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.APPROX/RANDOM.2021.38
URN: urn:nbn:de:0030-drops-147319
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14731/
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Golowich, Louis

Improved Product-Based High-Dimensional Expanders

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LIPIcs-APPROX38.pdf (0.7 MB)

Abstract

High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary combinatorial construction with near-optimal expansion is known. In this paper, we introduce an improved combinatorial high-dimensional expander construction, by modifying a previous construction of Liu, Mohanty, and Yang (ITCS 2020), which is based on a high-dimensional variant of a tensor product. Our construction achieves a spectral gap of Ω(1/(k²)) for random walks on the k-dimensional faces, which is only quadratically worse than the optimal bound of Θ(1/k). Previous combinatorial constructions, including that of Liu, Mohanty, and Yang, only achieved a spectral gap that is exponentially small in k. We also present reasoning that suggests our construction is optimal among similar product-based constructions.

BibTeX - Entry

@InProceedings{golowich:LIPIcs.APPROX/RANDOM.2021.38,
  author =	{Golowich, Louis},
  title =	{{Improved Product-Based High-Dimensional Expanders}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{38:1--38:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14731},
  URN =		{urn:nbn:de:0030-drops-147319},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.38},
  annote =	{Keywords: High-Dimensional Expander, Expander Graph, Random Walk}
}

Keywords: High-Dimensional Expander, Expander Graph, Random Walk
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)
Issue Date: 2021
Date of publication: 15.09.2021

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