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.

An Exploration of Auditory Brainstem Encoding of Stop Consonants in Infants and Implications for Language Outcomes

Current trends in speech-language pathology focus on early intervention as the preferred tool for promoting the best possible outcomes in children with language disorders. Neuroimaging techniques are being studied as promising tools for flagging at-risk infants. In this study, the auditory brainstem response (ABR) to the syllables /ba/ and /ga/ was examined in 41 infants between 3 and 12 months of age as a possible tool to predict language development in toddlerhood. The MacArthur-Bates Communicative Development Inventory (MCDI) was used to assess language development at 18 months of age. The current study compared the periodicity of the responses to the stop consonants and phase differences between /ba/ and /ga/ in both at-risk and low-risk groups. The study also examined whether there are correlations among ABR measures (periodicity and phase differentiation) and language development. The study found that these measures predict language development at 18 months.