Generalized Matrix Functions: Theory, Algorithms, and Applications

近年来，新技术的出现极大地影响了科学事业，就像日常生活一样，新技术使收集的数据量达到前所未有的水平。海量原始数字信息的持续创造需要新的、高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发挥作用的强大工具，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，以及开发新的技术。这项研究项目旨在产生新的数学知识体系和新的计算技术，以增强应对计算机视觉、压缩传感、控制等数据密集型科学和工程领域所产生的挑战的能力。定量金融、网络分析和综合以及医学成像是这项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，目标是利用由此产生的知识来开发新的、高效的计算机数据分析方法。博士生在计算数学方面的培训也是该项目的一个组成部分。主要研究人员旨在发展广义矩阵函数的理论，广义矩阵函数是一种基于矩阵(可能是矩形)的奇异值分解的矩阵函数。所得到的理论旨在成为发展有效地近似计算这类矩阵函数的数值方法的基础。重点将主要放在无法计算(全部)奇异值分解的大规模问题上。基于稀疏性和低阶近似的技术将与Krylov型方法(特别是Lanczos和Golub-Kahan算法)相结合，设计出解决各种涉及广义矩阵函数的问题的快速算法。这些算法将应用于低阶矩阵优化、离散反问题的正则化和有向网络的分析等问题。作为这项研究的副产品，将推导和分析用于计算标准矩阵函数的新算法，其中矩阵自变量仅以因式分解的形式可用。这项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.