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.

翻译：在离散线性系统求逆过程中，数据中的噪声可能被放大并导致无意义的解。为抑制此效应，求逆时需在数学上引入符合期望的解的特征，这一过程称为正则化。通过非负正则化参数控制所提供先验信息的影响。目前已有多种方法用于选取合适的正则化参数，同时也有多种求逆方法。本文推导了基于无偏风险估计和广义交叉验证的新方法，用于寻找谱窗正则化参数。这些估计量被扩展至多数据集情形（其中系统矩阵共享），从而确定正则化参数。研究表明，通过这些新估计量可针对多数据和多窗口情形学习谱窗正则化参数。结果表明，这些无需使用真实数据即可学习正则化参数的改进方法高效且有效，其性能与基于真实数据估计参数的学习方法相当。该理论发展在二维图像去模糊案例中得到了验证。结果证实，所获得的谱窗正则化参数估计可有效用于独立于训练数据之外的验证数据集，且无需已知数据。