Sparse signals recovered by non-convex penalty in quasi-linear systems
The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is n...
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| Vydáno v: | Journal of inequalities and applications Ročník 2018; číslo 1; s. 59 - 11 |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cham
Springer International Publishing
2018
SpringerOpen |
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| ISSN: | 1029-242X, 1025-5834, 1029-242X |
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| Abstract | The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the
ℓ
0
-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function
ρ
a
in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem
(
Q
P
a
λ
)
for all
a
>
0
. With the change of parameter
a
>
0
, our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods. |
|---|---|
| AbstractList | The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the [Formula: see text]-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function [Formula: see text] in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem [Formula: see text] for all [Formula: see text]. With the change of parameter [Formula: see text], our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods.The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the [Formula: see text]-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function [Formula: see text] in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem [Formula: see text] for all [Formula: see text]. With the change of parameter [Formula: see text], our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods. The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the [Formula: see text]-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function [Formula: see text] in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem [Formula: see text] for all [Formula: see text]. With the change of parameter [Formula: see text], our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods. The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the $\ell _{0}$ ℓ0-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function $\rho_{a}$ ρa in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem $(QP_{a}^{\lambda})$ (QPaλ) for all $a>0$ a>0. With the change of parameter $a>0$ a>0, our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods. The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the ℓ 0 -norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function ρ a in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem ( Q P a λ ) for all a > 0 . With the change of parameter a > 0 , our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods. Abstract The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the ℓ0 $\ell _{0}$-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function ρa $\rho_{a}$ in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem (QPaλ) $(QP_{a}^{\lambda})$ for all a>0 $a>0$. With the change of parameter a>0 $a>0$, our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods. |
| ArticleNumber | 59 |
| Author | Cui, Angang Li, Haiyang Peng, Jigen Wen, Meng |
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| Keywords | 93C10 Non-convex fraction function 34A34 78M50 Quasi-linear Iterative thresholding algorithm Compressed sensing |
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| References | Bruckstein, Donoho, Elad (CR3) 2009; 51 Candes, Romberg, Tao (CR1) 2006; 59 CR5 Geman, Reynolds (CR8) 1992; 14 CR7 Elad (CR4) 2010 Xu, Chang, Xu, Zhang (CR12) 2012; 23 CR9 Xing (CR11) 2003; 12 Ehler, Fornasier, Sigl (CR6) 2014; 12 Donoho (CR2) 2006; 52 Cui, Peng, Li, Zhang, Yu (CR10) 2018; 336 M. Elad (1652_CR4) 2010 F. Xing (1652_CR11) 2003; 12 E. Candes (1652_CR1) 2006; 59 M. Ehler (1652_CR6) 2014; 12 D.L. Donoho (1652_CR2) 2006; 52 Z. Xu (1652_CR12) 2012; 23 1652_CR9 A.M. Bruckstein (1652_CR3) 2009; 51 1652_CR7 A. Cui (1652_CR10) 2018; 336 1652_CR5 D. Geman (1652_CR8) 1992; 14 |
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| SubjectTerms | Analysis Applications of Mathematics Compressed sensing Iterative thresholding algorithm Mathematics Mathematics and Statistics Non-convex fraction function Quasi-linear |
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| Title | Sparse signals recovered by non-convex penalty in quasi-linear systems |
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