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

University Of California-Davis, Davis CA

Investigators

Abstract

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 exploit the mathematical properties and sparse data structure.