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Two-dimensional (2-D) image-matrix-based projection methods for feature extraction are widely used in many fields of computer vision and pattern recognition. In this paper, we propose a novel framework called sparse 2-D projections (S2DP) for image feature extraction. Different from the existing 2-D feature extraction methods, S2DP iteratively learns the sparse projection matrix by using elastic net regression and singular value decomposition. Theoretical analysis shows that the optimal sparse subspace approximates the eigensubspace obtained by solving the corresponding generalized eigenequation. With the S2DP framework, many 2-D projection methods can be easily extended to sparse cases. Moreover, when each row/column of the image matrix is regarded as an independent high-dimensional vector (1-D vector), it is proven that the vector-based eigensubspace is also approximated by the sparse subspace obtained by the same method used in this paper. Theoretical analysis shows that, when compared with the vector-based sparse projection learning methods, S2DP greatly saves both computation and memory costs. This property makes S2DP more tractable for real-world applications. Experiments on well-known face databases indicate the competitive performance of the proposed S2DP over some 2-D projection methods when facial expressions, lighting conditions, and time vary.