The low-rank matrix completion (LRMC) technology has achieved remarkable results in low-level visual tasks. There is an underlying assumption that the real-world matrix data is low-rank in LRMC. However, the real matrix data does not satisfy the strict low-rank property, which undoubtedly present serious challenges for the above-mentioned matrix recovery methods. Fortunately, there are feasible schemes that devise appropriate and effective priori representations for describing the intrinsic information of real data. In this paper, we firstly model the matrix data ${\bf{Y}}$ as the sum of a low-rank approximation component $\bf{X}$ and an approximation error component $\cal{E}$. This finer-grained data decomposition architecture enables each component of information to be portrayed more precisely. Further, we design an overlapping group error representation (OGER) function to characterize the above error structure and propose a generalized low-rank matrix completion model based on OGER. Specifically, the low-rank component describes the global structure information of matrix data, while the OGER component not only compensates for the approximation error between the low-rank component and the real data but also better captures the local block sparsity information of matrix data. Finally, we develop an alternating direction method of multipliers (ADMM) that integrates the majorization-minimization (MM) algorithm, which enables the efficient solution of the proposed model. And we analyze the convergence of the algorithm in detail both theoretically and experimentally. In addition, the results of numerical experiments demonstrate that the proposed model outperforms existing competing models in performance.

翻译：低秩矩阵补全（LRMC）技术在低层视觉任务中已取得显著成果。LRMC隐含一个基本假设，即现实世界中的矩阵数据具有低秩性。然而，实际矩阵数据往往不满足严格的低秩特性，这无疑对上述矩阵恢复方法构成了严峻挑战。幸运的是，存在可行的方案能够设计恰当且有效的先验表示来描述真实数据的内在信息。本文首先将矩阵数据 ${\bf{Y}}$ 建模为低秩近似分量 $\bf{X}$ 与近似误差分量 $\cal{E}$ 之和。这种更细粒度的数据分解架构使得信息的每个组成部分都能得到更精确的刻画。进一步地，我们设计了一种重叠组误差表示（OGER）函数来刻画上述误差结构，并提出了一种基于OGER的广义低秩矩阵补全模型。具体而言，低秩分量描述了矩阵数据的全局结构信息，而OGER分量不仅补偿了低秩分量与真实数据之间的近似误差，还能更好地捕捉矩阵数据的局部块稀疏信息。最后，我们开发了一种结合了优化-最小化（MM）算法的交替方向乘子法（ADMM），能够高效求解所提出的模型，并从理论和实验两方面详细分析了算法的收敛性。此外，数值实验结果表明，所提模型在性能上优于现有的竞争模型。