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Fourier Conjectures, Correlation Bounds, and Majority

Author Emanuele Viola

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Emanuele Viola
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA

Funding

  • Viola, Emanuele: Supported by NSF CCF award 1813930.

Acknowledgements

This paper includes the results in [Emanuele Viola, 2019]. I am grateful to Chin Ho Lee for pointing out the work [Eshan Chattopadhyay et al., 2020] to me, and to an anonymous reviewer for suggesting the use of hypercontractivity to bound 𝔼|g_{k}(x)| in the proof of Theorem 1 (alternatively one can reason along the lines of the proof of Theorem 4). A preliminary version of this paper had Theorem 7 only for d ≥ Ω(n^{1/3}), and the degree bound was O(d√{log n}). Jarosław Błasiok pointed out to us how to improve the proof to obtain Theorem 7. The proof in the preliminary version was similar, but rather than performing a case analysis, detected the two cases explicitly with an auxiliary polynomial, which led to d ≥ Ω(n^{1/3}). It also used the polynomials for {Maj} with polynomially-small error, as opposed to constant, which led to the extra √{log n} factor. Following these ideas, we also improved the results on the coin problem and h₂. We are very grateful to Jarosław Błasiok for letting us include the improved results!

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Emanuele Viola. Fourier Conjectures, Correlation Bounds, and Majority. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 111:1-111:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.111

Abstract

Recently several conjectures were made regarding the Fourier spectrum of low-degree polynomials. We show that these conjectures imply new correlation bounds for functions related to Majority. Then we prove several new results on correlation bounds which aim to, but don't, resolve the conjectures. In particular, we prove several new results on Majority which are of independent interest and complement Smolensky’s classic result.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Fourier analysis
  • polynomials
  • Majority
  • correlation
  • lower bound
  • conjectures

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References

  1. Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 136-150, 2015. URL: https://doi.org/10.1109/FOCS.2015.18.
  2. Nayantara Bhatnagar, Parikshit Gopalan, and Richard J. Lipton. Symmetric polynomials over Z_m and simultaneous communication protocols. J. of Computer and System Sciences, 72(2):252-285, 2006. Google Scholar
  3. Andrej Bogdanov and Emanuele Viola. Pseudorandom bits for polynomials. SIAM J. on Computing, 39(6):2464-2486, 2010. Google Scholar
  4. Joshua Brody and Elad Verbin. The coin problem, and pseudorandomness for branching programs. In 51th IEEE Symp. on Foundations of Computer Science (FOCS), 2010. Google Scholar
  5. Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty. Fractional pseudorandom generators from any fourier level. CoRR, abs/2008.01316, 2020. URL: http://arxiv.org/abs/2008.01316.
  6. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom generators from polarizing random walks. In CCC, volume 102 of LIPIcs, pages 1:1-1:21. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  7. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, and David Zuckerman. XOR lemmas for resilient functions against polynomials. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, ACM Symp. on the Theory of Computing (STOC), pages 234-246. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384242.
  8. Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom generators from the second fourier level and applications to AC0 with parity gates. In ACM Innovations in Theoretical Computer Science conf. (ITCS), pages 22:1-22:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.22.
  9. Thomas Cover and Joy Thomas. Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, 2006. Google Scholar
  10. Bill Fefferman, Ronen Shaltiel, Christopher Umans, and Emanuele Viola. On beating the hybrid argument. Theory of Computing, 9:809-843, 2013. Google Scholar
  11. Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, and Avishay Tal. AC^0[p] lower bounds against MCSP via the coin problem. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 66:1-66:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.66.
  12. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics: A Foundation for Computer Science, 2nd Ed. Addison-Wesley, 1994. URL: https://www-cs-faculty.stanford.edu/%7Eknuth/gkp.html.
  13. Nutan Limaye, Karteek Sreenivasaiah, Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. A fixed-depth size-hierarchy theorem for AC^0[⊕] via the coin problem. In ACM Symp. on the Theory of Computing (STOC), pages 442-453. ACM, 2019. Google Scholar
  14. Shachar Lovett. Unconditional pseudorandom generators for low degree polynomials. Theory of Computing, 5(1):69-82, 2009. URL: http://arxiv.org/abs/toc:v005/a003.
  15. Noam Nisan. Pseudorandom bits for constant depth circuits. Combinatorica. An Journal on Combinatorics and the Theory of Computing, 11(1):63-70, 1991. Google Scholar
  16. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  17. Alexander Razborov. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Akademiya Nauk SSSR. Matematicheskie Zametki, 41(4):598-607, 1987. English translation in Mathematical Notes of the Academy of Sci. of the USSR, 41(4):333-338, 1987. Google Scholar
  18. Ronen Shaltiel and Emanuele Viola. Hardness amplification proofs require majority. SIAM J. on Computing, 39(7):3122-3154, 2010. Google Scholar
  19. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In 19th ACM Symp. on the Theory of Computing (STOC), pages 77-82. ACM, 1987. Google Scholar
  20. Srikanth Srinivasan. On improved degree lower bounds for polynomial approximation. In FSTTCS, volume 24 of LIPIcs, pages 201-212. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. Google Scholar
  21. Srikanth Srinivasan. A robust version of hegedus’s lemma, with applications. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 1349-1362. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384328.
  22. Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. On the probabilistic degrees of symmetric boolean functions, 2019. URL: http://arxiv.org/abs/1910.02465.
  23. Charalampos E. Tsourakakis, Michael Mitzenmacher, Jaroslaw Blasiok, Ben Lawson, Preetum Nakkiran, and Vasileios Nakos. Predicting positive and negative links with noisy queries: Theory & practice. CoRR, abs/1709.07308, 2017. URL: http://arxiv.org/abs/1709.07308.
  24. Emanuele Viola. The complexity of hardness amplification and derandomization. Ph.D. thesis, Harvard University, 2006. Google Scholar
  25. Emanuele Viola. On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science, 5(1):1-72, 2009. Google Scholar
  26. Emanuele Viola. The sum of d small-bias generators fools polynomials of degree d. Computational Complexity, 18(2):209-217, 2009. Google Scholar
  27. Emanuele Viola. Challenges in computational lower bounds. SIGACT News, Open Problems Column, 48(1), 2017. Google Scholar
  28. Emanuele Viola. Matching Smolensky’s correlation bound with majority, 2019. URL: https://eccc.weizmann.ac.il/report/2019/175/.
  29. Emanuele Viola. New lower bounds for probabilistic degree and AC0 with parity gates. Theory of Computing, 2020. URL: https://eccc.weizmann.ac.il/report/2020/015/.
  30. Andrew Chi-Chih Yao. Probabilistic computations: Toward a unified measure of complexity (extended abstract). In IEEE Symp. on Foundations of Computer Science (FOCS), pages 222-227. IEEE Computer Society, 1977. Google Scholar
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