This paper characterizes the proximal operator of the piece-wise exponential function $1\!-\!e^{-|x|/\sigma}$ with a given shape parameter $\sigma\!>\!0$, which is a popular nonconvex surrogate of $\ell_0$-norm in support vector machines, zero-one programming problems, and compressed sensing, etc. Although Malek-Mohammadi et al. [IEEE Transactions on Signal Processing, 64(21):5657--5671, 2016] once worked on this problem, the expressions they derived were regrettably inaccurate. In a sense, it was lacking a case. Using the Lambert W function and an extensive study of the piece-wise exponential function, we have rectified the formulation of the proximal operator of the piece-wise exponential function in light of their work. We have also undertaken a thorough analysis of this operator. Finally, as an application in compressed sensing, an iterative shrinkage and thresholding algorithm (ISTA) for the piece-wise exponential function regularization problem is developed and fully investigated. A comparative study of ISTA with nine popular non-convex penalties in compressed sensing demonstrates the advantage of the piece-wise exponential penalty.

翻译：本文刻画了分段指数函数 $1\!-\!e^{-|x|/\sigma}$（具有给定形状参数 $\sigma\!>\!0$）的近端算子。该函数在支持向量机、零一规划问题和压缩感知等领域中，是 $\ell_0$ 范数的一种常用非凸替代。尽管Malek-Mohammadi等人[IEEE Transactions on Signal Processing, 64(21):5657--5671, 2016]曾研究过此问题，但遗憾的是，他们推导出的表达式并不准确，即在某种意义上缺少一种情形。通过利用Lambert W函数并对分段指数函数进行深入研究，我们基于他们的工作纠正了分段指数函数近端算子的表达式。此外，我们还对该算子进行了全面分析。最后，作为其在压缩感知中的应用，我们针对分段指数函数正则化问题提出并充分探究了一种迭代收缩阈值算法（ISTA）。将ISTA与压缩感知中九种流行的非凸惩罚函数进行对比研究，结果表明分段指数惩罚具有优势。