Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Precoder Design for Massive MIMO Downlink with Matrix Manifold Optimization

Rui Sun, Chen Wang, An-An Lu, Xiqi Gao, Xiang-Gen Xia

TL;DR

This work develops a unified matrix-manifold optimization framework for weighted sum-rate maximizing precoding in massive MIMO downlink under three power constraints: total power (TPC), per-user (PUPC), and per-antenna (PAPC). It proves each constraint induces a distinct Riemannian submanifold (sphere for TPC, oblique for PUPC and PAPC) and reformulates the constrained problem as unconstrained optimization on these manifolds. The authors derive complete Riemannian ingredients (gradient, Hessian, projection, retraction, vector transport) and propose three design methods—Riemannian Steepest Descent, Riemannian Conjugate Gradient, and Riemannian Trust Region—without large-matrix inversions, yielding favorable complexity and convergence. Numerical results show fast convergence and competitive WSR performance, with RCG often outperforming alternatives such as WMMSE, SLNR-based, and DCAM methods under similar complexity. Overall, the framework offers a scalable, geometry-aware approach to precoder design in large-scale MIMO systems.

Abstract

We investigate the weighted sum-rate (WSR) maximization linear precoder design for massive multiple-input multiple-output (MIMO) downlink. We consider a single-cell system with multiple users and propose a unified matrix manifold optimization framework applicable to total power constraint (TPC), per-user power constraint (PUPC) and per-antenna power constraint (PAPC). We prove that the precoders under TPC, PUPC and PAPC are on distinct Riemannian submanifolds, and transform the constrained problems in Euclidean space to unconstrained ones on manifolds. In accordance with this, we derive Riemannian ingredients, including orthogonal projection, Riemannian gradient, Riemannian Hessian, retraction and vector transport, which are needed for precoder design in the matrix manifold framework. Then, Riemannian design methods using Riemannian steepest descent, Riemannian conjugate gradient and Riemannian trust region are provided to design the WSR-maximization precoders under TPC, PUPC or PAPC. Riemannian methods do not involve the inverses of the large dimensional matrices during the iterations, reducing the computational complexities of the algorithms. Complexity analyses and performance simulations demonstrate the advantages of the proposed precoder design.

Precoder Design for Massive MIMO Downlink with Matrix Manifold Optimization

TL;DR

This work develops a unified matrix-manifold optimization framework for weighted sum-rate maximizing precoding in massive MIMO downlink under three power constraints: total power (TPC), per-user (PUPC), and per-antenna (PAPC). It proves each constraint induces a distinct Riemannian submanifold (sphere for TPC, oblique for PUPC and PAPC) and reformulates the constrained problem as unconstrained optimization on these manifolds. The authors derive complete Riemannian ingredients (gradient, Hessian, projection, retraction, vector transport) and propose three design methods—Riemannian Steepest Descent, Riemannian Conjugate Gradient, and Riemannian Trust Region—without large-matrix inversions, yielding favorable complexity and convergence. Numerical results show fast convergence and competitive WSR performance, with RCG often outperforming alternatives such as WMMSE, SLNR-based, and DCAM methods under similar complexity. Overall, the framework offers a scalable, geometry-aware approach to precoder design in large-scale MIMO systems.

Abstract

We investigate the weighted sum-rate (WSR) maximization linear precoder design for massive multiple-input multiple-output (MIMO) downlink. We consider a single-cell system with multiple users and propose a unified matrix manifold optimization framework applicable to total power constraint (TPC), per-user power constraint (PUPC) and per-antenna power constraint (PAPC). We prove that the precoders under TPC, PUPC and PAPC are on distinct Riemannian submanifolds, and transform the constrained problems in Euclidean space to unconstrained ones on manifolds. In accordance with this, we derive Riemannian ingredients, including orthogonal projection, Riemannian gradient, Riemannian Hessian, retraction and vector transport, which are needed for precoder design in the matrix manifold framework. Then, Riemannian design methods using Riemannian steepest descent, Riemannian conjugate gradient and Riemannian trust region are provided to design the WSR-maximization precoders under TPC, PUPC or PAPC. Riemannian methods do not involve the inverses of the large dimensional matrices during the iterations, reducing the computational complexities of the algorithms. Complexity analyses and performance simulations demonstrate the advantages of the proposed precoder design.
Paper Structure (27 sections, 8 theorems, 104 equations, 11 figures, 3 tables, 2 algorithms)

This paper contains 27 sections, 8 theorems, 104 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries for Matrix Manifold Optimization
  3. Problem Formulation and Riemannian Elements in Matrix Manifold framework
  4. WSR-maximization Precoder Design Problem in Euclidean Space
  5. Problem Reformulation on Riemannian Submanifold
  6. Riemannian Ingredients for Precoder Design
  7. TPC
  8. PUPC
  9. PAPC
  10. Riemannian Methods for Precoder Design
  11. Riemannian Gradient Methods
  12. Riemannian Trust Region Method
  13. Computational Complexity of Riemannian Methods
  14. Numerical Results
  15. Conclusion
  16. ...and 12 more sections

Key Result

Theorem 1

The Frobenius norm of $\forall\hat{ \mathbf{P}}\in\widehat{\mathcal{M}}$ is a constant and $\widehat{\mathcal{M}}$ forms a sphere. The Frobenius norm of each component matrix $\tilde{\mathbf{P}}_i$ of $\forall \tilde{\mathbf{P}}\in\widetilde{\mathcal{M}}$ is a constant and $\widetilde{\mathcal{M}}$

Figures (11)

  • Figure 1: The relationship between two manifolds Absil2009. $F$ is a smooth mapping from $\mathcal{N}_1$ to $\mathcal{N}_2$, and $\mathrm{D}F\left(\mathbf{X}\right)$ is a linear mapping from $T_{\mathbf{X}}{\mathcal{N}_1}$ to $T_{\mathbf{Y}}{\mathcal{N}_2}$.
  • Figure 2: Geometric interpretation of orthogonal projection, retraction and vector transport.
  • Figure 3: The illustration of the massive MIMO system.
  • Figure 4: Convergence comparison under TPC at SNR=20dB.
  • Figure 5: WSR performance when the RCG method converges under TPC.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Lemma 2
  • proof
  • ...and 6 more