Abstract

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With $O(r^{3}\kappa^{2}n\log n)$ random measurements of a positive semidefinite $n\times n$ matrix of rank $r$ and condition number $\kappa$ , our method is guaranteed to converge linearly to the global optimum.