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# Tight Bounds on the Fourier Spectrum of AC0

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LIPIcs.CCC.2017.15.pdf
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## Cite As

Avishay Tal. Tight Bounds on the Fourier Spectrum of AC0. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 15:1-15:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CCC.2017.15

## Abstract

We show that AC^0 circuits on n variables with depth d and size m have at most 2^{-Omega(k/log^{d-1} m)} of their Fourier mass at level k or above. Our proof builds on a previous result by Hastad (SICOMP, 2014) who proved this bound for the special case k=n. Our result improves the seminal result of Linial, Mansour and Nisan (JACM, 1993) and is tight up to the constants hidden in the Omega notation. As an application, we improve Braverman's celebrated result (JACM, 2010). Braverman showed that any r(m,d,epsilon)-wise independent distribution epsilon-fools AC^0 circuits of size m and depth d, for r(m,d,epsilon) = O(log(m/epsilon))^{2d^2+7d+3}. Our improved bounds on the Fourier tails of AC^0 circuits allows us to improve this estimate to r(m,d,epsilon) = O(log(m/epsilon))^{3d+3}. In contrast, an example by Mansour (appearing in Luby and Velickovic's paper - Algorithmica, 1996) shows that there is a log^{d-1}(m)\log(1/epsilon)-wise independent distribution that does not epsilon-fool AC^0 circuits of size m and depth d. Hence, our result is tight up to the factor \$3\$ in the exponent.
##### Keywords
• bounded depth circuits
• Fourier analysis
• k-wise independence
• Boolean circuits
• switching lemma

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## References

1. S. Aaronson. BQP and the polynomial hierarchy. In STOC, pages 141-150, 2010.
2. M. Ajtai. Σ₁¹-formulae on finite structures. Annals of Pure and Applied Logic, 24:1-48, 1983.
3. N. Alon, O. Goldreich, J. Håstad, and R. Peralta. Simple construction of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289-304, 1992.
4. L. M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput., 38(6):2220-2272, 2009.
5. P. Beame. A switching lemma primer, 1994.
6. R. B. Boppana. The average sensitivity of bounded-depth circuits. Inf. Process. Lett., 63(5):257-261, 1997.
7. M. Braverman. Polylogarithmic independence fools AC⁰ circuits. J. ACM, 57(5):28:1-28:10, 2010.
8. G. Cohen, A. Ganor, and R. Raz. Two sides of the coin problem. In APPROX-RANDOM, pages 618-629, 2014.
9. A. De, O. Etesami, L. Trevisan, and M. Tulsiani. Improved pseudorandom generators for depth 2 circuits. In APPROX-RANDOM, pages 504-517, 2010.
10. Y. Filmus. Smolensky’s lower bound. Unpublished Manuscript, 2010.
11. M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, apr 1984.
12. O. Goldreich and L. A. Levin. A hardcore predicate for all one-way functions. In STOC, pages 25-32, 1989.
13. P. Harsha and S. Srinivasan. On polynomial approximations to AC^0. In RANDOM 2016, pages 32:1-32:14, 2016.
14. J. Håstad. Almost optimal lower bounds for small depth circuits. In STOC, pages 6-20, 1986. URL: http://dx.doi.org/10.1145/12130.12132.
15. J. Håstad. A slight sharpening of LMN. J. Comput. Syst. Sci., 63(3):498-508, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1803.
16. J. Håstad. On the correlation of parity and small-depth circuits. SIAM J. Comput., 43(5):1699-1708, 2014.
17. R. Impagliazzo and V. Kabanets. Fourier concentration from shrinkage. In CCC, pages 321-332, 2014.
18. R. Impagliazzo, W. Matthews, and R. Paturi. A satisfiability algorithm for AC⁰. In SODA, pages 961-972, 2012.
19. R. Kaas and J. M. Buhrman. Mean, median and mode in binomial distributions. Statistica Neerlandica, 34(1):13-18, 1980.
20. J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In FOCS, pages 68-80, 1988.
21. E. Kushilevitz and Y. Mansour. Learning decision trees using the Fourier spectrum. SIAM J. Comput., 22(6):1331-1348, 1993.
22. N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform and learnability. J. ACM, 40(3):607-620, 1993.
23. M. Luby and B. Velickovic. On deterministic approximation of DNF. Algorithmica, 16(4/5):415-433, 1996.
24. O. Lupanov. Implementing the algebra of logic functions in terms of constant depth formulas in the basis &, ∨, ¬. Dokl. Akad. Nauk. SSSR, 136:1041-1042, 1961. In Russian.
25. Y. Mansour. An O(n^log log n) learning algorithm for DNF under the uniform distribution. J. Comput. Syst. Sci., 50(3):543-550, 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1043.
26. J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. on Computing, 22(4):838-856, 1993.
27. R. O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014.
28. R. O'Donnell and K. Wimmer. Approximation by DNF: examples and counterexamples. In ICALP, pages 195-206, 2007.
29. A. A. Razborov. Bounded arithmetic and lower bounds in boolean complexity. In Feasible Mathematics II, volume 13 of Progress in Computer Science and Applied Logic, pages 344-386. Birkhäuser Boston, 1995.
30. A. A. Razborov. A simple proof of Bazzi’s theorem. TOCT, 1(1), 2009.
31. R. Shaltiel and E. Viola. Hardness amplification proofs require majority. SIAM J. Comput., 39(7):3122-3154, 2010.
32. R. Smolensky. On representations by low-degree polynomials. In FOCS 1993, pages 130-138, 1993.
33. A. Tal. Shrinkage of de Morgan formulae from quantum query complexity. In FOCS, pages 551-560, 2014.
34. N. Thapen. Notes on switching lemmas. Unpublished Manuscript, 2009. URL: http://users.math.cas.cz/~thapen/switching.pdf.
35. L. Trevisan and T. Xue. A derandomized switching lemma and an improved derandomization of AC0. In CCC, pages 242-247, 2013.
36. A. C. Yao. Separating the polynomial hierarchy by oracles. In FOCS, pages 1-10, 1985.
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