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Fourier Bounds and Pseudorandom Generators for Product Tests

Author Chin Ho Lee

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Author Details

Chin Ho Lee
  • Northeastern University, Boston, USA

Funding

Supported by NSF CCF award 1813930. Work done in part while visiting Amnon Ta-Shma at Tel Aviv University, with support from the Blavatnik family fund and ISF grant no. 952/18.

Acknowledgements

I thank Salil Vadhan for asking about the Fourier spectrum of product tests. I also thank Andrej Bogdanov, Gil Cohen, Amnon Ta-Shma, Avishay Tal and Emanuele Viola for very helpful conversations. I am grateful to Emanuele Viola for his invaluable comments on the write-up, and Dean Doron, Pooya Hatami and William Hoza for pointing out a simplification and improvement of Theorem 8. Finally, I thank the anonymous reviewers for their useful comments.

Cite AsGet BibTex

Chin Ho Lee. Fourier Bounds and Pseudorandom Generators for Product Tests. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 7:1-7:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.CCC.2019.7

Abstract

We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest. As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths. We also extend our results to functions f_i whose range is [-1,1].

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • bounded independence plus noise
  • Fourier spectrum
  • product test
  • pseudorandom generators

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References

  1. Scott Aaronson. BQP and the polynomial hierarchy. In 42nd ACM Symp. on the Theory of Computing (STOC), pages 141-150. ACM, 2010. Google Scholar
  2. Miklós Ajtai. Σ \sp1\sb1-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983. Google Scholar
  3. Miklós Ajtai and Michael Ben-Or. A Theorem on Probabilistic Constant Depth Computations. In 16th ACM Symp. on the Theory of Computing (STOC), pages 471-474, 1984. Google Scholar
  4. Miklós Ajtai, János Komlós, and Endre Szemerédi. Deterministic simulation in LOGSPACE. In 19th ACM Symp. on the Theory of Computing (STOC), pages 132-140, 1987. Google Scholar
  5. Miklos Ajtai and Avi Wigderson. Deterministic simulation of probabilistic constant-depth circuits. Advances in Computing Research - Randomness and Computation, 5:199-223, 1989. Google Scholar
  6. Kazuyuki Amano. Bounds on the Size of Small Depth Circuits for Approximating Majority. In 36th Coll. on Automata, Languages and Programming (ICALP), pages 59-70. Springer, 2009. Google Scholar
  7. Roy Armoni, Michael E. Saks, Avi Wigderson, and Shiyu Zhou. Discrepancy Sets and Pseudorandom Generators for Combinatorial Rectangles. In 37th IEEE Symp. on Foundations of Computer Science (FOCS), pages 412-421, 1996. Google Scholar
  8. Andrej Bogdanov, Periklis A. Papakonstantinou, and Andrew Wan. Pseudorandomness for Read-Once Formulas. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 240-246, 2011. Google Scholar
  9. 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
  10. Mei-Chu Chang. A polynomial bound in Freiman’s theorem. Duke Math. J., 113(3):399-419, 2002. URL: https://doi.org/10.1215/S0012-7094-02-11331-3.
  11. Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Improved algorithms via approximations of probability distributions. J. Comput. System Sci., 61(1):81-107, 2000. URL: https://doi.org/10.1006/jcss.1999.1695.
  12. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom Generators from Polarizing Random Walks. Electronic Colloquium on Computational Complexity, Technical Report TR18-015, 2018. URL: https://eccc.weizmann.ac.il/report/2018/015.
  13. Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom generators from the second Fourier level and applications to AC0 with parity gates. In ITCS'19 - Proceedings of the 2019 ACM Conference on Innovations in Theoretical Computer Science. ACM, 2019. Google Scholar
  14. Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, and Avishay Tal. Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity. In ACM Symp. on the Theory of Computing (STOC), 2018. Google Scholar
