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 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{0}$\end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho_{a}$\end{document}ρa in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(QP_{a}^{\lambda})$\end{document}(QPaλ) for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>0$\end{document}a>0. With the change of parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>0$\end{document}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.


Introduction
In compressed sensing (see, e.g., [1,2]), the problem of reconstructing a sparse signal under a few linear measurements which are far fewer than the dimension of the ambient space of the signal can be modeled into the following 0 -minimization: where A ∈ R m×n is an m × n real matrix of full row rank with m < n, and b ∈ R m is a nonzero real vector of m-dimension, and x 0 is the 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 0norm. 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 : R n → R m , which is no longer necessarily linear, and reconstruct a sparse vector x ∈ R n from the measurements b ∈ R m given by Compared with the 0 -minimization under the linear circumstance, this nonlinear minimization is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the 0 -norm and the nonlinearity. In order to get a convenience for sparse signal recovery, in this paper, we set the nonlinear models have a smooth quasi-linear nature. By this means, there exists a Lipschitz map such that for all x ∈ R n . So, the 0 -minimization under the quasi-linear case can be mathematically viewed as the following form: In fact, the minimization (QP 0 ) under the quasi-linear case is also combinatorial and NPhard [6,7]. To overcome this problem, the authors in [6,7] proposed the 1 -minimization for the constrained problem and for the regularization problem, where x 1 = n i=1 |x i | is the 1 -norm of vector x. In [6,7], the authors have shown that the 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 λ 1 ) 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 λ 1 ) as the 1 -norm in linear compressed sensing.
In pursuit of better reconstruction results, in this paper, we propose the following fraction minimization: where is the fraction function which performs outstanding in image restoration [8], linear compressed sensing [9] and matrix rank minimization problem [10]. Clearly, with the change of parameter a > 0, the non-convex function P a (x) could approximately interpolate the 1 if x i = 0.
(11) Figure 1 shows the behavior of the fraction function ρ a (t) for various values of a > 0. 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 λ a ) 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.

Preliminaries
In this section, we give some preliminary results that are used in this paper.
Define a function of β ∈ R as and let Lemma 1 (see [9][10][11]) The operator prox β a,λ defined in (13) can be expressed as where g a,λ (γ ) is defined as where

Definition 1
The iterative thresholding operator G a,λ can be defined by where prox β a,λ is defined in Lemma 1.

Thresholding representation theory and algorithm for problem (QP λ a )
In this section, we establish a thresholding representation theory of the problem (QP λ a ), which underlies the algorithm to be proposed. Then an iterative fraction thresholding algorithm (IFTA) is proposed to solve the problem (QP λ a ) for all a > 0.

Thresholding representation theory
For any fixed positive parameters λ > 0, μ > 0, a > 0 and x ∈ R n , let and . If x * is the optimal solution of min x∈R n C 1 (x), then x * is also the optimal solution of min x∈R n C 2 (x, x * ), that is, Proof By the definition of C 2 (x, y), we have Theorem 2 For any λ > 0, μ > 0 and solution x * of min x∈R n C 1 (x), min x∈R n C 2 (x, x * ) is equivalent to Proof By the definition, C 2 (x, y) can be rewritten as which implies that min x∈R n C 2 (x, x * ) for any λ > 0, μ > 0 is equivalent to Combining Theorem 2, Theorem 1 and Lemma 1, the thresholding representation of (QP λ a ) can be concluded by where the operator G a,λμ is defined in Definition 1 and obtained by replacing λ with λμ.
With the thresholding representations (22), the IFTA for solving the regularization problem (QP λ a ) can be naturally defined as where

Adjusting the values for the regularization parameter λ > 0
In this subsection, the cross-validation method (see [9,10,12]) is accepted to automatically adjust the value for the regularization parameter λ > 0. In other words, when some prior information is known for a regularization problem, this selection is more reasonable and intelligent. Suppose that the vector x * of sparsity r is the optimal solution of the regularization problem (QP λ a ), and without loss of generality, set Then it follows from (14) that where t * a,λμ is obtained by replacing λ with λμ in t * a,λ . By t 2 a,λμ ≤ t 1 a,λμ , we have It follows that From (25), we obtain We denote by λ 1 and λ 2 the left and the right of the above interval, respectively: Since x * is unknown, and x k is the best available approximation to x * , so we can take in the kth iteration. That is, (26) can be used to automatically adjust the value of the regularization parameter λ > 0 during iteration.

Iterative fraction thresholding algorithm (IFTA)
Based on the thresholding representation (23) and the analyses given in Sect. 3.2, the proposed iterative fraction thresholding algorithm (IFTA) for regularization problem (QP λ a ) can be naturally described in Algorithm 1.
Remark 2 The convergence of IFTA is not proved theoretically in this paper, and this is our future work.

Numerical experiments
In the section, we carry out a series of simulations to demonstrate the performance of IFTA, and compare them with those obtained with some state-of-art methods (iterative soft thresholding algorithm (ISTA) [6,7]), iterative hard thresholding algorithm (IHTA) [6,7]. In our numerical experiments, we set where A 1 ∈ R 100×400 is a fixed Gaussian random matrix, x 0 ∈ R 400 is a reference vector, f : [0, ∞) → R is a positive and smooth Lipschitz continuous function with f (t) = ln(t + 1), η is a sufficiently small scaling factor (we set η = 0.003), and A 2 ∈ R 30×100 is a fixed matrix with every entry equals 1. Then the form of nonlinearity considered in (27) is a quasi-

Algorithm 1 Iterative fraction thresholding algorithm (IFTA)
Initialize: Given x 0 ∈ R n , μ 0 = 1-F(x 0 ) 2 2 (0 < < 1) and a > 0; while not converged do 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: where x k and x k-1 are numerical results from two continuous iterative steps and Tol is a given small number. The success is measured by computing where x * is the numerical results generated by IFTA, and Re is also a given small number. In all of our experiments, we set Tol = 10 -8 to indicate the stopping criterion, and set Re = 10 -4 to indicate a perfect recovery of the original sparse vector x 0 . Figure 8 shows the success rate of three algorithms in the recovery of a sparse signal with different cardinality. In this experiment, we repeatedly perform 30 tests and present average results and take a = 2.5. Figure 9 shows the relative error between the solution x * and the given signal x 0 . In this experiment, we repeatedly perform 30 tests and present average results and take a = 2.5.
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. In this experiment, we repeatedly perform 30 tests and present average results and take a = 2.5

Figure 9
Relative error between the solution x * and the given signal x 0 . In this experiment, we repeatedly perform 30 tests and present average results and take a = 2.5

Conclusion
In this paper, we take the fraction function as the substitution for 0 -norm in quasi-linear compressed sensing. An iterative fraction thresholding algorithm is proposed to solve the regularization problem (QP λ a ) 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.