Date of Award

Fall 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Wu, Yihong

Abstract

Random matrices play a central role in modern probability theory and high-dimensionalstatistics, both as fundamental mathematical objects and as powerful tools for inference. This dissertation presents several works on random matrix theory and its connections to statistical inference with correlated structures in high dimensions. On the random matrix side, we investigate the spectral properties of sample covariancematrices. Our approach is built upon the resolvent method, and we utilize local laws for various spectral analyses. Specifically, our contributions are three-fold: (1) we attain the first explicit rate of convergence for the largest eigenvalue of sample covariance matrices; (2) we derive the first quantitative estimate for the universality of the smallest singular value for random matrices at the hard edge; (3) we obtain the first sensitivity analysis for the principal component with respect to resampling of the data as the perturbation effect. These results deepen our understanding of the classical universality phenomenon. On the statistical side, we study inference of latent structures and correlations fromnoisy data. In particular, we focus on random graph matching of geometric models and empirical Bayes estimation in high-dimensional linear models. For these statistical inference tasks, we establish both positive and negative results: we derive optimal information-theoretic limits and, in parallel, develop algorithms that achieve these limits whenever feasible. These results solve open problems in the fields and also provide new directions to explore in the future.

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