## Abstract

This note is to study the proximity operator of h_{p}=‖⋅‖_{1}^{p}, the power function of the ℓ_{1} norm. For general p, computing the proximity operator requires solving a system of potentially highly nonlinear inclusions. For p=1, the proximity operator of h_{1} is the well known soft-thresholding operator. For p=2, the function h_{2} serves as a penalty function that promotes structured solutions to optimization problems of interest; the computation of the proximity operator of h_{2} has been discussed in recent literature. By examining the properties of the proximity operator of the power function of the ℓ_{1} norm, we will develop a simple and well-justified approach to compute the proximity operator of h_{p} with p>1. In particular, for the squared ℓ_{1} norm function, our approach provides an alternative, yet explicit way to finding its proximity operator. We also discuss how the structure of h_{p} represents a class of relative sparsity promoting functions.

Original language | English (US) |
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Article number | 101572 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 67 |

DOIs | |

State | Published - Nov 2023 |

## Keywords

- Proximity operator
- Relative sparsity
- Sparsity

## ASJC Scopus subject areas

- Applied Mathematics

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