Efficient l1-Norm-Based Low-Rank Matrix Approximations for Large-Scale Problems Using Alternating Rectified Gradient Method

Eunwoo Kim, Minsik Lee, Chong-Ho Choi, Nojun Kwak, Songhwai Oh

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm exaggerates the negative effect of outliers. Recently, to overcome this problem, various methods based on the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -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 realworld problems. In this paper, we propose two efficient low-rank factorization methods based on the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -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.