Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.

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.

Assessment and Life-Cycle Analysis of Recycled Materials for Sustainable Highway

Recycled materials replacing part of virgin materials in highway applications has shown great benefits to the society and environment. Beneficial use of recycled materials can save landfill places, sparse natural resources, and energy consumed in milling and hauling virgin materials. Low price of recycled materials is favorable to cost-saving in pavement projects. Considering the availability of recycled materials in the State of Maryland (MD), four abundant recycled materials, recycled concrete aggregate (RCA), recycled asphalt pavement (RAP), foundry sand (FS), and dredged materials (DM), were studied. A survey was conducted to collect the information of current usage of the four recycled materials in States’ Department of Transportation (DOTs). Based on literature review, mechanical and environmental properties, recommendations, and suggested test standards were investigated separately for the four recycled materials in different applications. Constrains in using these materials were further studied in order to provide recommendations for the development of related MD specifications. To measure social and environmental benefits from using recycled materials, life-cycle assessment was carried out with life-cycle analysis (LCA) program, PaLATE, and green highway rating system, BEST-in-Highway. The survey results indicated the wide use of RAP and RCA in hot mix asphalt (HMA) and graded aggregate base (GAB) respectively, while FS and DM are less used in field. Environmental concerns are less, but the possibly low quality and some adverse mechanical characteristics may hinder the widely use of these recycled materials. Technical documents and current specifications provided by State DOTs are good references to the usage of these materials in MD. Literature review showed consistent results with the survey. Studies from experimental research or site tests showed satisfactory performance of these materials in highway applications, when the substitution rate, gradation, temperature, moisture, or usage of additives, etc. meet some requirements. The results from LCA revealed significant cost savings in using recycled materials. Energy and water consumption, gas emission, and hazardous waste generation generally showed reductions to some degree. Use of new recycled technologies will contribute to more sustainable highways.