Generalized Matrix Functions: Theory, Algorithms, and Applications

Emory University, Atlanta GA

Investigators

Abstract

In recent years the scientific enterprise, like daily existence, has been greatly affected by the availability of new technologies that have enabled the collection of an unprecedented amount of data. This continuous creation of enormous amounts of raw digital information demands new, efficient ways to extract useful content and filter out the noise inherent in any data-gathering process. Mathematical and computational techniques of data analysis offer powerful tools that can be brought to bear, but new challenges arise on a daily basis. This requires the constant refinement and improvement of existing techniques, as well as the development of new ones. This research project aims to produce a new body of mathematical knowledge and new computational techniques that will enhance the ability to tackle challenges arising from data-intensive fields of science and engineering including computer vision, compressed sensing, control, and others. Quantitative finance, network analysis and synthesis, and medical imaging are other areas where the research can be expected to have an impact. The principal investigator will study a class of mathematical objects known as generalized matrix functions and aims to exploit the resulting knowledge to develop new, efficient computer methods for data analysis. Training of a PhD student in computational mathematics is also an integral part of the project. The principal investigator aims to develop the theory of generalized matrix functions, a type of matrix function based on the singular value decomposition of a (possibly rectangular) matrix. The resulting theory is intended to be the basis for the development of numerical methods for the efficient approximate evaluation of such matrix functions. The focus will be primarily on large-scale problems for which the (full) singular value decomposition cannot be computed. Techniques based on sparsity and low-rank approximations will be combined with Krylov-type methods (especially the Lanczos and Golub--Kahan algorithms) to design fast algorithms for solving a variety of problems involving generalized matrix functions. The algorithms will be applied to problems such as low-rank matrix optimization, the regularization of discrete inverse problems, and the analysis of directed networks. As a by-product of this research, new algorithms for the computation of standard matrix functions where the matrix argument is only available in factored form will be derived and analyzed. This research represents a new direction in numerical linear algebra and is expected to produce useful numerical tools for the solution of a variety of problems in data science and optimization.