# Accuracy of spike-train Fourier reconstruction for colliding nodes

## Abstract

We consider Fourier reconstruction problem for signals F, which are linear combinations of shifted delta-functions. We assume the Fourier transform of F to be known on the frequency interval [-N,N], with an absolute error not exceeding e \textgreater 0. We give an absolute lower bound (which is valid with any reconstruction method) for the “worst case” reconstruction error of F in situations where the nodes (i.e. the positions of the shifted delta-functions in F) are known to form an l elements cluster of a size h \textless\textless 1. Using “decimation” reconstruction algorithm we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for N*h of order of (2l-1)-st root of e the worst case reconstruction error of the cluster nodes is of the same order as h, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order of 2l-st root of e.

## Reference

A Akinshin, D Batenkov, Y Yomdin

“Accuracy of spike-train Fourier reconstruction for colliding nodes”(2015) DOI: 10.1109/SAMPTA.2015.7148965 arXiv:1502.06932

```
@Inproceedings{akinshin2015sampta,
abstract = {We consider Fourier reconstruction problem for signals F, which are linear combinations of shifted delta-functions. We assume the Fourier transform of F to be known on the frequency interval [-N,N], with an absolute error not exceeding e \textgreater 0. We give an absolute lower bound (which is valid with any reconstruction method) for the "worst case" reconstruction error of F in situations where the nodes (i.e. the positions of the shifted delta-functions in F) are known to form an l elements cluster of a size h \textless\textless 1. Using "decimation" reconstruction algorithm we provide an upper bound for the reconstruction error, essentially of the same form as the lower one. Roughly, our main result states that for N*h of order of (2l-1)-st root of e the worst case reconstruction error of the cluster nodes is of the same order as h, and hence the inside configuration of the cluster nodes (in the worst case scenario) cannot be reconstructed at all. On the other hand, decimation algorithm reconstructs F with the accuracy of order of 2l-st root of e.},
address = {Washington, DC},
author = {Akinshin, A and Batenkov, D and Yomdin, Y},
booktitle = {Sampling Theory and Applications (SampTA), 2015 International Conference on},
doi = {10.1109/SAMPTA.2015.7148965},
isbn = {978-1-4673-7353-1},
number = {264},
pages = {617--621},
publisher = {IEEE},
title = {Accuracy of spike-train Fourier reconstruction for colliding nodes},
url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7148965 https://publons.com/publon/14766954/},
year = {2015},
arxiv = {1502.06932}
}
```