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.

The conformational landscape of RNA translational regulators and their potential as drug discovery targets

RNA is an underutilized target for drug discovery. Once thought to be a passive carrier of genetic information, RNA is now known to play a critical role in essentially all aspects of biology including signaling, gene regulation, catalysis, and retroviral infection. It is now well-established that RNA does not exist as a single static structure, but instead populates an ensemble of energetic minima along a free-energy landscape. Knowledge of this structural landscape has become an important goal for understanding its diverse biological functions. In this case, NMR spectroscopy has emerged as an important player in the characterization of RNA structural ensembles, with solution-state techniques accounting for almost half of deposited RNA structures in the PDB, yet the rate of RNA structure publication has been stagnant over the past decade. Several bottlenecks limit the pace of RNA structure determination by NMR: the high cost of isotopic labeling, tedious and ambiguous resonance assignment methods, and a limited database of RNA optimized pulse programs. We have addressed some of these challenges to NMR characterization of RNA structure with applications to various RNA-drug targets. These approaches will increasingly become integral to designing new therapeutics targeting RNA.