Nonconvex fraction function recovery sparse signal by convex optimization

The problem of recovering a sparse signal from the linear constraints, known as the $\ell_{0}$-norm minimization problem, has been attracting extensive attention in recent years. However, the $\ell_{0}$-norm minimization problem is a NP-hard problem. In our latest work, a non-convex fraction function is studied to approximate the $\ell_{0}$-norm in $\ell_{0}$-norm minimization problem and translate this NP-hard problem into a fraction function minimization problem. The FP thresholding algorithm is generated to solve the regularization fraction function minimization problem. However, we find that there are some drawbacks for our previous proposed FP algorithm. One is that the FP algorithm always convergent to a local minima due to the non-convexity of fraction function. The other one is that the parameter $a$, which influences the behaviour of non-convex fraction function $\rho_{a}$, needs to be determined manually in every simulation, and how to determine the best parameter $a$ is not an easy problem. To avoid these drawbacks, here instead, in this paper we generate an adaptive convex FP algorithm to solve the problem $(FP_{a,\lambda})$. When doing so, our adaptive convex FP algorithm will not only convergent to a global minima but also intelligent both for the choice of the regularization parameter $\lambda$ and the parameter $a$. These are the advantages for our convex algorithm compared with our previous proposed FP algorithm.