Precoder Learning by Leveraging Unitary Equivariance Property

Incorporating mathematical properties of a wireless policy to be learned into the design of deep neural networks (DNNs) can reduce their hypothesis space, thereby improving learning efficiency. Multi-user precoding policies in multi-antenna systems possess 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, which has the potential to further reduce the DNN hypothesis space. We first demonstrate that unitary equivariance cannot be exploited in the same manner as permutation equivariance, i.e., solely through parameter sharing in the weight matrix, which prevents the learning of the optimal precoder. Recognizing this limitation, we develop a novel non-linear processing function for DNN layers that satisfies unitary equivariance, based on which we construct a joint unitary and permutation equivariant DNN architecture. Simulation results show that the proposed DNN not only outperforms existing learning methods in learning performance and generalizability but also reduces training complexity.