研究随机矩阵经归一化后的特征值统计规律，揭示其分布缩放法则与分解误差。
Statistics of Min-max Normalized Eigenvalues in Random Matrices
- 提出最小-最大归一化特征值的统计模型
- 发现累积分布的幂律缩放关系，验证理论预测
- 适用于数据科学中矩阵分解的误差分析
随机矩阵理论在纯数学、数学物理和机器学习等领域具有重要影响。从数据科学的实际应用出发，输入数据通常在处理前进行归一化。本文研究了随机矩阵经最小-最大归一化后特征值的统计特性。此前已有针对此类归一化特征值的有效分布模型。本文将其应用于评估累积分布的缩放规律，并推导了随机矩阵分解过程中产生的残差误差。通过数值实验验证了这些理论预测的准确性。
原文摘要 · Abstract (English)
Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus, this study investigates the statistical properties of min-max normalized eigenvalues in random matrices. Previously, the effective distribution for such normalized eigenvalues has been proposed. In this study, we apply it to evaluate a scaling law of the cumulative distribution. Furthermore, we derive the residual error that arises during matrix factorization of random matrices. We conducted numerical experiments to verify these theoretical predictions.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。