Recovering Sparse Nonnegative Signals via Non-convex Fraction Function Penalty

TL;DR

This paper introduces a non-convex fraction function approach to recover sparse nonnegative signals, providing theoretical equivalences, an iterative algorithm, and demonstrating improved performance over linear programming methods.

Contribution

The paper proposes a novel non-convex fraction function model and an iterative thresholding algorithm for sparse nonnegative signal recovery, with theoretical analysis and empirical validation.

Findings

Effective recovery of sparse nonnegative signals demonstrated.

The proposed method outperforms linear programming in experiments.

Theoretical equivalences establish the connection between different formulations.

Abstract

Many real world practical problems can be formulated as $\ell_{0}$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative signals to underdetermined linear systems. They have been widely applied in signal and image processing, machine learning, pattern recognition and computer vision. Unfortunately, this $\ell_{0}$-minimization problem with nonnegativity constraint is computational and NP-hard because of the discrete and discontinuous nature of the $\ell_{0}$-norm. In this paper, we replace the $\ell_{0}$-norm with a non-convex fraction function, and study the minimization problem of this non-convex fraction function in recovering the sparse nonnegative signals from an underdetermined linear system. Firstly, we discuss the equivalence between $(P_{0}^{\geq})$ and $(FP_{a}^{\geq})$, and the equivalence between $(FP_{a}^{\geq})$ and $(FP_{a,\lambda}^{\geq})$. It is proved that the optimal solution of the problem $(P_{0}^{\geq})$ could be approximately obtained by solving the regularization problem $(FP_{a,\lambda}^{\geq})$ if some specific conditions satisfied. Secondly, we propose a nonnegative iterative thresholding algorithm to solve the regularization problem $(FP_{a,\lambda}^{\geq})$ for all $a>0$. Finally, some numerical experiments on sparse nonnegative siganl recovery problems show that our method performs effective in finding sparse nonnegative signals compared with the linear programming.