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The goal of compressed sensing is to reconstruct a sparse signal under a fewlinear measurements far less than the dimension of the ambient space of thesignal. However, many real-life applications in physics and biomedical sciencescarry some strongly nonlinear structures, and the linear model is no longersuitable. 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 sparsesignal recovery, we set most of the nonlinear models have a smooth quasi-linearnature in this paper, and study a non-convex fraction function $\rho_{a}$ inthis quasi-linear compressed sensing. We propose an iterative fractionthresholding algorithm to solve the regularization problem $(QP_{a}^{\lambda})$for all $a>0$. With the change of parameter $a>0$, our algorithm could get apromising result, which is one of the advantages for our algorithm comparedwith other algorithms. Numerical experiments show that our method performs muchbetter compared with some state-of-art methods.

Non-convex Fraction Function Penalty: Sparse Signals Recovered from Quasi-linear Systems