Principal component analysis (PCA) is one of the most versatile techniques for unsupervised dimension reduction, which is implemented as a fundamental preprocessing method in multiple tasks of statistics and machine learning research because of its efficiency. Nevertheless, researchers have concentrated on the identification of outliers that do not conform to the low-dimensional approximation through statistical methods, e.g., outlier rejection, without giving insights on each data point with a dynamic ratio of signal-to-noise components in the high-dimensional regimes. To characterize the dynamic nature of the principal component information, we propose a Normalized Robust PCA with Adaptive Reconstruction Error minimization model, which considers both the adaptive normalization technique and flexible weights learning simultaneously. With this configuration, the principal component information constantly adjusts the degree of sparsity for activated samples. In other words, the signal component's discrimination and noise information restriction could work cooperatively. Empirical studies on one synthetic dataset and several benchmarks demonstrate the effectiveness of our proposed method over existing outlier rejection methods.

Robust principal component analysis (RPCA) is a technique that aims to make principal component analysis (PCA) robust to noise samples. The current modeling approaches of RPCA were proposed by analyzing the prior distribution of the reconstruction error terms. However, these methods ignore the influence of samples with large reconstruction errors, as well as the valid information of these samples in principal component space, which will degrade the ability of PCA to extract the principal component of data. In order to solve this problem, Fuzzy sparse deviation regularized robust principal component Analysis (FSD-PCA) is proposed in this paper. First, FSD-PCA learns the principal components by minimizing the square of ℓ2-norm-based reconstruction error. Then, FSD-PCA introduces sparse deviation on reconstruction error term to relax the samples with large bias, thus FSD-PCA can process noise and principal components of samples separately as well as improve the ability of FSD-PCA for retaining the principal component information. Finally, FSD-PCA estimates the prior probability of each sample by fuzzy weighting based on the relaxed reconstruction error, which can improve the robustness of the model. The experimental results indicate that the proposed model performs excellent robustness against different types of noise than the state-of-art algorithms, and the sparse deviation term enables FSD-PCA to process noise information and principal component information separately, so FSD-PCA can filter the noise information of an image and restore the corrupted image.