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Learning an efficient projection to map high-dimensional data into a lower dimensional space is a rather challenging task in the community of pattern recognition and computer vision. Manifold learning is widely applied because it can disclose the intrinsic geometric structure of data. However, it only concerns the geometric structure and may lose its effectiveness in case of corrupted data. To address this challenge, we propose a novel dimensionality reduction method by combining the manifold learning and low-rank sparse representation, termed low-rank sparse preserving projections (LSPP), which can simultaneously preserve the intrinsic geometric structure and learn a robust representation to reduce the negative effects of corruptions. Therefore, LSPP is advantageous to extract robust features. Because the formulated LSPP problem has no closed-form solution, we use the linearized alternating direction method with adaptive penalty and eigen-decomposition to obtain the optimal projection. The convergence of LSPP is proven, and we also analyze its complexity. To validate the effectiveness and robustness of LSPP in feature extraction and dimensionality reduction, we make a critical comparison between LSPP and a series of related dimensionality reduction methods. The experimental results demonstrate the effectiveness of LSPP.

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