Document Open Access Logo

Symmetric Distributions from Shallow Circuits

Authors Daniel M. Kane , Anthony Ostuni , Kewen Wu

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2026.53.pdf
  • Filesize: 0.78 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Sampling
  • distributions
  • locality
  • circuit complexity

Metrics

Abstract

We characterize the symmetric distributions that can be (approximately) generated by shallow Boolean circuits. More precisely, let f: {0,1}^m → {0,1}ⁿ be a Boolean function where each output bit depends on at most d input bits. Suppose the output distribution of f evaluated on uniformly random input bits is close in total variation distance to a symmetric distribution 𝒟 over {0,1}ⁿ. Then 𝒟 must be close to a mixture of the uniform distribution over n-bit strings of even Hamming weight, the uniform distribution over n-bit strings of odd Hamming weight, and γ-biased product distributions for γ an integer multiple of 2^{-d}. Moreover, the mixing weights are determined by low-degree, sparse 𝔽₂-polynomials. This extends the previous classification for generating symmetric distributions that are also uniform over their support.

Cite As Get BibTex

Daniel M. Kane, Anthony Ostuni, and Kewen Wu. Symmetric Distributions from Shallow Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.53

Author Details

Daniel M. Kane
  • UC San Diego, La Jolla, CA, USA
Anthony Ostuni
  • UC San Diego, La Jolla, CA, USA
Kewen Wu
  • Caltech, Pasadena, CA, USA

Funding

  • Kane, Daniel M.: Supported by NSF Medium Award CCF-2107547 and NSF CAREER Award CCF-1553288.

Acknowledgements

We thank Farzan Byramji and anonymous reviewers for helpful comments on an earlier draft of this paper.

