Collaborative Research: Random Matrices and Algorithms in High Dimension

能够处理海量数据的数值算法越来越重要，特别是对于人工智能(AI)软件中每天使用的那些算法。例子包括语音助手、手机面部识别，以及基于机器学习的金融欺诈检测。但许多算法只是启发式地应用，仍然知之甚少，这意味着缺乏理论保障。事实上，最近的人工智能应用表明，在没有适当验证的情况下，直接应用这些算法可能会产生人为的、误导性的信息。这项建议的广泛目的是加深我们对一类统计相关的随机矩阵模型的理解，这些模型用于对大数据集进行建模、分析和解释，并分析作用于这些模型的新的和经典的算法。预计这将产生新的见解和统计工具，并辅之以理论上的保证。这个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。提出的问题符合三个主要项目。第一个涉及随机矩阵模型的分析，该模型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和正交多项式联系起来，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，并进一步完善先前的分析，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中的算法。在第三个项目中，研究人员将使用新的随机矩阵估计、正交多项式理论中的新结果及其相关的Riemann-Hilbert理论，为经典和新的随机矩阵集合生成新的算法，最终产生新的可行的统计估计。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.