Unbiased predictive risk estimation of the Tikhonov regularization parameter: convergence with increasing rank approximations of the singular value decomposition
Unbiased predictive risk estimation of the Tikhonov regularization parameter: convergence with increasing rank approximations of the singular value decomposition
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吉洪诺夫正则化参数的无偏预测风险估计：随着奇异值分解的递增秩近似的收敛
DOI：
10.1007/s10543-019-00762-7
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发表时间：
2019
影响因子：
1.5
通讯作者：
Vatankhah, Saeed
中科院分区：
文献类型：
--
作者：
Renaut, Rosemary A.;Helmstetter, Anthony W.;Vatankhah, Saeed
截断奇异值分解可以与Tikhonov正则化相结合用于求解线性离散病态问题，并且需要估计一个正则化参数，该参数在数据函数的拟合大小和正则化项之间取得平衡。无偏预测风险估计量是在测量噪声为正态分布且方差已知的情况下寻找正则化参数的一种方法。本文提出了一种使用无偏预测风险估计量的算法，该算法可以从奇异值分解中自动找到正则化参数和使用的项数。该算法的基础是一个新的结果，证明了正则化参数与奇异值分解的项数收敛。对于分析，只要假设精确数据满足离散皮卡德条件，并且噪声完全污染了足够多的项的测量数据系数，这取决于噪声水平和系统的不适定性程度。给出了正则化参数的下界，从而实现了计算效率高的算法。将支持结果与广义交叉验证方法的结果进行了比较。二维算例仿真验证了理论分析和算法提高噪声水平的有效性，并表明截断奇异值分解得到的相对重构误差小于奇异值分解得到的相对重构误差。这是一篇发表在BIT数值数学上的文章的预印本。最终的认证版本可在https://doi.org/10.1007%2Fs10543-019-00762-7上获得。
The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between the sizes of the fit to data function and the regularization term. The unbiased predictive risk estimator is one suggested method for finding the regularization parameter when the noise in the measurements is normally distributed with known variance. In this paper we provide an algorithm using the unbiased predictive risk estimator that automatically finds both the regularization parameter and the number of terms to use from the singular value decomposition. Underlying the algorithm is a new result that proves that the regularization parameter converges with the number of terms from the singular value decomposition. For the analysis it is sufficient to assume that the discrete Picard condition is satisfied for exact data and that noise completely contaminates the measured data coefficients for a sufficiently large number of terms, dependent on both the noise level and the degree of ill-posedness of the system. A lower bound for the regularization parameter is provided leading to a computationally efficient algorithm. Supporting results are compared with those obtained using the method of generalized cross validation. Simulations for two-dimensional examples verify the theoretical analysis and the effectiveness of the algorithm for increasing noise levels, and demonstrate that the relative reconstruction errors obtained using the truncated singular value decomposition are less than those obtained using the singular value decomposition. This is a pre-print of an article published in BIT Numerical Mathematics. The final authenticated version is available online at: https://doi.org/10.1007%2Fs10543-019-00762-7.
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