摘要翻译：
在存在噪声的情况下恢复非线性退化的信号是一个具有挑战性的问题。本文通过最小化非凸最小二乘拟合准则与惩罚项之和来解决该问题。我们假设模型的非线性可以用有理函数来解释。另外，我们假设要寻找的信号是稀疏的，并且对$\ell_0$伪范数的有理逼近构成了一个适当的惩罚。由此得到的复合代价函数属于广义的半代数函数类。为了寻求全局最优解，可以将其转化为一个广义矩问题，并为此建立一个半定规划松弛层次。全局最优性是以增加维数为代价的，为了克服涉及变量数量的计算限制，必须仔细解决问题的结构。当非线性模型由卷积变换和成分非线性有理饱和组成时，一个实际感兴趣的情况是。然后，我们提出使用一个稀疏松弛，能够处理多达数百个优化变量。与将模型线性化的朴素方法相比，我们的实验表明，所提出的方法具有良好的性能。
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英文标题：
《Rational Optimization for Nonlinear Reconstruction with Approximate
  $\ell_0$ Penalization》
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作者：
Marc Castella and Jean-Christophe Pesquet and Arthur Marmin
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最新提交年份：
2018
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分类信息：

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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英文摘要：
  Recovering nonlinearly degraded signal in the presence of noise is a challenging problem. In this work, this problem is tackled by minimizing the sum of a non convex least-squares fit criterion and a penalty term. We assume that the nonlinearity of the model can be accounted for by a rational function. In addition, we suppose that the signal to be sought is sparse and a rational approximation of the $\ell_0$ pseudo-norm thus constitutes a suitable penalization. The resulting composite cost function belongs to the broad class of semi-algebraic functions. To find a globally optimal solution to such an optimization problem, it can be transformed into a generalized moment problem, for which a hierarchy of semidefinite programming relaxations can be built. Global optimality comes at the expense of an increased dimension and, to overcome computational limitations concerning the number of involved variables, the structure of the problem has to be carefully addressed. A situation of practical interest is when the nonlinear model consists of a convolutive transform followed by a componentwise nonlinear rational saturation. We then propose to use a sparse relaxation able to deal with up to several hundreds of optimized variables. In contrast with the naive approach consisting of linearizing the model, our experiments show that the proposed approach offers good performance.
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PDF链接：
https://arxiv.org/pdf/1808.00724