  This paper presents a novel loss function referred to as hybrid
ordinary-Welsch (HOW) and a new sparsity-inducing regularizer associated with
HOW. We theoretically show that the regularizer is quasiconvex and that the
corresponding Moreau envelope is convex. Moreover, the closed-form solution to
its Moreau envelope, namely, the proximity operator, is derived. Compared with
nonconvex regularizers like the lp-norm with 0<p<1 that requires iterations to
find the corresponding proximity operator, the developed regularizer has a
closed-form proximity operator. We apply our regularizer to the robust matrix
completion problem, and develop an efficient algorithm based on the alternating
direction method of multipliers. The convergence of the suggested method is
analyzed and we prove that any generated accumulation point is a stationary
point. Finally, experimental results based on synthetic and real-world datasets
demonstrate that our algorithm is superior to the state-of-the-art methods in
terms of restoration performance.
