Sparse signal reconstruction via the approximations of $ \ell_{0} $ quasinorm

Jun Wang; Xing Tao Wang

doi:10.3934/jimo.2019035

Article Contents

2020, Volume 16, Issue 4: 1907-1925. Doi: 10.3934/jimo.2019035

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Sparse signal reconstruction via the approximations of $ \ell_{0} $ quasinorm

Jun Wang and
Xing Tao Wang^,

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China

^* Corresponding author: Xing Tao Wang
^* Corresponding author: Xing Tao Wang

Received: June 2018

Revised: November 2018

Early access: May 2019

Published: May 2020

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Abstract

In this paper, we propose two classes of the approximations to the cardinality function via the Moreau envelope of the $ \ell_{1} $ norm. We show that these two approximations are good choices of the merit function for sparsity and are essentially the truncated $ \ell_{1} $ norm and the truncated $ \ell_{2} $ norm. Moreover, we apply the approximations to solve sparse signal recovery problems and then provide new weights for reweighted $ \ell_{1} $ minimization and reweighted least squares to find sparse solutions of underdetermined linear systems of equations. Finally, we present some numerical experiments to illustrate our results.

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Mathematics Subject Classification: Primary: 90C30, 65K05; Secondary: 49M37, 49M25.

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Figure 1. Comparison of success rates of finding the $ s $-sparse solution $ \mathbf{y} $ of $ \mathbf{b = Ax} $, where $ \mathbf{A} \in \mathbb{R}^{50\times 250} $ and $ \mathbf{y} \in \mathbb{R}^{250} $. For each $ s $-sparsity, $ 500 $ attempts were made

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Figure 2. Comparison of $ \Vert \mathbf{y}^{k}\Vert_{1} $ and weighted $ \Vert \mathbf{y}^{k}\Vert_{1} $ at each iteration $ k $ with $ \mathbf{A} \in \mathbb{R}^{50\times 250} $, $ \mathbf{y} \in \mathbb{R}^{250} $ and $ p = 0.618, s = 10 $

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Figure 3. Comparison of success rates of finding the $ s $-sparse solution $ \mathbf{y} $ of $ \mathbf{b = Ay} $, where $ \mathbf{A} \in \mathbb{R}^{50\times 250} $ and $ \mathbf{y} \in \mathbb{R}^{250} $. For each $ s $-sparsity, $ 500 $ attempts were made

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Figure 4. Comparison of $ \Vert \mathbf{y}^{k}\Vert_{2}^{2} $ and weighted $ \Vert \mathbf{y}^{k}\Vert_{2}^{2} $ at each iteration $ k $ with $ \mathbf{A} \in \mathbb{R}^{50\times 250} $, $ \mathbf{y} \in \mathbb{R}^{250} $ and $ p = 0.618, s = 10 $

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