Sparse non-negative signal reconstruction using fraction function penalty

Abstract

Many practical problems in the real world can be formulated as the non-negative ℓ0-minimisation problems, which seek the sparsest non-negative signals to underdetermined linear equations. They have been widely applied in signal and image processing, machine learning, pattern recognition and computer vision. Unfortunately, this non-negative ℓ0-minimisation problem is non-deterministic polynomial hard (NP-hard) because of the discrete and discontinuous nature of the ℓ0-norm. Inspired by the good performances of the fraction function in the authors’ former work, in this paper, the authors replace the ℓ0-norm with the non-convex fraction function and study the minimisation problem of the fraction function in recovering the sparse non-negative signal from an underdetermined linear equation. They discuss the equivalence between non-negative ℓ0-minimisation problem and non-negative fraction function minimisation problem, and the equivalence between non-negative fraction function minimisation problem and regularised non-negative fraction function minimisation problem. It is proved that the optimal solution to the non-negative ℓ0-minimisation problem could be approximately obtained by solving their regularised nonnegative fraction function minimisation problem if some specific conditions are satisfied. Then, they propose a non-negative iterative thresholding algorithm to solve their regularised non-negative fraction function minimisation problem. At last, numerical experiments on some sparse non-negative signal recovery problems are reported.