A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements

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 $O(r^3\hat{I}{º}^{2}n\log n)$ random measurements of a positive semidefinite $n\times n$ matrix of rank $r$ and condition number $\hat{I}º$, our method is guaranteed to converge linearly to the global optimum.

Fix a minor error in Appendix E