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Precoder Design for User-Centric Network Massive MIMO with Matrix Manifold Optimization

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

TL;DR

The paper tackles precoder design for user-centric network massive MIMO downlink under per-BS power constraints. It recasts the constrained Euclidean problem into an unconstrained optimization on a Riemannian submanifold formed by the feasible precoders, and develops a full Riemannian Conjugate Gradient (RCG) algorithm using projections, gradients, retractions, and vector transport. The approach avoids inverses of large matrices, achieves favorable computational complexity, and delivers superior weighted sum-rate performance compared to WMMSE and linear precoders, especially with a small serving cluster size. Simulations based on QuaDRiGa validate both the advantages of the UC N mMIMO architecture and the efficiency of the RCG method for practical 6G-like deployments.

Abstract

In this paper, we investigate the precoder design for user-centric network (UCN) massive multiple-input multiple-output (mMIMO) downlink with matrix manifold optimization. In UCN mMIMO systems, each user terminal (UT) is served by a subset of base stations (BSs) instead of all the BSs, facilitating the implementation of the system and lowering the dimension of the precoders to be designed. By proving that the precoder set satisfying the per-BS power constraints forms a Riemannian submanifold of a linear product manifold, we transform the constrained precoder design problem in Euclidean space to an unconstrained one on the Riemannian submanifold. Riemannian ingredients, including orthogonal projection, Riemannian gradient, retraction and vector transport, of the problem on the Riemannian submanifold are further derived, with which the Riemannian conjugate gradient (RCG) design method is proposed for solving the unconstrained problem. The proposed method avoids the inverses of large dimensional matrices, which is beneficial in practice. The complexity analyses show the high computational efficiency of RCG precoder design. Simulation results demonstrate the numerical superiority of the proposed precoder design and the high efficiency of the UCN mMIMO system.

Precoder Design for User-Centric Network Massive MIMO with Matrix Manifold Optimization

TL;DR

The paper tackles precoder design for user-centric network massive MIMO downlink under per-BS power constraints. It recasts the constrained Euclidean problem into an unconstrained optimization on a Riemannian submanifold formed by the feasible precoders, and develops a full Riemannian Conjugate Gradient (RCG) algorithm using projections, gradients, retractions, and vector transport. The approach avoids inverses of large matrices, achieves favorable computational complexity, and delivers superior weighted sum-rate performance compared to WMMSE and linear precoders, especially with a small serving cluster size. Simulations based on QuaDRiGa validate both the advantages of the UC N mMIMO architecture and the efficiency of the RCG method for practical 6G-like deployments.

Abstract

In this paper, we investigate the precoder design for user-centric network (UCN) massive multiple-input multiple-output (mMIMO) downlink with matrix manifold optimization. In UCN mMIMO systems, each user terminal (UT) is served by a subset of base stations (BSs) instead of all the BSs, facilitating the implementation of the system and lowering the dimension of the precoders to be designed. By proving that the precoder set satisfying the per-BS power constraints forms a Riemannian submanifold of a linear product manifold, we transform the constrained precoder design problem in Euclidean space to an unconstrained one on the Riemannian submanifold. Riemannian ingredients, including orthogonal projection, Riemannian gradient, retraction and vector transport, of the problem on the Riemannian submanifold are further derived, with which the Riemannian conjugate gradient (RCG) design method is proposed for solving the unconstrained problem. The proposed method avoids the inverses of large dimensional matrices, which is beneficial in practice. The complexity analyses show the high computational efficiency of RCG precoder design. Simulation results demonstrate the numerical superiority of the proposed precoder design and the high efficiency of the UCN mMIMO system.
Paper Structure (20 sections, 5 theorems, 79 equations, 9 figures, 3 tables, 1 algorithm)

This paper contains 20 sections, 5 theorems, 79 equations, 9 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. System Model
  4. Problem Formulation in Euclidean Space
  5. Problem Reformulation on Riemannian Submanifold
  6. Riemannian Ingredients for Precoder Design
  7. Orthogonal Projection
  8. Riemannian Gradient
  9. Retraction
  10. Vector Transport
  11. Riemannian Conjugate Gradient Precoder Design
  12. Conventional Conjugate Gradient Method
  13. Riemannian Conjugate Gradient Precoder Design
  14. Computational Complexity
  15. Numerical Results
  16. ...and 5 more sections

Key Result

Theorem 1

$\mathcal{M}$ defined in M_definition forms a Riemannian submanifold of $\mathcal{N}$ with the Riemannian metric

Figures (9)

  • Figure 1: An illustration of the UCN mMIMO system.
  • Figure 2: Geometric interpretation of orthogonal projection, retraction and vector transport.
  • Figure 3: The layout of the UCN mMIMO system.
  • Figure 4: The relationship between the WSR performance and the size of the serving cluster.
  • Figure 5: The comparison of the WSR performance between the RCG method and different precoding methods.
  • ...and 4 more figures

Theorems & Definitions (12)

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