Structure Preserving Reduced Rank Approximation: Theory, Algorithms and Software

结构保持降阶近似：理论、算法和软件

• 批准号: 9901992
• 负责人: Haesun Park
• 金额: $ 16.1万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-09-01 至 2002-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9901992&HistoricalAwards=false
• 关键词: Structure; Preserving; Reduced; Rank; Approximation

Determining a structure preserving reduced rank approximation to a given structured matrix has many important applications such as signal processing, image processing, system identification, and control processes. The goal of this project is to develop the mathematical foundation, and design fast and robust algorithms for computing this structured reduced rank approximation, and making the resulting software available for use in the application areas. In this research, all linear and nonlinear structures of matrices will be considered, where the matrices can be represented as a differentiable function of a parameter vector, which includes sparse, symmetric, banded with fixed bandwidth, Hankel, Toeplitz, Vandermonde, or combinations of these. In addition, the project will investigate a broad range of applications for which new techniques will prove effective, and for which the accuracy of solutions can be improved dramatically with the exploitation of both rank deficiency and problem structure. The basic approach used is the minimization of the norm of an appropriate error, while preserving the structure and specified lower rank of the approximating matrix. This is a difficult problem, both theoretically and computationally, because the structure preservation and rank reduction requirements make this a nonconvex minimization problem, which may have more than one local minimum. To overcome this difficulty, several techniques will be attempted, including the way in which the reduced rank condition is formulated, and the selection of good initial values. This approach leads to a structure preserving overdetermined system of equations, which can be solved by the Structured Total Least Norm (STLN) algorithm, developed previously by the principal investigator.

确定给定结构矩阵的保结构降秩逼近具有许多重要的应用，例如信号处理、图像处理、系统识别和控制过程。 该项目的目标是发展数学基础，设计快速和强大的算法来计算这种结构化的降秩近似，并使所得软件可用于应用领域。 在这项研究中，所有的线性和非线性结构的矩阵将被考虑，其中矩阵可以表示为一个参数向量的可微函数，其中包括稀疏，对称，带状固定带宽，汉克尔，Toeplitz，范德蒙，或这些组合。 此外，该项目将调查广泛的应用，新技术将被证明是有效的，并且解决方案的准确性可以通过利用秩亏和问题结构来显着提高。所使用的基本方法是最小化适当误差的范数，同时保持近似矩阵的结构和指定的低秩。这在理论上和计算上都是一个困难的问题，因为结构保持和秩减少的要求使得这是一个非凸最小化问题，它可能有一个以上的局部最小值。 为了克服这一困难，将尝试几种技术，包括降秩条件的制定方式，以及良好的初始值的选择。这种方法导致一个结构保持超定方程组，它可以解决的结构总最小范数（STLN）算法，以前开发的主要研究者。