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.

ESSAYS ON THE WTO ACCESSION, HOUSEHOLD INCOME AND MULTIDIMENSIONAL POVERTY: EVIDENCE FROM CHINA

In the course of integrating into the global market, especially since China’s WTO accession, China has achieved remarkable GDP growth and has become the second largest economy in the world. These economic achievements have substantially increased Chinese incomes and have generated more government revenue for social progress. However, China’s economic progress, in itself, is neither sufficient for achieving desirable development outcomes nor a guarantee for expanding peoples’ capabilities. In fact, a narrow emphasis on GDP growth proves to be unsustainable, and may eventually harm the life quality of Chinese citizens. Without the right set of policies, a deepening trade-openness policy in China may enlarge social disparities and some people may further be deprived of basic public services and opportunities. To address these concerns, this dissertation, a set of three essays in Chapters 2-4, examines the impact of China's WTO accession on income distribution, compares China’s income and multidimensional poverty reduction and investigates the factors, including the WTO accession, that predict multidimensional poverty. By exploiting the exogenous variation in exposure to tariff changes across provinces and over time, Chapter 2 (Essay 1) estimates the causal effects of trade shocks and finds that China’s WTO accession has led to an increase in average household income, but its impacts are not evenly distributed. Households in urban areas have benefited more significantly than those in rural areas. Households with members working in the private sector have benefited more significantly than those in the public sector. However, the WTO accession has contributed to reducing income inequality between higher and lower income groups. Chapter 3 (Essay 2) explains and applies the Alkire and Foster Method (AF Method), examines multidimensional poverty in China and compares it with income poverty. It finds that China’s multidimensional poverty has declined dramatically during the period from 1989-2011. Reduction rates and patterns, however, vary by dimensions: multidimensional poverty reduction exhibits unbalanced regional progress as well as varies by province and between rural and urban areas. In comparison with income poverty, multidimensional poverty reduction does not always coincide with economic growth. Moreover, if one applies a single measure ─ either that of income or multidimensional poverty ─ a certain proportion of those who are poor remain unrecognized. By applying a logistic regression model, Chapter 4 (Essay 3) examines factors that predict multidimensional poverty and finds that the major factors predicting multidimensional poverty in China include household size, education level of the household head, health insurance coverage, geographic location, and the openness of the local economy. In order to alleviate multidimensional poverty, efforts should be targeted to (i) expand education opportunities for the household heads with low levels of education, (ii) develop appropriate geographic policies to narrow regional gaps and (iii) make macroeconomic policies work for the poor.