# Structured low rank decomposition of multivariate Hankel matrices

## TL;DR

This paper introduces a new algorithm for decomposing multivariate Hankel matrices into low-rank components, leveraging algebraic structures and eigenvector methods, with improved numerical stability and error correction techniques.

## Contribution

It presents a novel multivariate generalization of the Pencil method using algebraic properties of Artinian Gorenstein algebras for low-rank Hankel matrix decomposition.

## Key findings

- The algorithm effectively decomposes multivariate Hankel matrices into polynomial-exponential series.
- The method is robust to noise and includes a rescaling technique for high-amplitude frequencies.
- A new Newton iteration improves accuracy by correcting errors in input moments.

## Abstract

We study the decomposition of a multivariate Hankel matrix H\_$\sigma$ as a sum of Hankel matrices of small rank in correlation with the decomposition of its symbol $\sigma$ as a sum of polynomial-exponential series. We present a new algorithm to compute the low rank decomposition of the Hankel operator and the decomposition of its symbol exploiting the properties of the associated Artinian Gorenstein quotient algebra A\_$\sigma$. A basis of A\_$\sigma$ is computed from the Singular Value Decomposition of a sub-matrix of the Hankel matrix H\_$\sigma$. The frequencies and the weights are deduced from the generalized eigenvectors of pencils of shifted sub-matrices of H $\sigma$. Explicit formula for the weights in terms of the eigenvectors avoid us to solve a Vandermonde system. This new method is a multivariate generalization of the so-called Pencil method for solving Prony-type decomposition problems. We analyse its numerical behaviour in the presence of noisy input moments, and describe a rescaling technique which improves the numerical quality of the reconstruction for frequencies of high amplitudes. We also present a new Newton iteration, which converges locally to the closest multivariate Hankel matrix of low rank and show its impact for correcting errors on input moments.