Document Open Access Logo

Boolean Function Analysis on High-Dimensional Expanders

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

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2018.38.pdf
  • Filesize: 0.49 MB
  • 20 pages

Document Identifiers

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.

Cite As Get BibTex

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

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).

Subject Classification

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

Metrics

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

Related Versions

References

  1. Boaz Barak, Parikshit Gopalan, Johan Håstad, Raghu Meka, Prasad Raghavendra, and David Steurer. Making the long code shorter. SIAM J. Comput., 44(5):1287-1324, 2015. (Preliminary version in 53rd FOCS, 2012). URL: http://dx.doi.org/10.1137/130929394.
  2. Boaz Barak, Pravesh Kothari, and David Steurer. Small-set expansion in shortcode graph and the 2-to-2 conjecture. (manuscript), 2018. Google Scholar
  3. Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean function analysis on high-dimensional expanders. (manuscript), 2018. URL: http://arxiv.org/abs/1804.08155.
  4. Irit Dinur, Yuval Filmus, and Prahladh Harsha. Low degree almost Boolean functions are sparse juntas. (manuscript), 2017. URL: http://arxiv.org/abs/1711.09428.
  5. Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In Proc. 58th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 974-985, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.94.
  6. Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. On non-optimally expanding sets in Grassmann graphs. In Proc. 50th ACM Symp. on Theory of Computing (STOC), 2018. (To appear). Google Scholar
  7. Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the 2-to-1 games conjecture? In Proc. 50th ACM Symp. on Theory of Computing (STOC), 2018. (To appear). Google Scholar
  8. Charles Dunkl. A Krawtchouk polynomial addition theorem and wreath products of symmetric groups. Indiana Univ. Math. J., 25:335-358, 1976. URL: http://dx.doi.org/10.1512/iumj.1976.25.25030.
  9. David Ellis, Yuval Filmus, and Ehud Friedgut. A quasi-stability result for dictatorships in S_n. Combinatorica, 35(5):573-618, 2015. URL: http://dx.doi.org/10.1007/s00493-014-3027-1.
  10. David Ellis, Yuval Filmus, and Ehud Friedgut. A stability result for balanced dictatorships in S_n. Random Structures Algorithms, 46(3):494-530, 2015. URL: http://dx.doi.org/10.1002/rsa.20515.
  11. David Ellis, Yuval Filmus, and Ehud Friedgut. Low degree Boolean functions on s_n, with an application to isoperimetry. Forum of Mathematics, Sigma, 5:e23, 2017. URL: http://dx.doi.org/10.1017/fms.2017.24.
  12. Shai Evra and Tali Kaufman. Bounded degree cosystolic expanders of every dimension. In Proc. 48th ACM Symp. on Theory of Computing (STOC), pages 36-48, 2016. URL: http://dx.doi.org/10.1145/2897518.2897543.
  13. Yuval Filmus. Friedgut-Kalai-Naor theorem for slices of the Boolean cube. Chic. J. Theoret. Comput. Sci., 2016(14), 2016. URL: http://dx.doi.org/10.4086/cjtcs.2016.014.
  14. Yuval Filmus. An orthogonal basis for functions over a slice of the Boolean hypercube. Electron. J. Combin., 23(1):P1.23, 2016. URL: http://arxiv.org/abs/1406.0142.
  15. Yuval Filmus, Guy Kindler, Elchanan Mossel, and Karl Wimmer. Invariance principle on the slice. In Proc. 31st Comput. Complexity Conf., volume 50 of LIPIcs, pages 15:1-15:10. Schloss Dagstuhl, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.15.
  16. Yuval Filmus and Elchanan Mossel. Harmonicity and invariance on slices of the Boolean cube. In Proc. 31st Comput. Complexity Conf., volume 50 of LIPIcs, pages 16:1-16:13. Schloss Dagstuhl, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.16.
  17. Ehud Friedgut, Gil Kalai, and Assaf Naor. Boolean functions whose Fourier transform is concentrated on the first two levels and neutral social choice. Adv. Appl. Math., 29(3):427-437, 2002. URL: http://dx.doi.org/10.1016/S0196-8858(02)00024-6.
  18. Howard Garland. p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Ann. of Math., 97(3):375-423, 1973. URL: http://dx.doi.org/10.2307/1970829.
  19. Tali Kaufman, David Kazhdan, and Alexander Lubotzky. Ramanujan complexes and bounded degree topological expanders. In Proc. 55th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 484-493, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.58.
  20. Tali Kaufman and David Mass. Good distance lattices from high dimensional expanders. (manuscript), 2018. URL: http://arxiv.org/abs/1803.02849.
  21. Tali Kaufman and Izhar Oppenheim. High order random walks: Beyond spectral gap. In Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer, editors, Proc. 20th International Workshop on Randomization and Computation (RANDOM), volume 116 of LIPIcs. Schloss Dagstuhl, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX/RANDOM.2018.46.
  22. Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, and Grassmann graphs. In Proc. 49th ACM Symp. on Theory of Computing (STOC), pages 576-589, 2017. URL: http://dx.doi.org/10.1145/3055399.3055432.
  23. Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in Grassmann graph have near-perfect expansion. (manuscript), 2018. Google Scholar
  24. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type A_d̃. European J. Combin., 26(6):965-–993, 2005. URL: http://dx.doi.org/10.1016/j.ejc.2004.06.007.
  25. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Ramanujan complexes of type A_d̃. Israel J. Math., 149(1):267-299, 2005. URL: http://dx.doi.org/10.1007/BF02772543.
  26. Ashley Montanaro and Tobias Osborne. Quantum boolean functions. Chic. J. Theoret. Comput. Sci., 2010, 2010. URL: http://dx.doi.org/10.4086/cjtcs.2010.001.
  27. Ryan O'Donnell and Karl Wimmer. KKL, Kruskal-Katona, and monotone nets. SIAM J. Comput., 42(6):2375-2399, 2013. (Preliminary version in 50th FOCS, 2009). URL: http://dx.doi.org/10.1137/100787325.
  28. Izhar Oppenheim. Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps. Discrete Comput. Geom., 59(2):293-330, 2018. URL: http://dx.doi.org/10.1007/s00454-017-9948-x.
  29. Rafael Plaza. Stability for intersecting families in PGL(2,q). Electron. J. Combin., 22(4):P4.41, 2015. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p41.
  30. Richard P. Stanley. Differential posets. J. Amer. Math. Soc., 1(4):919-961, 1988. URL: http://dx.doi.org/10.2307/1990995.
  31. Richard P. Stanley. Variations on differential posets. In Dennisa Stanton, editor, Invariant Theory and Tableaux, volume 19 of IMA Vol. Math. Appl., pages 145-165. Springer, 1990. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact