Robust principal component analysis via joint ℓ2,1-norms minimization

Principal Component Analysis (PCA) is the most widely used unsupervised subspace learning method, and lots of its variants have been developed. With so many proposed PCA-like methods, it is still not clear that which features are better or worse for principal components, especially when the data suffers from outliers. To this end, we propose Robust Principal Component Analysis via joint ℓ2,1-norms minimization, which provides new insights into two crucial issues of PCA: feature selection and robustness to outliers. Unlike other PCA-like methods, the proposed method is able to select effective features for reconstruction by using the ℓ2,1-norm regularization term. More specific, we first use a ℓ2,1-norm based transformation matrix to select effective features that can effectively characterize key components (e.g., the eyes and the nose in a face image), and then use an orthogonal transformation matrix to recover the original data from the selected data representation. In this way, the key components can be well recovered by using the effective features selected by a learned transformation matrix. On the other hand, we also impose ℓ2,1-norm on a loss term to select clean samples to recover its same class samples but with outliers. A simple yet effective optimization algorithm is proposed to solve the resulting optimization problem. Experiments on six datasets demonstrate the effectiveness of the proposed method.