  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.
