Noisy matrix completion has attracted significant attention due to its applications in recommendation systems, signal processing and image restoration. Most existing works rely on (weighted) least squares methods under various low-rank constraints. However, minimizing the sum of squared residuals is not always efficient, as it may ignore the potential structural information in the residuals.In this study, we propose a novel residual spectral matching criterion that incorporates not only the numerical but also locational information of residuals. This criterion is the first in noisy matrix completion to adopt the perspective of low-rank perturbation of random matrices and exploit the spectral properties of sparse random matrices. We derive optimal statistical properties by analyzing the spectral properties of sparse random matrices and bounding the effects of low-rank perturbations and partial observations. Additionally, we propose algorithms that efficiently approximate solutions by constructing easily computable pseudo-gradients. The iterative process of the proposed algorithms ensures convergence at a rate consistent with the optimal statistical error bound. Our method and algorithms demonstrate improved numerical performance in both simulated and real data examples, particularly in environments with high noise levels.

翻译：噪声矩阵补全因其在推荐系统、信号处理和图像修复中的应用而受到广泛关注。现有研究大多依赖各种低秩约束下的（加权）最小二乘法。然而，最小化残差平方和并不总是高效的，因为它可能忽略残差中潜在的结构信息。本研究提出了一种新颖的残差谱匹配准则，该准则不仅考虑了残差的数值信息，还融入了其位置信息。该准则是噪声矩阵补全领域首个从随机矩阵低秩扰动的视角出发，并利用稀疏随机矩阵谱性质的方法。通过分析稀疏随机矩阵的谱特性，并界定低秩扰动与部分观测的影响，我们推导出了最优统计性质。此外，我们提出通过构造易于计算的伪梯度来高效逼近解的算法。所提算法的迭代过程确保收敛速度与最优统计误差界一致。我们的方法与算法在仿真和实际数据示例中均展现出改进的数值性能，特别是在高噪声环境下。