Abstract

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 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 importance for subsequent inference.

摘要

我们精确地量化了统计误差对随机矩阵特征分解的数值逼近质量的影响，并且在温和的条件下，我们使用它来为随机矩阵的谱分解中的残差引入最佳数值容差。我们证明，当数值误差和统计误差具有相同的阶数时终止特征分解算法会导致计算节省而不会损失准确性。我们通过对模拟和真实网络的分析来说明我们的停止标准的实际后果。我们的理论结果和实际数据示例表明，统计误差和数值误差之间的权衡对于后续推理非常重要。