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

RI：小：非凸低复杂度模型的不确定性量化

• 批准号: 2100158
• 负责人: Yuxin Chen
• 金额: $ 45万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2022-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100158&HistoricalAwards=false
• 关键词: RI; Small; Uncertainty; Quantification; Nonconvex

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.

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算大量参数。 但是，为了支持现代科学发现和决策做出，不仅有必要寻求对参数的合理估计，而且更重要的是，对估计及其含义的可信赖解释。例如，我们可以为手头估计的质量提供什么保证？由于数据的不完善性，我们可以量化估计值的不确定性吗？ 为这些问题提供有效和定量的答案是确保：基于我们的估计做出的科学发现和决策的关键步骤，这是信息丰富且值得信赖的。然而，现有的统计工具箱在提供大规模估计方法的不确定性方面仍然高度不足，尤其是在那些数据样本的可用性受到严重限制的情况下。这限制了估计值的总体价值，并阻碍了科学和决策过程。一些示例应用领域包括：在计算机视觉和医学成像中的水脂分离中与关节形状匹配。受上述问题的促进，该项目的总体目标是开发新的基础理论，该理论以端到端方式整合统计评估和算法设计，从而为各种非convex低复杂模型提供最佳推论程序。拟议的项目将大规模优化技术与统计思维相结合，旨在开发一套新颖的分布理论，该理论可以为各种非凸低复杂模型提供有效的不确定性评估。具体来说，该项目包括以下研究。首先，基于新颖的非凸估计和偏低方法，开发一种原则方法来构建未知连续参数的最佳置信区间。其次，开发快速的非凸算法和有效的不确定性评估程序，以推理未知的离散变量。第三，研究凸松弛与非凸优化之间的紧密联系，从而使统一的不确定性定量框架可以适应这两种方法。所有研究推力都是由动机的，并最终将在具体的实用应用上进行测试。该项目将大大提高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NSF的法定任务，并被认为是通过该基金会的知识分子和更广泛的影响来评估的支持，并被认为是值得的。

项目成果

Spectral Methods for Data Science: A Statistical Perspective

• DOI: 10.1561/2200000079
• 发表时间: 2021-01-01
• 期刊: FOUNDATIONS AND TRENDS IN MACHINE LEARNING
• 影响因子: 32.8
• 作者: Chen, Yuxin;Chi, Yuejie;Ma, Cong
• 通讯作者: Ma, Cong

Tackling Small Eigen-Gaps: Fine-Grained Eigenvector Estimation and Inference Under Heteroscedastic Noise

• DOI: 10.1109/tit.2021.3111828
• 发表时间: 2021-11-01
• 期刊: IEEE TRANSACTIONS ON INFORMATION THEORY
• 影响因子: 2.5
• 作者: Cheng, Chen;Wei, Yuting;Chen, Yuxin
• 通讯作者: Chen, Yuxin

Nonconvex Low-Rank Tensor Completion from Noisy Data

• DOI: 10.1287/opre.2021.2106
• 发表时间: 2021-06-03
• 期刊: OPERATIONS RESEARCH
• 影响因子: 2.7
• 作者: Cai, Changxiao;Li, Gen;Chen, Yuxin
• 通讯作者: Chen, Yuxin