Computing the proximal operator of the q-th power of the ℓ1,q-norm for group sparsity

Rongrong Lin; Shihai Chen; Han Feng; Yulan Liu

doi:10.1016/j.acha.2025.101788

Computing the proximal operator of the q-th power of the ℓ_1,q-norm for group sparsity

Rongrong Lin, Shihai Chen, Han Feng, Yulan Liu^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

In this note, we comprehensively characterize the proximal operator of the q-th power of the ℓ_1,q-norm (denoted by ℓ_1,q^q) with 0 < q < 1 by exploiting the well-known proximal operator of |⋅|^q on the real line. In particular, much more explicit characterizations can be obtained whenever q = 1/2 and q = 2/3 due to the existence of closed-form expressions for the proximal operators of |⋅|^1/2 and |⋅|^2/3. Numerical experiments demonstrate potential advantages of the ℓ_1,q^q regularization in the inter-group and intra-group sparse vector recovery. © 2025 Elsevier Inc.

Original language	English
Article number	101788
Number of pages	15
Journal	Applied and Computational Harmonic Analysis
Volume	79
Online published	6 Jun 2025
DOIs	https://doi.org/10.1016/j.acha.2025.101788
Publication status	Published - Oct 2025

Bibliographical note

Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).

Funding

Lin was supported in part by the National Natural Science Foundation of China (12371103), Guangdong Basic and Applied Basic Research Foundation (2021A1515110680), and the Center for Mathematics and Interdisciplinary Sciences, School of Mathematics and Statistics, Guangdong University of Technology. Feng was supported in part by the Research Grants Council of Hong Kong (11303821 and 11315522). Liu was supported in part by the Guangdong Basic and Applied Basic Research Foundation (2023A1515012891).

Research Keywords

Group sparsity
Proximal gradient algorithms
Proximal operators
ℓ1,qq-norm
ℓq-norm

RGC Funding Information

RGC-funded

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title = "Computing the proximal operator of the q-th power of the ℓ1,q-norm for group sparsity",

abstract = "In this note, we comprehensively characterize the proximal operator of the q-th power of the ℓ1,q-norm (denoted by ℓ1,qq) with 0 < q < 1 by exploiting the well-known proximal operator of |⋅|q on the real line. In particular, much more explicit characterizations can be obtained whenever q = 1/2 and q = 2/3 due to the existence of closed-form expressions for the proximal operators of |⋅|1/2 and |⋅|2/3. Numerical experiments demonstrate potential advantages of the ℓ1,qq regularization in the inter-group and intra-group sparse vector recovery. {\textcopyright} 2025 Elsevier Inc.",

keywords = "Group sparsity, Proximal gradient algorithms, Proximal operators, ℓ1,qq-norm, ℓq-norm",

author = "Rongrong Lin and Shihai Chen and Han Feng and Yulan Liu",

note = "Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).",

year = "2025",

month = oct,

doi = "10.1016/j.acha.2025.101788",

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T1 - Computing the proximal operator of the q-th power of the ℓ1,q-norm for group sparsity

AU - Lin, Rongrong

AU - Chen, Shihai

AU - Feng, Han

AU - Liu, Yulan

N1 - Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).

PY - 2025/10

Y1 - 2025/10

N2 - In this note, we comprehensively characterize the proximal operator of the q-th power of the ℓ1,q-norm (denoted by ℓ1,qq) with 0 < q < 1 by exploiting the well-known proximal operator of |⋅|q on the real line. In particular, much more explicit characterizations can be obtained whenever q = 1/2 and q = 2/3 due to the existence of closed-form expressions for the proximal operators of |⋅|1/2 and |⋅|2/3. Numerical experiments demonstrate potential advantages of the ℓ1,qq regularization in the inter-group and intra-group sparse vector recovery. © 2025 Elsevier Inc.

AB - In this note, we comprehensively characterize the proximal operator of the q-th power of the ℓ1,q-norm (denoted by ℓ1,qq) with 0 < q < 1 by exploiting the well-known proximal operator of |⋅|q on the real line. In particular, much more explicit characterizations can be obtained whenever q = 1/2 and q = 2/3 due to the existence of closed-form expressions for the proximal operators of |⋅|1/2 and |⋅|2/3. Numerical experiments demonstrate potential advantages of the ℓ1,qq regularization in the inter-group and intra-group sparse vector recovery. © 2025 Elsevier Inc.

KW - Group sparsity

KW - Proximal gradient algorithms

KW - Proximal operators

KW - ℓ1,qq-norm

KW - ℓq-norm

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U2 - 10.1016/j.acha.2025.101788

DO - 10.1016/j.acha.2025.101788

M3 - RGC 21 - Publication in refereed journal

SN - 1063-5203

VL - 79

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

M1 - 101788

ER -

Computing the proximal operator of the q-th power of the ℓ_1,q-norm for group sparsity

Abstract

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Funding

Research Keywords

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GRF: Approximation Analysis for Feature Extraction by Deep Convolutional Neural Networks

GRF: Approximation Analysis of Classification with Deep Convolutional Neural Networks

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Computing the proximal operator of the q-th power of the ℓ1,q-norm for group sparsity

Abstract

Bibliographical note

Funding

Research Keywords

RGC Funding Information

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GRF: Approximation Analysis for Feature Extraction by Deep Convolutional Neural Networks

GRF: Approximation Analysis of Classification with Deep Convolutional Neural Networks

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Computing the proximal operator of the q-th power of the ℓ_1,q-norm for group sparsity