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

Journal of Singularities
volume 18 (2018), 409-426

Received: 3 January 2018. Accepted: 19 August 2018.

DOI: 10.5427/jsing.2018.18u

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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 from the noisy moment measurements. We provide an overview of some recent results 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 Singularity Theory.

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