TY - JOUR

T1 - A constructive approach for computing the proximity operator of the p-th power of the ℓ1 norm

AU - Prater-Bennette, Ashley

AU - Shen, Lixin

AU - Tripp, Erin E.

N1 - Funding Information:
This work was funded in part by AFOSR grant 21RICO035. The work of L. Shen was supported in part by the National Science Foundation under grant DMS-1913039 and DMS-2208385. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the U.S. Air Force Research Laboratory. Cleared for public release 23 August 2022: Case Number AFRL-2022-4045.
Funding Information:
This work was funded in part by AFOSR grant 21RICO035 . The work of L. Shen was supported in part by the National Science Foundation under grant DMS-1913039 and DMS-2208385 . Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the U.S. Air Force Research Laboratory. Cleared for public release 23 August 2022: Case Number AFRL-2022-4045.
Publisher Copyright:
© 2023 Elsevier Inc.

PY - 2023/11

Y1 - 2023/11

N2 - This note is to study the proximity operator of hp=‖⋅‖1p, the power function of the ℓ1 norm. For general p, computing the proximity operator requires solving a system of potentially highly nonlinear inclusions. For p=1, the proximity operator of h1 is the well known soft-thresholding operator. For p=2, the function h2 serves as a penalty function that promotes structured solutions to optimization problems of interest; the computation of the proximity operator of h2 has been discussed in recent literature. By examining the properties of the proximity operator of the power function of the ℓ1 norm, we will develop a simple and well-justified approach to compute the proximity operator of hp with p>1. In particular, for the squared ℓ1 norm function, our approach provides an alternative, yet explicit way to finding its proximity operator. We also discuss how the structure of hp represents a class of relative sparsity promoting functions.

AB - This note is to study the proximity operator of hp=‖⋅‖1p, the power function of the ℓ1 norm. For general p, computing the proximity operator requires solving a system of potentially highly nonlinear inclusions. For p=1, the proximity operator of h1 is the well known soft-thresholding operator. For p=2, the function h2 serves as a penalty function that promotes structured solutions to optimization problems of interest; the computation of the proximity operator of h2 has been discussed in recent literature. By examining the properties of the proximity operator of the power function of the ℓ1 norm, we will develop a simple and well-justified approach to compute the proximity operator of hp with p>1. In particular, for the squared ℓ1 norm function, our approach provides an alternative, yet explicit way to finding its proximity operator. We also discuss how the structure of hp represents a class of relative sparsity promoting functions.

KW - Proximity operator

KW - Relative sparsity

KW - Sparsity

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U2 - 10.1016/j.acha.2023.06.007

DO - 10.1016/j.acha.2023.06.007

M3 - Letter/Newsletter

AN - SCOPUS:85164244338

SN - 1063-5203

VL - 67

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

M1 - 101572

ER -