A convergent Gradient descent algorithm for rank minimization and semidefinite programming from random linear measurements

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(r3K2n log n) random measurements of a positive semidefinite n x n matrix of rank r and condition number K, our method is guaranteed to converge linearly to the global optimum.