Accuracy of noisy Spike-Train Reconstruction: a Singularity Theory point of view

TL;DR

This survey explores the Prony system's algebraic structure and its connection to Singularity Theory, focusing on the stability of noisy spike-train reconstruction, especially when nodes are close together.

Contribution

It provides an overview of the algebraic-geometric structures underlying error amplification in spike-train reconstruction and highlights potential applications of Singularity Theory.

Findings

Analysis of error amplification near node collisions

Description of algebraic structures like Prony and Hankel mappings

Connection between error behavior and Singularity Theory

Abstract

This is a survey paper discussing one specific (and classical) system of algebraic equations - the so called "Prony system". We provide a short overview of its unusually wide connections with many different fields of Mathematics, stressing the role of Singularity Theory. We reformulate Prony System as the problem of reconstruction of "Spike-train" signals of the form $F(x)=\sum_{j=1}^d a_j\delta(x-x_j)$ from the noisy moment measurements. We provide an overview of some recent results of [1-3, 6, 8, 9, 11, 12, 5] on the "geometry of the error amplification" in the reconstruction process, in situations where the nodes $x_j$ near-collide. Some algebraic-geometric structures, underlying the error amplification, are described (Prony, Vieta, and Hankel mappings, Prony varieties), as well as their connection with Vandermonde mappings and varieties. Our main goal is to present some promising fields of possible applications of Singulary Theory.