  We study the problem of robust matrix completion (RMC), where the partially
observed entries of an underlying low-rank matrix is corrupted by sparse noise.
Existing analysis of the non-convex methods for this problem either requires
the explicit but empirically redundant regularization in the algorithm or
requires sample splitting in the analysis. In this paper, we consider a simple
yet efficient nonconvex method which alternates between a projected gradient
step for the low-rank part and a thresholding step for the sparse noise part.
Inspired by leave-one out analysis for low rank matrix completion, it is
established that the method can achieve linear convergence for a general class
of thresholding functions, including for example soft-thresholding and SCAD. To
the best of our knowledge, this is the first leave-one-out analysis on a
nonconvex method for RMC. Additionally, when applying our result to low rank
matrix completion, it improves the sampling complexity of existing result for
the singular value projection method.
