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Sparse non-negative signal reconstruction using fraction function penalty

Sparse non-negative signal reconstruction using fraction function penalty

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Many practical problems in the real world can be formulated as the non-negative -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 -minimisation problem is non-deterministic polynomial hard (NP-hard) because of the discrete and discontinuous nature of the -norm. Inspired by the good performances of the fraction function in the authors’ former work, in this paper, the authors replace the -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 -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 -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.

http://iet.metastore.ingenta.com/content/journals/10.1049/iet-spr.2018.5056
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