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 $O( \mu r² \kappa² n \max(\mu, \log n))$ random observations of a $n1 \times n2$ $\mu$-incoherent matrix of rank $r$ and condition number $\kappa$, where $n = \max(n1, n2)$, the algorithm linearly converges to the global optimum with high probability.