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

AF：小型：协作研究：瑞利商型优化的数学理论和快速算法

基本信息

批准号：
1527104
负责人：
Ren-Cang Li
金额：
$ 13.9万
依托单位：
University of Texas at Arlington
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-07-15 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1527104&HistoricalAwards=false
关键词：
AF Small Collaborative Research Mathematical

项目摘要

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.

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轨迹比优化。该项目提高了对这一实际重要和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还涉及本科生在这个研究项目的各个方面。PI期望产生一个统一的RQ型优化的观点，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，利用的数学属性和稀疏的数据结构。