Riemannian Manifold Optimization for Advanced Wireless Communications: Fundamentals and Applications

TL;DR

The paper addresses the challenge of large-scale nonconvex optimization in next-generation wireless systems where geometric constraints arise naturally. It advocates Riemannian manifold optimization (RMO) as a constraint-preserving framework that operates on the problem's intrinsic manifold, using tools such as the Riemannian gradient and retractions to ensure feasibility at every iteration. It surveys fundamentals (manifolds, gradients, retractions, vector transport, and algorithm families like RGD, RCG, RTR, and quasi-Newton) and maps wireless problems onto manifolds including the complex circle, Stiefel, Grassmann, and HPD types, with case studies on FAS-assisted NOMA secure beamforming demonstrating superior performance and efficiency. The work also outlines future directions, including AI-enabled wireless systems, LEO satellite communications, and joint hardware–signal processing co-design, highlighting RMO's potential to enable real-time, geometry-aware optimization in complex 6G/6G+-scale networks.

Abstract

Next-generation wireless communications promise transformative technologies such as massive multiple-input multiple-output (MIMO), reconfigurable intelligent surfaces (RIS), integrated sensing and communication (ISAC), and fluid antenna systems (FAS). However, deploying these technologies is hindered by large-scale optimization problems with nonconvex constraints. Conventional Euclidean-space methods rely on approximations or relaxations, which degrade performance and incur substantial computational costs. Riemannian manifold optimization (RMO) offers a powerful alternative that directly operates on the manifold defined by the geometric constraints. This approach inherently satisfies the constraints at every optimization step, thereby avoiding the performance degradation and substantial computational costs. In this paper, we first elaborate on the principles of RMO, including the fundamental concepts, tools, and methods, emphasizing its effectiveness for nonconvex problems. We then introduce its applications in advanced wireless communications, showing how constrained problems are reformulated on their natural manifolds and solved using tailored RMO algorithms. Furthermore, we present a case study on secure beamforming in an FAS-assisted non-orthogonal multiple access (NOMA) system, demonstrating RMO's superiority over conventional methods in terms of both performance and computational efficiency.

Figures (5)

• Figure 1: Applications of RMO in advanced wireless communication systems. The figure illustrates several scenarios where RMO is applied to solve specific problems: (a) linear precoding, beamforming, and power allocation in massive MIMO systems; (b) constant modulus waveform design, phase design, and power allocation in ISAC-MIMO systems; (c) hybrid beamforming and link scheduling in ASGN; (d) phase shift design and power allocation in RIS-assisted massive MIMO systems; (e) secure beamforming and power allocation in FAS-assisted NOMA systems; and (f) signal detection, estimation, orthogonal pilot design, limited CSI feedback beamforming, and subspace signal design in mMTC systems.
• Figure 2: Illustration of a Riemannian optimization step. The figure depicts the core components of an iterative update in Euclidean space and on a manifold. Starting from the current iteration, the Euclidean gradient is first projected onto the tangent space to obtain the Riemannian gradient, which defines a search direction. A step is then taken within the tangent space, followed by a retraction that maps the resulting point back onto the manifold to produce the next iteration, ensuring that feasibility is maintained.
• Figure 3: An illustration of typical manifold structures in wireless communication. The figure showcases four representative classes of manifolds, each defined by specific geometric constraints, and maps them to key applications: (a) complex circle/sphere manifolds for precoding, beamforming, and constant-modulus waveform design; (b) the oblique and Stiefel manifolds for designing spreading codes and orthogonal pilots; (c) the Grassmann manifold for limited feedback, subspace-based estimation, and noncoherent signal design; and (d) the Hermitian positive definite manifold for secure precoding, RIS-assisted ISAC waveform design, and satellite link scheduling.
• Figure 4: Average secrecy rate versus SNR.
• Figure 5: Runtime to convergence versus SNR.