Matrix Recovery from Rank-One Projection Measurements via Nonconvex Minimization

06/28/2018

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by   Peng Li, et al.

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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 _XX_S_p^p for 0<p<1 under affine constraint, and stable recovery of low-rank matrix under ℓ_q constraint and Dantzig selector constraint. Our condition is also sufficient to guarantee low-rank matrix recovery via least q minimization _XA(X)-b_q^q for 0<q≤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.