通过Wilcoxon-Wigner随机矩阵检验潜在结构
Testing for latent structure via the Wilcoxon--Wigner random matrix of normalized rank statistics
AI总结：
本文提出基于Wilcoxon-Wigner随机矩阵的统计检验方法，用于大规模数据中潜在结构的检测，具有计算高效和鲁棒性强的特点。
摘要：
本文考虑在大规模对称数据矩阵中检验潜在结构的问题。目标是开发一种统计上合理的方法，具有广泛的应用性、计算效率高且对极端数据变化不敏感，从而克服现有方法的局限性。为此，我们引入并系统研究了某些对称矩阵，称为Wilcoxon-Wigner随机矩阵，其元素是从底层独立同分布的绝对连续随机变量样本中得到的归一化秩统计量。这些矩阵自然出现在统计学的一元问题矩阵化中，并在非参数统计、多元分析和数据降维之间形成概念上的接口。我们的结果表明，Wilcoxon-Wigner随机矩阵的主特征值及其对应的特征向量具有渐近正态波动，具有显式的中心化和缩放项。这些渐近结果使我们能够严格地进行参数无关和分布无关的谱方法，以解决两个假设检验问题，即社区检测和主子矩阵检测。数值示例展示了所提出方法的性能。在整个过程中，我们的发现与基于信号加噪声数据设置中独立条目对称随机矩阵的现有结果进行了对比。
英文摘要：
This paper considers the problem of testing for latent structure in large symmetric data matrices. The goal here is to develop statistically principled methodology that is flexible in its applicability, computationally efficient, and insensitive to extreme data variation, thereby overcoming limitations facing existing approaches. To do so, we introduce and systematically study certain symmetric matrices, called Wilcoxon--Wigner random matrices, whose entries are normalized rank statistics derived from an underlying independent and identically distributed sample of absolutely continuous random variables. These matrices naturally arise as the matricization of one-sample problems in statistics and conceptually lie at the interface of nonparametrics, multivariate analysis, and data reduction. Among our results, we establish that the leading eigenvalue and corresponding eigenvector of Wilcoxon--Wigner random matrices admit asymptotically Gaussian fluctuations with explicit centering and scaling terms. These asymptotic results enable rigorous parameter-free and distribution-free spectral methodology for addressing two hypothesis testing problems, namely community detection and principal submatrix detection. Numerical examples illustrate the performance of the proposed approach. Throughout, our findings are juxtaposed with existing results based on the spectral properties of independent entry symmetric random matrices in signal-plus-noise data settings.
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