Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality

Authors Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, Muli Safra



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Author Details

Esty Kelman
  • School of Computer Science, Tel Aviv University, Israel
Subhash Khot
  • Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, NY, USA
Guy Kindler
  • Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel
Dor Minzer
  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
Muli Safra
  • School of Computer Science, Tel Aviv University, Israel

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Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.26

Abstract

We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Fourier Analysis
  • Hypercontractivity
  • Log-Sobolev Inequality

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