Minimization of fraction function penalty in compressed sensing

TL;DR

This paper introduces a non-convex fraction function penalty for compressed sensing, proves its equivalence to  minimization under certain conditions, and develops an iterative thresholding algorithm that outperforms existing methods in sparse signal recovery.

Contribution

The paper establishes the theoretical equivalence between fraction function minimization and  minimization, and proposes a novel iterative thresholding algorithm with superior performance.

Findings

The  minimization and fraction function minimization are equivalent under certain conditions.

The paper derives a closed-form solution for the regularization problem.

The proposed FP algorithm outperforms soft and half thresholding algorithms in experiments.

Abstract

In the paper, we study the minimization problem of a non-convex sparsity promoting penalty function $$P_{a}(x)=\sum_{i=1}^{n}p_{a}(x_{i})=\sum_{i=1}^{n}\frac{a|x_{i}|}{1+a|x_{i}|}$$ in compressed sensing, which is called fraction function. Firstly, we discuss the equivalence of $\ell_{0}$ minimization and fraction function minimization. It is proved that there corresponds a constant $a^{**}>0$ such that, whenever $a>a^{**}$, every solution to $(FP_{a})$ also solves $(P_{0})$, that the uniqueness of global minimizer of $(FP_{a})$ and its equivalence to $(P_{0})$ if the sensing matrix $A$ satisfies a restricted isometry property (RIP) and, last but the most important, that the optimal solution to the regularization problem $(FP_{a}^\lambda)$ also solves $(FP_{a})$ if the certain condition is satisfied, which is similar to the regularization problem in convex optimal theory. Secondly, we study the properties of the optimal solution to the regularization problem $(FP^{\lambda}_{a})$ including the first-order and the second optimality condition and the lower and upper bound of the absolute value for its nonzero entries. Finally, we derive the closed form representation of the optimal solution to the regularization problem ($FP_{a}^{\lambda}$) for all positive values of parameter $a$, and propose an iterative $FP$ thresholding algorithm to solve the regularization problem $(FP_{a}^{\lambda})$. We also provide a series of experiments to assess performance of the $FP$ algorithm, and the experiment results show that, compared with soft thresholding algorithm and half thresholding algorithms, the $FP$ algorithm performs the best in sparse signal recovery with and without measurement noise.