Collaborative Research: Random Matrices and Algorithms in High Dimension

合作研究：高维随机矩阵和算法

• 批准号: 2306438
• 负责人: Thomas Trogdon
• 金额: $ 25.72万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306438&HistoricalAwards=false
• 关键词: Collaborative; Research; Random; Matrices; Algorithms

Numerical algorithms that can process huge amounts of data are increasingly important, notably for those algorithms used every day within artificial intelligence (AI) software. Examples include voice assistants, facial recognition for cellphones, and machine-learning-based financial fraud detection. But many algorithms are applied only heuristically and remain poorly understood, meaning that theoretical guarantees are missing. In fact, recent AI applications indicate that the direct application of these algorithms, without proper validation, may generate artificial, misleading information. The broad aim of this proposal is to deepen our understanding of classes of statistically-relevant random matrix models that are used to model, analyze and interpret large data sets and to analyze new and classical algorithms as they act on these models. It is expected that this will produce new insights and statistical tools, paired with theoretical guarantees. This award will also train junior researchers and help continue to build the community of researchers working in this field.The proposed problems fit into three main projects. The first concerns the analysis of random matrix models that extend the classical setting of sample covariance matrices. Then by connecting random matrix models and orthogonal polynomials via Riemann--Hilbert problems, the PIs will obtain new estimates and new conclusions about orthogonal polynomials for natural measures generated by these random matrices. Armed with the theoretical results, the second project concerns the direct application of the estimates from the first project, and further refinement of previous analyses, to understand the average-case behavior of numerical algorithms. The focus here is on algorithms from numerical linear algebra. In the third project, the investigators will use the new random matrix estimates, the new results in the theory of orthogonal polynomials and its associated Riemann--Hilbert theory, for both classical and new random matrix ensembles, to generate new algorithms, ultimately leading to new viable statistical estimators.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.

可以处理大量数据的数值算法越来越重要，特别是对于人工智能（AI）软件中每天使用的算法。 例子包括语音助手，手机的面部识别以及基于机器学习的财务欺诈检测。但是，许多算法仅被启发性地应用，并且对理解不足，这意味着缺失了理论保证。实际上，最近的AI应用程序表明，未经适当验证的这些算法的直接应用可能会产生人工误导的信息。 该提案的广泛目的是加深我们对统计上相关的随机矩阵模型类别的理解，这些模型用于建模，分析和解释大型数据集，并分析新的和经典的算法在这些模型上作用。 可以预期，这将产生新的见解和统计工具，并与理论保证配对。 该奖项还将培训初级研究人员，并帮助继续建立在该领域工作的研究人员社区。拟议的问题适合三个主要项目。 第一个涉及对扩展样品协方差矩阵的经典设置的随机矩阵模型的分析。然后，通过通过Riemann-Hilbert问题连接随机矩阵模型和正交多项式，PIS将获得有关这些随机矩阵产生的天然措施的正交多项式的新估计和新结论。 在理论上的结果中，第二个项目涉及第一个项目中的估计值的直接应用，以及对先前分析的进一步完善，以了解数值算法的平均案例行为。 这里的重点是来自数值线性代数的算法。 In the third project, the investigators will use the new random matrix estimates, the new results in the theory of orthogonal polynomials and its associated Riemann--Hilbert theory, for both classical and new random matrix ensembles, to generate new algorithms, ultimately leading to new viable statistical estimators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual优点和更广泛的影响审查标准。