eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
41:1
41:22
10.4230/LIPIcs.APPROX/RANDOM.2023.41
article
Low-Degree Testing over Grids
Amireddy, Prashanth
1
https://orcid.org/0000-0002-2713-8961
Srinivasan, Srikanth
2
https://orcid.org/0000-0001-6491-124X
Sudan, Madhu
1
https://orcid.org/0000-0003-3718-6489
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Department of Computer Science, Aarhus University, Denmark
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.41/LIPIcs.APPROX-RANDOM.2023.41.pdf
Property testing
Low-degree testing
Small-set expansion
Local testing