A multivariate generalization of Prony's method
It extends a classical signal processing technique to multivariate settings, offering theoretical guarantees for stable reconstruction, which is relevant for applications in sparse recovery and moment problems.
This paper generalizes Prony's method to reconstruct multivariate finitely supported complex measures from their moments, providing conditions for unique and stable reconstruction. Numerical experiments validate the theoretical results.
Prony's method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree. In this note, we give a generalization of this method to the multivariate case and prove simple conditions under which the problem admits a unique solution. Provided the order of the moments is bounded from below by the number of points on which the measure is supported as well as by a small constant divided by the separation distance of these points, stable reconstruction is guaranteed. In its simplest form, the reconstruction method consists of setting up a certain multilevel Toeplitz matrix of the moments, compute a basis of its kernel, and compute by some method of choice the set of common roots of the multivariate polynomials whose coefficients are given in the second step. All theoretical results are illustrated by numerical experiments.