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Junta Distance Approximation with Sub-Exponential Queries

Authors Vishnu Iyer , Avishay Tal , Michael Whitmeyer

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Algorithms
  • Complexity Theory
  • Fourier Analysis
  • Juntas
  • Normalized Influence
  • Property Testing
  • Tolerant Property Testing

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Abstract

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f:{±1}ⁿ → {±1}:  
1) We give a poly(k, 1/(ε)) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k'-juntas, where k' = O(k/(ε²)). 
2) In the non-relaxed setting, we extend our ideas to give a 2^{Õ(√{k/ε})} (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k).  Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [Michel Talagrand, 1994].

Cite As Get BibTex

Vishnu Iyer, Avishay Tal, and Michael Whitmeyer. Junta Distance Approximation with Sub-Exponential Queries. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 24:1-24:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.24

Author Details

Vishnu Iyer
  • University of California at Berkeley, CA, USA
Avishay Tal
  • University of California at Berkeley, CA, USA
Michael Whitmeyer
  • University of California at Berkeley, CA, USA

Acknowledgements

We thank Anindya De, Shafi Goldwasser, Amit Levi, and Orr Paradise for very helpful discussions.

References

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