Exploiting mathematical properties of wireless policies in deep neural network (DNN) design can improve learning performance and generalizability while reducing training complexity. Permutation equivariance and permutation invariance have been incorporated into DNN architectures. In this paper, we investigate conjugation equivariance (CE) and propose a general conjugation-equivariant neural network (CENN) framework for precoder learning. We first establish that the optimal policies for a unified class of precoding problems satisfy CE, i.e., when the channel matrices are conjugated, the conjugate of an optimal precoder remains optimal. We then show that, for DNNs with linear processing functions, enforcing CE restricts their ability to learn optimal precoding policies. To overcome this limitation, we develop a general nonlinear construction and prove that it converts an arbitrary base function into a CE processing function while preserving the base function's original equivariance properties. This construction enables existing equivariant networks to incorporate CE without adding learnable parameters. Simulations for fully digital and RIS-aided precoding show that the resulting CE-enhanced networks improve learning and generalization performance while requiring fewer training samples and shorter training time than their original counterparts.

This paper studies joint uplink (UL) and downlink (DL) resource optimization for interactive augmented reality (AR) services, where the live video captured by an AR device is uploaded to the network edge, and then the augmented video is subsequently downloaded. By modeling the AR transmission process as a tandem queuing system, we derive an upper bound for the probabilistic quality of service (QoS) requirement concerning end-to-end latency and reliability. The derived bound transforms the probabilistic QoS requirement into a tractable service-time condition that jointly characterizes the UL and DL service processes. Based on this condition, we formulate a weighted UL-DL transmit-power minimization problem and propose a learning-based framework to jointly optimize UL power allocation and DL beamforming. To enable gradient-based training, we further derive a differentiable upper bound for the service-time condition. Moreover, we design GNN-based policies for UL power allocation and DL beamforming, where the UL GNN exploits permutation equivariance (PE) and the DL GNN incorporates both PE and the optimal structure of wideband DL beamforming. Simulation results show that the proposed method satisfies the AR reliability requirement and reduces the weighted transmit power compared with baselines that optimize UL and DL resources separately.

This paper jointly optimizes the base-station (BS) continuous aperture array (CAPA) dimensions and beamforming functions to maximize energy efficiency (EE) of the downlink multiuser multi-CAPA system, where both the BS and the users are equipped with CAPAs. Since the beamforming functions are continuous current distribution over the BS CAPA, the resulting EE maximization problem is a nontrivial functional optimization problem that couples aperture sizing and beamforming design. To address this challenge, we propose a cascaded network architecture consisting of a graph neural network (GNN) and a functional-gradient based implicit neural representation (FGB-INR) to learn the BS CAPA dimensions and beamforming functions, respectively. Both networks exploit the permutation equivariance of the optimal optimization policy, and the update equations of FGB-INR are designed according to the functional-gradient structure of the EE objective. Simulation results show that the proposed method approaches the EE of the numerical method while substantially reducing inference latency. They also demonstrates that the functional-gradient structure in FGB-INR improves EE while reducing sample complexity and training time.

This paper studies the optimization of beamforming functions for multiuser multi-continuous aperture array (CAPA) systems, where both the base station and the users are equipped with CAPAs. We first derive a closed-form expression for the achievable sum rate, and then develop a functional weighted minimum mean-squared error (WMMSE) algorithm, which transforms the functional optimization problem into an equivalent parameter optimization problem by employing orthonormal basis expansion. Based on the functional WMMSE algorithm, we further propose BeamINR, an implicit neural representation (INR) method for learning continuous beamforming functions. BeamINR is designed as a graph neural network to exploit the permutation equivariance of the optimal beamforming policy, with an update equation designed according to the functional WMMSE iterations. Simulation results show that both the functional WMMSE algorithm and BeamINR outperform existing numerical and INR-based baselines. BeamINR approaches the sum rate of the functional WMMSE with substantially lower inference latency. Compared with INR-based baselines, BeamINR reduces training complexity and improves generalization to the number of users, CAPA sizes, and carrier~frequencies.