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.

RELATIONSHIP BETWEEN COLUMN DENSITY AND SURFACE MIXING RATIO FOR O3 AND NO2: IMPLICATIONS FOR SATELLITE OBSERVATIONS AND THE IMPACTS OF VERTICAL MIXING

Satellites have great potential for diagnosis of surface air quality conditions, though reduced sensitivity of satellite instrumentation to the lower troposphere currently impedes their applicability. One objective of the NASA DISCOVER-AQ project is to provide information relevant to improving our ability to relate satellite-observed columns to surface conditions for key trace gases and aerosols. In support of DISCOVER-AQ, this dissertation investigates the degree of correlation between O3 and NO2 column abundance and surface mixing ratio during the four DISCOVER-AQ deployments; characterize the variability of the aircraft in situ and model-simulated O3 and NO2 profiles; and use the WRF-Chem model to further investigate the role of boundary layer mixing in the column-surface connection for the Maryland 2011 deployment, and determine which of the available boundary layer schemes best captures the observations. Simple linear regression analyses suggest that O3 partial column observations from future satellite instruments with sufficient sensitivity to the lower troposphere may be most meaningful for surface air quality under the conditions associated with the Maryland 2011 campaign, which included generally deep, convective boundary layers, the least wind shear of all four deployments, and few geographical influences on local meteorology, with exception of bay breezes. Hierarchical clustering analysis of the in situ O3 and NO2 profiles indicate that the degree of vertical mixing (defined by temperature lapse rate) associated with each cluster exerted an important influence on the shapes of the median cluster profiles for O3, as well as impacted the column vs. surface correlations for many clusters for both O3 and NO2. However, comparisons to the CMAQ model suggest that, among other errors, vertical mixing is overestimated, causing too great a column-surface connection within the model. Finally, the WRF-Chem model, a meteorology model with coupled chemistry, is used to further investigate the impact of vertical mixing on the O3 and NO2 column-surface connection, for an ozone pollution event that occurred on July 26-29, 2011. Five PBL schemes were tested, with no one scheme producing a clear, consistent “best” comparison with the observations for PBLH and pollutant profiles; however, despite improvements, the ACM2 scheme continues to overestimate vertical mixing.