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We propose a simple, scalable, and fast gradient descent algorithm tooptimize a nonconvex objective for the rank minimization problem and a closelyrelated family of semidefinite programs. With $O(r^3 \kappa^2 n \log n)$ randommeasurements of a positive semidefinite $n \times n$ matrix of rank $r$ andcondition number $\kappa$, our method is guaranteed to converge linearly to theglobal optimum.

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