报告人： 胡彦林 博士

报告题目： Weighted Spectral Statistics of High-Dimensional Covariance and Correlation Matrices: Theory and Applications

时间： 2026年10月30日（周五）15:40—16:30

地点： 线上腾讯会议（会议号：826-857-372）

报告摘要：

In high-dimensional data analysis, variability along specified directions, the accuracy of principal subspace estimation, and the correlation structure of variable groups are closely connected to matrix-weighted spectral statistics. From this common perspective, this talk presents three lines of research on high-dimensional covariance and correlation matrices.

First, we establish central limit theorems for generalized linear spectral statistics of sample covariance matrices, characterize how the rank of the weight matrix affects centering and fluctuation scales, and develop eigenspace tests based on functional projection.

Second, we study precise asymptotic expansions and Gaussian fluctuations of principal subspace estimation errors, with applications to excess risk in principal component analysis, distributed principal subspace estimation, and hypothesis testing.

Finally, we develop a theory of generalized linear spectral statistics for sample correlation matrices, highlighting the effects of empirical marginal standardization and the fourth moments of non-Gaussian innovations on the limiting distributions. This theory yields methods for comparing group correlation strength and testing whether a multiple correlation coefficient is zero, together with procedures for selecting regularization parameters over a continuous interval using spectral statistics. Numerical simulations and real-data analyses illustrate the performance of these methods in high-dimensional structural inference.

报告人简介：

胡彦林，中国科学技术大学博士。本科毕业于中国科学技术大学数学学院，后在该校管理学院统计与金融系师从杨青特任教授开展统计学博士阶段研究。

主要研究方向为高维统计推断与随机矩阵理论，重点研究协方差与相关矩阵的广义线性谱统计量、主特征子空间估计及相应的假设检验。以第一作者在《The Annals of Statistics》和《Advances in Applied Probability》发表研究论文，内容涉及谱统计理论与风险度量。

曾获国家奖学金、中科院院长奖学金特别奖。近期工作涵盖主成分分析超额风险、分布式主子空间估计及高维相关结构检验，并结合生物组学数据开展应用研究。目前正开展高维知识蒸馏与随机特征模型的理论研究，关注蒸馏机制对预测风险的影响。