Accuracy of reconstruction of spike-trains with two near-colliding nodes

TL;DR

This paper analyzes how errors are amplified when reconstructing spike-train signals with nearly colliding nodes from their moments, revealing the geometric structure governing this error amplification.

Contribution

It introduces the algebraic curves that describe the geometry of error amplification in the reconstruction of near-colliding spike nodes.

Findings

Error amplification is governed by algebraic curves in parameter space.

Near-colliding nodes cause significant reconstruction errors.

The geometry of error amplification is characterized by invariant moment conditions.

Abstract

We consider a signal reconstruction problem for signals $F$ of the form $ F(x)=\sum_{j=1}^{d}a_{j}\delta\left(x-x_{j}\right),$ from their moments $m_k(F)=\int x^kF(x)dx.$ We assume $m_k(F)$ to be known for $k=0,1,\ldots,N,$ with an absolute error not exceeding $\epsilon > 0$. We study the "geometry of error amplification" in reconstruction of $F$ from $m_k(F),$ in situations where two neighboring nodes $x_i$ and $x_{i+1}$ near-collide, i.e $x_{i+1}-x_i=h \ll 1$. We show that the error amplification is governed by certain algebraic curves $S_{F,i},$ in the parameter space of signals $F$, along which the first three moments $m_0,m_1,m_2$ remain constant.