Matrix Recovery from Rank-One Projection Measurements via Nonconvex Minimization

Abstract

In this paper, we consider the matrix recovery from rank-one projection measurements proposed in [Cai and Zhang, Ann. Statist., 43(2015), 102-138], via nonconvex minimization. We establish a sufficient identifiability condition, which can guarantee the exact recovery of low-rank matrix via Schatten-$p$ minimization ${\min }_X\Vert X{\Vert }_{S_p}^p$ for $0<p<1$ under affine constraint, and stable recovery of low-rank matrix under ${\ell }_q$ constraint and Dantzig selector constraint. Our condition is also sufficient to guarantee low-rank matrix recovery via least $q$ minimization ${\min }_X\Vert \mathcal{A}(X)-b{\Vert }_q^q$ for $0<q\le 1$. And we also extend our result to Gaussian design distribution, and show that any matrix can be stably recovered for rank-one projection from Gaussian distributions via least $1$ minimization with high probability.