Fast and Provable Nonconvex Robust Matrix Completion

This paper studies the robust matrix completion problem and a computationally efficient non-convex method called ARMC has been proposed. This method is developed by introducing subspace projection to a singular value thresholding based method when updating the low rank part. Numerical experiments on synthetic and real data show that ARMC is superior to existing non-convex RMC methods. Through a refined analysis based on the leave-one-out technique, we have established the theoretical guarantee for ARMC subject to both sparse outliers and stochastic noise. The established bounds for the sample complexity and outlier sparsity are better than those established for a convex approach that also considers both outliers and stochastic noise.