Robust matrix factorization (RMF), which uses the $\ell_1$-loss, often outperforms standard matrix factorization using the $\ell_2$-loss, particularly when outliers are present. The state-of-the-art RMF solver is the RMF-MM algorithm, which, however, cannot utilize data sparsity. Moreover, sometimes even the (convex) $\ell_1$-loss is not robust enough. In this paper, we propose the use of nonconvex loss to enhance robustness. To address the resultant difficult optimization problem, we use majorization-minimization (MM) optimization and propose a new MM surrogate. To improve scalability, we exploit data sparsity and optimize the surrogate via its dual with the accelerated proximal gradient algorithm. The resultant algorithm has low time and space complexities and is guaranteed to converge to a critical point. Extensive experiments demonstrate its superiority over the state-of-the-art in terms of both accuracy and scalability.

In this paper, an algorithm for matrix factorization (MF) is proposed, which provide better robustness to outlier's behavior compared to state of the art algorithms for Robust MF (RMF). Another nice property of the new algorithm is that it is suitable for sparse data matrices which is a clear advantage when it is compared against previous approaches to RMF. Basically, the authors propose to replace the l1 penalty with a nonconvex regularizer and provide results for various selections of regularizers. Since the optimization in this case is not simple, the authors propose to use a Majorization Minimization (MM) optimization approach with a definition of a new MM surrogate. The paper provides a nice introduction to the problem of robust matrix factorization, providing a complete overview of the state of the art and giving good motivations for the development of a new algorithm. I found the paper well written, technically sounded and having a nice structure.