Sparse signals recovered by non-convex penalty in quasi-linear systems
- Angang Cui^{1},
- Haiyang Li^{2},
- Meng Wen^{2} and
- Jigen Peng^{1}Email author
https://doi.org/10.1186/s13660-018-1652-8
© The Author(s) 2018
Received: 11 December 2017
Accepted: 6 March 2018
Published: 14 March 2018
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 \(\ell _{0}\)-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 \(\rho_{a}\) in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem \((QP_{a}^{\lambda})\) 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.
Keywords
MSC
1 Introduction
In [6, 7], the authors have shown that the \(\ell _{1}\)-norm minimization can really make an exact recovery in some specific conditions. In general, however, these conditions are always hard to satisfied in practice. Moreover, the regularization problem \((QP_{1}^{\lambda})\) always leads to a biased estimation by shrinking all the components of the vector toward zero simultaneously, and sometimes results in over-penalization in the regularization model \((QP_{1}^{\lambda})\) as the \(\ell_{1}\)-norm in linear compressed sensing.
The rest of this paper is organized as follows. Some preliminary results that are used in this paper are given in Sect. 2. In Sect. 3, we propose an iterative fraction thresholding algorithm to solve the regularization problem \((QP_{a}^{\lambda})\) for all \(a>0\). In Sect. 3, we present some numerical experiments to demonstrate the effectiveness of our algorithm. The concluding remarks are presented in Sect. 4.
2 Preliminaries
In this section, we give some preliminary results that are used in this paper.
Lemma 1
Definition 1
3 Thresholding representation theory and algorithm for problem \((QP_{a}^{\lambda})\)
In this section, we establish a thresholding representation theory of the problem \((QP_{a}^{\lambda})\), which underlies the algorithm to be proposed. Then an iterative fraction thresholding algorithm (IFTA) is proposed to solve the problem \((QP_{a}^{\lambda})\) for all \(a>0\).
3.1 Thresholding representation theory
Theorem 1
Proof
Theorem 2
Proof
3.2 Adjusting the values for the regularization parameter \(\lambda>0\)
Remark 1
Notice that (26) is valid for any \(\mu>0\) satisfying \(0<\mu \leq \Vert F(x_{k}) \Vert _{2}^{-2}\). In general, we can take \(\mu=\mu _{k}=\frac{1-\epsilon}{ \Vert F(x_{k}) \Vert _{2}^{2}}\) with any small \(\epsilon\in(0,1)\) below. Especially, the threshold value is \(t_{a,\lambda\mu}^{\ast}=\frac{\lambda\mu}{2}a\) when \(\lambda=\lambda_{1,k}\), and \(t_{a,\lambda\mu}^{\ast}=\sqrt{\lambda\mu}-\frac{1}{2a}\) when \(\lambda=\lambda_{2,k}\).
3.3 Iterative fraction thresholding algorithm (IFTA)
Remark 2
The convergence of IFTA is not proved theoretically in this paper, and this is our future work.
4 Numerical experiments
The graphs presented in Fig. 8 and Fig. 9 show the performance of the ISTA, IHTA and IFTA in recovering the true (sparsest) signals. From Fig. 8, we can see that IFTA performs best, and IST algorithm the second. From Fig. 9, we see that the IFTA has the smallest relative error value with sparsity growing.
5 Conclusion
In this paper, we take the fraction function as the substitution for \(\ell_{0}\)-norm in quasi-linear compressed sensing. An iterative fraction thresholding algorithm is proposed to solve the regularization problem \((QP_{a}^{\lambda})\) 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. We also provide a series of experiments to assess performance of our algorithm and the experiment results have illustrated that our algorithms is able to address the sparse signal recovery problems in nonlinear systems. Compared with ISTA and IHTA, IFTA performs best in sparse signal recovery and has the smallest relative error value with sparsity growing. However, the convergence of our algorithm is not proved theoretically in this paper, and it is our future work.
Declarations
Acknowledgements
The work was supported by the National Natural Science Foundations of China (11771347, 91730306, 41390454, 11271297) and the Science Foundations of Shaanxi Province of China (2016JQ1029, 2015JM1012).
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modelling of signals and images. SIAM Rev. 51(1), 34–81 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springe, New York (2010) View ArticleMATHGoogle Scholar
- Theodoridis, S., Kopsinis, Y., Slavakis, K.: Sparsity-aware learning and compressed sensing: an overview. https://arxiv.org/pdf/1211.5231
- Ehler, M., Fornasier, M., Sigl, J.: Quasi-linear compressed sensing. Multiscale Model. Simul. 12(2), 725–754 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Sigl, J.: Quasilinear compressed sensing. Master’s thesis, Technische University München, Munich, Germany (2013) Google Scholar
- Geman, D., Reynolds, G.: Constrained restoration and recovery of discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 14(3), 367–383 (1992) View ArticleGoogle Scholar
- Li, H., Zhang, Q., Cui, A., Peng, J.: Minimization of fraction function penalty in compressed sensing. https://arxiv.org/pdf/1705.06048
- Cui, A., Peng, J., Li, H., Zhang, C., Yu, Y.: Affine matrix rank minimization problem via non-convex fraction function penalty. J. Comput. Appl. Math. 336, 353–374 (2018) MathSciNetView ArticleMATHGoogle Scholar
- Xing, F.: Investigation on solutions of cubic equations with one unknown. J. Cent. Univ. Natl. (Nat. Sci. Ed.) 12(3), 207–218 (2003) Google Scholar
- Xu, Z., Chang, X., Xu, F., Zhang, H.: L1/2 regularization: a thresholding representation theory and a fast solver. IEEE Trans. Neural Netw. Learn. Syst. 23(7), 1013–1027 (2012) View ArticleGoogle Scholar