  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.
