The first part of the paper proves the conjectures on inequalities in the Schatten p-quasi-norm of matrices. The second part of the paper uses the inequalities for proving a sufficient condition when the Schatten p-quasi-norm minimization can be used for low rank matrix recovery. More precisely, when the restricted isometry constant ${\delta }_{2s}<1$, there exists a real number $p_0<1$ such that any solution of the p minimization is the minimal rank solution for any $p\le p_0$. In addition, in the noisy setting, the estimate of the difference $\Vert X-X_0\Vert$ is also given which is useful for applications.

中文翻译：

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关于用于低秩矩阵恢复的Schatten p-准范数最小化

本文的第一部分证明了矩阵的Schatten p-拟范数不等式的猜想。当Schatten p-准范数极小化可用于低秩矩阵恢复时，本文的第二部分使用不等式证明了充分条件。更确切地说，当受约束的等轴测常数时${\delta }_{2s}<\mathrm{1个}$，存在一个实数 $p_0<\mathrm{1个}$使得p最小化的任何解都是任何$p\le p_0$。此外，在嘈杂的环境中，差异的估算$\Vert X-X_0\Vert$ 还给出了对应用程序有用的信息。