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.

Surveillance in Cyberspace: Applying Natural and Place Manager Surveillance to System Trespassing

Research on the criminological side of system trespassing (i.e. unlawfully gaining access to a computer system) is relatively rare and has yet to examine the effect of the presence of other users on the system during the trespassing event (i.e. the time of communication between a trespasser’s system and the infiltrated system). This thesis seeks to analyze this relationship drawing on principles of Situational Crime Prevention, Routine Activities Theory, and restrictive deterrence. Data were collected from a randomized control trial of target computers deployed on the Internet network of a large U.S. university. This study examined whether the number (one or multiple) and type (administrative or non-administrative) of computer users present on a system reduced the seriousness and frequency of trespassing. Results indicated that the type of user (administrative) produced a restrictive deterrent effect and significantly reduced the frequency and duration of trespassing events.

CHARACTERISTICS OF PRIVATE HIGH SCHOOLS FOR STUDENTS WITH EMOTIONAL DISABILITIES AND PERCEPTIONS OF THEIR PUBLIC SCHOOL CASE MANAGERS

Placement of students with disabilities in private special-education schools remains costly and controversial. This is particularly concerning, given the lack of research on the characteristics and quality of these restrictive settings. The purpose of this study was to identify the academic and vocational course offerings and behavioral supports provided in private special-education schools the serve high school students with emotional disabilities (ED). Second, the research examined the perceptions of the quality of services in these setting from the perspectives of public school case managers. Using a mixed-method design to collect data, 9 administrative heads of private special-education schools were surveyed, and 7 public school case managers were interviewed. Results indicated that (a) private special-education schools offer the basic academic core courses needed to meet graduation requirements, (b) vocational options for students enrolled in these schools are quite limited, (c) these schools provide a variety of behavioral interventions and supports, and (d) case managers are concerned with the lack of academic rigor and inconsistent programming at these schools but applauded the notion that students with ED are exiting with a high school diploma. Findings from this study may have policy implications for improving and developing programming options for high school students with ED.