A wonderful triangle in compressed sensing

In order to determine the sparse approximation function which has a direct metric relationship with the ℓ0 quasi-norm, we introduce a wonderful triangle whose sides are composed of Vert x Vert0, Vert x Vert1 and Vert x Vert∞ for any non-zero vector x ∈ ℝn by delving into the iterative soft-thresholding operator in this paper. Based on this triangle, we deduce the ratio ℓ1 and ℓ∞ norms as a sparsity-promoting objective function for sparse signal reconstruction and also try to give the sparsity interval of the signal. Considering the ℓ1/ℓ∞ minimization from a angle β of the triangle corresponding to the side whose length is Vert x Vert∞ - Vert x Vert1/Vert x Vert0, we finally demonstrate the performance of existing ℓ1/ℓ∞ algorithm by comparing it with ℓ1/ℓ2 algorithm.