Bias correction for Vandermonde low-rank approximation

The low-rank approximation problem, that is the problem of approximating a given matrix with a matrix of lower rank, appears in many scientific fields. In some applications the given matrix is structured and the approximation is required to have the same structure. Examples of linear structures are Hankel, Toeplitz, and Sylvester. Currently, there are only a few results for nonlinearly structured low-rank approximation problems. The problem of Vandermonde structured low-rank approximation is considered. The high condition number of the Vandermonde matrix, in combination with the noise in the data, makes the problem challenging. A numerical method based on a bias correction procedure is proposed and its properties are demonstrated by simulation. The performance of the method is illustrated on numerical results.