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Junta Distance Approximation with Sub-Exponential Queries

Authors Vishnu Iyer , Avishay Tal , Michael Whitmeyer



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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.

Cite AsGet 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

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].

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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