Precoder Learning by Leveraging Unitary Equivariance Property

TL;DR

This work investigates learning MU-MIMO precoders under a stronger inductive bias called joint unitary and permutation equivariance (UE-PE). It proves that linear weight-sharing designs derived from UE-PE cannot learn the optimal precoder and introduces a non-linear weighting mechanism that satisfies UE-PE, forming the UPNN. By deriving a weight structure $\mathbf{W}=\mathbf{\Omega}\otimes\mathbf{I}_N$ and then a nonlinear generalization with $\mathcal{G}(\mathbf{D})$, the authors demonstrate improved learning capability, generalization, and training efficiency over existing PE-based DNNs such as PENN and Edge-GNN. Empirical results show that UPNN outperforms baselines in sum-rate performance, requires fewer training samples, and generalizes better to unseen numbers of users and antennas, indicating strong practical benefits for efficient precoder design.

Abstract

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.

Figures (4)

• Figure 1: Parameter sharing structures for weight matrices of PENN, Edge-GNN, and that satisfying UE-PE.
• Figure 2: Learning performance vs the number of training samples with $N=8$ and $K=4$.
• Figure 3: Generalization performance to the number of users with $N=16$ and 500 training samples.
• Figure 4: Generalization performance to the number of antennas with $K=4$ and 500 training samples.