Sparse non‐negative signal reconstruction using fraction function penalty

Abstract

Many practical problems in the real world can be formulated as the non-negative ℓ 0-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 ℓ 0-minimisation problem is non-deterministic polynomial hard (NP-hard) because of the discrete and discontinuous nature of the ℓ 0-norm. Inspired by the good performances of the fraction function in the authors' former work, in this paper, the authors replace the ℓ 0norm 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 ℓ 0minimisation 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 ℓ 0-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.

A fast algorithm for sparse signal recovery via fraction function

“…, a > 0 is called the fraction function. In addition, the fraction function is also used to the matrix rank minimization problem [7,8] and sparse nonnegative signal recovery [9]. It is easy to verify that p a (t ) = 0 if t = 0 and lim a→+∞ p a (t ) = 1 if t = 0.…”

“…[6] substitute the $$\ell _{0}$$‐norm $$\Vert \mathbf {x}\Vert _{0}$$ by a nonconvex sparsity‐promoting penalty function, that is $$$$\begin{equation} P_{a}(\mathbf {x})=\sum _{i=1}^{n}p_{a}(\mathbf {x}_{i})=\sum _{i=1}^{n}\frac{a|\mathbf {x}_{i}|}{a|\mathbf {x}_{i}|+1}, \end{equation}$$$$where $$$$\begin{equation*} p_{a}(\mathbf {x}_{i})=\frac{a|\mathbf {x}_{i}|}{a|\mathbf {x}_{i}|+1}, \ a>0 \end{equation*}$$$$is called the fraction function. In addition, the fraction function is also used to the matrix rank minimization problem [7, 8] and sparse non‐negative signal recovery [9]. It is easy to verify that $$p_{a}(t)=0$$ if $$t=0$$ and $$\lim _{a\rightarrow +\infty }p_{a}(t)=1$...$…”