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This paper presents a novel loss function referred to as hybridordinary-Welsch (HOW) and a new sparsity-inducing regularizer associated withHOW. We theoretically show that the regularizer is quasiconvex and that thecorresponding Moreau envelope is convex. Moreover, the closed-form solution toits Moreau envelope, namely, the proximity operator, is derived. Compared withnonconvex regularizers like the lp-norm with 0<p<1 that requires iterations tofind the corresponding proximity operator, the developed regularizer has aclosed-form proximity operator. We apply our regularizer to the robust matrixcompletion problem, and develop an efficient algorithm based on the alternatingdirection method of multipliers. The convergence of the suggested method isanalyzed and we prove that any generated accumulation point is a stationarypoint. Finally, experimental results based on synthetic and real-world datasetsdemonstrate that our algorithm is superior to the state-of-the-art methods interms of restoration performance.

Robust Low-Rank Matrix Completion via a New Sparsity-Inducing Regularizer