References

  1. M. Ajtai. Σ^1_1-formulae on finite structures. Ann. Pure Appl. Logic, 24(1):1-48, 1983. URL: https://doi.org/10.1016/0168-0072(83)90038-6.
  2. Yaroslav Alekseev, Mika Göös, Konstantin Myasnikov, Artur Riazanov, and Dmitry Sokolov. Sampling permutations with cell probes is hard. In Proceedings of the 58th Annual ACM Symposium on Theory of Computing, 2026. Google Scholar
  3. Maryam Aliakbarpour, Eric Blais, and Ronitt Rubinfeld. Learning and testing junta distributions. In Conference on Learning Theory, pages 19-46. PMLR, 2016. URL: http://proceedings.mlr.press/v49/aliakbarpour16.html.
  4. Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. Improved bounds for the sunflower lemma. Annals of Mathematics, 194(3):795-815, 2021. Google Scholar
  5. Lászió Babai. Random oracles separate PSPACE from the polynomial-time hierarchy. Information Processing Letters, 26(1):51-53, 1987. URL: https://doi.org/10.1016/0020-0190(87)90036-6.
  6. Chris Beck, Russell Impagliazzo, and Shachar Lovett. Large deviation bounds for decision trees and sampling lower bounds for AC0-circuits. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 101-110. IEEE, 2012. URL: https://doi.org/10.1109/FOCS.2012.82.
  7. Adam Bene Watts and Natalie Parham. Unconditional Quantum Advantage for Sampling with Shallow Circuits. In Shubhangi Saraf, editor, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17:1-17:12, Dagstuhl, Germany, 2026. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.17.
  8. Lucien Birgé. Estimating a density under order restrictions: Nonasymptotic minimax risk. The Annals of Statistics, pages 995-1012, 1987. Google Scholar
  9. Anselm Blumer, Andrzej Ehrenfeucht, David Haussler, and Manfred K Warmuth. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM (JACM), 36(4):929-965, 1989. URL: https://doi.org/10.1145/76359.76371.
  10. Ravi B Boppana and Jeffrey C Lagarias. One-way functions and circuit complexity. Information and Computation, 74(3):226-240, 1987. URL: https://doi.org/10.1016/0890-5401(87)90022-8.
  11. Eshan Chattopadhyay, Jesse Goodman, and David Zuckerman. The space complexity of sampling. In 13th Innovations in Theoretical Computer Science Conference,(ITCS 2022), 2022. Google Scholar
  12. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, and David Zuckerman. XOR lemmas for resilient functions against polynomials. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 234-246, 2020. URL: https://doi.org/10.1145/3357713.3384242.
  13. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 670-683, 2016. URL: https://doi.org/10.1145/2897518.2897528.
  14. Gil Cohen and Leonard J Schulman. Extractors for near logarithmic min-entropy. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 178-187. IEEE, 2016. URL: https://doi.org/10.1109/FOCS.2016.27.
  15. Constantinos Daskalakis, Ilias Diakonikolas, Ryan O'Donnell, Rocco A Servedio, and Li-Yang Tan. Learning sums of independent integer random variables. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 217-226. IEEE, 2013. URL: https://doi.org/10.1109/FOCS.2013.31.
  16. Constantinos Daskalakis and Gautam Kamath. Faster and sample near-optimal algorithms for proper learning mixtures of Gaussians. In Conference on Learning Theory, pages 1183-1213. PMLR, 2014. URL: http://proceedings.mlr.press/v35/daskalakis14.html.
  17. Anindya De and Thomas Watson. Extractors and lower bounds for locally samplable sources. ACM Transactions on Computation Theory (TOCT), 4(1):1-21, 2012. URL: https://doi.org/10.1145/2141938.2141941.
  18. Ilias Diakonikolas. Learning structured distributions. Handbook of Big Data, 267:10-1201, 2016. Google Scholar
  19. 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.
  20. Lutz Dümbgen and Kaspar Rufibach. Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli, pages 40-68, 2009. Google Scholar
  21. Yuval Filmus, Itai Leigh, Artur Riazanov, and Dmitry Sokolov. Sampling and certifying symmetric functions. In Approximation, Randomization, and Combinatorial Optimization. (APPROX/RANDOM), 2023. Google Scholar
  22. Merrick Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Math. Systems Theory, 17(1):13-27, 1984. URL: https://doi.org/10.1007/BF01744431.
  23. 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. IEEE, 2010. URL: https://doi.org/10.1109/CCC.2010.29.
  24. Daniel Grier, Daniel M. Kane, Jackson Morris, Anthony Ostuni, and Kewen Wu. Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals. In Shubhangi Saraf, editor, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362 of Leibniz International Proceedings in Informatics (LIPIcs), pages 73:1-73:14, Dagstuhl, Germany, 2026. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.73.
  25. Aryeh Grinberg, Ronen Shaltiel, and Emanuele Viola. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 956-966. IEEE, 2018. URL: https://doi.org/10.1109/FOCS.2018.00094.
  26. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages 6-20, 1986. URL: https://doi.org/10.1145/12130.12132.
  27. Johan Håstad. Computational limitations for small depth circuits. PhD thesis, Massachusetts Institute of Technology, 1986. Google Scholar
  28. Russell Impagliazzo and Moni Naor. Efficient cryptographic schemes provably as secure as subset sum. Journal of cryptology, 9(4):199-216, 1996. URL: https://doi.org/10.1007/BF00189260.
  29. Mark R Jerrum, Leslie G Valiant, and Vijay V Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical computer science, 43:169-188, 1986. URL: https://doi.org/10.1016/0304-3975(86)90174-X.
  30. Daniel M Kane, Anthony Ostuni, and Kewen Wu. Locality bounds for sampling Hamming slices. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1279-1286, 2024. URL: https://doi.org/10.1145/3618260.3649670.
  31. Daniel M Kane, Anthony Ostuni, and Kewen Wu. Locally sampleable uniform symmetric distributions. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 1807-1816, 2025. URL: https://doi.org/10.1145/3717823.3718243.
  32. Daniel M Kane, Anthony Ostuni, and Kewen Wu. Symmetric distributions from shallow circuits. arXiv preprint arXiv:2511.14127, 2025. URL: https://doi.org/10.48550/arXiv.2511.14127.
  33. Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert E Schapire, and Linda Sellie. On the learnability of discrete distributions. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 273-282, 1994. URL: https://doi.org/10.1145/195058.195155.
  34. Bruce G Lindsay. Mixture models: Theory, geometry and applications. In NSF-CBMS Regional Conference Series in Probability and Statistics, pages i-163. JSTOR, 1995. Google Scholar
  35. Shachar Lovett and Emanuele Viola. Bounded-depth circuits cannot sample good codes. In 2011 IEEE 26th Annual Conference on Computational Complexity, pages 243-251. IEEE, 2011. URL: https://doi.org/10.1109/CCC.2011.11.
  36. Benjamin Rossman. The monotone complexity of k-clique on random graphs. SIAM Journal on Computing, 43(1):256-279, 2014. URL: https://doi.org/10.1137/110839059.
  37. Ronen Shaltiel and Jad Silbak. Explicit codes for poly-size circuits and functions that are hard to sample on low entropy distributions. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 2028-2038, 2024. URL: https://doi.org/10.1145/3618260.3649735.
  38. Leslie G Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134-1142, 1984. URL: https://doi.org/10.1145/1968.1972.
  39. Emanuele Viola. Bit-probe lower bounds for succinct data structures. SIAM Journal on Computing, 41(6):1593, 2012. Google Scholar
  40. Emanuele Viola. The complexity of distributions. SIAM Journal on Computing, 41(1):191-218, 2012. URL: https://doi.org/10.1137/100814998.
  41. Emanuele Viola. Extractors for Turing-machine sources. In International Workshop on Approximation Algorithms for Combinatorial Optimization, pages 663-671. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_56.
  42. Emanuele Viola. Extractors for circuit sources. SIAM Journal on Computing, 43(2):655-672, 2014. URL: https://doi.org/10.1137/11085983X.
  43. Emanuele Viola. Sampling lower bounds: boolean average-case and permutations. SIAM Journal on Computing, 49(1):119-137, 2020. URL: https://doi.org/10.1137/18M1198405.
  44. Emanuele Viola. New sampling lower bounds via the separator. In 38th Computational Complexity Conference (CCC 2023), 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.26.
  45. Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles. In 26th Annual Symposium on Foundations of Computer Science (sfcs 1985), pages 1-10. IEEE, 1985. Google Scholar
  46. Yannis G Yatracos. Rates of convergence of minimum distance estimators and kolmogorov’s entropy. The Annals of Statistics, 13(2):768-774, 1985. Google Scholar
  47. Huacheng Yu and Wei Zhan. Sampling, Flowers and Communication. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024), volume 287 of Leibniz International Proceedings in Informatics (LIPIcs), pages 100:1-100:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ITCS.2024.100.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact