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Low-Degree Testing Over Grids

Prashanth Amireddy,S. Srinivasan,M. Sudan

Published 2023 in International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

ABSTRACT

We study the question of local testability of low (constant) degree functions from a product domain $S_1 \times \dots \times {S}_n$ to a field $\mathbb{F}$, where ${S_i} \subseteq \mathbb{F}$ 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 $\Omega(1)$-far from having this property. In contrast, we show that there exist asymmetric grids with $|{S}_1| =\dots= |{S}_n| = 3$ for which testing requires $\omega_n(1)$ 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 $f : {S}_1 \times \dots \times {S}_n \to {G}$, 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 noise over large grids, which may be of independent interest.

PUBLICATION RECORD

  • Publication year

    2023

  • Venue

    International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

  • Publication date

    2023-05-08

  • Fields of study

    Mathematics, Computer Science

  • Identifiers
    DOI 10.48550/arXiv.2305.04983arXiv 2305.04983
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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