Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

EXAMINING THE IMPLEMENTATION CHALLENGES OF PROJECT-BASED LEARNING: A CASE STUDY

Title of Thesis: Thesis directed by: ABSTRACT EXAMINING THE IMPLEMENTATION CHALLENGES OF PROJECT-BASED LEARNING: A CASE STUDY Stefan Frederick Brooks, Master of Education, 2016 Professor and Chair Francine Hultgren Teaching and Learning, Policy and Leadership Department Project-based learning (PjBL) is a common instructional strategy to consider for educators, scholars, and advocates who focus on education reform. Previous research on PjBL has focused on its effectiveness, but a limited amount of research exists on the implementation challenges. This exploratory case study examines an attempted project- based learning implementation in one chemistry classroom at a private school that fully supports PjBL for most subjects with limited use in mathematics. During the course of the study, the teacher used a modified version of PjBL. Specifically, he implemented some of the elements of PjBL, such as a driving theme and a public presentation of projects, with the support of traditional instructional methods due to the context of the classroom. The findings of this study emphasize the teacher’s experience with implementing some of the PjBL components and how the inherent implementation challenges affected his practice.