In recent years, algorithms to recovery low-rank matrix have become one of the research hotspots, and more corresponding optimization models with nuclear norm have also been proposed. However, nuclear norm is not a good approximation to the rank function. This paper proposes a matrix completion model and a low-rank sparse decomposition model based on truncated Schatten p-norm, respectively, which combine Schatten p-norm with truncated nuclear norm, so that the models are more flexible. To solve these models, the function expansion method is first used to transform the non-convex optimization models into the convex optimization ones. Then, the two-step iterative algorithm based on alternating direction multiplier method (ADMM) is employed to solve the models. Further, the convergence of the proposed algorithm is proved mathematically. The superiority of the proposed method is further verified by comparing the existing methods in synthetic data and actual images.

近年来，恢复低秩矩阵的算法已成为研究的热点之一，并且已经提出了更多与核范数相对应的优化模型。但是，核范数并不是秩函数的良好近似。本文提出了一种基于截断的Schatten p-范数的矩阵完成模型和低秩稀疏分解模型，它们结合了Schatten p-标准与截断的核标准，因此模型更加灵活。为了解决这些模型，首先使用函数扩展方法将非凸优化模型转换为凸优化模型。然后，采用基于交替方向乘子法（ADMM）的两步迭代算法求解模型。此外，该算法的收敛性在数学上得到了证明。通过将现有方法在合成数据和实际图像中进行比较，进一步验证了该方法的优越性。