Group-Connected Riemannian Manifold Optimization for Reciprocal BD-RIS

Reconfigurable intelligent surfaces (RISs) provide a flexible means of engineering wireless propagation channels through configurable multiport scattering networks. Among their architectures, group-connected beyond-diagonal RISs (BD-RISs) offer a practical tradeoff between scattering flexibility and hardware complexity by partitioning the surface into inter-connected groups. In reciprocal and lossless implementations, each group scattering block must be symmetric and unitary. In this paper, we investigate sum-rate maximization for reciprocal group-connected BD-RIS-assisted multiple-input multiple-output (MIMO) systems under such structural constraints. In particular, we build on a recent result where the feasible space determined by the symmetry and unitary constraints was characterized as a Riemannian manifold, extending it into a product manifold of symmetric-unitary block manifolds, thus expanding the approach to manifold phase optimization in the group-connected setting. As a consequence, the proposed group-extended framework performs tangent-space projections and Takagi retractions block-wise, while preserving the coupling among all groups through the common MIMO equivalent channel. The formulation naturally includes single-connected and fully connected reciprocal BD-RIS architectures as special cases and enables a flexible performance-complexity tradeoff through the group size.