Minimization of Fraction Function Penalty in Compressed Sensing

In this paper, we study the minimization problem of a non-convex sparsity-promoting penalty function, i.e., fraction function, in compressed sensing. First, we discuss the equivalence of
$\ell _{0}$
minimization and fraction function minimization. It is proved that the optimal solution to fraction function minimization solves
$\ell _{0}$
minimization and the optimal solution to the regularization problem also solves fraction function minimization if the certain conditions are satisfied, which is similar to the regularization problem in a convex optimization theory. Second, we study the properties of the optimal solution to the regularization problem, including the first-order and second-order optimality conditions and the lower and upper bounds of the absolute value for its nonzero entries. Finally, we derive the closed-form representation of the optimal solution to the regularization problem and propose an iterative
$FP$
thresholding algorithm to solve the regularization problem. We also provide a series of experiments to assess the performance of the
$FP$
algorithm, and the experimental results show that the
$FP$
algorithm performs well in sparse signal recovery with and without measurement noise.