Statistical Analysis and Spectral Methods for Signal-Plus-Noise Matrix Models

Abstract

The singular value matrix decomposition plays a ubiquitous role in statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and understanding the geometric structure of singular values and singular vectors.

Chapter 2 of this dissertation presents an initial analysis of local (e.g., entrywise) singular vector (resp., eigenvector) perturbations for signal-plus-noise matrix models. We obtain both deterministic and probabilistic upper bounds on singular vector perturbations that complement and in certain settings improve upon classical, well-established benchmark bounds in the literature. We then apply our tools and methods of analysis to problems involving (spike) principal subspace estimation for high-dimensional covariance matrices and network models exhibiting community structure. Subsequently, Chapter 3 obtains precise local eigenvector estimation results under stronger assumptions involving signal strength, probabilistic concentration, and homogeneity. We provide in silico simulation examples to illustrate our theoretical bounds and distributional limit theory. Chapter 4 transitions to the investigation of singular value (resp., eigenvalue) perturbations, still in the signal-plus-noise matrix model framework. There, our results are leveraged for the purpose of better understanding hypothesis testing and change-point detection in statistical random graph analysis. Chapter 5 builds upon recent joint analysis of singular (resp., eigen) values and vectors in order to investigate the asymptotic relationship between spectral embedding performance and underlying network structure for stochastic block model graphs.