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

数据科学中的新兴应用通常涉及从高度不完整和嘈杂的测量集中估计大量参数。然而，为了使这些应用支持现代科学发现和决策制定，有必要不仅寻求对参数的合理估计，而且可能更重要的是，对这些估计及其含义的可靠解释。例如，我们可以对手头的评估质量提供什么保证？我们能否量化由于数据不完善而导致的估计的不确定性？为这些问题提供有效和定量的答案，是确保根据我们的估计作出的科学发现和决定是信息丰富和值得信赖的关键一步。然而，现有的统计工具箱在为大规模估计方法提供不确定性度量方面仍然非常不足，特别是在数据样本可得性严重有限的情况下。这限制了评估的总体价值，并阻碍了科学和决策过程。一些应用领域包括：计算机视觉中的关节形状匹配和医学成像中的水-脂肪分离。在上述问题的激励下，本项目的总体目标是开发新的基础理论，以端到端方式集成统计评估和算法设计，允许各种非凸低复杂度模型的最佳推理程序。该项目将大规模优化技术与统计思维相结合，旨在开发一套新的分布理论，为各种非凸低复杂度模型提供有效的不确定性评估。具体而言，本项目包括以下研究内容。首先，在新的非凸估计和去偏方法的基础上，开发一种原则性的方法来构建未知连续参数的最优置信区间。其次，开发快速非凸算法和高效的不确定性评估程序来对未知离散变量进行推理。第三，研究凸松弛和非凸优化之间的密切联系，从而建立一个统一的不确定性量化框架来容纳这两种方法。所有的研究重点都是由具体的实际应用驱动的，并最终将在具体的实际应用中进行测试。该项目将显著推进数据驱动应用中不确定性量化的基本技术，并将丰富数学优化、数据分析和统计建模的基础。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.