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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2023.83

Searching for Regularity in Bounded Functions

Authors: Siddharth Iyer and Michael Whitmeyer

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract

Given a function f on F₂ⁿ, we study the following problem. What is the largest affine subspace 𝒰 such that when restricted to 𝒰, all the non-trivial Fourier coefficients of f are very small? For the natural class of bounded Fourier degree d functions f: F₂ⁿ → [-1,1], we show that there exists an affine subspace of dimension at least Ω(n^{1/d!} k^{-2}), wherein all of f’s nontrivial Fourier coefficients become smaller than 2^{-k}. To complement this result, we show the existence of degree d functions with coefficients larger than 2^{-d log n} when restricted to any affine subspace of dimension larger than Ω(d n^{1/(d-1)}). In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of F₂ⁿ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.

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Siddharth Iyer and Michael Whitmeyer. Searching for Regularity in Bounded Functions. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{iyer_et_al:LIPIcs.ICALP.2023.83,
  author =	{Iyer, Siddharth and Whitmeyer, Michael},
  title =	{{Searching for Regularity in Bounded Functions}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{83:1--83:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.83},
  URN =		{urn:nbn:de:0030-drops-181351},
  doi =		{10.4230/LIPIcs.ICALP.2023.83},
  annote =	{Keywords: regularity, bounded function, Boolean function, Fourier analysis}
}

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DOI: 10.4230/LIPIcs.CCC.2021.24

Junta Distance Approximation with Sub-Exponential Queries

Authors: Vishnu Iyer, Avishay Tal, and Michael Whitmeyer

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

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

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)

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@InProceedings{iyer_et_al:LIPIcs.CCC.2021.24,
  author =	{Iyer, Vishnu and Tal, Avishay and Whitmeyer, Michael},
  title =	{{Junta Distance Approximation with Sub-Exponential Queries}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{24:1--24:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.24},
  URN =		{urn:nbn:de:0030-drops-142988},
  doi =		{10.4230/LIPIcs.CCC.2021.24},
  annote =	{Keywords: Algorithms, Complexity Theory, Fourier Analysis, Juntas, Normalized Influence, Property Testing, Tolerant Property Testing}
}

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Documents authored by Whitmeyer, Michael

Searching for Regularity in Bounded Functions

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Junta Distance Approximation with Sub-Exponential Queries

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