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.

INVESTIGATION OF MAGNESIUM ION CHARGE STORAGE MECHANISM IN MANGANESE DIOXIDE

Magnesium (Mg) battery is considered as a promising candidate for the next generation battery technology that could potentially replace the current lithium (Li)-ion batteries due to the following factors. Magnesium possesses a higher volumetric capacity than commercialized Li-ion battery anode materials. Additionally, the low cost and high abundance of Mg compared to Li makes Mg batteries even more attractive. Moreover, unlike metallic Li anodes which have a tendency to develop a dendritic structure on the surface upon the cycling of the battery, Mg metal is known to be free from such a hazardous phenomenon. Due to these merits of Mg as an anode, the topic of rechargea¬ble Mg batteries has attracted considerable attention among researchers in the last few decades. However, the aforementioned advantages of Mg batteries have not been fully utilized due to the serious kinetic limitation of Mg2+ diffusion process in many hosting compounds which is believed to be due to a strong electrostatic interaction between divalent Mg2+ ions and hosting matrix. This serious kinetic hindrance is directly related to the lack of cathode materials for Mg battery that provide comparable electrochemical performances to that of Li-based system. Manganese oxide (MnO2) is one of the most well studied electrode materials due to its excellent electrochemical properties, including high Li+ ion capacity and relatively high operating voltage (i.e., ~ 4 V vs. Li/Li+ for LiMn2O4 and ~ 3.2 V vs. Mg/Mg2+). However, unlike the good electrochemical properties of MnO2 realized in Li-based systems, rather poor electrochemical performances have been reported in Mg based systems, particularly with low capacity and poor cycling performances. While the origin of the observed poor performances is believed to be due to the aforementioned strong ionic interaction between the Mg2+ ions and MnO2 lattice resulting in a limited diffusion of Mg2+ ions in MnO2, very little has been explored regarding the charge storage mechanism of MnO2 with divalent Mg2+ ions. This dissertation investigates the charge storage mechanism of MnO2, focusing on the insertion behaviors of divalent Mg2+ ions and exploring the origins of the limited Mg2+ insertion behavior in MnO2. It is found that the limited Mg2+ capacity in MnO2 can be significantly improved by introducing water molecules in the Mg electrolyte system, where the water molecules effectively mitigated the kinetic hindrance of Mg2+ insertion process. The combination of nanostructured MnO2 electrode and water effect provides a synergic effect demonstrating further enhanced Mg2+ insertion capability. Furthermore, it is demonstrated in this study that pre-cycling MnO2 electrodes in water-containing electrolyte activates MnO2 electrode, after which improved Mg2+ capacity is maintained in dry Mg electrolyte. Based on a series of XPS analysis, a conversion mechanism is proposed where magnesiated MnO2 undergoes a conversion reaction to Mg(OH)2 and MnOx and Mn(OH)y species in the presence of water molecules. This conversion process is believed to be the driving force that generates the improved Mg2+ capacity in MnO2 along with the water molecule’s charge screening effect. Finally, it is discussed that upon a consecutive cycling of MnO2 in the water-containing Mg electrolyte, structural water is generated within the MnO2 lattice, which is thought to be the origin of the observed activation phenomenon. The results provided in this dissertation highlight that the divalency of Mg2+ ions result in very different electrochemical behaviors than those of the well-studied monovalent Li+ ions towards MnO2.