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

许多现代数据分析技术和机器学习中的应用都试图了解什么样的输入数据对输出的影响最大。瑞利商（RQ），或者更一般地说，RQ型目标函数，是捕捉这些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鲁棒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RQ型优化，以及可以执行多视图谱聚类的迹比优化。 该项目提高了对这一具有实际意义和面向用户的数学理论的理解，创建了在开源软件中嵌入的计算方法。 它不仅推进了数据科学中的数学理论和优化算法，而且还培养了计算机科学和计算数学研究生成功开展项目所需的跨学科知识和工具。PI还涉及到本研究项目的各个方面的本科生。PI希望产生一个统一的RQ型优化的观点，将它们重新表达为线性和非线性特征值问题，新的变分原理可以表征最优解。 这些新的原则应该暴露的基本问题的数值线性代数特征，支持开发的快速算法，利用的prosticalproperties和稀疏的数据结构。

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.