Sparse non‐negative signal reconstruction using fraction function penalty.

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 non‐negative 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. [ABSTRACT FROM AUTHOR]