  15. Sitan Chen, Thomas Steinke, and Salil P. Vadhan. Pseudorandomness for Read-Once, Constant-Depth Circuits. CoRR, abs/1504.04675, 2015. URL: http://arxiv.org/abs/1504.04675.
  16. Gil Cohen, Anat Ganor, and Ran Raz. Two Sides of the Coin Problem. In Workshop on Randomization and Computation (RANDOM), pages 618-629, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.618.
  17. Anindya De, Omid Etesami, Luca Trevisan, and Madhur Tulsiani. Improved Pseudorandom Generators for Depth 2 Circuits. In Workshop on Randomization and Computation (RANDOM), pages 504-517, 2010. Google Scholar
  18. Irit Dinur, Ehud Friedgut, Guy Kindler, and Ryan O'Donnell. On the Fourier tails of bounded functions over the discrete cube. Israel J. Math., 160:389-412, 2007. URL: https://doi.org/10.1007/s11856-007-0068-9.
  19. Dean Doron, Pooya Hatami, and William Hoza. Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas, 2018. ECCC TR18-183. Google Scholar
  20. Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. Efficient approximation of product distributions. Random Struct. Algorithms, 13(1):1-16, 1998. Google Scholar
  21. Michael A. Forbes and Zander Kelley. Pseudorandom Generators for Read-Once Branching Programs, in any Order. In 47th IEEE Symposium on Foundations of Computer Science (FOCS), 2018. Google Scholar
  22. Parikshit Gopalan, Daniel Kane, and Raghu Meka. Pseudorandomness via the Discrete Fourier Transform. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 903-922, 2015. URL: https://doi.org/10.1109/FOCS.2015.60.
  23. Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, and Salil Vadhan. Better Pseudorandom Generators from Milder Pseudorandom Restrictions. In IEEE Symp. on Foundations of Computer Science (FOCS), 2012. Google Scholar
  24. Parikshit Gopalan, Rocco A. Servedio, and Avi Wigderson. Degree and sensitivity: tails of two distributions. In 31st Conference on Computational Complexity, volume 50 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 13, 23. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016. Google Scholar
  25. Parikshit Gopalan and Amir Yehudayoff. Inequalities and tail bounds for elementary symmetric polynomials. Electronic Colloquium on Computational Complexity (ECCC), 21:19, 2014. URL: http://eccc.hpi-web.de/report/2014/019.
  26. Elad Haramaty, Chin Ho Lee, and Emanuele Viola. Bounded independence plus noise fools products. SIAM J. Comput., 47(2):493-523, 2018. URL: https://doi.org/10.1137/17M1129088.
  27. Pooya Hatami and Avishay Tal. Pseudorandom generators for low-sensitivity functions. In 9th Innovations in Theoretical Computer Science, volume 94 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 29, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. Google Scholar
  28. Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from Shrinkage. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 111-119, 2012. Google Scholar
  29. Russell Impagliazzo, Cristopher Moore, and Alexander Russell. An entropic proof of Chang’s inequality. SIAM J. Discrete Math., 28(1):173-176, 2014. URL: https://doi.org/10.1137/120877982.
  30. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for Network Algorithms. In 26th ACM Symp. on the Theory of Computing (STOC), pages 356-364, 1994. Google Scholar
  31. Nathan Keller and Guy Kindler. Quantitative relation between noise sensitivity and influences. Combinatorica, 33(1):45-71, 2013. URL: https://doi.org/10.1007/s00493-013-2719-2.
  32. Swastik Kopparty and Srikanth Srinivasan. Certifying polynomials for AC⁰[⊕] circuits, with applications to lower bounds and circuit compression. Theory Comput., 14:Article 12, 24, 2018. URL: https://doi.org/10.4086/toc.2018.v014a012.
  33. Chin Ho Lee and Emanuele Viola. More on bounded independence plus noise: Pseudorandom generators for read-once polynomials. In Electronic Colloquium on Computational Complexity (ECCC), volume 24, page 167, 2017. Google Scholar
  34. Chin Ho Lee and Emanuele Viola. The coin problem for product tests. ACM Trans. Comput. Theory, 10(3):Art. 14, 10, 2018. URL: https://doi.org/10.1145/3201787.
