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

Beyond-diagonal reconfigurable intelligent surfaces (BD-RIS) have emerged as an advanced RIS architecture that addresses the considerable optimization and hardware challenges associated with their design, particularly as their size increases. This paper introduces the extremely large BD-RIS (XL-BD-RIS) concept, which leverages near-field geometries to maintain performance at scale. We model the system using free-space Green functions, revealing that the resulting aperture fields reside in a low-dimensional subspace spanned by spherical-wave responses to terminal positions. Based on this insight, we develop a compact unitary modal matrix utilizing localization information that achieves the fully connected optimum using a number of reconfigurable entries dictated by the geometry, independent of surface size. Employing a weighted-MMSE Riemannian algorithm, we optimize both the beamformers and the modal matrix, demonstrating monotone convergence without increasing computational cost with surface size. Our numerical results show that a 24 × 24-element panel can reach optimal performance utilizing only around two hundred entries instead of the previously required three hundred thousand, marking a significant reduction in complexity compared to traditional methods.