Active RIS-Assisted Multi-User Multi-Stream Transmit Precoding Relying on Scalable-Complexity Iterations

This is the first investigation focused on delivering multi-stream information to multiple multi-antenna users employing an active reconfigurable intelligent surface (aRIS)-assisted system. We conceive the joint design of the transmit precoders and of the aRIS's power-amplified reconfigurable elements (APRES) to enhance the log-det rate objective functions for all users, which poses large-scale mixed discrete continuous problems. We develop a max-min log-det solver, which iterates quadratic-solvers of cubic complexity to maximize the nonsmooth function representing the minimum of the users' log-det rate functions. To mitigate the computational burden associated with cubically escalating complexity in large-scale scenarios, we introduce a pair of alternative problems aimed at maximizing the smooth functions representing the sum of the users' log-det rate function (sum log-det) and the soft minimum of the users' log-det rate function (soft min log-det). We develop sum log-det and soft max-min solvers, leveraging closed-form expressions of scalable (linear) complexity for efficient computation. This approach ensures practicality in addressing large-scale scenarios. Furthermore, the soft min log-det enables us to enhance the log-det rates for all users and their sum, ultimately improving the quality of delivering multi-user multi-stream information.