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且嘈杂的测量集中估计大量参数。 然而，为了让这些应用支持现代科学发现和决策，不仅需要寻求对参数的合理估计，而且也许更重要的是，需要对估计及其含义进行可靠的解释。例如，我们可以对现有估算的质量提供哪些保证？我们能否量化由于数据不完善而导致的估计的不确定性？ 为此类问题提供有效和定量的答案是确保以下方面的关键一步：根据我们的估计做出的科学发现和决策是信息丰富且值得信赖的。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，特别是在数据样本的可用性严重有限的情况下。这限制了估计的总体价值，并阻碍了科学和决策过程。一些示例应用领域包括：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。受上述问题的推动，该项目的总体目标是开发新的基础理论，以端到端的方式集成统计评估和算法设计，从而为各种非凸低复杂度模型提供最佳的推理过程。该项目将大规模优化技术与统计思维相结合，旨在开发一套新颖的分布理论，能够对各种非凸低复杂性模型进行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去偏差方法，开发一种原则性方法来构建未知连续参数的最佳置信区间。其次，开发快速非凸算法和有效的不确定性评估程序来推理未知的离散变量。第三，研究凸松弛和非凸优化之间的密切联系，从而使统一的不确定性量化框架能够适应这两种方法。所有的研究主旨都受到具体实际应用的推动，并最终将在具体的实际应用中得到检验。该项目将显着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 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。

项目成果

Nonconvex Low-Rank Tensor Completion from Noisy Data

来自噪声数据的非凸低阶张量补全

• DOI: 10.1287/opre.2021.2106
• 发表时间: 2022-03
• 期刊: Operations Research
• 影响因子: 2.7
• 作者: Cai, Changxiao;Li, Gen;Poor, H. Vincent;Chen, Yuxin
• 通讯作者: Chen, Yuxin

Bridging convex and nonconvex optimization in robust PCA: Noise, outliers and missing data

在稳健的 PCA 中桥接凸优化和非凸优化：噪声、异常值和缺失数据

• DOI: 10.1214/21-aos2066
• 发表时间: 2021-10
• 期刊: The Annals of Statistics
• 作者: Chen, Yuxin;Fan, Jianqing;Ma, Cong;Yan, Yuling
• 通讯作者: Yan, Yuling

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

解决小特征间隙：异方差噪声下的细粒度特征向量估计和推理

• DOI: 10.1109/tit.2021.3111828
• 发表时间: 2021-11
• 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

Spectral Methods for Data Science: A Statistical Perspective

数据科学的谱方法：统计视角

• DOI: 10.1561/2200000079
• 发表时间: 2021-01
• 期刊: Foundations and Trends® in Machine Learning
• 作者: Chen, Yuxin;Chi, Yuejie;Fan, Jianqing;Ma, Cong
• 通讯作者: Ma, Cong