Structured low rank decomposition of multivariate Hankel matrices

We present a new algorithm for the decomposition of a multivariate Hankel matrix Hσ as a sum of Hankel matrices of small rank. This decomposition corresponds to the decomposition of its symbol σ as a sum of polynomial-exponential series. By exploiting the properties of the associated Artinian Gorenstein quotient algebra Aσ, we obtain new ways to compute the frequencies and the weights of the decomposition from generalized eigenvectors of sub-matrices of Hσ. The new method is a multivariate generalization of the so-called Pencil method for solving Prony-type problems. We analyse its numerical behaviour in the presence of noisy input moments. We describe rescaling techniques and Newton iterations, which improve the numerical quality of the reconstruction and show their impact for correcting errors on input moments.