Structure Preserving Reduced Rank Approximation: Theory, Algorithms and Software

确定对给定结构矩阵保持降阶逼近的结构在信号处理、图像处理、系统辨识和控制过程中有许多重要的应用。这个项目的目标是发展数学基础，设计快速和健壮的算法来计算这种结构化的降阶近似，并使得到的软件可用于应用领域。在这项研究中，将考虑矩阵的所有线性和非线性结构，其中矩阵可以表示为参数向量的可微函数，参数向量包括稀疏、对称、固定带宽带状、Hankel、Toeplitz、Vandermonde或它们的组合。此外，该项目将调查广泛的应用，对于这些应用，新技术将被证明是有效的，对于这些应用，解决方案的准确性可以通过利用秩亏缺和问题结构来显著提高。所用的基本方法是最小化适当误差的范数，同时保持逼近矩阵的结构和指定的低阶。这在理论上和计算上都是一个困难的问题，因为结构保持和降阶的要求使其成为一个可能存在多个局部极小值的非凸极小化问题。为了克服这一困难，将尝试几种技术，包括表示降阶条件的方式，以及选择好的初始值。这种方法导致了一个保持结构的超定方程组，可以用主要研究者先前开发的结构化总体最小范数(STLN)算法来求解。

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.