Low-Rank Matrix Estimation From Rank-One Projections by Unlifted Convex Optimization
Abstract
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 d1× d2 matrix of rank-r, the estimator admits a practical subgradient method operating in a space of dimension r(d1+d2). 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 r2 (d1+d2) up to some logarithmic factors. This sample complexity improves on the existing results for nonconvex iterative algorithms.