In order to determine the sparse approximation function which has a direct metric relationship with the $\ell_{0}$ quasi-norm, we introduce a wonderful triangle whose sides are composed of $\Vert \mathbf{x} \Vert_{0}$, $\Vert \mathbf{x} \Vert_{1}$ and $\Vert \mathbf{x} \Vert_{\infty}$ for any non-zero vector $\mathbf{x} \in \mathbb{R}^{n}$ by delving into the iterative soft-thresholding operator in this paper. Based on this triangle, we deduce the ratio $\ell_{1}$ and $\ell_{\infty}$ norms as a sparsity-promoting objective function for sparse signal reconstruction and also try to give the sparsity interval of the signal. Considering the $\ell_{1}/\ell_{\infty}$ minimization from a angle $\beta$ of the triangle corresponding to the side whose length is $\Vert \mathbf{x} \Vert_{\infty} - \Vert \mathbf{x} \Vert_{1}/\Vert \mathbf{x} \Vert_{0}$, we finally demonstrate the performance of existing $\ell_{1}/\ell_{\infty}$ algorithm by comparing it with $\ell_{1}/\ell_{2}$ algorithm.

翻译：为了确定与 $\ell_{0}$ 拟范数具有直接度量关系的稀疏逼近函数，本文通过深入探究迭代软阈值算子，引入了一个美妙的三角。该三角由任意非零向量 $\mathbf{x} \in \mathbb{R}^{n}$ 的 $\Vert \mathbf{x} \Vert_{0}$、$\Vert \mathbf{x} \Vert_{1}$ 和 $\Vert \mathbf{x} \Vert_{\infty}$ 构成。基于此三角，我们推导出 $\ell_{1}$ 与 $\ell_{\infty}$ 范数之比作为稀疏信号重建的稀疏性促进目标函数，并尝试给出信号的稀疏度区间。通过从该三角的一个角度 $\beta$（对应于长度为 $\Vert \mathbf{x} \Vert_{\infty} - \Vert \mathbf{x} \Vert_{1}/\Vert \mathbf{x} \Vert_{0}$ 的边）来考虑 $\ell_{1}/\ell_{\infty}$ 最小化问题，我们最终通过与 $\ell_{1}/\ell_{2}$ 算法进行比较，验证了现有 $\ell_{1}/\ell_{\infty}$ 算法的性能。