A wonderful triangle in compressed sensing
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In order to determine the sparse approximation function which has a direct metric relationship with the ℓ_0 quasi-norm, we introduce a wonderful triangle whose sides are composed of ‖𝐱‖_0, ‖𝐱‖_1 and ‖𝐱‖_∞ for any non-zero vector 𝐱∈ℝ^n by delving into the iterative soft-thresholding operator in this paper. Based on this triangle, we deduce the ratio ℓ_1 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 ℓ_1/ℓ_∞ minimization from a angle β of the triangle corresponding to the side whose length is ‖𝐱‖_∞ - ‖𝐱‖_1/‖𝐱‖_0, we finally demonstrate the performance of existing ℓ_1/ℓ_∞ algorithm by comparing it with ℓ_1/ℓ_2 algorithm.
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