求解低秩矩阵恢复问题的非单调交替梯度方向法

低秩矩阵恢复问题作为一类在图像处理和信号数据分析等领域中都十分重要的问题已被广泛研究.本文在交替方向算法的框架下,应用非单调技术,提出一种求解低秩矩阵恢复问题的新算法.该算法在每一步迭代过程中,首先利用一步带有变步长梯度算法同时更新低秩部分的两块变量,然后采用非单调技术更新稀疏部分的变量.在一定的假设条件下,本文证明了新算法的全局收敛性.最后通过解决随机低秩矩阵恢复问题和视频前景背景分离的实例验证新算法的有效性,同时也显示非单调技术极大改善了算法的效率.

A non-monotone alternating directional method for matrix recovery problem

The low-rank matrix recovery problem, where the concerned matrix is separable into a low-rank part and a sparse part, has been frequently exploited in the fields of image processing and signal data analysis. In this paper, we propose an alternating directional method equipped with the non-monotone search strategy for solving the low-rank matrix recovery problem, where we apply a single step of the steepest gradient descent method to update variables associated with the low-rank part in parallel and the non-monotone search strategy to update the sparse structure matrix. Theoretically, we prove the global convergence of the proposed algorithm under some mild conditions. The efficiency and effectiveness of the proposed algorithm and superiority of the non-monotone search strategy for improving the performance of algorithms are demonstrated by solving some instances of random matrix recovery problems and background/foreground extraction problems.