Estimating the distance from testable affine-invariant properties

TL;DR

This paper demonstrates that for affine-invariant properties, if they are locally testable with few queries, then their distance from any function can be efficiently estimated using a small number of queries, extending to properties like cubic polynomials.

Contribution

It introduces a simple method to estimate the distance from affine-invariant properties using restrictions to low-dimensional affine subspaces, combining Fourier analysis and property testing techniques.

Findings

Distance estimation is possible with constant queries for affine-invariant properties.

The method applies to properties like cubic polynomials over finite fields.

The approach generalizes previous results from graph properties to algebraic function properties.

Abstract

Let $\cal{P}$ be an affine invariant property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over $\mathbb{F}_2$. Our test is simple: take a restriction of $f$ to a constant dimensional affine subspace, and measure its distance from $\cal{P}$. We show that by choosing the dimension large enough, this approximates with high probability the global distance of $f$ from $\cP$. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].