Incorporating mathematical properties of a wireless policy to be learned into the design of deep neural networks (DNNs) is effective for enhancing learning efficiency. Multi-user precoding policy in multi-antenna system, which is the mapping from channel matrix to precoding matrix, possesses a permutation equivariance property, which has been harnessed to design the parameter sharing structure of the weight matrix of DNNs. In this paper, we study a stronger property than permutation equivariance, namely unitary equivariance, for precoder learning. We first show that a DNN with unitary equivariance designed by further introducing parameter sharing into a permutation equivariant DNN is unable to learn the optimal precoder. We proceed to develop a novel non-linear weighting process satisfying unitary equivariance and then construct a joint unitary and permutation equivariant DNN. Simulation results demonstrate that the proposed DNN not only outperforms existing learning methods in learning performance and generalizability but also reduces training complexity.

Graph neural networks (GNNs) update the hidden representations of vertices (called Vertex-GNNs) or hidden representations of edges (called Edge-GNNs) by processing and pooling the information of neighboring vertices and edges and combining to exploit topology information. When learning resource allocation policies, GNNs cannot perform well if their expressive power is weak, i.e., if they cannot differentiate all input features such as channel matrices. In this paper, we analyze the expressive power of the Vertex-GNNs and Edge-GNNs for learning three representative wireless policies: link scheduling, power control, and precoding policies. We find that the expressive power of the GNNs depends on the linearity and output dimensions of the processing and combination functions. When linear processors are used, the Vertex-GNNs cannot differentiate all channel matrices due to the loss of channel information, while the Edge-GNNs can. When learning the precoding policy, even the Vertex-GNNs with non-linear processors may not be with strong expressive ability due to the dimension compression. We proceed to provide necessary conditions for the GNNs to well learn the precoding policy. Simulation results validate the analyses and show that the Edge-GNNs can achieve the same performance as the Vertex-GNNs with much lower training and inference time. Optimizing resource allocation such as link scheduling, power control, and precoding is important for improving the spectral efficiency of wireless communications.

Cooperative beamforming design has been recognized as an effective approach in modern wireless networks to meet the dramatically increasing demand of various wireless data traffics. It is formulated as an optimization problem in conventional approaches and solved iteratively in an instance-by-instance manner. Recently, learning-based methods have emerged with real-time implementation by approximating the mapping function from the problem instances to the corresponding solutions. Among various neural network architectures, graph neural networks (GNNs) can effectively utilize the graph topology in wireless networks to achieve better generalization ability on unseen problem sizes. However, the current GNNs are only equipped with the node-update mechanism, which restricts it from modeling more complicated problems such as the cooperative beamforming design, where the beamformers are on the graph edges of wireless networks. To fill this gap, we propose an edge-graph-neural-network (Edge-GNN) by incorporating an edge-update mechanism into the GNN, which learns the cooperative beamforming on the graph edges. Simulation results show that the proposed Edge-GNN achieves higher sum rate with much shorter computation time than state-of-the-art approaches, and generalizes well to different numbers of base stations and user equipments.