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影响最大。瑞利商 (RQ)，或者更一般地说，RQ 型目标函数，是捕获此信息的数学技术的基础。该项目对三种 RQ 类型优化进行深入的理论和算法研究：可以处理数据不确定性的鲁棒 RQ 优化、可以结合图像分割或数据聚类先验信息的约束 RQ 类型优化，以及可以执行多视图谱聚类的跟踪比率优化。 该项目提高了对这一实际重要且面向用户的数学理论的理解，创建了包含在开源软件中的计算方法。 它不仅推进数据科学中的数学理论和优化算法，而且还培养计算机科学和计算数学研究生成功承担项目所需的跨学科知识和工具。 PI 还让本科生参与该研究项目的各个方面。PI 期望产生 RQ 型优化的统一视图，将其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最佳解决方案。 这些新原理应该揭示潜在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 machinelearning try to learn what input data has the largest effects on theoutputs. Rayleigh Quotients (RQ), or, more generally, RQ-typeobjective functions, are the basis of a mathematical technique thatcaptures this information. This project conducts in-depth theoreticaland algorithmic studies of three RQ-type optimizations: robust RQoptimizations that can handle data uncertainty, constrained RQ-typeoptimizations that can incorporate prior information from imagesegmentation or data clustering, and trace ratio optimizations thatcan perform multi-view spectral clustering. This project improvesunderstanding of this practically important and user-orientedmathematical theory, creating computational methods that are embodiedin open-source software. It not only advances mathematical theoryand optimization algorithms in data science, but trains computerscience and computational mathematics graduate students ininterdisciplinary knowledge and tools necessary to undertake theproject successfully. The PIs also involve undergraduate studentsin 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 forwhich new variational principles can characterize the optimalsolutions. These new principles should expose the numerical linearalgebra characteristics of the underlying problems, supporting thedevelopment of fast algorithms that exploit the mathematicalproperties and sparse data structure.