An optimal condition of robust low-rank matrices recovery

In this paper we investigate the reconstruction conditions of nuclear norm minimization for low-rank matrix recovery. We obtain sufficient $\delta_{tr}<t/(4-t)$ with $0<t<4/3$ to guarantee robust $(z\neq0)$ or exact $(z=0)$ all rank $r$ matrices $X\in\mathbb{R}^{m\times n}$ from $b=\mathcal{A}(X)+z$ via minimization. Furthermore, not only show that when $t=1$, upper bound $\delta_r<1/3$ is same as result Cai and Zhang \cite{Cai Zhang}, but also demonstrate gained bounds concerning recovery error are better. Moreover, prove restricted isometry property condition sharp. Besides, numerical experiments conducted reveal method stable matrix.