Scalable RIS-aided hybrid beamforming for mmWave systems enables multiple sub-array architectures: A Geometric Mean Approach

This paper investigates a reconfigurable intelligent surface (RIS)-aided hybrid beamforming (HBF) framework for downlink multi-user millimeter-wave (mmWave) systems operating under practical hardware constraints. Given the severe path-loss disparity and blockage sensitivity inherent in mmWave propagation, RIS-assisted systems often suffer from pronounced user-rate imbalance, a challenge further exacerbated by the restricted spatial degrees of freedom in overlapped sub-array (OSA) hybrid architectures. To explicitly mitigate this disparity, we formulate a hardware-constrained optimization problem to maximize the geometric mean (GM) of user rates, thereby enforcing instantaneous fairness and service reliability. At the base station, an OSA-based HBF architecture is adopted to balance spatial flexibility and RF-chain efficiency, while the RIS operates with discrete phase shifts. Distinctive from conventional sum-rate-driven or idealized fully-digital designs, the proposed framework integrates GM-rate maximization with a manifold-theoretic optimization approach, enabling a realizable mapping from the fully-digital benchmark to a practical hybrid analog-digital structure. A two-block block coordinate descent (BCD) algorithm is developed, wherein the active precoder update admits an exact power-constrained solution and the passive phase optimization is executed on a Riemannian manifold to handle unit-modulus and discrete constraints. Numerical simulations demonstrate that the proposed approach effectively exploits OSA connectivity to achieve robust fairness across users. Specifically, an interference-aware warm-start strategy significantly accelerates convergence and yields an objective value improvement of up to 18.18% compared with existing methods, confirming both mathematical superiority and practical computational efficiency. Moreover, the results validate that low-resolution (e.g., 3-bit) phase shifters are sufficient to approach continuous-phase performance, underscoring the feasibility of the proposed framework for cost- and energy-efficient RIS-assisted mmWave deployments.