The Proximal Operator of the Piece-Wise Exponential Function

This letter 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 non-convex surrogate of the
$\ell _{0}$
-norm in support vector machines, compressed sensing, neural networks, etc. Although Malek-Mohammadi et al. [IEEE Transactions on Signal Processing, and 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. A comparative analysis of eleven sparse-promoting functions in compressed sensing demonstrates the effectiveness and efficiency of the piece-wise exponential function.