RIS-aided Beamforming Design for MIMO Systems via Unified Manifold Optimization

Reconfigurable Intelligent Surface (RIS) can enhance spectral efficiency in Multiple-Input Multiple-Output (MIMO) communication systems. A specific challenge in this context involves codesigning the nonconvex phase shifts for RIS and a precoding matrix with a complex sphere constraint, where these coupled variables are used to formulate the nonconvex objective of maximizing spectral efficiency. Most existing methods do not directly address the maximization of spectral efficiency problem but instead relax it into a sum-path-gain-maximization (SPGM) problem before solving it, which may degrade spectral efficiency due to large relaxation gap. We propose an efficient Unified Manifold Optimization (UMO) framework to directly solve the problem. This is achieved through utilizing the inherent constant modulus characteristic of the RIS and the complex sphere characteristic of the precoding matrix constraint. Specifically, we construct a unified manifold space (UMS) that can simultaneously satisfy the RIS and the precoding matrix constraints, enabling the problem to be rephrased as an unconstrained Riemannian problem over the UMS. Based on the UMS, we derive a parallel conjugate gradient algorithm for simultaneous optimization of a precoding matrix and RIS phase shifts. Simulation outcomes indicate that the proposed method excels when compared to current approaches in achieving spectral efficiency enhancement. Furthermore, our algorithm has lower computational cost than several existing techniques