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.

本文介绍了一种新的损失函数，名为混合常规-威尔士（HOW），以及与HOW相关的一种新的稀疏诱导正则化项。我们从理论上证明了该正则化项是准凸的，其对应的Moreau包络是凸的。此外，我们推导出该Moreau包络（即接近算子）的闭式解。与需要通过迭代来寻找相应接近算子的0<p<1的lp-范数之类的非凸正则化项相比，我们开发的正则化项具有闭式接近算子。我们将该正则化项应用于鲁棒矩阵完成问题，并基于交替方向乘法算法开发了一种高效算法。对所提出方法的收敛性进行了分析，并证明了任何生成的累积点都是一个驻点。最后，基于人工合成和真实数据集的实验结果表明，我们的算法在恢复性能方面优于现有方法。

基于新的稀疏诱导正则化器的鲁棒低秩矩阵补全