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

The Proximal Operator of the Piece-wise Exponential Function and Its Application in Compressed Sensing