Document Open Access Logo

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

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

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2021.26.pdf
  • Filesize: 495 kB
  • 17 pages

Document Identifiers

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

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.

Cite As Get BibTex

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

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

Funding

  • Kelman, Esty: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 835152).
  • Khot, Subhash: Supported by the NSF Award CCF-1422159, the Simons Collaboration on Algorithms and Geometry, and the Simons Investigator Award.
  • Safra, Muli: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 835152).

References

  1. William Beckner. Inequalities in Fourier analysis. Annals of Mathematics, 102:159-182, 1975. Google Scholar
  2. Aline Bonami. étude des coefficients de Fourier des fonctions de l^p(g). Annales de l'Institut Fourier, 20(2):335-402, 1970. URL: https://doi.org/10.5802/aif.357.
  3. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15(2):94-114, 2006. URL: https://doi.org/10.1007/s00037-006-0210-9.
  4. Nikhil R. Devanur, Subhash Khot, Rishi Saket, and Nisheeth K. Vishnoi. Integrality gaps for sparsest cut and minimum linear arrangement problems. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 537-546, 2006. Google Scholar
  5. Irit Dinur and Ehud Friedgut. Intersecting families are essentially contained in juntas. Combinatorics, Probability & Computing, 18(1-2):107-122, 2009. URL: https://doi.org/10.1017/S0963548308009309.
  6. Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals of mathematics, pages 439-485, 2005. Google Scholar
  7. Ronen Eldan and Renan Gross. Stability of talagrand’s influence inequality. CoRR, abs/1909.12067, 2019. URL: http://arxiv.org/abs/1909.12067.
  8. Dvir Falik and Alex Samorodnitsky. Edge-isoperimetric inequalities and influences. Combinatorics, Probability & Computing, 16(5):693-712, 2007. Google Scholar
  9. Ehud Friedgut. Boolean functions with low average sensitivity depend on few coordinates. Combinatorica, 18(1):27-35, 1998. Google Scholar
  10. Ehud Friedgut and Gil Kalai. Every monotone graph property has a sharp threshold. Proceedings of the American mathematical Society, 124(10):2993-3002, 1996. Google Scholar
  11. Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis, Ran Raz, and Ronald de Wolf. Exponential separation for one-way quantum communication complexity, with applications to cryptography. SIAM J. Comput., 38(5):1695-1708, 2008. URL: https://doi.org/10.1137/070706550.
  12. Leonard Gross. Logarithmic sobolev inequalities. American Journal of Mathematics, 97(4):1061-1083, 1975. URL: http://www.jstor.org/stable/2373688.
  13. J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In IEEE, editor, 29th annual Symposium on Foundations of Computer Science, October 24-26, 1988, White Plains, New York, pages 68-80, pub-IEEE:adr, 1988. IEEE Computer Society Press. Google Scholar
  14. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335-349, 2008. URL: https://doi.org/10.1016/j.jcss.2007.06.019.
  15. Robert Krauthgamer and Yuval Rabani. Improved lower bounds for embeddings intol_1dollar. SIAM J. Comput., 38(6):2487-2498, 2009. URL: https://doi.org/10.1137/060660126.
  16. S. Kudekar, S. Kumar, M. Mondelli, H. D. Pfister, and R. Urbankez. Comparing the bit-map and block-map decoding thresholds of reed-muller codes on bms channels. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 1755-1759, July 2016. URL: https://doi.org/10.1109/ISIT.2016.7541600.
  17. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. Google Scholar
  18. Ryan O'Donnell and Rocco A. Servedio. Learning monotone decision trees in polynomial time. SIAM J. Comput., 37(3):827-844, 2007. URL: https://doi.org/10.1137/060669309.
  19. Michel Talagrand. On russo’s approximate zero-one law. The Annals of Probability, 22(3):1576-1587, 1994. URL: http://www.jstor.org/stable/2245033.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact