In the computational sciences, one must often estimate model parameters from data subject to noise and uncertainty, leading to inaccurate results. In order to improve the accuracy of models with noisy parameters, we consider the problem of reducing error in a linear system with the operator corrupted by noise. Our contribution in this paper is to extend the elliptic operator shifting framework from Etter, Ying '20 to the general nonsymmetric matrix case. Roughly, the operator shifting technique is a matrix analogue of the James-Stein estimator. The key insight is that a shift of the matrix inverse estimate in an appropriately chosen direction will reduce average error. In our extension, we interrogate a number of questions -- namely, whether or not shifting towards the origin for general matrix inverses always reduces error as it does in the elliptic case. We show that this is usually the case, but that there are three key features of the general nonsingular matrices that allow for adversarial examples not possible in the symmetric case. We prove that when these adversarial possibilities are eliminated by the assumption of noise symmetry and the use of the residual norm as the error metric, the optimal shift is always towards the origin, mirroring results from Etter, Ying '20. We also investigate behavior in the small noise regime and other scenarios. We conclude by presenting numerical experiments (with accompanying source code) inspired by Reinforcement Learning to demonstrate that operator shifting can yield substantial reductions in error.
翻译:在计算科学中,人们必须经常从受到噪音和不确定性影响的数据中估计模型参数,从而得出不准确的结果。为了提高带有噪音参数的模型的准确性,我们考虑减少线性系统中与操作者因噪音而腐蚀的线性系统中的错误的问题。我们在本文件中的贡献是将椭圆操作者框架从Ettter, Ying'20扩展至一般非对称矩阵案。大致上,操作者转移技术是詹姆斯-斯蒂因估测器的一个矩阵类比。关键的洞察力是,将矩阵的反估转换为适当选择的方向将减少平均错误。在我们扩展过程中,我们询问一些问题 -- -- 即一般矩阵反向的源头是否总是会减少错误。我们表明,通常情况是这样的,但一般的非直线性矩阵有三个关键特征,使得在对称案中不可能出现对抗性实例。我们证明,当这些对抗性可能性被假定的噪音对准度误差所排除时,我们询问了若干问题 -- -- 即是否向一般矩阵的源值转移一般矩阵的源值,我们通过测量和递随附的递性结果的轨迹性实验结果,我们总是通过从对准性结果进行最优级的精确性原则的推算,我们通过测量结果的推算向其他的推算结果的推算,从而显示的推算结果的推算。