The Algorithm Of Approximating A Given Matrix With A Hankel Matrix Of Lower Rank And Weighted Approximations

Finding a low rank Hankel approximation of a given matrix is a critical task in many disciplines. The list of applications includes communication engineering, control engineering, pattern classification , model predigest and so on. This paper concerns the construction of a low rank matrix with Hankel structure that is nearest to a given matrix. Based on the special decomposition of Hankel matrix, we can transform the problem of approximating a given matrix with a positive semidefinite real Hankel matrix of lower rank to a smooth unconstrained optimization problem. For general low rank Hankel approximation of a given matrix, the problem is transformed to an equality constrained optimization problem. Based on the Lagrange function, we can transform the constrained optimization problem to an unconstained optimization problem. The problems can be solved by a quasi-Newton updating procedure. To improve the method, we make use of indirect approximation method to solve the optimization problem. For the weighted Hankel low rank approximation, we also transform the problem to an optimization problem. Using generalized Schmidt pairs, we propose an iterative algorithm for computing a weighted low rank matrix with Hankel structure that is nearest to a given matrix and give the convergence analysis of the method. Numerical experiments show that these methods are effective.