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Ellipsoidal Manifold Optimization for Distributed Antenna Beamforming

Minhao Zhu, Kaiming Shen

Abstract

This paper addresses the weighted sum-rate (WSR) maximization problem in a downlink distributed antenna system subject to per-cluster power constraints. This optimization scenario presents significant challenges due to the high dimensionality of beamforming variables in dense antenna deployments and the structural complexity of multiple independent power constraints. To overcome these difficulties, we generalize the low-dimensional subspace property--previously established for sum-power constraints--to the per-cluster power constraint case. We prove that all stationary-point beamformers reside in a reduced subspace spanned by the channel vectors of the corresponding antenna cluster. Leveraging this property, we reformulate the original high-dimensional constrained problem into an unconstrained optimization task over a product of ellipsoidal manifolds, thereby achieving significant dimensionality reduction. We systematically derive the necessary Riemannian geometric structures for this specific manifold, including the tangent space, Riemannian metric, orthogonal projection, retraction, and vector transport. Subsequently, we develop a tailored Riemannian conjugate gradient algorithm to solve the reformulated problem. Numerical simulations demonstrate that the proposed algorithm achieves the same local optima as standard benchmarks, such as the weighted minimum mean square error (WMMSE) method and conventional manifold optimization, but with substantially higher computational efficiency and scalability, particularly as the number of antenna clusters increases.

Ellipsoidal Manifold Optimization for Distributed Antenna Beamforming

Abstract

This paper addresses the weighted sum-rate (WSR) maximization problem in a downlink distributed antenna system subject to per-cluster power constraints. This optimization scenario presents significant challenges due to the high dimensionality of beamforming variables in dense antenna deployments and the structural complexity of multiple independent power constraints. To overcome these difficulties, we generalize the low-dimensional subspace property--previously established for sum-power constraints--to the per-cluster power constraint case. We prove that all stationary-point beamformers reside in a reduced subspace spanned by the channel vectors of the corresponding antenna cluster. Leveraging this property, we reformulate the original high-dimensional constrained problem into an unconstrained optimization task over a product of ellipsoidal manifolds, thereby achieving significant dimensionality reduction. We systematically derive the necessary Riemannian geometric structures for this specific manifold, including the tangent space, Riemannian metric, orthogonal projection, retraction, and vector transport. Subsequently, we develop a tailored Riemannian conjugate gradient algorithm to solve the reformulated problem. Numerical simulations demonstrate that the proposed algorithm achieves the same local optima as standard benchmarks, such as the weighted minimum mean square error (WMMSE) method and conventional manifold optimization, but with substantially higher computational efficiency and scalability, particularly as the number of antenna clusters increases.
Paper Structure (24 sections, 4 theorems, 99 equations, 8 figures, 1 algorithm)

This paper contains 24 sections, 4 theorems, 99 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Conversion to Manifold Problems
  4. Spherical Manifold
  5. Ellipsoidal Manifold
  6. Manifold Optimization Algorithm
  7. Preliminaries
  8. Remannian Gradient
  9. Retraction
  10. Product Manifold
  11. Reduced Riemannian Conjugate Gradient Method
  12. Geometric Elements
  13. Riemannian Gradient
  14. Conjugate Search Direction and Line Search
  15. Convergence Analysis
  16. ...and 9 more sections

Key Result

Proposition 1

For any stationary solution $\bm V^*$ to problem eq:original, the power constraint must be tight, i.e.,

Figures (8)

  • Figure 1: Illustration of the downlink channel model with distributed antenna clusters. The BS has $N_t$ transmit antennas divided into $C$ clusters, each with $L$ antennas and power constraint $P_c$. User $k$ with $N_r$ receive antennas is shown as an example.
  • Figure 2: Illustration of the feasible set in the original and low-dimensional reformulated problems. On the left-hand side, the feasible set forms a product of high-dimensional spherical manifolds in $\mathbb{C}^{N_t\times Kd_k}$. On the right-hand side, by leveraging the low-dimensional subspace property, the feasible set forms a product of high-dimensional ellipsoidal manifolds in $\mathbb{C}^{CKN_r\times Kd_k}$.
  • Figure 3: Illustrations of the Riemannian conjugate gradient method.
  • Figure 4: Network topology with hexagonal cell for $C=4$ clusters and $C=8$ clusters.
  • Figure 5: Cluster performance with $L=128, K=6, N_r=4, d_k=4$. The two figures in the top row show the convergence of WSR in iterations, and the two figures in the bottom row show the convergence in elapsed time.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4