The Proximal Operator of the Piece-wise Exponential Function and Its Application in Compressed Sensing
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This paper characterizes the proximal operator of the piece-wise exponential function 1-e^-|x|/σ with a given shape parameter σ>0, which is a popular nonconvex surrogate of ℓ_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.
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