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

Authors Tomer Adar , Eldar Fischer, Amit Levi



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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

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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

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.

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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References

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