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Minimization of fraction function penalty in compressed sensing. (arXiv:1705.06048v1 [math.OC])
来源于:arXiv
In the paper, we study the minimization problem of a non-convex sparsity
promoting penalty function
$$P_{a}(x)=\sum_{i=1}^{n}p_{a}(x_{i})=\sum_{i=1}^{n}\frac{a|x_{i}|}{1+a|x_{i}|}$$
in compressed sensing, which is called fraction function. Firstly, we discuss
the equivalence of $\ell_{0}$ minimization and fraction function minimization.
It is proved that there corresponds a constant $a^{**}>0$ such that, whenever
$a>a^{**}$, every solution to $(FP_{a})$ also solves $(P_{0})$, that the
uniqueness of global minimizer of $(FP_{a})$ and its equivalence to $(P_{0})$
if the sensing matrix $A$ satisfies a restricted isometry property (RIP) and,
last but the most important, that the optimal solution to the regularization
problem $(FP_{a}^\lambda)$ also solves $(FP_{a})$ if the certain condition is
satisfied, which is similar to the regularization problem in convex optimal
theory. Secondly, we study the properties of the optimal solution to the
regularization problem $(FP^{\lambda}_{a})$ 查看全文>>