License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.43
URN: urn:nbn:de:0030-drops-163849
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16384/
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Cohen, Gil ; Minzer, Dor ; Peleg, Shir ; Potechin, Aaron ; Ta-Shma, Amnon

Expander Random Walks: The General Case and Limitations

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Abstract

Cohen, Peri and Ta-Shma [Gil Cohen et al., 2021] considered the following question: Assume the vertices of an expander graph are labelled by ± 1. What "test" functions f : {±1}^t → {±1} can or cannot distinguish t independent samples from those obtained by a random walk? [Gil Cohen et al., 2021] considered only balanced labellings, and proved that for all symmetric functions the distinguishability goes down to zero with the spectral gap λ of the expander G. In addition, [Gil Cohen et al., 2021] show that functions computable by AC⁰ circuits are fooled by expanders with vanishing spectral expansion.
We continue the study of this question. We generalize the result to all labelling, not merely balanced ones. We also improve the upper bound on the error of symmetric functions. More importantly, we give a matching lower bound and show a symmetric function with distinguishability going down to zero with λ but not with t. Moreover, we prove a lower bound on the error of functions in AC⁰ in particular, we prove that a random walk on expanders with constant spectral gap does not fool AC⁰.

BibTeX - Entry

@InProceedings{cohen_et_al:LIPIcs.ICALP.2022.43,
  author =	{Cohen, Gil and Minzer, Dor and Peleg, Shir and Potechin, Aaron and Ta-Shma, Amnon},
  title =	{{Expander Random Walks: The General Case and Limitations}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16384},
  URN =		{urn:nbn:de:0030-drops-163849},
  doi =		{10.4230/LIPIcs.ICALP.2022.43},
  annote =	{Keywords: Expander Graphs, Random Walks, Lower Bounds}
}

Keywords: Expander Graphs, Random Walks, Lower Bounds
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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