Low-Rank Matrix Estimation from Rank-One Projections by Unlifted Convex Optimization

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