Description
- Abstract:
- Abstract of “Problems at the interface of high dimensional probability and random matrix theory”, by Xiaoyu Xie, Ph.D., Brown University, May 2024 In this thesis, we delve into three distinct problems within the realm of high-dimensional probability, each associated with different ensembles of random matrices. Our exploration is conducted through the lens of asymptotic analysis, providing insights into the different aspects of these matrices and related objects as the dimensionality increases. Specifically, we focus on the following three ensembles: 1. Heavy-tailed random matrices, characterized by their probability distributions with slowly decaying tails, which may have infinite variance and/or expectation. 2. Light-tailed random matrices, known for their exponentially decaying tails. 3. Random matrices uniformly sampled from the Stiefel manifold, which is a set of matrices with orthonormal columns equipped with a natural manifold structure and a rotationally invariant Haar measure. For heavy-tailed random matrices, we investigate the Grothendieck problem and the associated nonlinear operator norms, uncovering phenomena not observed in light-tailed matrices or when employing spectral norms. For light-tailed matrices, we establish the universality of normal fluctuations in eigenvector projections. Lastly, within the context of the Stiefel manifold, we explore quenched large deviation principles for projections of high-dimensional random vectors. This study illuminates the process of extracting information from high-dimensional data through their lower-dimensional projections.
- Notes:
- Thesis (Ph. D.)--Brown University, 2024
Citation
Xie, Xiaoyu,
"Problems at the interface of high dimensional probability and random matrix theory"
(2024).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://repository.library.brown.edu/studio/item/bdr:kyj983bg/
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....