Algebraic Geometry of Error Amplification: the Prony leaves

TL;DR

This paper explores the geometric structure of error amplification in solving Prony systems, especially when nodes are nearly colliding, revealing the role of Prony foliations and providing new parametrization results.

Contribution

It introduces the concept of Prony foliations and offers new explicit parametrizations of Prony leaves, advancing understanding of error behavior in near-collision scenarios.

Findings

Error amplification is governed by Prony foliations.

Explicit parametrization of Prony leaves is achieved.

Provides insights into error behavior near node collisions.

Abstract

We provide an overview of some results on the "geometry of error amplification" in solving Prony system, in situations where the nodes near-collide. It turns out to be governed by the "Prony foliations" $S_q$, whose leaves are "equi-moment surfaces" in the parameter space. Next, we prove some new results concerning explicit parametrization of the Prony leaves.