# Sparse signals recovered by non-convex penalty in quasi-linear systems

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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 \(\ell _{0}\)-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

Compressed sensing Quasi-linear Non-convex fraction function Iterative thresholding algorithm## MSC

34A34 78M50 93C10## 1 Introduction

*m*-dimension, and \(\Vert x \Vert _{0}\) is the \(\ell_{0}\)-norm of the real vector

*x*, which counts the number of the nonzero entries in

*x*(see, e.g., [3, 4, 5]). In general, the problem \((P_{0})\) is computational and NP-hard because of the discrete and discontinuous nature of the \(\ell_{0}\)-norm. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures [6], so that the linear model in problem \((P_{0})\) is no longer suitable. In this nonlinear case, we consider a map \(A: \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), which is no longer necessarily linear, and reconstruct a sparse vector \(x\in \mathbb{R}^{n}\) from the measurements \(b\in\mathbb{R}^{m}\) given by

*x*.

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

*The operator*\(\mathrm{prox}_{a,\lambda}^{\beta}\)

*defined in*(13)

*can be expressed as*

*where*\(g_{a,\lambda}(\gamma)\)

*is defined as*

*and the threshold value satisfies*

*where*

## 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

*For any*\(\lambda>0\)

*and*\(0<\mu<L_{\ast}^{-1}\)

*with*\(\Vert F(x^{\ast })x-F(x^{\ast})x^{\ast} \Vert _{2}^{2}\leq L_{\ast} \Vert x-x^{\ast} \Vert _{2}^{2}\).

*If*\(x^{\ast}\)

*is the optimal solution of*\(\min_{x\in\mathbb {R}^{n}}C_{1}(x)\),

*then*\(x^{\ast}\)

*is also the optimal solution of*\(\min_{x\in\mathbb{R}^{n}}C_{2}(x,x^{\ast})\),

*that is*,

*for any*\(x\in\mathbb{R}^{n}\).

## Proof

## Theorem 2

*For any*\(\lambda>0\), \(\mu>0\)

*and solution*\(x^{\ast}\)

*of*\(\min_{x\in\mathbb{R}^{n}}C_{1}(x)\), \(\min_{x\in\mathbb {R}^{n}}C_{2}(x,x^{\ast})\)

*is equivalent to*

*where*\(B_{\mu}(x^{\ast})=x^{\ast}+\mu F(x^{\ast})^{\top }(b-F(x^{\ast})x^{\ast})\).

## Proof

*λ*with

*λμ*. With the thresholding representations (22), the IFTA for solving the regularization problem \((QP_{a}^{\lambda})\) can be naturally defined as

### 3.2 Adjusting the values for the regularization parameter \(\lambda>0\)

*r*is the optimal solution of the regularization problem \((QP_{a}^{\lambda})\), and without loss of generality, set

*λ*with

*λμ*in \(t_{a,\lambda}^{\ast}\).

*λ*is

*k*th iteration. That is, (26) can be used to automatically adjust the value of the regularization parameter \(\lambda >0\) during iteration.

## 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

*η*is a sufficiently small scaling factor (we set \(\eta=0.003\)), and \(A_{2}\in \mathbb{R}^{30\times100}\) is a fixed matrix with every entry equals 1. Then the form of nonlinearity considered in (27) is a quasi-linear, and the more detailed accounts of the setting in the form of (27) can be found in [6, 7]. By randomly generating such sufficiently sparse vectors \(x_{0}\) (choosing the nonzero locations uniformly over the support in random, and their values from \(N(0,1)\)), we generate vectors

*b*. In this way, we know the sparsest solution to \(F(x_{0})x_{0} = b\), and we are able to compare this with algorithmic results. The stopping criterion is usually as follows:

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.

## Notes

### 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.

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