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.

TRADITION AND INNOVATION: ANALYSIS AND PERFORMANCE OF SELECTED COMPOSITIONS FOR VIOLONCELLO BY HEITOR VILLA-LOBOS

Brazilian composer Heitor Villa-Lobos (1887-1959) began his musical career as a cellist. When he was only twelve years old, it became imperativeupon the sudden and untimely death of his fatherthat the young Villa-Lobos earn money as a cellist to provide financial support for his mother and sisters. Villa-Lobos's intimate relationship with the cello eventually inspired him to compose great music for this instrument. This dissertation explores both the diversity of compositional technique and the evolution of style found in the music for cello written by Villa-Lobos. The project consists of two recorded recital performances and a written document exploring and analyzing those pieces. In the study of the music of Villa-Lobos, it is of great interest to consider the music's traditional European elements in combination (or even juxtaposition) with its imaginative and sometimes wildly innovative Brazilian character. His early works were greatly influenced by European Romantic composers such as Robert Schumann, Frédéric Chopin, and the virtuoso cellist/composer David Popper (whom Villa-Lobos idolized). Later, Villa-Lobos flourished in a newfound compositional independence and moved away from Euro-romanticism and toward the folk music of his Brazilian homeland. It is intriguing to experience this transition through an exploration of his cello compositions. The works examined and performed in this dissertation project are chosen from among the extensive number of Villa-Lobos's cello compositions and are his most important works for cello with piano, cello with another instrument, and cello with orchestra. The chosen works demonstrate the evolving range and combination of characteristic elements found in Villa-Lobos's compositional repertoire.

ANALYSIS AND SIMULATION OF ENERGY USE AND COST AT A MUNICIPAL WASTEWATER TREATMENT PLANT

The cost of electricity, a major operating cost of municipal wastewater treatment plants, is related to influent flow rate, power price, and power load. With knowledge of inflow and price patterns, plant operators can manage processes to reduce electricity costs. Records of influent flow, power price, and load are evaluated for Blue Plains Advanced Wastewater Treatment Plant. Diurnal and seasonal trends are analyzed. Power usage is broken down among treatment processes. A simulation model of influent pumping, a large power user, is developed. It predicts pump discharge and power usage based on wet-well level. Individual pump characteristics are tested in the plant. The model accurately simulates plant inflow and power use for two pumping stations [R2 = 0.68, 0.93 (inflow), R2 =0.94, 0.91(power)]. Wet-well stage-storage relationship is estimated from data. Time-varying wet-well level is added to the model. A synthetic example demonstrates application in managing pumps to reduce electricity cost.