RI: Small: Uncertainty Quantification for Nonconvex Low-Complexity Models

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Emerging applications in data science often involve estimating an enormous number of parameters from a highly incomplete and noisy set of measurements. In order for these applications to support modern scientific discovery and decision making, however, it is necessary to seek not merely reasonable estimations for the parameters, but perhaps more crucially, a trustworthy interpretation of the estimations and their implications. For instance, what reassurances can we offer about the quality of the estimates in hand? Can we quantify the uncertainty of our estimates due to the imperfectness of the data? Providing valid and quantitative answers to such questions is a crucial step in ensuring that: the scientific discovery and decision made based on our estimate are informative and trustworthy. Nevertheless, the existing statistical toolbox remains highly inadequate in providing measures of uncertainty for large-scale estimation methods, particularly in those scenarios where the availability of data samples is severely limited. This limits the overall value of the estimates and hampers scientific and decision-making processes. Some example application areas include: joint shape matching in computer vision and water-fat separation in medical imaging. Motivated by the above issues, the overarching goal of this project is to develop new foundational theory that integrates statistical assessment and algorithm design in an end-to-end manner, allowing for optimal inferential procedures for various nonconvex low-complexity models. Blending large-scale optimization techniques with statistical thinking, the proposed project seeks to develop a novel suite of distributional theory that enables valid uncertainty assessment for various nonconvex low-complexity models. Specifically, this project consists of the following research. First, develop a principled approach to construct optimal confidence intervals for unknown continuous parameters, on the basis of novel nonconvex estimation and de-biasing methods. Second, develop fast nonconvex algorithms and efficient uncertainty assessment procedures to reason about unknown discrete variables. Third, investigate the intimate connection between convex relaxation and nonconvex optimization, thus enabling a unified uncertainty quantification framework to accommodate both approaches. All research thrusts are motivated by, and will ultimately be tested on concrete practical applications. This project will significantly advance the fundamental techniques of uncertainty quantification in data-driven applications, and will enrich the foundations for mathematical optimization, data analytics, and statistical modeling.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.