Collaborative Research: Random Matrices and Algorithms in High Dimension

能够处理大量数据的数值算法正变得越来越重要，尤其是那些每天在人工智能（AI）软件中使用的算法。例子包括语音助手、手机面部识别和基于机器学习的金融欺诈检测。但是，许多算法只是启发式地应用，仍然很难理解，这意味着缺乏理论上的保证。事实上，最近的人工智能应用表明，这些算法的直接应用，如果没有适当的验证，可能会产生人为的、误导性的信息。本提案的广泛目标是加深我们对统计相关随机矩阵模型的理解，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，因为它们作用于这些模型。预计这将产生新的见解和统计工具，并提供理论保证。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。提出的问题可分为三个主要项目。第一部分是对随机矩阵模型的分析，它扩展了样本协方差矩阵的经典设置。然后，通过Riemann—Hilbert问题将随机矩阵模型与正交多项式连接起来，pi将得到由这些随机矩阵生成的自然测度的正交多项式的新估计和新结论。有了理论结果，第二个项目涉及直接应用第一个项目的估计，并进一步改进先前的分析，以了解数值算法的平均情况行为。这里的重点是数值线性代数的算法。在第三个项目中，研究人员将使用新的随机矩阵估计，正交多项式理论的新结果及其相关的黎曼-希尔伯特理论，用于经典和新的随机矩阵集合，生成新的算法，最终导致新的可行的统计估计。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。

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.