Generalized Matrix Functions: Theory, Algorithms, and Applications

近年来，科学事业和日常生活一样，受到新技术的极大影响，这些新技术使人们能够收集到前所未有的大量数据。这种大量原始数字信息的持续创造需要新的、有效的方法来提取有用的内容，并过滤掉任何数据收集过程中固有的噪音。数据分析的数学和计算技术提供了可以承受的强大工具，但每天都会出现新的挑战。这需要不断改进和改进现有的技术，同时也需要开发新的技术。该研究项目旨在产生一个新的数学知识体系和新的计算技术，以提高应对来自数据密集型科学和工程领域的挑战的能力，包括计算机视觉、压缩感知、控制等。定量金融、网络分析和综合以及医学成像是该研究有望产生影响的其他领域。首席研究员将研究一类被称为广义矩阵函数的数学对象，并旨在利用所得知识开发新的、有效的数据分析计算机方法。对计算数学博士生的培训也是该项目的一个组成部分。主要研究者的目标是发展广义矩阵函数理论，这是一种基于（可能是矩形）矩阵的奇异值分解的矩阵函数。由此产生的理论旨在成为发展这种矩阵函数的有效近似求值的数值方法的基础。重点将主要放在大规模问题上，这些问题的（完全）奇异值分解无法计算。基于稀疏性和低秩近似的技术将与krylov型方法（特别是Lanczos和Golub- Kahan算法）相结合，设计快速算法来解决涉及广义矩阵函数的各种问题。这些算法将应用于诸如低秩矩阵优化、离散逆问题的正则化以及有向网络的分析等问题。作为本研究的副产品，我们将推导和分析计算标准矩阵函数的新算法，其中矩阵参数只能以因式形式存在。该研究代表了数值线性代数的一个新方向，并有望为解决数据科学和优化中的各种问题提供有用的数值工具。

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.