Johannes Heiny: Phase transitions for high-dimensional random matrices with applications in statistical learning

Abstract

In this talk, we investigate the linear spectral statistics (LSS) of sample correlation matrices, which are used to understand the relationships between multiple variables in various fields, such as finance, biology, and social sciences. Sample correlation matrices, often derived from financial data, genomic studies, or environmental measurements, provide insights into the underlying relationships among variables.

We investigate the asymptotic behavior of LSS as the sample size and dimensionality increase, focusing on the contributions of eigenvalues to statistical inference with a particular emphasis on heavy-tailed scenarios. Using advanced techniques from random matrix theory and extreme value theory, we derive limiting distributions for LSS, which offer improved understanding of correlations in high-dimensional data. Our findings include specific conditions under which the Central Limit Theorem holds for LSS and the implications for hypothesis testing in multivariate settings. Moreover, we will analyze cases where LSS do not satisfy a Central Limit Theorem. This study contributes to the growing intersection of statistics, stochastic geometry and machine learning, fostering improved methodologies for extracting meaningful patterns from high-dimensional data.