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, ${\left[A\right(X\left)\right]}_i=〈A_i,X〉$ with $\text{rank}\left(A_i\right)=1$, $i=1,...,m$. Historical iterative hard thresholding sequence for low-rank matrix recovery and rank-one measurements was taken as $X^{n+1}=P_s(X^n-{\mu }_n{P}_t(A^{⁎}\text{sign}\left(A\right(X^n)-y)\left)\right)$, 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 ${\ell }_1/{\ell }_2$-RIP. Particularly, we discuss the rank-one Gaussian measurements under the tight upper and lower bounds on $E{\Vert A\left(X\right)\Vert }_1$, 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.