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(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.