Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent

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.