随机矩阵谱分解的数值容差
Numerical tolerance for spectral decompositions of random matrices
摘要：
我们精确地量化的质量的数值逼近随机矩阵特征分解的统计误差的影响，并在温和的条件下，我们用它来引入一个最佳的数值公差的残余误差的谱分解的随机矩阵。我们表明，终止的特征分解算法时，数值误差和统计误差是相同的顺序的结果在计算上的节省没有损失的准确性。我们还修复了一个无处不在的终止条件，在几个计算线性代数实现广泛使用的缺陷。我们说明了我们的停止标准与模拟和真实网络的分析的实际后果。我们的理论结果和实际数据的例子表明，统计和数值误差之间的权衡对于数据科学来说是非常重要的。
英文摘要：
We precisely quantify the impact of statistical error in the quality of a numerical approximation to a random matrix eigendecomposition, and under mild conditions, we use this to introduce an optimal numerical tolerance for residual error in spectral decompositions of random matrices. We demonstrate that terminating an eigendecomposition algorithm when the numerical error and statistical error are of the same order results in computational savings with no loss of accuracy. We also repair a flaw in a ubiquitous termination condition, one in wide employ in several computational linear algebra implementations. We illustrate the practical consequences of our stopping criterion with an analysis of simulated and real networks. Our theoretical results and real-data examples establish that the tradeoff between statistical and numerical error is of significant import for data science.
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