Adaptive Weighted Sparse Principal Component Analysis

In this paper, we propose an unsupervised feature selection method from the perspective of optimal reconstruction. The features selected by the proposed method can well represent the original data, and the effectiveness of the selected features is demonstrated by robust reconstruction and clustering. The proposed method emphasizes the joint $\ell_{2, 1}$-norms minimization on both reconstruction term and regularization term to make them be column-sparse. Relying on the column-sparse property of reconstruction term and regularization term, the proposed method is able to improve the robustness to outliers and select the effective features. The proposed objective function is nonconvex. Fortunately, it can be equivalently reformulated as a convex form (with change of variables) to capture a global optimization solution. In fact, the proposed method is related to the optimal mean robust principal component analysis (OMRPCA) because the proposed method is a sparse self-contained regression type of OMRPCA. Since OMRPCA essentially adds the adaptive weights for data samples, we call the proposed method adaptive weighted sparse principal component analysis (AW-SPCA). Experimental results demonstrate the effectiveness of AW-SPCA.