Robust Low-Rank Matrix Completion by Riemannian Optimization

Low-rank matrix completion is the problem where one tries to recover a low-rank matrix from noisy observations of a subset of its entries. In this paper, we propose RMC, a new method to deal with the problem of robust low-rank matrix completion, i.e., matrix completion where a fraction of the observed entries are corrupted by non-Gaussian noise, typically outliers. The method relies on the idea of smoothing the $\ell_1$ norm and using Riemannian optimization to deal with the low-rank constraint. We first state the algorithm as the successive minimization of smooth approximations of the $\ell_1$ norm, and we analyze its convergence by showing the strict decrease of the objective function. We then perform numerical experiments on synthetic data and demonstrate the effectiveness on the proposed method on the Netflix dataset.