Avoiding optimal mean robust PCA/2DPCA with non-greedy ℓ₁-norm maximization

Robust principal component analysis (PCA) is one of the most important dimension reduction techniques to handle high-dimensional data with outliers. However, the existing robust PCA presupposes that the mean of the data is zero and incorrectly utilizes the Euclidean distance based optimal mean for robust PCA with ℓ₁-norm. Some studies consider this issue and integrate the estimation of the optimal mean into the dimension reduction objective, which leads to expensive computation. In this paper, we equivalently reformulate the maximization of variances for robust PCA, such that the optimal projection directions are learned by maximizing the sum of the projected difference between each pair of instances, rather than the difference between each instance and the mean of the data. Based on this reformulation, we propose a novel robust PCA to automatically avoid the calculation of the optimal mean based on ℓ₁-norm distance. This strategy also makes the assumption of centered data unnecessary. Additionally, we intuitively extend the proposed robust PCA to its 2D version for image recognition. Efficient non-greedy algorithms are exploited to solve the proposed robust PCA and 2D robust PCA with fast convergence and low computational complexity. Some experimental results on benchmark data sets demonstrate the effectiveness and superiority of the proposed approaches on image reconstruction and recognition.