摘要翻译：
我们刻画了邻近算子，即映射向量到惩罚最小二乘优化问题解的函数。凸罚的邻近算子已被Moreau广泛研究并充分刻画。它们在实践中也被广泛地应用于非凸惩罚，如{\ell}0伪范数，然而将莫罗的刻画推广到这个设置似乎是文献中缺少的一个元素。我们将（凸或非凸）惩罚的邻近算子刻画为某些凸势的次微分函数。这是根据某些凸势对可能非凸罚的所谓Bregman邻近算子的更一般刻划的结果。作为我们分析的一个副作用，我们得到了一个检验来验证一个给定的函数是否是某个惩罚的邻近算子，或者不是。许多著名的收缩算子确实被证实是接近算子。然而，我们证明了窗口群套索和持久经验Wiener收缩--所谓社会稀疏收缩的两种形式--通常不是任何惩罚的邻近算子；例外情况是，当它们是组稀疏收缩的简单加权版本时，具有不重叠的组。
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英文标题：
《A characterization of proximity operators》
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作者：
R\'emi Gribonval (PANAMA, DANTE), Mila Nikolova (CMLA)
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最新提交年份：
2020
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分类信息：

一级分类：Mathematics        数学
二级分类：Classical Analysis and ODEs        经典分析与颂歌
分类描述：Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics
特殊函数、正交多项式、调和分析、Ode、微分关系、变分法、逼近、展开、渐近
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一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Signal Processing        信号处理
分类描述：Theory, algorithms, performance analysis and applications of signal and data analysis, including physical modeling, processing, detection and parameter estimation, learning, mining, retrieval, and information extraction. The term "signal" includes speech, audio, sonar, radar, geophysical, physiological, (bio-) medical, image, video, and multimodal natural and man-made signals, including communication signals and data. Topics of interest include: statistical signal processing, spectral estimation and system identification; filter design, adaptive filtering / stochastic learning; (compressive) sampling, sensing, and transform-domain methods including fast algorithms; signal processing for machine learning and machine learning for signal processing applications; in-network and graph signal processing; convex and nonconvex optimization methods for signal processing applications; radar, sonar, and sensor array beamforming and direction finding; communications signal processing; low power, multi-core and system-on-chip signal processing; sensing, communication, analysis and optimization for cyber-physical systems such as power grids and the Internet of Things.
信号和数据分析的理论、算法、性能分析和应用，包括物理建模、处理、检测和参数估计、学习、挖掘、检索和信息提取。“信号”一词包括语音、音频、声纳、雷达、地球物理、生理、（生物）医学、图像、视频和多模态自然和人为信号，包括通信信号和数据。感兴趣的主题包括：统计信号处理、谱估计和系统辨识；滤波器设计；自适应滤波/随机学习；（压缩）采样、传感和变换域方法，包括快速算法；用于机器学习的信号处理和用于信号处理应用的机器学习；网络与图形信号处理；信号处理中的凸和非凸优化方法；雷达、声纳和传感器阵列波束形成和测向；通信信号处理；低功耗、多核、片上系统信号处理；信息物理系统的传感、通信、分析和优化，如电网和物联网。
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一级分类：Mathematics        数学
二级分类：Functional Analysis        功能分析
分类描述：Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory
Banach空间，函数空间，实函数，积分变换，分布理论，测度理论
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英文摘要：
  We characterize proximity operators, that is to say functions that map a vector to a solution of a penalized least squares optimization problem. Proximity operators of convex penalties have been widely studied and fully characterized by Moreau. They are also widely used in practice with nonconvex penalties such as the {\ell} 0 pseudo-norm, yet the extension of Moreau's characterization to this setting seemed to be a missing element of the literature. We characterize proximity operators of (convex or nonconvex) penalties as functions that are the subdifferential of some convex potential. This is proved as a consequence of a more general characterization of so-called Bregman proximity operators of possibly nonconvex penalties in terms of certain convex potentials. As a side effect of our analysis, we obtain a test to verify whether a given function is the proximity operator of some penalty, or not. Many well-known shrinkage operators are indeed confirmed to be proximity operators. However, we prove that windowed Group-LASSO and persistent empirical Wiener shrinkage -- two forms of so-called social sparsity shrinkage-- are generally not the proximity operator of any penalty; the exception is when they are simply weighted versions of group-sparse shrinkage with non-overlapping groups.
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PDF链接：
https://arxiv.org/pdf/1807.04014