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.
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