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

Abstract

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 $O(\hat{I}¼r^2\hat{I}{º}^{2}n\max (\hat{I}¼,\log n))$ random observations of a $n_1\times n_2$ $\hat{I}¼$-incoherent matrix of rank $r$ and condition number $\hat{I}º$, where $n=\max (n_1,n_2)$, the algorithm linearly converges to the global optimum with high probability.