Learning Robust and Sparse Principal Components with the α-Divergence

In this paper, novel robust principal component analysis (RPCA) methods are proposed to exploit the local structure of datasets. The proposed methods are derived by minimizing the α-divergence between the sample distribution and the Gaussian density model. The α-divergence is used in different frameworks to represent variants of RPCA approaches including orthogonal, non-orthogonal, and sparse methods. We show that the classical PCA is a special case of our proposed methods where the α-divergence is reduced to the Kullback-Leibler (KL) divergence. It is shown in simulations that the proposed approaches recover the underlying principal components (PCs) by down-weighting the importance of structured and unstructured outliers. Furthermore, using simulated data, it is shown that the proposed methods can be applied to fMRI signal recovery and Foreground-Background (FB) separation in video analysis. Results on real world problems of FB separation as well as image reconstruction are also provided.