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

Abstract

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=〈Ai,X〉 with rank(Ai)=1, i=1,...,m. Historical iterative hard thresholding sequence for low-rank matrix recovery and rank-one measurements was taken as Xn+1=Ps(Xn−μnPt(A⁎sign(A(Xn)−y))), which introduced the “tail” and “head” approximations Ps and Pt, respectively. In this paper, we remove the term Pt 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 ℓ1/ℓ2-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.