Robust Independent Component Analysis via Minimum $\gamma $-Divergence Estimation

Independent component analysis (ICA) has been shown to be useful in many applications. However, most ICA methods are sensitive to data contamination. In this article we introduce a general minimum U-divergence framework for ICA, which covers some standard ICA methods as special cases. Within the U-family we further focus on the γ-divergence due to its desirable property of super robustness for outliers, which gives the proposed method γ-ICA. Statistical properties and technical conditions for recovery consistency of γ-ICA are studied. In the limiting case, it improves the recovery condition of MLE-ICA known in the literature by giving necessary and sufficient condition. Since the parameter of interest in γ-ICA is an orthogonal matrix, a geometrical algorithm based on gradient flows on special orthogonal group is introduced. Furthermore, a data-driven selection for the γ value, which is critical to the achievement of γ-ICA, is developed. The performance, especially the robustness, of γ-ICA is demonstrated through experimental studies using simulated data and image data.