Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty $1-e^{-|x|/\sigma}$ with a given shape parameter $\sigma>0$, which is treated as a popular nonconvex surrogate of $\ell_0$-norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted $\ell_1$ algorithm, which substitutes PiE with its first-order approximation, however, the obtained solutions only are the critical point. Based on the exact characterization of the proximal operator of PiE, we explore how the iteratively reweighted $\ell_1$ solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of $\sigma$, the initial value and the regularization parameter in the definition of the proximal operator. Moreover, the initial value can be adaptively and simply chosen to ensure that the iteratively reweighted $\ell_1$ solution belongs to the proximal operator of PiE.

翻译：计算具有给定形状参数$\sigma>0$的稀疏促进分段指数（PiE）惩罚$1-e^{-|x|/\sigma}$的近端算子——该惩罚被视为$\ell_0$范数的流行非凸替代——在通过支持向量机进行特征选择、图像重建、0-1规划问题、压缩感知等领域具有基础性意义。由于PiE的非凸性，长期以来其近端算子常通过迭代重加权$\ell_1$算法进行评估，该算法以PiE的一阶近似替代原函数，但所得解仅停留在临界点。基于PiE近端算子的精确刻画，我们探究了迭代重加权$\ell_1$解在某些区域如何偏离真实近端算子，这些区域可通过$\sigma$、初始值及近端算子定义中的正则化参数显式识别。此外，可自适应且简单地选择初始值，以确保迭代重加权$\ell_1$解属于PiE的近端算子。