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Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries

Authors Tomer Adar , Eldar Fischer, Amit Levi

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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Distribution testing
  • conditional sampling
  • sub-cube sampling

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Abstract

We study property testing in the subcube conditional model introduced by Bhattacharyya and Chakraborty (2017). We obtain the first equivalence test for n-dimensional distributions that is quasi-linear in n, improving the previously known Õ(n²/ε²) query complexity bound to Õ(n/ε²). We extend this result to general finite alphabets with logarithmic cost in the alphabet size.
By exploiting the specific structure of the queries that we use (which are more restrictive than general subcube queries), we obtain a cubic improvement over the best known test for distributions over {1,…,N} under the interval querying model of Canonne, Ron and Servedio (2015), attaining a query complexity of Õ((log N)/ε²), which for fixed ε almost matches the known lower bound of Ω((log N)/log log N). We also derive a product test for n-dimensional distributions with Õ(n/ε²) queries, and provide an Ω(√n/ε²) lower bound for this property.

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Tomer Adar, Eldar Fischer, and Amit Levi. Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.48

Author Details

Tomer Adar
  • Technion - Israel Institute of Technology, Haifa, Israel
Eldar Fischer
  • Technion - Israel Institute of Technology, Haifa, Israel
Amit Levi
  • University of Haifa, Israel

Funding

  • Fischer, Eldar: Research supported by an Israel Science Foundation grant number 879/22.

References

  1. Tugkan Batu, Eldar Fischer, Lance Fortnow, Ravi Kumar, Ronitt Rubinfeld, and Patrick White. Testing random variables for independence and identity. In Proceedings 42nd IEEE Symposium on Foundations of Computer Science, pages 442-451. IEEE, 2001. Google Scholar
  2. Tugkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D Smith, and Patrick White. Testing that distributions are close. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 259-269. IEEE, 2000. Google Scholar
  3. Arnab Bhattacharyya, Sutanu Gayen, Saravanan Kandasamy, and NV Vinodchandran. Testing product distributions: A closer look. In Algorithmic Learning Theory, pages 367-396. PMLR, 2021. Google Scholar
  4. Rishiraj Bhattacharyya and Sourav Chakraborty. Property testing of joint distributions using conditional samples. CoRR, abs/1702.01454, 2017. URL: https://arxiv.org/abs/1702.01454.
  5. Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda. Complexity of high-dimensional identity testing with coordinate conditional sampling. In The Thirty Sixth Annual Conference on Learning Theory, pages 1774-1790. PMLR, 2023. Google Scholar
  6. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci., 47(3):549-595, 1993. Google Scholar
  7. Clément L Canonne, Xi Chen, Gautam Kamath, Amit Levi, and Erik Waingarten. Random restrictions of high dimensional distributions and uniformity testing with subcube conditioning. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 321-336. SIAM, 2021. Google Scholar
  8. Clément L Canonne, Ilias Diakonikolas, Daniel M Kane, and Alistair Stewart. Testing bayesian networks. In Conference on Learning Theory, pages 370-448. PMLR, 2017. Google Scholar
  9. Clément L Canonne, Dana Ron, and Rocco A Servedio. Testing probability distributions using conditional samples. SIAM Journal on Computing, 44(3):540-616, 2015. Google Scholar
  10. Sourav Chakraborty, Eldar Fischer, Yonatan Goldhirsh, and Arie Matsliah. On the power of conditional samples in distribution testing. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pages 561-580, 2013. Google Scholar
  11. Xi Chen, Rajesh Jayaram, Amit Levi, and Erik Waingarten. Learning and testing junta distributions with sub cube conditioning. In Conference on Learning Theory, pages 1060-1113. PMLR, 2021. Google Scholar
  12. Xi Chen and Cassandra Marcussen. Uniformity testing over hypergrids with subcube conditioning. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4338-4370. SIAM, 2024. Google Scholar
  13. Oded Goldreich. Introduction to property testing. Cambridge University Press, 2017. Google Scholar
  14. Oded Goldreich and Dana Ron. On testing expansion in bounded-degree graphs. Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 68-75, 2011. Google Scholar
  15. Oded Goldreich and Dana Ron. Testing distributions of huge objects. TheoretiCS, 2, 2023. Google Scholar
  16. Leonid A Levin. One-way functions and pseudorandom generators. In Proceedings of the seventeenth annual ACM symposium on Theory of computing, pages 363-365, 1985. Google Scholar
  17. Liam Paninski. A coincidence-based test for uniformity given very sparsely sampled discrete data. IEEE Transactions on Information Theory, 54(10):4750-4755, 2008. Google Scholar
  18. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996. Google Scholar
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