A multivariate generalization of Prony's method

TL;DR

This paper extends Prony's method to multivariate signals, providing conditions for unique and stable reconstruction of measures supported on multiple points using moment matrices and polynomial root-finding.

Contribution

It introduces a multivariate generalization of Prony's method with theoretical guarantees for uniqueness and stability under specific conditions.

Findings

Stable reconstruction is guaranteed when moments are sufficiently large and points are well-separated.

The method involves constructing a multilevel Toeplitz matrix from moments.

Numerical experiments validate the theoretical results.

Abstract

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.