论文标题
通过非线性方程系统解决强大的矩阵完成问题
Solving the Robust Matrix Completion Problem via a System of Nonlinear Equations
论文作者
论文摘要
我们考虑了强大的矩阵完成的问题，该问题旨在恢复低等级矩阵$ l _*$和稀疏矩阵$ s _*$，从其总和$ m = l _*+s _*\ in \ mathbb {r}^r}^{m}^{m \ \ times n} $中。从算法上讲，可靠的矩阵完成问题被转化为解决非线性方程系统的问题，然后使用替代方向方法求解非线性方程。另外，该算法是高度可行的，适合大规模问题。从理论上讲，我们表征了$ l _*$可以通过观察到的$ m _*$的低等级近似来近似的足够条件。在适当的假设下，证明算法将线性收敛到真实解。数值模拟表明，该简单方法可以按预期工作，并且与最新方法相媲美。
We consider the problem of robust matrix completion, which aims to recover a low rank matrix $L_*$ and a sparse matrix $S_*$ from incomplete observations of their sum $M=L_*+S_*\in\mathbb{R}^{m\times n}$. Algorithmically, the robust matrix completion problem is transformed into a problem of solving a system of nonlinear equations, and the alternative direction method is then used to solve the nonlinear equations. In addition, the algorithm is highly parallelizable and suitable for large scale problems. Theoretically, we characterize the sufficient conditions for when $L_*$ can be approximated by a low rank approximation of the observed $M_*$. And under proper assumptions, it is shown that the algorithm converges to the true solution linearly. Numerical simulations show that the simple method works as expected and is comparable with state-of-the-art methods.
