Minimization of multi-penalty functionals by alternating iterative thresholding and optimal parameter choices
Minimization of multi-penalty functionals by alternating iterative thresholding and optimal parameter choices
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DOI：
10.1088/0266-5611/30/12/125003
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发表时间：
2014-12-01
期刊：
影响因子：
2.1
通讯作者：
Peter, Steffen
中科院分区：
文献类型：
--
作者：
Naumova, Valeriya;Peter, Steffen
受最近正则化理论、优化和信号处理的一些发展的启发，我们提出并分析了稀疏表示函数空间中的多惩罚正则化的数值方法。稀疏性先验是由高维数据的很大程度上预期的几何/结构特征驱动的，这些特征在典型的更各向同性的希尔伯特空间框架中可能无法很好地表示。在本文中，我们对能够正确建模和分离加性混合信号的多个分量的正则化器特别感兴趣。这种情况相当普遍，因为纯信号可能被加性噪声破坏。为此，我们考虑了一个由数据保真度项组成的正则化函数，其中信号和噪声是加性混合的，一个非光滑和非凸稀疏性促进项，以及一个惩罚项来建模噪声。我们提出并分析了一种基于简单迭代阈值步骤的迭代交替算法的收敛性，以实现函数的最小化。通过该算法，探讨了选择不同的正则化参数和惩罚范数对恢复纯信号和分离加性噪声质量的影响。对于给定的固定噪声水平，数值实验证实了与标准的单参数正则化方法相比，该方法的性能有显著提高。通过使用高维数据分析方法，如主成分分析，我们能够在期望解周围显示正则化解的正确几何聚类。最后，对于实验中考虑的压缩感知问题，我们提供了正则化规范和参数选择的指导方针。
Inspired by several recent developments in regularization theory, optimization, and signal processing, we present and analyze a numerical approach to multi-penalty regularization in spaces of sparsely represented functions. The sparsity prior is motivated by the largely expected geometrical/structured features of high-dimensional data, which may not be well-represented in the framework of typically more isotropic Hilbert spaces. In this paper, we are particularly interested in regularizers which are able to correctly model and separate the multiple components of additively mixed signals. This situation is rather common as pure signals may be corrupted by additive noise. To this end, we consider a regularization functional composed by a data-fidelity term, where signal and noise are additively mixed, a non-smooth and non-convex sparsity promoting term, and a penalty term to model the noise. We propose and analyze the convergence of an iterative alternating algorithm based on simple iterative thresholding steps to perform the minimization of the functional. By means of this algorithm, we explore the effect of choosing different regularization parameters and penalization norms in terms of the quality of recovering the pure signal and separating it from additive noise. For a given fixed noise level numerical experiments confirm a significant improvement in performance compared to standard one-parameter regularization methods. By using high-dimensional data analysis methods such as principal component analysis, we are able to show the correct geometrical clustering of regularized solutions around the expected solution. Eventually, for the compressive sensing problems considered in our experiments we provide a guideline for a choice of regularization norms and parameters.