M-estmators including the Welsch and Cauchy have been widely adopted for robustness against outliers, but they also down-weigh the uncontaminated data. To address this issue, we devise a framework to generate a class of nonconvex functions which only down-weigh outlier-corrupted observations. Our framework is then applied to the Welsch, Cauchy and $\ell_p$-norm functions to produce the corresponding robust loss functions. Targeting on the application of robust matrix completion, efficient algorithms based on these functions are developed and their convergence is analyzed. Finally, extensive numerical results demonstrate that the proposed methods are superior to the competitors in terms of recovery accuracy and runtime.

翻译：M估计器（包括Welsch和Cauchy函数）已被广泛用于增强对异常值的鲁棒性，但同时也降低了无污染数据的权重。为解决这一问题，我们设计了一个框架来生成一类仅降低异常值污染观测权重的非凸函数。该框架随后被应用于Welsch函数、Cauchy函数和$\ell_p$-范数函数，以生成相应的鲁棒损失函数。针对鲁棒矩阵补全的应用场景，我们基于这些函数开发了高效算法并分析了其收敛性。最后，大量数值实验结果表明，所提方法在恢复精度和运行时间方面均优于现有竞争方法。