In this paper, a non-convex optimization model of the low-rank and sparse matrix recovery from partial samples is proposed, and an inexact Newton-like algorithm with non-monotone scheme for solving the non-convex optimization model is suggested, in which the inexact Newton-like algorithm is used for the low-rank matrix part and the non-monotone scheme is used for sparse matrix part. It is proved that the iterative sequence of the new algorithm converges to a stationary point of the model. Finally, numerical experiments show that the proposed algorithm is far superior to the missing low-rank sparse decomposition (MLSD) algorithm in Azghani et al., 2019 and Algorithm 1 in Gu et al., 2016 in CPU time, which indicates that the new algorithm is more efficient.

本文提出了一种从局部样本中恢复低秩和稀疏矩阵的非凸优化模型，并提出了一种具有非单调方案的不精确类牛顿算法来求解非凸优化模型。对于低秩矩阵部分，使用非精确类牛顿算法，对于稀疏矩阵部分，使用非单调方案。证明了新算法的迭代序列收敛到模型的平稳点。最后，数值实验表明，该算法在CPU时间方面远远优于Azghani等人（2019年）和Gu等人（2016年）中缺少的低秩稀疏分解（MLSD）算法，这表明新算法算法效率更高。