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

## Abstract

We consider a signal reconstruction problem for signals $F$ of the form $ F(x)=\backslashsum_\j=1\^\d\a_\j\backslashdelta\backslashleft(x-x_\j\backslashright),$ from their moments $m_k(F)=\backslashint x^kF(x)dx.$ We assume $m_k(F)$ to be known for $k=0,1,\backslashldots,N,$ with an absolute error not exceeding $\backslashepsilon \textgreater 0$. 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 $x_\i+1-x_i=h \backslashll 1$. 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.

## Reference

Andrey Akinshin, Gil Goldman, Vladimir Golubyatnikov, Yosef Yomdin

“Accuracy of reconstruction of spike-trains with two near-colliding nodes”(2017) DOI: 10.1090/conm/699/14078 arXiv:1701.01482

```
@Inproceedings{akinshin2017spiketrains,
abstract = {We consider a signal reconstruction problem for signals \$F\$ of the form \$ F(x)=\backslashsum\_\j=1\\^\d\a\_\j\\backslashdelta\backslashleft(x-x\_\j\\backslashright),\$ from their moments \$m\_k(F)=\backslashint x\^kF(x)dx.\$ We assume \$m\_k(F)\$ to be known for \$k=0,1,\backslashldots,N,\$ with an absolute error not exceeding \$\backslashepsilon \textgreater 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 \backslashll 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.},
archiveprefix = {arXiv},
arxivid = {1701.01482},
author = {Akinshin, Andrey and Goldman, Gil and Golubyatnikov, Vladimir and Yomdin, Yosef},
language = {english},
pages = {1--17},
booktitle = {7th International Conference on Complex Analysis and Dynamical Systems (CA&DS VII)},
series = {Contemporary Mathematics},
volume = {699},
doi = {10.1090/conm/699/14078},
title = {Accuracy of reconstruction of spike-trains with two near-colliding nodes},
url = {https://publons.com/publon/34688522/},
year = {2017},
arxiv = {1701.01482}
}
```