Low-Degree Testing Over Grids

by   Prashanth Amireddy, et al.

We study the question of local testability of low (constant) degree functions from a product domain S_1 Ɨā€¦Ć—S_n to a field š”½, where S_iāŠ†š”½ 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 Ī©(1)-far from having this property. In contrast, we show that there exist asymmetric grids with |S_1| =ā€¦= |S_n| = 3 for which testing requires Ļ‰_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 Ɨā€¦Ć—S_n ā†’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.


page 1

page 2

page 3

page 4

āˆ™ 09/18/2017

Local decoding and testing of polynomials over grids

The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate ...
āˆ™ 04/18/2022

Low Degree Testing over the Reals

We study the problem of testing whether a function f: ā„^n ā†’ā„ is a polyno...
āˆ™ 08/29/2023

Adversarial Low Degree Testing

In the t-online-erasure model in property testing, an adversary is allow...
āˆ™ 09/21/2021

Learning low-degree functions from a logarithmic number of random queries

We prove that for any integer nāˆˆā„•, dāˆˆ{1,ā€¦,n} and any Īµ,Ī“āˆˆ(0,1), a bounde...
āˆ™ 11/26/2017

Agreement tests on graphs and hypergraphs

Agreement tests are a generalization of low degree tests that capture a ...
āˆ™ 02/17/2022

Improved Optimal Testing Results from Global Hypercontractivity

The problem of testing low-degree polynomials has received significant a...
āˆ™ 08/20/2021

Signal Detection in Degree Corrected ERGMs

In this paper, we study sparse signal detection problems in Degree Corre...

Please sign up or login with your details

Forgot password? Click here to reset