Estimating the distance from testable affine-invariant properties

Abstract

Let $\mathcal{P}$ be an affine invariant property of functions ${\mathbb{F}}_p^n\to [R]$ for fixed $p$ and $R$. We show that if $\mathcal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\mathcal{P}$ 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 $\mathcal{P}$. 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].