Asymptotics and Statistical Inference in High-Dimensional Low-Rank Matrix Models

High-dimensional matrix and tensor data is ubiquitous in machine learning and statistics and often exhibits low-dimensional structure. With the rise of these types of data is the need to develop statistical inference procedures that adequately address the low-dimensional structure in a principled manner. In this dissertation we study asymptotic theory and statistical inference in structured low-rank matrix models in high-dimensional regimes where the column and row dimensions of the matrix are allowed to grow, and we consider a variety of settings for which structured low-rank matrix models manifest. Chapter 1 establishes the general framework for statistical analysis in high-dimensional low-rank matrix models, including introducing entrywise perturbation bounds, asymptotic theory, distributional theory, and statistical inference, illustrated throughout via the matrix denoising model. In Chapter 2, Chapter 3, and Chapter 4 we study the entrywise estimation of singular vectors and eigenvectors in different structured settings, with Chapter 2 considering heteroskedastic and dependent noise, Chapter 3 sparsity, and Chapter 4 additional tensor structure. In Chapter 5 we apply previous asymptotic theory to study a two-sample test for equality of distribution in network analysis, and in Chapter 6 we study a model for shared community memberships across multiple networks, and we propose and analyze a joint spectral clustering algorithm that leverages newly developed asymptotic theory for this setting. Throughout this dissertation we emphasize tools and techniques that are data-driven, nonparametric, and adaptive to signal strength, and, where applicable, noise distribution. The contents of Chapters 2-6 are based on the papers Agterberg et al. (2022b); Agterberg and Sulam (2022); Agterberg and Zhang (2022); Agterberg et al. (2020a) and Agterberg et al. (2022a) respectively, and Chapter 1 contains several novel results