  Matrix factorization is a popular approach for large-scale matrix completion.
The optimization formulation based on matrix factorization can be solved very
efficiently by standard algorithms in practice. However, due to the
non-convexity caused by the factorization model, there is a limited theoretical
understanding of this formulation. In this paper, we establish a theoretical
guarantee for the factorization formulation to correctly recover the underlying
low-rank matrix. In particular, we show that under similar conditions to those
in previous works, many standard optimization algorithms converge to the global
optima of a factorization formulation, and recover the true low-rank matrix. We
study the local geometry of a properly regularized factorization formulation
and prove that any stationary point in a certain local region is globally
optimal. A major difference of our work from the existing results is that we do
not need resampling in either the algorithm or its analysis. Compared to other
works on nonconvex optimization, one extra difficulty lies in analyzing
nonconvex constrained optimization when the constraint (or the corresponding
regularizer) is not "consistent" with the gradient direction. One technical
contribution is the perturbation analysis for non-symmetric matrix
factorization.
