Accuracy of noisy Spike-Train Reconstruction: a Singularity Theory point of view
For researchers in signal processing and algebraic geometry, this paper provides a conceptual framework linking singularity theory to spike-train reconstruction, but it is a survey of existing results rather than presenting new findings.
This survey paper reviews the Prony system for reconstructing spike-train signals from noisy moment measurements, emphasizing the role of singularity theory in understanding error amplification when nodes near-collide. It synthesizes recent results on the geometry of error amplification and describes algebraic-geometric structures underlying the reconstruction process.
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 $F(x)=\sum_{j=1}^d a_jδ(x-x_j)$ 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.