Spectral methods are a staple of modern statistics. For statistical learning tasks such as clustering or classification, one can featurize the data with spectral methods and then perform the task on the features. Despite the success and wide use of spectral methods, certain theoretical properties of spectral methods are not well-understood. In this oral defense, we investigate the uniform convergence (as opposed to convergence in an ``average'' sense) of the spectral embeddings obtained from the leading eigenvectors of certain similarity matrices to their population counterparts. We will first introduce necessary preliminaries and review existing results on this topic, and then explain the motivations and benefits for studying the convergence in a uniform sense. After that, we present the two main results in our work. The first result is a general perturbation result for orthonormal bases of invariant subspaces that can serve as a general recipe for establishing uniform consistency type results. The second result in an application of the first result to normalized spectral clustering---by tapping into the rich literature of Sobolev spaces and exploiting some concentration results in Hilbert spaces, we are able to prove a finite sample error bound on the uniform consistency error of the spectral embeddings in normalized spectral clustering.

The material in this oral defense is based on the first chapter of Ruofei Zhao's thesis "Convergence and Consistency Results in Spectral Clustering and Gaussian Mixture Models".