Efficient l1-norm-based low-rank matrix approximations for large-scale problems using alternating rectified gradient method

Abstract

Low-rank matrix approximation plays an important role in the area of computer vision and image processing. Most of the conventional low-rank matrix approximation methods are based on the l(2)-norm (Frobenius norm) with principal component analysis (PCA) being the most popular among them. However, this can give a poor approximation for data contaminated by outliers (including missing data), because the l(2)-norm exaggerates the negative effect of outliers. Recently, to overcome this problem, various methods based on the l(1)-norm, such as robust PCA methods, have been proposed for low-rank matrix approximation. Despite the robustness of the methods, they require heavy computational effort and substantial memory for high-dimensional data, which is impractical for real-world problems. In this paper, we propose two efficient low-rank factorization methods based on the l(1)-norm that find proper projection and coefficient matrices using the alternating rectified gradient method. The proposed methods are applied to a number of low-rank matrix approximation problems to demonstrate their efficiency and robustness. The experimental results show that our proposals are efficient in both execution time and reconstruction performance unlike other state-of-the-art methods.