In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise variance are known, and the parameter choice involves minimizing a function whose solution depends only on the signal-to-noise ratio. Moreover, when neither noise variance nor data norm is given, we propose an iterative algorithm which alternates between a minimization step of finding the regularization parameter and an estimation step of estimating signal-to-noise ratio. Simulation studies on both small- and large-scale datasets suggest that the approach can provide very accurate and stable regularized inverse solutions and, for small sized samples, it outperforms discrepancy principle, balancing principle, unbiased predictive risk estimator, L-curve method generalized cross validation, and quasi-optimality criterion, and achieves excellent stability hitherto unavailable.

中文翻译：

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基于预测风险的Tikhonov正则化参数选择规则

在这项工作中，我们提出了一个新的标准，用于在噪声为高斯白噪声时在 Tikhonov 正则化中选择正则化参数。当数据范数和噪声方差都已知时，该标准最小化预测风险的下限，并且参数选择涉及最小化其解仅取决于信噪比的函数。此外，当既没有给出噪声方差也没有给出数据范数时，我们提出了一种迭代算法，该算法在寻找正则化参数的最小化步骤和估计信噪比的估计步骤之间交替。对小规模和大规模数据集的模拟研究表明，该方法可以提供非常准确和稳定的正则化逆解，对于小规模的样本，它优于差异原理、平衡原理、