A Universal Low-Dimensional Subspace Structure in Beamforming Design: Theory and Applications

Beamforming design plays a crucial role in multi-antenna systems, with numerous methods proposed to optimize key performance metrics such as spectral efficiency and power consumption. However, these methods often face two major challenges: high computational complexity and excessive communication overhead in distributed implementations. This paper addresses these challenges by analyzing a general beamforming optimization framework—referred to as the standard-form beam-forming problem—which encompasses various beamforming design tasks. We prove that any positive stationary point of this problem exhibits a low-dimensional subspace (LDS) structure, enabling the development of low-complexity and communication-efficient beamforming algorithms. As an illustrative example, we leverage the LDS structure to propose a computationally efficient beamforming algorithm for weighted sum rate maximization in coordinated multi-cell systems, with provable convergence to stationary points. Furthermore, we decentralize the algorithm for distributed coordinated beamforming, ensuring low interaction costs independent of the number of base station antennas. Notably, the proposed LDS structure is broadly applicable to a wide range of beamforming problems, including integrated sensing and communication (ISAC), intelligent reflecting surfaces (IRS), and beyond. Extensive numerical simulations validate the effectiveness and versatility of our approach, particularly the general applicability of the LDS structure.