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Manifold-Based Optimizations for RIS-Aided Massive MIMO Systems

Wilson de Souza Junior, David William Marques Guerra, José Carlos Marinello, Taufik Abrão, Ekram Hossain

TL;DR

This work advocates manifold optimization (MO) as a natural framework for RIS-aided massive MIMO optimization, where passive elements impose unit-modulus constraints and high dimensionality challenges hinder conventional methods. By leveraging MO on tailored product manifolds (e.g., complex circle $S^1$ and Stiefel $oldsymbol{W}$-spaces) and using Riemannian gradients and retractions, the paper outlines a practical, geometry-aware approach for active and passive beamforming, resource allocation, and network management. It provides a tutorial-style treatment, a problem-identification framework, and four real RIS-aided applications (fairness, energy efficiency, intra-cell pilot reuse, grant-free random access) with gradient-descent MO algorithms and performance comparisons to traditional methods. The results illustrate substantial gains in fairness, energy efficiency, and spectral efficiency, while offering reduced complexity and greater scalability for next-generation wireless networks. Overall, the MO toolkit presented here enables more efficient, interpretable, and scalable optimization in RIS-enabled mMIMO and XL-MIMO deployments.

Abstract

Manifold optimization (MO) is a powerful mathematical framework that can be applied to optimize functions over complex geometric structures, which is particularly useful in advanced wireless communication systems, such as reconfigurable intelligent surface (RIS)-aided massive MIMO (mMIMO) and extra-large scale massive MIMO (XL-MIMO) systems. MO provides a structured approach to tackling complex optimization problems. By leveraging the geometric properties of the manifold, more efficient and effective solutions can be found compared to conventional optimization methods. This paper provides a tutorial on MO technique and provides some applications of MO in the context of wireless communications systems. In particular, to corroborate the effectiveness of MO methodology, we explore five application examples in RIS-aided mMIMO system, focusing on fairness, energy efficiency (EE) maximization, intracell pilot reuse interference mitigation, and grant-free (GF) random access (RA).

Manifold-Based Optimizations for RIS-Aided Massive MIMO Systems

TL;DR

This work advocates manifold optimization (MO) as a natural framework for RIS-aided massive MIMO optimization, where passive elements impose unit-modulus constraints and high dimensionality challenges hinder conventional methods. By leveraging MO on tailored product manifolds (e.g., complex circle and Stiefel -spaces) and using Riemannian gradients and retractions, the paper outlines a practical, geometry-aware approach for active and passive beamforming, resource allocation, and network management. It provides a tutorial-style treatment, a problem-identification framework, and four real RIS-aided applications (fairness, energy efficiency, intra-cell pilot reuse, grant-free random access) with gradient-descent MO algorithms and performance comparisons to traditional methods. The results illustrate substantial gains in fairness, energy efficiency, and spectral efficiency, while offering reduced complexity and greater scalability for next-generation wireless networks. Overall, the MO toolkit presented here enables more efficient, interpretable, and scalable optimization in RIS-enabled mMIMO and XL-MIMO deployments.

Abstract

Manifold optimization (MO) is a powerful mathematical framework that can be applied to optimize functions over complex geometric structures, which is particularly useful in advanced wireless communication systems, such as reconfigurable intelligent surface (RIS)-aided massive MIMO (mMIMO) and extra-large scale massive MIMO (XL-MIMO) systems. MO provides a structured approach to tackling complex optimization problems. By leveraging the geometric properties of the manifold, more efficient and effective solutions can be found compared to conventional optimization methods. This paper provides a tutorial on MO technique and provides some applications of MO in the context of wireless communications systems. In particular, to corroborate the effectiveness of MO methodology, we explore five application examples in RIS-aided mMIMO system, focusing on fairness, energy efficiency (EE) maximization, intracell pilot reuse interference mitigation, and grant-free (GF) random access (RA).
Paper Structure (23 sections, 10 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 23 sections, 10 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND
  3. MOTIVATION AND CONTRIBUTION
  4. ORGANIZATION OF THE PAPER
  5. MANIFOLD FUNDAMENTALS
  6. ALTERNATIVES TO MANIFOLD OPTIMIZATION TECHNIQUE
  7. MANIFOLD OPTIMIZATION FRAMEWORK
  8. Optimization Over Smooth Surfaces
  9. Operators on Riemannian Manifold
  10. Challenges in Manifold Optimization
  11. COLLECTION OF MANIFOLDS
  12. MANIFOLD LEARNING METHODS AND STRUCTURES
  13. STEPS FOR IDENTIFY A MANIFOLD
  14. METHODOLOGY FOR FORMULATING AND SOLVING OPTIMIZATION PROBLEMS WITH MANIFOLDS
  15. PRACTICAL EXAMPLE AND GRADIENT DESCENT-BASED ALGORITHMS
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Application 1: (a) System model of the considered network; (b) Steps of the proposed algorithm; ( c) Simulation results.
  • Figure 2: Application 2: (a) System model for the considered network; (b) Steps of the proposed algorithm; (c) transmit power ($P_{\text{UT}}$) with $K=8$ varying: i) min ; ii) number of reflective elements $N$; iii) noise power at .
  • Figure 3: Application 3:(a) communication system assisted by multiple , each deployed on the facades of buildings. The users in the dotted circle area are served without the aid of any . They, as well as the users served by each , share the same set of pilot sequences in our investigated scenario. (b) Execution tasks of the two stages -aided pilot reuse method; (c) with $N=256$, $\tau_p=4$, and increasing: i. $M$, when $R=3$, and $K=16$ ; ii. $K$, $R$, and $\varsigma$, such that $K=\varsigma \, \tau_p = (R+1)\, \tau_p$, when $M=128$ antennas at the .
  • Figure 4: Application 4: (a) System model; (b) multi-beam reflection patterns designed via ; (c) performance of the protocol.