  There is a recent surge of interest in nonconvex reformulations via low-rank
factorization for stochastic convex semidefinite optimization problem in the
purpose of efficiency and scalability. Compared with the original convex
formulations, the nonconvex ones typically involve much fewer variables,
allowing them to scale to scenarios with millions of variables. However, it
opens a new challenge that under what conditions the nonconvex stochastic
algorithms may find the global optima effectively despite their empirical
success in applications. In this paper, we provide an answer that a stochastic
gradient descent method with variance reduction, can be adapted to solve the
nonconvex reformulation of the original convex problem, with a \textit{global
linear convergence}, i.e., converging to a global optimum exponentially fast,
at a proper initial choice in the restricted strongly convex case. Experimental
studies on both simulation and real-world applications on ordinal embedding are
provided to show the effectiveness of the proposed algorithms.
