In this paper, we propose two classes of the approximations to the cardinality function via the Moreau envelope of the $ \ell_{1} $ norm. We show that these two approximations are good choices of the merit function for sparsity and are essentially the truncated $ \ell_{1} $ norm and the truncated $ \ell_{2} $ norm. Moreover, we apply the approximations to solve sparse signal recovery problems and then provide new weights for reweighted $ \ell_{1} $ minimization and reweighted least squares to find sparse solutions of underdetermined linear systems of equations. Finally, we present some numerical experiments to illustrate our results.

中文翻译：

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通过以下近似来重建稀疏信号 \ begin {document} $ \ ell_ {0} $ \ end {document} 拟西诺

在本文中，我们通过$ \ ell_ {1} $范数的Moreau包络，提出了两类基数函数的近似值。我们显示这两个近似值是稀疏度优值函数的不错选择，本质上是截断的$ \ ell_ {1} $范数和截断的$ \ ell_ {2} $范数。此外，我们应用近似值来解决稀疏信号恢复问题，然后为重新加权的\\ ell_ {1} $最小化和重新加权的最小二乘提供新的权重，以找到欠定线性方程组的稀疏解。最后，我们提出一些数值实验来说明我们的结果。