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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 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.
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