Improved Optimal Testing Results from Global Hypercontractivity

TL;DR

This paper improves the query complexity for testing low-degree polynomials over finite fields by reducing the dependency on field size from tower-type to polynomial, using hypercontractivity and structural analysis of erroneous subspaces.

Contribution

It introduces a new approach that achieves polynomial dependence on field size in optimal testing of low-degree polynomials, extending to lifted affine invariant codes.

Findings

Dependency on field size is polynomial rather than tower-type.

Structural analysis of erroneous subspaces via hypercontractivity.

Method applies to lifted affine invariant codes.

Abstract

The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters: field size $q$, degree $d$ and proximity parameter $\delta$, and the goal is to design a tester making as few as possible queries to a given function, which is able to distinguish between the case the given function has degree at most $d$, and the case the given function is $\delta$-far from any degree $d$ function. A tester is called optimal if it makes $O(q^d+1/\delta)$ queries (which are known to be necessary). For the field of size $q$, the natural $t$-flat tester was shown to be optimal first by Bhattacharyya et al. for $q=2$, and later by Haramaty et al. for all prime powers $q$. The dependency on the field size, however, is a tower-type function. We improve the results above, showing that the dependency on the field size is polynomial. Our approach also applies in the more general setting of lifted affine invariant codes, and is based on studying the structure of the collection of erroneous subspaces. i.e. subspaces $A$ such that $f|_{A}$ has degree greater than $d$. Towards this end, we observe that these sets are poorly expanding in the affine version of the Grassmann graph and use that to establish structural results on them via global hypercontractivity. We then use this structure to perform local correction on $f$.