  35. Nutan Limaye, Karteek Sreenivasiah, Srikanth Srinivasan, Utkarsh Tripathi, and S Venkitesh. The Coin Problem in Constant Depth: Sample Complexity and Parity gates. In Electronic Colloquium on Computational Complexity (ECCC), volume TR18-157, 2018. Google Scholar
  36. Chi-Jen Lu. Improved pseudorandom generators for combinatorial rectangles. Combinatorica, 22(3):417-433, 2002. Google Scholar
  37. Yishay Mansour. An O(n^log log n) learning algorithm for DNF under the uniform distribution. J. Comput. System Sci., 50(3, part 3):543-550, 1995. Fifth Annual Workshop on Computational Learning Theory (COLT) (Pittsburgh, PA, 1992). URL: https://doi.org/10.1006/jcss.1995.1043.
  38. Raghu Meka, Omer Reingold, and Avishay Tal. Pseudorandom Generators for Width-3 Branching Programs. arXiv preprint, 2018. URL: http://arxiv.org/abs/1806.04256.
  39. Joseph Naor and Moni Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. on Computing, 22(4):838-856, 1993. Google Scholar
  40. Noam Nisan. Pseudorandom Generators for Space-bounded Computation. Combinatorica, 12(4):449-461, 1992. Google Scholar
  41. Noam Nisan and David Zuckerman. Randomness is Linear in Space. J. of Computer and System Sciences, 52(1):43-52, February 1996. Google Scholar
  42. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  43. Josip E. Pečarić, Frank Proschan, and Y. L. Tong. Convex functions, partial orderings, and statistical applications, volume 187 of Mathematics in Science and Engineering. Academic Press, Inc., Boston, MA, 1992. Google Scholar
  44. Omer Reingold, Thomas Steinke, and Salil P. Vadhan. Pseudorandomness for Regular Branching Programs via Fourier Analysis. In Workshop on Randomization and Computation (RANDOM), pages 655-670, 2013. Google Scholar
  45. Benjamin Rossman and Srikanth Srinivasan. Separation of AC⁰[⊕] formulas and circuits. In 44th International Colloquium on Automata, Languages, and Programming, volume 80 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 50, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2017. Google Scholar
  46. Rocco A. Servedio and Li-Yang Tan. Improved pseudorandom generators from pseudorandom multi-switching lemmas. CoRR, abs/1801.03590, 2018. URL: http://arxiv.org/abs/1801.03590.
  47. Ronen Shaltiel and Emanuele Viola. Hardness amplification proofs require majority. SIAM J. on Computing, 39(7):3122-3154, 2010. Google Scholar
  48. J. Michael Steele. The Cauchy-Schwarz master class. MAA Problem Books Series. Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 2004. URL: https://doi.org/10.1017/CBO9780511817106.
  49. John P. Steinberger. The Distinguishability of Product Distributions by Read-Once Branching Programs. In IEEE Conf. on Computational Complexity (CCC), pages 248-254, 2013. URL: https://doi.org/10.1109/CCC.2013.33.
  50. Thomas Steinke, Salil P. Vadhan, and Andrew Wan. Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs. In Workshop on Randomization and Computation (RANDOM), pages 885-899, 2014. Google Scholar
  51. Avishay Tal. Tight Bounds on the Fourier Spectrum of AC0. In Conf. on Computational Complexity (CCC), pages 15:1-15:31, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.15.
  52. Michel Talagrand. How much are increasing sets positively correlated? Combinatorica, 16(2):243-258, 1996. URL: https://doi.org/10.1007/BF01844850.
  53. Luca Trevisan and TongKe Xue. A derandomized switching lemma and an improved derandomization of AC0. In Computational Complexity (CCC), 2013 IEEE Conference on, pages 242-247. IEEE, 2013. Google Scholar
  54. Leslie G. Valiant. Short Monotone Formulae for the Majority Function. J. Algorithms, 5(3):363-366, 1984. Google Scholar
  55. Emanuele Viola. On approximate majority and probabilistic time. Computational Complexity, 18(3):337-375, 2009. Google Scholar
  56. Emanuele Viola. Randomness buys depth for approximate counting. Computational Complexity, 23(3):479-508, 2014. Google Scholar
  57. Thomas Watson. Pseudorandom generators for combinatorial checkerboards. Computational Complexity, 22(4):727-769, 2013. URL: https://doi.org/10.1007/s00037-012-0036-6.
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