Robust principal component analysis via optimal mean by joint ℓ_2,1 and Schatten p-norms minimization

Since principal component analysis (PCA) is sensitive to corrupted variables or observations that affect its performance and applicability in real scenarios, some convex robust PCA methods have been developed to enhance the robustness of PCA. However, most of them neglect the optimal mean calculation problem. They center the data with the mean calculated by the ℓ₂-norm, which is incorrect because the ℓ₁-norm objective function is used in the following steps. In this paper, we consider a novel robust PCA method that can pursue and remove outliers, exactly recover a low-rank matrix and calculate the optimal mean. Specifically, we propose an optimization model constituted by a ℓ_2,1-norm based loss function and a Schatten p-norm regularization term. The ℓ_2,1-norm used in loss function aims to pursue and remove outliers, the Schatten p-norm can suppress the singular values of reconstructed data at smaller p (0 < p < 1), so it is a better approximation to the rank than the trace norm. Experimental results on benchmark databases demonstrate the effectiveness of our proposed algorithm.