Adaptive iterative hard thresholding for low-rank matrix recovery and rank-one measurements.

In low-rank matrix recovery, many kinds of measurements fail to meet the standard restricted isometry property (RIP), such as rank-one measurements, that is, [ A (X) ] i = 〈 A i , X 〉 with rank (A i) = 1 , i = 1 ,... , m. Historical iterative hard thresholding sequence for low-rank matrix recovery and rank-one measurements was taken as X n + 1 = P s (X n − μ n P t (A ⁎ sign (A (X n) − y))) , which introduced the "tail" and "head" approximations P s and P t , respectively. In this paper, we remove the term P t and provide a new iterative hard thresholding algorithm with adaptive step size (abbreviated as AIHT). The linear convergence analysis and stability results on AIHT are established under the ℓ 1 / ℓ 2 -RIP. Particularly, we discuss the rank-one Gaussian measurements under the tight upper and lower bounds on E ‖ A (X) ‖ 1 , and provide better convergence rate and sampling complexity. Besides, several empirical experiments are provided to show that AIHT performs better than the historical rank-one iterative hard thresholding method. [ABSTRACT FROM AUTHOR]