Recent researches on data-driven and low-sparsity data injection attacks have been presented, respectively. To combine the two main goals (data-driven and low-sparsity) of research, this paper presents a data-driven and low-sparsity false data injection attack strategy. The proposed attacking strategy (EID: Eliminate-Infer-Determine) is divided into three stages. In the first step, the intercepted data is preprocessed by sparse optimization techniques to eliminate the outliers. The recovered data is then exploited to learn about the system matrix based on the parallel factorization algorithm in the second step. In the third step, the approximated system matrix is applied for the design of sparse attack vector based on the convex optimization. The simulation results show that the EID attack strategy achieves a better performance than the improved ICA-based attack strategy in constructing perfect sparse attack vectors. What is more, data-driven implementation of the proposed strategy is also presented which ensures attack performance even without the prior information of the system.