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Boolean Function Analysis on High-Dimensional Expanders

Authors Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha

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  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • high dimensional expanders
  • Boolean function analysis
  • sparse model

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Abstract

We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones).

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Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean Function Analysis on High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.38

Author Details

Yotam Dikstein
  • Weizmann Institute of Science, ISRAEL
Irit Dinur
  • Weizmann Institute of Science, ISRAEL
Yuval Filmus
  • Technion - Israel Institute of Technology, ISRAEL
Prahladh Harsha
  • Tata Institute of Fundamental Research, INDIA

Funding

  • Dikstein, Yotam: The research of the first and second authors was supported in part by Irit Dinur’s ERC-CoG grant 772839.
  • Filmus, Yuval: Taub Fellow - supported by the Taub Foundations. The research was funded by ISF grant 1337/16.
  • Harsha, Prahladh: Research supported in part by UGC - ISF grant. Part of the work was done when the author was visiting the Weizmann Institute of Science.

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