  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.
