Document Open Access Logo

New Direct Sum Tests

Authors Alek Westover , Edward Yu , Kai Zhe Zheng

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ITCS.2025.94.pdf
  • Filesize: 0.89 MB
  • 26 pages
HTML5 Logo HTML (experimental)
LIPIcs.ITCS.2025.94.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Linearity testing
  • Direct sum
  • Grids

Metrics

Abstract

A function f:[n]^{d} → 𝔽₂ is a direct sum if there are functions L_i:[n] → 𝔽₂ such that f(x) = ∑_i L_i(x_i). In this work we give multiple results related to the property testing of direct sums. 
Our first result concerns a test proposed by Dinur and Golubev in [Dinur and Golubev, 2019]. We call their test the Diamond test and show that it is indeed a direct sum tester. More specifically, we show that if a function f is ε-far from being a direct sum function, then the Diamond test rejects f with probability at least Ω_{n,ε}(1). Even in the case of n = 2, the Diamond test is, to the best of our knowledge, novel and yields a new tester for the classic property of affinity. 
Apart from the Diamond test, we also analyze a broad family of direct sum tests, which at a high level, run an arbitrary affinity test on the restriction of f to a random hypercube inside of [n]^d. This family of tests includes the direct sum test analyzed in [Dinur and Golubev, 2019], but does not include the Diamond test. As an application of our result, we obtain a direct sum test which works in the online adversary model of [Iden Kalemaj et al., 2022].
Finally, we also discuss a Fourier analytic interpretation of the diamond tester in the n = 2 case, as well as prove local correction results for direct sum as conjectured by [Dinur and Golubev, 2019].

Cite As Get BibTex

Alek Westover, Edward Yu, and Kai Zhe Zheng. New Direct Sum Tests. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 94:1-94:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.94

Author Details

Alek Westover
  • MIT, Cambridge, MA, USA
Edward Yu
  • MIT, Cambridge, MA, USA
Kai Zhe Zheng
  • MIT, Cambridge, MA, USA

Funding

  • Westover, Alek: Supported by MIT SPUR.
  • Yu, Edward: Supported by MIT SPUR.
  • Zheng, Kai Zhe: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA. Supported by the NSF GRFP DGE-2141064.

References

  1. Rudolf Ahlswede and Peter Gács. Spreading of sets in product spaces and hypercontraction of the markov operator. The annals of probability, pages 925-939, 1976. Google Scholar
  2. Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing low-degree polynomials over gf (2). In International Workshop on Randomization and Approximation Techniques in Computer Science, pages 188-199. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-45198-3_17.
  3. Prashanth Amireddy, Srikanth Srinivasan, and Madhu Sudan. Low-degree testing over grids. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.41.
  4. Mitali Bafna, Dor Minzer, and Nikhil Vyas. Quasi-linear size pcps with small soundness from HDX. CoRR, abs/2407.12762, 2024. URL: https://doi.org/10.48550/arXiv.2407.12762.
  5. Mitali Bafna, Srikanth Srinivasan, and Madhu Sudan. Local decoding and testing of polynomials over grids. Random Struct. Algorithms, 57(3):658-694, 2020. URL: https://doi.org/10.1002/RSA.20933.
  6. Omri Ben-Eliezer, Esty Kelman, Uri Meir, and Sofya Raskhodnikova. Property testing with online adversaries. In Venkatesan Guruswami, editor, 15th Innovations in Theoretical Computer Science Conference, ITCS 2024, January 30 to February 2, 2024, Berkeley, CA, USA, volume 287 of LIPIcs, pages 11:1-11:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ITCS.2024.11.
  7. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Optimal testing of reed-muller codes. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 488-497. IEEE, 2010. URL: https://doi.org/10.1109/FOCS.2010.54.
  8. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 73-83, 1990. URL: https://doi.org/10.1145/100216.100225.
  9. Siu On Chan. Approximation resistance from pairwise-independent subgroups. J. ACM, 63(3):27:1-27:32, 2016. URL: https://doi.org/10.1145/2873054.
  10. Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. Direct sum testing. In Proceedings of the 2015 Conference on innovations in theoretical computer science, pages 327-336, 2015. URL: https://doi.org/10.1145/2688073.2688078.
  11. Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1495-1524. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00088.
  12. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. URL: https://doi.org/10.1145/1236457.1236459.
  13. Irit Dinur and Konstantin Golubev. Direct Sum Testing: The General Case. In Dimitris Achlioptas and László A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 40:1-40:11, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.40.
  14. Irit Dinur and Or Meir. Derandomized parallel repetition via structured pcps. Electron. Colloquium Comput. Complex., TR10-107, 2010. URL: https://eccc.weizmann.ac.il/report/2010/107, URL: https://arxiv.org/abs/TR10-107.
  15. Irit Dinur and Omer Reingold. Assignment testers: Towards a combinatorial proof of the pcp-theorem. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 155-164. IEEE Computer Society, 2004. URL: https://doi.org/10.1109/FOCS.2004.16.
  16. Irit Dinur and David Steurer. Direct product testing. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 188-196. IEEE, 2014. URL: https://doi.org/10.1109/CCC.2014.27.
  17. Oded Goldreich and Shmuel Safra. A combinatorial consistency lemma with application to proving the PCP theorem. SIAM J. Comput., 29(4):1132-1154, 2000. URL: https://doi.org/10.1137/S0097539797315744.
  18. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. New direct-product testers and 2-query pcps. SIAM J. Comput., 41(6):1722-1768, 2012. URL: https://doi.org/10.1137/09077299X.
  19. Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on Boolean functions. Institute for Mathematical Studies in the Social Sciences, 1989. Google Scholar
  20. Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma. Sublinear-time computation in the presence of online erasures. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 90:1-90:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ITCS.2022.90.
  21. Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 403-412, 2008. URL: https://doi.org/10.1145/1374376.1374434.
  22. Dor Minzer and Kai Zhe Zheng. Adversarial low degree testing. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 4395-4409. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.154.
  23. Dor Minzer and Kai Zhe Zheng. Near optimal alphabet-soundness tradeoff pcps. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 15-23. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649606.
  24. Ryan O'Donnell. Analysis of boolean functions, 2021. URL: https://arxiv.org/abs/2105.10386.
  25. Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 475-484. ACM, 1997. URL: https://doi.org/10.1145/258533.258641.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact