Learning spectral windowing parameters for regularization using unbiased predictive risk and generalized cross validation techniques for multiple data sets
Learning spectral windowing parameters for regularization using unbiased predictive risk and generalized cross validation techniques for multiple data sets
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DOI：
10.3934/ipi.2023006
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发表时间：
2021-12
影响因子：
1.3
通讯作者：
Michael J. Byrne;R. Renaut
中科院分区：
文献类型：
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作者：
Michael J. Byrne;R. Renaut
在离散线性系统的反演过程中，数据中的噪声会被放大并导致无意义的解。为了克服这种影响，在反演过程中用数学方法实现被认为是理想的解的特征，这是一个称为正则化的过程。所提供的先验信息的影响由非负正则化参数控制。有许多方法用于选择合适的正则化参数，也有许多方法用于反演。提出了寻找谱窗正则化参数的无偏风险估计和广义交叉验证新方法。对这些估计量进行了扩展，用于在存在多个具有公共系统矩阵的数据集时寻找正则化参数。结果表明，这些新的估计器适用于多数据和多窗口，可以学习到谱窗正则化参数。结果表明，这些改进的方法不需要使用真实数据来学习正则化参数，是有效和高效的，并且与基于使用真实数据估计参数的学习方法相当。在二维图像去模糊的情况下，理论发展得到了验证。结果表明，得到的谱窗正则化参数估计可以有效地用于与训练数据分离的验证数据集，并且不需要已知数据。
During the inversion of discrete linear systems noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion, which is a process called regularization. The influence of provided prior information is controlled by non-negative regularization parameter(s). There are a number of methods used to select appropriate regularization parameters, as well as a number of methods used for inversion. New methods of unbiased risk estimation and generalized cross validation are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. It is demonstrated that spectral windowing regularization parameters can be learned from these new estimators applied for multiple data and with multiple windows. The results demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a learning method based on estimating the parameters using true data. The theoretical developments are validated for the case of two dimensional image deblurring. The results verify that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.