  We address the rectangular matrix completion problem by lifting the unknown
matrix to a positive semidefinite matrix in higher dimension, and optimizing a
nonconvex objective over the semidefinite factor using a simple gradient
descent scheme. With $O( \mu r^2 \kappa^2 n \max(\mu, \log n))$ random
observations of a $n_1 \times n_2$ $\mu$-incoherent matrix of rank $r$ and
condition number $\kappa$, where $n = \max(n_1, n_2)$, the algorithm linearly
converges to the global optimum with high probability.
