focusing (range resolvability).

Numerical Results

The paper validates the theory using a $24 \times 24$ ($M=576$) element panel in a near-field scenario ($d_{BS} = 0.16 d_F$).

• Performance vs. Entries:
  • The proposed modal design reaches the fully connected optimum with approximately 196 entries ($L=14$).
  • A mismatched DFT beamspace requires approximately 12,100 entries to reach 99% of the fully connected performance, representing a 60-fold entry-count penalty.
  • Group-connected architectures (e.g., $N_g=32$) require even more entries to approach the optimum and converge slower.
• Convergence: The modal design converges in roughly $2 \times 10^4$ outer iterations. While the DFT beamspace converges faster, it settles at a significantly lower objective value due to the mismatched basis.
• Range Resolvability: The study demonstrates that the proposed architecture can resolve two users at the same azimuth separated only by range, with the transition aligning with the depth-of-focus law ($2r_0^2/D$).
• Depth Dependence: As the BS moves deeper into the near field, the advantage of beyond-diagonal coupling over diagonal phasing increases (from 2.5 to 4.4 bits/s/Hz), while the required modal dimension $L$ remains small and tracks the geometry.

Significance and Claims

The paper claims to solve the "scalability vs. optimality" trade-off for XL-BD-RIS in the near field. Its primary significance lies in proving that fully connected performance is affordable at scale when the design is matched to the near-field geometry.

• Provable Optimality: Unlike group-connected architectures which offer empirical trade-offs, the proposed modal architecture provides a provable guarantee of attaining the fully connected optimum.
• Hardware Efficiency: It reduces the number of reconfigurable entries from $O(M^2)$ (e.g., ~330,000 for $M=576$) to $O((N+K)^2)$ (e.g., ~196), drastically reducing hardware complexity and control overhead.
• Geometry-Driven Design: The work establishes that the near-field is not a nuisance to be corrected but a resource that enables low-rank optimization, as the spherical wavefronts naturally confine the channel to a low-dimensional subspace determined by terminal locations.

The authors conclude that their framework provides a complete complexity-performance ladder where the modal architecture is the only entry that is both provably optimal and computationally scalable for extremely large surfaces.