This paper proposes a novel over-the-air computation approach for fast global model aggregation in federated learning. The goal is to improve communication efficiency and speed up the federated learning system by leveraging the superposition property of a wireless multiple-access channel. The key challenge is to maximize the number of involved devices while satisfying the mean-square-error (MSE) requirement to reduce model aggregation error. This is achieved through joint device selection and beamforming design, which is modeled as a sparse and low-rank optimization problem. A novel difference-of-convex-functions (DC) approach is developed to enhance sparsity and accurately detect the rank-one constraint. The DC algorithm with global convergence guarantees is further developed to solve the resulting DC program. The algorithmic advantages and performance of the proposed methodologies are demonstrated through extensive numerical results. The main contributions include: (1) a novel fast model aggregation approach for federated learning using over-the-air computation; (2) a sparse and low-rank modeling approach for efficient algorithms design; (3) a unified DC representation approach to induce both sparsity and low-rank structures; and (4) a DC algorithm with established convergence rate for nonconvex DC programs. The proposed DC approach for accurate feasibility detection and device selection outperforms state-of-the-art approaches in terms of prediction accuracy and convergence rate.This paper proposes a novel over-the-air computation approach for fast global model aggregation in federated learning. The goal is to improve communication efficiency and speed up the federated learning system by leveraging the superposition property of a wireless multiple-access channel. The key challenge is to maximize the number of involved devices while satisfying the mean-square-error (MSE) requirement to reduce model aggregation error. This is achieved through joint device selection and beamforming design, which is modeled as a sparse and low-rank optimization problem. A novel difference-of-convex-functions (DC) approach is developed to enhance sparsity and accurately detect the rank-one constraint. The DC algorithm with global convergence guarantees is further developed to solve the resulting DC program. The algorithmic advantages and performance of the proposed methodologies are demonstrated through extensive numerical results. The main contributions include: (1) a novel fast model aggregation approach for federated learning using over-the-air computation; (2) a sparse and low-rank modeling approach for efficient algorithms design; (3) a unified DC representation approach to induce both sparsity and low-rank structures; and (4) a DC algorithm with established convergence rate for nonconvex DC programs. The proposed DC approach for accurate feasibility detection and device selection outperforms state-of-the-art approaches in terms of prediction accuracy and convergence rate.