Document Open Access Logo

A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions

Authors Penghui Yao , Mingnan Zhao

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.134.pdf
  • Filesize: 0.82 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandom generators
  • polynomial threshold functions

Metrics

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

Abstract

Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-d polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length poly(k, d, 1/ε)⋅log n that can fool any function of k degree-d PTFs with probability at least 1 - ε. More specifically, we show that the summation of L independent R-moment-matching Gaussian vectors ε-fools functions of k degree-d PTFs, where L = poly(k, d, 1/ε) and R = O(log kd/ε). The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence.

Cite As Get BibTex

Penghui Yao and Mingnan Zhao. A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 134:1-134:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.134

Author Details

Penghui Yao
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, Nanjing 210023, China
  • Hefei National Laboratory, Hefei 230088, China
Mingnan Zhao
  • State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, Nanjing 210023, China

Funding

PY and MZ were supported by National Natural Science Foundation of China (Grant Nos. 62332009 and 12347104), Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302901), NSFC/RGC Joint Research Scheme (Grant No. 12461160276), Natural Science Foundation of Jiangsu Province (Grant No. BK20243060), and New Cornerstone Science Foundation.

References

  1. Louay M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM Journal on Computing, 38(6):2220-2272, 2009. URL: https://doi.org/10.1137/070691954.
  2. G. E. P. Box and Mervin E. Muller. A note on the generation of random normal deviates. The Annals of Mathematical Statistics, 29(2):610-611, 1958. URL: https://doi.org/10.1214/aoms/1177706645.
  3. Anthony Carbery and James Wright. Distributional and L^q norm inequalities for polynomials over convex bodies in ℝⁿ. Mathematical Research Letters, 8:233-248, 2001. URL: https://doi.org/10.4310/MRL.2001.v8.n3.a1.
  4. Eshan Chattopadhyay, Anindya De, and Rocco A. Servedio. Simple and efficient pseudorandom generators from Gaussian processes. In Proceedings of the 34th Computational Complexity Conference, Dagstuhl, DEU, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.4.
  5. Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. SIAM Journal on Computing, 39(8):3441-3462, 2010. URL: https://doi.org/10.1137/100783030.
  6. Ilias Diakonikolas, Daniel M. Kane, and Jelani Nelson. Bounded independence fools degree-2 threshold functions. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 11-20, 2010. URL: https://doi.org/10.1109/FOCS.2010.8.
  7. Parikshit Gopalan, Daniel M. Kane, and Raghu Meka. Pseudorandomness via the discrete fourier transform. SIAM Journal on Computing, 47(6):2451-2487, 2018. URL: https://doi.org/10.1137/16M1062132.
  8. Parikshit Gopalan, Ryan O'Donnell, Yi Wu, and David Zuckerman. Fooling functions of halfspaces under product distributions. In 2010 IEEE 25th Annual Conference on Computational Complexity, pages 223-234, 2010. URL: https://doi.org/10.1109/CCC.2010.29.
  9. Prahladh Harsha, Adam Klivans, and Raghu Meka. An invariance principle for polytopes. J. ACM, 59(6), 2013. URL: https://doi.org/10.1145/2395116.2395118.
  10. William B Johnson, Joram Lindenstrauss, and Gideon Schechtman. Extensions of Lipschitz maps into Banach spaces. Israel Journal of Mathematics, 54(2):129-138, 1986. URL: https://doi.org/10.1007/BF02764938.
  11. Daniel Kane, Raghu Meka, and Jelani Nelson. Almost optimal explicit johnson-lindenstrauss families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 628-639, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-22935-0_53.
  12. Daniel M. Kane. k-independent Gaussians fool polynomial threshold functions. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 252-261, 2011. URL: https://doi.org/10.1109/CCC.2011.13.
  13. Daniel M. Kane. A small PRG for polynomial threshold functions of Gaussians. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 257-266, USA, 2011. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2011.16.
  14. Daniel M. Kane. A structure theorem for poorly anticoncentrated Gaussian chaoses and applications to the study of polynomial threshold functions. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 91-100, 2012. URL: https://doi.org/10.1109/FOCS.2012.52.
  15. Daniel M. Kane. A pseudorandom generator for polynomial threshold functions of Gaussian with subpolynomial seed length. In 2014 IEEE 29th Conference on Computational Complexity, pages 217-228, 2014. URL: https://doi.org/10.1109/CCC.2014.30.
  16. Daniel M. Kane. A Polylogarithmic PRG for Degree 2 Threshold Functions in the Gaussian Setting. In David Zuckerman, editor, 30th Conference on Computational Complexity (CCC 2015), volume 33 of Leibniz International Proceedings in Informatics (LIPIcs), pages 567-581, Dagstuhl, Germany, 2015. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.567.
  17. Zander Kelley and Raghu Meka. Random restrictions and PRGs for PTFs in Gaussian space. In Shachar Lovett, editor, 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:24, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.21.
  18. Adam R. Klivans, Ryan O'Donnell, and Rocco A. Servedio. Learning geometric concepts via Gaussian surface area. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 541-550, 2008. URL: https://doi.org/10.1109/FOCS.2008.64.
  19. Pravesh K. Kothari and Raghu Meka. Almost optimal pseudorandom generators for spherical caps: extended abstract. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, pages 247-256, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746611.
  20. J. W. Lindeberg. Eine neue herleitung des exponentialgesetzes in der wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 15:211-225, 1922. URL: https://doi.org/10.1007/BF01494395.
  21. Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. SIAM Journal on Computing, 42(3):1275-1301, 2013. URL: https://doi.org/10.1137/100811623.
  22. Jorge Nocedal and Stephen J. Wright, editors. Quadratic Programming, pages 438-486. Springer New York, New York, NY, 1999. URL: https://doi.org/10.1007/0-387-22742-3_16.
  23. Ryan O'Donnell. Analysis of boolean functions. Cambridge University Press, 2014. URL: https://doi.org/10.1017/CBO9781139814782.
  24. Ryan O'Donnell, Rocco A. Servedio, and Li-Yang Tan. Fooling Gaussian PTFs via local hyperconcentration. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 1170-1183, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384281.
  25. Ryan O’Donnell, Rocco A. Servedio, and Li-Yang Tan. Fooling polytopes. J. ACM, 69(2), January 2022. URL: https://doi.org/10.1145/3460532.
  26. Alexander Razborov. A simple proof of Bazzi’s theorem. ACM Trans. Comput. Theory, 1(1), February 2009. URL: https://doi.org/10.1145/1490270.1490273.
  27. R. A. Servedio and L. Tan. Fooling intersections of low-weight halfspaces. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 824-835, Los Alamitos, CA, USA, October 2017. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2017.81.
  28. Rocco A. Servedio. Every linear threshold function has a low-weight approximator. In 21st Annual IEEE Conference on Computational Complexity (CCC'06), pages 18-32, 2006. URL: https://doi.org/10.1109/CCC.2006.18.
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