License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.41
URN: urn:nbn:de:0030-drops-188665
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18866/
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Amireddy, Prashanth ; Srinivasan, Srikanth ; Sudan, Madhu

Low-Degree Testing over Grids

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LIPIcs-APPROX41.pdf (0.9 MB)


Abstract

We study the question of local testability of low (constant) degree functions from a product domain 𝒮_1 × … × 𝒮_n to a field 𝔽, where 𝒮_i ⊆ 𝔽 can be arbitrary constant sized sets. We show that this family is locally testable when the grid is "symmetric". That is, if 𝒮_i = 𝒮 for all i, there is a probabilistic algorithm using constantly many queries that distinguishes whether f has a polynomial representation of degree at most d or is Ω(1)-far from having this property. In contrast, we show that there exist asymmetric grids with |𝒮_1| = ⋯ = |𝒮_n| = 3 for which testing requires ω_n(1) queries, thereby establishing that even in the context of polynomials, local testing depends on the structure of the domain and not just the distance of the underlying code.
The low-degree testing problem has been studied extensively over the years and a wide variety of tools have been applied to propose and analyze tests. Our work introduces yet another new connection in this rich field, by building low-degree tests out of tests for "junta-degrees". A function f:𝒮_1 × ⋯ × 𝒮_n → 𝒢, for an abelian group 𝒢 is said to be a junta-degree-d function if it is a sum of d-juntas. We derive our low-degree test by giving a new local test for junta-degree-d functions. For the analysis of our tests, we deduce a small-set expansion theorem for spherical/hamming noise over large grids, which may be of independent interest.

BibTeX - Entry

@InProceedings{amireddy_et_al:LIPIcs.APPROX/RANDOM.2023.41,
  author =	{Amireddy, Prashanth and Srinivasan, Srikanth and Sudan, Madhu},
  title =	{{Low-Degree Testing over Grids}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{41:1--41:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18866},
  URN =		{urn:nbn:de:0030-drops-188665},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.41},
  annote =	{Keywords: Property testing, Low-degree testing, Small-set expansion, Local testing}
}

Keywords: Property testing, Low-degree testing, Small-set expansion, Local testing
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)
Issue Date: 2023
Date of publication: 04.09.2023


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