Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Preliminary Validation of the "Plausibility of Intraracial Marriage Scale": A Study of Upwardly Mobile African American Women

Like many Americans, African American women often view marriage as an important life goal; however, it is likely that many also view it as one that is out of reach (Dixon, 2009; Wallace, 2014). The present study sought to address this issue by developing a new measure to assess African American women’s experiences in this regard in hopes of shedding light on how it impacts their dating and mating experiences. A total of 251 heterosexual African American women who were either enrolled in college or who had completed some form of postsecondary education completed an online survey assessing a variety of topics including their attitudes and emotions regarding a shortage of marriageable men, their standards to marry, and their willingness to date interracially. Results from this study established preliminary construct validity for a new scale assessing African American women’s attitudes and emotions about the shortage of “marriageable” African American men.