一般噪声矩阵系统的算子移位
Operator Shifting for General Noisy Matrix Systems
摘要：
在计算科学中，人们必须经常从受噪声和不确定性影响的数据中估计模型参数，从而导致不准确的结果。为了提高含噪声参数模型的精度，研究了算子被噪声污染的线性系的降阶误差问题。本文的贡献是将Etter，Ying '20中的椭圆算子移位框架推广到一般的非对称矩阵情形。粗略地说，算子移位技术是James-Stein估计量的矩阵模拟。关键的见解是，在适当选择的方向上移动矩阵逆估计将减少平均误差。在我们的扩展中，我们询问了一些问题-即，是否向原点移动一般矩阵逆总是减少错误，因为它在椭圆形的情况下。我们表明，这通常是这样的情况下，但有三个关键特征的一般非奇异矩阵，允许对抗的例子不可能在对称的情况下。我们证明，当这些对抗的可能性被消除的假设噪声对称性和使用的残差范数作为误差度量，最佳的移位总是朝着原点，镜像结果从Etter，Ying '20。我们还研究了在小噪音制度和其他情况下的行为。最后，我们通过展示受强化学习启发的数值实验（附带信源编码）来证明算子移位可以大幅减少错误。
英文摘要：
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.
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