Accuracy of noisy spike-train reconstruction: A singularity theory point of view

G. I.L. Goldman, Yehonatan Salman, Yosef Yomdin

Research output: Contribution to journalArticlepeer-review


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 aj δ(x−xj) from the noisy moment measurements. We provide an overview of some recent results of [1–3, 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 Singularity Theory.

Original languageEnglish
Pages (from-to)409-426
Number of pages18
JournalJournal of Singularities
StatePublished - 2018
Externally publishedYes


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