# Learning algebraic decompositions using Prony structures

## TL;DR

This paper introduces a unified algebraic framework that generalizes various Prony's method variants, enabling the decomposition of complex multivariate sums and polynomials with support on algebraic sets.

## Contribution

It provides a comprehensive algebraic structure connecting different Prony-based methods and extends their applicability to multivariate and algebraic support cases.

## Key findings

- Unified framework for Prony's method variants
- Applicable to multivariate exponential and polynomial sums
- Accounts for support on algebraic sets

## Abstract

We propose an algebraic framework generalizing several variants of Prony's method and explaining their relations. This includes Hankel and Toeplitz variants of Prony's method for the decomposition of multivariate exponential sums, polynomials (with respect to the monomial and Chebyshev bases), Gau{\ss}ian sums, spherical harmonic sums, taking also into account whether they have their support on an algebraic set.