Structure Preserving Reduced Rank Approximation: Theory, Algorithms and Software

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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.