Computing the proximity operator of the ℓ<sub>p</sub> norm with 0 < p < 1

Feishe Chen; Lixin Shen; Bruce W. Suter

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Computing the proximity operator of the ℓ_p norm with 0 < p < 1

机译：计算0 <1的ℓ_{p 范数的接近算子}

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Sparse modelling with the ℓp norm of 0 ≤ p ≤ 1 requires the availability of the proximity operator of the ℓp norm. The proximity operators of the ℓ0 and ℓ1 norms are the well-known hard- and soft-thresholding estimators, respectively. In this study, the authors give a complete study on the properties of the proximity operator of the ℓp norm. Based on these properties, explicit formulas of the proximity operators of the ℓ1/2 norm and ℓ2/3 norm are derived with simple proofs; for other values of p, an iterative Newton's method is developed to compute the proximity operator of the ℓp norm by fully exploring the available proximity operators of the ℓ0, ℓ1/2, ℓ2/3, and ℓ1 norms. As applications, the proximity operator of the ℓp norm with 0 ≤ p ≤ 1 is applied to the ℓp-regularisation for compressive sensing and image restoration.

机译：ℓp范数为0≤p≤1的稀疏建模需要ℓp范数的接近算子的可用性。 ℓ0和ℓ1范数的接近算子分别是众所周知的硬阈值估计和软阈值估计。在这项研究中，作者对ℓp范数的邻近算子的性质进行了完整的研究。基于这些性质，用简单的证明推导ℓ1/ 2范数和ℓ2/ 3范数的近似算子公式；对于p的其他值，通过充分探索ℓ0，ℓ1/ 2，ℓ2/ 3和ℓ1范数的可用接近算子，开发了牛顿迭代法来计算ℓp范数的接近算子。作为应用，将0≤p≤1的ℓp范数的接近算子应用于ℓp正则化，以进行压缩感测和图像恢复。

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《Signal Processing, IET》 |2016年第5期|557-565|共9页
作者
Feishe Chen; Lixin Shen; Bruce W. Suter;
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Syracuse University, USA;

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Computing the proximity operator of the ℓp norm with 0 < p < 1

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Computing the proximity operator of the ℓ_p norm with 0 < p < 1