[en] We investigate the performance of multi-user multiple-antenna downlink systems in which a base station (BS) serves multiple users via a shared wireless medium. In order to fully exploit the spatial diversity while minimizing the passive energy consumed by radio frequency (RF) components, the BS is equipped with M RF chains and N antennas, where M < N. Upon receiving pilot sequences to obtain the channel state information (CSI), the BS determines the best subset of M antennas for serving the users. We propose a joint antenna selection and precoding design (JASPD) algorithm to maximize the system sum rate subject to a transmit power constraint and quality of service (QoS) requirements. The JASPD overcomes the non-convexity of the formulated problem via a doubly iterative algorithm, in which an inner loop successively optimizes the precoding vectors, followed by an outer loop that tries all valid antenna subsets. Although approaching the (near) global optimality, the JASPD suffers from a combinatorial complexity, which may limit its application in real-time network operations. To overcome this limitation, we propose a learning-based antenna selection and precoding design algorithm (L-ASPA), which employs a deep neural network (DNN) to establish underlaying relations between the key system parameters and the selected antennas. The proposed L-ASPD is robust against the number of users and their locations, BS's transmit power, as well as the small-scale channel fading. With a well-trained learning model, it is shown that the L-ASPD significantly outperforms baseline schemes based on the block diagonalization and a learning-assisted solution for broadcasting systems and achieves higher effective sum rate than that of the JASPA under limited processing time. In addition, we observed that the proposed L-ASPD can reduce the computation complexity by 95% while retaining more than 95% of the optimal performance.