On choosing initial values of iteratively reweighted ℓ1 algorithms for the piece-wise exponential penalty.

Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty 1 − e − | x | / σ with a given shape parameter σ > 0 , which is treated as a popular nonconvex surrogate of ℓ 0 -norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, neural networks, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted ℓ 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 ℓ 1 solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of σ , 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 ℓ 1 solution belongs to the proximal operator of PiE. [ABSTRACT FROM AUTHOR]