  We study an estimator with a convex formulation for recovery of low-rank
matrices from rank-one projections. Using initial estimates of the factors of
the target $d_1\times d_2$ matrix of rank-$r$, the estimator admits a practical
subgradient method operating in a space of dimension $r(d_1+d_2)$. This
property makes the estimator significantly more scalable than the convex
estimators based on lifting and semidefinite programming. Furthermore, we
present a streamlined analysis for exact recovery under the real Gaussian
measurement model, as well as the partially derandomized measurement model by
using the spherical $t$-design. We show that under both models the estimator
succeeds, with high probability, if the number of measurements exceeds $r^2
(d_1+d_2)$ up to some logarithmic factors. This sample complexity improves on
the existing results for nonconvex iterative algorithms.
