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

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) = d j=1 ajδ(x − xj) from the noisy moment measurements. We provide an overview of some recent results of [1][2][3]7,8,10,11,29,53] on the "geometry of the error amplification" in the reconstruction process, in situations where the nodes xj 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.

Geometry and singularities of Prony varieties

“…A survey of the results on the geometry of error amplification, obtained in [1][2][3][4], is given in [19]. This survey stresses the role of Singularity Theory in study of Prony inversion, and, in particular, in study of the Prony varieties, and announces some new results in this direction.…”

“…This survey stresses the role of Singularity Theory in study of Prony inversion, and, in particular, in study of the Prony varieties, and announces some new results in this direction. However, these results on the geometry and singularities of the Prony varieties are stated in [19] without proofs.…”

“…In the present paper we start a systematic study of the topology, geometry and singularities of the Prony varieties, providing, in particular, proofs for most of the results announced in [19].…”

“…This paper is devoted to a detailed study of "Prony varieties", which play important role in some problems of Signal Processing (in particular, in Fourier reconstruction of "spike-train signals" -see Section 1.1 below). We believe that Prony varieties present a significant independent mathematical interest, especially from the point of view of Singularity Theory (compare [19]).…”

“…In particular, in the coarse of our study we provide proofs of most of the results announces in [19]. (However, we keep the present paper independent from [19], and give all the necessary definitions).…”