A Study of the Relationship Between Spectrum and Geometry Through Fourier Frames and Laplacian Eigenmaps

Abstract

This thesis has two parts. The first part is a study of Fourier frames. We follow the development of the theory, beginning with its classical roots in non-uniform sampling in Paley-Wiener Spaces, to its current state, the study of the spectral properties of finite measures on locally compact abelian groups. The aim of our study is to understand the relationship between the geometry of the supporting set of a measure and the spectral properties it exhibits.

In the second part, we study extensions of the Laplacian Eigenmaps algorithm and their uses in hyperspectral image analysis. In particular, we show that there is a natural way of including spatial information in the analysis that improves classification results. We also provide evidence supporting the use of Schrödinger Eigenmaps as a semisupervised tool for feature extraction. Finally, we show that Schrödinger Eigenmaps provides a platform for fusing Laplacian Eigenmaps with other clustering techniques, such as kmeans clustering.