AF: Small: Collaborative Research: Mathematical Theory and Fast Algorithms for Rayleigh Quotient-type Optimizations

机器学习中的许多现代数据分析技术和应用都试图了解哪些输入数据对输出的影响最大。瑞利商(RQ)，或者更广泛地说，RQ型目标函数，是捕捉这种信息的数学技术的基础。该项目对三种RQ型优化进行了深入的理论和算法研究：可以处理数据不确定性的健壮RQ型优化，可以结合图像分割或数据聚类的先验信息的约束型RQ型优化，以及可以执行多视角光谱聚类的跟踪比优化。这个项目提高了对这一实际重要的、面向用户的数学理论的理解，创造了体现在开放源码软件中的计算方法。它不仅推动了数据科学中的数学理论和优化算法的发展，而且还培训了计算机科学和计算数学研究生，使他们掌握成功开展该项目所需的跨学科知识和工具。PIs还包括本科生参与本研究项目的各个方面。PI期望产生RQ型优化的统一视图，将它们重新表述为线性和非线性特征值问题，新的变分原理可以表征其最优解。这些新原理应该揭示基本问题的数值线性代数特征，支持开发利用数学特性和稀疏数据结构的快速算法。

Many modern data analysis techniques and applications in machine learning try to learn what input data has the largest effects on the outputs. Rayleigh Quotients (RQ), or, more generally, RQ-type objective functions, are the basis of a mathematical technique that captures this information. This project conducts in-depth theoretical and algorithmic studies of three RQ-type optimizations: robust RQ optimizations that can handle data uncertainty, constrained RQ-type optimizations that can incorporate prior information from image segmentation or data clustering, and trace ratio optimizations that can perform multi-view spectral clustering. This project improves understanding of this practically important and user-oriented mathematical theory, creating computational methods that are embodied in open-source software. It not only advances mathematical theory and optimization algorithms in data science, but trains computer science and computational mathematics graduate students in interdisciplinary knowledge and tools necessary to undertake the project successfully. The PIs also involve undergraduate students in all aspects of this research project. The PIs expect to produce a unified view of RQ-type optimizations, reformulating them into linear and nonlinear eigenvalue problems for which new variational principles can characterize the optimal solutions. These new principles should expose the numerical linear algebra characteristics of the underlying problems, supporting the development of fast algorithms that exploint the mathematical properties and sparse data structure.