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Learning Optimal Linear Precoding for Cell-Free Massive MIMO with GNN

Benjamin Parlier, Lou Salaün, Hong Yang

TL;DR

This work tackles the challenge of computing the optimal linear precoder (OLP) for Downlink CFmMIMO within millisecond time scales by introducing OLP-GNN, a graph transformer that predicts the OLP from a channel graph. The method encodes each channel as a node with two edge types (AP- and UE-sharing relationships) and uses a 6-layer, 22.4k-parameter network to output a precoding matrix while enforcing per-AP power constraints through postprocessing. Trained on OLP targets generated by the slower B-SOCP approach, OLP-GNN achieves near-optimal spectral efficiency (SE) across LoS and NLoS scenarios and scales to larger systems, with median SE losses typically under a few percent and 95%-likely losses within ~6–12% depending on environment. The results demonstrate real-time feasibility (under 1–2 ms on GPUs) and reveal strong generalization to varying system sizes and propagation conditions, offering a practical pathway to deploy optimal linear precoding in future 6G CFmMIMO networks.

Abstract

We develop a graph neural network (GNN) to compute, within a time budget of 1 to 2 milliseconds required by practical systems, the optimal linear precoder (OLP) maximizing the minimal downlink user data rate for a Cell-Free Massive MIMO system - a key 6G wireless technology. The state-of-the-art method is a bisection search on second order cone programming feasibility test (B-SOCP) which is a magnitude too slow for practical systems. Our approach relies on representing OLP as a node-level prediction task on a graph. We construct a graph that accurately captures the interdependence relation between access points (APs) and user equipments (UEs), and the permutation equivariance of the Max-Min problem. Our neural network, named OLP-GNN, is trained on data obtained by B-SOCP. We tailor the OLP-GNN size, together with several artful data preprocessing and postprocessing methods to meet the runtime requirement. We show by extensive simulations that it achieves near optimal spectral efficiency in a range of scenarios with different number of APs and UEs, and for both line-of-sight and non-line-of-sight radio propagation environments.

Learning Optimal Linear Precoding for Cell-Free Massive MIMO with GNN

TL;DR

This work tackles the challenge of computing the optimal linear precoder (OLP) for Downlink CFmMIMO within millisecond time scales by introducing OLP-GNN, a graph transformer that predicts the OLP from a channel graph. The method encodes each channel as a node with two edge types (AP- and UE-sharing relationships) and uses a 6-layer, 22.4k-parameter network to output a precoding matrix while enforcing per-AP power constraints through postprocessing. Trained on OLP targets generated by the slower B-SOCP approach, OLP-GNN achieves near-optimal spectral efficiency (SE) across LoS and NLoS scenarios and scales to larger systems, with median SE losses typically under a few percent and 95%-likely losses within ~6–12% depending on environment. The results demonstrate real-time feasibility (under 1–2 ms on GPUs) and reveal strong generalization to varying system sizes and propagation conditions, offering a practical pathway to deploy optimal linear precoding in future 6G CFmMIMO networks.

Abstract

We develop a graph neural network (GNN) to compute, within a time budget of 1 to 2 milliseconds required by practical systems, the optimal linear precoder (OLP) maximizing the minimal downlink user data rate for a Cell-Free Massive MIMO system - a key 6G wireless technology. The state-of-the-art method is a bisection search on second order cone programming feasibility test (B-SOCP) which is a magnitude too slow for practical systems. Our approach relies on representing OLP as a node-level prediction task on a graph. We construct a graph that accurately captures the interdependence relation between access points (APs) and user equipments (UEs), and the permutation equivariance of the Max-Min problem. Our neural network, named OLP-GNN, is trained on data obtained by B-SOCP. We tailor the OLP-GNN size, together with several artful data preprocessing and postprocessing methods to meet the runtime requirement. We show by extensive simulations that it achieves near optimal spectral efficiency in a range of scenarios with different number of APs and UEs, and for both line-of-sight and non-line-of-sight radio propagation environments.
Paper Structure (18 sections, 20 equations, 4 figures, 1 table)

This paper contains 18 sections, 20 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. System Model and Notation
  4. Notation
  5. Cell-Free Massive MIMO
  6. Precoding Matrix and Downlink SINR Calculation
  7. Optimal Linear Precoding
  8. Zero Forcing and Maximum Ratio Precoding
  9. Graph Neural Network
  10. Graph Representation
  11. Data Preprocessing and Postprocessing
  12. Structure of the Neural Network
  13. Training and Loss Function
  14. Numerical Results
  15. Simulation Parameters and Performance Metrics
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Neighbors and outgoing edges of a typical node $\pi(m,k) = i \in V$
  • Figure 2: Structure of OLP-GNN. 'H' refers to the hidden attention layer, and 'L' is the final linear layer. The number between each layer represents the node feature size.
  • Figure 3: Cumulative distribution functions of the downlink SE for MR, ZF, OLP and OLP-GNN for different environments and graph sizes.
  • Figure 4: Number of FLOPs versus the number of edges for B-SOCP and OLP-GNN. Each point represents one rural NLoS scenario.