Title: Recovering Sparse Nonnegative Signals via Non-convex Fraction Function Penalty,
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.