An explicit characterization for the proximal operator of the capped $ \ell_q $ -norm
Abstract
The capped $ \ell_q $-norm, as a representative example of capped folded concave functions, has been widely adopted in sparse optimization problems due to its effectiveness in mitigating the bias issue associated with the $ \ell_1 $-norm. For the specific case of $ q = 1/2 $, the proximal operator of the capped $ \ell_q $-norm was characterized by Lili Pan and Xiaojun Chen [Group sparse optimization for images recovery using capped folded concave functions, SIAM J. Imaging Sci., 14(1):1-25, 2021]. However, applying their characterization generally requires comparing function values among several candidate points, which can be computationally cumbersome. To address this limitation, we derive an explicit characterization for the proximal operator of the capped $ \ell_q $-norm for each $ 0
Fields of Study & Subjects
Sparse and Compressive Sensing Techniques
Numerical methods in inverse problems
Medical Image Segmentation Techniques
Publication details
- Journal / source
- Mathematical Foundations of Computing
- Publisher
- American Institute of Mathematical Sciences (AIMS)
- Volume
- 13
- Issue
- 0
- Pages
- 39–61
- Publication date
- 1 January 2026